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= 1744 kN s/m; cs3 |
= 1981 kN s/m; cs4 |
= 2156 kN s/m; and cs5 |
= 2295 kN s/m. Hence, the following ci=csi-∑j=1i-1cj damping coefficients of the five dampers are obtained: c1 |
= |
cs1 |
= 1325 kN s/m; c2 |
= |
cs2 |
− |
c1 |
= 419 kN s/m;c3 |
= |
cs3 |
− |
c1 |
− |
c2 |
= 237 kN s/m; c4 |
= |
cs4 |
− |
c1 |
− |
c2 |
− |
c3 |
= 175 kN s/m; and c5 |
= |
cs5 |
− |
c1 |
− |
c2 |
− |
c3 |
− |
c4 |
= 139 kN s/m.The mean value of the 1000–2000 kN/m range indicated in Section was adopted as tentative choice of the stiffness of the first damper-disconnecting spring, i.e. kd1 |
= 1500 kN/m. By applying the relations reported in the same section, the following tentative stiffness values of the remaining springs are derived: kd2=c2c1·kd1=474kN/m; kd3=c3c2·kd2=268kN/m; kd4=c4c3·kd3=198kN/m; and kd5=c5c4·kd4=157kN/m. shows the cnl |
− |
δ analytical curve (4) drawn up to the hypothesized δmax,t value of 7.5 mm, in superimposition with the corresponding cs |
− |
δ curve produced by the 5-link model for the set of wi, ci and kdi values reported above. The five dots correspond to the δi values of 0.75 mm, 2.25 mm, 3.75 mm, 5.25 mm and 6.75 mm where the equality of cs(δ) to cnl(δ) is imposed. The correlation of the numerical cs |
− |
δ curve to the analytical model curve appears to be already satisfactory for the m |
= 5 version of the multi-link finite element model adopted herein. For greater values of m, the numerical curve layout tends to adhere even more strictly to the analytical one, because its horizontal segments tend to become smaller as m growths. However, the little improvements that could be achieved in the reproduction of the analytical damping coefficient-interpenetration depth graph for m |
> 5 do not justify, at least for this case study, the remarkable increase of the numerical solution times, which are an exponential function of the number of damper + gap + spring elements, as observed above.The non-linear dynamic analyses were carried out by assuming seven artificial ground motions as inputs, generated by SIMQKE-II software . The two spectra, scaled at the BDE and MCE levels of seismic action, with a 10% and 5% probability of being exceeded over a reference period of 50 years fixed for the building, respectively, are referred to the city of Florence and the soil conditions of the church site (classified as “C-type” according to ). The integration time step was fixed at 0.01 s, which guarantees accuracy to the numerical solution and sufficient contour definition to the response cycles of the 5-link elements.In particular, three selected cycles obtained from the time-history analysis carried out with the most demanding of the seven input ground motions scaled at the MCE amplitude level are compared in with the corresponding cycles derived from the non-linear viscoelastic analytical model defined by equations (6). The latter were obtained by imposing the δ(t) time-history deducted from the response of the finite element model to the analytical model implemented via Excel software, and by computing relevant total contact force Ft(t). The graphs in highlight a satisfactory superimposition of numerical and analytical cycle paths, with nearly identical contained areas and thus coinciding amounts of dissipated energy. The highest of the three numerical/analytical cycles plotted in is also the widest being derived from all the time-history analyses carried out. The effective reproduction of the analytical cycles is a consequence of the calibration of the numerical model parameters, kdi stiffness coefficients included, for which an empirical tentative choice criterion was proposed in Section . The efficacy of the five damper-disconnecting springs is demonstrated by the dampers closing completely in the restitution phase, as highlighted by the correct numerical simulation of the unloading paths of the analytical cycles in . In order to complete the numerical enquiry, kd1 was varied over the entire 1000–2000 kN/m choice range suggested in Section , and kd2 through kd5 according to the mutual proportions expressed therein. The influence on peak response quantities and on cycle contours proved to be negligible; this confirms that, for the range of structural dimensions and peak ground accelerations mentioned in Section , the recommended kd1 tentative choice range provides stable and satisfactory simulation capacities of the multi-link model.The interpenetration depth and contact force time-histories obtained from the analyses developed with the most demanding input accelerogram scaled at the BDE and MCE amplitudes are plotted in for the 5-link element connecting joints C and C’, for which the maximum values of both quantities were surveyed. These graphs highlight maximum δ values equal to 5.7 mm—BDE—and 7 mm—MCE—, and peak Ft contact forces of about 220 kN—BDE—and 285 kN—MCE. The response data concerning the BDE level highlight that pounding significantly affects the seismic response of the two buildings as early as this medium-to-high level of seismic action. The maximum penetration of 7 mm surveyed in the MCE-scaled analyses is very close to the δmax,t |
= 7.5 mm value tentatively assumed in the calibration process of the 5-link model. Therefore, the related wi values selected for the initial gap openings result to be correctly defined for this case study. Should the maximum penetration provided by the time-history analysis have been notably different from δmax,t, the calibration of the model would have been repeated by assuming this updated δ peak value, and by re-computing the gap openings accordingly.Similar maximum δ and Ft values are observed for the other four 5-link contact elements, and severe pounding response conditions are found at the MCE level of seismic action, which are reflected in the stress states of the tower members, and particularly of columns. Data about stress states are summarized in , where the maximum values of the bending moment around the x axis, Mx, the normal force, N, and the shear along y, Vy, are listed for the two internal rear columns (i.e. the most stressed ones), for MCE and BDE hazard levels and the different structural configurations examined in this study. The peak stress values, denoted with symbol PO in , are reached in impact joints B′ and D′. By comparing the PO values computed in pounding conditions with those corresponding to the case of theoretical non-pounding oscillations (NP in )—which could have occurred if the tower had been built with an at-rest separation gap wide enough to avoid collisions—percent increases of about 38% (BDE) and 37% (MCE) in bending moment, 71% (BDE) and 79% (MCE) in shear and 3% both for BDE and MCE in normal force are derived from , highlighting the heavy effects of pounding impacts in the dynamic response of the tower. The pairs of Mx |
− |
N values listed in are graphically reproduced by dots in the graphs of , where the Mx |
− |
N safety domain of the eight main columns of the tower referred to their mutual cross section in the top zone of the arcade (the impact zone), also drawn in , is plotted. The representative Mx |
− |
N point for pounding conditions (denoted again with symbol PO) is situated next to the boundary of the domain for the BDE (left graph in ), which causes the current safety margins of the most stressed columns to virtually annul for the normative basic hazard level. At the same time, the corresponding PO point for the MCE (right graph in ) is placed outside the safety domain. The point representative of the theoretical non-pounding response, named NP like in , remains within the domain in both cases, although at a relatively low distance from its boundary for the MCE.The results concerning the two external rear columns, not illustrated here for brevity’s sake, assess near-boundary bending moment–axial force stress states for BDE and unsafe states for MCE too. All rear columns failed to pass the shear resistance verifications, both for BDE (Vy |
= 216 kN) and MCE (Vy |
= 287 kN), as the estimated design resisting shear, VR, is equal to 150 kN for the stirrups spacing of 150 mm deducted from design drawings. This is a consequence of the high peak shear values achieved in the impact spots, due to the fact that each multi-link contact element reacts like a semi-rigid constraint at the horizontal translation when collision occurs. In non-pounding conditions, shear tests were passed only for BDE (Vy |
= 126 kN). However, the computed maximum shear for MCE, Vy |
= 160 kN, only slightly exceeds VR.The pounding effects on the church structure were not examined in detail in this study, because attention was essentially focused on the bell tower response. However, it can be noted that the high pounding-induced Vy values would be capable of causing local punching mechanisms of the façade in correspondence with the three internal spots, as early as at BDE level (punching does not occur in the two lateral spots due to the presence of the longitudinal walls of the central nave). Furthermore, the peak elastic stress values surveyed in the shell elements of the finite element model of the church in the façade zones situated over the high central window are significantly greater than the nominal tensile strength of masonry. Consequently, diffused cracks are likely to appear in this portion of the façade, starting from a BDE-scaled input seismic action. Although more careful evaluations should be carried out in order to formulate a definite assessment of pounding effects on the church, these data underline that severe local damage is expected in pounding conditions also on its façade, in addition to the assessed unsafe response of the bell tower. This prompts to adopt a pounding mitigation strategy to reach safe earthquake response conditions for the bell tower as well as to avoid damage to the church. A damped-interconnection based retrofit solution is offered to this purpose in the next section.The pounding mitigation strategy selected for this case study, recently proposed for application to adjacent frame R/C buildings The damping action of FV dissipaters is produced by a compressible silicone fluid flowing through the thin annular space found between the piston head and the internal casing, as illustrated by the schematic longitudinal section in . When the fluid is pressurized upon manufacturing by means of a static pre-load, the devices reach full re-centring capacity. This is the case of the class of FV dissipaters considered in this study. The analytical expression of the damping reaction force, FFV, exerted by a FV damper is where t |
= time variable; d |
= displacement; ḋ |
= velocity; cFV |
= damping coefficient; sgn[·] = signum function; |·| = absolute value; α |
= fractional exponent ranging from 0.1 to 0.2.In order to minimize the architectural intrusion of the pounding mitigation intervention, only two dissipaters were installed, and namely in the positions marked by the pairs of joints A–A′ and E–E′ in . The design analysis led to select the following properties of the two devices, for the mean value of the α exponent range, 0.15: cFV |
= 600 kN (s/m)0.15; maximum reaction force FFV,max |
= ±400 kN; stroke s |
= ± 50 mm; and maximum damping energy capacity Ed |
= 100 kJ. These properties correspond to a medium-sized pressurized damper in current production . As shown in these drawings, the dissipaters are housed in the terminal zones of the nave walls, at the façade corner. Here, a portion of masonry about 1.6 m long is demolished to allow their incorporation. Each dissipater is connected to the interfaced tower column by means of chemical anchorages, where four steel bars are inserted and screwed to the end steel plates of the device. On the wall side, the dissipater is joined to a R/C block, 0.8 m high and with base dimensions of 0.6 m (coinciding with the wall thickness) × 0.6 m (i.e. the residual length of the 1.6 m long demolished portion of the wall, not including the FV device length). Four threaded steel axes are encapsulated and grouted in the concrete casting. The connecting bars of the damper are screwed both to its end plate and to the axes embedded in the R/C block. A steel box composed of plates and profiles is built around the hollow created with the demolition of masonry, so as to restore continuity between the walls of the façade and the nave in the top corner area. The steel box hides the intervention in the inner side of the church. As shown in the photomontage image to the left in , a transom-window is mounted on the external side, so that the dissipater can be accessed for necessary inspection and control activities, and be removed if required in case of future laboratory tests.A new set of non-linear dynamic analyses was carried out to evaluate the benefits of the intervention. To this aim, the five contact elements were removed from the computational model and substituted with the two FV dampers, the finite element model of which—governed by relation (8)—is included in the basic library of SAP2000NL program, so as to reproduce the new structural configuration. The results are summarized in , which includes plotting of the response cycles of one dissipater and the time-history of relative displacements of the bell tower with respect to the church (measured again at the top C–C′ position in , where the maximum relative displacements are reached), obtained from the most demanding input ground motion scaled at MCE amplitude. The former graph highlights maximum positive and negative reaction forces equal to about 320 kN, determining a total force of about 640 kN for the two FV devices, against a total impact force of about 1400 kN in unprotected conditions, given by the sum of the peak pounding forces computed in the five contact joint pairs. On the church side, the reaction forces are safely absorbed by the R/C blocks built over the masonry walls, and properly diffused on the latter by means of the same blocks as well as of the steel boxes encasing the dampers. The time-history graph shows that the maximum value of the positive relative displacement, i.e. the component directed towards the façade, is equal to 19.5 mm. As this value is lower than the assumed minimum separation gap depth at rest, pounding does not occur up to the maximum considered earthquake level, thanks to the protective action offered by the dissipaters. As a consequence, the peak stress states, summarized in in the columns referred to the damped interconnection configuration (named DI), result to be 51% (Mx), 28% (N) and 49% (Vy) lower than the corresponding values computed from pounding response, for BDE; and 53% (Mx), 31% (N) and 55% (Vy) lower, for MCE. These values guarantee completely safe structural conditions of the tower columns, with wide safety margins up to MCE, as displayed by the positions of the DI dots within the Mx |
− |
N domain graphs in . The maximum shear is always lower than the design resisting shear VR. The peak stress states in retrofitted configuration are also noticeably lower than the values calculated for the theoretical non-pounding oscillations, bringing all tower columns to safer levels. Hence, the proposed damper interconnection strategy is not only capable to prevent pounding, but it also improves considerably the seismic response of the two coupled structures, as compared to the hypothesis of their non-pounding relative vibration. The estimated cost of the intervention is about 20,000 Euros, architectural finishes included.Another comparison was made with the response offered by a conventional rigid-connection retrofit intervention frequently adopted for pounding mitigation in adjacent frame building structures, details of which are illustrated in . In this case, five couples of steel tie bars are incorporated as connecting elements in the critical positions located by joints A–A′ through E–E′. In order to diffuse better the local normal stress states induced by the bars in seismic response conditions, a continuous R/C beam with cross section dimensions of 0.6 m (coinciding again with the wall thickness) × 0.6 m is built on its top, by demolishing a corresponding portion of the façade wall. The stress state data reported in for this alternative rehabilitation hypothesis (denoted by symbol RI) highlight relatively little benefits in terms of maximum bending moment, reduced by 22% (BDE) and 21% (MCE) as compared to pounding response conditions. At the same time, the effects of rigid connection, combined with the remarkable differences in height between the two structures, slightly increase, rather than decrease, the peak axial forces and shears. The representative RI dots in the Mx |
− |
N domain of remain within the safe domain of columns for both seismic hazard levels, but they are nearer to the boundary than the corresponding dots for non-pounding relative oscillations. Based on these results, this conventional pounding mitigation hypothesis shows remarkably poorer performance if compared to the damped-interconnection solution, in addition to greater architectural intrusion and cost (estimated at about 45,000 Euros).The assessment study carried out on the Chiesa del Sacro Cuore in Florence allowed evaluating the seismic pounding vulnerability of its modern heritage tower bell, representative of an important class of slender 20th century R/C structures built with separation gaps from adjacent buildings not wide enough to avoid collisions during their earthquake-induced motion. The multi-link viscoelastic finite element contact model devised for the development of the time-history analysis of pounding proved to satisfactorily reproduce Jankowski’s analytical model response. The damped-interconnection retrofit solution proposed to prevent pounding guarantees the required performance with little architectural intrusion and low cost. More specific remarks deriving from the results of the study are summarized below.The time-history response data show significant pounding effects between the two buildings at the normative basic design level of input seismic action, with virtually annulled safety margins for the tower columns in terms of combined bending moment-axial force stress states. Remarkably unsafe conditions are assessed at the maximum considered earthquake level. The rear columns fail to pass the shear resistance tests, for both BDE and MCE.For both hazard levels, bending moment grows by nearly 40%, and shear increases by about 70% (BDE) to about 80% (MCE), with respect to theoretical non-pounding conditions. This highlights that earthquake-induced impacts affect the response of the tower heavily.This holds true for the church building too, where pounding effects—although not evaluated in detail in this study—would certainly cause local punching mechanisms of the façade in correspondence with the three internal contact spots, and diffused cracks in the façade zones situated above the central window.Simple mathematical relations were proposed for the tentative choice of the initial openings wi of the gapci elements and the ci damping coefficients of the dampers constituting the multi-link viscoelastic contact model, and an empirical criterion was suggested for the preliminary estimation of the kdi stiffness values of the damper-disconnecting springs. This approach helped match Jankowski’s model response cycles while avoiding any subsequent iterative adjustment of the finite element model parameters in this case. However, the calibration process normally requires one or two iterations at most, starting from the initial set of parameter values fixed by the relations and criterion presented in Section , and demonstratively applied point-by-point in Section to this case study, for which the 5-link version of the model proved to bear the best balance between numerical simulation performance and computational effort.The proposed damped-interconnection retrofit solution allows preventing pounding, with reductions by about 50% in bending moment, by 30% in axial force and by 50% in shear, for the most stressed columns and both hazard levels, as compared to pounding response conditions. This determines completely safe stress states up to MCE. This enhanced performance is obtained thanks to the installation of only two medium-sized pressurized FV dampers in current production, with very little demolitions, practically null architectural intrusion, quick installation works and low cost.The conventional rigid-connection retrofit intervention designed to develop a comparative evaluation only slightly improves the seismic response of the bell tower, with reductions of the stress states of columns limited to about 20% in terms of bending moment, and no benefits on axial force and shear, as compared to pounding response conditions. In spite of this, the intervention is more invasive and expensive than the damped-interconnection.In addition to preventing structural collision, the installation of FV devices allows improving the seismic performance of the bell tower and the church substantially, also with respect to the non-pounding response likely to occur in the presence of an adequate separation gap between the two structures in current conditions.Based on these data, in the special case study examined in this paper, similarly to standard frame buildings analyzed at previous stages of this research, the proposed damped-interconnection retrofit solution confirms to be not only an effective pounding mitigation technique, but also a global seismic retrofit strategy for slender R/C structures featuring inadequate gaps at rest with respect to adjacent buildings.On solving the orientation gradient dependency of high angular resolution EBSDCurrent high angular resolution electron backscatter diffraction (HR-EBSD) methods are successful at measuring pure elastic strains but have difficulties with plastically deformed metals containing orientation gradients. The strong influences of these rotations have been systematically studied using simulated patterns based on the many-beam dynamic theory of EBSP formation; a rotation of only 1° can lead to apparent elastic strains of several hundred microstrains. A new method is proposed to correct for orientation gradient effects using a two-step procedure integrating finite strain theory: (i) reference pattern rotation and (ii) cross-correlation; it reduces the strain errors on the simulated patterns to tens of microstrains. An application to plastically deformed ferritic steel to generates elastic strain maps with significantly reduced values of both strains and residual errors in regions of rotations exceeding 1°.► Many-beam theory simulations show that HR-EBSD is sensitive to orientation gradients. ► Finite strain theory and rotation processing the reference EBSP solves the problem. ► New method succesfully applied to plastically strained IF steel.Over the last decade, electron backscatter diffraction (EBSD) capabilities have been extended towards high angular resolution techniques measuring small strains and lattice rotations with a sensitivity of about 10−4. These developments (the so-called “pattern shift technique”, or HR-EBSD) pioneered by Troost et al. The standard procedure relies on a comparison between a reference pattern and the current pattern by cross-correlation of a set of small Regions of Interest (ROI); the distortion field between the EBSPs, approximated by the discrete ROI shifts, is then used for the identification of the material deformation gradient tensor of the probed volume (see Following these validations, HR-EBSD is increasingly applied to plastically deformed materials When applying HR-EBSD to plastically deformed materials, it is desirable that both the elastic strain of the lattice cell and local rotations be measured with good accuracy. The former is typically used to quantify residual elastic stresses and the latter provides a measure of the lattice curvature from which an estimate of the geometrically necessary dislocations (GNDs) can be calculated The rotation axes investigated are the three sample axes denoted X, Y and Z, and three randomly chosen axes denoted A ∼(0.306, −0.728, −0.613), B ∼(−0.566, −0.424, 0.707) and C ∼(0.802, −0.534, 0.267) (vector coordinates are expressed in the sample reference frame). For each pattern, the elastic distortion parameters were randomly chosen in the typical elastic range [−10−3; 10−3]. To this end, an in-house parallelized implementation of the many-beam dynamical theory . This set of simulated patterns (4 sets of ∼180) will be made freely available on request to the author for validation purposes.These synthetic patterns were analysed, using the infinitesimal strain framework, with our own StrainCorrelator software (a)–(c) shows the absolute error made in recovering the strain tensor for each of the three rotation around the sample axes, as a function of the rotation angle (results obtained for the A, B or C rotation axes are similar and are provided as ). The angular range has been limited to those patterns where no unrealistic shift could be visually detected in the global displacement field, i.e. typically −4° to +4°. Outside this range, the error curves lose their apparent smoothness due to some rogue shifts perturbing the least-square fitting optimisation. The graph of d demonstrates that in the presence of even relatively small rotations of ∼1° the total error (defined by the L2-norm of the five independent components of the strain deviator) is of the order of 5×10−4. The most affected components depend on the rotation axis, but it is generally observed that for rotations of about 3° these errors lead to physically unreasonable strain values, with a total error exceeding the usual elastic limit of 2×10−3. should be taken as indications only, since similar analyses with other important parameters (ROIs and PC position) give different error values albeit with the same order of magnitude. Thus it has been found that the number and location of the ROIs influence the measured strain, with all other factors kept constant. a shows the strain value errors obtained for nine different ROI setups, as represented in b. The amplitudes, and sometimes the sign, of the error components appear to be dramatically affected by the ROI setup.Similarly, for a given ROI setup, the PC position also affects errors on the strain values, as illustrated in This extreme variability of the accuracy with respect to the geometrical configuration of both the ROI setup and projection parameters has not been observed in our previous tests dealing with infinitesimal strain simulations, i.e. less than 10−2 |
rad or 0.6° As an example of the limitations of small strain theory, consider a pure rotation of any crystal line element dX→ by an angle α around the X sample axis expressed as follows:dx→=R̲dX→withR̲=(1000cosα−sinα0sinαcosα)This pure rotation tensor (R̲−1=R̲T and det(R̲)=1) does not change the length of line elements. However, this tensor is expressed using the small strain theory as R̲=I̲+ε̲+ω̲, withε̲=(0000cosα−1000cosα−1)andω̲=(00000−sinα0sinα0), a change in length of a material line element is predicted, which is of the order 10−4 when the rotation angle is of the order 10−2 |
rad (i.e. ∼0.6°). This is a general consideration which indicates the maximum rotation angle that can be tolerated using small strain theory. As recalled in , HR-EBSD is insensitive to volumetric expansion and the above equation can be rewritten in terms of the deviator tensor as:thus diminishing the error made on strain by a factor of two-thirds. On the other hand, it is common practice to recover the full strain tensor by applying a plane stress traction free surface condition, and this justifies the use of Eq. In a first attempt to solve this problem, the least-square minimisation procedure has been modified in order to incorporate the polar decomposition of the deformation gradient tensor.In finite strain theory, the deformation gradient tensor is decomposed into the product of an orthogonal tensor and a positive definite symmetric tensor:U and V are called the right (resp. left) stretch tensor and R is a rotation tensor (R−1=RT and det(The Green-Lagrange strain tensor, defined as E̲=12(F̲TF̲−I̲)=12(U̲2−I̲) can also be written in terms of the displacement gradient tensor H̲=Grad̲u→ with F̲=I̲+H̲:In the case of HR-EBSD, it is clear that the (small strain—finite rotation) hypothesis should apply, since in the absence of any rotation the distortion of the EBSP is due to the elastic deviatoric strain of the lattice. Therefore, neglecting second order terms, the Green-Lagrange strain tensor is approximated by the infinitesimal strain tensor, from where we get the right stretch tensor as:For the case of HR-EBSD it has been found more convenient to implement the left side decomposition of F, since when measuring the strain of the “deformed” pattern, one is interested in the strain of the crystal in the “deformed” orientation, that is the finite rotation between the reference and the current pattern is applied first, followed by an elastic distortion of the rotated crystal. Further, as recalled in , HR-EBSD is only sensitive to the strain deviator tensor.minf(F̲)=∑{ROI}12‖Z⁎(F̲·r→i)·k→F̲·r→i−(r→i+q→i)‖2with F̲=(I̲+ε′̲)·R̲ where ε′̲=ε̲−(εkk/3)I̲ is the strain deviator and R̲ is now a finite rotation. The vectors r→i, k→ and q→i are defined in ; R̲ is conveniently parameterized by the Rodrigues vector:where α is the rotation angle and n→ is the unit direction vector of the rotation axis. can be solved by any non-linear solver such as Levenberg–Marquardt. The infinitesimal strain model and this new finite strain formalism were compared in the case of truly infinitesimal imposed rotations and were found to give the same results for a given set of shifts.The set of EBSPs used in the first part of the study was then re-analysed with this new physical model. Results are presented in The general trend is that the finite rotation formalism helps to reduce the error on the normal strain components (deviator) but does not improve the measurement of the shear components. Accordingly, the results obtained just using finite strain theory are also still very dependent on the geometry setup.Since the erroneous results and erratic behaviour of the least-square minimisation are not improved by the finite rotation formalism extension, the reason for this problem has to be related to the quality of the input data, i.e. the measure of the ROI shifts by cross-correlation. This is clearly shown in where the cross-correlation quality (i.e. the XCF peak intensity averaged over the set of ROIs) is presented as a function of the mean shift error (i.e. the angular difference between the measured shifts and the theoretical shifts calculated from the imposed deformation gradient tensor by the formula given in As the rotation angle increases, the mean shift error increases and the cross-correlation quality decreases. However, there is no direct relation between the cross-correlation quality and the mean shift error: the slope of the curves depends on the rotation axis.This can be explained by the fact that when the rotation becomes significant, the reference and deformed ROIs are less and less similar, due to the rotation or the stretch of the ROI patterns; the ROI patterns do not undergo a pure translation. In these conditions, the cross-correlation function intensity decreases and the peak shape is modified (rounded and distorted) as shown in . It follows that the peak position, which indicates a mean translation between ROIs, cannot account for the rotation and stretching of the pattern. Hence, the pattern distortion field becomes poorly approximated by the discrete grid of the ROI shifts, which in turn adversely affects the output of the minimisation procedure. It is interesting to note () that in the case of the Z rotation axis, although the cross-correlation quality decreases more rapidly than for the X or Y cases, the mean shift error is comparable to the Y case. Since a rotation around the Z axis produces mainly a rotation in the plane of the ROI patterns, with little distortion (the peak in 6d is more or less isotropically spread), it is a hint that the mean shift error is more strongly affected by distortion than by pure rotation.We therefore have to find a better method of measuring the distortion field of the deformed EBSP compared to the reference, in particular in the presence of a significant rotation.Since the finite rotation between the reference and the current pattern is roughly captured by the standard, Hough based, indexation of commercial software, this information can be used to “simulate” the effect of such a rotation on the reference pattern as sketched in . More precisely, knowing the orientations of the reference and current crystals from standard indexing, one can easily compute an approximate finite rotation, which is then applied to the reference pattern by standard image processing techniques (mapping and interpolation). In so doing, the features contained in the rotated reference image now more closely resemble the corresponding ones contained in the current pattern. Therefore, the residual distortion field between the rotated reference and the current patterns can be accurately determined by means of the usual sub-pixel cross-correlation. Care must be taken to avoid ROI locations on the edge of the useful part of the rotated reference image.From a mathematical viewpoint, the total deformation gradient tensor is again written in terms of a left side polar decomposition F̲tot=(I̲+ε̲)·R̲tot where the total finite rotation is the composition of two rotations R̲tot=δR̲·R̲0.R̲0 is the predominant approximate rotation computed from the orientations measured by standard EBSD and δR̲ is a small refinement rotation, which aims to correct the inaccuracy of standard indexation. It has been found convenient to work with the active rotation obtained as R̲0=g̲ref·g̲curT, where g̲ref and g̲cur are the orientations of the reference and current crystal, respectively. Thus, the total deformation gradient tensor can be written as the composition of two successive transformations: F̲tot=F̲i·F̲0, where the infinitesimal part F̲i is obtained from Eq. applied to the small shifts measured between the rotated reference and the current pattern. The procedure to simulate the effect of a finite rotation on the reference image involves standard mapping interpolation based on the transformation formula (Eq. . That is, for each pixel of the rotated pattern r→, its position r→′ in the reference image is obtained by applying Eq. with F̲=R̲0−1. As this initial position is generally sub-pixel, a bi-cubic interpolation is applied to compute its intensity in the reference image, which is then attributed to pixel r→ of the rotated pattern. Since only a single re-mapping operation is involved, this is a relatively simple and rapid technique.As explained in the previous paragraph, this novel cross-correlation approach has been developed to cope with rotations typically higher than 1°. This new procedure has been validated on the same set of synthetic patterns presented above, using the known pattern rotations. ) demonstrates that the error on the different strain components for finite rotations in the range [−15° +15°] is now of the order of 0.1 to 0.2×10−4, using the rotated reference image. This effectively solves the inaccuracy issue reported in the first part of the paper.To investigate the effect of the new algorithm on experimental data, an IF steel sample deformed in tension to ∼2.5% elongation has been analysed by EBSD, using a Zeiss Supra 55VP FEG-SEM operated at 20 kV with a probe current of ∼2 nA. The EBSD analysis was carried out using an HKL system (now Oxford Instruments) composed of a NordlysII camera and the Channel5 software suite. Patterns were recorded at full resolution (1344×1024 pixels), with a mean integration time of about 2 s per pattern. All experimental patterns were corrected for lens distortion effects according to the procedure detailed in shows the Band Contrast and the Euler maps (a and b) of the 150×25 μm scanned area (step size 0.5 μm).In the following we focus on a single grain outlined in 9b): this grain was chosen because it is relatively free of sample preparation artefacts (pits and surface roughness).Patterns belonging to the selected grain were subsequently analysed with our own StrainCorrelator software with and without the reference rotation step. The reference pattern was selected at the white spot location. This position was chosen so as to give a sufficiently pronounced orientation gradient (maximum angle of ∼2.5° as shown in c), in order to demonstrate the effect of the reference rotation technique (RRT).The spatial variation of the strain tensor components is summarised in a was obtained using the standard small strain version of StrainCorrelator while b was calculated with the newly introduced RRT. First it can be noticed that these strain components are in the −10−3 to 10−3 range (scale is in 10−4 units of strain), i.e. well inside the expected elastic regime for each of them. However, there are noticeable differences between 10a) and 10b): contrast is somewhat lower when the reference rotation technique is employed and this is particularly visible on the shear components and the normal component ε22.The effect of the reference rotation step can be further illustrated by plotting the spatial variation of the Von Mises stress (a and c) and the mean angular error (MAE, b and d). Histograms quantitatively show that both the Von Mises stress and MAE are lowered when the reference rotation step is enabled. When this step is disabled, some regions within the grain reach a Von Mises stress well above 400 MPa, which is to be compared to a yield stress less than ∼200 MPa.c, it is clear that these high stress regions are also characterised by the highest rotation angle and the highest mean angular error. This is in striking contrast with c and d, for which both the Von Mises stress and the MAE are much more homogeneous within the grain. This experimental example illustrates the capability of the reference rotation technique (RRT) to deal with intragranular orientation gradients characteristic of plastically deformed materials.This work aims to satisfactorily extend HR-EBSD analysis to the case of plastically deformed crystals and polycrystals, which are characterised by local orientation variations, typically up to 5° or, in extreme cases, 10°. The local rotations are due to the presence of rotated dislocation walls often referred to as sub-grain (low angle) boundaries or geometrically necessary dislocations (GNDs). It is therefore important to be able to separate out the respective influences of local (plastically induced) rotations from the local (elastic) strains. Current, small strain, HR-EBSD methods are valid for purely elastic strains but their direct application to plastically deformed crystals often leads to very high, and physically questionable strains. For example, in Ref. A method of analysing local strains and orientations by cross-correlation with iteratively generated kinematic—or “Bragg's Law”—simulations of EBSD patterns have been proposed by Kacher et al. In a recent paper, Britton and Wilkinson : it compares the shift error distribution with and without the RRT, for a 4° rotation about X. The standard method gives shift errors between 0.4×10−4 and 30×10−4 |
rad. For the same case the RRT method gives errors between 10−7 and 10−5 |
