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= 0.6944 ± 0.00086 nm (hydrated).The mean elastic strains determined in the axial and transverse direction (as detailed in where σ1 is the axial principal stress, σ2 and σ3 are the transverse principal stresses, ε1 and ε2 (=ε3) are the measured mean elastic strains, E is the elastic modulus and ν is the Poisson’s ratio of the apatite phase. An assumed Young’s modulus of 114 GPa for hydroxyapatite and a Poisson’s ratio of 0.27 were used in the calculations These principal stresses were then used to estimate the average von Mises effective stress (σeff) for the apatite phase using This effective stress (σeff) can be used to provide a measure of the degree to which the applied stress (σa) is partitioned to the apatite phase relative to the collagen phase. In a single-phase material where there is no load transfer and under uniaxial loading, the effective stress is equal to the applied stress (σeff |
= |
σa). In composite materials, the effective stress will be lower than the applied stress in the elastic regime for a given phase if it bears less of the applied stress. On the other hand, if the phase is stiff compared to the composite, it will exhibit an effective stress that is greater than the applied stress The synchrotron WAXS data enabled the average strain changes of the population of nanometre-sized apatite crystals of the trabeculae within the measurement volume to be determined as a function of applied strain. The observed changes were related to the different regimes of the macroscopic stress–strain curve and their associated deformation mechanisms. The applied loading strains will be referred to as macroscopic strains and internal strains in the apatite crystals determined by diffraction will be referred to as lattice strains. All of the applied stress values are reported with negative values to indicate compression. Based on architectural parameters calculated for bovine trabecular bone The assumption we have made is that the c-axis of the apatite crystals lie along the length of the trabeculae The macroscopic stress–strain curve for dehydrated bovine trabecular bone obtained during the diffraction experiments is shown in In dehydrated bone, lattice strains develop along the c-axis linearly in compression with applied stress whereas in the a-axis, the strains develop in tension (This may be as a result of non-parallel-aligned trabeculae (i.e. those which are perpendicular to the femur and load axis, associated with the a-axis) bending Prior to the macroscopic yield point, at an applied stress of approximately −10 MPa, the rate of strain accumulation increases in the c-axis (the decrease in slope can be seen in , Regime II). Therefore, there is a change in the load partitioning behaviour, and the trabeculae aligned parallel to the loading direction are bearing more load. This is confirmed with the increase in slope in the a-axis. The a-axis bears less load during this period (from −10 MPa to −14 MPa) as there is no significant increase in the a-axis strains with applied stress. The macroscopic yield ( it is evident that during the post-yield region (prior to the plateau region) c-axis lattice strains are relieved back to their initial state prior to deformation. In the a-axis, the strains also relax back to their initial value prior to deformation. At first sight this may appear anomalous because the onset of macroscopic plasticity in materials does not usually result in a complete relaxation of lattice strain, but rather some strain redistribution would be expected. However, this observation can be related to changes in the trabecular architecture under compression. It appears that the observed lattice strain “relaxation” is a result of brittle fracture of individual trabecular struts following bending and buckling. The sample is still bearing load (around −12 MPa) when the elastic strains are completely relaxed and return to zero. However, the X-ray beam traverses through around 2% of the total bone width to which load is being applied and hence the relaxation of elastic strains may be a result of global load redistribution following localized failure of individual trabeculae. It may also be as a consequence of the collagen phase in the trabecular structure taking up more of the load. During the plateau region, the applied stress remains constant at around −12 MPa.Due to the significant plateau of the macroscopic stress–strain curve (where applied stress remains constant), the axial lattice strain evolution is also plotted as a function of macroscopic strain (). During the initial portion of the plateau, at an applied macroscopic loading strain of 0.10 the c-axis lattice strains become tensile rather than compressive. This may, in part, be related to the bending of axially aligned trabeculae as they yield. Compressive lattice strains then begin to accumulate in the c-axis once more along the plateau region of the stress–strain curve. These compressive lattice strains are attributed to buckling of the trabeculae. The strains develop steadily under a constant applied stress, reaching values close to those observed during the linear elastic regime (around −2000 με). For clarity, changes in the lattice strains at macroscopic strains greater than 0.15 are not shown in . Along the a-axis, compressive strains accumulate under constant stress. The maximum amount of strain reached is half of that measured along the c-axis (−1000 με as compared to −2000 με along the c-axis). shows that from 0.15 to 0.16 macroscopic loading strain there is a slight decrease in the applied load, followed by a steady rise. During the region of load decrease, lattice strains relax to −700 με (i.e. Regime V in ). As load on the sample begins to rise, compressive lattice strains begin to accumulate once more. The local strain distribution in trabecular bone is inhomogeneous, meaning that a comparison with global apparent stress/strain behaviour is no longer straightforward ) can be divided into five main regions:The lattice strains develop linearly in the apatite crystals. Compressive strains develop in the c-axis whereas they develop in tension in the a-axis. The strains are of the same magnitude but of opposing directions for the two orientations.The strains accumulate at a faster rate in the c-axis up to the macroscopic yield point. There is no significant change in the a-axis with macroscopic strain.With the onset of buckling and localized damage, the lattice strains are ”relaxed” in the c-axis. Overall, there is some relaxation of the tensile strains in the a-axis. These changes occur post-yield and up to the initial plateau region of the macroscopic curve. The apatite crystals in the c-axis experience some tensile strain following relaxation of the compressive strains.Further along the plateau, there is a steady rise in the lattice strains in compression, in both the c- and a-axes. The loading rates are similar in both cases, but the magnitude of strains is approximately twice as high in the c-axis.The occurrence of localized fracture of trabeculae appears to be responsible for a drop in applied load. A drop in load leads to some relaxation of the lattice strains in the c-axis. As densification begins, load begins to increase, and compressive lattice strains begin to accumulate once more. The strain rate is identical to that in Regime IV. The strains along the a-axis are also relieved with the drop in load but then continue to increase in tension with the onset of densification.The macroscopic yield stress occurs at an applied stress of −6 MPa for the hydrated bone sample (In the loading direction, the c-axis lattice strains develop linearly in compression up to an applied stress of approximately −3.5 MPa (Regime I, ). In the a-axis, slight tensile lattice strains develop up to the same point (as observed for the dehydrated sample). Following this (Regime II), the c-axis slope becomes slightly steeper (strain accumulation rate decreases). Conversely, there is a decrease in the a-axis slope, as compressive lattice strains begin to develop in this orientation. Therefore, prior to the macroscopic yield, load is partitioned from trabeculae that are aligned along the loading direction to those that are non-parallel aligned. Just prior to the yield point, compressive lattice strains begin to develop in the a-axis much more rapidly. For an increase in applied stress of -0.5 MPa, lattice strains increase from −40 με (at an applied stress of −5 MPa) to −235 με (at an applied stress of −5.5 MPa). For the same period, strains are relieved along the c-axis. In a similar manner to the dehydrated sample, when the macroscopic yield occurs, lattice strains relax in both the c- and a-axes. The apatite crystals remain under some strain unlike the dehydrated sample where the strains completely relaxed. This may be as a result of less localized damage occurring to the trabeculae when hydrated. Although hydrated bone is less stiff The axial lattice strain evolution as a function of macroscopic strain for the hydrated sample is shown in The behaviour can be summarized as follows:Elastic strains develop linearly in the apatite crystals. The strains are compressive in the c-axis but are tensile in the a-axis. The strains are of a much lower magnitude in the a-axis.The strains accumulate at a slightly higher rate in the c-axis up to the macroscopic yield. In the a-axis, the strains become compressive.This region corresponds with the macroscopic yield. The c-axis strains are reduced by approximately 50%. Strains in the a-axis also develop in compression.This region corresponds with the plateau of the macroscopic stress–strain curve. After initial ”relaxation” of the strains (likely due to local collapse of trabeculae), strains in both the a- and c-axes develop in compression.Strains relax in the a-axis and develop in compression (due to buckling) along the c-axis.A comparison of the lattice strains in the dehydrated and hydrated sample shows that the magnitude of the maximum in c-axis strains are approximately similar (2100 με). An interesting observation is that load is partitioned to the a-axis close to the macroscopic yield in the hydrated sample, whereas a greater proportion of the load is partitioned to the c-axis in the dehydrated sample. A model is presented below to explain this observation. In addition, the tensile lattice strains that develop in the a-axis of the elastic regime are significantly less in the hydrated sample.The effective stress estimated for the apatite phase during the elastic regime is shown for both the dehydrated and hydrated bovine bone as a function of the applied stress in The slope of the plot for the hydrated sample is approximately three times greater than that for the dehydrated sample. This implies that in hydrated trabecular bone the apatite crystals bear significantly more stress than the collagen and thus act as a more effective reinforcing phase. Thus, the collagen phase is considerably unloaded relative to the apatite phase when hydrated. This is discussed in further detail later.The load partitioning methodology presented in this study was previously applied to antler trabecular bone As observed for antler bone, the apatite c-axis plays a dominant role in the load-bearing of bovine trabecular bone. Hence, during uniaxial compression axially aligned trabeculae bear more of the applied load, accounting for the higher strains measured in these c-axis crystallites. These experimental strain measurements are in good agreement with finite-element predictions presented by Van Rietbergen et al. It was noted that in the elastic regime, prior to the macroscopic yield point, the c-axis strain accumulation rate increased in the dehydrated sample whereas it decreased in the hydrated sample. Hence, based on the assumption stated above that the c-axis strains derive from trabeculae that are aligned along the loading direction and these are more important during elastic loading, it seems that in dehydrated bone load partitioning to non-parallel aligned struts is less efficient. This can be understood by considering a study by Yamada et al. The deformation mechanism of dry and wet trabecular bone will be discussed and related to load partitioning. First, consider deformation of individual trabeculae. For dehydrated bone, brittle fracture will be the dominant deformation mechanism, whereas for the hydrated bone, the trabeculae will primarily fail by plastic yielding ). Although microdamage will have occurred, the trabeculae must continue to bear the load. In contrast, with the hydrated sample, it seems that, when the parallel-aligned trabeculae begin to plastically yield, load is partitioned to non-parallel-aligned trabeculae. This explains the change in the slope of the a-axis (Regime II in ). Such a mechanism where failing struts transfer load to neighbouring struts in open-cell foams has also been hypothesized by Zhou and Soboyejo Now consider how hydration affects the nanoscale deformation of trabecular bone. For cortical bone, Braidotti et al. The tensile strains in the a-axis (during the elastic regime) for the axial direction are much lower in the hydrated sample. This is attributed to hydrated trabeculae deforming more easily when bending as a result of the collagen fibrils and the extrafibrillar matrix being more compliant. The apatite crystals then accumulate less strain during the deformation process. In the plateau region of the stress–strain curve, compressive lattice strains develop in both axes. This region of the curve is associated with buckling. Load appears to be partitioned equally between the two strut orientations during this region.An interesting comparison can be made between the lattice strain accumulation behaviour in the dehydrated antler ) the c-axis strains decrease quite significantly. Secondly, significant strains develop in the a-axis of the antler bone, whereas in the a-axis of the bovine bone the strains are not as substantial. Saturation of the c-axis strains in the antler trabecular bone is related to the toughness of this bone type. It is suggested that the antler trabeculae do not undergo as extensive microdamage as the bovine trabeculae owing to the lower mineral content that was discussed previously. Hence, the parallel-aligned trabeculae are still able to bear load. The substantial strains in the a-axis of antler bone that are not observed in the hydrated bovine bone are related to the significant buckling of parallel-aligned trabeculae in the former bone type. Considerable buckling means that strains which accumulate in the a-axis are both as a result of buckling of parallel-aligned trabeculae away from the loading direction as well as due to load being borne by non-parallel-aligned trabeculae. Bearing these observations in mind, we hypothesize that the antler trabecular bone fits the elastomeric open-cell model whereas bovine trabecular bone structures fits the plastic model The bulk lattice strain accumulation discussed thus far has been in terms of load partitioning to differently aligned trabecular struts. The estimate of von Mises effective stress () allows another important question to be addressed, i.e. the amount of load borne by the apatite phase if bone is considered a discontinuously reinforced composite of a rigid phase (apatite crystals) and a matrix (collagen) which serves to transfer loads between adjacent reinforcing particles We stated earlier that one of the main assumptions made in this study is that the c-axes of the apatite crystals are oriented along the length of the trabeculae. It is known that the apatite crystals grow with a specific crystalline orientation, with the c-axes being approximately parallel to the long axes of the collagen fibrils The apatite strain data presented can also be discussed in another way that ignores the relationship between the crystallite orientation and the X-ray diffraction direction ). This method assumes that the apatite crystallites are randomly arranged and therefore not aligned in any particular direction, and accounts for all alignments of crystals along the beam path. As i demonstrates in the dehydrated bone sample, tensile strains accumulate in both the axial and transverse directions up to an applied stress of −8 MPa, following which compressive strains develop in the axial direction and quite significant tensile strains accumulate in the transverse direction (close to 3000 με). Following the macroscopic yield, only tensile strains remain in both directions, with the axial strains close to zero. This suggests that, due to extensive damage (buckling of trabeculae), the trabeculae in the axial direction can no longer accommodate elastic strains. The high strains in the transverse direction may be as a result of considerable rotation of trabeculae away from the loading direction at the highest macroscopic loading strains.ii), apatite strains develop linearly in the axial direction up to an applied stress of −5 MPa (close to the macroscopic yield point) and reach a much higher magnitude than observed in the dehydrated sample (−880 με (hydrated) as compared to −140 με (dehydrated)). In the plateau region, some compressive strains remain in the axial and transverse directions (−200 με and 200 με, respectively). In the hydrated sample, the maximum tensile and compressive apatite strains are approximately the same, whereas in the dehydrated bone significantly higher transverse strains are observed. This can be related to microcracking and brittle fracture in the axial direction, meaning that the load is borne primarily by the transverse aligned apatite crystals. In contrast, the hydrated bone can transfer load more effectively, as described earlier, which means that apatite crystal strains which are lower in the hydrated bone are in good agreement with the von Mises analysis (During uniaxial compression of bovine trabecular bone, load is borne primarily by trabeculae aligned parallel to the loading direction, where compressive lattice strains accumulate. In dehydrated bone, we suggest that, close to the macroscopic yield, load is not effectively partitioned to trabeculae which are not aligned parallel to the loading direction as damage accumulates in the structure. Thus, this type of bone behaves like a brittle open-cell foam. In contrast, in hydrated bone, as the macroscopic yield is approached, load is transferred to non-parallel trabeculae. Thus, hydrated bone exhibits behaviour observed in plastically yielding foams.Certain figures in this article, particularly Figures 1, 3, 4, 6, 7, 8 and 9, are difficult to interpret in black and white. The full colour images can be found in the on-line version, at Flexural and shear hysteretic behaviour of reinforced concrete columns with variable axial loadThe importance of non-linear biaxial models that are applicable to the analysis of reinforced concrete members under cyclic and dynamic loads has been recognized. Variations in the axially applied force can influence strength, stiffness and deformation capacity of such members. In this study, an inelastic biaxial model based on plasticity theory, is proposed. This quadri-linear degrading model takes into account the effect of axial load variation on lateral deformation. The model predictions are examined against available experimental results. Using the developed model, the effect of various axial loading patterns on the lateral deformation of reinforced concrete columns is investigated.Reinforced concrete (RC) members with non-ductile reinforcement detailing experienced both flexural and shear failures during recent earthquakes. Under strong earthquake ground motion, structures are subjected to lateral loads, which impose biaxial flexural and shear forces on the columns. In the analysis of columns, accounting for the effect of biaxial loading on yielding, moment resisting capacity, inelastic deformation, and degradation of strength and stiffness of the member, is important. Considering the effect of these parameters is necessary to achieve realistic predictions of the seismic response of frame structures.There is a wealth of earthquake records that exhibit a vertical component with peak ground acceleration well in excess of the corresponding horizontal value Modeling RC elements subjected to biaxial flexural and shear loads with a variable axial load has received relatively little attention. Some analytical models for the non-linear analysis of RC frame structures have been proposed The objective of this study is to develop a lumped plasticity non-linear analytical model for RC members. The model developed in this study is based on the basic biaxial flexural model Predictions of the developed model are compared to test data to validate the reliability of the element. Having the model verified, the capability of the element to model the influence of axial force variations on the hysteretic behaviour of RC columns is demonstrated by investigating the effect of various axial load patterns on the lateral inelastic response of a slender RC column.The model development includes the formulation of the element and the determination of the yield surfaces and stiffness degradation behaviour of the flexure and shear subhinges. The effects of the axial load on the flexure and shear subhinges are accounted for. The final part of this section discusses how the model parameters are determined.The three-dimensional beam–column element is formulated to model inelastic hysteretic behaviour of RC beams and columns. The element is capable of representing the biaxial moment and shear–axial force interaction, stiffness and strength degradation, and variation of axial load. The element consists of an elastic element with lumped plastic hinge at each end. In the three-dimensional space, the element may be arbitrarily oriented in a global XYZ coordinate system as shown in For the elastic element, initial elastic flexural, torsional, axial, and shear stiffnesses need to be specified. The plastic hinge at each end of the element consists of three flexure plastic subhinges and one shear subhinge, as shown in . Each of the three flexural plastic subhinges is to represent a stage of the non-linear behaviour of RC member (i.e. concrete cracking, steel yielding or ultimate conditions), while the shear subhinge represents a shear failure. The force and moment matrix at each end consists of two parts; flexural forces and moments The interaction between the forces and moments at each subhinge is represented by a yield surface. The yield surfaces are YS1, YS2, and YS3 for the three flexure subhinges, and YSS for the shear subhinge. A quadratic function to describe the yield surface for each subhinge was used. For clarity, a 2D rather than 4D relationship is shown in for each flexure subhinge i. The quadratic yield surface function can be written in the form: is the current position of the yield surface for Fx, Mx, My and Mz at hinge i. For the shear subhinge, the failure surface is shown in The quadratic failure surface for the shear subhinge can be written as:where Fult is the element’s axial compressive capacity, Tult is the tensile capacity, and is the average axial force. The lateral shear capacities in y and z directions at an average axial force are Vyf and Vzf, respectively.All the plastic subhinges are assumed to be initially rigid, thus the initial stiffness will be the stiffness of the elastic element. Under the action of flexural and shear forces, the subhinges experience some flexibility, therefore a reduction in the element stiffness occurs. The flexibility of the flexure subhinges upon load reversal is divided into elastic, (non-recoverable) flexibilities. During shear failure, an isotropic contraction of the shear yield surface towards a residual shear surface that follows the shear subhinge flexibility , is assumed. The residual shear surface is represented by the shear capacity of transverse reinforcement.The flexibility matrix for the entire element, , can be obtained by the appropriate addition of the elastic element, , and the end hinges’ tangent flexibility matrices can be inverted to obtain the element’s total tangent stiffness matrix, at each end are calculated by appropriate summation of the flexibility of their constituting subhinges, such that:The flexure subhinge flexibility is divided into two parts; elastic and plastic flexibilities. Both flexibilities are initially zero. A certain event such as crack, yield or ultimate condition triggers a yield surface. Continuous loading follows kinematic strain hardening and the yield surface translates in the force space till it reaches the next yield surface , of such subhinge. Upon load reversal, a finite elastic flexibility, , is assigned to a triggered flexure subhinge (in addition to the plastic flexibility). Including elastic and plastic flexure subhinge flexibilities represents elastic stiffness degradation with reduced overall element stiffness (and consequently strength) when the element is subjected to reversed loading at the same displacement level. of a yielded flexure subhinge was derived by Chen and Powell where ṉ is the outward normal vector to the yield surface at the action point; and is the diagonal plastic stiffness matrix from the individual flexural action–deformation relationships for each force component, defined as:in which the plastic stiffness after yield for each force component is given by:in which the stiffnesses Ki are the initial input data representing the elastic beam stiffness.In a typical force–deformation relationship obtained from tests on concrete elements, it is observed that the level of degradation in stiffness increases as the ductility of the element increase or in case of repeated cycles at the same level of ductility. An example of such force–deformation relationship is that obtained from a tested cantilever column shown in Thus the elastic subhinge flexibility after unloading for each force component is shown in where Ks is the secant stiffness of the previous cycle and αm is an arbitrary degradation coefficient that ranges from 0 to 1 where 0 indicates no stiffness degradation. A practical range for αm was found to be from 0.03 to 0.1. The strength degradation depends on the plastic deformation of each subhinge in proportion to the total plastic deformation of all subhinges during the previous cycle. It is possible to assign different degradation coefficients αmy and αmz in each loading direction. The model can be extended to have a different coefficient αmi for each subhinge; this means that the degradation level for each subhinge can be different. This feature was not used in the current applications in order not to increase the number of parameters that need to be defined. The element in its quadri-linear degrading formulation implicitly includes the Bauschinger effect.Based on the tests and the theoretical verifications of Vecchio and Collins . The yield surface is defined by the element axial compressive Fult, and tensile Tult capacities, as well as the lateral shear capacities Vyf and Vzf at an average axial force Fav in y and z directions, respectively.When the shear cracks develop in a reinforced concrete member, inelastic shear deformation commences. The shear transferred by aggregate interlock and dowel action across the shear cracks will be reduced as the crack widens, therefore the element’s shear strength decreases. If large deformation is imposed, the shear will be resisted primarily by transverse steel reinforcement. In addition, slippage of the bars in tension results in significant pinching of the hysteretic loops. Based on the discussed physical behaviour, the shear force–deformation relationship (V–Δ) for the element in each direction as well as the stiffness degradation are assumed as shown in is initially nil till the shear yield function is equal to 1. The shear subhinge starts to contract and finite shear subhinge flexibility develops. An uncoupled shear force flexibility matrix, similar to that proposed by Ricles et al. where Kvy, Kvz represent plastic stiffness for shear subhinge in the y and z axes, respectively. During shear failure, which corresponds to contraction of shear subhinge yield surface, the shear forces remain coupled with the flexural forces through equilibrium. A minimum shear subhinge surface, as shown in corresponds to the residual shear resistance of the reinforcing ties in the concrete member in each direction Vy or zr. The effect of variation in axial force during the contraction of the shear subhinge yield surface is discussed in Upon reversing the load, the stiffness of the shear subhinge reduces when the shear force reaches zero. This is to model the pinching behaviour in the hysteretic response. The shear subhinge flexibility matrix during pinching is defined as:where αp is an arbitrary coefficient (from 0→1) with a practical range from 0.01 to 0.02; and is the initial shear flexibility matrixin which GA′ is the effective shear rigidity, and L is the element’s length.The pinching flexibility matrix is nil when the element total deformation is zero.After a complete cycle, shear stiffness degradation is introduced. Reduced initial shear stiffness according to the following equation was used:where αs is an arbitrary coefficient (from 0→1) with a practical value of approximately 0.03 and Ks is the secant stiffness as shown in There are two ways to incorporate the effect of axial load variation on the state of the yield surfaces of a hinge. The first approach is to consider the variation in axial deformation (extension and compression), while the second approach is to consider the variation of axial force as the control input. It is more convenient to use the first approach