《复合材料 Composites》课程教学资源（学习资料）第五章陶瓷基复合材料_fibrous molithic-22.pdf_大学文库

Availableonlineatwww.sciencedirect.com SCIENCE DIRECTO COMPOSITES SCIENCE AND TECHNOLOGY ELSEVIER Composites Science and Technology 64(2004)529-548 www.elsevier.com/locate/compsci Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments Zheng-Ming Huang Department of Engineering Mechanics, Tongji University, 1239 Siping Road, Shanghai, PR China Received I July 2001; accepted I August 2002 Abstract In my other paper(in this issue, the bridging micromechanics model has been combined with the classical lamination theory to redict the progressive failure strengths or the entire stress-strain curves of a set of typical polymer resin based composite laminates subjected to different biaxial loads. The predictions were performed only using the constituent fiber and resin properties and the geometric parameters of the laminates specified independently. Comparison of the predictions with the experimental measurements provided by the failure exercise organizers is carried out in this paper. As a whole, the overall correlation between the theory and the experiments is reasonable. Some additional comments regarding the applications of the bridging model to the simulation of ultimate behavior of fibrous laminates are provided. Comparison of the predictions of each other with and without thermal residual stresses is also made. It is demonstrated that for most of the present epoxy resin based composites, the effect of the thermal residual stresses is grossly insignificant. Thus, a general conclusion may be that in most cases the thermal residual stresses can be neglected for a thermoset polymer resin based composite C 2003 Elsevier Ltd. All rights reserved Keywords: Composite laminate 1. Introduction for the majority of the problems under consideration, the predictions with and without the thermal residual A recently developed micromechanics model, the stresses nearly coincide one another. However, a max bridging model [2], has been incorporated with the imum difference as high as 70% between the predicted classical lamination theory to predict the progressive ultimate strengths with and without the thermal residual failure behavior and ultimate strength of 14 multi- stresses for a laminate problem has also been recog- directional laminates subjected to biaxial loads [1]. nized. These results indicate that although in most cases Those laminates constituted the failure exercise problems the effect of thermal residual stresses can be neglected proposed by the organizers [3]. The predictions were cautions should also be kept in mind that the neglect made only using the constituent fiber and resin proper- may cause a significant error in some other case ties and the laminate geometric parameters provided by the organizers, without knowing any experimental information of the laminates. The prediction efficiency 2. Overall correlation is shown in this paper, by correlating the predictions with the experiments. Additional comments regardin ef. [3] specified in detail the information of all of the his correlation are thus made. Further predictions 14 exercise problems(referred to occasionally as Tes without considering any effect of thermal residual stres- Cases) including the laminate lay up sequences and ses on the laminate responses are carried out, and are angles, the fiber volume fractions, the lamina thick- compared with the experiments. It has been found that nesses, the constituent properties, and the stifness and strength parameters of the individual unidirectional Corresponding author. Tel: (UD) laminae. From these results, the input data required for the bridging model predictions were deter- 0266-3538/S- see front matter c 2003 Elsevier Ltd. All rights reserved doi:10.1016/S0266-3538(03)002227

Correlation of the bridging model predictions of the biaxial failure strengths of fibrous laminates with experiments Zheng-Ming Huang* Department of Engineering Mechanics, Tongji University, 1239 Siping Road, Shanghai, PR China Received 1 July2001; accepted 1 August 2002 Abstract In myother paper (in this issue), the bridging micromechanics model has been combined with the classical lamination theoryto predict the progressive failure strengths or the entire stress–strain curves of a set of typical polymer resin based composite laminates subjected to different biaxial loads. The predictions were performed onlyusing the constituent fiber and resin properties and the geometric parameters of the laminates specified independently. Comparison of the predictions with the experimental measurements provided bythe failure exercise organizers is carried out in this paper. As a whole, the overall correlation between the theoryand the experiments is reasonable. Some additional comments regarding the applications of the bridging model to the simulation of ultimate behavior of fibrous laminates are provided. Comparison of the predictions of each other with and without thermal residual stresses is also made. It is demonstrated that for most of the present epoxyresin based composites, the effect of the thermal residual stresses is grosslyinsignificant. Thus, a general conclusion maybe that in most cases the thermal residual stresses can be neglected for a thermoset polymer resin based composite. # 2003 Elsevier Ltd. All rights reserved. Keywords: Composite laminate 1. Introduction A recentlydeveloped micromechanics model, the bridging