rad, an improvement of two orders of magnitude. Thus, even if truly outlier shifts are progressively down-weighted by successive iterations, the minimisation procedure will still be fed with erroneous shifts (without RRT) and will therefore deliver erroneous strains and rotations. also helps to understand the variability of the results with respect to ROIs and PC position (reported in ) when RRT is not applied: different ROI sets give different shift error sets, and thus significant scattering of the strain error. In contrast, the shift error amplitude when RRT is applied is so small that fluctuations (noise) are of the same order of magnitude as the strain error (less than 10−5). This is shown in The first part of this work, using a large number of simulated patterns, shows that the error comes primarily from the concept of ROI shifts: in small strain theory it can be safely assumed that the pattern distortions at the ROI scale are well described by pure translations. However, in the presence of a significant rotation between the reference and the current crystal orientations, the ROI patterns are in fact distorted in a more complex way and these distortions have to be dealt with by a mathematically rigorous treatment. In particular, their associated rotations and/or stretches have to be compensated for in the HR-EBSD analysis using finite strain theory. When this is done, as proposed herein, the errors on the elastic strains can be reduced to about 10−4 compatible with theories of plastic yield in materials.It is suggested that standard small strain theory can be applied for rotations between reference and current orientations of less than ∼0.5–1°. The proposed RRT method is limited at small angles by the accuracy on the rotation axis, therefore the RRT step should not be applied blindly, but rather specifically in the case of rotations above 1°. In StrainCorrelator, both set of shifts are measured (i.e. first without RRT and then with RRT) and the solution, which gives the smaller mean angular error is selected. It is also advisable to use Forescatter imaging as a first qualitative detection of the presence of orientation gradients.It should be noted that in practice there are other potential error sources with this rotated reference technique, which need to be carefully analysed to prevent artefacts from cancelling out the benefits. The technique relies on a transformation mapping, which depends on the measured orientations and the Pattern Centre localisation and implicitly considers lens distortion corrected patterns It should also be expected that the accuracy of the method be adversely affected by experimental pattern characteristics such as blurred edges, signal-to-noise variations across the screen or excess and deficiency lines (i.e. band asymmetry). To circumvent the effect of edge blurring, one can predefine an Area of Interest limited to the usable part of the EBSP. Simple geometry can then be used to check that a given ROI is located in the AOI both in the “current” and “rotated reference” patterns. If not, this ROI can be relocated or simply removed from the ROI list. As already explained, when RRT is used, results are far less sensitive to ROI locations (fluctuations are within the noise) and there is therefore no penalty to remove ROIs located near the edge. Concerning band asymmetry When these conditions are respected it becomes feasible to determine elastic strain values by HR-EBSD in dislocated crystals. Moreover, this rotation-compensated reference method also opens up the possibility of accurately characterizing the local strains associated with dislocation substructures by HR-EBSD.This paper deals with high angular resolution electron backscatter diffraction (HR-EBSD) by cross-correlation, a technique developed to measure elastic distortion (strain and rigid body rotation) of crystalline materials. In the case of plastically deformed crystals, significant orientation gradients develop. We show that these orientation gradients are not satisfactorily handled by the standard HR-EBSD method based on small strain theory of elasticity; this method can only be applied when lattice rotations between reference and current patterns do not exceed about 1°. This is demonstrated by a systematic study on synthetic patterns where finite rotations (typical of plastic orientation gradients) are superimposed to small elastic distortions. It is shown that even rotations as small as ∼1° induce significant errors when recovering the elastic strain tensor. These errors are clearly attributed to the fact that the distortion of the EBSD pattern is poorly approximated by the ROI shifts measured by cross-correlation, due to complex rotation and stretch of the ROI pattern produced by crystal rotation.A new procedure, based on finite strain theory, is proposed to overcome this problem. The deformation gradient tensor is decomposed into the product of an infinitesimal distortion with a finite rotation, and an intermediate step is introduced in the cross-correlation procedure. This intermediate step aims at compensating the deleterious effect of the crystal finite rotation, by rotating the reference pattern towards the current pattern orientation, thus minimising the rotation/stretch of the ROI patterns.This novel procedure (reference rotation technique or RRT) is first validated on a synthetic set of dynamically simulated patterns. In this case, errors on the strain components drop below 0.2×10−4.Finally, we apply the RRT to experimental data obtained on a strained IF steel polycrystal. Results demonstrate that, without RRT, regions of unphysical high strain and stress are associated with the regions significantly rotated away from the reference. Further, these regions are also characterised by the highest mean angular error (that is the angular residuals between the measured and back-calculated shifts). When RRT is applied, these regions of (high stress—high error) simply disappear and residual stress values drop towards much lower values.It is assumed that the reader is already familiar with the geometry of an EBSD experiment. An EBSD pattern is nothing more than the gnomonic projection of the crystal lattice cell geometry (recalling that it projects a sphere onto a tangent plane, here the phosphor screen). As such, any elastic distortion of the probed material (point S in ) will be mapped onto the EBSP. The aim of HR-EBSD is to quantify the material distortion between a reference configuration and a “deformed” configuration, by comparing the EBSPs collected from these two probed volume.Consider an infinitesimal element dX→ in the reference crystal cell; this particular direction is mapped to point P on the phosphor screen (i.e. SP→=r→=α·dX→): the gnomonic projection is expressed in the screen reference frame as SP→=(xp−X⁎,yp−Y⁎,Z⁎)T where (xp,yp) are the coordinates of pixel P in the EBSP image (from the lower left corner) and (X⁎,Y⁎,Z⁎) are the pattern centre coordinates.From basic continuum mechanics, the reference element dX→ is transported to dx→=F̲·dX→ in the deformed configuration, where F̲ is the material deformation gradient tensor. The direction collinear to dx→ is now projected to point P′ such that SP′→=r→+q→=α′·dx→ (q→ denotes the translation or shift vector PP′→). Using the gnomonic projection, the scalar factor α′ can be determined from the fact that the third coordinate of SP′→ is equal to Z⁎ (point P′ is on the screen), hence α′=Z⁎/(dx→·k→) where k→ is the unit vector normal to the screen (a similar relation holds for the scalar factor α). Therefore, the mapping transformation from the reference EBSP to the current EBSP is simply written as:Multiplying numerator and denominator of the right hand term by the scalar factor α, it is equivalent to:is the fundamental transformation formula of HR-EBSD, which relates the shift q→ of a given direction r→ produced on the screen to the material deformation gradient tensor F̲.In current implementations of cross-correlation HR-EBSD, the mapping transformation is calculated on a discrete set of locations, and F̲ is obtained by least-square minimisation based on Eq. To describe the mapping transformation between the reference and deformed patterns, current implementations use cross-correlation (XCF) of small Regions of Interest (ROI) and assume that the shift q→ at the ROI centre r→ is adequately measured as the XCF peak position.It should be noted that in the case of a pure dilation F̲=λI̲, Eq. leads to r→′=r→, that is a dilation does not change the crystal directions and therefore does not produce shifts on the EBSP. Therefore HR-EBSD by cross-correlation is insensitive to dilation, and only the deviator part of the strain tensor can be determined by solving Eq. . The full strain tensor can be recovered by applying the out of plane normal stress traction free boundary condition.Supplementary data associated with this article can be found in the online version at SM1: input parameters for the synthetic patterns used in the validation section.On the analysis of vibro-acoustic systems in the mid-frequency range using a hybrid deterministic-statistical approachAt present, several numerical prediction techniques are available for the steady-state dynamic analysis of vibro-acoustic problems. Most of these techniques can be classified as either deterministic or statistical approaches. The use of the deterministic methods, like the Finite Element Method (FEM), is limited to the so-called low-frequency range due to the decreasing wavelengths and subsequent increasing model sizes and computational loads for increasing frequency. The underlying assumptions for the statistical methods, like the Statistical Energy Analysis (SEA), limit their use to the so-called high-frequency range. In-between the low- and high-frequency ranges there is a relatively wide mid-frequency-range, in which some of the subsystems fulfill the requirements for the statistical approach and some others do not (yet). Therefore, Shorter and Langley developed a deterministic-statistical framework combining FEM and SEA models. However, this method is limited in use due to the computational load of the FEM models. This paper proposes a framework for coupling Trefftz-based deterministic models with statistical SEA models. Instead of the FEM, the recently developed and computationally more efficient Wave Based Method (WBM) is being used. The presented method is verified in a numerical example.Due to the increased customer expectations and the more restrictive legal regulations, the vibrational and acoustical behavior of a product has become an important criterion in the product design process. Numerical simulation techniques have become an indispensable part in the product development cycle in order to meet this criterion. Most of the current numerical prediction techniques for steady-state dynamic analysis of vibro-acoustic systems can be classified as being based on either deterministic or statistical principles.The use of deterministic methods like the standard FEM In recent years, several techniques have been developed in order to improve the numerical efficiency and/or to reduce the numerical pollution errors and thus to extend the frequency range of the FEM. Amongst these, it is interesting to mention higher order techniques Although the aforementioned techniques make it possible to calculate the response of one single sample of a system up to higher frequencies, the methods do not take into account the influence of model uncertainties and variability in the numerical models which becomes important at those higher frequencies The techniques described up to now start from a deterministic point of view, which then can be modified in order to reduce the computational load and to include the effect of uncertainties and variability. Another group of methods, the energy based methods, start from a statistical point of view. The statistical behavior of the system is a priori included in the formulation of the method. The most known approach is the SEA In-between the low-frequency range where the whole system behaves deterministically and the high-frequency range with statistical behavior exists a mid-frequency range in which a mixed behavior is expected. Often a number of subsystems have a high modal density and overlap such that they already fulfill the requirements for the application of a statistical description, while some parts of the system still have a deterministic behavior. In view of this mixed dynamic behavior, Langley and co-workers have recently developed a hybrid deterministic-statistical modeling framework Given its favorable convergence properties, the embedding of the WBM in a similar hybrid WBM-SEA framework presents an attractive combination. Due to the indirect nature of the WBM, this method can however not be incorporated readily into the hybrid framework as proposed by Shorter and Langley. In this paper, a suitable extension of the framework is put forward in order to realize this goal. The intrinsic efficiency of the WBM and the ability to represent subsystems with a high-frequency behavior with the SEA result in a method which is able to tackle the mid-frequency region in an efficient and meaningful way.First, the basic principles of the hybrid approach, the WBM and SEA are briefly explained. In a second part, the WBM for acoustics is combined with SEA for structural vibrations in order to achieve a hybrid WBM-SEA method which is able to tackle vibro-acoustic problems. The numerical solution procedure is summarized in a third part. Then, the performance of the newly developed technique is demonstrated by means of the numerical analysis of the mid-frequency behavior of a statistical structure which is mounted on top of a deterministic acoustic cavity. The final section of the paper describes some conclusions.This section describes the basic principles of the different numerical prediction techniques used in this paper. First the hybrid approach is reviewed independently of the problem type(s) and deterministic method considered. The second part describes the basic concepts of the WBM for 3D acoustic problems. Finally, the very basics of SEA are recalled.This section reviews the hybrid approach as presented by Shorter and Langley in In a first step, the system is divided into subsystems with deterministic behavior and subsystems with statistical behavior. illustrates a problem consisting of two subsystems.The uncoupled matrix equation of an indirect (or direct) deterministic subsystem can be written aswith A the system matrix, w the vector of unknowns and b the load vector.The boundary ΓS of a statistical subsystem can be divided into a deterministic boundary ΓSd which is known precisely and which has a deterministic behavior, and a boundary ΓSs which has a statistical behavior. According to . The direct field describes the outgoing field as a result of the prescribed deformation of the deterministic boundary in the absence of the statistical boundary. The reverberant field satisfies blocked boundary conditions across the deterministic boundary and the prescribed boundary conditions across the statistical boundary, when added to the direct field. The diffuse field reciprocity relation with Ddir the direct field stiffness matrix, ω the circular frequency, n the modal density and 〈•〉 the ensemble average. If we introduce interface degrees of freedom qf describing the deformation of the deterministic boundary, the uncoupled equations of motion of a statistical subsystem can be written as:with f the generalized forces and frev,s the so-called blocked reverberant force on the connection degrees of freedom.For the sake of simplicity of notation, consider a system consisting of one deterministic and one statistical subsystem. The described methodology can easily be extended to larger systems. Combining the uncoupled Eqs. into a coupled matrix equation results in:with Cas and Csa the coupling matrices between the interface dofs qf and the degrees of freedom w of the model of the deterministic component. These coupling matrices will be specified for a vibro-acoustic coupling in Section with Dtot the total system matrix, q the vector of degrees of freedom, fext the external load vector and frev the reverberant field load resulting from uncertainty in the properties of the random boundaries of the statistical subsystem. Rewriting this equation in cross-spectral form and averaging over an ensemble of statistical boundaries gives with •H the Hermitian transpose and •−H the Hermitian transpose of the inverse of •, and with:〈Sff〉=Sffext+fextfrevH+〈frev〉fH+frevfrevH.From the statistics of the blocked reverberant force it follows that 〈frev〉 = 0 and as a result:〈Sqq〉=Dtot-1SffextDtot-H+Dtot-1SffrevDtot-H,in which we define the first term as Sqqdir and the second as Sqqrev. When combining Eqs. , the only remaining unknown is the vibrational energy E of the statistical subsystem. This can be retrieved by considering the power balance of the statistical subsystem:Pin: the power entering the statistical subsystem. In case of connection to a deterministic subsystem this is the power Pindir caused by external loading to the deterministic subsystem. Other sources, such as rain-on-the-roof excitation can be added.Poutrev: the power leaving the reverberant field for another subsystem.Pdiss: the power dissipated in the statistical subsystem, for which we assume here proportional damping.The incoming power can be determined from the imposed external loading, the power leaving the reverberant field and the dissipated power depend on the statistical subsystem energy E and will be described in Section leads to the vibrational energy E of the statistical subsystem and in turn to the ensemble average of the cross-spectrum of the unknowns 〈Sqq〉.The steady-state pressure field p at any position r, induced by a time-harmonic external point source excitation q with circular frequency ω, located at rq in the three-dimensional cavity domain V of a bounded acoustic system, is governed by the inhomogeneous Helmholtz equation:with ∇2•=∂2•∂x2+∂2•∂y2++∂2•∂z2,ka=ωca the acoustic wavenumber, ρa the density and ca the speed of sound of the fluid.Since the Helmholtz equation is a second-order differential equation, one boundary condition needs to be specified at each boundary location. The boundary Γa can be divided into four non-overlapping parts, Γa=Γp∪Γv∪ΓZ∪ΓSd. On each of the first three (deterministic) parts of the boundary, Γp, Γv and ΓZ, acoustic pressure, acoustic normal velocity or normal impedance boundary conditions are specified:with R•(r) the boundary residual on quantity • and p¯(r),v¯n(r) and Z¯n(r) the prescribed fields for acoustic pressure, normal velocity and normal impedance, respectively. Lv(•)=jρaω∂•∂n is the normal velocity operator. The boundary condition on the interface ΓSd with the statistical subsystem is defined in Section p(r)≈pˆ(r)=∑a=1nwΦa(r)wa+pˆq(r)=Φ(r)·w+pˆq(r),with w a (nw |
× 1) vector of unknown wave function contribution factors wa, and Φ(r) a (1 × |
nw) vector of wave functions Φa(r).pˆq(r) is a particular solution of the inhomogeneous Helmholtz equation with d(r, |
rq) the distance between a point at r and the acoustic source q at location rq.Φa(r)=Φr(r)=cos(kxrx)cos(kyry)e-jkzrz,Φs(r)=cos(kxsx)e-jkysycos(kzsz),Φt(r)=e-jkxtxcos(kyty)cos(kztz).In order for the wave functions to exactly satisfy the homogeneous Helmholtz Eq. , the wave number components kij need to satisfy:It can be proven that a convergent set of wave functions is obtained if the wave numbers are chosen as follows (kxr,kyr,kzr)=a1πLx,a2πLy,±ka2-a1πLx2-a2πLy2,a1=0,1,2…a2=0,1,2…(kxs,kys,kzs)=a3πLx,±ka2-a3πLx2-a4πLz2,a4πLz,a3=0,1,2…a4=0,1,2…(kxt,kyt,kzt)=±ka2-a5πLy2-a6πLz2,a5πLy,a6πLz,a5=0,1,2…a6=0,1,2…with Lx, Ly and Lz the dimensions of the (preferably) smallest rectangular box enclosing the considered domain, as illustrated in contains an infinite series of wave numbers and, as a result, there is an infinite number of wave functions. This series is truncated by defining an upper bound n• on each of the parameters a• through a frequency dependent truncation rule:with N the ‘truncation factor’. N typically ranges between 1 and 6. Applying this truncation rule results in the use of all wave functions with wavelengths larger than or equal to 1/N times the physical wavelength at each frequency of interest. Since the acoustic wavenumber is proportional to the frequency, this truncation rule results in a linearly increasing number of wave functions with increasing frequency. satisfy the governing dynamic equation, but not necessarily the specified boundary conditions. These conditions are enforced by minimizing the boundary residuals R• (Eqs. -∫Γpjρaω∂p˜(r)∂n·Rp(r)dΩ+∫Γvp˜(r)·Rv(r)dΩ+∫ΓZp˜(r)·RZ(r)dΩ=0.As commonly used in the FEM, the WBM adopts a Galerkin weighted residual formulation in which the weighting functions p˜(r) are expanded in terms of the same set of wave functions used in the field variable expansion . This leads to a matrix equation consisting of nw algebraic equations in the nw unknown wave contribution factors, which is typically written as Eq. As for the Boundary Element Method (BEM) The big advantage of the WBM is that the system matrices are substantially smaller in comparison with the element based techniques. This property, combined with the fast convergence, makes the WBM a less computationally demanding method for dynamic response calculations, which creates opportunities to tackle problems up to higher frequencies.In contrast to the deterministic methods such as the FEM, BEM, WBM, etc. SEA does not pose the problem in terms of a distribution of the field variable throughout the problem domain. SEA tries to calculate space- and frequency- averaged energy quantities for a statistical ensemble. Pioneering work was done by Lyon In a first step, the system is divided into a (small) number of subsystems. Each subsystem is defined as an element of an SEA model corresponding to a substantial energy storage location and should be chosen according to the principle of similarity The global SEA equations of a system are then obtained through an energy balance of each subsystem i, see with Pi,in the power input from the environment, Pi,diss the power dissipated in the subsystem and Pij the power flow from subsystem i to subsystem j.The time averaged energy flow from subsystem i to subsystem j is considered to be proportional to the difference in modal energy:with Ei and ni the time averaged total energy and modal density of subsystem i, ηij the SEA coupling loss factor, which is a measure for the transfer of energy from subsystem i due to coupling with subsystem j that is coupled to subsystem i and hij |
= |
ωniηij the ‘power transfer coefficient’ with ni the modal density of subsystem i. The coupling loss factors are most often determined based on the transmission coefficient of a junction. The determination of the transmission coefficient is however far from trivial and often based on analytical solutions for (semi-) infinite systems The time averaged dissipated power in subsystem i is:with ηi the damping loss factor and the modal overlap factor Mi of subsystem i: in matrix notation with the subsystem modal energies as unknowns leads to:M1+∑j≠1nh1j-h12…-h1n-h21M2+∑j≠2nh2j…-h2n⋮⋮⋱⋮-hn1-hn2…Mn+∑j≠nnhnjE1/n1E2/n2⋮En/nn=P1P2⋮Pnin which the system matrix is small and symmetric due to the reciprocity relation.This section describes the hybrid WBM-SEA for a coupled vibro-acoustic problem, with a 3D deterministic acoustic cavity that is coupled to a 2D elastic plate with statistical properties. The hybrid methodology described in Section is adopted, using a discretized frame with displacement degrees of freedom qf on the interface between the deterministic and statistical subsystem. This discretized interface frame is introduced in order to be able to use the diffuse field reciprocity relationship, Eq. describe the system, but the coupling matrices Csa and Cas and the different terms of the power balance equation still need to be specified.First the coupling of the acoustic load in the deterministic subsystem to the statistical subsystem (Csa) is determined by considering the response of the plate without the statistical boundaries to the pressure load associated with the acoustic wave functions. Secondly, the back-coupling (Cas) from the structure to the acoustic cavity is obtained by imposing normal velocity continuity at the interface. Finally, the power balance equation is elaborated. The concept is illustrated in Without loss of generality and for ease of notation, the method will be explained for a system consisting of only one deterministic and one statistical subsystem.Without loss of generality we consider the plate as being statistical with no deterministic boundaries except the coupling with the acoustic cavity. In case of other deterministic connections, the direct field formulation in this paragraph should be modified, but the applied approach remains the same. This choice means that we assume the edges of the plate to be the ‘statistical boundary’, having statistical properties. The direct field in the plate is defined as the field corresponding to a plate with the same deterministic boundary but considering the statistical boundary absent. In this case, that results in an infinite plate which is loaded on the deterministic interface with the acoustic cavity. We can thus write the displacement of the plate, and, as a result, also the displacement at the interface degrees of freedom qf, as the response of an infinite plate to the acoustic pressure on the interface and the reverberant field loading:with G(d) the Green’s function and Hdir a conductance matrix, relating the displacement at rj to the forces at rk, and rf the location vector of the interface points. Hdir is defined as:with for Kirchhoff plate bending the Green’s function defined as with kb the wavenumber, D the flexural rigidity of the plate and H0(2) the zeroth order Hankel function of the second kind.qf=∫ΓG(d(rf,r))(Φ(r)w+pˆq(r))dΓ+Hdirfrevor if we define the dynamic stiffness matrix Ddir=Hdir-1 and multiply Eq. Ddirqf=Ddir∫ΓG(d(rf,r))Φ(r)wdΓ+Ddir∫ΓG(d(rf,r))pˆq(r)dΓ+DdirHdirfrev-Ddir∫ΓG(d(rf,r))Φ(r)dΓw+Ddirqf=Ddir∫ΓG(d(rf,r))pˆq(r)dΓ+frev, corresponds to the lowest line in the matrix Eq. The presence of a vibrating structure along the boundary Γs of the acoustic cavity acts as a velocity excitation for the acoustic fluid. The corresponding residual (cfr. Eq. with u(r) the normal displacement field of the plate. This residual is added to the weighted residual formulation in the same way as the imposed velocity residual (second term in the weighted residual equation):∫Γp˜a(r)·Rva(r)dΓ=w˜T∫ΓΦT(r)Lv(Φ(r))wdΓ-∫ΓjωΦT(r)N(r)qfdΓ=w˜TCaaw+jωCasqf,with N(r) a vector containing shape functions interpolating the nodal displacements qf. At the numerical example in this paper, linear shape functions are used. The structural–acoustic coupling matrix Cas is found as:By applying the conservation of energy for a steady-state dynamic system, the power balance equation is defined, with the dissipated power Pdiss as in Eq. The time and ensemble averaged input power to the direct field of the statistical subsystem can be written as:Pindir=ω2∑jkIm{Ddir,jk}Sqq,jkdir=ω2∑jkIm{Ddir,jk}Dtot-1SffextDtot-H.The time averaged power leaving the reverberant field of the statistical plate subsystem and going into the acoustic cavity can be calculated by integrating the acoustic intensity over the interface surface:with •∗ the complex conjugate of •. For ease of notation we omit from now on the position dependency (r). Using the field variable expansion and with B a nw |
× 1 vector containing the normal derivatives of the wave functions and wrev a nw |
× 1 vector containing the wave function contribution factors due to the reverberant field loading, this can be written as:Using some simple mathematics which we omit here, this equation can be rewritten as:Poutrev=12Re-jρ0ω∫Γ∑jk(ΦBH)jk(wrevwrevH)jkdΓ=12Re-jρ0ω∑jk∫Γ(ΦBH)jkdΓwrevwrevHjk.Taking into account the ensemble average results in:Poutrev=12Re-jρ0ω∑jk∫ΓΦBHjkdΓSqq,jkrev=12Re-jρ0ω∑jk∫ΓΦBHjkdΓDtot-1SffrevDtot-Hjk=houtrevEn,with houtrev the power transfer coefficient for the energy leaving the reverberant field defined as:houtrev=12Re-jρ0ω∑jk∫ΓΦBHjkdΓDtot-14πωIm{Ddir}Dtot-Hjk. have the vibrational energy E of the statistical system as the only unknown quantity. Solving this equation leads to this vibrational energy, which then is used in Eq. to calculate the spectrum of the wave function contribution factors w and as a result also the spectrum of the pressure field in the acoustic cavity.This section summarizes the numerical solution strategy of the hybrid WBM-FEM which has been implemented in Matlab®. Except for step 1 each step is repeated for every frequency of interest.The system is partitioned in deterministic and statistical subsystems. For the statistical subsystem the deterministic and statistical boundaries are determined. On the interface between the deterministic and statistical subsystems an interface mesh is defined to represent the displacement field on the interface. This leads to the interface degrees of freedom.The wave numbers of the wave functions are determined using Eq. The direct field system matrices are composed:The direct field dynamic stiffness matrix Ddir of the statistical subsystem is calculated. In case of a plate connected to an acoustic cavity, this can be determined by inverting the conductance matrix Hdir, defined in Eq. The deterministic system matrix A and the loading vector b are calculated by integrating the residuals on the boundaries and interfaces of the deterministic subdomain using the weighted residual formulation . During the integration over the interface with the statistical subdomain, the coupling matrices Cas and Csa and the loading term f are calculated using, respectively, Eqs. . All numerical integrations performed are using Gauss–Legendre quadrature, which has been shown to be efficient for the WBM The total system matrix is assembled using Eqs. The energy of the statistical subsystem is determined:The total system matrix is inverted (inv-command in Matlab).The direct field input power Pindir, reverberant power transfer coefficient houtrev and modal overlap factor Mi are calculated using Eqs. is solved for the energy E of the statistical subsystem.The average field in the deterministic subsystem can be reconstructed using the average cross-spectrum of the unknowns 〈Sqq〉 (Eq. The hybrid approach described above provides a numerical modeling framework to combine an acoustical WBM model with a SEA model for a plate. In order to illustrate this approach, the method is applied to a simple verification example taken from literature . At one side a 1 mm thick aluminium plate (density ρ=2700 |
kg/m3, Young’s modulus E |
= 70 GPa, damping loss factor ηs |
= 0.01) with modal density n=LxLy4πρhD=0.227 modes/Hz compares the results obtained by three different methods in a frequency range of 0 to 400 Hz. Below 400 Hz the plate has 90 modes and the acoustic cavity 6 modes. Since the acoustic cavity has a low modal density and the plate a high modal density, we expect the typical mixed mid-frequency behavior, here with a deterministic acoustical behavior and a statistical structural behavior, within the considered frequency range. A hybrid WBM-SEA model is constructed in which a WBM model with truncation factor N |
= 2 for the acoustic cavity is coupled through 1189 interface nodes (≈6 interface degrees of freedom per wavelength in the plate at 400 Hz) to the plate which is considered to be statistical. The results are compared with the results in In order to assess the effect of damping, (a) and (b) show the energy in the plate for the same problem geometry but with a 0.5 mm thick aluminium plate, a unit velocity on a surface of 0.04 m2 centered at (0.2; 0.3; 0.5) and a lower acoustic damping, ηa |
= 0.001, for two different structural damping values ηs |
= 0.001 and ηs |
= 0.01. The figures contain the curves of hybrid WBM-SEA, hybrid FEM-SEA and 81 pure FEM calculations, of which the latter are performed for different combinations of clamped, free and simply supported boundary conditions.The WBM-SEA model consists of a WBM acoustic model with truncation factor N |
= 2 and 494 interface nodes. The FEM-SEA model has a FEM acoustic model of 6422 degrees of freedom (>20 elements per acoustic wavelength at 400 Hz), connected to the same interface as in the WBM-SEA model. The pure FEM vibro-acoustic model, see , has 31212 degrees of freedom (25704 acoustic dofs and 5508 structural dofs, >20 elements per acoustic wavelength and 6 elements per structural wavelength at 400 Hz). The frequency response calculation from 2 till 400 Hz (with a frequency step of 2 Hz) takes an average of 10.2, 1126 and 76 s per frequency line for the hybrid WBM-SEA, hybrid FEM-SEA and one single pure FEM calculation respectively. Only frequency-dependent operations are taken into account. This means that the calculation time for the hybrid WBM-SEA includes system construction and solution, for the hybrid FEM-SEA system construction without construction of the acoustic FEM matrices and solution and for the pure FEM only the solution of the system since the system matrix can be split into frequency independent matrices. The acoustic system matrix and the acoustic and structural system matrices in respectively the hybrid FEM-SEA and pure FEM are composed using MD-Nastran2010. The hybrid implementations and the solution of the FEM systems are performed single-threaded in Matlab® (R2010a) on a 2.67 Ghz Intel Xeon-based system with a Linux operating system.(a) and (b) show again the typical behavior of a mixed system in which one of the subsystems, the acoustic cavity, exhibits a deterministic behavior and dominates the mean response. The other subsystem, the aluminium plate, has a statistical behavior and determines the variation compared to the mean result and the level of damping. It is clear that a low structural damping, (a) (ηs |
= 0.001, modal overlap one at 350 Hz), leads to a wide spread on the results, but the mean response is still accurately predicted by the hybrid method. A higher structural damping, (b) (ηs |
= 0.01), and as a result higher modal overlap (= 10 at 350 Hz) leads to a smaller variance. It is important to note here that both the hybrid WBM-SEA and the hybrid FEM-SEA lead to the same results, but the hybrid WBM-SEA requires 110 times less time than the FEM-SEA.(a) and (b) show the energy in the acoustic cavity for a low and a high structural damping respectively. It is seen that the mean response is predicted very well by the hybrid WBM-SEA. For both damping values the variance on the acoustic energy is small, which is expected to be due to a weak coupling from the structure to the acoustical domain. Nevertheless, the higher damping value leads again to a smaller spread on the result.In the mid-frequency range there is a need for methods capable of tackling the typical mixed deterministic-statistical behavior. One of the possible approaches is to combine deterministic and statistical techniques. This paper proposes a hybrid coupling of the deterministic WBM with SEA.The ill-conditioning of the WBM system matrices requires accurate numerical integrations, but it does not prevent the WBM from being used successfully in a hybrid framework with SEA. It has been shown in previous publications that the small system matrices and the fast convergence of the method lead to more efficient calculations, as compared to the FEM, for problems of moderate geometrical complexity in the mid-frequency range. Therefore, the combination of these properties of the WBM with the statistical nature of SEA offers a promising method to tackle problems in the mid-frequency range.In order to couple the WBM with SEA an extension of the hybrid FEM-SEA framework of Shorter and Langley The hybrid WBM-SEA framework is shown to be able to predict the average response of a combined vibro-acoustic deterministic-statistic system, and this for several damping values. Both the hybrid WBM-SEA and the hybrid FEM-SEA give similar results, in which the deterministic acoustic modes are clearly visible, while the structural modes are not, since the effect of the latter is averaged in the statistical model. It is seen that for increasing damping and/or modal overlap the variance of the results decreases, which is a generally known property of statistical systems.Influence of high mixing intensity on rheology, hydration, and microstructure of fresh state cement pasteFrom both the fundamental and applied points of view, there is a growing interest in characterizing the rheological properties of concentrated suspensions, such as cement paste. When a cement paste is sheared, its intrinsic network structure will respond to the shear induced stresses. The influence of mixing intensity on the rheological properties of fresh state cement paste is studied. The results showed that contrary to what is popularly believed, when subjected to a high mixing intensity, the rheological properties of cement paste can increase once a certain threshold mixing intensity is achieved. This increase in rheological properties was associated with changes in the chemical nature and physical nature of the cement pastes.Due to the development of highly flowable concrete and increased use of chemical admixtures, renewed interest in predicting and understanding the rheological behavior of concrete has occurred. In its fresh state, concrete can be considered a suspension consisting of sand and gravel particles suspended in a paste matrix In 2011, a new standard, ASTM C1738, was introduced for preparation of cement pastes. This standard uses a high-shear mixer for preparation of cement pastes and was motivated from the work of Helmuth et al. Further research on the microstructure and rheological properties of fresh state cement paste under high mixing intensity conditions is needed to understand how this affects the fresh state behavior of cement-based materials. The aim of this paper is to examine the influence of high-shear mixing on the rheological properties and fresh state microstructure of Portland cement pastes. The fresh state microstructure was assessed with a focused beam reflectance measurement (FBRM) All cement pastes mixtures were prepared using deionized water and a cement conforming ASTM C150 Type I and Type II specifications. The chemical composition of the cement is shown in ; four different cement pastes with three different solid contents were evaluated. The mixture proportions in are expressed for one liter paste; however, for the actual preparation of the cement paste samples, 0.25 L of cement paste was prepared.The mixing protocol for the cement pastes is shown in . A Chandler 3060 high-shear mixer was used to prepare the pastes (see for a picture of the mixing blade geometry). As shown in , six key steps were conducted to homogenize the pastes. A relatively low mixing speed (500 rpm) was selected for the second step so that differences in the magnitude of the initial mixing intensity and secondary mixing intensity on the rheological behavior of the pastes could be examined. Five different mixing speeds (1000, 2000, 4000, 8000, and 12,000 rpm) were selected for Steps 4 and 6. A maximum mixing speed of 12,000 rpm was selected since this was the maximum mixing speed specified in the 2011 and 2013 versions of ASTM C1738 To evaluate the properties of fresh state cement pastes, flow curve rheological testing were conducted using with a commercial rheometer (Anton Paar MCR 301). The rheometer was equipped with a solvent trap to prevent excessive evaporation during the tests and a Peltier setting controlled the temperature of the samples at 23 ± 1 °C. A concentric cylinder geometry was used; the inner diameter of the cup and the outer dimension of the bob were 28.9 mm and 26.6 mm, respectively.Prior to conducting the flow curve test, a 0.1 s− 1 preconditioning process was applied for 30 s (which was enough time to reach a steady-state shear stress condition), followed by 30 s of resting. In order to obtain the flow curve, the shear rate was stepped down from 50 s− 1 to 10 s− 1 in increments of 10 s− 1 (see ). Each shear rate was maintained for 60 s since this was determined to be a sufficient amount of time to achieve a steady-state shear stress for each shear rate. The data were recorded every second, and the last 10 data points at a given shear rate were averaged to represent the average resultant shear stress at that shear rate. According to Ferraris et al. ) was used to characterize the mixtures. The apparent viscosity, μapp, was defined aswhere τ20 |
s-1 is the average resultant shear stress at 20 s− 1 of shear rate.A focused beam reflectance measurement (FBRM) system was used to evaluate the in situ particle conditions of the cement pastes The chord length of the particle is the length from one edge to the opposite edge on the scanning path rather than the diameter of the particles presents a schematic of the FBRM setup. As can be seen in the figure, the FBRM setup includes a beaker that contains the sample, the FBRM probe, and an impeller. After preparing the paste in the high-shear mixer, a portion of the paste was transferred to the FBRM beaker. To minimize the effect of sedimentation on the data, the paste was agitated with the impeller, and the probe was inserted into the sample to face the direction of flow at a downwards angle where Rb is the radius of the vane (in this case the radius of the FBRM impeller), Rc is the radius of the cup (in this case the radius of the FBRM sample container), and Ω is the angular velocity of the vane. From , it can be seen that Rb |
= 25 mm and Rc |