when there are considerable differences in the force–extension stiffnesses of each subhinge, thus keeping track of the appropriate axial force corresponding to a certain level of axial deformation. shows the axial load–deformation relationship for tied and spiral columns. From the figure, it can be postulated that the force–extension relationship is almost linear with constant initial stiffness up to the first peak load level. On the basis of this assumption, the second approach that considers the variation of axial force as the control input was adopted.Since the “event to event” solving technique rather than the “iterative” technique is used on the element level, the factor “FACM” which causes a certain flexure event to occur such as reaching a yield surface and change in stiffness, is calculated assuming a linear interpolation along the axial load path, as shown in . A curved path can be achieved if required, by subdividing the axial force variation increment into several linear sub-increments.The post-shear failure force–deformation response follows a softening branch as the failure surface contracts gradually. The rate of softening is specified by the plastic stiffness for the shear subhinge along the y and z axes Kvy and Kvz. After contraction of the failure surface, it is required that the shear force state remains on the failure surface. Considering a current state at the beginning of a load step, the following must hold:where FACS is the event factor for shear subhinge failure, Vfac is the contraction factor for shear subhinge shear force capacity (), Vy, Vz are the shear forces along the y and z axes, dVy, dVz are the incremental shear force reductions along y and z axes, dF is the increment of variation in axial force, Vyf, Vzf are the shear capacities along y and z axes; and dVyf, dVzf are the increments of shear capacity deterioration along y and z axes. shows the shear subhinge failure surface contraction for the cases of constant and variable axial loads. In the figure, the element is assumed subjected to cyclic lateral displacement from points 0 to 9 (loading 0→2, unloading 2→6, reloading 6→9). The shear subhinge failure event “FACS” is dependent on the contraction factor Vfac. shows the determination of the factor Vfac in case of constant axial load, while shows the change in the force state and consequently Vfac, in case of variable axial force as shown in the same figure where the axial force is varying through the points 0 to 9 (increasing 0→2, decreasing 2→6, increasing 6→9).In addition to the influence of axial load variation on the force state and the yield surfaces of the subhinges, it affects the flexibility matrix for the active flexural subhinges and consequently affects the element’s total flexibility and stiffness matrices as well. shows that the plastic flexibility matrix of a yielded flexure subhinge is dependant on the outward normal vector to the yield surface, ṉ, which in turn depends on the current force state.The input data includes the properties of the yield surface for each subhinge, in terms of strength and stiffness. The data for the flexure subhinges can be determined from the moment–rotation analysis about each principal axis. This analysis should take into consideration factors that affect the lateral deformation behaviour of RC elements, such as bond slip of tensile reinforcement.The rotation θ and deflection Δ of a reinforced concrete member at any point along its length is due to the distributed curvature of the member along its length and the lumped rotation near the fixation point due to the slippage of steel bars at the tension side of the member as well as the member shear deformation. The total rotation and deflection at any point along the RC member is the algebraic sum of these three components as illustrated in where Δtot is the total lateral tip displacement corresponding to the total rotation θtot; Δf and θf are the lateral deflection and rotation due to flexure; Δs and θs are the lateral deflection and rotation due to reinforcement bond slip; Δv and γv are the lateral deflection and rotation due to shear.The rotation θf of a column is calculated by integrating the distributed curvature, φ, along the member, while the deflection Δf is calculated by summing the static moment of the area under the column theoretical curvature diagram taken about the point of contraflexure. The theoretical moment–curvature diagrams were obtained assuming trilinear and parabolic stress–strain relationships for steel and concrete, respectively. Confined and unconfined concrete were modeled using the modified Kent and Park model The lumped rotation at the fixation point, θs due to reinforcement bond slip is defined as:The corresponding deflection at any point at distance x from support will be equal to “θsx” and the maximum tip deflection will be:The value of δs is determined from the anchorage slip analysis of the embedded reinforcement bar under monotonic pull, utilizing the model developed by Alsiwat and Saatcioglu Shear deflections were calculated using the shear stiffness expression derived by Park and Paulay where 1/Kv,45 is the shear deflection in one unit length due to one unit shear load; Es is the elastic modulus of shear reinforcement; n is the modular ratio; and ρs is the ratio of transverse steel volume to volume of concrete core.The shear subhinge Vy or zf, and Vy or zr data are based on the nominal shear capacity equation proposed by Priestley et al. The determination of the post-peak unloading stiffness Kvy and Kvz, in case of shear failure, was explored by a number of researchers. Ricles et al. In the current analysis, a simple approach was adopted for the determination of the unloading post-peak shear stiffness. It is assumed that the column will undergo post-peakshear stiffness degradation, Kvy and Kvz, which is equivalent to the loss of strength from Vy or zf to Vy or zr through a lateral displacement equivalent to twice the yield displacement. This assumption is valid for a moderate ductility type of response. On the other hand, a limited ductility response will be accompanied by brittle shear failure with a higher unloading stiffness, which is equivalent to the loss of strength from Vy or zf to Vy or zr through a lateral displacement equivalent to the yield displacement. For y direction:The proposed 3D element was examined by comparing the analytical predictions to experimental measurements for the cases of uniaxial flexural behaviour of cantilever RC columns under constant and variable axial loads tested by Abrams Two tests on RC cantilever columns, C1 and C8, under compressive axial loads and cyclic lateral displacements reported by Abrams . The average concrete compressive strength from cylinder tests is 44.1 MPa, and steel yield strength is 423 MPa. The axial load was kept constant during testing specimen C1. For specimen C8, the axial load was varied following a proportional pattern with respect to the lateral displacement. This included increasing the axial load for increased lateral push and decreasing it for increased lateral pull up to twice the yield deflection, after which the axial load was held constant. shows the input data used to model the two tested specimens. The degradation parameters, αm, αs and αp are necessary to model the post-peak behaviour of reinforced concrete. They are selected within the ranges described after . Variation of the degradation parameters within the recommended narrow ranges has a reasonably small impact on the results. The test measurements of the moment rotation relationship are shown in . The figure also shows the analytical results for both test cases using the developed model.Good agreement between the analytical predictions and the experimental results is observed. For the case of constant axial load, the analytically predicted strength and stiffness were fairly close to the experimental ones. For the case of variable axial load, the analytical yield and ultimate strengths as well as the loading and unloading stiffnesses at both the push and pull sides correlated well with the test results. The model was capable of representing the degradation in strength in the pull side due to the variation of axial load.In the current analytical model, pinching is pertinent to shear failure. In the case of combined variable axial and flexural loads, the pinching zone in the moment–rotation relationship was simulated using an equivalent average stiffness. Although pinching was not explicitly modeled in the case of axial and flexural loads, the use of equivalent average stiffness resulted in predicted energy dissipation capacities of the analyzed and tested columns to be close.A squat square fixed–fixed RC column, ATC-B, tested by Ramirez and Jirsa shows the specimen’s dimensions and details of reinforcement as well as the deflection history in N–S and E–W directions and the sequence of variation of axial load. shows the input data used to model the tested specimen.The experimental lateral load–displacement hysteretic response in the N–S and E–W directions is shown in . The specimen experienced initial flexural cracking followed by inelastic flexural deformations in both N–S and E–W directions. Shear failure occurred when displacement first reached 15.2 mm in the N–S direction leading to subsequent reduction in shear strength with a highly pinched hysteretic behaviour, as seen in . A value of −22 kN/mm (−126 kips/in.) was calculated for coefficients kvy and kvz to account for shear strength degradation. However, these values did not have much effect on the analytical results since the degradation in stiffness upon load reversal combined with variation of axial load dominated the shear strength capacity. Therefore, at each new cycle, the reduced stiffness led to a reduced shear strength capacity.The predicted response based on the analysis of the test specimen in the N–S and E–W directions is shown in . The analytical and experimental responses are shown to be in good agreement. The results of this analysis demonstrate that the element formulation is capable of modeling the hysteretic response of a reinforced concrete column with deterioration in shear capacity in the plastic flexural hinge zone due to excessive flexural ductility demand.Having the analytical model verified against experimental results, the next step was to study the effect of different axial load patterns on the response of laterally loaded columns. Testing of columns under varying axial load patterns is difficult. Very few experimental results are available in the published literature In this study, a slender RC cantilever column with given moment–rotation properties, was subjected to eight different axial load paths as shown in . The axial load paths were selected to cover different possibilities of axial load variation with respect to the lateral deformation.Path 1 had a constant axial load at −0.5 Pb (Pb is the balanced compressive axial load). Path 2 had a variable axial load between 0 and −Pb (i.e. ±0.5 Pb from the initially applied load of −0.5Pb). In path 3, the axial load was varied from 0 to −2 Pb. In path 4, the axial load was varied between 0 and −0.5 Pb. In path 5, the axial load was varied between ±0.5 Pb (i.e. compression and tension). Paths 2–5 were ‘in phase’ loading cycles; such that the axial load was proportional to the lateral load with maximum axial load coinciding with maximum lateral deflection. Path 6 was an ‘out of phase’ loading case, with a phase shift of quarter cycle between the axial load and lateral deformation variations. In the ‘out of phase’ loading of case 6, the maximum axial load coincided with zero lateral displacement. Paths 7 and 8 represent an axial load that was varied at twice the number of cycles with which the lateral load was varied. In the case of path 7, the axial load was varied such that no axial load applies at maximum lateral displacement while the maximum axial load was applied when the lateral displacement was zero. In the case of path 8, the axial load variation was such that the maximum axial load occurred at maximum lateral displacement while no axial load was applied at zero lateral displacement.The main purpose of the analysis of the column shown in is to investigate the effect of the axial load path on the energy dissipation capacity of the column. For this reason, a slender column is selected where shear effects are minimized.The moment–rotation relationships for the eight imposed variable axial load paths are shown in . It is recognized that there ought to be pinching of the loops under moment loading. The effect of pinching on the energy dissipation capacity is reduced by using equivalent average stiffness. In the analyzed problem, shear and pinching effects were selected to be of minor impact on the behaviour under investigation. The cumulative dissipated energy–displacement ductility factor relationship for the eight different axial load paths are plotted in . From the two figures, the following behaviour is observed:Comparing the cases of axial load paths 2–4 with path 1, it is observed that the lateral moment capacity of the column subjected to a variable compressive axial load corresponds to the force–moment interaction relationship for the column section. In effect, the maximum moment capacity occurs when the axial load reaches the balanced compressive axial load.Comparing the column behaviour when subjected to an axial load varying according to path 2 (0 to −Pb) to path 5 (±0.5 Pb), it is observed that the axial load with reversing sign (i.e. compression and tension) causes approximately 25% decrease in the lateral moment capacity of the column that is subjected to axial load that remains compressive. The decrease in the lateral moment capacity in the case of an axial load with reversing sign is accompanied by an increase in the unloading stiffness which results in accumulated energy dissipation capacity approximately equal to the case of an axial load of the same amplitude but remaining compressive.Comparing cases of axial load path 6 with path 2, it is observed that ‘out of phase’ loading (path 6) causes slight decrease in the moment capacity and approximately 15% decrease in the energy dissipating capacity of the RC column as compared to the ‘in phase’ loading. The ‘in phase’ loading of path 2 refers to the case when the maximum axial load is applied at the maximum lateral push while the minimum axial load is applied at the maximum lateral pull. The ‘out of phase’ loading of path 6 represents the case when the maximum and minimum axial loads coincide with zero lateral displacement.Comparing the column behaviour with an axial load following paths 7 and 8 with that of an axial load following paths 2–6, it is noted that applying two axial loading cycles for every one lateral load cycle will decrease the energy dissipating capacity of the columns.From the results of cases of axial load paths 7 and 8, it is observed that varying the compressive axial load such that the maximum axial load is applied at maximum lateral displacement and zero axial load is applied at zero lateral displacement will decrease the lateral moment capacity, stiffness and energy dissipating capacity of RC column as compared to the reverse load pattern (i.e. zero axial load at maximum lateral displacement and maximum axial load at zero lateral displacement). This can be attributed to the fact that increasing the axial load while increasing the lateral deformation will decrease the lateral stiffness of the column.A global element was developed to model the biaxial flexural and shear behaviours of RC columns subjected to varying axial load. The model included stiffness degradation upon load reversals. The behaviour of the element was verified using experimental results and was shown to be fairly accurate in predicting the response of axially loaded columns with an applied variable axial load.The effect of eight different axial load variation patterns on the response of laterally loaded RC columns was studied. The patterns covered different load paths (‘in phase’ or ‘out of phase’) with different axial load frequency of application as compared to lateral load (one or two axial load cycles for every cycle of lateral load) at different load levels (compressive 0 to −2 Pb or alternating compressive and tensile at ±0.5 Pb).It was concluded that the magnitude of the axial load did have a considerable effect on the lateral moment capacity of RC columns in accordance with the moment–axial force interaction relationship. Thus, it is important to accurately identify the possible expected levels of axial loads arising from different cases of loading due to horizontal and vertical earthquake ground motion components, as they will affect the section design.A cyclic axial load that is reversing sign (compression and tension) causes lower lateral moment capacity and energy dissipation capacity of the RC column as compared to a cyclic axial load that remains compressive.It was found that varying the compressive axial load such that the maximum axial load is applied at maximum lateral displacement and zero axial load is applied at zero lateral displacement will decrease the lateral moment capacity, stiffness and energy dissipating capacity of the reinforced concrete column as compared to the reverse load pattern (i.e. zero axial load at maximum lateral displacement and maximum axial load at zero lateral displacement). This conclusion is relevant since it will help to focus the seismic analysis on the worst-case scenario of combined axial and lateral loading.It was also concluded that increasing the frequency of the axial load cycles with respect to one cycle of lateral displacement (for example, two axial load cycles for every lateral loading cycle) will considerably decrease lateral moment capacity, stiffness and energy dissipating capacity of the column. This conclusion is important since the frequency content of the vertical ground motion component is usually higher than that of the horizontal component.Multiresolution molecular mechanics: Surface effects in nanoscale materialsSurface effects have been observed to contribute significantly to the mechanical response of nanoscale structures. The newly proposed energy-based coarse-grained atomistic method Multiresolution Molecular Mechanics (MMM) (Yang, To (2015), Nanostructures with decreased size and dimensionality such as nanowires, nanotubes, nanofilms and nanoparticles may exhibit increasingly unusual properties, i.e., optical, mechanical, electrical and thermal, compared to those of the macroscopic ones. These size-dependent material properties have been experimentally viewed in Nanotechnologies such as nanoscale resonant sensors, micro- and nano-electro-mechanical systems (M/NEMS), and stretchable nano-electronics, can be improved by a better understanding of size-dependent mechanical properties of nanostructures The key factor in which nanostructures are different from their macroscopic counterpart lies in that the increasing surface-to-volume ratio at nanoscale is significantly larger than that at macroscopic scale. The presence of abundant free surfaces at nanoscale alters the effective material behavior in both the elastic region and beyond. In particular, surface relaxation changes the local atomic configuration close to the surface due to the lack of bonding neighbors, which in turn alter the atomic interactions and thus affect the effective elastic moduli. Since the percentage of surface atoms increases with increasing surface-to-volume ratio that characterizes nanomaterials, the surface effects become significant with decreasing material dimensionality.The challenge in modeling surface effects for nano-sized structures is that, on the one hand, classical continuum theory that represents bulk material behavior cannot be applied directly but have to be modified to incorporate surface effects; on the other hand, traditional molecular mechanics is limited by the model size due to the demanding computational costs. A wide range of existing approaches have been proposed to solve this issue. In general, existing models may be classified into three groups: (I) enhanced classical continuum mechanics Some of the works based on enhanced classical continuum model depend on further development of the surface elasticity formulation proposed by Gurtin and Murdoch This has motivated the development of atomic-based continuum analysis. The well-known surface Cauchy–Born (SCB) theory proposed by Park in This aforementioned drawback calls for a powerful multiscale model that may not depend on the continuum framework such as the constitutive stress–strain relationship but pure atomistic description, which motivated the development of coarse-grained atomistic models In the present work, the recently proposed multiresolution molecular mechanics (MMM) is applied to study surface effects of nanostructures. In a general sense, MMM can be thought as a generalized nonlocal quasicontinuum approach that provides a unified mathematical framework from linear to any high-order interpolation shape functions in The remainder of this paper is organized as follows. Section reviews the methodology of MMM and illustrates how SRS and SRB are derived and can be consistently incorporated into SRMMM. Section defines error norms and specifically constructed error models to identify different error sources quantitatively. Section presents several numerical examples to demonstrate the benefits of the proposed approach, and Section concludes with closing remarks on the performance of SRMMM to capture surface effects.In this section, the general idea of MMM is reviewed by analogy to the well-known finite element method (FEM) in conventional continuum mechanics. In (a), a finite element with shape function ϕ is introduced to approximate the original continuum model and the potential energy Ec can be approximated as: where W is potential energy density, Ωc denotes the continuum domain, NQ represents the index set of NQ quadrature points and wic is the associated weight for a quadrature point i∈NQ. In the present work, calligraphic letter N represents index set and the corresponding Roman letter N defines its cardinality.In general, Gauss quadrature is widely employed to evaluate the energy integral in Eq. and there is a standard way to determine the optimal number of quadrature points needed and the associated position and weight for each quadrature point. Similarly, in (b), a finite element mesh is also employed to reduce the degrees of freedom of the original atomistic model and the potential energy Ea is approximated as: where NA is the index set of the NA atoms in the original atomistic model, Ei is the atomic site energy of an atom i∈NA, Ns denotes the index set of the chosen NS sampling or quadrature-type atoms and wia is the associated weight for a sampling atom i∈NS. The red dots in are either the conventional finite element nodes for continuum mechanics or the rep-atoms or nodal atoms for atomistic modeling. And the blue atoms represent either a quadrature point to evaluate an integral or a sampling atom to calculate a finite summation. Also, the gray atoms in (b) are called non-sampling atoms (NSAs). The reason is that these atoms are not the degrees of freedom (DOFs) of the coarse-grained system and their energy are not considered explicitly but sampled by the energy of blue dots (sampling atoms).As conventional finite element nodes and rep-atoms play similar roles in their respective modeling method, we do not differentiate the coloring approach between coarse-grained continuum and atomistic models, so is the same for quadrature and quadrature-type dots. This coloring scheme will be consistently employed in this work unless otherwise stated. We note that if a quadrature point coincides with a finite element node, as it may occur for high order elements, we will clearly point that out. In addition, any symbol with a subscript or superscript “c” employed for continuum model has the same physical meaning as the same symbol with a subscript or superscript “a” utilized for the atomistic system. is in developing a standard and systematic theory to determine the optimal number NS, weight wia and position for each sampling atom for general finite element shape functions, which is analogous to the Gauss quadrature employed in conventional FEM. A previous work by the authors in In conventional FEM, it is well-known that the order of employed quadrature rule is shape-function dependent For simplicity, external loads are not considered and linear elasticity model is assumed for continuum mechanics. Then the potential energy approximation E˜c in Eq. where Nnc is the index set of Nnc finite element nodes, uic is the nodal displacement vector for a node i∈Nnc and Kijc represents the nodal stiffness matrix for nodes i and j. Kijc can be formulated as: where Dc is a matrix of material constants that describes the continuum constitutive behavior, Bic is the strain matrix for node i and can be defined in three dimensional (3D) setting as:Bic=[ϕi,x000ϕi,y000ϕi,zϕi,yϕi,x00ϕi,zϕi,yϕi,z0ϕi,x], we note that the order of quadrature rule needed to evaluate E˜c or Kijc depends on the order of the product of the two shape function derivatives. As an example, let Πijc denote the product of shape function derivatives in one dimension (1D), as follows:, we point out that the distribution of E˜c is determined by the order of function bases in Πija. For example, we let ϕ be the 1D quadratic shape function. Then, by a simple calculation, one can see that the solid line in (a) describing the distribution of E˜c, i.e. the energy density of the FEM model can be defined as: where bci, i=0−2 are constants dependent on the nodal displacement vector uic,i∈Nnc and material constants in matrix Dc, xc is a material point position that continuously spans the continuum domain Ωc. Then E˜c can be evaluated as: Since fc(xc) is a continuous quadratic function, then the optimal number of quadrature points NQ is 2 and their corresponding weight wic in Eq. can be determined by Gauss quadrature rule, which shows how the employed shape function derivatives determine the order of selected quadrature rule in conventional FEM. Next, we will show how SRMMM is derived and related to the employed shape function differences.Similarly, for an atomistic model, a nonlocal linear spring potential is assumed for the interatomic interaction to facilitate demonstration of SRMMM. As such, the potential energy approximation E˜a in Eq. where Nα is the index set of Nα atoms that interacts with atom α∈NA, rα0 denotes the initial position vector of atom α and Da is the material constants or the local stiffness matrix that describes the interatomic interaction in each direction. For linear spring potentials in 3D, I is the 3 by 3 identity matrix. Note that Bia is different from Bic since there is no shape function directive involved and we do not have shear strain in the interatomic potential. Physically, this difference is due to the fact that the constitutive relationship in continuum mechanics generally contains six strains; however, the interatomic potential in atomistic modeling depends on pair distance difference in x, y and z directions.For lattice or crystal structures, the neighboring position vector rβ0 can be determined in terms of rα0, as follows: where C is a non-zero constant vector in terms of the lattice constant a0. If Eq. , we note that the evaluation of Kija is determined by the product of the shape function difference calculated at an atom α∈NA. For example, let ϕ be the one-dimensional quadratic shape function. Then I is 1 and Da is the pair-wise spring constant. Let Πija denote the shape function difference product as follows:Πija=[ϕi(rα0)−ϕi(rα0+C(a0))][ϕj(rα0)−ϕj(rα0+C(a0))] By a simple calculation (see Section 2.4 or Appendix A in (b) describing the distribution of E˜a, i.e. the atom-wise energy can also be defined as: where bai, i=0−2 are constants in terms of nodal displacement vector uia,i∈Nna and material constants in the employed linear spring potential, xa is an atomic position that discretely spans atomic domain Ωa. Then E˜a can be evaluated as: where xai is the atomic position of atom i∈NA.We note that the difference between the energy of a surface atom and that of a bulk atom is not distinguished so far for the sake of introducing the general idea of SRMMM. From , one can see that the similarity between E˜c and E˜a lies in that they have the same distribution order. The main difference between them is that fc(xc) is a continuous function (solid line) in terms of xc but fa(xa) is a set of discrete points in terms of xa that pass through the dashed line. As such, three sampling (or quadrature-type) atoms are required to exactly represent the dashed energy distribution line. Then the optimal number for sampling atoms NS is 3 and the corresponding weight wia in Eq. can be determined as in any curve-fitting process. For example, as the dashed line in (b) has a quadratic form, three sampling atoms (2 blue dots and the interior red node) are selected. Then the dashed line can be expressed as:, Φi satisfies the partition of unity and Kronecker delta property, as follows: Then the energy summation E˜a at the discrete atoms is calculated as:E˜a=∑j∈NAf(xaj)=∑j∈NA∑i∈NSΦi(xaj)Ei=∑i∈NS∑j∈NAΦi(xaj)Ei The above curve-fitting-like process to determine the optimal number of sampling atoms NS and the corresponding weight wia forms the MMM summation rule SRMMM, which clearly shows how SRMMM is related to the employed shape function differences. For more details about deriving SRMMM, we refer the interested reader to our previous work in It is worth to note that the above MMM