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w0), respectively and nˆ is the unit normal vector of the reference plane.The normal distance dn of point P1∗ from the reference plane isdn=nˆ·P01∗=nˆ·(x1+u1)-(x0+u0)(y1+v1)-(y0+v0)(z1+w1)-(z0+w0)=nˆ·x1-x0y1-y0z1-z0+nˆ·u1-u0v1-v0w1-w0The displacements of points P1 and P0 in Eq. dn=nˆ·x1-x0y1-y0z1-z0+nˆ([Ti][Di]-[Tj][Dj])Using the penalty method, a mathematical spring with stiffness Kn is placed between the point P1 and the reference plane in the direction normal to the contact face. The potential energy of the normal spring is given byBy minimizing the potential energy Πnc, the following matrices can then be added to submatrices [Kii], [Kij], [Kji], and [Kjj] in the global stiffness matrix:[Kii]=Kn[H]T[H][Kij]=Kn[H]T[Q][Kji]=Kn[Q]T[H][Kjj]=Kn[Q]T[Q]and the vectors [Fi] and [Fj] are calculated as follows and then added to the global force vector: is the projection of P1 on the contact face after the application of displacement increment. A shear contact spring is activated when the shear force is smaller than the shear resistance of a discontinuity to diminish the relative displacement of the two blocks in the form:where Fs is the shear contact force; φ is friction angle of the discontinuity; and Fn is the normal contact force. The shear displacement along the P0∗P2 is calculated asThe potential energy of the shear spring is given byΠsc=12Ksds2=12Ks(|P01∗|2-dn2)=12Ks(x1+u1)-(x0+u0)(y1+v1)-(y0+v0)(z1+w1)-(z0+w0)T·(x1+u1)-(x0+u0)(y1+v1)-(y0+v0)(z1+w1)-(z0+w0)-dn2Πsc=12Ksx1-x0y1-y0z1-z0T+[Di]T[Ti]T-[Dj]T[Tj]T×x1-x0y1-y0z1-z0+[Ti][Di]-[Tj][Dj]-12Ks(M+[H][Di]-[Q][Dj])2By expanding and minimizing the potential energy Πsc, the following matrices can be added to the submatrices [Kii], [Kij], [Kji], and [Kjj] in the global stiffness matrix:[Kii]=Ks[Ti]T[Ti]-Ks[H]T[H][Kij]=-Ks[Ti]T[Tj]-Ks[H]T[Q][Kji]=-Ks[Tj]T[Ti]-Ks[Q]T[H][Kjj]=Ks[Tj]T[Tj]-Ks[Q]T[Q]And the vectors [Fi] and [Fj] are calculated as follows and then added to the global force vector:[Fi]=-Ks[Ti]Tx1-x0y1-y0z1-z0+KsM[H]T[Fj]=Ks[Tj]Tx1-x0y1-y0z1-z0+KsM[Q]TWhen the state of the contact is sliding, a pair of equal and opposite frictional forces parallel to the sliding direction is applied on the contact face at the points P1 and P0. The magnitudes and directions of the frictional forces are obtained from a previous iteration. The Coulomb’s law is used to evaluate the dislocation movements of the block interfaces. When the shear force F conforms toa dislocation movement will happen and the friction force should be taken into account. In the above formula, N is the normal force on the contact boundary and, ϕ and C are the friction angle and cohesion, respectively. The frictional force is calculated from the normal contact compressive force from the previous iteration where dn is taken from the previous iteration. Let L be the direction of the frictional force () and nˆ be the unit vector pointing out of the block in a direction normal to the contact face. We haveTherefore, the potential energy of the friction force is given byd=[u1v1w1]-[u0v0w0]=[Di]T·[Ti(x1,y1,z1)]T-[Dj]T·[Tj(x0,y0,z0)]TThe relative derivatives of Πf with respect to dri and drj at the origin arewhich are added to the global force vector.Four examples are presented in the following sections to demonstrate the newly developed 3-D DDA algorithm.The first example involves a system of two blocks as shown in . The density, Young’s modulus and Poisson’s ratio values for each block are 2.6 × 103 |
kg/m3, 2 GPa and 0.2, respectively. The maximum time step is 0.2 s. b–e shows the progressive movement and multi-block interaction. It can be seen that no visible intrusion occurs in any of the contact cases including edge-to-edge and edge-to-face contacts, even after large movements of the blocks have occurred. illustrates the geometry of the first example simulating the calculation of a face-to-face contact between two blocks. In this example, block A is fixed, and block B falls down due to the gravity force. The blocks have the same size 5 × 2 × 2 m3. The values for the elastic modulus, Poisson’s ratio and mass density for each block are E |
= 2 GPa, v |
= 0.2, and ρ |
= 2600 kN/m3, respectively. The calculations were carried out for 65 steps with at-rest (i.e. zero initial velocity) conditions for block B. demonstrate how the newly developed 3-D contact algorithm can handle the face-to-face contact between blocks A and B when the two blocks come in contact with each other. contrast the results without and with including the contact algorithm (i.e. calculating and including contact forces between the faces), respectively.In this example, a single block rests on an inclined plane at angle α from horizontal. The contact surface friction angle is φ. Under the action of gravitational force, the displacement s of the block is determined analytically as a function of time t asThe inclination of the modeled plane is assumed to be 30°. Three contact surface friction angle values equal to 0°, 10° and 20° are included in the study. The accumulated displacements are calculated up to 2 s. A comparison between the analytical solutions from Eq. . It is seen that the 3-D DDA results show a satisfactory agreement with the time-dependent displacement functions predicted using the analytical solutions. show the magnitudes of absolute and relative errors of the 3-D DDA for this example, respectively. These figures indicate that the absolute value of the error for a sliding block increases at every time step, but the relative error with respect to block displacement decreases and the numerical solution approaches the analytical solution. It can be concluded that reducing the perturbation of the system during the first few time steps will improve the accuracy of the solution. Similar results have been obtained by Doolin and Sitar Example simulation of dominos involves a row of four dominos that fall over as illustrated in a. As the dominos fall, many simultaneous contacts can occur between them, which are modeled as individual collisions in b–e. After all dominos have fallen, they come to rest on the supporting surface. In this example, the values for the surface friction angle, the maximum displacement ratio allowed and the maximum time increment for each time step are 10°, 0.1 and 0.01 s, respectively.In this paper, a new algorithm is presented that can be used to search and calculate geometrical contacts in the 3-D DDA. This method computes the distance between two polyhedra by finding and tracking the closest points similar to the shortest link method (SLM) The success and accuracy of the newly developed 3-D program, involving a 3-D contact algorithm, are demonstrated through several examples involving two or more blocks that are brought in contact with each other in a 3-D domain. The uniqueness of the new method is its calculation of the contact plane and the unit normal vector of the plane. As a result, the contact calculations are simplified, which can be programmed and implemented in the 3-D DDA more easily.Establishment of Optimal Blade Clearance of Stainless Steel Rolling-Cut Shear and Test of Shearing Force ParametersThe purpose aims to improve the plate shearing section quality and metal yield by means of optimal blade clearance adjustment model of rolling-cut shear and accurate calculation of the maximum shearing force, as well as for the system optimization design, structure optimization design to provide important basis. A 3500 mm rolling shear of a large stainless steel factory was taken as the test object, and the blade clearance under different influence factors of the clearance value and the shearing section were tested. The optimized production accumulated data regression analysis and the rolling shear process of stainless steel shear test. The test results show that the optimal blade clearance adjustment module is a comprehensive function, which include steel plate thickness, material, temperature and shear plate volume. The shear stress at starting stage with the relative penetration depth increases affected by the above factors, and the fracture peak decreases rapidly after being cut into the roll phase constant.Biography: MA Li-feng (1977—), Male, Doctor, Associate ProfessorMulti-scale fatigue behaviour modelling of Al–Al2O3 short fibre compositesUsing very heterogeneous materials in structural parts submitted to cyclic loadings, this paper presents an elasto-plastic micromechanical model. After recalling the homogenisation principle based on a mean field theory, non-linear kinematic and isotropic strain hardening is introduced into the matrix. Validation is made on an Al–3.5%Cu/SiC particle composite, and an Al–Si7Mg/Al2O3 fibre composite is treated as a first application. Damage is introduced into the model using a fibre failure criterion. It is based on the evolution of the volume fraction of broken fibres as a function of the maximum principal stress in the fibre family. The damage law is identified by means of in situ tensile tests performed inside the scanning electronic microscope. The number of broken fibres is determined as a function of the applied load and the number of cycles. The model predicts the fatigue behaviour, the loss of stiffness, the volume fraction of broken fibres for different volume fractions, aspect ratios, distributions of orientation and distributions of strength of the fibres. The effect of the mechanical fatigue properties of the matrix is also studied.Introducing reinforcement into structural parts can improve the fatigue resistance of aluminium alloys. The shape, percentage and orientation of these inclusions have an influence on the mechanical behaviour of the composite. The use of these materials in the automotive or aerospace industry requires prediction tools during low cycle fatigue in respect to the microstructure. Several analytical methods have been used to obtain a “homogenised” behaviour based on the microscopic description of the constituents. At first, we will present the micromechanical model taking into account hardening in the matrix through an incremental scheme. It involves morphological properties of the reinforcement and provides mean stress and strain fields in the constituents. Then, the application of the model on a SiC particle reinforced Al–3.5%Cu composite will show the accuracy of the prediction in full-reverse cycling, in comparison with FEM calculations. Finally, an Al–Si7Mg/Al2O3 3-D-distributed fibre composite is studied. The microstructure is analysed and allows elasto-plastic predictions.The composite behaviour follows a classical approach in the elastic range, based on the works of Mori and Tanaka The stiffness tensor of the composite is obtained after homogenisation of the stress and strain tensors within the REV:where L0 is the matrix stiffness tensor, I the identity tensor and Sr the Eshelby tensor of the r-inclusion family which has a stiffness tensor Lr and volume fraction fr.and L account for the stiffness and the geometrical (through Sr) properties of the constituents.The microscopic scale is determined by means of the localisation tensors A0 and B0 in the matrix and Ar and Br in the r-inclusion, such that average strain and stress fields in the matrix areand average strain and stress fields in the r-inclusion areThis model has been chosen because of the materials of interest, which are composite materials with volume fractions from 0 to 20% of discontinuous reinforcements. Requirements in implementation aptitude and calculation velocity make us prefer this approach to the widely used self-consistent scheme.Reinforcements remain elastic and plasticity is considered exclusively within the matrix. Instead of a secant moduli method (Tandon and Weng In the Eshelby principle, the perturbation stress tensor in the inclusion is now with Str being the Eshelby tensor derived from the tangent properties of the matrix.The unknown tangent stiffness tensor of the matrix has to be determined, considering plastic flow properties.with n being the vector normal to the yield surface, h the plastic modulus andthe equivalent stress. s0 and x′0 are deviatoric stress and kinematic hardening tensors.An isotropic non-linear kinematic law as described by Chaboche The isotropic hardening component evolves with the cumulative plastic strain p as R(p)=R0+Q(1−exp(−bp)), while kinematic hardening has an instantaneous definition is the plastic strain rate of the matrix.Finally, when plastic flow occurs, solving the system (f=0, ) in the matrix gives the tangent stiffness tensor Lt0 (see ). This tensor is not isotropic and the homogenisation strictly needs to be made by means of a non-isotropic calculation of the Eshelby tensor to obtain the overall stiffness tensor of the composite. However, as a first approximation Lt0 will be assumed to be isotropic.The full-reverse cyclic fatigue loading is set by a strain increment on the REV. The first estimate of the local stress state in the matrix is found by an elastic localisation.The differentiated plastic equations are solved by iterations to determine and the plastic strain at t+dt. It is achieved following a local integration with a radial return method (Burlet and Cailletaud The tangent stiffness tensor of the REV is deduced according to , but the macroscopic strain direction has to be balanced by iterations to satisfy a unidirectional stress state.A first example is presented in this section, based on results of FEM calculations carried out by Llorca et al. The strain-hardening parameters of the matrix are identified on simulated hysteresis loops for the non-reinforced material and are given in For the composite, the simulation of the behaviour to stabilisation (5th cycle) leads to similar results as the FEM calculations of Llorca et al. . Good convergence is obtained for strain increments up to 5×10−2% (for example 20 points per cycle if ΔE/2=0.5%), but when the tangent modulus tends to zero (for higher strains and no strain-hardening), the isotropic approximation leads to instabilities. Calculations could be improved by an anisotropic form of Lt0 combined with the Eshelby tensor for spheroidal inclusions in an anisotropic media (Mura ) reveal the influence of the volume fraction on the monotonic behaviour of the composite. Stress–strain curves are in good agreement, but for reinforcement rates close to 20%, the Mori and Tanaka tangent moduli approach underestimates the material's strength calculated by Llorca's FEM model. This may be due to an overall plastic strain in the matrix, while local plastic strain remains localised at the top of the particles in the FEM calculations.However, cyclic stress–strain curves for both 13% and 20% composites () show that this gap vanishes at the stabilised state.Since the micromechanical model takes into account several reinforcement families, it is of interest to predict the behaviour of a more complex composite. As part of automotive applications, we studied a short fibre reinforced metal matrix composite manufactured by indirect squeeze casting.Alumina fibre preforms (ICI-Saffil RF grade) are impregnated by an Al–Si7Mg03 alloy under speed and pressure control. After a T6 heat treatment, the microstructure is very heterogeneous and made of α-Al dendrites, Al–Si eutectic phase including 1 μm diameter Si precipitates (grey) and small MgAl2O4 spinels concentrated at the fibre–matrix interface (). Little porosity has been seen at the clusters of fibres, resulting from bad impregnation.The passage real microstructure — REV is accomplished, considering only two constituents: a homogeneous isotropic matrix and Al2O3 fibres. Indeed, the mechanical properties of the matrix are deduced from an Al–Si7Mg03–T6 alloy that contains almost the same heterogeneity (). These results must be handled with care because of a different mould geometry for the pure alloy specimen, in which the heterogeneity distribution is different.Fibre orientation distribution is measured by quantitative image analysis. Even if preform processing (Fukunaga ). If θ is the angle of the fibre axis to the loading direction and φ to the observation plane, the fibres can be divided into families characterised by their orientation (θi,φi) and their volume fraction fi.Low cycle fatigue calculations are carried out on a composite REV involving 13 fibre families with a total volume fraction of 13.34%. Stress–strain curves are similar to those obtained for a 2-D randomly orientated system (φ=0°). The results for this simple REV are presented in . In comparison with an aluminium alloy, the higher stiffness and yield strength of the composite lead to a better general strength, but the total plastic strain amplitude is increased with increasing fibre volume fraction. Stabilised hysteresis loops in the matrix for different composites, as shown in , confirm an extra cumulated plastic strain for 13.34% and 20% composites.Experimental tests are to be conducted to confirm the general tendencies, but for higher cycle fatigue predictions, damage mechanisms will be included to maintain a correspondence between microstructure and mechanical properties.The plasticity of the matrix is not the only deformation mechanism. For an applied stress around the yield stress, damage mechanisms occur leading to an increase of the strain. So in order to complete the modelling, we will introduce the damage into our micro–macro model with a local failure criterion of the component. The microstructure evolution of the damaged composite is based on the evolution of the volume fraction of cracks.In order to determine the damage mechanisms occurring at the micro-scale in the composite, in-situ tensile tests were performed inside the scanning electronic microscope. During the first tensile loading, a damage mechanism based on the failure of the fibres mainly oriented in the load direction has been observed (We experimentally quantified the evolution of the number of broken fibres (We performed an interrupted fatigue test followed by microscopic observations. The evolution of the number of broken fibres with the number of cycles was measured. Secondly, cracks appeared in the matrix and propagated during fatigue (In order to improve the micro–macro modelling of the behaviour, we have introduced a damage law at the micro scale. By homogenisation, it is then possible to obtain the macroscopic behaviour coupling plasticity and damage.The damage law that adjusts best to the experimental data in respect of the number of broken fibres as a function of the applied strain presents a Weibull form (dashed and dotted lines in It should be noticed that this damage law is a function of the fibre volume and the maximum principal stress in the fibre. As the damage law is a function of the local maximum principal stress in the fibre, it takes into account the aspect ratio, the orientation and the mechanical properties of the fibre and the matrix.As the damage law at the micro scale is a function of the stress state in the fibre, we first simulate a tension test by means of the previously described model. We calculate the maximum principal stress in the fibre orientated parallel to the applied macroscopic stress. With the knowledge of the number of broken fibres orientated in the tensile direction as a function of the applied stress, it is then possible to calculate the volume fraction of broken fibres as a function of the maximum principal stress in the longitudinal fibre. As very few fibres of the total amount of fibres break, we assume that the local stress in the fibre is not modified by the increase of the number of broken fibres.With a value of m=1.1, the scatter of failure of one family of fibres is very large. This is due to the heterogeneity of the microstructure, in particular the local volume fraction of fibres. So for the same calculated maximum principal stress in longitudinal fibres, not all the fibres will fail. For a principal stress in the longitudinal fibre of σ=σu=773 MPa, 63% of the fibres are damaged. The mean volume of the fibre has been chosen equal to 1.41×10−6 mm3. It should be noted that this micro damage law takes into account the volume effect of the reinforcement on the failure probability. The longer the fibres, the higher their failure probability.The effect of damage on the microstructure of the composite has been taken into account. When a fibre breaks, it is replaced by an ellipsoid representing a crack. Its long axis is equal to the fibre diameter divided by , θ being the angle between the longitudinal axis of the fibre and the direction of the maximum principal stress in the fibre. The aspect ratio of the ellipsoidal crack, thickness over radius, has been taken as equal to 3.3×10−3. The stiffness tensor of this inclusion representing the crack is equal to zero. The volume fraction of broken fibres is given by the Weibull damage law previously determined and represented in The number of the remaining (intact) fibres is given by the equation:In order to simulate the tensile behaviour of the composite, an incremental procedure is adopted. As the damage begins, the relative volume fraction of cracks is introduced as a new microstructure. The remaining volume fraction of intact fibres is actualised following . A comparison between experiments and modelling is given in In fatigue with a compressive loading, we observe a closure effect of the cracks. In order to take this effect into account, we introduce a crack closure law, which is similar to the previous Weibull damage law. The initial value corresponds to the beginning of the unloading. Following this closure law, the volume fraction of cracks decreases, but at the same time, the volume fraction of intact fibres increases in a similar manner as it decreases in tension. No new damage occurs during the unloading and the compressive loading.During the reloading, we observe a reopening of the cracks. The adopted model for reopening the cracks deals with the reintroduction of the entire volume fraction of cracks at the moment when the matrix strain in the loading direction reaches zero. Then, the Weibull damage law can be applied once again, starting at an initial state of broken fibres given by the maximum stress previously applied.The quantity of intact fibres can be determined as a function of the number of cycles. It shows a stabilisation after 40 cycles for the studied composite (The effect of these broken fibres is pointed out by the loss of stiffness. The model predicts first the decrease of the stiffness tensor for an increasing applied tensile stress. It also predicts the decrease of Young's modulus as the number of cycles increases (The evolution of the stress state in the matrix is obtained with the number of cycles. Two mechanisms are in competition: the cyclic hardening, which induces an increase of the matrix stress, and the stress redistribution due to damage, which induces a decrease of the matrix maximum stress. For our material we obtain the stabilised cycle after about 40 cycles (, the effect of damage on the hysteresis loop is shown. It can be observed how important this effect is for an applied imposed strain. The hardening is less important and Young's modulus is calculated during the unloading after tension decreases. It also decreases for the unloading after compression, but to a lesser extent. The mean stress decreases as the number of cycles grows.As the model is analytical and needs only a PC type computer, it is very easy to change the data to analyse the effect of the microstructure parameters on the hysteresis loop in fatigue.We studied the effects of volume fraction, aspect ratio, distribution of the fibre orientation, elasto-plastic properties in fatigue of the matrix and the damage coefficients on the macroscopic fatigue behaviour of the composite.An increase of the volume fraction increases Young's modulus, the yield stress and the strain hardening. As the matrix is less solicited, the cyclic hardening is less important. Damage begins later and its evolution is quicker (The influence of the shape of the fibres, i.e. the length or more precisely the aspect ratio, is important between a spherical reinforcement and a short fibre. An aspect ratio of over 100 no longer influences the elasto-plastic behaviour, if the volume fraction remains low enough to avoid interaction and contact between the fibres (A 3-D distribution of fibre orientation is assumed for the implementation of our model. Different techniques can be used to determine this distribution. A direct method consists of a measurement by image analysis of the distribution of the aspect ratio of the ellipses resulting from the intersection with the edge plane. These measurements have to be done on the three faces of a cube.A quicker way to proceed is to determine this distribution of orientation by an indirect method using the propagation of the ultrasonic waves. The stiffness coefficients are a function of the density and the square of the velocity of the ultrasonic wave. By measuring these wave velocities in many directions of the material, it is possible to determine the complete stiffness tensor. By an inverse method, using the micro–macro model previously presented in the paragraph describing the elastic behaviour, it is possible to determine the distribution of fibre orientation.Using data from the model, the orientation distribution is discretised into 28 families of fibres. Each family corresponds to a prevailing direction, and is characterised by a particular volume fraction. shows the effect of three different types of fibre distribution (unidirectional (UD) and random in-plane distributions) on the predicted hysteresis loops.We compare three different aluminium matrices of different heat treatments. The 5 coefficients of the isotropic and cinematic hardening have been identified on the matrix alone using fatigue tests. One difficulty is to compare the mechanical properties of the matrix alone and the matrix in situ in the composite. The presence of reinforcement modifies the grain size of the matrix in situ. So, in order to be able to make a comparison, it might be necessary to modify the heat treatment of the composites compared to the matrix alone in order to obtain the same micro hardness. compare the influence of three different matrices with the plasticity properties given in In the previously described damage law, two independent parameters are present: m and σu. The Weibull parameter m corresponds to the scatter of the fibre failure in the composite. The smaller m is the more scattered is the state of stress and the alumina strength for one family of fibres. The scatter of stress is mainly due to the heterogeneity of the local interaction between fibres, which is not taken into account in our model. The fibres are not perfectly arranged; some can interact more than others.The parameter σu in the mean value of the alumina fibre strength gives a cumulative failure probability of 63%, i.e. it corresponds to 63% of broken fibres for a given family. This parameter is strongly related to the reinforcement material, alumina in this case.It is assumed that these two Weibull parameters are independent of the orientation of the fibre family. This hypothesis has been verified by Guo So, even if these coefficients have been experimentally determined for the longitudinal fibres, they are valid for the other directions as well.The scatter quantified by the parameter m has an important influence on the fatigue behaviour. The slow evolution of damage leads to a softer fatigue behaviour (The influence of σu is less important because changing σu means changing the strength of the fibres. So the skeleton of the composite is a little weaker for a low σu and the composite becomes softer (A micro-mechanical model has been proposed to predict the behaviour of composites randomly reinforced by discontinuous fibres under cyclic loadings. The main difficulty is to combine numerical solving of plasticity in the matrix with a micro–macro transition. However, hysteresis loops are in good agreement with FEM calculations and experiments for different reinforced aluminium composites. It can be used for predicting the effect of the volume fraction, the aspect ratio and the distribution of orientation of the fibres on the mechanical properties in fatigue. The choice of the matrix can be optimised. The fibre damage observed at the micro-scale has been modelled by introducing a statistical micro-damage law based on the evolution of the volume fraction of broken fibres as a function of the maximum principal stress in the reinforcement. The model predicts the evolution of the hysteresis loop in fatigue with the number of cycles. It also predicts the damage evolution based on the loss of the stiffness tensor related to the number of cracks developing in the material. This analytical model runs on PC type computers and allows optimisation of the microstructure of the composite to be carried out very easily. It predicts what will be the fatigue behaviour of the composite in the three dimensions for any kind of applied stress tensor.Microstructural study on ultra-high temperature erosion mechanism of infiltrated W-10 wt%Cu compositeIt is tried in this paper to study microstructural features related to erosion of infiltrated W-10 wt%Cu at ultra-high temperature against liquid alumina droplets. Also the effect of post-sintering annealing on the fracture mode and erosion resistance of the composite has been examined. For this purpose, the composite material has been prepared by compressing, sintering (also vacuum annealing) and infiltration of initial tungsten powder. The erosion test carried out by static firing of the bi-propellant engine (using two types of propellant: typically a fuel and an oxidizer) and the microstructure evolutions were studied by SEM and EDS. It is found that the main erosion mechanism of infiltrated W-10 wt%Cu composite, includes inter-granular fracture of tungsten skeleton and chipping. Extra sintering of skeleton takes place in outer copper-free skin of eroded composite. Also, melting and evaporation of copper occurs due to ultra-high temperature of erosion test and leads to formation of evacuated porosities. On the other hand, alumina droplets can penetrate into evacuated porosities. Also, vacuum annealing of sintered skeleton resulted into about 33% higher erosion resistance of composite. The obtained results were discussed based on weak points in infiltrated composite, transpiration cooling and microstructural evolution during erosion. This study provides an insight into correlating microstructural evolution with the erosion mechanism of infiltrated W-10 wt%Cu composite.Establishing a logical relationship between microstructural features and erosion mechanism is very important for designing and fabricating of suitable materials in desired service conditions. This importance would be critical in high temperature as well as high velocity abrasive particles, because simulating of actual erosion test condition in laboratories is very difficult and even impossible. W-10 wt%Cu composite is one of the most important materials with wide application of high temperature and sever erosive condition. The main application of W-10 wt%Cu composite is as erosion resistant materials using in nuzzle or vector of propellant motors which accomplished with very high temperature and high speed erosive liquid alumina particles The response of nominally brittle materials to liquid impact is cracking of the surface due to the direct deformation impact loading on the surface. This cracking is typically in the form of disconnected annular ring segments which eventually intersect under continued impingement and result in chips of material being removed Due to high volume fraction of tungsten, W-10 wt%Cu composites are fabricated only by infiltration route Since, W-Cu composites mainly fabricated by powder metallurgy route, they exhibit brittle behavior Despite some studies conducted about the erosion of hard-metals In this study, infiltration method has been employed to fabricate W-10 wt%Cu composite. For this purpose, initial powder particles with average particle size about 6 μm were used as raw material. Powder mixture compacted at pressure of 200 MPa by Cold Isostatic Press (CIP) to obtain green compact with suitable strength. To reach a porous tungsten skeleton with density of 80 ± 2 pct, green compact was sintered at 2150 °C for 180 min in hydrogen atmosphere. To examine the fracture mode and verify the final results, another specimen was annealed after sintering at 2200 °C for 120 min in vacuum atmosphere. Subsequently, two porous tungsten skeletons infiltrated by molten copper at 1300 °C for 120 min in hydrogen atmosphere. Thermal cycle for sintering, annealing and infiltration in this study is illustrated in . Apparent densities of sintered and infiltrated specimens were measured by the Archimedes water immersion method according to ASTM B328 standard. SEM was used for in detail studying of microstructure and fracture surfaces. Material loss of infiltrated W-10 wt%Cu composite due to liquid alumina particles erosion has been determined by static firing of the bi-propellant engine. Incident angle of 30° was selected to conduct the erosion test for 40 s with alumina droplets velocity about 1200 m/s. The temperature was about 3000 °C. Erosion was measured by means of a profile projector. Optical microscopy, SEM, EDS and quantitative metallography were used for studying the detailed microstructure of specimens.SEM micrographs of specimen (without annealing) after compressing, sintering and infiltration are shown in . As illustrated, prepared specimens reached to the densities of 11.6, 15.05 and 16.94 g·cm-3, respectively. Formation of necking between adjacent tungsten particles during sintering as well as the formation of interconnected porosity network is visible in SEM micrograph of b. The back scattered image of copper infiltrated tungsten block (without annealing shown in c) revealed predominately inter-granular fracture mode beside some trans-granular fracture surface. Several studies of fractography and the fracture mechanisms of W-based composites and heavy alloys c, post-sintering annealing in vacuum results in fracture mode change from predominate inter-granular mode to pure trans-granular mode.The optical micrograph form surface morphology of prepared sample after erosion test is shown in . As illustrated, the eroded surface of studying sample obviously consists of erosion pits and irregular protrusions. These irregular protrusions consist of solidified alumina droplets, tungsten debris and removed tungsten particles. Under the alumina containing gas stream action, liquid alumina and erosion products move along the exposed surface in a linear direction that results in formation of parallel grooves on the eroded surface. Also, the higher magnification SEM image of eroded surface is illustrated in b that confirms the mechanism of tungsten grain removal due to inter-granular fracture of the tungsten skeleton and chipping (b). It is shown that the erosion products consist mainly of removed tungsten particles and solidified alumina droplets. Also, the features observed in this test condition are typical of erosion debris and deposited alumina particles that almost cover the entire surface of the sample. The higher magnification of deposited alumina layer on the eroded surface and the result of EDS analysis are presented in d and e, respectively. Since erosion test temperature was more than boiling point of copper, infiltrated copper initiate to evaporate at regions near eroded surface. The results of EDS analysis in c obviously confirm copper evaporation mechanism at the near surface regions. In other words, the Cu weight percent obtained from EDS analysis is very lower than copper weight percent in initial prepared composite (about 10 wt%). Evaporation of copper led to the formation of evacuated porosities that may be filled by alumina droplets. Penetration of alumina droplets is shown in b. This mechanism can be led to closing the open porosity network and prevent more copper evaporation. Unlike to hard-metals (WC-Co) which display a mixture of brittle and ductile target behavior As mentioned above, the erosion of tungsten skeleton related to inter-particle fracture, chipping and grain removal of the surface. Based on previous studies, beside the porosities, W-W contact areas are the weakest point in infiltrated composite shows different types of remained vacant porosities after infiltration. Secondly, W-W interfaces acts as an absorption site for impurities such as oxygen, nitrogen, hydrogen, carbon and phosphorous Other porosity type that can be seen in b, is the intrinsic closed pores. These pre-existed pores created during production of powder particles and can be act as a stress concentration site and then, decrease strength of tungsten particles. The existence of these pores led to trans-granular fracture mode as well as grain removal during erosion. Un-infiltrated porosities are other porosities that illustrated in c. These porosities created due to incomplete infiltration of copper into tungsten skeleton. Addition of some transition elements such as Ni and Co to initial tungsten powder can improve the Cu infiltration into a porous tungsten skeleton and eliminate this type of remained porosities (c). Consequently, it can be said that using of Ni and Co in infiltrated W-10 wt.5Cu composite may be led to better erosion resistance of material shows SEM micrograph from upper surface of eroded specimen. Formation of a deposited alumina layer on the exposed surface is obvious. Also, penetration of alumina droplets into evacuated porosities can be seen. On the other hand, evacuated porosities due to copper evaporation are shown in the SEM image of . Liquid copper can flow through a heated surface made out of porous tungsten. The evacuating copper absorbs heat by convection and thus cools the material surface down. Using of copper as a coolant has the advantage that the heat of vaporization can be used as an additional cooling mechanism. Since liquid to vapor transition takes place at constant temperature, melted copper will not become hotter than their boiling temperature (~ 2700 °C). To prevent copper melt from evaporating within the porous tungsten skeleton, new liquid copper has to be supplied at a sufficiently mass flow rate to reach to exposure surface. Therefore, in the case of infiltrated W-10 wt% composite, continues transpiration cooling mechanism and continues supplying of molten copper are two key parameters for erosion resistance of composite.The other result that can be obtained from is extra sintering of the near surface skeleton. Extra sintering of copper-free skeleton in regions near the surface can be attributed to ultra-high temperature of erosion test. Sintering of tungsten skeleton initiates at 0.4Tm (~ 1400 °C) and complete at 0.8Tm (~ 2730 °C) The microstructure of eroded composite in the region between transpiration cooling region and internal parts of the specimen is illustrated in . In fact, this region is border line that separates these two different regions. As illustrated in , the near surface area of eroded specimen can be separated into 3 regions with different microstructural features. In region I, beside the deposition of alumina layer and extra sintering of tungsten skeleton, the copper phase was completely evaporated and exit from the skeleton. So, region I is only consist of a porous tungsten skeleton after the erosion test. In this region, all of the infiltrated coppers take apart in transpiration cooling mechanism. In fact, it is the porous skeleton in region I that resist against impacts of high velocity abrasive alumina particles. As shown in , region II consists of mixture of infiltrated and evacuated porosities. In this region, copper exists in a liquid state during erosion test and absorbs heat by convection. In fact, region II can supply required molten copper for continuing transpiration cooling mechanism in the region I. So, continues connectivity between molten coppers in two regions I and II is very critical. As mentioned previously in , Presence of closed porosities and un-infiltrated porosities may stop continues copper flowing outward to the region I and then, decrease erosion resistance of tungsten skeleton. Totally, region I and II were affected by the heat generated during erosion test and experience microstructural evolution. In region beneath the region II, no microstructural changes took place. In this region (region III), copper phase stays in its solid state. Detailed microstructure features of region III are shown in SEM micrograph of As the amount of molten copper increases, the convective cooling will be very efficient due to the large temperature difference of liquid. It means that, the erosion resistance of the composite will be increased by increasing the depth of heat affected regions (region I + region II). Based on previous published work about the effect of sintering activator on the erosion behavior of this composite, direct relationship between transpiration cooling depth and erosion resistance has been confirmed . The blue, red, green and violent indicates W, Cu, Al and O, respectively. As illustrated, W has a uniform distribution in the total cross-section area of eroded composite. Also, it can be seen from that the Cu content sharply decreased from internal regions to the transpiration cooling region layer near the exposure surface. On the other hand, Al and O exhibit inverse distribution manner compare to the Cu elemental map. In other word, concentration of O and Al is higher in regions near the surface that verify deposition of alumina layer as well as penetration of alumina droplets into evacuated porosities in transpiration cooling region (region I).SEM images from a polished cross-section of prepared composite after erosion test are shown in . Transpiration cooling mechanism, formation of evacuated porosities and three different regions are obvious in these SEM micrographs. Based on sintering theory, Final stage of sintering begins when most of the pores close Based on the different erosion mechanism discussed in this paper, different material loss stages (mechanism) during ultra-high temperature erosion of prepared composite is schematically illustrated in . It is found in this study that The erosion mechanisms of W-10 wt%Cu composite are in the sequence of melting of copper in regions near to exposure surface → evaporation of copper and formation of evacuated porosities → extra sintering of copper-free skeleton → tungsten grain removal due to inter-granular fracture and chipping → deposition of alumina layer on the eroded surface → penetration of alumina droplets into evacuated porosities. As the main result from this study, the erosion mechanism of infiltrated W-10 wt%Cu composite and microstructural related phenomena can be summarized as To examine the validity of proposed erosion mechanism and evaluating the effect of W-W interfaces strength on the erosion behavior of infiltrated W-10 wt%Cu, we prepared another composite material by means of vacuum annealing of sintered tungsten skeleton. In new specimen, tungsten porous skeleton was annealed after sintering at 2200 °C for 120 min and subsequently infiltrated at 1300 °C for 120 min. shows the column plot of relative surface profile loss (surface profile change divided by the initial surface profile of the specimen) after erosion for two studied composites (composite without annealing and composite with annealing). As illustrated, post-sintering annealing at 2200 °C in vacuum result into higher erosion resistance in infiltrated composite. In other words, annealed specimen exhibit about 33% better erosion resistance compare to specimen without annealing.Based on the proposed erosion mechanism in this study, the positive effect of post-sintering annealing on the erosion behavior of material can be attributed to microstructural change due to annealing. As mentioned previously, the strength of W-W interfaces plays critical role in resisting against impact of liquid alumina droplets. SEM micrographs from room temperature fracture surfaces of two prepared composites (with and without annealing) after sintering and also after infiltration are shown in . As illustrated, the fracture mode changes from predominantly W-W interface fracture to trans-granular cleavage fracture after the vacuum annealing. This fracture mode changing can be seen in both after sintering and after infiltration. In tungsten based alloys and composites, materials with trans-granular fracture mode show higher W-W interface strength rather than those with inter-granular one. It can be concluded that vacuum annealing of sintered skeleton can increase the bonding strength (W-W interface strength) between tungsten particles. As mentioned previously, During sintering preferential segregation of impurities takes place at W-W interfaces and H2 absorption takes place ) is the main material loss mechanism during high temperature erosion of infiltrated W-Cu composite; so, W-W strength increasing due to vacuum annealing led to better erosion behavior of material. As a result, this obtained result exhibit good agreement with proposed erosion mechanism in this work. In other words, material removal due to inter-granular fracture of tungsten and chipping is the most important erosion mechanism. Therefore, any processes which be able to increase W-W interface strength, can improve erosion resistance of infiltrated W-Cu composite.The infiltrated W-10 wt%Cu was subjected to liquid alumina particles at ultra-high temperature (3000 °C) and microstructural evolutions during sintering were studied to establish a logical relationship between microstructural features and erosion behavior of prepared composite. The obtained results exhibit that:The erosion mechanism of infiltrated W-10 wt%Cu in current test condition consists of inter-granular fracture of tungsten skeleton, grain removal, chipping, evaporation of copper phase, transpiration cooling, extra sintering of tungsten skeleton, deposition of alumina layers and penetration of alumina droplets into evacuated porosities.It is found that the erosion behavior of the composite has a mutual dependency to remained closed pores and un-infiltrated porosities. W-W separation takes place due to formation of diffusion related close porosity as well as segregation of hydrogen and impurities.Cross-section of eroded composite can be separated into three different regions based on transpiration cooling mechanism. Regions I and II near to exposure surface are heat affected and subjected to sever microstructural evolution. Evaporation of copper takes place in region I, and region II supplies required liquid for transpiration cooling mechanism.Post-sintering annealing of tungsten skeleton increase erosion resistance of infiltrated composite due to change of the fracture mode from inter-granular to trans-granular. This vacuum annealing can decrease the erosion rate of material from 19.7% to 11.2%. In fact, post-sintering annealing enhances W-W interface strength and delays tungsten particles separation during the erosion test.Using of a Cu-eutectic binder which may improve the consolidation of the composite; this has already been proven in other PM composite systems Studying the effect of post-sintering annealing temperature and time on the erosion resistance of infiltrated W-10 wt%Cu composite.Determination the depth of extra-sintering of tungsten skeleton during sintering and its effect on the transpiration cooling mechanism.Perspectives on the FESAC transformative enabling capabilities: Priorities, plans, and StatusIn early 2017, the Fusion Energy Sciences Advisory Committee (FESAC), an advisory committee to the United States Department of Energy, was charged with identifying transformative enabling capabilities (TECs) “that could promote efficient advance toward fusion energy, building on burning plasma science and technology.” A subcommittee with broad expertise was formed and sought feedback from scientific experts, including experts from outside the fusion community. Three workshops were conducted, and a report was approved by FESAC in 2018 that identified four of the “most promising” TECs: advanced algorithms, high-critical-temperature superconductors, advanced materials and manufacturing, and novel technologies for tritium fuel cycle control. In addition, one “promising” TEC was identified: fast flowing liquid metal plasma-facing components. This paper will give details on the promising TECs and an overview on considerations of these TECs in the United States since the publication of the report.The realization of commercially viable fusion energy would essentially solve the problem of dwindling energy supplies. Thus, the National Academy of Engineering identified “providing energy from fusion” as one of the 14 Grand Challenges for Engineering in the 21st century []. Although scientific and technological progress was steady in the last half of the 20th century, many of the international roadmaps still foresee a fusion power plant at least a generation away []. The desire to consider technologies and capabilities that might accelerate the development of a fusion energy source motivated the 2017 charge to the US Fusion Energy Science Advisory Committee (FESAC). FESAC was asked to identify the most promising transformative enabling capabilities (TECs) for the United States to pursue that could promote efficient advancement toward fusion energy, building on burning plasma science and technology. In 2018, the FESAC published a report outlining TECs []. The process for identifying these TECs and their descriptions are summarized in Maingi et al. []. This paper details perspectives on the progress and plans that have been made in the United States since the publication of the FESAC report. The perspectives in this paper are those of the authors and are not intended to represent the views of the United States Department of Energy.The FESAC report was produced by a subcommittee with expertise in a variety of scientific and technological areas. Input was sought via workshops and white papers, resulting in the identification of five TECs—four “most promising” first-tier TECs and one “promising” second-tier TEC [high-critical-temperature superconductors,novel technologies for tritium fuel cycle control, andfast flowing liquid metal plasma-facing components (PFCs), second tier.A summary of each of these areas is given in this section, with perspectives on the current plans and status in the following sections.The advanced algorithms TEC refers to a collection of mathematical methods that embody approaches addressing critical fusion problems in plasma control and optimized data analysis, while explicitly dealing with uncertainty and gaps in knowledge. They include control modeling and design, machine learning (ML) techniques, and integrated data analysis approaches. Although there is significant overlap in these fields, each area generally addresses uncertainty and knowledge gaps in different ways. For example, the domain of control mathematics includes a rich array of methods for developing control-oriented models that embody uncertainty, as well as methods for exploiting real-time feedback to ensure desired levels of performance in operational functions despite nonideal conditions. Machine learning and artificial intelligence (ML/AI) offer methods for exploiting large datasets to derive models, controllers, and other mathematical functions that can both quantify and manage uncertainty. Integrated data analysis (IDA) methods enable the combination of widely varying types of signals and data to quantify the Bayesian (or other statistical inference) justifiability of alternative interpretations. Both ML/AI and IDA can help maximize the extraction of knowledge from available data. These fields have seen varying levels of application and progress throughout the history of fusion research. Model-based control has been applied to magnetic fusion confinement devices since the earliest days of such experiments and experiencing significantly increased focus since the 1980’s []. Machine learning approaches and mathematical integration of data through statistical inference have experienced more recent application to fusion science, with earliest use in the 1990’s and a significant increase in the last two decades [High-temperature superconductivity (HTS) was discovered in the late 1980’s and has developed such that high-field magnets for fusion applications can now be considered. HTS conductors are attractive for their ability to achieve much higher fields than traditional low-temperature superconductors, which could enable smaller, high-field fusion devices, such as are being considered commercially []. The high-temperature capability could allow for the elimination of cryogens [] and be more tolerant of heating from radiative sources of a fusion reactor. In addition, HTS conductors could enable the use of demountable joints [], revolutionizing the installation and maintenance process for a reactor. Substantial interest from private industry and investors is growing, presenting a unique opportunity for rapid development of high-field fusion magnet technology based on HTS conductors.Advanced / emerging materials have the potential to enable break-through concepts for plasma-facing, blanket, and/or structural components in fusion reactors by offering unique advantages such as unconventional high-temperature capability. Materials of potential interest for fusion include (1) emerging materials of high interest in the general materials science and engineering (MSE) community, such as MAX phases, ultra-high-temperature ceramics (UHTCs), and high-entropy alloys (HEAs), (2) novel structural materials that may be specifically developed for fusion based on recent materials science advancements, such as castable nanostructured alloys (CNA) and MAX-phase ceramic matrix composites (CMC), and (3) refractory multi-functional composites that potentially enable extensive use of refractory metal like tungsten in fusion reactors by increasing the material’s performance over conventional monolithic refractory alloys. Additionally, the more “conventional” developmental materials including the nano-featured oxide dispersion-strengthened (ODS) steels and the alumina-forming corrosion resistant ODS FeCrAl alloys present significant potential of disruptive improvement in operating temperature regimes. These emerging and advanced materials and materials systems exhibit several clear and outstanding advantages over the current reference PFCs and blanket materials, presenting game-changing potentials for improved fusion energy systems.Advanced Manufacturing (AM) refers in general to a range of technologies that use non-conventional processing. Much like the changes seen in the field of ML, AM is rapidly evolving as a discipline and quickly becoming the industrial manufacturing route of choice for fabricating components with features not readily achievable by the conventional processing technologies. The novel features enabled by AM include complex geometries and transitional structures, often with materials or constituents that are refractory and/or hard to machine.Given that fusion reactors will consume substantial quantities of tritium (which will be scarce after the operation of ITER []), it will be necessary for such reactors to produce tritium. This is most likely to be done through neutron-lithium reactions in a breeding blanket surrounding the fusing plasma. The production of tritium (which may occur in either liquid or solid breeders) must be followed by the extraction of the tritium fuel. The tritium must then be processed, controlled, and accounted for. The size of the tritium inventory will have a significant impact on reactor licensing, safety, reliability, environmental impact, and cost. The expected extraction and processing rates for a reactor concept could exceed those required by ITER by over four times []. Thus, techniques to optimize the production, extraction, and processing of tritium are essential to shorten the path to bring any deuterium-tritium fusion concept to reality []. Minimization of tritium migration is being sought utilizing low permeation materials, controlled operating temperatures, specific system geometries (e.g. annular hot and cold fluid legs), and a detailed system of tritium detection and recovery.Liquid metal PFCs may prove to be an attractive alternative to handle both high steady state and transient plasma heat flux in a fusion reactor power plant, which would revolutionize control of the plasma-material interface. A continuously refreshed wall resolves some of the issues of erosion and material degradation that present with solid plasma-facing materials. However, along with the transformative potential, there are several significant challenges (fuel retention, corrosion of substrates, controlling flows, determining operational temperature windows, etc.) that led to this area being considered a “second-tier” TEC. While this second-tier TEC identified fast flowing liquid metals (on the order of m/s), slow flowing liquid metals (on the order of cm/s) were included as a part of the “advanced materials” TEC []. As both fast and slow flow are being considered in the United States currently, both will be discussed in this section.Modern tokamak control systems such as that of ITER [] provide examples of recent progress in the use of model-based algorithms and advanced architectures. illustrates elements of ITER’s plasma control system. The figure shows the role of complex online algorithms play in interpreting plasma state, exemplified by real-time Equilibrium Reconstruction and Forecasting functions, while Continuous Control and Exception Handling functions include model-based algorithms designed to optimize performance in the presence of noise, disturbances, and uncertainty. Integrated multi-actuator control algorithms, such as those governing current profile regulation, now commonly use physics-based and data-driven models []. Novel approaches have been developed to provide quantifiably high confidence control in ITER and other reactors []. Exception handling or off-normal and fault response systems can employ physics model-based control algorithms and increasingly make use of ML-based algorithms []. Machine learning research has been extensively applied to tokamaks over the last two decades for disruption prediction in order to trigger avoidance or mitigation action (e.g., to fire the Disruption Mitigation System, DMS, which in ITER is triggered by the Central Interlock System, CIS; both shown in ]. Integrated data analysis methods have continued to advance in the application of Bayesian probabilistic assessments to diagnostic measurements to quantify justifiable interpretation []. Recent applications include integration of interferometry and light detection and ranging (LIDAR) measurements in the Joint European Torus (JET) to produce a combined interpretation of electron density and temperature justified through Bayesian probabilities [] and Zeff determination from integrated x-ray tomography and charge exchange recombination spectroscopy [Since the FESAC TEC report, the fusion community has held several workshops to identify research needs for the effective application of ML/AI methods to fusion problems (e.g., []). These methods hold significant promise for accelerating or enabling solution of fusion problems, maximizing the knowledge extractable from the large cumulative dataset produced by US (and possibly worldwide) experiments and theory efforts. The most recent workshop, DOE Research Needs Workshop on Advancing Fusion with Machine Learning [], was cosponsored by the US Department of Energy (DOE) Office of Fusion Energy Sciences (FES) and Office of Advanced Scientific Computing Research (ASCR). Attendees from fusion and plasma sciences, applied math, and computer science met April 30–May 2, 2019, to discuss ML/AI applications to key fusion problems. Seven Priority Research Opportunity (PRO) areas were identified with high potential for ML/AI applications to produce transformative impacts on fusion energy.Control Augmentation with Machine LearningFor each PRO, the workshop identified relevant fusion problems, gaps in ML/AI methods that impair applicability to these problems, and research principles for maximizing the effectiveness of such applications. Detailed PRO assessments and discussion of foundational resources and activities for ML/AI applications are provided in the workshop final report [Application of the components identified in the Advanced Algorithms TEC to outstanding fusion problems can substantially enhance progress toward viable energy solutions. These powerful and related mathematical disciplines can provide solutions even when gaps exist in understanding or significant uncertainties exist in models or operational use. Focusing on algorithms based on dimensionless, regime- and geometry-independent quantities, as well as employing progressive training as fusion plasma performance advances, will enable effective ML/AI application to burning plasmas and beyond.To demonstrate the enabling capability of HTS conductors, in particular the REBa2Cu3O7-δ (REBCO, RE = rare earth elements) coated conductors, toward compact high-field fusion magnets, an aggressive development program of HTS fusion cables coupled with a comprehensive design study of compact high-field fusion magnets is the first critical step. The goal of such a program is to demonstrate a robust HTS fusion cable technology that can meet the needs of compact high-field magnets. The magnet parameters will set the performance targets for the cable development such as transport performance at relevant bending radii, background field, and operation temperature, while the actual cable performance will determine the realistic parametric space for magnet design. This iterative design and development process should be started as early as possible. Following the successful demonstration of HTS fusion cable technology, development and testing of toroidal field (TF) coils toward TF magnet assembly including demountable joints will require dedicated fabrication and test capabilities.In the past few years, the fusion magnet community has started developing high-current (50–100 kA) HTS fusion magnet cables based on multi-tape architecture []. Current work focuses on concept proof by testing cable samples to obtain first performance data. Of the few designs that have been tested, such as the stacked-tape cable, conductor on round core (CORC®) CICC and twisted stacked-tape cable, all have exhibited non-negligible degradation after a certain number of cyclic loads []. This is one of the main concerns regarding REBCO magnet technology as the magnet performance could deteriorate irreversibly. Although the mechanisms of degradation are likely related to the excessive strain that is applied to the brittle REBCO layer, more systematic tests are required to understand and address the issue and deliver a cable technology with robust performance.Another important area that the cable development program needs to address is the quench behavior of HTS cables and an effective quench detection strategy. The stored energy converts into heat during a quench that can catastrophically damage the conductor and magnet. Understanding the quench dynamics in a multi-tape cable structure, in the context of forced flow cooling, will be critical for the safe operation of high-field fusion magnets []. A normal zone in REBCO conductors, once initiated, does not propagate fast enough (on the order of mm/s) to allow quench detection based on voltage signals in a timeframe that is typically used in superconducting magnets []. The significant electromagnetic interference in a fusion magnet system can further complicate and challenge the quench detection. New techniques to detect a quench in HTS fusion cables must be developed such as the acoustic-, fiber-optic- and MEMS-based techniques []. The cable development program can provide an excellent test platform to develop these innovative techniques.To deliver cables that can meet the demanding performance needs for fusion applications, an effective cable development program will require a high-field test facility. The facility will allow cable performance to be measured and improved under conditions that are relevant to future magnet operations and provide critical early feedback to the development of cables and REBCO tapes. Several key performance issues to be addressed with a cable test facility include the following.Transport performance of HTS cables under various background fields and operation temperaturesLong-term stability of the transport performance under cycling Lorentz forcesQuench behavior of cables and effective detection and protection strategies; impact of the quenches on cable performanceElectrical and mechanical performance of electrical joints between cable segmentsCurrently, the SULTAN test facility at EPFL is one of the few operating facilities that are available for the fusion magnet community to test HTS cable samples []. PSI and its collaborators are planning to upgrade the SULTAN test facility to increase the background field from 10.9 to 15 T []. In the United States, the DOE FES, in collaboration with the Office of High Energy Physics, is exploring a cable test facility based on a 15 T dipole magnet []. The magnet could be commissioned as soon as 2025 () to test fusion cable samples and HTS accelerator insert magnets. In addition to the effort in the private sector, a dedicated fusion magnet and cable development program, as recommended by the recent NAS report [], will be critical to provide test samples to sustain the test facility.In addition to the high-field cable test facility, additional capabilities must include the following.Measure and understand the cable performance under strainDevelop demountable joints that can be performed by remote handling systemsMeasure and improve the radiation hardness of cables and other magnet materials such as insulation and impregnation materialsDevelop high strength steels, compatible with cryogenic operation, that will be able to withstand the forces imposed on high field magnets once assembledCultivate effective collaborations between national laboratories and private industry, which can significantly shorten the timeline for generation of fusion energyFor high-critical-temperature superconductors to be truly transformative, the conductor cost needs to be significantly reduced. This is possible due to the intrinsically low raw material cost for REBCO tapes []. Unprecedented orders of REBCO tapes from private industry have already been seen that challenge the present capability of manufacturers of REBCO tapes. The successful demonstration of the high-field fusion energy approach via REBCO conductors will further require the manufacturers to increase the volume and yield toward an ultimately lower cost.There are ample examples of new unconventional materials that are of high interest and are actively being studied by the MSE community for a broad spectrum of potential applications. UHTCs, MAX phases, and HEAs are among the widely studied emerging high-temperature materials that may be attractive for fusion thermo-structural components. Some of these novel materials offer attractive combination of properties for use in contact with liquid metals as well as other coolant and/or breeder systems.UHTCs include various borides and carbides, among which transition metal diborides have recently been extensively studied. ZrB2, for example, exhibits high strength (σ up to temperatures exceeding 2000 °C with a thermal conductivity kth]. Comparison of these properties with tungsten in clearly depicts the advantages of UHTCs in terms of the operating temperature window with the comparable thermal stress figure of merit (∼σ /kth). The operating temperature window for unalloyed tungsten is bound by the upper temperature recrystallization limit at ∼900 °C under irradiation and a lower temperature limit of ∼800 °C due to radiation embrittlement. Development challenges include improvement in ductility and fracture toughness and exploration of radiation effects.Refractory HEAs are another class of emerging materials of potential interest for PFCs. HEAs can consist of numerous possible combinations of elements, and very few systems have been explored thus far. Among them, for example, are the Nb-Mo-Ta-W system and its variants, which demonstrate good strength up to >1600 °C []. The high configurational entropy and reduced atomic self-diffusion are considered to potentially offer exceptional radiation tolerance in HEAs; promising radiation resistance results have been recently obtained on other HEA systems such as the Fe-Ni-Mn-Cr and Ni-Co-Fe-Cr systems []. Because of the potential for improved radiation resistance, HEAs allow access to the face-centered cubic crystal structure that generally offers enhanced ductility as compared with the body-centered cubic structure of tungsten and ferritic steels. Low thermal conductivity is considered a primary drawback of HEAs for thermo-structural applications, but they are known to approach the values for traditional low-entropy structural alloys as the temperature increases []. Although the currently known HEAs involve radiologically unpreferable elements, the concept of high entropy alloying offers potentials for high performance innovative materials of all classes from ferrous alloys to UHTCs.MAX phase materials are another option in this category, exhibiting a combination of metallic and ceramic properties. MAX phases are materials of high potential for nuclear applications at operating temperatures exceeding ∼500 °C, and they may exhibit improved radiation resistance compared with traditional ceramics at elevated temperatures due to a high-density nanolayered structure []. Similar to HEAs, the atomistically layered ternary MAX phases consist of numerous possible combinations of elements []. To date, only a few MAX phase systems have been studied for radiation effects to reveal that each ternary system has a characteristic temperature range in which the layered structure achieves dynamic recovery from radiation-induced atomic disorder. Recent work conducted in the US fusion materials science program demonstrated a lack of degradation after neutron irradiation to 20 dpa (displacements per atom) at 500 °C for Ti3SiC2 []. Properties of Ti3SiC2 are compared with other materials in Hydrogen permeation and retention are common challenges to innovative materials. In general, advanced materials with exceptional high temperature capabilities tend to retain hydrogen at relatively low temperatures. Moreover, nanostructured or advanced radiation-tolerant materials with heterogeneous nanostructures, complex alloy structures, and hydride-forming elements generally retain more hydrogen than less complex structures. However, the refractory capability of advanced materials may allow operation of the components at temperatures high enough to reduce the hydrogen interactions. Addressing the complex interactions with hydrogen in high temperature radiation environment is a critical consideration for emerging materials.The fusion materials community developed various materials in the past, including reduced activation ferritic/martensitic (FM) steels based on the 9Cr-1Mo heat-resistant FM steel and nuclear-grade SiC/SiC composites adapted from the refractory composite development for ceramic gas turbines. Work on new materials in non-nuclear areas gave the fusion materials community opportunities to modify and improve these materials to meet fusion-specific requirements such as reduced long-lived activation and enhanced tolerance against neutron irradiation. This approach is still being employed today.Castable nanostructured alloys (fusion CNAs) developed by the fusion program are a reduced-activation class of ferritic/martensitic steels designed with nanostructural features generated by their unique chemical tailoring and thermo-mechanical treatments []. These CNAs are manufactured using the traditional industrial steelmaking methods used for reduced activation ferritic martensitic (RAFM) steels, instead of the powder metallurgy route required for other advanced oxide-dispersion-strengthened (ODS) steels. Successful development of the fusion CNAs would enable access to new, low-cost, high-performance, industrial-scale RAFM steels with significantly improved high-temperature capability and radiation tolerance over the conventional RAFM steels [Several additional development opportunities are found in this category, including reduced-activation HEAs, enhanced radiation-tolerance UHTCs, and enhanced radiation-tolerance MAX-phase-matrix ceramic composites.Tungsten is the leading option for plasma-facing material despite several outstanding challenges. The lack of a viable method to ductilize the bulk form tungsten likely mandates its use in composite forms. The fusion materials community will have to carry out the tungsten composite development due to the general lack of leveraging opportunities with non-fusion applications. Fortunately, the MSE community has accumulated significant knowledge and experience with a wide variety of processing techniques for refractory ceramic composites and ceramic-metal composites (cermets), and many of these may be modified and applied to explore tungsten-based composite development. For example, continuous fiber, tungsten-matrix composites can be produced by multiple routes, including chemical vapor infiltration and powder sintering. Small-diameter tungsten fibers and SiC fibers are two prime candidate reinforcements. Distributed or semi-interconnected tungsten particulate composites with a ductile metal matrix (often referred to as ductile phase-toughened composites) may be produced through recently developed methods like rapid sintering. Unconventional processing techniques that may enable improved functional performance are found in the following section [The AM technologies have the potential to be applied to the fabrication of a broad variety of fusion in-vessel components including PFCs and blankets regardless of plasma physics configuration. A few examples help illustrate applications potentially useful to fusion system manufacturing.Composite materials are considered essential for solid PFC concepts because the refractory materials required for the plasma-facing surface do not possess adequate toughness at lower temperatures. One solution is a ductile phase-toughened (DPT) tungsten composite. Many of the AM technologies, including the laser, electron beam (EB), plasma arc lamp, binder jet, selected area gas phase deposition, and ultrasonic approaches, are suitable for fabricating such engineered composite materials due to their ability to enable rapid solidification, control phase compositions, and control grain growth. shows the microstructure of a refractory carbide–steel composite fabricated by binder jet AM [The reasons why AM is attractive for producing engineered refractory composites also apply to its transformative capability for fabricating functionally graded materials for transitional structures in fusion PFCs. For example, a seamless transition from a tungsten-rich composite plasma-facing surface to structural steel is feasible through either a low-temperature process of ultrasonic AM or an intermediate temperature rapid solidification process. shows an example of a developmental tungsten–steel composite fabricated by the room-temperature ultrasonic bonding technique and the roll bonding technique [AM techniques such as powder bed fusion and binder-jetting are capable of fabricating complex geometries that cannot be produced using conventional technologies. shows complex geometries that can be achieved in fabricating tungsten components, in this case using powder bed EB AM tungsten []. This capability can be harnessed to achieve very high and/or position-specific heat transfer, as well as position-specific variations in channel width to optimize heat transfer or tritium recovery in solid breeding blankets.Applications of AM are not limited to PFCs but extend to blanket structures, functional materials, and other components needed for fusion systems. Examples include cooling structures with complex geometries that allow enhanced cooling, such as an all-tungsten unibody finger divertor for helium impingement cooling and micro-finned or swirl cooling channels for divertors and blanket first walls. shows 3D printed tungsten divertor finger tiles with a near IR camera used for quantification of porosity and dimensional performance. Other examples could include solid breeding material with complex geometrical features to allow efficient and sustainable tritium recovery and improved heat transfer to reduce temperature gradients compared with conventional ceramic pebble bed configurations.Advantage of AM is not be limited to the ability to realize complex structures and enabling access to novel processing routes, but may extend to qualification of the materials and components. While qualification of AM components remains as a controversial issue due mainly to the difficulty in defining specifications of the material that can ideally be tailored for desired properties, the rapidly advancing in-situ process/quality monitoring technologies is presenting an innovative approach in qualifying components.The breeding blanket remains one of the most under-developed components in the fusion core. At the same time, this component must breed all required tritium, absorb > 85 % of the neutron power, and shield the outer components like superconducting magnets. In addition, the blanket has a plasma-facing surface and must remove the surface heat flux, withstand plasma particle erosion, withstand the vacuum that surrounds it, contain the bred tritium, resist plasma transients, and resist failure in accident scenarios. The tritium fuel cycle requires that the plasma be fueled and exhausted continuously, with some low percentage of fuel actually burned (∼1–10 %). The deuterium and tritium exhausted from the plasma chamber must be efficiently separated from other non-fuel components and returned to fueling to maintain a low tritium inventory. The tritium bred in the blanket, and any leakage into coolants, must be recovered and sent to storage/fueling. In addition to these primary tritium loops, there is a wide range of tritium handling, processing, and control throughout the plant. The US program is focused on RAFM steel structures and its advanced versions for higher strength at high temperatures and transmutant helium sequestration and helium coolant for high thermal conversion efficiencies.Major thrusts in the blanket and fuel cycle area are in the five primary areas. Tritium extraction is being considered from PbLi liquid metal breeder and helium coolant streams. Studies are ongoing to optimize the configuration and material of a vacuum permeator window []. Plasma exhaust separation of deuterium and tritium for direct recycling to plasma fueling is being examined with the use of a super permeable membrane and other techniques. shows the energy level diagram for a super-permeable membrane which is a combination of a very thin layer of very low hydrogen permeation material on a high hydrogen permeation material []. With sufficient energy a hydrogen atom or ion can penetrate the non-permeating material and enter the high permeating material. The mobility of hydrogen is extremely low in the direction of the low permeator and quite high in the direction of the high permeator. This difference establishes a large concentration gradient across the membrane driving hydrogen strongly toward the downstream side. Research continues to establish it precise properties, but the drive is expected to be sufficient to not require high vacuum pumping on the downstream side, although some pumping is clearly required to remove the hydrogen. Solid breeders for fusion blankets are also being explored. The critical issues being considered are the breeder basic material properties, irradiation response, macroscopic material and tritium behavior, and manufacturing possibilities (including AM techniques). PbLi compatibility with RAFM steels and other functional materials (e.g., SiC-SiC composite flow channel insert) is undergoing testing []. PbLi liquid metal simulations are being performed including blanket, manifolding and fringe field, corrosion and mass transfer, and design []. A separate program, the Fusion Energy Systems Studies [], is pursuing RAFM/helium–cooled blanket concept design and comparison, tritium behavior, process modeling, and plantwide control and accountancy.] are being explored to provide risk mitigation for solid PFCs because they can remove the plasma erosion/redeposition degradation and surface heat flux and reduce the peak nuclear damage and transmutation and, to some extent, the peak gradients on a solid PFC. The liquid metal candidates are Li, Sn-Li, and Sn. The various design concepts involve flow [] or capillary porous systems on the first wall [], and/or flow, capillary porous systems, and evaporative systems in the divertor []. Although other proposals exist (e.g., jets []), those listed appear to have a more substantial technical basis. That is, these LM concepts have developed a sufficient experimental basis that detailed engineering design can begin, albeit with continued experimentation on complex LM-plasma and LM-substrate behaviors, and LM MHD behaviors. A wide range of single to a few effect phenomena and properties can strongly influence the LM behaviors including impurities, core plasma content, segregation of LM alloys, EM forces, wetting of the substrate, hydrogen uptake, and nuclear properties []. The solid substrate material and its interaction with the LMs include corrosion (temperature, flow speed, and B-field), the need for insulation for LM magneto-hydrodynamics (MHD), and liquid metal embrittlement phenomena. Integrating a LM PFC into the fusion plant can be challenging and include the LM loop (tritium extraction, heat exchanger, cleanup, constituency control), penetrations for plasma heating and current drive or diagnostics, tritium handling, helium pumping from the divertor, pumping of LMs out of the plasma chamber, and flow geometries and magnetic fields (toroidal and poloidal). Because the LM PFC concepts are sufficiently distinct, they require a focused assessment [] rather than a generic sweep over all LM possibilities. A research sequence is suggested: develop an engineering model for the LM PFC concept to establish critical parameters and operational characteristics; examine the three leading LM candidates; establish a basic prototypic experimental platform for the given concept (e.g., chute for flowing) to access expected behaviors; establish other apparatus to explore physics that is unattainable in the prototypic platform; establish approaches to access plasma exposures (e.g., confinement facility, linear plasma device) to ascertain the critical plasma–LM interface behavior; and pursue a significant simulation activity in parallel with any design and experimental activities to validate these tools, interpret and plan experiments, and project to fusion regimes.The transformative enabling capabilities identified in the 2018 FESAC report hold the possibility of shortening the path to viable fusion energy. Research developments in the US fusion program show the promise of progress in each of these TEC areas. In 2018, the US National Academies of Science, Engineering, and Medicine published the Final Report of the Committee on a Strategic Plan for U.S. Burning Plasma Research [], which carried two key recommendations.The United States should remain an ITER partner as the most cost-effective way to gain experience with a burning plasma at the scale of a power plant.The United States should start a national program of accompanying research and technology leading to the construction of a compact pilot plant that produces electricity from fusion at the lowest possible capital cost.These recommendations align well with the TEC areas and should help promote advancing the development of these promising capabilities.Arnold Lumsdaine: Writing - original draft. Rajesh Maingi: Project administration, Conceptualization. Kevin G. Field: Resources, Investigation. Stephen Gourlay: Resources, Investigation. David Humphreys: Writing - original draft. Yutai Katoh: Writing - original draft. Charles Kessel: Writing - original draft. Xiaorong Wang: Writing - original draft.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Mechanism of plastic deformation of powder metallurgy metal matrix composites of Cu–Sn/SiC and 6061/SiC under compressive stressUnder compressive stress, the plastic deformation mechanism of the powder metallurgy (P/M) process metal matrix composite (MMC) varies with the bonding strength of interfaces. The strength of the bonds among the matrix particles, the particle size distribution, and the bonding strength between the matrix particles and the reinforcement dominate the mechanical behaviors of MMC. In this study, simple metal and reinforcement powder were simulated as globular particles in the structure. The different combinations of these globular particles were used to elucidate the mechanism of plastic deformation of metal matrix composites under loaded by compressive stress. Two types of deformation mechanisms operate, depending on the bonding strength of the grain boundary—“grain deformation” and “boundaries slip”. Additionally, the experimental results verified the accuracy of the plastic deformation mechanism proposed herein.Metal matrix composites (MMC) are broadly used in components of various pieces of industrial equipment. They typically include ceramic particles to improve their mechanical properties. The high strength to weight ratio of MMC enables it to be applied extensively in the aerospace industry. However, the dependency of the concentration of the reinforcement in the metal matrix composite on the type of processing affects the microstructure of the composite in a complicated way. Hence, different studies have reached different conclusions on the effect of reinforcing content on the mechanical properties and wear resistance of composites Research into MMC that has involved the uses of cold isotropic pressing (CIP) and spark sintering to make aluminum alloy composites, has shown that the hardness increased with the reinforcing content The plastic deformation mechanisms of MMC under compressive stress can be divided into two modes according to the strength of the matrix and the bonding strength of the grain boundaries—“grain deformation” or “boundary slip”. “Grain deformation” occurs when the bonding strength at the boundaries among the metal matrix composite particles exceeds that of the matrix particles. When MMC is under compressive stress in excess of its yield strength, the matrix particles plastically deform first. In contrast, the “boundary slip” occurs when the bonding strength of the grain boundaries is lower than the strength of the matrix particles. Under such a condition, when the loading compressive stress exceeds the yield stress of MMC, the shear strength of the grain boundaries is lower than the shear stress induced at the boundaries among the particles, causing the boundary sliding. depicts the deformation modes, “grain deformation” and “boundary slip”.The ratio of the MMC reinforcement particle size to the matrix particle size affects the characteristics of the interface of the alien material and also changes the mechanism of reinforcement. Three models can be applied to the strengthening of composites, according to the magnitude of the ratio of the reinforcement particle size to the matrix particle size.Model I: When the size of the reinforcing particles is approximately the size of the matrix particles, the distribution of reinforcing particles in MMC have a substitution type. If the bonding between the reinforcing particles and the matrix particles is weaker than the strength of the matrix particles, then “boundary slip” causes plastic deformation when the composites are under compressive stress. On the contrary, if the bonding strength between the reinforced particle and the matrix particle can be improved to exceeds that among the matrix particles, plastic deformation occurs by “grain deformation” when the composites are under compressive stress, as shown in Model II: When the ratio of the reinforced particle size to the matrix particle size exceeds 2, many matrix particles surround piled up reinforcing particles. If the reinforcing particles are uniformly distributed, then the diffusion effect at the interface between the two matrix particles during sintering will be weaker lightly. Furthermore, large reinforcing particles also have a large contact area with the matrix particles, so reinforcing particles can impede the sliding of matrix particles when the composites are under compressive stress. Under such conditions, the effect of the bonding strength of between the interface of the reinforcement and matrix on composites was unobvious. Therefore, the primary mechanism of plastic deformation of composites remained the “grain deformation” mode.Model III (interstitial-type reinforcement): When the reinforcing particles are much smaller than the matrix particles, some reinforcements are present in the voids among the matrix particles. Therefore, the total volume of the interstitial sites among the matrix particles had a critical volume. If the total volume of the reinforcements is less than the critical volume of voids, then all reinforcements will be inside these voids, as shown in (a). This result will cause the boundary of matrix particles along the gore to get well support and the deformation of these boundaries will be impeded. Consequently, under this condition, the strength of the composite can be increased effectively (as shown in , Case I). In contrast, if the volume of the total reinforcement content exceeds the critical volume, some reinforcements will be present away from the interstitial sites, between two matrix particles, as depicted in (b). Under such conditions, the effect of the reinforcements on the strength of the composite depends on the bonding strength of the interfaces among the matrix and the reinforcement particles.When the content of small reinforced particles exceeds the critical volume, some reinforcements form a bridge between the pairs of matrix particles. Moreover, the strength of the composites is determined by both the bonding strength between the two reinforcements and the bonding strength between the reinforcement and the matrix particle. The various combinations of the different interfaces that influence the strength of MMC will be as follows.Case II: When the bonding strength between the reinforcing particles and the bonding strength between a reinforcing particle and a matrix particle are less than the strength of bonding among the matrix particles, the composite strength is lower than the matrix material. Thus, plastic flow causes “boundary slip” at the weaker interface.Case III: The bonding between pairs of reinforced particles is weak, but the bonding between the reinforcing particles and matrix particles is strong. “Boundary slip” occurs at interfaces among the reinforcing particles when the composites are under compressive stress.Case IV: When the bonding strength between two reinforced particles is strong but the bonding strength between the reinforcing particles and the matrix particles is weaker than the strength of the matrix particles, a strong web structure is formed. Plastic deformation is simultaneously caused by the “boundary slip” and “grain deformation” mechanisms. “Boundary slip” occurs among the reinforcing particles and the matrix particles and “grain deformation” occurs among the matrix particles. Under such conditions, the increase in the strength of the composite is unclear and the brittleness increases with the content of reinforcements.Case V: When the bonding strength between two reinforcing particles and that between reinforcing particles and matrix particles are strong, “grain deformation” of the matrix particles causes the plastic deformation of the composite when the composite is under compressive stress. Under such conditions, the composite strength can be effectively increased.The preceding discussion indicates that, under compressive loading, the plastic deformation mechanism of powder metallurgy MMC can be systematically classified with reference to three models. Model III is associated with five cases of deformation, as depicted in The P/M process was used to make various specimens to identify mechanisms of plastic deformation. SiC particles (110 μm, 20 μm and 10 μm) were used as reinforcements and added to 6061 aluminum alloy (80 μm) powder with different weight percentages (5, 10 and 15%) to form an aluminum matrix composite. The mixed powder was then blended in a jar for 24 h to ensure uniformity. Cu–Sn alloy powder (40 μm) was used as the base powder to compare the effect of matrix strength and bonding strength. The copper base composite was blended in the same way as the aluminum base composite. After blending, mixed powders were formed into cylindrical specimens by the CIP process (210 MPa). Then, those specimens were sintered in a furnace in an atmosphere of N2 at a specific temperature (Al base: 600 °C, Cu base: 860 °C) for 2 h. A Rockwell hardness tester was employed to evaluate the hardness of these specimens, using the HRH hardness index under a load of 60 kg, and a probe in the form of a Ø=1/8 in. steel ball; the loading was maintained for 15 s. Finally, an optical microscope was used to observe the microstructure of composite specimens.After MMC of the different base metals was completed, hardness was tested and the microstructure observed to determine the accuracy of the proposed deformation models.Model I: Simulation of copper matrix (40 μm) reinforced by different amounts of SiC particles (20 μm). shows the relationship between the hardness and SiC contents of Cu–SiC MMC, and shows the microstructure of the specimens with different SiC contents. In fact, indicates that the hardest specimen is the Cu–SiC MMC with 5 wt.% SiC, and that the hardness of Cu–SiC MMC falls dramatically when the content of SiC exceeds 5 wt.%. Furthermore, the hardness of Cu–SiC MMC is lower than that of the Cu–Sn specimen when the SiC content exceeds 15 wt.%. Hence, the bonding between Cu and SiC is weaker than the strength of bonding among the Cu matrix particles. Moreover, when the SiC particle content was under 5 wt.%, the magnitude of the weak interfaces is insufficient to cause “boundary slip” deformation. Thus, the “grain deformation” mechanism dominates the plastic deformation behavior of Cu–SiC MMC with an SiC content of under 5 wt.%. Under such conditions, the SiC particles impede the plastic flow of the metal matrix and the Cu–SiC MMC is effectively strengthened. (b) shows the microstructure of Cu–SiC (5 wt.%), in which SiC particles are distributed uniformly in the matrix. However, when the SiC content exceeds 5 wt.%, the interfaces are weaker and the continuity of the well-bonded interfaces is destroyed, as shown in (c) and (d). Under such conditions, the “boundary slip” mechanism is induced, and slowly comes to dominate the plastic deformation, reducing the hardness of Cu–SiC MMC as the SiC content increases. Furthermore, when the Cu–SiC MMC has an SiC particle content of up to 15 wt.%, SiC particles cluster was in the microstructure, as depicted in (d). During sintering at 860 °C, SiC clustering weakens the interface between the two SiC particles (because the sintering temperature is too low to sinter SiC), such that the Cu–SiC (15 wt.%) MMC is less hard than the Cu–Sn specimen.Model II: Simulation of copper matrix (40 μm) reinforced by various amounts of SiC particles (110 μm). shows that the hardness of Cu–SiC (110 μm) MMC increases with the SiC content. Although the bonding strength at the interface between Cu and SiC is less than the strength of Cu particles, the SiC particles are larger than several Cu particles. Consequently, SiC particles can effectively obstruct “boundary slippage” at the interfaces between Cu particles, causing “grain deformation” to dominate the plastic deformation. In such a case, SiC can effectively strengthen Cu–SiC MMC when its surface is compressed. Furthermore, increasing the SiC content, more strongly impeded plastic flow, causing the hardness of Cu–SiC MMC to increase with the amount of reinforcing particles. This result was consistent with the other research presents the metallograph of Cu–SiC MMC with 15 wt.% SiC (110 μm). does not show clustering of the SiC particles. Additionally, large reinforcements cannot easily cluster during blending.Model III: Simulation of 6061 matrix (80 μm) reinforced by various amounts of SiC particles (10 μm). shows the relationship between the hardness and SiC content of Al–SiC MMC, and presents the microstructure of the specimens with various SiC contents. also indicates that the reinforcements in the Al–SiC MMC specimens were distributed among the interstices. also reveals that the hardest specimen was the Al–SiC MMC with 5 wt.% SiC, and the hardness of Al–SiC MMC was drastically reduced when the SiC content exceeded 5 wt.%. (b) shows that all of the reinforcements are inserted into the voids at the triangular space between the groups of matrix particles. Under such a condition, the reinforcements do not affect the strength of the bonding among the Al particles during sintering, and the boundaries of the matrix particles along the voids were sustained by the reinforcing particles. When a compressive load is applied to the Al–SiC (5%) MMC, the reinforcement can obstruct the “boundary slip” and “grain deformation”. Hence, Al–SiC (5%) MMC was the hardest of all of these specimens (Case I). However, when the SiC content of MMC exceeds 5 wt.%, the total volume of the SiC particles reinforcing particles exceeds the critical volume of voids, such that some reinforcements form a bridge between pairs of matrix particles; they are even present along the interfaces among the matrix grains forming a network structure, as shown in (c) and (d). Additionally, under such conditions, the bonding at the Al-interface is impeded and that of the Al–SiC MMC is weakened. (The sintering temperature of 860 °C is too low to sinter SiC particles.) Hence, when the specimen is compressively loaded, “boundary slip” dominates the deformation mechanism, reducing the hardness of the Cu–SiC MMC below that of a specimen without reinforcements (Case III). presents the cross-sectional microstructures of different Al–SiC MMC specimens in the direction of loading, following hardness testing. (a) shows that the plastic deformation of Al–SiC (5%) MMC was caused by the “grain deformation” of the Al matrix (model III, Case I), resulting in strengthening by reinforcement. In contrast, the reinforcements weakened the Al–SiC MMC when the SiC content exceeded 10 wt.%. (b) reveals that the plastic deformation of Al–SiC (15 wt.%) MMC was caused by the “boundary slip” mechanism, reducing its hardness (model III, case III). This result was consistent with the other research Some characteristic behaviors in response to the compressive loading of P/M MMC are summarized as follows.Plastic deformation of MMC under compressive loading proceeds by two mechanisms “grain deformation” and “boundaries slip”—according to the bonding strength among the different powder particles. Moreover, the characteristic behavior of MMC can be understood using three models, according to the size and content of reinforcement.When the reinforcements are more than twice as large as the matrix particles, the MMC is strengthened effectively.When the total volume of very small reinforcements is less than the critical interstitial volume, the MMC can be strengthened effectively.Study of oxide and α-Zr(O) growth kinetics from high temperature steam oxidation of Zircaloy-4 claddingOxidation kinetics of Zircaloy-4 cladding of fuel pins of Indian pressurized heavy water reactors (IPHWRs) under a simulated loss of coolant accident (LOCA) condition was investigated. The kinetic rate constants for the oxide and oxygen stabilized α-Zr phase growth were established from the isothermal metal-steam reaction at high temperatures (900–1200 °C) with soaking periods in the range of 60–900 s. Oxide and α-Zr(O) layer thickness were measured to derive the respective growth rates. The observed rates obeyed a parabolic law and Arrhenius expressions of rate constants were established. Percentage equivalent clad reacted (%ECR) was calculated using Baker-Just equation. Hydrogen estimation was carried out on the oxidized samples using inert gas fusion technique. The hydrogen pick up was found to be in the range 10–30 ppm. The measured values of oxide and α-Zr(O) layer thickness were compared with the results obtained using OXYCON, an indigenously developed model. The model predicts the oxide growth reasonably well but under predicts the α-Zr(O) growth significantly at thickness values higher than 80 μm.Loss-of-coolant-accident or LOCA is a design basis accident in a nuclear reactor. LOCA involves complex physical phenomena that take place sequentially during various stages of progression of the accident. During such a postulated accident in a pressurized heavy water reactor (PHWR), the Zircaloy cladding is likely to fail in two different modes; 1) Ductile mode by ballooning-deformation and rupture due to fission gas pressure in the temperature range ∼700–1000 °C and 2) Brittle mode by oxygen induced embrittlement as a result of significant steam oxidation of the cladding above 1000 °C. Clad ductility can be degraded as a result of microstructural modification due to the formation of inherently brittle oxide layer and the oxygen stabilized α-Zr (denoted henceforth as α-Zr(O)) by inward diffusion of oxygen into the cladding during the Zircaloy-steam reaction This paper focuses on the growth kinetics of oxide and α-Zr(O) in the as-received state of the Zircaloy-4 cladding. The purpose behind choosing the as-received Zircaloy-4 cladding is the fact that the power rating of fuel pin decreases with increasing burn up and the pins with thickest in-service oxide would not reach the highest temperature during LOCA The specific aspect investigated in this study was the double-sided isothermal oxidation of Zircaloy-4 cladding samples in the presence of flowing steam, in the temperature range relevant to postulated LOCA. A detailed post-test evaluation of microstructural evolution with oxygen ingress has been carried out. The oxidation kinetics has been established. An attempt has also been made to validate the indigenously developed model named OXYCON which was earlier validated against limited data generated from high temperature steam oxidation of Zircaloy-2 cladding in the temperature range 923 K–1073 K and 1323 K–1423 K The Zircaloy-4 fuel clad tubes used in this investigation were procured from Nuclear Fuel Complex, Hyderabad with the dimensional specifications corresponding to that of 220 MWe reactor fuel claddings. The data on the fuel pin cladding is presented in The elemental composition of the cladding material is given in . The clad tubes were fabricated by four stage pilgering with intermediate annealing of the vacuum arc melted ingots followed by stress relieving annealing in the final stage to relieve the internal stresses The steam oxidation studies were carried out on 5 mm wide rings of fuel cladding. Forty isothermal heating (in preheated steam) and quenching (in water) experiments were performed. A schematic diagram of the experimental setup used for the high temperature steam oxidation testing is shown in . The main components were the high temperature electrical resistance furnace having Kanthal wire as heating element, electric steam generator having water exhaustion rate of 4 lph, a steam pre-heater, control panel and a data acquisition system. The temperature of the clad tube specimen was measured by a Chromel-Alumel thermocouple spot welded on its outer surface which was in turn connected with the data acquisition system. Steam oxidation was carried out at 7 different test temperatures in the range 900–1200 °C and for soaking periods starting from 60 s till 900 s.The details of the temperature-time matrix used for the experiments are given in a. The overshooting of temperatures (for a very short duration compared to the soaking period) during each test of a particular temperature-soaking time combination is also presented in b. The furnace was heated up to the experimental temperature and the flow of the preheated steam (120 °C) in the furnace was adjusted to a constant level before introducing the specimens into the furnace. A heating rate of 30–40 °C/s to the test temperature could be achieved. The specimens were isothermally exposed in the furnace at the desired temperature in equilibrium steam flow condition for the required soaking period. The temperature was recorded during the experiment by an on-line data acquisition system. After the elapse of the required time inside the furnace under the steam flow, the specimens were removed and quenched rapidly in water at ambient temperature thereby achieving a cooling rate of 50–70 °C/s.In order to know the oxidation contribution during the heating phase, additional heating experiments were carried out where the specimens were withdrawn from the furnace as soon as the test temperature was attained and subsequently cooled in water.One typical time-temperature profile recorded by the data acquisition system for the experiments is shown in Transverse section of the specimens from all the experiments was mounted by hot mounting technique. The mounted specimens were subsequently subjected to mechanical grinding-polishing to a 600/1000 grit finish, followed by chemical polishing and etching in 10% HF-45% HNO3-45% glycerol before the examination using an optical microscope.Metallographically prepared specimens were examined using an optical microscope. A detailed microstructural study and quantitative image analysis of the various phases was carried out. A number of photomicrographs (20 on an average) were recorded at suitable magnifications for each tested sample. The measurement of the oxide (blackish grey layer on the inner and outer surfaces of the clad) and α-Zr(O) (white colored layer underlying the oxide on both the surfaces) thickness was carried out using an image analysis software. Photo-mosaics of microstructures across the clad tube wall thickness were also generated for a few selected samples by stitching a number of images together.The measurement of the oxide and α-Zr(O) layer thickness was carried out on both the inner and outer surfaces of the clad. The value reported for each sample is the average of six readings each taken on 20–25 image fields covering the entire circumference of the sample examined. The α-Zr(O) layer was observed to be uniform either for the specimens exposed to lower temperature for all soaking durations or to higher temperature only for shorter soaking periods. For steam oxidation at 1000 °C and above for longer soaking period of more than 5 min, the samples were seen to be having an irregular α/prior β boundary on account of the α-Zr(O) being non-uniform with finger like incursions. The measurement of the uniform α-Zr(O) and non-uniform α-Zr(O), i.e. uniform + incursions was carried out for these samples. The average (of inner and outer cladding surface) thickness of these phase layers were calculated from the measurements obtained from different image fields and henceforth will be designated as ξOX and ξα-Zr(O) for oxide and α-Zr(O) layer thickness, respectively. These data were subsequently considered for the study of the growth kinetics.The oxide scale and α-Zr(O) layer formed during the initial temperature transient phase of the heating was also estimated from the microstructural examination of the clad tube samples heated till the test temperature was obtained.The SEM examination was also carried out for few specimens especially for those heated at 900 and 950 °C in order to study the stratification of the microstructure and morphology of the matrix in greater detail which was not possible with the optical microscope. The metallographically prepared samples with deep etching were used for this purpose. The SEM examination was done at an accelerating voltage of 30 kV and images were recorded in BSE (back scattered electron) mode.The hydrogen content of the oxidized cladding was measured using an Inert Gas Fusion Analyzer. The samples were cut into 90–120 mg sample size by slow speed cut off machine after removing oxide layer if any by polishing. Prior to the hydrogen analysis, specimens were cleaned by Acetone and dried to remove the surface moisture that affects the hydrogen content significantly. The samples were then analyzed by inert gas fusion technique for their hydrogen content. The system was calibrated by dosing pure hydrogen gas and verified by analyzing certified Zirconium standards for hydrogen.The microstructural features observed in the steam oxidized Zircaloy-4 samples are consistent with the phase fields in Zr–O phase diagram (a and b. Finger like incursions of oxygen stabilized α-Zr inside the prior β-Zr phase is observed for specimens oxidized above 1000 °C and longer soaking durations. This is due to the fact that when the oxygen concentration in the prior or transformed β-Zr phase approaches the saturation at that temperature after a prolong exposure in steam, the growth of stabilized α-Zr phase in the form of local incursions into the transformed β-Zr phase occurs Zirconium oxide and α-Zr(O) are inherently brittle and the resistance of the fuel cladding to fragmentation under the influence of thermal shock and post LOCA handling of the fuel bundle depends on the ductile prior β-Zr phase.Photomicrographs of a selected region of the clad tube cross section observed under the optical microscope on a few representative samples, oxidized at 900 C for 10 min, 950 C for 5 min, 1000 C for 10 min, 1050 C for 1 min, 1150 °C for 5 min and 1200 °C for 1 min are shown in a–f. The corresponding % ECR values have also been mentioned in the figures. The %ECR calculation in the present study was carried out using Baker–Just correlation expressed as ECR = 2 × 873/W exp (−11550/T)√τ a–c for 3 selected specimens. The photomicrographs illustrate the fact that the evolution of the microstructure of the base material underlying the oxide layer took place during steam oxidation. Increase in the oxygen content of the alloy due to the diffusion of oxygen from oxide into the base metal during high temperature oxidation g. Hence, from the surface inwards sequentially, three layers were observed across the wall thickness of the clad tube pieces steam oxidized at 900 and 950 °C. (i) oxide layer (dark gray in color), (ii) underlying thin oxygen stabilised α-Zr and (iii) a layer of recrystallised α-Zr grains and prior β-Zr. to a relatively thicker uniform layer with a wavy pattern (d and f) and finally to a layer of non uniform thickness due to the formation of incursions penetrating into the matrix of prior β-Zr phase (e, 6a-c). The α-Zr(O) phase present in the microstructure can be categorized as isothermally grown compact α phase layer, α phase incursions and α phase islands (c). The prior β-Zr phase, observed between two layers of α-Zr(O) on the inner and outer clad surfaces is nothing but a Widmanstatten structure of α-Zr phase having an appearance of a basket weave pattern, c. Therefore, from the surface inwards sequentially, three distinct layers were observed across the wall thickness of the clad tube pieces steam oxidized at and above 1000 °C: (i) oxide layer (dark gray in color), (ii) underlying wavy/non-uniform α-Zr(O) layer (white in appearance) and (iii) a basket weave structure of prior β-Zr phase (light gray in appearance). Hence the steam oxidized clad became heterogeneous with a stratified microstructure (The effect of steam exposure at high temperature on the Zircaloy-4 cladding was evaluated metallographically to see the penetration of the brittle phases like oxide and α-Zr(O) into the ductile wall material. The oxidation kinetics of Zircaloy-4 cladding material in the temperature range 900–1200 °C was established from the evolution and growth of oxide scale and α-Zr(O) layer from microscopic observation on polished and etched samples.The oxidation contribution towards the growth of oxide scale and α-Zr(O) layer during the non-isothermal part of the heating was evaluated and was observed to be negligible for the experiments at 900 and 950 °C. At comparatively lower temperatures for all soaking durations and at higher temperatures where exposure times are ≥120 s, the contribution of the transient heating phase towards oxide and α-Zr(O) growth is 5–10%. Such contribution for higher test temperatures (1100, 1150 and 1200 °C) with exposure times ≤ 120 s is 10–15% as the history effect could not be practically fully eliminated. However, the small amount ‘preoxidation’ due to initial transient heating phase was assumed to be part of the isothermal oxidation and not excluded from the oxide and α-Zr(O) layer thickness data, thereby treating the heating cycles as approximations of “isothermal” exposures.A comparison with the literature shows that Jong Hyuk Baek et al. The results of oxide and α-Zr(O) thickness measurement are presented in . The standard deviation values for the thickness data have also been shown in the table. The values were observed to be significantly higher for 1000 °C due to unusually wavy nature of the α-Zr(O) layer. In case of α-Zr(O), two different sets of data w.r.t. α-Zr(O) phase were obtained viz; isothermally grown uniform α-Zr(O) and non-uniform α-Zr(O) layer (combination of uniform layer and incursions). Uniform α-Zr(O) layer was mainly considered for establishing the growth kinetics as the incursions either grow during heating The growth rate of the oxide or α-Zr(O) can be expressed by an equation of the form ξ = ktn, where ξ = Thickness of the layer, t = Time, k = Rate constant and n = Exponent.It is generally accepted that the mechanism governing this reaction is the diffusion of oxygen ion (anion) through the anion deficient ZrO2 lattice. Hence the growth of the layer can be considered to be following a parabolic rate law with the value of ‘n’ as 0.5 and the rate constant as kP, the parabolic rate constant. The above equation can then be re-written as ξ = kPt1/2.The oxide layer, uniform α-Zr(O) and the oxide layer + uniform α-Zr(O) layer thickness were plotted against the square root of time as shown in c, respectively. An increase in their thickness was observed with increase in exposure time at all temperatures. The rate of increase is higher at elevated temperatures. The kp values obtained from the slope of the best fit linear plots of ξ vs.t1/2for different temperatures are presented along with the regression factors in terms of R2 values in order to indicate the confidence level in . A reasonable conformity to a parabolic growth confirmed the applicability of the equation ξ = kpt1/2. Hence diffusion appears to be the rate controlling process in the oxide scale, α-Zr(O) and the combined layer growth.Since the conventional method of representing the parabolic rate constant as a function of temperature is the Arrhenius plot, the rate constant values were plotted in a natural logarithm scale as a function of the reciprocal of the absolute temperature assuming that a single process involving a constant activation energy is the rate-controlling one over the entire range of the experimental temperatures. c represent the plots of parabolic rate constants for the growth of oxide, α-Zr(O) layer and a combined growth of ξox+ξα-Zr(O). For oxide and α-Zr(O), an uncertainty in the KP values were determined from the standard deviation in the thickness measurement. Hence three plots of parabolic rate constants corresponding to mean, mean + sd and mean−sd values of oxide, α-Zr(O) and oxide+α-Zr(O) layer thickness are presented. The best fit lines of these plots were obtained by standard statistical method. The equations of the lines representing the temperature dependence of the parabolic rate constants for oxide, KOX (in μm/s0.5) for mean, mean + sd and mean−sd values of oxide thickness are as follows:The Arrhenius equations for the temperature dependence of the parabolic rate constants for uniform α-Zr(O), Kα (in μm/s0.5) for mean, mean + sd and mean−sd values of the thickness are as follows:The expressions for the parabolic rate constants as a function of temperature for oxide + uniform α-Zr(O), K OX+Uniform α (in μm/s0.5) corresponding to mean, mean + sd and mean−sd values are as follows:KOX+Uniformα(mean+sd)=55811.2exp(−13098/T)KOX+Uniformα(mean−sd)=56183exp(−13255/T)The Arrhenius plots KP vs. 1/T and the rate constant equations illustrate that oxide scale and oxygen stabilized α-Zr grew at different rates. The activation energies obtained are 27965, 24870 and 26155 Cal/mole for oxide, α-Zr(O) and combined growth, respectively.The investigations so far carried out for α-Zr(O) growth study focused on α-Zr(O) without the study of the influence of neglecting or considering the α-Zr(O) incursions. Hence an attempt had been made in the present study to investigate the influence of the incursions on the α-Zr(O) growth kinetics. presents the α-Zr(O) thickness and the parabolic rate constant values considering the uniform layer neglecting the incursions and the non-uniform layer including the incursions.The inclusion of the α-Zr(O) incursion in the data led to an expression of parabolic rate constant with a steeper slope having further implication of a higher activation energy.In the temperature range of interest during a postulated LOCA in PHWR, Zircaloy is highly reactive with steam. The high test temperatures during steam oxidation and the exothermicity of the zirconium/steam reaction introduce considerable experimental difficulties in determining the kinetics of the oxidation reaction which results in the growth of different phase layers like oxide and α-Zr(O). However, a comparison of the temperature dependence of the parabolic growth constants for oxide and α-Zr(O) established earlier with the present study was deemed necessary. A comparison of the Arrhenius expression of parabolic rate constants for oxide scale and α-Zr(O) layer growth has been carried out with Leistikow presents a comparison of the pre-exponential factor ‘A’ and activation energy values, ‘Q’ of the present study and other investigations. The dependence of the rate constant for oxide growth on the temperature was observed to be larger in the present investigation than the other findings. The discrepancy between the present and previous results could be due to the difference in the temperature ranges considered and the heating technique adopted for carrying out the tests. However the activation energy value of the present study closely matches with that of Cathcart et al. During the oxidation by steam, hydrogen is continuously generated on the surface of the cladding which is finally picked up by the bulk. The maximum hydrogen pick up in the samples during steam oxidation is presented in . The pickup is observed to be not very significant and in the range 10–30 ppm. This is also supported by the microstructural evidence as no significant hydride precipitation could be observed in the prior β-Zr phase. It is generally accepted that the crystallographic structure of zirconium dioxide produced during steam oxidation in the temperature range 800–1490 °C is either monoclinic, tetragonal or mixed type. Under otherwise similar condition of stress, impurity concentration and temperature, the ratio of monoclinic to tetragonal phase is governed by the stoichiometry of the oxide. The oxide formed in Zircaloys, on account of the presence of alloying elements having valences lower than that of zirconium, remains substoichiometric thereby having a higher volume fraction of tetragonal oxide in the growing oxide layer. As a result the fracture toughness of the oxide remains higher rendering microcracking of the layer difficult An indigenous model named OXYCON, developed for analyzing the oxidation and oxygen distribution in the PHWR cladding at high temperature during LOCA, has been used to analyze the growth of oxide and α-Zr(O) and oxygen concentration profile in the cladding. OXYCON calculates the thickness of ZrO2, α-Zr(O) and β-Zr layer and oxygen distribution in the cladding as a function of temperature and time. The model is applicable to oxidation above α/β transformation temperature (≥1000 °C). Following assumptions are made in the model:The diffusion coefficient of oxygen in oxide, α-Zr(O) and β-Zr phases is dependent only on temperature. The effect of oxygen concentration is negligible.Equilibrium concentration of oxygen exists at the phase boundaries.The volume expansion associated with oxidation of Zircaloy is normal to the sample surface i.e., in the radial direction.There is no temperature gradient across the cladding thickness.In the present version of OXYCON, the thickness of oxide and α-Zr(O) layers are calculated using parabolic rate