= 32.5 mm.In order to determine the heat signature of the pastes, after preparing the cement pastes with the Chandler high-shear mixer, 20 g of the cement paste was placed in a 3114/3236 TAM Air Thermal, Thermometric AB Isothermal Calorimeter for 20 h. For each mixture, at least two replicates were tested. The heat of hydration curves in are based on the results of a representative sample from each mix, while the data in are based on the average of the samples.Flame photometry was used to gain insight about the pore solution chemistry, particularly the potassium and sodium concentration since they contribute significantly to the ionic strength of the pore solution and previous work ), a 100 g sample of fresh state cement paste was put through a Buckner funnel to extract the pore solution. A 0.50 μm filter paper was used to extract the pore solution. Extraction was conducted by vacuum filtering using the process described by Hema shows the apparent viscosity values at 20 s− 1, and presents the flow curves for the cement pastes. The various curves in each sub-plot of correspond to the final mixing speeds that were used to prepare the paste. As expected, the apparent viscosities increased as the solid volume content increased. As the solid content increases, particle-to-particle interactions are likely to increase, and the increased interactions among the particles result in an increased resistance to flow Higher mixing energy has been associated with cement paste with lower viscosities and increased fluidities , it was seen that when the sample preparation mixing speed increased from 1000 rpm to 4000 rpm, the apparent viscosity did not increase (it generally decreased or remained the same for all the mixtures). However, when the mixing speed increased above 4000 rpm, various behaviors were observed. Between 4000 and 8000 rpm, the apparent viscosity was relatively unchanged for the mixtures without SP; however, a considerable increase in the apparent viscosity values were seen in the pastes containing SP as the sample preparation mixing speed increased from 8000 rpm to 12,000 rpm. Hence, it appears that mixing speeds within a range of 1000 rpm to 4000 rpm is sufficient to disperse the cement particles and that above a threshold mixing speed increasing the speed no longer results in decreasing the viscosity of the cement paste. Cement pastes are often characterized as being Bingham fluids display the results of the FBRM experiments and shows how agglomeration evolved over time when samples were prepared using different sample preparation speeds and different FBRM agitation speeds. It can be seen that in all cases, the global behavior of the samples trended towards agglomerate breakup when the agitation speed used during the FBRM experiment increased, and as the agitation speed increased generally the chord length of cement pastes decreased. As seen in , at agitation speeds of 150 and 300 rpm, the mean chord length of the cement pastes prepared with a final mixing speed of 12,000 rpm was generally greater than cement pastes prepared with a final mixing lower than 8000 rpm. However, as the FBRM agitation speed increased to 450 rpm (see plot c in ), the chord length of the cement pastes prepared using a final preparation speed of 12,000 rpm was often comparable to those prepared with lower sample preparation mixing speeds. The results illustrate the dynamic nature of agglomerate resilience. Samples prepared with the faster sample preparation speeds contained microstructures with larger agglomerates; this may be due to stronger interparticle bonds and/or faster rates of agglomeration. Regardless of the mechanism inducing the formation of the agglomerates, it can be seen that these large agglomerates were able to resist breakdown under the lower FBRM agitation speeds of 150 rpm and 300 rpm, but when enough power is applied to the system, even those agglomerates can be ruptured.Inspired by an approach described by Jarvis et al. in which the relationship between particle size and velocity gradient is used to determine a shear-based agglomeration strength ) is used as a measure to estimate the overall strength of the microstructure. The curves shown in represent three theoretical examples of a mixture that was prepared using three different sample preparation speeds (A, B, and C) and evaluated using the FBRM technique (note, the mixture composition and the FBRM testing approach is the same in the three examples). For ease of reading, we will refer to the mixture by the sample preparation speed that was used to prepare it (e.g., mixture “B” is the mixture that is prepared using sample preparation speed B). As the agitation speed increased from “1” to “2,” the average chord sizes of all the mixtures decreased. Mixture “A” had the largest initial average particle chord length of the three samples and its final average chord length was the same as mixture “B,” whereas mixture “C” had a slope that is equivalent to mixture “B,” but the average particle chord length of mixture C was smaller than that of mixture “B.” Since the slope of mixture “B” is the same as mixture “C” the microstructural response of these two mixtures to the agitation is similar, which suggests that the mixtures have equivalent agglomeration strength. However, since the average chord length of mixture “B” is greater than that of mixture “C,” the microstructure of mixture “B” is comprised of larger agglomerates than mixture “C”; thus, the overall state of agglomeration is considered to be higher than that of mixture “C.” Mixture “A” initially has agglomerates that are larger than the other two mixtures; however, as the agitation speed increased, the agglomerate size of mixture “A” decreased to an agglomerate size similar to mixture “B.” Thus, the agglomeration strength of mixture “A” was weaker than mixture “B,” and this is manifested by the slope of the curve represented by mixture “A” curve being steeper than that of the curve represented by mixture “B.” The sign of the slope indicates whether breakdown or agglomeration is occurring, where negative signs represent breakdown/disaggregation has occurred. The absolute magnitude of the slope is used to determine the agglomeration strength, and in the case of a negative slope higher values are associated with lower agglomeration strengths (i.e., weaker systems). Based on this concept, the agglomeration strength of the pastes was calculated by determining the slope of the line passing through the average chord length-FBRM agitation speed plot at 150 rpm and 450 rpm. shows the agglomeration strength of each cement paste as calculated and expressed in . It is readily apparent that the cement paste prepared at higher mixing speeds had weaker agglomeration strength. Furthermore, agglomeration strength decreased as the distance between agglomerates increased, regardless of whether this increased spacing was caused by reducing the number of particles in the system (e.g., compare 042 and 045 mixtures) or via polymeric dispersion (e.g., compare the 045 and 045_SP mixtures). Combining the FBRM chord length results () and the agglomeration strength results (), it is concluded that the cement paste prepared with very high mixing speeds contained larger agglomerates than their counterpart pastes prepared at the lower mixing speeds; however, these agglomerates were relatively weak. The results are consistent with work by Ferron The rheological properties of cement paste are influenced by its microstructure where ε0 is the permittivity of the vacuum, ε is the dielectric constant (relative permittivity) of the dispersion medium, R is the gas constant, T is the absolute temperature, F is the Faraday constant, and I is the ionic strength. As the ionic concentration is increased, the thickness of electrical double layers decreases, which can result in increased/stronger agglomeration. shows an illustration depicting how a high sample preparation mixing intensity may result in increasing agglomeration of cement particles. First, breakdown of cement particles occurs due to the mixing process, thereby increasing the specific surface area in contact with the mixing water. As the specific surface area increases, the ionic concentration of cement paste increases due to an increasing rate of ionic dissolution from the surface of the cement particles The cement pastes were prepared using the protocol shown in , and the hydration kinetics of the samples were monitored for 20 h after initial contact of the cement with the mixing water. shows that as the final sample preparation mixing speed increased, cement hydration was accelerated and the overall heat evolved during this time period increased (see ); this phenomenon occurred regardless of whether the mixture contained superplasticizer or not. Further analysis of the heat of hydration curves (see ) elucidates that the end of the acceleration period was more sensitive to the applied mixing speed than the end of the induction period. Hydration kinetics prior to the induction period is controlled by dissolution of ions from the cement grains (a), it can be seen that for the cement paste prepared at a final mixing speed of 1000 rpm, inclusion of the superplasticizer had a slight retardation effect. Interestingly, the delaying effect of superplasticizers was compensated by increasing the final mixing speed (compare (a), (b), and (c)). Retardation of cement hydration due to superplasticizers has been commonly reported and could be due to adsorption of the polymer molecules of the superplasticizer onto cement particles inhibiting subsequent topochemical reactions from occurring ) at all the sample preparation mixing speeds that were used in this research. However, as the sample preparation speed increased the difference between the apparent viscosity of the 045 and 045_SP decreased, which indicates that the cement-SP interaction has been affected.The pore fluid of cement is comprised of several ions (primarily, Na+, K+, OH−, Ca2 + and SO42 −) ). Thus the results are in agreement with the hypothesis that the increased mixing intensity increased the early-age dissolution process. The increased ion concentration in the mixture prepared at 12,000 rpm when compared to mixtures prepared at the lower mixing intensities is likely caused by increased deagglomeration of the cement grains due to initial rupturing of the flocs. This deagglomeration could then heighten the accessibility of the alkali ions to the pore solution. Increase in soluble alkali content has been shown to increase the rheological properties of pastes , in which the mixture containing the highest measured alkali ions displayed the greatest apparent viscosity. Ferron that mixing at 12,000 rpm may necessitate more superplasticizer to achieve a target fluidity condition than mixing at 4000 rpm.The results of this study contribute to our understanding of the mechanisms affecting the rheological behavior of cement pastes due to changes in the mixing speed. It was determined that after a certain mixing speed threshold, the rheological properties of fresh state cement paste increased. The results showed that pastes prepared at higher mixing speeds exhibited a larger mean chord size and higher rheological properties than their counterpart pastes prepared at lower mixing speeds. This trend was amplified in pastes containing superplasticizer. Changes in the physical and chemical nature of the pastes occurred when the mixing speed varied. Based on an agglomeration strength analysis using in situ particle size measurements, pastes that were prepared with high mixing speeds contained larger agglomerates that were more weakly linked together than pastes prepared at lower sample preparation speeds. This behavior was associated with changes in chemical reactions in the cement pastes, specifically with regards to hydration kinetics and soluble alkali concentration. As the mixing speed increases, increases in hydration kinetics and alkali content of the pore solution occurred. Such changes will likely lead to a decrease in the thickness of the diffuse double layer and promote agglomeration.Effects of the liquid phase on tensile elongation of Al–Bi alloyTensile tests were performed in Al–Bi alloys in order to show experimental evidences of elongation enhancement by a liquid phase and to understand its underlying causes. Peak elongation of 150–200% was obtained above the melting temperature (539 K) of Bi dispersoids, indicating that the liquid phase can enhance tensile elongation. It was found that homogeneous dispersion of liquid needs to be maintained during deformation. Otherwise, the liquid phase would agglomerate along perpendicular grain boundaries to the tensile axis, causing grain-boundary fracture. When elongation enhancement by the liquid phase was observed, the dislocation density was low and the cavity volume fraction increased slowly, resulting in the delay of cavitation failure. The role of the liquid phase on elongation enhancement was considered to be accommodation of stress concentration and delay in cavitation.Melting of a secondary phase in metallic alloys generally causes premature failure in a brittle manner In steel containing lead the loss of elongation caused by melting of lead was found to be recovered at higher temperatures It has also been reported that high-strain-rate superplasticity (HSRS) is related to the presence of the liquid phase along composite interfaces and matrix grain boundaries. In aluminum alloys and composites, HSRS was observed at very high strain rates of faster than 10−2 s−1However, the above notion of elongation enhancement by the liquid phase is totally different from the traditional concenpt of the LME and has been received with reservation. Mukherjee and his coworkers have claimed that the diffusive accommodation along the interface is a rate-controlling process for HSRS The controversies over the effects of the liquid phase stems from the fact that the structure and chemistry of the liquid phase under consideration are very complex and are not understood in detail to provide clear-cut evidences. The present paper is aimed at showing experimental evidences of the elongation enhancement by the liquid phase. We selected an Al–Bi alloy as a model system. Bi has almost no solubility in Al and is expected to precipitate as pure Bi. The melting point of Bi is 544 K that is higher than the half of the melting temperature of Al (933 K). Therefore, the aluminum matrix undergoes high-temperature deformation with various thermally activated processes, as in the case of the previous reports in Al–Zr–Pb alloys, lead containing steel, and superplastic materials. The possibility of the elongation enhancement by the liquid phase was examined by performing tensile tests at various temperatures above and below the melting point of Bi. The deformation mechanism was investigated by analyzing stress-strain curves and by observing the microstructure. We found that the homogeneously dispersed liquid can accommodate stress concentration, as exemplified by the reduction of the dislocation density, resulting in the delay in cavitation failure.Al–Bi alloys of three different concentrations (0.3, 1.3, and 1.7 mass% Bi) were prepared by melting pure Al (99.99%) and Bi (99.9999%) at 1073 K in an alumina crucible in air and by casting into a metal die. Before casting, the metal die was heated to 403 K to remove moisture on the die surface. The cast bar was machined to remove surface oxide layer. The bar was forged at room temperature to a thickness of 4 mm and cold-rolled to obtain a sheet of 2 mm in thickness. The chemical compositions indicated above were determined by an inductively coupled plasma (ICP) technique. Both the top and the bottom parts of the cast bars were examined by ICP. The average concentration from the top and the bottom parts was calculated and used in the present work. It should be noted that the concentration variation within a single bar was within +/−0.05%. Other cast bars having a large concentration variation were discarded.An example of the initial microstructure is shown in for the Al-1.3 mass% Bi alloy. In a bright-field TEM image, the Bi phase appears as dark particles and are dispersed homogeneously within the grain interior as well as along grain boundaries. There is no preferential site for the precipitation of Bi. The diameter of Bi is <2 μm. The average diameter of aluminum grains is ≈80 μm. Bi inclusions were found to melt at 539 K, as determined by a differential scanning calorimeter (DSC). This temperature was found to be lower than the melting point of Bi in literatures (TM=544 K). A slightly lower melting temperature was probably due to impurities introduced during fabrication or of hydrostatic pressure acted upon from the surrounding matrix, considering the fact that the volume of Bi decreases upon melting as in the case of melting of ice. DSC experiments also revealed a supercooling effects of liquid Bi in a wide temperature range from 544 to 350 K. As reported by Goswami and Chattopadhyay Tensile specimens were machined from the rolled sheets with a gage dimension of 10×3×2 mm3. Tensile tests were performed at constant cross-head speeds. The initial strain rate was varied in a range of 10−5–10−3 s−l. In order to understand the role of the liquid phase to the deformation and fracture behavior, the dislocation density was measured by analyzing transmission electron micrographs. The cavitation behavior was studied by measuring the cavity volume fraction to the total specimen volume using an Archimedes method. The fracture surface and the side surface of specimens were observed by a scanning electron microscope. shows elongation-to-failure as a function of deformation temperature for three different strain rates in the Al-1.3 mass% Bi alloy. Below the melting temperature of Bi (TM), elongation increases with increasing temperature at all strain rates. Above TM, however, elongation is dependent on strain rate. At a slow strain rate of 10−5 s−1, elongation decreases abruptly when the Bi inclusion melts. This result does not contradict with the conventional concept of LME. On the contrary, at the faster strain rates of 10−3 and 10−4 s−1, elongation increases continuously with increasing temperature even beyond TM up TO 565 K. The maximum elongation of ∼150% is observed above TM at a strain rate of 10−4 s−1, followed by a decrease in elongation. This result clearly shows an evidence for elongation enhancement by the liquid phase. shows scanning electron micrographs of the fracture surface deformed at 565 K. In accordance with the results in , grain boundary fracture is observed at a strain rate of 10−5 s−1. On the other hand, cavitaion fracture is observed at the faster strain rate of 10−4 s−1. Similar results are obtained at 500 and 700 K. The results at 10−3 s−l are basically the same as those at 10−4 s−l. shows elongation as a function of temperature for three Bi concentrations of 0.3, 1.3, and 1.6 mass% Bi. Tensile tests were performed at an initial strain rate of 1.3×10−4 S−l. At this strain rate, elongation enhancement by the liquid phase is observed above TM at all Bi concetrations. However, the peak elongation value and temperature depend on the Bi concentration. At 1.3 and 1.7% Bi, the maximum elongation is observed at 565 K, followed by a decrease at higher temperatures. But at 0.3% Bi, elongation of nearly 200% is obtained at 600 K. This dependence of the Bi concentration on elongation is found to be related to the distribution of the liquid Bi phase as shown in . The figure shows the side surface of samples before deformation and of samples deformed to fracture at 565 and 600 K. The tensile axis is in horizontal direction of the figures. The images are taken by SEM in a back scattered electron mode, in which the heavier Bi appears in a bright contrast. In both samples before deformation, Bi is uniformly distributed. When the samples are deformed in the 0.3% Bi alloy, the size of Bi grows with increasing temperature, but the distribution remains uniform. On the other hand, in the 1.6% Bi alloy, Bi tends to agglomerate along perpendicular grain boundaries to the tensile axis. This tendency of deformation-induced agglomeration becomes more prominent with increasing temperature from 565 to 600 K. At 600 K, the liquid phase along some perpendicular boundaries have a thickness of a few micrometers. It can be said, based on these results, that a large elongation is obtained when the liquid Bi phase remains homogeneously distributed during deformation without wetting a large area of grain boundaries. When agglomeration occurs during deformation and a thick liquid phase wets grain boundaries, bonding force of the boundaries decreases and fracture along the boundaries is expected as shown in In order to understand the reason for the elongation enhancement, deformation and fracture behaviors are characterized for Al-1.3 mass% Bi. shows the maximum stress value at various deformation temperatures plotted versus strain rate. Nominal stress-strain curve was converted to true stress-strain curves by using an iteration method, assuming a homogeneous deformation . At 300 and 400 K, m <0.1, suggesting that deformation is controlled by dislocation glide plasticity. The m value increases with increasing temperature and takes an approximately constant value of 0.27 above TM. This m value suggests that deformation occurs by dislocation glide controlled by climb.The temperature dependence of the maximum stress is shown in . The figure is plotted in an Arrhenius fashion. At low temperatures, a weak temperature dependence is observed in correspondence with the dislocation plasticity mechanism obtained from . Three straight lines having the same slope can be drawn above TM for different strain rates. The slope is equivalent to Qm/R where Q is the activation energy of deformation, m is the strain-rate sensitivity, and R is the gas constant. The activation energy is found to be 104 kJ mol−1. This value is less than the activation energy for self-diffusion of aluminum atoms (142 kJ mol−1 suggest that the deformation mechanism of the Al–Bi alloy is independent of strain rates at all examined temperatures and, thus, appears to be irrelevant to the elongation enhancement. The only obvious effect of Bi inclusion is the lower activation energy than in pure Al.Next, the role of the liquid phase in stress accommodation is studied by measuring the dislocation density in a fractured specimen of Al-1.3 mass% Bi. shows the dislocation density in the gage section of the specimens deformed to failure at various temperatures. The observed area is approximately 5 mm away from the fracture surface. The indicated results correspond to the two contrasting cases in at strain rates of 10−4 and 10−5 s−1. At both strain rates, below TM, the dislocation density is in the range of 1013 m−2 and gradually decreases with increasing temperature, independent of strain rate. Above TM, the temperature dependence of the dislocation density drastically changes with strain rate. When the loss of elongation is observed at 10−5 s−1, the dislocation density decreases only slightly with increasing temperature. In contrast, when the enhancement of elongation is observed at 10−4 s−1, the dislocation density decreases substantially by a facfor of 100 to 1000 above TM. Therefore, the presence of the liquid phase can reduce the dislocation density. We consider that the low dislocation density is due to the accommodation of stress concentration by the liquid phase. Therefore, stress-induced damage formation is expected to be retarded by the liquid phase. On the other hand, the result at 10−5 s−1 suggests that the agglomeration of the liquid phase along the perpendicular boundaries not only provides a weak fracture site, but also causes the depletion of the liquid phase from other boundaries. These depleted boundaries have a complete solid contact and lose an effective means for stress relaxation, resulting in the loss of elongation. The larger dislocation density above TM at 10−5 s−l can be consistently explained if the depleted boundaries are formed during deformation.In this alloy, cavitation damage is found to be a cause for fracture when elongation enhancement is observed, as shown in an SEM micrograph () of the fracture surface of the samples deformed at 565 K at 1.3×10−4 s−1. In , the development of the cavitation damage (cavity volume fraction) with strain is compared at a strain rate of 10−4 s−1 at temperatures both below and above TM. When Bi is solid at 500 K, cavitation becomes noticeable only after deformation to a strain of 0.2, followed by a rapid increase by further deformation. In contrast, when Bi is liquid at 565 K, cavitation occurs in the beginning of deformation, probably because of a weak cohesive energy of the liquid. However, the cavity volume increases at a slower rate than at 500 K, resulting in a much smaller cavity volume fraction at a large strain. Therefore, considering the cavitation damage being a major fracture mechanism at the faster strain rates, a larger elongation can be obtained at 565 than at 500 K because of a slow growth rate of cavities at 565 K.The temperature dependence of elongation clearly shows an evidence for the enhancement of the tensile elongation by the liquid Bi phase. The relationship is found between elongation and distribution of Bi. In order to obtain elongation enhancement, homogeneous dispersion of Bi needs to be maintained during deformation. There seem to be critical values for the volume fraction of Bi and for strain rate to maintain homogeneous dispersion. A large Bi volume fraction and a slow strain rate tend to induce agglomeration of Bi along perpendicular grain boundaries to tensile axis. In this case, the boundary cannot withstand traction force acting upon the perpendicular boundaries and grain-boundary fracture occurs. On the other hand, a small volume fraction of Bi and a fast strain rate tend to maintain homogeneous dispersion of the liquid phase. In this case, the liquid phase can effectively accommodate stress concetration and delay cavitation failure.At the moment, we consider that the agglomeration of the liquid phase is caused by localized stress variation within the grain interior and along grain boundaries. This stress variation may induce directional diffusion of Bi towards perpendicular boundaries to the tensile axis. Suzuki et al. It is interesting to note that there is no intermediate stage of Bi distribution atter deformation. Bi is either homogeneously distributed or agglomerated along grain boundaries depending on the Bi concentration and strain rate, suggesting the presence of the critical values. This also suggests that the localized stress variation can be totally different above and below the critical values. The macroscopic stress directionality may exist along the tensile axis when the enhancement is not observed. Whereas, the directionality may not exist when the enhancement is observed.Hydroelastic buoyant circular plate in shallow water: a closed form solutionThis paper considers a circular buoyant elastic plate of homogeneous stiffness floating in shallow water. The edge of the plate is free of shear forces and bending moments. The plate deflection is excited by a monochromatic ambient surface wave. A closed form solution of the corresponding linearized problem is obtained. Numerical examples, illustrating the analytic results for various physical parameters of the problem are presented.The recent interest in the design of offshore floating airports has sparked intensive studies of the hydrostatic behavior of such unconventional large marine structures (see, for instance, recent works in ). In many proposed engineering concepts, the floating airport is considered as a thin mat-like configuration extending in the horizontal direction for several kilometers with a thickness of less than 10 m. Due to its small bending rigidity this buoyant structure can be treated as a flexible thin plate which covers a part of the free surface and is subject to the action of incident waves. Floating airports should be located in the offshore zone where the water depth is relatively shallow. In this work the local water depth is estimated to be of the order of about ∼20 m. The typical wave length of the ambient wave affecting the plate is typically of the order of few hundred meters. Under these assumptions, both, the wave length and the horizontal dimensions of the plate are much larger than the water depth. In analyzing the plate hydroelasticity it is plausible therefore to use shallow water theory.The general shape of a floating airport may be rather complex both in the horizontal and in the vertical directions. However, for the preliminary analysis of its hydroelastic response it may be useful to use some schematic forms such as rectangular or circular plates of even draft and constant flexural rigidity. In this work we study a circular shape which was suggested as one of the alternative geometries for designing floating islands near the Israeli coast The number of closed form solutions describing the dynamic response of an elastic floating plate to an incident wave is rather limited, even under many simplifying assumptions. Stoker Some preliminary results of this work were recently presented by the authors at the “14th International Workshop on Water Waves and Floating bodies” A thin elastic circular plate of radius r0 covers a part of the free surface of shallow water of constant depth h (. In order to solve the problem analytically the following assumptions are adopted:there are no gaps between the thin elastic plate and the free surface;the edge of the plate is free of shear forces and bending moments;the plate flexural rigidity is constant;the fluid is assumed to be incompressible, inviscid and its motion irrotational;the amplitude of the incident wave, as well as that of the waves induced by the bending and twisting elastic plate, are assumed to be small;to describe the motion of the plate and fluid a linearized shallow water theory is invoked. |
Consider an incident progressive monochromatic wave with a wave elevation Wwpropagating in the positive x-direction and varying with time t (). Here the wave number k and the wave circular frequency ω are related aswhere g is the acceleration of gravity. The velocity potential of the wave is denoted by Φw.Divide the water domain into two regions: plate (r≤r0) and water region (r>r0). Below we denote all quantities pertaining to the plate region by a subscript “1”, whereas quantities related to the water region are denoted by the subscript “2”. Correspondingly, the velocity potential in the plate region is denoted by Φ1, and the velocity potential in the water region is denoted by The velocity potential in the water region is represented as a sum of two potentialswhere Φ2 is the velocity potential affected by the plate motion in the water region. The vertical displacement of the plate is denoted by W1, and that of the free surface by The later is represented as a sum of the incident wave elevation and the wave elevation W2 affected by the plateThe wave elevations W1,2 and Ww satisfy the shallow water equationwhere the Laplacian in polar co-ordinates The fluid pressure P1 in the plate region is defined by the linearized Bernoulli equationwhere ρ is the water density. In the water region the fluid pressure on the free surface is zero, and thus yield the governing equation for the potential Φ2The dynamic condition beneath the plate can be written as follows where m is the mass of the plate per unit area, and the equivalent flexural rigidity D is expressed in terms of the equivalent Young's modulus E, Poisson's ratio ν and the plate thickness d as follows: represents the linearized governing equation for the plate region.The water and plate regions have to be matched at the contour line r=r0. Following Stoker Far from the plate the potential Φ2 must vanish imply that the mass and energy flux are conserved within leading order terms retained in shallow water theory, and do not account for the free surface and plate deformation. Within such an approach, at the contour line r=r0 the pressure in the fluid and free surface elevation are discontinuous. This is considered as an unavoidable consequence of the shallow water approximation and the linearization procedure At the free edge of the plate the total vertical force V and bending moment Mr vanish where Qr is the shear force, and Mrθ the twisting moment (). The expressions for Mr, Qr and Mrθ in polar co-ordinates are the boundary conditions on the plate contour can be written as are the requiring boundary conditions in the plate region.Assuming steady state oscillations for both the plate and fluid, we represent all time dependent quantities aswhere the periodic time-dependent function is denoted by a capital letter and its amplitude is denoted by the same small letter. For steady oscillations the governing in the plate region yields a sixth-order partial differential equationThe coefficient b∗ is obviously positive. To verify the sign of a∗ note that initially the weight of the plate is balanced by its buoyancy hence the mass of the plate m per unit area is proportional to its draft d<h. For shallow water conditions, assuming kh=o(1), we obtain Δ=O(kh·kd)=o(1) implying that that the potential φ2 in the water domain satisfies the Helmholtz equationThe matching conditions between the two The boundary conditions for the potential φ1 at the plate edge are essentially the same as those given in Once the potentials φ1,2 are known, the wave elevations in the water and in the plate regions can be written asThe induced pressure, total vertical force V and bending moment acting on the plate may be readily obtained from by applying the corresponding equations to the amplitude w1. in the plate region can be represented aswhere zm (m=1,2,3) are the roots of the cubic equationUsing Cardan's formula they can be written in a closed form asNote that a and b are positive quantities. Since u is positive and υ is negative, it follows that z1 is real and negative whereas z2 and z3 are two complex conjugated quantities with positive real parts.The sought solution is symmetric with respect to the x-axis and can be represented as a Fourier serieswhere the coefficients ϕ1,m(n)(r) are unknown. Substituting and the complex square root is defined in the cut-plane excluding ]0,∞] in such a way that its real part is positive for Re(zm)>0. The linearly independent solutions of are the modified Bessel functions In(ξ) and Kn(ξ). The function Kn(ξ) is unbounded at the center of the plate, and thus, from a physical standpoint only the function In(ξ) should be considered as a physically meaning solutionwhere the constants Cm(n) have to be found from the boundary conditions. Once these constants are defined the potential φ1 can be written aswhere Hn(1)(kr) is the Hankel function. The amplitude function and the velocity potential φw can be also represented as a Fourier serieswhere ϵn=−in+1δng/ω, δ0=1 and δn=2 for n>0. Substituting we obtain a system of four linear algebraic equations with respect to the four unknowns Cm(n) (m=1,…,4):and α=1−ν. Once the solution of this system is obtained, all quantities of physical interest can be given in a closed form as and estimate the order of magnitude of Here the plate mass inertia effects are represented by the term ω2m, the hydrostatic effects are represented by the term ρg, and the effects of the dynamic pressure in the fluid are represented by the term ρgk2. Generally, if kh=o(1) then Δ=ω2m/ρg=O(kh)2. The linear shallow water theory, which we invoke here, takes into account small terms of the first order of magnitude O(kh). For the sake of consistency therefore the second order term Δ=O(kh)2 may be disregarded by assuming the plate mass inertia effects as well as hydrostatic effects can be neglected if the following relation is satisfied:Here we restrict our analysis for cases when the incident wave length is much smaller than the radius of the plate, i.e. (kr0)4≫1. Estimating the wave length as both, the mass inertia term and the hydrostatics term can be neglected ifFor instance, assuming S=10−4, or so, we obtain 4a3/27b2≃4.0×10−3 which is indeed a small quantity. However, even for much smaller values of the plate flexural rigidity when 4a3/27b2=O(1) and the hydrostatic term ρg must be taken into account, the mass inertia term may still be neglected as long as kh≪1, (kr0)4≫1 and and the plate's draft of the order of 6 m. For the above example This estimate is in agreement with the results of Lee and Newman , one can note that the case S=0 is singular since the reduced order of the governing equation does not allow us to satisfy all the imposed boundary conditions. For a finite (although very small) plate rigidity the corresponding boundary value problem is stiff. It means that the limit S→+0 may be nonuniform and, thus, can be hardly attained numerically. In this work in order to illustrate the convergence of our algorithm we disregard the plate mass inertia term in all numerical simulations even for very small values of the nondimensional parameter S<10−7. is straightforward. However, the solution of , although it is of low dimension 4×4, requires some precautions since for high modes (large n) it can be ill-conditioned. This peculiarity may lead to significant numerical errors in the solution of due to unavoidable inaccuracy in computing Bessel functions of complex argument with insufficient number of significant digits. In extreme cases the coefficients Cm(n) (m=1,…,4) may be dominated by a roundoff error leading to a very poor approximation of the Fourier coefficients in . Similar difficulties were reported by Hermans It is difficult to estimate the analytic convergence of , and for this reason it is not attempted here. Instead, we truncate the series by estimating the decay rate of the Fourier coefficients numerically, and perform numerical simulations for various number of modes.As a numerical example we consider a circular plate with the following parameters: plate radius and Poisson's ratio ν=0.15. For this particular case the reference stiffness parameter is S0=7.4×10−5.depicts the decay of the Fourier coefficients bw(r0) of for two distinct values of plate stiffness. It demonstrates the fact that an increase in the plate stiffness leads to a faster convergence of the Fourier series. Numerical simulations show that even in the case of an extremely low nondimensional stiffness, S∼10−8, it is sufficient to take into account only about 30 terms of the series in order to obtain sufficient accuracy.shows the plate deflection for a range of nondimensional stiffnesses S=(10S0−10−4S0). Case (a) corresponds to the maximal stiffness, resulting in a minimal plate deflection and maximal wave length of the plate deformations. The basic case (b) illustrates the fact that the incident wave penetrates into the plate with a curved wave front, and induces a relatively larger plate deformation with a shorter wave length compared to case (a). For smaller values of plate stiffness the plate deflection increases. For the nondimensional stiffness as small as S=10−4S0=7.4×10−9 the plate practically behaves as a free surface of zero stiffness. On the other hand, for values of the nondimensional stiffness larger than 10−3, the plate essentially behaves like a rigid body.The variation of the bending moment modulus over the plate is illustrated in . It confirms that the obtained solution indeed satisfies the imposed free–free edge condition.An exact closed form solution has been obtained for a circular thin elastic plate floating in shallow water. Numerical calculations based on plate radius show that for values of S<10−3 hydroelastic effects are important.For values of the nondimensional stiffness parameter S<10−8 the plate behaves like a flexible mat without stiffness. For values of S>10−3 the plate acts practically like a rigid body. The case of a very small rigidity was also the investigation mainly in order to prove that the obtained series are indeed convergent for relatively small number of modes: about 30 for S<10−8, about 20 for 10−7<S<10−6, and about 15 for S>10−6.For values of S>10−4 the computations may be performed using Fortran compiler, and the algorithm for computing Bessel functions On the plate contour, the plate and the free surface deflections may differ due to the imposed transition conditions requiring the continuity of mass and energy fluxes, but not the continuity of free surface and plate elevations. It is expected thus that the obtained solution may predict the proper hydroelastics response of the plate to an incident wave field over most of its area, except at the immediate vicinity of its free edge.Highly efficient and sustainable carboxylated cellulose filters for removal of cationic dyes/heavy metals ionsTreatment of textile wastewater containing highly toxic dyes and heavy metal ions raises serious health concerns that need to be addressed urgently. Carboxylated cellulose fabric filters were fabricated by 2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPO)-mediated oxidation and employed to remove heavy metal ions and dyes efficiently by adsorption from simulated wastewater. A series of characterizations including ATR, TGA, and SEM measurements were employed to investigate their structure and morphologies. The carboxylated cellulose fabrics exhibited excellent mechanical properties evidenced by tensile experiments and comparisons with other non-woven counterparts. Lead ion (Pb (II)) and methylene blue (MB) were introduced to investigate the filtration performance of carboxylated cellulose filters; high flux, high rejection ratio, and low pressure drop were achieved due to the internal micro-nanoscaled structure of the cellulose fabrics as revealed by XPS and elemental mapping characterizations. The maximum adsorption capacities for Pb (II) and MB were 81.30 and 76.92 mg/g, respectively, while the pressure drop for dynamic adsorption was as low as 0.07 kPa, indicating that the filtration performance was drivable by gravity. A spiral wound cartridge of cellulose fabrics was fabricated and a long-term filtration performance against multicomponent wastewater containing 2.0 mg/L Pb (II) and 2.0 mg/L MB, simultaneously, was demonstrated. The wastewater treatment capacities of 3.6 g of the filter were up to 21.7 L and 36.1 L, respectively, which was 10-times higher than that of the original cellulose fabrics; and the rejection ratios against MB and Pb (II) were 98.7% and 98.2%, respectively, which met the environmental criteria for wastewater disposal. The recycling and reusability of the filters were also demonstrated, indicating that the carboxylated cellulose fabrics can be considered as sustainable filters for wastewater treatment.Dyes, heavy metal ions, and other organic toxins are major contaminants of textile wastewater produced from paints, leather, textiles, and paper-making industries Typical adsorption filters are composed of porous materials, such as activated carbon Cellulose is the most abundant biomaterial on earth Here, we propose a new adsorption filter composed of carboxylated cellulose fabrics fabricated from controlled