analogy to FEM is introduced from an energy point of view. Though FEM is equivalent to an energy minimization problem under certain circumstances, FEM is, in general, based on the weak form of the original ordinary or partial differential equations, and on weighting the residual integral to zero by setting the trial function space also as the test function space. On the other hand, molecular mechanics lies in the energy minimization for statics considered here. The aforementioned analogy is only employed to better the understanding of the introduced MMM framework.In this section, we will explain why there is a need to distinguish the difference between the energy distribution of surface atoms and that of bulk atoms and hence decompose SRMMM into the bulk summation rule SRB and the surface summation rule SRS. is the index set of Nα neighboring atoms that interact with an atom α∈NA. Physically, the fundamental difference between surface atoms and bulk counterpart is that surface atoms are not fully coordinated and lack neighboring atoms. As such, NαS, the number of neighboring atoms of a surface atom α that belongs to the index set of surface atoms NAS, is different from NαB, the number of neighboring atoms that interacts with a bulk atom α that resides in the index set of bulk atoms NAB. As a result, the energy distribution of surface atoms, fSa(xa), will be different from that of bulk atoms, fBa(xa), even though they both reserve the quadratic form in Eq. for the one-dimensional quadratic shape function considered in Section . Mathematically, this is because the coefficient bai, i=0−2, in Eq. take different values for fSa(xa) and fBa(xa), respectively. This is schematically demonstrated in , where the energy of surface atoms (empty circles in rectangle boxes) does not follow the bulk energy distribution curve (dashed line).For the reasons discussed above, SRMMM is decomposed into SRB and SRS. Basically, SRB is employed to treat bulk atoms and SRS is used to treat surface atoms to accurately represent the energy distribution over the whole atomic domain Ωa. The methodology to determine the optimal number of sampling atom NS and their respective weights wia in SRB and SRS are exactly the same as for SRMMM in Section We note that the performance of SRB has been studied by a relatively comprehensive numerical examples in In this section, the employed SRS in 2D and 3D will be introduced for the respective 4-node bilinear quadrilateral, 8-node quadratic quadrilateral and 8-node hexahedral elements. for the 4-node quadrilateral element in 2D is given here for demonstration purpose. As such, the shape function ϕi for a node i∈Nna takes the following form: where x and y are the respective projection of position vector r=(xy) of an arbitrary atom in x and y directions, aiα, α=0−3 are constants in terms of the chosen nodal positions to construct ϕi. Let Cx and Cy be the projection of the non-zero constant vector C=(CxCy) in Eq. in x and y directions, respectively. Then ϕi(r+C) in Eq. ϕi(x+Cx,y+Cy)=ai0+ai1(x+Cx)+ai2(y+Cy)+ai3(x+Cx)(y+Cy) such that the shape function difference is expressed as: where biα, α=0−2 are given in terms of aiα, α=0−3 and constants Cx and Cy. Then, Πija in Eq. Πija=cij0+cij1x+cij2y+cij3xy+cij4x2+cij5y2 where cijα, α=0−5, are constants given in terms of biα and bjα, α=0−2. Finally, fa(xa) in Eq. where bai, i=0−5 are constants in terms of the nodal displacement vector uia,i∈Nna and material constants in the employed linear spring potential and the coefficients in the given bilinear shape function., we note that the optimal number of sampling atoms NSB for SRB is 6 and Φi in Eq. also has a fully quadratic form and can be determined by a curve-fitting-like process, as discussed in Section , such that wia is also identified correspondingly in Eq. For the employed 8-node quadratic quadrilateral element in 2D, fa(xa) can be derived following the same process as for the 4-node quadrilateral element. It is found that fa(xa) has a full cubic order plus some fourth order terms, as shown in Eq. fa(xa)=ba0+ba1x+ba2y+ba3xy+ba4x2+ba5y2+ba6x3+ba7x2y+ba8xy2+ba9y3+ba10x4+ba11x2y2+ba12y4 In this work, we will discard the uncompleted fourth order terms and assume fa(xa) takes the complete third order form:fa(xa)=ba0+ba1x+ba2y+ba3xy+ba4x2+ba5y2+ba6x3+ba7x2y+ba8xy2+ba9y3 As such, for the 2D quadratic quadrilateral element, NSB for SRB is 10, Φi in Eq. also has a fully cubic form and wia can be determined as in Eq. Similarly, for the employed 8-node hexahedral element in 3D, we find that fa(xa) possesses a full quadratic order plus some mixed high order terms, as shown in Eq. fa(xa)=ba0+ba1x+ba2y+ba3z+ba4xy+ba5xz+ba6yz+ba7x2+ba8y2+ba9z2+ba10x2y+ba11x2z+ba12y2x+ba13y2z+ba14z2x+ba15z2y+ba16x2y2+ba17x2z2++ba18y2z2+ba19x2yz+ba20y2xz+ba21z2xy Again, we will discard the mixed terms and assume fa(xa) takes the fully quadratic form:fa(xa)=ba0+ba1x+ba2y+ba3z+ba4xy+ba5xz+ba6yz+ba7x2+ba8y2+ba9z2 such that NSB is 10 and Φi and wia can be determined correspondingly, following the above discussed curve-fitting-like process.So far, the optimal number of sampling atoms NSB and their corresponding weight wia in bulk summation rule SRB are discussed and determined. Next, we will show how NSS, the optimal number of sampling atoms in SRS, and their corresponding weight wia are identified.For the rectangle triangular lattice in 2D, as shown in , we note that either x or y is constant for the respective surface edge and non-bulk edges (dashed lines). As such, fa(xa) in Eq. can be reduced to fewer terms. For instance, for the surface edge on the left hand side, y is constant such that, fSa(xa) can be reduced to: where εai, i=0−2 are constants in terms of bai in Eq. , it is clearly seen that NSS is 3 and wia in SRS can be calculated as in SRB. If a non-bulk layer is considered (), the same SRS can also be applied, as in the 2D example in Section schematically demonstrates the idea on how SRMMM (SRS + SRB) is employed for the four-node bilinear quadrilateral element with (For the 8-node quadratic quadrilateral element, the same analysis can be directly applied to such that fa(xa) in Eq. (b)) and wia in SRS can be determined accordingly.In order to distinguish the aforementioned sampling atoms from secondary sampling atoms (SSAs) that will be introduced later in this section, we name these sampling atoms primary sampling atoms (PSAs). In addition, the PSAs where SRS is applied are named surface PSAs (SPSAs) and PSAs where SRB is applied are called bulk PSAs (BPSAs).The same idea can be applied to a face-centered cubic (FCC) lattice in 3D. For each of the surface and non-bulk layers, either x, y or z is constant such that fa(xa) in Eq. can also be trimmed to fewer terms. For example, for the surface layer on the bottom, z is a constant such that fSa(xa) can be redefined as: where the summation rule SRMMM in 2D can be directly applied such that NSS and wia can be determined accordingly. We note that, mathematically, SRS in 3D is, in fact, the SRMMM (SRB + SRS) in 2D and SRS in 2D is actually the SRMMM (SRB + SRS) in 1D. This is understandable since, physically, it is obvious that the surfaces of 3D model are 2D and the edges of 2D are 1D. Note that the 2D surface or 1D edge could be curved. In the present work, we focus on regular shapes. Surfaces and edges with curvature will be considered in our future work.For positions of primary sampling atoms, based on our previous study in It is worth to note that the energy of corner atoms does not exactly follow fSa(xa) or fBa(xa) for the reason that the neighbors of corner atoms are located on more surfaces than that of surface or bulk atoms. Similarly, the energy of atoms that coincide with Nna finite element nodes (called nodal atoms) and energy of atoms near element edges (called element edge atoms) within a potential cut-off distance (rcut) also do not exactly follow fSa(xa) or fBa(xa) since neighbors of these atoms are located in more finite elements. Based on our previous results in . However, the energy of nodal atoms will be considered explicitly instead of being sampled by energy of primary sampling atoms. For this reason, we call the atoms whose energy are considered explicitly secondary sampling atom (SSA) to represent their own energies (such that the weight wia is 1 for each SSA) and we name the aforementioned sampling atoms (blue dots in ) whose energy is employed to sample the energy of non-sampling atoms (NSA) (gray dots in ) primary sampling atoms (PSA). A more detailed explanation and atom classification can be found in In conventional FEM, the selected order of quadrature rule to evaluate the energy (Eq. ) is generally determined by the accuracy of Gauss quadrature for linear elasticity. For nonlinear constitutive relationships, a relatively higher order of Gauss quadrature may be employed to improve accuracy. We note that reduced Gauss quadrature may have better performance for problems where internal constraints in the continuum theory being assumed, such as incompressibility or the Kirchhoff transverse shear constraints in bending analysis. Full integration or higher order Gauss quadrature may introduce the so-called shear or volumetric locking issues and thus may deteriorate the accuracy of FEM.Similarly, SRMMM has been introduced by assuming a nonlocal harmonic spring potential, as mentioned earlier. However, since any nonlinear interatomic potential can be well-approximated by a harmonic spring within its convex region, SRMMM is expected to be effective for nonlinear potentials, which have been clearly demonstrated in Note that for atomistic regions where interatomic potential is within its non-convex domain, full molecular mechanics can be employed since defects are likely to occur in that region. Since MMM is a fully coarse-grained atomistic model and no continuum description is introduced, full molecular mechanics can be directly and organically incorporated into MMM without any modification, as has been shown in In this section, the governing equations of MMM employing SRMMM (SRB + SRS) will be derived from the potential energy approximation E˜a through the variational principle. As mentioned in Section , we do not assume a specific form of the interatomic potential Ei employed below.Since atoms are classified into PSAs, SSAs and NSAs from the perspective of energy sampling, then E˜a in Eq. where NSSA and NPSA are the respective index of SSAs and PSAs. Note that wia=1 for each SSA and wia is determined in SRMMM except that Eq. Let Ua denote the displacement vector for all the Nna nodal atoms and fαext be an external force vector applied to an atom α∈NA. Following the variational principle for deriving the equilibrium equations of an MMM model, the negative derivative of E˜a with respect to the nodal displacement vector uia,i∈Nna, which are the degrees of freedom in MMM as in FEM, must be zero. Then we have:Fia(Ua)=−∑j∈NSSA∂Ej(Ua)∂uia−∑k∈NPSAwka∂Ek(Ua)∂uia+∑α∈NAϕi(r=rα0)fαext=0,∀i∈Nna where fαext is partitioned to a nodal atom as in FEM. Equation is the governing equation under the MMM framework for statics.In this section, different error sources will be defined and identified. In addition, error norms in displacement and energy fields, as inspired by their counterparts for classical FEM, are defined to quantify the performance of MMM.It is well-known that there are two different types of error in conventional FEM: (I) Discretization error that depends on element size and shape function order and (II) numerical integration error that depends on the quadrature rule utilized to evaluate the integrals involved. Similarly, there are also two error types in MMM: (I) Discretization error as in FEM and (II) sampling error that depends on the summation rules employed to evaluate the finite summations involved.As a simple example, let UFA be the displacement solution vector from full atomistic simulation (Model A in (a)) and UMMM be the one from MMM calculation (Model B in (b)). In addition, in order to quantify the discretization error, a special MMM model (Model C in (c)) is designed. In Model C, after using finite elements to coarse-grain the full atomistic model, the energy of each atom is considered explicitly such that each atom is taken as a second sampling atom (SSA), denoted as green dots in the present work. We note that the nodal atoms (red dots in ) are assigned the atom type SSA. Since they also represent the degrees of freedom, we stay with the color red for them. In Model C, there is only discretization error and no any other error is introduced. As such, Model C is, in general, the most accurate for a given discretization, but it is also the most expensive one. Since there is no primary sampling atom (PSA) in Model C, we denote the displacement field gained from it as U0p⁎sMMM with “0p” meaning no PSA and “⁎s” representing SSA everywhere.As such, for displacement field, the distance between Model A and Model B, eUtot can be decomposed as:UFA−UMMM=UFA−U0p⁎sMMM︸discretizationerror+U0p⁎sMMM−UMMM︸samplingerror where UFA−U0p⁎sMMM is the distance between Model A and Model C and is called the discretization error eUdisc;U0p⁎sMMM−UMMM is the distance between Model C and Model B and is called the sampling error eUsam. The different types of error are schematically shown in For a given discretization, the discretization error eUdisc is fixed. Then a good indicator to estimate the performance of a coarse-grained model or a summation rule is to identify the sampling error. As many other coarse-grained models do not clearly differentiate and quantify different error sources, we note that the identification of different error types is important, especially when spatial convergence is performed In this section, the two different error sources will be quantified by defining several appropriate error norms.The following error norms (L2 and H1) are widely used to measure the performance of classical continuum FEM eU=[∫Ωc(Uexact−UFEM)T(Uexact−UFEM)dΩc]1/2eE=[∫Ωc(εexact−εFEM)T(εexact−εFEM)dΩc]1/2 where eU and eE are the errors in displacement and energy fields, respectively, Uexact and εexact are the respective displacement and strain fields from exact solution.Similarly, the error norms defined in our previous work eUtotal=[∑i∈NA(uiFA−uiMMM)T(uiFA−uiMMM)∑i∈NA(uiFA)TuiFA]1/2=‖UFA−UMMM‖2‖UFA‖2eEtotal=[∑i∈NA∑j∈Ni((rijFA−rij0)−(rijMMM−rij0))T((rijFA−rij0)−(rijMMM−rij0))∑i∈NA∑j∈Ni((rijFA−rij0))T((rijFA−rij0))]1/2 where uiFA and uiMMM are the displacement vectors from full atomistic calculation and a MMM model, respectively, for an atom i∈NA;Ni is the index set of neighbors of the ith atom, rijFA and rijMMM denote the pair distance vector from each model and rij0 represents the initial pair distance vector. Note that all the errors are normalized by the solution from full atomistic model.In linear elasticity, the potential energy of a continuum model is a function of the strain field ε, and hence it makes sense that Eq. defines error in energy field. For an atomistic model, strain is not explicitly defined and thus Eq. cannot be directly employed for MMM. Instead, the pair distance vector is employed in the error measure, because the interatomic (pair) potential energy is a function of pair distance, which is analogous to the strain defined in continuum mechanics. We note that even for many-body potentials, the difference in each pair distance vector is also a good error indicator in potential energy. As such, the difference in strain vector field and in pair distance vector field are good error measures in energy field for the respective continuum and atomistic modeling.We note that in many of the previous works, errors are always measured in such a way that the displacement vectors U or pair distance vector rij from different models are first computed to generate a scalar, and then the difference between the scalars is employed as the error. For instance, let EtotFA and E˜tot be the respective total potential energy scalar from atomistic modeling and any other model generated by operations on the respective rij or U from each model, then the error is calculated as: We note that this is not a good error indicator because there are cases that the difference, for instance, in displacement field is significant but err is quite small, as schematically shown in . In contrast, the error norms defined in Eqs. Then using the error norms defined in Eq. , the discretization error eUdisc and sampling error eUsam defined in Eq. in displacement field can be deduced as:eUdisc=[∑i∈NA(uiFA−u0p⁎siMMM)T(uiFA−u0p⁎siMMM)∑i∈NA(uiFA)TuiFA]1/2=‖UFA−U0p⁎sMMM‖2‖UFA‖2eUsam=[∑i∈NA(u0p⁎siMMM−uiMMM)T((u0p⁎siMMM−uiMMM))∑i∈NA(uiFA)TuiFA]1/2=‖U0p⁎sMMM−UMMM‖2‖UFA‖2 Note that eUtotal, eUdisc and eUsam are all normalized by ‖UFA‖2 such that they are comparable to each other.Similarly, the discretization error eEdisc and sampling error eEsam in energy field are defined as:eEdisc=[∑i∈NA∑j∈Ni((rijFA−rij0)−(r0p⁎sijMMM−rij0))T((rijFA−rij0)−(r0p⁎sijMMM−rij0))∑i∈NA∑j∈Ni((rijFA−rij0))T((rijFA−rij0))]1/2eEsam=[∑i∈NA∑j∈Ni((r0p⁎sijMMM−rij0)−(rijMMM−rij0))T((r0p⁎sijMMM−rij0)−(rijMMM−rij0))∑i∈NA∑j∈Ni((rijFA−rij0))T((rijFA−rij0))]1/2 The errors defined in this section will be employed to quantify the performance of MMM to solve surface relaxation and bending problems in 2D and 3D in Section In this section, the aforementioned SRMMM (SRB + SRS) in Section will be employed in 2D and 3D to solve surface relaxation and beam bending problems. The accuracy of SRMMM will be quantified by the error norms defined in Section . It is worth to note that the proposed SRMMM can naturally capture both in-plane and out-of-plane surface stress components. This is due to the fact that the interatomic potential employed is directly utilized to calculate the interatomic force interactions among the surface sampling atoms, as in full atomistic modeling. As such, forces from neighbors located in out-of-plane and in-plane directions are automatically considered.In the following numerical example, the 2D triangle lattice () consists of 97 by 97 atoms (9361 atoms in total) interacting via the standard Lennard-Jones (LJ) potential, i.e., ULJ=4ε[(σ/r)12−(σ/r)6] where r is the interatomic distance with the initial nearest interatomic spacing r0=21/6 and potential parameters σ=ε=1. The interatomic interaction is modeled up to the second nearest neighbor interaction. For boundary conditions, the lower-left corner atom is fixed in both x and y directions and the upper-left corner atom is fixed in x direction to eliminate rigid body motion. The lattice model is coarse-grained by a regular quadrilateral mesh (dashed lines) with 16 4-node quadrilateral elements. The mesh has 25 nodes or nodal atoms and takes only 0.27% over the original full atomistic (FA) degrees of freedom (DOFs), corresponding to an element size hx=24r0 in x direction and hy=123r0 in y direction. The 2D lattice model is relaxed using a non-linear Newton solver.Since the second nearest neighbor interaction is considered, there are one surface edge and one non-bulk layer in the direction perpendicular to the bottom and top of the lattice, respectively. And there are one surface edge and three non-bulk layers in the direction perpendicular to the left and right of the lattice. As such, the surface summation rule SRS is employed for each edge and non-bulk layer and the bulk summation rule SRB is taken for the bulk region, as schematically shown in A comparison of the displacement field distributions from FA model and from the proposed MMM calculation is shown in . The figure clearly illustrates that the proposed MMM with SRMMM captures the displacement tendencies in x ((d)) directions exhibited from atomistic simulation, both qualitatively and quantitatively. In order to show the capability of MMM to capture corner and edge effects, the displacement fields of the upper-right corner and the middle of the edge on the right hand side, from the respective FA and MMM models, are tabulated in , where the accuracy of MMM is demonstrated.In addition, the different error types defined in Section , the discretization error is fixed for a given mesh. Thus the sampling error is a good (if not best) indicator about the performance of a proposed summation rule. In general, to minimize the total error eUtotal, the sampling error eUsam has to be minimized. As can be seen from , the introduced eUsam and eEsam are 0.04% and 0.0006% in displacement and energy field, respectively, with the proposed SRMMM. The small sampling errors obtained prove that MMM has great performance.ν(x,y)=−P6EI[3υy2(L−x)+(4+5υ)D2x4+(3L−x)x2] where L and D are the length and height of the beam, respectively. Parameters in Eqs. are taken as P=1000, E=3×105, υ=0.3 and I=D312. The Lennard-Jones potential defined in Section is taken here to describe the interatomic interaction. For boundary conditions, the atomic beam is fixed at the left lower and upper corners in x direction and at the middle of the left edge in y direction.The 8-node quadratic quadrilateral element is employed to coarse grain the beam with an element size hx=12r0 in x direction and hy=63r0 in y direction. hx and hy denote the nearest nodal spacing in each direction. The coarse-grained model has 23 nodes, corresponding to 0.95% over the original FA DOFs. In order to show the significance to capture surface effect, MMM with SRB only and with (SRB + SRS) are respectively applied to the coarse-grained model, as shown in (c) show the respective vertical bending distribution from FA, SRB only and (SRB + SRS). As seen in , (SRB + SRS) has much better agreement with FA results than SRB only. quantifies the defined different errors types from different sampling approaches. As observed in , the introduced sampling error eUsam from SRB only and (SRB + SRS) are 28.25% and 1.06%, respectively. This observation justifies the significance of capturing surface effect and that SRS can effectively capture surface effect with respect to the employed quadratic element for bending analysis.The numerical example considered next is the free surface relaxation of an FCC crystal lattice. The FCC lattice consists of 24 by 24 by 24 unit cells in the x, y and z directions, corresponding to 58,825 atoms as shown in . The same interatomic potential employed in 2D surface relaxation example is also used here. The atomic interaction is truncated after the third shell of the nearest neighbors. To eliminate rigid body motions, the boundary conditions in To coarse grain the FCC lattice, a regular hexahedral mesh (dashed lines) with 27 8-node hexahedral elements is introduced, as shown in . The mesh has 64 nodal atoms with an element size h=82r0 in each direction, which takes only 0.11% of the original full atomistic (FA) DOFs. Since the atomic interaction is modeled up to the third shell of nearest neighbors, there is one free surface and one non-bulk layer in the direction perpendicular to each cubic side surface, as shown in . Then SRS is applied to both the free surface and the non-bulk layer, as illustrated in (b). Note that SRS applied to the non-bulk layer is invisible in . The SRB employed is schematically shown in show the displacement distribution comparison between FA modeling and MMM simulation. As can be seen, MMM accurately captures the distribution trend in each (positive or negative) direction, similar to what has been observed in the 2D case. The expansion or contraction of a surface is clearly matched in x, y and z directions. In order to compare the distribution inside the cube, the models in are cut through by x=0, y=0 and z=0 planes, respectively. As shown in , very good agreement has been achieved. The purpose of showing the distribution pattern in each direction is to demonstrate that MMM has uniform accuracy in each direction and no direction preference is observed, as will be shown in The displacement fields for the corner at (L/2, L/2, L/2), the atom at the middle of the edge connecting corners at (L/2, −L/2, L/2) and (L/2, L/2, L/2), and the atom at the center of +x free surface are presented in to show the capability of MMM to capture the corner, edge and surface effects in 3D. The uniform accuracy of MMM in each direction is clearly demonstrated in To more rigorously understand the performance of MMM in 3D and to show the importance of accurately capturing surface effect, eUtotal, eUdisc, eUsam and eEsam from MMM with SRB only (. The introduced eUsam with SRB only and with (SRB + SRS) are 44.53% and 0.71%, respectively, which shows the effectiveness of the proposed SRS to capture surface effect. As what has been observed in 2D, the good performance of MMM lies in that SRMMM accurately represents and reproduces the energy distribution of coarse-grained atomistic model with solid mathematical foundation.It is worth to note that the employed 2D triangular and 3D FCC lattices are both centrosymmetric. The application of SRMMM to non-centrosymmetric lattices such as Silicon will be reported in our future publications. For example, techniques proposed in Compared to full molecular mechanics, the efficiency of MMM is indirectly measured by the number of DOFs employed in this work. The additional computational effort required in MMM, such as the calculation of weights for primary sampling atoms, makes the computational cost reduction not proportional to the reduction in DOFs. However, the efficiency of MMM is not affected significantly since the extra computational cost in MMM takes a small portion of the overall computational effort, when compared to the computational time required to, for instance, solve nonlinear governing equations. In fact, the efficiency of MMM based on linear elements has been reported in We have proposed the surface summation rule SRS within the framework of multiresolution molecular mechanics (MMM). Together with the previously developed bulk summation rule SRB, SRMMM is completed. Analogous to classical quadrature rules to evaluate energy integral in conventional continuum mechanics, SRMMM is based on mathematically deriving the order or distribution of finite energy summations involved in coarse-grained atomistic model and then determines the number, positions and weights for quadrature-type (sampling) atoms. SRS and SRB are designed correspondingly to treat surface and bulk regions respectively based on the observation that the surface energy distribution is different from bulk energy map mathematically. Physically, the difference is due to the fact that surface atoms lack bonding neighbors and are not fully coordinated compared with their bulk counterpart.To analyze the performance of any summation rule, different error sources are identified and the total error is decomposed into discretization error and sampling error. It is pointed out that sampling error is a good indicator of the accuracy of any proposed summation rule since discretization error is fixed for a given mesh. To quantify different error sources, rigorous errors norms, as inspired by the norms to quantify FEM accuracy in continuum mechanics, are defined.Numerical examples in 2D and 3D have been employed to verify and validate MMM with SRMMM. For both the 2D triangle lattices and 3D FCC model, the proposed MMM qualitatively and quantitatively captures the mechanical response as demonstrated in the two surface relaxations and one beam bending examples. The contraction or expansion of a surface is clearly matched compared to the FA simulation. A comprehensive comparison between displacement results from FA simulation and MMM shows similar and uniform accuracy of MMM in x, y and z directions. Corner and edge effects are also accurately captured. The effectiveness of SRMMM with respect to high order element is demonstrated by employing the quadratic quadrilateral element to solve a beam bending problem. Furthermore, the proposed SRMMM introduces very small sampling errors as quantified by the error norms defined in both the displacement and energy fields.In the future, several research directions will be pursued. First, the embedded atom (EAM) potential will be employed so as to apply MMM to realistic metallic materials. Note that the framework of MMM is potential-independent (as long as the potential can be well-approximated by a harmonic potential within the convex region). Second, we will apply MMM to models with complex geometries such as rough surfaces to study the effect of surface orientations. Finally, surface-defect interaction will be considered by using atomistic simulation to locally track defect at where it is needed.Melt preparation and investigation of properties of toughened Polyamide 66 with SEBS-g-MA and their nanocompositesPolyamide 66/SEBS-g-M alloys and their nanocomposites were prepared by melt compounding using a twin screw extruder. The mechanical properties, rheological behavior and morphology of these alloys and their nanocomposites were investigated. Morphological investigations with different methods show pseudo-one-phase type morphology for these prepared alloys at all percentage of rubber. Impact and tensile tests results showed that rubber toughened samples exhibit significantly more impact strength and elongation at break comparing to virgin Polyamide. Samples with 20% of rubber show impact strength about 15 times and elongation at yield several times more than those of virgin Polyamide. So, these rubbers modified polyamide can be consider as super toughened rubber. A general type organoclay at 4% and 8% has been used with rubber toughened samples to tolerate their modulus and tensile strength. Obtained results show that nanoclay could significantly increase modulus and tensile strength of rubber modified Polyamide 66 without considerable effects on impact strength. WAXD and SEM results show