model [2], has been incorporated with the classical lamination theoryto predict the progressive failure behavior and ultimate strength of 14 multidirectional laminates subjected to biaxial loads [1]. Those laminates constituted the failure exercise problems proposed bythe organizers [3]. The predictions were made onlyusing the constituent fiber and resin properties and the laminate geometric parameters provided by the organizers, without knowing anyexperimental information of the laminates. The prediction efficiency is shown in this paper, bycorrelating the predictions with the experiments. Additional comments regarding this correlation are thus made. Further predictions without considering anyeffect of thermal residual stresses on the laminate responses are carried out, and are compared with the experiments. It has been found that for the majorityof the problems under consideration, the predictions with and without the thermal residual stresses nearlycoincide one another. However, a maximum difference as high as 70% between the predicted ultimate strengths with and without the thermal residual stresses for a laminate problem has also been recognized. These results indicate that although in most cases the effect of thermal residual stresses can be neglected, cautions should also be kept in mind that the neglect maycause a significant error in some other case. 2. Overall correlation Ref. [3] specified in detail the information of all of the 14 exercise problems (referred to occasionallyas Test Cases) including the laminate layup sequences and angles, the fiber volume fractions, the lamina thicknesses, the constituent properties, and the stiffness and strength parameters of the individual unidirectional (UD) laminae. From these results, the input data required for the bridging model predictions were deter- 0266-3538/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0266-3538(03)00222-7 Composites Science and Technology64 (2004) 529–548 www.elsevier.com/locate/compscitech * Corresponding author. Tel.: +86-21-65985373. E-mail address: huangzm@mail.tongji.edu.cn (Z.-M. Huang)

Z.M. Huang/ Composites Science and Technology 64(2004)529-548 mined, as illustrated in Ref [1]. Those data can be clas- laminae subjected to different kinds of uniaxial loads sified into two groups(see Ref. [ID. The first group is (longitudinal tension, transverse tension, longitudinal the constituent fiber and resin properties and the second compression, transverse com mpression, and in-plane group is the laminate geometric parameters. In the pre- shearing) up to failure, i.e. the ultimate uniaxial loads dictions made in Ref. [1], all of the fiber materials wer The retrieval of the constituent strengths was accom considered as linearly elastic until rupture for which plished in such a way that when all the ultimate uniaxial only elastic and ultimate properties are required, loads had been applied to the individual UD lamina whereas the resins were regarded as elastic-plastic respectively, the comparative maximum tensile and materials. Thus, the material properties consisted of the compressive stresses generated in the fiber and the resin fiber and resin elastic constants, the resin plastic materials were taken as their respective tensile and parameters, and the tensile and compressive strengths of compressive strengths. For example, when the ultimate the fibers(along longitudinal direction) and the resins. longitudinal tension, transverse tension, and the in The second group data, i.e. the laminate geometric plane shearing were applied to the UD lamina respec- parameters, included the laminate lay-up arrangements, tively, the resin material would generate three different laying angles, and thickness of each lamina in the lami- maximum tensile stresses. The resin tensile strength was nates. While the second group data can be obtained defined as the largest of these three tensile stresses from the condition as-fabricated (in the predictions in has been found that the tensile strengths of both the Ref [1], those provided by the organizers were directly fibers and the resins were determined against the ulti employed), the first group data have to be prepared with mate longitudinal tension of the UD laminae. The fiber more care In Ref [I], the elastic properties of both the and the resin compressive strengths were obtained based fibers and the resins for all of the 14 problems were on the ultimate longitudinal and transverse compres taken exactly the same as those provided by the organi sions of the UD laminae, respectively. More details are zers [3], whereas the remaining constituent properties provided in Ref [1] were retrieved from the responses of the individual Ud The theoretical results taken from Ref. [1] are re- laminae. The resin plastic parameters were back calcu- plotted in curves designated as"theory with thermal lated against the stress-strain responses of the corres- residual stresses"in Figs. 1-14 for all of the 14 exercise ponding laminae subjected to in plane shearing up to problems, respectively, which are compared with the failure. Having determined the fiber elastic and the resin test results provided by the exercise organizers [4]. It is elastic-plastic parameters, the constituent strength data seen that except for the problems 3, 4, 6, and 8, the were retrieved against the ultimate strengths of the UD correlations between the theory and the experiments for heory(final failure, with thermal residual stresses o Test results Theory(final failure, with thermal residual stresses) y-directional stress(MPa) Fig 1. Measured and predicted biaxial failure stresses for 0 lamina subjected to combined yy and txy Material type: E-Glass/LY556/HT907/ DY063