TEMPO-mediated oxidation. The woven structure of cellulose fabrics guarantee the mechanical properties of the adsorption filter, while the carboxylate groups located on the exterior and interior of the fibers serve as functional sites to catch heavy metal ions and dyes from simulated wastewater. Two typical toxic species, lead ions and methylene blue, were employed to investigate the effectiveness of the carboxylated cellulose filters, and the separation efficiency in terms of flux, rejection, and usage. A spiral wound filtration cartridge was fabricated and a distinguished separation performance of the multicomponent wastewater containing both lead and methylene blue was demonstrated. The sustainability of the cellulose fabric filter was also demonstrated by recycling and reusing the filter many times as would be done in the context of wastewater treatment.Commercial white cellulose fabrics (0.01 g/cm2) were purchased from a local store (Beijing, China), followed by a pretreatment of boiling in water (~98 °C) for 3 h to remove additives, if any. 2,2,6,6-tetramethyl-1-piperidinyloxy (TEMPO) was bought from Aladdin and used, directly. Copper (II)-ethylenediamine complex (1.0 mol/L solution in water) was bought from Acros. Sodium bromide (NaBr), sodium hypochlorite (NaClO) aqueous solution, sodium hydroxide (NaOH), lead nitrate (Pb(NO3)2), methylene blue (MB), 4-(2-Pyridylazo)resorcinol (PAR), ethylenediamine tetraacetic acid disodium salt (EDTA-2Na), hydrochloric acid (36.5 wt%, HCl), and absolute ethanol were received from Aladdin and used without further treatments.Cellulose fabrics (2.0 g) were suspended in 200 mL of water containing 0.032 g of TEMPO and 0.20 g of sodium bromide. The oxidation reaction was started by adding certain amounts of NaClO aqueous solution. The pH value was maintained between 10.0 and 10.5 in the reaction by adding 0.5 mol/L of NaOH aqueous solution. The reaction continued for 3 h under stirring condition and then was terminated by adding 10 mL of absolute ethanol. The carboxylated cellulose fabrics were washed three times in pure water for 30 min each time and were dried in a vacuum oven at 60 °C before use.The carboxylated cellulose fabrics with the dimension of 40 cm × 12 cm were employed to prepare the cartridge. A transparent polyvinylchloride (PVC) tube with inner diameter of 1.8 cm was drilled with 25 holes distributed evenly within an 8-cm length. The carboxylated cellulose fabrics and a stainless steel web (150 meshes) with a dimension of 40 cm × 12 cm were deposited together and wrapped on the PVC tubes, tightly. The ending points of the fabrics and web were sealed with AB glue completely. The effective filtration area was 360 cm2 and the corresponding weight was 3.6 g. A plastic housing with a diameter of 4.8 cm was created and fitted with the cartridge.The content of carboxylate groups was investigated by conductometric titration. As far as details, carboxylated cellulose fabric (0.10 g) was suspended in 100 mL of water, and 0.1 mol/L of HCl aqueous solution was used to adjust the pH value of the suspension to 2.5–3.0. Sodium hydroxide standard aqueous solution with 0.025 mol/L was employed to neutralize the suspension where the content of carboxylate groups was calculated using the following Eq. The viscosity average degree of polymerization (DPv) was obtained by measuring the intrinsic viscosities of the solutions of pristine cellulose and carboxylated cellulose fabrics where mw and md is the mass of wet cellulose samples after centrifugation and dry samples, respectively.Heavy metal ion, Pb (II), and positively charged dye, methylene blue (MB), were employed to determine the adsorption capacity of the carboxylated cellulose fabrics through static adsorption. In a static adsorption experiment, the cellulose fabric sample (0.01 g) was suspended in a 20 mL-vial containing 10 mL of lead or MB solution with a particular concentration. The pH value of the lead solution apart from the MB solution was adjusted optimally to 5.0. The adsorption experiments were conducted at 25 °C and the concentrations of the solutions were determined by UV–vis spectrometry as well as by ICP (Agilent 7700). The recorded values were taken from the triplicate samples.MB with an initial concentration of 50 mg/L was used for the kinetic adsorption experiment. The equilibrium concentration of the MB solution was determined by a UV–vis spectrometer at 664 nm based on a calibration curve. The adsorption capacity (qe, mg/g) of MB adsorbed onto the fabric was calculated using the following Eq. where C0 is the initial concentration, Ce is the equilibrium concentration in the solution (mg/L), V is the volume of the solution (L), and m is the weight of the fabrics (g).The isotherms of MB adsorption were achieved with a series of MB solutions with concentrations ranging from 5 to 35 mg/L. The adsorption experiments were made to approach equilibrium by stirring at 25 °C for 40 min. The maximum adsorption capacity of the carboxylated cellulose fabrics was evaluated by fitting the isotherm with the Langmuir or Freundlich equation The static adsorption experiments of Pb (II) were carried out by following similar protocols. Carboxylated cellulose fabrics (0.01 g) were added into 10 mL of Pb (II) solution (pH = 5.0) with the initial concentration set at 60 mg/L for the kinetic study. The concentration of Pb (II) in solution was determined with a UV–vis spectrometer by reading the adsorption intensity at 523 nm. Adsorption isotherms of the adsorption experiments were conducted with initial concentrations of Pb (II) solutions ranging from 30 to 130 mg/L. The adsorption reached equilibrium status in 1 h at 25 °C under a stirring condition.Sodium chloride and calcium chloride, represented as competing monovalent and divalent ions, respectively, were employed to investigate the effect of coexisting ions on the adsorption capacity of the carboxylated cellulose filter against Pb (II). In detail, sodium chloride and calcium chloride were dissolved in 100 mg/L of lead nitrate aqueous solution, separately, with concentrations of 100, 200, 500, and 1000 mg/L. The carboxylated cellulose filter (0.01 g) was immersed in 10 mL of stirred aqueous solution for 1 h, following the typical static adsorption experimental protocol. The adsorption capacity and selectivity of the filter for Pb (II) was evaluated based on the measurements using a UV spectrometer.Dynamic adsorption experiments of the Pb (II) and MB adsorption were carried out using the carboxylated cellulose fabric disc with a 25-mm diameter. Pb (II) solution with concentration of 0.15 mg/L and 1.3 mg/L of MB solution were employed, separately, and a flow rate of 6.0 L/m2h was maintained in the filtration performance. Breakthrough curves were obtained by plotting the volume of permeate versus the concentration of residual dye and heavy metal ion, respectively.A spiral wound cartridge was prepared with the carboxylated cellulose fabrics and the filtration performance was evaluated. Basically, a custom-built microfiltration system composed of a peristaltic pump, pressure gauge, and cartridge housing was assembled as shown in The microfiltration performance was demonstrated by employing an aqueous solution containing 2.0 mg/L of Pb (II) and 2.0 mg/L of MB, and 10 mL/min of flow rate under an extremely low pressure of 3.0 kPa. The pH value of the solution was 6.0. The concentrations of Pb (II) ions and MB in the permeation solution were determined by the ICP and by the UV spectrometer, and breakthrough curves of permeation solution volume vs. concentration of Pb (II) and MB were plotted.The sustainability of the carboxylated cellulose fabric filter was evaluated by recycling the used filter through desorption performance. In the desorption process, the MB-adsorbed fabric filter was washed thoroughly with deionized water and then put into 1.0 mol/L of HCl solution (15 mL) for 1 h. The desorption process was repeated twice and the filter was washed with deionized water several times before starting a new adsorption experiment. The same procedure was applied to the desorption of Pb (II) from carboxylated cellulose fabrics. Then, the used cellulose fabric filter was put into 0.1 mol/L of EDTA-2Na solution (50 mL) for 1 h. The fabric filter was washed several times with deionized water and reused further in the next adsorption experiments. All of the recycled cellulose filters were reused at least five to ten times to confirm the nature of their sustainability.Thermogravimetric analysis of the cellulose fabrics were performed using the thermogravimetric analyzer DTG-60A (Japan). The samples were heated at 10 °C/min from 40 to 800 °C under 20 mL/min of nitrogen flow. A FT-IR spectrum was obtained in attenuated total reflectance (ATR) mode using a Nicolet 6700 FTIR spectrometer (USA) in the range of 4000–500 cm−1. X-ray diffraction (XRD) analysis of the samples was performed using an X-ray diffractometer (Bruker D8 ADVANCE Germany).The multifunctional tensile machine (UTM5205XHD sunthink China) equipped with a tensile fixture was used to measure the mechanical properties of the cellulose fabrics. Rectangular shaped samples were prepared with dimensions of around 300 mm (length) × 50 mm (width) × ~0.2 mm (thickness) for the cellulose fabrics. Stress-strain curve of the five replicates from each fabric were obtained at 20 mm/min stretching rate. Tensile strength and elongation at break of cellulose fabrics were determined from the obtained stress-strain curves. The average of these values was calculated.The morphology of the cellulose fabric was examined by scanning electron microscopy (SEM) (JEOL JSM-7800F Japan). The samples for cross-sectional view were prepared by breaking down the frozen samples after wetting with water and freezing with liquid nitrogen Cellulose fabrics were purified by pre-treatment with hot water and TEMPO-mediated oxidation was applied to fabricate carboxylated cellulose fabrics as shown in It is quite clear that the oxidation reaction occurred at the C6-position of cellulose where the hydroxymethyl group was converted partially to the carboxylate group. The degree of oxidation was mainly affected by the amount of oxidizing reagent, sodium hypochlorite in this case. Therefore, a series of carboxylated cellulose fabrics were fabricated by varying the amount of sodium hypochlorite from 0.01 mmol to 0.15 mmol per gram of cellulose fabrics and the chemical structure of the products were investigated by ATR as shown in All ATR spectra including that of the original cellulose fabrics and their oxidized products exhibited vibration peaks at 3330.7 cm−1, 2899.3 cm−1, and 1051.0 cm−1 which correspond to stretching vibrations of O–H, C–H, C–O groups, respectively. However, a new vibration peak at 1605.1 cm−1 appeared after oxidation which can be assigned to the stretching vibration of carboxylate groups, indicating that the oxidation reaction of hydroxymethyl groups was carried out, successfully. Moreover, the peak intensity of the carboxylate groups increased with increasing the amount of sodium hypochlorite in the reaction when all other conditions including the ratios of different reactants, reaction temperature and time remained the same. It was approaching a plateau after 0.05 mmol/(g of cellulose fabrics) of sodium hypochlorite was employed, evidenced by the titration experiments as shown in It is clearly seen that the content of the carboxylate groups approached the maximum value, 0.60 mmol/g, and remained unchanged when 0.05 mmol/(g cellulose fabrics) or more of sodium hypochlorite was used. Further increases in the amount of sodium hypochlorite did not increase the content of carboxylate groups. Instead, the mechanical properties of carboxylated cellulose fabrics reduced, drastically. The innovation of the fabrication of these carboxylated cellulose filters as compared with the conventional preparation by TEMPO-mediated oxidation The degrees of polymerization (DPv) and water retention values (WRVs) of carboxylated cellulose filters were also investigated to further understand the structure and morphological changes of cellulose fabrics after TEMPO-mediated oxidation, as shown in (C). It was clear that the DPv of original cellulose fabrics was about 2100 which was decreased drastically to 255 when 0.04 mmol/(g of cellulose fabrics) of NaClO was employed in the oxidation, and remained the same when 0.05 mmol/(g of cellulose fabrics) or more oxidation reagent was applied. However, the WRV of the carboxylated cellulose filters increased with increasing the amount of oxidation reagent even though the carboxylate content remained unchanged after 0.05 mmol/(g of cellulose fabrics) of NaClO applied. The remarkable increase in WRV was probably due to the formation of an internal porous structure of the carboxylated cellulose fabrics.The thermal decomposition of carboxylated cellulose fabrics was investigated and compared with that of cellulose fabrics as shown in The thermal decomposition behavior of cellulose fabrics can be divided into three intervals. The first interval was from room temperature to 274.5 °C where only 2.4% of weight loss was observed which could be attributed to desorption of moisture from the cellulose fabrics. The onset decomposition of cellulose fabrics started from 330.4 °C, and the interval lasted to 386.3 °C where the cellulose chain was broken down to intermediates followed by being converted into small molecules such as carbon dioxide, carbon monoxide, and water being released. The maximum decomposition rate was observed at 361.0 °C as indicated in ; the total weight loss percentage of this stage was 86.8%. The third interval was from 386.3 °C to 800 °C when the residual carbon was 8.7%. A different thermal decomposition history was observed for carboxylated cellulose fabrics. The onset decomposition was at 209.4 °C for carboxylated cellulose fabrics which was about 121 °C lower than the decomposition temperature of cellulose fabrics. A low decomposition rate was observed. This interval lasted until 253.3 °C which is mainly attributed to the decomposition of oxidized cellulose and the decarboxylation reaction which released carbon dioxide . This was considered as the decomposition of cellulose, formation of intermediates, and the release of volatile molecules from the carboxylated cellulose fabrics. When the temperature reached 800 °C, 22.7% of residuals remained which was higher than the 8.7% of the cellulose fabrics. Among the 14% of difference, about 1.2% (corresponding to about 0.53 mmol/g) was probably due to the left over sodium element that were introduced in the TEMPO-mediated oxidation process.The mechanical properties of carboxylated cellulose fabrics were investigated by tensile experiments and the ultimate tensile strength, along with that of other reported crosslinked nanofibrous filters It was quite clear that the ultimate strength of most filters is less than 10.0 MPa, while the carboxylated cellulose fabrics exhibited a maximum strength of 13.5 MPa. The original cellulose fabrics showed strength of 33.8 MPa. Nonwoven nanofibrous filters usually have very low mechanical properties unless second-step crosslinking or enhancing additives are applied. Carboxylated cellulose fabrics, however, demonstrated great mechanical properties in comparison with the natural cellulose materials and a nonwoven structure without further crosslinking or modification implying their potential for extensive use in practical applications. It should be noted that the mechanical properties of cellulose fabrics also depend on warp and weft count which is shown in the different tensile strengths of the fabrics. Another tensile strength of the carboxylated cellulose fabrics, 8.2 MPa, was also determined, which was comparable with that of most reported fibrous filters.The surface and cross-sectional morphologies were observed through SEM images as shown in There are no visible differences between the cellulose fabrics and carboxylated cellulose fabrics in the aspect of surface morphology as shown in (A) and (B). The fiber diameter of carboxylated cellulose fabrics was about 12 μm, and based on this, the specific surface area could be calculated to be 0.21 m2/g which was lower than that of the nanofibrous filters. However, the functionality (content of carboxylate groups by titration) was 0.53 mmol/g which was unexpectedly high even in comparison with the filters made by nanocellulose The internal structure of the cellulose fabric was porous before and after oxidation as evidenced by the SEM images shown in (C) and (D). These pores are micron and sub-micron sizes indicating that the oxidizing reagent along with water molecules could penetrate into the fabric, oxidize the inner walls of the pores, and convert the hydroxymethyl groups present there into carboxylate groups. Therefore, the carboxylated cellulose fabrics, though regarded as micro-filters, exhibited an impressive adsorption capacity which was even higher than that of some nanofibrous filters.Water contact angle measurements were conducted to determine the surface hydrophobicity of the carboxylated cellulose fabrics after oxidation. The results are shown in . It was seen that both cellulose fabrics and the carboxylated cellulose fabrics were hydrophilic. The water contact angle of cellulose fabrics decreased from 111.4° to 11.0° after 10 s due to the porous structure and the hydrophilic nature. The hydrophilicity of the cellulose fabrics was further enhanced by TEMPO-mediated oxidation evidenced by the change of water contact angle from 78.0° to 0° in 8 s after conversion of hydroxymethyl groups to carboxylate groups in the oxidation.It was very beneficial that enhanced hydrophilicity of the carboxylated cellulose fabrics was observed. This property enhances the penetration of water into the filter, and the advantage for microfiltration was obvious as compared with a conventional polypropylene (PP) filter which exhibits a water contact angle of ca. 135° As mentioned before, multi-component wastewater, such as textile wastewater, contains toxins including dyes and heavy metal ions which need to be removed and safely disposed of. Therefore, simulated wastewater with lead ion (Pb (II)) and methylene blue (MB) were employed to investigate the filtration efficiency of carboxylated cellulose fabrics. ATR and EDS measurements were employed to investigate the chemical structure of the carboxylated cellulose fabrics after adsorption of Pb (II) and MB as shown in It was very interesting to note that the vibration peak at 1605.1 cm−1 assigned to carboxylate groups in the spectrum of (B) moved to 1579.4 cm−1 in the ATR spectra (, left). This was probably due to the coordination of carboxylate groups with Pb (II) , right) where the signal of sodium ions (which came from the TEMPO-mediated oxidation of cellulose fabrics) in the spectrum (B) completely disappeared in the spectrum (C). Rather, a strong signal corresponding to lead was generated in the spectrum (C) (Moreover, a new vibration signal appeared at 1398.2 cm−1 in the spectrum of (D) compared with spectrum (B) which could be attributed to the bending vibration of methyl groups of MB A series of static adsorption experiments were performed to determine the adsorption capacity of the carboxylated cellulose fabrics where the initial concentrations of Pb (II) and MB were 58.3 and 50.7 mg/L, respectively. The pH values of the solutions were adjusted to the optimized ones which were 5.0 for Pb (II) solution The adsorption capacity of the carboxylated cellulose fabrics increased quickly at the beginning and approached equilibrium status in 60 min (). This indicates that the carboxylated cellulose fabrics have a relatively higher adsorption rate and are suitable for practical application. Furthermore, pseudo-first and pseudo-second order equations were employed to fit the kinetics of the adsorption where we found that the adsorption kinetics of either Pb (II) or MB followed the pseudo-second order model with the derivation of R2 > 0.99 (). This finding established that the adsorption mechanism was essentially a chemical adsorption. The adsorption rate constants of Pb (II) and MB are 0.0027 and 0.0037 g·mg−1mim−1, respectively, calculated through the pseudo-second order rate equation.The adsorption capability of the carboxylated cellulose fabrics was investigated using isotherms of Pb (II) and MB. They were used to correlate the equilibrium adsorption capacity with the equilibrium concentration through which maximum adsorption could be achieved. Pb (II) and MB solutions with different concentrations were used as the feed solutions to test the carboxylated cellulose fabric filters and the adsorption capacities were determined. Three types of thermal dynamic models, Langmuir, Freundlich, and Tempkin ) and to reveal the adsorption mechanism as depicted in It was quite clear that the Langmuir model could fit isotherms of both Pb (II) and MB, indicating that Pb (II) and MB have been adsorbed by a monolayer approach on the surface of the filter. Meanwhile, there is interaction between the adsorption sites on carboxylated cellulose fabrics and Pb (II) and MB as evidenced by the Tempkin model fitting; this confirmed the chemical adsorption mechanism revealed by the analysis of adsorption kinetics as discussed earlier. Moreover, the maximum adsorption capacities for Pb (II) and MB, as evaluated by the Langmuir model, were 81.30 and 76.92 mg/g, respectively. It is higher than the adsorption capacities of the reported values of most commercially available nanofiber/nanoparticle adsorbents, as shown in To substantially explore the adsorption mechanism of carboxylated cellulose fabrics, XPS analysis associated with the chemical state of elements was employed to reveal the interaction between the carboxylate groups, Pb (II) ions and MB which represented heavy metal ions and cationic dyes, respectively. (A) illustrated wide scan XPS spectra of cellulose fabrics, carboxylated cellulose fabrics, and carboxylated cellulose fabric filters after adsorption of Pb (II) and MB. The peaks at binding energies around 285.0 and 532.0 eV were assigned to the apparent peaks of C 1s and O 1s O (3.27%), indicating the successful oxidation of cellulose on the surface of cellulose fabrics, as shown in (C). Meanwhile, the relative amount of O–C–O component decreased from 8.96% to 7.17% after oxidation, which was probably due to the break of ether bonds in cellulose chains as evidenced by the remarkable decrease in DPv of the cellulose fabrics (O group shifted to high binding energy from 288.3 to 288.4 eV after adsorption with Pb (II). Meanwhile, the integral electron cloud intensity decreased, as shown in (C) and (D), which confirmed the chelating process between the lone pair electrons of oxygen atoms of carboxylate groups and vacant electron orbitals of lead ions (A)), corresponding to Pb 4f7/2 and Pb 4f5/2, also indicated the successful adsorption of Pb (II) in addition to the disappearance of Na 1s peak at 1071.4 eV after adsorption of Pb (II), as shown in The adsorption of MB onto the surface of carboxylated cellulose fabrics also induced the change of the chemical state of the elements as shown in (A). The inserted N 1s and S 2p peaks were attributed to the MB adsorbed on the surface of the carboxylated cellulose fabric filter. The C 1s peak of carboxylated cellulose fabrics can be deconvoluted into four peaks at binding energies of 288.3, 287.8, 286.4, and 284.8 eV assigned to carbon atoms in the form of O–CO, O–C–O, C–O, and C–C. These peaks moved to 288.4, 287.7, 286.3, and 284.7 eV, respectively, after adsorption of MB, indicating the electrostatic interaction between the carboxylate group and the cationic dye. A new peak at binding energy of 285.9 eV located in the spectrum of carboxylated cellulose fabric-MB could be assigned to C-N group of MB. Meanwhile, the reduced electron cloud intensity of Na 1s (1071.4 eV) in the spectrum of carboxylated cellulose fabric-MB compared to that of the carboxylated cellulose fabrics was attributed to the partial adsorption of MB, which was also evidenced by the EDS spectrum of carboxylated cellulose fabric-MB shown in In addition to the interaction between the carboxylated cellulose fabrics and target heavy metal ions and dyes, the density and distribution of carboxylate groups on cellulose fabrics also affect the adsorption mechanism. Therefore, EDS mapping images were recorded to analyze the distribution of carboxylate groups on oxidized cellulose fabrics, as shown in It was seen that the surface of carboxylated cellulose fabrics was covered with sodium, nitrogen, and sulfur elements after successful adsorption of MB (A). This indicated that MB molecules containing nitrogen and sulfur were adsorbed on the filter through a layered approach which was agreeable with the Langmuir model of monolayer adsorption (). The sodium elements on the surface of carboxylated cellulose fabrics were ascribed to some free carboxylate groups. As mentioned before, the carboxylated cellulose fabrics exhibited a micro-nanoscaled porous structure observed from the cross-sectional view of SEM images, which were expected to exhibit additional carboxylate groups for enhancing the adsorption capability confirmed by EDS mapping images of the cross-sectional cellulose fibers as shown in (B). Either sodium elements corresponding to carboxylate groups or nitrogen and sulfur elements corresponding to MB molecules were observed from mapping images, which verified our hypothesis that the internal micro-nanoscaled porous structure of the carboxylated cellulose fabric filter has carboxylate groups and could serve as additional adsorption sites for capture of MB molecules. Further evidence could also be obtained from the mapping images of carboxylated cellulose fabrics after adsorption of Pb (II) ions. Both top and cross-sectional views showed distribution of Pb elements as shown in (C), revealing that the adsorption occurred on the surface as well as in the internal of cellulose fabric fibers which enhanced the adsorption capability, drastically.To understand the effect of coexisting competing ions on the adsorption capacity of the carboxylated cellulose fabrics, sodium chloride and calcium chloride were dissolved in Pb (II) ion solution (62.5 mg/L), separately, with different concentrations. A series of static adsorption experiments were carried out and the results are illustrated in The monovalent ions, such as sodium (I) ions, exhibited a lesser effect on the adsorption of Pb (II), with the concentrations varying from 40 to 400 mg/L, as expected, and the adsorption selectivity of Pb (II) to sodium (I) was higher than 90.0%, as shown in . This will be the advantage of the carboxylated cellulose fabrics for practical applications. Furthermore, the divalent ions, e.g., calcium ions, decreased the adsorption capacity of the fabrics when different concentrations of calcium ions were involved. The adsorption capacity of the carboxylated cellulose fabrics was decreased from 44.02 to 30.02 mg/g when 400 mg/L of calcium coexisted in the solution, and the adsorption selectivity was down to about 70.0% which, however, still exhibited an adsorption advantage of the carboxylated cellulose fabrics to Pb (II) over that of Ca (II) probably due to the formation of carboxyl-Pb (II) complex.The filtration performance was demonstrated using MB and Pb (II) aqueous solutions that simulated textile wastewater. The flow rate remained 6.0 L/m2h at a pressure of about 0.07 kPa. A breakthrough curve was achieved after the filtration performance as shown in The carboxylated cellulose fabric filter exhibited excellent adsorption capability for MB and Pb (II) in the filtration of the first 200 mL of solution as evidenced by the rejection ratio which was higher than 90%, where the concentration of residual MB and Pb (II) in the permeate solution was quite low. This meets the criterion for wastewater disposal A spiral wound filtration cartridge housed in a plastic tube was fabricated using carboxylated cellulose fabrics and connected into a custom-built microfiltration system. The effective filtration area of the cartridge was 360 cm2 (effective weight: 3.6 g). A multicomponent aqueous solution containing 2.0 mg/L of Pb (II) and MB was employed as the feed solution; the operating pressure was 3.0 kPa and the flowrate remained at 10 mL/min throughout the filtration process. The breakthrough curves are shown in It was very exciting to see that the carboxylated cellulose fabric cartridge with 3.6 g of the fabrics could get rid of MB completely for 21.7 L of aqueous solutions, while Pb (II) ions with the same concentration coexisted. Further, the carboxylated cellulose fabrics exhibited a rejection ratio higher than 98.7%, and the pressure drop remained as low as 3.0 kPa under the flowrate of 10 mL/min. The filtration efficiency was so high that the rejection ratio remained at 92.3% even after 44.0 L of wastewater treatment. The original cellulose fabric cartridge failed to remove MB from the solution at the very beginning, as shown in . Further, the rejection ratio of the original cellulose fabric was only 78.6% after 2.5 L of filtration. A high rejection ratio of 98.2% for removal of Pb (II) was achieved by the carboxylated cellulose fabric cartridge after treatment of 36.1 L of wastewater when 2.0 mg/L of MB coexisted, as shown in . It was also concluded that Pb (II) ions were adsorbed favorably due to the stronger interaction of carboxylate groups with Pb (II) ions than with MB. The flowrate remained at 10 mL/min under the low pressure drop of 3.0 kPa throughout the filtration process, which indicated that the carboxylated cellulose fabric cartridge could be used as a highly permeable and long usage filter for wastewater treatment.The carboxylated cellulose fabric filters could be recycled by a simple post-treatment and reused many times without losing the adsorption capability as shown in The carboxylated cellulose fabric filter remained at a constant weight after 10 additional cycles of adsorption-desorption operations indicating the robust nature and sustainability of the filter. It was noted that the crystallinity of cellulose fabrics decreased minimally after oxidation due to the oxidation of the crystal surface of cellulose as evidenced by XRD measurement (). Therefore, the effects of oxidation under the optimum condition on the mechanical and other physical/chemical properties of cellulose fabrics should not be significant. Meanwhile, the adsorption efficiency of the filter was also kept at a level of 75% or higher for both of Pb (II) and MB adsorption even after 10 uses which guaranteed the practical application of the carboxylated cellulose filter in wastewater reclamation.Carboxylated cellulose fabric filters with adjustable functionality have been fabricated via TEMPO-mediated oxidation of cellulose fabrics and have been applied successfully in wastewater reclamation to remove cationic dyes and heavy metal ions. The carboxylated cellulose fabrics maintained their crystalline structure as in native cellulose, and therefore, exhibited excellent mechanical properties as evidenced by tensile experiments. The observation of surface and cross-sectional morphologies revealed that the oxidized fabrics exhibited micro and sub-micro structures which enhanced the functionality and water permeability of the filter as confirmed by water contact angle measurements. As a result, the carboxylated cellulose fabrics demonstrated great adsorption capability for Pb (II) and MB from simulated wastewater. In fact, the adsorption capacities of the carboxylated cellulose fabrics for Pb (II) and MB were 81.30 and 76.92 mg/g, respectively, which were comparable or even higher than that of most adsorbents. The adsorption mechanism was explored and it revealed that the adsorption of Pb (II) and MB occurred not only on the surface but also in the internal porous structure of the carboxylated cellulose fibers. To evaluate the filtration efficiency of the carboxylated cellulose fabric filter, dynamic adsorption performance was conducted with wastewater containing MB or Pb (II), respectively. The carboxylated cellulose fabric filter exhibited unexpected filtration performance by filtration of 200 mL of wastewater with higher than 90% rejection ratio and a reasonably high flux at a pressure as low as 0.07 kPa. A spiral wound filtration cartridge based on carboxylated cellulose fabrics was fabricated and long-term filtration performance was demonstrated where high rejection ratios of 98.7% and 98.2% were achieved for removal of MB and Pb (II); up to 21.7 L and 36.1 L of simulated wastewater containing MB and Pb (II), respectively, were filtered at an efficiency which was 10-times higher than that of original cellulose fabrics in the context of treatment of multicomponent wastewater. Moreover, the sustainability of the carboxylated cellulose fabric filters was also confirmed by 10 uses of the same filter through recycling and regeneration; the filter did not lose its adsorption capability even after the tenth cycle.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.Supplementary data to this article can be found online at The following are the Supplementary data to this article:Equivalent layered models for functionally graded platesFunctionally graded plates whose material properties vary continuously through the thickness are modelled as exactly equivalent plates composed of up to four isotropic layers. Separate models are derived for analysis using classical plate theory, first-order and higher-order shear deformation theory. For cases where Poisson’s ratio varies through the thickness, the integrations required to obtain the membrane, coupling and out-of-plane stiffness matrices are performed accurately using a series solution. The model is verified by comparison with well converged solutions from approximate models in which the plate is divided into many isotropic layers. Critical buckling loads and undamped natural frequencies are found for a range of illustrative examples.Functionally graded (FG) materials can be defined as those which are formed by gradually mixing two or more different materials, with the main aim of adapting their physical properties to the external environment. The variation of properties is required to be as smooth as possible in order to avoid phenomena such as stress concentrations which could lead to the development or propagation of fractures.Nature provides examples of materials whose physical properties vary gradually, but the concept of synthetically manufactured FG materials was first developed in Japan in the early 1980s Pioneering manufacturing techniques include powder metallurgy, physical and chemical vapour deposition, plasma spraying, self-propagating high temperature synthesis and galvanoforming Although the most important applications of FG materials have taken place in the aerospace industry, mechanical engineering, chemical plants and nuclear energy, they are now attracting attention in optics, sports goods, car components, and particularly in biomaterials by means of prostheses. Modern FG implants allow the bone tissues to penetrate between the metallic (often titanium) part and the bone by means of the hydroxyapatite (a transition porous material), forming a graded layup in which a suitable bonding is developed In engineering it is important to highlight the effects of FG materials in turbomachinery components such as rotating blades, since by varying the gradation it is possible to alter the natural frequencies in order to guarantee stability at particular spinning speeds. Finally it is worth mentioning smart applications, in which piezoelectric sensors and actuators are integrated into the FG material to control vibrations or static responses in structures Natural frequencies and critical buckling loads of FG plates have been tabulated by various authors The present paper includes the previously derived CPT models and examples introduces an equivalent (single layer) isotropic plate model for use with CPT under the assumption that Poisson’s ratio does not vary through the thickness of the plate. This assumption is relaxed in Section , where an exactly equivalent plate composed of two isotropic layers is derived for CPT by solving an inverse problem to satisfy six independent stiffness requirements. Section demonstrates that the extension of these models to FSDT is trivial, and then outlines extensions to HSDT giving equivalent plates with three and four layers, respectively. The numerical results in Section verify the proposed models, and also demonstrate its accuracy in finding critical buckling loads and natural frequencies of FG plates, using the different plate theories. Section summarises the conclusions and suggests further extensions to the method.Consider a FG plate of thickness h lying in the xy plane with the origin at mid-surface, having material properties which vary through the thickness (z) direction. Using standard notation, the plate constitutive relations of CPT are written aswhere the vectors N,M,ε0 and κ contain perturbation membrane forces per unit length, perturbation bending and twisting moments per unit length, perturbation mid-surface membrane strains, and perturbation curvatures and twist, respectively. The membrane, coupling and out-of-plane stiffness matrices are given byA=∫-h/2h/2E(z)Q(z)dzB=∫-h/2h/2E(z)Q(z)zdzD=∫-h/2h/2E(z)Q(z)z2dzQ11(z)=11-ν(z)2Q12(z)=ν(z)Q11(z)Q66(z)=12Q11(z)-Q12(z)Young’s modulus E(z), Poisson’s ratio ν(z) and density ρ(z) are assumed to vary through the thickness according to the rule of mixturesE(z)=Em+V(z)Eδν(z)=νm+V(z)νδρ(z)=ρm+V(z)ρδHere, subscripts m and r denote the properties of the metal and reinforcement components, respectively, and V(z) is a function representing the volume fraction of the reinforcement, which is assumed to follow the commonly encountered power lawThe non-negative volume fraction index n controls the variation of the properties of the FG plate, as illustrated in . As n approaches zero the plate consists essentially of reinforcement material, while as n approaches infinity it consists essentially of matrix material.If both materials have the same Poisson’s ratio νm=νr=ν0, thenAF=∫-h/2h/2E(z)dz=hEm+Eδn+1BF=∫-h/2h/2E(z)zdz=h22nEδn+1n+2DF=∫-h/2h/2E(z)z2dz=h312Em+3n2+n+2Eδn+1n+2n+3The presence of BF indicates coupling between the in-plane and out-of-plane behaviour.Now consider an isotropic plate of thickness h∗, Young’s modulus E∗, Poisson’s ratio ν∗=ν0, whose neutral surface is offset by δ∗ above the geometric mid-surface. The membrane, coupling and out-of-plane stiffness matrices are given byand is therefore equivalent to the FG plate of Eq. For vibration analysis an equivalent density ρ∗ is also defined, so that the mass per unit area ρ∗h∗ of the equivalent plate is equal to that of the FG plate, i.e. give an exact isotropic equivalence for FG plates with constant Poisson’s ratio. For cases where Poisson’s ratio varies through the thickness of the plate, an approximate solution was proposed When Poisson’s ratio varies through the thickness of the FG plate, Eq. cannot be simplified to the form of Eq. , because the integrations become more complicated on account of the terms in ν(z) appearing in the denominators of Eq. . The independent elements of A,B and D must be evaluated asA11=∫-h/2h/2E(z)1-ν(z)2dzA12=∫-h/2h/2E(z)ν(z)1-ν(z)2dzB11=∫-h/2h/2E(z)1-ν(z)2zdzB12=∫-h/2h/2E(z)ν(z)1-ν(z)2zdzD11=∫-h/2h/2E(z)1-ν(z)2z2dzD12=∫-h/2h/2E(z)ν(z)1-ν(z)2z2dz can be written in non-dimensional form asA‾11=A11Emh=I0+E‾δInA‾12=A12Emh=νmI0+νδ+E‾δνmIn+E‾δνδI2nB‾11=2B11Emh2=2I1-I0+E‾δ2In+1-InB‾12=2B12Emh2=νm2I1-I0+νδ+E‾δνm2In+1-In+E‾δνδ2I2n+1-I2nD‾11=12D11Emh3=12I2-12I1+3I0+E‾δ12In+2-12In+1+3InD‾12=12D12Emh3=νm12I2-12I1+3I0+νδ+E‾δνm12In+2-12In+1+3In+E‾δνδ12I2n+2-12I2n+1+3I2nare evaluated by means of the series solution , which is made from two layers of different isotropic materials. The figure shows the thicknesses h1,h2, Young’s moduli E1,E2 and Poisson’s ratios ν1,ν2 of the two layers. The thicknesses and Young’s moduli can be expressed in non-dimensional form as gives the independent elements of the stiffness matrices A,B and D in non-dimensional form asA‾11∗=A11Emh=K1+K2A‾12∗=A12Emh=ν1K1+ν2K2B‾11∗=2B11Emh2=L1+L2B‾12∗=2B12Emh2=ν1L1+ν2L2D‾11∗=12D11Emh3=M1+M2D‾12∗=12D12Emh3=ν1M1+ν2M2Suppose that the non-dimensional stiffnesses of a FG plate and the two layer plate of Section , respectively. Then, for analysis using CPT, the two plates are exactly equivalent ifA‾11∗A‾12∗B‾11∗B‾12∗D‾11∗D‾12∗T=A‾11A‾12B‾11B‾12D‾11D‾12Tβ1=α1+A‾12B‾11A‾11-B‾12β2=α2+A‾12B‾11A‾11-B‾12δ1=α1+A‾12D‾11A‾11-D‾12δ2=α2+A‾12D‾11A‾11-D‾12η1=L2K2=β1α1=b1-b2ξ1η2=-L1K1=-β2α2=-b1+b2ξ2ξ1=1α1ξ2=1α2b1=B‾11A‾11b2=B‾12-A‾12B‾11A‾11Taking the difference between the simultaneous Eqs. which, since the total thickness of the equivalent plate η1+η2h cannot equal zero, implies thatThe thicknesses of the two layers are now obtained from Eq. The Poisson’s ratios are obtained from Eqs. ν1=1A‾11b2b1-η1+A‾12ν2=1A‾11b2b1+η2+A‾12Finally, the Young’s moduli are found from Eqs. E1=b21-ν12η1η2+b1ν2-ν1EmE2=b21-ν22η2η1-b1ν2-ν1EmFor vibration analysis an equivalent density ρ∗ is defined, analogously to that of Eq. , such that the mass per unit area ρ∗h1+h2 of the equivalent plate is equal to that of the FG plate, i.e.First order (FSDT) and higher order (HSDT) shear deformation plate theories require the satisfaction of additional