that the Polyamide 66 nanocomposites are better exfoliated in presence of SEBS-g-MA. Some investigation on viscosity and melt flow index of PA66 and their alloys show that shear viscosity of PA66 nonlinearly decrease with incorporation of SEBS-g-MA and organoclay into it.Polymer layered silicate nanocomposites have attracted a great deal of interest over the last few years as a results of the potentially superior properties that these materials could exhibit relative to conventional composites. Among these improved properties are mechanical The incorporation of clays into polymer matrices has been known for more than 55 years. In 1950, Carter et al. Crystalline thermoplastic polymeric materials such as polypropylene and low carbon monomer polyamides are very sensitive to craze and crack propagations and so their toughness and impact resistance properties are relatively low. Polyamide 6 and 66 are relatively low cost engineering thermoplastics that have been found so many industrial applications including automotive industry, food packaging, electronic and many others. However, relatively low impact and bending resistance of this material especially in low temperature and high deformation rate, restricts application of these materials for many applications. Rubber toughening can be considered as an economical and effective method to tough and extend industrial applications of polyamide without important increasing of its price or decreasing its engineering properties However, most rubbers do not show necessary compatibility with polyamide and cannot correctly disperse in rubber. To get toughening effects from rubber in thermoplastic, normally rubber should be dispersed in micro scale where most of non-modified rubber can only produce dispersed particle with average diameter about 50 μm. So they should be modified to be able to produce micro scale particle and necessary toughening effects for Polyamide. Grafting of polar and reactive monomers such as maleic anhydride on many rubbers can provide the necessary compatibility for rubbers with polyamides Chow et al. have studied thermoplastic nanocomposites based on blends of Polyamide 6 (PA6) and polypropylene (PP) So in the present work, preparation of SEBS-g-MA toughened Polyamide 66 and their nanocomposites, and investigation of their physical–mechanical, morphological behaviors have been considered.The Polyamide 66 (Akulon S223-E) used in this study was a commercial product from DSM, Netherlands. The melt flow index (MFI at 275 °C, 2.16 kg), density, and melting point of PA66 were 38.7 g/10 min, 1140 kg/m3, and 260 °C respectively. SEBS-g-MA (Kraton FG1901X) containing 2 wt% MA was supplied by Kraton polymers, North America. The melt flow index (MFI at 230 °C, 5000 g) and density of SEBS-g-MA are 22 g/10 min and 910 kg/m3, respectively. Organoclay (Nanolin DK4) was a commercial product from ZHEJIANG FENGHONG clay chemicals CO, LTD (FCC) derivative of a Smectite with 35 Å D-Spacing, and Cation exchange capacity = 115–120 meq/100 g.Prior to each processing step, all polyamide containing materials and pure organoclay were dried in a vacuum oven for at least 24 h at 90 °C to avoid moisture induced degradation reactions. Composites were prepared by melt compounding using a intermeshing Counter-rotating twin screw extruder (Nanjing Giant (SHJ-20)) with a 21.7 diameter screw having depth of channel of 3.85 mm, L/D of 32, Max screw speed of 600 rpm, and flow rate of 2–8 kg/h. Toughened Polyamide 66 and its nanocomposites were obtained in mentioned extruder, operating at 265–275 °C and screw speed of 50 rpm.Alloys and nanocomposites were injection molded into test pieces by using an injection molder Monomat80 (Germany) after being dried at 90 °C for 24 h.The tensile and impact test specimens were molded according to ASTM D638 and ASTM D256, respectively. Processing conditions for materials and sample preparation are presented in Prior to testing, all specimens were dried in a vacuum oven at 90 °C for 24 h. Tensile tests were conducted according to ASTM D638 using HOUNSFIELD (H10KS) testing machine. Values for modulus, yield strength, and elongation at break were obtained at 5 mm/min. Notched charpy impact strength were measured using a 20 kJ impact tester from SANTAM Co, Iran. Hardness tests are done according to Shore D (ISO R 868) using a manual Dourometer.The PA66/SEBS-g-MA and PA66/SEBS-g-MA nanocomposites specimens were fractured at liquid nitrogen temperature and then fracture surfaces were gold coated and observed in a scanning electron microscopy (SEM) from Philips. X-ray diffraction of the organoclay and nanocomposites were performed in a Philips X’pert MPD (40 kW, 30 mA) in order to evaluate the evolution of the clay d001 reflection. Powder sample was scanned in reflection, whereas the injection-molded compounds were scanned in transmission mode in the interval of 2θ° |
= 2–10°. Data were collected for the organoclay and all samples with a step of 0.005°, time/step of 1 s and Cu Kα radiation.Many presented results in literature show that viscosity ratio of a blend components at processing condition has very important effects on the morphology of blend. For a blend with partially compatible components, the mean average particle size of distributed phase decrease with approaching viscosity of two phases. So apparent viscosity of rubber and Polyamide 66 are measured at different temperatures and shear rates, in order to find the temperature and shear rate at which two components of blend show the closet possible viscosity. At viscosity of Polyamide 66 and SEBS-g-MA and their viscosity ratio at 265, 275 and 285 °C measured by capillary viscometer are presented, respectively.Comparing these figures, we can see that at 265 °C viscosity of polyamide is much more than viscosity of SEBS-g-MA where at 275 °C their viscosity approaches and at 285 °C polyamide viscosity become much less than that of SEBS-g-MA. Also these figures show that for all temperatures with increasing the shear rate, viscosity of two phase approaches. shows that viscosity behavior versus shear rate for Polyamide 66 and SEBS-g-MA at 275 °C is so close and at shear rate above 100 s−1, they have almost the same viscosity. So, for dispersion of SEBS-g-MA rubber in Polyamide 66, it seems that temperature of extruder must be about 275 °C and screw rotation speed must be enough high to produce high shear rates.According to method described in the previous sections, we prepared PA66/SEBS-g-MA containing 5, 10, 15, and 20 wt% of SEBS-g-MA. The mechanical properties of these alloys were measured and presented in . Presented results show that introducing SEBS-g-MA in PA66 matrix significantly increase impact strength and elongation at break of matrix, where slightly decrease tensile modulus, tensile strength, and hardness of the matrix. To compensate reduced properties of Polyamide 66 due to addition of SEBS-g-MA rubber, organoclay nanocomposites of toughened polyamides with 4 and 8 wt% of organoclay are prepared. In , mechanical properties of different produced nanocomposites with 4 and 8 wt% of nanoclay are presented. Presented results show that modulus and strength of samples even at low loading of organoclay significantly increased which must be due to nanometeric dispersion of silicate layers in matrix. However, impact strength is somewhat reduced as organoclay is added, which have been reported previously by many researcher for nanocomposite , it can be seen that the hardness will decreased, but the clay content does not have considerable effect on hardness.Results from melt flow index measurements for virgin polyamide and toughened polyamide with SEBS and their nanocomposites with 4 and 8 wt% of nanoclay at 270 °C are presented in . Presented results show that MFI of SEBS-g-MA is only 33% less than that of virgin Polyamide. These results also show that organoclay does not much affect MFI of virgin Polyamide, but presence of SEBS-g-MA help that organoclay dramatically reduce flow ability of polyamide nanocomposites. It must be due to better exfoliation and dispersion of nanoclay particles in polyamide matrix in presence of SEBS-g-MA. It should be emphasized that presence of organoclay somewhat reduces MFI of Polyamide, but comparing results obtained from samples with 4 and 8 wt% of nanoclay without and with SEBS-g-MA shows that presence of SEBS-g-MA has synergistic effects on reduction of polyamide MFI. This phenomenon must be probably due to increasing interaction of polyamide and SEBS-g-MA phases at presence of nanoclay. compares X-ray diffraction patterns for the pure organoclay and nanocomposites containing 10 and 20 wt% SEBS-g-MA and 4 wt% organoclay. The pattern for the pure organoclay reveals a sharp intense peak around 2θ° |
= 2.52°, which is characteristic of the basal spacing of the modified layered silicate. The patterns for PA66/SEBS-g-MA nanocomposites show a low broad peak around 2θ° |
= 2.18°, which suggests that a portion of the organoclay must be exfoliated and the rest must be somewhat intercalated.Scanning electron microscopy (SEM) micrographs of the fracture surfaces are presented in a–f in the cases of 10 and 20 wt% of SEBS-g-MA with or without organoclay. SEM photos of these samples presented in show that particle size of rubber must have submicron size which cannot be observed with ultimate magnification of SEM photos. These SEM photos suggest that these blends must have a single phase or two co-continuous phase morphology, which must be results of excellent dispersion of rubber phase and excellent compatibility of SEBS-g-MA rubber with Polyamide 66. The excellent compatibility of SEBS-g-MA with polyamide certainly owes to grafted maleic anhydride groups which could physically and also chemically increase interactions of these two polymers. In this system, an interfacial reaction could occur between SEBS-g-MA and PA66 (SEBS-g-PA66), which can modify the interfaces from ‘bare interfaces’ to ‘occupied interfaces’. It seems that the interfacial tension decreases with increasing extent of the interfacial reaction.It seems that introducing organoclay in Polyamide 66/SEBS-g-MA blends causes that morphology of blend somewhat changes. In the cases of 10 wt% of SEBS-g-MA with 4 wt% of organoclay, few rubber particles with particle size less than 0.5 μm could be observed which could not be observed in the same blend without organoclay. In the case of 20 wt% of SEBS-g-MA and 4 wt% of organoclay, more particles with bigger size can be observed. So, it is evidence that incorporation of the organoclay in this blend reduces ability of dispersion of SEBS-g-MA in Polyamide 66 matrix. This could be probably due to change of viscosity ratio of two phases which reduces ability of dispersion of rubber in Polyamide 66.PA66/SEBS-g-MA and PA66/SEBS-g-MA/organoclay nanocomposites were prepared by melt compounding using an intermeshing counter-rotating twin screw extruder. The reduced modulus and strength of alloys with functional rubber addition was counteracted by incorporation of organoclay without significant negative effects on the impact strength. SEBS-g-MA has a viscosity close to Polyamide 66 melt at 275 °C and so produce submicron particle size or even fine co-continues phase in Polyamide 66. Comparison of mechanical properties of this rubber toughened polyamides with virgin polyamides shows an increase about 1200% and 240% for impact strength and elongation at break, respectively, which is a very interesting results and show excellent toughening of Polyamide 66 with SEBS-g-MA rubber. The SEBS-g-MA phase was finely dispersed at a scale smaller than 0.5 μm at all concentrations. In the other hand, by incorporation of organoclay tensile modulus and strength significantly increased even relative to other reports. Now, we have a product with improvement all mechanical properties relative to virgin Polyamide 66.Fibre-reinforced cemented paste backfillCompressive behavior and microstructural properties of tailings polypropylene fibre-reinforced cemented paste backfillThis experimental study was carried out to investigate the influence of polypropylene (PP) fibre on the compressive behavior and microstructural properties of tailings (classified as CL – lean clay) cemented paste backfill (CPB). The compressive behavior was determined by the unconfined compressive strength (UCS), and structural changes in CPB were evaluated by macrostructural failure analysis and microstructural tests with scanning electron microscopy (SEM) analysis. Orthogonal tests were designed to research the influences and significance of the cement content, solid mass concentration, fibre content and fibre length on CPB. The results indicate that although the fibre parameters are not as significant as the cement content and solid mass concentration for CPB strength, they are also important factors for improving the UCS of CPB, and the effect of PP fibre on the early (3 days curing time) and later (28 days curing time) stage strength of the backfill is much more obvious than that on the medium stage (7 days curing time) strength. And fibre can enhance the stiffness and ductility of CPB with increasing strain and E50. In addition, the fibre content is much more important than the fibre length, there is no linear relationship between UCS and the fibre parameters (fibre content and fibre length), and the best fibre parameter levels are a fibre content of 0.15% and a fibre length of 6 mm in this study. Moreover, macrostructural failure analysis and SEM microstructural tests indicated that PP fibre tend to bridge the cracks, which can reduce the porosity of the CPB matrix and improve its compactness, integrity and residual strength with some calcium silicate hydrate (C-S-H) gelling trapped on its surface.Fibre-reinforced cemented paste backfillCemented paste backfill (CPB) is an important development direction of mine backfill techniques and an innovative mine tailings processing method that can reduce the environmental and safety problems caused by the accumulation of tailings For the past few decades, several types of fibres, including steel, polypropylene, glass and carbon fibres, have usually been used to improve the stability of structures and ductility of concrete in civil engineering and construction industry In mining engineering, many researchers have also used fibres materials in the backfill and support measures field Therefore, this experimental study on tailings CPB was carried out to investigate the influence of PP fibre on the compressive behavior of CPB, and orthogonal tests were designed to research the influences and significance of the cement content, solid mass concentration, fibre content and fibre length on CPB. Macrostructural failure analysis and microstructural tests by scanning electron microscopy (SEM) were used to study the crack development and structural changes of CPB.The Fan Kou lead–zinc mine, located in the northeastern Guangdong Province of China, provided all the tailings for this study. The tailings were collected in ore processing plants, stored in barrels and transferred to the lab of Central South University. Different tailings samples were then taken for tailings size distribution determination by an LS particle size analyzer (LS13320, Beckman, USA), chemical composition measurement using an X-ray fluorescence spectrometer (XRF) (ZSX Primusll Ⅱ, Rigaku Corporation, Japan) and mineralogical composition analysis with an X-ray diffractometer (XRD) (Ultima Ⅳ, Rigaku Corporation, Japan). displays the particle size distribution of the tailings, and the results show that the mean particle size of the tailings is 39.30 μm. The tailings are medium with proportions of more than 47.15% for particles less than 20 µm in diameter lists the chemical composition measurement results (XRF) of the tailings, from which one can observe that the content of SiO2 is 27.10%, and the total content of main oxides (Al2O3, SiO2, Fe2O3, MgO, and CaO) amounts to 81.98%. The XRD patterns () show that the main mineralogical compositions of tailings are pyrite (FeS2), muscovite [(K,Na)(Al,Mg,Fe)2(Si3.1Al0.9)O10(OH)2], dolomite [CaMg(CO3)2], silicon oxide (SiO2) and calcium carbonate (CaCO3). The XRD results are the same as the XRF results. The geotechnical characterization of tailings is summarized in , and based on these results, the tailings are classified as CL – lean clay Monofilament polypropylene (PP) fibres (C3H6) were used to reinforce the CPB, and presents their main properties. These PP fibres have an average diameter of 19 µm, a density of 910 kg/m3, good dispersion, very high acid and alkali resistance and null moisture absorption and toxicity. Also according to , these PP fibres have a fracture strength of greater than 350 MPa and an elastic modulus of greater than 3.5 GPa, which makes it possible to improve the mechanical properties of CPB by mixing it with PP fibre. shows the shape of PP fibres of different lengths (3 mm, 6 mm, 9 mm, and 12 mm) used in this study. However, adding fibre will increase the cost of backfill. Therefore, the relatively small fibre contents in this study were 0.05%, 0.1%, 0.15% and 0.2% by mass of the sum of tailings and cement. Both the fibre length and content effect on the CPB will be tested.Portland cement P.O 42.5R (where “R” stands for initial high strength cement, with a minimum compressive strength of 22.0 MPa and 42.5 MPa at 3 and 28 days, respectively) was used as the binder. The CPB specimens with 14%, 17%, 20% and 25% cement content by mass of tailings were tested to compare the contributions of cement to the mechanical properties of CPB. Moreover, to reproduce the site conditions as accurately as possible, the tap water was utilized to prepare all the test specimens.Orthogonal tests are a method for designing a study to determine the influences of multiple factors at multiple levels . Tailings, cement, fibre and water with different proportions as shown in were mixed for preparing specimens. First, the dry tailings and cement were mixed in laboratory cement mortar blenders (JJ - 5, Wuxi Jianyi Experiment Instrument Co. Ltd, China) for 5 min, and fibres were added after the beginning of mixing to avoid the floating of fibres. Then, water was slowly added to the blenders and mixed for another 5 min. Sixteen mixtures and 144 triplicate specimens () were poured into cylindrical plastic molds 50 mm in diameter and 100 mm in height for fibre-reinforced and unreinforced specimens, respectively. The prepared specimens were sealed and cured in a standard curing box with a temperature of 22 ± 1 °C and ≥90% relative humidity for curing times of 3 days, 7 days and 28 days.Before the researching the mechanic properties of fibre-reinforced CPB, an important job is to evaluate the fluidity of backfill. The slump test was originally developed for the determination of the fluidity by the standard slump instrument which is a tapered cylinder with a height of 30 cm, an upper diameter of 10 cm, and a bottom diameter of 20 cm Unconfined compression tests are used frequently for concrete and cemented soils, and much experience regarding them has been accumulated. Additionally, the unconfined compression strength (UCS) is a main strength index used to evaluate the strength performance of CPB Scanning electron microscopy (SEM) analysis was used to research the influence of PP fibre on the microstructural properties of CPB specimens and to observe the connection method between the PP fibre and CPB matrix In this study, the slump of unreinforced backfill and fibre-reinforced backfill are 19.5–21.6 cm and 19.8–22.9 cm, respectively. It is demonstrated that fibre-reinforced backfill can be transported to the stopes smoothly. The value of the slump of unreinforced backfill and fibre-reinforced backfill are very close, and fibre dosen’t reduce the fluidity of the backfill. Moreover, fine tailings backfill has been reported having some beneficial effects of little wear for the pipeline, no segregation and reducing pipeline blocking The results of UCS tests are presented in . This figure demonstrates that the UCS of unreinforced CPB is low, the value of which for almost all specimens does not exceed 1.0 MPa with long curing periods except for the mixtures with a high cement content of 25%. This is close to the conclusion reached in previous research , the maximum UCS of unreinforced CPB at 3 days, 7 days and 28 days is 1.02 MPa, 1.41 MPa and 1.94 MPa, respectively. After adding PP fibre, the maximum UCS increased to 1.84 MPa, 1.57 MPa and 2.41 MPa at 3 days, 7 days and 28 days curing time. The percentages of maximum UCS improvement for the three curing times are 80.39%, 11.35% and 24.23%. All the mixtures of fibre-reinforced CPB preform strength increasing except few points, and the biggest growth even reach 469.57%. The average improvement of UCS at 3 days, 7 days and 28 days are 213.11%, 43.96% and 143.45%, respectively. This phenomenon is due to the reinforcement of the PP fibre. Flexible fibres will ensure good particle-to-particle contact, and PP fibre has high fracture strength (greater than 350 MPa). This makes the CPB denser and makes it exhibit a higher strength and better integrity to resist external pressure. also indicates that the mixture with comparatively higher cement content and solid mass concentration is stronger than the mixture with a lower cement content and has a better bonding effectiveness and strength. Based on , the longer the curing time is, the greater the strength of CPB; the enhancements of CPB at 3 days and 28 days are better than that at 7 days. This may be related to the hydration process in the CPB. In particular, the effect of PP fibre on the early and later stage strengths of the backfill is much more obvious than that on the medium stage strength.Two sets of representative stress-strain curves of T2 and T6 were presented in , and these curves also shown the influence of fibre on the stiffness and strain of CPB. The strain of mixture T2 (No Fibre), T6 (No Fibre), T2 (PP Fibre) and T6 (PP Fibre) at peak stress are 0.68%, 0.82%, 0.79% and 1.01%. Which indicated that the CPB matrix exhibited brittle failure in unconfined compression test with low strain, and the strain at peak stress had obvious improvement with fibre. Therefore, fibre can enhance the ductility of CPB. Cristelo et al. Macrostructural failure is the most intuitive way to observe the failure mode of CPB. As shown in , the failure modes of fibre-reinforced CPB and unreinforced CPB with the same cement content and solid mass concentration are completely different. After the compressive destruction, several tiny cracks appeared on the surface of the fibre-reinforced CPB specimen. Although it has exceeded the ultimate compressive strength, the fibre-reinforced CPB specimen maintains integrity and residual strength. Some large cracks and fracture zones formed in the unreinforced CPB specimen. Bulks even separated from the matrix. The fibres tend to bridge the cracks and prevent crack propagation, thus preventing premature failure and improving the strength and ductility of CPB.Orthogonal range analysis (Rj) is the most commonly used method in the analysis of the results of orthogonal tests Rj=max(Kj1‾Kj2‾,⋯,Kjm‾)-min(Kj1‾Kj2‾,⋯,Kjm‾)where Rj is the range of the test index as the level of column j changes, and the greater the value of Rj is, the greater the impact of the factor on the test index and the more important it is; Kjm represents the average value of the level of column j.Therefore, the order of the factors can be judged according to Rj. In this study, orthogonal range analysis was further conducted to research how the cement content (A), solid mass concentration (B), fibre content (C) and fibre length (D) affected UCS. The results showed that the values of UCS were affected by the pressing parameters in the following order: A > B > C > D. Based on the results of UCS, it is indicated that the fibre parameters are not the most important factors for strength, but they are also important factors for improving the UCS of CPB. Moreover, the fibre content is much more important than the fibre length., the single factor analysis of how the cement content and solid mass concentration affect the average UCS shows that the average UCS increases monotonically with the increase of A (cement content) and B (solid mass concentration) at each curing time. This is in line with the conclusion of The influence of fibre on the average UCS is shown in . For the fibre content, with much more fibre being added to the CPB, the average UCS exhibited a higher value at 3 days curing time and first decreased and then increased at 7 days and 29 days curing time. The fibre content level of 0.15% had the best effect on the UCS, and the UCS reached 1.40 MPa at 28 days curing time. For the fibre length, at 3 days curing time, the average UCS enhancement increased and then decreased with the increase of the fibre length. When the curing time reached 7 days and 28 days, the effect of the fibre length was not obvious. The best fibre length was 6 mm.These results suggested that the best fibre parameter levels are a fibre content of 0.15% and a fibre length of 6 mm. The results illustrate that more fibre content or a longer fibre length does not necessarily lead to a higher UCS. Because fibre can reinforce the interaction of the tailings particle-particle structure and improve the strength of CPB, it also generates a weak structure surface in the interior of CPB at the same time. When CPB is subjected to external compression, too many weak structure surfaces will lead to CPB failure before the compression reaches the peak stress, resulting in a decrease in CPB strength.SEM is a good way to understand the relationship between the UCS and microstructure of CPB specimens. SEM micrographs at the 1 mm level are presented in , showing a piece of fibre-reinforced CPB specimen after 28 days of curing with a fibre content of 0.15% and a fibre length of 6 mm. It is obvious that in the middle of the piece, a penetrating crack is formed, which divides the specimen into the two parts of A and B. These two parts are still part of a whole because of the PP fibre that goes through and connects them, similar to a bridge. Based on , the interconnection function of PP fibre is verified from two macro and micro aspects. There will be a dense fibre sheet in CPB specimens if sufficient fibre is used, and the fibre sheet can prevent CPB from being effectively destroyed.In general, after mixing tailings and cement, the coarse particles form the skeleton of the CPB matrix, and the fine particles and the hydration products fill in the voids. The major hydration reaction of calcium silicate hydrate (C-S-H) gelling can be expressed as Eq. 3CaO·SiO2+2CaO·SiO2+H2O→2CaO·SiO2·nH2O+CaOH2 shows SEM micrographs of interaction among the tailings, C-S-H and fibre at 50 µm. present the part of the fibre detached from the CPB matrix. PP fibre’s high fracture strength (>350 MPa) and high elastic modulus (>3.5 GPa) give it strong flexibility. When specimens were under pressure, fibre occur elastic deformation without breaking which also a process of energy absorption. Then some fibre were pulled out or broke after preventing the destruction and separation of CPB. As shown in , fibres were wrapped and fixed in particle-binder matrix under the action of high viscosity and high strength C-S-H, and the bare part of the fibres also shows a detailed views of C-S-H is trapped on the surface of the PP fibre. This help to give full play to the tensile strength of the PP fibre and increase the UCS of specimens.The SEM micrographs of CPB specimens of different cement contents (a: 25%; b: 20%; c: 17% and d: 14%) are shown in . With the decrease in cement, more pores appear, and less C-S-H gelling is generated. C-S-H gelling has been generally accepted as the major load-bearing phase in CPB shows that a high cement content is beneficial to the hydration reaction and CPB strength. shows that with the addition of PP fibre to CPB, the UCS strength is less than 2.5 MPa, and the strength of tailings CPB is not ideal. Based on , there is some pyrite (FeS2) in the tailings, and the sulfur content is 11.90%. The sulfate can undergo a chemical reaction in the water, make the slurry acidic and form sulfate radicals (Eq. ). Acid and sulfate attacks can restrain C-S-H gelling generation and increase the secondary production ettringite. Eventually decrease the strength of CPB (Eq. 3CaO·Al2O3+3CaSO4·2H2O+30H2O→3CaO·Al2O3·3CaSO4+32H2OFor the economic cost, there will be an increase in cost with fibre adding when the cement to tailings ratio is not changed. But, the increase of UCS based on PP fibre is also obvious shown in . If the purpose is to keep the UCS as unreinforced CPB, the adding of fibre can reduce the consumption of cement, and the cost to prepare backfill with fibre is cheaper than backfill without fibre. Moreover, decreasing cement content will increase the consumption of tailings, and which will help to further improve the recovery rate of tailings. In general, fibre-reinforced backfill has better overall benefits.In this study, a series of experiments were conducted to research the reinforcement of fibre on CPB. By the UCS tests, the strength of fine waste tailings backfill was confirmed low, and PP fibre can improve the UCS. Its increasing rate can even reach more than 4 times. And the result also indicated that the effect of PP fibre on the early and later stage strengths of the backfill is much more obvious than that on the medium stage strength. These are all benefited from the high fracture strength of PP fibre which can prevent backfill from being damaged by compression stress. Moreover, fibre can