constants. The equation for the growth of oxide and α-Zr(O) layer during a given time step were given as follows:If the OX1 and OX2 are the thickness of oxide at the beginning and end of the time step respectively, then OX2 = [(OX1)2 + 2 × Aoxide × exp (−Qoxide/RT) × △t]0.5Similarly, if α1 and α2 are the thickness of α-Zr(O) layer in the beginning and the end of time step, then α2 = [ (α1)2+ 2 × Aα × exp (−Qα/RT) × △t]0.5The values of constants in the above equations are taken from MATPRO A comprehensive illustration of the validation of the model against the measured oxide layer thickness is presented in . The agreement was observed to be satisfactory. shows the metallographically measured α-Zr(O) layer thickness against those calculated by OXYCON. The model predicts the α-Zr(O) growth reasonably well up to a thickness of 80 μm except a few scatter. However above 80 μm, the calculated values were lower than the measured values. A possible reason for the discrepancy could be the fact that at higher temperatures in comparatively longer-time oxidations, specimens exhibited large amount of significantly irregular incursions of α-Zr(O) that affected the observer's ability to define precisely the position of the α-Zr(O)/β-Zr interface. Hence making a distinction between isothermally grown uniform α-Zr(O) and α-Zr(O) incursions was rendered difficult. Although the incursions could be excluded from measurements in the samples exposed to relatively lower temperature or higher temperature with shorter oxidation duration, incursions could not be completely ignored for high temperature-longer soaking period experiments. Also the growth kinetics, in spite of being established from the average thickness of the outer and inner surface layers, the influence of the oxygen diffusion from outer clad surface on the layer growing from the inner surface and vice versa could not be completely ignored for a thin PHWR clad. Hence a higher volume fraction of α-Zr(O) was observed as a result of a longer steam exposure at higher temperatures. Furthermore, the discrepancy would have existed even below 80 μm for some time-temperature combinations and would have been still higher above 80 μm if non-uniform α-Zr(O) was considered for validation instead of uniform α-Zr(O) (Simulation of incursions needs a multidimensional analysis The high temperature steam oxidation kinetics of Zircaloy-4 used as fuel cladding material in Indian PHWR was studied in the temperature range 900–1200 C. Microstructural evolution was studied and the oxide and α-Zr(O) growth kinetics were established. The main findings of the study are given below:The microstructural features observed for the samples after steam oxidation were as follows:900 and 950 °C: Oxide layer, underlying thin oxygen stabilised α-Zr and a matrix of recrystallised α-Zr + prior β-Zr grains.1000°C and above: Oxide layer, underlying α-Zr(O) and a prior β-Zr phase beneath the α-Zr(O)The oxide scale, α-Zr(O) layer and a combined layer growth kinetics of steam oxidised Zircaloy-4 in the temperature range 900–1200 °C follow a parabolic rate law.The parabolic rate constants expressed by an Arrhenius equation in unit μm-sec−0.5 are as follows:KOX+Uniformα(mean+sd)=55811.2exp(−13098/T)KOX+Uniformα(mean−sd)=56183exp(−13255/T)The dependence of the oxide growth rate constant on the temperature in the present study was observed to be larger than the other findings. The present correlation tends to overestimate the oxide growth rate at high temperature (>1100 °C) and underestimate it below 1050 °C.For α-Zr(O) growth, the present expression of parabolic rate constant lies above the Liestikow's expression and Cathcart equation. Compared to Urbanic et al., the present correlation shows a higher growth rate at higher temperatures and lower growth rate at lower temperaturesAn indigenously developed model named OXYCON has been validated against the measured values of oxide and α-Zr(O) layer thickness. The agreement was satisfactory for oxide scale. For α-Zr(O) layer, the measured thickness was observed to agree with calculated values up to 80 μm, beyond that, there was significant under prediction by OXYCON.The hydrogen pick up in the cladding during oxidation was found to be low, in the range 10–30 ppm.Insight into the role of weak interactions on optoelectronic properties of LiGaTe2-chalcopyrite under pressure effect: DFT-D3, NCI and QTAIM investigationsWeak interactions are not negligible in layered structures and contribute strongly in their physical behaviors. Thanks to NL-vdW functionals and analysis methods of the electron density, their analysis is possible, which allows to better understand their effects on the different physical properties of solids. We have studied the role of weak interactions on structural properties, mechanical stability and optoelectronic behavior of LiGaTe2 under hydrostatic pressure effect. This study has shown that the contribution of weak interactions is not negligible in LiGaTe2. It has also been found that LiGaTe2 loses its mechanical stability by exceeding 5.64 GPa. NCI Analysis method has shown that weak repulsive interactions are more dominant than attractive ones and Li–Te bond has a strong ionic character, while Ga–Te bond is covalent. The optical properties for the different pressures have been studied by analyzing the variation of the dielectric function and that of the absorption coefficient.For several decades, semiconductor chalcopyrite compounds have attracted the attention of researchers because of their very varied physical properties. In addition to their very fascinating properties, their uncomplicated and inexpensive synthesis techniques are among the reasons for their wide use in several fields of application and several industries. They are mainly found in the industry of solar panels [Despite the large number of chalcopyrite materials that have been previously synthesized and exploited, the discovery of other new ones and the exploration of new physical properties is the current challenge of several researches. Recently, studies of Lithium-based chalcopyrite compounds have become a hot topic, as several properties of these materials have been recently determined or are under study.Among Lithium-based chalcopyrite compounds, LiMX2 (M = Al, Ga, In and X = S, Se, Te) have been studied experimentally and theoretically by several previous works. L. Isaenko et al. [] have synthesized LiGaX2 (X = S, Se, Te) using the Bridgman-Stockbarger technique from which they found that LiGaS2 and LiGaSe2 have an orthorhombic structure of Pna21 space group, while LiGaTe2 has a tetragonal structure with I-42d space group structure (chalcopyrite structure), they have also highlighted their nonlinear optical behaviors. V. A. Drebushchak et al. [] have experimentally studied the heat capacity behaviors of LiInS2, LiInSe2, LiGaS2, LiGaSe2, and LiGaTe2, from 180 to 460 K, using DSC in a scanning mode. A. V. Kosobutsky [] et al. have carried out a detailed Lattice dynamics study of LiAlTe2, LiGaTe2 and LiInTe2 compounds using density functional perturbation theory (DFPT). Using WC-GGA and mBJ potential, M. S. Yaseen et al. [] have studied the optoelectronic properties of LiGaX2 (X = S, Se, Te) in their tetragonal structure.LiGaTe2 is among the Lithium-based chalcopyrite compounds, which has attracted the attention of researchers thanks to its very fascinating optoelectronic behavior. In addition to the works cited above, A. Khan et al. [] have studied the structural, electronic and optical properties of this compound and also analyzed the type of bonding between the different atoms. A very detailed experimental study of the vibrational and optical behaviors of LiGaTe2 has been carried out by A. Yelisseyev et al. [Despite all previous studies on the physical properties of LiGaTe2, several informations are still unknown, in particular its elastic and optoelectronic behaviors under the effect of hydrostatic pressure and the role of weak interactions on these behaviors; as well as on its mechanical stability under pressure. This motivated us to take these points as those of the objective of the present work which deals, after this introduction and after the section of the calculation details and the theoretical background, the structural, elastic and optoelectronic properties under the effect of hydrostatic pressure, by highlighting the presence and the strong contribution of weak interactions on these different behaviors.The Nonlocal interactions are generally weak in some compounds and their contributions are negligible compared to atomic bonding, but in some other structures, in particular those which are in the form of atomic layers, these interactions are non-negligible, which largely contribute in the stability and the geometry of their crystal structure []. Chalcopyrites belong to this structure family, hence in this work, we highlight the role of these interactions in the geometry of LiGaTe2, and on its optoelectronic behavior and its mechanical stability under the effect of hydrostatic pressure. Therefore, we have started by testing several nonlocal van der Waals (NL-vdW) functionals [], which are based on LDA/GGA semi-local functionals and which are known for their good performance in the treatment of this kind of interactions [] (more details are in Supplementary Material (1)). Their structural results have been compared with those obtained by GGA-PBE [] taking as reference the average value of the previous experimental results, in order to know which of these NL-vdW functionals is most appropriate for the realization of the rest of this work and to show the incapacity of GGA-PBE functional alone to treat the properties envisaged in this work.This work has been carried out using FP-(L)APW + lo method [], which is implemented in WIEN2k code []. Therefore, the choice of input parameters must ensure an adequate precision of the results. RKmax = 8.5 has been adopted, which resulted in a number of 4403 of planewaves that is very consistent to ensure good accuracy of the results. Similarly, lmax = 10 and Gmax = 12 have been adopted for the maximum values of the largest G-vector in charge density Fourier expansion and for l-magnitude for the partial waves inside atomic spheres respectively. For the RMT radii of Lithium, Gallium and Tellurium, 1.55, 2.0 and 2.1 a.u have been chosen respectively. A large number of (1000) k-points for the mesh of the Irreducible Brillouin Zone (IBZ) [] has been chosen to ensure high precision of the results in a reasonable time. A limit value of 10−5 Ry for the total energy has been chosen as the convergence criterion for SCF-cycles.It has already been mentioned that in layered structures, the contribution of weak interactions (dispersion interactions) in their stability and geometry is -not negligible and should be taken into account. The structural properties and the stability of LiGaTe2-chalcopyrite have already been theoretically studied at zero pressure, but without taking into account the effect of these interactions. In this section, our objective is to highlight the effect of weak interactions on the mechanical stability and the chalcopyrite geometry of LiGaTe2 under the effect of hydrostatic pressure. Therefore, we have started by predicting the structural parameters (a0, c0 and xTe) at zero pressure using GGA-PBE functional and several non-local functionals (PBE-D3(BJ) []), which take into account these interactions, in order to compare the obtained results with the experimental ones, on the one hand, to highlight the significant presence of weak interactions and on the other hand, to choose the more appropriate functional to carry out the rest of this work.We recall that the prediction of the structural parameters of LiGaTe2-chalcopyrite structure is based on the analysis of the variation of c/a ratio as a function of the total unit-cell energy by a polynomial fit, in order to determine its optimal value and on the relaxation of its structure in order to determine the Tellurium internal parameter, which represents its position along x-direction (xTe). Once the optimized values of these two structural parameters are determined, the lattice constants (a0 and c0) at static equilibrium can be determined by adjusting the variation of the unit-cell volume as a function of its energy using one of the equations of states EoS (in this work, Murnaghan equation [The different structural results are shown in and are compared with the available experimental and theoretical values. We can see that the experimental found values are similar to each other, but those found theoretically are dispersed far from a0-experimental values. For c0, the found values in Refs. [] are very close to the experimental ones, but given the large underestimation of a0 values, the cell-volume is largely underestimated. For our results, the structural parameters found by GGA-PBE and optPBE-vdW are similar, hence, for a0, the found values are close to the experimental ones relatively to those found by the other theoretical works but for c0, the found values by these two functionals are largely overestimated. On the other hand, we note that the found values of a0 by DFT-D3(BJ), optB86b-vdW, optB88-vdW and rev-vdW-DF2 are very close to the experimental ones but the values of c0 are relatively less overestimated compared to those found by GGA-PBE and optPBE-vdW, which indicates that the found cell-volume of LiGaTe2 by these functionals are close to the experimental ones. To better illustrate these interpretations, we have shown in the values of the conventional cell-volumes, which are obtained by the different functionals and which are compared with the available theoretical values and with the average value of the different available experimental ones. From this figure, we can clearly see that the found results by NL-vdW functionals (DFT-D3(BJ), optB86b-vdW, optB88-vdW and rev-vdW-DF2) are the closest to the experimental values, which shows their performance and confirms the strong contribution of weak interactions in the geometry of LiGaTe2-chalocpyrite structure. On the other hand, the closest value to the average experimental one is that obtained by PBE-D3(BJ) NL-vdW functional, which indicates that it is the most suitable for describing these weak interactions and to study the rest of the properties considered in this work.After confirming the role of weak interactions in the study of the structural properties of LiGaTe2-chalcopyrite, we have studied in this section its mechanical stability under the effect of a hydrostatic pressure, which varies between 0 GPa and 10 GPa, using PBE-D3(BJ) functional in order to understand the elastic behavior of this compound under the effect of hydrostatic pressure and to determine the possible presence of a critical pressure Pc, which represents the limit of its mechanical stability.This study is based, firstly, on the determination of the elastic constants Cij, from where, we recall that a chalcopyrite (tetragonal) structure has six elastic constants (C11, C12, C13, C33, C44 and C66). Their determinations have been carried out using the theoretical model implemented in IRelast package [], which is compatible with WIEN2k code. The details of its theoretical background, the good performances and the detail of calculation are detailed in Ref. [, we represent the obtained elastic constants values Cij at zero pressure, taking into account the contribution of weak interactions by the use of PBE-D3(BJ) functional, from where we note that for the non-diagonal elastic constants (Cij with i ≠ j) and for C33, the found values are close to the theoretical available values, but for C44 and C66 constants, a large difference is obtained. We justify this difference by the sensitivity of the estimation of the diagonal elastic constants to the theoretical model and the adopted method, because their estimates are based on that of total energy. On the other hand, we note that the fact of taking into account the weak interactions can also be among the reasons of these differences for all the obtained elastic constants. Nevertheless, we justify the precision of our results by the bulk modulus values, which are obtained by Reuss, Voigt and Hill approximations, which are shown in and which are respectively obtained by Refs. [Hence, we can see that they are similar to each other and very close to the value obtained in the structural part by PBE-D3(BJ) functional using Murnaghan fit of E = F(V) variation (). This observation confirms the good performance of the adopted theoretical model and the accuracy of our results.For the other pressures, the same procedure of estimation has been used to predict the elastic constants of LiGaTe2 under hydrostatic pressures less than or equal to 10 GPa. shows the variation of the different elastic constants Cij of LiGaTe2 as a function of hydrostatic pressures less than or equal to 10 GPa, using PBE-D3(BJ) NL-vdW functional. We can see that all the variations are almost linear (or polynomial with a small deviation). This behavior indicates that the elastic constants increase with increasing pressure, but do not provide information on the mechanical stability of LiGaTe2 under this pressure range. Therefore, we recall that a chalcopyrite structure is mechanically stable under a given hydrostatic pressure if its elastic constants (at this pressure) satisfy the generalized mechanical stability criteria Mi, which are given by Ref. [The obtained values of the generalized stability criteria Mi for hydrostatic pressures less than or equal to 10 GPa are shown in , from which it can be seen that they increase with the increase of pressure except for M4 and M5 criteria, which decrease as a function of the hydrostatic pressure, which makes it possible to note that the mechanical stability of LiGaTe2 in its chalcopyrite structure decreases with the increase of pressure. On the other hand, we note that the value of M5 criterion becomes negative for a pressure of 7.5 GPa, which shows the presence of a critical pressure PC between 5 GPa and 7.5 GPa, from which LiGaTe2 loses its mechanical stability. To determine this critical pressure, we have plotted in the variation of M5 criterion as a function of the applied hydrostatic pressures, and the curve has been adjusted polynomially, from which it was found that the value of the critical pressure is about 5.64 GPa, which makes it possible to note that LiGaTe2 becomes unstable mechanically beyond of this pressure. These results allowed us to limit our study to 5 GPa for the realization of the rest of our work.At zero pressure, the values of the obtained elastic constants can be re-used to determine other mechanical quantities, which are of great importance such as Young's modulus E and that of shear G, using Reuss, Voigt and Hill approximations that are given by the following expressions [where; Sij representing the elastic compliance matrix elements, which is given by S, from where we notice that for a given mechanical quantity (Young's modulus or that of shear), there is a large difference found between the values which are obtained by the different approximations (Reuss, Voigt and Hill), this can inform us about a large elastic anisotropy of LiGaTe2. This latter ascertainment can be confirmed by the value of the universal elastic anisotropy index AU, which is given by Ref. [Hence a value close to zero, of this index, indicates the mechanical isotropy, while any far value indicates the elastic anisotropy. For LiGaTe2, a value of 2.83 has been found, which confirms that this compound is elastically anisotropic. To better understand the behavior of the elastic anisotropy of this compound under the effect of pressure, and for a detailed and directional analysis of this mechanical property, we have plotted the directional dependence surfaces of Young's modulus for the different pressures based on the direction cosines li and the calculated elastic compliance constants Sij using the following equation [1E=(l14+l24)S11+l34S33+l12l22(2S12+S66)+l32(1−l32)(2S13+S44)This study informs, on the one hand, about the mechanical isotropy or anisotropy of LiGaTe2, and on the other hand, about the directions for which, this compound is more elastically isotropic or anisotropic. We note that for an isotropic material, the Young's modulus value does not vary according to the different directions of space, and the directional dependence surface, in this case, is a sphere of E = Emin = Emax radius, but for an anisotropic material, this sphere is deformed, the more it is deformed the more large the elastic anisotropy. So, this property is mainly based on the analysis of the difference between the maximum value Emax and the minimum value Emin of Young's modulus, from where, for LiGaTe2, and for zero pressure, these two values are shown in , where we can see a large difference between them, which confirms once again the large elastic anisotropy of LiGaTe2 at zero pressure. The value of Young's modulus is the average value of all the points of the directional dependence surface, from where, the obtained average value Ea () is close to that obtained by Reuss approximation, which confirms the good performances of this theoretical model. In , we represent the three directional dependence surfaces of LiGaTe2 obtained for the different pressures, from which we can see that the increase of pressure increases the deformation of these surfaces, which shows that the pressure increases the elastic anisotropy of LiGaTe2. This increase in the elastic anisotropy is characterized by an increase in the difference between the minimum value and the maximum value of Young's modulus. This increase in elastic anisotropy as a function of hydrostatic pressure may be due to the decrease of the mechanical stability of LiGaTe2, which has been demonstrated previously in this work.The optoelectronic properties have been studied taking into account the structural parameters defined by the appropriate NL-vdW functional (PBE-D3(BJ)), which is mainly based on GGA-PBE semilocal functional. This functional largely underestimates the band-gap energy value []. Therefore, we have used several potentials that are known for their accuracy in estimating the band-gap energies of solids and have already been tested, and their success has been approved []. (more details are in Supplementary Material (2))The band-gap energy of LiGaTe2 has already been estimated experimentally, hence, it has been shown to be a semiconductor compound [], which highlights its optoelectronic importance. contains all the values of the band-gap energies, which are obtained by several methods, from which we can see that at zero pressure, the closest values to the experimental ones are those obtained by TB-mBJ. The obtained values by GLLB-SC are largely overestimated and those obtained by DFT-D3(BJ) are largely underestimated. These findings show that TB-mBJ is more appropriate to achieve the rest of the optoelectronic properties in this work. The obtained results show that LiGaTe2 has an indirect band-gap of Γ-H nature, this nature does not change under the effect of pressure. We also note that to our knowledge, there are no results on the behavior of the band-gap energy under the effect of pressure. We also note that the results of the fundamental band-gap, which are obtained by TB-mBJ show that it increases slightly for a pressure of 2.5 GPa then it decreases for a pressure of 5 GPa, this reduction may be due to the decrease in mechanical stability of LiGaTe2, which has been already determined in the previous part of this work. This behavior was also obtained by the other methods except that PBE-D3(BJ) showed a very moderate increase with the increase in pressure.To better understand the origin of the band-gap energy and the formation of the different bands near Fermi level, we have plotted in the electronic band structure in parallel with the partial density of states curves. For the electronic band structure, the total contributions of the states of each atom are shown with different colors and the width of the circles for each k-point at its corresponding energy (fat band plots) represents the rate of the contribution (the wider the width the greater the contribution), from which we can see that the valence band top is dominated by the Tellurium states while the conduction band bottom is dominated by a high contribution of Gallium states with a smaller contribution of those of Tellurium. We also note that the valence band top is located at Γ high symmetry point while the conduction band bottom is located at H high symmetry point, which confirms the results of , that the fundamental band-gap is indirect (Γ-H).For a detailed analysis, in the same figure and in parallel to the electronic band structure, we show the partial density of states curves, from which we can see that the valence band top is mainly dominated by p-states of Tellurium in strong hybridization with those p of Gallium, which indicates the presence of a strong covalent bond between Gallium and Tellurium and which will be proven in the next section of this work. The conduction band bottom is dominated by s-states of Gallium and p-states of Tellurium with a small contribution from all the rest of the states of all the atoms.] method has become a very powerful tool for analyzing, not only weak interactions, but also strong interactions (chemical bonds). This method is an analysis based on a combination of the electronic density (ρ(r)) and the reduced density gradient (s(r) or RDG), the latter is given by the following expression [Hence, low-s and high-ρ regions are those where the strong interactions are located while the low-s and low-ρ regions are those where the weak interactions are located []. This method also identifies the attractive/repulsive nature of the different interactions; hence, the attractive ones are characterized by a negative second electron density Hessian eigenvalue (λ2), while the repulsive ones are characterized by a positive λ2 [We have already found in the structural part of this work that the contribution of weak interactions is significant in LiGaTe2 structure, so it is important to analyze their nature (attractive or repulsive) and to identify the regions where they are located. As results, this study is based on the simultaneous analysis of the RDG diagram as a function of ρ(r) × sign(λ2) (]. These figures are also based on a color code; hence, the blue color regions represent those of strong attractive interactions, those in green color represent those of weak interactions, while those in red represent those of strong repulsive interactions. represents 2D plot of RDG as a function of ρ(r) × sign(λ2) of LiGaTe2 at zero pressure, with scutoff = 0.4, from which we can distinguish the presence of four regions: a) a region of low-ρ characterized by a positive λ2 which indicates a strong contribution of weak repulsive interactions. b) a region of low-ρ characterized by a negative λ2, which indicates a less strong contribution than the previous ones, which is that of the weak attractive interactions, which are of van der Waals type. c) a region of low-ρ but stronger than that of (b) region characterized by a negative λ2, which indicates attractive interactions (ionic bond). d) a region of high-ρ characterized by a negative λ2, which indicates strong attractive interactions (covalent bond).To better understand and analyze these different interaction regions in the conventional-cell of LiGaTe2, using the same scutoff value, we have plotted in (a, b, c and d) 3D RDG isosurfaces corresponding to each region, from which we can see that from (a), the weak repulsive interactions, which occupy the non-binding space are more dominant and occupy a larger space than that of the weak attractive interactions of (b), which are of van der Waals type, whose contribution is less dominant. On the other hand, (c) shows that RGD isosurfaces of (c) region are in the form of disc, which are in the region of bonding between Li-atoms (black balls) and Te-atoms (red balls) but closer to Li-atoms, which indicates the presence of a strong ionic character of Li–Te bonds. (d) shows that RGD isosurfaces of (d) region are located between Ga-atoms (silver balls) and Te-atoms (red balls), they are in the form of a cylinder, which is in the middle between Te-atoms and Ga-atoms (bonding regions), but the region opposite to Te-atoms is wider than the one opposite to Ga-atoms, which indicates the presence of a covalent (or polar with a strong dominance of the covalent) character of Ga–Te bond.To understand the effect of pressure on the different regions (a, b, c and d), we have plotted in RDG diagram as a function of ρ(r) × sign(λ2) for the different pressures, from which we can see that the (a, b and c) regions are slightly shifted to higher electron densities, while the (d) region is widely shifted, which indicates an increase in covalent nature of Ga–Te bonds and a slight increase in weak interactions (repulsive and attractive), while the increase in electron density for (c) region may indicate a slight decrease in the ionic character of Li–Te bonds.The quantum theory of atoms in molecules (QTAIM) is also a very powerful tool for analyzing the types of bonds between the different atoms that form solids, and this by analyzing charge transfers between bonded atoms. We recall that this method is based on the topological analysis of the electron density ρ(r), the latter makes it possible to identify the different critical points (located in rc and characterized by ∇ρ(rc) = 0) and their spatial distribution and classify them according to their types (Bond critical points BCP, Ring critical points RCP, Cage critical points CCP and Nuclear critical points NCP) []. This classification is based on the signs of the electron density Hessian eigenvalues (λ1, λ2 and λ3), which represent the nature of the electron density curvatures at the position of each critical point. This part of our work is based on these approaches, which are also implemented in Critic2 code [The BCPs indicate the positions of the various bonds in lattices, which makes it possible to identify their number and the bonded atoms, from where, for LiGaTe2, the different positions of the BCPs, for the different pressures, are shown in , from which we can see that there are only two types of bonds: Li–Te and Ga–Te, which confirms our study by NCI method. We also note that their positions vary slightly depending on the pressure, which shows that the pressure slightly changes the characters of Li–Te and Ga–Te bonds. On the other hand, the ratio between the distances between BCPs and the two nuclei of the two bonded atoms (rA/rB) can also be calculated, from which we note that for Li–Te and Ga–Te bonds, the ratio is greater than 1, which shows that BCPs are close to Li and Ga atoms because of the reduced number of their atomic numbers compared to that of Te. This ratio also changes slightly, and this is obvious because it is based on the positions of the BCPs.Once all the critical points have been determined, atomic basins (Ω-atom) and their topological volumes (VΩ) can also be determined by determining the surfaces of zero flux of the electron density gradient that separate atoms of the studied compound. shows the shapes of the different atomic basins of the different atoms which forms LiGaTe2. The charge of each atomic basin (QΩ) represents the topological charge of each atom and its calculation makes it possible to estimate the degree of charge transfer of each atom using the following expression [where OSΩ represents the nominal oxidation state of Ω-atom. contains the values of the topological volumes and charges as well as the charge transfer rate of each atom, from which we can see that the topological volumes of the different atoms decrease with the increase in the hydrostatic pressure due to the decrease in the lattice parameters. The contribution of charge transfers to this change is negligible compared to that of hydrostatic pressure. We also note that the charge transfer of Li-atom, which is bonded with Te-atom only, is very high (greater than 80%), which decreases slightly with the increase in hydrostatic pressure, which indicates that the Li–Te bond has a strong ionic character, which decreases slightly with increasing pressure. This result confirms that obtained by the NCI analysis. On the other hand, we note that the charge transfer of Ga-atom, which is also bonded with Te-atom only, is weak (less than 23%) which decreases slightly for a pressure of 2.5 GPa then it goes back up again slightly for a pressure of 5 GPa, which indicates that Ga–Te bond has a covalent character (or a polar character with a dominance of the covalent one), which changes slightly with the increase in pressure. The behavior of atomic bonding is in agreement with that deduced by the NCI analysis but its variation is different although it is slight. For the charge transfer of Te-atom, its value is close to 38%, because of its two bonding natures (ionic with Li and covalent with Ga), but its value indicates that covalent behavior is the most dominant.Once all the charge transfers of the different atoms are determined, we can estimate its average value relative to the whole lattice of LiGaTe2, which is called the topological ionicity index. Its expression is given by Ref. [From where we can see that its value is close to 46% () because of the mixed ionic-covalent nature of the bonds between the atoms that form this compound, but as we have already indicated, the covalent behavior is more dominant than that ionic. The pressure slightly changes this behavior, and its variation has the same behavior as that of Ga and Te atoms.LiGaTe2 is a semiconductor and the pressure effect slightly changes its band-gap energy, so it is important to study its optical properties and their behavior under the effect of hydrostatic pressure taking into account all the corrections that have been added in this study and which were not taken into account in previous studies. These properties are based on the analysis of the electronic transitions between the valence band top and the conduction band bottom by determining the dielectric function given by Ref. [where ε1 and ε2 represent respectively its real and imaginary parts []. Using TB-mBJ and the geometry determined by PBE-D3(BJ), the variation of the two parts of the dielectric function for the two optical directions (parallel//and perpendicularly ⊥ to z-axis) and for the different pressures are shown in , from which we can see that for the different pressures, the variations of parallel direction are different from those perpendicular, which confirms the birefringent nature of LiGaTe2 even under the effect of the pressure. The curves of the real part () make it possible to determine the values of the refraction index according to the two optical directions using the following expression [Hence, for a semiconductor such as LiGaTe2, its static value is given by: n(0) = √ε1(0). The static values according the two optical directions (n//(0) and n⊥(0)) also allow the determination of the birefringence which is given by Ref. [The different values of the refractive index for the different pressures and for the different optical directions as well as those of the birefringence are grouped in , from which we can see that the values of the refractive index found by PBE-D3(BJ) are slightly higher than those found by TB-mBJ and KTB-mBJ, but they are very close to the available theoretical ones. We also note that for the two optical directions, the values found by PBE-D3(BJ) show that the refractive index decreases slightly with the increase in pressure, while the values found by TB-mBJ and KTB-mBJ show the opposite, but the results found by the different methods show that the birefringence decreases with increasing pressure. On the other hand, the birefringence values found by the different methods are close to the theoretical and experimental values with slight differences, which show that the birefringence is positive which indicates that LiGaTe2 has positive uniaxial birefringence. Its small values show that light has almost the same behavior in both optical directions inside LiGaTe2. that the first rise for the various pressures corresponds perfectly to the value of the direct band-gap and their values are higher than that indirect, which confirms that the fundamental band-gap is of an indirect nature and confirms the obtained results of the electronic part. The first transitions for the different pressures are in the visible range while all maximum values belong to the ultraviolet range. On the other hand, we note that the variations are slightly offset under the effect of the pressure and are similar, which confirms that the band topology and the band-gap nature does not change with the increase of pressure.Knowing the variations of the two parts of the dielectric function for the different pressures makes it possible to determine the variation of the optical absorption, which is given by the expression of the absorption coefficient [The variations of the absorption coefficient for the two optical directions are shown in where we can see that for a given pressure, the parallel and perpendicular variations are slightly different, which confirms the weak birefringence of LiGaTe2. On the other hand, we note that the curves of the different pressures are similar, which confirms what has been mentioned before that the topology of bands does not change with the increase of pressure. On the other hand, the absorption begins with the value of the direct band-gap, which corresponds to the first small direct transition and it reaches its maximum values for energies close to ~8 eV. The values of the wavelengths corresponding to the maximum absorption values for the different pressures and for the two optical directions are shown in . The found values by the different methods are similar and show that αmax decreases slightly under the effect of pressure except the results of αmax⊥, which are obtained by KTB-mBJ and TB-mBJ, show that its values decrease for a pressure of 2.5 GPa then increase for a pressure of 5 GPa.The present work has been carried out by several theoretical approaches and several methods, in order to understand the role of weak interactions in the geometry of the chalcopyrite structure of LiGaTe2 as well as on its mechanical stability and its optoelectronic behaviors under the effect of hydrostatic pressure. Several findings and new results have been obtained, the most relevant are cited as follows:The study of structural parameters using several nonlocal van der Waals (NL-vdW) functionals has shown, on the one hand, that the contribution of weak interactions is significant, and, on the other hand, that PBE-D3(BJ) functional is more appropriate to study them.The mechanical stability under the effect of pressure has been studied by the analysis of the generalized mechanical stability criteria, after the estimation of the elastic constants, for the various pressures, from where it was found that LiGaTe2 becomes mechanically unstable by exceeding the pressure of 5.64 GPa. This study has also allowed the analysis of the elastic anisotropy, where it was found to increase with increasing pressure.The study of the electronic band structure and that of the density of states curves made it possible to understand the origin of the indirect nature of the band-gap and to identify the different states, which form the different bands.NCI analysis made it possible to locate and identify the type and the contribution of the various interactions (repulsive/attractive and weak/strong) in the LiGaTe2-cell. This method and that of QTAIM made it possible to identify the types of the different bonds between the different atoms that form LiGaTe2, from which it has been found that Li–Te has a strong ionic character while Ga–Te is covalent.The variation curves of the dielectric function and of the absorption coefficient for the different pressures have shown that LiGaTe2 has a low birefringence and a large absorption zone, which begins in the visible spectrum region.M. Bendjemai: Conceptualization, Methodology, Investigation, Resources, Writing - Original Draft, Validation, H. Bouafia: Supervision, Conceptualization, Methodology, Investigation, Resources, Writing - Original Draft, Validation, B. Sahli: Methodology, Investigation, Resources, Writing - Original Draft, Validation, A. Dorbane: Methodology, Investigation, Resources, Writing - Original Draft, Validation, Ş. Uğur: Methodology, Investigation, Resources, Writing - Original Draft, Validation, G. Uğur: Methodology, Investigation, Resources, Writing - Original Draft, Validation, S. Mokrane: Methodology, Investigation, Resources, Writing - Original Draft, Validation.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.The following is the Supplementary data to this article:Supplementary data to this article can be found online at Distribution and size of lava shields on the Al Haruj al Aswad and the Al Haruj al Abyad Volcanic Systems, Central LibyaThe Al Haruj Volcanic Province (AHVP) consists of two distinct volcanic systems. In the north is the system of Al Haruj al Aswad, covering an area of 34,200 km2, while in the south the system of Al Haruj al Abyad, covering an area of 7,850 km2. The systems have produced some 432 monogenetic volcanoes, primarily scoria (cinder) cones, lava shields, and maars. The density distribution of the volcanoes in each system, plotted as eruption points or sites, has a roughly elliptical surface expression, suggesting similar plan-view geometry of the magma sources, here suggested as deep-seated reservoirs. More specifically, the Al Haruj al Aswad magma reservoir has major and minor axes of 210 km and 119 km, respectively, and an area of 19,176 km2, the corresponding figures for the Haruj al Abyad reservoir being 108 km and 74 km, for the axes, and 6209 km2 for the area. We measured 55 lava shields on the AHVP. They are mostly restricted to the northern and southern parts of AHVP and date from late Miocene to (at least) the end of Pleistocene, while some may have been active into Holocene. In fact, although primarily monogenetic, some of the lava shields show evidence of (possibly Holocene) fissure eruptions in the summit parts. The early lava shields tend to be located at the edges of volcanic systems and with greater volumes than later (more central) shields. The average lava shield basal diameter is 4.5 km and height 63 m. There is strong linear correlation between lava shield volume and basal area, the coefficient of determination (R2) being about 0.75. When 22 Holocene Icelandic lava shields are added to the dataset, for comparison, the correlation between volume and basal area becomes R2 |
= 0.95. Numerical models suggest that the local stress fields favoured rupture and dyke injection at the margins of the source reservoirs during late Miocene – early Pliocene, in agreement with the distribution of the early, large-volume shields.The Al Haruj Intra-continental Volcanic Province (AHVP) is the largest part of the extensive volcanic field in Libya, which is regarded as a typical intraplate field (). The volcanic field developed during the late Miocene up to late Pleistocene and, presumably, continued to be active in Holocene. Its development has been linked to the tectonic evolution of the rifting of the Sirt Basin (). Age determinations of the AHVP have been made by many and using various techniques (, indicate an active period of volcanism from the late Miocene to late Pleistocene. suggest that the volcanic activity for the Al Haruj region commenced in the late Miocene and extended up to the pre-historic time, with the most recent lava flows (volcanic phases V, VI) being post-Neolithic in age. The older lava flows (Phases I, II, III and IV), however, were widely utilised for making Neolithic stone weapons.The AHVP is made up essentially of substantial alkaline basalts to transitional basalts as well as subordinate sub-alkaline basalts, whereas more differentiated volcanic rocks are absent ( have suggested a scheme by which to divide the basaltic lava sequence into several major volcanic facies or generations based on the characteristics of the lava flows, as observed on aerial photographs, as well as morphological features, field relations, and degree of weathering, as observed in the field. This scheme has already been shown to be useful by later studies (e.g. ). For the southern part of the AHVP, in this way, dating of main six volcanic sequences shows that the earliest one is from the end of Miocene and the youngest one from the end of Pleistocene (). Each volcanic phase is marked by multi-flow basaltic lobes (). As indicated above, however, volcanic activity in the AHVP appears to have continued into Holocene.The average total thickness of basaltic lavas in the entire AHVP is small compared with the areal distribution of the volcanic flows. More specifically, the average sequence thickness is around 145 m in the central parts and gradually decreases to a few meters in the peripheral parts, based on measurement magnetic anomalies (). We have crudely estimated the volume for each volcanic phase based on the average maximum thickness for each phase as measured in the field (B, C) suggest that the production rate for AHVP has generally decreased over time. For example, only 1730 km2, or about 11.5 km3, erupted during late Pleistocene-Holocene (0.30 Ma – 2.3 ka) whereas around 14,000 km2, or about 231 km3, erupted during the late Miocene - early Pliocene (7.94–3.67 Ma). Because some early lava flows are buried beneath subsequent flows, the areas and volumes of the early lavas are most likely underestimated compared with the areas and volumes of the later lavas, which are much better exposed. Thus the decline in lava production with time is clearly considerable, and may be greater than that indicated in There are a strong genetic relationships between most of the basaltic rocks from the various volcanic phases (e.g. ). These studies suggest that most of the rocks of the AHVP derived from a common parental mantle source with different degrees of partial melting.The volcano-morphological features and thickness of lava flows in the AHVP have suggested two different palaeovolcanic systems rather than a single system (e.g. ). The location of the exposed Eocene sedimentary rocks of the Wadi Thamat and Bishimah Formations coincides roughly to the boundary between two volcanic systems (. The northern volcanic system is called the Al Haruj al Aswad subprovince and is characterised by more than a hundred metre thick volcanic pile in the central part. The thickness, however, gradually decreases to a few metres towards the periphery of the system (). The area of the system, calculated using ArcGIS 10.1 during this study, is about 34,200 km2. The southern volcanic system is called the Al Haruj al Abyad subprovince, has a maximum thickness of about 20 m (), and covers an area of around 7,850 km2. The A Haruj al Aswad system consists mainly of lava shields rising from a lava plateau, while the Al Haruj al Abyad system is primarily composed of clusters of crater cones, of scoria spatter, and small lava shields which are aligned NW-SE to NNW-SSE (). Generally, lava shields on the AHVP appear older than the scoria cones ( suggest that the lava shields on the AHVP were primarily generated during two major eruptive stages separated by an interval of quiescence. The first eruptive stage is represented by most of the lava shields whereas the second stage is mainly marked by small cones primarily at the summits of the existing lava shields.While the geochemistry and geochronology of the volcanic systems of AHVP have thus received considerable attention in the past four decades (), the distributions of the volcanoes, as regards size and location, have received little attention, as have the associated sources, the magma reservoirs. In this paper, we focus on the volcanism and tectonics of both the volcanic systems on the Al Haruj region, with a focus on the distribution in size and location of the lava shields, and discuss our finding in the light of possible effects of the post-rift stage of the Sirt Basin on production of magma and eruption of lava. The AHVP is located on the south-western margin of the Sirt Basin (The paper has three interconnected aims. The first is to provide density maps of monogenetic volcanoes (crater cones, lava shields, and maars) so as to define the dimensions of the two main volcanic systems and their inferred source magma reservoirs. A second aim is to provide data on the size and areal distributions of the lava shields, and compare them, as regards volume, with similar shields in Iceland. For comparison, polygenetic shield volcanoes from the Galapagos and Hawaii are included in some of the plots, but the focus is on comparison with the monogenetic and, in many ways, similar lava shields in Iceland. The third aim is to use numerical modelling results to explain, crudely, the size, age, and areal distribution of the lava shields in the AHVP with references to the local stresses and loading history of the inferred source magma reservoirs.In this study, we have compiled a data set for the lava shields of the AHVP. Based on geological maps, most of lava shields belong to the first and the third volcanic phases except that four lava shields were generated during the second volcanic phase. The methodology used during the present study combines georeferenced existing geological maps () with multi-source high resolution remote sensing images available on Google Earth (2016) over the central part of Libya and digital elevation model (DEM) data collected by NASA Shutter Radar Topography Mission (SRTM). Google Earth imagery provides three-dimensional geospatial data through Keyhole Mark-up Language (KML). In the study area, the images have the high resolution of about 0.5–2.5 m (). This high resolution makes it possible to locate and precisely map the lava shields and calculate their heights, areas and diameters in the frame of the Geographic Information System (GIS), using the published geological maps as guidance for their ages. This method is a powerful tool for volcano-tectonic studies when size, remoteness, arid and inhospitable region of study induces difficulties of ground-based access (). From these results the volume and average slope angle of shields have also been estimated.The basal radius and height of each lava shield on the AHVP were extracted from very high-resolution satellite imagery, using the same criteria to mark the margin of the volcano as used by . The uncertainties or errors for our measurements are estimated at 10 ± 5%, partly based on comparison between measurement results on dykes and volcanic fissures obtained in the field and from satellite imagery (cf. ). The basal radius of a volcano is calculated using standard trigonometry. That is, the height of the volcano is the difference between elevation of the summit and the elevation of the base and then the resulting right triangle gives the radius of the lava shield. Similarly, the average slopes of the lava shields were obtained from trigonometry. The results show that the lava shields on the AHVP are, in comparison with Icelandic shields, gently sloping (cf. The areas of the lava shields were calculated using ArcGlobe. A polygon-shape file was created in ArcCatalog, and then a polyline-shape file for ArcGlobe imported into ArcMap to calculate the area for individual lava shields. Because ArGIS uses planimetric algorithms to calculate areas, it follows that ArcGIS must work with a projected coordinate system (PCS) rather than geographic coordinate system (GCS).The volumes of most lava shields were computed by approximating their shapes as truncated cones (). Here we consider lava shields as of cone or pyramid shape. Lava shields do not strictly have a cone shape, except the central parts (). However, the method can be used as a crude indicator of their volumes. More specifically, because of the small radius of the summit crater in most lava shields compared with their basal diameters, a very small error is introduced in the volume estimate even if the missing volume in the summit crater (the depression) is ignored. The approximation volume V of a lava shield is thus calculated as that of a circular cone, namely as follows:where r is the basal radius and h the maximum height of the lava shield. The volumes of lava shields are probably underestimates because a part of the lava flow field of each shield is buried beneath the lavas from later eruptions and thicknesses of the lava shields are usually imprecise (Frequency graphs and statistical analyses were made of lava shield location, height, basal radius, slope angle, area as well as volume to explore and evaluate the relationships between different lava shields and give a general view of the evolution of volcanism and tectonics in both volcanic systems. As regards location, the lava shields are essentially restricted to the north and south of the volcanic province and, as indicated, produced from the late of Miocene to Pleistocene () with possibly some activity in Holocene (). As explained, the lava shields themselves formed mostly from late Miocene to early Pleistocene, but there are indications of later fissure eruptions in their summit and central parts, some of which may be of Holocene age (). The early lava shields are generally located near the margins of the volcanic systems and the average volume of the early lava shields greater than that of later or more recent lava shields (Most lava shields on the AHVP are built up of pahoehoe lava flows (). They AHVP shields are generally steeper near the central summit crater than in the lava apron area, as is common for shields. Most of the lava shields on the AHVP are marked by well-developed cones of pyroclastic breccias, agglutinates, and lava spatters at their summits. Some of these appear to have been issued from later volcanic fissures (), as are commonly seen on lava shields elsewhere, such as in Iceland (Most of the lava shields formed during the early volcanic phases (late Miocene to early Pleistocene) and can be classified into three groups according to age (Late Miocene – early Pliocene lava shield (7.94–3.67 Ma);Late Pliocene – early Pleistocene lava shield (3.31–1.77 Ma);Early Pleistocene lava shield (1.77–1.22 Ma).Subsequently, eruptions occurred during the middle Pleistocene and into Holocene. These were mainly fissure eruptions, generating crater rows (of primarily scoria cones) up to several km long; one example is seen in . Some of the fissure eruptions, however, occurred in the summit areas of the lava shields, even if the lava shields are primarily monogenetic. The most recent eruption within the AHVP occurred 2300 years ago (). Hence the AHVP is likely to be still active, particularly the northern volcanic system.The first group (i) is composed of the largest lava shields of the AHVP. The average height is 69 m, basal diameter 5.5 km, and volume 0.46 km3. The sizes of these lava shields range up to large structures such as Qarat al Qala in the northernmost part of the Al Haruj al Aswad volcanic system (). A still larger one is Qarat Umm Gharaniq, one of the largest lava shields of the AHVP, whose height is 89 m, basal diameter 11.7 km, and volume 2.4 km3. Most of the large lava shields are located in the northernmost part of the Al Haruj al Aswad volcanic system, except that the lava shield Qarat Umm Tibesti, with a volume of 1.25 km3, is located in the northeast corner of the Al Haruj al Abyad volcanic system.The second group (ii) of lava shields consists of only 4 shields and is confined to the south part of Al Haruj al Abyad volcanic system. The Qarat Tawaylah lava shield (A), the southernmost of the Al Haruj al Abyad system, belongs to this type. The lava shield area is around 46.8 km2 and its height 49 m, yielding a volume of around 0.76 km3. Many lava shields in this group may have been buried by subsequent lava flows, particularly in the central parts of volcanic systems, which may partly account for their scarcity. By contrast, the third group (iii) is composed of shields that are widely distributed within both the volcanic systems. These lava shields have average height of 68 m, basal diameter of 3.4 km, and volume of 0.23 km3. They clearly look comparatively young (), but are smaller than the lava shields in the other two older groups (Lava shield volume and area show a reasonably strong linear correlation (). With R2 |
= 0.7459, the results mean that about 75% of the variation in volume of lava shields can be explained by, or predicted from, their variation in area. (Lava shield volume can, of course, also be taken as the independent variable and used to forecast or explain the variation in lava shield area.) Similar relationships are obtained for monogenetic lava shields in Iceland (); 22 Holocene Icelandic shields are included with the AHVP lava shields in . For comparison, we also include the large, polygenetic shield volcanoes on Hawaii (3 volcanoes) and Galapagos (12 volcanoes) together with the 22 from Iceland and 55 from the AHVP in . The radius and height of lava shields on the AHVP are clearly much closer to monogenetic lava shields in Iceland than polygenetic shield volcanoes on Hawaii and Galapagos (). On this logarithmic plot we have R2 |
= 0.6534, but the straight line is actually a non-linear function, so that the correlation (using non-transformed axes) is really non-linear.The typical monogenetic lava shields in Iceland consist of a central cone and a lava apron (). The central cone is roughly symmetrical and sloping in the range from 2° to 9° whereas the apron is typically of a roughly circular shape and extending for many kilometres with regional slopes from 0.2° to 5° (). Monogenetic lava shields, as in Iceland and AHVP, are believed to be formed in single eruptions, some of which may have lasted tens of years (). The eruption usually commences with a fissure but then gradually becomes focused at several points so as to generate several overlapping small shields. In final state, the eruption becomes concentrated on the one large point that forms the main shield that buries the overlapping smaller shields (A lava shield consists of multi-flow lobes or units, features that can be easily recognised in the field as well as from satellite imagery. Some of these features may have be identified by earlier authors (e.g. ) and given phase numbers to decipher whether multiple-flow lobes for each volcanic phase is the product of a single eruption, a multi-episode of a single eruption, or the result of totally different eruptions (). Based on field relations in the AHVP each volcanic phase is more likely the result of a single eruption than many eruptions. While each individual eruption begins with a volcanic fissure it may stop at any stage – so that a large lava shield is not a necessary the end stage (). Therefore, small overlapping cones may be initiated in an individual eruption from a segmented volcanic fissure as we have observed in many places in the Al Haruj al Abyad. For example, the 23-km-long Anquwd al Yasarat fissure, the longest one in the AHVP, produced five small lava shields and dozens of crater cones () rather than a single large lava shield.As indicated above, there was a gradual decrease in volume of lava shields and in the intensity of the volcanic phases through time. Also, the eruptive volumes in the Al Haruj al Aswad system are, on average, an order of magnitude larger than those in the Al Haruj al Abyad system (). This may suggests that the total volume during the lava shield phases produced by Al Haruj al Aswad is much than that produced during the same periods by Al Haruj al Abyad.The spatial distributions of 432 volcanic eruption sites have been examined and presented as an eruption-point density map, that is, as the number of eruption points per square kilometre (). For this task, ArcGlobe 10.1 was used to plot accurately the volcanic vents and polyline.shapefile was then imported to ArcGIS 10.1 to create the density map. Point density map calculates the number of points, here eruption points or sites, per square kilometre within a circular neighbourhood of a given radius and presents as intensity or magnitude contours. The spatial density map was drawn by ArcGIS 10.1 that used a search radius of 40 km to produce a more generalised output raster where all the volcanoes that fall within the neighbourhood (40 km) are used when calculating the density. Therefore the red spots in represent the maximum density which then gradually decreases (indicated by the colours) away from the centre. The 40 km search radius highlights the crustal-scale distribution (). Only volcanoes with clear circumferences are used in making the map (), so as to reduce the possibility of including non-volcanic features. The volcanoes used are lava shields, scoria cones, spatter cones, and a few maars (the latter occur mainly in the southernmost part of the AHVP). The similarity of clustering or density of the eruption points in the north and south volcanic systems indicates that the mechanism of magma generation was most likely similar (). In addition, many of the eruption points are aligned and, apparently, follow pre-existing fractures, possibly used partly as channels. More details on the magmatism in AHVP and the tectonic activity in the nearby Sirt Basin are provided by The spatial distribution of eruption points, presented as magnitude contours, can be subdivided into two main density subzones or subareas. One is in the northern part of the AHVP and mainly made up of lava shields, as well as some scoria cones, including the youngest volcanoes in the area. The second subarea is in the southern part of the AHVP and contains mainly scoria cones or crater rows associated with volcanic fissures and small lava shields. The central area between these subareas has a very low density of eruption points () and marks the boundary between the two volcanic systems identified in . The suggestion as to the boundary between the systems is supported the earlier studies (e.g. ) which indicate that the Al Haruj Volcanic Province (AHVP) consists of two different paleovolcanic systems. We suggest that each system is supplied with magma from a specific magma reservoir. The density maps may be used as a crude indication of the geometry and location of the source magma reservoirs.The basalts of the AHVP show broad similarities, with only minor geochemical differences being recognised (). The primary melts may have been generated by 1–5% melting of a garnet-peridotite mantle region. The depth of origin of the partial melts may then have extended to the depths of 80–150 km ( suggest that the melts of the AHVP were (and are) generated at depths of 70–74 km with a much higher melt fraction, namely between 13.5 and 18.9%.The estimated equilibrium pressure values from studied peridotite xenoliths indicate that the magmas erupted at AHVP derive from magma reservoirs at temperatures of 850–950° |
C, pressures of 1–1.8 GPa, and thus from depths of 35–55 km ( suggest that fractional crystallisation of the primitive magma occurred at depths of 25–39 km and temperatures of 1215–1360° |
C. The location of the reservoirs would then be in the transition zone between the lowermost crust and upper mantle, as is also the depth of many magma reservoirs which form part of the partially molten ‘magma layer’ in Iceland (). More detailed geophysical studies may help provide a better constrain on the magma layer in general and that of the reservoirs of the AHVP in particular.The low-density magma, the lighter magma, tends to move upwards to the regions of minimum potential energy (), and gradually forms layers (compartments) with high magma proportion, even totally molten pools, in the upper portions of the reservoirs (cf. ). Therefore, the magma content or proportion in the partially molten of upper parts magma reservoirs may be much greater than that in the lower parts (The high degree of thermal maturity of shallow sediments as seen in oil wells in the western margin of the Sirt Basin indicates that more than one kilometre of sedimentary sequence was uplifted and eroded during the late Miocene (). Regional uplift during extension is commonly the effect of the excess pressure in and doming of a magma reservoir (). The late Miocene uplift in the whole west part of the Sirt Basin is supported by the burial history of wells (QQQ1-11; aa1-11, Zallah trough) in northern part of the area of investigation (). Thus the crustal doming during the late Miocene and increased melt migration into the reservoirs may have led more elongated magma reservoirs in the vertical section (thicker reservoirs) and accumulation the lighter magma (basaltic magma) in the upper portions. As a result the locally thicker crustal parts between two magma reservoirs may have acted as crustal barrier so as to hinder magma to transport between reservoirs (The main reason for uplift of the western part of the Sirt Basin during the late Miocene is still widely disputed, however. Some authors (e.g. ) suggest that lithospheric doming was most likely associated with the mantle upwelling as found in many areas in Europe, where the asthenospheric rise caused decompression partial melting at various levels ( suggest that extensive volcanism in the AHVP and larger parts of North Africa may be the result of diapiric upwelling plumes or hot fingers in the upper mantle. By contrast, suggest that the relatively low temperatures obtained from studies of peridotite xenoliths do not support the existence of hotspot under the Al Haruj region during the time of the main volcanism but rather the presence of a relatively cold lithosphere. They suggest that the volcanism of the AHVP was related to the extension of the lithosphere, that is, passive rifting. The magmatism of the AHVP, however, post-dates the extension peak of the Sirt Basin. It follows that the thinning of lithosphere may not have had a direct relation to the melt generation. It is also noteworthy that the main volcanism took place at the south-western periphery of the Sirt Basin, which has a stretching factor of 1.1, rather than in the central part, which underwent significant thinning and has a stretching factor of 1.4. These values of stretching are derived from back-stripping geophysical results by The rough elliptical surface expression of the distribution of eruption points or volcanoes in the density map may be taken as an indication of ellipsoidal geometry of the magma reservoirs beneath the Al Haruj region (). This follows from the observation that volcanic fields and volcanic systems tend to reflect the plan-view geometries of the source reservoirs (), a conclusion that follows from stress-concentration considerations (). It is well known that magma reservoirs may extend in a direction parallel to the maximum compressive stress σ1 and perpendicular to the maximum tensile stress σ3 (). So if we take the reservoirs to be oblate ellipsoids, as a first approximation (), the minor axis would be oriented NE-SW, that is, parallel to minimum principal compressive stress while major axis would be trending NW-SE, that is, in a direction parallel to maximum principal compressive stress (The areas and lengths of the axes of the two magma reservoirs have been calculated through ArcGIS 10.1. We use the measured area of the Al Haruj al Aswad volcanic system, 19,176 km2, and its major and minor axes of 210 km and 119 km, respectively. Also, for the Haruj al Abyad volcanic system, the area is 6209 km2 and the major and minor axes are 108 km and 74 km, respectively. The volume of reservoir may be estimated if the thickness is known and vice versa, using the equation (where Vb and C are the bulk volume and the half thickness of reservoir respectively, while A is the area.Mechanical behaviour of magma reservoirs can be modelled as poroelastic (). Thus the volumes of the reservoirs during individual eruptions in the AHVP may be roughly estimated by the following equation (where Ve is the volume of eruptive material in a single eruption, φ is fraction porosity of the reservoir, pe is the excess magmatic pressure in the reservoir, βm and βb are magma compressibility and bulk compressibility of the reservoir, respectively.Here we assume that the magma reservoirs were generally located at an average depth of around 35 km, which corresponds to the boundary between the lower crust and upper mantle (). The structural lithosphere of the Sirt Basin has been subdivided into several layers including upper mantle, lower crust, upper crust and uppermost crust. The latter involves Palaeozoic, Mesozoic and Cenozoic sediments (). The Moho discontinuity is estimated at around 35–40 km deep in southeast of the Sirt Basin while it is at around 26 to 33 km in the centre and northeast parts of the basin (). Thus, it seems plausible that the magma reservoirs are located at the boundary between the lower crust and upper mantle.Elastic rock properties of the uppermost crust of the Sirt Basin have been measured on extracted drill cores samples as well as through downhole studies (). The mechanical properties of the deeper crustal parts were inferred from a global model for the Earth's crust based on seismic refraction data (). The compression (Vp) and shear seismic velocities (Vs) of the main three crustal layers and the underlying mantle are given in Table (). The values of Young's modulus for the upper crust, lower crust and the underlying mantle were calculated by using the compression and shear seismic velocities and densities from global model for the Earth's crust data based on the following equation (where Ed is dynamic Young's modulus, Vp is compression wave velocity, Vs is shear wave velocity and ρ the density of the rock. The static Young's modulus is commonly half the dynamic Young's modulus according to laboratory measurements (). Therefore, the values of static Young's modulus in Table (2) are equal to half value of the dynamic Young's modulus.Young's modulus for the layer 7 (the lower crust), which is most likely to host the magma reservoirs, is about 60 GPa (). The bulk modulus K for this layer can be calculated from the following equation (where E is the static Young's modulus and ν is the Poisson's ratio whose common value for most solid rocks is around 0.25. Therefore, the compressibility β of layer 7 (the reciprocal of bulk modulus) is about 2.50 × 10− 11 |
Pa− 1 and compressibility of basaltic magma at 1100–1300° |
C is about 1.25 × 10− 10 |
Pa− 1 (). At the time of reservoir rupture, the excess magmatic pressure is equal to the tensile strength of the host rock. Therefore, we may substitute excess pressure by tensile strength of the host rock in the Eq. where T0 is the tensile strength of the host rock.The average unconfined compressive strength of the rocks in the Sirt Basin measured in laboratory tests is 36 MPa (). Tensile strength has a well-known relation with unconfined compressive strength, but the relation depends somewhat on the rock type ( propose a relation between unconfined compressive strength (USC) and tensile strength T0 of limestone on the form:This relation is very close to the well-known assumption that the ratio of compressive strength to tensile strength is approximately 10 (). Accordingly, the average tensile strength of rocks in the Sirt Basin is around 4.2 MPa. This conclusion is in good agreement with the common in-situ tensile strength of typical solid rocks of 4–5 MPa (, and inserting the appropriate values for the parameters, the bulk volume of the reservoir is obtained from the following equation:As we see here the volume of the reservoir needed to produce a given eruptive volume Ve depends upon the assumed magma fraction φ. For instance, if we assume the magma reservoir was totally molten (φ |
= 1) the volume of reservoir would be smaller (≈ 1588Ve) than if the same reservoir had only 0.25 magma fraction(≈ 4233Ve). |
Although the volume of a single eruption in the AHVP is often difficult to estimate, particularly since many individual volcanic phase or products could be formed in more than one volcanic eruption (as discussed earlier), the lava shields are generally believed to be monogenetic (). That is, the formation of each lava shield is thought to be related to one eruption, although some of the eruptions may have lasted for several years or more. Thus, most lava shields on the Al Haruj region may have assumed as formed in single eruptions where the whole reservoir may have supplied magma during the eruption.The average volume of latest Miocene – early Pliocene lava shields on the Al Haruj al Aswad volcanic system is greater than the volume of same age lava shields on the Al Haruj al Abyad volcanic system, about 0.61 km3 and 0.34 km3, respectively. For a rough estimate of the average volume of a corresponding feeder dykes, the average length and thickness (opening) of volcanic fissures is 364 m and 1 m, respectively (), and its height (dip dimension), using the estimated magma source, is about 35 km depth. Thus the estimated average volume of the feeder dykes is about 0.013 km3. Combining the feeder-dyke volume and the extrusive volumes, the average volume leaving the reservoir Ve for the two volcanic systems becomes 0.623 km3 and 0.353 km3. Hence the area of individual reservoir has been estimated from ArcGIS 10.1 and volume of reservoir may be estimated from Eq. for individual volcanic eruption. Using these, the thickness of individual magma reservoir may be roughly estimated from Eq. If the uppermost part of the reservoir for the Al Haruj Aswad volcanic system were totally molten, then, from Eq. , the volume of that part of the reservoir which supplies magma during eruptions would be 990 km3. If, however, the reservoir was partially molten with a porosity or magma fraction of 0.25 (cf. ), the bulk volume of reservoir would be 2637 km3. The average volume of early lava shields in the Al Haruj al Abyad volcanic system is around 0.34 km3 and the estimated average volume of the feeder dykes, assumed the same as for the northern volcanic system, about 0.013 km3. It follows then that the volume of the corresponding reservoir would have to be 561 km3 if the φ |
= 1 and 1494 km3 if φ |
= 0.25. The thickness of the reservoir is estimated from Eq. as about 78 m for the Al Haruj al Aswad volcanic system and as about 136 m for the Al Haruj al Abyad volcanic system when the reservoirs are assumed completely molten. By contrast, the thicknesses of the reservoirs for the north and south volcanic systems are around 206 m and 361 m, respectively, when the reservoirs are assumed partially molten (φ |
= 0.25). Notice that although we give the average thicknesses of the reservoirs to a metre, these are crude estimates and depend strongly on the assumed melt fraction φ and other factors, discussed below.The estimated thickness values for the reservoirs are similar to those of thick sills, such as are exposed in many continental areas. It should be noted, however, that the reservoirs are likely to be much thicker. The thickness values indicated here, and below, refer primarily to the thickness of the uppermost parts of the reservoirs supplying magma to the individual eruptions (). Under certain conditions, a larger part of the reservoir (that is, a much thicker layer) would contribute to the eruption – for example during major caldera or graben subsidence into the top of the reservoir (). So, generally, the reservoirs may be much thicker than the above values – of the order of kilometres, as discussed further below.There may also be an uncertainty in the assumption that each individual lava shield represents one magma flow from the reservoir. It is possible that several lava shields were formed simultaneously from a single reservoir, in which case the reservoir volume, in particular the reservoir thickness, would be increased significantly. For example, the Anquwd al Yasarat volcanic fissures appear to be erupted during a single eruptive cycle () and fed from the same feeder dyke. This follows because the distances between the nearby segments tips are very small in comparison with the segment lengths. The segments are therefore functioning mechanically as single continuous fracture (). Hence we may also estimate the volume and thickness of the associate magma reservoir assuming that several lava shields were generated in a single long-lasting magma flow from the source reservoir, indicating a larger reservoir thickness than in the estimate above (the estimated reservoir area is essentially constant given the density distribution of the eruption points, , so only the thickness changes when the volume erupted during a single magma flow from the reservoir increases as would be the case if several lava shields were formed in a single flow out of the reservoir).For these particular volcanic fissures (Anquwd al Yasarat) the volume of eruptive materials is estimated at about 4.17 km3 using ArcGIS 10.1 (). The corresponding reservoir area would remain the same, but a thickness changes. For φ |
= 0.25 the thickness would be 4265 m and for φ |
= 1.0 (totally molten) the thickness would be 1600 m. These are both plausible thicknesses.Whatever the magma fraction the reservoirs, only their uppermost parts normally supply magma to an eruption. This follows from general models on magma chambers/reservoirs () and is also in agreement with most basaltic rocks in the AHVP being moderately fractionated, with MgO value in the range 7 to 9% (The finite element method (FEM) is the most common numerical technique used for solving differential and partial differential equations (). Numerical models based on the finite element method can provide quantitative information on the local stresses field distribution around magma reservoirs. Numerical models can thus improve our understanding the volcanotectonic activity in areas such as the present one, in particular as regards region the stress-conditions for magma reservoir rupture, dyke injection and propagation with the potential to feed eruptions. Comsol Multiphysics (5.1), the finite element program used here, is a commercial multipurpose software (We made several numerical models in order to explore the potential effects of uplift or doming of the western part of the Sirt Basin on the stress fields around magma reservoirs in the Al Haruj region. We conclude that the local stresses around the magma reservoirs are favourable to rupture and injection dyke during the late Miocene – early Pliocene boundary conditions.The lithosphere of the Sirt Basin has been subdivided into eight mechanical layers, comprising the upper mantle, lower crust, upper crust as well as uppermost crust with the Palaeozoic, Mesozoic and Cenozoic sedimentary rocks (), as already mentioned. Thus, the crust and underlying mantle of the western part of the Sirt Basin are here modelled as eight layers with various thicknesses, densities, and Young's moduli (). All the layers are assumed to have the same typical Poisson's ratio, 0.25, for solid rocks.The magma reservoir was modelled as elliptical with a major axis of 193 km and located at a crustal depth of 35 km, namely at the crust-mantle boundary. The reservoir is, in some of the models, subject to a constant internal magmatic excess pressure (pe) of 5 MPa, which corresponds to the typical in-situ tensile strength (T0) of the host rocks in the Sirt Basin. There is, in addition, the tensile loading of 5 MPa associated with the regional stress field. Some models have, in addition, regional doming-related pressure at the bottom of the crust. All models are fastened at the lower boundary to avoid any rigid-body rotation and/or displacement. The numerical models then provide the magnitudes of the maximum principal tensile stresses as coloured contours (B), we focus on the effect of doming due to excess magmatic pressure at the bottom of the crust. The doming is known to have been about 1000 m. Here the doming is generated by 20 MPa excess pressure at the depth of the crust-mantle boundary and extends under the entire western part of the Sirt Basin. More specifically, an entire 850 km wide zone is subject to doming at the bottom of lower crust. In addition there is 5 MPa extension at the boundary of the model, reflecting the extension associated with the Sirt Basin. By contrast, in the second model the uplift or doming is confined to the reservoirs of the volcanic systems themselves and their immediate surroundings (). Thus, the main difference between the models is that the model in B relates to general large-scale crustal doming over a wide area, including the western part of the Sirt Basin. By contrast, the model in shows the local effect of excess pressure, expansion (and dome local doming) in the reservoirs themselves.The local stress field in the western part of the Sirt Basin was influenced particularly by doming in the late Miocene. The doming induces stresses which, in turn, depend on the mechanical properties of the crustal layers. The resulting variations in the ambient stress field in magnitudes and orientation are commonly referred to as stress perturbations and give rise to specific patterns referred to as the local stress field (). The local stress field magnitudes reach several tens of megapascals (MPa) (B). We infer that the local palaeo-stress field controlled the generation and reactivation fractures in the area, in particular as observed in seismic lines of the western part of the Sirt Bain (cf. ), we assume that the original doming stresses, resulting in compressive stresses at the crust over the 850 km wide area, have been somewhat reduced. This relaxation may have taken millions of years. After the relaxation of compressive stresses in the bottom of crust, at the crust-mantle boundary, the magma reservoir itself induces local tensile stresses around its margins. The induced tensile stresses around reservoir margins result in increased chances of reservoir rupture and injection of dykes, particularly from its margins to feed eruptions ( A). The eruptions may have been sustained in frequency until the middle Pleistocene ( B) and then declined in number and volume since the middle Pleistocene. Subsequently, the volcanic activity in the Al Haruj al Abyad system diminished much in middle Pleistocene whilst the magmatism and volcanism still continued at least Holocene near the centre of the magma reservoir of the Al Haruj al Aswad system, producing much smaller fissure eruptions with more evolved basaltic rocks than the earlier eruptions (i.e. This study focuses on monogenetic volcanoes in the AHVP area in the western part of the Sirt Basin in Libya. We show that the volcanoes form two density peaks or groups that can be distinguished as two separate volcanic systems. Each systems contains numerous monogenetic volcanoes, particularly crater cones, lava shields, and (some) maars. Each system is thought to be fed by a deep-seated and very extensive magma reservoir.The lava shields on the AHVP are mostly gently sloping, based on the classification of . They are generally more gently sloping but otherwise roughly similar to many Holocene monogenetic lava shields in Iceland in terms of areas and volumes, although the largest shields in Iceland are considerably larger than those of the AHVP. By contrast, the lava shields of AHVP are many times smaller than the polygenetic shield volcanoes on Hawaii and Galapagos. The AHVP lava shields are regarded as being mostly monogenetic, that is, formed in single eruptions, some of which may have been of long duration.The spatial distributions of the lava shields are essentially restricted to the north and south parts of the volcanic province. They were primarily formed in the period from late Miocene to early Pleistocene, but some volcanic activity may have continued into Holocene. In particular, there are indications of small fissure eruptions – some of which may be of Holocene age – in the summits of some lava shields. The early lava shields are mostly located at the margins of the volcanic systems and generally of larger volumes than later-formed shields – which are also geographically more widely distributed. |
The volumes of an individual lava shields and areal distribution of volcanic phases gradually decrease with time. More specifically, the volumes of individual lava shields and sizes of the areas covered by volcanic phases in the Al Haruj al Aswad volcanic system is, on average, greater than similar eruptions and phases in the Al Haruj al Abyad volcanic system. It is clear that the total volume of lavas produced over a given time period in the northern volcanic system is much more than the total volume of lavas produced, during the same time, in the southern volcanic system.Most of the eruption sites or points are concentrated in two areas within the AHVP. These distributions suggest the existence of associated separate magma reservoirs () beneath the volcanic systems, namely one reservoir beneath each system. Thus, we propose that the AHVP may be divided two distinct volcanic systems, each with its own source magma reservoir, a suggestion also made by other authors (i.e. ). The two proposed volcanic systems are those of Al Haruj al Aswad and Al Haruj al Abyad. the magma fraction of the reservoir presumably played a significant role in controlling the eruptive volumes at the surface. Due to the uncertainty about individual volcanic eruptions in the AHVP, we have made two assumptions to estimate bulk volumes and thicknesses of magma reservoirs that supplied magma to the eruptions of the AHVP. First, that each individual lava shield represents a single eruption. Second, that crater rows (volcanic fissures) and overlapping small lava shields represent single eruptions. Although the volume and thickness of parts of the reservoirs that supply magma to eruptions are quite different depending on which assumption is used, general considerations suggest that in nearly all the eruptions in the AHVP only the uppermost part of each reservoir supplied magma to the eruption. This means that the potential of the reservoirs – the maximum eruptive volumes that they can generate – is much greater than the typical volume (cf. Several numerical models were made in order to investigate local variation in the stress field resulting from general doming of the area, as well as local loading by magma reservoirs through internal excess magmatic pressure and external tensile loading. We also include the effects of the mechanical properties of the layers that constitute the crustal segments forming the roofs of the magma reservoirs. Our finding in the first model, based on large-scale regional doming at the lower crust-upper mantle boundary, reveals that compressive stresses are generated in the lower part of the crust (tensile stresses in the upper part), and that these compressive stresses would tend to hinder dyke initiation and propagation to the surface. Thus, this type of loading is clearly not favourable for rupture of magma reservoir and dyke injection and eruption.Following the doming and, then, gradual relaxation of compressive stresses in the bottom of crust – partly because of the effect of remote regional tensile rifting stress - excess magmatic pressure led to induced local tensile stresses around margins of magma reservoir. The relaxation time was presumably millions of years but eventually relaxed the compressive stresses around the magma reservoir. The relaxation seems to have continued during most of the Tortonian time to early Messinian time. Subsequently the volcanic activity began in the Al Haruj region at the Messinian time (A). The eruptions continued until the early Pleistocene (B), after which they gradually declined in volume and number, particularly since the middle part of the Pleistocene. The Al Haruj al Abyad volcanic system became more or less inactive in middle part of the Pleistocene while activity continued near the centre of the volcanic system of Al Haruj al Aswad, although with gradually smaller eruptive volumes, issued primarily in fissure eruptions, and producing more evolved basaltic rocks (Enhanced removal of Cr(VI) by polymer inclusion membrane based on poly(vinylidene fluoride) and Aliquat 336New polymer inclusion membranes (PIMs) based on poly(vinylidene fluoride) (PVDF) (polymer matrix), tricaprylmethylammonium chloride (Aliquat 336) (ion carrier) and 2-nitrophenyloctylether (2NPOE) (plasticizer) were successfully elaborated by casting evaporation method and used in selectively facilitated transport of Cr(VI) ions in an acidic aqueous medium. Obtained PIMs are dense and homogeneous and are characterized by intermolecular interactions of the membrane components (i.e. polymer matrix, ion carrier and plasticizer). The presence of ion carrier and plasticizer enhances the membrane flexibility and its hydrophilic character. The decrease of the PVDF melting point is ascribed to the strong electrostatic interactions between liquid compounds (i.e. ion carrier and plasticizer) and polymer chains. PVDF-based PIM with only 20 wt% of Aliquat 336 ensures almost complete transport of Cr(VI) ions from the donor to acceptor phase. Moreover, the addition of 5 wt% of plasticizer significantly increases the transport flux. Also, Cr(VI) ions are selectively recovered (~97%) from a mixture containing other heavy metal ions (Cd(II), Pb(II), Fe(III), Zn(II), Cu(II), Ni(II), Co(II)) with such PIM. Elaborated PVDF-based PIMs reveal improved transport properties compared to other polymer-based PIMs, exhibiting high stability (more than 190 h) and lifetime durability and so they are suitable for long term application.Chromium (Cr) is a common heavy metal pollutant in water, where it mainly exists in two stable oxidation states, namely, hexavalent chromium (Cr(VI)) and trivalent chromium (Cr(III)). The hexavalent chromium is chosen as a target molecule because of its high hazard Various methods have been developed for the Cr(VI) removal from wastewater, e.g. filtration membrane, ion exchange, reverse osmosis, precipitation, electrochemical treatment, solvent extraction, adsorption/biosorption, and others Polymer inclusion membranes (PIMs) have recently attracted attention for the separation of metal ions and small organic compounds from their aqueous solutions PIMs are generally prepared by casting a solution containing a base polymer, an extractant (ion carrier) and a plasticizer. After the solvent evaporation, a stable and flexible thin film is formed. Owing to the compatibility with the majority of extractants, cellulose triacetate (CTA) and poly(vinylchloride) (PVC) are the most used polymers for the PIM elaboration To solve the problem of stability in aqueous and alkaline medium of CTA- and PVC-based PIMs, new more stable membranes should be designed. Poly(vinylidene fluoride) (PVDF) is a semi-crystalline polymer with repeated unit of . It has a glass transition temperature Tg of around −39 °C and exhibits high mechanical strength, good chemical resistance and thermal stability as well as excellent aging resistance, which is very important for the industrial application of separation membranes During the last decade, new PIMs were elaborated using PVDF and its copolymers – poly(vinylidene fluoride-co-tetrafluoroethylene) (PVDF-co-TFE) and poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF-co-HFP) The physical and chemical properties of PIMs based on PVDF and its copolymers strongly depend on the preparation techniques. For example, O’Bryan et al. revealed that the properties of PVDF-co-HFP/Aliquat 336 membranes obtained by the phase inversion technique and by solvent evaporation casting technique were different In contrast with well characterized CTA- and PVC-based PIMs, the research dealing with PVDF-based PIMs is mainly focused on the optimization of the membrane composition for the efficient extraction of the desired species. And to our knowledge, the physical and chemical properties of dense PVDF-based PIMs are not investigated concerning their structure and extraction ability. Therefore, PIMs based on PVDF, Aliquat 336 and 2NPOE are designed and characterized in the present work. The composition of elaborated membranes was optimized in terms of efficient removal of Cr(VI) from aqueous solutions and the extraction and removal performance was compared with the behaviour of common PIMs based on CTA and PVC. The influence of the ion carrier and plasticizer on the membrane properties was studied by thermal (differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA)), surface (contact angle, scanning electron microscopy (SEM) and infrared spectroscopy) and mechanical analysis. The obtained results were correlated with the membrane transport behaviour. In addition, the durability study was carried out in order to detect the leakage of the liquid phase (i.e. ion carrier and plasticizer) from the membrane and/or any changes of the membrane structure during the storage for 18 months.CTA pellets with 43–49 wt% acetyl, PVC powder (Mw = 43 000 g/mol), Aliquat 336 (≥97% purity), 2NPOE (≥99% purity), chloroform (99.0–99.4% purity) and hydrochloric acid were received from Sigma-Aldrich. PVDF powder was purchased from Alfa Aesar. 1.5-diphenylcarbazide (DPC) (≥97% HPLC purity) obtained from Fluka chemika, N,N-dimethylformamide (DMF) (≥99.8% purity), acetic acid (≥99.5% purity) and ammonium acetate (≥98% purity) provided by Biochem, THF (99.9% purity) received from VWR chemicals, potassium chromate K2CrO4 (99.5% purity) obtained from Acros and sodium hydroxide NaOH (99% purity) purchased from Merck were used without further purification. All water used in the work was milli-Q water (Milli-Q Water System, Millipore, resistivity = 18 MΩ⋅cm at 25 °C).For clarity, XPVDF/YAliquat 336/Z-2NPOE refers to the membrane with a certain amount of PVDF, Aliquat 336 and 2NPOE. The values of X, Y and Z represent the PVDF, Aliquat 336 and 2NPOE amount, respectively, expressed in wt%.The FTIR spectra were recorded on Thermo Nicolet Is50 FTIR spectrophotometer equipped with a diamond crystal in attenuated total reflectance (ATR) mode. A series of 256 scans was collected for each membrane over the 4000–400 cm−1 spectral range with a 4 cm−1 resolution.The Cr(VI) concentration in transport experiments was analyzed in acidic medium using an UV–Visible spectrophotometer at 540 nm (ThermoFisher Evolution 220).The mixture metal ions concentration was determined using inductively coupled plasma spectrometer (ICPThermo ICP 7000 Series).Mechanical measurements were performed on Instron 5543 equipped with the 500 N load cell. Tensile tests were carried out using normalized specimens (type 5A) according to ISO 527–2. The specimens were stretched to the main axis at 1 mm/min at room temperature (25 ± 1 °C) and hygrometry (34 ± 4% relative humidity). At least five samples were tested for each membrane composition.The water contact angle θ was measured using a sessile drop method by Multiskop goniometer (Optrel, Germany). For this, water was dropped on the membrane using a micro syringe and photographed with a black and white CCD camera (0.8 magnification lens). At least five drops were analyzed at different membrane locations at room temperature (25 ± 1 °C) and hygrometry (34 ± 2% relative humidity).DSC measurements were performed by means of the Polyma apparatus from Netzsch. All the experiments were performed in an aluminum pan with pierced lid in the nitrogen atmosphere from −50 °C to 200 °C at the heating rate of 10 °C/min. The results were analyzed with Proteus software.The membrane thermal behavior was studied using the TGA analyzer (TGA Q500, TA Instruments, USA). The TGA measurements were carried out from 30 °C to 800 °C in the nitrogen atmosphere. The heating rate and the nitrogen flow rate were 10 °C/min and 90 mL/min, respectively.For structural characterization, the membrane morphology was observed with a Zeiss EVO 40 scanning electron microscope. Samples were fractured in liquid nitrogen and covered with a carbon layer before observations.All experiments were carried out using a Teflon permeation cell composed of two equal (200 mL) cylindrical compartments (diameter: 6 cm, height: 8.5 cm) at room temperature (25 ± 1 °C). The membrane (diameter: 4 cm) was sandwiched between these two compartments. A Cr(VI) solution in 0.1 M HCl was used as the donor phase and the acceptor phase solution was buffered by a solution of acetic acid and ammonium acetate at pH 5 or by a solution of 0.1 M NaOH. Each solution was mechanically stirred at 1000 rpm. The samples were taken periodically from the both compartments with a micropipette and analyzed.The kinetic transport parameters were calculated according to:where Cdonor is the Cr(VI) concentration in the donor phase at time t and C0donor is the initial Cr(VI) concentration in the donor phase (i.e. at t = 0).The relationship lnCdonorC0donor versus time was linear, that was confirmed by the values of the determination coefficient (R2) close to 1 (i.e. 0.99). The permeability coefficient P was calculated as follows:where V is the volume of the aqueous donor solution, and A is the active membrane area. The initial flux Ji is determined using:The percentage of metal ions transported from the donor to acceptor phase (i.e. recovery factor) is defined as:R(%)acceptorphase=CtacceptorC0donor·100,where Ctacceptor is the Cr(VI) concentration in the acceptor phase at time t.The percentage of metal ions accumulated in the membrane phase (i.e. accumulation factor) is defined as:A(%)membranephase=CtmembraneC0donor·100,where Ctmembrane is the Cr(VI) concentration in the membrane phase at time t equal to Ctmembrane=C0donor-Ctdonor+Ctacceptor.The percentage of metal ions removed from the donor phase (so-called removal factor) is determined according to:R(%)donorphase=C0donor-CtdonorC0donor·100In order to reveal possible interactions of membrane different components (i.e. polymer, ionic liquid and plasticizer), FTIR analysis was carried out and the obtained spectra are shown in . The spectra for pure membrane components are given for comparison. The spectrum of Aliquat 336 is characterized by two bands at 2922 cm-1 and 2853 cm−1 attributed to the CH groups, and the quaternary ammonium group signals are detected at 1466 cm-1 and 1377 cm−1OH groups at 3373 cm−1 indicates the presence of some trapped moisture within the Aliquat 336 sample NO2 group is characterized by the band at 1522 cm−1, the CN group is detected at 1350 cm−1, and the bands between 1020 cm−1 and 1170 cm−1 attributed to the C group of the PVDF main chain and the band located at 1401 cm−1 is attributed to the CIt is known that one of the attractive characteristics of PVDF is its polymorphism, i.e. the possibility to form different crystalline structures, and depending on the crystallization conditions PVDF may present at least four different crystalline structures: the orthorhombic α, β and δ phases and the monoclinic γ phase. Besides, the crystalline phases of PVDF can be characterized by the infrared absorption bands between 400 cm-1 and 1000 cm−1b). The amorphous phase is characterized by two bands at 479 cm−1 and 874 cm−1In order to quantify the β phase content F(β) in PVDF-based membranes the following equation was used where Aα and Aβ represent the absorbance at 766 cm−1 and 840 cm−1 corresponding to the α and β phases, respectively; Kα and Kβ are the absorption coefficients at the respective wavenumber equal to 6.1·104 and 7.7·104 cm2/mol, respectively. The β phase content is estimated to be 80.6% for pure PVDF membrane. This result is in good agreement with the result found by Ferreira et al. for PVDF membrane prepared by the solvent evaporation method using DMF as a solvent, exactly F(β) was found to be 65–80% depending on the polymer weight content and the solvent evaporation temperature The spectra of the 80PVDF/20Aliquat 336 and 75PVDF/20Aliquat 336/5-2NPOE membranes do not reveal the appearance of new peaks associated to new absorption bands (). This fact indicates that there are no covalent bonds between the membrane components. This result allows us to suppose that only weak interactions (such as van der Waals or hydrogen bonds) exist between the PIM components and that the liquid compounds (i.e. Aliquat 336 and 2NPOE) are physically immobilized in the PVDF matrix. However, the band at 1401 cm−1 which corresponds to the C-F group in pure PVDF is shifted to 1403 cm−1 and 1404 cm−1 for 80PVDF/20Aliquat 336 and 75PVDF/20Aliquat 336/5-2NPOE, respectively (a). Such slight shift towards higher wavenumber indicates possible intermolecular interactions of membrane different components, thus intermolecular interactions between the C-F negatively charged groups of PVDF and the positively charged ammonium group of Aliquat 336 exist. The similar result was already found in PIMs based on CTA NO2 group from 1522 cm−1 in pure 2NPOE to 1526 cm−1 in 75PVDF/20Aliquat 336/5-2NPOEis also observed, suggesting intermolecular interactions between the NO2 group of 2NPOE and the ammonium group of Aliquat 336.As to the PVDF crystalline phase, it is not changed significantly in case of PVDF-based PIMs. And really, in both PIMs (i.e. 80PVDF/20Aliquat 336 and 75PVDF/20Aliquat 336/5-2NPOE) the β phase is dominant with the F(β) content of 83.1% for the 80PVDF/20Aliquat 336 membrane and 82.0% for the 75PVDF/20Aliquat 336/5-2NPOE membrane. The observed slight difference in the F(β) content for PIMs compared to the pure PVDF membrane (F(β) = 80.6%) can be attributed to the ability of the ionic liquid to favour the formation of the β phase Mechanical resistance is one of the key factors for the membrane use as PIM should be flexible and able to be applied in transport modules, i.e. PIM should possess low Young’s modulus value, and rather high elongation at break and tensile strength values. To study the influence of the ionic liquid and plasticizer presence on the mechanical behaviour of the PVDF-based membranes, tensile tests were carried out. The stress-strain curves of the pure PVDF and blend membranes are shown in a. The average values of the Young’s modulus, tensile strength and elongation at break determined from these curves are gathered in . As one can see, pure PVDF has a Young’s modulus of about 1877 MPa, that agrees well with the values found in literature for the PVDF membranes prepared using acetone and DMF as solvents The plasticizing effect of Aliquat 336 on the behaviour of some polymers, such as PVC and CTA, has been already reported in literature ). The tensile strength value is also reduced from ~ 42 MPa to ~ 12 MPa for the pure PVDF and 70PVDF/30Aliquat 336 membranes, respectively (). This decrease in tensile strength is accompanied by a high increase of elongation at break – from 12% for the pure PVDF membrane to 187% for the 70PVDF/30Aliquat 336 membrane. This fact confirms the plasticizing effect observed in case of the ionic liquid presence in membranes. Further addition of 5% of plasticizer (2NPOE) almost doubles the elongation at break values (). Moreover, the stiffness is reduced and the flexibility increased, thus promoting elongation at break and, therefore, such membranes can be used in aqueous media without any risk of tearing. The similar effect is observed for the PVDF-based membranes with other ionic liquids In order to investigate the effect of the polymer matrix on the elastic behavior of the membranes, the mechanical properties of PVDF-based PIMs were compared to the behaviour of PVC- and CTA-based PIMs which are widely used in the studies of the metal ions transport. All membranes contain the same quantity of ion carrier (i.e. Aliquat 336) (20%) and plasticizer (2NPOE) (5%). The tensile behavior of PIMs based on CTA, PVC and PVDF is presented in b and the values of Young's modulus, stress at break, and elongation at break are given in . According to these data, PVDF- and PVC-based PIMs have rather similar mechanical properties, whereas CTA-based PIM is much more resistant. The Young’s modulus of CTA-based PIM is four to five times higher than that of the membranes based on PVC and PVDF. Compared to PVC- and PVDF-based PIMs, the elongation at break value is reduced by 20 times and the strength at break value is almost 3 times higher for CTA-based PIM. This result agrees well with literature as the higher mechanical resistance of CTA-based PIM compared to PVC-based PIMs is revealed The water contact angle measurements were performed in order to estimate the membrane wettability as the hydrophilic/hydrophobic balance is a decisive parameter for the transport performance and stability of PIMs. The membrane hydrophilic nature is favorable for the membrane wettability as the hydration process will be fast in this case. However, the hydrophobic behavior assures the membrane stability as it will delay the carrier leakage.Pure PVDF is known for its hydrophobic character . The pure PVDF membrane reveals the water contact angle value around 71°. Besides, this value is almost constant up to 3 min of measurements. The obtained value is close to values found in literature for the dense PVDF membranes elaborated with DMF The addition of Aliquat 336 to the membrane increases strongly the membrane hydrophilicity (a). For example, the membrane containing 10% of Aliquat 336 has the same contact angle value as the pure PVDF membrane at t = 0. However, this value decreases up to 33° after 180 s of the water drop deposition. When the Aliquat 336 content in the PVDF matrix exceeds 10%, the water contact angle is reduced dramatically and it is ~ 26° at t = 0 and slows down to ~ 10° after 180 s for the PVDF-based membranes containing 20% and 30% Aliquat 336. Thus, the character of PVDF-based PIMs turns from a slightly hydrophilic to a very pronounced hydrophilic one. This behaviour is also observed for the membranes based on CTA and PVC In order to study the influence of the polymer matrix on the PIMs wettability, the measurements for PVC- and CTA-based PIMs were also carried out (b). As expected, the pure CTA membrane is the most hydrophilic one. As it was already reported in literature, CTA-based PIM is more hydrophilic than PVC-based one if the amount of Aliquat 336 is rather low and does not exceed 40% b). This result may be explained by the homogeneous distribution and saturation of the PVDF surface with only 20% of Aliquat 336 (as the water contact angle does not vary for PIMs with 20 and 30% of Aliquat 336 as shown in a) as compared with PVC- and CTA-based PIMs for which at least 40% of Aliquat 336 is required for the surface saturation One of the PIM important characteristics is the membrane microstructure as it allows us to control the carrier distribution and it affects the membrane transport efficiency. Moreover, the morphology and the structure of the PVDF-based membranes depend strongly on the preparation method. For example, the phase inversion technique by immersion in the non-solvent allows obtaining the membranes with an asymmetric porous structure that weakens the membrane mechanical properties As one can see from the obtained SEM images (), the membrane surface in contact with the glass plate is dense and smooth, while the membrane surface in contact with air presents a granular structure with some pores, whatever the membrane composition. Similar results were reported for the pure PVDF membranes casted from different solvents ). It should be also mentioned that if the glass contacting surfaces in the PVDF and PVDF/Aliquat 336 membranes are perfectly smooth, the surface becomes slightly rough in the presence of 2NPOE. The growth of spherulites and the addition of 2NPOE ensure the higher membrane roughness and promote the formation of surface pores, thus providing a greater interfacial surface area and improving the metal ion extraction and transport The SEM cross section images of all membranes reveal homogeneous and dense structure without macro-voids (), meaning that the pores present on the membrane surface are not interpenetrated. The obtained results testify to the good compatibility of membranes compounds (i.e. PVDF, Aliquat 336 and 2NPOE).It was shown that imidazolium-based ionic liquids modified the PVDF crystallization kinetic and slowed down the growth of the crystalline phase where ΔH and ΔHm are the melting enthalpy of PVDF-based PIM and 100% crystalline PVDF, respectively. ϕ is the total mass fraction of Aliquat 336 and 2NPOE in PIM. The value of ΔHm has been estimated to be 104.7 J/g The melting temperature and enthalpy as well as crystallinity degree for PVDF-based PIMs are summarized in . The endothermic peak at 164.7 °C present on the DSC curve of pure PVDF (not shown) corresponds to the melting temperature Tm of the crystalline phase. The melting point depression (i.e. the shift of the Tm value to the lower temperatures) is observed for PVDF-based PIMs. Such decrease may be attributed to the strong electrostatic interactions between the ionic liquid (Aliquat 336) and the crystalline phase of polymer that leads to lower cohesive energy of the crystalline phase and, so, a lower thermal energy is sufficient for the crystalline phase melting It can be seen that the incorporation of Aliquat 336 in PIMs leads to the decrease of the membrane crystallinity degree. Such behaviour has been already observed for the PVDF membranes with other ionic liquids Thermogravimetric analysis (TGA) was performed in order to study the thermal stability of the elaborated membranes. The obtained TGA curves are presented in a. For the sake of clarity, the thermogravimetrical derivative curves (DTG) are also shown (As one can see from TGA thermogram of Aliquat 336, the weight loss is observed from the beginning of the heating that can be linked to the presence of certain moisture within the ionic liquid. This fact confirms the FTIR result, where a bonded a). The trapped moisture alters significantly the stability of ionic liquids b), thus confirming the PVDF interactions with the membrane liquid compounds, i.e. ion carrier and plasticizer.Due to its hydrophilic character, high flexibility and surface roughness, the 75PVDF/20Aliquat 336/5-2NPOE membrane should be efficient for the Cr(VI) removal. In order to verify this supposition, all elaborated membranes were tested for the Cr(VI) transport in the acid medium. For the results reproducibility, the tests were repeated at least three times for the membranes prepared from the same casting solution as well as for the membranes prepared with the other casting solution.Prior to measurements of PVDF-based PIMs, the pure PVDF membrane and the PVDF membrane containing only the plasticizer (i.e. 2NPOE) were tested. In the absence of the ion carrier, no transport from the donor to acceptor phase was detected. This result allows us to conclude that the membrane without the ion carrier serves as a barrier to the metal ion permeation.Cr(VI) ions are transported into the membrane from the donor phase to the acceptor phase by the diffusion phenomenon under the effect of a driving force created by the gradient of chemical potential between these two phases . As one can see, the decrease of the Cr(VI) concentration in the source phase is accompanied by an increase of its concentration in the receiving phase indicating that the metal ions are successfully transported from the source to receptor phase with a removal factor greater than 99%. However, when the receiving medium consists of 0.1 M NaOH, the membrane changes its color and in some time (after one cycle, i.e. 12 h) becomes black (see insert to a). Besides, it is fragile and brittle at the end of the experiment, thus preventing its reuse for further transport cycles. The PVDF chemical stability in the acidic medium can be considered as excellent b, the recovery factor of ~ 90% is obtained during the Cr(VI) extraction in the buffer medium. Moreover, the use of acetic acid/ammonium acetate buffer solution makes it possible to keep the membrane in its initial state as no deterioration can be seen (see insert to b). However, some quantity of Cr(VI) is accumulated inside the membrane at the end of the experiment giving a yellow color to the membrane.So, the acetic acid/ammonium acetate buffer solution (pH 5) makes it possible to avoid the chemical deterioration of the PVDF membrane and, thus, this solution is used as a receptor phase for further study. Since Cr(VI) is well separated by the elaborated membranes and, thus, the Cr(VI) concentration becomes high enough in the receiving phase, it later can be reused in another cycle in chemical industry. Also, the Cr(VI) ions can be recuperated by their precipitation after the reduction into the cationic form of Cr(III) In order to determine the influence of the ion carrier content on the membrane transport efficiency, PIMs with different Aliquat 336 concentration ranging from 0% to 30% were tested for the Cr(VI) removal during 12 h (a). As one can see, the membrane without the carrier (i.e. pure PVDF) did not remove Cr(VI). This result confirms that the ionic liquid (Aliquat 336) acts as an ion carrier. Besides, the membrane with only 10% of Aliquat 336 does not transport Cr(VI) ions into the acceptor phase and the extracted chromium is accumulated inside the membrane testifying that the carrier concentration is below the percolation threshold limit In the case of the PVDF-based membranes with the Aliquat 336 concentration above the percolation threshold (i.e. above 10%), a gradual increase of the extraction percentage (both removal and recovery factors) and of the flux is observed (). This result indicates that the presence in PVDF-based PIM of only 20% of Aliquat 336 is sufficient to ensure the complete removal of Cr(VI) ions from the donor phase. The increase of the Aliquat 336 concentration to 30% induces only the transport flux increase. It should be mentioned that in case of traditional PIMs (i.e. based on PVC and CTA), in order to ensure the effective Cr(VI) transport more than 20% of Aliquat 336 (in some cases even more than 40%) or more than 30% of plasticizers must be incorporated into the membrane b) due to the accumulation of some quantity of Cr(VI) (about 10%) in the membrane phase.It is known that the addition of plasticizer can improve the permeability properties of PIMs by enhancing the compatibility of the membrane components b). As one can see from the obtained results, the plasticizer improves the back extraction of Cr(VI) (recovery factor >95%). At the same time, the equilibrium time is reduced and the transport flux is enhanced from 3.8 µmol/(m2·s) for 80PVDF/20Aliquat 336 to 5.6 µmol/(m2·s) for PIM containing only 5% of 2NPOE. This result can be explained by the fact that the plasticizing action of 2NPOE improves the Cr(VI) ions mobility ). Turgut et al. studied the influence of 2NPOE on the Cr(VI) transport using PVDF-co-HFP and imidazolium bromide room temperature ionic liquid Also, the influence of the Cr(VI) initial concentration on the transport flux was investigated. It was revealed that the increase of the transport flux was observed with the Cr(VI) concentration increase. Besides, this dependency was linear. For example, for the donor phase containing 10 mg/L of Cr(VI) the flux of 5.58 ± 0.20 µmol/(m2·s) was observed, whereas the flux was 12.28 ± 0.30 µmol/(m2·s) when the concentration became four time higher, i.e. 40 mg/L. These experimental results are in good agreement with Eq. In order to study the effect of the polymer matrix on the Cr(VI) transport properties, the performance of PVDF-based PIM containing 20% of Aliquat 336 and 5% of 2NPOE was compared with the behaviour of membranes containing the same amount of the ion carrier and plasticizer and based on CTA and PVC. It has been already shown that CTA-based PIMs are more efficient than PVC-based PIMs during facilitated transport of chromate ions for PIMs based on three studied polymers. As it can be seen, PVDF-based PIM is the most efficient as more than 98% of Cr(VI) is removed in only 6 h with a back extraction factor higher than 95% and the transport flux of 5.6 µmol/(m2·s). The efficiency of CTA-based PIM is lower than that of PVDF-based PIM. In this case, the removal factor of 97% of Cr(VI) requires 12 h of the transport experiment, the recovery factor is only 41%, and the transport flux equal to 2.6 µmol/(m2·s), i.e. two times lower compared to the PVDF matrix, is obtained. In addition, no Cr(VI) transport was observed through PVC-based PIM (only 1.25% goes to the receiving phase), probably as in this case the ion carrier concentration was below the percolation threshold limit. After 12 h of experiment, only 80% of Cr(VI) ions from the feed phase were removed by PVC-based PIM and were accumulated in the membrane.One can conclude that with such a low concentration of the ion carrier and plasticizer (20% and 5%, respectively) the nature of the polymer matrix has a great influence on the PIM transport properties. Indeed, the use of CTA and PVC requires a higher amount of plasticizer to ensure the mobility of the target molecule complex and efficient facilitated transport ), PVC-based PIM is more plasticized compared to CTA-based PIM due to a higher plasticizing effect of Aliquat 336 on the PVC matrix b). As to PVDF-based PIM, its mechanical properties are close to those of PVC-based PIM (), it possesses the most hydrophilic character among three studied membranes (b), and its Tg is very low (about −39 °C), meaning that the polymer is in its rubbery state characterized by the high polymer chains mobility. All these features promote the transfer of the molecule target/ion carrier complex even without a plasticizer. O’Bryan et al. reported that PVDF-co-HFP-based PIMs exhibited significantly higher rates of extraction and back extraction for SCN- ions and long-term stability compared to PVC-based PIMs Generally, industrial activity generates effluents containing a mixture of heavy metal ions. Therefore, in addition to be effective and possess high permeability properties, PIM should be also selective in order to transport only specific species. In order to study the selectivity of PVDF-based PIMs towards Cr(VI) ions, the membrane with optimal composition, i.e. 75PVDF/20Aliquat 336/5-2NPOE, was used. A mixture solution containing Cr(VI), Co(II), Cu(II), Zn(II), Ni(II), Pb(II), Fe(III), Cd(II) with identical initial concentration of 10 mg/L was prepared in 0.1 M HCl. The experiments were conducted for 8 h. As one can see from the obtained results (), only Cr(VI) ions are transported with high efficiency (96.9%). Moreover, Co(II), Cu(II), Zn(II), Ni(II) and Pb(II) ions are not transported through this PIM. Cd(II) and Fe(III) may be transported at the same time with Cr(VI) but to a much less extent – 2.7% for Cd(II) ions and 0.3% for Fe(II) ions. The permeability of these ions may be explained by the formation of anionic complexes of Cd(II) and Fe(III) with Cl- anionsIn addition to the high transport efficiency, one of the most important parameters for large-scale application of PIM is its ability to be reused during several cycles without performance loss. Therefore, the stability of 75PVDF/20Aliquat 336/5-2NPOE PIM was evaluated on the basis of the recovery factor (Eq. ) determined after 24 sequential cycles (1 cycle = 8 h). These experiments were performed with the same membrane, while both the donor and acceptor phases were renewed after each cycle. The evolution of the recovery factor of the Cr(VI) transport is shown in a. No significant difference of the Cr(VI) recovery was observed during first 16 cycles (i.e. for 128 h). During that time, the high recovery factor (~98%) remained constant. However, after this time a gradual decrease of the removal factor was observed – after 19 cycles (i.e. 152 h) the recovery factor was reduced to ~ 81% and after 24 cycles (i.e. 192 h) this value was 54%. This result testifies to the improvement of the recovery capacity and stability of the PVDF-based PIM compared to the membranes based on CTA and PVC The stability of PVDF-based PIM is also evaluated by the method proposed by O’Bryan In order to quantify the possible leakage of the liquid components (i.e. ion carrier and/or plasticizer) from PVDF-based PIM, the 75PVDF/20Aliquat 336/5-2NPOE membrane was immersed in different media used as the donor and acceptor phases in the case of the Cr(VI) transport measurements (namely, milli-Q water, 0.1 M HCl and buffer solution of acetic acid/ammonium acetate at pH 5) and its weight was monitored (b). During the membrane immersion in water its weight decreased rapidly during first hours because of the ion carrier and/or plasticizer leakage, but starting from 50 h, the membrane weight was stabilized. 