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 all of the other 10 problems are reasonable. The corre- Problem 1 (Ud E-glass/LY556/HT907/DY063 lamina lation for the problem 3 is slightly less satisfactory. subjected to combined oyy and oxy loads). The theoretical However, relatively large differences have been found and experimental data for this problem are graphed for the problems 4, 6, and 8. Discussions on the correl- in Fig. 1. It is seen that the predicted strength envel- ations of the specific problems are given below ope in variation trend from the tensile to the Theory (final failure, without Theory (final failure, with thermal residual stresses) ○O 可三苏 00 500 500 x-directional stress(MPa) Fig. 2. Measured and predicted biaxial failure stresses for 0 lamina subjected to combined xr and txr. Material type: T300/91 1750 1250 75 1 O Theory (final failure, without thermal residual str Theory(final failure, with thermal residual stresses X-directional stress(MPa) Fig. 3. Measured and predicted biaxial failure stresses for [5] laminate subjected to combined Oxx and orr. Material type: E-Glass/MY750 epoxy
all of the other 10 problems are reasonable. The correlation for the problem 3 is slightlyless satisfactory. However, relativelylarge differences have been found for the problems 4, 6, and 8. Discussions on the correlations of the specific problems are given below. Problem 1 (UD E-glass/LY556/HT907/DY063 lamina subjected to combined yy and xy loads). The theoretical and experimental data for this problem are graphed in Fig. 1. It is seen that the predicted strength envelope in variation trend from the tensile to the Fig. 2. Measured and predicted biaxial failure stresses for 0 lamina subjected to combined xx and xy. Material type: T300/914C. Fig. 3. Measured and predicted biaxial failure stresses for [5]s laminate subjected to combined xx and yy. Material type: E-Glass/MY750 epoxy. Z.-M. Huang / Composites Science and Technology 64 (2004) 529–548 531
Z -M. Huang/Composites Science and Technology 64(2004)529-548 compressive transverse stress component is similar to with a compressive transverse stress component the measured one. However, discrepancy in magnitude involved. This is mainly attributed to the fact that the exists. In general, the prediction overestimated a point resin tensile strength used is somewhat"higher"than on the failure envelope with a tensile transverse stress the in situ resin tensile strength whereas the resin com- component involved whereas underestimated a point pressive strength used is somewhat"lower"than the in Theory(initial failure, with 1000 thermal effect) Theory final failure =1.5a+34.4 o Test results(final fa Theory(final failure, without thermal effect 1000 500 500 1000 1000 y-direction stress(MPa) Fig. 4. Measured and predicted biaxial failure stresses for[90/#30%] laminate subjected to combined oyy and axx. Material type: E-Glass/LY556 Theory(initial failure, with thermal effect Theory(final failure, with hermal effect) Test results(final failure) Theory(final failure, without 300 thermal effect) X-direction stress(MPa) Fig. 5. Measured and predicted biaxial failure stresses for [90/+30] laminate subjected to combined Oxr and Txy. Material type: E-Glass/LY556
compressive transverse stress component is similar to the measured one. However, discrepancyin magnitude exists. In general, the prediction overestimated a point on the failure envelope with a tensile transverse stress component involved whereas underestimated a point with a compressive transverse stress component involved. This is mainlyattributed to the fact that the resin tensile strength used is somewhat ‘‘higher’’ than the in situ resin tensile strength whereas the resin compressive strength used is somewhat ‘‘lower’’ than the in Fig. 4. Measured and predicted biaxial failure stresses for [90/30]s laminate subjected to combined yy and xx. Material type: E-Glass/LY556 epoxy. Fig. 5. Measured and predicted biaxial failure stresses for [90/30]s laminate subjected to combined xx and xy. Material type: E-Glass/LY556 epoxy. 532 Z.-M. Huang / Composites Science and Technology 64 (2004) 529–548
Z.M. Huang/ Composites Science and Technology 64(2004)529-548 situ resin compressive strength for this lamina. As men- strengths were retrieved from its longitudinal an tioned earlier, the fiber and resin tensile strengths were transverse compressive strengths respectively. The used retrieved from the longitudinal tensile strengths of the resin tensile strength, 56. 5MPa, was higher than the D lamina, whereas the fiber and resin compressive resin tensile strength, 33 1 MPa, retrieved from the theory (initial failure, with thermal theory(final failure, with thermal 1000o Test results(final failure s theory(final failure, without I effect a200 8 600 1000 1400 y-direction stress(MPa) Fig. 6. Measured and predicted biaxial failure stresses for (90/+45/0] laminate subjected to combined orr and oxx. Material type: AS4/ 3501-6 1000 o Theory (y-strain, without thermal residual stresses) a Theory (x-strain, without thermal residual stresses Test results (y-strain) Test results(x-strain 800 theory (y-strain, with thermal residual stresses) Theory(x-strain, with thermal residual stresses) 折 (Predictions without thermal effect) 1 ply failure information SS=4463MPa, plies=(o), mode=MT SS=508MPa, plies=(45), mode=MT 3rd ply failure inforI SS=659MPa, plies=(90), mode=FT 0 Strain (% redicted oyy vs Eyy and oyy vS Exr curves for[900/=45 /0], laminate under uniaxial tension(oy/ oxx=1/0). Material type AS4/3501-6 Only the failure information without thermal effect is shown, in which"TT"=strength. ""MT"=resin tensile failure, and"FT"=fiber
situ resin compressive strength for this lamina. As mentioned earlier, the fiber and resin tensile strengths were retrieved from the longitudinal tensile strengths of the UD lamina, whereas the fiber and resin compressive strengths were retrieved from its longitudinal and transverse compressive strengths respectively. The used resin tensile strength, 56.5MPa, was higher than the resin tensile strength, 33.1 MPa, retrieved from the Fig. 6. Measured and predicted biaxial failure stresses for [90/45/0]s laminate subjected to combined yy and xx. Material type: AS4/3501-6. Fig. 7. Measured and predicted yy vs. "yy and yy vs. "xx curves for [90/45/0]s laminate under uniaxial tension yy=xx ¼ 1=0 . Material type: AS4/3501-6. Onlythe failure information without thermal effect is shown, in which ‘‘TT’’ =strength, ‘‘MT’’=resin tensile failure, and ‘‘FT’’=fiber tensile failure. Z.-M. Huang / Composites Science and Technology 64 (2004) 529–548 533