improve the strain at the peak strength of CPB significantly, and the improvement of stiffness enhanced the ability of the backfill to resist elastic deformation. At the same time, this effect is enhanced due to close interaction between PP fibre and C-S-H with the SEM analysis. With high fracture strength of PP fibre, the development of the cracks in the backfill are suppressed. And these results suggested that the best fibre parameter levels are a fibre content of 0.15% and a fibre length of 6 mm in this study. Moreover, PP fibre will not deteriorate the workability of backfill. However, with regard to UCS, there still have to choose between mechanical reinforced and fibre cost in engineering applications. Therefore, future studies will investigate the effects of reinforcement with fibres of different parameters in CPB and other mechanical properties of CPB. Moreover, the influence of the sulfur content on tailings should also be investigated.Work hardening behavior of dual phase copper–iron alloy at low temperatureIn-situ neutron diffraction measurements were performed on a cold-rolled copper-iron (Cu–Fe) alloy during tensile tests at 293 K and 150 K. The alloy was composed of face-centered cubic Cu and body-centered cubic Fe. The roles of Cu and Fe on the deformation behavior of alloys were discussed and clarified. The strength and work-hardening rate of the alloy increased with decreasing test temperature. Furthermore, the phase stress of Fe increased considerably with decreasing test temperature; however, the response of this stress to the applied true stress exhibited no dependence on the temperature. The phase stresses of Cu changed only slightly with decreasing test temperature. However, the Cu phase stress response to the applied true stress increased with decreasing test temperature, indicating an increase in the work-hardening rate. The strengthening of Fe and the increase in the work-hardening of Cu contributed to an increase in the strength and work-hardening rate of the Cu–Fe alloy at low temperatures.Dual-phase alloys consisting of phases with different crystal structures are characterized by unique properties, resulting from a combination of constituent phase properties, and have been employed in many industrial components. Cold-rolled Cu–Fe alloys are typical dual-phase alloys consisting of Cu with a face-centered cubic (FCC) structure and Fe with a body-centered cubic (BCC) structure. These alloys have been used in many industrial products, such as electromagnetic shield construction materials, information tags, and conductive spring materials, because of the high conductivity derived from Cu and magnetism derived from Fe.] revealed that Cu–Fe alloys exhibit an excellent balance of strength and ductility at temperatures below room temperature. The temperature dependence of the mechanical properties is closely related to the crystal structure []: the yield stress and tensile strength of BCC materials increase with decreasing temperature [], but change or increase only slightly for FCC materials []. In addition, elongation decreases with decreasing temperature in the case of BCC materials [], but increases in the case of some FCC materials [], we attributed the excellent strength-ductility balance of the Cu–Fe alloy at low temperatures to two independent but mutual contributions from Cu and Fe; Cu contributed only to an increase in elongation, and Fe contributed only to that in strength. However, the deformation behavior of Cu and Fe in the Cu–Fe alloy differs from that of single-phase materials. The results revealed that Cu deformed simultaneously with Fe, and Fe retained sufficient deformability at temperatures below its ductile-to-brittle transition temperature. To confirm the occurrence of independent (but mutual) contributions by Cu and Fe, we investigated the stress evolution and deformation behavior of the individual constituents at room temperature and low temperatures.In-situ neutron diffraction during tensile deformation represents an effective method for the simultaneous evaluation of the deformation behavior exhibited by multi-constituent phases in various dual-phase alloys []. The roles of individual constituent phases during deformation can be clarified by evaluating the phase stresses induced by changes in the lattice constant during deformation. In a dual-phase steel consisting of ferrite (softer phase) and martensite (harder phase) [], ferrite undergoes plastic deformation prior to macroscopic yielding, and martensite accommodates elastic deformation after macroscopic yielding. Regarding duplex stainless steel consisting of ferrite (harder phase) and austenite (softer phase) [], the material undergoes significant work-hardening, owing to the preferential work-hardening of austenite during deformation. Austenite contributes only modestly to the strength of multiphase low-alloy steels containing bainitic ferrite, metastable austenite, and deformation-induced martensite []. The strength is mainly a result of the bainitic ferrite and deformation-induced martensite, and the stress contribution of metastable austenite is small as its phase fraction decreases with the martensitic transformation []. Through its transformation to martensite, metastable austenite plays a role in improving elongation []. Determining the role of constituent phases in deformation is therefore essential for elucidating the deformation behavior of multiphase alloys.In this study, the deformation behavior of individual constituent phases, stress partitioning behavior between the phases and among the grains, and changes in the crystal orientation in a cold-rolled Cu–Fe alloy were investigated. To this end, in-situ neutron diffraction measurements were performed during tensile tests at and below room temperature. The stress contributions of the constituent phases to the strength and the role of these phases on the mechanical properties of the alloy were discussed.A commercial cold-rolled Cu-40 mass%Fe alloy (CFA-60) was investigated. The Cu–Fe alloy described in Ref. [] was annealed at 1123 K for 1800 s followed by water-cooled. The configuration of the tensile test specimen is shown in a. Using an electric discharge machine, a plate-type specimen (gauge length: 25 mm, width: 4 mm, and thickness: 1.9 mm) was prepared such that the longitudinal direction (loading axis direction; LD) was parallel to the rolling direction (RD). The specimen surface was ground with various grits of silicon carbide emery paper up to 2400 grit, and then polished using a buff. The microstructure was characterized via scanning electron microscopy (SEM) coupled with electron backscatter diffraction (EBSD) performed on the normal direction (ND) section. The sample for microstructure observation was taken from the specimen. The sample surface on the ND was ground and polished using a colloidal silica slurry. The EBSD data were collected from a 150 μm2 area (step size: 0.2 μm) and were then analyzed with the OIM™ analysis 7.0.1 software. Data points with a confidence index value of <0.5 were eliminated as noise.In-situ neutron diffraction measurements were performed during the tensile tests at 293 K and 150 K using TAKUMI [], a time-of-flight diffractometer dedicated to the investigation of engineering materials at the MLF of J-PARC. The test temperature was controlled with a liquid nitrogen cooling system [] and was monitored using two thermocouples attached to the grip parts of the specimen. Schematics of the experimental setups at 293 K and 150 K are shown in b and c, respectively. The tensile loading direction was aligned horizontally 45° with respect to the incident neutron beam. The macroscopic strains were measured using an extensometer at 293 K and a strain gauge at 150 K. Neutron diffraction data for the scattering vector parallel to the LD were collected at the loading detector, while data for the scattering vector perpendicular to the LD were collected at the transverse detector. For the tensile loading test, the transverse direction (TD) was oriented horizontally at 293 K (c). A 5 mm (width) × 5 mm (height) incident beam slit was employed, and a pair of radial collimators (viewing width: 5 mm) was adopted. These neutron diffraction experiments were performed at a neutron beam power of 300 kW. The tensile test at 293 K was conducted in step-increasing load control mode (holding time: 300 s) in the elastic region and continuously with a constant crosshead speed (initial strain rate: 3.3 × 10−5 s−1) in the plastic region. The tensile test at 150 K was conducted continuously with two crosshead speeds (strain rate up to 1% strain: 1.3 × 10−5 s−1, after 1% strain: 3.3 × 10−5 s−1). Moreover, the neutron diffraction measurement data for the plastic region were sliced in 300 s or 600 s, and the tensile tests were performed when the ultimate tensile strength was reached.The lattice spacings of several hkl planes in LD were obtained using a single peak-fitting method available in the Z-Rietveld software []. The phase fractions of Cu (fCu) and Fe (fFe) were obtained by averaging the phase fractions determined from the Rietveld refinements applied by the software. To minimize the influence of texture, refinements were applied to the diffraction patterns for the RD, TD, and ND. The hkl lattice strains in the LD (εi,11hkl, i = Cu and Fe, 11 refer to direction parallel to LD) were estimated as follows:where, di,11hkl is the lattice spacing of the hkl plane obtained during deformation, and di,11,0hkl is the reference lattice spacing of the hkl plane before deformation. The di,11,0hkl value at 150 K was the lattice spacing measured at 150 K before deformation. A tensile stress of about 20 MPa was applied and maintained to be constant during cooling to 150 K to avoid increases in the external force due to the heat shrinkage. After the test temperature reached 150 K, the di,11,0hkl value at 150 K was measured while maintaining 20 MPa. Phase stresses can be evaluated from the phase average lattice strains (phase strains), in accordance with Hooke's law. Phase strains are often estimated using the average lattice constants determined from the Rietveld refinement or the Pawley method; however, the absorption of intergranular stresses during loading tests is sometimes difficult []. In this study, εCu,11311 and εFe,11211 were adopted for evaluating of the phase stresses associated with Cu and Fe, respectively, because the phase strains determined in the plastic region were less accurate than these values. Moreover, the lattice strain values estimated from the 211 plane of the BCC and the 311 plane of the FCC structures reflect the average macroscopic elastic strains of BCC and FCC polycrystalline materials, respectively []. Ideally, the conversion from elastic strains to stress values requires six strain tensors. Two strain tensors, i.e., the strain tensors parallel and perpendicular to LD, are sufficient for uniaxial tensile loading. In this study, one lattice strain tensor of εi,11hkl was used for the phase stress conversion, because family grains contributing to εi,11hkl differ from family grains contributing to εi,22hkl (22 refers to direction perpendicular to LD), and the geometries for neutron diffraction measurements of the strain tensors perpendicular to LD at 293 K and 150 K were different. When the εi,22hkl is not available, the phase stresses in the loading direction of Cu (σCu,11) and Fe (σFe,11) can be determined using Eqs. Here, ECu311 and EFe211 denote the diffraction elastic constants of Cu-311 and Fe-211, respectively, and are nearly equal to the respective macroscopic elastic constants of Cu []. Therefore, previously reported values of these constants [). Inverse pole figures before and after deformation were analyzed using “MAUD” [] software package from the diffraction patterns for the directions parallel and perpendicular to LD. shows a secondary electron image, and a phase map obtained via SEM with EBSD measurements of the Cu–Fe alloy considered in this study. The alloy consisted of Cu, Fe, and some inclusions (indicated by white arrows in ], the inclusions were identified as iron oxides through energy dispersive X-ray spectroscopy. The Cu and Fe grains formed a layered structure that extended along the RD. According to our previous study [], both Cu and Fe consist of fine grains (diameter: <1 μm). In addition, Cu precipitated inside Fe, and vice versa. However, small precipitates were hardly observed in b due to the measurement was carried out with low magnification, SEM image (a) is similar with the SEM image in the previous study []. Thus, in accordance with this previous study, small white precipitates observed in the Fe grains (a). Although our present Cu–Fe alloy is a commercial one and the fabrication process is not sure, the formation mechanism of layer structure can be supposed as follows: first, dendritic crystallization occurs preferentially in Fe phase because of its higher coagulation temperature than Cu phase. The Fe dendrites should align along a certain direction relating to heat distribution. The crystallization of the Cu phase should occur between Fe dendrites, resulting in alternating layer structures of Fe and Cu phases. Next, the dendrites rotate by rolling and extends in the processing direction. Consequently, the layer structure was observed in the present alloy.The initial diffraction patterns for the direction parallel or perpendicular to LD (before tensile deformation) at 293 K and 150 K are presented in . Peaks corresponding to Cu, Fe, and the inclusion (low-intensity peaks indicated by black arrows in the figure) were detected. The peak was attributed to wüstite (FeO) based on Rietveld refinement. The initial phase fractions of Cu, Fe, and FeO were 58.0 mass% (54.8 vol%), 41.5 mass% (44.5 vol%), and 0.5 mass% (0.7 vol%), respectively. The peak intensities of Cu-111 and Fe-110 parallel to the RD and TD were high, reflecting a rolling texture. In this study, the εi,11hkls values were evaluated only for Cu and Fe, as the extremely low peak intensities of FeO prevented the accurate evaluation of these quantities.a shows the nominal stress (σN)–nominal strain (εN) curves of the Cu–Fe alloy at 293 K and 150 K. The values of the yield stress (0.2% proof stress) and tensile strengths are summarized in . The 0.2% proof stress increased with a decrease in the test temperature. In the σN–εN curve corresponding to 293 K, a stress plateau region, which may be associated with local deformation, was observed after yielding and continued to exist up to a strain of 1.6%. At 150 K, stress fluctuations, which may also be associated with local deformations, were observed in the region after yielding and lasted up to a strain of 3.2%. These regions are hereafter referred to as “local deformation regions.” At 150 K, work-hardening also occurred in the local deformation region, although the work-hardening rate was low. b shows the curves of work-hardening rate (dσT/dεT) and the true stress (σT) with respect to the true strain (εT) at 293 K and 150 K. The tensile strength and work-hardening rate after the local deformation region increased substantially with decreasing temperature. The increase in the work-hardening rate induced by lowering the temperature will be discussed in a subsequent section.a shows the σCu,11, and σFe,11 values as a function of the applied true strain (εapp). The neutron diffraction data for the local deformation region at 293 K was extracted or binned to be one diffraction pattern for the direction parallel or perpendicular to LD, because the local deformation occurred for only about 5 min in the tensile loading with the strain rate used in this study (about 3.3 × 10−5 s−1). Cu yielded significantly earlier than Fe at both test temperatures. Moreover, the σCu,11s values were obtained at yielding (σCuYSs) values of 41 MPa and 44 MPa at 293 K and 150 K, respectively. The values of σCuYS and σFe,11 at yielding (σFeYS) are compared in . The σFeYS values were considerably higher than the σCuYS values, and increased significantly when the test temperature was lowered. At 293 K, after local deformation, the σCu,11 and σFe,11 values increased with increasing applied true strain. In the local deformation region at 150 K, the σCu,11 value decreased, whereas the σFe,11 value continued to increase with applied true strain; however, these trends oppositely changed at a true strain of 3.2%. These trend changes in the local deformation region at 150 K may be associated with the passage of local deformation, such as Lüders bands. The length of the specimen (25 mm) was substantially larger than the that of the measured area (~7 mm), which was located in the middle region of the specimen. Therefore, some local deformations might occur outside the region monitored for the neutron diffraction measurements. A distribution of lattice strains within the parallel section of the tensile test specimens was reported for medium manganese steels with large ranges of Lüders deformation bands, by a mapping measurement using synchrotron X-ray diffraction with a fine beam []. In addition, Lüders deformation has been confirmed in fine-grained pure Cu and single-phase Fe materials []. We considered that local deformation such as Lüders bands may occur in the present Cu–Fe alloy because the grain sizes were quite fine []. After local deformation at 150 K, the σCu,11 and σFe,11 values increased with increasing applied true strain. The σCu,11 values above 7% strain increased only slightly with decreasing test temperature, indicating that the work-hardening of Cu at 150 K may be higher than that at 293 K.b shows the σCu,11 and σFe,11 values as a function of the applied true stress (σapp). PCu and PFe in b indicate macroscopic stresses when Cu and Fe yielded, respectively. The phase stresses of Cu and Fe corresponding to the PCu and PFe values are denoted as σCuYS and σFeYS, respectively. Furthermore, the 0.2% proof stresses were substantially larger than the PCu values and nearly equal to the PFe values at both test temperatures. σCu,11s and σFe,11s increased with σapp at the beginning of tensile testing. When σapp exceeded PCu, the increase in the σCu,11 value with σapp became small, whereas the increase in σFe,11 became large, owing to the preference for plastic deformation of Cu. These trends continued until σapp reached to PFe. For true stress values exceeding PFe, the increase in σCu,11 became large, whereas the increase in σFe,11 consequently became small as Fe was also plastically deformed.a and b shows the changes in εCu,11hkl and εFe,11hkl for several hkl planes in the LD as a function of σapp at 293 K and 150 K. The variations in the εi,11hkl values reflect elastic anisotropy. At both temperatures, the dependence of the εCu,11hkl and εFe,11hkl values on the applied stress were considerably similar to those of the σCu,11 and σFe,11 values shown in b. Among the εCu,11hkls values, εCu,11111, εCu,11222, and εCu,11220 were similar and minimum, while εCu,11200 and εCu,11311 were maximum and intermediate, respectively. Among the εFe,11hkls values, εFe,11110 and εFe,11220 were similar and minimum, while εFe,11200 and εFe,11211 were maximum and intermediate, respectively, for a given applied stress. Furthermore, at around PCu the values of εCu,11111, εCu,11222 and εCu,11220 decreased, whereas the value of εCu,11200 increased, indicating that plastic deformation occurred preferentially in the 111, 222, and 220 family grains (see c and d). εCu,11hkls exhibited similar trends after PCu had been surpassed. In Fe, at around PFe, plastic deformation occurred preferentially in the 110 and 220 family grains. These results indicate that intergranular stress partitioning occurred in Cu and Fe, possibly owing to differences in the diffraction elastic constant and plastic deformability between hkl family grains []. The responses of εCu,11111s and εCu,11222s to the applied stress can be used to determine whether stacking faults were introduced into Cu during loading at 150 K. When stacking faults occur during deformation, the lattice strains of equivalent index planes, such as 111 and 222 or 200 and 400, exhibit different responses to the applied stress, as observed in high-entropy alloys []. However, in this study, significant overlap of the εCu,11111 and εCu,11222 values occurred in the entire deformation region (b). This suggests that only a low density of stacking faults was generated in the Cu component of the Cu–Fe alloy at 150 K.The stress contributions of Cu and Fe to the strength (σicont,i= Cu and Fe) can be evaluated by weighting σCu,11s, and σFe,11s with the fCu and fFe values, respectively. The results presented in a and b shows the σCucont and σFecont values as a function of the applied true strain at 293 K and 150 K, respectively. The sum of the σCucont and σFecont values at both test temperatures, although slightly lower because of neglecting the FeO contributed stress, were nearly equal to the σapp as indicated by the black solid lines in the figure. Thus, the lattice strain and phase fraction evaluations and the stress conversions employed in this study were sufficiently accurate. At both test temperatures, the σFecont values were considerably higher than the σCucont values for the entire deformation region, although the fFe value was lower than the fCu value. The σCucont values decreased slightly after yielding and in the local deformation region, because of a preference for plastic deformation in Cu, but then increased monotonically. Furthermore, the σFecont values increased significantly when the test temperature was lowered, but the σCucont values increased only slightly.The effects of lowering the temperature on the stress contribution were further investigated by normalizing the σCucont or σFecont values to the σapp (as shown in c). The stress contribution of Fe to the strength was dominant at both test temperatures. The stress contribution of Cu decreased slightly when the test temperature was lowered, whereas the contribution of Fe increased considerably. This indicates that the large increase in strength at 150 K resulted mainly from a strengthening of Fe induced by lowering the temperature. After the local deformation region (after 1.6% strain at 293 K and 3.2% strain at 150 K), the stress contribution of Fe seemed to decrease slightly with increasing applied true strain. In contrast, the stress contribution of Cu increased. This suggests that the work-hardening behavior of Fe and Cu was modified by lowering the test temperature.The true stress–true strain curve in the plastic region can be described by the Hollomon equation:where, K is the strength coefficient (constant), and n (n-value) is the work-hardening exponent. Equation can be rewritten in the logarithmic form as follows:The work-hardening exponent of each constituent phase can be evaluated from Eq. shows the logarithmic values of σCu,11s and σFe,11s plotted against the logarithmic value of the applied true strain for strain values ranging from 3.2% to 18%. The linear fits and the n-values of Cu and Fe at both test temperatures are also shown in . The fitted n-values are comparable to those of some fine-grained ferritic steels tensioned at room temperature (0.1–0.3) [] and fine-grained pure Cu tensioned at room temperature (0.2–0.4) []. The n-values of Cu were substantially larger than those of Fe, indicating that the work hardening rates of Cu were higher than those of Fe. The n-value of Fe changed only slightly when the test temperature was lowered, but the n-value of Cu increased to >0.5. These results show that the increase in the work-hardening rate of Cu at 150 K increased the stress contribution of Cu to the strength during deformation, thereby increasing the work hardening of the Cu–Fe alloy at 150 K.The full width at half maximum (FWHM) values in the LD of several hkl peaks of Cu and Fe during deformation were evaluated and normalized their peak position. The FWHM values of several hkl peaks of Cu (FWHMCuhkl) and Fe (FWHMFehkl) were considered with respect to the σapp at both test temperatures, as shown in a and b, respectively. The FWHMCuhkl values started to increase slightly with σapp after PCu and then significantly after PFe. The FWHMFehkl values remained unchanged from the beginning until around PFe, and then increased when the applied stresses exceeded PFe. These trends can be explained by considering the preferential occurrence of plastic deformation, as previously described in Section . The increment of the FWHMCuhkl values was higher than that of the FWHMFehkl values (as discussed in Section ), and hence the work-hardening rate of Cu was higher than that of Fe. The FWHMFehkl values obtained for temperatures of 293 K and 150 K differed only slightly, but the FWHMCuhkl values after PFe increased considerably with decreasing test temperature. The FWHM value is typically related to the dislocation density. However, the FWHM value measured under loading can also be induced by non-uniform distribution of stresses such as phase and intergranular stresses. Therefore, changes in the integrated intensities of several hkl peaks of Cu and Fe were also evaluated to confirm that the changes in the FWHM values resulted from the changes in dislocation density.The relative integrated intensities in the LD of several hkl peaks corresponding to Cu (ICuhkl) and Fe (IFehkl) were evaluated. During this evaluation, the hkl integrated intensities obtained during deformation were normalized to those obtained before deformation and to the specimen cross-section reductions during deformation. The results exhibited in a and b shows the relative integrated intensities of the 111 and 110 peaks corresponding to Cu (ICu111) and Fe (IFe110), respectively, for the LD as a function of σapp. After PCu and PFe, the ICu111 and IFe110 values started to increase with increasing applied stress, indicating the development of a tensile texture. An evolution of stacking faults during deformation at low temperatures had previous been observed in some FCC alloys []. This was confirmed by the decrease in the absolute value of the slope describing the relationship between the integrated intensity of the hkl peak in the LD and the applied stress during tensile deformation []. However, the ICu111 and IFe110 values increased monotonically with applied stress after PFe (see ), suggesting that no stacking faults were introduced into the Cu–Fe alloy during tensile testing in this study. Lines for approximating the linear relations between the ICu111 and IFe110 values and σapp are inserted in . For the line approximating the ICu111–σapp relationship, the slope of ~0.0032 increased to ~0.0045 when the temperature decreased from 293 K to 150 K. In contrast, the slope of the line describing the IFe110–applied true stress relationship decreased slightly from 0.0026 to 0.0025. This may be a result of the fact that the amount of dislocation glide and the level of plastic deformation in Cu increased with decreasing test temperature, leading to an increase in the Cu work-hardening rate. The slight decrease in the slope of the IFe110–σapp plot may be a consequence of a decrease in the ductility of Fe with decreasing temperature. The increase in strength and the decrease in ductility of Fe at 150 K could the result of a slight increase in the load shared by Cu, thereby increasing the amount of dislocation glide occurring in Cu. shows inverse pole figures before and after deformation of Cu and Fe. At 293 K (a and b), the probability of 111-Cu and 110-Fe for the RD which is parallel to the LD increased after deformation. At 150 K, the similar trends in the texture changes were observed. The increase in the probability of 111-Cu by deformation was however larger at 150 K than at 293 K. The initial texture of Fe in the specimen used for the tensile test at 150 K was slightly stronger than that at 293 K, however, the increase in the probability of 110-Fe by deformation was almost similar regardless of the test temperature. Thus, the degree of plastic deformation of Cu can be considered to be more pronounced at 150 K, supporting our finding that the amount of dislocation glide in Cu increased by lowering temperature.], the strain distribution in the microstructure on the surface of a Cu–Fe alloy was investigated using a digital image correlation method. The results revealed that the strain distribution became inhomogeneous with decreasing temperature, which may be caused by restricting of slip systems in Fe. The critical resolved shear stress of Fe was also reported to increase significantly with decreasing temperature [], which may indicate that the plastic deformability of Fe decreases as the temperature decreases. These results support our findings in this study. In addition, layer structure may also contribute to good elongation of Cu–Fe alloy at low temperature. A multilayered steel combining as-quenched martensitic steel and high-ductility steel showed good elongation due to suppress of brittle fractures of martensite [], Fe in Cu–Fe alloy has enough ductility even at 8 K. Furthermore, in another previous study [], it reported that Cu–Fe laminated sheets showed excellent tensile properties at low temperature may resulted from strain accommodation in Fe layers by soft Cu layers. Therefore, the good elongation of Cu–Fe alloy at 150 K may associate with not only increasing work-hardening of Cu but also suppressing of brittle fracture of Fe by fine grained and multilayered microstructure of Cu–Fe alloy.The stress partitioning behavior and stress contribution to the strength of Cu and Fe in a cold-rolled Cu-40 mass%Fe alloy during tensile deformation at 293 K and 150 K were investigated using in-situ neutron diffraction. The roles of Cu and Fe in the deformation behavior at both test temperatures of the alloy are discussed. The main results are summarized as follows:The strength and work-hardening rate of the Cu–Fe alloy increased with decreasing test temperature.During deformation, the phase stress of Fe was considerably higher than that of Cu. By lowering the temperature, the phase stress increased significantly in Fe, but increased only slightly in Cu. The strengthening of Fe induced by lowering the temperature contributed substantially to the increase in strength during the 150 K deformation.The work hardening rate of Cu was higher than that of Fe, and increased with decreasing test temperature, leading to an increase in the work hardening rate of the Cu–Fe alloy during the 150 K deformation.The strengthening of Fe and the increasing work-hardening of Cu contributed to an increase in the strength and work hardening rate of the Cu–Fe alloy at low temperatures.Changes in the FWHMs and integrated intensities of the Cu peaks suggest that the increase in the work-hardening rate of Cu at 150 K deformation stemmed from the increase in the activity of dislocation glides due to lowering the test temperature.Takayuki Yamashita: Conceptualization, Formal analysis, Investigation, Data curation, Writing – original draft, Writing – review & editing, Visualization, Project administration, Funding acquisition. Norimitsu Koga: Conceptualization, Investigation, Writing – review & editing, Visualization, Funding acquisition. Takuro Kawasaki: Resources, Investigation, Writing – review & editing, Visualization. Satoshi Morooka: Resources, Investigation, Writing – review & editing, Visualization. Shohei Tomono: Conceptualization, Investigation, Visualization. Osamu Umezawa: Conceptualization, Resources, Writing – review & editing, Supervision, Project administration. Stefanus Harjo: Resources, Writing – review & editing, Supervision, Project administration, 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.Nitride precipitation in compositionally heterogeneous alloys: Nucleation, growth and coarsening during nitridingA theoretical approach is proposed to take into the account nucleation, growth and coarsening of nitrides in alloys featuring heterogeneous nitrogen content. It is based on physical considerations and accounts for both bulk nitrogen diffusion, which results from the nitriding process and nitrides precipitation kinetics. It predicts local information such as average particle density, radius and volume fraction of nitrides as a function of depth. The work presented in this paper leads to two important conclusions. First, the precipitation rate is not so high that precipitation can be considered as infinitely faster than the diffusion of nitrogen into the bulk. Second, the precipitation state at a given depth depends on the local interaction between nucleation, growth and coarsening phenomena, themselves depending on the local nitrogen content. Finally, the precipitation in alloys with heterogeneous nitrogen content induced by nitriding is radically different from classical precipitation in a single phase where the driving force for precipitation is consumed.