75PVDF/20Aliquat 336/5-2NPOE membrane lost ~ 10% of its initial weight that represents ~ 40% of the mass of its liquid components. It should be noted that the stability of PVDF-based PIM in water is much better than the stability of PVC-based PIM studied by Kagaya et al. as in that case the loss of 30% of the membrane initial weight was observed, i.e. more than 80% of the weight of Aliquat 336 a) and strong electrostatic interactions between the liquid compounds and the PVDF polymer chains () are the proofs of good miscibility and compatibility of PVDF with Aliquat 336.When the membrane is immersed in 0.1 M HCl, its weight loss is reduced and presents less than 3%. Moreover, when the membrane is immersed in buffer solution of acetic acid/ammonium acetate (pH 5) its weight remains constant (b). As the loss of the liquid phase from PIMs depends on the medium nature and its ion concentration In order to increase the membrane stability several methods can be proposed The stability of 75PVDF/20Aliquat 336/5-2NPOE membrane was also determined by the mechanical properties. For this purpose, the stress-strain curves for the fresh membrane, after one cycle and 24 cycles of the Cr(VI) permeability measurements were obtained (not shown). After one cycle of the Cr(VI) transport, the membrane mechanical properties stayed practically unchanged (i.e. elongation at break = 302%, strength at break = 15.6 MPa, Young's modulus = 438.9 MPa) compared to the fresh membrane (). This result confirms that the membrane remains in its initial state, i.e. the liquid components loss is negligible. However, after 24 cycles, the membrane tensile strength increases up to 20.2 MPa, its Young's modulus also increases to 750.9 MPa, and its elongation at break is reduced to ~ 25%. Such increase of the membrane mechanical behavior is certainly due to the loss of a certain quantity of membrane liquid components (To study the stability and durability of PVDF-based PIMs over time, the membranes were stored several months at room temperature (24 ± 2 °C) and relative humidity (34 ± 5%) and the transport experiments were carried out periodically. The evolution of the Cr(VI) concentration in the donor and acceptor phases for PVDF-based PIMs is plotted in . It can be seen that PIMs preserve their transport efficiency after 9 and 18 months of storage whatever the membrane composition. This result testifies to the membrane durability. The performed TGA analysis reveals that the aged membranes keep their liquid compounds – about 25% for 75PVDF/20Aliquat 336/5-2NPOE membrane and about 30% for 70PVDF/30Aliquat 336 membrane. The membrane elastic behaviour was not really changed after 18 months of storage as the following values were obtained: Young's modulus of 417.5 MPa, strength at break = 16.8 MPa, elongation at break = 259.5% for 75PVDF/20Aliquat 336/5-2NPOE and Young's modulus = 258 MPa, strength at break = 13.1 MPa, elongation at break = 235.4% for 70PVDF/30Aliquat 336. These values are very close to the mechanical behaviour of the fresh membranes (). Visually, the surfaces of the aged membranes are similar to those of the fresh membranes and are not oily, which means that the liquid compounds were not leaked out from the membrane during storage. The water contact angle measurements carried out for the aged membranes didn’t show any variation in water contact angle values compared to the fresh membranes indicating the good stability of PIMs over time. In fact, the PIM liquid compounds are trapped in a polymer matrix, thus inhibiting the leaching of the ion carrier. Moreover, as demonstrated by FTIR and thermal analysis, the intermolecular and strong electrostatic interactions between the liquid components and the polymer chains (such as van der Waals or hydrogen bonds) contribute to maintain the liquid compounds inside the membrane, thus increasing its stability and durability.PVDF-based PIMs containing Aliquat 336 as an ion carrier and 2NPOE as a plasticizer designed for the Cr(VI) transport were successfully elaborated by the solvent evaporation technique and the membrane composition was optimized in terms of mechanical and transport properties. As revealed by microscopy, thermal and FTIR analysis, the dense and homogeneous membranes with strong electrostatic interactions between the different components (i.e. base polymer, ion carrier and plasticizer) were obtained. It was found that the ion carrier quantity had a strong influence on the membrane mechanical properties, hydrophobic/hydrophilic character and, consequently, on the Cr(VI) transport performance. Moreover, the addition of small amount of the plasticizer (2NPOE) (only 5 wt%) improved significantly the initial flux of the Cr(VI) ions transport – from 3.8 µmol/(m2·s) for 80PVDF/20Aliquat 336 to 5.6 µmol/(m2·s) for 75PVDF/20Aliquat 336/5-2NPOE. It was found that the membrane based on 75% PVDF, 20%Aliquat 336 and 5% 2NPOE selectively transported Cr(VI) ions (96.9%). Besides, the increase of the initial Cr(VI) concentration in the feed phase had also a positive influence on the transport flux. The performance of PVDF-based PIM was compared to PIMs based on conventional polymers, such as CTA and PVC, and much better transport properties, lifetime stability and durability were revealed. Also, the results of the aging study testified to the fact that PVDF-based PIM preserved its elastic behavior and transport capacity towards Cr(VI) ions up to 18 months. Thus, the results presented in this work indicate the excellent selectivity and stability of PVDF-based PIM, making it a promising candidate in areas of ion separation and water treatment.The present work contains new results which are of both fundamental and practical significance for chemistry of PIMs and the Cr(VI) extraction. The influence of the membrane composition on the selective transport of Cr(VI) ions was evaluated in terms of the membrane stability and Cr(VI) recovery ability and efficiency. Besides, the behaviour of elaborated PIM was compared with that of supported membranes based on conventional polymers, such as CTA and PVC.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Mechanical property profiles of microstructures via asymptotic homogenizationMechanical property profiles of microstructures via asymptotic homogenizationMicrostructure is the microscopic scale of a material, which can strongly influence physical properties, such as elasticity, strength, hardness, etc. Recently, many research efforts have been made to design the topology of microstructures carefully and govern their material properties according to the specific application of these materials in industrial practice. However, most works target specific properties that are tailored to particular applications. An integral image for one microstructure concerning multiple mechanical properties is in high demand. In this paper, we present a microstructure analyzer based on asymptotic homogenization. A mechanical property profile (MPP) is proposed to depict commonly used mechanical properties, including Young’s modulus, shear modulus, Poisson’s ratio, compressive strength, shear strength, the spatial distribution of effective Young’s modulus surface, and the worst-case von Mises stress distribution. Specifically, we utilize a radar chart to visualize the five scalar properties to help with an intuitive understanding of the microstructure. Moreover, we apply joints reinforcement on truss-based microstructures to enhance their strength. The results show that our analyzer and MPPs have the potential to help users understand and select the desired microstructure from alternatives and help them improve specific microstructure properties in microstructural design.Microstructures show superiority compared with traditional materials in many applications. For biomedical implants, the porous structures According to different criteria, there are different classifications for these microstructures. If we adopt form as a criterion, the microstructures can be classified into beam-based and sheet-based structures. According to the generation method, they can be generated by an implicit function like TPMS or primitives assembly like a truss. They can also be generated through the topology optimization method.Both topology and relative density play an essential role in the mechanical property of microstructures. In the case of structure design, it is necessary to know which microstructure best fits the design purpose. So, a comprehensive description of every candidate microstructure is needed.In this paper, we focus on the analysis of microstructures with different topologies affecting the mechanical properties of materials. We collect some representative microstructures of various kinds, including foam, beam-based, and sheet-based structures, and intercept units with periodic boundaries as representative volume elements (RVE). They are modelled following manually designed rules, procedural modelling, or implicit functions. Considering that the analysis of microstructural properties using physical experiments is expensive and time-consuming, we apply the asymptotic homogenization method to predict the macroscopic scale mechanical behaviors of microstructure units. We analyze the homogenization results in terms of several common-used mechanical properties, including Young’s modulus (E), Poisson’s ratio (v), shear modulus (G), compression strength (σys), and shear strength (τys).Then we establish a comprehensive mechanical property profile for each microstructure, in which a radar chart is introduced to depict the above mechanical properties of microstructure, as well as the isotropy of Young’s modulus and worst-case stress distribution. Finally, we attempt to apply joints reinforcement on truss-based microstructures to improve the strength.The contributions in this paper are as follows:We propose a convenient microstructure analyzer to compute mechanical properties of microstructures based on asymptotic homogenization, avoiding users from performing physical experiments for different applications.We present a mechanical property profile for each microstructure that depicts five commonly used properties, the spatial distribution of effective Young’s modulus surface, and the worst-case von Mises stress distribution. We involve a radar chart to view the five property values in one image to better understand the performance of different microstructures.We apply joints reinforcement on truss-based microstructures as a structural adjustment operation, such that their strength properties are improved.Microstructure generation. Due to the unique benefits of high strength-to-weight ratio, outstanding energy absorption, and minimizing material requirements, many microstructures with different forms and tailored properties are designed. Some of them are bionic, such as the porous structures inspired by the honeycombs or human bones The existing microstructures in the literature can be divided into three main groups, manually designed microstructures, procedural modelling microstructures and implicit function defined microstructures.Most manually designed microstructures consists of trusses. Body-centered cubic (BCC) and face-centered cubic (FCC) are two classical structures that have been widely used in microstructure region. Inspired by atoms arrangement in the crystal, BCC and FCC structures are constructed by connecting nodes on the corner, the center of the face, and the center of cubic using trusses Procedural modelling microstructures commonly have more complex topologies with designated properties compared with manually designed microstructures. Panetta et al.proposed elastic textures, a set of parametric, tileable, printable, cubic patterns In recent years, some work has considered the design of aperiodic structures. Compared with the uniform unit cell structure, due to the uneven unit cell, aperiodic structure is easier to achieve isotropic properties in the macroscopic level, i.e., physical properties in different directions are not equivalent but approximately close. However, the design of stochastic structure requires more complex and global calculations, and the geometric expression of large-size requirements becomes one of the challenges. To express randomness, the most commonly used geometric structure is the Voronoi structure. Martínez et al. used beam and surface elements of Voronoi structures to represent the stochastic microstructure by procedural modeling method Another group of microstructures is those generated by implicit functions. Triply periodic minimal surfaces (TPMS) are described with periodic surfaces composed of simple trigonometric functions, and the mean curvature of the surface is zero. For the ability of providing precise description of a variety of physical structures such as silicates, bicontinuous composites, lyotropic colloids, TPMS have been applied for microstructure design recently Homogenization.As an efficient approach to characterize the effective mechanical properties of microstructures, homogenization theory has been developed since the 1970s Microstructure analysis.As microstructures show extraordinary mechanical properties, a lot of work have been done to study the mechanical properties of microstructures using physical experimental, analytical and computational methods.Conventional physical experimental method is to design and fabricate microstructure specimens and collect data from physical tests. Compression and tensile tests are basic methods to test the uniaxial mechanical properties of the material. The responses are recorded and used to calculate the stress-strain curves from which stiffness and strength properties can be derived. Previous work discussed the relation between mechanical properties and factors such as base material, topology and relative density. Al-Ketan et al. Physical experimental method provides realistic analysis results of microstructures. Multiple duplicate specimens were tested to decrease experimental error Analytical study of the mechanical behavior of microstructures is thriving. The aim is to derive analytical relationships between relative density and mechanical properties. In general, there is a power-law relationship between the relative density of a microstructure and its elastic modulus. The coefficients of power-law depend on the topology of microstructure Computational method is another approach to obtain mechanical properties of microstructures. Finite element analysis (FEA) on microstructures could provide more accurate predictions on mechanical properties compared with analytical methods. Maskery et al. As computational method gives consideration to both comprehensive analysis result and efficiency, we analyze microstructures in computational way like The acquisition of mechanical properties of microstructures is the prerequisite for any structural application. Thus, we construct an analyzer to obtain mechanical properties, including Young’s modulus, Poisson’s ratio, shear modulus, compression strength, shear strength, the anisotropy of Young’s modulus, and von Mises stress distribution under worst-case loading. Some representative volume element (RVE) methods are proposed to avoid complex computation of full-scale simulation of microstructures. The RVE methods can obtain effective mechanical properties by analyzing a unit cell of microstructure. We adopt asymptotic homogenization theory to construct our analyzer.AH assumes that each field quantity depends on two different scales: one on the macroscopic level x, and the other on the microscopic level, y=x/ϵ. ϵ is a magnification factor that scales the unit cell’s dimensions to the material’s sizes at the macroscale. The field quantities, such as displacement u, strain ε, and stress σ, vary smoothly in the macroscopic level and are periodic at the microscale Consider heterogeneous object Ωy filled with a periodic microstructure. The elastic response of an object under macroscopic external load f is governed by the linear elastic equation:where C is a fourth-order elasticity tensor of base material, uy denotes the displacement of periodic microstructure, and ε(u)=12(∇u+(∇u)T) is the Cauchy strain tensor.Taking the derivative of the asymptotic expansion of displacement field and neglecting high order terms, the microscopic strain tensor can be defined as:where ε¯ is the average or macroscopic strain, and ε* is the fluctuating strain varying periodically at the microscale level, which characterizes the structure. The fluctuating strain can be obtained from a microscopic equilibrium equation, which is written as:where ε^ is the virtual strain. The above equation can be discretized and solved by the finite element method. Once the fluctuating strain ε* is solved, the homogenized constitutive matrix CH can be calculated by the following equation:where |Ωy| denotes volume of periodic microstructure Ωy. More details refer to The Young’s modulus, shear modulus, and Poisson’s ratio can be extracted from the constitutive matrix’s inverse, which is the homogenized compliance matrix SH. More specifically, it has,SH=(1Ex−vyxEy−vzxEz000−vxyEx1Ey−vzyEz000−vxzEx−vyzEz1Ez0000001Gyz0000001Gzx0000001Gxy),where Ei is Young’s modulus along axis i, Gij is the shear modulus in direction j on the plane whose normal is in order i, and vij is the Poisson’s ratio that corresponds to a contraction in direction j when an extension is applied in order i.The anisotropic elastic property can be depicted by plotting Young’s modulus along different orientations 1Eijk=S11H−2(S11H−S12H−12S44H)×(ℓi12ℓj12+ℓj22ℓk32+ℓi12ℓk32),where Eijk is the Young’s modulus in the [ijk] direction. ℓi12,ℓj22,ℓk32 are the direction cosines of direction [ijk]. Then we can calculate the effective Young’s modulus along any direction and plot Young’s modulus surface in a 3D space.An elasto-perfect plasticity analysis is performed to get the yield strength of microstructures. The procedure refers to where ε¯0 is initial macroscopic strain and Δε¯ is increment of macroscopic strain. Then the macroscopic strain is applied to RVE under periodical boundary conditions. After solving the fluctuating strain ε* from . The microscopic stress σ is expressed as:The von Mises yield criterion is adopted to determine the yield of the element.Once the von Mises stress of element reaches the yield strength of base material, it is regarded as yielded. The corresponding local elastic tensor of the yielded element is set to zero. The macroscopic stress σ¯ can be calculated by the following equation:With incremental macroscopic strain and corresponding macroscopic stress, we can fit the curve of macroscopic stress versus strain. The compressive yield strength along axis σys and shear strength τys can be obtained separately at the peak of corresponding macroscopic strain-stress curve, where the material collapse phenomenon occurs.The stress distribution varies according to different loads. To figure out the maximum stress concentrations under the load of a certain magnitude, we apply a worst-case stress analysis to microstructures.Under the assumption of small deformation and elastic material behavior, there is a linear relation between the macroscopic strain ε¯ and microscopic strain ε through a local structural tensor A:We apply six independent unit strains to and use the solution to construct A. Then the microscopic stress σ under a particular macroscopic stress σ¯ is described as:, we obtain the von Mises stress distribution under a particular macroscopic stress. When von Mises stress reaches maximum, the corresponding macroscopic stress is defined as the worst-case loading. This loading can be calculated by solving a tensor eigenvalue problem. More details are referred to We redevelop homogenization Matlab code from We collect several microstructures of diverse types and analyze their mechanical properties, to figure out the performance differences in mechanics between these microstructures. Microstructures in the collection can be classified into three groups. The first group is manually designed microstructures (a–e). They commonly have simple and regular topologies. The second group is microstructures generated by procedural modelling like Voronoi foam j–l). They are continuous and smooth surfaces.According to the mechanical properties obtained from the analyzer introduced by , we build an MPP for each microstructure. At first, we introduce a radar chart to depict five mechanical properties as shown in the second row of . The form of a radar chart is intuitive and user-friendly for comparing microstructures. However, for a microstructure, only a radar chart cannot depict a comprehensive profile of its common mechanical properties. We, therefore, apply two auxiliary plots to help refine the microstructure profile (see the third and fourth rows of The first auxiliary plot is Young’s modulus surface, which is used to describe Young’s modulus distribution in different orientations. For a point p on Young’s modulus surface, both the distance to center c and color of the point denote the Young’s modulus along cp→. Combined with color bar, the values of Young’s modulus in different orientations are expressed more clearly. By default, we make an assumption that a microstructure is “isotropic” due to its periodic symmetry. But we observe that the term “isotropic” of a microstructure usually refers to a particular case of orthotropic material whose properties are equal in the orthogonal directions, distinguished from truly isotropic material symmetric in all orders. In fact, Young’s modulus of most microstructures varies considerably in different orientations. Therefore, Young’s modulus surfaces are essential since the 3D spatial representations of effective Young’s modulus has no way to be reflected in the radar chart.Another auxiliary plot is the von Mises stress distribution, which is complementary to the yield strength. In practice, the microstructure yield strength demonstrated in the radar chart is only able to help the user to select a stronger microstructure. To improve microstructure’s mechanical properties in strength, the plot of stress distribution must be taken into account as a guideline for structural optimization. Due to the symmetry of some microstructures, the worst-case von Mises stress distribution and corresponding loading are not unique. For example, shows the worst-case von Mises stress distributions of FCC and corresponding loadings. The stress distributions are rotationally symmetric and have the same maximum von Mises stress. Considering the simplicity, we plot one of the worst-case von Mises stress distribution with the corresponding loading in MPPs of symmetric microstructures.As the effect of the geometry is less evident at high relative densities lists the MPP for each microstructure with 10% relative density, from which we reach the following observations.a) shows the largest Young’s modulus along the axis. Due to the simplicity of topology, the cubic structure can be generated with thicker rods compared with different structures. Thick rods can resist deformation caused by uniaxial loads. The von Mises stress distribution under worst-case loading reveals that joints are vulnerable regions of cubic structure. At the same time, this characteristic weakens other properties, including shear modulus, shear strength, and Young’s modulus along different orientations.b) consists of 8 rods connecting center and 8 vertexes of a cube. Young’s modulus surface shows the anisotropy of the structure. On the contrary of simple cubic structure, BCC shows large Young’s modulus along the diagonal and small modulus along the axis. BCC also shows the ability to resist deformation caused by shear loading. The stress distribution shows that the ends of 8 rods is the weakest region.c) is a modified version of a simple cubic structure. Compared to a, the added rods make the mechanical performance more balanced. Shear modulus, shear strength, and Young’s modulus in different orientations are improved with the cost of reduction of uniaxial Young’s modulus, which means this structure will keep stability in complex loading cases. The von Mises stress distribution implies that rods’ joints on the face of cube is the weakest region.d), the widely used microstructure has better performance on resisting deformation caused by shear loading. Young’s modulus along the diagonal is larger than modulus along the axis.e) has a distinctive topology compared with above mentioned microstructures. As the radar chart implies, this structure has relatively small Young’s modulus, thus showing high elasticity. The Young’s modulus surface shows that it is closer to isotropy.f) has the most complex topology. Both small values of E and G show that the structure has elasticity instead of high stiffness. Due to the stochasticity in the microscopic level, Voronoi foam shows nearly elastic isotropy. The joints of structure also emerge stress concentration.The analysis results of several simple truss-based microstructures have been shown above. Next, more complex truss-based microstructures are analyzed. For the first two elastic textures (g and h), complex combinations of rods bring better performance on elasticity. The worst-case stress distribution indicates that the special design avoids stress concentration at joints. The third structure (i) behaves like the cubic structure, which has a high E value and a relatively small G value. From the perspective of structure, it can be regarded as a modified cubic structure by adding support rods. The additional joints and rods increase the G value.Apart from truss-based microstructures, we also analyze several sheet-based TPMS microstructures. These sheet-based structures show high stiffness and strength properties which accords with the conclusion in j) has large shear modulus and shear strength. The Young’s modulus along the diagonal is as twice as it along the coordinate axis. The weakest region appears at the boundary of the structure. TPMS-IWP (l) show isotropy on Young’s modulus. Considering shear modulus, shear strength, and compressive strength, they can be regarded as well-rounded microstructures.By analyzing the above structures, we find that some are oriented towards lightweight and high-strength applications, and the other part pursues better elastic properties. The physical properties of various truss-based structures conform to the law of Maxwell number For users, if the goal is to have high stiffness and strength while being lightweight, the best choice is the TPMS structure in the sheet-based type. The sheet-based structure has better stiffness and isotropy than the truss-based structure of the same relative density On the other hand, users should consider choosing suitable truss-based structures for applications that pursue high elasticity based on the visualization results. They have a greater degree of freedom to express elasticity and are easier to measure whether they exhibit elastic properties through the Maxwell number.Apart from microstructures with 10% relative density, our analyzer is also able to generate MPPs of microstructures with graded densities. As the von Mises stress distribution under worst-case loading indicates, the joints of microstructure are likely to emerge stress concentrations. Elastic textures g–i) do not encounter such problem because they are designed with reinforced joints constructed by the convex hull. We follow such practice For truss-based microstructures, the structure are described with vertexes and edges. We generate bar element for each edge. At each vertex, a convex hull is constructed with the points provided by the end of bar element of every associated edge. It should be note that, the reinforcement of joints is performed under a fixed relative density. Compared with the original, the microstructure with reinforced joints has thinner rods. shows reinforced joints constructed by convex hull in Voronoi foam. demonstrates the comparison between microstructure with reinforced convex hull joints and the original. Both the microstructures are generated with 10% relative density. The analysis results show reinforced joints are helpful with optimization of stress distribution and strength improvement. In some examples, this practice has side effects of the weakening stiffness. Users are able to determine whether strengthen microstructure joints or not according to goals of physical performance.A uniaxial compression test of three microstructures is conducted according to the standard GB/T 8813-2008 to verify the analysis results. We fabricate microstructure specimens using a DLP 3D printer with photopolymer resin. The Young’s modulus is 1850MPa, Poisson’s ratio is 0.3, and yield strength is 75MPa. The physical size of each specimen is 50mm*50mm*50mm, and the manufacturing error is controlled in 1%. We utilize a WDW-10M universal test machine with a 10kN load cell for the compression test. The loading speed is set to 5mm/min. shows three microstructures fabricated using grey resin and the corresponding collapsing cases under uniaxial compression. shows strain-stress curves obtain from physical experiment and simulation. The gradient is calculated at linear stage of strain-stress curve as Young’s modulus, and stress at the peak of curve is regarded as compressive yield strength. The Young’s modulus and compressive yield strength obtained from experiment and simulation are summarized in The simulation results predict Young’s modulus and compressive yield strength reasonably. But there still exist differences between experimental data and simulation due to several reasons. First, the experimental errors introduced by printing quality, properties of base material, the accuracy of sensors are difficult to avoid. In addition, Pasquale et al. In this work, we construct a usable microstructure analyzer which is able to show the mechanical properties from multi-aspects. A group of microstructures with different topologies are analyzed and the mechanical property profiles are depicted to reveal the advantages, weaknesses, and differences. The main conclusions drawn from MPPs are:With 10% relative density, simple microstructures like simple cubic, BCC show extremely elastic anisotropy. They have good stiffness along specific orientations. Extra rods added on simple microstructures could make them more balanced.As the complexity of topology increases, microstructures show better elasticity and balanced mechanical properties. In general, sheet-based microstructures have greater stiffness and strength compared with truss-based structures.As the increase of relative density, the stiffness and strength rise, and the elasticity of microstructure changes towards isotropy. The weakest region of microstructure does not change, from the worst-case von Mises distribution.According to analysis results, we conduct structural adjustment with special consideration of the weakness. The reinforced microstructures display better mechanical performance. However, this adjustment seems to be intuitive. We notice stress concentration on the connection, then strengthen the joints using a convex hull. In fact, with the help of our analyzer, we can conduct a precise structural optimization of microstructures to improve their mechanical performance. In this work, we focus on mechanical property profiles of different types of microstructures, and we will explore potential applications of our analyzer on structural optimization in future work. As we adopt an elasto-perfect plasticity model, the prediction of deformation behavior after yielding may lose a little accuracy. Despite this, the simple and fast model is enough for our analyzer, designed to figure out the differences between microstructures and provide guidance for structural optimization.Peiqing Liu: Methodology, Software, Validation, Writing – original draft. An Liu: Methodology, Software. Hao Peng: Methodology, Writing – original draft. Lihao Tian: Methodology, Writing – review & editing. Jikai Liu: Writing – review & editing. Lin Lu: Conceptualization, Writing – review & editing, Supervision.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Fabrication of Ni–Al2O3 composite microcomponent by electroformingThick nickel–alumina (Ni–Al2O3) composite microcomponents containing dispersed alumina nanoparticles were fabricated by a combination of UV lithography and electroforming, using Shipley BPR100 resist to fabricate the micromoulds. Tests show that the microhardness of composite microcomponents is significantly improved over that of electroformed pure nickel components and that the structures maintain good fidelity. The method has potential for the fabrication of hard wearing microcomponents for micromachines and other applications.Since the advent of LIGA technology, electroplated nickel has been the predominant material for fabricating metallic microcomponents. However, it is increasingly clear that Ni microcomponents cannot meet the developing demands of mechanical properties, corrosion resistance, etc. of a variety of applications Compared with sintering and hot pressing, electroforming is a high volume production technique for metallic components. Combing conventional UV lithography, microstructures of different shapes can be realized through electroforming in patterned photoresist moulds; the basic processing steps of this technology are illustrated in . In our study, a thick layer of negative tone photoresist, Shipley BPR100, was used to make micromoulds for electroforming. Fumed alumina nanoparticles were dispersed in the bath of a commercial nickel plating solution for composite electroforming, in anticipation that the nanoparticles incorporated in the nickel matrix will enhance the mechanical properties of Ni–Al2O3 nanoparticle microcomponents.A silicon wafer (100 mm diameter, 500 μm thickness) was used as the substrate and a conductive seed layer of Cr, followed by a layer of Au, were evaporated onto the polished surface. Before lithography, the substrate was cleaned using acetone and isopropyl alcohol (IPA) to remove the organic impurities, and then pre-baked at 150 °C for 60 s. After cooling to room temperature, the substrate was coated with one layer of Shipley BPR100 photoresist, which has a single-spin film thickness ranging from 40 to 130 μm, depending on coater speed. The photoresist was soft baked, then exposed to UV light and developed with agitation in 1:24 Eagle 2005 and deionized water, at 39 °C for 5–30 min depending, on the film thickness. This was followed by post bake at 105 °C for 5 min. The thickness of the finished micromould was measured using a Taylor Hobson Form Talysurf-120 L stylus metrology tool.The Ni–Al2O3 nanoparticle composite microcomponents were prepared by codepositing Al2O3 nanoparticles dispersed in a nickel sulfamate electrolyte solution with the growing nickel during electroforming. A commercial nickel sulfamate plating solution (PMD Chemicals Ltd., UK) was used; the composition was nickel 85–95 g/L, nickel chloride 8–12 g/L and boric acid 25–35 g/L. All chemicals were of analytical reagent grade. Fumed α–Al2O3 nanoparticles (SpectrAl™ 81) of high purity were treated before being added into the plating solution for nanocomposite electroforming. Mechanical stirring and ultrasonic agitation were used to disperse the nanoparticles. (Future work will include an investigation of chemical dispersion). A surfactant, Sodium dodecyl sulphate (SDS), was used to improve the wettability of the electrolyte on the surface of BPR100 photoresist mould and to facilitate the removal of bubbles from the cathode surface.All electroforming experiments were performed in a glass beaker on a hotplate, with magnetic stirring. Before each run of experiments, the pH value of the solution was adjusted to 4.1 by adding sulphamic acid and nickel carbonate, before the silicon substrate was placed vertically in the bath as a cathode. The current density, which was in the range 20–100 mA/cm2, and the bath temperature was maintained at 51 °C. shows the SEM micrograph of the resist mould for a mechanical microlinkage after lithography. As can be seen from the figure, the pattern is well defined, and there is no residue of BPR100 photoresist left on the bottom of the exposed area after developing. The 120 μm thick microlinkage mould was made of a single photoresist layer using the processing parameters showing above and micromoulds of thickness over 200 μm were also fabricated using the double coating method.The treatment to remove moulds from the substrate varies with the properties of the photoresist. However, BPR100 micromoulds can be stripped away easily without any damage to the microcomponent by merely immersing the substrate in a Microposit 1165 solution at 50 °C for a few minutes. The SEM micrograph of an electroformed microlinkage, shown in , confirms the integrity of outline shape and uniformity of the electrodeposited composite layer.The electroformed Ni–Al2O3 nanoparticle composite microlinkage was ultrasonically agitated in DI water for 30 min to remove the loosely attached Al2O3 nanoparticles, before examination by SEM and EDS. reveals the surface morphology of the microlinkage shown in . The topology of the pyramid structures is not as clearly defined as those present in electroforms from a pure nickel plating bath and peaks are rounded off. It is also observed that the SDS not only prevents the formation of gas pitting on the surface of microcomponents, but also affects the dispersion of nanoparticles in the bath.The backscattered electron SEM micrograph in shows that Al2O3 nanoparticles are homogeneously distributed in the composite. EDS examination shown in and X-ray mapping confirm that the dark regions shown in are nanoparticle rich and the bright regions are almost particle-free. The content of Al2O3 nanoparticles in the composite is affected by the bath composition and electroforming conditions. As expected, the higher the concentration of Al2O3 nanoparticles in the bath, the more nanoparticles are incorporated into the composite electroform with no observed tendency to precipitation.Previous studies have shown that the inclusion of second-phase micro and nanoparticles in a ductile metal matrix could improve the mechanical strength and thermal properties through a composite effect shows the SEM micrograph of a micro indenture on the surface of a microgear electroformed from a bath containing 4 g/L Al2O3; the microhardness is HV0.2 561.52.An initial experimental study of the electroforming of Ni–Al2O3 composite microcomponents has been completed. High quality Shipley BPR100 micromoulds were fabricated using photolithography technology at the depths desired for micromechanical components, e.g. for microengines and other machines. A process was developed for electroplating of alumina nanoparticle-nickel composite microcomponents of good pattern and structural integrity and the effects of different process conditions were explored. Analysis on composition and microhardness shows that the distribution of nanoparticles is even and that microhardness is significantly improved in comparison with electroformed pure Ni components.Boudinage in multilayered rocks under layer-normal compression: a theoretical analysisThis paper presents a dynamic analysis of boudinage in multilayers of alternate brittle and ductile layers under layer-normal compression. Based on the mode of fracturing of individual brittle layers, boudinage is classified into three types: tensile fracture boudinage (Type 1), shear fracture boudinage (Type 2a) and extensional shear fracture boudinage (Type 2b). The layer-thickness ratio, Tr (=tb/td), and the strength ratio, F (=T/2ηε), between the brittle and the ductile units are the principal physical factors determining the type of boudinage. Type 1 boudinage develops rectangular boudins and occurs when Tr is low (<4.5) or F is high (>0.8). In contrast, Type 2a boudinage takes place when Tr is high (>8.5) or F is low (<0.5). The intermediate values of these factors delimit the field of extensional shear fracture boudinage. The square of fracture spacing or boudin width in Type 1 boudinage is linearly proportional to layer-thickness, whereas that in Type 2 boudinage shows a non-linear relationship with layer-thickness. The aspect ratio (Ar) of all the types of boudins is inversely proportional to layer-thickness ratio (Tr). However, Type 1 and Type 2 boudins, have contrasting aspect ratios, which are generally greater and less than 1, respectively.Boudinage structure is a common extensional features, especially in rocks with a layering of contrasting lithologies. Comparable structures, described as foliation boudins (), are also observed in homogeneous foliated rocks, apparently showing no competence contrast (). The development of foliation boudins is attributed to the presence of pre-existing fractures () or interlocking pinching-and-swelling instability ( have demonstrated that during layer-normal compression extensional structures of different orders develop sequentially in the multilayer, similar to the development of different orders of folds in multilayers under progressive layer-parallel shortening.) are considered to develop by tensile fracturing of brittle layers at right angles to layering and where the layers suffer little or no ductile deformation before or after rupturing. This is also supported by the mechanical models on tensile failure of brittle objects embedded in a ductile matrix (Tensile fracture boudins may also assume rhomboidal shape, if the brittle units in the multilayer make an angle with the bulk tension direction or are deformed by layer-parallel simple shear. In such cases tensile fractures form oblique to the layering, giving rise to rhomboidal boudins (). Other workers have shown that the rhomboidal shape may be the result of post-boudinage deformation of rectangular boudins due to a layer-parallel shear component (Field observations suggest that in many cases the rhomboidal shape of boudins may be linked to development of parallel shear fractures in the competent layers oblique to the layering (). The boudins in such cases may assume a trapezoidal shape if the shear fractures are non-parallel, forming a horst-and-graben geometry. A domino-type structure results when the rhombic boudins undergo rotation and offsetting during progressive layer-parallel extension () indicate that the orientation and the spacing of parallel oblique-fractures in the brittle layer are the principal physical factors that could control the kinematics of rhombic boudins.It is thus understood that boudinage involves both tensile and shear fracturing of the competent units in a multilayer. Rock deformation experiments () indicate that tensile fracturing occurs at low confining pressure, whereas shear fracturing prevails at relatively high confining pressure. However, such a correlation may not hold for a natural system. For example, shear fracture and tensile fracture boudins are often observed in the field at a single outcrop or even a hand specimen (e.g. fig. 17.1 in ), suggesting that the two modes of fracturing may prevail at the same confining pressure. Apparently, in addition to confining pressure, there appear to be other factors governing the mode of fracturing of brittle layers (cf. Under layer-normal compression of multilayers, extensional features develop affecting the individual layers as well as the entire multilayer in the course of progressive deformation (). In certain circumstances closely spaced competent layers may respond to the stress as a whole and rupture by through-going fractures, a phenomenon that corresponds more to faulting than boudinage (Lloyd, 1999, personal communication). The purpose of the present paper, however, is to investigate theoretically the mechanical basis of different styles of first order boudinage (i.e. fracturing of brittle layers as single units) affecting the individual layers of a multilayer, in relation to mode of fracturing under layer-normal compression. A few experiments were performed on multilayer models of plasticine (brittle) and putty (ductile) under layer-normal compression to test the theoretical model. Based on the mode of fracturing of individual brittle layers, three different types of boudinage have been recognized: (1) tensile fracture boudinage (Type 1), giving rise to rectangular boudins, (2) shear fracture boudinage (Type 2a), producing rhombic or trapezoidal boudins without any intervening separation zone, and (3) extensional shear fracture boudinage (Type 2b), forming rhombic or trapezoidal boudins with intervening separation zones (. The study also reveals that each type of boudinage produces boudins with a characteristic range of aspect ratios.Modeling of fracture development in a stiff layer embedded by a softer medium hinges on the formulation of stress transfer to the stiff layer from the matrix. The different models as used in the mechanics of composite materials and applied to geological systems () include (1) the shear-lag model, (2) the stress-perturbations model and (3) the energy balance model (see for a review of these models and references).In the present analysis, the mechanical model is framed to represent boudinage of a multilayer with alternate competent (brittle) and incompetent (ductile) layers of uniform thickness under layer-normal shortening (. The incompetent units are modeled with a Newtonian viscous rheology. The bulk deformation is considered to be a pure shear with an overall extension parallel to layering. The transfer of stress to the competent unit from the incompetent medium is formulated with the shear-lag model ( have shown that deformation of a multilayer can take place with or without interface-slip. In our model, we assume that there is no slip at the interfaces between the brittle and ductile units. In addition, the model is idealized by assuming that the layer-normal shortening occurs mostly in the ductile unit and the change in layer thickness of the brittle unit is negligibly small. During deformation the flowing ductile material exerts tangential and normal traction to the surfaces of the brittle layers () that, under specific conditions, may eventually lead to boudin formation, the type of boudins being determined by the mode of fracturing of the brittle layers. We adopt the Griffith criterion for the dynamic analysis of failure of brittle layers (cf. ). According to the Griffith criterion, the tensile stress σxx and the compressive stress σyy in brittle layers have to satisfy the following conditions for failure (where T is the tensile strength of brittle layer. represent the conditions for tensile and shear failure, respectively.In the next section we present the mathematical derivations of the tensile and the compressive stresses on the brittle layers in a multilayer, following the approaches adopted by earlier workers (e.g. Let tb and td be the thicknesses of brittle and ductile units, respectively. We choose a Cartesian coordinate, xy, with the x-axis located along the central line of the ductile layer between any two brittle layers (). The multilayer is deformed under pure shear with the principal shortening normal to layering. In a non-slip boundary condition the instantaneous velocity at all points on the brittle–ductile interface (y=td/2) assumes a constant value. The velocity components along x and y directions are given by u=0 and v=εtd/2, respectively, where ε is the rate of bulk shortening of the ductile unit. The velocity functions for the flow in the ductile unit, that satisfy the above mentioned boundary conditions have the following expressions (To find the stresses in the brittle unit, induced by the flow in the ductile material, we require the strain-rate components in the ductile layer. Differentiating shows that the flow in the viscous layer develops a layer-parallel shear, and thereby exerts a shear stress on the surface of the brittle layer (). The shear stress at any point on the layer interface, i.e. at y=td/2, is:η is the coefficient of viscosity of the ductile material. The shear stresses are symmetrically disposed on either side of the brittle layer, which, as a consequence, suffers a tensile stress (). We now arbitrarily choose a layer segment within x=−l and l (). Under the condition of dynamic equilibrium:where σxx is the tensile stress at x=0. After substituting τ from The above equation is the expression for the tensile stress in the brittle layer, which is similar to that shown by Brittle layers in the multilayer also experience a compressive stress from the reaction to the flow in the enclosing viscous layers. We determine the compressive stress on brittle layers in the following way. The rate of energy required for the deformation of an infinitesimal volume in the ductile layer is:The energy required per unit time, to induce ductile flow in the layer segment in between x=0 and l is then:Substituting the strain components from If the compressive load on the brittle layer is σyy, the rate of work required for bulk shortening in the ductile layer is:, we get the magnitude of compressive stress:The analysis reveals that both the tensile and the compressive stresses on the brittle layer are functions of the length of the layer segment under consideration. For a critical value of l, the stresses in ), and result in rupturing of the brittle layer. The failure takes place by tensile fracturing if the stresses satisfy of the Griffith criterion. On the other hand, if the stresses fulfil the condition of , the failure will occur either by shear fracturing or extensional shear fracturing.In the analysis of mode of brittle failure we rewrite , for convenience, in terms of dimensionless quantities as:. F is the ratio of the tensile strength of the brittle unit to the flow strength of the ductile unit, and is designated as a rheological factor in the subsequent discussion. In we add bulk confining pressure, p, as the failure takes place in response to the total stress (deviatoric stress+isotropic stress). stress space, the failure condition of (i.e. tensile failure) defines a straight-line segment, whereas that of (i.e. shear failure) describes a parabolic line segment (. These two line segments together delimit the field of stability from that of failure, and meet each other at the point (F, −3F). This, in other words, means that if the failure is by tensile fracturing and if the failure is by shear fracturing. It can be shown from that the stress condition on the parabolic failure curve, marking the transition between extensional shear failure and shear failure, will satisfy the condition: . The bottom line is that there are three different segments in the failure curve defining three regimes of failure: tensile failure, shear failure and extensional shear failure. show that layer-thickness ratio controls the tensile and the compressive stresses on the brittle layer. For a given layer-thickness ratio Tr, with increase in Ar the changes in the compressive and tensile stresses describe a linear regression in the stress space. The line meets the failure curve at a critical value of Ar (). The gradient of the regression line increases with increasing layer-thickness ratio. A regression line, therefore, meets one of the three regimes of the failure curve depending on the layer-thickness ratio.The layer-thickness ratio in the multilayer that marks the transition between Type 1 and Type 2 boudinage is obtained from Similarly, the transition between Type 2b and 2a boudinage is marked by: delimit the fields of the three types of boudinage in Tr–F space (. The equations reveal that tensile fracture boudinage is possible when Tr<4.5 and F>0.8. In contrast, shear fracture boudinage may take place at all ranges of Tr if F<0.5, or at all ranges of F if Tr>8.5. Extensional shear fracture boudinage, on the other hand, occurs when Tr is less than 8.5 and F>0.5. In general, tensile fracture boudinage is favored at relatively smaller values of layer-thickness ratio, and is progressively replaced by extensional shear fracture boudinage and shear fracture boudinage with increasing layer-thickness ratio in the multilayer. This explains occurrence of different types of boudins in layers of different thicknesses in a single multilayer over an outcrop or hand specimen. It also appears from that in multilayers with little mechanical contrast (F<0.8) between individual layers, the dominant mode of deformation would be through shear fracturing irrespective of layer-thickness ratio. The theoretical result is consistent with the experimental observations of To verify the control of layer-thickness ratio on the mode of boudinage in individual brittle layers we also conducted a series of experiments with analogue physical models. The models consisted of alternate layers of commercially available plasticine (brittle unit) and putty (ductile unit). The interfaces between the plasticine and putty layers were smeared with kerosene oil to prevent interlayer slip during deformation. We deformed the model by a layer-normal compression in a pneumatically driven vertical piston machine. The model was confined by two parallel, vertical glass plates, fixed by two horizontal pistons, and it was allowed to extend in one horizontal direction. A number of experiments were performed on multilayers with different layer-thickness ratios. Models with a layer thickness ratio of 0.15 were observed to be boudinaged by tensile fractures, giving rise to rectangular boudins (. In contrast, models with a layer-thickness ratio of 2.6 showed rupturing of the brittle layers dominantly by shear fractures (). In the domains of parallel shear fractures, the boudins underwent rotation and offsetting, giving rise to a typical domino structure, while in the domains of non-parallel shear fractures normal slip along the oppositely dipping fractures produced a horst-and-graben structure (). For moderate layer-thickness ratios, boudinage was often associated with extensional shear fractures, producing separation zones between the boudins.For a given layer-thickness ratio in the multilayer, the compressive stress and the tensile stress in the brittle layer reach the failure condition for a particular ratio between the length of the layer-segment and the thickness of the brittle layer (l/tb=Ar). This ratio represents the dominant aspect ratio of boudins in a brittle layer. From , we have in the regime of Type 1 boudinage: show that, irrespective of the mode of boudinage, the aspect ratio of the boudin is inversely proportional to the layer-thickness ratio in the multilayer; however, the aspect ratio of Type 1 boudins decreases more strongly with increasing layer thickness ratio than that of Type 2 boudins (. The equations also reveal that the different types of boudins observed in a multilayer of fairly uniform mechanical contrast between layers are likely to have characteristic ranges of aspect ratios. Tensile fracture boudinage will be favored by relatively thinner layers and the boudins will have relatively large aspect ratios generally greater than 1 (). In contrast, shear fracture boudinage would prevail in relatively thicker layers and the boudins will have relatively low aspect ratios, generally less than 1 (). This is consistent with natural and experimental observations (). Extensional shear fracture boudins, however, will have intermediate values of aspect ratio.The theoretical results indicate that the aspect ratio of tensile fracture boudins is more sensitive to both layer-thickness ratio and mechanical contrast between layers in the multilayer than shear fracture and extensional shear fracture boudins (. However, the aspect ratio of both types of boudins becomes less dependent on the rheological factor with increasing layer thickness ratio (It must now be clear that the boudin width, as modeled in this paper, essentially represents the spacing of the fractures developing within the brittle layers in a multilayer in the process of boudinage. The relationship between layer-thickness and fracture spacing has been an area of major interest to material scientists including the earth scientists. Several types of relationships have been obtained using different stress transfer models (). However, these studies mainly deal with tensile fractures. The analysis presented in this paper can be utilized to understand the relationship between layer-thickness and fracture spacing, not only for tensile (Mode 1) fractures but also for shear (Mode 2) fractures. For tensile fractures the relationship can be deduced from the factor F, a dimensionless quantity, is simply the ratio between the tensile strength of the brittle material and the flow strength of the ductile material with viscous rheology. The rheological factor defined by in their model based on elastic rheology, on the other hand, is a complex function of the elastic constants. Nevertheless, the relationship between fracture spacing and layer thickness remains similar in both the models. suggests that the width of boudins is linearly proportional to . This is consistent with equation (26) of For a fixed value of F, the spacing of tensile fractures is controlled by two parameters: the thickness of the brittle layer (tb) and the thickness of the ductile layer (td). For a constant thickness of ductile layers in the multilayer, fracture spacing is a function of the square root of the thickness of the brittle layer (). However, the spacing becomes progressively insensitive to variation in layer-thickness as the intervening ductile layers become thinner and thinner (). This has been well illustrated in the field examples documented by It may be noted that the relationship between fracture spacing (Lc) and layer-thickness (tb) as discussed above, holds only for tensile (Mode 1) fractures and is likely to be different if the fracturing is by shear failure. The spacing of shear (Mode 2) fractures, derived from It is important to note that, whereas the square of fracture spacing in Type 1 boudinage in is linearly proportional to layer-thickness (tb) in the multilayer, that in Type 2 boudinage in varies nonlinearly with the layer-thickness (. The variation of fracture spacing in Type 2 boudinage shows a decreasing gradient with increasing layer-thickness. Thus, the square of spacing of shear fractures becomes less sensitive to layer-thickness when the latter is large. In addition, with decrease in thickness of ductile layers (td), the spacing of shear fractures becomes more insensitive to layer-thickness than that of tensile fractures.In Lc2–tb space, for any layer-thickness of the brittle layer, the spacing of shear fractures is less than that of tensile fractures. Both Mode 1 and Mode 2 fractures lower their spacing as the thickness of the intervening ductile layers is decreased. But the spacing of Mode 2 fractures shows larger departures, and tends to be independent of the thickness of brittle layer (The outcome of the present paper is summarized along the following points. (1) The type of boudinage in multilayered rocks is controlled largely by the layer-thickness ratio and the mechanical contrast between the brittle and ductile units. (2) Multilayers with low layer-thickness ratios (<4.5) are virtually boudinaged by tensile fracturing, while those with large layer-thickness ratios (>8.5) by shear fracturing of the brittle layers (). For moderate layer-thickness ratios, boudinage is likely to be by extensional shear fracturing. (3) The square of boudin width or fracture spacing in tensile fracture boudinage is linearly proportional to layer-thickness, whereas that in shear fracture boudinage is non-linearly proportional to layer-thickness (). (4) The aspect ratio of tensile fracture boudins tends to be greater than 1. In contrast, the aspect ratio of shear fracture boudins is relatively low, generally less than 1 (). (5) In a layered sequence with variable thickness of brittle layers, the aspect ratio of tensile fracture boudins may show a wide scatter, but that of shear fracture boudins in a similar situation will not vary significantly (Free vibration of FGM layered beams by various theories and finite elementsThe Carrera Unified Formulation (CUF) is used to perform free-vibrational analyses of functionally graded (FG) structures. CUF is a hierarchical formulation for obtaining refined structural theories that account for variable kinematic description. These theories can be obtained by expanding the unknown displacement variables over the beam section axes by adopting any kind of function. The number of the terms in the expansions is a free parameter of the analysis. For Taylor-like expansions, the linear case can result in classical beam theories. For the first time in the 1D CUF framework, the Finite Element method is used to solve the governing equations of functionally graded beams which are derived in a weak form by means of the Principle of Virtual Displacements. These equations are written in terms of f¨undamental nuclei.¨ Their forms do not depend on the expansions used. Several structures are considered, including a sandwich beam with FG core, laminated beams, thin- and thick-walled boxes as well as sandwich cylinders. The results are shown in terms of natural frequencies and compared with those available in existing literature.In recent years, functionally graded materials (FGMs) have gained great attention in many engineering fields, especially when the structures are subjected to high temperatures. The smooth variation of the mechanical properties along preferential directions is achieved by changing the volume fractions of two or more constituent materials, which are generally ceramic and metal(s). By doing this, the well-known problems of the classical composite materials, such as the discontinuities of the stress distributions at the interfaces and the low resistance to temperature shocks, can be avoided. In addition, the behaviour of these materials strongly depends on the gradation law which is used and for this reason, many theoretical models are conceived in order to describe the FG structures properly. Among the several beam theories available in the literature, in In addition to the above papers, many two-dimensional theories have appeared in recent literature for the study of plates and shells containing FGMs. For example, Reddy and co-workers provided the first results on the vibration of FG plates and cylindrical shells in The main scope of this work is to evaluate several refined beam finite elements. These higher-order theories are obtained by means of the Carrera Unified Formulation (CUF). CUF is a well-known procedure, which was initially conceived for the development of refined plate and shell theories In the present work, by using the Finite Element method several FG structures under various boundary conditions are considered. The displacement components are defined by using a variety of functions such as polynomial, trigonometric, hyperbolic and exponential. The study focuses on laminated and sandwich structures as well as thin-walled boxes, in which the mechanical properties vary according to the typical power gradation law. The results are reported in terms of natural frequencies and they are compared with those existing in the literature.The CUF states that the displacement field, u(x,y,z,t), is an expansion of generic functions, Fτ(x,z) for the vector displacement, uτ(y):where T is the number of terms of the expansion and, according to the generalized Einstein’s notation, τ indicates summation. The main advantage of the present approach is that it considers both type and number of functions Fτ(x,z) as input parameters. Therefore, the number of higher-order theories which can be generated is theoretically infinite. In previous papers, encouraging results were obtained by using several advanced theories based on trigonometric, exponential, polynomials and miscellaneous expansions for the study of composite structures. For a thorough and clear description of the new displacement theories, the authors suggest ux=ux1+xux2+zux3+x2ux4+xzux5+z2+e1xaux6+e1zbux7+sin3πxaux8uy=uy1+xuy2+zuy3+x2uy4+xzuy5+z2+e1xauy6+e1zbuy7+sin3πxauy8uz=uz1+xuz2+zuz3+x2uz4+xzuz5+z2+e1xauz6+e1zbuz7+sin3πxauz8Further expansions are evaluated in the results section. The use of the 2D polynomials xizj allows one to write the so-called Taylor-like expansions (i and j are positive integers). For example, the second-order displacement field is:ux=ux1+xux2+zux3+x2ux4+xzux5+z2ux6uy=uy1+xuy2+zuy3+x2uy4+xzuy5+z2uy6uz=uz1+xuz2+zuz3+x2uz4+xzuz5+z2uz6ux=ux1+xux2+zux3+x2ux4+xzux5+z2ux6+x3ux7+x2zux8+xz2ux9+z3ux10uy=uy1+xuy2+zuy3+x2uy4+xzuy5+z2uy6+x3uy7+x2zuy8+xz2uy9+z3uy10uz=uz1+xuz2+zuz3+x2uz4+xzuz5+z2uz6+x3uz7+x2zuz8+xz2uz9+z3uz10 are indicated as TE2 and TE3, and the numbers are the highest exponents of the expressions. A remarkable feature of these expansions is that classical beam theories are obtainable as particular cases. It should be noted that classical theories require reduced material stiffness coefficients to contrast Poisson’s locking and, for this reason, for classical and first-order models Poisson’s locking is corrected according to Carrera et al. ux=ux1+xux2+zux3+x2ux4+xzux5+z2ux6+(-1)kζkux7Zuy=uy1+xuy2+zuy3+x2uy4+xzuy5+z2uy6+(-1)kζkuy7Zuz=uz1+xuz2+zuz3+x2uz4+xzuz5+z2uz6+(-1)kζkuz7Zwhere ζk=2zk/hk is a non-dimensional layer coordinate and hk the thickness of the k-layer. The exponent k changes the sign of the zig-zag term in each layer. In this case, the acronym becomes TE2zz.Adopting the classical Finite Element technique, it is possible to consider arbitrary shaped cross-sections and boundary conditions. The generalized displacement vector is defined as follow:where qτi is the nodal displacement vector:The Lagrange shape functions Ni are listed in where Lint and Line are the strain energy and the work of inertial loadings, respectively, whereas Lext is the work of external loadings and δ stands for virtual variation. If the work of external loadings is null, Eq. δLine-δLint=∫V(δuTρu¨)dV-∫V(δ∊pTσp+δ∊nTσn)dV=0in which the vector u¨ is the acceleration vector. In order to separate the components which lay on the cross-sections (“p”) from those on the planes orthogonal to them (“n”), both stresses and strains are grouped into the following vectors:σp=σzzσxxσzxT,∊p=∊zz∊xx∊zxTσn=σzyσxyσyyT,∊n=∊zy∊xy∊yyTUnder the hypothesis of linear analysis and linear elastic material, the strain–displacement relations and the Hooke’s law are, respectively:The differential operators Dp,Dny and DnΩ are shown in Cppk=C∼11kC∼12k0C∼12kC∼22k000C∼66k,Cpnk=00C∼13k00C∼23k000,Cnnk=C∼55k000C∼44k000C∼33Considering the FGM, since the Young’s modulus (E) and the Poisson’s ratio (ν) are functions of the coordinates (x, |
y, |
z), the coefficients C∼ijk(x,y,z) vary with the position according to the following formula:C∼11k=C∼22k=C∼33k=E(1-ν)(1+ν)(1-2ν)C∼44k=C∼55k=C∼66k=E2(1+ν)The apex “k” refers to the generic piece of the structure, which can be made up of any kind of isotropic material. Clearly, this allows us to investigate composite structure and non-homogeneous sections. The virtual variation of the strain energy is rewritten in a compact format using Eqs. The Kijτs is the stiffness matrix in the form of the fundamental nucleus which is shown in the following formula:Kijτs=Ilij◁DnpTFτIC∼npkDpFsI+C∼nnkDnpFsI+DpTFτIC∼ppkDpFsI+C∼pnkDnpFsI▷Ω+Ilij,y◁DnpTFτIC∼nnk+DpTFτIC∼pnkFs▷ΩIΩy+Ili,yjIΩy◁FτC∼npkDpFsI+C∼nnkDnpFsI▷Ω+Ili,yj,yIΩy◁FτC∼nnkFs▷ΩIΩyIlij,Ilij,y,Ili,yj,Ili,yj,y=∫lNiNj,NiNj,y,Ni,yNj,Ni,yNj,ydy, the virtual work of inertial loads in Eq. where q¨ is the nodal acceleration vector and the density ρk can change similarly to E and ν. The last equation can be written in the following compact manner:where Mijτs is the mass matrix in the form of fundamental nucleus. Its components are:Mxxijτs=Myyijτs=Mzzijτs=Ilij◁(FτρkIFs)▷Mxyijτs=Mxzijτs=Myxijτs=Myzijτs=Mzxijτs=Mzyijτs=0The expressions of the fundamental nuclei do not vary with the changing of the approximation order of the displacement theories indeed, so far, no assumption have been made on the models accuracy. In this way, it is very easy to obtain refined beam models with a completely automatic procedure.Finally, the undamped dynamic problem is written as it follows:where q is the vector of the nodal unknowns. Introducing harmonic solutions, q=q0eiωt, it is possible to compute the natural frequencies (ωk) by solving the eigenvalues problem:Although only a part of results can be presented in this section, all considered expansions are summarized in in which acronyms are introduced. In order to understand their meaning, we may refer to the following short expressions:functions with single trigonometric factor:sx=sinmπxacx=cosmπxasz=sinnπzbcz=cosnπzbsd=sinmπxzabcd=cosmπxzabfunctions with two trigonometric factors:cc=cosmπxacosnπzbcs=cosmπxasinnπzbsc=sinmπxacosnπzbss=sinmπxasinnπzbchx=cosh(mx)shx=sinh(mx)chz=cosh(nz)shz=sinh(nz)If Taylor’s expansions have been added to displacement fields, their order has been specified by the subscript. For the sake of clarity, in the following is presented the example for the first component of the displacement field E22:ux=costux1+sinπxaux2+cosπxaux3+sinπzbux4+cosπzbux5+sin2πxaux6+cos2πxaux7+sin2πzbux8+cos2πzbux9+xux10+zux11+x2ux12+xzux13+z2ux14Anyway, as said before, the reader can refer to In order to assess the theory, we consider a square cross-section beam. The structure is made of alumina and steel, whose properties are E1=390GPa, ν1=0.25,ρ1=3960kg/m3 and E2=210GPa, ν2=0.31,ρ2=7800kg/m3, respectively. With the purpose of enabling a general application of results, they are reported in the following non-dimensional form:where a is the cross-section dimension and ω is the angular frequency. E,ν and ρ vary according to the power gradation law in Eq. along either z-axis (α1=0) or both z and x directions (α1≠0 and α2≠0).Several length-to-thickness ratios (L/a=5,10,100) are considered and the results are compared with those available in , our results obtained by Taylor-like models strongly agree with the references despite slight differences concerning the axial mode. Furthermore, show the results related to the E2, E4 and E9 expansions which are explicitly expressed in Eqs. u(x,y,z,t)=1u1(y,t)+sinmπxau(4m-2)(y,t)+sinmπzbu(4m-1)(y,t)+cosmπxau(4m)(y,t)+cosmπzbu(4m+1)(y,t)u(x,y,z,t)=1u1(y,t)+e(mx)u(2m)(y,t)+e(mz)u(2m+1)(y,t)u(x,y,z,t)=cosmπxacosnπzbum,n(y,t)+cosmπxasinnπzbum,n(y,t)+sinmπxacosnπzbum,n(y,t)+sinmπxasinnπzbum,n(y,t)where the indexes m and n indicate summation and go from 1 to N for sinusoidal, cosinusoidal and exponential terms and from 0 to N for trigonometric terms with two factors. The last columns show the number of degrees of freedom (DoF) of the displacement models. It must be underlined that the Taylor-like expansions appear more effective than both trigonometric and exponential series in the computation of the natural frequencies. Indeed, for example, TE5 provides results closer to the reference than to E9 with less than half of DoF.The second assessment regards the dynamic behaviors of thin and thick FGM boxes. Both geometrical and material properties (see ) considering different values of α1 and α2. The angular frequencies (rad/s) related to the flexural modes in z- and x-direction and the torsional modes are indicated as ωz,ωx and ωt, respectively. For the thin box, the results in show that the flexural frequencies remain nearly constant and greater than the references, regardless of order and type of expansion. On the contrary, the enrichment of the displacement fields lead the beam elements to detect the torsional frequencies at very different values. Since the structure walls are very thin, the torsional modes can be affected by the shell-like deformations of the cross-section and, for this reason, there can be strong discrepancies in frequency computation (see the theory E9-4). In contrast, for the thick boxes (), the results obtained with the present kinematic models agree with the 3D solutions for all considered conditions. In the case in point, shell deformations are lessened by the considerable thickness of the boxes and, for this reason, even TE5 theory ensures a good accuracy in the computation of the torsional frequencies. Some slight discrepancies between the present results and those proposed in In the following case, sandwich beams are considered. These structures consist of two isotropic faces perfectly bonded to a functionally graded core. The mechanical properties of the core vary according to the linear rule of mixture in which the volume fraction of each of the constituent materials is determined by using the power law distribution (Eq. ). The core is made up of Zirconia and Aluminum whose properties are shown in together with those of the face sheets. The face layer and core thicknesses are 3 and 14 mm, whereas the overall length and width are 200 and 20 mm respectively. The natural frequencies are shown in in which those related to the bending modes in z-direction as well as the first axial mode are compared with the solutions found in In the final example a FGM sandwich cylinder is studied. Referring to , the ratios length-to-mean radius (L/R) and mean radius-to-thickness (R/h) are assumed to be 5 and 10, respectively. The FG core is perfectly bonded to two isotropic faces, whose thicknesses are equal to h1=h3=0.1h. The Young’s modulus and the mass density of the core vary according to the following formula:where h2 is the core thickness, ζ is the generic coordinate (-h22<ζ<h22) and the exponent α assumes the values 3, 5 and ∞. In the case in point, E1 and E2 are assumed to be 70 and 380 GPa, whereas the mass densities ρ1,ρ2 are 2702 and 3800 kg/m3, respectively. The Poisson’ s ratio ν is constant and equal to 0.3. For sake of clarity, shows how the properties E and ρ change along the thickness. The results in are reported in terms of non-dimensional frequency parameter ω¯=ωhρ2E2 and compared with those presented in illustrate how the percentage error ((ω¯-ω¯ref)×100ω¯ref), relating to the various modal shapes, varies with respect to the number of degrees of freedom (DoF). For both values of the exponent α (3 and ∞), it is possible to note that higher order models are needed to reach acceptable results for all modes of deformation and indeed, by using either the tenth-order Taylor expansion or the E9-4 series, errors remain lower than 5%.In this work, one-dimensional finite elements based on various displacement theories have been employed to perform free vibration analyses of the functionally graded beams. The implementation of the finite elements in accordance with the Carrera Unified Formulation made it possible to consider a great variety of structures and boundary conditions. In the light of the results obtained, the following remarks can be made:the invariant properties of CUF make it possible to derive infinite numbers of one dimensional models, without the need for adhoc assumptions;when thick sandwich beams as well as thin-walled structures are treated, higher-order theories are mandatory in order to ensure a good accuracy in the computation of the flexural and torsional frequencies;the variable kinematic models are able to predict with excellent accuracy the shell-like frequencies for different gradation laws.In conclusion, the extension of 1D-CUF to the dynamic study of functionally graded structures makes it possible to solve problems which generally require bi- or three-dimensional solutions. Even if the choice of the model is problem-dependent, the possibility to conceive a huge variety of kinematic theories appears to be a remarkable advantage of the proposed formulation. The inclusion of CUF in genetic algorithms could be a suitable approach for comparing several theories for many structural problems. Moreover, future works could investigate the static behaviour of FG structures both in terms of displacements and stress distributions as well as the effect of temperature on fixed and rotating FG beams.Improved core model of the indentation for the experimental determination of mechanical properties of elastic-plastic materials and its application, (1985)) takes into account the additional pressure and shear stresses jump at the boundary with the hydrostatic core where it is formed from the material of the elastic-plastic zone. This causes additional volumetric deformations and additional pressure in the core that increases the magnitude of the constant in the Tabor ratio. The effects of the proposed improvement of the Johnson model: additional volumetric deformations and pressure in the core, changes in the Tabor constant and related values of yield strength as well as characteristic size of the elastic-plastic region, were determined on a wide class of materials with different elastic-plastic properties. An analysis of the concept of characteristic (representative) strain is given for these materials.A modified model of the indentation by elastic indenters is proposed, in which the sample elastically-plastically deforms and the material obeys the Mises flow condition. The model generalizes and refines the well-known simplified Johnson's core model () with an elastically deformable indenter, based on the experimental concept of the formation of a hydrostatic core adjacent to the indenter and the analysis of the spherical cavity expansion in an elastic-plastic material made by , determines and investigates the dependence of the Tabor parameter () С = НМ/YS (where НМ is the Meyer hardness, YS is the yield strength of the sample) on the elastic modulus, yield strength, and indenter geometry. The details of this study and historical background for this problem until 1969 are contained in . The concept of the core model and the Tabor-Johnson relations were investigated and tested by numerical methods by many researchers, see, for example, showed good agreement between the model calculations for perfectly elastic plastic materials and experimental data for hardening materials using the concept of characteristic (representative) deformation on the uniaxial compression curve proposed by , (1996). An original core model is given by , where an analytical review of works on the subject is presented and the model of indentation of a cone into a sample from a material subject to finite strain plasticity (in contrast to incremental plasticity of Johnson's model) is discussed, taking into account compressibility and arbitrary strain hardening outside the core. In this case, the previously obtained by ) solution to the problem of the expansion of a spherical cavity is applied, which determines the functional dependences of the radial displacement and pressure at the boundary of the cavity on the Mises (Tresca) effective stresses at its boundary and the known dependence of the total deformation on the Mises (Tresca) effective stresses characterizing the properties of the material, in particular, its hardening. Also, as in , (1985) model, the radial displacement of the boundary of the cavity is determined by the volume of the embedded part of the indenter under the assumption of incompressibility of the core material. A constant non-hydrostatic stress state is assumed in the core, in which the Mises (Tresca) effective stresses are continuous across the boundary of the core and are associated with hardness and pressure in the cavity. The case of small deformations is considered and investigated in detail.Such simplified concepts of the core models do not require large computational resources, program services of monopolies and have no alternative in the absence of fundamental physical models and extensive material data for the realization of more accurate approaches. Analysis of the literature shows that the interest to the analytical simplified approaches does not weaken and is extended to more complex problems in particular, penetration ()). This trend will continue in the near future.The model proposed in this paper is also based on the concept of the formation of a hydrostatic core under the indenter and the expansion of a spherical cavity in the material, refines the consequences of this concept and allows determine the Tabor parameter, yield strength, characteristic relative size of the elastic-plastic zone in the sample, volumetric deformation of the core, and effective angle of indenter under load. A methodology is presented for determining these parameters with known hardness and elastic constants of the indenter and the sample., (1985) the proposed model differs by taking into account the following: 1) volumetric strain (compression) of the material during core formation; 2) additional pressure in the core due to a jump in shear stresses at its boundary, as well as 3) the history of formation of the core. The additional pressure is determined by the curvature of the core boundary and the jump in hoop stresses on it (equal to the yield strength of the material) and further will be referred to as the “shell” effect (the material surrounding the core is the shell for the core). In , (2017), the effect of compression of the core material due to the pressure difference in the core and in the surrounding elastic-plastic zone was also taken into account, however, without consideration of the “shell” effect.The results of applying the model to a large group of materials and their comparison to finite-element simulations and previous model are presented.Based on a study of a wide (in terms of plasticity properties) group of materials, the analysis of the concept of characteristic (representative) strain εr during indentation of materials, its structure and mechanical meaning (as average of the total linear strain in the contact area in the indentation direction, εr=⟨εt⟩) is given. It is shown that the Tabor ratio C = HM/YS is determined by the volumetric stress-strain state of the sample under the indenter and mainly depends on the characteristic size of the elastic-plastic zone. This dependence, in practice, is functional, monotonously increasing with the growth of this size., (1985) the basis of this new approximate model of indentation with an embedded core is Hill's solution () for the problem of expansion of a spherical cavity in an elastic-plastic medium with Mises flow condition. Also the absence of friction is assumed in the contact area. In contrast to , (1985), elastic compression (elastic volumetric strain) of the material from which the core is formed during indentation is taken into account, as well as the additional pressure in the core caused by a jump in shear stresses across its boundary. The change in the geometry of the indenter due to its elastic deformation is also taken into account.The load on the indenter is assumed such that in the sufficiently brittle materials normal fracture cracks (mode I) do not form, causing material discontinuity and significantly changing the stress-strain state accepted in the model. In examples of its application to specific materials given at the end of the paper, this condition was strictly controlled. For more brittle materials (with a large ratio of compressive to tensile strength), where during indentation a significant fracturing by mode I cracks is observed, other models are used, see, for example, . For the case of large loads on the indenter with significant fracturing of the material, hardness model of , etc. can be used, where a core is formed under the indenter from the fractured compacted material, under which the region of porous elastic material formed by mode I cracks is located. shows a diagram of the contact interaction of a conical indenter and a sample in a spherical coordinate system 0rθϕ in which a hydrostatic core of radius “c” is formed, the dashed line shows the undeformed indenter. The following notation is accepted: 0 ≤ r ≤ c is the region of the core; c ≤ r ≤ bS is the spherical layer of the sample where elastic-plastic deformations occur; r ≥ bS is the region of elastic deformation of the sample. Strains are assumed to be sufficiently small and do not affect the density of the material.With the continuous penetration of an elastic indenter, the core increases at the expense of the material of the elastic-plastic zone of the sample. This occurs at its boundary, where the material of this zone is compressed by a pressure in the core which is greater than the pressure in the elastic-plastic zone (when crossing the boundary of the core, a jump in pressure and volume deformation takes place; shear stresses that are absent in the hydrostatic core also are discontinuous; radial stresses are continuous). During such penetration, the material of the elastic-plastic zone is additionally compacted at the core boundary by pressure ΔpS (due to a pressure jump ΔpS at this boundary, see below) and joins the core material. For a pressure jump ΔpS at the boundary of the core and hardness HM = P/(πc2), we have:, p. 99), pressure in elastic-plastic zone of the sample at the boundary r = c + 0 with the core is equal:and pressure in the core (and at its boundary r = c – 0) is equal:where σr(r) and σθ(r) = σϕ(r) are radial and hoop stresses, respectively, and p = HM is the pressure in the core equal to Meyer hardness HM.Taking into account the assumptions made, the mass conservation equation (an analog of the incompressibility condition (volume conservation at constant density) of the core (6.34) in ) for the sample with an increase in its approach with the indenter by dh = dhi + dhS takes the form:where εS is the volumetric strain in the sample caused by the pressure difference in the core and at the boundary of the elastic-plastic zone of the sample ΔpS=HM−2YSln(bS/c); KS=ES3(1−2νS) is the bulk modulus of elasticity of the sample material; uS(r) is the radial displacement at a point of the sample, depending on its coordinate r, and h = hi + hS is the approach of indenter and sample. Here and further the indices i and S refer to the indenter and the sample, respectively. The remaining notations in We now establish the relationship between dhS, the radius of the contact region c, and its increment dc, associated with an increase in the approach h by dh., for the indenter angle ψ under load, we have the relation:where 1Ei∗=1−νi2Ei; Ei, νi are the Young modulus and Poisson ratio of indenter. Therefore, for dhS we have (see Then, on the basis of relation (1), a differential equation that determines the dependence (see ) of radial displacement uS(c) at the core boundary on the size of the contact region c (core radius) and corresponds to equation (6.34) in 2πc2(1+εS)duS(c)=πc2dh=πc2(cotψ)dc,εS=−HM−2YSlnbScKS., (1985) this equation takes into account the elastic compressibility εS of the material during core formation. relates the radial displacement uS(c) of the boundary between the elastic-plastic region and the core with the size of the contact region c, that is characteristic size of the indenter penetration into the sample.Based on relation (3), the equation that determines the relative size of the elastic-plastic zone bS/c in the sample (see ) and corresponds to equation (6.35) in (1+εS)[6(1−νS)(bS/c)3−4(1−2νS)]=(EScotψ)/YS,where ES is the Young modulus, νS is the Poisson ratio, and YS is the yield strength of the sample (subscript S). From , (1985) this equation differs by taking into account the elastic compressibility of the material during core formation. for the thickness uS(c) of the incremented part of the core with radius c, accurate to a value of the order of ∝0.5cotψ⋅c|εS|≪uS(c) (since the value of εS depends on c and is assumed to be constant when integrating the equation), we obtainHence, for the initial radius of the core c0, we haveSince |σr−σθ|=YS everywhere in the elastic-plastic region and the radial stresses are continuous at its boundary with the core (σr(c + 0) = σr(c – 0) = –HM), then at the boundary of the core and the elastic-plastic region the hoop stresses σθ have a jump σ = HM + σθ = YS and are equal tothat is, tensile stresses equal to the yield strength are added to the core hoop stresses.Thus, when a core is formed during the indenter penetration, an additionally stretched in the circumferential direction (by stresses σ = YS) layer of the material of the elastic-plastic region is attached to the core, which creates additional compression of the core (“shell” effect, see ). The tensile force in this layer corresponding to the increment of the elastic-plastic region duS(c) (see (3)) isand the pressure in the core is incremented by:Hence, with an increase in the spherical layer joining the core from radius c0=c−uS(c)=c(1−cotψ2(1+εS)) to radius c (which takes into account the history of the formation of the core), for additional pressure pR in the core we have:pR=cotψ(1+εS)YSlncc0=cotψ(1+εS)YSln[1−cotψ2(1+εS)]−1,and for it with cotψ << 1 we get the estimatepR=−cotψ(1+εS)YSln[1−cotψ2(1+εS)]−1≈cot2ψ2(1+εS)2YS.Thus, for total pressure in the core p = HM, we have:p=HM=YS(23+2lnbSc)+pR=YS(23+cotψ(1+εS)ln[1−cotψ2(1+εS)]−1+2lnbSc),Therefore, if the elastic characteristics of the sample and indenter material (ES, νS, Ei, νi), as well as the hardness HM are known, then equations and the equation for pressure in the core (12) form the following system of transcendental equations:{cotψ=cotγi−2HMEi∗,(1+εS)((bSc)3−2(1−2νS)3(1−νS))=EScotψ6(1−νS)YS,HM=YS(23+cotψ(1+εS)ln[1−cotψ2(1+εS)]−1+2lnbSc),εS=−HM−2YSlnbScKSthe solution of which with respect to real unknown quantities (cotψ, bS/c, YS) approximately determines the stress-strain state in the sample in accordance with the proposed model, as well as the Tabor constantC=HMYS=(23+cotψ(1+εS)ln[1−cotψ2(1+εS)]−1+2lnbSc).Such a record of the equations of the model is convenient, since it clearly demonstrates its mechanical meaning.If we introduce the notation: αS=2(1−2νS)3(1−νS), βS=ES6(1−νS)HM, θS=HMKS and dimensionless unknowns x=bSc, y=YSHM, z=cotψ, then the system of equations with respect to unknowns x, y, z takes the form:{z=cotγi−2HMEi∗,(1−θS(1−2ylnx))(x3−αS)=βSzy,1=y(23+z(1−θS(1−2ylnx))ln[1−z2(1−θS(1−2ylnx))]−1+2lnx),where z is in the range [0; cotγi], x is the relative size of the elastic-plastic zone.If x, y, z is the solution to system (15), then for the Tabor constant C we have the valueC=1y=23+z(1−θS(1−2ylnx))ln[1−z2(1−θS(1−2ylnx))]−1+2lnx.By simple transformations, system (15) can be represented as{z=cotγi−2HMEi∗,(1−θS(1−2ylnx))(x3−αS)=βSzy,1=y(23+y(x3−αS)βSln[1−y(x3−αS)2βS]−1+2lnx)., the quantity z can be considered as a parameter, and the quantities x(z), y(z) – as functions of this parameter. take into account the elastic compressibility of the material during core formation, as well as the additional pressure in the core (due to the “shell” effect that arises when the indenter is embedded), and, thus, the proposed modified model (equation , (15), (16))) generalizes and develops the model discussed in , (2017) within the framework of the concept of a hydrostatic core.The results are presented for the case of penetration of a circular cone with an angle at the apex 2γi. The transition from pyramidal indenters to equivalent conical ones (and vice versa) can be performed using the condition of equal projection of the areas of imprints left by different indenters for the same penetration volume (the same penetration depth for pyramidal and conical indenters). This condition leads to the following relation between the sharpening angles of the equivalent conical and pyramidal (trihedral and tetrahedral) indenters:where γi, γV, γB – respectively, the angles of sharpening of the indenters: conical, tetrahedral (e.g., Vickers, γV = 68°) and trihedral (e.g., Berkovich, γB = 65°).Calculations according to the new refined model were performed for a number of materials studied earlier in by a diamond (Ei = 1200 GPa, νi = 0.07) Vickers indenter (with an aperture angle of faces 2γ = 136° and an equivalent cone angle 2γi = 140.6°). The initial experimental data for HM, ES, νS, Ei, νi, are also taken from shows the relative dimensions of the elastic-plastic zone x, volumetric deformation in the core εS, yield limit YS and Tabor constant C, as well as the relative deviation of these quantities δx, δεS, δYS, and δC (in %) for the present model from the model of , the materials of each group are arranged by the decreasing value of the ductility characteristics δH (δH=εpεt,εt=lnsinγV,εp=εt−εe<0,εe=−1+νS1−2νSHMES,where εe, εp, εt are, respectively, elastic, plastic and total average in the contact area linear (in the direction of the force P, see , the value εt=lnsinγV≈−0.076,γV=68° is a constant value and is determined by the geometry of the contacting bodies (indenter and sample) before their deformation. If for sufficiently developed plastic deformations we neglect the compressibility (volumetric strain ε11+ε22+ε33=0) of the contacting bodies, then the value εt=−0.076 determines with good accuracy the average linear compressive strain in the contact region in the direction of the force acting on the indenter (see ), similar to uniaxial tension-compression. Therefore, this value can be considered as a representative (characteristic) strain εr=εt=−0.076 under uniaxial compression, as was proposed by Tabor and Johnson (). This deformation corresponds to both the yield strength YS in and the hardness value HM = CYS as the average value of the contact pressure under the indenter. In this case, the value of C is determined by the complex volume stress-strain state of the sample under the indenter, which differs significantly from uniaxial compression, see below. The structure of representative strain εr=εe+εp is approximately determined (estimated) by the formula: εe=(1−δH)εr,εp=δHεr.If the average total linear deformation and its components in the contact region of the indenter and the sample are determined from the deformed contact pair scheme according to the formulas by εt=εe+εp,εp=−ln1+cot2γSR<0,cotγSR=cotγi−2HME∗,where 2γSR is the angle at the apex of the residual imprint in the sample after unloading of the indenter, εe, εp, εt are, respectively, elastic, plastic and total average in the contact area linear (in the direction of the force P, see ) strains, then value εt will not be the same (constant) for all materials, including materials in . However, its average value ⟨εt⟩=-0.060 (for all materials in ), which has a small standard deviation σ=0.003, is almost constant, since all the experimental values in the table were obtained by the Vickers diamond indenter. In this case, the corresponding plasticity characteristic δ‾H=εp/εt, where the values εp,εt are calculated by , differs from the value δH indicated in , moreover δ‾H=εp/εt<δH, and the difference (δH−δ‾H) is larger for less plastic materials, with lower values of δH. , in essence, are analogues of formulas that determine the longitudinal strain of a rod by its deformed cross section under uniaxial compression. shows the functional dependencies С(x) and δH(x)Cx=23+A+2lnx,δH=1+AlnsinγV⋅1+ν‾S1−2ν‾SCx231−ν‾Sx3−21−2ν‾S,A=0.0688±0.00062,ν‾S=0.283±0.003,γV=68°,which, with practical accuracy, approximate the results obtained in on the basis of the proposed core model of the indentation. The formula for plasticity characteristic δH is obtained by simple algebraic transformations from its definition (18) and equations ) of system (15), and the formula for C(x) is obtained from the third equation of system (15). In this case the constants A and ν‾S in were determined by the method of least squares. Dependences (20) show that the Tabor parameter C=HM/YS and the relative characteristic size x of the elastic-plastic zone practically functionally depend on the plasticity characteristics δH of the material and are monotonically increasing functions of this quantity (see also ). This means that the Tabor ratio is determined by the volumetric stress-strain state of the material under the indenter, which differs significantly from uniaxial. shows a comparison of the elastic-plastic zones obtained by the proposed model and those calculated by the FEM (finite element method) in ) for power-law strain-hardening elastic-plastic materials. When determining x from , the Tabor ratio C was calculated using the formula C=1.440+0.264ln(ES/σr) from ), where FEM calculations are presented for samples with ratios: ES/σr=36;63;521. Since the elastic-plastic zone in the FEM model has a complex boundary geometry, the characteristic size x is compared with the corresponding average FEM size ⟨x⟩ (i.e., the complex boundary in FEM is approximated by a sphere). This comparison indicates their quite satisfactory correspondence: x≈⟨x⟩, including the case ES/σr>400 when equation Given the mechanical meaning of the quantity εt, its practical constancy for a given indenter geometry, and the existence of functional dependencies C(x),δH(x), it is natural, following the concept of characteristic deformation of , to consider the value εt as a characteristic (representative) strain on the uniaxial compression curve, i.e. to assume εr=εt. We emphasize that the modulus of this quantity is significantly less than 1 (i.e., the small strain hypothesis is valid for the expansion model of spherical cavity of ) and is close to the values (0.07, 0.08) indicated in , and related mainly to the geometry of the Vickers indenter. For conical and pyramidal indenters, propose an empirical quantity |εr|≈0.2cotγi and the use of various angles γi to obtain stress-strain characteristics of materials, including hardening. The noted constancy of the quantity εr=εt (as the average linear strain in the contact region of the indenter and the sample, normal to this region), as well as the dependences δH(x), C(x), are associated more with the indenter geometry (i.e., with the hardness measuring instrument) than with the Tabor ratio C=HM/YS. showed that the introduction of the characteristic strain value εr is rather arbitrary (this is confirmed by the wide range of suggested values for εr, see the review in ) and is connected with the fact that the model results for elastic-perfectly plastic materials can be applied to materials with hardening with a good degree of approximation.An improvement of the widely used in practice , (1985) approximate core model is presented, the basic concept of which is the formation in the sample under the indenter of a spherical hydrostatic core, which is confirmed by experimental observations. This concept makes it possible to significantly simplify the modeling of the elastic-plastic penetration of the indenter when measuring hardness using the well-known model of of spherical cavity expansion in an elastic-plastic medium.The proposed modification formulates the conditions for the coupling of the core and the elastic-plastic zone at their boundary and takes into account additional consequences of the concept of a hydrostatic core, on the basis of which refinements to the , (1985) model were made: 1) elastic compression of the elastic-plastic zone material at the boundary with the core, from which the core is formed during indentation (due to a pressure jump at the core boundary); 2) formation of additional pressure in the core (“shell” effect), caused by a jump in hoop stresses at the core boundary where the plastic flow occurs by the value of the yield limit YS, as well as due to the curvature of the core boundary. For a large group of materials with different level of plasticity the estimate of the corrections and their influence on the calculated yield strength and relative size of the elastic plastic zone formed under the indenter was made. It was shown that taking into account the “shell” effect leads to an increase in the size of the elastic-plastic zone (from 0.7% to 3.46%), as well as Tabor constant (from 2.1% to 14.77%). Volumetric deformation in the core is more significant for less ductile materials having a lower plasticity characteristic.For the considered group of materials, a detailed analysis of the concept of characteristic (representative) strain εr and the Tabor-Johnson relationship C=HM/Ys is given, confirming its validity. To this strain corresponds both the yield strength YS in and the value of hardness HM = CYS, where C is a function of the relative characteristic size x of the elastic-plastic zone under the indenter. The plastic εp=εrδH and elastic εe=εr(1−δH) components of the strain εr are determined by the ductility characteristic δH.Boris Galanov: Conceptualization, Methodology, Investigation, Formal analysis. Sergei Ivanov: Investigation, Formal analysis, Software, Writing – original draft, Writing – review& editing. Valeriy Kartuzov: Project administration, Funding acquisition, Writing – review& editing.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Penetrability prediction of microfine cement grout in granular soil using Artificial Intelligence techniquesIn connection to permeation grouting present study is aimed to investigate the penetrability of microfine cement (MC) grout in granular soil. Series of laboratory based sand column grouting tests were undertaken to characterize the penetrability of MC grout in granular soil in terms of rheological properties of grout suspension (i.e. yield stress, τ0 and plastic viscosity, µ), properties pertinent to sand and grout material such as fine sand content (FC), relative density of sand (RD) and uniformity coefficient of sand (Cu) and groutability ratio (N2) and grouting procedure (i.e. grouting pressure, P), using permeation grouting technique. Ten (10) different sand types having d10 ranging from 0.17 mm to 2.53 mm and Cu ranging from 1.35 to 5.76 were grouted in laboratory with MC grout suspensions under two different relative densities (i.e. 30% and 70%). MC grout suspensions were prepared with four different water to cement (w/c) ratios viz. 0.8, 1, 2 and 3. Rheological tests of the MC grout suspensions prepared with different w/c ratios were performed to evaluate the flow properties (τ0 and µ). Subsequently, artificial neural network (ANN) and support vector machine (SVM) based penetrability prediction models were developed to correlate penetrability with τ0, µ, FC, RD, Cu, N2 and P. Sensitivity analysis and neural interpretation diagram (NID) was employed to identify the key variables in penetrability prediction, to measure its effect and to explain and extract understandable knowledge from the proposed model.In deep soil stabilization, permeation grouting with microfine cement is a widely utilized approach for soil improvement in construction engineering. Permeation grouting is the process of filling voids in a soil or rock mass with a grout fluid at a low injection pressure to decrease the permeability and to improve the shear strength, while not destroying the original structure of the soil or rock (). Conventionally, groutability ratios such as N1 and N2, etc. have been used to predict the groutability of granular soils with particulate grouts (). Groutability ratios i.e. N1 and N2 are defined in Eqs. where d15 and d10 are the diameter of soil passing 15% and 10% of total soil mass respectively, d85 and d90 is the diameter of grout passing 85% and 90% of total grout mass respectively., a soil is groutable when N1 > 25 or N2 > 11 and ungroutable if N1 < 11 and N2 < 6. But such criteria i.e. Eqs. do not consider rheology of the grout, other relevant properties of grouted medium (i.e. relative density, fine sand content, gradation) and grouting pressure (). Recently, a number of approaches were proposed to relate groutability with above mentioned parameters that provides reasonably good groutability estimations for cement based grouts (). These approaches characterize fresh properties of cement grout suspension either in terms of w/c ratio or apparent viscosity of the grout (). Grout w/c ratio does not reflect the effect of fineness of cement and superplasticizer whereas apparent viscosity is the viscosity corresponding to a certain shear rate. Cement grout as a fluid is most often assumed to behave like a Bingham fluid with good accuracy () and its actual rheological behaviour can be represented only in terms of yield stress and plastic viscosity which are considered as fundamental flow properties. In addition, yield stress is a crucial parameter in grouting applications since it is related to stoppage of grout flow through a porous medium and the post grouting performance after a certain resting period (). Therefore, investigating groutability in the light of rheological parameters i.e. yield stress and plastic viscosity in particular and other relevant properties (such as properties of grouted medium i.e. gradation characteristics, relative density and fine sand content, grouting pressure besides effective grain size of grout and soil) in general, presents a research scope.The term “groutability” typically refers to “penetrability” and “injectability” of a grout through a soil (). “Penetrability” corresponds to the permeation distance the suspensions can travel during injection and is evaluated by monitoring the penetration distance of a grout at a predetermined maximum pressure (). “Injectability” is related to the ability of the grout to enter into the voids of a given soil (For successful implementation of any grouting project, estimation of penetrability of grout suspension is of key importance and should be done prior to grouting as it determines the efficacy of soil improvement as well as overall cost of the project. A reliable prediction of the penetrability of cement suspensions can help in selecting proper grouting materials and also enables one to assess the distance and sequence of grouting boreholes in a more realistic manner, thus minimizing, the uncertainties in the design and execution of grouting operations (). However, groutability is affected by several parameters i.e. grout rheology, effective grain size of grout and soil, properties of soil media and grouting pressure. Therefore, due to high dimensionality of the grouting problem and complex interaction between the variables, estimation of groutability and hence penetrability is a difficult job. Nevertheless, in such situations, use of Artificial Intelligence (AI) techniques such as ANN and SVM for prediction of penetrability will be an interesting choice as these techniques have produced successful results in different knowledge domains (), and in particular in the field of Geotechnical engineering (). In fact, development of AI technique based models using data collected from geotechnical works can serve as a useful basis for the design of future projects. The underpinning objective of using these techniques is that data contains useful trends and rules which can be extracted from the data through application of these techniques. Contrary to traditional statistical approaches, AI techniques have significant capacity/supremacy to deal with huge amount of data, characterized by high-dimensionality and intricacy. Furthermore, the developed models based on these techniques can be easily updated when new data are available. For brevity of the present study, details of AI techniques (i.e. ANN and SVM) are not discussed and can be found in Typically injected pore volume of a grout at a given length of soil column in laboratory have been used to asses groutability by various researchers (). In the present study, in addition to penetrability (which is assessed by measuring the permeation distance or length), emphasis is also given to quantify injectability and is obtained by measuring the injected pore volume of a grout. Although, primary objective of the present study is to investigate penetrability aspect of groutability, but due consideration has been also given to the injectability aspect while investigating penetrability, by applying correction factor to penetrability due to injectability, as further highlighted in subsequent sections.The main objectives of the present study are:To investigate the penetrability of MC grouting in granular soil considering the effect of parameters such as rheological properties of grout suspension i.e. yield stress and plastic viscosity, effective grain size of sand and grout, relative density of sand, fine sand content, gradation of sand and grout injection pressure;To develop ANN and SVM models to characterize penetrability in terms of yield stress and plastic viscosity of grout, effective grain size of sand and grout, relative density of sand, fine sand content, gradation of sand and grout injection pressure;To conduct sensitivity analysis to evaluate relative importance of experimental parameters in influencing the penetrability of MC grout in granular soil.In order to study the penetrability of microfine cement (MC) grout in granular soil, locally available sand with predominantly sub-rounded or nearly rounded grain shape was used in the present study for the preparation of different soil types. At first five different sand fractions with grain size range 4.75–2.36 mm, 2.36–1.18 mm, 1.18–0.600 mm, 0.600–0.300 mm and 0.150–0.300 mm denoted as R1, R2, R3, R4 and R5 respectively were prepared. Using the above mentioned sand fractions by mixing them in various proportions, five different composite sand types (designated as CS1, CS2, CS3, CS4 and CS5) were also produced. As a whole ten different clean dry sand types were prepared in the laboratory to study the effect of effective size, gradation and fine sand content of sand on the penetrability of cement grouts. The engineering properties of different sand types used in the study are presented in . It may be observed that all the composite sand types i.e. CS1–CS4 except CS5 corresponds to same grain size upper limit, but lower limit is varied to produce sand with different uniformity coefficients (), which are higher than the ones of sand fractions R1–R5 (). Grain size distributions of the five composite sands (. More specifically, CS1 and CS2 contain only coarse and medium fractions (i.e. R1, R2 and R3) without any medium fine or fine fractions (i.e. R4 and R5). Composite sand types CS3 and CS4 contain all the fractions i.e. R1–R5 and in particular CS4 contains all sand fraction in equal proportion and thus have smoothest gradation with highest value of Cu. Fines proportion i.e. percentage R5 in CS3 and CS4 are 10% and 20% respectively. Sand type CS5 do not contain any coarse fraction unlike the remaining composite sand types which contain the coarser fractions and amount of fines content is highest (i.e. 30%) in CS5 among the composite sand types. Exact proportions of the sand fractions comprising each composite sand type are shown in . All sands were grouted at a loose condition (at RD = 30%) and were dry prior to grouting. Sand CS4 and R4 was also grouted in a relatively dense condition (RD = 70%) to evaluate the effect of relative density (RD) of sand types on the groutability/penetrability of MC grout. Since all the sand samples were grouted in dry condition, therefore, effect of degree of saturation is not considered as a parameter in the present research.Commercially available microfine cement (MC), a blend of Portland cement and slag powder was used as grout for injection purpose. Chemical composition and grain size distribution of MC used in the present study as supplied by the manufacturer are given in . Characteristic grain sizes such as d90 and d85 of MC are found to be 0.009 mm, 0.085 mm respectively (as seen in Fig). Blaines fineness i.e. specific surface area of MC is 907 m2/kg (data supplied by the manufacturer). In terms of gradation and specific surface area, cement used in the study is considered as “microfine” because they satisfy the requirements of EN 12715 (), which characterize microfine cements by a specific surface area >800 m2/kg and d95 < 20 μ. A polycarboxylate based superplasticizer was used as high range water reducing admixture (HRWRA). Ordinary tap water was used to prepare fresh grout with microfine cement. All suspensions tested during this investigation were prepared using potable water as it is considered appropriate for preparing cement-based suspension grouts (). The different w/c ratios of the suspensions considered in the study are 0.8, 1, 2 and 3 by weight of dry MC. A fixed superplasticizer dosage of 1.4% by weight of dry cement () was used to enhance the flow properties of MC grouts for all w/c ratio values. Superplasticizer dosage of 1.4% was adopted based on the similarity of MC characteristics (grain size distribution and specific surface area) and superplasticizer type used by with that of present study. Suspension preparation required a total mixing time of 20 min with a high speed mixer. Rated rpm for the high speed mixer as supplied by the manufacturer is 2000 rpm at no load. However, rpm of the mixer eventually decreases when it was used for mixing the cement grout. Also, the mixing speed varies during mixing time as microfine cement was gradually added during the total mixing time. It is to be mentioned here that during mixing of the cement grout with water, mixing speed was not measured. For preparation of suspensions, first the appropriate superplasticizer dosage was mixed with water thoroughly. Then the specified amount of cement solids were added in small increments with 70% of the superplasticizer mixed water until all the cement solids were added while agitating the grout suspension continuously for 10 min. After the addition of remaining 30% water, grout suspensions attains the specified w/c ratio and was subjected to continuous agitation for another 10 min before commencement of the grouting process. This remaining water will scrap off any grout solid that is sticking to the grout chamber. Furthermore, to ensure sufficient dispersibility of microfine cements, a circulation pump as shown in is also connected to the grout chamber to circulate the grout from the bottom to the top of the grout chamber. The circulation pump will also account for the portion of the grout below the mixer blade where there is a likelihood of deposition of solid cement grains. Since agitation of the grout suspension with specified water w/c ratio (after the addition of all cement solids and water) was done only for 10 min, thus, rheological parameters variation with time such as yield stress τ0(t) and plastic viscosity μ(t) within such a short period of time is ignored.Bohlin Visco 2000, a rate and temperature controlled viscometer was used to measure the rheological properties of the grout mixes. For evaluation of the rheological parameters, stepwise decreasing shear rate method is used. The stepwise decreasing shear rate sequence is the most commonly used method for measuring the rheological parameters of cement based mixes (). In this method, grout samples are subjected to stepwise increasing shear rate sequence followed by a stepwise decreasing shear rate sequence. Stepwise decreasing shear rate, also known as down curve was used for drawing the flow curve. Down curves of the rheological tests were considered to investigate whether mixes under investigation behave like a Bingham or a shear thinning or a shear thickening material. Reason for taking down curves of the rheological test is that thixotropic effect can be excluded in the down curve. In the present study, the grout samples were subjected to stepwise increasing shear rate sequence (17 “sec−1” to 490 “sec−1” in 3 min) followed by a stepwise decreasing shear rate sequence (490 “sec−1” to 17 “sec−1” in next 3 min). Plug flow, if any, is corrected by point elimination method (). Rheological parameters reported are the average of three respective readings.Schematic of laboratory setup used for injecting sand columns with MC grout suspensions is shown in . The transparent grouting column (sand column) used in the study had a thickness of 5 mm with an internal diameter of 14 cm and a height of 70 cm. The sand column was prepared by placing a 5 cm thick filter layer at both top and bottom of the sand column and filling the remaining length (60 cm) with dry sand in desired relative density. A gravel layer consisting of pea gravel (corresponds to size range 6.3–4.75 mm) was used as filter material to ensure uniform distribution of MC grout suspensions during injection. Dry, clean sands were pluviated with a funnel into the sand column. During pluviation, the free drop height was adjusted as per the desired target relative densities (i.e. 30% and 70%). A scale (flexible measuring tape) was attached to the transparent sand column to measure the penetration distance of the grout in the sand column.A plunger pump was used for injecting the MC grout suspensions through the sand column. The rate of discharge of the pump was regulated to be constant and was equal to 120 L/h. A diaphragm pressure gauge (with a maximum pressure of 1000 kPa) was connected to the injection line between the pump and the grout injection inlet at the base of the permeation setup to monitor the grout pressure continuously. The pump inlet line was connected to the grouting chamber in which the MC grout suspension was subjected to continuous agitation to prevent separation of grout solid and water. In addition to this, a circulation pump is also connected to the grout chamber to circulate the grout from the bottom to the top of the grout chamber. A graduated glass tube is attached to the grout chamber to monitor the grout level before the start and after the completion of the injection test. Injection process continued until a grout volume equals to two times of the void volume of sand column is injected or a maximum grouting pressure of 500 kPa is reached, whichever occurs earlier. A container is placed at the outlet of the sand column to collect the grout effluent, if any, during testing.Measurement of grout quantity expressed in terms of void volume injected (VVI) into the sand column was determined by noting down the initial and final reading of the grout level indicator before the start and after the end of injection process respectively using the following equation:I = initial reading of the grout level indicator attached to the grouting tank,F = final reading of the grout level indicator attached to the grouting tank,C = allowance for grout volume contained in the pipe (connecting grouting tank with sand column) and filter layer,Vv = theoretically calculated void volume of the sand column.Typical flow curve of MC grout suspensions prepared with different w/c ratio = 0.8 is shown in (a). The down curve does not follow the up curve and forms a loop. This is due to the thixotropic breakdown of the sheared material. Down curve of the plot almost follows a straight line which indicates that Bingham model can be fitted for MC based grout suspensions. Typical down curve of MC grout suspension with w/c = 0.8 is shown in (b). It may be observed that down curve follow Bingham model with very high degree of accuracy (R2 = 0.98). Typical flow curves and down curves of MC grout suspension with w/c = 1, 2 and 3 are presented in . Rheological properties obtained from fitted down curves (three down curves are drawn for each w/c ratio, and average value of fitting results of three down curves are reported as rheological parameters) of MC grout suspension with w/c = 1, 2 and 3 are presented in , decrease in R2 value with increase in suspension w/c ratio is observed (possibly due to grout bleeding during rheological testing, however, further investigation is needed to establish this). From the results in , it may be concluded that with a reasonable accuracy all the suspensions (i.e. w/c = 1, 2 and 3 with R2 = 0.94, 0.89 and 0.84 respectively) can be fitted as Bingham model. It is worthwhile to mention here that as suspension w/c ratio increases from 0.8, effect of thixotropy becomes marginal unlike in thick suspensions (i.e. w/c = 0.8) with high solid concentration where distinct effect of thixotropy is observed.In the present study, injectability of the grout was characterized as “good” when a grout volume equals to two void volumes of sand column could be injected under a maximum injection pressure of 500 kPa, “moderate” when the injected void volume is greater than one void volume of the sand column and “poor” when even one void volume of the sand column couldn’t be injected under a maximum pressure of 500 kPa. A total of 48 no’s of sand columns were grouted in laboratory with different combination of experimental parameters. Parametric study of the different parameters such as d10, τo, µ, Cu, P, RD and FC on penetrability are presented in discussions given below.Effect of sand grain size i.e. d10 on maximum injection pressure for different w/c ratio is presented in (a). It may be observed that in general grouting pressure decreases irrespective of w/c ratio of the grout with increase in d10 of the grouted sand. This is due to the reason that as d10 increases, groutability ratio N1 and N2 increases which are the pre-requisite for successful grouting. It is also observed that with decrease in grout w/c ratio grouting pressure increases for a particular d10 value of the grouted sand. In (a), some data points corresponds to cases where penetration length was less than full penetration i.e. 60 cm (connected by dotted lines) under a maximum pressure of 500 kPa. This implies that increase in pressure alone is not sufficient to obtain full penetration. For instance, at grout suspension w/c = 3, full penetration could not be obtained (penetration length = 11 cm) with d10 = 0.17 mm under a maximum pressure of 500 kPa (a) and this may be due to the presence of excessive amount of fine sand content (i.e. sand R5, FC = 100%) resulting in lower groutability ratios and thus lower groutability potential of the sand (i.e. N1 = 23 and N2 = 19). Nevertheless, this is not the case always for partial penetration. In (a), it may be observed that for grout suspension with w/c = 0.8 when grouted with sand having d10 = 0.64 mm (sand R3, FC = 0%, N1 = 84 and N2 = 71), penetration length observed was 33 cm under a maximum pressure of 500 kPa (b) and may be attributed to rheological blocking of the flow of grout. During grout propagation in to the soil media under pressure, increase in propagation length decreases the pressure gradient. As a result grout flow rate decreases with decrease in pressure gradient. Subsequently, the applied shear rate on the grout flow decreases and at one point the grout propagation stops as shear stresses developed by the flow attains the yield stress of grout. Despite the size relationship for sand fraction R3 (i.e. FC = 0%, N1 > 25 and N2 > 11) are sufficient for satisfactory permeation, but relatively high yield stress and plastic viscosity for w/c = 0.8 prohibits full penetration of the grout.Yield stress represents the threshold value of stress that the grout must be subjected to initiate the flow whereas plastic viscosity determines the flow resistance of the grout in the soil media once the flow is initiated. Effect of yield stress and plastic viscosity of grout suspensions with penetrability is presented in (b) & (c). It may be observed that for all the soil types and RD values considered in (b) & (c), there is a significant decrease in penetration length with increase in both yield stress and plastic viscosity (i.e. decrease in w/c ratio), indicating the crucial role of yield stress and plastic viscosity in influencing the penetration length of grout. In order to study the effect of relative density, sand columns prepared at two different relative densities i.e. 30% and 70% were injected with different grout w/c ratios. Soil type CS4 and R4 was chosen and the penetration results against two different relative densities for soil type R4 is shown in (d). For soil type CS4, effect of relative density is discussed in subsequent discussions, as occurrence of filtration phenomenon is observed in that case. In (d), substantial difference in grout penetrability was observed at w/c ratios (i.e. w/c = 1 and 2) compared to that of higher water cement ratios (i.e. w/c = 3). Nevertheless, with increase in w/c ratio, difference between grout penetrability values obtained at RD = 30% and 70% decreases except at w/c = 0.8. At w/c = 0.8, effect of rheological parameters i.e. yield stress and plastic viscosity was found to be prevailing to limit the grout penetration rather than relative density. At w/c = 3, difference in penetrability was minimal indicating that effect of relative density becomes marginal as grout approaches from Bingham fluid (low w/c ratio) to Newtonian fluid (high w/c ratio). This situation is schematically explained in represents the area available for a suspension (Bingham fluid) whereas closer to the contact point between the grains, only a Newtonian fluid will be able to penetrate. This implies that a larger available area of the grouted medium is reachable for a gout suspension with high w/c ratio compared to grout with a low w/c-ratio, which behaves more like a Bingham fluid (). When a grout suspension with higher w/c ratio is used, the grout behaves more like a Newtonian fluid and thus, explains why effect of relative density diminishes at higher grout w/c ratios. Effect of fine sand content (fraction finer than 0.3 mm) of sand on the penetrability is presented in (a). It may be observed that for all values of w/c ratio, as FC increases, penetrability decreases. Nonetheless, effect of FC in sand is most significant for w/c = 1 as only 10% increase in FC of sand resulted in almost 3-fold decrease in penetrability. At 0% FC, full penetration was achieved with VVI equals to = 1.15. Whereas at 10% FC, penetration length decreased to 21 cm with VVI = 0.32. With further increase in FC, both penetration length and VVI decreased. The result indicated that 10% FC can affect penetrability and VVI significantly for w/c = 1. Effect of FC in penetrability is least significant for w/c = 0.8, where the penetrability is more affected by rheological parameters as discussed earlier. Nevertheless, it may be also observed that for w/c = 2 & 3, relatively much higher penetrability and VVI was obtained under identical conditions. Again this may be related to the fact that at higher w/c ratios, suspensions behave more like a Newtonian fluid (i.e. yield stress tends to be very low) rather than a Bingham fluid. Yield stress is the factor responsible for controlling the stoppage of grout flow by constraining the propagation of grouts through soils. This explains why grouts with higher w/c ratio have higher penetrability.On careful inspection of grout propagation through the sand media, it was observed that for some cases distinct filtration took place where clogging of cement particles takes place resulting in penetration of grout water only leaving behind the cement grains. This is also can be linked to the observation that in some cases a much faster grout propagation rate was noticed as grout water can propagate at a faster rate because of its very low plastic viscosity. The effect of filtration on penetrability and injectability of grout for soil type CS4 is shown in (b), some apparent inconsistency in penetrability is observed, as penetrability at RD = 70% was greater than that of at RD = 30%. This may be attributed to the occurrence of filtration phenomenon which is evident from (c), void volume injected (VVI) is shown for each case corresponding to the penetration values as shown in (b). It is to be noted here that measured VVI in (c) is much less than that of target VVI expected for penetration obtained at w/c = 2 and 3 in (b) (As the length of sand column is 60 cm, theoretically it may be assumed that target VVI = 1 for 60 cm penetration without filtration). Effect of filtration on the penetrability of soil type CS5 considering the aspect of VVI is shown in (d), along with penetrability, measured VVI and target VVI (as discussed earlier) is also plotted. Results in (d) suggest that both measured VVI and target VVI increases with w/c ratio. Nevertheless, a large deviation of measured VVI from target VVI was noticed at higher w/c ratios (i.e. w/c = 2 and 3) indicating the occurrence of filtration phenomena at higher w/c ratios.(c) & (d), since, large difference between measured VVI and target VVI is observed, it is evident that the measured penetration distances for these cases not solely represent the penetration of cement grout with specified w/c ratio. Rather, a part of the measured penetration distance consists of penetration of grout water (or grout with much higher w/c ratio compared to the specified w/c ratio) leaving behind the grout solid. Therefore, considering this aspect of filtration on penetrability, each penetrability value obtained in the present study was corrected for filtration using VVI as follows:As mentioned above, theoretically it may be assumed that target VVI = 1 for 60 cm penetration without filtration. Thus, if penetration distance takes any value between 0 and 60 cm, target VVI is accordingly proportioned between 0 and 1.However, in the present investigation for some cases apparent inconsistency between penetration distance and VVI measured was observed. For instance, say observed penetration distance is 50 cm (target VVI = 0.83) and measured VVI is 0.5, this suggests that theoretical penetration distance should be 30 cm rather than the observed 50 cm which is due to the filtration where grout water (or grout with high w/c ratio) has penetrated up to 50 cm.Since, for the each case in the present study, both penetration distance and VVI was measured, so correction for filtration was done for the individual cases, if any, using the following equation.Corrected penetrability=observed penetrability×measured VVItarget VVIIt is to be noted that above correction factor is applied only for the cases where difference between theoretical and observed VVI is large (>15%). A small difference between theoretical and observed VVI may be attributed to the inherent uncertainties that results from theoretical calculation of void volume and actual void volume that exists in sand column.For some of the grouted sand columns, VVI was greater than 1 (one) and thus grout effluent was collected subsequently at the outlet of the sand column. In order to study the effect of filtration for these cases, density of the grout sample at inlet and outlet of the sand column were determined. A wide variation among the densities collected at the inlet and outlet of the sand column implies the occurrence of filtration whereas a small variation can be ignored for practical application purposes. (a) shows the variation of ρinletρoutlet with groutability ratio N1. In general, with increase in N1 and w/c ratio, decrease in filtration tendency was observed. For soil type R4 (N1 = 44) and CS3 (N1 = 46) and in particular for R4 (N1 = 44) at 30% and 70% RD, a high value of ρinletρoutlet (1.75 & 2.05 at w/c = 3 and 1.97 at w/c = 2) was observed indicating the occurrence of filtration phenomenon. Rest of the cases, observed ρinletρoutlet value was less than 1, suggesting absorption of grout water by the dry sand in the sand column. Variation of ρinletρoutlet with another groutability ratio N2 is presented in (a) & (b), it may be observed that the conventional groutability criterion i.e. N1 > 25 & N2 > 11 is not sufficient to achieve successful grouting as significant amount of filtration is observed irrespective of the w/c ratio of grout suspension.In the present study, range of groutability ratio N1 and N2 used for the different sand types are 23–330 and 18–278 respectively. This suggests that almost all the sands should be groutable as per the criterion i.e. N1 > 25 and N2 > 11. However, groutability predictions based on the groutability ratio N1 and N2 are in agreement with experimental observations for some cases and for remaining cases groutability prediction based on N1 and N2 are not conformed experimentally. This is due to the fact that these groutability prediction formulas don’t consider other factors such as rheology of the grout, FC and other relevant properties of sand, grouting pressure and filtration of the grout. Similar observations are also reported by . Notably, effect of these parameters becomes less significant with increase in N2 and diminishes as N2 approaches a large value. For instance, for sand type R1 with high N2 (N1 = 300, N2 = 278) irrespective of the w/c ratio a very good penetrability was achieved under a maximum pressure much less than 500 kPa. Again for sand type CS3 (N1 = 46, N2 = 20), penetrability was poor at w/c = 0.8 and 1 (due to high yield stress and plastic viscosity), moderate at w/c = 2 and good at w/c = 3 under a maximum pressure of 500 kPa. Similarly, for sand type R5 (N1 = 23, N2 = 19), penetrability was poor irrespective of w/c ratio under a maximum pressure of 500 kPa as FC in the sand is 100%. In view of the above discussion, a groutability criterion with N1 > 45 and N2 > 40 seems to be more appropriate for groutability prediction in terms of N1 and N2.A total of 48 no’s penetrability data points corresponding to various combinations of the experimental variables were used as database for the development of AI based models i.e. ANN and SVM model. From the results and discussion in the preceding section, it is evident that all the factors studied above affected penetrability of MC grout. Therefore, for the development of AI based models i.e. ANN and SVM model, yield stress (τo), plastic viscosity (µ) of grout, groutability ratio N2 (ratio of d10 of soil to d90 of the grout), coefficient of uniformity (Cu) (as it accounts for soil gradation), relative density (RD), fine sand content (FC), maximum injection pressures (P) was considered as input and penetration length of the grout as the output of the model. Groutability ratio N2 is considered as it includes effective grain size i.e. d10 of soil alongside d90 of the grout, which is an important factor affecting penetrability ( have also used groutability ratio N2 in their proposed empirical approach for prediction of groutability.In the present study, formulation of AI models was carried out with 70% of the data as training data and remaining 30% as testing data. A statistically consistent training and testing data set were generated in order to ensure similar patterns of data in both training and testing subset (. Once the available data have been divided into their subsets (i.e. training and testing), both input and output variables in training and testing subsets were pre-processed by normalizing to fall in the range [−1, +1].ANN model used in the present study was characterized by Feed-forward network (back-propagation algorithm) with tan-sigmoid (hyperbolic tangent) transfer function at hidden layer and a pure linear transfer function at output layer. In order to determine the optimum number of neurons or nodes in the single hidden layer, numbers of neuron in the hidden layer were varied from 1 to 12 in a trial and error basis. For each value of neuron in the hidden layer, ANN model performance is evaluated and compared to find the optimum number of neurons. Bayesian regularization (BR) back propagation training algorithm is used for its better generalization to the training data. Training algorithm used in BR is Levenberg-Marquardt. In the present study, training algorithm Levenberg-Marquardt is implemented in the batch training mode.For development of SVR model to predict penetrability of MC grout, three different kernel functions are adopted and are given as follows; Polynomial kernel function (POLY) K〈xk·x〉=(〈xk·x〉+1)d, Radial basis kernel function (RBF) K〈xk·x〉=exp(-‖xk-x‖2/2σ2) and Exponential radial basis kernel function (ERBF) K〈xk·x〉=exp(-‖xk-x‖/2σ2), where, d and σ are user defined kernel parameters; d = degree of polynomial function; σ = width of RBF function. To develop the SVR models constant ε in the loss function and penalty parameter C must be defined by the user. Optimal values of C and ε were largely determined by trial and error (). In the present study, optimal values of C |