constitutive relations beyond those of Eq. , and the use of additional stiffness coefficients beyond those listed in Eq. is supplemented by the constitutive relationbetween the transverse shear stresses T and transverse shear strains γ. Because the two components of the FG plate are isotropic, the additional stiffness coefficients of As are obtained easily aswhere f is the shear correction factor. Thus no additional integrations are required beyond those already used to obtain A11 and A12 under the assumptions of CPT. is replaced by the constitutive relations where M∗,κ∗,T∗ and γ∗ denote higher order moments, curvatures, transverse shear stresses and transverse shear strains, respectively. Additional integrations are required to obtain the additional stiffness matrices E,F and H, while the stiffness coefficients of Ds and Fs corresponding to transverse shear effects are obtained asAs a result there are more independent stiffness parameters than those listed in Eqs. . Therefore an equivalent plate has to comprise more than two layers, as outlined in the following discussion.To obtain the stiffness matrices of the FG plate, Eq. E=∫-h/2h/2E(z)Q(z)z3dzF=∫-h/2h/2E(z)Q(z)z4dzH=∫-h/2h/2E(z)Q(z)z6dzIf Poisson’s ratio remains constant through the thickness of the plate, then extension of the arguments of Section EF=∫-h/2h/2E(z)z3dzFF=∫-h/2h/2E(z)z4dzHF=∫-h/2h/2E(z)z6dzThere are therefore six independent stiffness properties, AF,BF,DF,EF,FF and HF. An equivalent three layer plate, see (a), can therefore be found by selecting six parameters, namely the thicknesses h1,h2,h3 and Young’s moduli E1,E2,E3 of the three layers, so that the six independent stiffness properties of the equivalent plate match AF,BF,DF,EF,FF and HF of the FG plate. Each layer is assumed to have the same Poisson’s ratio as the two components of the FG plate.If Poisson’s ratio varies through the thickness of the plate, extension of the arguments of Section leads to twelve independent stiffness properties, namely those of Eq. E11=∫-h/2h/2E(z)1-ν(z)2z3dzE12=∫-h/2h/2E(z)ν(z)1-ν(z)2z3dzF11=∫-h/2h/2E(z)1-ν(z)2z4dzF12=∫-h/2h/2E(z)ν(z)1-ν(z)2z4dzH11=∫-h/2h/2E(z)1-ν(z)2z6dzH12=∫-h/2h/2E(z)ν(z)1-ν(z)2z6dz(b), can therefore be found by selecting twelve parameters, namely the thicknesses h1,h2,h3,h4, Young’s moduli E1,E2,E3,E4 and Poisson’s ratios ν1,ν2,ν3,ν4 of the four layers, so that the twelve independent stiffness properties of the equivalent plate match those of the FG plate given in Eqs. The problems of selection of the layer parameters for the HSDT analyses of this section are not readily amenable to analytic solution. However, given suitable trial solutions, they can be solved numerically using the MATLAB software summarises the forms of the equivalent plates for the various cases considered in Sections . Given any FG plate, the integrations given in these sections enable calculation of the independent stiffness parameters listed in the penultimate column of . Then solution of between 3 and 12 simultaneous non-linear equations gives the layer parameters listed in the final column of the table, which define an equivalent plate, comprised of between one and four layers and having the same independent stiffness parameters. Hence structural analyses, such as the calculation of critical buckling loads or undamped natural frequencies, can then be performed using existing forms of analysis for plates composed of isotropic layers.Alternatively, approximate solutions can be obtained by dividing the FG plate into a large number nl of isotropic layers, each of thickness h/nl, with the material properties for the ith layer given by Eq. and performing analysis for the resulting laminated plate. Under the assumption that the results from such analysis converge uniformly towards exact results as nl is increased, a set of results fnl/4,fnl/2,fnl from the layered model can be used where fnl represents the result found for the approximate model with nl layers. Such predictions are taken as comparators for the equivalent isotropic and two layer models presented in this paper.Such analysis is available in the exact strip software VICONOPT The first example considered here is a FG plate of thickness h=2mm, composed of an aluminium matrix reinforced by ceramic material. The aluminium and ceramic have Young’s modulus Em=70GPa and Er=121GPa, respectively, and both materials have the same Poisson’s ratio νm=νr=ν0=0.25. shows plots of the Young’s modulus E∗, thickness h∗ and neutral surface offset δ∗ of the equivalent isotropic plate, obtained using Eqs. for different values of the volume fraction index n. (a) demonstrates how the equivalent Young’s modulus varies from that of the reinforcement material when n=0 to that of the matrix material as n approaches infinity, in agreement with which shows that these represent extreme cases where the FG plate is composed of a single isotopic material.For these extreme cases, the equivalent isotropic plate is identical to the FG plate. Therefore it has thickness h∗=h and its neutral surface coincides with the geometric mid-surface, i.e. δ∗=0, as shown in (b) shows that the thickness of the equivalent plate can be either larger or smaller than that of the FG plate, by up to about 3% for this example, depending on the value of n. Also, from (c), the neutral surface of the equivalent plate is always offset above its geometric mid-surface, by up to 5% of the true thickness h. It can be shown analytically from Eqs. demonstrates that the stiffness matrices obtained using VICONOPT by the approximate layered approach, with nl=128 and extrapolated by Eq. , converge exactly to those of the equivalent isotropic plate. also lists the Young’s modulus E∗, thickness h∗ and neutral surface offset δ∗ of the equivalent isotropic plate.A simply supported square aluminium–ceramic FG plate of length a=100mm, thickness h=2mm and the material properties given in Section was loaded in uniform longitudinal compression. For seven values of the volume fraction index, ranging from n=0 (i.e. pure ceramic) to n=∞ (i.e. pure metal), the software VICONOPT was used to find the critical buckling load, firstly by using an analytical model of the equivalent isotropic plate defined in Section , and secondly by using an approximate layered model with up to nl=128 isotropic layers. The analytical results can be regarded as exact for this example because both materials have the same Poisson’s ratio. shows that the results from the layered model converge towards the analytical results as nl is increased, giving 4–6 significant figures of accuracy when nl=128. shows that these extrapolated predictions from Eq. are even closer to the analytical results, demonstrating the correctness of the equivalent isotropic model. illustrates that the stiffness matrices for such a plate with thickness h=2mm, obtained by VICONOPT using the approximate layered approach with nl=128 and extrapolated by Eq. , converge to those of the equivalent two layer plate proposed in this paper, rather than to those of the previously proposed isotropic plate for the isotropic plate. This accuracy is confirmed by , where the Poisson’s ratios of the component materials have been artificially adjusted to give a more extreme variation between them, namely νm=0.15 and νr=0.4.The software VICONOPT was used to find the fundamental natural frequencies of simply supported square FG plates of length a=100mm with the composition described in Section , for seven values of the volume fraction index, ranging from n=0 (i.e. pure reinforcement) to n=∞ (i.e. pure matrix). As in Section , results from an analytical model of the equivalent isotropic plate were compared with those from an approximate layered model with up to nl=128 isotropic layers, and with extrapolated predictions from the latter. shows agreement to approximately 4 significant figures between the analytical and approximate results.However, in contrast to the example of Section shows that this accuracy cannot be significantly improved by extrapolation, indicating that as nl is increased the layered model converges to natural frequencies which differ slightly from those of the equivalent isotropic plate. The reason for this is the approximation introduced by Eq. to obtain a constant value of Poisson’s ratio for the equivalent isotropic model.The natural frequencies ωij of a simply supported square isotropic plate of length a and thickness h, with Young’s modulus E, Poisson’s ratio ν and density ρ, are given by the expression where i and j represent the number of half-waves in the longitudinal and transverse directions, respectively. Hence the natural frequencies can be written in the normalised form shows that this relationship is satisfied exactly for FG plates with extreme values of the volume fraction index n=0 and n=∞. The remaining cases in were obtained using the equivalent isotropic model, and show slight discrepancies from the analytical results due to the approximation in the representation of Poisson’s ratio by Eq. . However the discrepancies can be avoided by using the equivalent two layer model rather than the equivalent isotropic model.VICONOPT was used to find the first three undamped natural frequencies of a thick Ti-AlOx FG plate with the same material properties as the plate of . The plate had length and width 100mm and thickness h=10mm. list the results obtained by the approximate layered approach with nl=128 and extrapolated by Eq. , and by the equivalent isotropic and two layer plates, using CPT and FSDT (with shear correction factor κ=5/6) respectively. Both tables show that the approximate layered results converge to those of the two layer plate, making this an ideal representation of the FG plate whichever theory is chosen. The significant difference between most of the corresponding results in illustrates the necessity to account for transverse shear deformation in the vibration analysis of moderately thick plates.VICONOPT is currently unable to carry out buckling or vibration analysis based on HSDT. However the stiffness calculations of Section have been coded and verified by the fourth author in an extensive parametric study An equivalent two layer model has been developed for a functionally graded (FG) plate whose material properties, including Poisson’s ratio, vary continuously through the thickness. The model allows critical buckling loads and undamped natural frequencies of a FG plate to be obtained using existing methods based on classical plate theory (CPT) or first order shear deformation plate theory (FSDT) for plates composed of isotropic layers. Analytic expressions have been derived for the thickness, Young’s modulus, Poisson’s ratio and density of the component layers. The model gives an exact analogy with the FG plate, so extending and improving a previously proposed isotropic plate model which however remains exact if the matrix and reinforcement materials have the same Poisson’s ratio.The two layer plate model has been verified by using the software VICONOPT to obtain the stiffness properties for a simply supported FG plate. The correctness and accuracy of the model have been confirmed by comparing these results with well converged solutions from an approximate model in which the plate is divided into isotropic layers. Agreement has also been demonstrated in the calculation of undamped natural frequencies, using both CPT and FSDT, for a thick FG plate.Because the equivalent model can be used directly with established software such as VICONOPT, the analysis is readily extended to FG plates with different loading and support conditions, and to prismatic panels containing such plates, e.g. aircraft wing and fuselage panels. Attention will also be given to FG plates whose volume fraction varies in different ways through the thickness of the plate.CPT and FSDT give satisfactory results for thin and moderately thick plates, respectively, but should be replaced by a more accurate higher order shear deformation theory when analysing thicker plates of the dimensions used in composite aircraft panels. The methods proposed in this paper have therefore been extended to obtain equivalent plates composed of three and four isotropic layers, which exhibit the additional material properties needed to fully represent the FG material when using higher order theories. Critical buckling and undamped vibration studies for such plates will be perfomed using alternative HSDT software for plates composed of isotropic layers.Composite failure prediction of single-L joint structures under bendingIn this article, the single-L peel joint is investigated under bending deformation. The angled L-piece is the critical component regarding composite delamination failure. The failure load and location were predicted by a numerically stabilised damage model and validated by experimental results. The dependence of the failure strength based on the moment arm and applied displacement was investigated. Numerical optimisation was performed regarding geometry and lay-up of the L-piece. Furthermore, the failure under a combination of bending and tensile displacement was investigated numerically. The method was shown to be versatile and successful in predicting the experimental outcome.Adhesive composite joints nowadays play an important role in aerospace, wind turbine and ship designs. Accurate failure predictions are required for efficient joint design and to utilise the advantages of adhesive bonding, such as a more uniform stress distribution in the joint area, or less weight of the total structure when compared to mechanical fasteners Adhesively bonded composite structures are especially prone to delamination failure as a result of a high gradient of peel stresses at the end of overlap regions or through-thickness load transfer. The through-thickness strength of composites is usually low compared to the in-plane strength due to the absence of load-bearing fibres across the bonded surfaces.Of the possible adhesive joint designs, the single-L peel joint was chosen for validation purposes in this study. The primary function of this out-of-plane joint is to transmit flexural, tensile and shear loads between two sets of panels meeting at the joint by an angle connection (see The joint characteristics are similar to in-plane joints, such as single-lap, but additionally the peel properties of the adhesive and the radius of the load transferring L-piece become important Advanced adhesive application techniques required to guarantee repeatable joint performancePractical joint geometry, which requires accurate modelling for industrial applicationDifferentiation between adhesive and composite failure modes possible to test models for composite failure predictionRange of optimisation possibilities in terms of composite lay-up, and variations of joint geometryThe objective of this paper is to provide detailed insight into the design and failure of composite peel joints under bending deformations as shown in . Cross-ply lay-ups formed from unidirectionally reinforced carbon fibre prepregs are considered for the composite adherends, although an identical approach can be utilised for glass and other fibre reinforced composite lay-ups. The joints are tested by varying the moment arm under bending deformation. Failure prediction with a numerical composite damage model is validated with these experimental results. Numerical optimisation is undertaken by varying composite parameters (i.e. stacking sequence) and geometry (i.e. thickness and radius of the L-piece). Finally, the effects of combined bending and tension are considered theoretically, combining results for purely tensile loading (published in a separate paper The adhesive joint is manufactured out of three parts: A base plate, an angled L-piece and a rib (). The adhesive bond line thickness is 0.75 mm. The peel fillet can be varied in size and shape. Adhesive control therefore plays an important role for the design and performance of peel joints. A process of adhesive injection was applied to improve the joint quality and fillet control during preparation, described in detail elsewhere describes this set-up. The test results in composite failure in the L-piece and is therefore composite strength-limited.The adherend laminates were made from high-tensile strength, uni-directional carbon fibre prepreg with an epoxy matrix (HTA/977-2). shows the material properties of the unidirectional lamina.The dimensions and lay-ups of the three composite parts were as follows:Base: 300×70×10 mm, Stacking: [45/0/−45/903/45/0/−45/903/45/904/−45/902]sRib: 150×70×8 mm, Stacking: [45/90/−45/03/45/90/−45/04/45/02]sL-piece: 50×42.5×3.7 mm with a 9 mm internal radius, Stacking: [45/0/−45/90/45/0/−45/0]sThe stacking sequence of each part was optimised with respect to current industrial guidelines The curved composite specimens have a tight radius of 9 mm to minimise space. For aerospace applications, peel joints are constructed with radii down to 5 mm. Ige The Hysol EA 9394/C-2 adhesive was chosen due to its excellent compressive properties, which results in high stresses in the L-piece during deformation without adhesive failure. It is toughened with alumina particles and gives good performance under high and low temperatures, which makes it ideal for aerospace applications. The adhesive was modelled by assuming isotropic properties which require only the knowledge of the Young's modulus and Poisson's ratio. For larger deformations the true stress/strain data in The base plate was clamped against a fixed back plate on its ends over a length of 50 mm by steel plates, equidistant from the centre of the joint (see ). These clamps restricted all movement in both x- and y-direction.The tests focused on studying the joint behaviour in detail by monitoring local strain fields, using strain gauges. These measurements enabled the identification of correct boundary conditions for the finite element model and the study of local changes in strain distribution after the onset of damage. The strain gauges were applied at specific joint locations in the middle of the joint width as indicated in . Related experimental tests showed that while peel fillet details do not play an important role for the failure mechanism during the test, they can influence the stiffness of the structural response significantly. The fillet radius was therefore kept constant at 3 mm for this investigation.The flexural loading was applied at the two distances of 61 and 70 mm, measured from the back of the base plate, under displacement control (1 mm/min). The test details and key results are summarised in shows the difference in load-displacement response, failure strength and the local strain gauge details. The joint is able to withstand a high amount of bending according to the final failure displacement. However, a composite structure is normally designed for first ply failure which occurs at a much lower displacement by matrix cracking and can lead to prior failure of the structure if other loadings occur. This has to be considered for the final design. All specimens exhibited nonlinearity within the global load-displacement curves prior to delamination failure. The local strain gauge patterns proved to be more sensitive to local structural changes than the global load-displacement curve. shows the strain readings at 2 locations, together with the deviation of these strains from a purely linear response.Schematics of the numerical model can be seen in . The critical area of failure is highlighted, and later plots will be restricted to this area of the joint. The figure also illustrates the use of different coordinate systems throughout the structure.The composite material is assumed to be elastic at the beginning of the analysis. A user subroutine UMAT within ABAQUS was developed to describe the material behaviour under damage development in a three-dimensional stress state. Details about the implementation are explained in supply all the necessary composite material data for the layered modelling approach and the adhesive.Special attention had to be paid to applying the correct boundary conditions at the base plate to simulate the experimental behaviour. The bending test was simulated by idealised rigid clamping conditions applied on a 145 mm long back plate, which means that all movement in the base piece was restricted from 145 mm outwards as in the experimental set-up. It was established during numerical simulation trials that the bending test does not show large dependencies regarding the clamping conditions as most of the bending occurs in the L-piece which is determined by structural details (e.g. the fillet size, stacking sequence or thickness). The boundary conditions are verified in . The influence of a maximum rib misalignment (e.g. due to spring back) of 2° was also investigated and found not to influence the results on the global or local scale. The effect was consequently neglected for the bending test simulations. Displacement boundary conditions were applied on the rib to mimic the experimental test under displacement control and to enhance numerical stability during damage progression.An elastic damage model for composite failure, which considers the three-dimensional stress state within the joint, was developed Most researchers seem to agree that Hashin's piecewise linear failure criteria (σ22+σ33)2Yt2+σ122+σ132+σ232−σ22σ33Sc2=11Yc[(Yc2Sc)2−1](σ22+σ33)+(σ22+σ33)24Sc2σ122+σ132+σ232−σ22σ33Sc2=1Index 1 stands for the fibre direction, 2 for the transverse direction and 3 denotes the through-thickness direction. For the present analysis, the “1” direction is associated with the y-direction of the local coordinate systems as shown in , “2” with z and “3” with x. This unusual convention is required due to the fixed orientation of the cylindrical coordinate system in a 2D model as explained earlier. Xt, Xc are the tensile and compressive strength in the fibre direction, Yt, Yc are the tensile and compressive strength in the transverse direction and Sc is the in-plane shear strength of the ply.Composite failure is characterised by the formation and evolution of microcracks and cavities with pronounced directionality. These effects cause primarily stiffness degradation and only small permanent deformations remain in the stress-free body after unloading as long as the material is not close to complete deterioration. Consequently, anisotropic elastic damage mechanics can be applied where damage tensors instead of a scalar value are proposed to account for the anisotropy of damage.To derive the constitutive tensor of the damaged material, two main approaches using damage mechanics have been applied throughout the literature: (1) Derivation of the strain energy function including damage and (2) physical arguments related to the material properties.The stiffness matrix can be ascertained by determining the derivative of the Helmholtz free energy function, which is expressed as a polynomial in terms of strain and damage components, with respect to the strain tensor. This approach was mostly applied to understand stiffness degradation due to matrix cracking The second method is based on physical arguments and information on the dependencies between effective elastic properties and individual damage variables. Separate functions ψi are introduced directly in the compliance matrix itselfThey are related to the engineering constants and therefore rely on experimental data and intuitive judgement when considering the effect of individual failure modes on out-of-plane or anisotropic properties. This approach has mostly been applied to study the effect of combined failure modes in composites Following the physical approach as introduced by Matzenmiller et al. [S(ψi)]d=[1ψ1E1−ν21E2−ν31E30001ψ2E2−ν23E20001ψ3E30001ψ4G1200sym1ψ5G1301ψ4G23]Regarding the evolution of the damage, two degradation approaches are commonly used: (1) continuous degradation and (2) constant degradation of the elastic properties. The continuous degradation approach is the physically more reasonable option to describe damage development; however, it requires at least twice as many variables as the constant degradation approach as the rate of degradation needs to be specified. It furthermore requires regularisation for mesh size independence, as mesh refinement leads to vanishing energy dissipation, and the deformation localises in an area of decreasing volume Although the method of constant degradation was found to show converging results for mesh refinement, it can suffer from increment-size dependence. Varying damage patterns were established on a model problem solved with a differing number of increments. In general, more elements failed for larger increment size due to overstepping of the damage onset in several elements during the first iteration. It can be concluded that failure has to be minimised for the first iteration of a new increment, and the procedure has to be implemented with an increment size control algorithm to give reliable results with 1.05 for the following numerical predictions. This restart criterion was applied to guarantee early detection of any new failure onset at the integration points. A flowchart of the program structure, illustrating the restart procedures, is shown in . For the chosen problem, the restart option restricting failure to one element was found to be rarely necessary as the direction of damage growth for composite structures is regulated by the individual composite layers with different material properties. The layers therefore exhibit different stress distributions throughout the model thickness, and damage instabilities manifesting themselves as jumps from one layer to the next were not observed.The last problem to be mentioned for all types of degradation is the dependency of damage development on the mesh alignment, where the element boundaries can define the direction of damage growth and therefore introduce artificial damage patterns. In our case the mesh is aligned with the underlying composite layers and thereby also helps to confine the damage to one material layer. A different approach to test the influence of mesh alignment was not possible, as different layer properties require an element boundary along the layer interface.Comparisons within the region of elastic deformation give an indication of the general validity of the finite element model, i.e. the boundary conditions, modelling details and material data. The experimental stiffness was within 10% for all test cases with the same fixed boundary conditions shown in . Furthermore, average experimental strain gauge values (see ) were compared to numerical values. The fit with the numerical data was excellent at all six strain gauge locations. The results are shown in for an arbitrary elastic displacement of 1.76 mm (600 N).Regardless of the value of the degradation factors, the weakest link of the structure is the L-piece under bending displacement, as various failure modes are encountered in this component prior to final failure. Preliminary analyses without progressive damage calculation showed that the upper side clearly indicates matrix cracking under tension, while the lower side of the L-piece undergoes matrix cracking under compression. Consequently, values from strain gauges 3 and 4 (see ) can be used to correlate the degradation factors ψ2 and ψ4 for matrix cracking under tension and compression, as these strain gauge values are the critical predictions close to the areas of damage detection. The remaining properties are not affected by matrix cracking. shows the comparison with numerical results for strain gauges 3 and 4 to establish whether the numerical method predicts the onset and location of failure correctly. While the left figures display the strain gauge values themselves, the right figures demonstrate the more sensitive deviation of strain from linear material behaviour. The numerical analysis without damage development is itself nonlinear, due to the elastic-plastic material behaviour of the adhesive. These factors contribute to strain nonlinearity; however, the additional effect of composite damage can be observed by including the curves for a model without material degradation after composite damage. It can easily be seen that this strain increase exhibits less nonlinearity, and does not follow the experimentally observed strain patterns very well.The following degradation factors for damage development were then found to produce well fitting numerical curves:Tensile matrix cracking (T)/compressive matrix cracking (C)ψ2T=0.2,ψ2C=0.4,ψ4T=0.2,ψ4C=0.4;ψiT,C=1.0(i=1,3,5)No sudden jumps are observed in the numerical strain gauge values as might be expected due to the constant degradation factors applied for the damage modelling. However, degradation was performed on an integration point scale and the numerical strain values were always averaged over several elements to model the behaviour of the full strain gauge length. Similar observations were made for the lower moment arm of 61 mm. Overall, the fit on the local scale implies that the failure locations for sub-critical damage are modelled correctly and also at the correct onset load, which is an excellent result for the relatively simple degradation model.The degradation values above are identical to the ones supplied for typical CFRP systems by Camanho et al. However, the failure mode of compressive and tensile fibre breaking was not encountered in the present analysis.The composite damage model starts to predict composite failure at about 1.8 mm displacement (700 N). The simulated load-displacement curves for the above degradation values with prediction of failure modes encountered are shown in . Comparison with the experimental curves in (a) and (b) for the aligned L-piece show very good agreement with respect to the stiffness degradation during the test. Damage plots for various load stages as indicated in . These figures indicate failed elements as predicted by the different stress-based composite failure criteria. An element is marked as failed once the average value of its integration points exceeds the failure value. During the first stage, only matrix cracking under tension in the upper layers of the L-piece is predicted. It is interesting to notice that first matrix cracking occurs in the 45° plies on top, which is different than a plate under in-plane loading, where 90° plies represent the weakest layer. The 45° plies carry higher transverse stresses during bending, which explains this prediction. The second stage shows the onset of compressive matrix cracking in the outer L-piece layers. The third stage including the delamination predictions is discussed in detail in Section 3.5.Tong−TsaiCriterion:σ112−σ11σ33Xt2+σ33Z+σ132R2=1NorrisCriterion:σ112−σ11σ33XtXc+σ332Z2+σ132R2=1Tong−NorrisCriterion:σ112−σ11σ33XtXc+σ33Z+σ132R2=1Here, Z is the through-thickness strength, and R denotes the interlaminar shear strength. The failure criteria are only invoked for through-thickness stresses greater than zero.Failure predictions using various failure criteria from Eqs. . The failure loads of all test cases are most closely determined by the Tong–Norris criterion (Eq. ). The Tong-Tsai and the Norris criterion are quite close for some of the experimental test cases, but actually predict delamination around the 0° layer on the compressive side of the L-piece, which is the wrong location. However, although the Tong–Norris criterion gives the most accurate predictions regarding location and the closest predictions regarding strength, it over-predicts the failure loads by about 17% for both test cases. This numerical outcome is quite satisfactory, especially as the over-prediction is consistent for both test cases. The main reason for the over-prediction is most likely the fact that residual stresses, which can develop during manufacture, were neglected in the analysis. Furthermore, material strength values depend on the quality of the composite and can be scale sensitive, which is especially true for the through-thickness strength In anticipation of using the model to develop conservative design charts, a strength adjustment is proposed, to give a closer correlation with experimental failure loads. This calibration is justified as the failure criteria are mostly empirical, and can be achieved by appropriate adjustments to the strength parameters. This also indicates the sensitivity of the results to the values of these parameters. Three possible adjustments are proposed to fit the numerical data to the experimental data (): (1) reduction of fibre strength properties for tension and compression by 15% each, (2) reduction of through-thickness strength by 55% and (3) combined reduction of fibre (10%) and through-thickness strength (20%). The smaller reduction needed in fibre properties is due to the fact that the tensile term in Eq. contributes more to the failure value. Typically, this term contributes around 75% and the through-thickness term around 25%; the influence of the interlaminar shear term is negligible.It was found with these adjustments that the failure locations stay consistent around the 0° ply for correction 1 and 3, while the failure location changes to the middle of the L-piece for correction 2. This failure location was not observed experimentally, and a 50% drop in through-thickness strength below the assumed transverse ply strength does not seem reasonable. Correlation 1 and 3 are therefore identified as suitable to calibrate the model to the measured failure data, and both give similar structural results. visualises the predicted location of the delamination which was experimentally established to mostly occur between the second (0°) and third ply (45°). An x–y plot in the interface between the layers is the more accurate way of showing the delamination failure value; however, the whole layer has failed, and the failure index in both interfaces above and below the 0° layer reaches the value 1 in close increments of the analysis. Delamination failure occurs over several elements (i.e. a length of more than 2 mm) during one increment. Correlation 1 has also been used to produce design charts for the joint structure as shown in The experimental investigations showed a dependence of the failure strength on the moment arm of the bending displacement. To classify this dependence, the original joints were modelled with a large variation of the moment arm. The adhesive radius was kept at 3 mm. Two additional L-piece radii of 5 and 17 mm were also investigated numerically. The influence of these parameters is shown in Investigation shows that failure is due to a roughly constant failure moment within the L-piece, as power law fits are found to agree well for failure strength predictions with an exponent close to −1:The length of the moment arm larm in this equation must then be measured from the middle of the L-piece where delamination failure is predicted (see With respect to the radius, the through-thickness stresses become more prominent in the failure analysis with tighter curvature. All radii predict failure around the outer 0° layer, but for the radius of 5 mm, which represents the smallest radius aircraft manufacturers tend to choose, failure in the middle of the L-piece is predicted quite soon after first delamination failure in the outer layers. Practically, this means that L-pieces with smaller radii might show delamination failure between the middle layers rather than outer layers for bending deformations. More matrix cracking is also observed in the 90° layers for the 5 mm radius before the delamination strength is reached.The percentage of layer orientations was kept unchanged throughout the investigation of stacking sequence. The order of the layers, however, can be changed within the framework of an optimised lay-up: 45° layers should be placed on the outside of a structure to minimise impact damage, and ±45° layers should be separated to minimise interlaminar shear stresses in between the layers. The original stacking sequence of [45/0/−45/90/45/0/−45/0]s was already optimised with respect to these design rules. Two other lay-ups are possible within the same restrictions:The position of the 90° layer is consequently changed from the middle of the stacking sequence to the inside of the structure and to its outside. shows the influence of the change of the lay-up on the onset of failure for the 9 mm radius. Regarding delamination, the onset of failure moves to the fourth ply for variation 2, which is the outer most 0° ply for this configuration. However, hardly any difference can be observed regarding the failure load compared to the original stacking sequence. For variation 1 on the other hand, the failure location remains the same, but the changed stacking sequence leads to an increase in failure strength by 20%. For fibre buckling failure, both variations show an increase in failure strength; variation 2 again proves to be the better choice of stacking sequence. As a rule of thumb, it can be concluded that 90° plies should be placed as close as possible to the symmetry line of the lay-up if curved structures are loaded under bending.Sub-critical damage does not immediately lead to ultimate failure of a composite, but for conservative designs, safety factors are applied with respect to first ply failure. There is no reason why this design rule should not be valid for composite joints and in this case joint quality can be measured by comparing the onset of sub-critical failure modes to the final failure strength. The joint designs, as displayed earlier in the design charts, are investigated in First failure is given by tensile matrix cracking, while second failure is established as compressive matrix cracking for all joints. Final failure is predicted by delamination. The values for each joint design are largely independent of the moment arm and hence only one value is given for each design. For most of the designs, first failure occurs at around 30% of the final failure load. This value is reduced only with a variation of the 90° ply position in the lay-up, as was the case for stacking variation 2. The second failure ratio is also reduced when compared to the other joint designs, which identifies this design as clearly inferior.The numerical variations above are straightforward to confirm experimentally. However, combined loading conditions (bending and tensile displacement on the rib), are most likely to be the loading case experienced under real practical conditions for this type of joint, but are more difficult to achieve experimentally. Tensile loading was investigated in an earlier publication and found to result in adhesive failure, which was successfully predicted Bending and tensile contributions are applied within one loading step, with fixed displacements in a specified proportion. However, at this point it is important to recognize that these combined displacement conditions now influence each other, and superposition from the two separate loading cases does not apply. For pure tensile deformation, fixed displacements are given for all nodes through the thickness, so that the end of the L-piece remains vertical. This corresponds to the experimental set-up. This condition must be retained for all combined analyses for a consistent failure envelope, including the case in which only a bending displacement is applied. For the previous pure bending analyses and experiments, the end of the rib was not constrained (see ). The current analysis requires ux=0 under nominally pure bending and will therefore differ, since the boundary condition now introduces a counter moment and a (small) axial tension, which will modify the failure displacement and load. This type of constraint interaction is likely to occur as well under practical loading conditions.In plotting the results, for the x-axis a ratio of combined loading is defined in terms of the tensile and bending displacements. 0% relates to the case of pure bending failure (4.93 mm bending displacement), while 100% corresponds to pure tensile failure (2.4 mm tensile displacement). Intermediate cases are defined by the ratios of the numerical displacement values normalised with respect to their values under tension and bending alone:with s and p being the values of applied displacement during the analysis. 