► A novel model of nitride precipitation occurring during nitriding is proposed. ► It accounts for bulk nitrogen diffusion and nitride precipitation kinetics simultaneously. ► It predicts volume fraction of precipitated nitride better than previous models ignoring precipitation kinetics. ► It is also capable of calculating local data such as average precipitate density and radius at any time and any depth.The attractive properties of microalloyed steels combined with the presence of nitride-forming elements make them excellent candidate materials for nitriding applications in the aeronautic and automotive industries. Basic understanding of the modification of these materials during nitriding, as well as its consequences on their mechanical properties is a key issue for optimizing the process.During nitriding at a given temperature, nitrogen atoms diffuse from its surface to the bulk. Diffusing nitrogen interacts with iron and/or alloying elements to form nitrides. A characteristic microstructure will show a compound layer, composed of iron nitrides (γ′-Fe4N and ε-Fe2N) covering a diffusion layer, where alloying element(s) precipitate with nitrogen as thin nanometer scale nitrides Several attempts have been made to describe the diffusion of nitrogen in pure iron However, few models are available to describe the nitrogen depth profile when precipitation occurs, despite the fact that the general formalism in simple phases is well established. The most widely used model is a model originally applied to internal oxidation The model previously proposed by Sun and Bell In this paper, the alloy is compositionally heterogeneous because nitrogen diffuses from the surface to the core. A new approach, which couples nitride precipitation kinetics and nitrogen diffusion to the bulk is proposed. The kinetics of nitride precipitation is described through nucleation, growth and coarsening phenomena in a nitrogen gradient. These precipitation stages are reached at different treatment time across the thickness of the sample. A comparison between the previous models and some experimental data is made and discussed.The model presented here aims at describing the evolution of the precipitation state during a nitriding treatment in a thin sheet made of an alloy containing nitride-forming elements. During this treatment, two phenomena occur. On one hand, there is diffusion of nitrogen from the reactive atmosphere into the material; on the other, there is precipitation of nitride particles as the nitrogen concentration progressively increases in the solid solution. Consequently, the model takes into account both phenomena over time and through the depth of the considered material. The two other dimensions of the considered sheet are assumed to be infinite. In view of the symmetry of the problem, only one half of the sheet is simulated, the other half being assumed identical.Even though simultaneous in practice, diffusion and precipitation are separated in each elementary time steps of the calculation. They are simulated by combining already existing models describing each process individually. The nitrogen diffusion problem is solved by integrating the equation of Fick's second law. The precipitation reaction is described using the method proposed by Deschamps and Bréchet For our approach to be valid, several requirements need to be met by the modeled system:The solubility of nitrogen in the considered system must be sufficiently low so that nitrogen self-interaction remains negligible and that its diffusion coefficient DN can be considered constant throughout the whole matrix phase.The diffusivity of the nitride-forming solute is negligible compared to that of nitrogen so that long distance diffusion of the solute needs not to be accounted for.Nucleation of precipitates occurs homogeneously according to the classical theory.In the following, the model is exemplified using the case of chromium nitride (CrN) precipitates in a ferritic iron (α-Fe) matrix containing chromium. It demonstrates the ability of the method to describe the precipitation of any nitride Me1yNz in a matrix Me2 (α-Fe here) containing the alloying element Me1.The precipitation state is quantitatively given by the mean particle density N and the mean radius R of these precipitates as well as the remaining concentrations of nitrogen XN and solute XMe1 in the solid solution. Once the nitrogen diffusion profile has been computed for the current time, all these parameters are determined in each differential space element as depicted in A suitable formalism for multi-component diffusion must represent fluxes in terms of chemical potential where Lkl are the Onsager coefficients and μVa is the chemical potential of vacancies, μk is the chemical potential of the species k and depends on the concentration of other diffusing species. Conservation of vacancies is assumed to approach local equilibrium andTo obtain a generalized expression of Fick's first law, chemical potential gradients must be transformed into concentration gradients. By means of the chain rule of derivation and introducing a matrix notation, In the case of nitrogen diffusion in Fe–Me1 (Me1=Si, V, Mn…) It is reasonable to suppose that DNN∇→XN≫DNMe1∇→XMe1, therefore The transport equation is obtained from Fick's second law:Nitrogen diffusion is assumed to be concentration independent since the matrix can be regarded as a dilute solution. As a consequence, the thermodynamic factor can be supposed equal to unity. It can be writtenSince the treated problem is one-dimensional, it can be writtenEach time step begins with the calculation of the diffusion profile according to the nitrogen surface concentration value set by the user. This is achieved by numerically integrating . This equation is discretized by integration over steps of time Δt and space Δx. Thus, the nitrogen concentration of the nth space element during the ith time element (XN)ni can be obtained using Euler backward method((XN)ni−(XN)ni−1)Δx2=(2(XN)ni−(XN)n+1i−(XN)n−1i)DNΔthence the whole nitrogen concentration profile can be accessed provided that the profile of the previous time step is known. This problem can be reworked as(2DNΔx+ΔxΔt−DNΔx0⋯0−DNΔx⋱⋱⋱⋮0⋱⋱⋱0⋮⋱⋱⋱−DNΔx0⋯0−DNΔx2DNΔx+ΔxΔt)(X1iX2i⋮Xn−1iXni)=ΔxΔt(X1i−1X2i−1⋮Xn−1i−1Xni−1)It can be noticed that the matrix on the left of the expression is a tri-diagonal matrix and as such enables the problem to be conveniently solved by the Tri-Diagonal Matrix Algorithm Given the usual size of nitride precipitates during nitriding treatments (several nanometers to several hundreds of nanometers), it is then relevant to account for the solubility increase at low precipitate radii known as the Gibbs–Thomson effect. To this end, a solubility product Ks,GTMey1Nz has to be calculated for each time step at every depth and under the assumption of a dilute solid solution, it is given by where KsMey1Nz is the solubility product of infinitely large Me1yNz precipitates, VMey1Nz is the molecular volume of Me1yNz and γ is the surface tension of the Me1yNz/α–Fe interface. Precipitation of nitrides only begins when it translates into a decrease of the Gibbs energy of the system, or in other words when the driving force for nucleation Δg is negative. This driving force is given byConsequently, the precipitation calculation is only carried out in space elements where the condition Δg<0 is met, one at a time. All the variables discussed in this section belong to the same space element. The nitride particles of each space element are assumed to be either in a nucleation-growth or growth-coarsening regime. To determine the current regime, a condition is set so as to determine, which one dominates the other. Nucleation dominates coarsening when there are more critical nuclei formed than particles dissolved during the considered time step.The activation energy ΔG for the formation of a spherical nucleus of nitride from the elements in solid solution is given as the sum of volume and surface termsThere exists a particular value of R named the critical radius R⁎ for which ΔG reaches a maximum ΔG*. Particles exhibiting such a radius are in equilibrium with the matrix. It is expressed asIt corresponds both to the nucleation radius in the nucleation regime and to the radius over which particles keep growing and below which they dissolve in the coarsening regime. Combining Actually, for a particle to grow effectively, it needs to reach a radius R′ slightly superior to R* so that ΔG(R′)=ΔG*+kBT, otherwise it is dissolved by thermal agitation. R′ can be approximated by It is now possible to compute the variation over time of particle density in the considered space element. This variation is given by the nucleation rate that is computed using a standard Becker–Döring law:{dNdt|Nucleation=N0Zβ⁎exp(−ΔG⁎kBT)(1−exp(−tτ))Z=VMey1Nz2πR⁎2γkBTβ⁎=8πR⁎2DMe1XMe1aFe4τ=2πβ⁎Z2where N0 is the number of substitutional sites per unit volume in ferrite, Z the Zeldovich factor, β* the critical nucleus absorption rate of solute atoms, τ the incubation time and aFe the lattice parameter of ferrite.The actual precipitate population is assumed to be adequately modeled by a collection of particles having all the same average radius. Hence, in the following formalism, equations known to be valid for a single precipitate are applied to the whole population at the same time. Consequently, the variation of the mean radius is obtained by summing contributions owed to both creation of new nuclei and growth of preexisting particles. It can be expressed asdRdt=(DMe1(XMe1−XMe1eq)R(yVFeVMey1Nz−XMe1eq))︸Growth+(R′−R)NdNdt︸Nucleationwhere XMe1eq is the solute concentration at the precipitate interface and VFe=aFe3/2 is the atomic volume of iron in ferrite. This concentration has to satisfy both the equation of local equilibrium and stoichiometry compatible fluxes for the diffusing species. Consequently, it is obtained by solving the following system:{Ks,GTMey1Nz=(XMe1eq)y(XNeq)zyDN(XN−XNeq)=zDMe1(XMe1−XMe1eq)For generic Me1yNz nitride, solving this system implies finding the root of a polynomial of degree y+z (2 for the example), which belongs to the interval [0; X0]. If y+z≥5, there is no analytic root available but a suitable root can still be easily and rapidly found using Brent's method As the solid solution is progressively depleted under the effect of precipitation, nitride particles can experience two situations.Either their mean radius is much larger than the critical radius R*, in which case they undergo pure growth and the evolution of the precipitation parameters is given by{dRdt|Growth=DMe1(XMe1−XMe1eq)R((yVFe/VMey1Nz)−XMe1eq)dNdt|Growth=0Or their mean radius is equal to the critical radius. In this situation, the conditions of the standard Lifshitz–Slyozov–Wagner law are met:{dRdt|Coarsening=427XMe1eq(yVFe/VMey1Nz)−XMe1eqR0DMe1R2R0=2γVMey1NzkBTR=R⁎However, the decrease of particle density due to the dissolution of critical particles still needs to be derived in the second case in order to fully determine the precipitation parameters. To this end, the mass balance of the system can be written as follows considering the nitride forming solute mass conservation throughout the precipitation reaction:XMe10︷Initialmatrixsolutecontent=XMe1(1−43πNR3)︷Matrixsolutecontentafterprecipitation+yVFeVMey1Nz43πNR3︷PrecipitatessolutecontentThe differentiation of this equation leads to the particle dissolution rate during pure coarsening:dNdt|Coarsening=∂N∂XMe1ssdXMe1dt|Coarsening︸≈0+∂N∂RdRdt|Coarsening=−3NRdRdt|CoarseningIn order to reproduce the continuous shift from growth to coarsening, a coarsening fraction fCoarsening is introduced so that{dRdt=(1−fCoarsening)dRdt|Growth+fCoarseningdRdt|CoarseningdNdt=fCoarseningdNdt|CoarseningThis fraction can actually be modeled arbitrarily by any function verifying fCoarsening→0 when R⪢R* and fCoarsening=1 when R=R*. It can notably be taken asThe precipitation of nitride induces solute depletion in the surrounding matrix. Before proceeding to the next time step, it is crucial that such depletion is precisely assessed. In order to do so the volume fraction of precipitates (fv)ni=4πNR3/3 is introduced as well as (Δfv)ni=(fv)ni−(fv)ni-1, the variation of this fraction over the ith time step. It enables deriving the mass balance for both Me1 and nitrogen:{(XMe1)ni=(XMe1)ni−1(1−(fv)ni−1)−(yVFe/VMey1Nz)(Δfv)ni1−(fv)ni(XN)ni=(XN)ni−1(1−(fv)ni−1)−(zVFe/VMey1Nz)(Δfv)ni1−(fv)niThe procedure then restarts all over again using these profiles along with the previously obtained precipitation parameters vectors Ni, Ri and fvi so as to compute the values corresponding to the i+1th time step.Nitriding a ferrous alloy containing chromium leads to a very intense nanoscale precipitation of chromium nitride. These nitrides perfectly suit the purpose of illustrating the model presented herein. The measurement of the precipitation state as a function of time and depth is a tedious process that is why there is little available data in the literature. We have decided to use the data acquired by Sennour et al. gives the parameters, which were used in the present simulation work. It also contains the input value of certain required physical properties of Cr and CrN., the calculated volume fractions given by the model proposed in this paper (defined as “kinetic model” herewith) have been compared with the experimental data from Sennour et al. The evolution of the mean radius of nitride particles can be estimated from direct integration of Eq. where k represents the supersaturation and defines the thermodynamic driving force for precipitation. For comparing the diffusion length of nitrogen and the precipitation rate, it is interesting to study the evolution of the ratio l2/R2, which can be expressed asThe maximum value of k is around 0.1, DNand DCr are defined in and are, respectively, equal to 7.5×10−12 |
m2 |
s−1 and 4×10−12 |
m2 |
s−1. As a conclusion, the precipitation rate is not high compared to the diffusion rate of nitrogen. The precipitation reaction appears to be a limiting factor and its kinetics cannot be disregarded.The evolution of both particle density and mean particle radius has been studied as a function of depth as shown in and compared to experimental data. In general, the density of CrN particles in the material is high compared to the one obtained for classical isothermal precipitation in a single phase. The density of particles is higher and the mean radius is smaller close to the surface than near the material core. In order to understand such an evolution it is necessary to analyze the evolution of precipitation state as a function of time and depth.Since there exists a nitrogen activity gradient between the reactive atmosphere and the interior of the material, nitrogen mass transport occurs according to Fick's law and thus nitrogen content is time and space dependent. This implies that at a given time, solute content and precipitation state both vary with depth. The system total mass fluctuates over the course of the treatment as nitrogen progressively transfers from the atmosphere to the material. Therefore it is necessary to define thermodynamic equilibrium according to the initial alloy solute content and atmosphere nitrogen activity instead of local composition. Let us now consider two different depths; one very close to the surface (50 μm) and one deeper, relatively far away from the surface (350 μm). The evolution of density, volume fraction and mean radius of CrN precipitates at these positions are given in At 50 μm, the evolution of the particle density shows an increase of the density due to the nucleation of nitrides at first and then a decrease of particle average radius induced by coarsening. The coarsening stage starts close to the equilibrium state as shown by the evolution of the volume fraction of particles. The kinetics of precipitation is rapid and both the density and the volume fraction of nitrides are quite high (respectively, 4×1023 and 1.5%). This behavior is explained by the high driving force happening close to the surface where nitrogen content in solid solution is high and roughly constant because of the incoming nitrogen from the nitriding atmosphere. In fact, both the equilibrium volume fraction of particles as well as the time at which coarsening begins depend on chromium content in solid solution. During nitriding the nitrogen intake is continuous and as a result, nitrogen is not the limiting reagent unlike isothermal precipitation treatment. When almost all chromium is consumed by the precipitation reaction, then the system is locally close to an equilibrium state and coarsening starts. It is important to note that when the nucleation rate is very high (between 700 s and 2000 s) the mean radius of particles does not raise and even decreases. This is due to the intense nucleation of small precipitates while preexisting particles grow slowly. When the density of CrN particles is constant (after 2000 s), the volume fraction increase is solely due to a radius variation; the system locally undergoes pure growth. The subsequent density decline characterizes the coarsening stage. However, even though the mean radius follows the standard LSW law, its evolution differs radically from the classical case of isothermal precipitation in a single phase because the limiting element is the substitutional one (Cr) here and not the interstitial one (N).At 350 μm away from the surface, precipitation starts after a longer time because nitrogen atoms need more time to reach this inner location. Particle density is notably inferior because of the lower nitrogen content available. The nucleation driving force is comparatively small and the average radius evolution is almost completely dictated by growth. As a direct consequence, the mean radius of particles at 350 μm is higher at the final stage than at 50 μm. Since a lot more time is required for the same amount of nitrogen to diffuse that deep, chromium remains much longer in solid solution and the system aside from equilibrium.The obtained material is heterogeneous and a gradient of nitride precipitation is observed from the surface to the bulk as displayed in . Close to the surface the density is higher and the mean particle radius is smaller. As explained previously, it is related to the high driving force for nucleation induced by the high level of nitrogen in solid solution and slow coarsening caused by the complete solute depletion of the matrix.In order to highlight the abovementioned discrepancy observed in terms of precipitation in the case of nitriding and classical single phase precipitation, a comparison of two systems is conducted as illustrated in . The first one is an “open system” designating a system subjected to a continuous nitrogen intake (it is the case during nitriding). The second one is a “closed system” where a certain initial value of nitrogen concentration is set (the surface nitrogen activity chosen for the “open system”, which is actually the nitrogen solubility limit in ferrite) with no additional nitrogen added as the precipitation progresses. It is the case of classical isothermal precipitation in a single phase. In the “open system”, the CrN particle density decreases very slowly no lower than 96% of its maximum value. Conversely, in the “closed system” the CrN particle density decreases much faster below 80% of its maximum value. It may be one of the reasons why the nitrides observed after nitriding are so small in size and so numerous, even after a prolonged treatment Despite the notable improvement of the agreement between experimental data and modeling brought by the method presented herein, some discrepancies still remain. In particular, the evolution of the mean radius as a function of depth displayed in c is qualitatively well described but not quantitatively. Such a behavior may possibly originate from unspecified fitting parameters such as interface energy, which is difficult to assess experimentally. The final results of the model depend drastically on its value. But for simplicity's sake, it was supposed to be constant whatever temperature, size and total nitrogen. It appears to be a strong assumption since interfacial energy can reasonably be supposed to depend on the degree of coherency between CrN particles and matrix and then evolves with parameters such as time and temperature. As a consequence, an improvement of the interface energy description is certainly needed.From a physical point of view, some effects are not taken into account by the model. First, the nitrogen content of the matrix (far away from CrN precipitates) is alleged to be equal to the value defined by the thermodynamic equilibrium in the Fe–N–Cr ternary. However, it was shown that Fe–Cr alloys have a capacity for larger nitrogen uptakes than expected The second point is that the CrN particles were considered as spherical. It is a crude assumption that allows a more practical solving of the diffusion equations. But, it was shown that the CrN precipitates are rather in full agreement with platelets lying in the {001} bcc matrix about 1 nm thick and 15 nm in length The last point concerns the exact composition of nitrides. It was supposed to be CrN according to the thermodynamic calculations at the given temperature and nitrogen composition. Indeed, in our conditions, CrN is more stable than Cr2N. But, it was shown The developed model gives the precipitation state at any time and any depth in the material during nitriding treatment. It gives local information regarding the precipitation state (volume fraction, average particle density and mean radius). Therefore, it is possible to predict the strengthening induced by the nitride precipitation as a function of depth. Indeed, the contribution of strengthening provided by the bowing of dislocations particles has been investigated extensively for spherical particles and depends on the mean distance between precipitates, which is related to both the density and the mean radius of particles In this paper, a simple and highly flexible model fully based on physical considerations was proposed so as to model the precipitation of any nitride in a ferritic matrix during nitriding. It accounts for all three steps of precipitation, namely nucleation, growth and coarsening. It was achieved by combining “long-distance” nitrogen mass transport according to Fick's law with precipitation kinetics. Not only does this model predict the volume fraction of precipitated particles of nitride, it also gives access to much more precise local data such as average radius and particle density at any depth in the material and any moment of the thermochemical treatment when provided with adequate input data.Results generated with this model lead to three important conclusions:The precipitation rate cannot be considered as high compared to the diffusion rate of nitrogen to the center of the material. As a result, the previous models proposed in the literature, which assumes thermodynamic equilibrium to be reached instantaneously overestimate the volume fraction of nitrides. Furthermore, our model accounting for precipitation kinetics provides local information such as mean radius and density at any depth and allows predicting the hardening induced by precipitation.The precipitation in alloys with heterogeneous nitrogen content induced by nitriding is radically different from classical precipitation in a single phase where the driving force for precipitation is consumed. One of the major dissimilarities is that the coarsening step during nitriding is strongly slowed down because in this configuration the limiting reagent is the nitride forming alloying element instead of nitrogen, which is continuously supplied by the reactive medium.The precipitation state at a given depth depends on the local interaction between nucleation, growth and coarsening phenomena, themselves depending on the local nitrogen content. As a consequence, the obtained material is heterogeneous and a gradient of nitride precipitation can be observed in the material.Finally, it is thought that this model can be used as such to treat similar carbide precipitation during carburizing. It could also be evolved so that it would gain the ability to simulate simultaneous precipitation of multiple nitrides over the course of the same nitriding treatment or nitrocarburizing treatment. In a similar fashion, it is highly likely that it could also model internal oxidation problems. Actually, the present paper gives an example in the context of ferritic steel thermochemical treatment but the physical basis of the formalism implies that it could be transposed to much a wider array of materials and processes.Highly ductile fiber-reinforced concrete (HDC)Flexural strengthening of over-reinforced concrete beams with highly ductile fiber-reinforced concrete layerThis study investigates the effectiveness of highly ductile fiber-reinforced concrete (HDC), which is characterized by its high ultimate compressive strain ability, in improving the failure mode and deformational characteristics of over-reinforced concrete beams. The flexural behavior of over-reinforced concrete beams strengthened with HDC was experimentally investigated. The variables included the tensile longitudinal steel reinforcement ratio, thickness of the layer, strengthening materials, and strengthening methods. The crack pattern, failure mode, load–deflection responses, flexural capacity, ductility, and strain were investigated. The experimental results indicated that applying HDC to strengthen the compression zone of an over-reinforced concrete beam is a highly effective method to change its brittle failure and improve the ductility. HDC-strengthened beams showed an increase in flexural capacity and deformation compared to normal concrete-strengthened beams with equal strengthening thickness. A better deformation capacity can be achieved by using a larger thickness of the strengthening layer, ensuring the coordination between the strengthening layer and the existing beam. The effectiveness of HDC in strengthening over-reinforced concrete beams was improved as the tensile longitudinal steel reinforcement ratio increased. A simplified calculation method for the flexural capacity of HDC-strengthened beams was proposed, which is in good agreement with the experimental results. The compression zone relative depth for the balanced failure could be notably improved by applying a larger thickness to strengthen the compression zone of over-reinforced concrete beams than the critical thickness of the HDC layer, which can be