and ε were determined through trial and error. Different combinations of d, σ, C and ε were tried on training dataset and for each combination of these parameters; performances of testing dataset were recorded.The performance of ANN and SVM models were reported in terms of three statistical parameters namely, root mean squared error (RMSE), mean absolute percentage error (MAPE) and linear correlation coefficient (R). Based on the results of parametric study of number of hidden layer neuron against ANN model performance, it was concluded that ANN model produced optimum generalization and testing performance in terms of R, RMSE and MAPE with only Nh = 1 i.e. 1 neurons in the hidden layer. Furthermore, keeping the number of hidden nodes to a minimum, provided that reasonable performance is achieved, is always better, as it: (a) reduces the computational time needed for training; (b) helps the network achieve better generalization performance and avoid the problem of overfitting and (d) allows the trained network to be analyzed more easily (). With regard to SVR model, optimal values of kernel and loss function parameter corresponding to each kernel function, as determined from parametric study, is summarized in Optimum performance of ANN and SVR models are summarized in (a)-(d) shows the SVR predicted against experimentally observed penetrability of ANN and SVR models with respect to different kernel functions for training and testing data respectively. From (a)-(d), it may be observed that, both ANN and SVR models except SVR-ERBF kernel have demonstrated a fair degree of accuracy in predicting the penetrability. ANN, SVR-POLY and SVR-RBF predicted values are less scattered with respect to line of equality with a very high correlation coefficient value of 0.9849, 0.9920 & 0.9932 and 0.9832, 0.9831 & 0.9817 for training and testing data respectively. It may be observed that ANN, SVR-POLY and SVR-RBF predicted testing values lies well between lower and upper prediction bounds corresponding to 99% confidence interval. Confidence interval refers to the likelihood of predicted values to lie in the region bounded by them over a range of experimental values. Results in (a)-(d) also signify about very good generalization performance of ANN, SVR-POLY and SVR-RBF kernels in terms of R which is indicated by marginal difference between training and testing R value. Although ERBF showed very high correlation among experimental and predicted values (i.e. R value as high as 1.0) for training dataset, however, SVR-ERBF model has overfitted the training data which was indicated by high R value for training data (i.e. R = 1.0) and relatively low R value (i.e. R = 0.9344) for testing data, resulting in poor generalization of testing data. Comparison of RMSE and MAPE values obtained from ANN, SVR-POLY and SVR-RBF models for training and testing data as observed in indicated that prediction performance as well as generalization ability of ANN, SVR-POLY and SVR-RBF models were better than SVR-ERBF model.It is to be noted here that although both ANN and SVR has performed equally good in predicting penetrability, however, ANN model has resulted in a simple network i.e. with only one neuron in the hidden layer compared to high dimensional nonlinear model of SVR (i.e. SVR-POLY and SVR-RBF). This suggests that interpretation of ANN model will be more realistic and easy in connection to penetrability prediction (). Furthermore, it is also shown that ANN model has truly generalized the grouting mechanism as illustrated in subsequent sections (sensitivity analysis of ANN model).During training of the ANN model, variation of performance function (MSE) with number of epochs (i.e. iterations) is presented in (a). It is observed that performance function i.e. MSE decreases with number of epochs. MSE reached a constant value of 13.01 (RMSE = 3.61) for training data and RMSE = 4.38 for testing data at an epoch approximately equal to 12 and no further decrease in performance function is observed with further increase in epochs. ANN training stopped at an epoch 29 and is due to the fulfilment of stopping criterion other than MSE reaching zero i.e. zero error. Since training stopped at an epoch 29, thus, performance of the network at an epoch (29 − 1 =) 28 is saved and is reported. (b) plots the state of ANN model training with respect to epochs. It may be observed that as MSE attained a constant value at a certain epoch (a), performance gradient also attained a constant value at that particular epoch i.e. 12 suggesting no further improvement of network performance with number of epochs. Same trend is also observed with effective number of parameters and sum squared parameter. One feature of BR algorithm is that it provides a measure of how many network parameters (weights and biases) are being effectively used by the network. In this case, the final trained network uses approximately 10 parameters (exact value is 9.19 as shown in the b) out of the 10 total weights and biases in the 7-1-1 (7 inputs, 1 neuron in the hidden layer and 1 output) network. This effective number of parameters should remain approximately the same, no matter how large the number of parameters in the network becomes. This assumes that the network has been trained for a sufficient number of iterations to ensure convergence. It is evident from (b) that ANN training stopped due to Marquardt adjustment parameter (mu) exceeding the maximum default value of mu which was set to be 1 × 1010 and suggests that the algorithm has truly converged (). It is also reported that the algorithm has converged if the sum squared error (SSE) and sum squared weights (SSW) are relatively constant over several iterations () which is also observed in the present case.ANN has often been labelled as “black box” because it doesn’t provide any meaningful insight into the relative influence of input variables in the prediction process. This lack of explanatory power is a major concern since the interpretation of developed models is desirable for gaining knowledge of the relationships driving grouting mechanism. However, methods like Garson’s algorithm and Connection weight approach have been reported to yield successful interpretation of ANN models when applied to geotechnical engineering problems (). Hence, present study employs these methods to assess the affect of input parameters in penetrability prediction and to extract comprehensible knowledge from the proposed model. Both the methods use optimized weight vector to recognize important input parameters, details of which is discussed in . Using the optimized weight vectors of the ANN model (), importance and relative ranking of different input variables of ANN model determined by Garson’s algorithm and Connection weight approach is presented in . It may be observed that the order of ranking given by both the methods is same. Groutability ratio, N2 is ranked first among all the variables, which is quite evident as it is the preliminary criterion for qualitative estimation of grouting. reported that an important parameter affecting penetrability is the ratio between the grout grains and the aperture available for grouting. Fine sand content (FC in %) is second most important parameter and confirms the crucial role of FC as reported in other research investigations. reported that soils with FC greater than 20% may not be groutable with particulate grouting. This implies that groutability of a soil depends upon FC% in the grouted media, even if size criterion between the soil and grout is well satisfied. Plastic viscosity (µ) was the third most important parameter influencing penetration and VVI followed by coefficient of uniformity (Cu), injection pressure (P) and yield stress (τo). Relative density (RD) was found to be least influential parameters affecting penetration. This may be attributed to the fact all the soil types considered in the study are poorly graded sand and thus effect of improving packing density by increasing RD has marginal impact in influencing penetrability. Although, yield stress (τo) was ranked as second least influential parameter by Connection weight approach and Garson’s algorithm but for practical reasons it may be regarded as third most influential parameter just like plastic viscosity. This is due to the reason that effect of plastic viscosity and yield stress cannot be isolated while investigating penetrability. During grout propagation in to the soil media under pressure, increase in propagation length decreases the pressure gradient. Grout flow rate which depends upon viscosity of the grout suspension also decreases with decrease in pressure gradient. Consequently, the applied shear rate on the grout flow decreases and at one point the grout propagation stops as shear stresses developed by the flow attains the yield stress of grout. This suggests that both yield stress and plastic viscosity has a combine effect in affecting penetrability. Reason behind apparent inconsistency in ranking exhibited by Connection weight approach and Garson’s algorithm with regard to yield stress may be related to the lesser number of data points with significant yield stress value (i.e. such as for w/c = 0.8) present in the database.In addition to sensitivity analysis, using the weights shown in , neural interpretation diagram (NID) is also presented (). NID is used for visual analysis of connection weights among the neurons and details of which can be found in , grey line indicates negative connection weight and black line indicates positive connection weight. Relative magnitude of connection weight is represented by the thickness of the line. Inverse effect and direct effect of input variables on penetrability is signified by grey circle and black circle respectively. As seen in , NID showed negative or inverse effect of τo, µ, RD and FC on penetrability which is evident from experimental results and is a justified representation of grouting mechanism. Nonetheless, it is not clear from the present study why effect of Cu on penetrability is positive. In general as Cu increases gradation of soil improves which results in better packing density of the sand and consequently, groutability of the sand decreases.While both ANN and SVM can predict penetrability of MC grout with a good accuracy, it is not convenient to use ANN and SVM directly in field applications. For field application purpose, empirical equations are more convenient for practicing engineers with limited or no knowledge of ANN/SVM. Both ANN and SVM can be explicitly used to express the equation in the trained model in functional form through a series of matrix operations. However, the trained models contain many weights and biases together with transfer functions (in case of ANN) and kernel functions (in case of SVM) and the final equations become very complicated and cumbersome to use. Fortunately, in the present study optimum number of neurons in the hidden layer was found to be only 1 (one) for the trained ANN model. This implies that resultant ANN model is a very simple one and can be used directly to obtain penetrability prediction formula in a simplistic equation form without involving a series of matrix operations (as it contains only one set of weights/bias corresponding to a single neuron). The optimal ANN model configuration is 7-1-1 i.e. 7 inputs, 1 neuron in the hidden layer and 1 output with total number of weights and biases equal to 10. Furthermore, as observed in (b) that effective number of parameters for ANN model is approximately equals to 10, suggesting that all the weights and biases of ANN model are significant. Out of this 10 weights and biases, 8 are weights (i.e. 7 weights corresponding to 7 inputs and 1 wt corresponding to output) and 2 are biases as shown in . Thus, these weights and biases will appear as coefficients and intercepts, in a linear equation relating input parameters to the penetrability. The relationship between the output and input as per the developed network architecture of ANN model is given by the following equation:Penetrabilityn1×1=bo1×1+∑k=17wk1×1×tanhbk1×1+∑i=17wik1×7Xin7×1where penetrabilityn is the predicted penetrability value normalized in the range [−1 to 1], bo is the bias at the output layer, wk is layer weight matrix between hidden layer and output layer where k is the number of neurons in the hidden layer, bk is the bias matrix connected to hidden layer, wik is the input weight matrix between the input and hidden layer, where, i is the number of inputs, Xin is the normalized input variable matrix in the range [−1, +1]. Normalized values of experimental (input) parameters can be obtained from following equations using the data given in Normalized values of penetrability can be determined from following steps:A=-0.177842∗τ0n-0.99774∗μn-0.047526∗RDn+0.23125∗Cun-2.166707∗FCn+3.35557∗N2n+0.18335∗Pn+1.14569where Cn is the normalized penetrability and can be de-normalized by the following equation:In the present study, an attempt has been made to investigate penetrability of MC grouts in granular soil. Effect of parameters such as yield stress and plastic viscosity of grout, fine sand content, relative density and gradation of soil and grouting pressure were studied in laboratory. In addition to this effect of filtration on penetrability is also presented. A comparative study of all the parameters mentioned above with penetrability is discussed. Following general conclusions may be derived from the present study:Investigation results of grouting granular soil with MC revealed that the penetrability of grout is affected by τo and µ of the grout suspension. Other parameters such as N2, FC, RD, Cu and P also have important role in affecting the penetrability.Investigation results of rheological tests conducted on MC grout suspensions showed that down curve of MC grout suspensions with w/c ratio = 0.8, 1, 2 and 3 shows a linear relationship between shear stress and shear rate and follows Bingham equation with a very good accuracy.Estimation of penetrability based on the groutability ratio N1 (N1 > 25) and N2 (N2 > 11) are not sufficient to ascertain groutability as it is hardly realized experimentally. This is due to the fact that these groutability prediction formulas don’t consider other factors such as rheology of the grout, fine sand content, FC and other relevant properties of sand, grouting pressure and filtration of the grout. Rather, a groutability criterion with N1 > 45 and N2 > 40 seems to be more appropriate for groutability prediction in terms of N1 and N2.Result of ANN and SVR model showed that both the models are almost equally good for penetrability prediction in terms of performance parameters i.e. R, RMSE and MAPE.ANN model for penetrability prediction developed in the present study includes only one neuron in the hidden layer, resulting in a simple network. Furthermore it was shown that all the connection weights in ANN model are significant. This means that ANN model has truly generalized the penetrability mechanism and has understood the complex relationship between penetrability and factors relating to it.Sensitivity analysis of ANN model for penetrability prediction depicted N2 ratio as the most influential parameter affecting penetrability followed by FC, µ, Cu, P and τo. RD was least influential parameter affecting penetrability. Furthermore, results obtained from sensitivity analysis were consequently substantiated with previous research findings as well as experimental observations from the present study.Neural interpretation diagram (NID) of ANN model for penetrability prediction demonstrated negative or inverse effect of τo, µ, FC and RD on penetrability which is a justified physical representation of grouting mechanism.Using the connection weights and biases of developed ANN model, an empirical equation is proposed for estimation of penetrability. The developed equation will serve as a useful basis for preliminary estimation of penetrability for permeation grouting applications.Analytical correlation of hardness and scratch adhesion for hard filmsAn analytical expression has been derived in order to correlate the critical load of scratch adhesion Lc with hardness and film thickness for thin hard films on softer substrates, based on the model proposed by Burnett, Rickerby and co-workers (B–R model) for interpretation of composite hardness measurements. The equation was used to calculate the correlation of Lc with Hv(sub) (substrate Vickers hardness), Hv(comp) (composite Vickers hardness), and t (film thickness) for AISI 316 stainless steel, HSS M2 and WC substrates coated with TiN films of different thickness. The calculated results were compared with experimental data for TiN films produced under similar conditions by PACVD and PAPVD deposition techniques. The present analysis yields quantitative results which agree to a satisfactory degree with experimental data in all cases considered and contribute to the understanding of the physical processes which control the deformation of the film–substrate mechanical system under scratch indentation conditions.One of the most important functional requirements for a coated product is that the film should not detach from the substrate material during the service lifetime of the product. Film detachment is usually discussed within the general context of film adhesion. This approach, however, extended, is not entirely correct Given the importance of the substrate hardness and film properties on the film failure resistance under severe deformation conditions, the understanding of the physical mechanisms responsible for correlation between these variables with the critical load of scratch adhesion, Lc, taken as a reference for the film failure resistance, is of relevance for design purposes of coated materials which have to perform under those conditions. Empirical results indicate that the critical load of scratch adhesion is a linear function of the substrate hardness, of the composite hardness above a given indentation depth, and of the film thickness In this paper, the volume law-of-mixtures model originally proposed by Sargent The present analysis yields quantitative results which agree to a satisfactory degree with experimental data in all cases considered and contributes to the understanding of the physical processes, which control the deformation of the film–substrate mechanical system under scratch indentation conditions. presents the plastic zone morphology considered in the B–R model for the case of hard coatings on softer substrates. According to the B–R model, the composite Vickers hardness Hv(comp) corresponding to the case illustrated in where Hv(f) is the bulk film material Vickers hardness, Hv(sub) the corresponding hardness value for the substrate, Vv(f) is the film plastic zone volume, Vv(sub) is the substrate plastic zone volume if allowed to expand without any film adhesion constraint, χ is an empirical factor required to account for the reduction of the substrate plastic zone volume as a result of its adhesion to the film, and V=Vv(f)+χ3Vv(sub) is the total deforming volume.For indentation of a bulk material, the plastic zone radius R, may be written as R/ro≈(E/H)1/2, where ro is the characteristic length of the indentation volume (i.e. ro∼), and E and H are the Young's modulus and hardness of the material, respectively and the B–R equation may be rewritten in terms of Vv(f)/χ3Vv(sub) as:The B–R model assumes that indentation elastic recovery effects in the hardness values may be expressed in terms of the corresponding indentation size effect (ISE) indexes as Hv(dv)=Hv(dv=1)dvm−2, where Hv(dv=1) is the hardness at unit indentation diagonal and m is the corresponding material ISE index. Both Hv(dv=1) and m are characteristic constants of the indented material. According to this definition, the following two relationships may be written involving the substrate and film hardness values:where dv′ is the Vickers indentation diagonal for which the measurement is being performed, and dv′ may be taken as a reference diagonal value for which elastic recovery and film effects are no longer strongly dominant, as illustrated schematically in is only valid in the elastic–plastic indentation regime. For fully plastic indentation conditions, the hardness values become independent of the indentation diagonal. It will be assumed for the present analysis that Hv(dv≥dv′)≈Hv(plastic)≡constant.Scratch experiments are normally conducted using a Rockwell C indenter, with a spherical diamond tip radius of 200 μm. Scratch indentation is thus equivalent to a spherical indentation process under sliding conditions. Since the indentation dimensions for a given load are practically the same for sliding and static conditions . Since in the fully plastic indentation regime both Hv and Hs have constant values, then:where Hs(sub) is the substrate fully plastic scratch (spherical) hardness value which, since it is a constant, may be defined in terms of the critical load Lc, and the critical scratch channel width ds(crit) as Hs(sub)≡4 Lc/π. From these considerations, the second may finally, be written in a form suitable for calculation:Once the materials that form the film–substrate system and the film thickness are defined, F(dv) depends only on dv for a selected constant reference dv′. F(dv) represents the ratio between the composite and the substrate Vickers hardness and its numerical value tends asymptotically to unity as dv increases to the point where plastic deformation effects are dominant in the deformation (i.e. dv≥21 μm for the experimental conditions considered). have been used to calculate the correlation of Lc with Hv(comp), Hv(sub) and film thickness t, for the same conditions considered by the authors in previous experimental work . The E (Young's modulus) values indicated in are those reported by Burnett and Rickerby for the materials of interest for ds(crit) were taken from experimental data and the corresponding correlation for TiN films of different thickness produced by PAPVD on stainless steel AISI 316 and HSS M2 substrates in the current experiments is presented in , the values used for all substrate materials were ds(crit)=0.10 mm (PAPVD) and ds(crit)=0.060 mm (PACVD).The results provided by the model are particularly sensitive to the values of the quantities indicated above. Thus, some comments on the uncertainties involved and on their effects on the accuracy of the calculations performed are appropriate at this point. The experimental data for the substrates used have a spread of almost 10% about the mean selected values; this may account for deviations up to 20% in the calculations which depend on ds2, both in the elastic–plastic and fully plastic indentation regimes. For the specific case of the elastic–plastic indentation regime and particularly for low indentation diagonal values (dv≪dv′) the parameter χ, which appears as χ3 in as well as the ISE index m, which appears in exponential form, constitute significant sources of uncertainty because of their inherent empirical nature. An additional source of uncertainty for the same conditions is the fully plastic hardness value for the TiN coating, for which no direct experimental measurements are available. The fact that the discrepancy between experimental and calculated data does not exceed 30% under the worst circumstances and is below 15% in most cases, supports the idea that the selected values are a reasonably satisfactory approximation for the calculations performed. for scratch indentation may be interpreted in terms of the simplified mechanism illustrated schematically in , which is compatible with the experimentally observed facts. Considering the case of a hard and brittle film on a softer substrate, it may be imagined that the composite will deform plastically under the effects of the indenter tip until, at a given critical deformation of the film surface at the edge of the indentation, where the deformation is largest, tensile cracks will develop and propagate inwards leading to film failure. Since the deformation geometry of the film surface for a given indentation diameter is independent of the film thickness and of the substrate properties as shown in , it follows that the critical scratch channel width ds(crit) will be a constant determined only by the film material and its microstructure properties, in agreement with the results presented in Having established this general behaviour, it is now appropriate to compare and analyse the results predicted using with those obtained experimentally by the authors for the correlation of Lc with the different variables of interest ] (dv/dv′)msub−2. This equation has been used to calculate the correlation Hv(sub) vs. Lc. Considering ds(crit)≡constant for a given film material and deposition conditions, independently of the substrate material and film thickness, as discussed previously, and considering dv′≡21 μm, a linear correlation of the form Hv(sub)(dv=const.)∝Lc may be expected provided the ISE indexes m are nearly equal for the different substrate materials , this is the case for the substrates under consideration and thus, a strict linear correlation is to be expected for the present analysis. predicts well the slope modification of the correlation as dv increases.. Given the results and considerations of , the question of whether there is a linear correlation of the form Hv(comp)(dv=const.)∝Lc for a specific hard coating on different substrate materials depends on whether the function F(dv) remains nearly constant for all the substrates considered. Using , F(dv) has been calculated for TiN films of 4-μm thickness on the reference substrate materials of interest. The results are presented in These results indicate that for a small size indentation diagonal (i.e. dv<20 μm), F(dv) does not have a constant value. However, for larger indentation diagonal values, F(dv)≈1 and for those conditions a linear correlation of the form Hv(comp) (dv=const.)∝Lc is approached. The value of F(dv) is controlled largely by the factor Vv(f)/χ3Vv(sub), which reflects the relative weight of the plastic zone volumes of the film and the substrate. This leads to the physical argument that for small size indentations, the film and substrate plastic volumes are comparable (Vv(f)/χ3Vv(sub)∼1, or even ≫1), which determines that film effects contribute significantly to the film–substrate system deformation. For larger size indentations, however, the substrate plastic volume becomes much larger than the corresponding film volume (Vv(f)/χ3Vv(sub)≪1) and thus, film effects become negligible in the deformation behaviour of the film–substrate system. Under those conditions the film plays no significant role in the indentation deformation and it is just the substrate that controls the deformation process. Under those conditions with F(dv)≈1, and a strict linear correlation prevails between the composite hardness and the scratch critical load.To illustrate further the arguments presented above, was used to calculate the correlation Hv(comp) vs. Lc for TiN films of 4-μm thickness on different substrate materials for different indentation diagonals. The results obtained are compared with experimental data in , where it may be observed the lack of a linear correlation for a low Vickers indentation diagonal value (i.e. dv=14 μm). Again in this case, there is a good agreement between calculated and experimental data. These results are qualitatively similar to those observed for surface hardened substrates by the authors A complementary conclusion from the previous analysis is that for relatively large indentation diagonals, film thickness effects do not affect significantly the F(dv) value, which is close to unity under those conditions. Thus, the correlation Hv(comp) vs. Lc is insensitive to film thickness variations for indentation conditions approaching the plastic deformation regime. The film thickness effects on the composite hardness calculated from and confirm the results presented by Ichimura and Rodrigo for F(dv)∼1 (i.e. dv>20 μm) and considering that for those conditions (dv/dv′)msub−2≡1 for all the substrates considered, it is possible to write as an approximation:Thus, the slope of the correlation is ∼1.82/. The value of the slope was calculated from for PACVD and PAPVD TiN films. The correlation Hv(comp) vs. Lc calculated for the two slope values considered is shown in , compared to experimental data for the same conditions indicate that in the reference experiments If it is assumed that the B–R model holds for spherical indentation, it is possible to rewrite where Vs(f) and Vs(sub) may be calculated using the plastic boundary radius for spherical indentation, Rs. To this effect, we shall assume, as for the case of Vickers indentation in . If Vind. is calculated assuming a spherical indentation with an indenter of radius ri, which produces contact circle of diameter ds at the surface of the composite material, then the plastic boundary radius may be written using purely geometrical considerations:Considering the high loads associated to scratch indentation near the critical load, it may be further assumed that both Hs(f) and Hs(sub) in will have constant values typical of the fully plastic indentation regime. For the same reason, H in may be considered as the plastic hardness value of the given material. Furthermore, since in the plastic indentation regime Vs(f)≪Vs(sub) as discussed in , the term Vs(f)/χ3Vs(sub)≪1 for most cases of practical interest and may be neglected in first approximation in the denominator of the first multiplying factor in . Thus, for critical load conditions and with the indicated simplifications, For a given film–substrate system, the factor ≈constant and, in those conditions, both Rs(f) and Rs(sub) have constant values. Thus, , the function F(ds)=2πri3/3−(π/3)[ri2−(ds/2)2]1/2 [2ri2+(ds/2)2] should be calculated for ds=ds(crit) for a given Lc vs. ds(crit) correlation. to determine the value of the constants defined by , the values of Lc as a function of t were calculated using are in a good general agreement with the experimental data. It may be observed that, although the critical load increases with the film thickness, as generally accepted, the rate of increase is very small. The reason for this result is the same which was discussed in in connection with the contribution of the film and substrate plastic volumes to the overall deformation behaviour. For large indentations, film effects become negligible in the deformation of the film–substrate system, which is precisely what is observed in An analytical expression has been derived in order to correlate the critical load of scratch adhesion Lc, with hardness and film thickness for thin hard films on softer substrates, based on the model proposed by Burnett, Rickerby and co-workers (B–R model) for interpretation of composite hardness measurements. The equation was used to calculate the correlation of Lc with Hv(sub) (substrate Vickers hardness), Hv(comp) (composite Vickers hardness) and t (film thickness) for AISI 316 stainless steel, HSS M2 and WC substrates coated with TiN films of different thickness. The calculated results were compared with experimental data for TiN films produced under similar conditions by PACVD and PAPVD deposition techniques.As a general conclusion, it may be stated that the obtained analytical expression yields results in overall satisfactory agreement with the reference experimental data, and contributes to the understanding of the physical reasons for the experimentally observed behaviour of the correlation in all cases considered. Furthermore, it validates the B–R model and suggests that it can also be applied to spherical indentation conditions.Some specific conclusions of practical interest, which result from physical arguments associated to the analysis performed are worth mentioning. The most relevant conditions for achieving a high critical load for film failure under indentation conditions are high substrate hardness and high quality film microstructure. On the other hand, film thickness and film hardness are relevant variables for determining the composite hardness and the deformation behaviour of the film–substrate system at low indentation conditions. At high indentation conditions such as those that are typical of scratch indentation near the critical load, film effects are negligible.Design and fabrication of a CH/Al dual-layer perturbation target for hydrodynamic instability experiments in ICFA polystyrene (CH)/aluminum (Al) dual-layer perturbation target for hydrodynamic instability experiments in inertial confinement fusion (ICF) was designed and fabricated. The target was composed of a perturbed 40 μm Al foil and a CH layer. The detailed fabrication method consisted of four steps. The 40 μm Al foil was first prepared by roll and polish process; the perturbation patterns were then introduced on the surface of the Al foil by the single-point diamond turning (SPDT) technology; the CH layer was prepared via a simple method which called spin-coating process; finally, the CH layer was directly coated on the perturbation surface of Al foil by a hot-press process to avoid the use of a sticker and to eliminate the gaps between the CH layer and the Al foil. The parameters of the target, such as the perturbation wavelength (T) and perturbation amplitude (A), were characterized by a QC-5000 tool microscope, an alpha-step 500 surface profiler and a NT1100 white light interferometer. The results showed that T and A of the target were about 52 μm and 7.34 μm, respectively. Thickness of the Al foil (H1), thickness of the CH layer (H2), and cross-section of the dual-layer target were characterized by a QC-5000 tool microscope and a scanning electron microscope (SEM). H1 and H2 were about 40 μm and 15 μm, respectively, the cross-sectional photographs of the target showed that the CH layer and the Al foil adhered perfectly with each other.In inertial confinement fusion (ICF) experiments, when the ignition target capsule is irradiated by a high-intensity laser facility, such as NIF, Nova, Omega and LMJ, the hydrodynamic instability occurred for the spatial distribution uniformity of laser intensity and the roughness of target capsule The CH/Al dual-layer perturbation target was designed to research the RT instability in ICF. The CH polymer, a low-Z material composed of C and H elements, with a density of 1.03 g/cm3, exhibits high energy deposition and good machinability, chosen as the preferred outer ablator material shows the schematic of the CH/Al dual-layer perturbation target.Perturbation patterns were introduced on the surface of Al foil by the SPDT technology The CH layer was prepared by a spin-coating process. Solid CH (purity: 99.9%) was dissolved in chloroform and stirred with a magnetic bar for 4 h, then vibrated in an ultrasonic bath for 2 h. After this, the CH solution filtered with a qualitative filter paper to obtain a transparent, uniform CH solution. In order to obtain the CH solution with appropriate concentration for spin-coating, the solution was continuing stirred with a magnetic bar to make the solvent chloroform volatilized gradually. For CH layer preparation, the CH solution was dropped upon a circular, smooth quartz glass with a diameter of 35 mm and which was fixed by the adsorption devices of the spin coater. First, the glass was spinning with a low-speed about 100 rpm for 3 s to uniform the CH solution on the glass. And then, the spinning speed of the glass was quickly accelerated to about 2500 rpm and kept for 30 s to make the solvent chloroform volatilized and form a CH film on the glass. Finally, took off the CH film from the quartz glass, the CH layer was obtained. shows the schematic of this spin-coating process. Thickness of the CH layer was controllable via adjust the concentration of the CH solution and the spinning speed of the quartz glass. shows the schematic of the hot-press process.The parameters of the target, such as the perturbation wavelength (T), perturbation amplitude (A), were characterized by a QC-5000 tool microscope (Metronics QC-5000, resolution: ±0.001 mm), an alpha-step 500 surface profiler (Tencor Corporation, Alpha-step500) and a NT1100 white light interferometer. Thickness of the Al foil (H1), thickness of the CH layer (H2), and cross-section of the dual-layer target were characterized by QC-5000 tool microscope and scanning electron microscope (SEM, Philips-XL30FEG) as well. Surface roughness of the CH layer surface was also measured by alpha-step 500 surface profiler (Tencor Corporation, Alpha-step500). shows the surface of the Al foil before and after polish, respectively. In (a) the surface of the Al foil was rough, while in (b), the surface was smoother. A smooth surface of the Al foil was necessary to introduce the perturbation patterns. (c) and (d) shows the quantitative surface roughness of the Al foil before and after polish, respectively, scanning length of them were both 1 mm. In (c), the maximum fluctuation was over 1 μm, the surface average roughness (Ra) and the root mean square roughness (Rq) were 212.72 nm and 261.38 nm. While in (d), the maximum fluctuation was under 0.1 μm; Ra and Rq were 16.8 nm and19.5 nm. Compared these two results, it indicated that the polish process reduced the surface roughness substantially.After been polished, the perturbation patterns were introduced on the surface of Al foil by the SPDT technology. shows the enlarged view of the Al foil after been perturbed, (a) was the surface of the perturbed Al foil which shows a uniform T about 50 μm, and (b) was the cross-section of the perturbed Al foil which shows not only a uniform T about 50 μm but also a uniform A about 8 μm. In the turning process, as the scale of perturbation patterns was small, a sharp edge cutter with curvature radius of 2 μm was chosen to cut the Al foil. Under this condition, the perturbation patterns with T/A of 5–10 could obtain.. In this figure, uniform T and A of the perturbation patterns were shown clearly, and could read precisely as 52 μm and 7.34 μm, respectively. The perturbation patterns were very similar to the sinusoidal fluctuate, which meet the requirements of the hydrodynamic instability experiments in ICF well.. In this figure, A and T were uniform as well, and were 7.97 μm and 52 μm, respectively. The result of T was agreed with the result which was measured by the surface profiler, while the result of A was some deviation to the result measured by the surface profiler. As the surface profiler was a contact and more precise measuring method, and the white light interferometer have some deviation especially when the magnification was large, the result measured by the surface profiler was considered to be the credible one, so the perturbation amplitude A was 7.34 μm. What's more, this little deviation had no great influence for observing the three-dimensional image of the perturbation patterns. The perturbation patterns were very close to the sinusoidal fluctuate.Thickness of the CH layer was controllable via adjusting the concentration of the CH solution and the spinning speed of the quartz glass. Based on large amounts of experimentations, experiential relationship of the thickness of the CH layer and the concentration of the CH solution, the spinning speed of the quartz glass was shown in . The spin-coating process was the same as Section , 3 s for 100 rpm and 30 s for each high spinning speed listed in . The results indicated that the concentration of the CH solution higher or the spinning speed of the quartz glass lower, the CH layer obtained thicker. Uniform (in 1 mm length, the maximum fluctuation was under 0.5 μm, the surface average roughness (Ra) and root mean square roughness (Rq) both under 0.15 μm) CH layer could obtain was about 8–33 μm.The CH layer was coated on the perturbed Al foil directly by a hot-press process. shows the picture of CH/Al dual-layer perturbation target, (a) was the face-on view. As the CH layer was transparent, the perturbation patterns on the surface of the Al foil can be seen clearly, this picture almost has no difference with (b)–(d) shows the cross-section view of the CH/Al dual-layer perturbation target; (b) shows the cross-section image measured by the QC-5000 tool microscope. In this picture, the CH layer and the perturbed Al foil adhered with each other, the perturbation patterns were introduced into the interface between the CH layer and the Al foil as the seed for the growth of RT instability successfully. More detailed cross-section images measured by SEM were shown in (c) and (d). These two pictures indicated that the interface between the two layers has no gaps, these two layers combined well, and the perturbation patterns introduced into the interface between these two layers successfully. H1 and H2 were 40 μm and 15 μm, respectively.Surface roughness of the CH layer surface has important influence on the hydrodynamic instability experiments in ICF, so a surface profiler was used to characterize the surface roughness of the CH layer. The characterizing method was the same to which described in Section . The scanning length and scanning speed were 1000 μm and 100 μm s−1, respectively. The results were shown in , the surface average roughness (Ra) and root mean square roughness (Rq) of the CH layer surface were measured to be 11.47 nm and 13.62 nm, respectively, far less than the wavelength of the driven laser (530 nm). Which revealed that the CH layer surface of the dual-layer target have good surface smoothness, meet the requirements of the hydrodynamic instability experiments well.In the hot-press process, the Al foil was heated to 150 °C and pressed by 2.5 kg weight, the amplitude and wavelength of the perturbation patterns might have changed after been hot-pressed. In order to evaluate the effect, that hot-pressing the CH onto the perturbed Al foil has on the amplitude and wavelength of the perturbation, the CH/Al dual-layer perturbation target was cleaned by chloroform to remove the CH layer, the amplitude and wavelength of the perturbation were measured by a surface profiler again. The measuring method was the same as Section . In this figure, uniform T and A of the perturbation were 52 μm and 7.31 μm, respectively, just very little and ignorable change compared with before hot-pressing the CH onto it. As the Al with a melt point of 660 °C and an elasticity modulus of 70 Gpa, while the hot-press process with a temperature of 150 °C, this temperature was not high enough to make Al soft and the pressure not large enough to make Al distort either. So the amplitude and wavelength of the perturbation almost have no change after been hot-pressed.A CH/Al dual-layer perturbation target was designed and fabricated for the hydrodynamic instability experiments in ICF. The target was composed of a perturbed Al foil and a CH layer. Uniform sinusoidal perturbation was introduced into the interface between the Al foil and the CH layer by SPDT technology. The CH layer was coated on the Al foil directly by a hot-press process. T, A, H1 and H2 of the CH/Al dual-layer perturbation target were 52 μm, 7.34 μm, 40 μm and 15 μm, respectively. The cross-section images of the target showed that the CH layer coated on the perturbed Al foil perfectly. Surface roughness of the surface of CH layer was far less than the wavelength of the driven laser, the whole target meets the requirements of the hydrodynamic instability experiments well. As the process of coated the CH layer on the Al foil avoid the use of a sticker, eliminate the gaps between the CH layer and the Al foil, and almost have no influence to the amplitude and wavelength of the perturbation, it proved to be a simple and useful method for target fabrication and maybe much broader application in the future.Numerical simulation of reinforced concrete beam/column failure considering normal-shear stress interactionFibre beam elements have vast applications to simulate the behaviour of reinforced concrete beam/column members. If normal-shear interaction is ignored, separated normal and shear constitutive laws are usually used in fibre beam elements, which is invalid to simulate shear failure in beam/column members with small or medium shear span-to-depth ratios. In this paper, a fibre beam element is treated as a degenerated solid element, and a unified concrete constitutive model is proposed for the degenerated solid element. Different from the separated constitutive models, the unified concrete constitutive model incorporates normal and shear behaviour based on three dimensional stress–strain states and, thus, normal-shear interaction is naturally considered. Moreover, compressive failure, shear failure and tensile failure are accounted for. Therefore, beam/column members with a wide range of shear span-to-depth ratios can be simulated with the degenerated solid element considering normal-shear interaction. Lastly, a variety of examples are presented to demonstrate the applicability and reliability of the proposed method by comparing numerical predictions with experimental results.tangential material matrix with a dimension of 6 × 63 × 3 matrix extracted from tangential material matrix D6eccentricity parameter of out-of-roundnessinvariants of the principal stress tensor componentsvalue defining the onset of plastic flowrelevant to the slope of the softening functionbrittleness index of concrete, relevant to the ductility and post-peak stress–strain relationshipincremental stress vector of Δσy, Δσz and Δσyzincremental stress vector of Δσx, Δσxy and Δσxzincremental strain vector of Δεy, Δεz and Δεyzincremental strain vector of Δεx, Δεxy and Δεxzcylindrical coordinates of hydrostatic length in the Haigh–Westergaard coordinatesWhen simulating the behaviour of reinforced concrete (RC) beam/column members, a three dimensional (3D) fibre beam element formulation is often preferred in order to obtain satisfactory accuracy with acceptable computational cost At each fibre cross-section, there are one normal stress component along the longitudinal (fibre) axis and two orthogonal shear stress components along the strong and the weak axes, respectively. For simplicity, the constitutive laws for the normal stress and shear stresses are independent of each other. Many of the widely accepted and validated concrete constitutive laws published for concrete under uniaxial compression, such as the Modified Kent and Park model For the prediction of shear failures in RC beams, empirical shear models based on experimental studies have been adopted in many previous numerical studies on fibre beam element formulations Alternatively, in order to consider the interaction between normal and shear stresses in fibre beam elements, Petrangeli et al. In principle, any stress–strain state at any material point is three dimensional, which consists of six components for both stress and strain. Apparently, the ideal way to account for the interaction between normal and shear stresses in concrete fibres is to employ plasticity-based constitutive laws for concrete. In this paper, based on the concept of degenerated solid element, the fibre beam element is treated as a special 3D solid element to consider the interaction between the normal and shear behaviour of concrete so that a 3D plasticity-based concrete model can be adopted as the concrete constitutive law. An obvious advantage of this approach is that the stress state in the fibre beam element is determined from a 3D material stress–strain state and, thus, the couple effect of normal and shear stresses in concrete fibres can be fundamentally accounted for by imposing certain constraint equations due to stress simplification resulting from transformation of beam elements from solid elements. In addition to the plasticity model to simulate concrete deformation in compression, a fracture model should also be incorporated to simulate concrete tensile behaviour in RC beam members Generally, there are two fundamental ways to simulate tensile failure of concrete by differentiating whether cracks are discrete or smeared On the other hand, when omitting localised cracking simulation, the cracks in the smeared crack model are considered to be spread across the elements by changing their constitutive equations. The smeared crack model is similar to the concept of crack band model which is in good agreement with experimental studies on fracture and size effect and is also convenient to be programmed Some other concrete models are also proposed in order to combine the merits of the discrete and smeared crack models, such as cohesive segments method To avoid complicating concrete constitutive relations in the beam element formulation and stabilize convergence when finding the optimum return point on the active failure surfaces The outline of this paper is summarised as follows. In Section , the concept of degenerated solid element is proposed and applied to fibre beam element formulations with corresponding constraint equations regarding the stress simplifications from a 3D solid element. In Section , a unified concrete model is proposed for the degenerated solid element in the 3D space. That is, the compressive concrete constitutive law is proposed based on a three-parameter failure function and a concise one-parameter potential function, while a fixed crack approach to model concrete tensile behaviour is adopted and modified. Besides, the proposed material properties are calibrated for concrete plasticity and fracture models. Lastly, several numerical examples, including one-element RC member test, an RC column and 14 RC shear beams, are presented to validate the proposed concrete plasticity model combined with fracture model incorporated into fibre beam–column element formulations.Theoretically speaking, numerical simulations using 3D solid elements are capable of predicting the deformation and failure behaviour of RC beam–column members, provided that an appropriate plastic-fracture model is employed to describe the concrete compressive and tensile behaviour. While in principle any stress–strain state at any material point is three dimensional, for simplicity and efficiency in numerical simulations, various simplified formulations have been proposed to deal with specific stress or strain states in certain geometric configurations. For example, the beam element is derived from the idealisation that one typical dimension (length) is much larger than the other two typical dimensions (width and depth), while the shell element is similarly derived from the idealisation that one typical dimension (thickness) is much smaller compared to its longitudinal and transverse dimensions. Consequently, the stress and strain states of the degenerated and idealised elements are simplified as well.With simplifications of stress and strain states, the degenerated elements, such as beam and shell elements, can be treated as degenerated solid elements but with corresponding constraint equations stemming from 3D solid elements In 3D solid element formulations, the incremental concrete stress vector Δσ6 can be obtained in Voigt’s notation by multiplying the incremental concrete strain vector Δε6 and elastic material matrix D6 as shown in Eq. Δσ6=ΔσxΔσyΔσzΔσyzΔσxzΔσxy=D11D12D13D14D15D16D21D22D23D24D25D26D31D32D33D34D35D36D41D42D43D44D45D46D51D52D53D54D55D56D61D62D63D64D65D66ΔεxΔεyΔεzΔεyzΔεxzΔεxy=D6Δε6With concrete assumed as an isotropic material, the tangential material matrix D6 can also be expressed as a positive-definite fourth-order tensor Dwith the so-called minor and major symmetriesand the conventional mapping of the first and second pairs of indicesTherefore, the entries in the tangential material matrix D6 are explicitly given as follows:D6=λ+2μλλ000λλ+2μλ000λλλ+2μ0000002μ0000002μ0000002μ, the two independent material constants λ and μ are called the Lamé constants and are given aswhere E is the Young’s modulus and ν is the Poisson’s ratio. The Lamé’s second constant μ is also known as the shear modulus and is usually denoted as G.As an example of degenerated elements, the constraint equations resulting from solid elements to fibre beam elements are proposed based on the concept of degenerated solid element. The most significant attribute is that, the interaction between the normal and shear behaviour of concrete in the fibre beam element is considered.Different from a 3D solid element, the incremental strain components Δεx, Δεxy and Δεxz in a displacement-based fibre beam formulation () are always known according to the incremental deformations induced by a load increment. On the other hand, the incremental stress components Δσy, Δσz and Δσyz are always equal to zero at all the load increments and iterations due to the beam idealisation from the idealised stress state of a solid element. Thus, based on the constitutive law of a solid element formulation given in Eq. , the other three unknown incremental strain components Δεy, Δεz and Δεyz can be calculated by solving the constraint equations given in Eqs. Δσun=ΔσyΔσzΔσyz=000,Δεun=ΔεyΔεzΔεyz,D3=D22D23D24D32D33D34D42D43D44,D‾3=D21D25D26D31D35D36D41D45D46andΔε¯3=ΔεxΔεxzΔεxy that zero stress components are taken as a set of constraint equations to calculate the non-zero strain components and provide the unknown strain vector Δεun. Besides, the residue stress vector M is also taken into account, which results from known incremental strain vector Δε¯3 and material sub-matrix D‾3 associated with the unknown incremental strain components Δεy, Δεz and Δεyz. The entries of material sub-matrix D3 and D‾3 can be obtained from those of tangential material matrix D6 with an array size of 6 × 6 as shown in Eq. It is noteworthy that the unknown incremental strain vector Δεun can be easily calculated when the element is in the elastic state because the material sub-matrix D3 can be directly derived from the initial elastic material matrix D6. However, when the element is in the plastic state, the tangential material matrix D6 has to be updated according to the derivation as follows.Firstly, the new stress state in the plastic model is computed by using a predictor–corrector formula where g is the potential function and plastic multiplier dλ is given in Eqs. Therefore, the equivalent stress–strain relationship in the plastic state can be obtained in Eq. Dep=D:I-∂g∂σ⊗∂f∂σ:D∂f∂σ:D:∂g∂σ-∂f∂αδ:∂g∂σBased on the above discussion, the constitutive laws for 3D solid elements can be applied to the fibre beam element in the plastic state. With the zero stress vector Δσun taken as constraint equations, the incremental strain vector Δεun can be obtained in an iterative approach as shown in , where all the symbols and entry sequence are the same with those in Eqs. . In a general case for a displacement-based beam formulation, the available information at a certain load increment is the equilibrium stress vector σ3eq and elastic and plastic strain vectors ε3eq,e and ε3eq,p, respectively, at the last load increment. The basic idea of the iterative approach shown in is to adjust the magnitude of strain components Δεun to make the corresponding stress vector Δσun equal to zero to meet the beam element simplification at each iteration and load increment. If the predicted stress components of Δσun do not satisfy the predefined tolerance (Tol.), then the stress vector Δσun has to be utilised to correct the prediction of the strain vector Δεun until the prescribed tolerance is satisfied. To ensure the accuracy of stress–strain relationship, the material matrix D6 should be updated either when concrete becomes plastic or when a crack appears at a certain concrete fibre as discussed in the next section.Failures in RC fibre beam elements can be classified into three types: (1) compressive (2) shear and (3) tensile. Compared with the other failures, the first type of failure due to compression behaves in a ductile way. A simple but reliable plasticity-based concrete model Apparently, the shear and tensile failures share an identical attribute, that is, concrete fails due to the maximum principle stress exceeding the tensile strength and behaves in a quasi-brittle way. As the main purpose of the present study is to demonstrate the application of degenerated solid element into fibre beam elements and the advantages of the proposed plasticity-based model for concrete compressive behaviour, rather than using other advanced but complicated fracture models as discussed in Section , only a simple fixed smeared crack model In the implementation of failure checking in the finite element program,: the first step is to detect if brittle failure has occurred in every fibre of beam elements, and subsequently, compressive failure is tracked.For simplicity, the plasticity model in Haigh–Westergaard stress space is introduced, which is defined by the cylindrical coordinates of hydrostatic length (ρ), deviatoric length (ξ) and Lode angle (θ) as shown in These coordinates are functions of the invariants (I1, J2, J3) of the principal stress tensor components (σ1 |
> |
σ2 |
> |
σ3) according to Eqs. ρ=2J2whereJ2=(σ1-σ2)2+(σ2-σ3)3+(σ3-σ1)26θ=13cos-1332J3J23/2whereJ3=σ1-I1/3σ2-I1/3σ3-I1/3The well-known three-parameter failure surface proposed by Menétrey and Willam f(ξ,ρ,θ)=1.5ρk·fc2+mρ6·k·fcr(θ,e)+ξ3·k·fc-c=0where k and c are the concrete hardening and softening functions, respectively. The term m is the friction parameter and r is the elliptic function, both of which are defined in Eqs. r(θ,e)=41-e2cos2θ+2e-1221-e2cosθ+2e-141-e2cos2θ+5e2-4e1/2 that define the shape and size of the loading surface in the stress space are the mean uniaxial concrete compressive strength (fc), the mean uniaxial concrete tensile strength (ft) and the eccentricity parameter of out-of-roundness (e). As shown in , concrete hardening and softening are controlled by functions k(εvp) and c(εvp), respectively, where the parameter εvp is the plastic volumetric strain. The hardening function proposed by Papanikolaou and Kappos k(εvp)=k0+(1-k0)1-εv,tp-εvpεv,tp2(εvp<εv,tp)1(εvp≥εv,tp)where εv,tp is the plastic volumetric strain at uniaxial concrete strength (onset of softening), and k0 is the value that defines the initial yield surface that bounds the initial elastic regime c(εvp)=1(εvp<εv,tp)11+εvp-εv,tpt22(εvp≥εv,tp)where parameter t controls the slope of the softening function where αp0 is a material parameter defined as brittleness index of concrete, and the function Q is expected to reveal the effect of the principle stress on concrete ductility. As an extension of the flow rule in Q(σ1,σ2,σ3)=(1+σ1/ft)(σ1>0)1(σ1=σ2=0,σ3<0)0.45e8σ1/fc+0.05(σ1≤0,σ2<0,σ3<0)It is seen that the extended flow rule is applicable to the cases associated with shear stresses because of the introduction of tensile principle stress (σ1 |
> 0) in Eq. , where εm is the ultimate strain in tension and εp is the normal plastic strain perpendicular to the crack. Besides, a small residual tensile stress after εm is assumed to ensure numerical stability. It should be noted that the unloading path does not return to the origin but the strain–stress state (εp, 0), so the residual plastic strain upon unloading is considered.Different shapes of the tension stiffening part in the tensile stress–strain relationship have been proposed, such as linear, bi-linear, multi-linear lines and also an exponential curve. Nevertheless, as most practical applications are not very sensitive to the exact shape, it mainly depends on practical convenience , where the tension stiffening constant α depends on the relative percentage of steel reinforcement in the cross-section and is usually taken as 0.5–0.7. In the present study, the value of α is taken as 0.6. However, it is more rigorous to determine the ultimate tensile strain εm based on fracture energy, which will be discussed when suggesting material properties for the proposed plasticity and fracture model in Section The cracked shear modulus Gc is assumed to be a function of the current tensile strain. Taking 1, 2 and 3 as the three principle directions of the stress state in a 3D solid element (), when the concrete cracks in 1-direction, the incremental shear stress–strain relationship is expressed aswhere the cracked shear moduli at the crack plane are given aswhere G is the shear modulus of intact concrete.Once the tensile stress in the 2-direction reaches the tensile strength, a second crack plane perpendicular to the first one is assumed to form and the incremental shear stress–strain relationship in 1-direction becomesG12c=max0.0,12G12tG13c=max0.0,14G1-250ε1-ε1pOnce the normal strain at a certain crack plane (e.g. 1-direction) is negative, the crack is deemed to have closed and the corresponding cracked shear moduli (e.g. G12c and G13c) will revert to the intact shear modulus G.With concrete cylinder strength as the only required input material parameter, all the parameters with suggested values related to the proposed constitutive laws are summarised in for concrete cylinder strengths ranging from 20 MPa to 100 MPa. It should be noted that the brittleness index of concrete αp0 is calibrated with the results of uniaxial, biaxial, and triaxial compressive curves obtained from experimental studies in the literature. More details of the verification of the suggested values related to the proposed constitutive laws can be found in For a given concrete cylinder strength, the relevant material parameters can be obtained by linear interpolation from . The suggested values of brittleness index of concrete αp0 follow the trend that the post-peak behaviour is more ductile (smaller value of αp0) with decreasing concrete cylinder strength. To accurately describe the ductility and post-peak stress–strain relationship of concrete used in the model, αp0 can also be specified by users when the concrete properties can be obtained from uniaxial compression tests.As for the ultimate tensile strain εm, it should be determined by Eq. where the crack bandwidth h is taken as 35 mm, which is within the range of 1.5dmax to 4.0dmax obtained from experimental studies for various concretes In order to validate the softening law in the proposed fracture model (), some widely cited experimental studies . As most practical applications are not sensitive to the exact shape focus on the validation of the tensile softening part, the tensile strength and the corresponding strain from the experimental studies are, therefore, adopted directly when conducting the analysis based on the proposed fracture model.To verify the proposed concrete model in a degrade element with constraint equations and rule out inaccuracy from beam geometric nonlinearity, a validated 3D co-rotational beam element ) is predefined to be 10−6 in the present study.Several numerical examples are given herein to validate the proposed concrete plasticity model combined with a fracture model in a fibre beam–column element formulation. First, a one-element RC member is utilised to emphasise the performance of the implemented plasticity-based concrete model and the fracture model at the material level subjected to compression, tension and shear, respectively. Then, the prediction of an RC column subjected to pure compression is conducted and compared with the experimental study. Lastly, 14 RC shear beams from different publications are employed to illustrate the accuracy of the proposed unified concrete plasticity model. Meanwhile, the limitations of a fibre beam element formulation to predicting shear failure of RC members are discussed.The combined constitutive relationships of the plasticity model and the fracture model for concrete have been successfully validated at the material level and the concrete cylinder strength is 25.3 MPa. For the cases of pure compression and tension, a comparison of the simulation results with the calculation results at the material level is shown in For the case of shear, the load-deformation response is strongly related to the dimension of the model and, therefore, the simulation result based on a one-element concrete member cannot be compared with the calculation result at the material level. In fact, compared with the load-deformation response, the crack pattern is more meaningful and emphasised herein. For the numerical model with one beam element, the concrete member shown in Since there are two Gaussian points along the beam element, the crack direction for each fibre around the beam cross-section is plotted in is compatible with the stress state subjected to the applied shear force and the induced bending moment. Due to the uniform shear distribution and linear normal stress distribution about the neutral axis, the normal stress component at the extreme bottom and top fibres is dominant compared with shear stress components. Therefore, the crack direction is almost perpendicular to the fibre cross-section. However, the shear stress components dominate the stress state at the fibres adjacent to the neutral axis and the crack direction is almost 45° with respect to the fibre axial direction.A confined square RC column tested by Sheikh and Uzumeri , the RC column is 305 mm × 305 mm × 1960 mm, with 19 mm of concrete cover. The utilised concrete cylinder strength is 31.9 MPa which is the same value used by Červenka and Papanikolaou . The #5 steel reinforcement properties consist of a Young’s modulus of 200 GPa, a hardening modulus of 922.76 MPa, a Poisson ratio of 0.3 and a yield strength of 371.7 of MPa. Five fibre beam elements were utilised to mesh the column. As shown in , the predictions by the proposed unified concrete plasticity model are compared with experimental study Shear tests on single-span reinforced concrete beams with and without shear reinforcement were conducted by Leonhardt and Walther . Since the beam dimensions, loading and boundary conditions are symmetric, five fibre beam elements were utilised to mesh one half of the beam. Based on the loading position, the shear span-to-effective depth ratio is equal to 3.0., the analysis results are compared with the simulation by 3D solid elements . On the other hand, it should be noted that the rigid-plane assumption imposes an idealised constraint relationship between reinforcement and the surrounding concrete in both axial and transverse directions. Correspondingly, the simulated reinforcement by fibre model in the analysis provides more bending and shear resistance. In fact, after the opening of cracks, the yielded reinforcement is difficult to behave perfectly to resist local shear loading as there is no directional constraint as assumed in the fibre beam element. Thus, it is fairly reasonable to ignore the contribution of the hardening part of steel reinforcement and the analysis results before the hardening of reinforcement (as indicated by the black circles) could provide better predictions in terms of shear resistance and ductility as shown in Clearly, the accuracy of the simulation by the proposed constitutive laws is acceptable. Due to the rigid-plane assumption in the fibre beam element, shear failure in the model cannot lead to termination of the simulation. However, the crack pattern can be captured by the proposed constitutive laws and there is an obvious descending trend in the load–displacement response as shown in . Since the bottom reinforcement has not fractured yet, the obvious descending trend is treated as the occurrence of shear failure.A classical set of RC beams with the variations of simple span length, concrete strength, beam width and stirrup details were tested by Bresler and Scordelis The geometry, loading, boundary condition and steel reinforcement details are illustrated in . The depth of all the specimens is 560 mm. Five fibre beam elements were utilised to mesh the beam. In addition, equivalent steel fibres were assigned to the location of steel reinforcement as shown in . The properties of concrete and steel reinforcement and geometric dimension are listed in The comparisons of experimental studies and numerical simulations by the proposed unified concrete plasticity model and a uniaxial concrete model (Modified Kent and Park model . There is good agreement between the predictions of the proposed concrete constitutive laws and experimental results for all the specimens with different reinforcement details and shear span-to-depth ratios. Compared with the predictions by the uniaxial concrete model, the results predicted by the proposed unified concrete plasticity model are generally better and shear failure can be captured for specimens with shear span-to-depth ratios of 4.0 and 5.0 as shown in (a) and (b). However, the proposed model is not significantly superior to the uniaxial model in the predictions of the third series of specimens as shown in (c), as the failure mode is a combination of shear and flexural failures due to a greater shear span-to-depth ratio.A statistical analysis of the prediction results of ultimate shear strength for each series of the RC beams is given in . The overall average of the ratios of the peak values from predictions and experiments for all the 12 specimens is 81.6% with a standard deviation of 0.126. If the OA series without transverse reinforcement is not included in the statistical analysis, the overall average of the ratios is 86.5% with a standard deviation of 0.099. The reason for the discrepancy is that the rigid-plane assumption in the fibre beam element results in additional pseudo lateral constraint for the beam cross-section, which bears greater resemblance in behaviour to RC beams with transverse reinforcement. Therefore, the fibre beam formulation with the proposed concrete model is more applicable to shear failure simulation of RC members with transverse reinforcement.As a summary of the numerical validations, all the presented examples are compared with experimental studies and available 3D solid element simulations. However, there are two minor intrinsic disadvantages for the 3D fibre beam element. Firstly, the rigid-plane assumption for the beam element cross-section results in fictitious continuities between fibres, which in reality should be discontinuous after the occurrence of concrete cracking. This will make the predictions of shear strength by fibre beam elements larger than the experimental results.Secondly, the boundary and loading conditions in a 3D fibre beam element are applied at the centroid of the beam cross-section, because the beam element is still a line element. However, in laboratories, the boundary condition and loading point are applied at the top and bottom surfaces in shear beam tests. This will result in discrepancy between experimental studies and numerical predictions. Nevertheless, based on the comparisons for all the examples, the predictions by the 3D fibre beam element along with the proposed unified concrete plasticity model and a simple fracture model are reliable and reasonably accurate.The interaction of normal and shear stresses is usually neglected in the conventional fibre beam elements, but it plays an important role in predicting shear failures of RC beam–column members. In order to accurately simulate the stress interaction in the 3D space and predict shear failures in RC beams, the concept of degenerated solid element is proposed and the corresponding constraint equations are derived for 3D fibre beam elements. As for the concrete constitutive law of degenerated solid element, a unified concrete plasticity model is proposed. Based on the failure and potential functions in terms of plasticity theory, the unified concrete plasticity model can predict the shear behaviour of beam–column members with small and medium shear span-to-depth ratios (not less than 3.0). To better validate the proposed unified concrete plasticity model in more complicated examples, the classical Hinton concrete fracture model is modified for a 3D beam formulation to consider opening and closing of cracks. All the presented examples show that the predictions by the 3D fibre beam element along with the unified concrete plasticity model and the modified fracture model are reliable and satisfactorily accurate, even though there are some assumptions in the beam element formulation which result in inaccuracy compared with 3D solid element simulations and experimental studies.As a degenerated element, the 3D fibre beam element is an application of the concept of degenerated solid element with some assumptions for certain degrees of freedom. Obviously, the proposed degenerated solid element and the unified concrete plasticity model could be further applied to other types of elements, such as shell element, in a similar approach as discussed in Section Structural and elastic properties of LiBH4 for hydrogen storage applications► The elastic constants of LiBH4 are determined for the first time in this study. ► The LiBH4 is harder than any MBH4 compounds. ► The calculated Debye temperature was found higher compared to that of MBH4 series. ► The calculated Poisson’s ratio is smaller than the values reported for most MBH4. ► The LiBH4 compound is relatively stable against shear.Structural and elastic properties of LiBH4 in the orthorhombic structure were investigated using both the norm-conserving pseudopotentials and full potential within the general gradient approximation (GGA) in the frame of density functional theory. The Orthorhombic LiBH4 phase is found to be mechanically stable at ambient pressure. The calculated linear bulk moduli are found to be in good agreement with the experimental values reported in the literature. Shear and Young’s moduli as well as Poisson’s ratio for ideal polycrystalline LiBH4 are also calculated. According to the obtained results, LiBH4 can be classified as brittle material. A Debye temperature of 1272 K was also calculated using theoretical elastic constants.The extensive use of hydrogen as fuel for practical applications such as fuel cell for electric and hybrid vehicles, requires the progress of a safe, reliable and cheap method for its transportation. However, an efficient way to store hydrogen, in particular for mobile applications, remains a challenge Several research work has been devoted to understand the structural, electronic, thermodynamics and the hydrogen storage properties of LiBH4). The unit cell parameters of this structure are a |
= 7.17858(4), b |
= 4.43686(2), c |
= 6.80321(4) Å Nonetheless, little information about structural and elastic properties of this material is available. Except the bulk modulus for the LiBH4 is studied theoretically and experimentally The knowledge of the elastic constants of complex hydrides compounds is necessary for many practical applications related to the mechanical properties of a solid, and closely related to many fundamental solid-state properties, such as equation of state, specific heat, thermal expansion, Debye temperature, melting point, etc.In this work, bulk modulus and the elastic constants of the LiBH4 were calculated for the first time using the density functional theory (DFT) with general gradient approximation (GGA) functional. Moreover, shear and Young’s moduli, Poisson ratio and Debye temperature were also calculated.In the present work, two different approaches to examine the elastic properties of the LiBH4, were used. In order to estimate the bulk modulus and structural properties, the most recently developed Vienna package WIEN2k The convergence of the performed calculations with respect to the plane-wave cut-off and k-point mesh, were carefully tested. An energy cutoff of 50 Hartree and a 6 × 6 × 6 grid for k-point were used.The elastic constants tensor was subsequently obtained using the linear-response method, implemented in the ABINIT code. The linear response is used to calculate the second derivative of the total energy with respect to the strains.The atomic positions were relaxed; this relaxation lowered the total energy. The final structure obtained within the GGA approximations is given in . The relaxed structure is in good agreement with the reported structure obtained from experiments The bulk modulus, B0, and its pressure derivative B0′ can be deduced from the variation of the total energy with the volume of the lattice. In the present study, the total energies are calculated at several volumes around the equilibrium and are fitted to the Murnaghan’s equation of state using FP-LAPW methods. The relaxed atomic positions given in (FP-LAPW column) were used. The structural optimization, total energy curves for LiBH4 obtained from GGA calculations are shown in . It is clear that the equilibrium structural parameters obtained from GGA calculations are in good agreement with the corresponding experimental values (see By fitting the total energies to the Murnaghan equation of states, it is found that the bulk modulus and its pressure derivative for LiBH4, in the Orthorhombic structure, to be 50.95 GPa and 3.126, respectively. The calculated bulk modulus value (50.95 GPa) is in good agreement with the experimental value reported by Talyzin et al. The elastic properties define the behavior of a solid that undergoes stress, deforms, and then recovers to return to its original form. To investigate the mechanical stability description of the LiBH4 structures, a set of zero-pressure elastic constants was determined from the stress of the strained approach using ABINIT code, taking into account the ionic relaxations in response to strain perturbations.The calculated elastic constants are listed in . For stable orthorhombic crystals, the nine independent elastic constants Cij should satisfy the well-known Born stability criteria C11>0,C22>0,C33>0,C44>0,C55>0,C66>0[C11+C22+C33+2(C12+C13)]>0(C11+C22-2C12)>0(C11+C33-2C13)>0(C22+C33-2C223)>0The calculated elastic constants Cij satisfy Born stability criteria. Thus, the orthorhombic phase of LiBH4 is mechanically stable at ambient pressure. From , it can be seen that C11, C22 and C33 have, approximately, the same value, that means the atomic bonding between nearest neighbors along the (1 0 0), (0 1 0) and (0 0 1) planes, respectively, have the same strength.It is noticed also that the calculated elastic constants of LiBH4 compound are somehow large compared to the calculated values of some related borohydrides MBH4 (M = Na, K, Rb and Cs) compounds If single crystal samples are not available, the measure of the individual elastic constants Cij cannot be possible. Instead, the polycrystalline bulk modulus (B) and shear modulus (G) may be determined. Frequently, for such calculations, two main approximations are used, namely Voigt (V) BV=(1/9)[C11+C22+C33+2(C12+C13+C23)]GV=(1/15)[C11+C22+C33+3(C44+C55+C66)-(C12+C13+C23)]In terms of the Reuss approximation, bulk (BV) and shear (GV) moduli can be expressed by:BR=1/[S11+S22+S33+2(S12+S13+S23)]GR=15/[4(S11+S22+S33)-4(S12+S13+S23)+3(S44+S55+S66)]where, Sij are the elastic compliance constants listed in (The compliance matrix Sij is the inverse of the matrix Cij).The elastic moduli of the polycrystalline material can be approximated by Hill’s average and for shear moduli will be presented by:Both Young’s modulus (E)and Poisson’s ratio (ν) are given by the following equations The calculated values of bulk, shear and Young’s modulus, as well as Poisson’s ratio of the orthorhombic LiBH4 structure are summarized in for both Voight, Reuss and Hill approaches. The calculated bulk moduli, from the elastic constants, agree well with those directly obtained from the fitting of the Murnaghan EOS given above using FP-LAPW and in the range of the experimental value reported by Talyzin et al. In the same idea the calculated shear moduli from single crystal elastic constants of LiBH4 is larger than that of the MBH4 calculated previously The calculated Poisson’s ratio (measures the stability of a crystal against shear) is considerably smaller than the values reported for most MBH4 borohydrides The elastic anisotropy of crystals can apply great effects on the properties of physical mechanisms, such as anisotropic plastic deformation, crack behavior, and elastic instability. Hence, it is important to calculate elastic anisotropy structural hydrides in order to improve their mechanical durability for mobile application (hydrogen storage) The bulk moduli along the a-, b- and c-axis can be calculated from the previously calculated compliance constants SijBa=1/[S11+S12+S13]Bb=1/[S12+S22+S23]Bc=1/[S13+S23+S33]The obtained values of Ba, Bb, and Bc of LiBH4 are listed in . It is noted that the bulk modulus along a-axis is larger than the bulk modulus along b-axis and c-axis, implying that the a-axis is the least compressible.The calculated values of the anisotropy of the bulk modulus along a- and c-axis with respect to the b-axis are:Note that, when a value of one indicates an elastic isotropy, while any departure from one represents an elastic anisotropy. It is clear that LiBH4 is slightly elastically anisotropic.The shear anisotropic factors provide a measure of the degree of anisotropy in the bonding between atoms in different planes. Therefore, the shear anisotropic factor for the {1 0 0} shear planes between 〈0 1 1〉 and 〈0 1 0〉 directions, is defined as For the {0 1 0} shear planes between 〈1 0 1〉 and 〈0 0 1〉 directions it is And for the {0 0 1} shear planes between 〈1 1 0〉 and 〈0 1 0〉 directions it is The calculated shear anisotropic factors are reported in . The deviation of the anisotropy factors from unity is a measure for the elastic anisotropy. The shear anisotropy results indicate that the an elastic anisotropy for {0 0 1} shear planes between 〈1 1 0〉 and 〈0 1 0〉 directions is larger than those for the {1 0 0} shear planes between 〈0 1 1〉 and 〈0 1 0〉 directions and the {0 1 0} shear planes between 〈1 0 1〉 and 〈0 0 1〉 directions. Unfortunately, there is no experimental data to compare with the values obtained in this study.A set of fundamental physical properties may be estimated using the calculated elastic constants. Among them is the Debye temperature (TDebye), which is linked to many physical properties such as specific heat, elastic constants, and melting point where h is Planck’s constant, k is Boltzmann’s constant, NA is Avogadro’s number, ρ is the density of molecule, M is the molecular weight and n is the number of atoms in the molecule.The average sound velocity (vm) in polycrystalline materials is given by where vt and vl are the transverse and longitudinal elastic wave velocities of the polycrystalline materials and are given by Navier’s equation The calculated ρ, vl, vt, vm and TDebye are listed in . The results predicted that TDebye of LiBH4 is higher than that of MBH4 related compounds calculated theoretically Based on the first principles calculations, the mechanical and structural properties of LiBH4 compound were investigated. The calculated lattice parameters are in good agreement with the experimental data. Strain energies for nine different distortions of LiBH4 compound using GGA in the theoretically optimized crystal structure in order to calculate elastic constants, have been also calculated. To the best of our knowledge, there are no available experimental data about the elastic constants of LiBH4 compound. The results obtained in this study could provide a useful reference for future studies. Using Hill’s approximation, the ideal polycrystalline aggregates bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio, are calculated. In the case of LiBH4 compound, the value of B/G is smaller than 1.75, and therefore LiBH4 can be classified as brittle material. The calculated Poisson’s ratio is considerably smaller than the values reported for most MBH4 borohydrides. This indicates that LiBH4 compound is relatively stable against shear. The calculated Debye temperature was found higher compared to that of MBH4 series, which reveals that LiBH4 is harder than any MBH4 compounds.First-principles investigation of mechanical, electronic, dynamical, and thermodynamic properties of Al3BCIn this paper, Al-12 wt%B4C composites without and with 3 wt%Ti were fabricated via the stir-casting process, the results show that Al3BC was the main component of the reactant at the interface of B4C–Al composites. The first-principles was utilized to conduct an in-depth study of the mechanical, electronic, and thermodynamic properties, and the lattice dynamics of Al3BC. The optimized lattice parameters were agreed well with the experimental values. Al3BC had good thermal stability, but the dynamic stability was poor. The bulk modulus, shear modulus, Young's modulus, Poisson's ratio, B/G, and hardness of Al3BC are 153 GPa, 137 GPa, 317 GPa, 0.16, 1.17, and 26.43 GPa, respectively, these values indicate that Al3BC is a brittle material with high hardness and that brittle failure is the main failure form. Al3BC had a slight elastic anisotropy. It also had semiconductor properties, and its chemical bonding was covalent, metallic, and ionic.The Al–B–C system contains a variety of compounds, such as Al3BC3, Al8B4C7, and Al3BC. Al3BC is an aluminum-rich ternary compound in the Al–B–C system and was first observed by Halverson et al. in the B4C/Al composite material []. Viala et al. reported the composition of Al3BC and identified that it has a hexagonal unit cell with a = 3.491(2) Å and c = 11.541(4) Å []. Meyer and Hillebrecht determined the crystal structure of Al3BC. Also, Al3BC has close packed Al atoms (layer sequence of ABACBC) with isolated boron atoms positioned in all of the octahedral voids between layers A and C, whereas isolated carbon atoms occupied trigonal voids in layer B [B4C-reinforced aluminum matrix composites have been extensively used in the nuclear power industry in recent years because of its excellent neutron absorption capacity []. The composite material is used primarily for transportation and storing spent fuel []. Al3BC is the main product of the interfacial reaction of B4C–Al composite materials and has significant influence on mechanical properties such as tensile strength and the elongation rate of composite materials; thus, it has attracted much attention. Some studies have suggested that the interfacial reaction product Al3BC is harmful to the mechanical properties of B4C–Al composites [] reported that the interfacial reaction to form Al3BC can improve the wettability of the B4C/Al interface and increase the bonding strength of the interface between the reinforced particles and the matrix.In addition, Al3BC is a promising new ceramic material that has excellent properties, such as high hardness, low density, high thermal stability, and good wear resistance; it is a candidate material for use as an aluminum matrix composite reinforcement [] used an intermediate alloy, which contains particles of Al3BC, in the AZ263 magnesium alloy, and the results showed that Al3BC is a heterogeneous good nuclear matrix for magnesium alloys. Zhao et al. [] used Al3BC as a reinforcing particle in an aluminum matrix composite to prepare a composite material that has high room temperature strength, high elastic modulus, and good mechanical properties at high temperature. Al3BC crystals are used to reinforce the A356 alloy, which is commonly used in aerospace and automotive industries [However, because it is difficult to obtain sufficient amount of Al3BC through experiments, there has been no research reported on the properties and performance of Al3BC, and this is mainly because of the following two reasons: (1) Al3BC can react with acidic and alkaline solutions [], and it is difficult to obtain pure Al3BC crystals from the Al–B–C system; (2) the size of Al3BC is very small, usually in the micron or even nanometer scale, and thus, it is difficult to measure through experimental methods. Therefore, the first-principles method was used to calculate and analyze the mechanical properties of Al3BC in this work. In addition, the crystal structure, electronic structure, and thermodynamic properties and the lattice dynamics of Al3BC were studied in detail.In this work, for the purpose of avoiding the interference of other elements, high purity Al (99.95%) was invoked as the matrix material. B4C particles (99.7%, 7 μm), Ti powders (99.7%, 35 μm), and K2TiF6 flux (99.0%) were selected for this investigation. The stirring casting method was used to prepare Al-12 wt%B4C composites without and with 3 wt% Ti, and the details of the preparation steps are described in our former studies []. Microstructures of the prepared composite materials were investigated using a Quanta 200 SEM and a Tecnai G2 F20S-TWIN TEM.In this work, the first-principles method based on density functional theory (DFT) was used to study the crystal structure, mechanical properties, electronic structure, lattice dynamics, and thermodynamic properties of Al3BC. The exchange correlation functional was the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) scheme []. The charge interaction between an electron and an ion was treated using ultrasoft pseudopotentials []. The 3s23p1, 2s22p1, and 2s22p2 electrons were used as the electron configurations for Al, B, and C, respectively. In order to guarantee the accuracy of the results and shorten the calculation time, the convergences of energy with respect to both cut-off energy and k-points sampling were tested. The convergence curves of cut-off energy and k-points were shown in . According to the test results, a cut-off energy of 500eV and a k-point grid of 19 × 19 × 5 were used. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme was used in the geometry optimization []. The self-consistent field (SCF) tolerance was 2.0 × 10−6eV/atom. In the convergence tolerance option, the energy was 1.0 × 10−6 eV/atom, the maximum Hellmann-Feynman force was 0.05 eV/Å, the maximum displacement was 0.002 Å, and the maximum stress was 0.1 GPa. Because the elastic constant of a material is related to stress and strain, the elastic constant was determined via linear fitting of the strain function. The step length of each strain was 6, and the maximum strain amplitude was 0.003. The phonon dispersion of Al3BC was calculated using the PHONON code [ shows SEM and TEM images of the as-prepared Al-12 wt%B4C composite. As shown in (a), the reaction between Al and B4C intensifies, the B4C particles gradually decompose, and a large amount of interfacial reaction product Al3BC is generated around B4C. With the addition of Ti, a TiB2 layer formed at the interface between the Al matrix and the B4C particles ((b)). The TiB2 layer tightly surrounded the surface of the B4C particles, acting as a protective layer to reduce the decomposition rate of B4C. However, Al3BC was still produced at the interface between Al matrix and B4C particles. (c) shows the TEM image of the micro-structure of the B4C/Al interface with Ti addition, it can be seen the interfacial reaction product Al3BC is attached to the B4C particles, and the TiB2 layer is on the outside. Also, the size of the interfacial reaction product Al3BC is on the nanometer scale. Wang et al. [] used a liquid stirring method to prepare an Al-31 wt% B4C composite containing 3.5 wt% Ti and shows that Al3BC is also produced at the interface, which is consistent with the results observed in this paper. The characteristic (112‾0) atom planes with 0.175 nm interplanar spacing of Al3BC were detected from high-magnification view (], the potential interfacial reactions that can take place during fabrication are as follows:3K2TiF6 + 13Al → 3Al3Ti + 3KAlF4 + K3AlF6Al3BC belongs to the hexagonal crystal system (space group: P63/mmc, No. 194). For the Al3BC structure, the original atomic positions were Al1 4f (1/3, 2/3, 0.07564) site, Al2 4e (0, 0, 1/4) site, B 2a (0, 0, 0) site, and C 2d (2/3, 1/3, 3/4) site []. The optimized Al3BC model is shown in shows the variation of total energy with lattice constants under GGA-PBE functional. The lower the energy, the more stable the crystal structure. The optimal lattice constants calculated according to the relationship between energy and lattice constant are shown in . The difference between the calculated value and the experimental value is low. For Al3BC, the difference for a was 0.17%, and the difference for c was 0.138%. This indicates that the calculated results are consistent with the experimental data.The structural stability of Al3BC can be predicted from the formation enthalpy (ΔH) []. If the formation enthalpy is negative, then the structure is thermally stable. Also, when the formation enthalpy is more negative, the thermal stability is better. The formation enthalpy formula is as follows [ΔHAl3BC=μAl3BCbulk−(3μAlbulk+μBbulk+μCbulk)where ΔHAl3BC is the formation enthalpy of Al3BC. μAl3BCbulk, μAlbulk, μBbulk, and μCbulk are the chemical potentials of Al3BC, Al, B, and C, respectively. The chemical potential of Al3BC is equal to the total energy of a single molecular formula after structural optimization, that is, μAl3BCbulk=EAl3BCbulk. Similarly, the chemical potentials of Al and B can be obtained in the same way. The chemical potential of the element C is equal to the energy of a single atom in the C bulk phase after relaxation, and it can be calculated using diamond. The formation enthalpy of diamond is 0.025 eV/atom higher than that of graphite.The calculated results of the formation enthalpy are listed in , the formation enthalpy of Al3BC was negative, and this shows that Al3BC is thermally stable.Mechanical properties are the key parameters of materials. The mechanical properties of a single-crystal and of polycrystalline materials are closely related to the second-order elastic constants of a single-crystal []. Al3BC belongs to the hexagonal system and has five independent second-order elastic constants, namely, C11, C12, C13, C33 and C44. The value of C66 can be calculated using (C11–C12)/2. The calculated results for the second order elastic constant Cij of Al3BC are listed in . The criteria for the mechanical stability of the hexagonal structure can be expressed as [C11>0,C44>0,C11−C12>0,(C11+C12)C33−2C132>0From the calculations, the second-order elastic constant of the Al3BC crystal satisfies the above conditions and was stable under the perturbation of the elastic strain.Also, second-order elastic constants reflect the bonding strength and deformation resistance of crystals in specific directions []. For Al3BC, C11 was higher than C33, and this means that Al3BC compresses more easily in the direction of the c-axis than it does in the direction of the a-axis. C12 and C13 were relatively small, and this indicates that Al3BC tends to shear along the b- and c-axes of the crystal when a large force is applied along the a-axis of the crystal. C44 represents the resistance of the (001) plane to shear deformation. C66 represents the resistance of the [110] orientation to shear deformation. As seen from , the shear stress in the [110] orientation is slightly less than that of the (001) plane. C11+C12>C33, and this indicates that the bonding strength in the (001) plane is stronger than that along the c-axis; it also indicates that the tensile modulus in the (001) plane is greater than that along the c-axis.Cauchy pressure (C12–C44) can be used as an indicator to assess whether a material is brittle or ductile. A positive Cauchy pressure indicates ductile behavior, and a negative Cauchy pressure indicates that the material is brittle. The negative Cauchy pressure of Al3BC (−75 GPa) indicates that it is a brittle material. Cauchy pressure can also be used to evaluate bonding properties in materials []. If the Cauchy pressure is positive, and the crystal is dominated by metallic bonding, on the contrary, the crystal is dominated by covalent bonding. As seen from the calculation results, the Cauchy pressure of Al3BC is negative, and thus, covalent bond is dominant in the Al3BC crystal.To explore mechanical properties, the bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (ν) of Al3BC were calculated, and the results are listed in . Bulk modulus is a very stable material constant, and it describes the compressive resistance of a material under volumetric elastic deformation. Shear modulus is closely related to hardness and describes the resistance to shear. Young's modulus measures the stiffness of an isotropic elastomer, which is a physical quantity that describes the deformation resistance of solid materials. When the Young's modulus is higher, the material is less likely to deform, and the stiffness of the material is greater. The bulk modulus and shear modulus of Al3BC can be obtained using the Voigt-Reuss-Hill approximation algorithm and the elastic constants after optimization []. The calculation formulas for each modulus are as follows:BR=[(C11+C12)C33−2C132]/[C11+C12+2C33−4C13]GR=115[4(2S11+S33)−4(2S13+S12)+3(2S44+S66)]−1where B and G are respectively the bulk modulus and shear modulus in the Voigt-Reuss-Hill approximation. BV (GV) and BR (GR) are determined from the crystal symmetry. Sij is the elastic compliance constant and is the inverse of the second-order elastic constant (Sij=Cij−1). The calculation results obtained from the formulas are shown in . The results show that Al3BC has higher bulk modulus and shear modulus, indicating that it has stronger resistance to volume deformation and to shearing.The ratio (δ) of the bulk modulus to C44 is used as the machinability index to evaluate the machinability of a material []. For Al3BC, the value of δ (1.08) is greater than 1, and this indicates that Al3BC has good machinability. According to the Pugh rule, the ratio of the bulk modulus to the shear modulus is usually employed to distinguish whether a material is ductile or brittle []. When B/G > 1.75, a material exhibits ductility, and when B/G < 1.75, a material shows brittleness. As seen in , the B/G value of the Al3BC structure was calculated to be 1.17, which indicates that it is a brittle material. Also, the brittle or ductile behavior of a material can also be determined from Poisson's ratio. If the value of Poisson's ratio is greater than 1⁄3, a material is ductile, and if it is less than 1⁄3, a material is brittle []. The equations for Poisson's ratio and Young's modulus are respectively as follows [, the value of Poisson's ratio for Al3BC is 0.16; this is less than 1⁄3, and so, Al3BC is a brittle material. This is consistent with the B/G and Cauchy pressure results discussed above. Also, the failure form of the material can be judged from Poisson's ratio []. Brittle failure occurs when the value of Poisson's ratio is lower than 0.26, and ductile failure occurs when the value of Poisson's ratio is greater than 0.26. For Al3BC, the value of Poisson's ratio is 0.16, and so, the failure form of Al3BC is brittle failure. Poisson's ratio can also be used to predict the properties of bonds in crystals. When v = 0.1, crystals are covalently bonded, and when v = 0.33 crystals have metal bonding []. For Al3BC, the value of Poisson's ratio is 0.16; thus, there is both covalent bonding and metallic bonding in the crystal, but covalent bonding dominated. The value of the Young's modulus was relatively large, and this indicates that the material is not prone to deformation. Zhao et al. [] utlized through nanoindentation technology to measure the Young's modulus of Al3BC and reported a value of 332 ± 17 GPa, which is consistent with the Young's modulus calculated in this study.Hardness refers to the local resistance of a material to the invasion of external objects, and this is closely related to the wear resistance of materials. If the hardness is high enough, it can improve the wear resistance of aluminum matrix composites [, can be used to predict Vickers hardness.The result show that Al3BC has a Vickers hardness of 26.43 GPa, which indicates that it is a material that has high hardness. ElasticPOST code [] was used to carry out three-dimensional (3D) characterization of Al3BC hardness, as shown in . The two-dimensional (2D) projection of its hardness is shown in . The maximum value of hardness (Hmax) is 27.8 GPa, the minimum value of hardness (Hmin) is 24.9 GPa, and Hmax/Hmin = 1.12. Zhao et al. [] used nanoindentation technology to measure the hardness of Al3BC and reported a value of 24±1 GPa, which is consistent with the hardness calculated in this study.Anisotropy is very important in the study of mechanical properties of materials. According to the above research, 3D characterization of E, B and G, as shown in . From it, Al3BC is anisotropic because the shapes of E, B, and G are all deviated from the sphere. The 2D projections of the E, B and G on the xoy, xoz and yoz planes are demonstrated in . For Al3BC, the maximum value of E (Emax) is 322.7 GPa, the minimum value of E (Emin) is 288.5 GPa, and Emax/Emin = 1.12. The Bmax is 160.2 GPa, the Bmin is 139.2 GPa, and the Bmax/Bmin is 1.15. The Gmax is 141 GPa, the Gmin is 134.9 GPa, and the Gmax/Gmin is 1.05.In addition, anisotropy can also be expressed as the anisotropy index AU [When AU is zero, a crystal is completely isotropic, and when AU is positive, a crystal is anisotropic. AS seen from , the value of AU is 0.0087 for Al3BC, and this indicates that there is a small amount of anisotropy in the elastic property. Compression anisotropy is expressed as a percentage index AC. The shear elastic anisotropy is expressed as a percentage index AS []. The equations for these indices are as follows:. For Al3BC, the value of AC (0.13%) is relatively large, and this indicates that its compression anisotropy is strong. However, value of AS (0.05%) is relatively small, and this indicates that its shear anisotropy is weak.The shear anisotropy factor is determined from the number of independent shear elastic tensors. The number of shear anisotropy factors is equal to the number of independent shear elasticity tensors. Al3BC has three shear anisotropic factors (A1, A2 and A3). A1 is the shear anisotropy factor of the (100) shear planes between the [010] and [011] directions, A2 is the shear anisotropy factor of the (010) shear planes between the [001] and [101] directions, and A3 is the shear anisotropy factor of the (001) shear planes between the [010] and [110] directions. A1, A2 and A3 can be calculated from the following equations [. When the deviation of the shear anisotropy factor value from 1 is greater, the anisotropy of a crystal is more obvious. A3 (1.67) has the greatest deviation from 1, and so, the (001) shear plane of Al3BC has the greatest elastic anisotropy.The ratio kc/ka can also be used to study the elastic anisotropy, where ka is the compression anisotropy along the a-axis, and kc is the compression anisotropy along the c-axis [) is greater than 1, and this means that the compressibility along the c-direction is greater than the compressibility along the a-direction.It is well known that the electronic properties of crystals are closely related to their electronic structures. The band structure of Al3BC along the high-symmetry direction was calculated to understand the electronic properties of Al3BC. Band structure is an effective means of studying electronic properties, and it can be used to demonstrate the properties of electron transport in crystals [ shows the band structure of Al3BC. The dotted red line at the value of 0 is the Fermi level (Ef). For Al3BC, the conduction band is above the Fermi level, and the valence band is below the Fermi level. It is clear from the band structure diagram that there is a band gap of 0.482 eV, and this indicates that Al3BC is a semiconductor material.The density of states reflects the aggregation of electrons and can be used as an indication of the strength of bonds formed between atoms []. Therefore, to better understand the stability of the structure and the properties of the electron, the total density of states (TDOS) and partial density of states (PDOS) for the 3s and 3p orbitals of Al, the 2s and 2p orbitals of B, and the 2s and 2p orbitals of C in the Al3BC were calculated (). The TDOS of Al3BC is roughly distributed in the region of −15–10eV. The red vertical dotted line at the value of 0 is the Fermi level. Al3BC has TDOS value at the Fermi level, and this indicates that Al3BC has certain metallicity. This is consistent with the conclusion that Al3BC is a semiconductor material, according to the band structure analysis. The orbital hybridization of the nonmetallic atoms B and C was the main contribution to the TDOS of Al3BC below the Fermi level, whereas the orbital hybridization of metallic atoms and nonmetallic atoms was the main contribution to the TDOS above the Fermi level. Below the Fermi level, there are several distinct peaks in the TDOS. Peaks from −15eV to −10.5eV were mainly derived from C-2s and Al-3p orbital hybridization. In the −7.5~0eV region, there was also significant hybridization between B-2p, C-2p, and Al-3p, and the B-2p and C-2p orbitals were the main contribution to the TDOS. Above the Fermi level, the hybridization of Al-3s, Al-3p, B-2p and C-2p orbitals are the main contribution to the TDOS of Al3BC, and Al-3s and Al-3p are dominant.An electron density diagram can be used to describe the distribution of charge. shows the electron density diagram and electron density difference diagram of Al3BC. The red region indicates the highest charge density, and the blue region indicates the lowest. As shown in (a), the charge density around the atom C is the highest, followed by that around the atom B, and the charge density around the atom Al is the lowest. The absence of charge accumulation around the metal atom Al proves to be an obvious charge depletion phenomenon, and the interaction between adjacent metal atoms is metallic bonding. There was a large charge accumulation around the C atom, and this corresponds to the highly charged red area around the C atom. The charge density difference diagram intuitively reflects the transfer and redistribution of charge [(b), there was a large amount of charge distributed between the C and Al atoms, and the charge distribution has certain directivity, indicating that the nonmetallic atom C and metallic atom Al form a covalent bond. There was an obviously more regional charge accumulation between the B and Al atoms, and this indicates that the two atoms form an ionic bond. Differences in bond strength can be analyzed synthetically using charge distribution and atomic spacing. For this purpose, the atomic distance between each atom was also measured. It was found from the measurements that the atomic distance between the B and Al atoms is greater than that between the C and Al atoms; in contrast, the charge density between the B and Al atoms is much smaller than that between the C atom and Al atoms. Thus, the strength of the Al–C band was greater than that of the Al–B band.Mulliken population analysis is a method that using the linear combinations of atomic orbitals (LCAO) basis set to determine properties such as atomic charge, bonding, and charge transfer. The technique described by Sanchez-Portal et al. [] was used to carry out population analysis with the projection of plane wave states onto a localized basis. Population analysis of the resulting projected states was subsequently performed using the Mulliken formalism []. Mulliken population analysis results for nonmagnetic materials include Mulliken charge, Hirshfeld charge, bond length, and Mulliken bond number.The overlap population provides an objective standard for bonding between atoms and can also be used to predict the covalent or ionic properties of bonds. A high value for the bond population indicates a covalent bond, whereas a low value indicates an ionic interaction. A positive population overlap indicates bonding, and negative population overlap indicates antibonding []. Further measurements of ionic properties can be made from the effective ionic valence, and are defined as the difference between the formal ionic charge and the Mulliken charge on the anion species. A Mulliken charge of 0 indicates a pure ionic bond, and for a value greater than 0, a higher value indicates higher that the bond has higher covalent character. The calculated effective valence and Mulliken charge are given in . Mulliken bond number, bond length, and bond overlap population of Al3BC are listed in , the chemical bonds of Al3BC are mainly covalent bonds. From largest to smallest, the covalency of these four bonds follow the order: Al2–C > Al1–B > Al1–C > Al1–Al2. The ionicity of the bonds is reflected by the bond overlap populations. The ionicity of the bond can be calculated from the equation: fh=1−exp[−|pc−pu|/pu] [], where pu refers to the overlap population of a u-type bond and pc refers to the overlap population of a bond in a pure covalent crystal. (In this calculation, we assume that pc is equal to 2 for a pure covalent bond.) A value of fh = 0 indicates a pure covalent bond, whereas fh = 1 indicates a pure ionic bond. The calculated fh values of Al1–C, Al2–C, Al1–B, and Al1–Al2 were 0.9991, 0.0253, 0.2212, and ≈1, respectively. Then, from largest to smallest, the order of ionicity level for the four bonds is: Al1–Al2>Al1–C > Al1–B > Al2–C. The metallicity of the bond can be calculated using the equation fm=pu′/pu [], where pu′ is the metallic population. The calculated fm values of Al1–C, Al2–C, and Al1–B were 0.3256, 0.0417, and 0.0509, respectively. Then, from largest to smallest, the order of metallic level of the three bonds is: Al1–C > Al1–B > Al2–C.A phonon dispersion curve can be used to analyze the dynamic stability of Al3BC. The phonon dispersion curve of Al3BC along the high symmetric directions (G-A-H-K-G-M-L-H) is shown in ], the unit cell of Al3BC has 10 atoms and produces 30 phonon branches, including 3 optical modes and 27 acoustic modes. The absence of an imaginary phonon frequency indicates the dynamic stability of Al3BC in the ground state. As seen from the phonon dispersion curves, Al3BC has two imaginary phonon frequencies, and this indicates that its dynamic instability. As seen in the phonon spectrum, there is no phonon gap between phonon modes and optical modes, and this is related to the mass difference between each atom in Al3BC. Some optical and phonon modes are very close in frequency, and this means that the energy between these modes is easily transferred []. The total and partial density graphs of phonon states (PHDOS) are shown at the right side of the phonon dispersion curves. The PHDOS can be divided into two regions. In the middle and low frequency region (−4–12.5 THz), the total density of states of phonons was mainly contributed by Al atoms. Because the B and C atoms are lighter than Al atoms, the high frequency region (>12.5 THz) was mainly affected by movements of the B and C atoms.Thermodynamic properties are important properties of crystals. Thermodynamic properties include free energy F(T), enthalpy H(T), entropy S(T), and heat capacity CV(T). shows the thermodynamic properties of Al3BC calculated using a quasi-harmonic approximation at zero pressure [(a), at a temperature of 100K, the free energy, enthalpy, and entropy are close to 0 because the internal energy (U) of Al3BC is small and the particles are relatively orderly. However, when the temperature reaches 100K, the distance between adjacent particles in Al3BC gradually increases with an increase in temperature; that is, the crystal undergoes thermal expansion at this point, and the electrostatic force between particles becomes smaller, the disorder degree of particles increases, and the entropy increases accordingly. Therefore, the product of T × S increases with an increase in temperature T. Also, the thermal motion of the particle in Al3BC is continuously increasing, the kinetic energy of the particle is continuously increasing, and the internal energy of the crystal increases. The definition of enthalpy is: HU + PV (where P is the pressure of the system, and V is the volume.). At zero pressure, PV = 0, which means that as the temperature increases, enthalpy increases. Both enthalpy and entropy increase, but the increase of T × S is faster than that of enthalpy. Free energy can be expressed as FH-T × S, and thus, free energy decreases with an increase in temperature. Between 0 and 400K, the heat capacity (CV) originates from the superposition of the crystal lattice vibration energy and the electron thermal motion energy. Thus, the heat capacity (CV) increases sharply with an increase in temperature ((b)). However, when the temperature reaches 400K and continues to rise, the thermal motion energy of the electron can be ignored. Only the crystal lattice vibration energy contributes to the heat capacity (CV); this increases slowly and finally approaches a limiting value.In this paper, Al-12 wt%B4C composites were manufactured by the stir-casting process, the first-principles method was used to study the crystal structure, mechanical properties, electronic properties, lattice dynamics, and thermodynamic properties of Al3BC. On the basis of the results, the following conclusions can be drawn:For the B4C–Al composites, a large amount of Al3BC were generated at the interface without Ti addition, and a certain amount of Al3BC were also produced at the interface attached to the B4C particles with Ti addition during the stir-casting process.From calculating the formation enthalpy, it was found that the thermal stability of Al3BC was better, but the phonon dispersion shows that the dynamics stability of Al3BC was lower.For the mechanical properties of Al3BC, the calculated value of bulk modulus, shear modulus and Young's modulus were all large, indicated that Al3BC had strong resistance to volume deformation, shear failure, and unlikely to undergo elastic deformation.The calculation results for Cauchy pressure, Poisson's ratio, B/G value, and Vickers hardness (H) show that Al3BC was a brittle material with high hardness and that the main failure mode was brittle failure. The compressibility of Al3BC in the c-axis direction was greater than the compressibility in the a-axis direction. Also, Al3BC had a slight anisotropy, and its (100) shear plane had the largest elastic anisotropy.From the analysis of band structure, densities of states, charge density, charge density difference, and Mulliken population, it was found that Al3BC was conductive and contains covalent bonds, metal bonds, and ionic bonds. Of these, Al–C covalent bonds were the dominant bonding, and there was no B–C covalent bond formation.Above 100K, the enthalpy and entropy increased, and the free energy decreased. After the temperature reached 400K, the heat capacity (CV) increased slowly and eventually approached a limiting value.Qiyao Hu: Conceptualization, Investigation, Writing-Reviewing and Editing, Funding acquisition. Wenbo Guo: Software, First-principles calculations, Validation, Formal analysis. Peng Xiao: Microstructure analysis, Data curation, Resources. Junping Yao: Methodology, Visualization, Supervision, Project administration.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Nitride coatings improve Ti-6Al-4V alloy behavior in creep testsThis study aimed to evaluate the effect of plasma-assisted PVD TiN and TiAlN/TiAlCrN coatings on the Ti-6Al-4V alloy under creep conditions at 600 °C. Microstructural and surface coating analysis were investigated using XRD, optical and scanning microscopy and isothermal oxidation. The results showed that the TiN coated sample showed the lowest secondary creep rate values for stresses greater than 222 MPa. Below 222 MPa, the TiAlN/TiAlCrN coated sample presented a secondary creep rate value higher than the TiN coated sample. This behavior was due to TiAlN/TiAlCrN oxidation resistance that, according to isothermal oxidation tests, did not suffer oxidation at 600 °C. The stress exponent analysis indicated that the main creep mechanism was controlled by climbing dislocations. The fracture mode is similar for all conditions studied and displayed transgranular fracture with decohesion of intergranular regions. Finally, the coated samples contribute to improved oxidation resistance, reduced damage tolerance parameter (λ) and increased lifetime of Ti-6Al-4V alloy.Ti-6Al-4V was the first alloy developed in the US in 1954. This alloy consumes 80% of titanium world production compared with other titanium alloys. Titanium and titanium alloys are applied in various industrial sectors including aircraft manufacturing, which presents the greatest demand However, titanium and its alloys have difficulties in many applications, which require good tribological properties such as abrasion and wear resistance and a low friction coefficient. Another problem is titanium chemical reactivity, which is temperature dependent. Interaction with other substances (such as hydrogen, oxygen and nitrogen) occurs more readily at high temperatures. Superficial treatments can be a solution because they improve the mechanical properties of titanium through a ceramic compost deposition without modifying the bulk microstructure The ceramic coatings or thermochemical treatment application had good results in fatigue, wear and creep tests Thus, this paper aims to study Ti-6Al-4V alloy creep behavior, at 600 °C, subjected to TiN and TiAlN/TiAlCrN coatings, deposited by plasma-assisted PVD. The study correlates steady-state creep rates, time to fracture, and isothermal oxidation resistance. Microscopy and DRX analyses were applied to confirm the numerical parameters tendency. Moreover, fracture mode and damage tolerance evaluation completed this study to present a whole examination of Ti-6Al-4V mechanical properties at high temperatures, with and without protective nitride coatings.Ti-6Al-4V alloy with Widmansttäten morphology and the chemical composition is given in TiN and TiAlN/TiAlCrN based coatings (coated condition) were deposited on a total of 10 (ten) specimens. The coatings were deposited by cathodic arc plasma assisted physical vapor deposition (PVD). The chamber has six cathodes, or targets, which act as a highly-emitting area and present negative voltage in relation to the chamber wall. Targets released the cathode material in the nitrogen plasma cloud, where the reaction took place, generating nitrides that were deposited on the substrate surface. summarizes the deposition conditions of both coatings The coatings and crept samples' fractures were observed by backscatter imaging on a JEOL 1450-VP scanning electron microscopy (SEM). Uncoated and coated cylindrical samples (d=6 mm, l=2 mm) were subjected to isothermal oxidation tests in the air at 600 °C, between 23 and 168 h. The samples were weighed on an electronic balance before and after the oxidation process. The oxides formed on the surface were identified by XRD. The X-ray diffraction (XRD) analyses were performed in an Empyrean model Panalytical device. The tests were carried out with Cu Kα radiation using a nickel filter, voltage of 40 kV, current of 30 mA, 2θ varying from 20° to 90° in steps of 0.02° and a time span of 10 s. Optical profilometry was used to determine the average roughness (Ra) of the treated samples' surfaces; measurements were taken in an area of 1.27×0.95 mm2 and with 10× magnification.Creep tests were carried out using cylindrical creep specimens with 39 mm gauge length and 6 mm diameter (machined according to ASTM139/95), on an Instron Model M3 Creep and Stress Rupture Tester with 30 kN capacity under constant load in the range from 190 to 300 MPa at 600 °C. The temperature was monitored using a thermocouple placed at the center of the specimen. The strain was measured by a high temperature extensometer. shows Ti-6Al-4V alloy with Widmanstätten microstructure and coarse grains. This alloy has a dual phase microstructure composed of α phase (lighter) and β phase (darker). This configuration is possible due to aluminum presence, which stabilizes the α phase, and vanadium, which stabilizes the β phase.The Widmanstätten morphology type is important as it prevents the dislocation slip, which needs to change direction to keep moving from one region to the other. Grain size was determined using an intercept method. The average grain size corresponds to 1.25±0.05 mm, whereas the lamellae thickness was 11 µm. The average roughness (Ra) of uncoated Ti-6Al-4V is equal to 1.03 µm. This result is according to prior work that studied the creep behavior of Ti-6Al-4V with Widmanstätten microstructure show TiN and TiAlN/TiAlCrN coated Ti-6Al-4V surfaces and cross-sections, respectively.b shows the cross section of a typical multilayer coating formed by layers based on TiAlCrN (lighter) and TiAlN (darker). A multilayer coating aims to ensure greater protection to the substrate against wear, abrasion, and corrosion. The aluminum and chromium presence added to overlapping layers increase oxidation resistance and mechanical properties, besides decreasing the layer defects and making its microstructure denser and more compact a shows β Ti right below TiN layer (without stabilizing α Ti). b displays a layer of about 9 µm, called α case, below the TiAlN/TiAlCrN layer. This α-phase layer was most likely formed by aluminum plus nitrogen action present in the TiAlN/TiAlCrN composition layer that stabilized α Ti. The coating's surface () presented defects such as micro-cracks, porosity, inclusions, grain boundaries, and some spallation areas. The average roughness values increased with PVD treatment. These defects in conjunction with increased roughness may contribute to failure process. shows the isothermal oxidation curves of uncoated and coated Ti-6Al-4V alloy at 600 °C. Mass measurements were made at times ranging from 23 to 168 h.The oxidation rate (angular coefficients) was calculated using linear regression of experimental data of the studied materials. At uncoated condition, this value was 0.13 mg cm−2 |