mined, as illustrated in Ref. [1]. Those data can be classified into two groups (see Ref. [1]). The first group is the constituent fiber and resin properties and the second group is the laminate geometric parameters. In the predictions made in Ref. [1], all of the fiber materials were considered as linearlyelastic until rupture for which onlyelastic and ultimate properties are required, whereas the resins were regarded as elastic–plastic materials. Thus, the material properties consisted of the fiber and resin elastic constants, the resin plastic parameters, and the tensile and compressive strengths of the fibers (along longitudinal direction) and the resins. The second group data, i.e. the laminate geometric parameters, included the laminate lay-up arrangements, laying angles, and thickness of each lamina in the laminates. While the second group data can be obtained from the condition as-fabricated (in the predictions in Ref. [1], those provided bythe organizers were directly employed), the first group data have to be prepared with more care. In Ref. [1], the elastic properties of both the fibers and the resins for all of the 14 problems were taken exactlythe same as those provided bythe organizers [3], whereas the remaining constituent properties were retrieved from the responses of the individual UD laminae. The resin plastic parameters were back calculated against the stress-strain responses of the corresponding laminae subjected to in plane shearing up to failure. Having determined the fiber elastic and the resin elastic–plastic parameters, the constituent strength data were retrieved against the ultimate strengths of the UD laminae subjected to different kinds of uniaxial loads (longitudinal tension, transverse tension, longitudinal compression, transverse compression, and in-plane shearing) up to failure, i.e. the ultimate uniaxial loads. The retrieval of the constituent strengths was accomplished in such a waythat when all the ultimate uniaxial loads had been applied to the individual UD lamina respectively, the comparative maximum tensile and compressive stresses generated in the fiber and the resin materials were taken as their respective tensile and compressive strengths. For example, when the ultimate longitudinal tension, transverse tension, and the inplane shearing were applied to the UD lamina respectively, the resin material would generate three different maximum tensile stresses. The resin tensile strength was defined as the largest of these three tensile stresses. It has been found that the tensile strengths of both the fibers and the resins were determined against the ultimate longitudinal tension of the UD laminae. The fiber and the resin compressive strengths were obtained based on the ultimate longitudinal and transverse compressions of the UD laminae, respectively. More details are provided in Ref. [1]. The theoretical results taken from Ref. [1] are replotted in curves designated as ‘‘theorywith thermal residual stresses’’ in Figs. 1–14 for all of the 14 exercise problems, respectively, which are compared with the test results provided bythe exercise organizers [4]. It is seen that except for the problems 3, 4, 6, and 8, the correlations between the theoryand the experiments for Fig. 1. Measured and predicted biaxial failure stresses for 0 lamina subjected to combined yy and xy. Material type: E-Glass/LY556/HT907/ DY063. 530 Z.-M. Huang / Composites Science and Technology 64 (2004) 529–548

Z -M. Huang/Composites Science and Technology 64(2004)529-548 Test results(x-strain) Test results (y-strain) Theory (y-strain, without hermal residual stresses) A Theory (x-strain, without Theory (y-strain, with thermal residual stresses Theory (x-strain, with thermal 0 梦0 residual stresses Fig. 14. Measured and predicted ayy vS. Eyy and oyy vS. Exx curves for [+45@]s laminate under biaxial stresses(oyyoxx=1/-1) Material type E-Glass/MY750 reason is the same as the above mentioned If the resin This is because the lamina strengths in these directions compressive strength used is higher, the predicted max- were used to determine the constituent strengths. The imum shear strength will occur at a larger applied com- resin tensile strength was also retrieved from the lamina pressive transverse stress. longitudinal tensile strength, resulting in a higher pre- dicted transverse tensile strength for the lamina. It is Problem 2 (UD T300/BSL914C lamina subjected to seen that the predicted failure envelope on the fourth combined oxr and xy loads). This is the only composite quadrant is approximately a rectangle, whereas the made of T300/BSL914C material system amongst the 14 measured data could be better represented by a mono exercise problems. Correlation between the theory and tonically increased curve as the longitudinal tensile the experiments for this lamina is grossly satisfactory, as stress component decreased from its maximum to zero shown in Fig. 2. The figure clearly indicates that the However, compared with those for problem 2, not experimental deviation for composite strengths can be enough measured data for this problem have been quite large. For the present lamina when the long- available. More