calculated by the proposed approach.Highly ductile fiber-reinforced concrete (HDC)The demand for repair and strengthening of existing reinforced concrete (RC) beams has increased owing to the degradation of structural materials, change of usage, increase in service load, aging, and aggressive environments. The selection of an applicable strengthening material and method is essential in strengthening operations. Researchers have extensively studied the performance of commonly employed strengthening techniques in the tension zone Fiber-reinforced cementitious composites provide a feasible method for strengthening existing RC structures. Highly ductile fiber-reinforced concrete (HDC) The failure of over-reinforced concrete beams is sudden due to compressive concrete crushing before the tensile steel reinforcement has yielded. Various concrete structure design codes restrict the use of over-reinforced concrete beams to avoid brittle failure. However, a large amount of tensile steel reinforcement is sometimes applied to obtain a relatively small structural depth. For existing RC beams, over-reinforced concrete beams may occur in certain applications when the load increases owing to a change in the building function and the beam depth is restricted. Furthermore, an over-reinforced section may occur as the strength of concrete decreases with the increase in service time, or when the strength of the poured concrete does not meet the requirements. To improve the flexural capacity of RC beams, Al-Hassani The main ideas of the research on over-reinforced concrete beams focus on setting an external confinement or adding a layer in the compression zone to avoid brittle compression failure. The external confinement mainly involves the use of steel plates and other materials to provide restraint on the compressive concrete. Another way is to apply high-strength concrete to the compression zone of the beam. Yulita et al. In this study, based on the high ultimate compressive strain of HDC, the application of HDC to strengthen the compression zone of over-reinforced concrete beams was proposed to change their brittle failure and improve their deformational characteristics. Ten RC beams, including three control beams, two normal concrete-strengthened beams, and five HDC-strengthened beams were prepared and tested. The effects of the tensile longitudinal steel reinforcement ratio, strengthening thickness, strengthening materials, and strengthening methods on the flexural behavior of over-reinforced concrete beams were evaluated. The properties of the beams were investigated based on crack pattern, failure mode, load–deflection responses, flexural capacity, ductility, and strain analyses. Eventually, a simplified calculation method for flexural capacity was proposed.In this study, all the existing beams were cast using normal concrete (NC) of C30 lists the detailed properties of PVA fibers provided by the manufacturers. summarizes the obtained strength of NC and HDC in the test. The cube compressive strength of the NC is measured by three cube specimens with a side length of 100 mm, and the uniaxial compressive strength of NC is converted by fc=0.76fcu/1.05. As depicted in this figure, the average ultimate compressive strain of HDC is 0.006, which is 2.73 times that of normal concrete. Moreover, the residual strength of HDC is 10–20% when there is a large compression deformation. In addition, the HDC exhibits prominent tensile strain-hardening and multiple cracking behaviors under uniaxial tension. The mechanical properties of the steel rebars listed in were determined by uniaxial tensile tests of steel bars at room temperature, according to GB/T 228.1-2010 The research program contained three control beams, two RC beams strengthened by an NC layer in the compression zone, and five RC beams strengthened by an HDC layer in the compression zone. The control beam and the NC-strengthened beam were designed as over-reinforced concrete beams. The HDC-strengthened beam was designed to be an under-reinforced concrete beam. The proposed strengthening method produced an improvement in failure because the design strain values in the concrete and main steel reinforcements were within the range of ACI 318-19 , the dimensions of the three control beams were 150 mm in width, 200 mm in depth, and 2100 mm in length. The other seven beams were strengthened by the enlargement of the depths. Based on the plane section assumption, the critical thickness of the HDC layer, corresponding to the ultimate state where the ultimate compressive strain of the HDC and compressive concrete is reached simultaneously, was approximately 30 mm. Based on reference , where CB denotes control beams, NB denotes NC-strengthened beams, and HB denotes HDC-strengthened beams.The shear connector was set at the interface between the existing beam and the strengthening layer to ensure that the performance of the strengthening materials can be fully utilized. The shear connector was obtained by welding two 10-mm-diameter HRB400 steel reinforcements. The total shear strength of the shear connectors in the shear span is determined by the stress distribution of simple supported composite beams subjected to a two-point symmetrical load where Ass is the cross-sectional area of the shear connector, f is the tensile strength of the shear connector, b and d are the width and thickness of the strengthening layer, respectively, and fdc is the tensile strength of HDC fdc=50N/mm2The shear strength (Nv) of the shear connector used in this study is calculated according to the following equation where Ed is the elastic modulus of HDC Ed=1.35×104N/mm2The interval (t) of shear connectors is determined by the following equation:According to the above formula, nf=2 and t= 233.33mm. The shear connectors were set conservatively in this study to improve the role of HDC. The number and interval of the shear connectors were of 7 and 250 mm, respectively, along the span of the beam to ensure the coordination work between the layer and the existing beam. The depth of the shear connector was set to 60 mm, which satisfies with the requirement from GB 50017-2003 that the depth of the shear connector should be not less than four times the diameter of the shear connector . According to the design requirements, the shear connector is inserted into the compressive concrete of the existing beam after the concrete is in the initial setting. After curing the existing beam for 28 days, the upper surface (compression side) was roughened with depths of approximately 2.0–5.0 mm, and the concrete debris was cleaned using high-pressure water guns. Subsequently, the HDC or NC layer was cast. The test beam was water cured for 28 days and then naturally cured until the tests.A two-point symmetrical load at a distance of 400 mm was applied using a 500-t electrohydraulic servo testing machine with a bending platform, as shown in . The test adopted a displacement control method with a loading rate of 0.2 mm/min. A linear variable differential transformer (LVDT) was placed at the specimen midspan to measure the deflection. Strain gauges with a base length of 3 mm were attached to the tensile longitudinal reinforcement at midspan and at the applied loads (one at each point load) to test the steel reinforcement strain. Strain gauges with a base length of 100 mm were installed on one side along the height of the beam midspan to monitor the concrete strain. During the test, the appearance and propagation of cracks in each specimen were observed and recorded.For the control specimens (CB-1, CB-2, and CB-3), the first vertical crack occurred on the tension side of the constant moment zone at the early stage of the loading process. As the load increased, the number of cracks in the constant moment zone increased as the width and length increased. When the load reached the peak load, the compressive concrete at the top was crushed, and a width of the main cracks of up to 1.0 mm was observed in the tensile zone. The tensile longitudinal steel reinforcement remained elastic, and the failure did not provide sufficient warning. Obviously, the control specimens exhibited brittle flexural compression failure.The compression zone of specimen NB-1 was strengthened with 30-mm-thick NC, which has a failure process similar to that of the control specimens with brittle flexural compression failure.The compression zone of specimen NB-2 was strengthened with 50-mm-thick NC. At the early stage of loading, the observations and crack distribution were similar to those of the control beams. The tensile longitudinal steel reinforcement yielded at a load of 153.45 kN. As the load increased, the length and width of cracks extended rapidly, while the number of cracks was almost constant. The concrete in the compression zone was crushed with a slight surface spalling at the peak load (154.90 kN). Meanwhile, the width of the main cracks reached 3.0 mm. For the two NC-strengthened beams, no debond cracks were observed at the interface between the NC layer and the existing beam.At the initial stage of loading, the experimental phenomenon of the specimens (HB-1, HB-2, HB-3, and HB-4) was similar to that of the control specimens. After the tensile longitudinal steel reinforcement yielded, cracks extended upwards and widened rapidly, while the number of cracks essentially remained unchanged as the load increased. The neutral axis was shifted up with the propagation of cracks. The mid-span deflection of the specimens significantly increased with the load increment. As the peak load was reached, the HDC layer and compressive concrete were crushed in sequence with the main cracks reaching a width of up to 4–4.5 mm in the tension zone. The HDC layer maintained a good connection with the existing beam, without the appearance of debonding cracks.In contrast to the other strengthened beams, specimen HB-5 was strengthened by an HDC layer in the compression zone without shear connectors. For specimen HB-5, the first crack occurred in the tension zone of the midspan. As the applied load increased, the number, width, and depth of cracks increased. When the load reached 115 kN, debonding cracks were observed at the interface between the HDC layer and the existing beam in the shear span. The debonding cracks of the interface propagated from the beam-end to the midspan direction and widened with the increment in load. The tensile steel reinforcement yielded at a load of 134 kN. A noticeable sound, which corresponded to the HDC layer debonding from the existing beam and crushing in the shear span, was observed at the peak load (143.50 kN). After that, the load drastically dropped to 120 kN (83.6% of the peak load), indicating that specimen HB-5 failed with a width of the main cracks of up to 2.0 mm in the tension zone.The crack distribution and failure modes of the specimens are illustrated in . The load–deflection responses of the beam specimens are shown in Three control specimens (CB-1, CB-2, and CB-3) and an NC-strengthened specimen, NB-1, experienced brittle flexural compression failure, owing to their high percentage of tensile longitudinal steel reinforcement. The NC-strengthened specimen NB-2 experienced a similar balanced-reinforced concrete beam failure with poor ductility. The HDC-strengthened specimens with shear connectors (HB-1, HB-2, HB-3, and HB-4) exhibited a typical flexural tension failure, which corresponded to the HDC or compressive concrete crushing after yielding of the longitudinal tensile steel bars. This indicates that HDC can effectively improve the failure mode of over-reinforced concrete beams. The HDC-strengthened specimen without shear connectors (HB-5) had flexural failure followed by separation of the HDC layer in the zone between the load and the support, which is attributed to the lower elastic modulus of HDC compared to that of NC Compared with the control specimens, the HDC-strengthened specimens exhibited ductile failure with wider and higher cracks. For specimens HB-2 and HB-4, the relative depth of the compression zone was significantly decreased, and the number and length of cracks were higher than those of specimens NB-1 and NB-2, owing to the higher ultimate compressive strain ability of HDC compared to that of NC. During the test, no indication of debonding between the HDC layer and the existing beam was observed in the other strengthened specimens except specimen HB-5, demonstrating that the shear connectors effectively prevented debonding. Furthermore, the shear connectors should be set between the HDC layer and the existing beam to ensure their coordination in the case of an HDC layer with a certain thickness, which needs to be determined by further research.The load–deflection responses of control specimens (CB-1, CB-2, and CB-3) and HDC-strengthened specimens (HB-1, HB-2, and HB-3) with different tensile longitudinal steel reinforcement ratios are shown in (a). It should be noted that the descent stage of the load–deflection curve becomes steeper with increasing tensile steel reinforcement ratio, which is consistent with the regular pattern where the ductility decreases as the tensile steel reinforcement ratio increases. Compared with the control specimens, the improvement of the flexural capacity and midspan deflection of HDC-strengthened specimens increased with increasing steel reinforcement ratio. Because the HDC layer was crushed, the load–deflection curve of HB-3 sharply dropped in the post-peak.(b) illustrates the load–deflection curves of NC-strengthened specimens (NB-1 and NB-2) and HDC-strengthened specimens (HB-2 and HB-4) with identical strengthening thicknesses. This figure shows that the increase in flexural capacity and deformation depended on the strengthening materials and the thickness of the NC or HDC layer. HDC-strengthened specimens showed an obvious increase in the flexural capacity and deformation compared to NC-strengthened specimens with equal strengthening thickness, owing to the higher compressive strength and ultimate compressive strain ability of HDC in relation to those of NC. The effectiveness of HDC or NC in improving the flexural capacity and deformation increased with increasing strengthening thickness. For specimen HB-4, the HDC crushing caused a sudden drop in load, and then the concrete endured all the pressure stress, which corresponds to the slow rise of the post-peak curve.The load–deflection curves of HB-2 and HB-5 with different strengthening modes are depicted in (c). Specimens HB-2 (with shear connectors) and HB-5 (without shear connectors) exhibited different load responses due to the change in failure mode. Slippage was observed between the HDC layer and the existing beam in HB-5 after reaching the peak load. Therefore, the midspan deflection of HB-5 is larger than that of HB-2, corresponding to the better yield platform of HB-5 in (c). However, the load–deflection curve of HB-5 decreased rapidly after the peak point. The location of the failure of HB-5 is in the shear span, and it is difficult to predict in the design stage. The HDC layer in the constant moment zone was not crushed, indicating that the high ultimate compressive strain ability of HDC has not been fully utilized. In summary, shear connectors should be set to avoid separation between the HDC layer and the existing beam when the thickness of the HDC layer is not less than 30 mm. lists the major test results, including the first crack, yielding, peak, and ultimate point. The yielding point is obtained with the secant stiffness method proposed by Park . The peak point corresponded to the maximum load. The ultimate point was defined as the post-peak point where the external load dropped to 85% of the peak load. Pcr, Py, Pp, and Pu denote the first crack, yielding, peak, and ultimate load, respectively. Δcr, Δy, Δp, and Δu are the deflections at the first crack, yielding, peak, and ultimate point, respectively., the peak loads of specimens HB-1, HB-2, and HB-3 improved by 29.0%, 37.6%, and 42.5%, while the corresponding deflections increased by 10.3%, 24.4%, and 35.3% compared to those of specimens CB-1, CB-2, and CB-3, respectively. The improvement in the peak load and corresponding deflection is attributed to the higher compressive strength and higher ultimate compressive strain ability of HDC in relation to those of concrete used in the existing beam, respectively. It should be noted that the effectiveness of HDC in strengthening over-reinforced concrete beams was improved with an increase in the tensile longitudinal steel reinforcement ratio. It is evident from that the HDC-strengthened specimens exhibited an obvious increase in peak load and deflection in comparison to the NC-strengthened specimens with identical strengthening thicknesses. This is due to the higher ultimate compressive strain ability of HDC and the lower elastic modulus of HDC compared to those of NC The deflection ductility index (μΔ) is defined as the ratio of deflection at the ultimate load (Δu) to that at the yield load (Δy). The ductility index ratio (R) is the ratio of the ductility index of the strengthened beam to that of the corresponding control beam. lists the values of the ductility index and ductility index ratio. Generally, μΔ decreases as the tensile longitudinal steel reinforcement ratio increases. μΔ increased as the strengthening thickness increased for strengthened specimens with the same tensile longitudinal steel reinforcement ratio. Compared with NB-1 and NB-2, the ductility index of HB-2 and HB-4 was improved by 11.4% and 28.9%, respectively, owing to the high ultimate compressive strain ability of HDC. This result indicates that the effectiveness of HDC in improving the ductility was better than that of NC. The ductility index of HB-5 (without shear connectors) is higher than that of HB-2 because slippage appeared between the HDC layer and the existing beam in specimen HB-5 at the shear span.The strain distributions through the depth at the mid-span section of specimen HB-3 are plotted in . This figure shows that the strain distributions were approximately linear over the beam height. Furthermore, the strengthened beams had no slip at the HDC layer-to-concrete interface, except that a slight slip occurred in HB-5, indicating that the shear connectors were effective for obtaining satisfactory bonding performance. The discussion of the other specimens was omitted because they have similar strain distributions. shows the load–strain relationship of the tensile steel reinforcements of the specimens prior to the peak load. The increase in the tensile steel reinforcement strain directly reflects the development of vertical cracks of the beams in the constant moment zone. The measured maximum tensile strain in the tensile steel reinforcements of the specimens is listed in , it can be observed that the tensile steel reinforcements of the specimens yielded, except for the control specimens (CB-1, CB-2, and CB-3) and the NC-strengthened specimen NB-1. With the increase in load, the tensile steel reinforcement strains of the control beams and NB-1 exhibited two stages (the first two stages), while that of the HDC-strengthened beams and NB-2 had an additional stage on its basis, as follows: (1) elastic stage before point A: the tensile steel reinforcement strain increased linearly prior to the cracking of NC in the tension zone; (2) AB stage: the growth rate of the tensile steel reinforcement strains accelerated as the load increased; the tensile steel reinforcements still worked in the elastic state; (3) yielding of the tensile steel reinforcements (after point B): wide main cracks were formed at point B, and the tensile steel reinforcements reached the yield strength; the tensile steel reinforcement strains increased rapidly, and the load–strain curves tended to be horizontal. The maximum tensile strain in the tensile steel reinforcements of HB-5 is not provided because the strain gauge was damaged at 94.1% of the peak load (after point B).To simplify the theoretical model, the basic assumptions for the strengthened specimen are as follows: (1) the plane section remains plane; (2) the tensile strength of concrete is neglected; (3) there is no debonding between the strengthening material and the existing beam before the peak state; (4) the strengthened beam fails when one of the strengthening materials and the concrete reach the ultimate compressive strain.The stress–strain relationship of the steel reinforcement adopts an ideally elastic–plastic curve.The stress–strain relationship of NC is recommended by GB 50010-2010 where ε0 and εcu are the peak and ultimate compressive strain of NC, respectively.Based on the uniaxial compressive stress–strain curve of HDC measured by the test (b). The uniaxial compressive stress–strain relationship can be determined byσc=αcfdcβcεd0εc0⩽εc⩽βcεd0σc=αcfdc+fdc-αcfdcεd0-βcεd0εc-βcεd0βcεd0⩽εc⩽εd0where βcεd0 and εd0 are the strains corresponding to the compressive stiffness change point and peak compressive strain of HDC, respectively; fdc is the compressive strength of HDC (MPa); and αc=0.95 and βc=0.90 are parameters related to the strength and ductility of HDC, which determine the compressive stiffness change point The constitutive model parameters of NC and HDC are listed in The stress and strain distributions of the over-reinforced concrete beams are shown in . The force and bending moment equilibrium equation can be expressed asA linear relationship between σs and ξ is used to simplify the calculation, as follows:where α1 and β1 are the equivalent rectangular stress block coefficients of concrete, and these coefficients are applied in the case of compressive concrete crushing, in which the maximum strain of the concrete reaches the corresponding ultimate strain; ξ is defined as the relative depth of the compression zone, which is the ratio of the compression zone depth (x) to the distance from the centroid of the tensile longitudinal steel reinforcements to the upper edge of the compression zone (h0), ξ=x/h0When the HDC layer provides compressive stress together with the concrete in the compression zone, the derivation of the equivalent rectangular stress coefficients (αd1, βd1) of the HDC layer is as follows:Based on the assumption that the plane section remains plane, the compressive strain at y above the neutral axis is given bywhere εc and εdc are the compressive strains at the top edge of the NC and HDC, respectively, x is the depth of the compression zone, and y is the distance from any point on the compression zone to the neutral axis, as shown in Taking the derivative of both sides of Eq. , the resultant force of the HDC layer (C) is expressed byC is used to take the moment of the neutral axis, which can be obtained asycC=b∫x-dxσcεcydy=b(xεdc)2∫εcεdcσcεcεcdεc, the distance from the acting point of C to the neutral axis (yc) can be expressed asBecause the centroid of the actual stress distribution is the same as that of the equivalent rectangular stress block, the distance between the resultant force point of the HDC layer and the centroid of the tensile rebar is equal. Therefore, the following equation can be obtained:Based on the plane section assumption, βd1 is obtained as the following equation by substituting Eq. βd1=2εdcεdc-εc1-∫εcεdcσcεcεcdεcεdc∫εcεdcσcεcdεcThe area of the actual stress distribution is equal to that of the equivalent rectangular stress block.When εc=εcu and εdc=εd0 simultaneously, the force equilibrium equation is given byThe thickness of the HDC layer in the critical state (db) can be obtained by Eq. Therefore, the calculation of αd1βd1 is divided into three cases corresponding to the three ultimate states, as listed in Referring to the calculation method of bending of the T-section reinforced concrete beam For NC-strengthened specimens, if xu⩽d, the compressive strength of the concrete in the existing beam is ignored; if xu>d, it is considered that the concrete in the strengthening layer has reached the peak compressive strain, and the concrete in the existing beam was calculated according to the equivalent rectangular stress block α1fcbx-d+αd1fdcbβd1d=fyAsα1fcbx-dh0-d-x-d2+αd1fdcbβd1dh0-d2=Mα1fcbx-d+fcjbd=fyAsα1fcbx-dh0-d-x-d2+fcjbdh0-d2=Mwhere αd=0.8589 and βd=0.6919 are the equivalent rectangular stress coefficient of HDC The compression zone relative depth for balanced failure (ξb) is the ratio of the compression zone depth for the balanced-reinforced concrete beam failure (xb) to the distance from the centroid of the tensile steel reinforcements to the upper edge of the compression zone (h0), ξb=xb/h0. The calculation of the balanced-relative depth of the compression zone ξb can be divided into three cases corresponding to the above three ultimate states, as listed in The calculated and experimental values of the flexural capacity are listed in . Ftu is the peak load, Fcu,s is the value of the flexural capacity using the simplified calculation, and Fcu,g is the value of the flexural capacity using the general calculation based on GB50010-2010 that the simplified calculated values of the flexural capacity are in good agreement with the experimental values and the general calculated values, which indicates that the proposed calculation method has clear theoretical significance and reference value. lists the calculated value of ξ and ξb. The failure mode of the HDC-strengthened specimens can be obtained on the basis of the relative magnitudes of ξ and ξb. The corresponding failure modes of the obtained ξ are in good agreement with the test results, indicating that the proposed calculation methods are reliable. The ξ of HDC-strengthened specimens significantly decreased compared to that of the corresponding NC-strengthened specimens owing to the high compressive strength and high ultimate compressive strain ability of HDC. Compared with CB-3, the ξb of HB-3 was reduced by 5.4%. This occurs because the strengthening thickness of HB-3 is less than the critical thickness of the HDC layer (d<db), which corresponds to the ultimate state where the concrete reaches the ultimate compressive strain while the HDC has not reached the peak compressive strain. That is to say, the HDC has not been fully utilized in the case of d<db. On the contrary, the other HDC-strengthened specimens significantly increased the ξb owing to the high ultimate compressive strain ability of HDC. Therefore, the ξb can be notably improved by applying a larger thickness than the critical thickness of the HDC layer to strengthen the compression zone of over-reinforced concrete beams.This study aims to determine the effectiveness of HDC, which is characterized by a high ultimate compressive strain ability, in improving the failure mode and deformational characteristics of over-reinforced concrete beams. The flexural behavior of over-reinforced concrete beams strengthened with HDC was investigated. The tensile longitudinal steel reinforcement ratio, strengthening thickness, strengthening materials, and strengthening methods served as the test variables. The experimental results were observed and discussed based on crack pattern, failure mode, load–deflection responses, flexural capacity, ductility, and strain analyses. In addition, a simplified calculation method for flexural capacity was proposed and verified through comparison with the test values. From the above discussions, the following conclusions can be drawn:The HDC material is applied to strengthen the compression zone, and it performed effectively in improving the failure mode of over-reinforced concrete beams. All the HDC-strengthened specimens experienced flexural failure with large flexural cracks and favorable ductility owing to the high ultimate compressive strain ability of HDC, whereas the NC-strengthened specimens exhibited a nearly brittle failure mode.The increase in flexural capacity and deformation depended on the strengthening materials and the thickness of the NC or HDC layer. HDC-strengthened specimens showed an obvious increase in flexural capacity and deformation compared to NC-strengthened specimens with equal strengthening thickness. The improvement in flexural capacity and deformation is due to the higher compressive strength and higher ultimate compressive strain ability of HDC compared to those of NC, respectively. Better deformational characteristics can be achieved by using a larger thickness of the strengthening layer, ensuring the coordination between this layer and the existing beam.A considerable increase in ductility is obtained by adding an HDC layer to the compression zone of over-reinforced concrete beams, and the increase ranges from 7% to 61% depending on the longitudinal reinforcement ratio and the thickness of the HDC layer. The effectiveness of HDC in strengthening over-reinforced concrete beams was improved as the longitudinal reinforcement ratio increased.Shear connectors should be set between the HDC layer and the existing beam to ensure their coordination in the case of an HDC layer with a certain thickness, which needs to be determined by further research. In this study, debonding occurred in the HDC-strengthened beam in which a 30-mm HDC layer was applied to the compression zone without shear connectors.The formula for calculating the equivalent rectangular stress coefficient of the strengthening layer is suggested based on the equivalent principle of equal force and same force acting point. Furthermore, a simplified calculation method for the flexural capacity of HDC-strengthened specimens is proposed, and the calculated values agree well with the experimental values. This calculation method can be applied to predict the flexural capacity of reinforced concrete beams strengthened by other fiber-reinforced cementitious composites in the compression zone.An approach for predicting the compression zone relative depth for the balanced-reinforced concrete beam failure of HDC-strengthened beams was presented, and the calculated results have good consistency with the test results. The compression zone relative depth for the balanced failure can be notably improved by applying a larger thickness to strengthen the compression zone of over-reinforced concrete beams than the thickness of the HDC layer in the critical ultimate state, in which the concrete reaches the ultimate compressive strain and the HDC reaches the peak compressive strain simultaneously.Mingke Deng: Conceptualization, Methodology, Resources, Writing - review & editing, Supervision, Project administration, Funding acquisition. Min Zhan g: . : Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Writing - original draft, Writing - review & editing, Supervision, Project administration. Fudong Ma: Writing - review & editing. Fangyuan Li: Writing - review & editing. Hongzhe Sun: .