h−1, and the TiN coated condition oxidation rate was equal to 0.14 mg cm−2 |
h−1. Despite having the same oxidation rate at 600 °C, the TiN coated condition presented a smaller oxide amount per cm2 of the sample. The TiAlN/TiAlCrN coated condition did not show evidence of oxidation at 600 °C. show XRD analysis of uncoated, TiN and TiAlN/TiAlCrN coated Ti-6Al-4V alloys after isothermal oxidation. For each experimental condition, analyses at 23 and at 168 h were made., at 600 °C, uncoated Ti-6Al-4V alloy is oxidized to TiO2. According to Guleryuz and Cimenoglu The XRD patterns of the TiN coated condition () show TiO2 and an oxynitride formation, with TiNyOx formula. Polyakova and Hubert shows that the TiAlN/TiAlCrN coated Ti-6Al-4V alloy is resistant to oxidation at 600 °C for periods up to 168 h (as the mass gain measurements revealed). The overlapping of the diffraction patterns (a) shows peaks with the same intensity and at the same position. Zhu et al. The TiO2 based layer formed on Ti-6Al-4V alloy surface has a protective characteristic and should prevent titanium diffusion at metal/environment interface. However, this layer is composed of more than one compound (TiO, TiO2, Al2O3, V2O5, etc.) resulting in a heterogeneous compound. This compositional difference generates internal stress, causing cracks, spalling, and substrate exposure to the environment To complement the oxidation study, the samples were analyzed via scanning electron microscopy. shows the cross-section images of the samples tested at 600 °C under isothermal oxidation.a displays the formation of an oxidized layer, 1.3 µm of thickness, TiO2 based. This value is according to Kumar et al. that studied the oxidation behavior of pure titanium at 650 °C, for 48 h. The author reported the layer thickness values ranging from 3.5 to 19 µm b and c show the TiN oxidation behavior. At t=23 h (b), the coating was partially modified (oxidized) and still presented a compact microstructure. At t=168 h (c), the coating was completely oxidized and had a porous appearance. The oxide layer thickness developed from TiN coating were 3.2 µm (gray layer) at t=23 h, and 6.2 µm at t=168 h. TiAlN/TiAlCrN coated sample remained stable and the thickness presented a value equal to the initial (6 µm). shows a representative curve of deformation (ε) as a function of time (t) obtained at 600 °C for the material in uncoated, TiN and TiAlN/TiAlCrN coated conditions, at 222 and 300 MPa. shows the main experimental creep parameters obtained at 600 °C, based on the ε×t curves in uncoated, TiN, and TiAlN/TiAlCrN coated conditions.Titanium alloys creep life can be prolonged (ranging from 20,000–120,000 h) and, for this reason, accelerated tests give an idea of the alloy's creep behavior in a short period of time The curves obtained for all experimental conditions present the primary, secondary and tertiary stages defined. Based on , an increasing behavior of instantaneous deformation with the initially applied stress was observed. However, coated samples exhibit ε0 values ranging between 6% and 42% lower than uncoated condition due to higher elastic stiffness induced by PVD treatment. shows the dependence of the steady-state creep rate with applied stress at uncoated, TiN, and TiAlN/TiAlCrN coated Ti-6Al-4V alloy. analyses show that the steady-state creep rate (ε̇s) increases with the applied stress for all conditions studied. The steady-state creep rate is a primary parameter in the creep resistance analyses. Increasing ε̇s values means lower creep resistance (or mechanical resistance). The percentage difference between the steady-state creep rate values at uncoated and coated conditions was 69% lower than TiN coated condition, and 56% lower than TiAlN/TiAlCrN coated condition. However, the TiAlN/TiAlCrN coated condition at 222 MPa showed a steady-state creep rate value 70% lower than at uncoated condition. Therefore, the steady-state creep rate (consequently, the deformation process) decreased due to the anti-oxidation effect that the ceramic material on surface provides. The PVD coated samples have a ceramic layer that is able to protect the metallic bulk from creep damage. also gives an estimate of the alloy stress exponent n (the slope is n) at uncoated and coated conditions, exposed to creep at 600 °C. The stress exponents n=4.83 at uncoated condition, n=5.63 and 3.58 at TiN coated condition, and n=6.86 and 4.69 at TiAlN/TiAlCrN coated condition are appropriate for pure metals and hardened solid solution alloys The stress exponent, n, can indicate the main deformation mechanism that controlled the creep process. Zhu et al. In general, it is observed that the n value change is related to deformation mechanisms alterations. The regions, indicated as A and B in coated conditions (), presented n ranging from: 3.58 to 5.3 for TiN coated condition and 4.69 to 6.86 for TiAlN/TiAlCrN coated condition. However, these values are in the same range where the dominant mechanism is the non-conservative dislocations movement displays the dependence of time to fracture on applied stress. The Eq. relates these parameters using a power law Time to fracture decreased with increasing stress for all studied conditions. The coated samples increased 2.9 times in TiN coated condition and 2.1 times in TiAlN/TiAlCrN coated condition, when related to time to fracture of uncoated condition. The improvement in steady-state creep rate and in time to fracture is attributed to the ceramic layer presence, which is oxidation resistant. correlate time to fracture and the applied stress and can be used for data extrapolation to predict the creep lifetime at a given temperature Samples ductility (εf) decreased with as applied stress increased. This behavior is related to surface degradation resulting from environment action that induces superficial crack nucleation.It was observed that TiAlN/TiAlCrN coated samples showed better behavior than the TiN coated samples at low levels of stress. This fact can be explained based on the oxidation resistance of each coating. During the oxidation tests in the present work, it was found that TiAlN/TiAlCrN coated samples were oxidation resistant at 600 °C. When the stress decreased, the coatings mechanical strength became a secondary parameter and the oxidation resistance controlled the alloy's creep resistance. Thus, at 190 MPa, TiAlN/TiAlCrN coated samples presented lower steady-state creep rate and longer lifetime. The Ra parameter is important because higher superficial roughness increases stress concentrators on the surface. The increase in stress concentration promotes crack nucleation and consequently decreases lifetime. Although PVD treatment increased superficial roughness of Ti-6Al-4V alloy, in this case, oxidation resistance was a more determinant parameter. The ceramic materials present in the coated condition are more resistant to oxidation at 600 °C than uncoated Ti-6Al-4V (as seen in isothermal oxidation tests). Another analysis that was done was to compare Ra values of TiN and TiAlN/TiAlCrN coated samples. The average roughness of TiN coated condition is lower than TiAlN/TiAlCrN coated one. This difference explains why TiN coated Ti-6Al-4V alloy was more resistant to creep at stresses higher than 222 MPa, even though TiN layer was thinner and less stable at 600 °C. The lower Ra value of TiN coated condition guaranteed higher mechanical strength at higher stresses. show samples at uncoated, TiN and TiAlN/TiAlCrN coated conditions, crept between 190 and 300 MPa. show that the fracture mode is similar for all conditions studied. This fracture mode is a transgranular type (microcavities nucleation and coalescence of dimples structure) with decohesion of intergranular regions independent of deformation level. The dimple structure formation is related to slipping systems of titanium crystal structure. In the α – Ti phase, which presents HCP crystal structure, the slipping systems are related to the basal (0001), prismatic {101̅0}, and pyramidal {101̅1} planes and to the directions family 〈112̅0〉 present intergranular decohesion that produce intergranular fracture initiated from slipping grain boundaries and/or intergranular diffusion. The coating aspect after deformation is also shown in . The images confirm the hypothesis discussed regarding TiAlN/TiAlCrN coating mechanical resistance at different stress levels. At 300 MPa (), there is coating delamination, suggesting poor mechanical strength. At 190 MPa (), there is no coating delamination from the metal sample, suggesting better mechanical behavior.The Monkman-Grant relation correlates steady-state creep rate and time to fracture (MG) expressed by Eq. where k and CMG are constants dependent on material, applied stress and temperature. This relationship relates the secondary creep stage contribution to the total creep strain and has been used for lifetime prediction based on short-term tests. The literature reported k≈1 for a different variety of materials, demonstrating that time to fracture is inversely proportional to steady-state creep rate associates steady-state creep rate and time to fracture, at uncoated, TiN, and TiAlN/TiAlCrN coated conditions.) is normalized by the corresponding strain at fracture and has been used by various authors for different materials in order to obtain better experimental data fitting. shows the k, CMG, p and CMMG parameters obtained from Eqs. in this present work. The experimental data adjustments, based on Eq. , were better fitted than data adjusted based on Eq. , because p≈1. The CMMG values (between 0.088 and 0.69) are of the same magnitude as the values reported by Roy These results allow an estimate based on the damage tolerance (λ) defined in this work by Eq. The parameter λ is important for evaluating the material susceptibility to localized crack formation and ability to accumulate damage along the tertiary creep stage. In general, different damage mechanisms can contribute to the material degradation under constant load and high temperature conditions. Among these are microcavity nucleations along grain boundaries arising from sliding or diffusion processes; necking; coalescence and/or partition of particulate hardened solution materials; dislocation structure change with dimples formation. Such mechanisms decrease the alloy's creep resistance at elevated temperatures. In addition, the interaction with the oxygen and the oxidation products formation can reduce the effective area support acting as a damage mechanism, carrying out to a catastrophic failure The λ values range from 1 to 20 for alloys applied at elevated temperatures. When 1<λ<2.5, it means that the material is inclined to w or r cavitation type formation followed by low εf values. The combination of these effects is undesirable and dangerous due to catastrophic failure imminence shows the λ values obtained for each condition studied.In the uncoated condition, λ>10 was obtained, suggesting that the microstructure has undergone degradation resulting from recovery processes with high dislocation activity and plastic deformation (εf) above 20%. The area reduction ranging from 20% (300 MPa) to 44% (222 MPa) led to a ductile fracture mode with a well-developed tertiary creep stage. However, the alloy affinity with oxygen reduced effective area support, in turn reducing the lifetime.The most critical cases were detected for the TiN and TiAlN/TiAlCrN coated conditions with λ<2.5. These values agreed with studies reported by Singh et al. In this present work, TiN coated condition showed higher damage tolerance than TiAlN/TiAlCrN coated condition, between 222 and 300 MPa. The damage tolerance calculated at 190 MPa for the TiAlN/TiAlCrN coated condition was better than the TiN coated condition, with 5% reduction in ε̇s and 8% increase in lifetime. This behavior suggests that better oxidation resistance of TiAlN/TiAlCrN coating contributed to decrease stress concentrators on the surface, allowing εf values above 20% and area reduction from 15.4% (300 MPa) to 20.5% (190 MPa).The low λ values indicate that coatings deposition on Ti-6Al-4V alloy was critical in reducing plastic deformation capacity related to cavitation phenomenon susceptibility. The εf values range between 0.06 and 0.24 for TiAlN/TiAlCrN coated condition and between 0.07 and 0.18 for TiN coated condition. The ductility loss was from 2 to 4 times that of the uncoated alloy. The area reduction ranged between 14.8 (300 MPa)<Ø<20.8% (190 MPa). Thus, reaching the tertiary creep stage, PVD coated samples showed inability to accumulate damage.The behavior of Ti-6Al-4V superficially treated in creep conditions was studied. The main conclusions are in following:PVD treatment increased the average roughness of the Ti-6Al-4V alloy;The TiN coated alloy presented the same oxidation rate of the uncoated Ti-6Al-4V alloy at 600 °C;The TiAlN/TiAlCrN coated alloy did not show any visible evidence of oxidation at 600 °C;The oxidized layer formed on uncoated alloy is TiO2, while the oxidized layer formed on TiN coated alloy is a mixture of TiO2 and an oxynitride TiNyOx;The TiN coated alloy showed the lowest steady-state creep rate at stress levels higher than 222 MPa;At higher levels of stress, the brittleness and roughness of the film are the most important characteristics, thus the TiN coated alloy appeared to be more creep resistant. As the stress decreased (σ<222 MPa), the oxidation resistance became more important, thus the TiAlN/TiAlCrN coated alloy was more resistant to creep deformation;Damage tolerance values (λ) contributed to indicate that TiN and TiAlN/TiAlCrN coatings reduced plastic deformation and showed inability to accumulate damage.Simulations of thermomechanical behavior of composite refractory liningsThe wall of refractorized vessels are composites, made of metallic casing often containing tubes and a refractory material for the protection against the high temperature environment. The objective of the present paper is to model a two layer shell with a thermomechanical behavior equivalent to those of the 3D lining and that render possible the finite element analysis of the complete vessel. A smeared crack model is used for the damage analysis of the refractory material. The equivalent shell is made up of an exterior orthotropic layer and an interior isotropic damageable layer. The set of thermal and mechanical parameters of the equivalent shell is obtained by an inverse method in conjunction with finite element analyses of the 3D panel subjected to an appropriate set of loadings. Some validation analyses show that the identified parameters lead to shell behavior, which is in good agreement with those of the 3D wall. In the case of a simplified cyclone, it is shown that the equivalent shell permits to compute the thermomechanical behavior of a complete refractorized vessel and especially to follow the damaged zones of the refractory.The use of substances at high temperature requires the utilization of refractory linings in order to protect the metal structure of the apparatus. Notably in the steel industry, metal in fusion subjects the hot part of the wall at 1600°C, and the exterior part of the lining has to be at a temperature less than 350°C The geometry of the lining made of metallic tubes and of refractory (). The objective of this paper is to propose a method for the definition of shells with a thermomechanical behavior, which are equivalent to 3D composite refractory lining and that will render possibly the finite element computation of a complete refractorized structure. The shell assumption is justified by the small thickness of the refractory lining in comparison with the other dimensions of the structure. The shell elements under consideration are composed of two layers. The exterior layer is orthotropic and elastic; the interior layer is isotropic and damageable. The thermal and mechanical parameters of the equivalent shell are obtained by an inverse approach using 3D simulations of a representative refractory lining element subjected to elementary loadings. Some validation examples are then presented that show that, in case of different loadings, there is a good agreement between the 3D analyses and the shell simulations. The computation of a whole refractorized cyclone is then presented. It is discretized with a restricted number of finite elements. The present method is simple and avoids some difficulties related to homogenization techniques In this work, we consider a refractory lining composed of a metallic wall made of circular cross-sectional tubes at the exterior and an interior layer made with a refractory material (). This one is generally cast in order to follow the shape of the casing. The connection of the two components is often made by anchors. A complete connection is assumed and the damage of the refractory due to these anchors is neglected. If it can be important for some anchors' shapes For simplicity, the mechanical behavior of the casing steel is assumed to be elastically linear. The study can be extended to elastoplasticity, but the number of parameters defining the equivalent shell would be somewhat larger. Usually, large plasticity zones in the metal casing are avoided in service life.The mechanical behavior of the refractory is assumed to be quasi-brittle damageable. The model used to describe this mechanical behavior is a ‘smeared crack’ model. The damage, due to micro cracks assumed to be smeared, is translated as a local stiffness loss of the material . The normal to the crack surface is oriented in the principal stress direction. The posterior crack can only appear in directions orthogonal to the first crack. The behavior after cracking follows the Hillerborg theory . The mechanical behavior is defined by the Young's modulus E, the stress of damage beginning σt and the softening curve. It will be assumed to be linear in the present study and will be given by the slope denoted by a. The parameter identification has been achieved in for two refractory materials used in thermal combustors. In the present work, the values obtained in the case of a refractory made of silicon carbide (SiC) will be used. The values obtained through four point bending tests In order to validate the model, some experimental tests have been performed on refractory panels including metal parts and subjected to thermal loadings. Some pull-out tests have also been performed. It has been shown that these tests could be correctly simulated by finite element analyses using the model given above The equivalent shell must have geometrical, thermal and mechanical properties that lead to results as close as possible to those given by the 3D model of the lining (. Due to its small number of degrees of freedom, it can be used to compute a whole refractorized structure. The use of a simplified element within the shell assumption is justified by the ratio between the refractory lining thickness and the complete refractorized structure. The structures under consideration in this study (parts of thermal combustors, steel production reactors) have dimensions of some ten meters whereas the lining thickness is one or some ten centimeters. A two-layer shell will be used. It is appropriate to use a single layer shell, but it is not be possible for this shell to behave closely as that of the 3D model in all cases. In order to increase the number of parameters and also in order to describe in a closer way the physics of the 3D problems, the shell under consideration has two layers with different behaviors. The exterior layer (A) (relatively to hot temperature product containing) will be assumed orthotropic elastic and the interior layer (B) will have an isotropic damageable elastic behavior (. The geometrical parameters are those of the two layers of the shell. There are 10 mechanical parameters: two elastic parameters and two damage parameters for the interior layer (B), six elastic parameters for the orthotropic layer (A). The thermomechanical behavior needs the thermal conductivities (λi), the specific heats (Ci) and the thermal expansions (αi) for both materials. Globally, the equivalent shell definition needs 18 coefficients (. In order to determine these coefficients, a set of elementary loadings is applied to a 3D model of refractory lining. The same calculations made on the shell permit to determine the behavior coefficients by an inverse approach. The coefficients of the shell layers are those minimizing the difference with the 3D analysis.For a set of adequate tests (that will be precised below), a set of observed quantities (loads on the kinematics boundary conditions, displacements, temperatures) is chosen. These quantities are observed for different loadings (prescribed loads and displacements) and denoted ϵs (s taking the value 1 to S). The computed quantities for the 3D model are denoted as F3D(ϵs) and the same quantities obtained by the shell model are denoted as FC(ϵs). The residual vector r is the difference between these quantities. For the observed quantity number i (i taking values 1 to n)where m is the set of p parameters to be identified. The least square methods are efficient and simple enough to implement In order to keep a physical meaning to the identified parameters, it is necessary to prescribe some constraints. These q constraints are written as follows In the present case, the set of coefficients has to be positive. These constraints are included in the error function that becomeswhere ωj/Cj(m) are penalty functions and ωj are positives weights.The penalty functions have an important contribution to E∗(m) only when the set of parameters gets close of the admissible frontier. To start the optimization problem, an initial set of parameters m(0) is chosen within the admissible zone.The minimization of the error function is made by the Levenberg–Marquardt method which is used to solve the non-linear least square problems At iteration k, an increment of the parameters dm(k) is calculated bywhere λ(k) is the positive Levenberg–Marquardt parameter at iteration k and J is the jacobian matrix of E∗f and H are the first and the second derivatives of penalty functions, respectively, in regard to parameterswhere i takes the values 1 to n (number of observed quantities), α and β take the values 1 to p (number of parameters), j takes the values 1 to q (number of constraints).The calculation of the jacobian matrix J needs the finite element results for the set of parameters m and for p sets of parameters where the value of mα is perturbedThe derivation of penalty functions f and H areThe Levenberg–Marquardt parameter λ determines both the direction and the value of the increment dm. λ(k) is set large enough in order to obtain E∗(k+1) less than E∗(k).λ(k) is then decreased when the set of parameters gets closer to the solution.The total number of shell finite element analyses necessary for the identification is given by the product of the number of coefficients to be identified by the number of iterations. This number is important and it has been necessary to render this procedure automatically. This has been done by writing an identification program. As an input data file, it needs the chosen observed variables obtained by the 3D analyses. The initial parameters are a second input data file. These values must be chosen very carefully from previous analyses. They are often responsible for the success or not of the identification procedure. Initial values that are too far from the solution can lead to locally minimums that do not correspond to the sought parameters or to a lack of convergence.From a set of parameters, the program starts the shell analyses. The n observed quantities, solutions of these analyses, are compared to the values given by the 3D analyses. If the parameters are not stabilized, then the shell finite element analyses are performed for the p perturbed parameter sets. A parameter increment dm(k) is obtained by the Levenberg–Marquardt algorithm (. The procedure is stopped when the identified parameters do not change any more.The representative part of the refractorized structure is represented in . Only a quarter of the structure is meshed for symmetry reasons. Its dimensions are 0.81 m×0.81 m and its thickness is 0.1 m. The choice of this representative part is a compromise between a sufficiently small thickness, in order to avoid boundary effects to be too large, relative to the shell solution and a reasonable size of the non-linear 3D analyses. The mesh is made up of 4600 eight node elements (. There are 20,034 degrees of freedom. The mechanical and thermal properties of the steel casing and refractory are given in It could be envisaged to define a sufficient set of analyses to simultaneously identify all the set of parameters, with a single loop of the identification method presented above. It is nevertheless more efficient to separate as much as possible, the parameters that can be identified independently or one after the other. This is the reason why a first set of analyses will be made in order to determine the elastic coefficients of the two layers; a second test set will give the thermal coefficients. From these results, it will be possible to identify the dilatation coefficients from a thermomechanical test. Finally, the damage parameters are calculated for a fourth time.Two specific constraints have been shown by these different stages. The first concerns with the thickness of the two layers of the shell. They are prescribed by the thermal analysis and correspond to the thickness of the tubes, and to the complementary thickness. These thicknesses are used in the mechanical analyses. Another constraint is due to the damage of the refractory (cf. ). After the beginning of the refractory cracking, the slope of the tension curve is mainly due to the metal casing stiffness. This remark allows to much reduction of the solution space, since the mechanical properties of the orthotropic elastic layer will have to be those of the metal casing alone.The elastic properties of the two layers can be identified independently from the other parameters. Accounting for the previous remark, the modulus in the two directions of the orthotropic layer are given by the metal casing alone. The identification of the six other coefficients is made from tension tests in both directions, four point bending tests in both directions and an in-plane shear test.The objective is to determine the Young's modulus and Poisson's ratio. A displacement is prescribed on one face and symmetric conditions are applied on the opposite face (. The observed quantities are the load reaction in the direction of the prescribed displacement, the in-plane displacement of a node perpendicular to the prescribed displacement and the out of plane displacement. In the case of a tension in the tube direction (), the stresses are homogenous in each material (. This tensile test leads to a curvature oriented in the perpendicular direction (. This curvature is due to the difference between the Poisson's ratio of the metal and the refractory.The Von Mises stress field due to a tension in the direction perpendicular to the tubes (. A curvature is induced in the tube direction (. It is due to both the tube geometry and the mechanical property difference. The load displacement curves in the tension directions are given in . The stiffness of the structure is larger in the tube direction than in the perpendicular direction where the tubes are easily deformed.Four point bending tests in tube direction and in the perpendicular direction are performed (. Their principal goal is to introduce the influence of out of plane shear modulus. The analysis is made for prescribed displacements. The observed quantities are the load on the support and the maximum out of plane displacement in both cases i.e. four observed quantities for these two tests.In-plane displacements are prescribed on each edge in order to obtain a pure shear state in the panel plane (. The goal of this test is to introduce the influence of the in-plane shear modulus. The observed quantities are the load on the face in both directions and the out of plane displacement.The set of the five elastic tests presented above are used simultaneously in an inverse identification loop as presented in in order to determine the six elastic parameters. This is possible since 13 quantities are observed in these five tests. These tests have been chosen in order that any coefficient has a major influence on at least one analysis. gives the evolution of error as defined in , with the number of iterations. There is a high rate of decrease in error for the first 12 iterations then the decreasing rate is much less until the convergence criterion is satisfied. The set of coefficients that can be obtained by the method is not always unique. It can be different for different initial values. This is due to the existence of local minimums in the case of a non-linear problem such as the present study. The retained values are those for which the physical significance has appeared to be more realistic. The calculated elastic properties of the orthotropic exterior layer (A) are: and for the isotropic interior layer (B), The test performed is a transient thermal analysis. Two different temperatures are prescribed at the internal face of the refractory (hot face, 850°C) and on the interior surface of the tubes (cooled surface, 350°C). These values are typical of a thermal combustor. The used shell finite elements allow to define the temperature for different positions in the thickness. A large number of points (nine points per layer) is used in order to achieve a good accuracy of the thermal identification. Consequently averages have been done for different positions in the thickness in the 3D analyses because the temperatures are not homogenous in the thickness of the structure (. As it has been mentioned in the previous section, the association of the temperature to points located in the thickness imposes to fix the thickness of the two layers. The thermal point position must coincide exactly with those of the lines of the average determined in the 3D model. The identified thermal properties are valid for heat transfer mainly in the direction of the lining thickness. The anisotropy of the metal casing leads to anisotropic in-plane thermal properties. In the present study, it is assumed that the heat transfer is mainly done in the lining thickness direction that is true for practical applications. The identification process converges very fast. The values of the temperatures along the thickness are compared for 3D analysis and shell analysis in . The identified values for the thermal conductivity and the specific heat of the exterior layer (A) are This stage uses the previously determined values of elastic and thermal properties. The panel is subjected to a uniform temperature in all the structure. Accounting for the different properties of the metal casing and of the refractory, the panel is subjected to bending (. The observed quantities are the three displacement components of the point of the mean plane on the free corner of the panel. The thermal expansion of the exterior layer (A) is and the one of the interior layer (B) is The last two parameters to be identified are the stress of damage beginning and the softening slope coefficient. The analyses performed are tensile tests in the direction of the tubes and in the perpendicular direction. The analyses are presented in but the elastic properties are those previously estimated and the refractory has a behavior including damage such as presented in compares the load–displacement curve in the two directions for the 3D model and for the shell model. The identified stress of damage beginning is The objective of these computations is to show that the identified parameters allow to correctly analyze the mechanical response of the refractory lining subjected to loadings, different from those used for identification. The results obtained by the 3D analysis are compared with the shell computation made independently.In this analysis, the panel is clamped on all its edges. It is subjected to a uniform pressure on the upper face. The comparison between the 3D and shell analyses shows that the difference concerning maximum displacement at the center of the plate and loads on the edges is less than 8%. shows the vertical displacement computed in both cases.In this validation of thermal analysis, a 850°C temperature is prescribed on the internal surface of the refractory for 15 s. The comparison of the temperatures obtained by the two analyses are given The in-plane shear test that has been described in is performed, accounting for the damage of the refractory layer. The loads associated to the shear directions are compared in . The cracked zones are shown for both analyses in In order to show the interest of the simplified shell approach and its possible utilization for the structural analysis of a refractorized structure, the simulation of the behavior of a cyclone under thermomechanical loads is performed. The model used is a very simplified one compared to a real cyclone. The mesh used is regular and made only with quadrangles (. The orthotropic axe orientations are those of the tubes shown in . The cyclone is suspended at its top. A 900°C interior temperature is raised in 50 h. The mesh is made of 6449 nodes for 36118 d.o.f. The damaged zones of the interior and exterior faces of the refractory lining are given in . The final temperature cannot be reached. The cracked zones are too important and a damaged state of the structure leads to a stop in the computation. This is a correct analysis for the cyclone under consideration where the refractory lining is monolithic. In reality, the cyclone involves some expansion joints that permit to avoid such a large damage. Some studies are currently in progress in order to include these joints that can be very numerous in the equivalent shell mechanical behavior.A possible approach for the structural analysis of refractory vessels has been presented. A two-layer shell can describe the composite wall made up of a metallic casing including tubes and a refractory. The properties of the exterior orthotropic layer and of the interior isotropic damageable layer are obtained by an inverse approach. The loadings applied to the 3D panels are appropriate if their solutions account for all the parameters. For an efficient identification of the coefficients, it appears it was necessary to separate as much as possible the computations of the sets of parameters that can be made separately or successively. The determined characteristics lead to finite element analyses in good agreement with the 3D computations and to lower numerical computational time. This allows analyzing complete refractorized vessels.If the global method is given, some improvements are necessary for realistic analyses of industrial structures. The first point appeared in the cyclone computation example. The damage zones that have been computed in service conditions are very large. It is necessary to introduce the effects of expansion joints that reduce the cracking. A study is currently in progress in these directions. It has been assumed in the present work that there is a total connection between the metallic casing and the refractory. In reality, anchors ensure punctual connections. The global properties of the lining are modified if the sliding and friction between the casing and the refractory are to be taken into account. Finally the variations with the temperature in several quantities would be necessary, although the numerical investigations are very difficult.Towards the strength-ductility synergy of Al2O3/Al composite through the design of roughened interfaceMetal matrix composites (MMCs) are widely used for structural application due to their higher specific modulus and strength compared to their corresponding matrix []. The reinforcement-matrix interface is considered as a critical factor that affects the mechanical properties of composites and has been intensively studied []. A strong interface bonding not only contributes to efficient load-transfer from matrix to reinforcements [], but also inhibits the crack propagation []. As it is recognized that interface bonding is affected by three factors: interface chemistry, asperities, and residual stress []. A chemical reaction at the reinforcement-metal interface may lead to a strong bonding between matrix and reinforcement, but a brittle reaction product can be highly detrimental to the performance of the composites []. Surface modification of reinforcements, such as surface treatment [], can resist the brittle reaction and improve the wettability between matrix and reinforcement, and thus maximize the stress transfer. The surface asperities of the reinforcement allows for mechanical interlocking and offers additional means to tailor the mechanics of the interface []. Mechanical interlocking is one of the simple and promising ways to improve interfacial bonding and achieve the strength-ductility synergy of composites.Previous studies have explored the effect of the surface asperities of reinforcements on the properties of MMCs. For example, in the nanoceramic-decorated fiber or carbon nanotubes (CNTs) reinforced metal matrix composites [], such interface morphology produces strong anchoring effect of nanoparticles to resist the sliding of fiber or CNTs under tensile loading. Monocrystalline Al3C4 nanorods that tightly conjoined the FLG platelets with Al matrix ensured an efficient load transfer at the FLG-Al interface []. The Al matrix composites reinforced by graphene nanosheets (GNS) decorated with Ni nanoparticles (Ni NPs@GNS) exhibited an excellent strengthening efficiency while preserving a good ductility []. Despite the proven beneficial effect of these reinforcements decorated with metal or ceramic nanoparticle on the composite's properties, these methods to generate surface roughness provide limited control over the distribution, size and shape of the asperities.Nacre is considered as a typical example which highlights the role of mechanical interlocking at the highly sophisticated interfaces. The breaking of mineral bridges, shearing resistance contributed by nano-asperities in nacre would suppress the deformation between platelets and contribute to the additional energy dissipation []. Recently, inspired by nacre, two-dimensional Al2O3 platelets have been used as building blocks to design the nacre-like polymer [] matrix composites. Unfortunately, the weak interface bonding between Al2O3 and matrix limited the full benefit from the reinforcements. Attempts have been made to replicate the asperities of platelets in nacre-like Al2O3 platelet reinforced composites. Libanori et al. [] have prepared roughened platelets by sintering Al2O3 platelets with silica nanoparticles at 1050 °C. The results showed the highly roughed and aligned platelets increase the toughness and the flexural strength of epoxy by 110% and about 57%. However, the nacre-inspired interfacial interlocking structure is rarely reported in MMCs, and a comprehensive understanding towards the roughed platelet/metal interface is lacking to account for the contribution of the mechanical interlocking to mechanical properties of nacre-like MMCs.In the work, Al2O3 platelets were decorated with SiO2 submicron-spheres by a high-shear pre-dispersion process to prepare the roughened SiO2-decorated Al2O3 (SiO2@Al2O3) platelets. Then the sintering at high temperature of 1050 °C was used to control the roughness on the surface of SiO2@Al2O3 platelets. The mechanical properties of SiO2@Al2O3/Al composites showed an improved strength and ductility compared to Al2O3/Al composite. A roughened interface with asperities was designed to improve the load transfer efficiency between the platelets and Al matrix. The effect of surface roughness on the strengthening and toughening mechanism of the composites were discussed.The Alumina (Al2O3) platelets with diameter of ~8.6 μm and thickness of ~2 μm were supplied by Zhengzhou Sanhe Co., Ltd. (China), as shown in a. The amorphous silica (SiO2) submicro-spheres with the diameters of 300 nm were provided by Andi Metal Materials Co., Ltd. (China), as shown in b. Flake Al powders with the diameter of ~21 μm and thickness of ~400 nm were used as matrix materials, as shown in The SiO2@Al2O3 platelets were fabricated by a high-shear dispersion process, as shown in . The 99.5 vol% Al2O3 platelets and 0.5 vol% SiO2 powders were firstly blended for 2 min at the low-speed of 300 rpm and then processed at the high-speed of 1500 rpm for 10 min in a high-speed mechanical powder processor (Nobilta, Hosokawa Micron Corporation, Japan).Then, the SiO2@Al2O3 platelets were sintered at 1050 °C for 1 and 2 h in air atmosphere in a muffle furnace. SiO2@Al2O3 platelets after different sintering time (1h, 2h) are named as SiO2@Al2O3-1, SiO2@Al2O3-2, respectively.Tensile specimens with a cross-section of 2 × 4 mm2 and a gauge length of 12 mm were machined along the extrusion direction. Three tensile tests for each set of samples were carried out with a Zwick Z100 testing machine equipped with an automatic contacting extensometer (Zwick makroXtens) at a constant strain rate of 5 × 10−4 s−1 at room temperature. Vickers hardness tests were performed using HBRV-187.5 hardness tester from Shanghai LianEr Testing Equipment Co., Ltd, by applying a load of 294.2 N, for a dwelling time of 15 s, and 5 hardness tests were made for each set of samples.For metallographic analysis, the specimen surface was prepared using conventional metallographic techniques and then went through ion polishing (model of ion polisher: Gatan Ilon). XRD was performed on an X-Ray diffractometer (XRD-6100, Shimadzu Co. Led., Japan). The powder and extruded samples for TEM observation were prepared by Ga+ focused ion beam (FIB, GAIA3) and Ar+ ion thinning system (PIPS II). The powders, microstructure and fracture morphologies of composites were characterized using a field emission scanning electron microscopy (FE-SEM, TESCAN, RISE) and a high-resolution transmission electron microscopy (HR-TEM, Talos F200X G2). Crack propagation near fracture surface was observed in an Xradia 520 Versa X-Ray microscopy (XRM, Zeiss) with 0.7 μm voxel size. shows the SEM images of SiO2@Al2O3 platelets before and after sintering. During high-shear dispersion process, a severe but tailorable shear stress was induced to de-agglomerate the SiO2 and coated them onto the surface of Al2O3 platelets. As shown in a, SiO2 submicro-spheres are uniformly distributed on the surface of Al2O3 platelets. The surface coverage (sc) of SiO2 on the surface of Al2O3 platelets is about 2.5% and the calculation equation is shown as follows:where h is the thickness of Al2O3 platelet, VSiO2 is the volume fraction of SiO2 in Al2O3 platelet, dSiO2 is the diameter of SiO2. The sintering at high temperature of 1050 °C is used to ensure the strong interfacial bonding between Al2O3 and SiO2. b, c shows the SEM images of SiO2@Al2O3 powders after different sintering time. As the sintering time increases, SiO2 submicro-spheres begin to fuse and the contact area between Al2O3 and SiO2 increases, resulting in the formation of SiO2-hemispheres. When the sintering time increases to 2h, all of SiO2 submicro-spheres on Al2O3 platelet are transformed into SiO2-hemielliposid morphology, which is also verified by EDS line-analysis in the SEM image. Kleebe et al. [] found a dissolution of Al2O3 into the amorphous SiO2 and the SiO2-hemielliposid was composed of Si(Al)-glass at the low sintering temperature. As shown in , the element mapping shows that SiO2-hemielliposid contains a small fraction of aluminum at the SiO2–Al2O3 interface. The phenomenon suggests a strong diffusion bonding between SiO2 and Al2O3 after sintering. As illustrated in e, as the sintering time increases, the roughness on the surface of SiO2@Al2O3 platelets gradually decreases. Therefore, different roughness on the surface of SiO2@Al2O3 platelets can be obtained via controlling the sintering time. shows the SEM images of Al2O3/Al and SiO2@Al2O3/Al composite. All the composites show a similar structure as the Al2O3/Al composites (a), in which Al2O3 platelets are parallel-aligned along extrusion direction (ED) and uniformly distributed in the matrix. In SiO2@Al2O3/Al composite (b), SiO2 submicro-spheres are observed inside Al matrix and the smooth Al2O3–Al interface is presented. It suggests that the Van der Waals force is not strong enough to ensure the adherence of SiO2 on Al2O3 platelets during composites’ fabrication. In c, d, some asperities are observed at the interface between Al and Al2O3 due to the strong bonding between SiO2 and Al2O3. It is qualitatively explained that the roughness at the interface of SiO2@Al2O3-1/Al composite (c) is higher than that of SiO2@Al2O3-2/Al composite ( shows the bright-field STEM (BF-STEM) and high-resolution TEM (HRTEM) images of Al2O3/Al composites. HRTEM image in b shows that there is a Al2O3 nanolayer (n-Al2O3) between Al matrix and Al2O3. No gap or void is observed in the interfacial regions, indicating a smooth mechanical bonded Al2O3/n-Al2O3/Al interface. In addition to extra Al2O3 platelets, some broken n-Al2O3 are also distributed in the Al matrix. The effect of those broken n-Al2O3 in Al matrix fabricated by Flake PM has been reported in previous study [ show that the BF-STEM, the corresponding element mapping and HRTEM images of SiO2@Al2O3/Al and SiO2@Al2O3-1/Al composites. Similar to the SEM results, the SiO2 submicro-sphere is observed in Al matrix of SiO2@Al2O3/Al composites and distributed at the grain boundaries. In SiO2@Al2O3-1/Al composite (b), the SiO2-hemisphere is adjacent to the Al2O3 and Al matrix and a large number of dislocations are detected in Al matrix near SiO2-hemisphere. These asperities limit the deformation of Al matrix near SiO2-hemisphere during extrusion, resulting in the strain hardening and the formation of dislocations. HRTEM (), two kinds of SiO2-hemiellipsoid characteristics are observed. In addition to the SiO2/n-Al2O3/Al interface in area I (b), HRTEM image and FFT pattern in area II (e) show that SiO2 is directly connected to Al grain. In c, the clean SiO2–Al interface in area II presents a curved shape, resulting in the formation of nano-asperity. Although the n-Al2O3 would inhibit the reaction between Al and SiO2, it is still observed in that parts of SiO2-hemiellipsoid are transformed into γ-Al2O3. These phenomena suggest that a small amount of SiO2 is reacted with Al, and the SiO2 is not completely consumed under the sintering temperature of 560 °C for 2h. The partial interface reaction between SiO2 and Al not only maintains the morphology of asperity, but also contributes to the interfacial bonding at SiO2–Al interface. Based on the above microstructural features, the roughened SiO2@Al2O3–Al interface is formed in the SiO2@Al2O3-1/Al and SiO2@Al2O3-2/Al composites. shows the tensile properties of Al matrix, Al2O3/Al and SiO2@Al2O3/Al composite. summarizes the mechanical properties of Al matrix, Al2O3/Al and SiO2@Al2O3/Al composites. The average yield strength (YS), ultimate tensile strength (UTS) and elongation of Al2O3/Al composite are 169 MPa, 230 MPa and 14.8%. The strength and hardness of three types of SiO2@Al2O3/Al composites increase after the Al2O3 platelets are roughened. As shown in b, if the SiO2@Al2O3 platelets are not sintered, the YS and UTS of SiO2@Al2O3/Al composite only slightly increase. When sintered after 1h, the composite's YS and UTS reach peak values of 188 MPa and 250 MPa along with an increased elongation of 16.2%. As sintering time further increase to 2h, YS, UTS and elongation begin to decrease to 182 MPa, 244 MPa and 15%. It is furthermore noted in that compared with other Al matrix composites reinforced with other particles (SiC, Al2O3, B4C), the as-prepared SiO2@Al2O3-1/Al composite exhibit the best combination of the strength and ductility.In metal matrix composites, strength increment is generally ascribed to a synergy of Hall-Petch strengthening (ΔσH−P), load-transfer strengthening (ΔσL−T), coefficient of thermal expansion (CTE) mismatch strengthening (ΔσCTE), and Orowan looping strengthening (ΔσOrowan). However, the micro-sized reinforcements are always located at the grain boundaries and ΔσOrowan is negligible in the study. Grain reinforcement strengthening part is also negligible for micro-sized reinforcements []. Therefore,ΔσL−T and ΔσCTE are considered as the main strengthening mechanism of the Al2O3/Al composites compared to Al matrix. The yield strength of the composites, σc, in the study can be expressed as:where σm is the yield strength of Al matrix. The strength contribution of CTE can be expressed as [ΔσCTE=1.25Gb(Vr1−Vr12ΔCTE×ΔTb)1/2(1d‾)1/2where G is the shear modulus of the matrix (25.4 GPa), b is the burgers vector of matrix (0.286 nm), Vr is the volume fraction of Al2O3 (10 vol%), ΔCTE is the difference between the matrix and the reinforcement (15.9 × 10−6 K−1), ΔT is the difference in temperature between the hot processing and room temperature (400 K). d‾ is the equivalent diameter of Al2O3 platelets (6 μm). According to equation , the increment of ΔσCTE is 19.6 MPa in Al2O3/Al composites. The effect of SiO2-hemisphere or hemielliposid on ΔσCTE is negligible due to the low volume fraction (0.05 vol%) in the Al matrix. a shows the comparison of the increment to yield strength due to different strengthening mechanism. The ΔσL−T of SiO2@Al2O3/Al composites is higher than that of Al2O3/Al composite. Therefore, enhanced load transfer is the main reason for strength improvement of SiO2@Al2O3/Al composite with a roughened interface.According to the modified shear-lag model under pull-out model for the Al2O3 platelet-reinforced composite [], the effect of load transfer strengthening was closely related with Al2O3/Al interfacial structure and bonding strength. When the Al2O3/Al interfacial bonding strength is strong enough to avoid interface debonding, the maximum load-transfer strengthening effect (ΔσL−TMax) can be estimated by:where S and A are respectively the interfacial area (2l2+2lh) and cross-section (lh) of Al2O3. l, h are the length and thickness of Al2O3, and τm is the shear stress of Al matrix (σm/3). The maximum load transfer effect for the smooth Al2O3–Al interface is predicted as 27 MPa. The load transfer efficiency is defined by the ratio between the Al2O3 load-transfer strength and the maximum load-transfer strength for the smooth Al2O3–Al interface. The load transfer efficiency in Al2O3/Al, SiO2@Al2O3-1/Al, and SiO2@Al2O3-2/Al composites are respectively calculated as 68%, 138%, 116%. At the smooth Al2O3–Al interface, load transfer depends on the interfacial shear stress between reinforcement and matrix. The good interface bonding can improve the interfacial shear stress, resulting in the enhanced load transfer efficiency. If the interfacial shear stress is higher than the shear stress of Al matrix, interface debonding would not occur and the load transfer is maximized at the smooth Al2O3–Al interface. Compared to SiO2@Al2O3-1/Al composite (), the chemical reaction occurs at parts of SiO2–Al interface, resulting in the effective interface bonding in the SiO2@Al2O3-2/Al composite (). But the experimental results show that SiO2@Al2O3-1/Al has larger load transfer efficiency and better strength. It is speculated that the contribution of reaction bonding at SiO2–Al interface to load-transfer strength is limited. The improvement of load-transfer strength is mainly originated from platelet-matrix interlocking rather than chemical bonding. At the roughened interface, the relative sliding between asperities and Al matrix requires the high normal stress to overcome the platelet-matrix interlocking, which is another way to further increase the load transfer efficiency. As the roughness decreases, the relative force between asperities and Al matrix then decreases, and thus the load transfer efficiency also decreases. Therefore, load transfer capacity is enhanced by platelet-matrix interlocking at the roughened SiO2@Al2O3/Al interface., and the mechanical performances of composite can be improved largely.In general, the improvement of strength often comes with the sacrifice of ductility for PRMMCs []. In the study, by designing a roughened SiO2@Al2O3–Al interface, the strength-ductility synergy of the SiO2@Al2O3/Al composite is achieved compared to Al2O3/Al composite. The previous studies and computer simulations [] have reported that the toughening mechanism of platelet reinforced composite is attributed to the extrinsic mechanism, including platelet pull-out, crack deflection, bridging, and crack tip blunting. In order to further explore the underlying mechanism of improved tensile ductility, the fracture morphology was characterized by XRM, shown in . It is observed from sectional view in b, c that microcracks are mainly concentrated in the front of Al2O3 platelets and the microcrack plane is perpendicular to the extrusion direction (ED). It suggests that the cracks first form at the front of Al2O3 platelets, rather than the Al2O3–Al interface parallel to ED. During tensile deformation process, once the stress concentration at crack tip of the front of Al2O3 platelets exceeds the interface bonding strength, the cracks will propagate along the smooth and weak Al2O3–Al interface, resulting in interface debonding in Al2O3/Al composite (a). The rapid crack propagation would lead to the pre-failure of composites, and its ductility is also dramatically reduced. On the contrary, cracks that propagate along the interface would be arrested by the asperities at the SiO2@Al2O3–Al interface. As shown in b, the well-bonded interface between SiO2 and Al would impede the opening of crack, further preventing unstable propagation of cracks. Meanwhile, it is found that many dense dimples exist around the roughened Al2O3 platelet, demonstrating the excellent plastic deformation of Al matrix. The surface roughness contributes to a more efficient dissipation of elastic energy within the material and a more extensive deformation of Al matrix before material failure occurs [, the green 3D rendering contains the main cracks (fracture surface) and microcracks (below the fracture surface). Compared to Al2O3/Al composite, the main cracks in the SiO2@Al2O3/Al composite display a more tortuous path. In f, the microcracks per unit volume below fracture surface in SiO2@Al2O3-1/Al composite is twice than that in Al2O3/Al composite. All of these phenomena indicate that more additional energy can be dissipated during tensile deformation, resulting in the improved ductility in SiO2@Al2O3-1/Al composite. Therefore, the roughened interface design can not only improve the load transfer capacity through platelet-matrix interlocking, but also enhance the ductility of the composites by extrinsic toughening mechanism. Meanwhile, the design strategy is also feasible for Al-alloy matrix. For Mg-containing Al alloys, the formation product (MgAl2O4) among Al–Mg–SiO2 system could be formed []. The particle size and sintering condition need to be adjusted to obtain the optimal roughened interface. On the other hand, for Al alloys containing different alloying elements, different kinds of oxide can be selected to rough the surface and bridge the metal matrix and reinforcements.The roughened SiO2@Al2O3 platelets are fabricated by high-shear dispersion process and subsequent sintering at the high temperature. A strong diffusion bonding between SiO2 and Al2O3 is formed after sintering. Different roughness on the surface of SiO2@Al2O3 platelets are obtained via controlling the sintering time.As the roughness increases, the strength and ductility of SiO2@Al2O3/Al composites are simultaneously improved. The YS and UTS of SiO2@Al2O3-1/Al composite reach peak values of 192 MPa and 250 MPa along with an increased elongation of 16.2%. Compared to the smooth Al2O3–Al interface, the load transfer efficiency is improved by 138% due to platelet-matrix interlocking at the roughened SiO2@Al2O3/Al interface.The improved tensile ductility of SiO2@Al2O3/Al composite is attributed to the extrinsic toughening mechanism. The well-bonded interface between SiO2 and Al would impede the opening of crack, further preventing unstable propagation of cracks. Compared to Al2O3/Al composite, the increased microcracks per unit volume below fracture surface and the more tortuous crack path in SiO2@Al2O3-1/Al composite indicate that more additional energy can be dissipated during tensile deformation.Zhiming Zhang: Conceptualization, Investigation, Data curation, Writing – original draft, Writing – review & editing. Genlian Fan: Investigation, Methodology, Writing – review & editing. Zhanqiu Tan: Investigation, Writing – review & editing. Haitao Zhao: Investigation. Yanjin Xu: Methodology. Dingbang Xiong: Supervision. Zhiqiang Li: Supervision, Project administration, Writing – review & editing, Funding acquisition.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Development and properties of Ti–In binary alloys as dental biomaterialsThe objective of this study is to investigate the effect of alloying element indium on the microstructure, mechanical properties, corrosion behavior and in vitro cytotoxicity of Ti–In binary alloys, with the addition of 1, 5, 10 and 15 at.% indium. The phase constitution was studied by optical microscopic observation and X-ray diffraction measurements. The mechanical properties were characterized by tension and microhardness tests. Potentiodynamic polarization measurements were employed to investigate the corrosion behavior in artificial saliva solutions with and without fluoride. In vitro cytotoxicity was conducted by using L929 and NIH 3T3 mouse fibroblast cell lines, with commercially pure Ti (CP–Ti, ASTM grade 2) as negative control. All of the binary Ti–In alloys investigated in this work were found to have higher strength and microhardness than CP–Ti. Electrochemical results showed that Ti–In alloys exhibited the same order of magnitude of passivation current densities with CP–Ti in artificial saliva solutions. With the presence of NaF, Ti–10In and Ti–15In showed transpassive behavior and lower current densities at high potentials. All experimental Ti–In alloys showed good cytocompatibility, at the same level as CP–Ti. The addition of indium to titanium was effective on increasing the strength and microhardness, without impairing its good corrosion resistance and cytocompatibility.► The addition of In into Ti can increase the mechanical property. ► Ti-In alloys exhibited similar passivation behavior with CP-Ti. ► Ti-In alloys had good cytocompatibility comparable with CP-Ti. ► Ti-10In and Ti-15In showed transpassive baheviour with the addition of NaF.Titanium and titanium alloys have been widely used in biomedical devices due to their low density, excellent corrosion resistance and unique biocompatibility Aiming to improve the above mentioned deficiencies of CP–Ti, several binary titanium alloys, such as Ti–Cu There was also an attempt to add indium into titanium alloy The purpose of this study is to investigate the addition of indium on the microstructure and mechanical properties of experimental Ti–In alloys, including Ti–1In, Ti–5In, Ti–10In and Ti–15In (in atomic percentage) alloys. Electrochemical techniques and in vitro cytotoxicity tests were also employed to assess the corrosion resistance and biocompatibility of Ti–In alloys.Sponge titanium and bulk indium (99.95% and 99.995% in purity, from Beijing Mountain Technical Development Center for Non-Ferrous Metals) were used to fabricate the experimental alloys by arc-melting in argon atmosphere. The ingots of 30 g each were re-melt four times to ensure chemical homogeneity. The experimental samples were cut by electro-discharge machine directly from the as-cast ingots, with 1 mm × 2 mm × 60 mm in size for tensile tests and 10 mm × 10 mm × 1 mm for electrochemical and cytotoxicity tests. As reference specimens, titanium samples with the same size were cut from a sheet of commercially available grade 2 pure titanium.The square piece samples were mechanically polished with SiC papers and chemically etched with a solution containing 2–5 ml HF, 10–15 ml HNO3 and 80–85 deionized water. Optical microstructure photograph of binary Ti–In alloys were obtained by using Zeiss Axiovert 200 MAT microscope. Tensile tests of solution treated samples were conducted on an Instron 3365 universal testing machine with a strain rate of 0.01 min− 1 (3 samples for each test group). Before tensile testing, solution treatments were conducted at 1123 K for 1.8 ks to avoid composition segregation during cast procedure, and then the samples were water quenched to room temperature. Microhardness measurements were carried out on Shimadzu HMV microhardness tester, and five tests were conducted for each cast specimen. X-ray diffraction (XRD) for phase analysis was conducted by using an X'pert Pro diffractometer at 40 kV and 100 mA.A saturated calomel electrode (SCE) reference electrode and a platinum counter electrode were used for the corrosion test. A CHI 650C workstation was used to perform the open circuit potential and potentiodynamic tests. The test temperature was maintained at 37 °C and the scan rate of potentiodynamic tests was 1 mV/s. The composition of electrochemical test solution was as follows: NaCl (0.4 g/L), KCl (0.4 g/L), CaCl2 (0.78 g/L), NaH2PO4·H2O (0.69 g/L), Na2S·9H2O (0.005 g/L), KSCN (0.3 g/L), and Urea (1 g/L). All the chemical reagents were of analytical purity. When NaF (2 g/L) was added, CaCl2 was eliminated to avoid CaF2 precipitation. The final pH of the solution was 5.2. After potentiodynamic tests, corroded surfaces of CP–Ti, Ti–5In and Ti–15In were examined by scanning electron microscope (Mx2600FE, Camscan, UK).MTT colorimetric assay was employed to evaluate the in-vitro cytotoxicity of binary Ti–In alloys. In this method, the optical density of the culture solution, which is measured spectrophotometrically, is considered to represent the number of viable cells. Briefly, cellular enzymes could reduce MTT (3-(4,5-dimethyl-2-thiazolyl)-2, 5-diphenyl-2H-tetrazolium bromide, a yellow tetrazole) to insoluble formazan, which gives a purple color. Therefore the number of viable cells could be obtained by measuring the color of the culture solution. Detailed description of this method could be found in Inductively coupled plasma atomic emission spectrometry (Leeman, Profile ICP-AES) was employed to measure the concentrations of Ti and In ions dissolved into the cell culture medium after 72 h incubation. An average of three measurements was taken for each group.The ion release content and cell viability data were analyzed statistically using one-way ANOVA method at the 95% confidence level (SPSS for Windows 16.0; SPSS, Chicago, IL). shows the representative optical micrograph of as-cast Ti–5In alloy, which was identical with the microstructure of other Ti–In alloys investigated in this study. A typical lamellar casting microstructure was found with clear grain boundaries, indicating a uniform single phase. The XRD patterns of as-cast Ti–In alloys shown in (B) confirmed that the single phase was hexagonal α-Ti phase. All the experimental Ti–In alloy samples were composed entirely of hexagonal α-Ti phase without precipitates or second phase, which indicated that the alloying element indium below 15 at.% did not change the phase constitution of CP–Ti.The tensile test results of solid solution treated CP–Ti and Ti–In alloys are showed in . The yielding strengths (YS) and ultimate tensile strengths (UTS) of Ti–In alloys were significantly higher (p < 0.05) than that of CP–Ti, and the elongations were much higher as well, except for Ti–10In alloy. With the increase of indium, both YS and UTS of Ti–In alloys increased. Compared to CP–Ti (YS 262 MPa, UTS 393 MPa), YS increased to 414 MPa (by 58%), and UTS increased to 550 MPa (by 40%) for Ti–5In alloy; for Ti–15In alloy, YS increased to 539 MPa (by 105%), and UTS increased to 761 MPa (by 93%). presents Vickers hardness and bending modulus of CP–Ti and as-cast Ti–In alloys. As it is shown, the hardness of Ti–In alloys increased significantly with the increase of indium content. All of the Ti–In alloys exhibited higher micro-hardness than CP–Ti. In contrast, the bending modulus, all around 90 GPa, had nothing with indium content statistically. shows the variation of the open circuit potentials (OCP) of the experimental Ti–In alloys with immersion time in two solutions, artificial saliva and artificial saliva with 0.2% NaF addition. As it is shown, the addition of NaF yielded great difference on corrosion potential of the same Ti–In alloy. In the fluoride-free artificial saliva solution, the corrosion potential increased gradually at the early stage of immersion, and then varied slightly with immersion time after 1000 s. All the Ti–In alloys stabilized at more negative potentials than CP–Ti after 7200 s immersion. In the fluoride containing artificial saliva solution, the corrosion potentials decreased significantly in two steps to minima below − 1 V before 2000 s. The first step was a progressive decrease of corrosion potential with time, indicating the progressive damage to the natural protective oxide of the experimental Ti–In alloys. The second step was a sudden drop of corrosion potential, indicating a sudden destruction of the original protective surface. Following this initial activation, the corrosion potential of the Ti–In alloys increased slightly for a while and stabilized after 7200 s immersion. It was noted that Ti–In alloys showed lower corrosion potentials than CP–Ti in fluoride-containing solution, which is consistent with that for the fluoride-free solution.(C) and (D) shows the potentio-dynamic curves of CP–Ti and Ti–In alloys in fluoride-free and fluoride-containing saliva solutions. In the fluoride-free solution, both CP–Ti and Ti–In alloys showed similar polarization behavior. After cathodic polarization, all samples exhibited passivation behavior with almost constant passivation current densities regardless of the increasing potential. The passivation current densities of CP–Ti and Ti–In alloys were nearly identical, which were within the range of 7 and 10 μA/cm2 at 2.5 V, indicating the same order of magnitude of corrosion resistance.In the fluoride-containing solution, the experimental materials showed different passivation behavior compared with fluoride-free solution. They all exhibited clear active-passive transition peaks just above the corrosion potentials. Following this transition, CP–Ti, Ti–1In and Ti–5In alloys showed passivation behavior within a large potential range up to 2.5 V. Passivation current densities in this range varied from 100 to 372 μA/cm2, two orders of magnitude higher than that in the fluoride-free solution. It is interesting to note that Ti–10In and Ti–15In alloys exhibited similar potentio-dynamic curves, which showed secondary active–passive transition peaks at around 0.5 V. Above this potential, the experimental alloys were passivated until 2.5 V, with passivation current densities of 87 μA/cm2 and 92 μA/cm2 for Ti–10In and Ti–15In alloys, respectively. The contrast of passivation behavior between the two solutions suggests the detrimental effect of F− to the corrosion resistance of the Ti–In alloys.Surface morphology of corroded CP–Ti and Ti–In alloys are showed in . As can be seen, CP–Ti, Ti–5In and Ti–15In alloys exhibited different surface characteristics. CP–Ti showed severely corroded morphology, exposing porous structure of substrate with oxide layer totally removed. The other two Ti–In alloys were much less corroded, with partial oxide layer left on the surface. Moreover, these two Ti–In alloys surface differed from each other either. The substrate of Ti–5In alloy was also porous but with finer pores. This was just opposite to Ti–15In alloy, whose remaining oxide layer was porous and the substrate corroded uniformly. The different surface morphology of these two samples presented different corrosion mechanisms, probably due to the different content of indium and the oxides formed on the surface. shows the Ti ion released into the cell culture medium (DMEM solution at 310 K) after 72 h incubation. It's evident that there is little difference among all of the tested samples. The average concentrations of Ti were as follows: 10.00 ng/cm2 for CP–Ti, 8.58 ng/cm2 for Ti–1In alloy, 9.90 ng/cm2 for Ti–5In alloy, 8.36 ng/cm2 for Ti–10In alloy and 11.73 ng/cm2 for Ti–15In alloy. Statistical analysis results showed that there was no significant difference between any two sample groups (p > 0.05). The concentrations for indium ion were below the detection limitation, indicating very little amount of In ions released, partially due to the low content of In in the Ti–In alloys substrate. shows the viability of mouse L-929 and NIH3T3 fibroblasts, expressed as the absorbance ratio of the Ti–In alloy culture medium groups to that of CP–Ti (negative control) with error bars representing standard deviations, and DMSO was used as positive control. It can be seen that, compared to CP–Ti group, the L-929 cell viabilities of all Ti–In alloy groups kept at around 100% after 1 day culturing. But they showed a little decrease after 2 days except for Ti–1In alloy group, and then the viabilities increased slightly for Ti–10In and Ti–15In alloy groups, in contrast to Ti–1In and Ti–5In alloy groups after 4 days culturing. The viability of positive control group decreased constantly with the culture duration, exhibiting significant cytotoxicity. One-way ANOVA results showed no significant difference between CP–Ti and Ti–In alloy groups (p > 0.05) after 4 days culturing. The cell viability of Ti–15In alloy was significantly lower than that of CP–Ti group (p < 0.05) after 4 d culturing, according to LSD multiple comparison results, but still above 90% compared with CP–Ti group.In comparison with L-929, the viabilities of NIH3T3 for CP–Ti and Ti–In alloy groups show somewhat of difference. After 1 day culturing, the viabilities increased to above 110%, then decreased to around 80%, and then increased again to some extent equal to that after 1 day. The viabilities of the positive control group were all below 15%. Statistical analysis results suggest that the viabilities of Ti–In alloy extracts were significantly higher than that of CP–Ti extract after 4d (p < 0.05, one-way ANOVA). Significant difference does not exist among Ti–In alloy groups.In order to provide a device or facility, certain specific property requirements pertaining to both the material and the product must be fulfilled. The material-specific properties include mechanical, physical, chemical and physicochemical properties, whereas biological properties such as biocompatibility are of additional importance. Intrinsic mechanical properties of metals and alloys are critically related to the microstructure, and can be tailored by means of mechanical processing, heat treatment and alloying. However, biocompatibility is generally related with the corrosion property of the material, since metal ions often release into the adjacent environment, during the corrosion process, and affect the tissues around it. Therefore, this study investigated the mechanical properties and cytotoxicity as well as the microstructure and corrosion behavior of experimental Ti–In alloys to examine their feasibility of using as dental alloys.According to the Ti–In phase diagram and Gulay's work Ohkubo pointed out that cast CP–Ti may be too soft and flexible for circumferential and bar retainers . The strength improvement was more effective than other alloying elements like Cu and Zr. By converting weight percentage to atomic percentage which was used in this work, YS and UTS of Ti–5Cu (corresponding 3.8 at.% Cu) increased by 45% and 13% With the increased yield strength and the same elastic modulus, Ti–In alloys can be deformed to a larger strain without yielding. Therefore, this feature may increase the retention force by affording a larger elastic strain, and without permanent deformation or fracture for dental applications, such as removable partial denture frameworks, bridges and clasps. But it should be pointed out that the mechanical performances of a prosthodontics do not only depend on the mechanical properties, but also on the geometric design.As is known, oxide layer on the surface of Ti provides protection and corrosion resistance. Corrosion occurs when the passive oxide film decomposed to soluble products, especially in the presence of fluoride , that the addition of NaF altered the electrochemical behavior of CP–Ti and Ti–In significantly, as shown in the decreased OCP and increased corrosion current density. This phenomenon was also found in pure titanium Besides, Ti–10In and Ti–15In exhibited secondary active–passive transition as indicated in the potentiodynamic results. This situation demonstrates the “transpassive” dissolution of one oxide and leaving a stable one on the surface. The dissolution reaction under the corresponding potentials usually yields an increase of current ) suggested that both Ti–5In and Ti–15In had much more protective oxides on the surface, which increased the corrosion resistance to some extent.In vitro cell viability tests showed that Ti–In alloys have the comparable cytocompatibility with CP–Ti. The lowest cell viability was more than 90% for L-929 fibroblast and 80% for NIH 3T3. Due to the limitation of this work, the decrease in NIH 3T3 viability on the second day is not well understood, further investigations are needed. However, after 4 days of culturing, the NIH 3T3 viabilities of all the Ti–In alloys were almost 110%, indicating good cell proliferation. Thus, the addition of indium did not alter the cytocompatibility of Ti. Investigation of metal cations cytotoxicity used in dental cast alloys showed that the TC50 (toxic concentration by which 50% of the cells in a culture are killed) of In+ 3 for L-929 fibroblasts was 2310 μM (265.23 mg/L) In this work, basic material properties of Ti–In alloys were evaluated primarily, with commercially pure Ti as reference. The binary alloys showed better mechanical strength, higher hardness, and equal in vitro cytotoxicity than CP–Ti. But further in vitro investigations such as casting procedure, wear resistance and porcelain compatibility are necessary to examine the potential of Ti–In as prosthodontic dental material.The addition of indium to titanium from was found to be effective to increase the mechanical strength and microhardness without changing the phase constitution. Ti–In alloys exhibited similar passivation behavior with CP–Ti and the same order of magnitude of passivation current densities in artificial saliva solutions without fluoride. With the presence of NaF, Ti–10In and Ti–15In showed transpassive behavior and lower current densities at high potentials. In vitro cytotoxicity tests showed that Ti–In alloys had good cytocompatibility comparable to CP–Ti.Constitutive model of concrete damaged by freeze–thaw action for evaluation of structural performance of RC elementsThe coupled environmental-mechanical damage model extended by the authors to include the degradation effects on structural behavior of RC structures due to freezing-and-thawing cycles (FTC) is reformulated and generalized in the present work in order to better simulate the different aspects of the physical phenomenon. In particular the model is modified to consider separately the effect of FTC respectively on compressive and tensile strength and a new relationship to properly evaluate the equivalent number of FTC is proposed. To validate the model, an experimental campaign carried out on simply supported beams subjected to FTC is simulated. By comparing obtained numerical results with experimental evidence, the model is proved to be suitably accurate in reproducing the main aspects observed during tests: failure load, ultimate displacement, and failure mode. Actually the enhancement of freeze and thaw – mechanical model gives the base for the definition of a reliable numerical tool for analysis of RC structures subjected to FTC.Durability of RC constructions gained increasing interest during last decades and, especially in countries with cold climate, freezing-and-thawing cycles (FTC) represent one of the most dangerous phenomena for RC structures. Indeed a high number of RC constructions are built in wet environments, such as bridge piers, bridge slabs, off-shore platforms, etc. making FTC a serious problem for their durability. Moreover in many cases interaction between different mechanisms takes place: for instance in bridge slabs surface cracking and/or scaling due to FTC accelerates carbon dioxide, chloride, and oxygen diffusion processes which may induce reinforcement corrosion.The problem of FTC has been studied since the middle of the last century from the material point of view. Although different questions remain unanswered, nowadays the main mechanisms that lead to degradation due to FTC have been identified and just about clarified (e.g. It is widely accepted that two different types of frost damage can be distinguished (e.g. Some researchers investigated the degradation of compressive and tensile strength due to frost degradation. Shang and Song It is worth noting that very little attention has been devoted to studying structural performance of RC elements subjected to FTC. For instance Hassanzadeh and Fagerlund In this work an innovative coupled environmental-mechanical damage model proposed by the authors in In the first part of the paper the coupled environmental-mechanical constitutive damage model is briefly recalled and extended by introducing two different environmental damage variables, i.e. the positive and the negative ones, in order to properly reproduce the different effects of the environmental attack on compressive and tensile behavior. Then the new freeze and thaw degradation (FTD) model is proposed, from the general formulation up to the parameters definition, taking advantage of a number of experimental works available in literature. In particular, different validation examples are considered in order to show the suitability of the model in reproducing the main effects of frost degradation on concrete properties. Finally eight beams tested by Hassanzadeh and Fagerlund In the present section the coupled damage model developed by the authors for concrete is briefly recalled. Readers can refer to The constitutive model is formulated within the framework of continuum damage mechanics. It is based on an isotropic plastic-damage model with two distinct damage scalar variables: one for tension and one for compression (e.g. The constitutive law is described by the following equation:where σ is the usual Cauchy stress tensor, σ¯+ and σ¯- are the positive and negative parts of the effective stress tensor obtained via a spectral decomposition. The effective stress tensor is defined as:with C0 the fourth-order elastic stiffness tensor and ɛe the elastic part of the total strain tensor, which is split in an elastic and plastic part.where r+ and r− denote respectively positive and negative threshold monitoring the size of expanding damage surface and τ¯+ and τ¯- are the equivalent stresses defined as following:with E the concrete Young modulus; σoct and τoct the octahedral normal and shear stresses respectively, and the scalar K a material property that accounts for the increase of compressive strength due to biaxial compression. In particular K depends on the ratio R0 between 2D and 1D compressive strengths according to:The adopted evolution laws for damage variables are:d+=1-r0+r+expA+1-r+r0+d-=1-r0-r-1-A--A-expB-1-r-r0-where r0+ and r0- describe the initial threshold variables, r+ and r− are the current threshold variables and A+ , A− and B− are model parameters.σ=(1-denv+)(1-d+)σ¯++(1-denv-)(1-d-)σ¯-=(1-d∗+)σ¯++(1-d∗-)σ¯-where d∗+ and d∗- are respectively the positive and negative coupled damage variables. Independently from their specific definition, the environmental damage variables are represented by an increasing function with time, which means ḋenv+≥0 and ḋenv-≥0. The introduction of two environmental damage variables, respectively the positive and the negative ones, allows to represent in an independent way the effects of attack in tension and in compression, both in terms of stiffness and strength reduction, , in agreement with experimental evidences for example in case of frost degradation.Finally the model assumes that the damage criterion also describes the plastic surface so that the development of material damage is simultaneous with the accumulation of irreversible strains according to the following relation:where H is Heaviside step function, ḋ=ḋ++ḋ- and β≥0 is a plastic strain coefficient that is assumed to be a model parameter, and symbol 〈·〉 denotes the MacAuley brackets, equal to the positive part of its argument. For more information about β, readers can refer to In this work the model originally proposed in The main relations which define the proposed FTD model are the following:where 〈·〉 are MacAuley brackets, ad-, aε, aR, ad,1+, ad,2+ are model parameters, εpeakd and R0d are respectively the values of peak strain εpeak and R0 for degraded concrete, while Neq represents a parameter related to both the actual number of FTC and to the freeze–thaw conditions as it will be described in the next paragraph. Finally Nc denotes the value of Neq which corresponds to a sudden change in the evolution of denv+, according to experimental evidence (e.g. Shang and Song In particular the parameters ad-, aε and aR depend on concrete characteristics (for example porosity, air content etc.) reflecting the experimental evidence of increasing of frost-resistance together with quality of concrete. Eqs. , where the effects of different values of model parameters ad-, aε and aR are evidenced, showing as an increase of such parameters affects the values of the negative damage parameter denv-, the peak strain εpeak and the ratio R0 between 2D and 1D compressive strengths.a where it is indicated the meaning of the parameters. Moreover b shows the effect of different values of model parameters ad,1+, ad,2+ and Nc. In particular the two parameters ad,1+ and ad,2+ control slope of first and second branch respectively, while Nc controls the transition point of denv+ evolution.Concerning Neq, it is equal to the actual number N of the FTC if concrete is subjected to frost deterioration conditions assumed for the “Procedure A” described in ASTM C 666 In this work a specific relation for evaluation of Neq is proposed:where N is the actual number of FTC, while χ and γ are parameters that are assumed to depend on the experimental conditions according to which the FTC are performed., where also the effect of the model parameters χ and γ is depicted. In particular the parameter γ allows to take into account possible non-linear relation between the degradation grade and the actual number of FTC, N, e.g. In this section, the relationships defining all the parameters introduced in Eqs. are proposed, on the basis of a number of experimental campaigns. In particular, in addition to the experimental tests carried out by Shang and Song the compressive strength reduction due to frost action may be written as:where fcd and fc are respectively the values of compressive peak stress in degraded and sound conditions (Starting from the experimental results for the three series of specimens, the values of ad- are evaluated by regression analysis for each type of concrete as shown in In similar way, regarding the increasing of peak strain due to frost action, according to the values of aε are evaluated by regression analysis of the experimental results for the different concrete series, . The results of the regression analyses are summarized in , where the values of ad-and aε for the three series of concrete are provided together with the corresponding values of the cylindrical compressive strength fc. If these results are reported respectively in fc |
− |
ad- plane and fc |
− |
aε plane it can be observed that they are well represented by a line, as shown respectively in , where the corresponding regression lines are reported. Therefore the following linear relationship between ad- and fc is proposed:with fc [MPa] the cylindrical compressive strength, Similarly, for aε the following relation between aε and fc is evaluated, It is worth noting that the range of validity of Eqs. is restricted to the case of ordinary concrete (i.e. approximatively from C20 to C45), for which the experimental results are available, for this reason in outside this range the lines representing these equations are dotted.As previously stated, in addition to the experimental results on concrete with compressive strength 34.2 MPa described by Shang and Song in For each type of concrete, respectively named Shang2, Shang1 and Shang3, the values of ad- is predicted according to . It is worth noting that the value of ad- obtained for Shang2 with such a predictive method slightly differs from the one reported in The FTC procedure was applied according to GBJ82-85 . Due to the limited data available, in Eq. it is assumed γ |
= 1, while the parameter χ is evaluated in order to obtain the same strength degradation as with ASTM C666 Procedure A. Such a calibration leads to χ |
= 1.36 which is used for the other types of concrete, namely Shang 2 and Shang 3. Obtained results are summarized in where errors between predicted and experimental results are reported for both Shang 2 and Shang 3 cases. It can be seen that error is limited in the range −8%, +5%, so validating the proposed model.Concerning the effect of FTC on peak strain, the only available data regard concrete with fc= 34.2 MPa (i.e. Shang 2). Parameter aε is predicted by means of Eq. obtaining aε= 0.0076, while the value of χ is clearly the one obtained in the calibration phase, i.e. 1.36. Comparison between predicted and numerical results is shown in in terms of ratio between degraded and sound peak strain, demonstrating that FTD model predicts suitably well the increasing of peak strain with increasing number of FTC, at least in the range of experimental data.Since to authors’ knowledge the only complete experimental campaign about the effect of FTC on biaxial strength of concrete has been published by Shang and Song in It is worth noting that since the experimental tests refer only to one value of compressive strength, in this relationship no explicit dependence of aR on fc is introduced.As previously mentioned, reduction of elastic modulus is experimentally observed with increasing of frost damage. In particular both static elastic modulus and dynamic elastic modulus decrease with increasing of internal frost damage. Petersen et al. where Esd and Es are respectively damaged and sound static modulus of elasticity, while Edynd and Edyn are respectively damaged and sound dynamic modulus of elasticity. The ratio between damaged and sound dynamic elastic modulus is also known as RDME (Relative Dynamic Modulus of Elasticity) and it is widely accepted as reference parameter for assessing the amount of frost damage and to evaluate the frost resistance of concrete. Actually the dynamic modulus of elasticity can be easily monitored due to the non-destructiveness of the test being based on ultrasonic measurements. the reduction of static elastic modulus due to frost action may be written as:which is non negative and it is equal to compressive strength reduction expressed by Eq. . Indeed, in the proposed FTD model it is assumed that the environmental damage variable affects in the same way elastic modulus and strength., the reduction of dynamic modulus of elasticity may be written according to the following equation:leading to an explicit relation between RDME, ad-and Neq.To validate the proposed formulation, the experimental tests performed by Duan et al. where it is possible to see that the proposed model describes adequately well the RDME evolution with increasing internal frost damage. The average error is about +9% while maximum error is +16.9%.As a further validation, the experimental results obtained by Shang and Song , the model effectively captures the RDME evolution with increasing frost damage: the average error is of about +3 % while maximum error is equal to 12.4%.Finally the experimental tests performed by Hanjari et al. . In fact, according to the proposed model, the actual number of FTC N to which the specimen was subjected needs to be transformed to the equivalent number Neq through Eq. in order to consider the different FTC conditions. To this aim parameters χ and γ are calibrated considering two of the available experimental results: in particular one test with low internal damage and one test with medium–high internal damage are selected. As an example a depicts the results obtained assuming the two tests indicated with the red circle marker.Then a sensitivity analysis is performed in order to investigate the influence of the two selected experimental tests on the evaluation of χ and γ. In particular seven tests with low level of internal damage and ten tests with medium level of internal damage are considered. summarizes the obtained results in terms of mean error between predicted and observed RDME and standard deviation. The mean error ranges between about −15% and +21%, standard deviation is generally variable between 30% and 40%.To further investigate the response of the model, an optimization procedure based on Ordinary Least Squares approach is used. In particular the parameters χ and γ that best fit the experimental results are obtained minimizing the following cost-function:where RDMEi are the observed n experimental data, while RDMEinum are evaluated according to Eq. (indicated with bold character). As it is possible to see, mean error in this case is equal to −0.9%, while standard deviation is about 31%. This is due to the large scatter the experimental data present.As already mentioned, the proposed relation for capturing the evolution of environmental positive damage variable due to FTC, expressed by Eq. , allows to profitably represent the experimental data, according to which a sudden tensile strength reduction is evidenced at the beginning of the frost phenomenon. In the experimental results obtained by Shang and Song , while, due to the lack of experimental results for low values of Neq, no reliable estimation is performed for the model parameters ad,1+ and Nc. For this reason an alternative simplified relationship is here proposed which relates denv+ to denv-. Several experimental results obtained by different researchers are considered and plotted in in terms of the corresponding values of environmental damage variables denv+ and denv-. These results suggest a linear relation between the two damage variables, which is here expressed by means of linear regression as: relates directly compressive strength degradation with tensile strength degradation, according to other researchers (e.g. only the second part of the bilinear function, neglecting the initial phase of frost-damaging process characterized by Neq<Nc, where few experimental results are generally available. In the following application, the simplified Eq. is adopted for the evaluation of denv+.In this section, the proposed coupled environmental-mechanical damage model with FTD formulation is applied to evaluate the load carrying capacity of some of the RC beams tested by Hassanzadeh and Fagerlund It is worth noting that in the experimental program, the degraded beams were vacuum treated, submerged in water and then subjected to two freeze–thaw cycles, In the present work eight beams are analyzed with the proposed model. In particular beams with shear reinforcement are selected (R1, R2, R3, R4 and D1, D2, D3, D4). Series 1 and 3 (large beams) geometry is characterized by 4.4 m span and two concentrated loads with a shear span of 1.7 m. The cross section is (0.2 × 0.5) m2 for both series. Reinforcement of beams 1 consists in 4Φ20 longitudinal bars and 28Φ8 stirrups. Reinforcement of beams 3 consists in 6Φ20 longitudinal bars and 32Φ8 stirrups. In geometry of large beams Series 1 and 3 is depicted.Series 2 and 4 (small beams) are characterized by 3.0 m span and two concentrated loads with a shear span of 1.0 m. The cross section is (0.2 × 0.3) m2 for both series. Reinforcement of beams 2 consists in 3Φ20 longitudinal bars and 18Φ8 stirrups. Reinforcement of beams 4 consists in 5Φ20 longitudinal bars and 22Φ8 stirrups.Concrete used for the specimens is characterized by a compressive strength of fc |
= 37.6 MPa and a splitting tensile strength of 4.1 MPa.For more information about geometry of the specimens, reinforcement arrangement, material properties and experimental procedure, readers can refer to Researchers report only the characteristic yield strength of the steel they adopted in their investigation, which is equal to 590 MPa. A preliminary reverse analysis for one of the beams permitted to estimate the mean value fy |