50% combined loading therefore identifies the ratio of 1:1 (s=4.93 mm, p=2.4 mm), while 25% for example corresponds to a ratio of 3:1 (s=4.93 mm, p=0.8 mm). The y-axis on the failure plots indicates the overall failure load F calculated from the net force components Fx and Fy as F=Fx2+Fy2. It is important to bear in mind that the end constraint ux=const means that a moment is also being induced, so that the net forces alone do not fully describe the resulting structural loading. shows the switch of failure mechanism from composite failure to adhesive failure as a function of combined loading. It can be interpreted in that this switch occurs at high tensile loading contribution (displacement ratio 65%). From the y-axis under pure bending (0%) it can also be seen that the effect of imposing the zero displacement end constraint ux is an increase in failure load of 280% compared to the original unconstrained value. The figure shows that combined loading tends to increase the failure load above the values for pure tensile or bending failure. From a practical point of view for joint design, it is important that combined loading does not lead to lower joint strength than is determined by the two simple loading conditions of tension and bending. The increase in strength with large tensile contributions is due to the fact that the bending displacement introduces compressive stresses within the adhesive, which partly counteract the tensile stresses on the joint. In contrast, for small tensile contributions the failure load is initially fairly insensitive to combined loading.An alternative view of the contributions to failure from tension and bending is obtained by plotting the failure envelopes for displacement and force. (c) and (d) show the tensile and bending components, with the net displacement or force at failure in vectorial form. Under combined loading the displacement components increase in proportion, giving straight lines on . Since the load-displacement plots under pure tension and bending were found to remain close to linear up to the point of failure, it is reasonable to assume that under combined loading the force components do the same, which justifies the straight lines on (d). It can be seen that the resulting angles of the net load vectors to the vertical are greater than for the corresponding net displacement vectors, due to the additional end constraint. Note also that all failure displacements and loads lie on or outside a conservative elliptical failure envelope based on pure bending and tension. The figures also show the displacement and load at failure for bending without end constraints, which illustrate that the counter moment reduces the displacement and increases the load at failure, as expected.The method introduced for combined loading works well for the single-L peel joint prediction. As failure introduced by combined loading proves to be outside an elliptical failure envelope, only the two limiting test cases of bending and tension need to be established experimentally and/or numerically. An elliptical failure envelope can then be used for design predictions. However, as a note of caution, it has to be made clear that the design philosophy relies on the two failure modes of adhesive failure and composite failure. A similar double-L joint configuration (i.e. a joint with two L-pieces back to back on either side of the rib) was investigated under bending and found to fail by composite-adhesive interface failure between the L-piece and the base The finite element model was found to correlate well with the bending tests. Damage and failure prediction was done in ABAQUS by including the developed user subroutine as a UMAT. Different composite failure modes were recognized and the area of first occurrence could be determined reliably. The simple progressive damage modelling showed very good agreement with the experimental data for local strain fields influenced by extensive composite failure before final delamination failure. The delamination criterion could be calibrated to account for a strength decrease, which was most likely caused by residual stresses.The applicability of the code to predict failure for similar joint designs was explored. It was shown that the code was able to evaluate the sensitivity of the single-L joint to changes in parameters, such as L-piece curvature, thickness, or stacking sequence, and combined loading. This insight into the failure of angled joint structures is necessary to improve design strategies for composite joining.A novel isogeometric topology optimization framework for planar compliant mechanismsIn this article, we focus on a design problem of planar compliant mechanisms within the framework of isogeometric topology optimization. An integrated model is developed to identify the optimal deformation transferring path for precise motion output. The model comprises two coupled computational layers: the upper layer for geometry representation and the lower layer for sensitivity calculation. In the upper layer, the structural geometries are described explicitly by parameterized level-set surfaces, which is quite different from the implicit description way of topology optimization, giving great advantages to improve the identification accuracy of elaborate structures (e.g., flexible hinges), which are usually encountered in compliant mechanism systems. By moving, deforming, overlapping and merging these level-set surfaces, the generated shape can be projected onto the lower layer which is discretized using a NURBS patch, and NURBS-based isogeometric analysis is adopted to calculate the structural sensitivity which is then fed back to the upper layer for driving new iteration. The proposed method has the higher ability to search the optimal topology with complex kinematic behavior with respect to the conventional topology optimization methods in terms of computational effectiveness and numerical robustness. Design formulation of compliant mechanisms is constructed under the framework of the proposed method, in which the Jacobian and stiffness matrices of compliant mechanism are optimized simultaneously to achieve kinematic and stiffness requirement respectively. Three typical benchmark problems (e.g., displacement inverter, amplifier, and redirector) are tested to demonstrate these advantages.Compliant mechanisms are a type of continuous and monolithic mechanisms, which can transfer motion, force, or energy from the input position to the output position through its elastic deformation The research and development of compliant mechanism design approaches have greatly expanded its application scope and field. These design approaches mainly include two types, one is the kinematics-based approach, which usually uses the pseudo-rigid body model for analysis, and is mainly used in the design of lumped compliant mechanisms. Shao et al. , which aims to find out the optimal material distribution in the given design domain to achieve the desired performance under specific loads and boundary conditions.Since the pioneering work of Bendsøe and Kikuchi , topology optimization has been extensively explored and several methods have been developed over the past decades. The most well-established methods are solid isotropic material with penalization (SIMP) method , which have become important tools in the design of compliant mechanisms. Under the framework of SIMP method, Zhu et al. In recent years, researchers have proposed several of explicit geometric description frameworks which describe the topologies by geometrical primitives and obtain optimal topologies by directly controlling shape parameters. Among these explicit methods, geometric projection methods Isogeometric analysis (IGA), with higher numerical stability and accuracy, is expected to solve the above problems. Isogeometric approximations can be traced back to the work of Grebennikov et al. Due to the superiorities of IGA, it has been applied in the design of compliant mechanisms. In 2014, Radaelli and Herder combine isogeometric analysis and level set to successful design flexoelectric micro and nanostructures, although the method based on boundary evolution also cannot obtain parameterized boundaries. In conclusion, it is still difficult to couple the explicitly expressed compliant mechanism with isogeometric analysis for topology optimization design.In this present work, a new integrated model named IGA-based explicit topology optimization (IGA-ETO) is presented to deal with aforementioned issues in implicit or FEA-based compliant mechanism design methods. This integrated model comprises two coupled computational layers; the upper layer uses parameterized components to describe the geometry / topology, and the lower layer uses IGA for structural response and sensitivity analysis. The method inherits both advantages of the explicit expression and IGA. Besides, the objective function formulation of compliant mechanisms in this work is established based on Jacobian and stiffness matrices, which treats the design as a multiobjective problem, in order to achieve kinematic and stiffness requirement respectively. Finally, several examples of compliant mechanisms are set to demonstrate the accuracy, computational effectiveness and numerical robustness of the proposed method.The rest of this pater is organized as follows: introduces the proposed novel 2-layer IGA-ETO topology optimization method conceptually and mathematically. The corresponding numerical solution aspects are discussed in . Numerical examples and the experiment are studied in . Finally, some concluding remarks are provided in Traditional topology optimization methods are usually based on the framework of ground structure, including the most typical pixel-based one and node point-based one where the same set of meshes are used for both geometric description and numerical calculation. On the basis of this framework, the optimal structure is obtained by gradually removing the unnecessary parts from the design domain which is full of material. In this study, we introduce a different solution framework for topology optimization, where the components are used as building blocks to form topologies.Let us consider the topology optimization of a compliant mechanism shown in (a), which is designed as an actuator of MEMs that converts input force to reverse output force with the objective function minimized under available volume constraint. A well-known optimal design structure is shown in (b). It can be observed from this figure that the structure is constituted by only ten “structural components”. Here we define a structural component as an object which is constituted by an amount of solid material occupying a specific volume in the design domain. Components can be parameterized mathematically by specifying their geometric characteristics such as shape, length, thickness, direction, mutual connection relationship, and so on. In fact, a limited number of components can sufficiently form any structure with any type of topology. Thus, we can obtain structures with different topologies, including those optimal ones, by manipulating the geometry parameters of a limited number of components. Also, compared with traditional approaches, limited components mean limited control variables which avoids the excessively large computation.Based on the component idea, for a specific topology optimization problem, the optimal structural topology can be obtained by finding the geometric characteristic parameters of the components, including the shape, length, orientation, relative positions and so on.In general, the topology optimization problem can be expressed as:Findd=(d1,...,dnc)TMinimizeI=I(d)s.t.gi(d)≤0,i=1,...,l,d⊂Udwhere nc denotes the total number of components involved in the optimization problem. d = (d1,..., dnc)T denotes the key parameters of each components which are used for design variables. I is the objective function of the problem. Ud are admissible sets that d belongs to. gi(d),i=1,...,l, are the constraint functions in the optimization process.There are two aspects worth noting about this issue. The first is the geometric description. The second is the evaluation of the obtained topological structure which is represented by the geometry, including the analysis of physical characteristics and the solution of the objective function, etc. For the geometric description, we hope to complete it in a straightforward and numerically achievable way based on the idea of the above components. For the numerical calculation, we want to build the relevant analysis model by an accurate and efficient method. Owing to the high accuracy and robustness of IGA, we use it for analysis instead of traditional finite element method, which is exactly our key breakthrough.In order to connect the geometric description process and the analysis process, a 2-layer computing framework is proposed for topology optimization problem, as indicated in . The upper layer is the geometric description layer and the parameter-driven components are employed to perform the topology description of the structure in a continuous and closed approach. The lower layer is the isogeometric analysis layer. IGA is adopted to evaluate and analyze the performance of the current components-based structure, and then drive the update iteration of parametric components.A projection approach is used to model the parameterized components in the IGA's discrete NURBS elements. The NURBS model is constructed by three types of NURBS elements, that is, the inner elements with solid material, the outer elements with weak material and the boundary elements, referring to . The solid element is filled with material. Meanwhile, in order to avoid numerical singularities, the weak element with very little material is used to mimic voids in the design domain. The difficulty originates from the expression of the boundary element, and its mathematical implementation will be mentioned in Therefore, a complete explicit topology optimization framework from the geometry expression to the IGA calculation is established. Following this framework, two advantages can be obtained. On the one hand, this explicit component enables the design of structure to be shape-controllable according to the actual demands. On the other hand, the use of IGA can improve the accuracy of structural performance evaluation and physical field solution, thereby ensuring the reliability of the results and the stability of the algorithm.In this subsection, the parameterized structural components aforementioned will be discussed mathematically with the level-set function.Considering one component constituted by solid material occupying a specific volume in the design domain, the region Ω of the structure in design domain ΩD can be described by the following level-set function:{ϕ(x)>0,ifx∈Ω,ϕ(x)=0,ifx∈∂Ω,ϕ(x)<0,ifx∈ΩD∖ΩTherefore, the outline of components can be described using the level-set function. In order to parametrize the components, that is to introduce the geometry parameters of component into the level-set function, we define its center at (x0, y0), with length L, slant angle θ, and widths wA, wB, and wC of sampling points A, B and C, respectively. Points A and B are the endpoints of the basic component, and point C is the midpoint. Then the level- set function ϕi(x,y) of the ith component leads to:ϕi(x,y)=(cosθi·(x−xi0)+sinθi·(y−yi0)Li/2)6+(−sinθi·(x−xi0)+cosθi·(y−yi0)Wi(x’))6−1Wi(xi′)=wiA+wiB−2wiCLi2(xi′)2+wiB−wiA2Lixi′+wiC2 where xi’ is the x-coordinate of the point (x, y) in the local coordinate system which takes point C as the origin and central axis as the x-axis respectively. Now we have described a component using the level-set function, with 7 design variables.The assembly of different components is implemented by:where ϕsum is the level set of the complete structural, H(x) is Heaviside function, and b is a large constant (e.g., b = 10000). With the assembly method in , the structure is translated into a level-set model. Hence, the parameterized level-set based explicit geometry description method is schematically shown in Consider a 2D plane stress problem, the optimization problem in Findd=(d1,⋯,dnc)TMinimizeI=I(d,u)s.t.∫ΩDH(ϕ(x;ΩD))D:ɛ(u):ɛ(δu)dV=∫ΩDH(ϕ(x;ΩD))b·δudV+∫Γtt¯·δudS,∀δu∈Uad∫ΩDH(ϕ(x;ΩD))dV≤V¯d⊆Udu=u¯,onΓu, ε(·) denotes the linear strain vector, and D represents the elasticity matrix of the ith component. Further, b and t¯ denote the body force density and the surface traction on the Neumann boundary Γt, respectively; u and δu are the displacement field on {u|u∈H1(ΩD),u=u¯onΓu} and the corresponding weight function on Uad={δu|δu∈H1(ΩD),δu=0onΓu}, respectively; u¯ is the given displacement on the Dirichlet boundary, and we take u¯=0 in the following discussion for simplicity. The symbol V¯ represents the upper bound of volume of the available solid material.In a 2D plane stress problem, the elastic constant matrix D can be written mathematically as follows:where E and ν are the Young's modulus and Poisson's ratio respectively.Non-Uniform Rational B-Spine (NURBS), developed from B-splines, is the basic geometric representation employed in computer-aided design and isogeometric analysis. A knot vector is a non-decreasing sequence of real numbers in one direction in the parameter space, written as Ξ = {ξ1,…, ξn + p + 1}, where ξi is the ith knot, i = 1, 2, …, n + p + 1 is the corresponding index, n and p are the number of the basis functions and its polynomial order. If the knot vector Ξ is non-equally spaced, it is non-uniform, otherwise it is uniform. If the first and last knot values in a knot vector repeats p + 1 times, then it is called an open knot vector. In the standard of CAD systems and IGA, non-uniform and open knot vectors are selected.With a given knot vector Ξ = {ξ1,…, ξn + p + 1}, the B-spline curve C(ξ) takes the form:where Pi is the controls points and Ni, p is the B-spline basis function, according to the cox-de Boor recursion formula, it can be constructed beginning with constants basis recursively from p = 0:Ni,p(ξ)=ξ−ξiξi+p−ξiNi,p−1(ξ)+ξi+p+1−ξξi+p+1−ξi+1Ni+1,p−1(ξ)In order to solve the representation of conic curves, NURBS basis function is developed from B-splines by introducing positive weights ωi as follows:Here some important properties of this function are briefly listed:Local support: The p-order NURBS basis function affects at most p + 1 knot spans;Continuity: Ri,p(ξ)is p-k order continuous where the multiplicity of knot is k;Knot insertion: New knots can be inserted into the initial knot vector, without changing the NURBS curve/surface geometry.Now, the problem is expanded to 2D. For a NURBS patch located on [ξ1,ξnξ+p+1]×[η1,ηnη+q+1], the 2D NURBS basis function Ri, j(ξ, η) with degree p in ξ-direction and degree q in η-direction can be defined:Ri,j(ξ,η)=Ri,p(ξ)Rj,q(η)=Ni,p(ξ)Nj,q(η)ωi,j∑k=0nξ∑l=0nηNk,p(ξ)Nl,q(η)ωk,lUsing control points Pi, j, which contain spatial coordinate information and form a control grid in two directions, a NURBS surface can be established as follows:A NURBS surface's construction process is shown in The basis used to build the structure geometry should also serve as the basis for discreting the solution space. This is the basic and natural idea that isogeometric analysis holds. The idea of using the same basis for the representation of both the geometry and unknown solution space originates from the isoparametric concept. It is quite a common idea in both IGA and traditional FEA. However, the fundamental difference between IGA and FEA lies in the sequence. In traditional FEA, to analyze a given geometry model, the polynomial basis (usually Lagrange basis) is first chosen to discretize the unknown solution space and subsequently to approximate the given geometry. For IGA, a basis that can exactly represent the given geometry is selected firstly, and that is the NURBS basis. Then the basis is used for representation of unknown solution space, so that the geometric features are preserved and time-consuming mesh generation process can be omitted. The basic analysis process and characteristics of IGA is shown in Considering a general structural analysis problem in a domain Ω bounded by S, its weak form of the equivalent integral equation leads to:∫Ωɛ(u):D:ɛ(δu)dV=∫Ωb·δudV+∫Γtt¯·δudS,∀δu∈UadUsing the Galerkin method, a pre-specified set of NURBS function R are used for both u and δu, we can write:where uAand δuB denote the nodal displacement and the variations of nodal displacement, and nnp is the total number of shape functions. In a 2D problem, uA = {uxA, uyA}, since each control point has two unknowns - the x and y displacements.These approximations are substituted into is the differential operator. As δuB are arbitrary, where B is the strain-displacement matrix. Then we define the following:The global stiffness matrix K is a sparse matrix assembled from NURBS element stiffness matrices. Here we use “NURBS element” to distinguish it from traditional finite element. Due to the local support of the NURBS basis function, NURBS elements display stronger continuity, and hence, exhibit higher accuracy and excellent robustness in analysis.For one NURBS element, we denote the number of the local shape functions over it as nen, so the single element stiffness matrix is nen×nen. The element stiffness matrix of the eth NURBS element, ke0, can be acquired through two mappings, that is from physical space Ωe to parameter space Ω^e, and then to parent space Ω¯e where the Gaussian quadrature can be used. The process is shown as follows:ke0=∫ΩeBTDBdΩ=∫Ω^eBTDB|J1|dΩ^=∫Ω¯eBTDB|J1||J2|dΩ¯=∑i=1n∑j=1mAi,jB(ξi,ηj)TDB(ξi,ηj)|J1||J2|where Ai,j is the weight of Gaussian points. Through traversal, we obtain the stiffness matrices of all NURBS elements, and then assemble to obtain the overall stiffness matrix.Although the analysis steps are similar to the traditional FEA, the NURBS basis is introduced into the analysis framework. Its merits over Lagrange interpolating polynomial functions have also been introduced into IGA, such as higher-order continuity, local support, and hkp-refinement. It is worthy of emphasizing that the local support and continuity of NURBS enable IGA to be more accurate and stable.To unify the smooth boundaries described by level-set and discrete NURBS element, a projection method is introduced previously. Three types of material in NURBS elements are used to construct the IGA model. The solid and weak elements have obvious physical meaning, that is, the presence or almost absence of materials, while the boundary element requires additional processing to ensure the calculation results more accurate. Here we consider multiplying the Young's modulus of the material by a factor to indicate different material, which is between 0 and 1 and is related to the parametrized level-set based geometry. Mathematically, the Young's modulus of a NURBS element can be calculated as follows:Ee=[∑igp=1ngpAigp(∑icp=1ncpRicp(ξigp)H(ϕicpe))]peE0where E0 is the Young's modulus of the solid material at each control point, ngp is the number of Gaussian points in every NURBS element, and pe is the penalty coefficient which is often set to Pe ≥ 2. H(ϕicpe)icp = 1, 2, …, ncp is the Heaviside function value of the icpth control point in the eth element.In this paper, the IGA-ETO method is presented to design compliant mechanisms, and we use Jacobian-based method to model the compliant mechanisms. For simplicity, only single design domain problem is considered, which is sufficient to verify the effectiveness of the proposed method. For more information on compliant mechanism modeling, please refer to reference For the design of a compliant mechanism with single-input and single-output, the system equilibrium equation shown in can be established. According to the physical meaning of compliance, a unit dummy load is alternately applied to the DOFs of the input and output, and displacement vectors are obtained by solving . Then the elements of the input and output degrees of freedom in the displacement vectors are extracted to form a compliance matrix Cm. Thus, a reduced matrix that only describes the compliance relationship between the degrees of freedom of the input and output can be established. Further, the matrix inversion operation is performed to obtain the mechanism stiffness matrix Km=Cm−1. At this time, a system equilibrium equation which only focuses on the degrees of freedom of the input and output can be established as follows:Assuming that there is no external load applied on the output, i.e., Fo= 0, by solving the second equation in , the relationship between Ui and Uo, i.e. Jacobian matrix J, can be established:At this time, the Jacobian matrix J is a scalar, but it will expand into a matrix in complex problems. According to the physical meaning of compliance similarly, dummy unit forces Fin and Fout are used to solve , and obtain the corresponding displacements Uin and Uout. The stiffness of the input and output degrees of freedom can be characterized as:Combining the Jacobian matrix, input and output stiffness, the objective function can be expressed as:Findd=(d1T,...,dncT)TMinimizeI(x)=−(e−(J−J*)2)wf︸f·(kinkout)ws︸ss.t.F=Kd∫ΩDH(ϕ(x;ΩD))dV≤V¯d⊆U(d)u=u¯,onΓuwhere f and s represent the kinematic and stiffness requirement, J* is the desired Jacobian matrix. wf and ws are weight factors and satisfy wf + ws= 1.Solving the sensitivity of the objective function is an essential part of the topology optimization design. According to the analysis in the previous section, the sensitivity of the objective function is unified to calculate the sensitivity of Km, which is determined by the sensitivity of the elements in Cm.∂Cm(j,k)∂x=−UjT∑e=1NE{Pe·[∑igp=1ngpAigp(∑icp=1ncpRicp(ξigp)H(ϕicpe))]Pe−1×[∑igp=1ngpAigp(∑icp=1ncpRicp(ξigp)∂H(ϕicpe)∂x)]ke0}Ukwhere K is the isogeometric global stiffness matrix of the structure, and ke0 is the isogeometric element stiffness matrix of eth NURBS element. NE denotes the total number of NURBS elements in the lower layer. Besides, Fj(Fk) is the dummy unit force used to calculate Cm, and Uj(Uk) is the corresponding displacement.Besides, the sensitivity of the volume constraint function:∂V∂x=∑e=1NE(∑igp=1ngpA¯igp(∑icp=1ncpH(ϕicpe)∂x))In the present work, the method of moving asymptotes (MMA) optimizer is used to update the parameters in optimization iterations.The flowchart of the proposed IGA-ETO method for the design of compliant mechanisms is shown in As for a certain design problem shown as the design model, first we do the initialization work by defining the relevant parameters needed for the problem. Then we construct the two-layer framework by performing the bottom calculator via IGA and use parameterized level-set components to describe the geometry explicitly. Thus, we obtain an initial structure of the compliant mechanism.Then the numerical analysis which includes the objective function and its sensitivity is performed to evaluate the topology of the structure, as well as to provide guidance for the next optimization.Using optimization method, the parameters for the components are updated and the new structure is then obtained. By repeating this process, the structure is optimized step by step, and gradually approaches the structure we desire.In this section, several topological optimization examples of compliant mechanisms are presented to verify the effectiveness and robustness of the IGA-ETO method. The artificial material properties for these examples are described as: Young's modulus is E = 1Gpa and Poisson's ratio is υ=0.3. As mentioned before, only 2D plane stress is considered in this paper. All the numerical examples are carried out on a computer with Intel Core i5 – 8400 (2.81 GHz) CPU, 8.00GB RAM, and MATLAB R2016b.The design domain of the displacement inverter is shown in , which is used as a common compliant mechanism in many situations. The upper and lower corners of the left side are fixed, and an external load is applied on the middle of the left side. It is expected that the output point in the middle of the right will be reversely shifted. Due to symmetry, only the lower half of the mechanism is considered in the design process.The SIMP method and the IGA-ETO method are used to design the displacement inverter respectively. In the SIMP method, the design domain is discretized by 200 × 100 finite elements and the filter radius rmin= 1.2. While in IGA-ETO method, the prescribed design domain is discretized by a coarse NURBS mesh (12 × 6), which is different from the traditional finite element mesh and is refined 4 time (192 × 96 NURBS meshes are actually involve in analysis) using h-refinement. Only the degree of freedom in the x- direction of the output point is considered, thus the desired Jacobian matrix i.e. the displacement magnification is a scalar, and its value is J*= − 3. Besides, the weight factor wf = 0.5 and the material usage is restricted to 20%. The design results of the two methods are shown in and the red bar is a length scale, which represents 5 mm.The results obtained by the SIMP and IGA-ETO methods are as follows, the objective functions are −3.547 and −3.553, the Jacobian matrix are − 2.79 and − 2.81, respectively. From the two design results, both methods have designed effective mechanisms, but it is clear that the objective function I of the result obtained by the IGA-ETO method is smaller, and the value of the Jacobian matrix J = − 2.81 is closer to the desired value J*, which indicates that the displacement inverter designed by the IGA-ETO method has better kinematic performance than that of the SIMP method. The reason is that the higher-order continuity of IGA makes the solution process more accurate, which is also the biggest advantage of IGA.In addition, the IGA-ETO method inherits almost all the advantages of the explicit method, and the optimization results can be directly interfaced with CAD. It can also be seen from the design results that the result of SIMP has “gray elements” and many microstructures, which are difficult to post-process and manufacture. The difference is that in the IGA-ETO result, the material distribution is more uniform and the boundaries are clearer, which overcomes the related shortcomings of SIMP. The uniform material distribution reduces the stress concentration. As is shown in , the maximum stress of the IGA-ETO's result is lower than half of the SIMP's result.The design domain of the displacement amplifier is the same as that of the displacement inverter, both of which are single-input and single-output problems, as shown in . The prescribed design domain is discretized by a coarse NURBS mesh (12 × 6) which will be refined later and the desired Jacobian matrix J*= 3. Besides, the weight factor wf = 0.6 and the material usage is also restricted to 20%.An important advantage of IGA is that the NURBS mesh can be optionally refined according to requirements. We set the initial coarse NURBS mesh as 12 × 6 in this example, and then the coarse mesh is refined 2, 3, and 4 times, respectively. The sizes of the refined meshes that are actually involved in analysis are 48 × 24, 96 × 48, and 192 × 96. The corresponding design results are obtained through different refinement levels within 732, 569, and 968 iterations, as shown in . The operation time for each iteration of three refinement schemes are approximately 6.28 s, 17.18 s, and 69.43 s respectively. The three refinement schemes obtained similar compliant mechanisms and the material layouts of the design results are very close, which illustrates the algorithm stability of the IGA-ETO method under different refinement schemes. But in more detail, as the time of refinements increases, the optimized design results become smoother. So, this general rule is also applicable to the IGA-ETO method.Comparing the output data of the design results in , the objective function decreases as the number of refinement increases, which is also consistent with what we expected. At the same time, the kinematic capacity (value of the Jacobian matrix element) of the mechanism is constantly approaching the desired value, which indicates that refinement can promote topology optimization design. As for stiffness requirements, the input and output points of each mechanism have a certain stiffness kin, kout to ensure the movement function of the mechanism. Although different refinement levels can cause small changes in input and output stiffness, small changes will hardly affect the stiffness guarantee of the mechanisms.This example illustrates the application of the IGA-ETO method in the design of single-input and dual-output compliant mechanisms. The function of a displacement redirector is sketched in . The input port i is at the middle of the left side and the input causes two output displacement at ports o1 and o2, respectively. In the design domain, the upper left and lower left corners are fixed. The weight factor wf = 0.5 and in this case, the y-direction degrees of freedoms of the output ports o1 and o2 are considered, so the Jacobian matrix J of the displacement redirector is a 2 × 1 matrix, whose desired form is shown in the following equation:In order to demonstrate the robustness of the proposed approach, the parameters in the displacement redirector design problem, such as the geometric configuration of the initial components, the time of refinement of the NURBS mesh, and upper bound of the available solid material volume, are set to different values. In the control example, the initial coarse NURBS mesh is set to 12 × 12, and this mesh will be refined four times later; the upper bound of the available solid material volume is set to 30%. Besides, the initial layout consists of 72 components at the beginning of the design. Therefore, the control and experimental examples are designed under different parameter settings.After 902, 648, 848, and 797 iterations, the control and experimental examples reached iterative stability, indicating that the optimal material layout was obtained. Although the parameters in the displacement redirector examples, such as the initial components, the time of refinement of the NURBS mesh and even the upper bound of the available solid material volume, change to some extent, the optimal topology structures obtained are similar. This is clearly seen from , demonstrating the robustness of the proposed approach. The values of important parameters of the design results under different conditions are shown in . There are small differences in their values, but the design results are all valid. The iteration history of the objective function and material volume of the control example is shown in . After iterating to a certain number of steps, the convergence condition is reached.The effectiveness and robustness of the IGA-ETO method has been proven through the design of several compliant mechanisms with different kinematic functions. It is further necessary to discuss the influence of the initial layout of the components in the upper layer on design results. Since each component only contains seven shape parameters and the component itself does not have the ability to change genus, it is essential to initially configure multiple components in the design domain for the design of compliant mechanism. Therefore, the difference in the number of components will change the initial layout, and this change may be the most obvious and direct factor affecting the final design results. In order to discuss the influence of the initial layout of the components in the upper layer on design results, a set of numerical examples with different number of components are set up on the basis of ensuring that other parameters and settings remain unchanged. The design domain is still the displacement redirector shown in . Besides, in order to evaluate the time spent on each iteration step, we define the total iteration time as T, and the number of iteration steps as N, so the average time AT = T/N.The initial layout of the structure will change as the number of components changes. However, according to , when the number of components increases from 40 to 72, the objective function value changes slightly. The objective function value only decreased by 6.2% when n is nearly doubled, which indicates that as the number of components increases, the marginal revenue brought by the increase continues to decrease and approaches zero. Therefore, the proposed IGA-ETO method has low dependence on the initial layout, which can bring better stability to the design. In addition, it can be seen that the calculation cost increases linearly as the number of components increases, which is caused by the linear increase in the number of variables. Usually in design problems, we expect stronger deformability and topological change ability, therefore, we can add components appropriately to obtain better design results.Based on the result above designed by IGA-ETO method in (a), a sample of displacement redirector is obtained through 3D printing, as shown in . The thickness of the sample is 4 mm, the thinnest place is 0.8 mm, and the accuracy of the 3D printer used is 0.05 mm, which meets the experimental requirements. TPU is used as the sample material, which has excellent processing performance and resilience.During the experiment, the stepper motor provides the required input displacement uin at the input, and the mechanism generates the output displacements uout1 and uout2 through deformation transfer. The industrial camera is used to complete the image recognition and acquisition, and input the acquisition information to the OpenCV-based post-processing algorithm, and obtain accurate displacement datum after processing. The relationships between the input and output displacements of the simulation (obtained in ANSYS) and experiment are shown in According to the Jacobian matrix of the design result obtained by IGA-ETO method, a "theoretical input-output relationship" utheo is drawn, and as the simulation values of uout1 and uout2 are equal, they are uniformly represented by uout. It can be seen from that for both simulation and experiment, ideal output displacements are generated at the outputs, which meets the basic design requirements of a displacement redirector. Especially with a small displacement input of 0 ~1.386 mm, the input-output relationship of the experiment results basically coincides with the theoretical relationship line, the error is within 5% of the allowable, and the mechanism has a good kinematic function. As input increases, the input-output relationship line gradually deviates from the theoretical straight line, and the slight difference between uout1 and uout2 are caused by a combination of manufacturing errors, measurement errors, and non-linearity. In summary, in addition to excluding various objective errors, experiments have verified the effectiveness of the kinematic function of the displacement redirector with a small input displacement, thus proving the feasibility of the proposed design method.In the present paper, we propose an explicit topology optimization method based on isogeometric analysis to design compliant mechanisms. The 2-layer IGA-ETO method possesses the properties of the high accuracy of NURBS basis functions for IGA and parameterized explicit boundary described by components. By superposing the level set functions of the components in the design domain, a clear structural boundary is obtained. And structure with explicit boundary is projected into the IGA layer discrete with NURBS element to complete the calculation, evaluation and updating. Thus, the optimal design results are obtained through a series of iterations.Several numerical examples demonstrate the effectiveness and reliability of the IGA-ETO method. Compared with the classic FEA-based topology optimization method under the same mathematical modeling of compliant mechanisms, the IGA-ETO method obtains a design result with a smaller objective function value, which means that the IGA-ETO result is better. In addition, the kinematics function of the compliant mechanism designed by the IGA-ETO method is also closer to the expected value, and there is almost no loss of kinematics function in the post-processing process, which is exactly what we expect. IGA's unique refinement scheme enables element refinement to proceed easily, and finer division means more potential for optimal design results. Besides, the similar design results under the perturbation of multiple optimization parameters in the displacement redirector example demonstrate the robustness of the proposed method. It is worth noting that the above design results all give clear and explicit boundaries, which can be easily connected to CAD. In the experimental part, the optimization results were directly used for modeling and manufacturing without extra post-processing, and the desired mechanism performance was obtained. In summary, the proposed IGA-ETO method introduced IGA as a powerful analysis tool into the field of topology optimization design of compliant mechanisms, which can achieve reliable design results.Moreover, the design domains currently used are just simple square design domains, multi-patch coupling technology need to be considered when the design domain is more complex. In addition, local refinement should be implemented so as to improve local calculation accuracy while ensuring computational efficiency. These are what we need to further study in the next work.The sensitivity of J can be calculated as follows:∂J∂x=K22−1∂K22∂xK22−1K21−K22−1∂K21∂x=K22−1L1∂Km∂xL1TK22−1K21−K22−1L1∂Km∂xL2Twhere L1 and L2 are two constant selection matrices, which can extract K22 and K21 from the mechanism stiffness matrix Km. According to , the sensitivity of kin is calculated by , while the sensitivity of kout can be obtained by the same way.∂kin∂x=−(Fin)T∂Uin∂x·(kin)2=(Uin)T∂Km∂xUin·(kin)2At this time, the sensitivity of the objective function is unified to calculate the sensitivity of Km, which is determined by the sensitivity of the elements in Cm.