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.Material parameters in void growth model for G20Mn5QT cast steel at low temperaturesWith superior structural performances, G20Mn5QT cast steel has been extensively used in fabricating complex joints in steel constructions. Fracture failure of G20Mn5QT cast steel is then an important issue, especially for low-temperature situations. In this paper, mechanical properties of G20Mn5QT cast steel at low temperatures were studied by tensile coupon tests and Charpy impact tests. The effects of the low temperature on mechanical properties and the ductile-brittle transition temperature of G20Mn5QT cast steel were obtained based on the test results. The void growth model (VGM) and the stress modified critical strain (SMCS) model, which are two typical micromechanical fracture models, were calibrated for G20Mn5QT cast steel by experiments and complementary FEA on smooth notched tensile (SNT) specimens at four temperature levels, 20°C, −20°C, −40°C and −60°C. For each temperature level, the material parameters in VGM and SMCS calculated based on test results of all six different specimens are in very good consistence. This consistency indicates the applicability of VGM and SMCS for predicting the ductile fracture of G20Mn5QT cast steel, even at the temperature of −60°C, which is much lower than the transition temperature of the material. The calibrated material parameters representing the resistance to ductile fracture of the material decrease linearly with the drop of the temperature. The characteristic length of G20Mn5QT cast steel was also determined by observation of fractured surfaces of test specimens with the scanning electron microscope, which has the tendency to decrease with the drop of the temperature.Cast steel is an ideal solution for complex joints in steel constructions. With cast steel joints, geometric discontinuity can be avoided by smooth transition segments to reduce the stress concentration, and the welding can be arranged outside the joint region to minimize its adverse heat effects on the structural performance of the joints. Sometimes, cast steel joints are applied in regions of extreme environments such as low temperature. For example, a large amount of cast steel joints was adopted in steel constructions in north China. If these joints were exposed outdoors, they would experience the minimum mean temperature as low as −40°C in winter. This situation determines that these cast steel joints must have sufficient low-temperature ductility to ensure the safety of the structures.For steel joints and connections without sufficient ductility, the ductile fracture may occur under the action of monotonic loadings With the superior welding performance, G20Mn5QT cast steel In this paper, mechanical characteristics of G20Mn5QT cast steel at low temperatures were studied. Tensile coupon tests and Charpy impact tests were carried out on G20Mn5QT cast steel at different temperatures. Two micromechanical fracture models, VGM and SMCS, were calibrated for G20Mn5QT cast steel at four temperature levels by tests and complementary FEA on SNT specimens. The effects of low temperature on mechanical properties, Charpy impact toughness, and the calibrated fracture parameters were discussed for G20Mn5QT cast steel.The ductile fracture of steel was considered as the result of nucleation, growth, and coalescence of microscopic voids in micromechanics of fracture. The relationship between the void growth and the combined effect of the stress triaxiality and the equivalent plastic strain has been established according to analytical derivations in references In VGM, the growth of microvoids controls the ductile fracture of materials. When voids grow to exceed the critical size, the ductile fracture occurs. Void growth index (VGI) was then defined to describe the void growth quantitatively. The micromechanical fracture criterion can be expressed for VGM aswhere R is the radius of voids; R0 is the initial void radius; c is a constant; T is the stress triaxiality and expressed as T = σm/σe (σm is the hydrostatic stress, and σe is the von-Mises stress); dεp=2/3dεpij·dεpij is the incremental equivalent plastic strain; εp is the equivalent plastic strain and εpcritical is its critical value for the initiation of the ductile fracture; η is a material parameter.In many practical cases, with the increase of loading, the plastic strain rises rapidly while the stress triaxiality remains nearly a constant. If the variation of stress triaxiality is ignored, SMCS can be setup based on VGM. SMCS Index and a material parameter α were defined to express the fracture criterion of SMCS as shown in Eqs. The parameters η and α represent the resistance to ductile fracture of the material and can be calibrated through SNT tests and complementary FEA.The ductile fracture of steel is the behavior of a certain volume of materials. A length scale, termed as the characteristic length (l*) All the tensile tests were conducted on an electronic universal testing machine with a capacity of 200kN. The test setup was shown in . In order to create a low-temperature environment, a cold chamber was manufactured with the injection of liquid nitrogen. The cold chamber was made up of two layers of the polyurethane board, with a thickness of 80 mm. In order to maximize the effect of thermal insulation, two layers of the polyurethane board were interlaced at the edge and corner of the cold chamber. The gaps between polyurethane boards were sealed with silica gel. Liquid nitrogen was injected into the chamber through a precise valve. Two PT100 thermocouples were mounted on each test specimen at two different locations to measure the real-time temperature of the specimen. The inflow of liquid nitrogen was controlled by adjusting the valve so that the target temperatures could be obtained and maintained. Four temperature levels, TEMP = 20°C, −20°C, −40°C and −60°C, were set for the tensile tests on G20Mn5QT cast steel. The thermocouples were connected to the acquisition equipment, and the temperature of the specimen could be monitored through a computer. An extensometer was put on the tension coupon in the cold chamber to measure the deformation and control the load rate.The temperature change process of a tensile specimen tested at −60°C is shown in . The first 11 minutes is the cooling process, during which the specimen was cooled from ambient temperature to −60°C. In order to ensure the internal temperature of the specimen reach −60°C, the same as the external temperature, the target temperature (-60°C) was maintained for another 10 minutes. After that, the formal loading process began from the 21st minute.With the superior performance in weldability, G20Mn5QT cast steel is very suitable for fabrication of complex steel joints. Mechanical properties of G20Mn5QT cast steel stipulated in reference A 30 mm thick G20Mn5QT cast steel plate was cast for the manufacturing of test specimens in this paper. Mechanical properties of this cast steel plate at low temperatures were determined by tension coupon tests. The geometric details of the tension coupons were shown in . A total number of 12 tension coupons were machined for the tests at the four temperature levels. The tension coupons were put into the cold chamber and screwed into two end sleeves, which were clamped by the testing machine, as shown in (b). An extensometer was put on the tensile coupon in the cold chamber to measure the deformation of gage length. The loading rate was 1.0 mm/min, which was controlled through the deformation measured by this extensometer. Before the loading application, the upper chuck of the testing machine should be loosened to release the contractions of the coupons during the cooling process.Nominal stress-strain curves were determined by tension coupon tests, as illustrated in . Note that the extensometer was taken off before the deformation of the test coupon reached its measuring range. Consequently, the stress-strain relationship beyond ε = 0.15% was not obtained directly by these tests. True stress-strain curves were determined based on the test data, as shown in The diameters at the fractured surfaces of the test coupons were measured, and the true stress and strain were determined by Eqs. in which, d0 and df is the diameter of the coupons before tests and at the fractured surface respectively. The true stress-strain curves were then extended linearly to the fracture of the coupons in , based on which relationships between the true stress and plastic strain was established for G20Mn5QT cast steel.Mechanical properties of G20Mn5QT cast steel determined by tensile coupon tests were summarized in where it can be observed that all of them vary with the temperature, as shown in . The elastic modulus E, yield strength fy and ultimate strength fu increase almost linearly by 19.6%, 16.6% and 7.9% respectively with the drop of the temperature from 20°C to −60°C; while the elongation δ and reduction of cross section ψ decrease by 18.3% and 24.2% respectively. Obviously, the strength of G20Mn5QT cast steel increases and the ductility deteriorates at low temperatures.Charpy impact toughness tests were conducted for G20Mn5QT cast steel at 20°C, 0°C, −20°C, −40°C and −60°C respectively on a pendulum impact tester with a standard strike energy of 300 J. Fifteen Charpy impact specimens were machined, three for each temperature, from the G20Mn5QT cast steel plate. The Charpy impact specimen and its geometry details were shown in . Charpy impact specimens were cooled to a certain testing temperature in the mixture of ethyl alcohol and liquid nitrogen. Specimens were kept in the mixture long enough to make sure they were cooled completely. The temperature was monitored by PT100 thermocouple mounted on the test specimen. The mixture was placed very close to the test machine and the specimen was taken out of the mixture only after the test machine was ready. It took less than 5 seconds to take the Charpy impact specimen out of the cooling mixture, put it on the testing machine quickly and complete the test.The Charpy impact energies of G20Mn5QT cast steel at different temperatures were listed in . The Charpy impact energy obtained by tests at 0°C meets the requirements in . An obvious decrease of the Charpy impact energy can be observed with the drop of the temperature, especially when the temperature reaches 0°C, which means the deterioration of the toughness of G20Mn5QT cast steel.. The ductile-brittle transition temperature is −12.3°C, while the width of the temperature transition zone is 13.3 °C.Material parameters were calibrated for VGM and SMCS by SNT tests at four temperature levels, 20°C, −20°C, −40°C and −60°C. Twenty-four SNT specimens were machined from the G20Mn5QT cast steel plate, as shown in . Three different notch radii (R = 1.5, 3.125 and 6.25 mm) were considered to provide different triaxial stress conditions.The tests were carried out on the electronic universal testing machine. The test setup was the same as the one severed for tension coupon tests shown in . After the temperature of the specimen in the cold chamber dropped to the test temperature, the uniform axial tensile load was applied and the specimen was stretched continuously until the fracture at the neck. The deformation measured by the extensometer was adopted to control the loading at the rate of 0.2 mm/min. Load versus deformation curves were illustrated for all test specimens in . Points with sudden slope change on these curves were identified as ductile fracture initiation. Two tests were conducted for each notch radius and each temperature, as shown in . Tensile loads Pf and specimen deformations Δf at fracture initiation were obtained for all SNT specimens.Fractographic features of the SNT specimens were studied through the scanning electron microscope (SEM). Typical micrographs of the fracture surfaces at different temperatures were presented in . During the ductile fracture, the coalescence and cleavage of microvoids generated microscopic plateaus and valleys, which can be easily distinguished from the micrographs. With the drop of the test temperature, the fracture surfaces tend to be more even, and less plastic deformation developed on fracture surfaces.It is generally accepted that, for particular materials, the characteristic length (l*) should be the size of a microvoid colony , for G20Mn5QT cast steel at different temperatures. It should be noted that the obtained characteristic lengths for different temperatures are all at the order of 0.1 mm, the same as those of many other structural steels The nonlinear FEA of SNT specimens was conducted by the general FE software ABAQUS . Finite element models were discretized with the CAX4R element implemented in ABAQUS, which is a type of 4-node bilinear axisymmetric element. In the notch region, ultra-fine mesh with element size being the characteristic length in was adopted to capture the ductile fracture. Coarser meshes were adopted in regions far away from the notch to save the computational cost.In finite element analysis, the constitutive relationship of the material was set according to the results of the tension coupon tests at different temperatures as shown in . The Mises yield criterion was adopted to model the plastic behavior of the material.Material parameters in the two micromechanical fracture models were calibrated for G20Mn5QT cast steel at the four temperature levels, TEMP = 20°C, −20°C, −40°C and −60°C, based on results of SNT tests and complementary FEA.The study on SNT specimens tested at the temperature of −20°C was taken as an example to illustrate the calibration procedure. The distribution of the stress triaxiality and equivalent plastic strain over the critical section were obtained by FEA for the specimens with the three different notch radii, as shown in . The stress triaxiality is higher at the section center than on the surface. The equivalent plastic strain changes little over the section. VGI and SMCS Index were calculated over the critical section by Eqs. . The maximum value of both indexes is at the center of the critical section, indicating that the ductile fracture tends to initiate there. Variations of the stress triaxiality against the equivalent plastic strain were plotted for the three specimens in . The stress triaxiality increased sharply at the beginning of the loading process when the equivalent plastic strain was neglectable small. Then, the equivalent plastic strain kept rising with the load. However, the stress triaxiality remained almost unchanged. Therefore, it is acceptable that VGM can be simplified to SMCS by assuming the stress triaxiality is constant during the prediction of ductile fracture. These conclusions apply for the case at the temperature of −20°C, as well as the cases at other temperatures.The deformations of the specimens, Δf, at the fracture obtained by SNT tests were as shown in . Apply these deformations to FE models of the corresponding test specimens and determine the equivalent plastic strain and stress triaxiality at the center of the critical section of each specimen by FEA. It can be seen that, for SNT specimens with the same notch radius, the equivalent plastic strains at the fracture decrease with the drop of the temperature. Then, η in VGM was calculated by Eq. . Average values of η and α were calculated for all the four temperature levels, as listed also in . It can be seen that, for each temperature level, the values of η and α calculated based on test results of the six different specimens are all in very good consistence. The largest deviation between a calibrated value and the average is less than 15%. This indicates that VGM and SMCS are applicable for ductile fracture prediction even at the temperature of −60°C, which is much lower than the transition temperature of the material.Material parameters, η and α, have already been calibrated for G20Mn5QT cast steel at ambient temperature in reference for the temperature of 20°C. Note that the yield stress of the G20Mn5QT cast steel in reference Ductile fracture of SNT specimens with notch radii 1.5, 3.125 and 6.25 mm was predicted with the calibrated VGM and SMCS at the four different temperatures. VGI and SMCS index at the center of the notch section were calculated based on FEA. The deformations at fracture were predicted for all specimens, as shown in The predicted deformations at fracture (ΔfFEM) and those obtained by tests (Δftest) were compared for all SNT specimens in . It can be seen that the predictions agreed well with test results and all data fell inside the 15% margin lines. The agreement verified the rationality of the micromechanical fracture models calibrated in this research.All the calibrated η and α were plotted against the temperature as shown in . Both material parameters have the tendency to decrease with the drop of the temperature. This is consistent with the basic concept about the resistance of structural steels to the ductile fracture.The average values of the material parameters at the same temperature level were also plotted against the temperature in . It can be observed that both the average values of η in VGM and α in SMCS decrease almost linearly with the drop of the temperature. Therefore, the relationships between the material parameters in the micromechanical fracture models and temperature were established with linear regression analysis, as shown in Eq. Based on the proposed equations, material parameters in the two micromechanical fracture models for G20Mn5QT cast steel at different low-temperature levels can be determined for ductile fracture prediction conveniently.Structural steel tends to lose its ductility with the drop of temperature, which makes the fracture failure of steel an important issue for steel construction at low temperatures. In this paper, mechanical properties of G20Mn5QT cast steel at different low temperatures were studied by tension coupon tests and Charpy impact tests. The void growth model (VGM) and the stress modified critical strain (SMCS) model were calibrated for G20Mn5QT cast steel by tests and complementary FEA on SNT specimens at four temperature levels, 20°C, −20°C, −40°C and −60°C. The following conclusions were obtained:(1) The elastic modulus, yield strength and ultimate strength of G20Mn5QT cast steel increase almost linearly by 19.6%, 16.6% and 7.9% respectively with the drop of the temperature from 20°C to −60°C; while the elongation and reduction of cross section decrease by 18.3% and 24.2% respectively.(2) The Charpy impact energy of G20Mn5QT cast steel decreases significantly with the drop of the temperature, especially when the temperature is below 0°C, which means the deterioration of the toughness of G20Mn5QT cast steel. The ductile-brittle transition temperature is −12.3°C, while the width of the temperature transition zone is 13.3 °C.(3) Material parameters, η in VGM and α in SMCS, were calibrated for G20Mn5QT cast steel at four temperature levels, 20°C, −20°C, −40°C and −60°C. For each temperature level, the values of η and α calculated based on test results of all six different specimens are in very good consistence. This consistency indicates the applicability of VGM and SMCS for ductile fracture prediction even at the temperature of −60°C, which is much lower than the transition temperature of the material. The calibrated material parameters for G20Mn5QT cast steel decrease linearly with the drop of the temperature.(4) Fractographic features of the SNT specimens were studied through the scanning electron microscope (SEM). With the drop of the test temperature, the fracture surfaces tend to be more even, and less plastic deformation developed on fracture surfaces. Microscopic plateaus and valleys can be easily distinguished from the micrographs. The characteristic length of G20Mn5QT cast steel has the tendency to decrease with the drop of the temperature.Yue Yin: Conceptualization, Methodology, Supervision, Writing - original draft, Writing - review & editing. Shuai Li: Methodology, Formal analysis, Investigation, Writing - original draft. Qinghua Han: Conceptualization, Project administration, Funding acquisition, Writing - review & editing. Mengfei Li: Investigation, Visualization, 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.Rotation-field annealing effect on magnetoimpedance of Co-based amorphous ribbons▶ In this paper, a new annealing method, called the RFA method, is introduced to improve the magnetoimpedance (MI) effect in Co-based amorphous ribbons; e.g. VAC6025. This method includes applying a sufficiently large rotating field hR, with a frequency F from 1 to 100 Hz, when the ribbon sample has been at the annealing temperature TA (from 160 to 280 °C) for 1 h, and then is field-cooled to room temperature. This RFA method can not only give the highest magnetoimpedance sensitivity (4.5%/Oe) or figure of merit (1.8% MHz/Oe), but also preserve the good mechanical (or ductile) property for VAC6025.A new annealing method, called the RFA method, is introduced to improve the magnetoimpedance (MI) effect in Co-based amorphous ribbons; e.g. VAC6025. This method includes applying a sufficiently large rotating field hR, with a frequency F from 1 to 100 Hz, when the ribbon sample has been at the annealing temperature TA (from 160 to 280 °C) for 1 h, and then is field-cooled to room temperature. The best achievements are that: [I] after the F |
= 1 or 50 Hz and TA |
= 200 °C RFA treatment, the maximum MI ratio (ΔZ/Z)m |
= 72% and the maximum operating frequency fm |
= 0.6 MHz; [II] after the F |
= 50 Hz and TA |
= 160 °C RFA treatment, the maximum field sensitivity Sm |
= 4.5%/Oe and the highest figure of merit FOM = 1.8% MHz/Oe; [III] since both of the optimal TAs above are much lower than the crystallization temperature (TX |
= 500 °C), the samples remain ductile after RFA. In conclusion, we believe that the RFA treatment can greatly improve the MI performance of the VAC6025 ribbons, since hR, operating at TA and/or below, can effectively randomize the magnetic anisotropy, and thereby reduce the effective (or average) anisotropy energy 〈K〉. This is equivalent to stating that the RFA method is effective in enhancing the permeability or the ΔZ/Z ratio of any amorphous ribbon sample, including VAC6025.Zero-magnetostriction Co-based amorphous ribbon, which is known to exhibit giant magnetoimpedance (GMI) effect Nearly zero magnetostrictive Co-based amorphous ribbon, i.e. VAC6025 (Co66Fe4Mo2B11.5Si16.5), was purchased from Goodfellow Cambridge Ltd. Each rectangular sample, with dimensions of 21 mm (L: length) × 2.2 mm (w: width) × 0.0225 mm (t: thickness) was cut from the ribbon spool. The as-cast sample was a fresh sample without any annealing treatment. All the other samples were subjected to specific field-annealing treatments. The RFA method includes annealing each ribbon sample at TA |
= 160, 200, 240, and 280 °C in a vacuum of 1 × 10−6 |
Torr. Note that TX of VAC6025 is 500 °C , there were four kinds of field-annealing treatments after the T |
= |
TA condition was reached. First, we had h |
= 0 Oe for the 1 h annealing. Second, we applied a dc (or static) longitudinal field h |
= |
h∥ along the length of the sample during annealing, followed by field-cooling. Third, a dc transverse field h |
= |
h⊥ was applied along the width during annealing, followed by field-cooling. Fourth, a rotating field h |
= |
hR |
× e−iΩt |
= |
h∥(t) + |
ih⊥(t), where Ω |
= 2πF and F |
= 1, 5, 10, 50, and 100 Hz was applied parallel to the plane of the sample during annealing, followed by field-cooling. We roughly estimated the longitudinal and transverse demagnetizing fields, HD(L) and HD(W) at each TA as shown in , to ensure that h∥, h⊥, and hR (about 40 Oe) were sufficiently larger than those HDs.The magnetoimpedance (MI) measurements were carried out using a HP4284A precision LCR Meter (in the f range 1 kHz–1 MHz) and an Agilent E4991A RF impedance/material analyzer (1 MHz–1 GHz), respectively at room temperature. The ac probe current sent through the sample was 4.8 mA. The impedance of the testing leads was nullified for each f. A pair of Helmholtz coils was used to apply an external longitudinal (H∥) or transverse (H⊥) field on the ribbon sample, as shown in (a). A typical Z versus H∥ plot of the h |
= 0 sample at f |
= 10 MHz is shown in (b). |H∥| increases from Z0 to Zp, and then decreases to ZS. This dip (Zp |
− |
Z0) phenomenon becomes more apparent when f is higher. The MI ratio is defined as ΔZ/Z |
≡ (Zp |
− |
ZS)/ZS, and the sensitivity as S |
≡ (dZ/dH)b(1/Zb) at the inflection (or biasing) point B, as shown in (b). The dc electrical resistivity (ρ) was measured with the standard four-probe method. The magnetic easy- and hard-axis hysteresis loops of each ribbon sample were measured by a vibration sample magnetometer., the idea of employing the RFA method is to reduce the effective 〈K〉 in the ribbon sample. Notice that due to the difference between HD(L) and HD(W), the easy-axis (EA) of the sample after the RFA treatment is along L, as in the case after the h∥ treatment, but with 〈K〉 reduced in the former case. , where HC is the easy-axis coercivity, HK is the hard-axis anisotropy field, and Ku |
≡ 〈K〉 is the uniaxial anisotropy energy, should confirm the arguments just made. Obviously, in the EA formation mechanism it is the internal annealing field, hi |
≡ |
h |
− |
HD, that matters. Further, Refs. In short, the permeability, μC or μT, is always inversely proportional to 〈K〉. From MI theory To summarize, three predictions can be made from Eq. . First, by reducing 〈K〉 with the RFA method as evidenced from , and thereby enhancing μC or μT, we can greatly improve the ΔZ/Z ratio. Second, μC |
> |
μT from Eq. , ΔZ/Z in the case of H |
= |
H∥ and h |
= |
h∥ (the dc longitudinal anneal) is larger than that in the case of H |
= |
H∥ and h |
= |
h⊥ (the dc transverse anneal). Third, in the low f region, Hb is proportional to HC |
+ |
HD(L) in the H |
= |
H∥ case and to HK |
+ |
HD(W), where HK |
= 〈K〉/[2πMS] is the anisotropy field, in the H |
= |
H⊥ case. In the high f region, the eddy-current effect enhances HC or HK so that Hb is an increasing function of f for both cases. Because we have HC |
< |
HK and HD(L) < |
HD(W) for all the samples, the biasing field Hb in the H⊥ case is always larger than that in the H∥ case for each sample. In addition, from a simple geometrical relationship in the ΔZ/Z versus H∥ curve, it is easy to tell that (dZ/dH)b should be inversely proportional to Hb. As a result, S in the H∥ case is also always larger than that in the H⊥ case for each sample. Therefore, in of this paper, we concentrate on the H |
= |
H∥ results only. shows ΔZ/Z plotted as a function of f for the five kinds of samples: as-cast, h |
= 0, h |
= |
h∥, h |
= |
h⊥, and h |
= |
hR at F |
= 50 Hz. All the curves follow the same trend: ΔZ/Z first increases; then as f increases, ΔZ/Z reaches the maximum (ΔZ/Z)m at f |
= |
fm |
= 0.6 MHz (); thereafter ΔZ/Z decreases monotonically. This is in agreement with the GMI theory that fm is roughly equal to the sample relaxation frequency ωC/2π, about 1 MHz , we see that (ΔZ/Z)m in the F |
= 50 Hz case is as high as 72%. This ratio is the highest, which is sufficient to fulfill our first prediction in the last paragraph. Next, (ΔZ/Z)m |
= 52% for the static h |
= |
h∥ case is larger than (ΔZ/Z)m |
= 48% for the static h |
= |
h⊥ case. This fact confirms our second prediction. (ΔZ/Z)m |
= 58% for the h |
= 0 case also shows that even with the zero-field annealing, the self-demagnetization field will automatically force the easy-axis parallel to L., we show Hb plotted as a function of f for the (same) five kinds of samples (as in ): the data in red are for the H |
= |
H⊥ case, while those in black are for the H |
= |
H∥ case. (For interpretation of the references to colour in this figure, the reader is referred to the web version of the article.) There are two features in . First, for all the samples, in the f range from 50 kHz to 0.1 GHz of , Hb in the H∥ case is always smaller than the corresponding Hb in the H⊥ case. Second, Hb increases monotonically, as f increases for both cases. These facts assert the third prediction. shows the rotating frequency F effect on (ΔZ/Z)m. Once again, when F |
= 0, which corresponds to the static h |
= |
h∥ case, (ΔZ/Z)m is the lowest. Hence, confirms that the RFA method with F in the range 1–100 Hz can improve (ΔZ/Z)m with respect to the other static field-annealing methods. In addition, there are double peaks in the (ΔZ/Z)m versus F plot: one occurs at F |
= 1 Hz, and the other at F |
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