= 670 MPa, which is the same value assumed by Hanjari et al. The numerical analysis is performed with Finite Element Code OpenSees , where fc,el is the elastic limit stress under uniaxial compression. It is worth noting that the value adopted for the fracture energy Gfd of damaged concrete was measured by Hassanzadeh and Fagerlund, the results for beams R1 and D1 are depicted in terms of comparison between experimental and numerical load–displacement curves. A ductile failure is numerically predicted for the sound beam according with experimental evidence. This is due to reinforcement ratio that is lower than the balanced one. Moreover failure load and displacement at failure are well captured by the proposed model. Due to change of mechanical properties of concrete induced by frost-degradation, the mechanical reinforcement ratio for beam D1 becomes greater than the balanced one, suggesting a brittle failure due to crushing of compressed concrete. This is confirmed by experimental observation and is profitably captured by numerical model. Moreover, also in this case, both failure load and displacement at failure are effectively reproduced by the FTD model. An overestimation of initial stiffness can be observed. This may be due from one side to the fact that the model assumes the same elastic modulus degradation as compressive strength degradation, on the other side to the assumption of perfect bond between reinforcement and concrete. In fact frost degradation may affect also bond behavior, as found by some authors depicts comparison between experimental and numerical load–displacement curves for beams R3 and D3. Due to the mechanical reinforcement ratio of beam R3, a brittle failure due to crushing of concrete is expected. This is confirmed by both experimental and numerical results. Moreover it is possible to observe that the proposed model properly predicts both failure load and displacement at failure. Also beam D3 exhibits brittle failure due to crushing of compressed concrete and the numerical model confirms to well simulate failure load and displacement at failure with a slight overestimation of initial stiffness by numerical model.Concerning small beams, series 2 and 4, the mechanical reinforcement ratio is respectively equal to balanced mechanical ratio and greater than balanced mechanical ratio. Beam R2 may fail either by yield of steel or crushing of concrete. Beam R4 is expected to fail by crushing of concrete.Numerical analysis evidences that beam R2 fails by crushing of concrete shortly after yielding of reinforcement. Failure load and ductility are well predicted by numerical model as it can be seen in where the comparison between numerical and experimental results in terms of load displacement curve is depicted. Beam D2 fails due to crushing of concrete in upper zone of the beam. This is confirmed by both numerical and experimental results. It can be seen in that the model profitably predicts failure load and peak displacement and also in this case an overestimation of the initial stiffness can be noticed. illustrates numerical and experimental results for series 4. Failure mode of beam R4 is correctly captured by numerical model, although a discrepancy on failure load is evidenced. It is worth noting that an analytical section analysis provides a theoretical value of failure load equal to 285 kN, consistently with numerical response. It should be pointed out that the experimental tests were performed in a period of time that varies between 5 months and 2 years, as the authors reported in A Freeze-and-Thaw Degradation (FTD) model, based on an extensive reformulation of the coupled mechanical-environmental damage model developed by the authors in a previous work Moreover effect of frost on tensile strength is specifically considered, by introducing new relationships for the positive damage parameter denv+. In this way the different consequences of frost degradation in terms of tensile and compressive strength are properly simulated.Concerning the structural performance of RC elements degraded by FTC attack, it is worth noting that it may be considerably compromised and failure mode may change from ductile to brittle. Such a behavior is well reproduced by the proposed FTD model by the numerical simulation of a number of four-point bending tests of RC beams subjected to frost degradation, In conclusion, the proposed FTD model denotes an important enrichment of the original coupled environmental – mechanical model, which has proved to be able to account for deterioration due chloride attack and carbonation (e.g. Cavitation erosion and corrosion behaviour of laser surface alloyed MMC of SiC and Si3N4 on Al alloy AA6061Laser surface alloying of SiC/Si3N4on AA6061 aluminium alloy was carried out using a 2-kW CW Nd-YAG laser. Different ratios of SiC and Si3N4 powders were mixed and the layers were preplaced by pasting. Subsequent laser surface melting of the layers gave a surface metal matrix composite (MMC). The microstructure, chemical compositions and phase identification of the modified layers were examined using scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDX) and X-ray diffractometry (XRD), respectively. The laser surface alloyed MMCs of AA6061-SiC and AA6061-Si3N4 consisted of small amounts of Al4C3/Al4SiC4and AlN, respectively. For specimens alloyed with 100% Si3N4 (AA6061-Si3N4), the cavitation erosion resistance Re was improved by three times as compared to the AA6061 alloy whereas there was no significant improvement in Re of the specimen alloyed with 100% SiC (AA6061-SiC). The surface hardness of the specimens alloyed with SiC/Si3N4 was increased seven times. All the laser alloyed specimens with SiC/Si3N4 showed a decrease in pitting resistance and absence of passivity owing to the aluminium–ceramic interfaces, which favour pit initiation or hindered passivity.Cavitation erosion is a progressive loss of material from a solid due to the impact action of the collapsing bubbles or cavities in the liquid near the metal surface. Damage due to cavitation erosion can be found in high-speed impellers, valves, pump casings and turbine blades in the marine industry and ultrasonic mixers in the food and pharmaceutical industries, etc. It is well known that aluminium alloys have the lowest cavitation erosion resistance among the engineering alloys In order to improve the surface engineering properties of Al alloys by laser surface modification, most published works deal with the corrosion resistance The incorporation of ceramic reinforcement into the matrix of aluminium alloys by laser surface alloying (LSA) is a promising technology to produce surface MMC that function as cavitation erosion resistant coatings while the bulk properties of the substrate can be retained. By preplacing ceramic powders on the aluminium alloys, the laser energy absorbed by the ceramic can be efficiently transferred to and melt the alloy surface forming a layer of metal matrix composite (MMC) at the surface under specific conditions. The surface MMC Al-SiCp on Al alloys was successfully fabricated by laser surface alloying and cladding and the hardness and wear resistance was found to be increased significantly by eight and 10 times, respectively The as-received aluminium alloy AA6061 (T6) was in the form of plate with dimensions of 40×40×10 mm. Its nominal chemical compositions in wt.% is: 0.4–0.8 Si; 0.7 Fe; 0.15–0.4 Cu; 0.15 Mn; 0.8–1.2 Mg; 0.04–0.35 Cr; 0.25 Zn; 0.15 Ti and Al balance. Reagent grade powder of silicon carbide (SiC) and silicon nitride (Si3N4) were used with an average particle size of 40 μm. The physical properties of the materials are shown in The surface of the substrate AA6061 was ground by a 220 grit silicon carbide paper. A slurry was prepared by mixing the ceramic powders with a binder (4 wt.% polyvinyl alcohol, PVA). Different ratios of SiC and Si3N4 powders in weight percentage were mixed, as shown in . The slurry was pasted on the surface of the Al alloy and then dried at 120°C for 30 min. Finally the pasted specimens were polished by a 1000 grit silicon carbide paper to make a uniform layer of ∼0.1 mm thick, as measured by a micrometer. Subsequent laser surface treatment was then carried out using a 2-kW CW Nd-YAG laser and the laser-treated specimens were denoted by A, B, C, D and E. The volume fraction of all phases present in the specimens were estimated by image analysis (shown in A laser power of 1200 W at the workpiece with a beam size of 3 mm in diameter was used for laser surface alloying of ceramics on AA6061 whereas a smaller beam size (2 mm) was used for laser surface melting of monolithic AA6061 because of the high reflectivity of the ground surface. The beam scanning speed used was 5 mm/s. Argon flowing at 20 l/min from a 6-mm inner diameter copper tube was used as shielding gas. The alloyed surface was achieved by overlapping successive melt tracks at 50% of the track width. After laser surface treatment, the microstructure and chemical compositions of the laser alloyed specimens were analysed by scanning electron microscopy (SEM), optical microscopy (OM), energy dispersive spectroscopy (EDX) and X-ray diffractometry (XRD). The radiation used in XRD was Cu Kα with nickel filter and generated at 40 kV and 35 mA. The scan rate used was 1.5°/min.The laser alloyed specimens for cavitation erosion and electrochemical corrosion tests were cut into squares of dimensions 13×13 mm. The specimens were polished to constant surface roughness using 1 μm diamond paste and then cleaned, degreased, dried, and weighed before and after each subsequent cavitation erosion and pitting corrosion test. Cavitation erosion experiments were carried out using an ultrasonic-induced cavitation facility with a-550 W ultrasonic probe conforming to ASTM Standard G32-92 . The distance between the stationary specimen and the vibrating stainless steel stud was kept at 1 mm. The surface of the stainless steel stud was polished by 1-μm diamond paste in every intermittent period of 30 min in order to keep a constant surface roughness. The vibration frequency and peak-to-peak amplitude were 20 kHz and 30 μm, respectively. The specimens were subject to a series of cavitation erosion tests in 3.5% NaCl solution and the temperature of the solution was kept at 23°C by a chiller. Each cavitation erosion test was completed after 4 h, which included eight intermittent periods of 30 min each.The weight loss of materials of the specimens by cavitation erosion was measured by an electronic beam balance with an accuracy of ±0.1 mg. The erosion loss and erosion rate was calculated:where ΔW is the weight loss in each time interval in mg, Δt is the time interval in hours and A is the surface area of the specimen in cm2. In addition, the normalised cavitation erosion resistance ReN (with respect to as-received AA6061) is defined as:The ReN of the specimens is only a rough estimation since the densities of AA6061, the ceramics and the intermetallic phases are a little bit different.Cyclic potentiodynamic polarisation scans were carried out using an EG&G PARC 273 corrosion system according to ASTM Standard G61-86 The scanning direction was then reversed when an anodic current density of 5 mA/cm2 was reached and continued until the loop closed at the protection potential or until the corrosion potential was reached.The surface MMCs with different ratio of SiC and Si3N4 were successfully fabricated on aluminium alloy matrix by the laser surface alloying process. The thickness of laser-alloyed layers of various specimens is 80±10 μm. Cross-sectional and surface appearances of the laser-alloyed specimens A, D and E are shown in . All the specimens were of high integrity, free of porosity and cracks. The laser-alloyed layers formed a good metallurgical bond with the substrate alloy. The corresponding XRD spectra of AA6061, laser melted and alloyed specimens are shown in a shows the cross-section of specimen A (100 wt.% SiC). The polygonal SiC particles were uniformly dispersed and tightly bound to the aluminium matrix. The average particle size of SiC in the alloyed zone is ∼15 μm after alloying. For specimen E (100 wt.% Si3N4), clusters of Si3N4 and AlN were scattered randomly in the aluminium matrix (b). For specimens B, C and D, the ceramic phases SiC, Si3N4, Al4C3, Al4SiC4, and AlN were found by XRD. The typical appearance of specimen D is shown in c. All these specimens showed a high volume fraction of ceramic phases reinforcement (40–61 vol.%) in the surface MMC.Since a high energy density (0.16 MJ/m2) was used, SiC was partially dissolved in the melt and reprecipitated during solidification. Small amounts of Al4C3 and Al4SiC4 were observed and identified in the XRD spectra (c–e) and was consistent with the findings of Hu and Baker a (ii)]. These compounds were formed from the reactions From the above reactions, the excess silicon dissolved in the solid solution of the matrix. The line scans of chemical compositions of specimens A and E (across the aluminium matrix and the ceramic phases) by EDX are shown in . In the aluminium matrix and SiC particle, the compositions were rich in Al and Si, respectively. Similarly, Si3N4 decomposed during the irradiation of the laser beam. Then Si and N were alloyed into the aluminium matrix and aluminium nitride (AlN) was formed. The dissolution of SiC/Si3N4 causes the formation of new phases, which may affect the surface properties such as cavitation erosion and corrosion resistance.The incorporation of SiC and Si3N4 into the laser-alloyed layers resulted in an increase of microhardness considerably. Average surface hardness were measured by a load of 200 g and load time of 15 s (). The microhardness of the layers was found to be in the range of 350–400 Hv and increased by approximately seven times as compared with that of the as-received AA6061. shows the graphs of cumulative erosion loss as a function of time for the laser-alloyed specimens eroded in 3.5% NaCl solution at 23°C. The data of the laser melted AA6061, which was similar to that of as-received AA6061, was used for comparison purposes. The erosion rates and normalised cavitation erosion resistance ReN of specimens, at the end of the 4-h test period, are shown in . The cavitation erosion resistance of all laser-alloyed specimens was improved as compared with that of monolithic AA6061. In addition, the cavitation erosion resistance of specimen E was the highest. The ranking of cavitation erosion resistance is:In addition, the ReN increases with the increase in wt.% of Si3N4in the alloyed layer, as shown in The ReN of the laser alloyed specimens is dependent on the ratio of SiC and Si3N4 present in the MMCs. According to Wei et al. where r is the total cavitation erosion rate, vi is the volume fraction of phase i and ri is the cavitation erosion rate of phase i. As the total volume fraction of ceramics (SiC/Si3N4) is constant in the aluminium matrix, the difference in ReN is attributed to the ratio of ceramics phases. According to Wilson and Ball ). By the rule of mixture, the cavitation erosion resistance of the specimens increases with an increase in Si3N4 content despite the higher hardness of SiC.Cavitation damage of the soft AA6061 occurred by plastic deformation at the surface leading to roughening, work hardening, necking, ductile fracture and removal of material finally. The erosion damage is initiated at delineated grain boundaries. The surface of the as-received AA6061 was damaged by ductile fracture (b) after being exposed to cavitation erosion for 4 h.For the specimens alloyed with SiC and Si3N4, their deformation mechanism is more complicated. In the initial stage, the ceramics SiC and Si3N4 in the matrix alloy were almost unaffected. The tiny pores and pits at the interface between the matrix and the ceramics provided sites of stress concentration, and erosion was initiated there. Since the aluminium matrix is much softer (56 Hv) than the ceramics, work-hardening and fracture occurred in the matrix preferentially. With the continuation of the damage, pits, work hardening and fracture as well as craters in the matrix increased. The adjacent craters merged into a large defect and the surface roughness increased. The craters did not occur in the hard ceramics phases because of their high fracture toughness and hardness. After the matrix around the ceramic phases were eroded, the ceramic phases such as SiC or Si3N4 were removed away from the surface as individual debris (a). In the present study, a high volume fraction of ceramic phase reinforcement in the MMCs was obtained and a less volume fraction of matrix which encased the ceramic particles resulted in rapid loss of ceramic phases. After exposure to cavitation for 4 h, the MMC layer of specimen A was almost eroded away and the damaged surface of the aluminium matrix by ductile fracture was observed (b). On the other hand, the surface of E was less damaged as compared with A (b). Some cluster of Si3N4, which remained in the matrix, was almost unaffected.The much larger interfacial area between the ceramic phase and the soft Al matrix of the cluster shaped Si3N4 phase (as in specimen E), as compared with that of the relatively simple polygonal shape of the SiC particles (as in specimen A), has played an important role in the cavitation erosion resistance. The large interface of Si3N4 clusters provides a better locking mechanism between the hard ceramic and the soft matrix. The interfacial strength between the Si3N4 clusters and the Al matrix is much better than that of the SiC particles. This results in a lower rate of erosion in specimen E. In fact, the results showed that the higher the Si3N4 content in the surface MMC, the more locking interfaces between the ceramic and the matrix, the better the erosion resistance.Potentiodynamic polarisation curves of the as-received, laser-melted and laser-alloyed specimens in deaerated 3.5% NaCl solution at 23°C are shown in . The corrosion potential, pitting potential and protection potential are summarised in The as-received and laser-melted AA6061 suffered from severe pitting corrosion. a shows the pit morphology of the as-received AA6061. For the as-received AA6061, the precipitation Mg2Si acted as the initiation sites for pitting corrosion. For the laser-melted specimen, the corrosion potential was increased by 32 mV because the precipitation was dissolved in the resolidified surface, which had high homogeneity. There is no significant change in pitting potential for the laser-melted specimen.The aluminium matrix of specimens A and E was uniformly corroded as shown in b,c. For laser-alloyed specimens with high reinforcement volume fractions, their corrosion potentials increase, pitting resistances decrease significantly and passivity is absent as compared with the monolithic AA6061 (T6). This is because the matrix–ceramic (AA6061-SiC/Si3N4 and AA6061-Al4C3/Al4SiC4/AlN) interfaces were the main source of pit initiation sites, due to the presence of impurities and weakness of passive films The laser surface alloyed MMCs AA6061-SiC and AA6061-Si3N4 consisted of a small amount of carbide and nitride phases Al4C3/Al4SiC4 and AlN, respectively.The cavitation erosion resistance of specimens laser-alloyed with Si3N4 was increased by three times compared with as-received AA6061 because Si3N4 has a high fracture toughness.The cavitation erosion resistance of specimens laser-alloyed with SiC and Si3N4 increased with the increase in wt.% of Si3N4.The surface hardness of the laser-alloyed specimens increased by approximately seven times as compared with that of as-received AA6061.The laser-alloyed specimens showed a decrease in pitting resistance and absence of passivity owing to the existence of matrix–ceramic interfaces, which favoured pit initiation or hindered passivity.The cavitation erosion resistance and pitting corrosion resistance of the surface MMCs studied cannot be improved simultaneously.Interface and micromechanical characterization of tensile strength of bio-based composites from polypropylene and henequen strandsThe contribution of a reinforcement to the tensile strength of a composite can be evaluated by different micromechanics models. Nonetheless, one of the main difficulties is the evaluation of the intrinsic properties of the reinforcements. The literature shows experimental and model-based methodologies to estimate such intrinsic properties, and few are based on single fiber tensile test in combination with a Weibull analysis. This paper proposes using henequen strand to prepare polypropylene-based composites. Henequen fibers show a high cellulose content that allows obtaining strong interfaces when a coupling agent is added to the composite formulation. The novelty of this work is based on a simplified methodology to evaluate the intrinsic tensile strength of the reinforcements and its contribution to the tensile strength of the composite. A percentage of coupling agent that returns the highest tensile strength is identified with a strong interface. Then, typical values for a coupling factor and interfacial shear strength are used with a modified rule of mixtures and a modified Kelly and Tyson models to obtain the orientation factors. The prediction of composite behavior from fiber properties is necessary to anticipate the correlation between experimental and the back-calculated parameters.Composites consisting of lignocellulosic fibers and synthetic thermoplastics have received substantial attention in scientific literature as well as industry primarily due to improvements in process technology and economic factor (). The use of fiber-reinforced plastic composite arises due to the increasing environmental consciousness such as difficulties of convenient removal after the end of life time, which is relatively stable and difficult to separate and recycle (). Moreover, the substitution of the traditional engineering fibers (mineral fibers) by natural fibers as composite reinforcement offers a possible alternative to decrease the consumption of petroleum-based products (The primary advantages of using natural fibers as fillers/reinforcements in plastics are their low densities, non-abrasive, high filling levels possible resulting in high stiffness properties, high specific properties, easily recyclable, unlike brittle fibers the fibers will not fractured when processing over sharp curvatures, biodegradable, wide variety of fibers available throughout the world, low energy consumption and low cost. These benefits are not likely to be ignored by the plastics industry for use in the automotive, building, appliance and other applications.Most often, the fiber properties and fiber structure are influenced by several conditions and they vary depending on the area of growth, the climate and the age of the plant. The tensile strength of natural fibers which is of main importance regarding reinforcing efficiency also depends on the test gauge length of the specimens. This dependence, strength vs. test gauge length, informs about the homogeneity or number of defects on the fiber surface or of the material in general (The traditional source of agro-based fiber has been wood, however, there is a wide variety of agro-based fibers to consider for utilization (), such as bundles of strands from annual plants, and agroforestry fibers from trimming. Other large sources of fiber can come from recycling agro-fiber based products such as paper and paper, waste wood and point source agricultural residues (Among all different natural fibers from annual plants, henequen fibers (Agave fourcroyedes) are leaf fibers, native from Yucatan Peninsula, Mexico. Henequen strands contain higher cellulose content than other natural fibers or wood fibers. The bibliography reports cellulose contents for henequen strands of about 60% (). Also in the literature, the intrinsic tensile properties of henequen fibers have been reported with tensile strength of 500 ± 70 MPa, Young’s modulus of 13.2 ± 3.1 GPa, and density of 1200 kg/m3 (). The thermal degradation has shown onset degradation temperature around 320 °C (). However, due to the high cellulose content, henequen fibers exhibit distinct hydrophilic character that prevents their compatibility with common hydrophobic polymeric resins. Different modification reactions have been applied onto natural fibers to overcome this and to improve the fiber-matrix adhesion, such as: the use of coupling agents (silanes, titanates, zirconates, triazine compounds), alkyl succinic anhydride, stearic acid, the graft copolymerization of monomers onto the fiber surface, and the use of maleic anhydride copolymers (). Also the pre-impregnation of fibers with the polyolefin solution has been proved to improve the interfacial fiber-matrix adhesion (). Henequen fibers reinforced thermoplastic have been investigated and both, fiber surface treatments () have been shown to increase the quality at henequen fiber/matrix interface. From the previous techniques, copolymers from maleic anhydride with polyolefins has led to the most promising results, in terms of improving the stress transfer and the subsequent mechanical properties of composites (In this paper henequen strands were used as reinforcement for a PP-based composite. In order to obtain a strong interface between the fibers and the matrix, a coupling agent was proposed. Different amounts of such coupling agent were used to prepare composites and test them under tensile loads. It was possible to stablish a percentage that returned the highest composite tensile strength and thus the strongest interface. A chemical characterization of the henequen fibers stablished a high increase of the cellulose contents from the primary to the secondary wall, increasing the presence of hydroxyl groups in the surface of the reinforcements. Likewise, the henequen filaments were submitted to single fiber tensile test. The test were repeated at four different gauge length to stablish the relationship between length and intrinsic tensile strength. Furthermore, a Weibull analysis of the results was carried out to obtain a theoretical characteristic strength. The literature shows that the intrinsic values obtained from raw fibers and fibers inside the composite can be noticeably different (). Thus a modified rule of mixtures and a modified Kelly and Tyson model were used, to validate the results. Composites with strong interfaces have coupling factors and interfacial shear strengths values available in the literature. The micromechanics models returned values for an orientation factor in line with the upper bounds proposed in the literature. The paper proves that a micromechanics analysis based on intrinsic properties obtained from single fiber tensile tests returned sensible results. The paper proposes a simplified model to preview the properties of a composite from a reduced set of experimental data.The composites were prepared using polypropylene homopolymer (PP) (Isplen PP090 G2M) kindly provided by Repsol-YPF (Tarragona, Spain) and with a density of 0.905 g/cm3. This same PP homopolymer has being reinforced with reinforcement contents up to 50 wt% in previous works (). Polypropylene functionalized with maleic anhydride (MAPP) (Epolene G3015) was acquired from Eastman Chemical Products (San Roque, Spain). Henequen strands (HS) obtained from agave (Agave fourcroydes) was supplied by Centro de Investigación Científica de Yucatán (CICY) (Mérida, Mexico). These fibers showed a density of 1.480 g/cm3. Other reactants were used for fiber delignification and they are summarized as follows: sodium hydroxide (Merck KGaA, Darmstadt, Germany), anthraquinone (Badische Anilin & Soda Fabric AG, Germany) used without any further purification.The natural fibers were prepared for chemical characterization according to the TAPPI standard T257. The samples were dried at 105 °C to constant weight.The ash content was determined by incineration of the sample at 525 °C following TAPPI standard T211. On the other hand, the extractive content (TAPPI standard T204) was determined by means of a Soxhlet extraction using an ethanol/toluene mixture. Subsequently, from an extractive free sample, the lignin content was determined according to TAPPI standard T222, precipitating the lignin with sulfuric acid. Finally, the cellulose and hemicellulose content was determined by high performance anion exchange chromatography (HPAEC) according to the methodology described by Hausser et al. (The tensile strength test of the henequen strands was performed following ASTM D3822-01 standard. Mechanical force – deformation curves were obtained using the universal testing machine INSTRON 5500R, fitted with a 5 kN force cell. The diameter of the fibers was determined by optical microscopy. Microscopy images were obtained and the width of the fibers was evaluated as a mean value of 3 measures.Usually lignocellulosic fibers are brittle materials that show a high dispersion on the values of their intrinsic tensile strength. This is inherent to this kind of fibers and is expected due to the variability of the morphology of its surface and the presence of defects in such surface. It is necessary, therefore, to use statistical analysis to evaluate the mean tensile strength of such natural fibers. The statistical analysis was carried out using R version 3.4.4. A two parameters Weibull model was proposed to compute the mean tensile strength of the henequen strands. The probability of failure can be calculated as:Where, β is known as the Weibull modulus, and it is a measure of the dispersion of the strength values. The higher the Weibull modulus, the shorter the scatter of the strength values is. In the same equation, σ and η are the measured fiber tensile strengths, and the characteristic strength (or scale factor) of the fiber, respectively.The analysis was performed for four different gauge lengths, and the effect of the henequen strands length on the intrinsic tensile strength of the strands was stablished as the linear regression of the Weibull strengths against the fiber lengths. This provided a probable intrinsic tensile strength for the strands (The components of the reinforced material (PP, HS/FG and MAH-PP) were compounded using a Brabender Plastograph™. The mixing procedure was carried out at 180 °C, at 80 rpm for 8 min. Afterwards, the composite was collected and pelletized (Retsch™ SM 100 pelletizer, equipped with a set of knifes and a grid) to a nominal particle size of 6 mm, suitable to be injected. Before any further processing, the pellets were placed in an oven to remove any remaining moisture until constant weight was achieved. Composite specimens for mechanical characterization under tensile loads according to ASTM D638 standard were injection-molded in a Meteor-40 injection-molding machine (Mateu & Solé) equipped with three heating areas working at 175°, 175° and 190 °C, being the highest one corresponding to the nozzle. The first and second pressures were 120 and 37.5 bars, respectively.To determine the influence of lignin and extractive contents to the matrix-strand bonding, a delignification process was carried out. A dispersion of henequen strands (HS) (20 g l−1 of consistency) was treated with sodium hydroxide (70 g l−1) and anthraquinone (1 wt% regarding HS content). The reaction was conducted at 95 °C and atmospheric pressure for 2 h. This treatment allowed the elimination of a part of lignin, hemicellulose and extractives. Finally, the strands were washed with water and dried until a constant weight was achieved.The specimen materials were conditioned at 23 °C for 48 h prior testing according with ASTM D618. Afterwards, the composites were tested using a Universal testing machine (Instron™ 1122) at a speed rate of 2 mm/min, according to ASTM D790 standard specifications for tensile test.The fracture-surface area of the tensile test specimens was observed by scanning electron microscopy (SEM) (Zeiss DMS 960). This observation allowed a qualitative assessment of the degree of adhesion between the matrix and the reinforcement. The methodology used to determine the polarity of the henequen strands and the delignified henequen strands was based on the titration of a finely powdered suspension of the material with MGCh (methyl-glycol-chitosan) (Wako Chemical GMBH, Neuss, Germany) as cationic reagent to interact with the polar groups of the surface of the fibers (). The cationic reagent was added in excess and the excess that did not interacted with the surface of the substrate (material) was titrated with a solution of potassium poly(vinyl sulphate), using blue of o-toluidine (TBO) as indicator. The values of polarity are shown in terms of micro-equivalents of MGCh per gram of material.To isolate the strands after processing, different pieces of the mechanical testing specimens were subjected to a Soxhlet extraction in decahydronaphthalene, at 189 °C for 3 h. Once the strands were separated from the PP, they were filtered, washed and dispersed in distilled water. HS length and diameter were characterized by means of an optical microscope (Leica, model DMR-XA). The obtained pictures were analyzed and measured using commercial software (SigmaScan Pro5).A schematic flow-chart of the bio-based composite preparation and characterization processes is shown in ) were used as fiber reinforcement for PP composites.Compared to stone groundwood, hemp strands have considerably higher cellulose content and somehow more hemicelluloses (pentosanes); instead, the percentage of lignin in the cell wall is much less, while the extractives and ashes content is at similar level than stone groundwood. This composition is similar to that of other annual plant fibers like abaca; as shown in It is worth noticing the lignin distribution at the fiber cell wall for these two different lignocellulosic sources, depicted in the graphs from . While the lignin content decreases gradually from the middle lamella to the primary wall, and so on to the secondary wall for stone groundwood, in the case of henequen strands, lignin concentrates mostly at the middle lamella, with little content at the primary cell wall, and a still less at the secondary wall. This indicates that the external boundary of henequen strands has a lignin-like character, with an internal side strong in cellulose polymer.The raw henequen strands were more than 1 m long in length, with average diameters of 220.8 ± 106.45 μm (). These lengths allowed the study of single fiber tensile strength at different gauge lengths. Other reinforcements, like wood fibers, hemp strands… show initial lengths in the range from 0.1 to 3 mm that do not allow this kind of study (). The determination of henequen strands morphology is really important to further stablish the aspect ratio of the reinforcing fibers. Nonetheless, it is known that the morphology of the fibers changes noticeably during the mixing and mold injection processes (). Besides, the initial length of the fibers must be reduced to the order of 1 mm to allow its mixing.Tensile mechanical properties (ultimate tensile strength σt and elongation at break εmax) of the polypropylene matrix and the corresponding composites with a 40 wt% of henequen strands, at different MAPP content are shown in . Tensile strength and elongation at break of PP - 40 wt% henequen composites as function of MAPP content; in red, the maximum strength and the elongation at break for the PP matrix.Properties of short fiber composites are determined by the fiber volume fraction, the fiber dispersion (distribution), the strength of fiber/matrix interface and by the fiber aspect ratio (length/diameter) (). The tensile strength was higher for the composites compared to the plain matrix, even without coupling agent, and increased with the MAPP content up to 4 wt/wt%. Further MAPP content decreased the tensile strength, as observed in . The elongation at break, instead, decreased with the addition of henequen filaments, resembling a rigid composite material behavior. The addition of a more rigid phase decreased the ability of the composite to break under deformation. However, the presence of the coupling agent, enhanced the tensile strength of the composite in parallel with its elongation at break. The Young’s modulus of the composites at every MAPP content was very similar, with a mean value of 5.2 ± 0.1 GPa. It is accepted that the presence of coupling agents in the composites formulation has little impact on its Young’s modulus (). Thus, for materials with almost the same Young’s moduli but increasing tensile strengths, increasing strains at break are to be expected, as a direct consequence of Hooke’s law. Similar behaviors have been reported in the literature for PP reinforced with natural fibers as abaca, hemp, sisal or stone groundwood (The tensile strength of the composite is the mechanical property most affected by the strength of the fiber/matrix interface, since effective tensile stress transfer only occurs in composites with good interfacial quality (). In a fiber-reinforced composite, the fiber/matrix interface transfers an applied load to the matrix to the nearest fibers and continues from fiber to fiber via the matrix and the interface. A poor interface is also a drawback because of the differences of thermal expansions of fiber and matrix (). Thus, adhesion between fiber and matrix is a major factor in determining the response of the interface and its integrity under stress.However, without coupling agent the tensile strength of the composite improved by 27% approximately as compared with the plain polypropylene. This increase was observed despite the lack of compatibility of both constituents. The different surface polarity between the polypropylene and henequen filaments was confirmed by the analysis of surface cationic adsorption measured by colloidal titration. In this methodology, substrates adsorb an amount of methyl glycol chitosan (MGCh) cationic electrolyte, in a way that the more polymer is adsorbed onto the substrate surface, the higher is the surface polarity. For the present case, polypropylene adsorbed 4.5 microequivalents of MGCh per gram, while henequen filaments adsorbed 20.7 microequivalents of MGCh per gram. Despite the different polarity, the tensile strength augmented with the addition of 40 wt% of henequen strands. The fiber-matrix wettability is dominated not only by the interface compatibility but also by other factors such as the fiber surface morphology and porosity (). The irregular surface morphology of henequen strands favors the mechanical anchoring with the PP matrix, and the porosity of henequen strands promotes the inter-diffusion of the PP matrix along the fiber. This justifies the increase of the tensile strength of composites reinforced with henequen strands. Similar results have been observed in composites reinforced with other natural fibers (By the addition of 2–4 wt% of the coupling agent, the tensile strength improved by 55 and 71% with respect to the non-reinforced matrix, and by 23 and 35% with respect to the uncoupled composite. These increases are in the line of those observed for stone groundwood reinforced PP or hemp strands reinforced PP (). For the elongation at break, the elongation capacity is larger with coupling agent. This is in line with the literature that shows that the coupling agent has little impact on the Young’s moduli. Thus, materials with the same Young’s modulus will show higher strains at break the higher the tensile strength is (The overall improvement is attributed to the compatibility effect of the coupling agents. It is assumed that the maleic groups of MAPP undergo covalent linkages with the hydroxyl groups of cellulose that improve the stress transfer between both constituents. Other chemical interactions, such as hydrogen bonding and Van de Waals forces can also take place, although their contribution to the stress transfer will be less significant (In order to study the chemical interaction between MAPP coupling agent and the hydroxyl groups of the henequen filaments, several samples from dry henequen strands were submitted to a controlled modification reaction with MAPP reagent under continuous refluxing of toluene. After completing the reaction, the materials were totally dried and subsequently washed in a soxlhet apparatus with xylene, to remove any un-covalently linked reagent on the strands surface.The ATR-FTIR of the treated and un-treated strands is shown in A. Characteristic absorption peaks around 2900 and 2950 cm−1 of the aliphatic C–H of the polyolefin fraction of MAPP, and the presence of a new absorption band at 1460 cm−1 corresponding to the associated bending vibration of the same aliphatic bonds were found in the MAPP-modified henequen filaments. The confirmation of the chemical bonding between the MAPP coupling agent and the henequen strands was evidenced in the region from 1700 to 1800 cm−1. B shows the spectra amplification (in descendant order): MAPP spectrum, henequen strands, MAPP treated henequen strands and the digital subtraction of the un-treated strands from the treated one. In the MAPP spectrum, the absorption band at 1775 cm−1 corresponded to the vibration of the carbonyl group of the anhydride function and the absorption band at 1713 cm−1 was associated to the tensile vibration of the carboxylic groups in the MAPP structure, due to the hydrolysis of some anhydrides groups. As a result, the subtraction of the spectrum of un-treated henequen strands from the treated one gave three new absorption bands drawn at 1776, 1752 and 1717 cm−1. Thus, while the absorption bands at 1776 and 1717 cm−1 were assigned to the presence of un-reacted MAPP in the treated strands, the absorption peak at 1752 cm−1 was assigned to the ester group, consequence of the chemical reaction of the anhydrides of MAPP and the hydroxyl groups of the henequen strands. Felix and Gatenholm studied the esterification reaction of some cellulosic fibers with waxes like MAPP and reported of absorption bands at these frequencies (). Therefore, the high increments of the tensile strengths found in henequen strands/PP composites comprising 2 and 4% of MAPP can be attributed to the chemical reaction between the MAPP and the hydroxyl groups of the henequen strands.SEM microphotographs of the fractured sections of the samples submitted to tensile test, the pulling out of the fiber for the uncoupled composites demonstrates the poor adhesion between fiber and matrix (A); instead, a much better wettability is evidenced for the coupled composites (B). The uncoupled composites present a very clean hollow, without any remaining residual fiber. It can also be observed a partial hollow with the same characteristics. Presumably the fracture propagated along the interface, the weakest phase of this composite. (). On the other hand, the coupled composite shows a fiber with a fragile fracture, showing that the fracture preceded from the matrix to the fiber, being the interface stronger.The tensile strength of the composites comprising 6 or 8 wt% of MAPP decreased progressively about 8.5 and 12.6%, with respect to the highest obtained value, respectively. Different phenomena contribute to these decrease. On one side, there is a lower amount of PP matrix in the final formula, and also the number of possible bonds with the surface hydroxyl groups becomes saturated. On the other hand, the molecular weight of the maleated PP is lower than that the matrix, and this increases the dispersity of the final properties of composite materials (). The number of surface hydroxyl groups of the henequen fibers is limited, due to the presence of the other components (lignin, extractives and waxes) that reduce the surface-hydroxyl group accessibility. This fact was experimentally supported by a delignification reaction. Henequen fibers were subjected to a NaOH/anthraquinone treatment according to the procedure described previously. The reaction yield was 86.8% (equivalent to a weight loss of 13.2%, corresponding to extractives, waxes and partly lignin). The treated fibers were used as reinforcement of PP-composites up to a 40 wt% content and a 4 wt% of MAPP. The tensile strength obtained for the composite was 51,3 ± 0,7 MPa, representing 1.18 times the tensile strength of the analogous composite reinforced with non-treated henequen fibers.The objective of micromechanical study is to determine the contribution of each constituent to the macro-property of the composite. All the phases in the composite (matrix, reinforcement, and interface) are involved during the stress transfer progression in the composite material. A first approach to the micromechanics modelling was carried out using a modified rule of mixtures (mRoM), in Eq. Where σtC is the composite tensile strength, fc is a coupling factor, σtF is the intrinsic strength of henequen strands, VF the volume fraction of reinforcing fibers, and σtm* the tensile strength of the matrix at the composite failure.The intrinsic strength of henequen strands was determined from four set of approximately 60 fibers undertaking single tensile test. Single henequen strands were tested at different gauge length of ¼”, ½”, ¾” and 1″ (6.35, 12.70, 19.05 and 25.40 mm) respectively. The experimental values were submitted to Weibull analysis, and all the results are presented in As expected, the experimental intrinsic tensile strengths of the fibers showed a high standard deviation. The strength of a brittle natural fiber is highly dependent on its length, because long fibers show higher probability of finding surface imperfections; therefore, the mean strength at failure of single filaments decreased with the gauge length in the test (). The specific intrinsic tensile strengths computed using the experimentally measured density (1.48 g/cm3), are far below from other reinforcements considered of high quality; cotton or jute, for instance showed specific strengths between 900 and 500 MPa/g cm−3 at gauge lengths between ¼” and 1″, respectively, and glass fibers have specific tensile strengths of 580 MPa/g cm−3 for the same gauge length (). From the linear representation of the probability of failure versus the applied stress (), the Weibull shape factor (β) and the characteristic fiber strength (η) were determined (Weibull shape modulus (β) is a characteristic parameter of the material and quantifies the scatter of the intrinsic tensile strengths; lower β values indicate higher scattering of the tensile strength results. The values were similar at every gauge lengths, and also similar to other natural fibers (). The characteristic fiber strength values (τ) were superior to the experimental ones, and remained similar at all gauche lengths. Still, longer fibers showed lower intrinsic tensile strengths, as expected, due to the presence of defects in their surfaces.The characteristic tensile strength of henequen filaments (η) behaved linearly with the fiber length according to Eq. , with a regression correlation of R2 = 0.99., it is possible to estimate the characteristic strength of the henequen strands by knowing the average length of the reinforcing fibers into the composite.The distribution of the fiber lengths after and before composite mixing are shown in The mean arithmetic length of the henequen fibers before and after mixing were 652.05 μm and 485.04 μm, respectively. It was clear that the strands were submitted to severe attrition phenomena that decreased its mean length 34.4%. The figure shows how the long percentage of long fibers decreased while the short ones increased. Based on the morphology, and according to Eq. , the theoretical characteristic fiber strength of henequen strands would be 450.7 MPa. In the present case, the intrinsic tensile strength for these henequen is in the range described in the literature, that shows values between 620 and 440 MPa (Nonetheless, the obtained value allowed computing the coupling factors of the composites (From our experience, and from the literature (), coupling factor values around 0.2 are equivalent to strong fiber/matrix interfaces in the case of semi-aligned short fiber-reinforced composites (). This was accomplished thanks to the presence of 4 wt% of MAPP, and the interface of such composites can be qualified as strong (In connection to the quality of the interface, the interfacial shear strength (τ) is the parameter related to the ultimate stress transfer along reinforcement’s surfaces. The interfacial shear strength (τ) accounts for the ability of the interface to transmit loads from the matrix to the fiber (). In the literature, this parameter has been defined according to Von Mises (VMC) and Tresca criteria (TC), Eqs. , respectively. These criteria stablish an upper and lower limit, respectively, for the interfacial shear strength of composites with strong interfaces:where σm is the tensile strength at break of the matrix. According to this, a strong interface will show interfacial shear strengths between 15.9 MPa (Von Misses) and 13.8 MPa (Tresca), with a mean value of 14.9 MPa. The morphology of the fibers, mainly their mean length and diameter, has a remarkable role in the value of the interfacial shear strength. The henequen fibers extracted from the composites had mean weighted length (LF) of 618.15 μm and an average diameter (DF) of 25.4 μm. Knowing the definition of the critical fiber length (Lc) from the shear lag model, this parameter is impacted by the tensile strength of the composite (σtC), the fiber diameter (DF) and the shear strength (τ) (Eq. Thus, the critical fiber length can be obtained from the interfacial shear strengths by Von Mises and Tresca criteria. The obtained critical lengths range from 360 to 414.7 μm, respectively. Therefore, the experimental mean length of the reinforcing fibers is above the Lc. For supercritical values of the mean fiber length, a more complex equation to preview the interfacial shear strength of composite interfaces can be applied (Eq. ), including the intrinsic fiber tensile strength (σtF), the fiber morphology, and the length-and-interface factor (χ2) (The length-and-interface factor (χ2) relates to the coupling factor (fc) by means of the fiber orientation factor (χ1) according to fc=χ1∙χ2 (). Therefore, fiber orientation factors (χ1) can be deduced from the coupling factor (fc) and the length-and-interface factor (χ2), for each shear strength value (τ).The value of the coupling factor (fc) is higher than the expected for these kind of composites, with usual values in the range from 0.16 to 0.2 (). Thus the use of a more sophisticated model was proposed. In this sense the modified Kelly-Tyson equation has shown (σtC=χ1·∑i=0i=Lcτ·liF·ViFDF+∑j=Lcj=∞σtF·VjF·1-σtF·DF4·τ·ljF+1-VF·σtm*, the DF and lFi,j terms represent the fiber diameter and fiber lengths, respectively, and χ1 is the orientation factor, introduced to modify the original Kelly and Tyson model developed for totally aligned reinforcements ().The equation uses the fiber length distribution presented in . It is clear that the increase of short fibers can increase the percentage of subcritical reinforcements inside the composite, decreasing the strengthening capabilities of the reinforcement. Thus the research of mixing methods that reduce the fiber shortening phenomena is of importance to obtain competitive composites. The literature shows how changing a mixer can increase noticeably the aspect ratio of the reinforcements and the tensile strength of the composites (Using the previewed interfacial shear strengths and the intrinsic tensile strength of the henequen fibers obtained by means of the Weibull analysis, it is possible to solve the equation and obtain a value of the orientation factors (In short fiber injected composites, the expected value for the fiber orientation factor are found between 0.25 and 0.35 () provide values around the upper limit. The solution to the Kelly and Tyson modified equation provided by Bowyer and Bader was used to obtain the value of the interfacial shear strength from the experimental data values around 15 MPa were obtained (All in all, the theoretical Weibull characteristic strength deduced for henequen strands (σtF=450.7MPa) is appropriate, although it is probably underestimated since higher values of intrinsic strength of the reinforcing fibers would lead to coupling factors and to fiber orientation factors within the range. According to other authors (), it is highly desirable to be able to predict composite properties from data on fiber properties, however, more studies need to be completed for a proper relation between plant fiber properties and the composite behavior.Bio-based composite materials from henequen strands as reinforcing elements and polypropylene as matrix were produced. Henequen strands show higher cellulose content and intrinsic tensile strength in the range of 500 MPa. In the current work, the authors stablish a relationship between the length of the reinforcements and its characteristic strength. Henequen strands, with a mean length of 485.04 μm, showed a theoretical Weibull characteristic strength of 450 MPa. The tensile properties of Henequen strands-polypropylene composites were evaluated and a strong interface was obtained by using 4 wt% of coupling agent. The chemical bonding between Henequen strands and the matrix was proved and assessed by FTIR. A modified rule of mixtures and a modified Kelly-Tyson equation were used to determine the contribution of each phase to the macro-property. The interface was rated as strong and values for a coupling factor and an interfacial shear strength for such interfaces were proposed attending to the literature. Several micromechanics models were used to obtain values for the orientation factor. The obtained values were inside the range considered plausible for semi-aligned short fiber reinforced composites. Nonetheless the values showed a deviation towards the high boundaries. Thus more research is needed to stablish if the Weibull analysis can underestimate the real intrinsic strength of the reinforcements. The values were obtained from raw reinforcements, and it is possible that the preparation processes impact its morphology and intrinsic properties. The use of a more complex micromechanics models is proposed to assess the quality of the obtained intrinsic tensile strengths.Influence of an additional elastic stress on dry wear behaviour in reciprocating testsThis paper is concerned with an experimental investigation to assess the influence of the additional elastic bulk stress state, as imposed by an external load, on dry reciprocating wear. Results obtained from dry reciprocating wear tests on an Al7175 alloy, in contact with a 34CrNiMo6 steel pin, are presented. It is shown that the additional elastic bulk stress has a substantial influence on the wear behaviour of the materials used in this study. It is, therefore, pertinent to take it into consideration in wear assessment. A physical model, based on micro and macro surface roughness, capable of interpreting the experimental obtained results has been developed. The model proposed is a modification of the Archard's model for the prediction of wear volume, in order to take into account elastic bulk stresses imposed by external loading.In practical applications, when two components are in contact, besides the local contact stresses, one of the components may be under an additional stress state, either in tension or in compression. Perhaps the most common way in which additional stress states exist in a component is when residual stresses are introduced in the component surface, due to plastic deformation. It is also possible to have an additional stress state in a component when there is thermal deformation of the component, as is the case of disc brakes. An additional stress state may also exist if an external load is applied on the component, thus introducing in it elastic tensile or compressive stresses. Independently of the way the additional stress state is introduced in the material it is reasonable to expect that this additional stress state may affect its wear behaviour.While technical literature presents different studies relating wear with additional stresses as introduced with plastic deformation (residual stresses) As wear is sometimes directly related to hardness or, in other cases, with residual stresses (directly or indirectly assessed), a brief state of the art of those relationships is subsequently presented. Traditionally, wear is correlated to surface hardness of the materials , some studies relate directly wear to the residual stresses ). Residual stresses are commonly related to hardness and, therefore, wear relations based on hardness are used.Most of the researchers consider important to express the material response in wear as a function of a basic material parameter as hardness The basic model that is usually used to describe the wear mechanism is the Archard's linear wear equation where V is the volumetric material loss of the body, K the wear coefficient (it is dimensionless and always less than unity), H is the hardness of the softer body in the contact, Fn is the applied normal load and S is the sliding distance.There are two ways of introducing stresses on the material without changing its chemical composition: (i) introducing residual stresses by plastic deformation of the material surface by using shot peening, laser peening, etc; (ii) introducing an additional stress state by externally loading the component. In the latter case the component is under an elastic additional stress state.Research works concerning the relation between hardness and the stress state of the material, when the stress state is in the elastic domain as introduced, for example, by imposing an external load to a component, are scarce. In the studies of Kokubo and Sines and Carlson as reposted in Ref. Studies on the effect of an additional stress state (stress introduced by an imposed external load) on the wear behaviour of a component are very limited. To the authors knowledge only one study, conducted by Mitchell and Shrotriya As the influence of additional elastic stress states on wear behaviour is still far from being understood, the present experimental investigation has been carried out. In this study, the influence of an additional elastic stress state, imposed by an external load to the specimen, on dry reciprocating wear and on hardness was assessed. Results obtained from dry reciprocating wear tests on Al7175 alloy, in contact with a 34CrNiMo6 steel pin, are presented.It is shown that the additional elastic stress may have a substantial influence on the wear behaviour of the component, but it has not been possible to correlate the hardness values with the additional elastic stress state. A physical model, based on the micro and macro contact surface roughness is presented. This model seems to be able to explain the results obtained. In order to predict the wear volume due to the additional elastic stress state, a modification of the Archard's model is also proposed.Reciprocating pin-on-plate wear tests has been carried out. The specimens used in the wear tests were machined from Al7175 alloy bars and the pins from 34CrNiMo6 steel bars. The mechanical properties of these alloys are given in . The microstructures of these alloys are shown in , and were obtained by SEM in the Laboratory of Materials Characterization, University of Minho. Equipment: Leica Cambridge S360. shows the specimen used in the reciprocating wear tests. It has a simple shape with filets in both sides where the axial load is applied. The cross section is a rectangle. The pin has a cylindrical shape with 4 mm radius and spherical contact surface.A reciprocating pin-on-plate tribometer PLINT-TE67 was used to evaluate the wear characteristics of Al7175/34CrNiMo6 contact pairs. In order to apply the additional elastic stress state (either in tension or in compression) on the specimen, a new device that used a screw to introduce the load and a load cell to measure its value, was specifically designed and constructed to work in conjunction with a pin-on-plate testing machine. A picture of this device is presented in shows schematically the tests performed in this work: (i) applied normal load (Fn) on the pin and alternative displacement (d) of the specimen (the specimen is not subjected to an additional elastic stress state); (ii) applied normal load on the pin, alternative displacement of the specimen and additional elastic tensile stress on the specimen; (iii) applied normal load on the pin, alternative displacement of the specimen and additional elastic compressive stress state on the specimen. Experiments were carried out with a normal load of 150 N and a relative displacement amplitude of 2 mm. All tests have been stopped after a sliding distance of 43 m. The frequency of the tests was kept constant for all tests at 1 Hz. The additional elastic stress state of the specimen was 208 MPa under tension and −208 MPa under compression. The applied peak contact pressure was 132 MPa obtained by the Hertzian equation. Five tests were performed for each type (i, ii, and iii) of tests. All the tests were performed in laboratory environment. The wear volume of the specimen was obtained as the product of the area of the central profile of the wear scar by the scar dimension perpendicular to the sliding direction The pin is pressed against the specimen by applying pneumatically the normal load which induces the contact pressure between the pin and the specimen. The top of the pin was fixed to the load arm with a chuck (holding device consisting of adjustable jaws).The states of compression or tension of the specimen were accomplished using screws and nuts placed in both sides of the wear device (see ). The desired value of the axial load applied to the specimen was set up by means of a load cell mounted on the specimen, in association with a data acquisition system (Spider 8) conveying information to a personal computer (program used–Catman 3.1). Another data acquisition system associated to the tribometer recorded different testing variables including sliding distance and friction coefficient. The surface of specimens and pins were polished with abrasive paper, finished with diamond spray (1 μm), and then cleaned ultrasonically in alcohol as to provide a standard surface.It is already known that the level of the residual stress depends on the manufacturing process and the type of material, and it may or may not be significant. In order to ensure that the material would be entirely free from initial residual stresses induced during manufacturing and processing, the specimens were heat-treated prior to the test, according to standard heat treatments suggested for the material by its supplier. The heat treatment procedures were conducted in a furnace with Argon-inert atmosphere. The specimens were maintained for 1 h at around 250 °C and then let to cool.In order to have a good characterization of the contact surface, after each test the specimens were ultrasonically cleaned to eliminate as much debris as possible. Then the following independent techniques were used: shows the high resolution images (obtained by SEM) of the wear surfaces for specimens under three additional stress states: compression, zero stress and tension.A Mahr Perthen profilometer was used to determine the wear scar volume in all specimens. Five measurements of profile in the longitudinal and transversal directions of each specimen were carried out. Depth, length and area of the wear scar were obtained as the average of the five measurements. These values were introduced in AutoCAD in order to calculate the wear volume.Five measurements of the roughness, Ra, in the longitudinal and the transversal directions on each wear track were performed. The average roughness Ra is the average deviation of the profile from the centre line over a length, L, of assessment The measurements of the average roughness, Ra, were performed according to the DIN 4768 standard. The form and the waviness are suppressed with a waviness filter. Waviness will place a mean line into the measured profile. Evaluations based on this line do not take into account the waviness contained therein and accept, as planned, only the remaining roughness.Vickers hardness tests were conducted according to standards, using a macro-hardness tester with a Vickers pyramidal indenter. Vickers hardness was measured for all three additional elastic stress conditions, from tension to compression.Five hardness readings were taken for each stress condition from which the average value was calculated. Two series of hardness tests were carried out with to different normal loads, 15 and 30 N.Numerical simulations were carried out in order to assess what would be the effect of the additional elastic stress on the deformation of the specimen and consequent average roughness of the surface. The COSMOSWorks code has been employed for the numerical simulations. Some linear analyses were performed applying an additional stress level on the bulk material of ±208 MPa, as introduced by the external load. The additional elastic stress level is the same used on wear tests. The model was meshed with linear tetrahedral elements (4 corners node connected by six straight edges). The global size of the element is 0.2 mm. This mesh size proved to be adequate for the linear analysis.The material used in this study, Al7175 alloy, experienced a substantial change on the wear behaviour when the stress on the specimen was changed from compression to tension. The results of the reciprocating wear tests are presented in presents the experimental wear volume as a function of the additional elastic stresses. The wear volume decreased about 43% when the specimen was under tensile stresses and increased about 19% when the specimen was under compressive stresses, as compared to no additional elastic stress state. From the results presented on it can be seen that the additional stress state, both in tension and in compression, have a substantial influence on wear, being therefore appropriate to consider it in the wear predictions. It is worth nothing that the additional stress level as introduced by the external load is ±208 MPa. As the yield strength of the material is 461 MPa (), it means that the specimen is in the elastic domain, the stress level being about 45% of the yield strength.In order to quantify the wear volume, the Archard's model (Eq. ) was modified in order to take into account the effect of an additional elastic stress, either in tension or in compression, imposed by an external load. The modified Archard's model can be expressed in the following mathematical form:where: σadd is the additional elastic stress and σr is the tensile or rupture strength.The wear coefficient, K, was defined in this work as the wear volume, V , divided by the normal load, Fn, and also divided by the sliding distance, S. The wear coefficient was determined for the zero load case and was used in compression and tension cases. The wear coefficient may change for other additional elastic stress states but in this case the same wear coefficient was used for all the cases as no significant difference on hardness was obtained between different stress states ( and it can be seen that there is a good correlation between experimental and predicted values.In order to physically explain the wear behaviour when the material is under an additional stress state, in the elastic domain, the following aspects will be discussed subsequently: hardness; wear mechanism; additional stress state (tensile or compressive) with plastic and elastic deformation; micro-roughness as a function of grain size and orientation and macro roughness as a function of the material finishing process. shows the surface hardness as a function of the additional bulk stresses. The main point to highlight is that hardness did not change significantly with the additional stress state. It can be seen a small raise of the hardness value when the material is subjected to compression. Under tension no substantial change in hardness was observed. Moreover, the changes are very small when compared to the standard deviation. It should be mentioned that the measurements of hardness were not taken on the wear track but on the fresh surface specimen, only as a function of the additional elastic stress state.Thus, it is considered that no correlation can be drawn between additional elastic stress and hardness. This effect has also been observed by Kokubo and by Sines and Carlson Thus, a first aspect to take into consideration is that when elastic additional stress states exist in the material they may substantially interfere in its wear behaviour, as obtained in this study and in Ref. Scanning electron microscopy studies showed a similar wear mechanism for all three test cases (). The worn area is of an oval shape, being possible to observe that the material is plastically deformed in the sides in all cases.SEM was also used to determine if there was material transfer from one surface to the other. No transfer was found between matting materials in none of the three tests. So observed differences in wear volume can not be attributed to the material transfer effect.Additional stress states can be introduced with severe surface plastic deformations, introducing surface residual stresses, and with elastic deformations due to external loads or due to thermal deformations.Results with residual plastic compressive stresses show that they are in general beneficial to wear behaviour Thus, the presence of residual stress states as obtained with severe plastic deformations of the material, generally gives rise to a beneficial effect of compressive stresses on wear.) show the same tendency of those of Ref. A first point to highlight at this moment is that the results are completely different if the additional stresses are elastic or plastic. Under compressive plastic additional stresses, as observed with residual stresses, the wear behaviour is improved A physical model that may be able to explain the wear behaviour under additional elastic stress states, as obtained in the present paper, will be subsequently presented. The physical model is based on both grain shape and orientation, and on micro and macro-roughness.Aluminium, or any other metallic material, under an additional stress state (either tensile or compressive) will experience deformation changes in grain shape. This may provide an explanation for its different wear behaviour under different additional stress states, as will be subsequently explained.Results show that there is higher wear resistance under tensile stresses (lower wear volume—) and a lower wear resistance under compressive stresses (higher wear volume). When the material is either under tensile or compressive stress, there occurs deformation of the crystalline network (based on Hooke's law). Under tensile stresses the grains will elongate in the loading direction and will be aligned with the direction of the wear track. In the perpendicular direction grains will decrease in size (Poisson′s coefficient) A physical model of wear based on grain orientation when the material is subjected to an additional elastic stress state is shown schematically on . Under tensile stresses the grains structure will behave like fibers, resulting in a very strong structure. Belomte et al. Under compressive stresses, as schematically shown in , there are more interfaces or bonds, due to the compressive stresses some grains are also pulled-out being then easily removed by the sliding pin, this resulting in a poor wear resistance. Thus, this process increases simultaneously the number and the height of the micro-asperities making it easier for the pin to remove more material, thus creating a bigger depth along the wear track and increasing the wear volume in comparison with the other stress levels—no additional stress and tensile stress cases. Therefore, the number of micro-asperities as well as their height changes with an additional stress state, being smaller under tensile additional stresses and higher under compressive additional stresses, as compared with the zero load case. Thus the effect of micro-roughness would be in accordance with the wear results in this study where wear increases for elastic compressive stresses and diminishes for elastic tensile stresses.This effect of grain size deformation on roughness due to additional elastic stress states may be considered a micro-roughness effect because the asperities are at the grain size level. Roughness due to machining or other finishing process originates a macro-roughness effect. Both the macro-roughness effect, along with the micro-roughness effect may be able to explain the difference in wear due to the additional elastic stress state, as will be subsequently shown.A physical model of classical sliding wear has been presented in Ref. However, in reciprocating wear tests the phenomena may be different. Accepting that roughness is an important variable during the whole wear test, it is shown on what would be the effect of additional stresses on the changes in shape of the asperities. Under tensile stresses (b) the asperities will be elongated in the loading direction (based on elastic deformation theories—Hooke's law). Roughness will then decrease and this will lead to an improved wear resistance. Under compressive stresses (c) the asperities will be sharper giving rise to deeper peaks and valleys this resulting in an increase in wear volume.In order to validate the physical model presented in , a numerical simulation on a surface with asperity peaks similar to the roughness peaks on ). The deformation scale is amplified 1665 times in order to highlight the evolution of roughness. d presents the results of the numerical simulation (the deformed shape). It can be clearly seen that under compressive stresses the peaks height increases while when the material is under tensile stresses the height of the peaks decreases.In order to see if in real tests this effect would be observed, measurements of roughness after tests performed with different additional elastic bulk stresses (positive and negative) were performed. shows the roughness of the wear track (both the longitudinal and the transversal directions) as a function of the additional bulk stresses after 43 m of sliding distance. It can be seen that the roughness values are higher in tests performed with bulk additional compressive stresses then in tests performed with bulk additional tensile stresses. These results show that during the wear tests the roughness value on the contact region, not only is higher due to the initial compressive additional stresses in the specimen, as shown on , but remains higher during the whole process, thus facilitating material removal, and therefore decreasing the wear resistance. Under additional tensile stresses the specimen's roughness decreases, remaining lower during the whole test thus increasing the wear resistance.Thus, the macro-roughness effect is also in accordance with the wear results in this study where wear increases due to the additional elastic compressive stresses and diminishes with additional elastic tensile stresses.It is interesting to observe that either based on micro-roughness effects, on macro-roughness effects or on both effects at the same time, the evolution of wear with additional elastic stress states can be explained based on the previous physical model. Whether the micro-roughness or the macro-roughness is predominant on wear behaviour will depend on the surface finishing state of each testing case.From the present investigation of the wear behaviour of the Al 7175 alloy when subjected to additional elastic stress states, the following conclusions can be drawn:Additional stress states in the elastic domain may have a substantial influence on the wear behaviour of the studied aluminium alloy. Wear increases for elastic compressive stresses and diminishes for elastic tensile stresses. Thus, the additional stress state should be taken into consideration in the wear evaluation. This effect is not taken into account in existing models.The wear behaviour of the Al7175 alloy under different additional elastic stress states (either tensile or compressive) may be explained by micro-roughness (grain shape changes) and /or by macro-roughness states. The application of compressive stresses increased the roughness of the wear scar, while the tensile stresses decreased it.A modification in the Archard's model has been proposed in order to incorporate the effect of and additional stress (either tensile or compressive), giving a good correlation between predicted and experimental wear volumes.A particle-element contact algorithm incorporated into the coupling methods of FEM-ISPH and FEM-WCSPH for FSI problemsNumerical simulation of FSI problems is one of the most important topics in computational fluid dynamics. In this paper, a particle-element contact algorithm is incorporated into coupling methods of FEM-ISPH and FEM-WCSPH for solving FSI problems. The objective of contact algorithm is to adjust positions and normal velocities of slave particles and master nodes by conservation of linear momentum and angular momentum. Compared with particle–particle contact algorithm, which is based on contact force of Monaghan boundary condition, the calculation of contact force is not required in the present contact algorithm. Moreover, correction algorithms of Yildiz et al. are used for both WCSPH and ISPH to treat noises in fluid field and improve the accuracy of numerical simulations. Numerical examples investigate the comparison of particle-element contact algorithm and commonly used particle–particle contact algorithm, and it indicates that the present contact algorithm is effective for FSI problems.Fluid–Structure Interaction (FSI) is an important problem in computational mechanics. For example, the drive to model nonlinear flutter response has spawned in computational aeroelastics (). The response of cardiac, arterial and respiratory systems is crucial in the computational biomechanics (). In recent years, many numerical methods have been proposed for these FSI problems with complex geometry (). Most of these numerical methods have been proposed for FSI problems by Eulerian approach in fluid medium and Lagrangian approach in solid medium (), such as Arbitrary Lagrangian Eulerian (ALE) method (). However, ALE method is time consuming to track the moving interface. Because Lagrangian meshless methods can naturally handle moving interface with large deformations of fluid, pure Lagrangian method may be attractive for FSI problems.Smoothed Particle Hydrodynamics (SPH) is a Lagrangian meshless method, which has been originally developed by . It has been successfully employed in engineering problems, such as astrophysics, fluid mechanics, solid mechanics and etc. ) There are two principal variants of SPH to impose the incompressibility constraint of fluid, namely Incompressible SPH (ISPH) and Weakly Compressible SPH (WCSPH) methods. ISPH is based on velocity-divergence-free projection method (). In this method, pressure term in the conservation of momentum equation is obtained by solving a pressure Poisson equation. Velocity-divergence-free projection method has been reported to suffer from the accumulation of density error () have proposed a stable algorithm to obtain velocity-divergence-free field and constant density by solving the Poisson equation twice in each time step. Furthermore, Artificial Particle Displacement (APD) technique has been employed to treat the particle clustering and accumulation of density errors (). The scheme of APD can significantly improve accuracy by modifying the particle distributions without any further computational cost.Compared with ISPH method, WCSPH is easy to program (). In WCSPH, random oscillation of pressure is presented due to numerical noises. have illustrated that ISPH method produces more accurate pressure fields than the WCSPH method for FSI problems. However, have compared ISPH with WCSPH for free surface water flows, they have concluded that both WCSPH and ISPH can obtain the same accurate results. have applied APD to WCSPH and have compared it with ISPH, they have concluded that WCSPH can provide accurate results of pressure. Recently, have developed an improved WCSPH using Moving Least Square approach (MLS) for density re-initialization, and they have concluded that the improved WCSPH is more accurate and stable than ISPH for incompressible flows. have combined the density correction algorithm, the APD algorithm and Monaghan′s XSPH velocity variant algorithm for SPH method to improve accuracy of violent free surface flows.For FSI problems, the pressure and viscous stresses of the fluid can cause a considerable deformation on the solid boundary, which in turn affects the pressure, velocity and stress in fluid. Because of the advantage of FEM for solving structural dynamics and SPH for simulating free-surface fluid dynamics, FEM has been coupled with SPH (FEM-SPH) to investigate FSI problems. FEM-SPH model was proposed by to study on structure-structure impact problems. have presented an alternative FEM-SPH model for the dynamic impact problem. FEM-SPH model has also been applied to fluid–structure impact problems by and to free-surface flow interaction with elastic structures by ), the calculation of contact force is required and it is very sensitive to handle the interaction of interface. For particle–particle contact algorithm, used contact potential or Monahan boundary condition to treat contact force in SPH.In this paper, particle-element contact algorithm based on master–slave scheme is incorporated into the coupling methods of FEM-ISPH and FEM-WCSPH for solving FSI problems, which is originally proposed by for explosion and impact problems. The advantage of this contact algorithm is that the contact force is not required in calculation. Moreover, in order to treat the noises in fluid field and to improve the computational accuracy of SPH, the correction algorithms proposed by are used, i.e., the treatments of density correction, Monaghan’s XSPH velocity variant and APD algorithms are used for WCSPH, the treatments of Monaghan’s XSPH velocity variant and APD algorithms are used for ISPH. Finally, the present contact algorithm is verified and compared with commonly used particle–particle contact algorithm for FSI problems.In this work, the FEM model is based on updated Lagrangian formulations for large-deformation structure (). The principle of virtual power of FEM can be written as∫Ω∂(δvi)∂xjσjidΩ−∫ΩδviρbidΩ−∫Γtδvit¯idΓ+∫Ωδviρu¨idΩ=0where δvi is virtual velocity, xj is the coordinate, σji is Cauchy Stress tensor, bi and t¯i is body force and surface force, respectively. Furthermore Γt is the boundary of traction, and dΩ is the area of element. The governing equation of is constructed in the current configuration.Using the shape function of polynomial interpolation, ∫Ω∂NI∂xjσjidΩ−∫ΩNIρbidΩ−∫ΓtNIt¯idΓ+∫ΩNIρNJu¨iJdΩ=0∀I∉Γvwhere Γv is the boundary of velocity in the current configuration, and NI is the shape function of node I. Then, FEM formulation is given aswhere MIJ is the mass matrix, fiIext and fiIint is the vector of equivalent external force and internal force for the node I, respectively.In this paper, governing equations of incompressible fluid are the conservation of mass and linear momentum, which are expressed in Lagrangian form and given as followingApplying particle approximation of SPH, discretization of governing Equations can be written aswhere v→ is the velocity, p is the pressure, g→ is the acceleration of gravity, ρ is the density, m is the mass, and υo is the viscosity of fluid. W is smoothing kernel function with a smooth length h, and cubic Spline kernel function () is used in this paper. ∏ is the Monaghan artificial viscosity (), which is used to approximate the viscous stresses of fluid, andΠij={−απc¯ijρ¯ijhv→ij⋅r→ijr→2ijv→ij⋅r→ij<0,0v→ij⋅r→ij≥0where απ is the free parameter depending on problems, r→ is position vector, c¯ij=(ci+cj)/2 is the average speed of sound, ρ¯ij=(ρi+ρj)/2 is the average density, v→ij=v→i−v→j and r→ij=r→i−r→j is the relative velocity and position of particles, respectively. The expression of viscous term is proposed by (υo∇2v→)i=∑j=1Nmj(υoi+υoj)r→ij⋅∇iWijρj(r→2ij+η2)v→ijwhere η=0.1h is a parameter to avoid zero denominator.The ISPH method, in which pressure and velocity are proposed as primitive variables, is utilized in this paper. In this method, density of particles is constant, and standard projection method is used to solve the velocity-pressure coupling problems ( is split into two parts. The first part is predicting step based on viscosity of fluids and gravity forcesAnd the second part is correcting step based on pressurewhere v→i* is the intermediate velocity, which can be obtained by solving the conservation of momentum without pressure gradient. Intermediate velocity can be projected on divergence-free space, and the divergence of Considering incompressible condition, the divergence of v→in+1 is equal to zero, then is the Poisson equation. In order to obtain the pressure of particle, pi=[ρi∇⋅v→i*Δt+∑j=1N2mjρjpjr→ij∇iWij|r→ij|2+η2]/[∑j=1N2mjρjr→ij∇iWij|r→ij|2+η2]The velocity is calculated by pressure gradientIn the WCSPH method, the pressure is calculated by the equation of state (). In this work, the following equation of state is used for fluidThe speed of sound for particle must be chosen carefully to ensure that the fluid is very closely incompressible. The speed of sound proposed in Ref. where φ is a problem dependent coefficient, vmax is the maximum value of fluid velocity, L0 is a characteristic length, FB is the magnitude of body force, and δ is the factor of relative incompressibility or density variation.The precise calculation of density field is very critical for WCSPH. Without the density correction treatment, pressure field can oscillate excessively due to numerical noise. In this paper, a density correction algorithm proposed by where ρ^i is the corrected density, and σ is a constant which is set to unit in this paper. proposed and utilized XSPH velocity variant algorithm for free surface flows, which later had become one of the widely used and well-known numerical remedies for improving particle distribution. The expression isHere, ρ¯ij=(ρi+ρj)/2 is the average density and ε is a constant. In order to avoid the error due to the density fluctuation in velocity variant, it is modified by Homogeneity of particle distribution is quite significant to the accuracy of both ISPH and WCSPH models. Highly irregular particle distribution can break down the calculation. In order to prevent the clustering of particles, APD algorithm is developed in literature (). In this method, trajectory of particles can be disturbed by adding relatively small artificial displacement, which is defined aswhere δrik is the APD vector, β is a problem dependent parameter and is set to 0.01 for all simulations in this paper, ro is the cutoff distance, N is the number of neighbors for particle i in its supporting domain.Monaghan’s XSPH velocity variant and ADP algorithms are combined by , in which XSPH algorithm is used only for the particles on free surface, and APD algorithm is employed for fully populated flow regions. Moreover, XSPH velocity is utilized for both advancing the position of particles and replacing the fluid velocity. In this paper, correction algorithms of are used for both WCSPH and ISPH to treat the noises in fluid field.For the FSI problems in this paper, structure and fluid is simulated by FEM and ISPH methods or by FEM and WCSPH methods, respectively. Because these methods are based on Lagrangian description, interface between fluid and structure can be easily handled by contact algorithm. In this section, particle-element contact algorithm is developed and incorporated into FEM-ISPH and FEM-WCSPH methods.Particle-element contact algorithm is based on Johnson's master–slave scheme (). Slave particles of fluid are considered as circles with specified radius in 2D problems, as shown in . Considering fluid of incompressibility in ISPH and nearly incompressibility in WCSPH, the specified radius of particles is a constant and is equal to the half of initial particle spacing. Segments of structure surface are designated as master segments which are composed of two master nodes. Particle-element contact algorithm contains searching algorithm and contact algorithm. Using searching algorithm, fluid particles that contact with surface of structure are recognized. Contact algorithm can adjust the positions and velocities of the fluid particles and master segment nodes.Searching algorithm is determined by box test and crossover test (, a box test is used, x1, x2, y1 and y2 are the coordinates of master nodes N1 and N2, and coordinates of particle Ns are xs and ys. The first check for each fluid particle is to determine which master segments are candidates for interaction. The box test in indicates that fluid particle Ns can be associated with master segment N1–N2 only if it is contained within a rectangular box that extends a distance de beyond master segment N1–N2. Distance de is defined aswhere Vref is the maximum relative velocity between particle and master segment, and Δt is the time increment.If fluid particle passes the box test, it is subjected to crossover test, as shown in . This is performed to determine whether the fluid particle has crossed over the master segment by the following equationswhere δcro is the crossover distance between particle and master segment, A and B are direction cosines that are normal to external surface of the master segment, and l is the length of the segment. If is satisfied, then fluid particle can touched or crossed over the master segment line., the simplified determination of master segment or master node is given as follows:If particle falls within the region 1, it will interact with master segment that the greatest crossover occurs in all candidate of master segment.If particle falls within the region 2 for two adjacent master segments, it will interact only with the single node that is common for two segments.Otherwise, it does not interact with any master segment.Then, in the searching process, there are three possibilities for each fluid particle: (1) It interacts with a master segment. (2) It interacts with a master node. (3) It has no interaction with the structure surface. Based on conservation of linear momentum, conservation of angular momentum and the normal velocity of fluid particle is equal to the normal velocity of master segment at the fluid particle position, the positions and velocities of the master nodes and fluid particles are adjusted through the process of iterations to make sure that the each fluid particle is placed exactly at the segment. As shown in , if particle interacts with master segment, the velocities of particles and master nodes can change bywhere ΔV1m and ΔV2m is normal velocity change of master nodes N1 and N2 at the mth iteration, respectively, ΔVsm is the normal velocity change of particle at the mth iteration, ΔVcm is the normal velocity change of contact point of the segment at the mth iteration, and Ms, M1 and M2 is the mass of the particle and master nodes, separately. Furthermore, R1=l1/l,R2=l2/l, and δcrom is the normal deflection at the mth iteration. Adjustment coefficient αp is calculated bywhere Niter is the total number of iterations, and m is the current iteration number.The normal velocity change can be written as followingIf the particle interacts with the master node N1, based on the conservation of linear momentum, the normal velocity change during the iterations can be written as followingIt should be noted that only the normal velocity is adjusted in particle-element contact algorithm, and the contact interface is treated as smooth boundary.Particle-element contact algorithm is verified and compared with commonly used particle–particle contact algorithm for the analysis of FSI problems, in which particle–particle contact algorithm use Monahan boundary condition to treat the contact force (). For convenience in the numerical examples, FEM-SPH using the particle-element contact algorithm is marked as FEM-SPH-1, and FEM-SPH using the particle–particle contact algorithm is marked as FEM-SPH-2.The case of still water in a tank is used to test the convergence and to check the pressure near the wall. The tank with 4 m in length and height contains water of height H=1.95 m. Particle spacing is Δx=0.1m. The speed of sound is c0=80m/s. At the beginning of simulation, the pressure is assumed hydrostatic and the water is at rest. Velocity L2 error norm ( shows the hydrostatic pressure after 1.0 s at cross section in the middle of the domain (x=2m), and the pressure calculated by FEM-ISPH is more accurate than that calculated by FEM-WCSPH near the boundary. For the pressure of FEM-ISPH, the results obtained by the particle-element contact algorithm is more accurate than that obtained by particle–particle contact algorithm near the boundary. shows the particle distribution and pressure field of different methods after 1.0 s. It can be seen that particle-element contact algorithm is effective and accurate on the left and the right corner of the tank. However, for the particle–particle contact algorithm, which is commonly used in SPH, corners of still water in the tank are unreasonably arched due to the contact force obtained by Monaghan boundary condition. shows the convergence of velocity L2 error norm for different methods, it indicates from the results of FEM-ISPH and FEM-WCSPH that higher convergence of the particle-element contact algorithm are obtained than that of the particle–particle contact algorithm. It also can be seen from the velocity L2 error norm that the particle-element contact algorithm can obtain better accuracy than that obtained by particle–particle contact algorithm when the same particle resolution is used for different coupling algorithms.The deformation of an elastic plate due to time-dependent water pressure has been studied on experimentally and numerically by , in which a part of water is isolated by a gate with a rigid upper part and a deformable lower part of rubber. The initial configuration of problem is illustrated in . The geometric dimensions are L=0.079 m, H=0.14 m, A=0.1 m and S=0.005 m, separately. The density of water is ρ=1000 kg/m3. The material properties of elastic plate are density ρ=1100 kg/m3, Young modulus E=1.2×107 N/m2 and Poisson ratio υ=0.4. The speed of sound is c=15 m/s. Artificial viscosity with free parameter απ=1.0 is used in this case. Parameter ε=0.08 is used in Monaghan’s XSPH velocity variant algorithm. The gravitational force acts downwards with g=9.8 m/s2 and air is neglected in simulations.In FEM-ISPH and FEM-IWCSPH models, the initial particle spacing is 0.2 cm, which is corresponding to 3500 particles. And the number of quadrilateral element is 395 and 3550 for elastic plate and rigid part, respectively. shows comparison water profiles obtained by experiment and simulations. The water profiles of different methods are similar with those of experiment, and the smooth pressure fields are obtained by different contact algorithms. Furthermore, the influence of the particle resolution on different coupling algorithms is also investigated in this benchmark tests, as shown in , in which the abbreviation FPS denotes fluid particle spacing, and the particle spacing 0.002 m, 0.00133 m, 0.001 m is corresponding to 3500, 7875 and 14000 particles, separately. Compared with experimental results, the results obtained by different particle resolutions indicates that coarse particles can produce numerical results with satisfactory accuracy, and it also means that present particle-element contact algorithm is effective and accurate for simulation of FSI problems.In order to study on FSI problems with numerical results of free surface, as shown in , the collapse of water column with an elastic obstacle is discussed. For this well-known example, the initial geometry of the model is according to paper (), in which the width and height of water column is L=14.6 cm and 2 |
L, respectively. And the density of water is ρ=1000 kg/m3. The obstacle is placed at the bottom of wall at a distance L to the right of water column. The height and width of obstacle is h=8.0 cm and w=1.2 cm, respectively. The material properties of obstacle are density ρs=2500 kg/m3, Young modulus E=106 |
N/m2 and Poisson ratio υ=0. The speed of sound is c=20 m/s. Artificial viscosity with the free parameter απ=0.4 is used. Parameter ε=0.01 is used in Monaghan’s XSPH velocity variant algorithm. The gravitational force acts downwards with g=9.8 m/s2 and air is neglected in simulations. For SPH simulations, the initial particle spacing is 0.25 cm, which is corresponding to 6728 particles. The number of quadrilateral element is 486 and 1802 for elastic obstacle and rigid box, respectively. shows the numerical results obtained by the FEM-ISPH and FEM-WCSPH with different contact algorithms. It can be seen that water profiles and pressure are similar with the PFEM (), and smooth pressure can be obtained by different contact algorithms. shows the displacement at the upper left corner of baffle, and it indicates that numerical results in this paper are in good agreement with the reference solutions () before 0.4 s. All the results vary wildly after 0.4 s and the present results are mostly between PFEM results (). It shows that FEM-ISPH and FEM-WCSPH with the particle-element contact algorithm can obtain reasonable results compared with those of PFEM and SPH. The influence of the particle resolution on different coupling algorithms is shown in , in which the abbreviation FPS denotes the fluid particle spacing, and the particle spacing 0.00292 m, 0.00195 m, 0.00146 m is corresponding to 5000, 11250 and 20000 particles, separately. shows that the higher particle resolution is used, the better accuracy of FEM-ISPH and FEM-WCSPH results are obtained. It also means that present particle-element contact algorithm is effective and stable for FSI problems.In this paper, the particle-element contact algorithm is incorporated into FEM-ISPH and FEM-WCSPH models for solving FSI problems, in which treatments of density correction, Monaghan’s XSPH velocity variant and APD algorithms are used for WCSPH, and treatments of Monaghan’s XSPH velocity variant and APD algorithms are used for ISPH. Compared with the commonly used particle–particle contact algorithm, the advantage of this contact algorithm is that the contact force is not required in calculation, and it is effective and accurate for benchmark test of still water in a tank. For the benchmark test of an elastic plate subjected to time-dependent water pressure and collapse of water column with an elastic obstacle, smooth pressure can also be obtained with correction algorithms proposed by . The performance of FEM-ISPH and FEM-WCSPH models indicates that particle-element contact algorithm should be an attractive technique for solving FSI problems. Trans. Nonferrous Met. Soc. China 23(2013) 631−635 Effect of solution treatment and artificial aging on microstructure and mechanical properties of Al−Cu alloy Jae-Ho JANG 1,2 , Dae-Geun NAM 1 , Yong-Ho PARK 2 , Ik-Min PARK 2 1. Korea Institute of Industrial Technology, Busan 618-230, Korea; 2. School of Material Science and Engineering, Pusan National University, Busan 609-735, Korea Received 21 May 2012; accepted 5 November 2012 Abstract: In order to achieve good mechanical properties of Al−Cu alloys such as high strength and good toughness, precipitation hardening and artificial aging treatment were applied. As defined by the T6 heat treatment, the standard artificial aging treatment for Al−Cu alloy followed heat treatments of solution treatment at 510−530 °C for 2 h, quenching in water at 60 °C and then artificial aging at 160−190 °C for 2−8 h. The effects of solution treatment and artificial aging on the microstructure and mechanical properties of Al−Cu alloy were studied by optical microscopy (OM), scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDS), transmission electron microscopy (TEM) and tensile test. The results of solution treatment indicate that the mechanical properties of Al−Cu alloy increase and then decrease with the increase of solution temperature. This is because the residual phases dissolve gradually into the matrix, and the fraction of the precipitation and the size of the re-crystallized grain increased. Compared to the solution temperature, the solution holding time has less effect on the microstructure and the mechanical properties of Al−Cu alloy. The artificial aging treatments were conducted at 160−180 °C for 2−8 h. The results show that the ultimate tensile strength can be obtained at 180 °C for 8 h. Ultimate tensile strength increased with increasing time or temperature. Yield strength was found as the same as the ultimate tensile strength result. Key words: Al−Cu alloy; solid solution treatment; artificial aging; microstructure; mechanical property 1 Introduction Aluminum alloys are widely used in aerospace and automobile industries due to their low density, good mechanical properties and corrosion resistance [1,2]. Copper is a potent precipitation strengthening agent in aluminum alloy. Cu addition up to 5.0% (mass fraction) leads to alloys with very high strength and good toughness when subjected to natural or artificial aging [3,4]. And 2xxx series aluminum alloys have only recently become commercially available and are under development as potential precipitation hardened materials. But the conventional aluminum alloys cannot meet the higher and higher work environment because of the coarsening of its strengthening precipitates θ′ (CuAl 2 ). It is reported that the microstructure and mechanical properties of Al−Cu alloys are sensitive to structure of ingot, heat treatment and subsequent deformation condition [5,6]. In order to obtain improved mechanical properties, aluminum alloys are often subjected to different heat treatments [7−10]. During solution treatment, the alloys are exposed to high temperature corresponding to the maximum safe limits relative to the lowest melting point for each specific composition. By doing so, the soluble phases formed during solidification can be re-dissolved in the matrix. And Al−Cu alloys can be strengthened by precipitation of several metastable phases, which are produced by an artificial aging. The objective of this study is to investigate the effects of solution treatment and artificial aging on microstructure and mechanical properties of Al−Cu alloy. The relationship between the microstructure and mechanical properties were discussed. 2 Experimental The experimental Al−Cu alloys such as 2011 alloy were prepared with Al−Cu ingot by vertical continuous casting. The chemical composition of the alloy is listed in Table 1. The ingot was homogenized at 500 °C for Corresponding author: Dae-Geun NAM; E-mail: [email protected] DOI: 10.1016/S1003-6326(13)62509-1 Jae-Ho JANG, et al/Trans. Nonferrous Met. Soc. China 23(2013) 631−635 632 Table 1 Chemical composition of ingot used in this study with specification for 2011 Al alloy (mass fraction, %) Alloy Si Fe Cu Bi Pb Zn Al Specification 0.4 0.7 5.0−6.0 0.2−0.6 0.2−0.6 0.3 Bal. Al 2011 ingot 0.38 0.48 5.42 0.32 0.26 0.22 Bal. 3 h. As defined by the T6 heat treatment, the Al−Cu alloy followed heat treatments of solution treatment at 510−530 °C for 2 h, quenching in water at 60 °C and then artificial aging at 160−190 °C for 2−10 h. Room temperature mechanical properties tests were performed on Instron 5985 universal testing machine. Metallographic microscope (GX51−223B, Olympus) was employed for metallographic microstructure analysis. The specimens were prepared through a conventional mechanical polishing followed by etching with Keller reagent (2 mL HF, 3 mL HCl, 5 mL HNO 3 and 190 mL water) for OM and SEM observations. And TEM observations were performed on transmission electron microscope (TECNAI F20). The specimens for TEM observation were prepared by the standard twin-jet electropolishing method using 80% methanol and 20% nitric acid solution at −25 °C. 3 Results and discussion Figure 1 demonstrates the microstructures of as-cast Al−Cu alloy. A typical billet structure consisting of dendritic α phases is given in Fig. 1(a). Serious dendritic segregation exists in the billet. And large amounts of intermetallic phases in the interdendritic region are presented. According to the EDX analysis, the secondary phases are CuAl 2 (Fig. 2). Most residual phases are expected to be dissolved into the matrix by solution treatment at high temperature in order to get the solute atoms as much as possible [11]. However, higher temperature increases the possibility of melting of the residual phases. Thus, it is necessary to optimize the solution temperature. Figure 3 demonstrates the curves of mechanical properties of the aged Al−Cu alloy solution treated at different temperatures. It is obvious that the tensile strength of the alloy increases with increasing the solution temperatures, reaches a maximum value of 342 MPa at 525 °C, and then decreases with further increasing temperature (Fig. 3(a)). The elongation increases and then decreases with increasing the solution temperature. And the elongation value is 31% when the alloy is solution treated at 525 °C. But the yield strength has no change as temperature increases. As a result, the optimum solution temperature is 525 °C for Al−Cu alloy in this test. Figure 4 presents the optical microstructures Fig. 1 Microstructures of as-cast Al−Cu alloy: (a) OM; (b) SEM Fig. 2 EDX analysis of secondary phase in Fig. 1(b) of the aged Al−Cu alloy solution treated at different temperatures. It can be seen from Fig. 4 that complete recrystallization occurs in all the alloys. Fine equiaxed grains are observed in the specimen solution treated at 515 °C (Fig. 4(a)). With increasing the solution temperature, the size of the recrystallized grains increases (96.4 μm, Fig. 4(b)). As the temperature increases to 525 °C, the grain is particularly large (108.1 μm, Fig. 4(c)). And some melting compounds both in the grain boundaries and triple conjunctions are observed (Fig. 4(d)), which means that the specimen is overbunt. The solid solution was carried out at 525 °C for 2 h in order to dissolve the solute, mainly Cu present in the alloy which is responsible for hardening. The main purpose of the solution heat treatment is to obtain a supersaturated solid solution. But in order to maintain Jae-Ho JANG, et al/Trans. Nonferrous Met. Soc. China 23(2013) 631−635 633 Fig. 3 Mechanical properties of Al−Cu alloy solution treated at different temperatures for 2 h Fig. 4 Optical microstructures of aged Al−Cu alloy solution treated at 515 °C (a), 520 °C (b), 525 °C (c) and 530 °C (d) for 2 h this desired condition at low temperatures, water quenching is needed. The artificial aging stage consists of further heating the alloy at relatively low temperatures (160−190 °C), and it is during this stage that the CuAl 2 precipitation of dissolved elements occurs. These precipitates are responsible for the hardening of the material which is commonly observed in artificial aging. Figure 5(a) shows the ultimate tensile strength of Al−Cu alloy after aging. From Fig. 5(a), the increase of temperature is to be expected to dissolve the intermetallic phases into the Al matrix, thus strengthening it. The artificial aging temperatures used were 160, 170, 180 and 190 °C. The ultimate tensile strength shows the maximum after aging at 180 °C for 8 h. Tensile strength increases with increasing aging time or temperature but decreases after aging for 10 h. However, when aging at 190 °C, a decrease in ultimate tensile strength is observed, which means that the specimen is overaging with increasing aging temperature. Figure 5(b) shows that the yield strength was found as the same as the ultimate tensile strength result. Artificial aging at 160 °C does not reveal an appreciable improvement in yield strength, probably because up to this aging temperature, the GP zones and/or precipitates formed may not be sufficient enough to reflect noticeable changes in the yield strength. The same behavior is observed at 170 °C although the values attained are not as high as those obtained by aging at 180 °C. At 190 °C the decrease in yield strength, characteristic of overaging behavior, may be noted, while at 190 °C, overaging Jae-Ho JANG, et al/Trans. Nonferrous Met. Soc. China 23(2013) 631−635 634 Fig. 5 Mechanical properties of Al−Cu alloy subjected to artificial aging occurs with increasing aging time. The tensile strength values reflect the conditions of the bulk casting, since any casting defects, such as inclusions and pores, may affect the values obtained by the test. Microhardness, in turn, reflects the conditions of the Al matrix, since the test is carried out only in the matrix, and only in a small area of the bulk specimen [11−13]. In Fig. 5(c), an increase of the microhardness values obtained may be seen with increasing the aging time when aging at 160 °C or 170 °C. The hardness peak is attained when aging at 180 °C for 8 h. When aging at 180 °C a drop and a slow recovery of the microhardness with aging time are observed, indicating that the strengthening effect of the artificial aging has already been reached at lower aging temperatures. When the artificial aging temperature is increased to 190 °C, the hardness values show a decrease over time, as to be expected. It will be observed that the best combination of properties, i.e. high tensile strength and microhardness, is achieved when aging at 180 °C for 8 h. The corresponding properties consist of an ultimate tensile strength of 406 MPa, yield strength of 239 MPa and microhardness of HV124. The precipitation of Cu phases is event responsible for the improvement in properties when applying the artificial aging treatments. When aging the Al−Cu alloy at 180 °C for 8 h, the mechanical properties are the highest values. The precipitation (θ′) behaviors after aging 8 h at 180 °C are shown in Fig. 6(a). Al−Cu alloy has now begun to nucleate heterogeneously on dislocations, while the homogenously nucleated GP zones have developed into within the dislocation-free volumes of the material. The microstructure was found to consist of a high density CuAl 2 (θ′) in the background matrix. Strong (001) Al streaks can be seen in the diffraction pattern shown in Fig. 6(b). A weak set of four CuAl 2 reflections could also be seen in the (220) plane. Fig. 6 TEM image (a) and diffraction pattern (b) of Al−Cu alloy after artificial aging at 180 °C for 8 h, near [001] zone 4 Conclusions Al−Cu alloys were successfully prepared by vertical continuous casting with different process parameters, which are mainly temperature and time, in solution Jae-Ho JANG, et al/Trans. Nonferrous Met. Soc. China 23(2013) 631−635 635 treatments and artificial aging. It was found that some massive residual phases of CuAl 2 in Al−Cu alloy can be re-dissolved into matrix by solution treatment. As the solution treatment temperature increased to 525 °C, the recrystallized grain size and the fraction of the precipitation in the Al−Cu alloy increased, which also leads to increase in the mechanical properties, especially tensile strength and hardness, of the Al−Cu alloy materials. The overburnt temperature of the Al−Cu alloy was set to 530 °C, which was obtained from hardness results with corresponding microstructure changes. It can be, therefore, suggested that the optimized solution temperature and time were 525 °C and 2 h, respectively. The tensile strength and hardness of the Al−Cu alloy showed 342 MPa and HV 94. Furthermore, the mechanical properties were increased to 406 MPa for tensile strength and HV124 for the Vickers hardness by artificial aging treatment at 180 °C for 8 h after solution treatment. The improvement of mechanical properties is most likely due to the precipitation of hard CuAl 2 phase in the Al−Cu alloy. However, further researches are necessary to investigate on the corrosion resistance and machinability for application of aerospace and automobile parts. References [1] BAKAVOS D, PRANGNELL PB, BES B, EBERL F. The effect of silver on microstructural evolution in two 2xxx series Al-alloys with a high Cu:Mg ratio during ageing to a T8 temper [J]. Mater Sci Eng A, 2008, 491(1−2): 214−223. [2] SOFYAN B T, RAVIPRASAD K, RINGER S P. 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Effect of Ag content and heat treatment on the stress corrosion cracking of Al−4.6Cu−0.3Mg alloy [J]. Mater Chem Phys, 2005, 91(2−3): 454−462. [8] UNLU N, GABLE B M, SHIFLET G J, STARKE E A Jr. The effect of cold work on the precipitation of Ω and θ in a ternary Al−Cu−Mg alloy [J]. Metall Mater Trans A, 2003, 34(12): 2757−2769. [9] XIAO D H, SONG M. Superplastic deformation of an as-rolled Al−Cu−Mg−Ag alloy [J]. Mater Des, 2009, 30(2): 424−426. [10] WANG J, Y I D, SU X, YIN F. Influence of deformation ageing treatment on microstructure and properties of aluminum alloy 2618 [J]. Mater Charact, 2008, 59(7): 965−968. [11] SONG M, HE Y, XIAO D, HUANG B. Effect of thermomechanical treatment on the mechanical properties of an Al−Cu−Mg alloy [J]. Mater Des, 2009, 30(3): 857−861. [12] PFOST D, CRAWFORD P, MENDEZ J, BARROSA R, QUINTANILLA F, FLORES F, BOJORQUEZ B, GEALOGO P, FOYOS J, ES-SAID O S. The effect of solution treatment and rolling mode on the mechanical properties of 2090 Al−Li alloy [J]. Mater Proc Technol, 1996, 56(1−4): 542−551. [13] SENKOV O N, SHAGIEV M R, SENKOVA S V, MIRACLE D B. Precipitation of Al 3 (Sc, Zr) particles in an Al−Zn−Mg−Cu−Sc−Zr alloy during conventional solution heat treatment and its effect on tensile properties [J]. Acta Mater, 2008, 56(15): 3723−3738. 固溶处ç�†å’Œäººå·¥æ—¶æ•ˆå¯¹ Al−Cuå�ˆé‡‘显微组织和 力å¦æ€§èƒ½çš„å½±å“� Jae-Ho JANG 1,2 , Dae-Geun NAM 1 , Yong-Ho PARK 2 , Ik-Min PARK 2 1. Korea Institute of Industrial Technology, Busan 618-230, Korea; 2. School of Material Science and Engineering, Pusan National University, Busan 609-735, Korea 摘 è¦�：对 Al−Cuå�ˆé‡‘进行æž�出强化和人工时效处ç�†ä»¥èŽ·å¾—优异的力å¦æ€§èƒ½ï¼Œå¦‚高的强度ã€�好的韧性。其çƒå¤„ç�† 工艺æ�¡ä»¶ä¸ºï¼š510~530 °C固溶处ç�† 2 hï¼›60 °C水淬；160~190 °C人工时效 2~8 h。采用光å¦æ˜¾å¾®é•œã€�扫æ��电镜ã€� 能谱分æž�ã€�é€�射电镜和拉伸实验对ç»�固溶和人工时效处ç�†çš„ Al−Cuå�ˆé‡‘的组织和力å¦æ€§èƒ½è¿›è¡Œè¡¨å¾�。固溶处ç�†å®ž 验结果表明，Al−Cuå�ˆé‡‘的力å¦æ€§èƒ½éš�ç�€å›ºæº¶å¤„ç�†æ¸©åº¦çš„å�‡é«˜å…ˆå¢žåŠ ，然å�Žé™�低。这是由于 Al−Cuå�ˆé‡‘的残余相 é€�æ¸�溶解进入基体ä¸ï¼Œä»Žè€Œå¯¼è‡´æž�出相的数é‡�å’Œå†�结晶晶粒尺寸ä¸�æ–å¢žåŠ ã€‚ç›¸è¾ƒäºŽå›ºæº¶å¤„ç�†æ¸©åº¦ï¼Œå›ºæº¶å¤„ç�†æ—¶é—´ 对 Al−Cuå�ˆé‡‘çš„å½±å“�较å°�。人工时效处ç�†å®žéªŒç»“果表明，å�ˆé‡‘ç»� 180 °C时效 8 h，å�¯ä»¥èŽ·å¾—最大的拉伸强度。å�ˆ 金的最大拉伸强度和屈æœ�强度éš�ç�€æ—¶æ•ˆæ—¶é—´çš„延长和温度的å�‡é«˜è€Œå�‡é«˜ã€‚ 关键è¯�：Al−Cuå�ˆé‡‘；固溶处ç�†ï¼›äººå·¥æ—¶æ•ˆï¼›æ˜¾å¾®ç»„织；力å¦æ€§èƒ½ (Edited by Sai-qian YUAN) automobile parts. References [1] BAKAVOS D, PRANGNELL PB, BES B, EBERL F. The effect of silver on microstructural evolution in two 2xxx series Al-alloys with a high Cu:Mg ratio during ageing to a T8 temper [J]. Mater Sci Eng A, 2008, 491(1−2): 214−223. [2] SOFYAN B T, RAVIPRASAD K, RINGER S P. Effects of microalloying with Cd and Ag on the precipitation process of Al−4Cu−0.3Mg (wt.%) alloy at 200 °C [J]. Micron, 2001, 32(8): 851−856. [3] EDDAHBI M, JIMENEZ J A, RUANO O A. Microstructure and creep behaviour of an Osprey processed and extruded Al−Cu−Mg− Ti−Ag alloy [J]. J Alloys Comp, 2007, 43(1−2): 97−107. [4] LUMLEY R N, POLMEAR I J. The effect of long term creep exposure on the microstructure and properties of an underaged Al−Cu−Mg−Ag alloy [J]. Scripta Mater, 2004, 50(9): 1227−1231. [5] XIAO D H, WANG J N, DING D Y, YANG H L. Effect of rare earth Ce addition on the microstructure and mechanical properties of an Al−Cu−Mg−Ag alloy [J]. J Alloys Comp, 2003, 352(1−2): 84−88. [6] FERRAGUT R, DUPASQUIER A, MACCHI CE, SOMOZA A, LUMLEY R N, POLMEAR I J. Vacancy–solute interactions during multiple-step ageing of an Al−Cu−Mg−Ag alloy [J]. Scripta Mater, 2009, 60(3): 137−140. [7] CHANG C H, LEE S L, LIN J C, YEH M S, JENG R R. Effect of Ag content and heat treatment on the stress corrosion cracking of Al−4.6Cu−0.3Mg alloy [J]. Mater Chem Phys, 2005, 91(2−3): 454−462. [8] UNLU N, GABLE B M, SHIFLET G J, STARKE E A Jr. The effect of cold work on the precipitation of Ω and θ in a ternary Al−Cu−Mg alloy [J]. Metall Mater Trans A, 2003, 34(12): 2757−2769. [9] XIAO D H, SONG M. Superplastic deformation of an as-rolled Al−Cu−Mg−Ag alloy [J]. Mater Des, 2009, 30(2): 424−426. [10] WANG J, Y I D, SU X, YIN F. Influence of deformation ageing treatment on microstructure and properties of aluminum alloy 2618 [J]. Mater Charact, 2008, 59(7): 965−968. [11] SONG M, HE Y, XIAO D, HUANG B. Effect of thermomechanical treatment on the mechanical properties of an Al−Cu−Mg alloy [J]. Mater Des, 2009, 30(3): 857−861. [12] PFOST D, CRAWFORD P, MENDEZ J, BARROSA R, QUINTANILLA F, FLORES F, BOJORQUEZ B, GEALOGO P, FOYOS J, ES-SAID O S. The effect of solution treatment and rolling mode on the mechanical properties of 2090 Al−Li alloy [J]. Mater Proc Technol, 1996, 56(1−4): 542−551. [13] SENKOV O N, SHEffect of solution treatment and artificial aging on microstructure and mechanical properties of Al–Cu alloyIn order to achieve good mechanical properties of Al-Cu alloys such as high strength and good toughness, precipitation hardening and artificial aging treatment were applied. As defined by the T6 heat treatment, the standard artificial aging treatment for Al-Cu alloy followed heat treatments of solution treatment at 510–530 °C for 2 h, quenching in water at 60 °C and then artificial aging at 160–190 °C for 2–8 h. The effects of solution treatment and artificial aging on the microstructure and mechanical properties of Al-Cu alloy were studied by optical microscopy (OM), scanning electron microscopy (SEM), energy dispersive X-ray spectroscopy (EDS), transmission electron microscopy (TEM) and tensile test. The results of solution treatment indicate that the mechanical properties of Al-Cu alloy increase and then decrease with the increase of solution temperature. This is because the residual phases dissolve gradually into the matrix, and the fraction of the precipitation and the size of the re-crystallized grain increased. Compared to the solution temperature, the solution holding time has less effect on the microstructure and the mechanical properties of Al-Cu alloy. The artificial aging treatments were conducted at 160–180 °C for 2–8 h. The results show that the ultimate tensile strength can be obtained at 180 °C for 8 h. Ultimate tensile strength increased with increasing time or temperature. Yield strength was found as the same as the ultimate tensile strength result.Electrical discharge machining of Inconel 825 using cryogenically treated copper electrode: Emphasis on surface integrity and metallurgical characteristicsIn the present work, analysis of surface integrity and metallurgical characteristics of the machined Inconel 825 work surface has been carried out in relation to Electrical Discharge Machining (EDM) using Cryogenically Treated Tool (CTT) in comparison with Non Treated Tool (NTT). Degree of severity of surface cracking as well as formation of white layer onto the EDMed Inconel 825 work surface has been investigated herein. The process physics of EDM using CTT has been explained with scientific relevance to EDAX, XRD, residual stress as well as micro-hardness test data of the test samples. For a constant setting of process parameters [Peak current (IP) = 10A; Pulse-on time (Ton) = 100 μs; Duty factor (τ) = 85%], surface crack density has been found relatively less (∼73%) for the EDMed Inconel 825 work surface obtained by using CTT, as compared to the case of NTT. However, relatively thick white layer (∼26%) has been attributed to the EDMed Inconel 825 specimen obtained by using CTT, as compared to the case of NTT (for a common parameters setting: IP |
= 6A; Ton |
= 300 μs; τ=85%). Additionally, effects of cryogenic treatment of tool electrode have also been discussed emphasizing aspects of tool life, extent of carbon deposition at the bottom and edge of the electrode, and tool shape retention capability. As compared to NTT, carbon (possibly carbide) layer (deposited at the edge of the tool electrode) of relatively low thickness value (∼75%) has been observed for CTT.Nickel-based super alloy Inconel 825 is widely used in aerospace, nuclear, and chemical industries because of their excellent mechanical and chemical properties at elevated temperatures. Difficulty is faced whilst machining of Inconel 825 because of its poor thermal conductivity, high toughness, high hardness, and high work hardening behaviour. Moreover, it contains highly abrasive carbide particles which tend to stick on the tool surface, resulting inferior surface finish. Enormous heat is generated during machining causing reduction in tool life. Since, traditional machining processes are unsuitable for ‘difficult-to-cut’ materials and high temperature resistant alloys (Inconel 825, in the present case), Electrical Discharge Machining (EDM) has become an appropriate option. However, excessive tool wear, imperfections of corner size accuracy of the machined component, cost of post machining operations, fatigue failure of the machined component etc. are some of the major concerns in EDM operations. In order to improve machining performance during EDM of such ‘difficult-to-cut’ materials, cryogenic processing (or cryogenic treatment) of electrode material is hereby recommended.Generally, the parts can be cryogenically treated either by (i) Shallow Cryogenic Treatment (SCT) or by (ii) Deep Cryogenic Treatment (DCT) During DCT, the temperature is gradually reduced to −184 °C at a cooling rate of 1 °C/min and the part component is kept in the cryogenic processor container for about 24 h durations. The temperature is then gradually raised to the room temperature again. The parts are then tempered under similar conditions to incur stress relief. By conducting the cool-down cycle in gaseous nitrogen, temperature can be controlled accurately, and thermal shocks to the material are avoided. Cryogenic treatment can also be performed without tempering process; as reported by Extensive literature review has been articulated at this stage to retrieve potential benefits of cryogenic processing of tool materials in order to improve thermal and electrical properties, wear resistance etc. to a remarkable extent, thus to achieve satisfactory machining performance in perspectives of product quality as well as productivity. Literature survey has depicted considerable effort put by past researchers to study aspects of machinability of Inconel super alloys during electrical discharge machining; however, application potential of cryogenically treated tool electrode has been found an unexplored area of research.Inconel 825 is a Nickel (Ni)-Iron (Fe)-Chromium (Cr) based super alloy with additions of Molybdenum (Mo) and Copper (Cu). As compared to Inconel 825, Inconel 718 contains Niobium (Nb) which imparts strength and high temperature resistance. The Mo and Cu in Inconel 825 provide substantially improved corrosion resistance in reducing environments (when compared to conventional austenitic stainless steels). Inconel 825 thus finds its application in chemical processing, pollution-control equipment, oil and gas well piping, nuclear fuel reprocessing, acid production, and pickling equipment.It is well understood that machinability characteristics of Inconel (different grades) super alloys are almost similar; thus, these alloys are included in the category of ‘difficult-to-cut’ materials. However, depending on their chemical composition; their property, application, performance of the machined surface may differ. Since, work has already been reported to a remarkable extent on machining of Inconel 718; aspects of machining and machinability of Inconel 825 has not been sufficiently addressed in exiting literature resource. Hence, this study has considered Inconel 825 as work material.In addition to that, it has been noticed that most of the past research have considered Material Removal Rate (MRR), Electrode Wear Rate (EWR), and surface roughness (Ra) etc. (of the EDMed Inconel end product) as the major focus towards evaluating machining performance Cryogenic processing of tool material has been reported to examine improvements in MRR, EWR etc.; whilst it is felt that effect of the same on Surface Crack Density (SCD), White Layer Thickness (WLT), chemical composition of the EDMed work surface, metallurgical characteristics of the machined surface, residual stress and micro-hardness etc. need to be investigated in detail. To this context, the specific objectives of the present work have been delineated herein.To study application potential of using CTT whilst EDM on Inconel 825 as compared to the case of EDM using NTT.To study the effects of cryogenic treatment of electrode material on crystallize size, dislocation density, extent of grain refinement etc. (as compared to ‘non-treated’ electrode material).To study surface integrity (morphology and topography) of the EDMed Inconel 825 obtained by using CTT as compared to NTT. Topographic measures of the EDMed work surface viz. Surface Crack Density (SCD), crack opening width, and White Layer Thickness (WLT) are indented to be studied in detail to articulate potential benefits of using cryogenically treated tool electrode.To examine chemical constituents, metallurgical aspects (phases present: matrix and precipitates, degree of grain refinement, crystallize size, dislocation density etc.), residual stress and micro-hardness of the EDMed Inconel 825 end product obtained by using CTT as compared to NTT.To investigate the effect of cryogenic treatment of electrode material on (a) tool shape retention capability, (b) aspects of tool wear, and (c) extent of carbon deposition at the bottom surface of the tool during execution of EDM operation on Inconel 825.Finally, to investigate whether cryogenic processing of electrode material proves beneficial to improve ease of electrical discharge machining of Inconel 825.Before conducting EDM experiments, the tool electrode have been cryogenically treated in order to improve their properties. For DCT, the workpiece and tool have been cooled down (ramp-down) to approximately −185 °C at cooling rate 1 °C/min, and held for 24 h and then gradually heated back at the same rate i.e. at cooling rate 1 °C/min to the ambient temperature (ramp-up). The DCT cycle adapted herein has been depicted in . The snapshot of the set up used for cryogenic treatment of tool electrode has also been presented herein (). SEM micro-graphs revealing surface structure of tool material: Non-Treated and Cryogenically Treated (at Magnification X500) have been provided in EDM experiments have been carried out on Die Sinking EDM (Make: Electronica, Modal ELEKTRA EMS-5535, Pune, India) setup (specifications of the setup has been furnished in ). A pure (99%) Copper [thermal conductivity 401 W/(m °C) at 20 °C; melting point 1082.78 °C; boiling point: 2567 °C] rod of circular cross section(ϕ20) has been used as a tool electrode. Commercially available grade Rustlick™ EDM-30 (ITW Professional Brand) (Specific Gravity: 0.80 @ 25 °F; Viscosity: 36 SSU @ 100 °F (38 °C); Flash Point: 200° F; Dielectric Strength: 45KV) has been used as dielectric medium. Polarity has been kept positive (i.e., workpiece positive) throughout experimentation.Experiments have been carried out using three controllable process parameters: peak discharge current (Ip), pulse-on-time (Ton), and duty factor (τ); each varied at three discrete levels as per availability of factorial setting in the particular EDM setup used herein. The domain of experiment as selected for the present work has been shown in The machining duration has been kept constant (10 min) for each of the experimental runs.A snapshot of EDMed Inconel 825 specimens has been provided in . After EDM operations, topographical measures of the EDMed work surface viz. Surface Crack Density (SCD) and White Layer Thickness (WLT) have been measured for each of the experimental schema.In order to measure surface crack density, the top surface morphology of the EDMed workpiece has been studied using Scanning Electron Microscopy (SEM) (SEM Model: Jeol JSM-6480LV, and Jeol JSM-6390LV; Country: Japan) at a magnification of 500×. For each specimen, randomly three sample areas have been viewed under SEM and corresponding total crack lengths have been measured using PDF-XChange Viewer software. The total crack length on each SEM micrograph has been divided by the micro-graph area to obtain the value of SCD. For a particular specimen, SCD has been determined at three distinct locations; and the average of these three has been taken as the representative SCD for that particular sample. Hence, the term ‘SCD’ must be interpreted as ‘average SCD’ or ‘relative SCD’ throughout the text.The white layer (also called recast layer) is the result of the re-solidification of the melted material (not completely flushed off from the EDMed surface by the dielectric fluid). For determining WLT, moulds have been prepared by taking wire-EDMed cut pieces of the EDMed Inconel 825 specimens using cold mounting compound powder and liquid. After cold mounting, the specimens have been polished by a series of finer grades of emery papers followed by diamond polishing using Hifin diamond compound, Hifin fluid ‘OS’ and Selvyt cloth. The polished specimens have then been etched by Kalling’s reagent (solution of CuCl2 of 5 g, HCl 100 mL, and Ethanol 100 mL) for about 19 s duration of emersion. In order to measure the WLT, the micro-graph obtained under Scanning Electron Microscope (SEM) (Model: Jeol JSM-6480LV and Jeol JSM-6390LV; Country: Japan) at a magnification of ×500 has been viewed. The thickness of the white layer has been measured by ImageJ Software at five different locations across each cross-sectioned specimen; and, an average value has been considered for further analysis. Hence, the term ‘WLT’ must be understood as ‘average/relative WLT’ throughout the text.In addition to that, residual stress and micro-hardness tests have been carried out on few selected specimens using XRD Texture Measurement Machine [Model No: D8 ADVANCE with DAVINCI design, Make: BRUKER, Country: Germany] and Vicker’s Micro-hardness tester [Model No. LECO LM 810; with Load 25 gf and dwell time 10 s], respectively.Effect of cryogenic treatment of Copper electrode has been studied in perspectives of metallurgical information obtained from XRD analysis, residual stress, and micro-hardness test data. XRD spectra () of ‘non-treated’ Copper has revealed existence of cubic crystal system; peak patterns have been found almost exactly matching to that of Cu with impurities from 0.001–0.01%, Ag, Al, Bi, Fe, Si, and Zn (Reference Code: 04-0836). It has also been observed that no significant phase change has been attributed due to cryogenic treatment of Copper (As compared to ‘non-treated’ electrode material, deep cryogenic treatment has resulted substantial grain refinement; which has attributed to smaller crystallite size and higher dislocation density (). The computation of crystallite size has been carried out using Debye Scherrer formulation. The theoretical basis and formulations for computing dislocation density could be found in the reporting by This has been found in good agreement with the micro-hardness test data. Experiment has revealed that cryogenically treated Copper specimen has shown relatively high micro-hardness value (Range ∼114.2HV to 119.8HV) as compared to that of ‘non-treated’ tool material (Range ∼94.1HV to 99.3HV). Effect of cryogenic treatment has thus been interpreted in view of reduced degree of crystal imperfections, voids as well as dislocations; and consequently, evolution of residual stress of lesser magnitude [(−79.4 ± 37.3) MPa] as compared to that of ‘non-treated’ tool material [(−181.2 ± 114.7) MPa].Therefore, in the present case, it has been assumed that CT must have improved electrical as well as thermal conductivities of the tool material; thus improving ease of machining of Inconel 825. Moreover, it has been observed that cryogenic treatment of tool material has resulted reduced tool material consumption (lesser tool wear) as compared to NTT. These have been explained in later sections.As compared to ‘normal’ Inconel 825, the EDMed work surface of Inconel 825 has exhibited poor surface integrity (in purviews of morphology and topography) whilst executing EDM using NTT as well as CTT. The inferior surface morphology has resulted formation of crater marks, globules of debris, melted metal deposition, pockmarks or chimneys, surface cracks and white layer. However, the intensity of such surface irregularities has been found different for the case of EDM on Inconel 825 using cryogenically treated Copper tool as compared to that of normal tool electrode (SEM micro-graphs revealing existence of surface cracks on the EDMed Inconel 825 work surface obtained through EDM using (a) NTT, and (b) CTT, for a constant parameters setting i.e. [IP |
= 10A; Ton |
= 100 μs; τ = 85%] have been depicted in . It has clearly been understood that use of CTT has been noticed fruitful since intensity of surface cracks developed on the machined surface of Inconel 825 has been found relatively less as compared to the case of NTT. Surface cracks are highly undesirable whilst the EDMed part component is subjected to practical application field; reduced crack density thus supports application potential of cryogenic treatment of the tool material in the context of EDM on Inconel 825. The explanation for reduced crack density has been explained well by SEM micro-graphs revealing existence of white layer on the EDMed Inconel 825 work surface obtained using NTT as well as CTT, for a constant parameters setting [IP |
= 6A; Ton |
= 300 μs; τ = 85%] have also been depicted in . It has been noticed that white layer thickness has assumed higher values for the case of EDM using CTT as compared to the case of NTT. This may be due to the enhancement of heat dissipation capacity of the electrode material (after cryogenic treatment) which results in increased heat transfer rate through the bulk of the electrode material. Due to increased rate of heat transfer, molten material gets uniformed cooled at a faster rate and deposited smoothly onto the top surface of the machined zone. Hassle-free deposition of the molten material (resolidication) in turn increases the thickness of the white layer. Uniform deposition of the molten material also results in evolution of lesser residual stress and thus reduces probability of formation of surface cracks to some extent.Energy Dispersive X-Ray Spectroscopy (EDAX) elemental spectra revealing chemical composition (wt%) of ‘normal’ Inconel 825, EDMed work surfaces of Inconel 825 obtained through using NTT, and CTT (for a constant setting of EDM parameters i.e. Ip |
= 10A, Ton |
= 300 μs, τ = 85%) have been shown in , respectively. It has been observed that carbon content has been increased during EDM due to carbon enrichment onto the machined zone while dielectric cracking has been incurred. The EDM oil (used as dielectric medium) being a hydrocarbon; during spark discharge, pyrolysis of dielectric fluid has taken place. Thus, carbon got deposited on the machined zone leading to increase in carbon content of the EDMed surface as compared to the unaffected (normal) parent material.EDAX analysis has exhibited that as compared to ‘normal’ Inconel 825 (0.08 wt% C), carbon content of EDMed Inconel 825 work surface has been increased to 4.2% (for the case of EDM using NTT), and 3.2% (for the case of EDM using CTT). It has been observed that as compared to EDM with NTT; use of CTT has caused lesser extent of carbon enrichment onto the machined zone which has been seemed beneficial.Since, the topmost layer of the EDMed surface is detrimental for the service life of the part component when subjected to fatigue loading. This is because surface cracks do initiate at this surface. Lesser carbon content of this layer should correspond to lower hardness value; thus, imparts relatively more resistance to surface cracking.The decrement of carbon content onto the top surface of EDMed Inconel 825 (obtained by using CTT), as compared to NTT, can be explained by the fact that cryogenic processing of electrode tool substantially improves electrical as well as thermal properties of the tool material. Hence, during EDM using CTT, the increased rate of heat dissipation through the tool electrode in turn decreases energy density at the spark gap. This may supress the tendency of carbide precipitation on the machined zone. Thus, EDMed Inconel 825 work surface obtained by using CTT has exhibited relatively less carbon content.For micro-hardness tests, the points of indentations at three distinct locations (approximately at the middle position along the thickness of white layer measured from the top surface) for a particular sample (transverse cut section of EDMed workpiece through WEDM route) have been depicted in As compared to ‘normal’ Inconel 825 (micro-hardness in the range of 235.0HV to 242.1HV), EDMed Inconel specimen (obtained at parameters setting: IP |