experiments are required, especially to tudinal load was small, the difference in the measured determine the data points of the failure envelope on the failure strengths can be as large as 80% of a relative other three quadrants error. Thus, sufficient experimental data should be obtained when they are used to verify theoretical pre- Problem 4 l(900/+300/90)E-glass/LY556/HT907/ dictions DY063 laminate subjected to combined orr and o loads]. Relatively large discrepancy exists between the Problem 3 (UD E-glass/MY750/HY917/DY063 lamina theory and the experiments for this problem, as indi subjected to combined yy and orr loads). Results of this cated in Fig. 4. This is one of the few exercise problems problem are shown in Fig. 3. It may be noted that for which the bridging model predictions did not corre- according to the organizers [4] the problem 3 was actu- late reasonably with the measured data. The dis- ly not an exact UD lamina, but an angle plied lami- crepancy became the largest when the combination of 5°,i.e.[±5° Is. Thus, the the applied biaxial loads, o. andσ near predicted curve(for the laminate) plotted in Fig. 3 is linear line as shown in the slightly different from the corresponding figure shown in Oxx=1.50yy+34.4(MPa). The laminate Ref [1]. Similarly to Problem l, the correlation in the response could not be accurately es longitudinal (x-directional) tension and compression procedure of Ref. [1]. see the discussions given for and the transverse (y-directional)compression is good. problem 10 for more detail. Furthermore, the resin

reason is the same as the above mentioned. If the resin compressive strength used is higher, the predicted maximum shear strength will occur at a larger applied compressive transverse stress. Problem 2 (UD T300/BSL914C lamina subjected to combined xx and xy loads). This is the onlycomposite made of T300/BSL914C material system amongst the 14 exercise problems. Correlation between the theoryand the experiments for this lamina is grosslysatisfactory, as shown in Fig. 2. The figure clearlyindicates that the experimental deviation for composite strengths can be quite large. For the present lamina when the longitudinal load was small, the difference in the measured failure strengths can be as large as 80% of a relative error. Thus, sufficient experimental data should be obtained when theyare used to verifytheoretical predictions. Problem 3 (UD E-glass/MY750/HY917/DY063 lamina subjected to combined yy and xx loads). Results of this problem are shown in Fig. 3. It maybe noted that according to the organizers [4] the problem 3 was actuallynot an exact UD lamina, but an angle plied laminate with a plyangle of 5, i.e. [5]s. Thus, the predicted curve (for the laminate) plotted in Fig. 3 is slightlydifferent from the corresponding figure shown in Ref. [1]. Similarlyto Problem 1, the correlation in the longitudinal (x-directional) tension and compression and the transverse (y-directional) compression is good. This is because the lamina strengths in these directions were used to determine the constituent strengths. The resin tensile strength was also retrieved from the lamina longitudinal tensile strength, resulting in a higher predicted transverse tensile strength for the lamina. It is seen that the predicted failure envelope on the fourth quadrant is approximatelya rectangle, whereas the measured data could be better represented bya monotonicallyincreased curve as the longitudinal tensile stress component decreased from its maximum to zero. However, compared with those for problem 2, not enough measured data for this problem have been available. More experiments are required, especiallyto determine the data points of the failure envelope on the other three quadrants. Problem 4 [(90/30/90) E-glass/LY556/HT907/ DY063 laminate subjected to combined yy and xx loads]. Relativelylarge discrepancyexists between the theoryand the experiments for this problem, as indicated in Fig. 4. This is one of the few exercise problems for which the bridging model predictions did not correlate reasonablywith the measured data. The discrepancybecame the largest when the combination of the applied biaxial loads, yy and xx, was near to the linear line as shown in the figure, i.e. when xx=1.5yy+34.4 (MPa). The resulting laminate response could not be accuratelyestimated using the procedure of Ref. [1], see the discussions given for problem 10 for more detail. Furthermore, the resin Fig. 14. Measured and predicted yy vs. "yy and yy vs. "xx curves for [45]s laminate under biaxial stresses yy=xx ¼ 1= 1 . Material type: E-Glass/MY750. Z.-M. Huang / Composites Science and Technology 64 (2004) 529–548 537