∂Cm(j,k)∂x=FjT∂Uk∂x=FjT∂(K−1Fk)∂x=−FjTK−1∂K∂xK−1Fk=−UjT∂K∂xUk=−UjT∑e=1NE{Pe·[∑igp=1ngpAigp(∑icp=1ncpRicp(ξigp)H(ϕicpe))]Pe−1×[∑igp=1ngpAigp(∑icp=1ncpRicp(ξigp)∂H(ϕicpe)∂x)]ke0}UkBesides, the sensitivity of the volume constraint function:∂V∂x=∑e=1NE(∑igp=1ngpA¯igp(∑icp=1ncpH(ϕicpe)∂x))Dynamic pressurization: novel method for measuring fluid permeabilityThe permeability of an aerogel can be determined during the process of supercritical drying by measuring the dilatation of the body as the pressure in the autoclave is changed. When the pressure is reduced in the autoclave by venting the supercritical fluid, the higher pressure inside the aerogel causes it to expand; the strain relaxes as the fluid flows out of the gel. We present an analysis of the the kinetics of this volumetric strain and apply it to a silica aerogel. The linear strain of the gel was measured with an LVDT mounted inside the pressure vessel. The permeability was obtained over a range of pressure from about 8 MPa to ambient pressure, and was found to vary inversely with the pressure, as expected from the Klinkenberg model. The permeability obtained by extrapolation to infinite pressure agreed with the independently measured liquid permeability obtained by beam-bending.Aerogels are materials with extremely high porosity (typically >90%, often >99%), but with mean pore sizes in the mesoporous range (i.e., radii ). Usually, aerogels are made by supercritical drying (SCD) of gels The Wurzburg group measured gas transport in aerogels by imposing stepwise changes in pressure, then measuring the kinetics of relaxation of the pressure in the chamber as gas diffused into the sample ) that the experimental conditions were actually isothermal. In the present paper, we present a simplified analysis of the deformation produced by an isothermal jump in pressure and use it to obtain the permeability as a function of pressure. Our values of D are close to those obtained by Gross Of course, the experimental method, called dynamic pressurization (DP), is not limited to aerogels. Any material that is sufficiently compliant to be deformed by small pressure differentials could be examined using DP. With the analysis presented below, one can estimate the maximum deformation to be expected, given the size of the pressure jump and the elastic modulus of the body, then determine whether that strain would be measurable with the equipment available.Measurements were performed on a silica gel made from tetraethoxysilane with the B2 recipe A beam-bending experiment was performed in ethanol on a gel rod with a span/diameter ratio > 10. When a fixed deflection is applied to the rod of gel, a pressure gradient is created in the liquid within the pores. As liquid flows to equilibrate the pressure within the gel, the force that must be applied to sustain a constant deflection decreases with time. Fitting the measured force to the theoretical curve where ηF is the viscosity of the pore fluid, R is the radius of the rod, D∞ is the liquid permeability, and Hp is the longitudinal modulusSCD was performed by placing the gel in an autoclave and flushing with liquid CO2, then raising the temperature and pressure above the critical point, and slowly depressurizing. If there were no shrinkage, the aerogel density would be 135 kg/m3, which corresponds to ρ=0.068 and a porosity of 93%. However, syneresis and shrinkage during SCD increased the density to 200 kg/m3, or ρ=0.10, as determined by measuring the aerogel dimensions and weighing. The longitudinal sound velocity in the aerogel was measured using two 180 kHz piezotransducers. The acoustic wavelength was about 1 mm, so this measurement yields the longitudinal modulus, Hp.The BET surface area of an aerogel was measured by nitrogen adsorption (Micromeritics 2010).For the dynamic pressurization experiments, a piece of the aerogel 66 mm in length and 6.9 mm in diameter was mounted in a sample holder made from stainless steel wires and suspended from the lid of the autoclave, as shown in . On top of the sample, a small piece of thin aluminum sheet was placed to protect it from being damaged. The core of a linear variable differential transformer (LVDT) displacement sensor was supported by a thin piece of stainless steel wire standing on the aluminum sheet. The LVDT coil was mounted on the outside of a piece of 6.35 mm high-pressure tubing connected to the autoclave lid and sealed at the upper end. This allows the axial extension or compression of the gel to be measured without the need for electrical connections to the interior of the pressure vessel. The wire and core together weighed less than 3 g, exerting a stress of about 1 kPa on the gel cross-section. This caused a bias axial strain of −0.1%, small enough to be neglected. The autoclave was heated with a water pipe coil connected to a recirculator running at 55 °C. After sealing the autoclave, the input control valve was pulsed open for less than a second at a time and pressure and strain data were logged continuously at a rate of 9 Hz. The sample deformation typically relaxed within about 10 s; after about 20 s the next step was initiated. Between pressure steps, the temperature in the vessel was also recorded by means of a thermocouple. This was repeated until the pressure in the vessel had reached the maximum pressure of 10.2 MPa, then the exhaust valve was used to control pressure steps of opposite direction until the autoclave returned to ambient pressure. Only the pressurization data are used in the following analysis, because the sample cracked during depressurization. In total, about 240 steps were recorded; each step was fitted separately. The compressibility and viscosity of CO2 were calculated from the equations recommended by Vukalovich and Altunin In this section we analyze the deformation of a cylindrical porous body as a result of changes in the pressure in the surrounding vessel. The approach, which closely follows that of The constitutive equation for an isotropic, saturated porous body can be written in the following form:In cylindrical coordinates, the total stresses (which represent the sum of the forces on the solid and liquid phases) are σr, σθ, and σz; the corresponding strains are εr, εθ, εz. The pressure in the fluid is PF; Kp=Ep/[3(1−2νp)] is the bulk modulus and εs is the spontaneous baseline strain. We found that there was a reproducible strain of the gel, shown in , that depended only on the pressure. The origin of the strain is not certain, but the dried gel had a noticeable odor of ethanol, so we suspect that the surface energy of the network was influenced by an adsorbed ethanol/CO2 solution. Compliant materials with high surface area typically show significant strains during adsorption The volumetric strain of the gel is given by Eq. (28) of where PA is the pressure in the autoclave and β=(1+νp)/[3(1−νp)]. The angle brackets represent the volumetric average for any function f(r,θ)where u=r/R and R is the radius of the cylinder. The volumetric strain rate iswhere the superscript dot represents the partial derivative with respect to time.The continuity equation is given by Eq. (10) of where ηF is the viscosity and ρF is the density of the fluid; the flux, J, of fluid within the body is assumed to obey Darcy’s law where D is the permeability (units of area). If it is permissible to ignore the spatial variation of ρF within the body, then Eq. This approximation will be justified shortly.which is assumed to be a function only of time (i.e., the spatial variation of density within the sample is neglected). The spontaneous strain, ϵs, is approximated in a narrow pressure interval bywhere ϵ0 and α are constants. With these approximations, Eq. where the reduced time is defined by θ=t/τD, and the hydrodynamic relaxation time is defined asThe cylinder is assumed to be long compared to its radius, so that axial flow can be neglected; therefore Eq. For the sample used in the experimental study, Kpα≈0.02, so it is a good approximation to neglect the terms involving α. This reduces Eq. The initial condition in the experiments does not correspond to equilibrium, because the pressure is drifting upward or downward before and after each jump. For the moment, though, let us assume that the initial pressure in the sample, PF(u,0), is equal to that in the autoclave, PF(u,0)=PA(0). The boundary condition is PF(1,θ)=PA(θ). Subject to these conditions, Eq. is solved by applying the Laplace transform with respect to θ, which is defined by where s is called the transform parameter. The solution of Eq. The Laplace transform of the relaxation function φ with respect to θ isHere, Bn is a root of the Bessel function of the first kind of order zero, J0(Bn)=0. The transform of h is must be inverted numerically; this was done using the Stehfest algorithm in the Numerical Inversion package in Mathematica® when θ approaches zero, φ is a well-behaved function that is closely approximated byThis approximation provides the correct behavior as . The exact function and this approximation are shown in Returning now to the question of the initial condition in the experiments, let us see what the steady-state pressure is inside the sample when the pressure changes continuously. If Therefore, if the pressure at the start of the experiment (i.e., just before the jump) has been changing steadily at a rate corresponding to A, the initial condition isWith this condition, the solution of Eq. constitute the central result of this paper. Experimental data for the axial strain of the sample are fit to these equation with τD as the only free parameter; using an independent measurement of the elastic modulus, the permeability is found from Eq. If an instantaneous step change in pressure of ΔPF is made, then Eq. indicates that the instantaneous strain will beThe preceding analysis can be used for any material that is sufficiently compliant to allow strains of this magnitude to be measurable for practical values of ΔPF., along with a polynomial fit. For the calculations, piecewise polynomials were used over pressure intervals of 0–2, 2–4, 4–6, 6–8, 8–9, and 9–10.3 MPa., which can be written in terms of the fluid pressure, as follows:where CF is the compressibility of the fluid. For an ideal fluid, CF=1/PF, and shows that this relationship holds true for CO2 at pressures up to about 2 MPa. If the fluid is ideal, thenwhere n=number of moles, Mw=molecular weight, V=volume, and Rg=ideal gas constant. In this case, Eq. Fits to all the data were repeated using Eq. The load relaxation obtained in the beam-bending experiment is shown in . The fit indicates that Poisson’s ratio is 0.20, Young’s modulus is 1.87 MPa, so the bulk modulus is 1.0 MPa, and the permeability is D∞=13.5 nm2.The acoustic velocity was 170 m/s and the bulk density of the sample was 207 kg/m3, so the longitudinal modulus was c11=Kp/β=5.97 MPa. Since νp=0.2, then β=0.5, and Kp=3.0 MPa.. The fits were performed separately on each pressure jump, using the procedure explained in . The pressure steps were relatively small, so the viscosity of the fluid, ηF, was treated as constant within each data set, as were the parameters α and τD. The quality of the fits, illustrated in , is generally excellent for all pressures up to about 6 MPa. At pressures approaching the critical point of CO2 (7.5 MPa), the calculated compressibility is unreliable, so those data were not satisfactorily fit. The fits (not shown) were equally good when the fluid properties were assumed to be ideal; that is, when The permeability of the aerogel can be calculated from the fitted values of τD using Eq. together with the known properties of the fluid and the results of acoustic measurements on the aerogel (Kp=3.0 MPa and ν=0.2). The permeability is predicted to depend on the pressure in the fluid according to This equation is based on a simplified analysis of diffusion in a cylindrical capillary tube with radius rc. The average distance from the wall at which the last intermolecular collision occurs is cλ, where λ is the mean free path of the gas; it is expected that c≈1. The mean free path is related to the diameter of the molecule, σ, by ; the pressure is adjusted by atmospheric pressure, Patm=0.1 MPa, because the autoclave records pressure relative to Patm. For all the data in the range , the fit yields D∞=8.90±0.17 nm2 and k=2.76±0.07 MPa. The molecular diameter of carbon dioxide obtained from viscosity measurements indicates that the mean pore size of the aerogel is rc≈13 nm.Another estimate of the pore radius, rp, can be obtained from the liquid permeability using the Kozeny equation where κw is called the Kozeny constant, but is actually a weak function of relative density indicates that rp=9.5 nm, which is about 30% smaller than rc.Data in the pressure range below 2 MPa were fit assuming ideal behavior for the fluid, leading to the results shown in . For this range, the fit yields D∞=7.3±1 nm2 and k=3.5±0.6 MPa, so Eq. indicates that the mean pore size of the aerogel is rc≈9–12 nm; from D∞ we estimate rp≈8–9 nm. Thus, the assumption of ideality yields reasonably accurate results.Before supercritical drying, the permeability of the wet gel was measured using the beam-bending method indicate that the pore radius in the wet gel is rp≈11 nm. After supercritical drying the gel had contracted so that its density increased to ρ=0.10 (so κw≈2.3). Previous studies where ρs is the skeletal density; for silica aerogels, ρs≈2.0 g/cm3. Given that the pore volumes of the gel and aerogel are 6.85 and 4.5 cm3/g, respectively, the pore radius in the aerogel is expected to be rp≈7.3 nm. In view of the large adjustment in the pore volume, this compares favorably with the value (9.5 nm) found using D∞ from indicates that the permeability of a wet gel with the structure of the aerogel (rp=7.3 nm, ρ=0.1) should be D∞≈5.3 nm2, which is reasonably close to the value (8.9 nm2) found from The BET surface area of the aerogel was found to be S=920 m2/g. Using the pore volume of Vp=4.5 cm3/g, the hydraulic radius is found to be rh=2Vp/S≈9.8 nm.which yields rc≈12 nm, using the data from owing to deformation of the apparatus; however, the values extrapolated to zero pressure were valid, and those values yielded a pore radius that was about 70% of the hydraulic radius and 40% of the half-chord length from SAXS. Similarly, in By measuring the linear strain of an aerogel during depressurization of the autoclave, it is possible to determine the permeability as a function of the fluid pressure. A simple modification of the autoclave (viz., installation of an LVDT in the pressure chamber) allows this measurement to be done routinely during supercritical drying. The analysis involves fitting Eqs. to the strain data with a single free parameter, the characteristic time, τD. We have performed this calculation using data obtained for a silica gel with a relative density of 0.10, using the true properties of fluid CO2 or treating the fluid as an ideal gas. The fits are excellent and the results of the two calculations are surprisingly similar. The dependence of permeability on pressure obeys Klinkenberg’s model; moreover, the permeability obtained by extrapolating the data for the fluid to infinite pressure agrees reasonably well with the permeability measured in liquid. Comparable estimates of the mean pore radius are obtained, based on the hydraulic radius (9.8 nm), liquid permeability (7.3–9.5 nm), and gas diffusion (12–13 nm).There are two extreme possibilities as to which modulus is appropriate for the fluid. If the pressure changes during the experiment are slow compared to the thermal relaxation time of the sample, the isothermal bulk modulus of the fluid has to be used. On the other hand, if the processes in question are fast enough to ensure thermal isolation of the sample – no heat is exchanged between the sample and the surrounding fluid on the time scale of the experiment – then a composite adiabatic modulus is appropriate. This is a composite modulus, instead of the pure adiabatic modulus of the fluid, because the fluid locally exchanges heat with the skeleton (the thermal relaxation time of heat transfer between the pore fluid and the skeleton is very short, of the order of ns). In this case, the bulk modulus of the fluid can be calculated as follows. The general definition of the bulk modulus isAn adiabatic process implies dQ=dU−dW=0, that iswhere Cv is the combined heat capacity of a certain volume containing skeleton and fluid. It is given bywhere Cvf and Cvg are the specific heats of the fluid and the skeleton, respectively and mf and mg are the corresponding masses in the volume. From we can deduce the connection between temperature and pressure changes for adiabatic processeswhich allows us to write the adiabatic bulk modulus asHere we have introduced the mass density of the fluid ρf and the apparent density of the aerogel, ρg. This is a general expression that can be used to calculate the composite adiabatic bulk modulus of the fluid in gel pores from the equation of state of the fluid, which enters through the derivatives of p and the fluid density. Note that this is also the modulus that determines the contribution of the fluid to the sound velocity. Using the ideal gas equation of state, Eq. becomes identical to the modulus derived for that purpose earlier is position-dependent within the gel, but it is reasonable to neglect this as long as the temperature and pressure variation within the gel is small. For the numerical fitting in If the thermal relaxation time is of the same magnitude as the hydrodynamic relaxation time, then the thermal diffusion problem must be introduced into the analysis. This is not necessary here, since the thermal relaxation is about a factor of 10 slower than the hydrodynamic relaxation for aerogels.A function y(u,t) can be expanded in a series of Bessel functions asThe Bessel transform of y is defined by leads to an expression for the coefficients an(t):, the following integral leads to a simple result:The measured values of pressure in the autoclave were too noisy to use for numerical differentiation, so they were smoothed with a cubic spline Since the data collection rate was relatively high, it is reasonable to assume that the average fluid pressure, 〈PF〉, varies linearly within each time intervalWith this approximation, the integral in Eq. This function is given to better than 1% accuracy byThe exact function and the approximation are shown in This equation allows the fluid pressure to be evaluated at time θn using values obtained in previous time steps. The sum in zn exists only when n>2. If we number the intervals such that θ1=0, thenAll successive pressures are found from Eq. . The strains are then calculated from Eq. . To test the ideal gas approximation, CF is replaced by 〈PF(θ)〉 in Eq. To calculate the correct value of α for evaluation of Eq. , it is necessary to take account of the portion of the initial strain that results from the steady-state pressure. For that purpose, least-squares lines were fit through the first five points of each data set to find the change in autoclave pressure with time. The slope of this line, dPA/dt, was used to estimate ASimilarly, the slope was found for the last five points in the set and another value of A was obtained, and the average of the two was used for the calculation.To evaluate the baseline strain, we assume that the steady-state conditions exist at the start of the data set, so Eq. Steady-state is assumed to exist at the end of each data set, so another value of α is determined from the final strain value, and the average of the two values of α is used for the fitting.All fits were performed using the experimental values (see Interface damage, bi-axial failure envelopeOn the prediction of the bi-axial failure envelope of a UD CFRP composite lamina using computational micromechanics: Effect of microscale parameters on macroscale stress–strain behaviorA computational micromechanics based Finite Element (FE) analysis methodology is presented to predict the bi-axial failure envelope of a unidirectional (UD) carbon-epoxy composite ply. In order to estimate the effect of various microscale parameters that are influencing the macroscopic stress–strain behavior, under individual load cases, detailed numerical studies are conducted using a 3D RVE (Representative Volume Element) model. The constituent epoxy matrix plastic deformation in the RVE is captured using the linear Drucker-Prager plasticity model.The effect of the fiber–matrix interface damage, followed by frictional sliding of the constituent materials on the computed interface tractions is captured using the cohesive zone damage model combined with the Coulomb friction law, which is implemented into Abaqus using VUMAT. From the detailed FE analysis of the RVE under individual load cases, it is observed that the predicted macroscopic stress–strain behavior is sensitive to the fiber–matrix interface properties as well as the in-situ epoxy stress–strain behavior. Hence, using a coupled experimental-computational micromechanics approach the interface and the in-situ epoxy material properties are calibrated and validated. Using the calibrated interface and in-situ epoxy material properties, the bi-axial (transverse tension/transverse compression – in-plane shear) failure envelope of a UD composite ply is estimated. Comparing the predicted damage profiles and the failure envelope with the experimental results leads to good agreement and validates the proposed numerical methodology.Interface damage, bi-axial failure envelopeAttributed to the excellent specific strength and stiffness Polymer Matrix Composites (PMCs) find extensive structural applications in advanced mechanical industries such as energy, aviation, and aerospace. Unlike other structural materials, PMCs are hierarchical with three distinct length scales namely, microscale, mesoscale, and macro-scale. Pertaining to each length scale, there exist specific damage initiation and propagation mechanism.In general, composite structures are designed to undergo multi-axial load conditions during their service life. Hence, over the years numerous ply/laminate level failure models have been proposed to predict the multi-axial failure envelope. The aforementioned failure criteria can be broadly categorized into stress based The mechanical properties of fiber-reinforced composites depend on the fiber, matrix, fiber distribution, orientation, and void content. Besides the aforementioned properties, at the micro-scale, the nature of the fiber–matrix interfacial bonds and the mechanisms of load transfer between different constituent materials play an important role Attributed to the above mentioned complex microstructural parameters, phenomenological or macro-scale damage models cannot accurately describe the complex failure behavior of Fiber Reinforced Polymer (FRP) matrix composites The above mentioned research work has shown the effect of RVE size, matrix plasticity model, and various fiber/matrix interface parameters on the predicted stress–strain and damage behavior of the UD FRP composite plies. Even though several researchers tried to predict the ply level stress–strain behavior and the bi-axial failure envelopes of a UD composite ply using computational micromechanics, often these results are not directly compared (one-on-one) with the experimental results. This can be attributed to the fact that there is a lack of reliable fiber/matrix interface data in the open literature. Moreover, related to the matrix material behavior, as stated by Fiedler et al. In general, for the micromechanical FE simulations, the bulk epoxy material properties are used Similar to the aforementioned observations, based on the microcompression tests conducted using epoxy micropillars, Wang et al. Considering the goal of the current numerical study i.e. estimating the bi-axial failure envelope of a UD composite ply, while understanding the influence of microscale parameters on the macroscopic stress–strain behavior, the following numerical methodology is employed. i) an RVE model is generated using the fibers distribution and the placement of the fibers observed in a microscopic image of an actual UD composite ply; ii) initially, Abaqus built-in cohesive zone module is used to capture the fiber/matrix interface damage initiation and propagation. However, under the applied transverse compression load, to facilitate the load carrying capability of the fiber and matrix surfaces after the interface element damage, friction combined with a cohesive damage model In order to accomplish the micromechanical FE analysis, an RVE model with random fiber distribution needs to be created. The chosen fiber placement in the RVE must be statistically representative of the fiber distribution that is observed in an actual composite ply. Also, the chosen RVE should be large enough so that the average properties of this volume element are independent of its size and position within the material a, a section of a composite ply is selected in such a way that the local fiber volume fraction in the chosen field of view is similar to the macroscopic fiber volume fraction (60%) of the composite.b, the obtained non-periodic fiber distribution is converted into a periodic microstructure using the following procedure. The fibers highlighted in red circles (b) belong to the periodic microstructure. However, the edge fibers marked by the blue circles (B1, B2, and T1) hinder the periodicity of the microstructure. Hence, to facilitate the periodicity, corresponding to B1, B2, and T1 fibers, B1′, B2′, and T1′ are created on the opposite face of the RVE (b). Finally, the fiber marked by the yellow circle is removed from the selected field of view. Consequently, the generated RVE CAD model (c) is a perfectly periodic microstructure. Here it should be emphasized that the above mentioned RVE CAD model generation process is completely automated with the help of a python script. For the aforesaid python program, the input data need to be provided in the form of individual fiber centroids as well as the fiber diameter.d, the generated RVE model has a square shape with an equal length and width of 53.4 µm, and a thickness of 1.65 µm. From the RVE thickness point of view, previous studies of Totry et al. In order to accomplish the micromechanical simulations, the RVE geometrical model is translated into an FE mesh using continuum 3D elements in Abaqus. The FE mesh of the RVE model (d) consists of a majority of C3D8R elements along with few C3D6R elements. The generated RVE model consists of IM7 carbon fiber as reinforcement and the epoxy matrix (Hexcel 8552) as a binder material. For the RVE modeling purpose, the carbon fiber diameter is taken as 6.6 µm c). Moreover, the application of PBCs to the RVE model is automated with the help of a MATLAB script. Here it is important to emphasize that the authors aware of the shortcomings of the applied PBCs to the RVE model such as predicting the higher crack density compared to the experimental observations. However, comparing the predicted average stress–strain curves obtained using PBC and uniform displacement boundary conditions The fibers in the RVE do not contribute to the transverse damage, hence, the fibers are considered as elastic and transversely isotropic materials (). Consequently, the following sections present a detailed description of the material models used for the interface and the epoxy matrix.For simulating the damage initiation and subsequent damage propagation of the interface, three dimensional eight noded cohesive elements (COH3D8) having a thickness of 0.1 µm were inserted between the fiber and matrix in the RVE (d). The constitutive response of these elements is governed by the bi-linear traction-separation law For the current numerical studies, a cohesive zone model that consists of interface damage combined with the Coulomb friction law is implemented into Abaqus/explicit using VUMAT as proposed by In the above equation tn, t1, and t2 are the normal (Mode I) and shear (Mode II & Mode III) stresses at the interface, while the corresponding displacements are denoted by δn, δ1, and δ2 respectively. Besides, Kn, K1, and K2 are the values of interface stiffness corresponding to Mode I, Mode II, and Mode III respectively. Here, it should be mentioned that for the current numerical studies the values of normal and shear stiffness are assumed to be equal (i.e. Kn = K1 = K2 = K). The interface damage is assumed to be governed by the effective mixed mode displacement λ which is defined by the following expression λ=K1δ12+K2δ22+Kn〈δn〉2K1δ12+K2δ22+Kn〈δn〉2For interface damage initiation, the damage activation function F(λ) is defined as follows As soon as the above mentioned damage activation function reaches zero, the interface damage will be initiated. In the above equation ρd is a threshold function, and H(λ) is a function of effective displacement which evolves monotonically after damage initiation. These functions are updated at every time step ‘t’ during the numerical analysis, which are defined as follows:, λ0 and λf are the effective mixed mode displacement jumps corresponding to damage initiation and final failure of the interface respectively (b). Whereas λ is the instantaneous effective displacement computed at every time step during the analysis. Using the quadratic nominal stress criterion λ0=δn0δ101+α2(δ20)2+(αδn0)2,δn>0(δ10)2+(δ20)2,δn≤0where, α = mode-mixity ratio =(δ1)2+(δ2)2δnIn addition, λf is calculated using the Benzeggagh-Kenane (BK) mixed-mode fracture criteria λf=2Kλ0Gnc+Gsc-Gncα21+α2η,δn>0(δ1f)2+(δ2f)2,δn≤0In the above equation, Gnc is the critical normal fracture energy and Gsc is the critical shear fracture energy. Moreover, η represents the BK power-law coefficient and is taken as 1.45 In the above equation, τu represents the undamaged part of the interface stresses. Whereas ‘D’ is a scalar damage variable, which evolves monotonically from zero (no damage) to one (complete damage) and is defined as follows can fully estimate the damage initiation and evolution at the fiber–matrix interface under mixed mode load conditions. However, under specific load cases such as transverse compression, to capture the effect of friction between the fiber and matrix surfaces on the interface stresses, the above discussed cohesive zone model should be combined with a proper friction law. According to Alfano et al. a) can be divided into a damaged (τd) and an undamaged part (τu). Also, the relative displacements corresponding to both the parts across the REA are assumed to be equal and constant (δu = δd = δ) In the above equation the first term [1-Dτu] defines the cohesive damage law, while Dτddenotes frictional stresses on the damaged interface. Similar to the assumption of REA interface stresses, the displacements corresponding to the damaged part of the REA is divided into an elastic (δde) and inelastic (δdi) (Eq. ), with δdi corresponding to the frictional sliding.The following constitutive equation relates the displacements (δde=δ-δdi) to corresponding tractions (τd) during the softening phase (Here, it should be noted that the utilization of the Macaulay bracket in the above equation implies that compressive normal displacement does not affect the interface damage model. In the above equation, δ1di and δ2di represent the inelastic sliding occurring due to friction (b) in the damaged part of the REA. The evolution of inelastic frictional sliding is governed by the following equation ‘ϕ’ is a friction function and λ corresponds to the displacement jump defined previously by Eq. . The friction function ϕ is defined as follows:Finally, the constitutive equation for the interface damage combined with the Coulomb friction law takes the following form:τ=1-DK1000K2000Knδ1δ2δn+DK1000K2000Knδ1-δ1diδ2-δ2di-〈-δn〉Summarizing the above described damage methodology, c shows a detailed flow-chart of the implemented interface damage model combined with the Coulomb frictional law. In order to validate the implemented VUMAT code, initially, the predicted numerical stress–strain curves without frictional effects are compared to the stress–strain curves obtained using Abaqus built-in cohesive zone model.It is a well documented fact that the yield behavior of the polymeric materials is sensitive to the hydrostatic pressure. As a consequence, the yield stress in tension becomes different from the yield stress in compression where p is the hydrostatic stress, β is the slope of the linear yield surface in p-t stress plane, d is the cohesion strength, r is the third invariant of the deviatoric stress, q is the von Mises equivalent stress, t is the flow stress and K is the ratio of yield stress in triaxial tension to triaxial compression. In the above equation, the K value accounts for stress path variation of shear strength under given hydrostatic pressure and determines the shape of the yield function in the deviatoric stress plane. Apart from q and K, the flow stress ‘t’ consists of the Lode’s angle as a ratio of the third to the second invariant of the deviatoric stress For the current FE simulations, the input material properties for the Drucker-Prager plasticity model, such as β and K values are calculated using the relation between the Mohr-Coulomb and the Drucker-Prager yield criteria. According to the Mohr-Coulomb criteria, the cohesion stress c (c≠d) and the angle of internal friction (ϕ) are related to the tensile (σmt) and compressive strengths (σmc) of the material by the following expression Here it should be noted that, in the absence of the neat 8552 epoxy (Hexcel) stress–strain curves, for the current numerical study, bulk epoxy stress–strain curves are used from a similar aviation grade material (8551–7 from Hexcel) , the internal material friction angle ϕ and the cohesion yield stress c of the epoxy were obtained as 7.78° and 56.72 MPa respectively. As an additional validation, the computed cohesion stress value is compared to the experimental shear strength of the epoxy (57 MPa) With the Mohr-Coulomb internal material friction angle (ϕ), the slope of the yield surface (β) for the Drucker-Prager plasticity model is computed using the below equation The above equation gives the β value of 15°. Similarly, the ratio of the triaxial tension to compression (K) is computed as 0.89 using the below equation Using the above discussed yield criterion, the matrix material starts to deform plastically when the yield surface is reached. Further loading produces the plastic flow in the epoxy material. Under the Drucker-Prager plasticity condition, the plastic flow (G) is defined by the following equation where ψ represents the dilatation angle in the p-t stress plane. Under the conditions of the nondilatant flow rule, the ψvalue is set to zero. This is due to the fact that volume changes in the yielding or post-yielding regimes have been reported to be minor for polymer materials Using the above discussed RVE modeling methodology, to avoid convergence issues caused by the interface damage and the matrix plastic deformation, the following FE simulations are accomplished in Abaqus / explicit using the direct integration method. In order to obtain a thorough understanding of various explicit solver parameters that can influence the outcome of the quasi-static computational micromechanical results, readers are referred to the recent research work of Garoz et al. For calibrating the chosen element for the FE mesh and the plasticity model for the matrix, bulk epoxy material FE simulations are accomplished under tension, compression, and shear loads. For the computational purpose, a cubic geometry (1 mm3) is modeled using C3D8R elements. Moreover, kinematic (displacement) boundary conditions are applied to the FE model (), the following input data is provided to the Drucker-Prager plasticity model:i) within the linear elastic limit, Young’s modulus and Poisson's ratio listed in are used; ii) beyond the elastic limit, true stress vs. true plastic strain values are derived from the experimental stress–strain curves b, under various load conditions, the predicted stress–strain behavior matches very well with the experimental In order to understand the effect of various material properties influencing the transverse tensile stress–strain and damage behavior of an RVE, the following important aspects related to the micromechanical FE simulations are addressed: i) initially, a parametric study is conducted to calibrate the strength and fracture energy values used for the fiber–matrix interface; ii) using the calibrated interface properties, a detailed damage analysis is conducted for the RVE under transverse tension. Here it should be highlighted that the above mentioned numerical studies are accomplished using bulk epoxy tensile stress–strain curve (b); iii) finally, in order to estimate the effect of in-situ epoxy material properties on the predicted transverse tensile strength, RVE FE simulations are accomplished using in-situ epoxy material properties obtained from the nanoindentation testing on a carbon-epoxy composite ply Attributed to the difficulties in obtaining the interface normal strength and fracture energy values (mode I) using simple experimental testing, reliable mode I interface data is not available in the open literature. Hence, in the current research work, the mode I interface properties are evaluated through reverse engineering based on correlating the obtained RVE average stress–strain behavior to the experimental macroscopic stress–strain behavior As explained in the above section, the cohesive elements that are used for representing the interface mechanical behavior follow the bi-linear traction separation law. Hence, in order to ensure the continuity of the stress–strain fields in the linear elastic region, an initial stiffness of 106 MPa/mm is assigned to the cohesive elements. For the parametric study of the interface properties, it is assumed that the transverse tensile strength of a composite ply is mainly influenced by mode I strength and fracture energy values compared to mode II interface properties a). The initial guess value for the mode I fracture energy is taken as 0.04 N/mm. In addition, mode II interface strength and fracture energy values for the initial calculations are taken as 96 MPa a, for the chosen mode I fracture energy (0.04 N/mm) the interface strength value of approximately 60 MPa predicts the transverse tensile strength of the RVE that is in close agreement with the experimental strength. During the parametric study, it is observed that the transverse tensile strength and failure strain of a UD composite ply is a combined effect of both mode I interface strength and fracture energy values. Hence, in order to calibrate the mode I fracture energy, the interface strength value is varied around 60 ± 5 MPa and the mode I fracture energy value is varied from 0.001 N/mm to 0.02 N/mm. From the aforementioned parametric study, the mode I interface strength of 57 MPa and fracture energy of 0.002 N/mm predict the average transverse tensile strength and failure strain of the RVE that is in good agreement with the experimental stress–strain curve. As shown in b, keeping the calibrated mode I interface strength of 57 MPa constant, the effect of interface fracture energy on the transverse tensile stress–strain behavior is shown. From (a, b) one can observe that the transverse tensile strength and failure strain of an RVE is sensitive to the chosen mode I interface strength and fracture energy values.The current section is devoted to present a detailed micro-scale damage analysis of the RVE under the applied transverse tensile load. Firstly, as shown in a, a very good correlation is observed between the average stress–strain curves predicted using Abaqus built-in CZM to the cohesive zone damage model implemented using VUMAT. During the transverse tensile load application process, initially, at around 54 MPa average stress the interface damage initiation is observed (b). As expected, the observed interface damage is mainly mode I dominant (b). The initiation and propagation of the interface damage accelerate the plastic deformation of the epoxy strips that are bridging the neighboring interface decohesions (c). As soon as the local equivalent stress in the epoxy strips that are connecting the neighboring interface damage reaches approximately 100 MPa (c), RVE strain softening behavior is triggered. The aforementioned stress value corresponds to the tensile yield limit of the epoxy matrix (b). The initiation of the strain softening process increases the epoxy strips plastic deformation and leads to the final failure of the interface elements (For validating the current numerical methodology, the obtained FE interface damage profile (a) is compared to the experimental micro-scale damage profile ((a, b) compares the obtained FE circumferential decohesion angles (54–70°) to the experimental values (64–70°) and a good correlation is observed. In addition, as reported by Paris et al. The above discussed micromechanical FE simulations are accomplished using the bulk epoxy non-linear stress–strain behavior obtained from the macro-scale experiments. Hence, implicitly it is assumed that the constituent matrix material of the RVE behaves similarly compared to the bulk epoxy material. “While it is unlikely that the fiber properties would change, there is a distinct possibility that the polymer properties could chemically change as a result of consolidation and/or cure process”. Moreover, apart from the process induced effects, regarding the size effects, the micro-scale epoxy fiber tests conducted on RTM 6 resin (Hexcel) by Hobbiebrunken et al. ). The above discussed failure process of the epoxy matrix is completely different from the brittle fracture behavior that is commonly observed from the macro-scale tension tests compares the in-situ epoxy matrix material properties measured from nanoindentation tests . Considering the in-situ epoxy material properties (), RVE FE simulations are accomplished under transverse tensile load and the obtained results are compared to the average stress–strain curve predicted using the bulk epoxy material properties. Due to the non-availability of the non-linear