= 10A, Ton |
= 300 μs, τ = 85%) has exhibited higher hardness values falling in the range of (568.0HV to 632.8HV), (827.6HV to 863.0HV), for the following two cases of EDM using NTT, and CTT, respectively.In this context, it may be noted that EDAX analysis has been made on the surface of EDMed Inconel 825; whilst, micro-hardness has been measured (for the transverse-cut section of EDMed specimen) approximately at the mid-depth (from the top surface) of the white layer thickness. Moreover, from XRD test, it has been observed that EDMed Inconel 825 using CTT has corresponded to relatively less crystallite size (more refined grain structure) as compared to the case of EDM using NTT. Hence, more micro-hardness in case of EDM using CTT as compared to NTT is well justified.Literature has claimed that CT of material removes imperfections (voids, dislocations, stress lines etc.) to a major extent and thereby provides a more refined grain structure. As a consequence, chance of fatigue failure is substantially reduced Residual stress generated within the EDMed work surface is basically the effect of non-homogeneity of heat flow and metallurgical transformations or the localized inhomogeneous plastic deformation that occurs during EDM operation In the present work, it has been observed that ‘normal’ Inconel 825 has corresponded to the compressive residual stress (−413.0 to −261.4 MPa). The residual stress of tensile nature has been attributed to the EDMed specimen of Inconel 825 (obtained at constant parameters setting: IP |
= 10A, Ton |
= 300 μs, τ = 85%) for the case of EDM using NTT (868.7–1492.3 MPa), and CTT (712.2–958.0 MPa), respectively. Thus, the EDMed Inconel 825 specimen obtained by using CTT has exhibited lesser residual stress as compared to the case of NTT. Evolution of relatively less residual stress is indeed beneficial since it reduces severity of surface cracking. This helps in improving fatigue life of the EDMed end product.X-Ray Diffraction (XRD) spectra inferring metallurgical aspects of ‘normal’ Inconel 825, and EDMed work surface of Inconel 825 obtained by using NTT, and CTT (for a constant setting of EDM parameters i.e. Ip |
= 10A, Ton |
= 300 μs, τ = 85%) have been provided in In this work, XRD measurements have been performed using a Panalytical X’Pert PRO diffractometer with CoKα radiation (λ = 1.7906A0). The XRD patterns have been indexed with X’Pert HighScore Plus software containing PDF-2 files database.As compared to unaffected (normal) Inconel 825 (), the peak pattern has appeared almost similar (approximately at similar positions; slightly sifted rightward along °2θ axis) for the surface of EDMed Inconel 825 obtained by using NTT, and CTT (, respectively). Considering Full Width Half Maximum (FWHM) of the peaks, it has been observed that grain refinement has taken place during EDM operation for the Inconel 825 work surface as compared to the ‘normal’ Inconel 825 parent material and that too irrespective of the condition of the tool material used (viz. NTT and CTT) during execution of EDM operation. Occurrence of peaks at similar positions has affirmed that no remarkable phase alteration has taken place as a result of EDM.XRD analysis has revealed that ‘normal’ Inconel 825 basically has consisted of Iron-Nickel and Chromium based cubic solid solution matrix with precipitates of Copper Nickel [Reference Code: 47-1406 and Chemical Formula: Cu0.81Ni0.19]. However, the peak pattern of unaffected Inconel 825 has exactly matched to that of Iron Nickel [Reference Code: 47-1405 and Chemical Formula: Fe0.64Ni0.36] and also Iron Nickel [Reference Code: 18-0646 and Chemical Formula: FeNi]. of the EDMed Inconel 825 work surface obtained by using NTT (at parameters setting: Ip |
= 10A, Ton |
= 300 μs, τ=85%) has identified presence of matrix of Iron Nickel [Reference Code: 47-1405 and Chemical Formula: Fe0.64Ni0.36] with precipitates of Copper Nickel [Reference Code: 47-1406 and Chemical Formula: Cu0.81Ni0.19] and Nickel Aluminum Titanium Carbide [Reference Code: 19-0035 and Chemical Formula: Ni3(Al, Ti)C]. Such carbides are expected to be deposited along the grain boundaries of the specimen. Carbide formation can be attributed due to carbon enrichment onto the machined zone during pyrolysis of the dielectric medium. Similar information has been retrieved from the XRD spectra of EDMed Inconel 825 work surface obtained by using CTT (at parameters setting: Ip |
= 10A, Ton |
= 300 μs, τ=85%) as shown in The variation of crystallite size(L),and dislocation density(δ) for (1) ‘normal’ Inconel 825, (2) EDMed work surface of Inconel 825 obtained by using NTT, and (3) EDMed work surface of Inconel 825 obtained by using CTT has been depicted in . Considering a particular crystallographic plane [2 0 0] corresponding to the highest intensity peak selected from the peak pattern of XRD spectra, it has been observed that as compared to (1) ‘normal’ Inconel 825 (which has corresponded to L |
∼ 60.686 nm and δ |
∼ 3.9327 × 10−4), (2) EDMed work surface of Inconel 825 obtained by using NTT and at parameters setting: IP |
= 10A; Ton |
= 300 μs; τ=85%, and (3) EDMed work surface of Inconel 825 obtained by using CTT and at similar parametric setting have exhibited sufficient grain refinement followed by decrease in crystallite size and consequently, increase in dislocation density. However, the degree of grain refinement has been found relatively high for the case of (3) EDMed Inconel 825 work surface obtained by using CTT(L∼12.9593nm;δ∼86.4878×10-4) as compared to the case of (2) EDMed Inconel 825 work surface obtained by using NTT (L∼43.1854nm;δ∼7.7853×10-4).While comparing micro-hardness of the cryogenically treated Copper tool to that of normal tool electrode; it has been found that cryogenic treatment has improved hardness of the tool material due to substantial gain refinement and internal stress relief (Refer to ). Results have indicated that ‘normal’ Copper has shown micro-hardness values falling in the range of 94.1HV to 99.3HV; whilst cryogenically treated Copper electrode has exhibited micro-hardness values adhering to the range the range 114.2HV to 119.8HV. This has been found in good agreement of the comment made by Lal et al., Leskovsek et al., Molinari et al. Macroscopic view of the edge of tool electrode (NTT and CTT, both) after EDM operations on Inconel 825 has been depicted in (a) and (b), respectively. Carbon deposition has been observed at the bottom surface as well as around the edge of the tool electrode during EDM operation.This can be explained by the fact that during electrical discharge, pyrolysis of the dielectric medium takes place. Due to the pyrolysis of dielectric fluid (also called dielectric cracking), carbon atoms come out and get deposited onto the bottom/edge of the tool (and also on the work surface) forming a blackish layer. However, as compared to normal tool, cryogenically treated tool electrode has exhibited presence of very thin layer of the deposited carbon. As observed under optical microscope [Model No: OM-19; Make: Radical Instruments; Country: India], the thickness of the deposited layer (after EDM experiments using NTT) has appeared as ∼0.4871 mm ((a)); whilst the deposited layer has been found very tiny (∼0.1203 mm) after EDM operations with CTT (Cryogenic treatment of tool electrode has thus found advantageous due to less carbon deposition (may be deposited material is in the form of Copper Carbide) at the bottom surface as well as along the edge of the electrode. This may be due to the fact that through cryogenic treatment, the electrical and thermal properties of tool material are substantially improved On the contrary, relatively thick carbon layer has been found set down both at the bottom surface and also along the edge of NTT after EDM. Formation of such layer creates a barrier for the heat to be transmitted through the electrode material and thus more heat is required to execute the same for the progress of EDM operation. It imposes untoward effect on the tool electrode with excessive tool wear and reduced tool life. Shape retention capability of the tool electrode is also adversely affected.The phenomenon of formation of thin carbon layer at the bottom surface and edge of the cryogenically treated tool electrode has also been supported by the information (wt% of C) obtained from the EDAX elemental spectra of the bottom surface of the tool electrodes (). It has clearly been noticed that as compared to NTT which has corresponded to 39.37 wt% Carbon at the bottom surface; the bottom surface of the CTT has corresponded to lesser extent of carbon content (35.51 wt%) owing to the formation of very thin deposited layer of Carbon (or possibly carbides).The conclusions drawn from the aforesaid research have been summarized below.Deep cryogenic treatment of Copper tool electrode has attributed to decrease in crystallite size; resulting a more refined grain structure as compared to that of ‘non-treated’ Copper. Cryogenic treatment of the electrode material has resulted reduced (∼12%) crystallite size and increased (∼28%) dislocation density as compared to NTT material. Moreover, cryogenic treatment has reduced residual stress and crystal imperfections; thus ensuring improved tool life and improved tool shape retention capability; and also reduced tool wear.Top surface morphology of the EDMed work surface of Inconel 825 has exhibited presence of crater mark, globules of debris, spherical deposition, pock marks or chimneys, and surface cracks. However, the intensity of aforesaid surface irregularities has been found relatively less for the case of EDM with CTT as compared to the case NTT. Reduced crack density (and also the crack opening width) for the case of EDM using CTT could be found beneficial for the fatigue life of the EDMed part component whilst subjected to service. For a constant setting of parameters: IP |
= 10A; Ton |
= 100 μs; τ=85%, surface crack density has been found relatively less (∼73%) for the EDMed Inconel 825 work surface obtained by using CTT, as compared to the case of NTT.The white layer has been found relatively thick onto the top surface of the EDMed Inconel 825 specimen obtained by using CTT. Increased heat conduction rate and thereby reduced tool wear has resulted decrease in energy density at the discharge gap. Due to smooth deposition of molten material, thicker white layer has been formed. Results have indicated that relatively thick white layer (∼26%) has been attributed to the EDMed Inconel 825 specimen obtained by using CTT, as compared to the case of NTT (for a common parameters setting: IP |
= 6A; Ton |
= 300 μs; τ=85%).As compared to ‘normal’ Inconel 825 parent material, EDAX elemental spectra of EDMed work surface has exhibited higher Carbon content (wt%). This has been attributed due to the carbon enrichment onto the work surface during pyrolysis of dielectric fluid. However, for the case of EDM using CTT, carbon enrichment on the work surface has been found relatively less as compared to the case of EDM using NTT.As compared to ‘normal’ Inconel 825, the average micro-hardness (obtained at the transverse-cut section of the EDMed specimen; approximately at the mid-depth of the thickness of while layer measured from the top surface) has been found more for EDM using NTT and CTT both. This may be explained due to thermo-electrical effect of EDM process that has resulted considerable grain refinement (decease in crystallite size) within the work material. However, for the case of EDMed specimen obtained using CTT, average micro-hardness has been found the highest. As compared to the EDMed work surface of Inconel 825 obtained by using NTT (at parameters setting: IP |
= 10A; Ton |
= 300 μs; τ=85%), the EDMed Inconel 825 work surface obtained by CTT has exhibited relatively less crystallite size (∼70% reduced) due to a more refined grain structure. Hence, EDMed Inconel 825 obtained by using CTT has exhibited higher hardness values as compared to the case of NTT.As compared to ‘normal’ Inconel 825, the residual stress has been found more for the EDMed specimen obtained using NTT and CTT both. However, for the case of EDM with CTT, evolution of residual stress within the EDMed specimen has been found relatively less in magnitude as compared to the case of EDM using NTT.XRD spectra of EDMed Inconel 825 work surfaces have identified presence of Nickel-Iron solid solution with precipitates of carbides of varied extent. As compared to the EDMed surface obtained by using NTT; use of CTT has caused relatively more grain refinement. This has further been found in good agreement to the decrease in crystallite size and consequently the increase in dislocation density.In comparison with NTT, use of CTT has resulted relatively tiny layer of deposited carbon at the bottom as well as edge of the tool electrode. This in turn has facilitated increased rate of heat transfer through the bulk of the electrode material. This is expected to cause reduction of tool wear and hence substantial improvement of tool shape retention capability. As compared to NTT, deposited Carbon (possibly carbides) layer of relatively less thickness (∼75%) has been observed at the edge of the tool electrode for CTT.Research can be extended further in the following directions as stated below.Effect of using CTT (during EDM on Inconel 825) may be studied in perspectives of volumetric material removal rate, tool wear rate, roughness average of the EDMed work surface.Effects of SCT as well as DCT of the tool material may be studied during EDM of Inconel super alloys in perspective of overall machining performance.Corner shape accuracy of the machined hole may be studied as effected by using cryogenically treated tool electrode.Effect of using cryogenically treated tool as well as work material may be experimentally studied with respect to the case of using NTT and normal (non-treated) workpiece.Measurements of residual stresses in a welded orthotropic steel deck by the hole-drilling method considering stress biaxialityWelding residual stress (WRS) significantly affects the fatigue cracking of orthotropic steel deck. In this paper, the influence of strain release coefficients (SRC) calibration methods on the WRS measurement results of the hole-drilling method was studied; then the WRSs of an orthotropic steel deck were measured by the hole-drilling method considering stress biaxiality. High-value WRSs are greatly overestimated, when the SRC are calibrated via the formula method and experiment. The uniaxial FEM calibration ignoring the stress biaxiality still leads to a peak WRS error of 11.48%. Considering the influence of stress biaxiality on SRC, the average peak WRS is 496.3 MPa, appearing at the weld centerline on the deck; as the distance from the weld centerline increases, the WRSs gradually decrease until to the compressive stress; the WRS distribution of the deck top surface is approximately M-shaped with two tensile WRS areas around the weld; the WRSs on the deck bottom surface have a similar distribution trend to those on the top surface; the WRSs of the U-rib are obviously smaller than those of the deck. The transverse WRS distribution is similar to that of longitudinal WRS, while its value is far less than longitudinal WRS.released strains along the corresponding directionhalf-width of the wire grid for the strain gaugelength of the wire grid for the strain gaugedistances from hole center to both ends of the strain gaugearea integral parameters of the released strain based on the grid area of the strain rosettethe larger and smaller applied external stressesOrthotropic steel deck (OSD) has been widely employed in long-span steel bridges due to the high load-carrying capacity, light weight and short construction period, etc. Several researchers have performed measurements for the WRSs in OSD. Kitada et al. In the WRS measurement process of the hole-drilling method, the hole edge may yield locally due to the stress concentration. The error induced by plastic deformation around the hole can exceed 30% Due to the plastic deformation around the hole edge under high stress state, the standard ASTM E837-13a The hole-drilling method is a semi-destructive WRS measurement method proposed by Mathar . Then a hole with specified depth is drilled at the strain rosette center. The material removal within the hole causes the release of WRS, resulting in a new stress field. The strain rosette around the hole will monitor the released strain during the drilling, and the WRS can be calculated according to the strain-stress relationship. Based on Kirsch’s equation σ1,2=ε1+ε34A∓(ε1-ε3)2+2ε2-ε1-ε324Btan2β=2ε2-ε1-ε3ε1-ε3where ε1, ε2, and ε3 are the released strains along the corresponding direction measured by the strain rosette; σ1 and σ2 are the principal stresses; A and B are the SRC, which are determined by material properties, hole diameter and strain rosette size; and β is the principal stress direction angle.Factors affecting the values of the SRC include material parameters, hole diameter, and strain rosette size, etc. For the case of through hole, if the material is in the linear elastic range, the calculation formulas of the SRC based on the Kirsch theoretical solution are A=-α·1+ν2E·a2r1r2B=-12lbE1+νa2S2-32a4S3+1-νa2S1where E and ν are the Young’s modulus and Poisson’s ratio of the material; a is the radius of the hole; b and l are the half-width and length of the wire grid for the strain gauge; r1 and r2 are the distances from hole center to both ends of the strain gauge; α = 1.065, which is the correction coefficient; S1, S2 and S3 are the area integral parameters of the released strain based on the grid area of the strain rosette, which can be calculated by Eq. S1=2tan-1xbr1r2S2=-2bxx2+b2r1r2S3=-23bx(x2+b2)2r1r2However, the calculation formula is only applicable to the case where the material is in the linear elastic range. In elastic mechanics, the stress concentration factor of plane circular hole is 3 ), and then the released strains ε1, ε2, and ε3 caused by hole-drilling are measured. The SRC can be derived from Eq. A=ε1+ε32(σ1+σ2)B=ε1-ε32+2ε2-ε1-ε322(σ1-σ2)If a uniaxial stress field is applied to the specimen along the direction parallel to a strain gauge, then relationships of σ1 = σ, σ2 = 0, and β = 0 can be got. Eq. Besides, the SRC can also be calibrated by FEM. The principle of FEM calibration is the same as the experiment calibration, except that the released strain is obtained by finite element simulation. Actually, the calibration experiments in references are generally uniaxial tensile test The plastic deformation of the hole edge caused by the stress concentration will lead to smaller SRC. When this factor is not considered, it will lead to the overestimation of high-value WRS. This problem can be solved by the grading calibration of SRC with several different methods. To study the influence of SRC calibration methods on the WRS measurement results, three different methods were used to calibrate the SRC respectively, and the final WRS calculation results were compared.In the case of Q345qD steel and TJ120-1.5-Φ1.5 strain rosette, the specific parameters of the SRC calculation formula are listed in . Substituting these parameters into Eq. , the SRC results were A = −3.324 × 10−7 MPa−1 and B = −6.690 × 10−7 MPa−1.Grading calibration for SRC can be performed by uniaxial tensile test according to standard GB/T 31310–2014 shows the geometry details of the test specimen required in standard for the hole with diameter of 1.5 mm. Three measurement points were located respectively on the left, middle and right parts of the specimen. When specimen was tensioned under different calibration stresses, the hole-drilling method was applied to the measurement points. Then the released strain could be measured in specified stress field of σ = 0.3σy, 0.5σy, 0.7σy and 0.9σy. The SRC were obtained by substituting measured strains and stress field parameters into Eq. . For the Q345qD steel and TJ120-1.5-Φ1.5 strain rosette, the SRC results of grading calibration experiment are listed in , which are also shown in previous work that the SRC obtained by the calibration experiment decrease with the calibration stress increasing. The distribution ranges of A and B are −3.413 × 10−7 MPa−1 to −3.078 × 10−7 MPa−1 and −7.014 × 10−7 MPa−1 to −5.450 × 10−7 MPa−1, respectively.In order to study the effect of stress biaxiality on SRC, a grading calibration for SRC was performed by the FEM in biaxial stress state. The finite element model was established by ABAQUS, as shown in . To improve the calculation efficiency, the length of the finite element model was simplified to 100 mm according to Saint-Venant’s Principle and the transitional mesh division was used. Based on many trial calculations, the element size used in the area near the hole was 0.15 mm. The model had 83,996 C3D8 elements and 88,895 nodes. The strain rosette in the model had the same geometry details with that used in the experiment calibration. Uniform loads were applied on the four sides of the model to simulate the biaxial WRS field. The center point of model bottom surface was fixed in X, Y and Z directions to avoid the rigid body displacement of the model. The material of the model was Q345qD steel, with Young’s modulus of 206 GPa and the Poisson’s ratio of 0.3. The stress-strain relationship used in the calculation is shown in . To simulate a general biaxial stress field, the stress ratio k = σe,2/σe,1 considered in the calculation ranged from −0.8 to 0.8. Both σe,1 and σe,2 were the applied external stresses, and σe,1 was larger than σe,2. The element remove technique was used in the hole area to simulate hole-drilling process. After the drilling, the average strain of the surface nodes covering the strain gauge area was extracted as the released strain. The SRC were obtained by substituting the released strain and stress field parameters into Eq. Some typical results of FEM calibration are shown in . The magnitude of the biaxial stress is quantified by the von Mises equivalent stress σeq=σ12-σ1σ2+σ22. When the stress level is less than 0.5σy, the SRC are almost constant. When the stress level exceeds 0.5σy, the SRC begin to be affected by the plastic deformation of the hole edge, and decrease with the increase of stress level. For the same stress level, the SRC decrease with the increase of stress ratio. This phenomenon can be attributed to the difference of plastic deformation in different stress states. As shown in , with the increase of stress ratio, the plastic strain gradually shifts from ε2 direction to ε1 direction, which leads to the relative decrease of ε1. According to Eq. , this decrease of ε1 leads to the decrease of SRC. As the stress level increases, the difference of SRC (A and B) between biaxial FEM calibration (k = −0.8 to 0.8) and uniaxial FEM calibration (k = 0) gradually increases. When the stress level reaches the value of σy, A and B at the stress ratio of 0.8 decrease by 8.12% and 43.64% respectively compared with that at the stress ratio of 0. Therefore, it is necessary to consider the biaxiality of WRS when calibrating SRC, which can effectively improve the measurement precision of the hole-drilling method.The SRC obtained by different calibration methods are shown in . In linear elastic range, the SRC obtained by uniaxial FEM calibration are very close to the formula values. This proves that the FEM simulation process is reasonable. When the stress level exceeds 0.5σy and continues to increase, the SRC calibrated by FEM gradually decrease and tend to stay away from the formula values. The SRC of experiment calibration are greater than those of FEM calibration, but their growth trends are similar. This discrepancy may be due to the inherent uncertainties of hole-drilling method The OSD specimen used in this paper was composed of one U-rib and one deck. The specimen size is detailed in , which is designed according to the actual structural size commonly used by steel bridges in China , the U-rib and deck were connected by submerged arc welding. The rib-to-deck welded joint consists of two weld passes on the interior and exterior side of the U-rib respectively. During the welding, the interior side of the U-rib was pasted with ceramic backing, so the welding torch could be used on exterior side to complete the interior weld. After the interior welding, the exterior welding was also conducted to realize the full-penetration welding of the OSD. According to standards GB 50017-2017 are 6.3 mm and 9.3 mm respectively, which meet the requirements of standards. The welding parameters are listed in . The base material of specimen was Q345qD steel and the welding wires of exterior and interior weld were H08MnA (Ф3.2) and ER50-6 (Ф1.6), respectively. The linear heat energy input q can be calculated by Eq. where η is the energy efficiency of submerged arc welding, which is generally 77–90% The WRSs of the specimen were measured along three different paths, which were located at external surfaces along the specific cross sections (see ). To avoid the influence of arc starting and extinguishing at both ends of the weld on WRSs, the measurement sections were selected away from the specimen ends. As shown in (a), the coordinates of the measurement sections were y = 300 (Path 1), y = 500 (Path 2), and y = 700 (Path 3). The measurement path was a closed loop which consist of top and bottom surface of the deck, and the external surface of the U-rib. The number and location of measurement points on these measurement paths were the same, which are shown in (b). Because of the large WRS gradient in the heat-affected zone, the density of the measurement points near the weld was large, with a spacing of 10 mm. The measurement points far away from the weld were relatively sparse, and the spacing increased gradually to the maximum value of 100 mm. Each measurement path had a total of 90 measurement points, including 39 points on the deck top surface, 28 points on the deck bottom surface, and 23 points on the external surface of the U-rib.The measurement of WRSs was conducted by the hole-drilling method, as shown in . Firstly, the surface of the measured area was polished and cleared to remove the oxide layer and dirty things. After the surface treatment was completed, the strain rosette was pasted on the measurement point in the specified direction. Then a hole with the diameter of 1.5 mm and depth of 1.8 mm was drilled at the center of the strain rosette. The optical alignment instrument was used to avoid eccentric drilling. When the released strains were stable, they could be saved and used to calculated the WRSs based on Eq. The WRS calculation results will be different when the SRC are calibrated by different methods. To study the influence of the SRC calibration methods on WRS measurement results, the WRSs of the OSD specimen were calculated based on four different SRC calibration methods, and the results were also compared. The SRC used in WRS calculation were obtained by two-dimensional linear interpolation between the calibrated SRC according to the stress level and stress ratio.The WRS along the welding direction is longitudinal WRS (LWRS), and that perpendicular to the welding direction is transverse WRS (TWRS). show the LWRSs on the deck top surface calculated via four different SRC calibration methods. Since the influences of stress level and biaxiality on SRC are caused by different distribution of plastic deformation, the difference of the WRS calculated via different SRC calibration methods is prominent in the high stress area. From these results, when the WRSs are small, the differences of the WRSs calculated with different SRC are small, and the WRS curves almost coincide. With the WRSs increasing, the differences of them increase gradually. At high stress levels, the plastic deformation caused by stress concentration will lead to additional plastic strain detected by strain gauge. The SRC formula values based on elastic mechanics cannot characterize plastic strain. In this case, if they are still used to calculate the WRS, the plastic strain will cause additional stress in the calculation result, so the total WRS will be overestimated. The peak WRSs calculated with the formula values of SRC are 30–40% higher than that calculated with biaxial FEM calibration.Since the SRC decrease with the increase of stress ratio, the SRC calibrated in uniaxial stress field by experiment are larger than the SRC of biaxial stress field with positive stress ratio, which lead to the overestimation of WRSs calculated with experiment calibration. Besides, the experiment calibration can only reach 0.9σy due to the loading limit in standard GB/T 31310-2014 For the FEM calibration, the biaxiality of WRS has a great influence on the SRC, which furtherly affects the measurement results of WRS. Similar to the experiment calibration, uniaxial FEM calibration will also cause a certain overestimation of high-value WRS due to its larger SRC values than that of biaxial stress field. As shown in , the high-value LWRSs calculated with uniaxial FEM calibration are greater than that calculated with biaxial FEM calibration. In the WRS peak areas (a) and (b) of path 1 (see ), the LWRSs calculated with uniaxial FEM calibration are at most 50.1 MPa (10.54%) and 33.4 MPa (7.83%) greater than that calculated with biaxial FEM calibration. As shown in , the LWRSs of path 2 and path 3 calculated via different SRC calibration methods are similar to that of path 1. In the WRS peak areas (a) and (b) of path 2 (see ), the LWRSs calculated with uniaxial FEM calibration are at most 54.5 MPa (11.20%) and 12.5 MPa (2.78%) greater than that calculated with biaxial FEM calibration. These values are 66.8 MPa (12.69%) and 43.9 MPa (8.98%) in the WRS peak areas (a) and (b) of path 3 (see ). The average maximum difference between the LWRSs calculated with uniaxial and biaxial FEM calibration of three measurement paths is 11.48%. These deviations in stress peak areas prove the overestimation of high-value WRS induced by uniaxial FEM calibration.As mentioned in the introduction, the WRSs in thin and medium plates are usually biaxial stresses. The SRC obtained by uniaxial tensile test are inadequate for the application to deformation induced under biaxial stress states The WRS distribution of the deck top surface is shown in . The average WRS curve is obtained by averaging the data of three measurement paths. During the welding, the non-uniform heating and cooling induced spatial variations of thermal expansion and shrinkage along with phase transformations in the solid state, leading to a non-uniform WRS field around the weld. The LWRS distribution (see (a)) is nearly symmetrical about the center line of the specimen. This distribution is approximately M-shaped with two tensile WRS peak areas around the weld. The maximum tensile LWRSs on three measurement paths are 475.4 MPa, 487.1 MPa and 537.8 MPa respectively, which generally appear at the weld centerline (x = −150 mm), and the average value is 496.3 MPa. These peak stresses exceed the yield strength of the material, which may be due to the multiaxial nature of WRS and the strain hardening of the material caused by multi-pass welding The highly concentrated welding heat source creates an extremely uneven temperature field. The thermal history of different positions near the weld varies greatly, resulting in a large stress gradient in the heat affected zone. As the distance from the weld increases, the LWRSs of deck top surface decrease rapidly and the maximum stress gradient reaches 24 MPa/mm. On the exterior weld side, the tensile LWRSs are in the range within 60 mm from the weld, and they are in the range of 40 mm on the interior weld side. Tensile and compressive LWRS areas are alternately distributed. The average peak compressive LWRS is −232.0 MPa, which appears near the middle line of the deck (x = 50 mm). Since the weld shrinkage was restrained by the surrounding base material during cooling, the tensile WRS was induced in the vicinity of weld, while the compressive WRS was emerged correspondingly in the remaining areas. The WRS distribution obtained in this paper generally conforms to this law.TWRS distribution of deck top surface in (b) is similar to LWRS, except that the stress values are greatly reduced. It may be due to the larger longitudinal temperature gradient caused by the movement of welding heat source than the transverse temperature gradient, and the larger longitudinal stiffness of OSD specimen than the transverse stiffness. The maximum tensile TWRS is 261.2 MPa and maximum compressive TWRS is −168.1 MPa.During the thin (14 mm) deck welding, the temperature gradient along the deck thickness was not obvious, which induced the similar WRS along the thickness direction. As shown in (a), the LWRS distribution of the deck bottom surface is similar to that of the top surface, and also has a large tensile LWRS near the weld. As the distance increases, the tensile LWRSs gradually decrease to compressive stress with a large stress gradient. The LWRS values of the deck bottom surface are close to that of the corresponding positions on the top surface. The average peak LWRS is 419.9 MPa. The tensile LWRS area is within about 50 mm away from the weld, and the remaining areas are almost the compressive LWRS areas. The compressive LWRSs gradually increase until they reach a stable stress level of about −220 MPa. In the area near deck edge, the compressive LWRSs gradually decrease until close to zero. Compared with the LWRSs, the TWRSs on the deck bottom surface (see (b)) are also greatly reduced. The tensile TWRSs are quite small, with an average peak value of 70 MPa. The compressive TWRSs in the stable zone are about −120 MPa and the maximum compressive TWRS is −216.6 MPa.The WRS distribution of the external surface of the U-rib is shown in . The WRSs of the U-rib are also approximately symmetrically distributed, and the peak values appear at both ends of the U-rib, which are near the weld. The LWRSs of the U-rib (see (a)) are obviously smaller than that on the deck, and the average peak LWRS is 336.5 MPa. The tensile LWRSs on the U-rib are distributed in the area within 30 mm from the weld, while the range of the deck is about 40 to 60 mm. The smaller WRS level and distribution area of the U-rib may be due to its smaller stiffness than deck caused by smaller plate thickness. After reaching the peak compressive stress of about −130 MPa, the LWRSs increase to a tensile stress of about 60 MPa, and finally turn into a compressive stress of about −80 MPa at the centerline of the U-rib. Except for the area near the weld, the LWRSs in most areas of the U-rib are small, within 100 MPa. The TWRSs on the external surface of U-rib (see (b)) are predominantly compressive stress, and the peak value is −210.6 MPa.In short, the tensile WRSs are always around the weld, while the remaining areas are compressive WRS areas. It can be verified that the welding WRSs present a self-equilibrating distribution with alternant tension and compression areas. The peak WRSs generally appear at the weld centerline on the deck top surface, and the average peak WRS is 496.3 MPa, exceeding the yield stress of the material. The tensile WRSs are distributed in a narrow area near the weld with a large stress gradient. The TWRS distribution trend is similar to that of LWRS, while TWRS peak value is far less than LWRS, and about half of LWRS.The WRS measurement results also show that the stress ratios of most areas on the OSD are positive, which can reach about 0.6 in the high stress area, and even larger in the low stress area. This phenomenon proves that the WRSs in OSD are usually in biaxial stress state, and biaxial SRC calibration need to be considered for hole-drilling method.In this paper, the influence of SRC calibration methods on the calculation results of WRSs was studied, and the welding WRSs of OSD were measured by the hole-drilling method considering stress biaxiality. Based on the results, the main conclusions are summarized as follows:When the WRS value is small, the WRSs calculated with different SRC calibration methods are nearly the same. High-value WRSs are greatly overestimated, when the SRC calibrated via the formula method and experiment. The uniaxial FEM calibration ignoring the stress biaxiality still leads to a peak WRS error of 11.48%. Therefore, it is necessary to calibrate the SRC in biaxial stress state to improve the precision of the hole-drilling method.The LWRSs present tension-compression distribution in a self-equilibrating state. The LWRS distribution of the deck top surface is approximately M-shaped with two tensile WRS areas around the weld. Considering the influence of stress biaxiality on SRC, the peak LWRSs generally appear at the weld centerline with average value of 496.3 MPa. The tensile LWRSs are distributed in a narrow area around the weld, and the remaining areas are compressive LWRS areas. The LWRSs on the deck bottom surface have a similar distribution trend to those on the top surface. The LWRSs of U-rib are obviously smaller than that of the deck.Compared to LWRSs, TWRSs have the similar distribution trend. However, TWRSs are far less than LWRSs, with peak value of about half of LWRS. The maximum tensile TWRS is 261.2 MPa, which appears on the deck top surface. The maximum compressive TWRS is −216.6 MPa, appearing on the deck bottom surface.The measurement results show that the WRSs in OSD are in obvious biaxial stress state. The stress ratios are generally positive and can reach about 0.6 at some points in the WRS peak area. This proves the necessity of the biaxial SRC calibration for hole-drilling method.Yadong Li: Conceptualization, Methodology, Resources, Writing - original draft, Writing - review & editing, Project administration. Jun Wu: Methodology, Software, Data curation, Formal analysis, Writing - original draft, Writing - review & editing. Bin Qiang: Conceptualization, Writing - original draft, Writing - review & editing. Siting Zhou: Investigation. Weiqin Liu: Visualization. Changrong Yao: Conceptualization, Supervision, Project administration.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.The internal loads, moments, and stresses in rod-like particles in a low-speed, vertical axis mixerA discrete element method (DEM) model is used to predict the internal load and moment distribution within rod-like particles in a low-speed, vertical axis mixer. The internal loads and moments are combined with small deformation beam bending theory to determine the internal stress distributions. Parametric studies using the model examined the influence of particle aspect ratio, blade rotational speed, and material properties. The spatial distributions of loads and moments, averaged over all particles and time steps, are symmetric about the particle center-plane with a maximum at the particle center-plane. In addition, the largest average maximum absolute principal stress is observed to occur along the particle circumference at the center-plane of the particle. These results indicate that particle failure is not only most likely to occur at the center-plane of the particle, but the failure will begin at the particle’s circumference. The largest average loads, moments, and maximum absolute principal stress increase with particle aspect ratio. The frequency distributions of maximum absolute principal stress at the high stress range are fit well with a Weibull distribution. Increasing blade speed, bed height and particle–particle friction coefficient generally lead to an increase in internal loads, moments, and stresses. The largest maximum absolute principal stresses occur at the base of the mixer and in front of the blades near the mixer circumference where the bed depth is the greatest.Vertical-axis mixers, such as high shear granulators and agitated filter dryers (AFDs), are commonly used in the processing of particulate materials. In an AFD, active pharmaceutical ingredient (API) particles produced by crystallization are filtered, washed, and dried. During the drying process, an impeller blade agitates the bed to increase the drying rate and enhance drying uniformity. Unfortunately, the rotating blade can also frequently lead to unwanted particle breakage or attrition, which can impact the properties of the bulk powder, such as particle size distribution, flowability, bulk density, segregation behavior, and dissolution rate, which, in turn, can lead to deleterious manufacturing and product performance.Numerous experimental and computational studies have been performed to investigate the influence of impeller speed, fill level, mixer size, blade angle, particle size distribution, and inter-particle friction and cohesion on particles in a mixer. These studies have provided excellent insight into how particles move in such systems; however, the work on particle attrition, especially quantitatively predictive work, is limited. Previous experimental work has generally involved the reporting of trends or empirical correlations. Computational work has typically assumed spherical particles, although many particles, such as the API materials typically processed in AFDs, are rod-like in shape. In addition, only a handful of studies have investigated the loading state within individual particles. Having such information would be particularly useful when investigating particle breakage.In this paper, a novel approach is proposed for predicting the internal load, moment, and stress distributions within individual rod-like particles. This model is used to examine how these distributions change as particle aspect ratio, material properties, and operating conditions vary for particles in a vertical-axis mixer. These distributions are particularly useful when combined with standard material failure theories to predict particle breakage rates and resulting size distributions.There have been numerous studies, both experimental and computational, concerning particle movement and mixing in vertical-axis mixers. A detailed review can be found in the background section of . Here, the focus is on particle attrition in vertical-axis mixers, with additional discussion concerning the modeling of rod-like particles.Particle attrition in vertical axis mixers has been studied using both experimental and computational methods. The experimental studies are presented first followed by the computational studies. examined the change in mean particle size within an AFD for two types of material: cubic crystals of KCl and needle-shaped crystals of -threonine. They found that for both materials, attrition only became significant when the moisture content in the materials was below a threshold value, i.e., when the material was nearly dry. They also observed that for KCl, the size decrease was not sensitive to agitator speed, but the size decrease for the -threonine was. For both materials, size decreased with the number of agitator revolutions and approached a constant value after a large time, most likely because the crystals either became sufficiently rounded in the case of cubical KCl, or the -threonine needles broke to a small enough aspect ratio that the loads were no longer sufficient to continue breakage. developed an attrition cell for assessing API sensitivity to mechanical stress in an AFD. The attrition cells were similar in design to a scaled-down AFD. A small amount of material was characterized, placed in a narrow cylinder, and a movable lid was placed on the material’s surface. A load was applied to the lid to mimic the weight of overlying material. The material in the cylinder was then agitated by a rotating blade, similar to what would occur in an AFD. After a specified amount of time, the material was characterized again and an attrition measure was calculated. Lamberto et al. utilized the change in mean particle size to characterize the attrition of materials and ranked the materials with a breakage classification (hard, medium, or easy to break), which was used as a guide for how to process at larger scale. performed similar experimental work. They assessed materials utilizing an FT4 powder rheometer with a rotating agitator located at the end of a vertical shaft, which also applied a specified normal load. This device was similar in design to the attrition cell discussed in the previous paragraph. Compared with , this group not only ranked materials according to attrition potential, but also investigated the influence of different operating parameters on material attrition, including blade speed, agitation time, sample volume, and applied normal force. They reported that increasing blade speed, agitation time, or normal force led to more attrition, with normal force being the most significant factor. The attrition cell data showed reasonable agreement with data from a pilot plant AFD. described a combined discrete element method (DEM)-experimental approach to predict the extent of paracetamol particle breakage due to impeller agitation. Spherical particle DEM simulations of a small-scale, agitated dryer were performed to determine the normal stresses and shear strains in various parts of the particle bed. The data were combined with the attrition correlation of to predict total attrition within the dryer, with the correlation parameters fit from annular shear cell experiments. The predictions showed reasonable agreement with a corresponding agitated dryer experiment. In addition, Hare et al. reported that in both the simulations and experiments, the total bed attrition was independent of impeller speed for a given number of impeller revolutions, i.e., the attrition was a function of total strain and not of strain rate. This result was consistent with the findings of . Moreover, the simulation demonstrated that more than 50% of the attrition occurred in the bottom third of the bed, toward larger radial distances, and in the vicinity of the mixer blade.More fundamental studies on multi-particle attrition frequently relied on the use of shear cells rather than vertical axis mixers. In most instances, these studies utilized spherical or angular particles as opposed to rod-like particles. In general, these studies measured bulk attrition rates and particle size distributions and attempted to relate these quantities to material or operating parameters (e.g., In addition to these bulk level studies, there have been investigations focused on particle attrition and breakage at the particle level using computational methods. Of particular interest in these studies is that the simulated particles can be damaged directly in the simulations. For example, performed 2D DEM shear cell simulations using initially square particles that were comprised of a collection of bonded Delaunay triangles. When the load on a bond exceeded a critical value, the bond was broken and thus the particle could attrit over time. ) examined particle, or more aptly, agglomerate, breakage due to crushing and impact using a similar technique. These DEM agglomerates were roughly spherical in shape, with the bonded elements consisting of smaller spheres. As with Potapov and Campbell, when the bond load exceeded a critical value, the bond would break. The approach of using “glued spheres” with breakable bonds is now a common method in DEM for studying attrition and breakage problems (see, for example, Few previous DEM studies have incorporated rod-like particles. Due to the additional computational time required for contact detection, most researchers perform DEM simulations using spheres rather than more realistic particle shapes. Those simulations that do incorporate more complex shapes generally rely on the use of glued sphere approximations, which provide the simplicity of sphere–sphere contact detection and the flexibility of generating a wide variety of particle shapes, but at the cost of artificially rough particle surfaces (), smaller coefficients of restitution (), and the increased computational overhead of tracking a larger number of component spheres. used a DEM glued-spheres model to simulate the movement of rigid, rod-like particles in a Freeman powder rheometer. Although they did find that the rheometer torque increased when non-spherical particles were simulated, they could not discern a clear trend with aspect ratio (defined as particle length over cross-sectional diameter) over a range of aspect ratios between one and two. They did not investigate the effect of particle roughness caused by the use of glued spheres, but they did report that increasing the inter-particle friction increased the impeller torque. developed a combined DEM and experimental methodology for population balance (PB) modeling of rigid, rod-like particle breakage under uniaxial compression. They used a glued-spheres approach, similar to , but their particles could break when the force between neighboring glued spheres exceeded a critical value. They used this model to predict the form of the PB breakage kernel and the daughter distribution functions. The distribution function parameters were fit using either experimental or DEM data. Of particular interest for the present work is the observation that particle breakage, on average, occurred in the middle of the rod-like particles, regardless of aspect ratio. The authors did not investigate the influence of particle roughness. performed DEM simulations of simple shear flows using rod-like particles to investigate particle alignment and bulk stresses. They utilized both true cylindrical particles as well as glued-sphere particles in order to investigate the influence of particle surface roughness, among other factors. For dilute flows, the flow stresses were largely dependent on the particle aspect ratio rather than the surface roughness, with larger aspect ratio particles generating smaller stresses. In dense granular flows, however, the stresses depended on the particle aspect ratio, particle friction, and surface roughness, with large aspect ratio, frictional, and rough particles producing the largest stresses. Indeed, glued sphere particles in which the spacing between component spheres was one diameter produced flow stresses more than an order of magnitude larger than simulations using smoother particles. These findings emphasize the significance of both the aspect ratio and surface roughness on flow behavior and stresses. developed and validated a DEM model using sphero-cylindrical particles to examine the velocity, solid fraction, and particle orientation fields in a vertical axis mixer. They found that particle trajectories within the bed were similar to those that were reported previously for spheres, with a vortex circulating in the direction opposite of the blade rotation on horizontal planes of the bed. However, some important differences were also observed. For example, increasing the aspect ratio of particles tended to increase their degree of alignment and decrease their velocities relative to the mixer blade. Larger particle aspect ratios also decreased the overall bed solid fraction as well as the solid fraction uniformity. The particles also aligned such that their major axes were offset by 10–15 degrees from the flow streamlines. In addition to the strong correlation between the particle principal orientation and velocity vectors, regions of larger velocity gradient magnitude were found to result in smaller solid fractions and smaller degrees of alignment between particles.Although not in vertical mixers, some researchers have examined the packing of rod-like particles using numerical methods, including Monte Carlo and DEM simulations. investigated the influence of aspect ratio on packings of sphero-cylinders (not glued sphere particles) in unit cells or columns. All reported that beds comprised of particles with an aspect ratio approximately equal to 1.5 had the largest solid volume fraction for mono-disperse systems with periodic boundaries.There have been no prior studies investigating rod-like particle breakage at the particle level in vertical axis mixers using smooth particles. Prior studies utilizing rod-like particles have either examined breakage from a bulk level, e.g., examining trends or developing empirical correlations, have focused on particle kinematics and dynamics rather than breakage, or have made use of glued-sphere DEM models, which have been shown to produce large stresses when compared to smooth particles.This paper describes a novel approach for predicting the internal loads, moments, and stresses acting within rigid, rod-like particles. A DEM model is developed consisting of (smooth) sphero-cylindrical particles agitated in a slowly rotating, two-bladed, vertical-axis mixer. The loads and moments within particles are calculated from force and moment balances while the internal stress state is found using beam bending theory. The model is used to examine the number frequency and spatial distributions of loads, moments, and stresses as functions of particle aspect ratio, fill level, impeller blade speed, and inter-particle friction coefficient in order to better predict under what conditions particle breakage will occur.A DEM computer simulation is used to investigate the internal loads, moments, and stresses in rod-like particles in a vertical-axis mixer. The DEM model used in this study is identical to the one previously described in and the reader is referred to that study for details. For convenience, a summary of the system and particle geometry, contact force models, and parameters are provided in the following paragraphs.The particles in the simulation are modeled as rigid, sphero-cylinders () with an aspect ratio of AR=L/d where L is the length of the particle and d is the diameter of the particle’s cylindrical and hemispherical components. Using this definition, a sphere has an aspect ratio of AR=1. This geometry is meant to mimic the rod-like API particles typically encountered in an AFD process, yet retain the simplicity of sphero-cylinder contact detection (). A snapshot from one of the simulations is shown in . Note that the particles in these simulations do not break.The particles are initially generated at random locations with random orientations and no translational or rotational speeds in the vertical space above the impeller blades. No overlaps are allowed during this initial creation phase. Once all of the particles are generated, they are allowed to fall under the action of gravity until the particles come to rest. Although the resulting particle orientations in this initial state are not measured, they appear to show little coordinated structure. Furthermore, the initial bed depth is nearly constant.A damped Hertzian spring normal force model () and a sliding tangential force model () are employed to compute the particle–particle and particle–boundary interaction forces. The particles are treated as cohesionless. found that attrition only became significant when material was almost dry, i.e., liquid bridge cohesive forces were negligible. The model parameters are given in , with the justification for the parameters provided in . Of particular note is the fact that the elastic moduli are several orders of magnitude smaller than the actual values of real materials. As is the case with most DEM simulations, small elastic moduli are used in order to increase the integration time step used in the algorithm and thus reduce the computational time required to complete the simulations. In the previous work of , it was shown that the impeller torque predicted by the model increased with the modulus value, with a realistic modulus value producing a torque approximately equal to the experimentally measured value. The implications of using smaller elastic moduli values on particle internal stresses are considered in A novel approach is used to calculate the internal load and moment distributions within individual rod-like particles. At a given instant in time, the dynamics of a particle can be characterized by the translational velocity vP, acceleration aP, rotational acceleration ω̇P, and rotational velocity ωP of the particle’s center of mass (point P), as shown in a. Note that the rotational acceleration and velocity are given in the particle-fixed frame of reference (xyz)P. The forces acting on the particle include surface forces FP,i due to contact with other particles and boundaries, as well as a gravitational body force acting at the particle’s center of mass GP.To determine the internal loads and moments within the particle, it is assumed that the particle is geometrically rigid, as is the case in essentially all DEM simulations (particle deformations are accounted for in the force models). Taking advantage of the rod-like geometry of the particle, the internal loads and moments are determined on cross-sectional areas having a unit normal pointing along the axis of the particle, i.e., the z direction in . The resultant loads and moments on a particular cross-section C–C′ (b) are determined by applying Newton’s Second Law to one of the two particle segments (e.g., segment B) resulting from the cross sectioning. From Newton’s Third Law, the loads and moments on the other segment (segment A) are equal and opposite in sign to segment B’s values. The cross-sections are made only in the cylindrical portion of the particle; hence, internal loads and moments are not reported for the hemispherical end-caps.From rigid body kinematics, the translational acceleration of segment B’s center of mass is given by,where rB/P is the vector pointing from the center of mass of the original particle (point P) to the center of mass of segment B (point B). The rotational velocity and acceleration about segment B’s center of mass are identical to the rotational velocity and acceleration of particle P,The forces acting on segment B include the external surface forces acting on the segment (FB,i⊆FP,i), as well as the internal forces that segment A exerts on segment B, which include an internal normal force N=Ne^z and an internal shear force V=Vxe^x+Vye^y. In addition, segment A exerts an internal bending moment M=Mxe^x+Mye^y, and internal twisting moment T=Te^z on segment B. Lastly, a gravitational body force GB also acts on segment B, with its value being proportional to the volume of segment B.The accelerations, external surface forces, and gravitational body force acting on segment B are all known. The internal loads and moments may be determined using Newton’s second law and Euler’s equations of motion,IBω̇B+ωB×(IBωB)=(∑ri/B×FB,i)+M+T+(rCC′/B×V),where IB is segment B’s principal moment of inertia matrix in the particle’s frame of reference,Note that the moments of inertia include the hemispherical end cap of the segment. The parameters ri/B and rCC′/B are the vectors pointing from the center of mass of segment B to the location of external surface force FB,i and the cross-section CC′ where internal shear force V is applied. Eqs. are a collection of six equations with six unknowns (N, Vx, Vy, Mx, My, and T) and thus the system can be solved for the internal loads and moments. The internal loads and moments are calculated at seven evenly distributed locations along the length of each particle’s cylindrical portion () starting and ending at the interfaces between the hemispherical end-caps and the cylindrical portion of the particle. A larger number of internal locations could be used, but at increased computational cost.In order to present spatial distributions of the internal loads and moments over all particles, the mean value of a quantity at cross-section location z is given bywhere αiq is the quantity of interest (load or moment) for particle q at time instance i. The quantity Q is the total number of particles in the system. Because the system is axisymmetric and at steady state, the quantity α is averaged over a large number of time instances τ, each having a duration equal to the inverse of the simulation data output frequency. The number of time instances τ is equal to the output frequency (=80 samples recorded per blade revolution) multiplied by the number of blade revolutions over which the measurements are made (=three blade revolutions). Thus, in the given work, averages are made over τ=240 samples, with each sample reporting α values averaged over a time period of 0.025 s at an impeller speed of 30 rpm.The stresses internal to the particles may be estimated from the internal loads and moments. In order to calculate these stresses, the particle is assumed to behave as a beam (). Two beam theories are commonly referred to when calculating internal beam stresses and deflections: Euler–Bernoulli and Timoshenko. Euler–Bernoulli theory is most appropriate for slender, i.e., high aspect ratio, beams while Timoshenko beam theory, which allows for shear deformation of the beam, is more relevant for short beams. The difference in the two theories appears in beam deflection predictions when large beam deformation occurs, but both theories use the same expressions to relate the resultant forces and moments to the internal stress distribution acting on a cross-section of the beam. Recall that the particle is assumed here to remain geometrically rigid so deflections are neglected. Both beam theories give the internal stress state on a given cross-section as (where x and y are the distances from the long axis of the particle, R is the particle radius, and N, V, M, and T are internal loads and moments on the given cross-section. In order to balance spatial resolution with computational efficiency, the stress components are calculated at a discrete number of points at each cross section as shown in Although the stress components may be of interest independently, what is typically of more interest for particle breakage is a measure of stress that is related to a failure model. For example, with materials that fail in a ductile manner, it is the equivalent or Mises stress that is usually of interest,where σ1, σ2, and σ3 are the principal stresses of the stress tensor with σ1≥σ2≥σ3. If the Mises stress equals or exceeds the yield strength of the particle material, σy, then failure is assumed to occur (If the material fails in a brittle manner, then one common failure model, known as the maximum stress criterion, states that failure occurs when either the maximum principal stress or absolute value of the minimum principal stress exceeds the ultimate strength of the material Su (In this work, particle bending contributes most to the stresses, so |σ1 | and |σ3| are nearly equal in value. Since for most materials the ultimate tensile strength is smaller than the ultimate compressive strength, particle failure in tension may be reasonably assumed. It is important to note that in these simulations the particles do not break. In an actual system, if particle breakage occurs, the distribution of stresses will likely change due to changes in force transmission through the particle bed (Since it is either the maximum Mises stress or maximum absolute principal stress that is of interest in these two failure models, frequency distributions by number of these stresses are calculated over all particles. The frequency distribution by number is given bywhere σ is the maximum stress of interest and qi(σ−1/2Δσ, σ+1/2Δσ) is the number of particles at time instance i with a maximum stress value within the given range. Thus, f(σ)Δσ is the fraction of particles within the range σ–1/2Δσ≤σ<σ+1/2Δσ averaged over the duration of the time instance (=0.025 s at a blade speed of 30 rpm, a sampling frequency of 80 samples per blade revolution, and a total of three blade revolutions). Complementary cumulative distributions of these stresses,are also generated from the frequency distributions since these are more convenient for determining what fraction of particles exceed a particular stress value. In addition to frequency distributions, spatial distributions of the stresses within a particle are generated (using Eq. In order to have confidence that the loads and moments predicted by the current algorithm are accurate, comparisons are made to results generated from a previously published glued-spheres DEM model, which simulates the rod-like particles in a fundamentally different manner. The predicted stresses are also compared to the stresses generated from a finite element method (FEM) simulation of a single particle subject to an external load.The glued-sphere DEM model used for comparison is the one described in . In that model, a rod-like particle is generated by elastically bonding multiple spheres in a straight line. The use of spheres makes the particle surface bumpy, which has been shown to produce significantly larger bulk stresses than smoother particles produce (). In order to provide a better comparison to the current model, which has a smooth surface, the component spheres are spaced one sphere radius apart rather than one sphere diameter (measured from sphere center to sphere center).The internal loads and moments in the glued-sphere model are found using the relations described in using a bond stiffness of 1 GPa so that the fibers are stiff, mimicking the rigid sphero-cylindrical particles. The internal loads and moments in the present model are determined at the same locations as the bonds in the Frequency distributions of loads and moments over each internal cross-section generated using each model are compared for rod-like particles of AR=3 in a small, vertical-axis mixer using the parameters listed in . The system geometry is similar in shape to , but the dimensions are smaller to reduce computational time. The coefficients of restitution and particle-base sliding friction are also different than those in . A larger particle-base sliding friction coefficient is used here to prevent solid body rotation of the particle bed, which is more likely to occur as the system size decreases. The coefficient of restitution is also different, but that parameter has little influence on the results. Results for both a static and dynamic case (blade rotation speed of 30 rpm) are considered. plots the distributions of internal normal force, shear force, bending moment, and twisting moment from both the glued-spheres and sphero-cylinder models for the static bed case. The two models show excellent agreement, although the sphero-cylinder model predicts a slightly larger number of larger shear forces and twisting moments. plots the same data, but for a dynamic bed. Again, the normal force distribution comparison is excellent between the two models, but the present model predicts more small magnitude twisting moments and shear forces and more large magnitude bending moments. The sources of the discrepancies are not certain; however, the fact that the glued-sphere particles can vibrate while the rigid sphero-cylindrical particles cannot is likely to contribute. For example, a twisting moment can only exist in the sphero-cylindrical particle model when a tangential force is present. As soon as the force is zero, the twisting moment is also zero. However, in the glued-spheres model, when the tangential force disappears, the particle will vibrate and continue to generate internal twisting moments until damping causes the vibration and twisting moment to go to zero. Hence, the glued-spheres particle is expected to have a larger fraction of larger twisting moments.The slight deformation in the glued-sphere particles may also contribute. For example, deformation of an elastic fiber would cause its central axis to no longer remain straight and, thus, contact normal forces would also contribute to the twisting moment in addition to frictional forces. For the rigid sphero-cylindrical particles, only the frictional forces can contribute to twisting. This effect would be more pronounced during the dynamic case when particles are subject to larger deformation and thus produce a larger fraction of twisting moments in the glued spheres model. In addition, bending of the fibers would cause a reduction in the moment arm and produce a larger fraction of smaller bending moments. Note that the stresses leading to particle breakage will be dominated by the bending moment and shear load, so the differences in twisting moment are of less practical concern. These factors suggest that the sphero-cylinder model is best suited for nearly rigid, highly damped particles.To ensure that the assumption of a beam model (Eqs. ) is appropriate when calculating the internal stresses, a finite element method (FEM) model is compared to the present model for a simple, static three point bending scenario (see, for example, ). The conditions for the two models are listed in , the errors between the two models relative to the FEM predictions for the maximum principal and maximum Mises stresses are smaller than 10% for particle aspect ratios greater than approximately AR=4.5. The DEM simulations consistently over-predict the stresses when compared to the FEM results. The increase in the relative errors with decreasing aspect ratio is not surprising since a key assumption used in beam bending theory, i.e. that the beam cross-section remains rigid on its plane and does not get distorted, becomes less reasonable. Note that the calculation of the internal loads and moments is independent of the beam bending stress model, thus the accuracy of those quantities is unaffected by aspect ratio. In practice, the decrease in the accuracy of the stress predictions at smaller aspect ratios is likely to be of less concern since experiments have demonstrated that it is the larger aspect ratio particles that are more likely to fail (The DEM simulations are used to investigate internal particle loads, moments, and stresses for the vertical axis mixer geometry shown in . The baseline simulation parameters are given in . The simulations are allowed to first come to steady state, which takes less than two blade revolutions, before data are collected. Data are then gathered over three blade revolutions, corresponding to more than 240 output time samples. The load, moment, and stress calculations are all performed in post-processing, although they could be performed at each time step to model in-situ particle breakage, for example, but with increased computational cost.A discussion of the observed flow patterns is not given here since they are described in detail in . Instead, the current focus is on the distribution of internal loads, moments, and stresses as functions of the particle aspect ratio, fill level, blade speed, and particle–particle friction coefficient.The following sections discuss the influence of particle aspect ratio, bed depth, blade speed, particle–particle friction coefficient, and elastic modulus. The average distribution of loads, moments, and stresses within a particle are presented in the first section to determine where breakage within a particle is most likely to occur. In the remaining sections, distributions of the maximum absolute principal stress are presented for the range of parameters investigated. plots the spatial distribution of the internal loads and moments at each of the seven locations within the particle for four different particle aspect ratios. The mean value, calculated using Eq. , is shown. Four observations stand out.First, when averaged over many particles and many time instances, the spatial distributions are symmetric about the particle center-plane (z=0) with a maximum occurring there. This behavior is not unexpected since the particles are symmetric about their center of mass. Although the spatial distribution for an individual particle may not be symmetric due to asymmetric external loading, the distribution becomes increasingly symmetric as more particles are included in the average.Second, the largest loads and moments are located at the particle center-plane (z=0). The normal load remains nearly constant along the particle length, but it does increase slightly toward the particle center. The fact that maximums occur at the particle center-plane implies that if a particle breaks, it will do so, on average, at the particle’s center-plane. This observation is consistent with ’s results from glued-sphere, rod-like particle DEM simulations of a compression cell in which particles broke predominately in their center regardless of their aspect ratio. Recent work by in which breakable, glued-sphere, rod-like particles were agitated within an attrition cell produced similar results.The third observation is that the loads and moments increase with increasing particle aspect ratio, and the magnitude of the difference between the values at the particle center-plane and the particle edge (z/(L−d)=±1/2) also increases. This behavior is not unexpected since a simple force balance demonstrates that the spatial distribution of the shear load and moments will be strongly dependent on the aspect ratio. For example, for a pinned–pinned beam subject to a uniform load per unit length of p, with the distance between the beam supports being L, the bending moment at the beam’s center is,where d is the diameter of the cylindrical and hemispherical portions of the particle. Furthermore, larger aspect ratio particles with friction are expected to have increased loading due to an increased number of contacts (). The normal force is less sensitive to the aspect ratio since it only depends on the magnitude of the external loads parallel to the z direction of the particle. The largest of these loads are likely due to contact with the end-caps of the particle (); however, tangential loads acting along the cylindrical portion also contribute, and as the length of the cylindrical portion increases (i.e., the aspect ratio increases), additional contacts can contribute to the normal load. The fact that the increasing aspect ratio results in larger loads and moments implies that larger aspect ratio particles are more likely to break as compared to smaller aspect ratio particles. This result is consistent with previous work (e.g., The fourth observation from the plots is that the normal forces are in compression, on average, and that the normal force magnitudes are smaller, but of the same order of magnitude as the shear force. The bending moments, however, are approximately an order of magnitude larger than the twisting moments. This latter observation makes sense when considering that the moment arm that can cause a twisting moment is on the order of the particle diameter d whereas the moment arm causing a bending moment is on the order of the particle length L. plots the maximum values of the loads and moments from , i.e., the loads and moments at z=0, as a function of the aspect ratio. The normal and shear forces and twisting moment appear to increase nearly linearly with aspect ratio over the range of aspect ratios investigated. However, upon closer inspection the trends are observed to have a slight negative curvature indicating that the rate of increase of the maximum values for these quantities decreases slightly within increasing aspect ratio. The cause for this trend is not definitively known, however, it could be due to the fact that the loads acting on the particles change as the particles align more consistently with increasing aspect ratio (). The trend in the maximum bending moment, in contrast, appears to have a positive curvature, which is consistent with the dependence of the bending moment on the length of the particle (Eq. The maximum absolute principal stress, max(|σ1|, |σ3|), averaged over all particles and time steps, is observed to occur along the particle circumference (r/d=1/2) at the center-plane (z=0), as shown in for a particle with an aspect ratio of AR=7. The same trend is observed at all aspect ratios as well as for the average Mises stress σM. Having the largest maximum absolute principal stress or Mises stress along the circumference at the particle center-plane is consistent with the load and moment distributions as well as with the fact that the largest strains in beam bending occur farthest from the particle’s neutral axis (i.e., the centerline). These results indicate that particle failure is not only most likely to occur at the center-plane of the particle, but the failure will begin at the particle’s circumference.The average center-plane (z=0), circumferential (r/d=1/2) maximum absolute principal stress and Mises stress are plotted in as a function of particle aspect ratio. Note that the maximum absolute principal stress and Mises stress data are nearly identical. This similarity is reasonable when considering the relative magnitudes of the principal stresses. For example, for a point in tension at the center-plane, circumference, σ2 and σ3 are much smaller in magnitude compared to σ1, which means, according to Eq. , the Mises stress will be nearly equal to σ1. Consequently, the remaining stress plots will present only maximum absolute principal stress information since Mises stress values are nearly identical.The largest average maximum absolute principal stress (and Mises stress) increases with increasing particle aspect ratio, with a positive curvature similar to that observed for the bending moment (refer to (c)). The bending moment contributes the most to the maximum absolute principal stress at the center-plane circumference and, thus, the stress trend should reflect the bending moment trend.Of particular interest in particle breakage studies are the frequency and complementary cumulative distributions by number (Eqs. ) of the largest maximum absolute principal stress (or Mises stress) over all particles and averaged over all time steps. This information, coupled with the ultimate (or yield) strength of the particle, can be used to determine what fraction of particles will break in a given time period. The complementary cumulative distribution gives the fraction of material with the largest maximum absolute principal stress (max(|σ1|, |σ3|)) larger than a given value, which is of particular interest for breakage studies. plots the max(|σ1|, |σ3|) frequency and complementary cumulative distributions for the four particle aspect ratios examined. Note that the particle elastic modulus used in these simulations is E=1 MPa, which is two to three orders of magnitude smaller than most real materials. The impact of this assumption is discussed in . As an example, if particles with AR=9.2 have an ultimate strength of 0.1 MPa, then nearly 5% of the particles would experience stresses sufficiently large to cause breakage. Note that, on average, σ1,max occurs at the particle’s center-plane circumference, but for any given particle σ1,max may occur in some other location depending on the external loading conditions.For convenience in comparing distributions, the frequency distributions are fit to a (two-parameter) Weibull distribution,f(σ1,max)=abσ1,maxb−1exp(−aσ1,maxb)(σ1,max>0),where a and b are fit parameters. The corresponding complementary cumulative distribution isIn particular, this fit is made for stresses larger than the point of maximum curvature in the distribution in order to include just the large stress information (refer to a). Weibull distributions are used in a wide range of fields, including those involving reliability predictions and even post-milling particle size distributions. used a three parameter exponential fit to their force data for particles in a piston geometry. In their 2D simulations of a shear cell, fit their particle forces to a power law expression at small forces and an exponential relation at larger values. The Weibull distribution used here at larger stresses is similar in form to the one used by Estrada et al. In all cases, a Weibull distribution fits the large stress data well with coefficient of determination (R2) values greater than 0.996 for all cases. An example fit is shown in The Weibull fit parameters for the frequency distributions corresponding to different aspect ratio are shown in . Both fitting parameters decrease with increasing particle aspect ratio; however, the parameter a decreases inversely with aspect ratio from ~35 down to ~7 while the parameter b decreases in more of a linear manner from ~0.52 to ~0.43. Over these ranges, the change in a has a more significant influence on the frequency distributions than parameter b. Note that the mean and median values for max(|σ1|, |σ3|) should be calculated directly from the DEM-generated data since it contains the full distribution. The Weibull distributions are only for the larger stress data.Since it is ultimately σ1,max that is of most interest in practice, the average load and moment plots are not presented for the remainder of the parametric studies and instead only the σ1,max frequency and complementary cumulative distributions, and the corresponding Weibull fit parameters, are shown. plots the frequency and complementary cumulative distribution functions for simulations performed with an aspect ratio of AR=7 and five different bed depths (H/HB=1.0, 1.5, 2.0, 3.0, and 4.0, where HB is the height of the mixer blade). The corresponding Weibull fitting parameters are given in . The mean and median of the max(|σ1|, |σ3|) distributions increase with dimensionless bed height, as is expected since it has been observed that the stresses in an AFD geometry are due largely to the weight of the material in the device (). The fitting parameter b is nearly independent of H/HB and approximately equal to 0.5, and the parameter a decreases in a nearly linear manner with increasing H/HB. Hence, increasing the bed depth is expected to result in a larger fraction of broken particles (assuming the stresses exceed the ultimate strength).In addition to bed depth, the influence of blade speed is also reported. The frequency and complementary cumulative distributions for max(|σ1|, |σ3|) are shown in , with the corresponding Weibull fits in Increasing the blade speed from 2 rpm to 90 rpm increases the largest maximum absolute principal stresses. Both fitting parameters a and b decrease slightly over this range. reported that the extent of breakage was independent of impeller speed for a given number of impeller rotations. reported that breakage increased with impeller speed, but this was for a fixed agitation time, which would imply a larger number of impeller rotations for larger impeller speeds. The current work does indeed show that breakage will generally increase with impeller speed independent of the number of blade revolutions (assuming the ultimate strength is exceeded), but the effect is small when compared to the influence of particle aspect ratio and bed depth. At these small impeller speeds, internal particle stresses appear to be governed primarily by the weight of the material on top of it as opposed to impacts with the blade., increasing bed depth results in larger stresses for underlying particles due to the increased weight of material above them. presents field plots of the average maximum absolute principal stress at three depths within the bed and indeed the largest stresses occur at the bottom of the mixer, upstream of the blades, and near the mixer circumference where the bed is the deepest. These findings are consistent with the work of . The stresses near the base of the container for the static case, which has a nearly level surface, has a more uniform spatial distribution. Another possible contribution to the increased stresses for the dynamic case may be the interference between particles as they re-orient to flow around the blade. This effect, however, must be small in comparison to the depth effect since significant re-orientation occurs in the valley downstream of the blade (The influence of particle–particle friction coefficient is also examined. The frequency and complementary cumulative distributions for max(|σ1|, |σ3|) for these cases are shown in , with the corresponding Weibull fits in The mean max(|σ1|, |σ3|) increases slightly as particle–particle friction coefficient increases from 0.11 to 0.55. The effect is not particularly significant, however, as suggested by the nearly constant fitting parameter a. Although the parameter b increases over this range, it has a much smaller impact on the distribution than parameter a. That an increasing particle–particle friction coefficient increases the stresses is not unexpected since it will result in larger tangential forces for a given normal load. observed that increasing inter-particle friction led to larger attrition rates for their spherical particles. More interestingly, however, is that increasing inter-particle friction above a critical value has been observed to cause rod-like particles to rotate significantly relative to one another as opposed to remaining aligned (). This rotation would be expected to cause an abrupt increase in σ1,max due to the expected large loads caused by the interference between particles as they rotate. This effect, however, is not observed in the current work. It could be that a sufficiently large inter-particle friction coefficient is not used or that the presence of a free surface boundary condition allows the bed to more easily accommodate particle rotation as opposed to the fixed volume fraction boundary conditions used by Of particular interest in the current work is the influence of particle elastic modulus on the predicted stresses. DEM simulations generally use elastic modulus values orders of magnitude smaller than those encountered for real materials. The reason for using small values is that the simulation integration time step decreases as the elastic modulus increases. Hence, in order to have computationally efficient simulations, smaller elastic moduli are used., have shown that, at least for particle kinematics, having an artificially small elastic modulus does not significantly affect DEM predictions, unless the elastic modulus is too small (“too small” depends on the flow conditions). However, there has been no examination of how small elastic modulus values affect internal particle stresses. It is reasonable to expect that stress magnitudes will be strongly influenced by the elastic modulus since the inter-particle forces are a function of elastic modulus.A dimensional analysis of the DEM simulation variables reveals the following dimensionless relationship,σEp=fcn(Ld,EpρpgH,νp,Dd,HHB,HBd,ω2Dg,EbEp,νb,μpp,μpb,εpp,εpb),where, in addition to the previously defined variables, E is the elastic modulus, ν is the Poisson’s ratio, ρ is a mass density, g is the acceleration due to gravity, and ε is a coefficient of restitution. The subscripts “p” and “b” indicate particle and boundary, respectively. If the dimensionless quantities remain constant on the right hand side of the equation, then the stresses are expected to scale linearly with the elastic modulus. Note that both Eb and Ep are generally decreased by several orders of magnitude in DEM simulations so that the ratio is nearly constant between the DEM model and the real system. The remainder of the independent dimensionless terms are easily kept constant between DEM and the real scale except for the cylinder-to-particle diameter ratio, D/d, and the ratio of the characteristic elastic force to the force due to the bed weight, Ep/(ρpgH). The lack of maintaining full geometric similarity by letting D/d vary is a topic of significant interest for DEM (see, for example, ). Of particular interest here, however, is the latter term (note that the validation experiments and DEM simulations here had identical D/d values). If the DEM gravitational acceleration and bed depth remain identical to the real system values, then the particle density should be decreased by several orders of magnitude if the particle elastic modulus is also decreased by the same amount. Unfortunately this would result in a significantly smaller DEM integration time step, which is precisely what is attempting to be avoided by decreasing the elastic modulus. In most DEM simulations, the particle density is maintained at its real value so the Ep/(ρpgH) term is not held constant between the DEM and real scales. As a result, it cannot be expected that the stresses should scale linearly with the elastic modulus.In order to better understand how to scale the stresses appropriately as the elastic modulus changes, several simulations were performed with varying elastic moduli (particle and boundary elastic modulus values were identical, all other parameters remained constant). The complementary cumulative distributions for these simulations are plotted in along with the corresponding mean values. The Weibull fit parameters for the distributions are given in . As expected, a simple linear trend with elastic modulus is not observed. Instead, the mean stresses appear to increase in a nearly logarithmic manner with elastic modulus.In order to be of most use for predicting breakage, one would need to know the ultimate strength (or yield strength) of the particles in order to determine what fraction of particles would be expected to break. Since ultimate strengths and elastic modulus values for API crystals are not readily available, the properties of acetal, the material used in the kinematics validation experiments described in , are used here as a point of comparison. The elastic modulus and ultimate tensile strength of acetal are E=3.1 GPa and SU=75.8 MPa, respectively (). Using this critical value and making use of the data in , the simulations predict that particle breakage should not occur, which is consistent with what was observed in the corresponding validation experiments (). Experiments using material with a smaller ultimate stress will be performed in a follow up study so predictions from the current model can be directly compared to experimental breakage data.This work presents a new model for predicting the internal loads, moments, and stresses within rigid rod-like particles. Newton’s laws are applied to a section of the particle to determine the internal loads and moments, while beam bending theory is used to determine internal stresses. When the stresses are coupled with a failure model, particle breakage can be predicted. The model is verified against a previously published glued-spheres model and finite element method stress predictions. In the present work, the model particles are not allowed to break; however, the DEM simulation could be modified to break particles at predicted failure locations so that the evolution of particle size distribution can be studied. The advantages of using sphero-cylinders over glued-sphere particles are that (a) the particle surfaces are smooth and thus do not generate the large stresses observed with the artificially rough surfaces of widely spaced glued spheres, and (b) contact detection between sphero-cylinders can be faster than that for glued-spheres when the particle aspect ratio is large.Applying the model to a vertical axis mixer, the internal loads and moments are predicted for a range of particle aspect ratios. The average internal load and moment distributions are symmetric about the particle center-plane, with maximums at the particle center. The average loads and moments increase with increasing particle aspect ratio. The average normal load, which is compressive, shows little variation over the length of the particle. It has a magnitude of the same order as the shear load. The bending moment, however, is at least an order of magnitude larger than the twisting moment. The shear load and bending moment contribute the most to the internal particles stresses. The predictions suggest that, on average, particles will break at their center-plane, which is consistent with previous studies.Although the stress tensor at various locations within the particle can be predicted from beam bending theory, only the maximum absolute principal stresses within the particle are presented since this quantity is used within a common brittle material failure model. The Mises equivalent stress, which is often used in ductile failure models, is found to be nearly equivalent to the maximum absolute principal stress value. The two stresses are similar in magnitude because the normal stress due to bending contributes most to the stresses. The largest maximum absolute principal stress (and Mises stress) is found to occur along the circumference of a particle’s center-plane, on average, indicating that that is the initial location of particle failure.Frequency distributions by number of the largest maximum absolute principal stress per particle are generated for variations in particle aspect ratio, bed depth, impeller speed, and particle–particle friction coefficient. At larger stresses, the distributions are fit well by a (two parameter) Weibull distribution. found that a similar distribution fit their shear force data well at larger force values. Knowing that the data is fit well by a particular distribution form makes it easier to compare distributions. Complementary cumulative distributions generated from the frequency distributions make it more convenient to determine what fraction of particles would be expected to break for a given operating condition.Increasing particle aspect ratio, bed depth, impeller speed, and particle–particle friction coefficient all act to increase the fraction of larger maximum absolute principal stresses. Particle aspect ratio has the most significant effect, followed by bed depth. Impeller speed and particle–particle friction coefficient have smaller influences on the stresses. The exception is when the bed transitions from being static (no impeller rotation) to dynamic. Flowing particles exhibit significantly larger stresses than static particles. The change in bed depth as particles flow around the blade appears to be the source of the difference. However, once flowing, changing the speed, at least over the current range, does not cause a significant increase in the largest maximum absolute principal stresses. The limited blade speed dependence is reasonable considering that the particle–particle and particle-wall interactions depend on their relative velocities, not the absolute velocities. As has been found in previous studies (e.g., ), the internal particle stresses are more influenced by the weight of the supported material.One parameter of particular interest in DEM studies involving forces and stresses is the particle elastic modulus. Since this value is typically made artificially small in DEM studies by several orders of magnitude, its effect on the stress distributions was studied. Due to imperfect scaling of the ratio of the characteristic elastic force to the characteristic hydrostatic force, the stresses do not scale linearly with the elastic modulus. Instead, the stresses appear to increase in a logarithmic manner as the elastic modulus increases. This information is particularly useful for scaling small elastic modulus DEM simulation results.A mortar-finite element approach to lubricated contact problemsIn this paper, a new numerical approach, based on mortar discretization of interface fields, is proposed to solve lubricated contact problems between deformable solid bodies. The Reynolds approximation to the thin film fluid equations is used to describe the fluid behavior on the interface, while the equations of small strain elasticity (to which finite elements are applied) are assumed to govern the solid body behavior. The result of this approach is a monolithic scheme in which the fluid and solid mechanics are solved simultaneously, without the need for iterative or staggered approaches between the fluid and solid phases.Key features of the approach are that the fluid film thickness, directly related to the deformation of the solid phase, is computed from a least squares projection based on dual basis functions (as pioneered by Wohlmuth [B.I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer-Verlag, Heidelberg, 2001]). The free boundary problem for the fluid phase, corresponding to the boundary between lubricated and cavitated regions, is regularized with a penalty method and thus resolved as part of the solution. Several numerical examples are given to demonstrate the approach.Lubricated contact problems constitute an important class of contact problems, and are found widely in engineering applications. In a lubricated contact problem, the relative motion of two solid surfaces causes fluid flow between them, which in turn generates pressure (normal traction) and shear stress (frictional traction) which act to deform the solid bodies. Assuming smoothness of the solid surfaces, the fluid phase of this procedure is often considered to be governed by the classical Reynolds equation, which can be derived from the Navier–Stokes and continuity equations based on some classical assumptions (see Early numerical approaches to the lubrication problem, usually between rigid boundaries, were usually based on finite difference approximation. Christopherson (see In many situations, elastic deformations of the lubricated surfaces become significant and they dramatically change the fluid film thickness. Consequently, deformations of the solid phase have to be taken into account to obtain an accurate solution for the pressure field and shear stresses. Elastohydrodynamic lubrication problems are highly nonlinear because of the dependence of the fluid film thickness on deformations of the solid phase, the positive pressure constraint, and the dependence of viscosity on pressure. Many researchers (see, for example, Recently, some researchers have been attempting to solve more challenging lubricated contact problems arising in metal forming applications, such as rolling. These problems usually involve extremely large deformations, high loading, and noticeable thermal effects. The surface roughness has to be considered because the film thicknesses are extremely thin. In this case, the averaged Reynolds equation (see, for instance, In this paper, we investigate the use of a new modeling approach to lubricated contact between deformable bodies, building upon a mortar framework for the solution of contact problems in mechanics, investigated heretofore primarily in the case of frictional dry contact (see, e.g. The outline of the presentation is as follows. In Section , we give a basic problem description and review the equations of motion for the system (in both the solid and intervening fluid phases). In the presentation, no limitation has been placed on the solid phase constitutive laws (although small strain response is assumed for simplicity), and the Barus law (see discusses an important facet of the treatment of the Reynolds approximation for the fluid layer – namely, a regularization which enables resolution of the so-called Reynolds boundary, dividing the lubricated and cavitated regions. Section addresses the introduction of the numerical discretization, and in particular, shows how the mortar framework can be adapted to naturally encompass the lubrication conditions prevailing on the interface. Section gives a brief summary of the linearization procedure used to solve the nonlinear equations of evolution in a Newton–Raphson implicit framework, while Section discusses an alternative, discontinuous Galerkin (DG) approach to treatment of the Reynolds fluid equations. Finally, Section gives a set of numerical examples which outline the performance of the numerical method.A lubricated contact problem, with two deformable solid bodies and a thin fluid film between them, is shown in . The two solid bodies occupy the open sets Ωs(1) and Ωs(2), where the subscript s represents the solid phase. The closure of Ωs(i)(i=1,2) is denoted as Ω¯s(i), and is the union of the open set with its boundary ∂Ω(i).The surfaces ∂Ωs(i) are divided into Γsu(i), where displacements are prescribed; Γsσ(i), where tractions are prescribed; and Γsc(i), where the contact (at this point, assumed to be either dry or lubricated) occurs. In keeping with a common nomenclature, we will designate Γsc(1) and Γsc(2) as the slave and master surfaces, respectively, although these terminologies will not retain their same meanings as in traditional (i.e., non-mortar) contact mechanics strategies. One assumes Γsu(i),Γsσ(i) and Γsc(i) satisfyΓsσ(i)∩Γsu(i)=Γsσ(i)∩Γsc(i)=Γsu(i)∩Γsc(i)=∅The solid phase of a lubricated contact problem is governed by the usual solid continuum mechanics equations. Here, for simplicity, we assume infinitesimal deformations for the solid phase, although it is possible to extend the formulation to large deformation cases with proper treatment of geometric nonlinearities. In formulating the general numerical approach to be followed, no limitations are placed on the constitutive laws: the material response may be either elastic or inelastic, rate dependent or independent.In lubricated contact problems, the thickness of lubricant films, although allowed to vary spatially, is assumed to be much smaller than the other characteristic dimensions of the problem. Given this circumstance, the fluid phase of a lubricated contact problem will be taken to be governed by the Reynolds equation, which can be derived from the Navier–Stokes and continuity equations based on the thin-film assumption (see, e.g. , the lubricated contact domain Ωf, taken as the part of the slave surface Γsc(1) exclusive of the dry contact region Ωd, is divided into two different subdomains, Ωfl and Ωfc. Ωfl is the subdomain where the lubricant is continuous and has positive pressure, while Ωfc is the subdomain where the lubricant has cavitations and is ruptured (the fluid film can only support a negligible amount of tension before rupture). ΓfD is the surface where the Dirichlet boundary conditions (on pressure) are prescribed for the Reynolds equation, while ΓfN is the boundary between the dry contact domain and the lubrication domain and where Neumann boundary conditions (on flow) are prescribed. ΓfR, which is called the Reynolds boundary, is the boundary between Ωfl and Ωfc. Based on these definitions, we haveFurthermore, in this paper, dry contact is not considered for simplification of the formulation. As a result, we haveFor small deformation lubricated problems, the major unknowns are the displacement fields u(i) (i |
= 1, 2) in the solid bodies, and the pressure field p in the fluid film. To define the weak form later in this paper, we first define solution and weighting spaces C(i) and V(i), consisting of solutions u(i) and their variations u∗(i) according toC(i)=u(i):Ω¯s(i)→Rndm|u(i)∈H1(Ωs(i)),u(i)=u¯(i)inΓsu(i)V(i)=u∗(i):Ω¯s(i)→Rndm|u∗(i)∈H1(Ωs(i)),u∗(i)=0inΓsu(i),where H1(·) represents the Sobolev space of functions with square integrable derivatives.For the fluid phase, we define solution and weighting spaces P and Q consisting of solutions p and their variations p∗ according toConsidering small displacements, the displacement fields of the two contact bodies are denoted by u(i),(i=1,2), and a linear relationship holds between strain and displacement,The balance of linear momentum is written the usual manner,where f(i) denotes the body force in body (i), and the Cauchy stress σ(i) is related to the strains ε(i) viaTraction and motion boundary conditions are stated in the standard manner asσ(i)n(i)=t¯(i)inΓsσ(i),u(i)=u¯(i)inΓu(i),where t¯(i) and u¯(i) denote the prescribed tractions and displacements associated with body (i), and n(i) is the outward normal to ∂Γsσ(i).Given these notations, the virtual work principle for the solid phase of the two body lubricated contact problem can be written asG(u,p,u∗):=∑i=12G(i)u(i),p,u∗(i)=∑i=12∫Ωs(i)∇u∗(i):σ(i)dΩ-∫Ωs(i)u∗(i)·f(i)dΩ-∫Γsσ(i)u∗(i)·t¯(i)dΓ-∑i=12∫Γsc(i)u∗(i)·t(i)dΓ=0=Gint,ext(u,u∗)+Gc(u,p,u∗)=0., Gint,ext(u,u∗) is the sum of the virtual work arising from the internal and external forces, while the notation Gc(u,p,u∗) denotes the virtual work associated with the lubricated contact tractions. From , we note that the solid and fluid phase are coupled by the displacement fields and the pressure field, through the contact virtual work Gc(u,p,u∗).In this formulation, the contact virtual work is represented asBased on the thin film assumption, the inertia force of the fluid film is negligible. Consequently, we havewhich implies that the contact virtual work in Gc(u,p,u∗):=-∫Γsc(1)t(1)(X)·u∗(1)(X)-u∗(2)(Y¯)dΓ,where Y¯ is the position of the contact point associated with X. We note that in the mortar formulation of the problem, to be discussed below, the identity of Y¯ for each X is never directly determined, given the non-locality of the numerical formulation.The equation governing the pressure distribution in thin film lubrication is known as the Reynolds equation, and can be derived from the Navier–Stokes and continuity equations. As presented by ∂(ρh)∂t+∇˜·-ρh312μ∇˜p+ρ(V∼(1)+V∼(2))2h=0inΩfl,where ρ is the current mass density of the lubricant, h is the fluid film thickness (directly related to the deformation of the solid phase), μ is the viscosity of the lubricant, V∼(1) and V∼(2) are (tangential) projections of slave and master surface velocity vectors onto the contact surface, and p is the fluid pressure – the primary unknown of Eq. . In reality, the fluid viscosity is directly related to the fluid pressure in many settings, and different viscosity–pressure relations have been proposed in the literature. For one such case, the so-called Barus viscosity (see, e.g. where μ0 is the viscosity when p=0 and α is a material property., ∂∂t is the spatial time derivative, while ∇˜ is the surface gradient operator on the slave surface, which is defined aswhere τα denotes the contravariant basis vectors on the slave surface. V∼(1) and V∼(2) can be expressed as:where n is the surface normal on the slave surface.The volume flow rate per unit width, a measurement of lubricant flux over the contact interface, may be defined as:, q˜ is also a surface vector defined on the slave surface. On the Neumann boundary, ΓfN, shown in , the film thickness h goes to zero. As a result, one hasAs mentioned before, Dirichlet boundary conditions may also be prescribed on the outer boundary asFor the cavitation domain Ωfc, the fluid pressure is close to the atmosphere pressure, which is negligible compared to the fluid pressure in the lubrication domain Ωfl. Consequently, we haveTo satisfy the continuity of mass flow across the boundary ΓfR (also known as the free boundary), the Reynolds boundary condition is defined aswhere n˜ is the outward normal vector of ΓfR. Based on , one can easily prove that the mass flow continuity is satisfied across the Reynolds boundary. However, the position of the Reynolds boundary is undecided which is one of the major difficulties in solving the Reynolds equation directly.If one assumes steady state flow and incompressibility (constant density) for the fluid phase, one may further simplify Eq. to obtain the steady state Reynolds equation asIt is this field equation that will be considered for the lubricant domain, in the formulation we now propose.The fluid phase of the lubricated contact problem is described in the last section by Eqs. . This problem is difficult to solve since the existence of the cavitation region and the position of the Reynolds boundary is generally not known beforehand, and must be determined as part of the solution. This free/moving boundary introduces new nonlinearities into the problem. Several different approaches have been proposed to solve this problem. , the original problem is transferred to an equivalent complementary problem, which can be solved using a standard penalty regularization method. is equivalent to the following complementary problem:p=0and∇˜·-h312μ∇˜p+V∼(1)+V∼(2)2h⩾0,inΩfc.Therefore, in the whole fluid phase domain Ωf (both lubrication and cavitation regions), we may writewith p and q˜ satisfying boundary conditions The solution to the original lubrication problem given by can be obtained by solving the equivalent problem , subject again to the boundary conditions . In this work, taking advantage of the classical nature of the complementarity constraints , a penalty regularization of these conditions is considered and then a weak form is set up to solve the problem with the finite element method. The detailed procedure is given in the next two subsections.To make the problem more amenable to finite element procedures, it is desirable to cast the inequality into an equivalent equality. This can be accomplished in two ways, either by applying a penalty regularization (see where 〈·〉 is the Macauley bracket, representing the positive part of the operand. Eq. implies that a negative pressure field will be penalized by a large number εp, causing p≈0 in the region that p<0. Furthermore, the condition is satisfied exactly. As a result, the conditions are approximately satisfied by the penalty regularization To obtain an approximate finite element solution for the fluid pressure field, it is desirable to build a weak form of the problem . The weak form of the penalty regularized problem may be obtained using standard procedures asGf(u,p,p∗):=∫Ωfp∗∇˜·-h312μ∇˜p+V∼(1)+V∼(2)2h-εp〈-p〉dΩ=0.Note that Gf is also a function of u because the film thickness h (alternatively, the gap) is dependent on the displacement of the solid phase. Using integration by parts, one obtainsGf(u,p,p∗):=-∫Ωf-h312μ∇˜p+V∼(1)+V∼(2)2h·∇˜p∗dΩ-∫Ωfεp〈-p〉p∗dΩ+∫ΓfNp∗n˜·q˜dΓ=0,Since the flux q˜ satisfies the Neumann boundary condition Gf(u,p,p∗):=∫Ωfh312μ∇˜p·∇˜p∗dΩ-∫Ωfεp〈-p〉p∗dΩ-∫ΩfV∼(1)+V∼(2)2h·∇˜p∗dΩ=0.This is a highly nonlinear equation because of the dependence of h on the solid phase deformations and because of the nonsmoothness caused by the Macauley bracket. For this reason, a proper linearization is needed after the spatial discretization, so that Newton–Raphson nonlinear iteration strategies can be robustly applied.With the formulation given in the previous sections, the lubricated contact problem can be summarized as the following variational problem: are coupled by the traction fields applied on the solid phase by the lubricant, and through the dependence of the film topology and thickness with the deformations associated with the solid phase. In general, the contact traction t(1) consists of contributions from both the lubricated contact region (Ωf) and the dry contact region (Ωd); however, the latter is not considered in this paper for convenience. In the following part of this paper, we replace t(1) with λ to agree with the notational convention used in prior presentations of the mortar framework (see, for example, To exemplify the proposed approach, we use two dimensional bilinear or three dimensional trilinear finite elements to spatially discretize the contacting bodies, and consider the mortar approximations that should be used on the interface to describe the contact of these bodies. Consider the spatial discretization of the bodies Ωs(i) using the finite set of elements E(i)h:where Ωse(i) denotes the subdomain occupied by element e. The discretization of contact surfaces Γsc(i)h occurs over subsets of ∂Ωs(i)h. Note that the slave surface Γsc(1)h coincides with the lubrication domain Ωfh here, since dry contact is not considered.Finite-dimensional subspaces of solution and weighting spaces in are denoted C(i)h and V(i)h, and may be defined viaC(i)h=u(i)h:Ω¯s(i)h→Rndm|u(i)h∈C0(Ω(i)h),∀e∈E(i)h,u(i)h(Ωe(i)h)∈PN(Ωe(i)h),u(i)h=u¯(i)inΓsu(i)hV(i)h=u∗(i)h:Ω¯(i)h→Rndm|u∗(i)h∈C0(Ω(i)h),∀e∈E(i)h,u∗(i)hΩe(i)h∈PNΩe(i)h,u∗(i)h=0inΓsu(i)h,where PNΩe(i)h is the set of all polynomials on Ωe(i)h of order ⩽N. The mortar and nonmortar fields, u(i)hΓsc(i)h⊂X(i)h, and the variations of these fields, u∗(i)hΓsc(i)h∈W(i)h, are subsets of C(i)h and V(i)h and are obtained by restriction of to contact surfaces Γsc(i)h. The discretized mortar multiplier space, which is physically interpreted as the space containing contact tractions, is defined on the nonmortar side (i.e., the slave surface) asMh=λh|λh∈C0Γsc(1)h;∀e∈Gh,λhΓsce(1)h∈PNΓsce(1)h,where Gh is the set of nonmortar element surfaces making up the slave surface.Similarly, finite-dimensional subspaces of solution and weighting spaces in are denoted Ph and Qh, and may be defined viaPh=ph:Ω¯fh→Rndm-1|ph∈C0(Ωfh),∀e∈Gh,ph(Ωfeh)∈PN(Ωfeh),ph=0inΓfDhQh=p∗h:Ω¯fh→Rndm-1|p∗h∈C0(Ωfh),∀e∈Gh,p∗h(Ωfeh)∈PN(Ωfeh),p∗h=0inΓfDh.We notice here Qh,Ph and Mh are defined on the same set of elements because dry contact is not considered in this paper. The film thickness at every point of the slave surface is directly related to the deformations of the two contact bodies, and must also be expressed in a discretized form. The discretized film thickness space, physically interpreted as the space containing admissible distributions of film thickness, is defined on the nonmortar side asHh=hh|hh∈C0Γsc(1)h;∀e∈Gh,hhΓsce(1)h∈PNΓsce(1)h.With these definitions, we may now define the discretized version of contact virtual work by developing shape function expansions for the above contact surface fields and substituting into . With the isoparametric interpolation scheme, the multipliers, the deformation fields and their variations on the contact surface may be discretized. The discretized contact virtual work can then be expressed asGcm(uh,ph,u∗h)=-∑A∑B∑CλA·nAB(1)u∗B(1)-nAC(2)u∗C(2),where the mortar integrals nAB(1) and nAC(2) are expressed asnAB(1)=∫Γsc(1)hNA(1)(ξ(1)(X))NB(1)(ξ(1)(X))dΓ,nAC(2)=∫Γsc(1)hNA(1)(ξ(1)(X))NC(2)(ξ(2)(Y¯(X)))dΓ., as well as in the sequel, the notations NA(i),NB(i), etc. refer to the restrictions of the finite element shape function associated with node A,B,… to the contact surface. The integrations indicated in are performed by subdivision of the elements into so-called mortar segments; this is particularly important in the case of since the integrand involves shape functions from both sides of the contacting boundary. A mortar segment is defined as a subportion of the contact surface containing only one element from each side of the interface; the specific procedure used to identify these segments and perform the integration over each segment is discussed in . Importantly, the approach is to be contrasted with those used, for example, by The normal and tangential portions of the contact operator are now exposed by splitting λA into normal and frictional partsThe definitions of λAN and λAT will be given in the next subsection based on fluid phase quantities. As a point of connection with prior mortar implementations of dry frictional contact, these variables would be governed directly by impenetrability conditions and suitable integration of a friction law, as discussed in The fluid pressure field and its variation may be interpolated asph(X)=∑J∈ηNJ(1)(ξ(1)(X))pJ=∑J∈η-ηgNJ(1)(ξ(1)(X))pJ+∑J∈ηgNJ(1)(ξ(1)(X))gJ,where η is the set of all nodes on the discretized slave surface, ηg is the set of nodes where Dirichlet boundary conditions are prescribed, and gJ is the boundary value of p at node J. Since p=0 at the Dirichlet boundary, we haveGf(uh,ph,p∗h)=∑I∈η-ηgpI∗∑J∈η-ηg∫Ωfh(hh(X))312μNI,α(1)(ξ(1)(X))NJ,β(1)(ξ(1)(X))mαβdΩpJ-∫ΩfhNI(1)(ξ(1)(X))εp〈-∑J∈η-ηgNJ(1)(ξ(1)(X))pJ〉dΩ-∫ΩfhNI,α(1)(ξ(1)(X))τα(X)·V∼(1)h(X)+V∼(2)h(X)2hh(X)dΩ=0,where hh(X),V∼(1)h(X) and V∼(2)h(X) are discretized forms of film thickness and surface velocities. They are defined as:V∼(i)h(X)=∑A=1nsNA(1)(ξ(1)(X))V∼A(i),i=1,2,, VA(1) is the spacial velocity of the slave node A. τAα and τAα are covariant and contravariant basis vectors, and V∼A(2) will be defined in the following subsection. The mαβ in are the so called contravariant components of the metric tensor, defined in the usual way aswhere τα and τβ are surface tangential contravariant basis vectors.hA=hA0-uA(1)-1n¯AA(1)∑C=1nmn¯AC(2)uC(2)·nAV∼A(2)=1n¯AA(1)∑C=1nmn¯AC(2)VC(2)·τAα⊗τAα,where nA is the surface normal vector defined on the slave node A,VC(2) is the spacial velocity of the master node C, and n¯AA(1) and n¯AC(2) are dual mortar integrals. They are defined asn¯AB(1)=∫ξ(Ωfh)N¯A(1)(ξ(1)(X))NB(1)(ξ(1)(X))dξn¯AC(2)=∫ξ(Ωfh)N¯A(1)(ξ(1)(X))NC(2)(ξ(2)(Y¯(X)))dξ,where ξ(Ωfh) is the isoparametric space corresponding to the domain Ωfh (or Γsc(1)h), and N¯A(1) is the dual basis function of NA(1). For example, the dual basis functions of a two-node (node 1 and node 2) one dimensional linear element are defined aswhere N1 and N2 are the standard basis functions on node 1 and node 2, respectively. As a result, is a diagonal matrix since n¯AB(1)=0 for A≠B. This property makes the inversion of the mortar matrix much cheaper than that of the standard definition of the mortar matrix. On the other hand, one still has the optimal convergence properties associated with the mortar formulation, as shown by The normal and tangential (i.e. shear stress applied by the fluid film) tractions in ∇˜pA is the surface gradient of pressure at the slave node A, which introduces another difficulty for lubricated contact problems since the pressure gradient is not continuous at each node in general. Some smoothing procedure has to be adopted to get ∇˜pA. In this work, the dual basis projection technique is applied again to compute the pressure gradient at each slave node.The smoothed surface gradient of pressure is then written asTo define it, we first write the unsmoothed surface gradient of pressure, computed directly by taking the gradient of which is not continuous across the element boundaries because of the NA,α(1) term. We then employ least-squares projection by enforcingwhere N¯A(1) is the dual basis of NA(1) and ϖA is an arbitrary vector at node A. One may substitute ∇˜pA=∑C=1ns∫ξ(Ωfh)N¯A(1)NC,α(1)ταdξpCn¯AA(1).By employing dual basis functions, the smoothing procedure does not require inversion of a non-diagonal matrix, and is much cheaper than would be the case with standard basis functions.The discretized version of the lubricated problem, shown in , is a nonlinear problem. In applying a Newton–Raphson method to solve this problem, we employ consistent algorithmic linearization to iteratively solve the nonlinear problem through a series of linear ones. Specifically, given an iteration i for the solution, denoted by (uih,pih), we solve linearized systems for displacement and pressure increments (Δuh,Δph) according toGint,ext(uih,u∗h)+ΔGint,ext(uih,u∗h)+Gc(uih,pih,u∗h)+ΔGc(uih,pih,u∗h)=0where Δ(·) denotes the directional derivative of the operand. Following solution of for Δuh and Δph, the solution is updated viaThe linearization of Gint,ext(uih,u∗h) is not presented here but it can be computed by following usual procedures depending on the material model utilized. By computing the directional derivative of ΔGcm(uh,ph,u∗h)=-∑A∑B∑CΔλA·nAB(1)u∗B(1)-nAC(2)u∗C(2),where the linearization of contact traction can be obtained from linearizing ΔλA=ΔλAN+ΔλAT=-ΔpAnA-αμAhAV∼A(1)-V∼A(2)ΔpA+μAhA2V∼A(1)-V∼A(2)ΔhA-ΔhA2∇˜pA-hA2∑C=1ns∫ξ(Ωfh)N¯A(1)NC,α(1)ταdξΔpCn¯AA(1),ΔhA=-ΔuA(1)-1n¯AA(1)∑C=1nmn¯AC(2)ΔuC(2)·nA(no summation onA),, the linearization of Gf(uh,ph,p∗h) can be expressed asΔGf(uh,ph,p∗h)=∑I∈η-ηgp∗I∑J∈η-ηg∫Ωfh3hh212μΔhhNI,α(1)NJ,β(1)mαβdΩpJ-∑J∈η-ηg∫Ωfh(hh)312μ2ΔμNI,α(1)NJ,β(1)mαβdΩpJ+∑J∈η-ηg∫Ωfh(hh)312μNI,α(1)NJ,β(1)mαβdΩΔpJ+∑J∈η-ηg∫ΩfhNI(1)εpH(-ph)NJ(1)dΩΔpJ-∫ΩfhNI,α(1)τα·V∼(1)h+V∼(2)h2ΔhhdΩ=0,where H(·) is the heaviside function, and Δμ and Δhh are computed as. Given the linearization procedure, the global stiffness can therefore be written as:Note that the pressure degrees of freedom are only defined on the slave nodes.In our research, the continuous Galerkin method was first implemented in our research program. However, as will be seen, some spurious oscillations in the pressure field are observed when we use piezo-viscous fluid materials, so that a method to reduce these oscillations is desirable. As shown in , the finite-dimensional subspaces of solution and weighting spaces for DG method are denoted PDGh and QDGh, and may be defined viaQDGh=p∗h∈L2(Ωfh)|p∗h(Ωfeh)∈PN(Ωfeh),∀e∈Gh,where L2(Ωfh) represents the space of square integrable functions on Ωfh, and PN(Ωfeh) denotes the set of all polynomials on Ωfeh of order ⩽N. shows an example of solution spaces for continuous and discontinuous Galerkin methods, respectively.In the following, we denote by Γint=⋃e1⋃(e2⩾e1)Ω¯fe1h∩Ω¯fe2h the union of interior boundaries of the elements. Then, the DG formulation for the Reynolds equation of the fluid phase can be written as:GDGf(uh,ph,p∗h)=∫Ωfhhh312μ∇˜ph·∇˜p∗hdΩ-∫ΓfDp∗hhh312μ∇˜ph·ndΓ+∫ΓfDphhh312μ∇˜p∗h·ndΓ-∫Γint〚p∗h〛hh312μ∇˜ph·ndΓ+∫Γint〚ph〛hh312μ∇˜p∗h·ndΓ+∫ΓfDτp∗hpdΓ+∫Γintτ〚ph〛〚p∗h〛dΓ-∫Ωfhp∗hεp〈-ph〉dΩ+∫Ωfh∇˜·V(1)+V(2)2hhp∗hdΩ=0,where the double bracket 〚·〛 denotes the jump of the operand across an element boundary Ω¯fe1h∩Ω¯fe2h, and 〈·〉 denotes the average value of the operand across the element boundary. Also, n is the normal vector on the element boundary that has positive dot product with a pre-defined direction, e.g. the positive direction of the x-axis, and τ is the stabilization parameter. The larger τ is, the smaller the jump of the solution across each element boundary.In this paper, as an alternative to the basic formulation described in the last