538 Z -M. Huang/Composites Science and Technology 64(2004)529-548 plastic parameters used may also have played an failure information of the predictions without the ther- important role. Fig. 21 of Ref [1] clearly indicated that mal residual stresses was indicated The similar infor for this laminate structure subjected to the combined mation with the thermal effect has been reported in Ref. Oyy and Oxx loads, different use of the resin plastic [1](Fig. 24 of Ref. [lD. It may be noted that the third parameters resulted in very large difference in the pre- ply failure of the laminate without the thermal residual dicted failure envelope. The present resin plastic prop- stresses was due to the fiber tensile failure, which is dif- erties were retrieved from the in-plane shear response of ferent from the failure mode, i.e. the resin tensile failure, the UD lamina, which could be somewhat different identified in Ref [l] where the thermal residual stresses from those measured directly on monolithic resin speci- were taken into account. It is also noted that after the mens. It would be useful by comparing the predictions first ply failure the predicted stress-strain curve in the x based on the resin properties retrieved from the lamina direction began to change its slope to more ductile, overall response and measured from the monolithic whereas the curve in the y-direction had little variation pecimens. However, no measured data from the in slope. On the other hand, the second ply failure monolithic specimens were available and no attempt for resulted in an apparent stifness reduction in the y'- his comparison has been made. directional response, but gave rise to a stifness increase in the x-directional curve. the same features in stiffness Problem 5 1(900/=+300/90) E-glass/LY556/HT907/ variation have been also observed with the results with- DY063 laminate subjected to combined oxx and oxy out the thermal residual stresses, as indicated in loads]. Relatively good correlation between the theory and the experiments for this problem has been found, as shown in Fig. 5. The theory under-predicted the uni- Problem 8 I(0/= 45 /90)s AS4/3501-6 laminate sub xial tensile strength, which is dominated by the lamina jected to combined oxx and Oyy loads (oyy/oxx=2) longitudinal tensile strength Quite large difference of the theory from the test result exists for this problem, as shown in Fig. 8. This is the Problem 6 I(00/-+45/90)s AS4/3501-6 laminate sub- problem for which the poorest correlation has been jected to combined orr and oxr loads]. The predicted and found. It can be clearly seen from Fig 8 that the influ test results for this problem are graphed in Fig. 6, and ence of thermal residual stresses for this problem was poor correlation has been found. This may be the only significant. Large deviation exists between the predic- exercise problem for which large discrepancy exists tions with and without the thermal residual stresses. The without apparent reasoning. It is not very clear why prediction procedure without the thermal residual significant difference could exist between the theory and stresses is described in the next section. In Ref [1],an he experiments for this problem. The difference for the exact formula for calculating the thermal residual stres part of the failure envelope where biaxial compressions ses generated in the laminate was elaborated and uti- were applied is understandable. During the measure- lized throughout. However, that formula can be applied ment, the material buckling was observed with the only to an ideally fabricated laminate. No stress con biaxial compressions that would reduce the composite centration due to any fabrication defect can be incor load carrying ability distinctly, whereas the prediction porated. More over, when the predicted thermal [1] did not account for the material buckling. Moreover, residual stresses have been large enough so that the the present problem involves a symmetrically arranged resin material in the composite has been caused to yield laminate (quasi-isotropic in the laminate plane) sub- the resin instantaneous compliance matrix should be jected to all the possible in-plane biaxial loads. Ideally, defined more carefully. One should know whether the the failure envelope of this laminate must be symmetric resin is in the status of loading or unloading when a with respect to some in-plane axis. The predicted curve subsequent increment of a mechanical load is applied. If shown in Fig. 6 confirms this symmetry. Nevertheless, it is in unloading, the compliance matrix should be spe from the given experimental data, this symmetry cannot cified using Hooke's law. Otherwise, the matrix is be seen. It would be useful if more experiments had been defined using Prandtl-Reuss theory. However, the performed, especially with load combinations of knowledge on loading or unloading is usually not negative oyy and positive o, straightforward in the simulation because of the multi axial stress state generated in the resin material(both Problem 7 I(0/+45/90%)s AS4/3501-6 laminate sub- the thermal residual stresses and the mechanical load jected to uniaxial xr load The correlation between the generated stresses were generally multiaxial for any theoretical and test results for this problem is better exercise problem where a laminate is involved) than that for the preceding problem, but still, not very It remained a question in Ref [1] that according to satisfactory, as shown in Fig. 7. It is noted that without which criterion the material should be regarded as considering thermal residual stresses, the prediction loading or unloading when both the existing and the seemed to be much better. In the figure, the progressive additional stresses are multiaxial. Due to this difficulty