stress–strain curve for the in-situ epoxy matrix, elastic-perfectly plastic material behavior is assumed. As shown in , the predicted transverse tensile strength of the RVE using in-situ epoxy material properties is in reasonable agreement with the strength value predicted by the RVE using bulk epoxy non-linear stress–strain curve. Attributed to the elastic-perfectly plastic material properties used to represent the in-situ epoxy material behavior, the predicted stress–strain curve has a higher slope and hence slightly lower failure strain compared to the experimental results.From the above discussed comparative study, one can observe that even though the in-situ epoxy matrix tensile strength is 22 MPa higher than the bulk epoxy tensile strength, its influence on the transverse tensile strength of the RVE is negligible. The aforementioned phenomenon can be attributed to the fact that, once the interface damage initiates, the epoxy strips that are bridging the neighboring interface damage act as a load transfer material between the two halves of the RVE (a). Attributed to the size of these epoxy strips, even a minuscule increment in the applied external load causes the stress concentration to increase sharply and reaches the tensile yield limit of the epoxy, and consequently triggering the RVE strain softening behavior. Hence, from the above discussed interface as well as the matrix strength studies, it can be concluded that the transverse tensile strength of a UD composite ply is mainly controlled by the brittle failure behavior of the fiber–matrix interface In order to understand various microscale parameters that are influencing the average compressive stress–strain behavior of an RVE, detailed numerical studies are conducted to address the following aspects: i) due to the availability of experimental mode II interface values for carbon-epoxy composite In the process of estimating the transverse compression stress–strain behavior of a UD composite ply using micromechanical simulations, initially, RVE FE analysis is accomplished using bulk epoxy non-linear compression stress–strain curve (b). Regarding interface material properties, already calibrated mode I strength (57 MPa) and fracture energy (0.002 N/mm) values are kept constant. The mode II interface strength value is taken from the experimental micro-bond tests conducted on an IM (Intermediate Modulus) carbon–epoxy (Hexcel) material system a, the implemented cohesive zone damage methodology using VUMAT predicts similar stress–strain behavior compared to Abaqus built-in CZM. Moreover, one can observe that the predicted compressive strength of the RVE is approximately 40% lower compared to the experimental value (a). A detailed study of the local stress–strain and damage profiles leads to the following observations: i) at around 130 MPa average compressive stress, the interface damage initiates. As observed in the microscale transverse compression experiment of a UD CFRP composite ply b), the current numerical methodology predicts the plastic strain localization in the epoxy matrix around the fiber–matrix interface damage (c); ii) followed by the interface damage, at around 150 MPa average compressive stress the accumulated plastic strain in conjunction with the interface damage started to form a dominant shear band; iii) finally, in accordance with the epoxy matrix internal material friction angle (∅2≅40) combined with the RVE local fibers placement, the plastic shear band forms across the RVE at an angle of 490 perpendicular to the load application direction (a). Moreover, the predicted plastic shear band angle is within the limits of the experimental shear band formation angle (θc=53±30) observed from the transverse compression tests conducted on a UD carbon-epoxy composite ply (a, the discrepancy between the predicted and the experimental compressive stress–strain curves can be attributed to the following two aspects of the constituent materials of the RVE: i) in-situ epoxy stress–strain behavior; ii) friction between the fiber/matrix interface. Consequently, in order to investigate the effect of epoxy material properties, RVE FE simulations are accomplished using in-situ epoxy elastic-perfectly plastic material data (a, the predicted compression strength of the RVE using in-situ epoxy compression strength (176 MPa) without friction is approximately 215 MPa. Considering the experimentally reported variation in the in-situ epoxy strength (176 ± 17 MPa), the friction coefficient between the fiber–matrix interface is varied from 0 to 0.4. As shown in b, the in-situ epoxy strength of 183 MPa combined with the interface friction of 0.4 predicts the RVE compressive strength that is in good agreement with the experimental strength.From the above discussed transverse compression simulations, it can be concluded that the in-situ compression strength of the epoxy matrix directly controls the shear band formation stress levels, and consequently influences the predicted average transverse compression strength of the RVE b, attributed to the assumed elastic-perfectly plastic material behavior for the in-situ epoxy matrix, the overall stress–strain behavior of the RVE is not in good agreement with the experimental stress–strain curve. In the absence of the experimental non-linear stress–strain curve for the in-situ epoxy matrix, the above discussed FE simulations are accomplished using elastic-perfectly plastic in-situ epoxy data. Hence, the in-situ non-linear compression stress–strain curve for the epoxy matrix is estimated by matching the RVE average stress–strain response to the experimental compressive stress–strain response of the UD composite ply. Using mode II interface strength of 96 MPa and fracture energy of 0.107 N/mm, along with the estimated in-situ epoxy compression stress–strain curve (c), the predicted average compressive stress–strain behavior of the RVE (d) is in reasonable agreement with the experimental results. Finally, combining the estimated in-situ epoxy curve (c) with a friction coefficient of 0.4, the predicted RVE compressive strength and failure strain are in good agreement with the experimental results. As shown in d, incorporating the friction coefficient increases the compression strength of the RVE by approximately 13 MPa. Finally, as shown in c, one can observe that the predicted in-situ compression strength value of the epoxy matrix is approximately 60% higher than the bulk epoxy compression strength, which is in line with the projected compression strength value for in-situ epoxy As discussed above, the RVE ductile stress–strain behavior under the applied transverse compression load is not only influenced by the in-situ matrix properties but also depends on mode II interface strength and fracture energy values. Hence, using the predicted in-situ epoxy stress–strain curve (c), a detailed parametric study is conducted. For the current numerical studies, mode II interface strength value is varied from 40 MPa to 120 MPa, while keeping the mode II fracture energy constant at 0.107 N/mm. As shown in a, the interface strength value has a significant influence on the predicted failure stress, strain, and the overall ductile stress–strain behavior of the RVE. Moreover, from a one can observe that the interface strength values above 100 MPa predict almost similar stress–strain response. Besides the mode II interface strength, interface fracture energy can influence the average stress–strain behavior. In order to investigate the influence of interface fracture energy on the average compressive stress–strain behavior, the experimentally measured interface strength of 96 MPa is kept constant, and the interface fracture energy value is varied from 0.05 N/mm to 0.2 N/mm. As shown in b, except at lower interface fracture energies (<0.05 N/mm), all the predicted stress–strain curves with other fracture energies almost converged to the experimental stress–strain curve. This can be justified using the fact that the transverse compression failure behavior of a UD composite ply is characterized by a limited interface failure combined with the excessive local plastic deformation of the epoxy matrix around the fiber–matrix interface damage locations In order to predict the in-plane shear stress–strain response of a UD composite ply using micromechanical FE simulations, numerically calibrated mode I and experimental mode II interface strength and fracture energy values (a, the predicted in-plane shear strength of the RVE using bulk epoxy shear stress–strain curve is approximately 43% lower than the experimental shear strength. Moreover, one can observe that the predicted RVE average shear strength (57 MPa) (a) is similar to the computed bulk epoxy cohesion strength. Hence, as explained above (RVE under transverse compression), in order to predict the in-situ shear stress–strain behavior of the epoxy, RVE in-plane shear simulations are accomplished by matching the predicted average stress–strain response to the experimental in-plane shear stress–strain behavior. b shows the in-situ shear stress–strain behavior of the epoxy in comparison to the bulk epoxy shear stress–strain behavior. Finally, c shows the comparison between the experimental To achieve the goal of the current research work, using the interface mechanical properties listed in along with the estimated in-situ epoxy stress–strain curves, the RVE is subjected to bi-axial loading in σ22-τ12 plane. Each point in the predicted failure envelope () corresponds to a particular ratio of transverse (tension/compression) to in-plane shear displacement. In addition, the predicted failure envelope is marked based on the observed dominant failure mode under the applied bi-axial load. As shown in , the effect of the interface friction is negligible in the tension dominated region of the failure envelope. However, in the shear dominated failure region, the interface friction coefficient shows a minor influence. In contrast, one can observe that the increase in friction coefficient from 0.2 to 0.4 causes the increase in the bi-axial strength of the RVE in the compression dominated region. Nevertheless, increasing the friction coefficient from 0.4 to 0.6 does not show a significant influence on the bi-axial strength of the RVE. Finally, (a, b) shows the failure profiles of the RVE in the combined transverse tension-in-plane shear and transverse compression-in-plane shear dominated stress space respectively.A comprehensive micromechanics based FE analysis methodology is presented to predict the bi-axial failure envelope of a UD composite ply in the matrix dominated directions. A coupled numerical-experimental methodology is employed to predict and calibrate the interface and in-situ epoxy properties. In order to include the frictional sliding between fiber–matrix interface, cohesive zone damage methodology combined with the Coulomb frication law is implemented into Abaqus/explicit using VUMAT. Under individual load cases, the obtained damage profiles and stress–strain curves are compared to the experimental results. Finally, the thoroughly validated numerical methodology under uniaxial load cases is extended to predict the bi-axial failure envelope of a UD composite ply. Detailed micromechanical studies related to various material parameters that are influencing the strength and failure strain predictions of the RVE lead to the following conclusions:From the above presented numerical methodology, one can observe that the chosen matrix material properties have a minor influence on the predicted transverse tensile strength of the RVE. Hence, using the bulk epoxy non-linear stress–strain behavior as an anchor point, mode I interface strength and fracture energy values are calibrated.Under the applied transverse tensile load, the predicted homogenized strength and failure strain, as well as the brittle failure behavior of the RVE are mainly influenced by mode I interface strength and fracture energy values.In contrast, transverse compression and in-plane shear stress–strain behavior of the RVE is mainly influenced by the in-situ epoxy stress–strain behavior. Hence, keeping the experimental mode II interface strength and fracture energy values as anchor points, the in-situ compression and in-plane shear stress–strain response of the epoxy matrix is estimated.By observing the failure behavior of the RVE under biaxial loads, it can be concluded that under the combined transverse tension - in-plane shear loads, the failure behavior of the RVE is mainly controlled by the mode I interface strength values.Under transverse compression - in-plane shear loads, the yield behavior of the RVE is mainly controlled by the in-situ strength of the epoxy matrix.The friction between the fiber–matrix interface shows a considerable influence on the predicted strength values under biaxial loads, especially, under transverse compression-in-plane shear dominated stress space.In the current computational studies, crack tunneling is not considered. Also, the influence of the random distribution of the interface properties on the predicted stress–strain behavior is not considered. Moreover, the observed experimental final failure in a UD composite ply is a localized phenomenon and not periodic. Hence, along with the advantages of PBCs, one should be aware of the limitations of periodic boundary conditions that are applicable to the RVE models for predicting the strength of a UD composite ply.Finally, after conducting a detailed computational study regarding the influence of various microscale parameters on the macroscopic stress–strain behavior, the future research work in this area should focus on the following aspects: a) understanding the in-situ epoxy yield behavior under shear as well as combined loads; b) experimental evaluation of mode I interface properties.A. Sharma: Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation. S. Daggumati: Conceptualization, Supervision, Investigation, Writing - original draft, Project administration, Funding acquisition. A. Gupta: Methodology, Software, Formal analysis. W. Van Paepegem: 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.Supplementary data to this article can be found online at The following are the Supplementary data to this article:Nowadays, environmental sustainability is a big issue in product development. Industries all over the world are particularly urging for natural materials due to increasing normative asking for a reduced environmental footprint. Such need boosted the interest in natural fiber composites, which are now presented as a viable alternative to traditional materials over a wide spectrum of applications Many works in the literature focused on the numerical modeling of low-velocity impacts on composite structures The works on low-velocity impacts on NFRPs presented in the literature we detail the materials utilized, the relative manufacturing process, and the experimental procedures. Results from the experimental low-velocity impacts are detailed in Section . The experimental results about the impacts on GFRPs are then utilized to validate a numerical scheme whose implementation is detailed in Section and whose results are presented in Section . Conclusions are finally given in Section evidencing that the specific tensile strength and stiffness of GFRP approximately doubles the FFRP ones. However, a FFRP laminate is about 24% thicker than a GFRP of similar weight and such increment has a strong influence on the flexural stiffness and strength. Naming t the thickness of the GFRP laminate, then its bending stiffness is DG=EGt3/12 while the thickness of FFRP with similar weight equals approximately 1.24t and its stiffness is DF=EF(1.24t)3/12 and their ratio is DGDF≈0.8. The bending moments to failure are instead given by MG=σ1Gt2/6 and MF=σ1F(1.24t)2/6, and their ratio is MGMF≈1.8. These rough estimations indicate that under bending loads FFRPs might be stiffer than GFRPs, but the latter is still 80% stronger.Basing on these considerations we choose to perform a comparison between the low-velocity impact response of an E-Glass/epoxy and a Flax/epoxy laminate by manufacturing two panels having similar stacking sequence and areal weight and, therefore, comparable flexural stiffness.Specimens have been manufactured through a light resin transfer moulding (LRTM) process. The LRTM process has been chosen as it allows to produce tailor-made specimens with controlled thickness (hence fiber volume fraction) and a superior surface finishing on both sides. The process shares similarities with the vacuum bagging process, as the resin flows into the dry fibers due to the pressure gradient introduced by the vacuum, but instead of having a flexible vacuum bag, a rigid mould is utilized. The dry fibers are stacked into a mould with a cavity of desired depth (which equals the desired laminate final thickness) and a rigid counter mould pairs with it. The central cavity presents a peripheral channel allowing the resin to easily flow around the perimeter of the laminate, and a central outflow connected to a resin trap. The infusion thus starts from the external perimeter to wet the fibers up to the center. Notably, differently to vacuum bagging, no infusion mesh is utilized. This makes the infusion process very slow, thus limiting the processing materials to low-viscosity and long pot-life resins.A vacuum channel is further milled on the mould to secure the closure, and to prevent air to enter from the sides. As the inflow valve is open to let the resin into the mould the internal vacuum is lost, but the full closure is ensured by the external vacuum channel. To further ensure the full sealing of the mould, two silicon seals have been produced and inserted in the mould: one outer to the central cavity, to ensure no resin flows towards the external vacuum channel, and a second seal in the outer of the vacuum channel, to further ensure the sealing. The mould has been machined appositely for the purpose out of a 500 × 500mm nylon board 25 mm thick through a CNC milling machine, whereby a glass plate 18 mm thick serves as counter mould. Glass is chosen as it allows to visually control the infiltration process. shows an image of the GFRP panel during the infusion process, where the central portion of the mould is white as fibers are still dry. The panel’s final dimension is 310 × 310mm, allowing to realize 6 specimens with dimensions 150 × 100mm, according to ASTM D7136 suggestions. Specimens have been cut out of the panel through CNC milling by utilizing a 2 mm diamond bit.The mould has been carefully cleaned and two applications of PVA have been utilized as release agent. The vacuum has been applied through a pump with a nominal 4 mbar of absolute pressure and the vacuum has been held for more than 30 min prior to infusion. The infiltration process lasted about 30 min for both GFRP and FFRP panels.The two laminates share the same matrix system: an Epoxy L20 resin with 181 EPH hardener (Hexion). The matrix system has a pot life of 190 min and should cure for 24 h at ambient temperature followed by 15 h at 60°C to reach a nominal Tg of 120°C. We followed such curing cycles as suggested by the distributor, but we comment that DSC analyses revealed a Tg of 82°C instead of the 120°C declared by the manufacturer. However, this is in line with other experimental reports in the literature Quasi-isotropic laminates have been produced utilizing the stacking sequences: [0,90,45,−45,45,−45,90,0]s for the GFRP and [0/90,±45,0/90,0/90,∓45,0/90] for the FFRP. The GFRP has been manufactured utilizing UD layers with an areal weight of 220 gr/sqm. We comment that the so-called UD layer is not perfectly UD as it sows a 10% of fibers in the weft direction. Flax fibers market do not offer a large choice and we had to stand the only areal weight available to consumers, that is a 600 g/sqm for the biaxial ±45 non-crimp fabric, and 300gr/sqm for the 0/90 twill layer. The final thickness and weight of the specimens are 3.0 mm (80 gr) and 3.7 mm (75 gr) for the GFRP and FFRP panels, respectively.Low-velocity impacts have been conducted through a drop-weight apparatus according to ASTM D7136 standard shows a simplified image of the drop weight apparatus reporting the basement, the specimen, the four lever clamps, the impactor, and the laser beam sensor. The impact height is varied in the range 30–300 cm to obtain the impact energies of 5 J, 10 J, 15 J, 20 J, and 30 J for the GFRP, and 3 J, 5 J, 10 J, 15 J, and 20 J for the FFRP specimens.The minimum impact energy has been chosen as a function of the delamination threshold load, which has been in a first approximation predicted utilizing Davies and Zhang where E is the homogenized flexural modulus of the laminate and GIIc is estimated in 0.44 N/mm, which leads to a delamination threshold load of approximately 2.0 kN. We thus modeled the impact through a spring-mass system as suggested by It is constantly reported in the literature that low-velocity impacts on composite structures are very repeatable. Therefore, a single specimen has been tested at each impact height. However, we performed two repetitions of the impact from the height of 100 cm to assess the intra-specimens variability. The two tests showed extremely similar results both in terms of impact dynamics and in terms of visible damage. The results of a single test are thus reported in the following.The impact dynamics for the five impacts in terms of force over time, force–displacement curve, and absorbed energy over time are reported in . The change in damage mechanism influences the impact dynamics in terms of force and displacement over time. The smooth shape of the time traces of the impact forces presented in indicates that no major damage is introduced within the specimen, except for the one at the higher impact energy, which shows a force deviating from the sinusoidal trend as it exceeds 6.5 kN, to decrease with some high-frequency oscillations. Such behavior should be ascribed to fiber breakage. The force–displacement curves show a sharp corner as the impactor reaches the maximum penetration depth, that is, the force reaches its peak when the displacement is maximum. Such trend varies at the highest impact energy, since it instead shows a plateau with increasing displacement at an almost constant force of 6.5 kN, indicating that substantial damage is being introduced within the laminate at such a load level. The unloading curve is well separated from the loading curve further suggesting that a larger amount of damage has been introduced, as the area below the curve indicates the energy absorbed by the material. As a consequence, the penetration at the end of the impact shows a marked increase at such an energy level.The most relevant information which can be gained from the curves presented in . Nominal impact energies are utilized as headings as these will be utilized in the following of the article when referring to a given test. The actual impact velocities and energies do not perfectly match with the ones which can be estimated from the nominal impact height due to air and rail frictions. All the values reported in except for the headings are therefore the measured ones and not the nominal ones.The absorbed energy EA is calculated by subtracting the residual or rebound energy ER from the impact energy EI calculated from the impact and rebound velocities, which are measured through the laser beam sensor. The displacement over time is computed through numerical integration of the impact force P(t) asa(t)=g-P(t)M,v(t)=v0∫0ta(τ)dτ,δ(t)=∫0tv(τ)dτbeing g the gravity acceleration, M the falling mass (1.3 kg), and v0 the initial velocity. It is important to state that the residual indentation reported in , which refer to the impactor displacement at the end of the impact does not necessarily correspond to permanent indentation as plate dynamics cannot be neglected.Notably, the energy absorbed during the impact through material damage appears not to be influenced by the impact energy, being approximately 25% of the total. Such value shows a sudden peak at the higher impact height, where almost 50% of the energy is absorbed by the material. Indeed, such behavior should be ascribed to a variation of the damage mechanisms: while at the lower impact energies matrix cracking and delamination dominates (with minor fiber kinking on the impacted layer), major fiber breakage is attained on the rear face at the higher drop height.Differences in the damage mechanisms can be easily gained by a visual inspection of the specimen, being the GFRP material translucent. The pictures of the front and back sides of the impacted specimens are reported in . The visible damage increases with the impact energy. All specimens are characterized by a highly damaged region right below the contact surface, and visible internal delaminations extend from the impact point. These are self-similar in shape but increase in size with the impact energy. On the impacted side, minor matrix cracks and some fiber kinking arise at the highest impact energies. Overall, the major damage locates in the rear face, where exposed fibers are visible even at the lowest impact energy, and such kind of damage increase with the impact speed. The presence of exposed fibers should be ascribed to matrix cracking and its shape is approximately circular. Fiber breakage is clearly visible at the higher impact energy, where major fiber breakage is attained on the back surface. It is also clear from the 45°-oriented crack that the fiber damage extends to the inner plies.The drop tests on the FFRP specimens have been conducted from the nominal impact heights of 30, 50, 100, 150, and 200 cm. A single specimen has been tested at each impact height. However, we performed two repetitions of the impact from the height of 100 cm to assess the intra-specimens variability. The two tests showed extremely similar results both in terms of impact dynamics and in terms of visible damage. Results of a single test are thus reported in the following. shows the images of the front and rear faces of the impacted laminates. From a visual inspection of the specimens, it is clear that fiber breakage is attained at all the impact heights. Cracks originate on the back face of the impacted specimen and their size increase with the impact height to eventually form a cross-shaped damage. At the lower impact heights, no damage is visible in the front face, while a marked indentation imprint is observed for energies above 15 J. At the higher impact height, the impactor completely penetrated the laminate, and the laminate shred into four diamond-shaped leaves separated by the cross-shaped fracture.The impact dynamics for the five impacts in terms of force over time, force–displacement curve, and absorbed energy over time are reported in . The time traces of the impact forces largely differs from the ones relative to the impacts on GFRPs presented in , showing marked oscillations as soon as the impact force exceeds approximately 1.5 kN. It is further evident from the force–displacement curve that at a such load level a major damage mechanism is introduced within the laminate, with the force showing a plateau which slightly increases as the impactor penetrates into the target. Such behavior should be ascribed to fiber breakage. As a consequence, the unloading portion of the curves is well separated from the loading path, and with it goes the permanent indentation. Not surprisingly, the energy absorbed during the impact, which is defined as the area below the force–displacement curve is very large. At the lowest impact height, the absorbed energy is more than 40% of the impact energy, and such ratio increases with the impact energy, to eventually get to 100% at the higher impact height since specimen penetration is attained in this case. The most relevant information which can be gained from the curves presented in This section introduces a numerical modeling scheme that is utilized to get deeper insights into the damage mechanisms involved with low-velocity impacts on natural fiber composites. The numerical model is implemented within the LS-DYNA environment and the numerical scheme is at first validated against the experimental results on GFRPs, being these translucent and thus allowing to check if all the damage mechanisms are captured correctly, with particular interest given to delaminations and fiber breakage. The validated model is then utilized to simulate the impacts on FFRPs.The numerical model of the drop tests has been implemented following a common scheme which has been already utilized for the analysis of low-velocity impacts on composite structures The impactor is located at the top centre of the laminate with an offset of 0.1 mm from the surface. It is modeled as a rigid hemispherical shell with an imposed final mass of 1.3 kg and its initial velocity varies coherently with the experimental values reported in . Segment-based contact has been defined between the impactor and the top layer of the laminate model by imposing in the contact options and scaling the contact stiffness by imposing to prevent premature element erosion beneath the impact as suggested in The linear-elastic composite material model based on the Chang/Chang failure criteria () has been adopted. The elastic material behavior of each ply is calculated based on the input of Young’s modulus, shear modulus, and Poisson’s ratio. Damage occurs as soon as one of the four criteria by Chang/Chang is met (tensile and compressive failure in the fiber and matrix direction). The localized stiffness reduction due to internal damage can cause an excessive element distortion leading to premature erosion and inherent difficulties in numerical convergence of the numerical solution. Such an issue has been overtaken by introducing large failure strains in . This way elements are not eroded as soon as the failure stresses are reached but they can carry a fraction of the failure stress (here imposed to 5%) until the failure strains are reached. The respective layer is then assigned with zero stiffness properties. Such an approach has been found to substantially improve the stability of the numerical model. Following the suggestions reported by Vannucchi in summarize the remaining properties of the composite materials implemented within The more ply interfaces are modeled through cohesive layers the more the delamination prediction is accurate The model is comprised of stacked layers of fully integrated shell elements with an offset equal to each ply thickness and we set an hourglass control () compatible with such element type (formulation n. 16). Contact elements with tiebreak option have been utilized within each pair of shell plies to model delamination initiation and growth. A failure law based on the cohesive zone model is utilized imposing in the contact card. Within this model, damage initiates as soon as the normal () failure stresses are met and is a linear function of the distance between the mating surfaces. The contact is then released as soon as a critical distance is met (). Within this scheme, the energy release rates GIC and GIIC are approximated byTo maintain the fracture onset and propagation stable during the solution we followed the numerical scheme proposed in The numerical model has been at first validated against the experimental results of the GFRP specimens. In particular, the experimental results in terms of visible damage and impact dynamics have been utilized to tune the numerical model through a trial and error approach in terms of post-failure material properties. As mentioned in Section , it has been found that the numerical stability drastically improves if elements are not eroded right after failure but can maintain a minimum bearing strength (here imposed to 5% of the failure strength), to be eventually eroded when the strain attains a value equal to 0.2. It has been found that residual strengths larger than 5% lead to an overestimation of the contact force and an underestimation of the damage, while the role of the large eroding strain is to improve the stability of the contacts in the numerical model. Comparison of numerical and experimental results concentrated on matching not only the peak impact force but also the energy absorbed during the impact, as it was observed that substantial variations on the mechanical properties of the material and interlaminar interface might have a minimal effect on the time evolution of the contact force, but largely influence the force–displacement curve and energy absorption. At the end of the optimization of the numerical model, which resulted in the properties presented in , the numerical results are found to match very well with the experimental data both in terms of impact dynamics, presented in . The latter presents the predicted damage in terms of fiber breakage (left) and delaminated areas (right). A summary of the information from the numerical simulation and the relative comparison against the experimental results is presented in . Delaminations match very well with the experimental data both in terms of overall extension and shape, and the predicted fiber breakage is in accordance with the experiments, with fiber breakage distributed across all layers right below the impact point and a more extended fiber breakage located mainly on the back-face which is attained also at the lowest impact energy and increases with it. Circular matrix damage on the back face compatible with the extension of the exposed fibers is also predicted through the numerical model.The validated numerical model has been finally utilized to simulate the impacts on the FFRP panels to gain some more insights on the damage mechanisms attained within the laminate. Since the two laminate share the same matrix system and curing cycles, the interlaminar properties tuned to match the experimental results on the GFRPs have been utilized to simulate the interlaminar strength of the FFRP panels.The numerical results are in good agreements with the experimental findings both in terms of impact dynamics, shown in , and in terms of damage. A summary of the information from the numerical simulation and the relative comparison against the experimental results is presented in shows the contour plot of the predicted fiber and matrix damages. The fiber damage is in line with the experimental observations, with a cross-shaped damage centered on the impact point whose size increases with the impact energy. The extension of the predicted fiber breakage is maximum at the back-face and reduces moving through the thickness, to eventually be negligible on the front face if it wasn’t for the damage right below the contact point. Minor delaminations are predicted for all the impact heights, as these are found to generate during the initial stage of the impact. These are located right below the impact point, are very small in size, and do not increase with the impact height. also reports the contour plot of the matrix damage extension. Notably, the model predicts a four leaf-shaped matrix damage which increases with the impact energy and is located on the outer layers. Such shape is in line with the morphology of the four leaves which resulted as the specimen shred under the higher impact energy shown in , fracture morphology which is also captured by the numerical model, as visible in . The model is further capable to predict not only the full penetration at the higher impact height but also the marked circular residual imprint experimentally observed at the impact energy of 15 J, as reported in Overall, the numerical predictions are very similar to the experimental results, even if the experimental peak force and the absorbed energy, hence the impact damage, are slightly overestimated. Such good accordance between experiments and numerical results was actually unexpected. In fact, we have to stress that the material model implemented in the numerical model is linear elastic; such approximation largely differs from the quasi-static material behavior 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.On-chip ultrasonic manipulation of microparticles by using the flexural vibration of a glass substrateAs biotechnology develops, techniques for manipulating and separating small particles such as cells and DNA are required in the life sciences. This paper investigates on-chip manipulation of microparticles in small channels by using ultrasonic vibration. The chip consists of a rectangular glass substrate with a cross-shaped channel (cross-section: 2.0 × 2.0 mm2) and four lead zirconate titanate transducers attached to the substrate’s four corners. To efficiently generate the flexural vibration mode on the chip, we used finite element analysis to optimize the configurations of the glass substrate and transducers. Silicon carbide microparticles with an average diameter of 50 μm were immersed in the channels, which were filled with ethanol. By applying an in-phase input voltage of 75 V at 225 kHz to the four transducers, a flexural vibration mode with a wavelength of 13 mm was excited on the glass substrate, and this flexural vibration generated an acoustic standing wave in the channel. The particles could be trapped at the nodal lines of the standing wave. By controlling the driving phase difference between the two pairs of transducers, the vibrational distribution of the substrate could be moved along the channels so that the acoustic standing wave moved in the same direction. The trapped particles could be manipulated by the two-phase drive, and the transport direction could be switched at the junction of the channels orthogonally by changing the combination of the driving condition to four transducers.In vitro assay evaluations are powerful tools for examining individual cells and DNA molecules and for predicting the reactions of these entities in the human body. Most biological tests, including hematological examination and gene analysis, produce diagnoses based on and deduced from the average values and trends of the measured results, which can be obtained from a huge number of samples (for example, a blood sample includes many blood cells). However, these average results include unintended values from other intact cells and may dilute and blind the differences among individual cells, and they pose a risk in missing diseases in their early stages or missing diagnoses of diseases with minimal manifestation, such as blood cancers Based on these needs, techniques for manipulating and separating small particles such as cells and DNA are required in the life sciences. There have been several reports on labs-on-a-chip with reactors, sensors, and microchannels, in which small objects can be selectively transported and distinguished Ultrasonic techniques are promising candidates for noncontact manipulation in microchannels and enable positional control over small objects trapped by an acoustic standing-wave field shows the configuration of the chip we used for ultrasonic manipulation of microparticles. The chip consists of a rectangular glass substrate (30.0 × 30.0 × 3.8 mm) with small cross-shaped channels (cross-section: 2.0 × 2.0 mm) and four piezoelectric lead zirconate titanate (PZT) ultrasound transducers (10.0 × 10.0 × 1.0 mm, C-203, Fuji Ceramics, Fujinomiya, Japan) polarized in the thickness direction. The PZT transducers (PZT 1–4) are attached with epoxy to the four corners of the glass substrate. The channels are filled with ethanol, which we chose by considering the surface tension and the shape of the liquid surface: liquids with higher surface tension, such as water, may form a cylindrical surface of liquid along the channel, affecting the acoustic field. To increase the vibration amplitude, we optimized the configuration of the glass substrate, transducers, and channel by using finite element analysis (FEA; ANSYS 11.0, ANSYS Inc., Canonsburg, PA, USA). The material constants used in the simulation are summarized in shows the out-of-plane vibration displacement amplitude distribution in the thickness direction (z direction) of the glass substrate at 273 kHz calculated by FEA. In-phase continuous sinusoidal electric input voltages were applied to all the four transducers (PZT 1–4) as the boundary conditions, and the vibrational amplitude was normalized by its maximum value at the center of the plate. Exciting the four transducers generated the asymmetric flexural vibration mode on the transducers and the glass substrate, and the loop positions with larger vibration amplitude aligned along each channel. Because the flexural vibration has several resonance frequencies over 20 kHz, the vibration mode of the glass substrate and the manipulation characteristics can be controlled by the input signal.When the flexural vibration is excited along the channel, the sound wave radiates from the substrate to the liquid, generating an acoustic standing-wave field in the channel. Particles that are much smaller than the wavelength of a sound wave can be trapped along the nodal (or anti-nodal) lines of the acoustic standing wave by the acoustic radiation force, which is a quasi-static force generated by the difference in average acoustic energy densities between two media over a period To manipulate the particles in the channel in one dimension, the four transducers were excited with a two-phase drive; PZTs 1 and 3 () were excited with the input voltage V |
= |
V0cosωt, and PZTs 2 and 4 were excited with V |
= |
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