section, the DG method was implemented for the fluid phase while retaining a continuous Galerkin method for the solid phase. In other words, only ph and p∗h are discontinuous across element boundaries, while hh and displacement fields remain continuous. In , pA is calculated as the average of the pressure values around the node A.In this sections, some numerical examples are presented to show the performance of the proposed formulations.In this example, a lubricated contact problem is analyzed to test the performance of the formulation, in a manner that isolates the lubricant behavior. As illustrated in This is a two dimensional problem for the fluid phase (solid phase is rigid) because we do not ignore the side leakage (otherwise it is a one dimensional problem for the fluid phase). shows the relative error of h on the slave surface, defined as:where h˜i is the film thickness computed with the dual mortar projection and hi is the actual film thickness. As shown in , the maximum value of the relative error is less than 2.0×10-4. is the cavitation region. We can also see in the figure that the Reynolds boundary is clearly resolved. plots the unsmoothed and smoothed surface gradients of pressure in x and y direction, respectively. We find the pressure gradients are well smoothed by the dual basis least square smoothing.In this subsection, an elastohydrodynamically lubricated line contact problem is explored. As illustrated in This problem has been tested by many researchers with different numerical methods (see, for instance, the works by In many events, the fluid viscosity is largely related to the pressure in the fluid film. In this example, both the isoviscous and Barus viscous are considered. As mentioned by The material for the cylinder is linear elastic with Young’s modulus E=2.1×105MPa and Poisson’s ratio ν=0.3. The initial viscosity for the fluid is μ0=4.0×10-8MPas and the piezo-viscous constant α in is chosen as 0.005MPa-1. The speed v of the ground is 1157.5mm/s. This is a two dimensional problem since side leakage is ignored and the deformation of the solid phase is considered.The load is applied as prescribed displacement at the top of the half cylinder until the total reaction force reaches the desired value. The final loads (reaction forces in the vertical direction) in this example are all about 374 N/mm. Only half of the cylinder is discretized and the finite element mesh is shown in show plots of the pressure profiles for lubricated contact problems with and without shear stress, respectively. The pressure profile in is similar to the results obtained by the semi-analytical approach adopted by other researchers (because shear stresses were ignored in these research works when calculating deformations of the solid phases). However, shows an obvious effect of the shear stress on the pressure profile, meaning that viscous stresses cannot be neglected if the modeling of viscous effects is felt to be important for accuracy in prediction of film thickness and pressure profiles. Another effect of the shear stress is that it results in more oscillations in the pressure profile as can be seen in . These oscillations can be reduced if a more refined mesh is used, as plotted in Also, pressure spikes near the outlet of the lubrication zone are observed in . Pressure spikes, discussed extensively by presents the pressure profile, without the pressure spike, for an isoviscous material with viscosity μ=4.0×10-8MPas.The final loads that can be reached with our formulation are usually lower than those with semi-analytical approaches, especially for piezo-viscous material with steep pressure–viscosity relations. This is because the semi-analytical methods can obtain more accurate film thickness than a complete numerical formulation, therefore fewer oscillations are observed in semi-analytical methods. However, the advantage of our numerical approach is that a complete numerical formulation can be applied to a much broader class of problems, with arbitrary choices for the geometry and material properties of the solid phase.In highly loaded problems, the film thickness in the high pressure region is usually several orders of magnitude smaller than the size of the finite element mesh. In this case, the film thickness computed, with the mortar approach, for curved solid surfaces has some numerical errors that are direct results of geometric discretizations. These numerical errors will cause oscillations in the pressure profile for the piezo-viscous fluid.As mentioned previously, the DG method in Section has also been implemented and applied to solve this problem. However, even with higher order DG elements, the results were relatively insensitive to the numerical method, and still showed oscillations. A possible explanation is that the solid phase is still discretized with continuous linear elements, so the discretization error still exists and it causes error in the calculated fluid film thickness.A three dimensional lubricated contact problem is tested in this example. As shown in , the space between two cylinders is filled with lubricant. The geometric parameters in are: R1=21.15,R2=15.15,R3=15,R4=10, and d=0.015. The length of the inner cylinder is 15.75 and the length of the outer cylinder is 15. The inside surface of the inner cylinder is given an angular speed ω=500. The outside surface of the outer cylinder is fixed. The Young’s modulus of both cylinders is E=2.1×1011, Poisson’s ratio is ν=0.3, and the viscosity of the fluid is μ=0.04. Instead of using a steady state assumption, a quasi-static rotation is applied to the inner surface of the internal cylinder. The rotating angle in one time step is 0.05°. The time interval for one time step is ▵t=0.00016449. presents the finite element mesh for the two cylinders. The inside surface of the outer cylinder is chosen as the slave surface and the outside surface of the inner cylinder is chosen as the master surface.Only one load step is applied for this problem. The computed pressure distribution in the fluid film is plotted in The fluid pressure reaches its maximum at about the location where the fluid film thickness is at its minimum. The pressure profile is not symmetric because of the cavitation region on the tension side. The pressure on the cylinders tends to increase the fluid film thickness because of the deformation of the solid phase.In this paper, a general mortar approach has been proposed for solving lubricated contact problems with deformable solid phases. The fluid film thickness is computed from a dual mortar projection. The free boundary problem for the fluid phase is regularized with a penalty method. The solid and the fluid phases are solved in a coupled equation system. The proposed formulation can be applied to general lubricated contact problems. Some oscillations are observed for piezo-viscous fluid under high loadings (as is true also for published semi-analytical approaches), but they are reduced with refinements of the mesh.The formulation has been demonstrated in both two and three dimensions, for the relatively simple case where the contacting bodies are linearly elastic. It is asserted that the groundwork has been laid for accurate simulation of a much more general class of lubricated contact problems than has been possible to date.TiAl/RGO (reduced graphene oxide) bulk composites with refined microstructure and enhanced nanohardness fabricated by selective laser melting (SLM)This work for the first time investigated the effect of laser scan line spacing on the microstructure, phase evolution and nanohardness of Ti-48Al-2Cr-2Nb/RGO (reduced graphene oxide) metal matrix composites (MMCs) fabricated by selective laser melting (SLM). The results show that with increasing the laser scan line spacing from 80 to 140 μm, the average grain size generally decreases from 10.13 to 8.12 μm. The SLM-processed Ti-48Al-2Cr-2Nb/RGO parts are dominated by high-angle (>15°) grain boundaries (HAGBs) and α2 (Ti3Al) phase. With the increase in laser scan line spacing, the contents of HAGBs and α2 phase both decrease. Due to instantaneous high temperature during the SLM process, some RGO sheets transform to amorphous carbon. The nanohardness of SLM-processed Ti-48Al-2Cr-2Nb/RGO parts increase from 8.13 ± 0.39 GPa to 9.85 ± 0.46 GPa when increasing the laser scan line spacing from 80 to 140 μm, which is much higher than that of the traditional casting TiAl counterparts (4.98 ± 0.10 GPa).Titanium aluminide (TiAl)-based alloys are recognized as high promising materials for making light-weight structural components in new generation aerospace engines, due to their attractive properties such as low density, high specific strength and good oxidation resistance at elevated temperatures []. However, the conventional processing routes of TiAl-based alloys are prone to cause cracks owing to their inherent low ductility and poor hot deformability []. In the past several decades, with the development of manufacturing technologies and deeper understanding of TiAl-based alloys, some thermo mechanically technologies such as isothermal forging and hot extrusion were introduced to fabricate TiAl-based parts with satisfied microstructure and mechanical properties []. However, the high processing temperature resulted in extremely high capital investment for these processing methods []. Besides, the thermo mechanically fabricated TiAl-based parts possess heterogeneous microstructure and limited structural complexity [In recent years, more scientists focus their attentions on TiAl-based metal matrix composites (MMCs) in order to control the microstructure and improving the mechanical properties of TiAl-based alloys. Chen et al. [] systematically studied the TiC/Ti2AlC reinforcement in TiAl-based alloy fabricated by a metallurgical method. Their results suggested the maximum strength and strain were improved by 38.82% and 121.37% as compared to the unreinforced parts. Zhou et al. [] investigated the microstructures and mechanical properties of TiAl/Ti2AlN MMCs processed by reaction hot pressing. Their findings illustrated that the TiAl/Ti2AlN MMCs showed superior compressive strength of 1112 MPa at 800 °C. Although these conventional reinforcements, such as TiC, Ti2AlC and Ti2AlN, can improve the mechanical properties of TiAl-based alloys. However, the strengthen effect was greatly weakened due to the segregation of these reinforcements.RGO (Reduced graphene oxide), a two-dimensional structure of sp2 bonding carbon, has attracted more and more attentions in recent years due to its outstanding high strength and elasticity modulus, remarkable electron mobility and super high thermal conductivity []. Therefore, RGO has been regarded as a promising novel reinforcement for TiAl-based alloys [] preliminarily studied RGO reinforced TiAl-based matrix fabricated by spark plasma sintering. Due to the conformal anti-wear protective layer on the sliding contact interfaces caused by RGO, the RGO/TiAl MMCs showed a >4 times of magnitude higher wear resistance compared with the pure TiAl parts. However, previous investigations on RGO reinforced TiAl-based alloys were mostly based on the traditional process, showing several restrictions: 1) high processing temperature leading to the decomposition of RGO, 2) poor bonding strength between the RGO and TiAl-based alloys, 3) limited structural complexity of fabricated RGO/TiAl components []. Thus, the manufacturing technology of RGO/TiAl MMCs should be further developed.Selective laser melting (SLM), a kind of rapid development additive manufacturing technologies, is capable of fabricating near fully-dense complex metal parts directly from 3D CAD models []. In comparison to the traditional manufacturing technologies, SLM displays several advantages, such as capability of producing complex structures, high material use efficiency and high level of flexibility []. Moreover, the decomposition effect of RGO can be suppressed due to the fast heating and cooling rates of SLM process. Hence, SLM implicates great potential of manufacturing TiAl/RGO MMCs. Recently, some researchers have studied the effect of SLM parameters on the microstructures and properties of TiAl-based alloys. Vilaro et al. [] investigated the SLM and direct metal deposition (DMD) fabricated Ti-47Al-2Cr-2Nb alloy. Their results suggested that due to the rapid melting and cooling effect of SLM and DMD, ultra-fine microstructure can be achieved, resulting in high microhardness. Gussone et al. [] investigated the influences of energy density and preheat temperature on chemical composition and microstructure of a novel γ-TiAl alloy. The results suggested that the tensile properties of SLM-processed sample at 850 °C were comparable to those of conventionally produced TiAl counterparts. More recently, our previous work also studied the effects of SLM parameters and heat treatment on the microstructure, phase evolution and nanohardness of Ti-45Al-2Cr-5Nb alloy [Gas atomized Ti-48Al-2Cr-2Nb powder and RGO sheets were supplied by Shanxi Rongtian Aviation Equipment Co. Ltd. and Arke Inc., respectively. A solvent precipitation method was introduced to prepare the Ti-48Al-2Cr-2Nb/RGO MMCs (the RGO content was kept as 0.125 wt%). illustrated the morphology of Ti-48Al-2Cr-2Nb powder and RGO, respectively. It can be seen that the Ti-48Al-2Cr-2Nb powder showed a spherical or near spherical shape while RGO was a typical nano-sheet structure. (c) implicated the particle size distribution of Ti-48Al-2Cr-2Nb/RGO MMCs. The average particle size was calculated to be 25.9 μm which was suitable for SLM fabrication. Before the SLM process, the Ti-48Al-2Cr-2Nb/RGO MMCs was sieved (200 mesh) to reduce the agglomerations and improve its fluidity.An EOS-M280 type SLM equipment (EOS Co. Ltd., Germany) was introduced to build up cuboid specimens (10 × 10 × 5 mm3) on pure titanium substrates. During the SLM process, the chamber was filled with argon atmosphere (99.99%) to protect the Ti-48Al-2Cr-2Nb/RGO MMCs from oxidation. The scanning strategy used for the built up samples was shown in . The arrows implicated the movement of the laser beam. Between two successive layers N and N + 1, the laser beam showed a 90° rotation angle. Based on the previous investigation [], the SLM parameters were set as follows: the laser power P was 250 W, the scanning speed V was 800 mm/s and the layer thickness was 50 μm. To systematic study the effect of laser scan line spacing on the microstructure, phase and nanohardness of Ti-48Al-2Cr-2Nb/RGO samples. Four different levels of laser scan line spacing were selected, namely, 80 μm, 100 μm, 120 μm and 140 μm, respectively. For simplification, the samples were donated as E1, E2, E3 and E4.A Mastersizer-3000 type laser diffraction particle size analyzer (Worcestershire, UK) was introduced to study the particle size distribution of Ti-48Al-2Cr-2Nb/RGO MMCs. X-ray diffraction (XRD) measurement was carried out on a XRD-7000S type equipment (Shimadzu, Japan). The diffraction angle (2θ) was ranged from 20° to 110° with a step size of 0.02°. It was worth noting that all the XRD measurements were carried out on the top view of the samples. An Automet-300 type automatic polishing machine (Buehler, America) was used to mechanical polish all the samples. Then the mechanically polished specimens were electrolytic polished by a LectroPol-5 type (Struers, Denmark) machine at 20 V for 25 s for EBSD analysis. EBSD measurement was conducted on a HKL Nordlys orientation system (Oxford Instruments, United Kingdom) mounted on a JSM-7600F type scanning electron microscope (JEOL, Japan). The step size of EBSD test was set at 500 nm. Then the raw EBSD data was interpreted by the HKL Channel-5 analysis software package. Transmission electron microscope (TEM) was conducted on a JEM-2100 type (JEOL, Japan) equipment. A Hysitron triboindenter (Berkovich Indenter TI750, Hysitron, American) equipped with a diamond Berkovich tip was used to test the nanohardness of E1, E2, E3 and E4. The indentation load and holding time were kept as 3500 μN and 2 s. Twenty indents were conducted on each sample and then the test results were averaged.To study the effect of laser scan line spacing on the grain size, orientation and texture of Ti-48Al-2Cr-2Nb/RGO parts. EBSD measurement was carried on the top view of E1, E2, E3 and E4. The “region A” in (b) showed the color-coded stereographic triangle inversed pole figure (IPF). In the top view of E1 ( (c)), it can be seen most of the grains possessed red-green-blue color. When taking IPF into consideration, it can be concluded that the EBSD map of E1 revealed a combination of (0001), (101¯1) and (112¯1) orientations. As the laser scan line spacing increased from 80 μm to 100 μm, the grain size was slightly refined as shown in (d). However, the red region increased greatly, while the green-blue areas decreased significantly. Therefore, the grains orientation in E2 showed a strongest (0001) direction. At an even higher laser scan line spacing of 120 μm, the grain size in E3 was further refined ( (e)). Besides, most of the grains showed an equal red-green-blue color, which suggested the grains possessed (0001), (101¯1) and (112¯1) orientations. When the laser scan line spacing increased to the maximum of 140 μm, the grain size of E4 reached to the minimum as shown in (f). While their orientations still kept as (0001), (101¯1) and (112¯1) directions unchanged. Therefore, we can draw a conclusion that with increasing the laser scan line spacing from 80 μm to 140 μm, the grain size of E1, E2, E3 and E4 generally refined while E2 showed strongest (0001) orientation.To quantitative analysis the effect of laser scan line spacing on the average grain size of E1, E2, E3 and E4, channel-5 software was introduced to detailed analysis the EBSD images of (c)–(f), with the results were summarized in . As the laser scan line spacing was 80 μm (E1), the grain size was well distributed in a range of 2.25 μm to 19.25 μm, and the average grain size was calculated to be 10.13 μm. When the laser scan line spacing increased to 100 μm (E2), the grain size ranged from 2.25 μm to 16.75 μm. Besides most of the grains size were <12 μm. The average grain size was figured out to be 9.65 μm, which was slightly smaller than E1. At an even higher laser scan line spacing of 120 μm (E3), the grain size was further refined with the average grain size was determined to be 8.84 μm. When the laser scan line spacing increased to maximum of 140 μm (E4), the grain refinement effect reached the maximum, and the average grain size was calculated to be 8.12 μm.For the purpose of facilitating analysis the grain boundary misorientation, low-angle grain boundaries (LAGBs, <15°) and high-angle grain boundaries (HAGBs, >15°) were fetched in to separate the grain boundaries of E1, E2, E3 and E4 [(c)–(f), “white grain boundary” and “black grain boundary” represented the LAGBs and HAGBs, with their statistical data results were shown in . Obviously, all the samples were dominated by HAGBs. The HAGBs content of E1, E2, E3 and E4 were calculated to be 88%, 81.08%, 78.22% and 76.57%, respectively. It was well known that the Ti-48Al-2Cr-2Nb/RGO MMCs powder piled up layer by layer to fabricate 3D metal parts during the SLM process. However, due to the effect of high-energy laser beam, when a new layer was finished fabricating, the previously solidified layer(s) would be partially or totally remelted again. This phenomenon was similar to temper heat treatment, which would cause recrystallization and then the HAGBs aroused. As is known, due to the rapid movement of laser beam, rapid melting and solidification will take place in Ti-48Al-2Cr-2Nb/RGO MMCs. As a result, sub-grains would appear in the SLM-fabricated Ti-48Al-2Cr-2Nb/RGO samples []. The grain boundary in sub-grains firstly shown as LAGBs. Then the LAGBs gradually transformed to HAGBs due to the remelting and solidification phenomenon of SLM process []. However, the remelting and solidification time was so short that the recrystallization cannot be fully accomplished, and thus resulting in some residual LAGBs in the grain boundary. Moreover, the effect of laser scan line spacing on the deposited layer could be determined using the overlap ratio, which was interpreted by Eq. (1) [where: L(%) is the overlap ratio, Hs is laser scan line spacing and D is the deposited layer width. Obviously, with increasing the laser scan line spacing, the overlap ratio tended to decreased, which would further shorten the recrystallization time. Therefore, the content of HAGBs further decreased with the increase of laser scan line spacing.(c)–(f), the color distribution of SLM-processed Ti-48Al-2Cr-2Nb/RGO samples varied with the increase of laser scan line spacing. Hence, laser scan line spacing showed a great influence on the local crystallographic texture of E1, E2, E3 and E4. In this chapter, the most common {0001} pole figures (PFs) from the top view of E1, E2, E3 and E4 (the {0001} orientation was a relatively better direction along the scanning direction for SLM process) were detailed analyzed and shown in . Based on the measured PFs and the orientation distribution function (ODF), the texture index of {0001} orientation along the scanning direction could be exactly calculated by Eq. (2) [where: f is the orientation distribution, g is the Euler space coordinates. Along the {0001} orientation, samples E1, E2, E3 and E4 showed different texture index as can be seen in (a)–(d). Based on Eq. (2), the texture index of E1 from the {0001} orientation was calculated to be 14.28. While E2 exerted the highest texture along the {0001} orientation with its texture index was figured out to be 33.86. As compared to E2, E3 and E4 showed a much weaker {0001} texture, and their texture index were calculated to be 16.53 and 12.47. Accordingly, it was concluded that with the increase of laser scan line space, the {0001} texture index of E1, E2, E3 and E4 highly consistence with the findings about the variation of the grain orientations ( implicated the phase diagram of Ti-Al alloy. As can be seen from the phase diagram, there existed two peritectic reactions L + β → α, α + γ → γ, two eutectic reactions α → α + γ, γ → α2 + γ, and two ordering transitions α → α2, β → B2 during the solidification of Ti-48Al alloy. Therefore, the phases evolution of Ti-48Al-2Cr-2Nb alloy during the SLM process can be summarized as: L → L + β → β + α → α + γ + β → α2(Ti3Al) + γ(TiAl) + B2 [(b) showed the XRD spectra of E1, E2, E3 and E4 (top view). As can be seen from the XRD patterns, only the α2, γ and B2 diffraction peaks had been detected. Two strong diffraction peaks at 39.1° and 41.3° were determined as the matrix of (0002)α2 and (202¯1)α2. With increasing the laser scan line spacing, the intensity of (202¯1)α2 at 41.3° slightly increased while the intensity of (0002)α2 peak at 39.1° decreased significantly. Moreover, the intensities of (200)γ, (202)γ, (620)β, (311)γ and (202¯4)α2 basically remain unchanged. Likewise, the intensities of the (110)γ, (220)β, (310)β and (520)β increased with the increase of laser scan line spacing. As discussed before, the laser energy density decreased with the increase of the laser scan line spacing. Therefore, the higher laser scan line spacing resulted in shorter duration time of the liquid phase, with which the peritectic reaction L + β → α would be hindered []. Therefore, the content of β phase increased with the increase of the laser scan line spacing. However, the eutectic reactions rate of α → α + γ and γ → α2 + γ were very sensitive to the cooling rate []. As mentioned above, with the increase of the scan line spacing, the laser energy density decreased. Therefore, on the one hand, the eutectic reactions of α → α + γ and γ → α2 + γ would be promoted, and thus increased the content of α2. On the other hand, the peritectic reaction of L + β → α would be suppressed, and thus reduced the content of α2. The two opposite effects lead to the decreased content of α2 at 39.1° and increased content of α2 at 41.3°. In addition, the cooling rate was very large (about 106 K/s) during the SLM process. Consequently, the inhibition effect of L + β → α was greater than the promotion of the α → α + γ and γ → α2 + γ. Therefore, the content of α2 phase tended to decrease with increasing the laser scan line spacing. Due to the stimulation effect of α → α + γ and γ → α2 + γ reactions, the content of γ phase increased with the increase of laser scan line spacing. However, to our surprise, there was no RGO can be identified by XRD measurement.In order to study the distribution and morphology of RGO in the SLM-processed Ti-48Al-2Cr-2Nb/RGO parts, as well as acquired more detailed information about α2, γ and B2. Further characterizations were carried on E1 by TEM and HRTEM. displayed the bright-field image of E1, from which we can see that there existed a few RGO sheets, γ and B2 phases. The RGO, γ, B2 were randomly distributed in the α2 matrix. Besides, the B2 was supposed to derive form the uncompleted transformation of L + β → α []. The selected area diffraction pattern (SADP) corresponding to (b). Obviously, SADP illustrated the polycrystalline structure of E1. The different diffraction rings represented different phases []. From the SADP, it was found that the radius ratios of the rings were figured out to be 1:1.21:1.67:1.94. According to the SADP image and the standard XRD reference codes (reference codes number: 00-014-0415, 03-065-5414, 03-065-7479 and 00-049-1717). The inter-planar spacings were calculated to be 0.141 nm, 0.171 nm, 0.236 nm and 0.274 nm, which represented the (204¯0)α2, (112)γ, (210)B2 and (420)RGO []. According to the bright-field image, XRD results and phase diagram, it was concluded that the β phase firstly transformed to α and γ phases, then the γ-related, B2-related and RGO sheets phases precipitated within the α2 matrix [In order to further study the RGO sheets, phase evolution mechanism and orientation relationship of α2, γ and B2. HRTEM was introduced and conducted on the RGO sheets (blue arrow), circled areas of “zone A” and “zone B” in (a). The corresponding HRTEM images were shown in (c) showed the HRTEM morphology of RGO sheets, with its inter-planar spacing was calculated to be 0.351 nm, which exactly match the standard XRD reference code (00-049-1717). However, to our surprise, there existed some amorphous carbon ((d)) around the RGO sheets. Due to the transient high temperature inside the molten pools during the SLM process, a small amount of RGO sheets would be decomposed and transformed to the amorphous carbon. The insert of (d) implicated the SADP of amorphous carbon. Obviously, it belonged to an amorphous diffraction ring, which was in good agreement with the HRTEM pattern of (e) and (f) indicated the HRTEM images of “zone A” and “zone B” in (a), respectively. The inter-planar spacings were calculated to be d211 = 0.337 nm (B2), d110 = 0.278 nm (γ) and d0002 = 0.231 nm (α2), which exactly match the standard XRD pattern (reference codes number: 03-065-4421, 03-065-5414 and 00-009-0281). It could be seen in (e) that the RGO sheets, B2 and amorphous carbon were coexisted in a range of tens of nano-meters. Besides, the B2 phase was surrounded by RGO sheets and amorphous carbon. From the HRTEM image of (f), the orientation relationships of α2 and γ can be expressed as: (110)γ//(0002)α2 []. As the (202¯0)α2 and (110)γ were mainly derived from the (210)β []. Thus, a more detailed phases evolution mechanism of the SLM-processed Ti-48Al-2Cr-2Nb/RGO MMCs could be concluded as: (220)β first transformed to (110)γ. Then the (110)γ transformed to (202¯0)α2. Lastly, residual B2 phase, incompletely transformed γ, RGO sheets and amorphous carbon randomly distributed in the α2 matrix. illustrated the representative nanoindentation load-displacement curves of E1, E2, E3 and E4 (top view). Obviously, the indentation depth of E1 was about 143.91 nm, which was the deepest among all the tested samples. As the laser scan line spacing increased, the indentation depth gradually decreased. When the laser scan line spacing reached to the maximum of 140 μm, the indentation depth reduced to the minimum of 131.17 nm. While the nanohardness and Young's modulus of E1, E2, E3 and E4 increased form 8.13 ± 0.39 GPa and 155.62 ± 7.78 GPa, to 9.85 ± 0.46 GPa and 179.02 ± 8.90 GPa, respectively ( (b)). Moreover, it was worth noting that all the tested samples (E1, E2, E3 and E4) showed a much higher nanohardness than the traditional casting fabricated TiAl counterparts (4.98 ± 0.10 GPa) []. The reasons could be attributed to the following three reasons: 1) During the SLM process, rapid movement of laser beam resulted in very large thermal gradient. Therefore, SLM-processed parts showed refined grain size than casting counterparts. Based on the Hall-Petch law, the refined grains lead to the increase of nanohardness. 2) Due to the large thermal gradient of SLM process (about 106 K/s), a large and complex residual stress would retain in SLM-processed samples []. However, a reasonable level of residual stress favored the enhancement of hardness []. 3) Due to the stochastic distribution of RGO sheets and amorphous carbon in the α2 matrix, the nanohardness further increased. Hence, the SLM-processed samples showed much higher nanohardness than the casting counterparts. Moreover, it can be clearly found in (b) that the nanohardness and Young's modulus both increased with the increase of the laser scan line spacing. There were some reasons for this: Firstly, according to the XRD results (), the content of B2 phase increased with the increase of the laser scan line spacing. B2 was the hardest phase among the α2, γ and B2 []. Thus, nanohardness increased with the increase of laser scan line spacing. Secondly, based on the EBSD orientation images ( (c)–(f)), the average grain size generally refined with the increase of laser scan line spacing. Based on the Hall-Petch law mentioned above, the nanohardness increased with the significant refinement effect of grain size.In this work, RGO sheets reinforced Ti-48Al-2Cr-2Nb alloy had been processed by SLM (the RGO content was kept as 0.125 wt%). The effect of laser scan line spacing on microstructure features, crystallographic texture, phases evolution and nanohardness have been systematically investigated. The main conclusions were as follows:During the SLM process, the average grain size of Ti-48Al-2Cr-2Nb/RGO samples is generally refined with the increase in the laser scan line spacing. As the laser scan line spacing is kept at 100 μm, the grains show the strongest (0001) orientation.The grain angle boundaries distribution of SLM-fabricated Ti-48Al-2Cr-2Nb/RGO parts are dominated by HAGBs. The content of HAGBs decreases with the increase in the laser scan line spacing.With increasing the laser scan line spacing, the content of α2 phase decreases while the contents of γ and B2 increase. In the SLM-processed Ti-48Al-2Cr-2Nb/RGO parts, γ, B2, RGO sheets and amorphous carbon phases are randomly precipitated within the α2 matrix.The phases evolution mechanism of α2, γ, B2, RGO sheets and amorphous carbon can be summarized as: (220)β firstly transforms to (110)γ. Then the (110)γ transforms to (202¯0)α2. Finally, B2, γ, RGO sheets and amorphous carbon stochastic distributed in the α2 matrix. The orientation relationship between α2 and γ phases can be expressed as: (0002) α2//(110) γ.With the increase in laser scan line spacing from 80 to 140 μm, the nanohardness and Young's modulus increase from 8.13 ± 0.39 GPa and 155.62 ± 7.78 GPa to 9.85 ± 0.46 GPa and 179.02 ± 8.90 GPa, respectively.Effect of cast process and microalloying on the fracture toughness of Zr-based bulk amorphous alloysSEM micrographs of the tilt- and suction-cast samples where near the notch root after bending test; (a) Zr53Cu30Ni9Al8 and (b) (Zr53Cu30Ni9Al8)99.25Si0.75 are tilt-cast samples, (c) Zr53Cu30Ni9Al8 and (d) (Zr53Cu30Ni9Al8)99.25Si0.75 are suction-cast samples. Note that the tilt-cast Zr-53 BMG without Si possesses the best fracture toughness among these BMGs in this study.In order to clarify the effects of tilt- or suction-cast processing and microalloying of minor Si on the three-point bending fracture toughness of the Zr-based bulk metallic glasses (BMGs), the tilt-cast and suction-cast Zr53-based and Zr48-based BMGs with and without Si were studied. The results revealed that the tilt-cast specimens present much better fracture toughness than the suction-cast counterparts as a result of their lower degrees of residual porosity. In addition, the microalloying of Si to 0.75 at.% in these BMGs would impose negative effects on their fracture toughness. The highest fracture toughness of 120 MPa |
m was achieved for the tilt-cast Zr53Cu30Ni9Al8. Moreover, the morphology of fracture surface, the calculated fracture energy, and the Poisson’s ratio show the similar trend to prove that the tilt-cast Zr-53 BMG without Si possesses the best fracture toughness among these BMGs in this study.Most of bulk metallic glasses (BMGs) exhibit attractive characteristics for scientific research and engineering application owing to the unique properties such as large elastic strain limit, high hardness, high strength, excellent corrosion resistance, and near perfect as-cast surface For most Zr based BMGs with sufficiently high GFA, the cooling rate can be lowered to less than 100 K/s, and the rapid quenching cast method can be shifted from rapid suction/injection casting to conventional tilt casting. From our previous experience, during rapid suction casting, tiny defects such as bubbles or porosity can be formed when the liquid metal rapidly flows into the mold, deteriorating the BMG mechanical properties In this study, two factors are considered on the fracture toughness of the Zr-based BMGs, namely, the cast method and the minor Si alloying. The influence is examined in terms of various parameters such as the morphology of fracture surface, the Poisson’s ratio, the ratio of shear/bulk modulus, and the calculated fracture energy.The alloy ingots based on the compositions of (Zr53Cu30Ni9Al8)100−xSix (x |
= 0 or 0.75) and (Zr48Cu36Al8Ag8)100−xSix (x |
= 0 or 0.75), all in atomic percent, were firstly prepared by arc melting of the appropriate mixture of pure elements, including Zr (99.9 wt% purity), Cu (99.99 wt% purity), Ni (99.9 wt% purity), Al (99.99 wt% purity), Ag (99.99 wt% purity), and Si (99.99 wt% purity), under a Ti-gettered argon atmosphere. The glassy plate samples were fabricated by using vacuum suction casting or tilt casting into the water-cooled Cu mold to form alloy plates. Sample for the notch toughness tests were taken from the as-cast bulk amorphous alloys plates, as shown schematically in , with a sample dimension of B (thickness) = 2.5 mm, W (width) = 7.5 mm, and S (span) = 36 mm.The cast BMGs were characterized by differential scanning calorimetry (DSC) with a heating rate of 40 K/min, X-ray diffraction (XRD, Bruker D8A, Cu Kα radiation) and the Mitutoyo HM-200 series micro hardness tester. The notched plate specimens were tested in compression under an initial strain rate of 0.3 mm/min at room temperature by using a MTS HT-9102 universal testing machine. Both ends of the specimens were polished to make them parallel to each other prior to the compression test. In parallel, the ultrasonic measurements were conducted to extract the values of Poisson ratio ν, shear modulus μ, bulk modulus B, and elastic modulus E. The notch root and fracture surface of the deformed specimens were examined by scanning electron microscopy (SEM, Hitachi S-3500N). shows the X-ray diffraction patterns of the (Zr53Cu30Ni9Al8)100−xSix(x |
= 0, 0.75) and (Zr48Cu36Al8Ag8)100−xSix (x |
= 0, 0.75) amorphous alloy plates, 2.5 mm in thickness. There is no resolvable crystalline peak over the 2θ range of 20–80°, only a broad hump is observed in the range of 30–50° for all of the BMGs in this study.The DSC scans of the (Zr53Cu30Ni9Al8)100−xSix (x |
= 0, 0.75) and (Zr48Cu36Al8Ag8)100−xSix (x |
= 0, 0.75) amorphous alloy plates are shown in . All of the plate samples exhibit a clean glass transition followed by a supercooled liquid region and then exothermic reaction due to crystallization. According to the DSC data listed in , the values of GFA index, γ [=Tx/(Tg |
+ |
Tl)] . It is suggested that the small Si atoms (radius of 0.111 nm) . The suction-cast samples show about 5–10 times higher porosity volume fractions than the tilt-cast samples. A typical comparison of cross-sectional images between the tilt- and suction-cast samples is shown in To understand how the toughness would change with the minor Si element and under different manufacture processes (tilt and suction casting), the notched bending specimens were tested for extracting the fracture toughness. shows the typical curves of load versus crack opening displacement (COD) for these notched plate samples under three-point bending loading. According to ASTM standard E399 faw=3aw1/21.99-aw1-aw(2.15-3.93aw+2.7aw221+2·aw1-aw3/2where P is the load, S is the span between the two support rollers, B is the specimen thickness, W is the specimen width, and a is the initial crack size. The extracted KQ values of the eight plates are summarized in . The tilt-cast Zr53Cu30Ni9Al8 specimen possesses the best fracture toughness KQ, reaching an average reading of 122 MPa |
m, a very high toughness reading for BMGs. Note that the plate specimen width is still not large enough to justify the value to KIC; the toughness can only be classified as KQ. The exact values should be higher than the reported valid KICNext, to examine the effect of Si addition, KQ of the tilt-cast Si-bearing (Zr53Cu30Ni9Al8)99.25Si0.75 BMG decreases from 122 MPa |
m of the base alloy to 107 MPa |
m, or ∼14% drop. For other counterparts, minor Si addition can even lead to 30–100% degradation, a significant toughness drop.In addition to the KQ values, in accordance with ASTM E1820 standard where Apl is the area under the force (PQ) versus displacement record, b0 is the ligament length equal to W |
− |
a, E is the elastic modulus of the materials, and K is the KQ for the current case. The calculated J values are also listed in . Again, the tilt-cast Zr53Cu30Ni9Al8 specimen possesses the highest rupture energy of 155 kJ m−2. From the rupture energy J values, tilt casting always yield higher readings than suction casting. Also, Si addition consistently results in much lower J. It is revealed here that although minor Si addition would slightly improve the glass forming ability, the negative role in appreciably degrading the fracture toughness should put into overall consideration. depicts the SEM characterization of the fracture path and shear band evolution near the notch tip after the three point bending test. A series of SEM micrographs of fractured specimen are presented in for the suction-cast samples. For the most ductile Zr53Cu30Ni9Al8 plates ((a)), there are multiple shear bands with a curved and deflected fracture path at the notch root. Note that shear bands in the tilt-cast Zr53Cu30Ni9Al8 plate ((a)) appear in a finer and denser manner, in comparison with that of the suction-cast counterpart ((a)). In contrast, for the most brittle (Zr48Cu36Al8Ag8)99.25Si0.75 samples ((d)), there is hardly any noticeable shear band and the fracture path is straight.It is of interest to compare the alloy composition effect on the fracture surfaces of these (Zr53Cu30Ni9Al8)100−xSix (x |
= 0, 0.75) and (Zr48Cu36Al8Ag8)100−xSix (x |
= 0, 0.75) bulk amorphous alloys plates after three point bending testing. For all of these BMGs, the fracture surfaces are characterized with different degrees of vein-like morphology. We thus focus on the two extreme cases, the least tough (Zr48Cu36Al8Ag8)99.25Si0.75 and toughest Zr53Cu30Ni9Al8 bulk amorphous alloys plates. The SEM fractographs of the toughest Zr53Cu30Ni9Al8 display the Taylor’s fluid meniscus instability (FMI) near the notch root in (1a) and multiple vein-like patterns in (1c), for the tilt and suction-cast samples, respectively. The vein patterns are characteristic of the shear banding deformation, consistent with . In contrast, SEM images of the least tough (Zr48Cu36Al8Ag8)99.25Si0.75 do not exhibit any FMI near the notch root in (2c), for the tilt and suction-cast samples, respectively. clearly reveal the distinct fracture manners in consistent with the measurements of KQ and J in With the sufficient data on the Poisson’s ratio ν and shear/bulk modulus μ/B, it is interesting to explore the relationship between fracture toughness and ν or μ/B displays the plots of the Poisson’s ratio ν versus μ/B for the current eight samples, including the tilt-cast (b) alloys. The trends in both plots are basically parallel and consistent, with the toughest Zr53Cu30Ni9Al8 BMG having the highest ν and the lowest μ/B, and the least tough (Zr48Cu36Al8Ag8)99.25Si0.75 BMG possessing the opposite. The current results are fully consistent with the theory In this study, it is clearly demonstrated that there are three factors influencing the BMG fracture toughness, (i) the alloy composition, (ii) the cast method, and (iii) the microalloying of small Si atoms (by only 0.75 at.%). Firstly, the Zr53Cu30Ni9Al8 BMG has been well studied for its phase separation tendency Secondly, tilt and suction casting do yield different micro-porosity degrees due to its casting process. The suction-cast samples show much higher porosity volume fractions than the tilt-cast samples. Accordingly, the much higher cast porosity degree would lead to the much lower toughness in the suction-cast samples, in agreement with the report of Yokoyama et al. Finally, it is unexpected to see that the fracture toughness of the Si-containing BMGs would exhibit such degraded fracture toughness, though GFA is very slightly improved. Since the covalent atomic radius of Si (0.111 nm) is smaller than each element atom of this Zr-based BMG, such as Zr (0.160 nm), Al (0.143 nm), Cu (0.128 nm) and Ni (0.125 nm) Based on the results of XRD, DSC, fracture toughness, ruptures energy, Poisson’s ratio, shear modulus, bulk modulus and SEM observed for the current eight BMG samples, the effects of alloy composition, cast method and microalloying Si on the fracture toughness and ruptures energy can be summarized below.The extracted fracture toughness, rupture energy, Poisson’s ratio, shear modulus, bulk modulus, fracture path, and fractographs all reveal the consistent results, showing that the tilt-cast Zr53Cu30Ni9Al8 BMG owns the most ductile and tough bending properties. In contrast, the suction-cast (Zr48Cu36Al8Ag8)99.25Si0.75 BMG possesses the most brittle character.Based on the current eight samples, the theoretically predicted relationship of ν and μ/B is well followed experimentally.Phase separation could effectively result in the ductile behavior in the Zr53Cu30Ni9Al8 BMGs, tilt casting could effectively reduce the micro-porosity in the cast BMGs, and Si minor addition would degrade appreciably the fracture toughness. These three factors are all important in preparing promising BMGs for applications.Applying surface free energy method for evaluation of moisture damage in asphalt mixtures containing date seed ashPresent study deals with applying surface free energy theory in evaluating the possibility of using date seed ash as a bitumen modifier of hot mix asphalt mixtures against moisture damage. For this purpose, pure bitumen as well as ash containing samples was employed to produce mixtures having two different aggregate types (namely, limestone and siliceous). Prediction of surface energy was well supported by the laboratory method (AASHTO T-283) and proves the potential use of ash as a modifier. The achieved results indicated that using date seed ash will improve TSR of limestone and siliceous mixtures (i.e. 12.65% and 20%, respectively).Moisture damage of asphalt mixtures is one of the most common damages that occur in asphalt pavements Studies of moisture damage can be performed by applying various methods, such as experimental laboratory testing methods, field tests or analytical methods. Among these, experimental techniques present suitable methods for modeling field conditions. However they do not include studies related to micro-mechanism of moisture damage Surface energy of a material is briefly defined as the amount of work performed to develop a unit area of new surface of a material in vacuum There are different methods of measuring surface energy of asphalt mixture components. In some researches, surface free energy of bitumen was measured by using the Wilhelmy plate method and surface free energy components of aggregates using the universal sorption device (USD) Several methods have been adopted to reduce moisture damage in asphalt mixtures. A common method is to add anti-stripping additives such as hydrated lime, fly ash, cement powder and also nano-particles including nano-hydrated lime, nano-fibers, nano-clay etc. to the mixture Date is one of the oldest plants in earth In this study, date seed ash was added to bitumen as a replacement of filler. The main objective of the work was to evaluate moisture susceptibility of HMA mixtures containing date seed ash (DSA). This was performed by assessing moisture susceptibility via laboratory and analysis approaches and the results were discussed.Surface energy is defined as the work which is required for creating a unit area of a new surface of the material in vacuum. A variety of theories have been proposed to estimate surface free energy components where γtotal and γAB refer to total surface energy and acid-base component, respectively. Other terms are as described previously.Van Oss theory is carried out for obtaining surface energy of solids using at least three probe liquids (i.e. the liquid whose surface energy is known) ΔGL,Sadhesion=WL,Sadhesion=2γLLWγSLW+2γL+γS-+2γL-γS+where ΔGL,Sadhesion and WL,Sadhesion represent Gibbs free energy of adhesion and work of adhesion between solid and liquid interface, respectively. Indexes L and S stand for liquid and solid parts. Based on the Eq. , Van Oss developed the relationship between work of adhesion, surface free energy components and contact angle between liquid and solid interface θ, and suggested the following equation WL,Sadhesion=γLtotal(1+cosθ)=2γLLWγSLW+2γL+γS-+2γL-γS+, with substituting bitumen from solid sample (using three probe liquids) and replacing the three angles correspond to probe liquids, a three-equations, three-unknown system would be achieved. With solving the three-equations, three-unknown system, the parameters related to surface energy of bitumen can be calculated where x1, x2, x3 are square roots of the unknown surface free energy components of bitumen.It is to be stressed that the relations presented so far are utilized for studying the adhesion energy in two-phases of bitumen and aggregate. As the substitution of bitumen with water is thermodynamically favored, bitumen is replaced with water of hydrophilic nature. The amount of energy required to render such possible substitution can be defined as in Eq. In this equation, W, A, and S are the indexes related to water, asphalt, and aggregate particles, respectively.Bitumen used in this research was 60–70 pen bitumen from Isfahan Refinery. Major characteristics of this bitumen are reported in The ash used in this research was prepared in two phases. The first phase included burned date seeds that were converted into coal. The next phase consists of placing seed coal in furnace at a temperature between 600 and 800 °C for 1.5–2 h. As it can be observed in , the ash was grayish; white; considering that size of ash particles may be different from each other, they were passed through a No. 200 sieve before being added to bitumen. Scanning electron microscopy of ash is also presented in . It can be seen that DSA particles have granular grading. Chemical analysis of date seed ash used in this research is presented in . As it can be seen, the key compound in DSA was silicon dioxide. The activated structure of silicon can have an effective role in chemical stability and improvement of bitumen absorption Two aggregate types, namely, limestone and siliceous aggregates were used from Yazd province. Chemical composition of these aggregates is reported in . Grading was also selected based on Iranian Asphalt Pavement Regulation; with the maximum nominal size of 19 mm Mixes of DSA and bitumen were prepared at different DSA levels (i.e. 0, 5, 10, 15, 20, and 25%). Mixing method consisted of, heating bitumen up to 160 °C. Then, the ash was added and the mixture was prepared at 600 rpm for 10–15 min. Then, in order to achieve a homogeneous mixture, mixing speed was increased to 2000 rpm and continued for 1 h as suggested by Köfteci et al. In this research, all traditional tests on bitumen were carried out on the bitumen containing bitumen and various percentages of DSA. The results are reported in . To calculate thermal sensitivity of bitumen, Penetration index (PI) values were also determined by Eq. In the above equation, pen and S.P represent penetration value and softening point temperature, respectively. As aforementioned, PI stands for Penetration index.This method is consisted of dropping a liquid of volume 5–10 μl with specified surface energy vertically on a horizontal solid surface (considered material) and measuring the contact angle between the tangents on solid–liquid interface and droplet and by solving the resulting three-equations in a three-unknown system, surface energy of the solid object can be calculated.In this research, surface energy of bitumen and aggregates was measured by applying sessile drop method. Three probe liquids were used as described in and the experiment was carried out at ambient temperature of 25 °C.Samples were prepared according to ASTM Standard. For the two aggregate types, their filler of 6%, optimum bitumen content of mixes were determined based on four parameters of maximum stability, maximum density, 4.8% air void, and 80% void filled with bitumen. Marshall Flow and VMA were checked for all mixtures to meet the criteria of the Iranian code for Highway paving , with increasing DSA, penetration values of the bitumens were decreased at a constant rate (i.e. 65 for the base bitumen and 50 for B-DSA-25, representing a 23% reduction). Increasing in SiO2 causes bitumen stiffer As mentioned before, surface energy is composed of three acidic, alkali, and disperse components. With measuring the drop contact angles using three probe liquids of methylene, water, and glycerin, the results reported in were achieved. Results of contact angle of these drops and bitumen are presented. for example shows the contact angle of water on bitumen surface.According to the above table, it is obvious that, the contact angles were decreased by increasing the amounts of DSA and small p-values confirm that the contact angels were significantly different.This issue was repeated for all the three probe liquids. Based on the reported contact angles and Eq. , all the components related to surface energy of the modified bitumens were calculated. The results are presented in . It is evident that, with increasing DSA, surface energies were increased but at an almost moderate rate. This is strongly because of significant changes in disperse and alkali component. However, changes in acidic component are not considerable. Acidic component varies between 0.14 and 0.62 the studied area. However, disperse and alkali component changed from 24.56 to 0.017 for base bitumen to 32.1 and 3.276 for B-DSA-25, respectively.A large portion of asphalt molecule is a combination of a vast variety of high boiling hydrocarbons. Some are aromatic carbons, some are aliphatic and some others contain both However there can be seen one or more heteroatoms like nitrogen, sulfur and oxygen in a large proportion of asphalt molecules. In an asphalt molecule which contains both aromatic rings and aliphatic chain, aromatic carbons are in a strong bond with hydrogen, while some certain types of carbon in aliphatic chain are susceptible to oxidation.The introduction of DSA into the asphalt mass, results in particles to come closer to the susceptible aliphatic carbon. Because of the presence of oxidizing compounds in DSA composition, an oxygen atom replaces two hydrogen atoms connected with the aliphatic carbon and because of this, oxidation number of carbon adjacent to the aromatic ring increases. This approves that, oxidation has been occurred. In fact, oxidation results in broken aliphatic chain and formation of carboxylic acid as it can be seen in is considered as a weak acid and thus by entering of alkali and soil alkali metals to the environment, H+ is replaced by alkali metals like Na+ or Ca2+. These two ions are capable of acting as a strong alkali cation and therefore they can form sodium or calcium salts by creation of ionic bond with the weak acidic anion as shown in As described earlier, three components of surface free energy (i.e. acidic, alkali and disperse) were measured for all types of modified bitumens. In addition to chemical analysis, proportion of disperse energy per total energy was also calculated in order to evaluate the effect of DSA on different types of bitumens. For simplicity, this ratio was abbreviated as DT (disperse per total).Although total surface energy increased at an almost moderate rate with the increased amount of DSA, the amount of DT was decreased. shows DT changes, compared with the amount of DSA additive. As it can be observed from this figure, when DSA additive is increased to more than 10%, the DT decrease occurred at a higher rate. As a result of this, bitumen dispersion part was decreased, compared with its polar component.As described before, indirect tensile strength (ITS) is the best definition of HMA strength when subjected to tension. Aging of bitumen combined with penetration of water during frost thaw cycles, affect adversely on the strength of asphalt mixtures. TSR testing results of mix samples containing modified bitumens of different DSA percentages were obtained based on modified Lottman test. The results are depicted in that DSA plays a positive role in the ITS values of asphalt mixtures, particularly in range of 10–15%. For example, for the limestone specimen with 15% DSA, the ITS value is 15% higher than control specimen in unsaturated and 10% in saturated condition. that the tensile strength ratios (TSRs) of mixes with limestone aggregate are relatively greater than the control specimens. In particular, TSR value of the mixture with 10% DSA is 13% higher than the control specimen. For siliceous mixtures, except the specimen with 25% DSA, all the rest have more TSR value in comparison with control one.In the threefold system of bitumen, aggregate and water, the tendency of bitumen to separate from aggregate surface could be examined. This quantity can be explained as the amount of released energy when water is present and can be named EPW (Energy released in Presence of Water). First, surface energy components of aggregates should be assessed. shows the energy parameters for both types of aggregates studied in this research. EPW was determined for both aggregate types. It can be seen from that EPW was first increased and then decreased for both types of mixtures.As explained before, adhesion energy between bitumen and aggregate can be known as an index between bitumen and aggregates. With reference to , it can be seen that with increasing DSA, surface energy of bitumen is increased. As a result of this, adhesion energy between bitumen and aggregates was also increased. This could be due to the fact that, with increasing DSA, the bitumen polarity and especially its alkali feature will increase.With TSR values and comparing these with the amount of surface energy of the threefold state presented in , it is clear that changing process of surface energy in the threefold state and TSR was completely similar; with increasing DSA, these were first increased and then decreased., the ratio of acid-base to total surface energy of DSA has been plotted against DSA content. As it can be observed, with increasing the amount of DSA, the share of polar part of bitumen from total surface energy is increased. This increasing trend appears at percentages greater than 10%. Consequently, the disperse component of surface energy was decreased. With decreasing the disperse share from total surface energy of bitumen, bitumen operation tended towards being polar. On the other hand, according to the polarity of water, free surface energy between water and bitumen was increased. As the difference between bitumen-aggregate and water-aggregate surface energy decreased, the sensitivity of asphalt mixture to moisture was decreased too. shows variation between differences of bitumen-aggregate and water-aggregate surface energy against percentage of DSA. As it can be observed, a quadratic equation was fitted between the available data. This amount decreased first and then it was increased. By differentiating this equation, the optimum percent of DSA for improving moisture sensitivity resistance was determined 12.92 and 13.57 for limestone and siliceous aggregates, respectively.In general, with adding DSA to bitumen, surface free energy of bitumen is increased; however, for the amounts greater than 15%, the role of polar component increased and bitumen operation moved towards polar material. It seems that augmentation of bitumen’s polarity occurred because of increase in alkali component. The most important factor in the increased alkali component of the modified bitumen was the presence of compounds of alkali metals.With increasing DSA in bitumen of asphalt mixtures, TSR values were first increased up to 0.78 for siliceous and 0.89 for limestone aggregates and then with increasing the amount of DSA it showed decreased TSR values for both aggregate types. The optimum amount of ash content for siliceous and limestone aggregates were 11.44 and 7.95, respectively. The high amount of DSA’s optimum percent for siliceous aggregates could be attributed to the remarkable lack of alkali elements such as calcium in siliceous aggregates, compared with limestone aggregates. As a result of this, with limestone mixtures, DSA could reach the optimum amount earlier and with less DSA percentages.Surface free energy between bitumen-aggregate continuously increased with augmenting the DSA content. However, in presence of water, this parameter was first raised and then it diminished. It is theoretically speculated that least moisture damage is obtained in DSA contents where the SFE of bitumen-aggregate interface is maximized. Accordingly, this hypothesis could readily be deduced by comparing TSR results with that of SFE in the presence of water ( that consistency of TSR results with SFE ones is more appropriate in siliceous samples than limestone ones. This is probably due to the existence of silica compounds in DSA. The increasing trend of surface energy in bitumen-aggregate and bitumen-water systems is evaluated in and it can be seen that surface energy in bitumen-water system fairly follows a constant rate. However this trend for bitumen-aggregate pair is accompanied with a sharp decrease in the range of 10–20% DSA. Consequently, a large discrepancy between surface energy of bitumen-water and bitumen-aggregate is observed in the aforementioned range. According to Eq. , this discrepancy leads to augmentation of adhesion energy of bitumen-aggregate in presence of water.Based on the research performed on evaluating the effects of DSA in asphalt mixtures, the following conclusions were obtained:Addition of DSA to asphalt mixes resulted in increased surface free energy of bitumen.When the amount of DSA in mixes was exceeded 15%, bitumen acted as a polar material.With increasing DSA percentage in bitumen, TSR was first increased up to 0.78 for siliceous and 0.89 for limestone and then decreased for both aggregates types.Optimum percentage of DSA in asphalt mixes containing limestone and siliceous aggregates for coping with moisture damage was determined to be 11.44 and 7.95, respectively.Using 10% DSA in asphalt mixtures, TSR value was increased by 12.6% for limestone. Similarly, an increase of 15% in TSR value was observed in siliceous aggregates having 20% DSA.Fourier p-elements for curved beam vibrationsSeveral Fourier p-elements for in-plane vibration of thin and thick curved beams are presented. Fourier trigonometric functions are used as enriching functions to avoid the ill-conditioning problems associated with high order polynomials. The element matrices are analytically integrated in closed form. With the additional Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly improved. Furthermore, the elements with enriching shape functions can avoid membrane and shear locking. The vibration of a thin ring, whose exact solutions are available, is analyzed by the present elements. The present elements can compute accurately high natural modes as the higher mode shapes synchronize with the Fourier functions nicely. The free vibration analysis of a number of hinged circular arches with various subtended angles and the tapered cantilever arches having uniform and non-uniform cross-section is carried out as numerical examples. The condition numbers for polynomial p-elements and Fourier p-elements are compared to show the superior numerical stability of the method.change of curvature in thin curved beam elementshear strain in thick curved beam elementCurved beam elements have generated great interest among researchers over the last decades. It is not only because curved beams do exist in many structures, but also the study of curved beam elements provides insight into various aspects of shell element behavior. In the displacement-based finite element method (FEM), the polynomials used to approximate the displacement fields are chosen to satisfy the requirements of state of constant strain and the rigid-body motion. Furthermore, they need to avoid membrane and shear locking which lead to excessively induced stiffness in the deep thin regimes. Many simple elements cannot satisfy all of these requirements For the free vibration problems, Krishnan et al. It is also well known that the convergence of p-version elements is more rapid than that of h-version elements with respect to the number of d.o.f. Several Fourier p-elements both for thin and thick curved beams are presented in this paper. The element matrices are integrated analytically and one element can predict many high order frequencies accurately. By introducing the Fourier d.o.f., the elements are free of locking and can predict rigid-body modes without difficulty. The convergence of the present elements is very fast with respect to the number of Fourier trigonometric terms. By comparison with other curved beam elements, it is shown that the present elements produce higher accurate modes per degree of freedom., respectively. These functions produce zero values, and the desired derivatives, at each node.. According to the classical thin shell theory, the extensional strain ε, rotation φ, and change of curvature χ of the arch are expressed in terms of the displacements and their derivatives as where R is the radius of the arch, u and w are the tangential and normal components of the displacement at s, respectively, s is the curvilinear co-ordinate. The strain energy and kinetic energy expressions becomewhere EA, EI and ρA are the axial stiffness, the flexural stiffness and the mass per length of the beam. For the element THINij (i≥0, j≥1), u and w are related to nodal d.o.f. through interpolation (shape) function as satisfying Ci and Cj continuity, respectively, in which ξ=s/L with L is the length of the curved beam element; ue and we are the vectors of nodal d.o.f. of tangential displacements and normal displacements respectively. ui and wj can be rewritten as. For undamped free vibration, the Lagrange equation iswhere ω is the vibration frequency of the curved beam, Since the expressions of the shape functions are simple, analytical integrations of the right components of the two equations in Eq. can easily be obtained and many trigonometric terms can be involved in the shape functions to predict very high modal frequencies by one element. Several thin elements are presented herein, and each element is denoted by THINij whose shape functions for u and w satisfy Ci and Cj continuity, respectively.The circular arch element taking into account of the effect of transverse shear deformation is shown in . The extensional strain ε, flexural strain κ, and shear strain component γ in the curvilinear co-ordinate system are expressed in terms of the displacements and their derivatives as where θ represents the rotation of the cross-section. The strain energy and kinetic energy expressions becomewhere G is the shear modulus, A is the cross-sectional area of the beam and k is the shear correction factor. u, w and θ are related to the nodal d.o.f. through interpolation function aswhere Ni is the C0 shape functions mentioned in and δe={u1,w1,θ1,u2,w2,θ2,…,ui,wi,θi,…}T, in which ui, wi and θi (3≤i≤p+2) are the internal d.o.f. of the element with p trigonometric terms. Substituting Eqs. into the Lagrange equation yields the following stiffness and mass matricesThe coefficients of the stiffness matrix and the mass matrix are obtained straightforwardly and are given in where σ=I/A and φ=−R/(R2+σ). The displacement field for θ isThe normal (radial) displacement field w is expressed by a cubic polynomial plus a Fourier series as yield the following equations governing the tangential displacement and the section rotationThe element d.o.f. are expressed in terms of ai (0≤i≤6) aswhere u1, w1, θ1 are the three d.o.f. at s=0; u2, w2, θ2 are the three d.o.f. at s=L; wr is the internal d.o.f. of radial displacements and expression for H is in , and the coefficients in the expressions of the displacements can be expressed in terms of the element d.o.f. as is a single row matrix with the same components as those of the (i+1)-th row in G. Substituting Eq. where δe={u1,w1,θ1,u2,w2,θ2,…,wr1,wr1,…wrp}T in which wri (1≤i≤p) are the internal d.o.f. of the element and p is the number of trigonometric terms involved. is the shape functions for w, and for its first six components, is the component at the i-th row and the j-th column in G. Similarly, according to Eqs. , one can also get the Fourier p formulation of u and θ. Then, the stiffness matrix and mass matrix of the element can be obtained similarly to THICK-1.where λmax and λmin are, respectively, the largest and smallest eigenvalues of the coefficient matrix. A large condition number indicates that the FEM solutions may be ill-conditioning. The CN values of the mass matrices of both elements are compared in with various p which is the additional terms in shape functions fi(ξ). The CN of the Fourier p-element is smaller than those of the hierarchical element. Furthermore, using double precision, the mass matrix of the hierarchical element will be ill-conditioning when p=41 and any software will fail to find eigenvectors gives the percentage errors of the first 500 frequency parameters using 498 trigonometric terms compared with the exact solutions. The percentage errors of the frequencies obtained by Fourier p-element are less than 0.3% except the last two. So the solutions computed by the Fourier p element are accurate. The natural frequencies computed by both elements respectively with p=20, are compared in along with the exact solutions. The Fourier p-element using trigonometric shape functions is indeed more effective in predicting the medium- and high- frequency modes than the element using orthogonal Legendre polynomials as the shape functions.. The problem is analyzed by several elements in the literature . Using one element, the quarter of the ring is analyzed by six thin Fourier p-elements, respectively. The solutions are compared with those of QQ6 and C-CQ3 which are the two most accurate elements in the literature shows the number of total d.o.f. needed to converge to the exact natural frequencies ω2=58.04 Hz and ω3=137.69 Hz. It can be found that the convergence of THIN02 and THIN12 is fastest and any Fourier p-element is converging faster than QQ6 and C-CQ3. Since the ring with R/h=320 is very thin, it is clear that the present Fourier p-elements are free of locking.Two curved beam elements with a subtended angle of 90° are used to predict the modes of the whole ring. The lowest ten frequencies of the ring computed by seven present Fourier p- elements with p=10 are listed in along with the exact solutions. The convergence of some modes computed by the elements with C0 shape functions is somewhat slow, but the solutions of other elements are quite accurate. All elements can capture the rigid-body modes except C0 elements as the two displacement fields are coupled and C1 is the minimum requirement. With 100 trigonometric terms, the natural frequencies of the ring are computed by THIN11. The 51st to 56th non-zero frequencies are: ω1=2901.417 Hz, ω2=2901.422 Hz, ω3=3120.767 Hz, ω4=3120.775 Hz, ω5=3348.094 Hz, ω6=3348.103 Hz, respectively. The six contour mode shapes are presented in . Those high order frequencies calculated by only two elements agree well with the exact solutions shows the hinged circular arches with subtended angle α. The following numerical values , respectively, and the comparison with other existing elements The application of the present thin and thick curved beam elements to the tapered cantilever arches shown in having uniform and non-uniform cross-section is carried out. If the cross-section of the arches is non-uniform, its thickness varies linearly along the length, from h1 at the fixed end to h2=h1/2 at the free end. The shear correction factor k=0.83 and the Poisson’s ratio v=0.3. Two types of the arches with αR/r1=10 and 20, where r1 is the radius of gyration of cross-section and , are analyzed by one THICK-1 element with p=10 and five THICK-2 elements with p=5. The first three frequency coefficients ( along with the available results. The results show good agreement with the existing ones Several efficient Fourier p thin and thick curved beam elements to predict rigid-body modes without locking are presented. Fourier trigonometric functions are used as enriching functions to avoid ill-conditioning problems. With the additional trigonometric shape functions, the accuracy of elements is greatly improved and one element can predict many modes.To examine the convergence and accuracy of the present elements, the vibration of a thin circular ring is analyzed as the first case study. It is concluded that the shape functions of the displacements for the thin elements should satisfy C1 continuity at least in order to achieve fast convergence. The following two case studies, one is the hinged arches with various subtended angles, the other is the tapered cantilever arches, show that the present Fourier p-elements lead to excellent convergence characteristics and accurate prediction.The stiffness matrix coefficients of element THICK-1 areThe mass matrix coefficients of element THICK-1 areif m=3(i−1)+1 and n=3(j−1)+1, or m=3(i−1)+2 and n=3(j−1)+2, with α and β (α, β=0, 1) denote the order of the derivatives, f are the C0 Fourier shape functions shown in , and i, j=1, 2,…,p+2 with p is the trigonometric terms.Effects of particle clustering on the plastic deformation and damage initiation of particulate reinforced composite utilizing X-ray CT data and finite element modelingIn this paper, a new simulation technique which can include microstructural inhomogeneity of particulate reinforced composites is proposed to accurately study deformation pattern and damage mechanism in these composites. Three dimensional microstructures constructed from XCT images incorporated into finite element modeling codes with minimal approximation to capture the effects of cluster size, local volume fraction of particles in the cluster and the distance between clusters as relevant statistical quantities describing the microstructural inhomogeneity of particulate reinforced composites. A quantitative parameter as degree of clustering is defined to consider particle clustering effect. The results indicate that the damage growth rate of composite with higher degree of clustering is significantly higher than those composites with lower degree of clustering. It is found that for region with higher degree of clustering and bigger size of clusters, the von Mises stress is higher at the same loading condition and the growth rate of plastic flow is considerably higher than the other region with lower degree of clustering. Moreover, the dislocation description of deformation in two-phase materials rationalize particle clustering effect on the yield behavior of the particulate reinforced composites and the flow stress in these composites. The macroscopic stresses that lead to the initial yielding in the matrix decrease when clusters closely proximate with bigger size and higher degree of clustering.A thorough understanding of how a heterogeneous material's microstructure affects its macroscopic properties is of great importance in design and development of high performance heterogeneous materials. This is particularly challenging given the multiphase and heterogeneous nature of most high-performance composites. Modeling and prediction of the overall elastic–plastic response and local damage mechanisms in heterogeneous materials, in particular particle-reinforced composites, is a very complex problem. Various theoretical and numerical methods are proposed to clarify the relationship of the microstructure and the macroscopic property, of which the finite element (FE) analysis is an important effective method []. The FE analysis primarily requires the development of methods to automatically generate geometrical or mesh models to actually take into account complex microstructures of heterogeneous materials. However, due to the very irregular shape and complex distribution of phases, the incorporation of the information about microstructures into the models is one of the challenges in computational mechanics [Finite element method (FEM)-based models are often applied to the so called representative volume element (RVE), thus assuming that the microstructure of the composite can be reproduced by assembling a large number of such elements. However, this can be a serious limitation when dealing with complex and highly heterogeneous composites microstructures, such as randomly dispersed particulate systems. Therefore, an approach able to consider the actual microstructure of the composite is useful in order to accurately predict the overall properties. The RVE models have been developed from the simplest two dimensional (2D) versions, which assume rather specific shape and distribution of phases, to complex three dimensional (3D) models, which take into account most characteristics of the real shape and distribution of material components [Recent developments in high-resolution 3D imaging techniques, such as focus ion beam (FIB)/scanning electron microscopy (SEM) or X-ray computed tomography (XCT) launched a larger interest in performing 3D simulations on the actual microstructure of different materials []. The combination of XCT and image analysis proved to be a powerful tool to characterize heterogeneous material's microstructure in 3D, providing information that cannot be obtained with traditional 2D microscopy techniques. XCT has been used as a basis for obtaining microstructurally realistic FE models. XCT has ability to detect the internal structure as small as 1 μm, which provides a feasible method to build models based on actual microstructure []. These models considering the inherent morphology, clustering and arrangement of phases, with minimal microstructural approximations, are frequently adopted to analyze the macroscopic behaviors of heterogeneous materials.] have shown that 3D microstructures constructed from serial sectioning can be incorporated into commercial FEM codes to model the deformation behavior of composites. Maire et al. [] discussed different possible methods for using tomography results as inputs for numerical models, especially FE meshes. They classified three different ways to produce meshes reflecting the actual architecture of a cellular material: meshes generated from a Voroni description of microstructure, voxel/element meshes, and tetrahedral meshes of the actual shape of the cellular architecture. Yu et al. [] proposed a method to identify micro scale fiber bundle configuration of a needle-punched carbon-carbon composite through micro-CT image processing. Then a FE model is built and used to predict mechanical properties and simulate progressive damage of this composite. Li et al. [] used both the micro-CT observations and FE analyses to investigate the impact shear damage mechanisms and energy absorption of the 3D braided composite.It has been shown that the mechanical behavior and properties of particulate reinforced composites are highly dependent on the real microstructure of the composite and particle distribution and volume fraction. In most particulate reinforced composites, the particles are not uniformly distributed. Instead, these materials contain local regions where the particles are clustered. It is well established experimentally that damage nucleation in polymer- and metal-matrix composites occurs in regions of the microstructure that contain high local volume fraction of reinforcements []. Thus, the accurate simulation of deformation and damage initiation in composites requires new simulation techniques which can include inhomogeneous reinforcement distributions.In this study, the effects of clustering and critical microstructural characteristics of particulate reinforced composite on deformation pattern and damage mechanism have been investigated using 3D FE modeling. Quantitative analysis of the critical characteristics with regard to the effect of particle clustering have been obtained using XCT data in the first part of this study []. The relevant statistical quantities describing the microstructural inhomogeneity of particulate reinforced composites (Ni60Nb40/Mg) have been used as input for FE modeling with minimal microstructural approximation. The proposed new simulation technique can include microstructural inhomogeneity of particulate reinforced composites to accurately study deformation pattern and damage mechanism in these composites using XCT data incorporated into FE models. Moreover, particle-clustering effect on yield behavior and flow stress of these composites has been qualitatively studied using dislocation description of deformation in two-phase materials.Amorphous alloy powder with composition Ni60Nb40 (at. %) was prepared by mechanically alloying powder mixtures of elemental Ni and Nb metals. The powder mixture was milled at room temperature in air for 87 h, using a Retsch PM400 planetary ball mill with a ball-to-powder ratio of 3:1 and milling speed of 200 rpm. To produce Mg-composites, elemental Mg-powder (99.6% purity) was blended with volume fraction 5% of Ni60Nb40 powder for a duration of 1 h and consolidated at room temperature at 450 MPa for 1 min. The compacted cylindrical billets of 36 mm diameter were microwave sintered at 100% power level for 12.5 min so as to achieve a temperature of 550 °C (based on prior calibration). The sintered billets were soaked at 400 °C for 1 h, and hot extruded at 350 °C to produce rods of 8 mm diameter. Rods extruded at 750 psi and 600 psi were used for further analysis.In order to understand the mechanical behavior of particulate reinforced composites, it is important to investigate the size, morphological characteristics, and distribution of inclusions in the material. If a group of reinforcement particles is closely packed such that the mechanical properties of the material near the particle group differ from the surroundings, the particle group is referred to as a “cluster” []. In the following, XCT imaging is used to provide a precise picture of the microstructural heterogeneity and clustering in these composites.For high-resolution XCT imaging a cylindrical (diameter ≈ 1 mm, height ≈ 1 mm) region was selected from the rod sample. According to the measuring procedure introduced in Ref. [], the sample was imaged using Xradia MicroXCT-400 tomograph (Zeiss Xradia, Concord, California, USA) with 0.59 μm pixel size. The X-ray tube voltage and power were 40 kV and 4 W, respectively, and a 0.37 mm glass filter was applied. Total of 1800 projection images over 360 degrees of rotation were acquired with 50 s exposure time per angular position. The projection images were reconstructed using utility software provided by the manufacturer of the tomograph, resulting in a 3D volume image where the pixel values are roughly proportional to the local X-ray attenuation coefficient []. In order to characterize the clustering based on the reconstructed images, the reinforcement regions shown in the image (a) must be classified into individual particles and clusters. Comparison of the reconstructed images to, e.g., scanning electron microscope images of similar clustered material (b) suggests that the resolution of the reconstructed images is not high enough to fully differentiate individual particles inside particle clusters, but individual clusters are easily differentiated from each other []. The clusters are shown in the images as large connected reinforcement regions with varying X-ray attenuation coefficient. As the images show the individual particles inside the clusters mostly connected to each other, geometrical conditions like distance of individual particles are not effective for classification purposes. Instead, a condition based on the local X-ray attenuation coefficients inside each reinforcement region was used, as described below.To facilitate the classification, the reinforcement regions were segmented using the Phansalkar method []. The segmented regions were classified into clustered and individual particles based on observation that a cluster consisting of multiple smaller particles has many local X-ray attenuation coefficient maxima inside it, each of the maxima roughly originating from a single reinforcement particle inside the cluster. The count of local maxima was determined for each particle and reinforcement regions with more than three maxima were classified as clusters. All other particles were classified as individual reinforcement particles. This classification process resulted in a visually plausible result as shown in a–b for two specimens chosen from different rods extruded at 600 psi and 750 psi, respectively.The amount of clustering can be naturally expressed by dividing the total volume of particles Vf into two parts Vfc and Vfm where Vfc is the volume of clustered particles, Vfm is the volume of individual particles, and Vf=Vfc+Vfm. The degree of particle clustering in particulate reinforced composites quantitatively describe as follows:where ζ is the volume ratio of clustered particles over the total particles in the matrix.For a composite, the volume fraction of particles f1 is defined by:Where V is volume of composite. Denote f1c as volume fraction of particles agglomerated in the cluster regions and f1m volume fraction of individual particles in the matrix. We can give the relationships of f1, f1c and f1m versus the degree of clustering, ζ, as below:Two types of 5 vol % Ni60Nb40/Mg composites with different microstructure and degree of particle clustering were studied in this work to highlight the capability of the proposed new simulation technique which can include microstructural inhomogeneity of particulate reinforced composites. Based on the proposed definition, the clustering parameter, ζ are respectively 0.9 and 0.51 for these composites. This means that for composite extruded at lower extrusion pressure, around 90% of the particles are agglomerated in the clustered regions and 10% of particles are uniformly dispersed in the matrix. While for composite extruded at higher extrusion pressure, around 51% of the particles are clustered and 49% of particles are found as individual particles in the matrix.In order to make the model closer to reality, 3D RVE should be generated on the basis of real microstructure of the samples. It is necessary to find first the cluster spatial coordinates that describe the real spatial distribution of clusters in the matrix. Firstly, the bounding sphere for each cluster, i.e. the smallest sphere that contains all the cluster pixels, was determined using Welzl method [] that calculates the optimal bounding sphere of a set of points in linear time using a randomized linear programming type algorithm. Thus, the spatial coordinates of clusters are determined in the samples based on the center coordinates of the bounding sphere for each cluster. Secondly, the equivalent spherical diameter, deq (), was determined, which equals the diameter of sphere having the same volume of individual cluster, Vfci, as described in Eq. Vfci is determined as total count of pixels for each individual cluster.a–b shows the normal and log- normal probability distribution of the equivalent spherical diameter and bounding sphere diameter for sample extruded at different extrusion pressure. The equivalent spherical diameter distribution has closely the same bounding sphere diameter distribution. However, the differences between the equivalent spherical diameter and bounding sphere diameter for sample extruded at higher pressure with lower concentration of cluster regions are higher. It should be noted that the diameter of the bounding sphere for each cluster describes the maximum extension of the cluster; therefore, these results reveal clusters have different morphology related to various extrusion pressure. This is correlated with results obtained in previous study using X-ray nanotomography that suggested there may be two types of clusters in the present composites; relatively spherical ones for sample extruded at lower pressure and more ellipsoidal ones for sample extruded at higher pressure [a–b, the equivalent spherical diameter, deq, is compared to the bounding sphere diameter for sample extruded at 600 psi and 750 psi, respectively. The comparison reveals an approximately linear behavior, but exhibits enhanced scatter in larger cluster sizes. For further modeling and analyses in this study, the equivalent spherical diameter, deq, was employed due to the proportionality of deq and the bounding sphere diameter, which leads to acceptable results for statistical examination and modeling. These microstructural data sets can be incorporated into FE models to predict the onset of local damage mechanisms and the deformation pattern of particulate reinforced composites.On the basis of bounding spheres and equivalent diameter of clusters, two RVEs representing clusters in 3D microstructure of a–b. Two sub-RVEs have been extracted from the whole RVE for composites with different extrusion pressure. These typical RVEs are chosen from the center of the whole RVE. The selected RVEs contain the equivalent clusters while in order to define RVEs based on the real microstructure of the composites in each case the individual particles should be added to the RVEs. This addition of individual particles to each RVE occurred based on the value of proposed clustering parameter, ζ, and f1m, as presented in section c–d shows the finite element RVEs generated for 3D microstructure of a–b on the basis of real microstructure of particulate reinforced composite with minimal microstructural approximation. This approach is applicable for different type of particulate reinforced composite materials. In this paper, two types of 5 vol % Ni60Nb40/Mg composites with different microstructure and degree of particle clustering were studied to highlight the capability of the proposed new simulation technique which can include microstructural inhomogeneity of particulate reinforced composites using XCT data as input for FE modeling with minimal microstructural approximation.The material properties of the composite constituents are presented in . The Young's modulus of the Ni60Nb40 amorphous particles is measured by nanoindentation method. In the composite medium, the matrix undergoes elastic-plastic deformation and the reinforcement deforms elastically. Mg matrix and the Ni60Nb40 amorphous particles are considered to behave as elasto-plastic and elastic-perfectly plastic materials, respectively. The multi-linear isotropic hardening Mises plasticity law is used for the plastic behavior of the model.A free mesh containing 3D 8 node solid element and suitable for modeling irregular meshes was used in this study to have an accurate model of microstructure. Symmetric boundary conditions are applied on three faces while constraint equation is applied to the nodes on the other two faces to have equal displacements in the Y and Z directions at all nodes and fixed in the other directions. The loading is applied in various steps in the form of displacement type on the right side of the model (displacement control solution). Loading steps strategy is a requirement to control the numerical convergence of nonlinear elastic–plastic solution.a–b shows the equivalent plastic strain in the matrix at final loading step for typical RVEs presented in c–d, respectively. To obtain more information about particle clustering effect on plastic deformation under loading, the evolution of equivalent plastic strain, εeq, at different strain state for composites extruded at different extrusion pressure with various degree of particle clustering is depicted in a–b. It can be seen that the particle clustering influences the damage behavior of composites. These histograms are obtained from plastic strain in the elements of the models. The in-situ growth behavior of equivalent plastic strains are depicted in a–d. It is found the damage growth rate of composite with higher degree of clustering is significantly higher than the other case with lower degree of clustering as shown in d. Moreover, the distribution of the equivalent plastic strain in the elements in the matrix, for strain state 0.1, is plotted in for composites extruded at 600 psi and 750 psi, respectively. This distribution for composite with higher degree of particle clustering shows a correspondingly higher fraction of elements with lower plastic strain than the other composite with lower degree of particle clustering. The plastic deformation is constrained in the matrix regions between clusters. This distribution illustrates particle clustering has a considerable influence on the plastic flow on the matrix. It should be noted that in-situ growth behavior of equivalent plastic strains and histogram of the evolution of equivalent plastic strain provide more detailed information that cannot be reached by general distribution of equivalent plastic strain presented in In two phased material such composites (or alloys), one component (often dispersed as particles in composites) deform less than the other, or not equal at all, so the gradients of deformation form with a wavelength equal to the spacing between the phases or particles. Such composites (or alloys) are plastically non-homogeneous, because gradients of plastic deformation are imposed by the microstructure. When a plastic crystal is deformed, dislocations are generated, move, and are stored; this storage causes the material to work-harden. Dislocations become stored for two main reasons: they accumulate by trapping each other, or they are required for the compatible deformation of various parts of the crystal constrained within its surroundings. The dislocations that are mutually trapped are referred to as statistically stored dislocations [] and, as yet, there is no simple argument to estimate their density, ρs. The dislocations that are stored due to incompatibility in deformation are called geometrically necessary dislocations (GND), ρGND. The statistically stored dislocations is a characteristic of material, that is, of the crystal structure, shear modulus, stacking-fault energy, etc. The GND is a characteristic of the microstructure, that is, the geometric arrangement and size of phases or particles. We use the concept of GND to rationalize particle clustering effect on the yield behavior of the particulate reinforced composites and the flow stress in these composites.Strengthening of reinforcement particles on yield strength (σym) of particulate matrix arises due to: (i) Orowan strengthening (ΔσOrowan), the stress increase needed to move a dislocation through an array of impeding particles, (ii) stress contribution due to statistically stored dislocations introduced by the thermal expansion mismatch between the matrix and reinforcement (ΔσCTE) and (iii) generation of GND to accommodate the plastic deformation mismatch between the matrix and particles (ΔσGND) []. In this study, the stress contribution due to statistically stored dislocations introduced by the thermal expansion mismatch between the matrix and reinforcement is neglected. The strength of a reinforced matrix is given by Refs. [where σymr and σym are the yield strength of reinforced and unreinforced matrices, respectively. Δσ represents increment in yields of reinforced matrix and is estimated as follows:ΔσOrowan, can be estimated using the following equation [where β is a constant, μm, bm are the shear modulus of metal matrix and its Burgers vector. λ is the interparticle spacing of the second phase particles, which is given bywhere f and D are the volume fraction and the diameter of the second phase, cluster, respectively.ΔσGND is calculated using the following equation [where α is a geometric factor and the density of GNDs can be estimated by Ref. [with γm the local shear strain in the matrix.When the particles agglomerate in clusters, the diameter of the second phase increases. Thus, the effects of aggregation of particles as clusters on the strength of a reinforced matrix and the composite can be correlated with the decrease in the extent of Orowan and GND strengthening with increasing the cluster size as a second phase in the matrix. The results of tensile property measurements conducted on the 5 vol % Ni60Nb40/Mg composites extruded at different extrusion pressures under tensile loading are listed in . Experimental findings show that there is a strong relationship between strength of composite and the local volume fraction and distribution of the reinforcement particles []. These results are in good agreement with the dislocation description of deformation in two-phase materials rationalizing particle clustering effect on the strength and yield behavior of the particulate reinforced composites. Hence, the conclusion from this section can be a valuable result for researchers in material and manufacturing to optimize mechanical properties of composites based on the knowledge of the relationship between the microstructure and the macroscopic response.The first part of this study reveals that the critical characteristics with regard to the effect of particle clustering are cluster size, local volume fraction of particles in the cluster and the distance between clusters []. When multiple clustered reinforcement regions are in close proximity, they behave as a large single cluster. Thus, it is important to elucidate the effects of the proximity of clusters on the local damage initiation and flow stress of particulate reinforced composites. To study the effects of particle clustering on the yield behavior of the particulate reinforced composites and the flow stress of the matrix in these composites, the sub-RVE has been extracted from the whole RVE for composite extruded at 600 psi with the highest degree of clustering (ζ=1) as depicted in . This sub-RVE was chosen since the effects of the cluster size and proximity of clusters clearly were illustrated.The macroscopic stresses that lead to the initial yielding in the matrix are obtained for sub-RVEs chosen from specimen extruded at 750 psi, and 600 psi with different degree of clustering based on this FEM simulation, and they are presented in a–c shows the von Mises stress distribution in the matrix at strain state 0.009 in the beginning of loading step and near to the initial yielding in the matrix for typical RVEs presented in , respectively. In this case, the frequency distribution of von Mises stress in the matrix is depicted in . This figure highlight the effects of particle clustering and cluster size on the distribution of von Mises stress. It can be seen there is a uniform distribution in the case that degree of clustering and cluster size are lower as shown for composite extruded at 750 psi. In this case, around 95% of von Mises stress distributed in the interval between 70 and 80 MPa; however, for two other cases with higher degree of clustering and bigger cluster size there is not a uniform distribution. The macroscopic von Mises equivalent stress is plotted versus the non-dimensional loading time in . It is found that for region with higher degree of clustering and bigger size of clusters, the von Mises stress is higher at the same loading condition and the growth rate of von Mises stress is considerably higher than the other region with lower degree of clustering. The von Mises stress expresses the degree of easiness for plastic deformation. This figure implies that plastic flow on the matrix occurs earlier and easier in the case that clusters closely proximate with bigger size and higher degree of clustering.Moreover, the hydrostatic stress distribution of sub-RVE extracted from the composite extruded at 600 psi with the highest degree of clustering (ζ=1) is depicted in . It can be seen that the peak values of hydrostatic stress which are important factor for debonding and void formation happen in clusters and regions that clusters are in close proximity. As depicted in , the plastic flow is prohibited in these regions and consequently by correlation with hydrostatic stress there is a great tendency towards debonding and crack initiation failure mode. It is worth to note that high level of clustering may cause some bonding problems. The clustered regions can be potential sites for void nucleation, which gives rise to earlier commencement of failure. However, this aspect should be investigated in more detail and thus the effect of clustering and critical microstructural characteristics of particulate reinforced composite on interface debonding and void formation will be further investigated in the following part of this study.In this study, the effects of clustering and critical microstructural characteristics of particulate reinforced composite on deformation pattern and damage mechanism have been investigated using 3D FE modeling. 3D microstructures constructed from XCT images incorporated into FEM codes with minimal approximation to capture the effects of cluster size, local volume fraction of particles in the cluster and the distance between clusters as relevant statistical quantities describing the microstructural inhomogeneity of particulate reinforced composites (Ni60Nb40/Mg). A quantitative parameter as degree of clustering is defined to consider particle clustering effect. The results indicate that the damage growth rate of composite with higher degree of clustering is significantly higher than those composites with lower degree of clustering. The in-situ growth behavior of equivalent plastic strains as a new approach provides more detailed information about particle clustering effect on plastic deformation. The dislocation description of deformation in two-phase materials rationalize particle clustering effect on the yield behavior of the particulate reinforced composites and the flow stress in these composites. Results show the macroscopic stresses that lead to the initial yielding in the matrix decrease when clusters closely proximate with bigger size and higher degree of clustering. It is found that for region with higher degree of clustering and bigger size of clusters, the von Mises stress is higher at the same loading condition and the growth rate of von Mises stress is considerably higher than the other region with lower degree of clustering. The regions that clusters located in close proximity are potential sites for debonding and void formation since the hydrostatic stress have peak values and the plastic flow is constrained there.Extrusion of nanocomposite Al90Fe5Nd5 powders and characterization of the consolidated materialIn this work, a new Al90Fe5Nd5 nanocomposite atomized powder was extruded applying different processing parameters and the resulting bulk material was microstructurally and mechanically characterized. Successful extrusion was obtained for an extrusion ratio of 7:1 and extrusion temperatures of 370 and 450 °C. Yield strengths up to 726 MPa were obtained. A linear relationship between hardness and yield strength was found, which would lead to a yield strength as high as 900 MPa for the 25–50 μm powder extruded at 370 °C.Aluminium alloys based mainly in Al-RE, Al-ETM-LTM and Al-RE-TM systems (RE=rare earth, ETM and LTM=early and late transition metals, respectively) have been investigated in the last decade to produce amorphous and nanocomposite materials. These are of great interest for the transport industries considering the high tensile strengths that can be achieved: 1000 MPa with amorphous Al alloys The Al90Fe5Nd5 (at.%) powder was produced by helium atomization of pure Nd and an Al–Fe master alloy. The powder processing and its characterization have been described elsewhere . Prior to the extrusion, the encapsulated powder batches were submitted to heating at the extrusion temperature for 45 min.The Al90Fe5Nd5 extruded bars were characterized by X-ray diffraction (XRD) using Cu–Kα radiation. Metallographic sections were observed by scanning electron microscopy (SEM). Mechanical characterization was carried out by hardness and compression tests. Vickers hardness measurements were performed on cross sections of all extruded bars applying a mass of 5 kg. Compression tests were performed at 0.5 mm min−1 using a universal testing machine along the extrusion direction of cylindrical specimens of materials extruded at 450 °C, which were deformed until fracture occurred.Industrial thermomechanical processing, such as hot extrusion, for consolidation of Al-base powders involves the use of high temperatures. Thus, bulk nanocomposite Al-base alloys with high mechanical properties are difficult to obtain because the nanostructure is usually unstable under the processing conditions. The Al90Fe5Nd5 alloy powder, however, presents a high thermal stability, with a large temperature difference between the first (240 °C) and the second (360 °C) crystallization events Apart from thermal factors, successful consolidation also depends on the flow stress of the powder particles at the extrusion temperature (ET), because if this is too high the extrusion pressure (EP) required for consolidation will exceed the capacity of current industrial and laboratory presses. An indication of the flow stress of the powder particles could be obtained through their Vickers hardness (HV). Hardness values for the Al90Fe5Nd5 powder are given in Ref. . Comparison with data from the literature shows that Al-base powder with similar hardness has been successfully extruded. Ohtera et al. To ensure the retention of maximum metastable structure of the Al90Fe5Nd5 powder, ideal extrusion conditions would include an ET below the first crystallization event and a powder size range of <25 μm, which contains the largest amount of amorphous phase, 42 vol.% Extrusion conditions established for the first extrusion trial were an ET of 330°C, i.e. just below the second crystallization event, powders in the 50–100 μm size fraction and an extrusion ratio (ER) of 5:1. shows the XRD pattern of this first bar. This pattern is similar to that found in the as-atomized powder of this size range for comparison, and shows the presence of Al, a ternary metastable phase and a small amount of Al11Nd3. shows a SEM micrograph of a longitudinal section of this material. Two types of microstructures are observed: bright small precipitates, probably of the metastable ternary phase, in a dark αAl matrix; and small, featureless, undeformed Al90Fe5Nd5 particles which did not suffer phase transformation during processing. These small hard particles come from the high agglomeration capacity of Al, as referred to in shows the maximum EP of this first extrusion trial. This value is below the press limit, indicating that a decrease in particle size range or ET would be possible. However, this bar did not present good consolidation, and thus, the next extrusion trial was performed employing a higher ER, and, to avoid the risk of excess EP, a higher ET.In the second extrusion trial, the ER was increased to 7:1 and the ET to 370 °C, i.e. just above the second crystallization peak. The XRD pattern of this extruded bar, , is similar to that of the previous extrusion. The lack of evidence of any difference between these extrusion conditions can be due to a masking of any potential changes by bar texture, or to negligible microstructural change occurring in the second crystallization event, as pointed out by some authors shows a SEM micrograph of this bar. Although some pores could still be found between small particles, these deformed more than those in the first extrusion (compare , was higher than that of the first trial, but still below the press limit.The smaller powder size range of 25–50 μm was selected for the third extrusion trial, while keeping the ET at 370 °C and the ER at 7:1. The XRD pattern of this bar is shown in , where, in addition to peaks corresponding to Al, to a metastable phase and to Al11Nd, a small signal of the ternary Al10Fe2Nd stable phase is present. Comparison with the as-atomized powder pattern of the same size range, also shown in , indicates that the extrusion thermomechanical process promoted precipitation of the ternary phase. shows a longitudinal micrograph of this bar. No significant difference with the one extruded from the 50 to 100 μm size range powder at the same ET has been observed by SEM. On the other hand, contrary to what occurred in the previous extrusion trials, a very high EP has been achieved in this case, , which precludes diminishing particle size range without changing extrusion parameters accordingly.For subsequent extrusion trials, ET was increased to 450 °C, i.e. below the third crystallization event, while keeping ER at 7:1. The first extrusion at this high temperature was performed with powder in the size range of 50–100 μm. The XRD pattern of this material, , presents only peaks from the equilibrium Al, Al11Nd3 and Al10Fe2Nd phases, indicating that any amorphous or metastable phase disappeared during extrusion. shows the good quality of this extruded bar in terms of absence of visible pores, with all particles deformed, including the smallest ones. Its microstructure is more diffuse than that of the bars extruded at lower temperature, which is due to the phase transformations that took place during processing at 450 °C. As the required EP was low, , the particle size range was decreased for the following extrusion. presents the XRD pattern of the 25–50 μm bar extruded at 450 °C. In this case, peaks from all the equilibrium phases and a small signal from the metastable phase can be seen. Comparison with the as-atomized powder, , shows that the effect on this size range of such a high ET was to promote precipitation of all the stable intermetallic phases, but not enough so as to decompose the metastable phase. The microstructure of this bar, , is finer than that of the former 50–100 μm bar, and presents few isolated pores. EP for this material was also very low, , affording the possibility of extruding powder in the smallest size range.Powder in the <25 μm size range was finally extruded at 450 °C. The XRD pattern of this bar, , is similar to that of the 25–50 μm bar, , with peaks from all the equilibrium phases and a signal of the metastable phase, which again indicates that some amount of metastability remained after consolidation at 450 °C. The microstructure of this smallest size range bar, , is much finer than that of the other bars, reflecting that the structural refinement due to the difference in original powder size remained effective after extrusion. Very few pores were detected as dark regions, which are difficult to distinguish from the majority of the dark areas, corresponding to Al. Maximum EP for this material was only 630 MPa, summarizes the phases that are present in each material as a function of extrusion condition and powder size fraction, together with those of the as-atomized powders Mechanical characterization of the consolidated Al90Fe5Nd5 alloy was carried out through Vickers hardness and compression tests, the latter only on the bars extruded at 450 °C. Results are collected in , together with hardness values of the as-atomized powder Results of yield strength (s0.2) of specimens extruded at 450 °C are also collected in . s0.2 increases with decreasing powder size fraction, which may be attributed to a finer microstructure and a higher metastability. shows s0.2 values as a function of hardness. A direct relationship between these two properties can be observed. Moreover, extrapolating the straight line in in order to estimate s0.2 corresponding to 250 HV, which is the hardness of the 25–50 μm bar extruded at 370 °C, a s0.2 as high as 900 MPa is found. Unfortunately, the high brittleness and hardness of this material did not allow the machining of samples for compression tests. The higher HV, and estimated s0.2, of the 25–50 μm bar extruded at 370 °C, compared with the <25 μm bar extruded at 450 °C indicates that a larger powder size fraction extruded at lower ET could be more interesting than a smaller size fraction extruded at higher ET. However, not only was the 25–50 μm material very brittle, but it also required a very high EP, , rendering it unsuitable from a technological point of view. An alternative to obtain better properties at viable EP would be to extrude the smallest size fraction powder, <25 μm, at 450 °C, while applying a higher ER, which would refine its microstructure.The yield strength of the Al90Fe5Nd5 extruded alloys can be compared, for example, with the 845 MPa of an AlNiMm extruded alloy The fracture surfaces of the compression specimens are mixed, with zones where cracks propagated through the powder particles and others where they propagated along the powder boundaries. This fracture mode gives rise to a step-like morphology, (a), typical of powder metallurgical materials. At higher magnification, areas without plastic microdeformation, (b), and others with dimpled plastic microdeformation, (c), can be observed. The size of the dimples grows as the powder particle size fraction increases.Nanocomposite Al90Fe5Nd5 atomized alloy powder was successfully extruded. The microstructural scale of the extruded bars decreased as powder size decreased, indicating that the structural refinement resulting from the difference in powder size was still effective, even though the original microstructure changed during extrusion. Increasing the ET or powder size fraction results in a loss of metastability and a softening of the material. s0.2 increases as the powder size fraction decreases, with a maximum of 726 MPa obtained for the <25 μm powder. Higher s0.2, without a critical increase in the EP, can be expected if the <25 μm powder is extruded at 450 °C but at a higher ER than 7:1. A maximum HV of 250 has been obtained with the 25–50 μm powder extruded at 370 °C, which combines a fine microstructure with a significant amount of metastable phases. A linear relationship between HV and s0.2 has been found, which predicts a s0.2 as high as 900 MPa for this bar. Fracture surfaces show a powder metallurgical morphology, with the size of dimples growing as the powder size fraction increases.Instabilities in power law gradient hardening materialsTension and compression instabilities are investigated for specimens with dimensions in the micron range. A finite strain generalization of a higher order strain gradient plasticity theory is implemented in a finite element scheme capable of modeling power law hardening materials. Effects of gradient hardening are found to delay the onset of localization under plane strain tension, and significantly reduce strain gradients in the localized zone. For plane strain compression gradient hardening is found to increase the load-carrying capacity significantly.For thin metal plates or free standing thin films that play an important role in applications ranging from microelectro-mechanical systems to coatings the mechanical properties are not well described by conventional plasticity theory. The material models must account for observed size-effects (), and a variety of such material models have been developed by incorporating gradient effects in the constitutive and equilibrium equations.Instabilities under tensile or compressive loading are important limitations of the load-carrying capacity for thin plates or films. For the plane strain tension test (), or the plane strain compression test (), the critical stresses for bifurcation into diffuse modes have been determined, dependent on the ratio of the wave-length to the plate thickness, and the regimes have been identified, where the governing equations are elliptic, parabolic or hyperbolic. Size-effects have been incorporated into these bifurcation analyses by , who used a finite strain version of the gradient plasticity model proposed by . In addition to the two material moduli μ and μ∗ appearing in the previous bifurcation analyses, this gradient theory of plasticity also incorporates a characteristic material length scale l. It has been found that for very small values of the material length the value of the lowest bifurcation stress is practically unaffected by nonlocal effects, but when the wave-length decays towards the material length, the bifurcation is more and more delayed by the nonlocal effects.The nonlocal plasticity model used in the present paper is a finite strain generalization recently developed by for the strain gradient plasticity theory by . For plane strain specimens in tension or compression the focus here is on determining how much the onset of instability is delayed, or the load-carrying capacity increased, when the specimen size is as small as the characteristic material length, or even smaller. In these analyses the presence of initial geometrical imperfections is accounted for. The bifurcation analyses mentioned above have shown that both for tension and compression instabilities occur in symmetric modes as well as anti-symmetric modes. However, for longer wave-lengths symmetric modes give the lowest bifurcation stress in tension, leading to necking, and anti-symmetric modes give the first bifurcation in compression, leading to column buckling. Therefore, these will be the only cases considered in the present studies.Necking in plane strain tension was also analyzed by for a linear hardening solid. Their numerical implementation of the gradient plasticity theory did not give good convergence for power law hardening materials. This is resolved here by using higher order elements in the numerical implementation, and the results shown are for power law hardening.The material behavior is modeled by a finite strain generalization proposed by for the strain gradient plasticity theory by . An updated Lagrangian formulation is used to model the strain gradient effects at finite strains based on the work of In the strain gradient plasticity theory by gradient hardening is introduced through the gradient of the plastic strain ratePlastic work in the material is due to an effective plastic strain, EP, defined in terms of the conventional definition of effective plastic strain, ϵ˙P2=23ϵ˙ijPϵ˙ijP, and three invariants of homogeneous degree two of the gradient of the plastic strain rate, ρijk. Denoting these invariants by I1, I2 and I3, the effective plastic strain is defined by the incremental relationwhere l1, l2 and l3 are material length parameters introduced for dimensional consistency, and the numerical coefficients of the invariants are chosen such that the material length parameters in the theory have similar meaning as the length parameters in the strain gradient theory of The plastic strain increment is defined according to the usual relation for J2 flow theorywhere Sij is the deviator of the Cauchy stress σij, and σ(e)=32SijSij is von Mises’ effective stress.Expressing the plastic strain rate as the product of its magnitude, ϵ˙P=23ϵ˙ijPϵ˙ijP, and its direction, mij=32Sij/σ(e), gives the following expression for ρijkwhich shows how ρijk depends on the conventional effective plastic strain and its gradient, as well as on the direction of the plastic strain increment and its gradient, mij,k. Introducing this relation in the expression for the effective plastic strain (Eq. E˙P2=ϵ˙P2+Aijϵ˙,iPϵ˙,jP+Biϵ˙,iPϵ˙P+Cϵ˙P2where the tensors Aij, Bi and C depend on the three material length parameters l1, l2, and l3 as well as on the spatial gradients of the plastic strain increment direction (for details see Within the framework of the present theory, a single parameter theory closely related to the strain gradient theory of can be formulated. This is done by defining the measure of the effective plastic strain aswhere l∗ is a new material length parameter (). This single parameter version is not a special case of the general theory based on Eq. , even though it does fit within the same theoretical framework.The principle of virtual work for the strain gradient plasticity theory used is expressed in the current configuration by∫V(σijδϵ˙ij+(Q-σ(e))δϵ˙P+τiδϵ˙,iP)dV=∫S(Tiδu˙i+tδϵ˙P)dSwhere ϵij is the total strain, Q is the work conjugate to the plastic strain, and τi is the higher order stress which is work conjugate to the gradient of ϵP. The volume and the surface of the solid are denoted V and S, respectively, and the outward unit normal is denoted Ni. Due to the higher order nature of Eq. boundary conditions have to be applied for the higher order traction t |
= |
τiNi or for the plastic strain ϵP, in addition to the conditions on the tractions Ti |
= |
σijNj or the displacements ui. The higher order boundary conditions must be applied to the exterior of the plastically deforming parts of the solid, which in general includes internal elastic–plastic boundaries.With J denoting the determinant of the metric tensor in the deformed geometry the Kirchhoff type stresses are defined byUsing the current state as reference we have J |
= 1 (updated Lagrangian formulation), and the principle of virtual work (Eq. ) can be expressed in incremental form as follows:∫V(ς∇ijδϵ˙ij-σij(2ϵ˙ikδϵ˙kj-e˙kjδe˙ki)+(q˙-σ˙(e)ς)δϵ˙P+ρ∨iδϵ˙0,iP)dV=∫S(T˙0iδu˙i+t˙0δϵ˙P)dSHere, the spatial derivative of the effective plastic strain ϵ˙0,iP is evaluated in the reference coordinate system, and T˙0i and t˙0 are the nominal traction rates. This relation is identical to the expressions derived in when excluding higher order terms. The symbol ()∇ denotes the Jaumann rate and ()∨ denotes the convected rate.The finite strain generalization of the constitutive equations for the stress-measures work-conjugate to the total strain, the plastic strain, and the plastic strain gradient, respectively, areς∇ij=Rijkl(ϵ˙kl-ϵ˙Pmkl)=ς˙ij-ω˙ikσkj-σikω˙jkwhere e˙ij=u˙i,j, ω˙ij is the anti-symmetric part of e˙ij andwhere, ET is the tangent modulus and E is Young’s modulus. It is noted that the hardening modulus is evaluated at EP rather than at ϵP as it would be in a conventional theory.For the power-law hardening material we use the following expression for the tangent moduluswhich deviates very little from that corresponding to a standard power-law. In this expression ϵ0 |
= |
σy/E is the uni-axial yield strain, with σy denoting the initial yield stress.The constitutive equation for the generalized effective stress in since σ˙(e)ς=mijς∇ij. Further details on the present strain gradient plasticity theory can be found in The numerical solutions are obtained using a finite element method where nodal effective plastic strain increments, ϵ˙nP, appear directly as unknowns on equal footing with the nodal displacement increments, D˙n. The structure of the finite element method as it is used in the present study has been used by have used it to model the small strain theory by have used it to model finite strain effects in plane strain sheet necking for linearly hardening materials.The displacement increments, u˙i, and the effective plastic strain increments, ϵ˙P, are interpolated within each element between the nodal displacement increments, D˙n, and the nodal effective plastic strain increments, ϵ˙nP, respectivelyHere, Nin and Mn are shape functions, and k and l are the number of nodes used for the displacement interpolation and the effective plastic strain interpolation, respectively.Triangular elements with k |
= |
l |
= 3 were used by . Here, it has been found that much better convergence is obtained for power law hardening materials, when eight node serendipity elements are used for interpolating both the displacements and the effective plastic strain, so that k |
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