plastic parameters used mayalso have played an important role. Fig. 21 of Ref. [1] clearlyindicated that for this laminate structure subjected to the combined yy and xx loads, different use of the resin plastic parameters resulted in verylarge difference in the predicted failure envelope. The present resin plastic properties were retrieved from the in-plane shear response of the UD lamina, which could be somewhat different from those measured directlyon monolithic resin specimens. It would be useful bycomparing the predictions based on the resin properties retrieved from the lamina overall response and measured from the monolithic specimens. However, no measured data from the monolithic specimens were available and no attempt for this comparison has been made. Problem 5[(90/30/90) E-glass/LY556/HT907/ DY063 laminate subjected to combined xx and xy loads]. Relativelygood correlation between the theory and the experiments for this problem has been found, as shown in Fig. 5. The theoryunder-predicted the uniaxial tensile strength, which is dominated bythe lamina longitudinal tensile strength. Problem 6 [(0/45/90)s AS4/3501-6 laminate subjected to combined yy and xx loads]. The predicted and test results for this problem are graphed in Fig. 6, and poor correlation has been found. This maybe the only exercise problem for which large discrepancyexists without apparent reasoning. It is not veryclear why significant difference could exist between the theoryand the experiments for this problem. The difference for the part of the failure envelope where biaxial compressions were applied is understandable. During the measurement, the material buckling was observed with the biaxial compressions that would reduce the composite load carrying ability distinctly, whereas the prediction [1] did not account for the material buckling. Moreover, the present problem involves a symmetrically arranged laminate (quasi-isotropic in the laminate plane) subjected to all the possible in-plane biaxial loads. Ideally, the failure envelope of this laminate must be symmetric with respect to some in-plane axis. The predicted curve shown in Fig. 6 confirms this symmetry. Nevertheless, from the given experimental data, this symmetry cannot be seen. It would be useful if more experiments had been performed, especiallywith load combinations of negative yy and positive xx. Problem 7 [(0/45/90)s AS4/3501-6 laminate subjected to uniaxial xx load]. The correlation between the theoretical and test results for this problem is better than that for the preceding problem, but still, not very satisfactory, as shown in Fig. 7. It is noted that without considering thermal residual stresses, the prediction seemed to be much better. In the figure, the progressive failure information of the predictions without the thermal residual stresses was indicated. The similar information with the thermal effect has been reported in Ref. [1] (Fig. 24 of Ref. [1]). It maybe noted that the third plyfailure of the laminate without the thermal residual stresses was due to the fiber tensile failure, which is different from the failure mode, i.e. the resin tensile failure, identified in Ref. [1] where the thermal residual stresses were taken into account. It is also noted that after the first plyfailure the predicted stress-strain curve in the xdirection began to change its slope to more ductile, whereas the curve in the y-direction had little variation in slope. On the other hand, the second plyfailure resulted in an apparent stiffness reduction in the ydirectional response, but gave rise to a stiffness increase in the x-directional curve. The same features in stiffness variation have been also observed with the results without the thermal residual stresses, as indicated in Fig. 7. Problem 8 [(0/45/90)s AS4/3501-6 laminate subjected to combined xx and yy loads yy=xx ¼ 2 ]. Quite large difference of the theoryfrom the test results exists for this problem, as shown in Fig. 8. This is the problem for which the poorest correlation has been found. It can be clearlyseen from Fig. 8 that the influence of thermal residual stresses for this problem was significant. Large deviation exists between the predictions with and without the thermal residual stresses. The prediction procedure without the thermal residual stresses is described in the next section. In Ref. [1], an exact formula for calculating the thermal residual stresses generated in the laminate was elaborated and utilized throughout. However, that formula can be applied onlyto an ideallyfabricated laminate. No stress concentration due to anyfabrication defect can be incorporated. More over, when the predicted thermal residual stresses have been large enough so that the resin material in the composite has been caused to yield, the resin instantaneous compliance matrix should be defined more carefully. One should know whether the resin is in the status of loading or unloading when a subsequent increment of a mechanical load is applied. If it is in unloading, the compliance matrix should be specified using Hooke’s law. Otherwise, the matrix is defined using Prandtl-Reuss theory. However, the knowledge on loading or unloading is usuallynot straightforward in the simulation because of the multiaxial stress state generated in the resin material (both the thermal residual stresses and the mechanical load generated stresses were generallymultiaxial for any exercise problem where a laminate is involved). It remained a question in Ref. [1] that according to which criterion the material should be regarded as loading or unloading when both the existing and the additional stresses are multiaxial. Due to this difficulty, 538 Z.-M. Huang / Composites Science and Technology 64 (2004) 529–548

《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_fibrous molithic-22

《复合材料 Composites》课程教学资源（学习资料）第五章陶瓷基复合材料_fibrous molithic-22