Experimental Characterization of Damage in Random Short Glass
Transcription
Experimental Characterization of Damage in Random Short Glass
Experimental Characterization of Damage in Random Short Glass Fiber Reinforced Composites MARIE-LAURE DANO* AND GUY GENDRON Department of Mechanical Engineering Laval University, Quebec City Quebec, G1K 7P4, Canada FRANC¸OIS MAILLETTE AND BENOIˆT BISSONNETTE Department of Civil Engineering Laval University, Quebec City Quebec, G1K 7P4, Canada ABSTRACT: This article presents the results of an experimental program carried out to characterize the mechanical behavior of random short glass fiber reinforced composites. Tensile, compressive, and shear tests are first performed. The results show that the material is characterized by in-plane isotropy and that it exhibits a damageable elastic behavior in tension and a brittle linear elastic behavior in compression. Then, a series of tests are conducted to evaluate the elastic stiffness tensor of the damaged material. The experimental results reveal that damage induces anisotropy. The results of the experimental program are used to identify and validate a continuum damage mechanics model that has been developed to predict the material mechanical behavior. KEY WORDS: damage characterization, short glass fiber composites, testing, anisotropy, cracks, behavior. INTRODUCTION S composites have become an attractive material for many industrial applications. For example, short glass HORT FIBER REINFORCED *Author to whom correspondence should be addressed. E-mail: [email protected] Journal of THERMOPLASTIC COMPOSITE MATERIALS, Vol. 19—January 2006 0892-7057/06/01 0079–18 $10.00/0 DOI: 10.1177/0892705706055447 ß 2006 SAGE Publications Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 79 80 M.-L. DANO ET AL. fiber reinforced polymers are being used to make bus body panels, urban train seats, and recreative vehicle parts. To advantageously use these materials and optimize the design process, a reliable prediction of their mechanical behavior is essential. A continuum damage mechanics model was proposed [1] to predict the mechanical behavior of random short fiber reinforced composites. The model development is discussed in detail in [1], but for purposes of completeness, the main equations are briefly presented here. The model uses two phenomenological internal damage variables, D1 and D2, to define the elastic compliance tensor of the damaged material: 2 Cd 1=Eð1 D1 Þ2 1 6 ¼6 4 =Eð1 D1 Þð1 D2 Þ 3 =Eð1 D1 Þð1 D2 Þ 0 1=Eð1 D2 Þ2 0 0 1=Gð1 D1 Þð1 D2 Þ ð1Þ 0 7 7 5 The residual strains and the unilateral behavior relative to crack closure are taken into account. The residual strains are assumed to be a function of the two damage variables and are expressed as: ( "p11 ¼ D1 D2 "p22 ¼ D1 þ D2 ð2Þ where and are two material parameters associated with the permanent strain effects. The evolution laws of the damage variables D1 and D2 are established within a classical thermodynamic framework using the associated thermodynamic forces Y1 and Y2. These forces are determined by derivation of the thermodynamic potential with respect to the associated damage variable: Yi ¼ @U @Di i ¼ 1, 2 ð3Þ The evolution laws are assumed to be coupled to account for a possible interaction between the two principal damage directions. Therefore, for quasi-static loading, the evolution laws for the two damage variables are written as: Di ¼ f YSi ðtÞ i ¼ 1, 2 Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 ð4Þ Experimental Characterization of Damage in Composites 81 where f is a function intrinsic to the material and YSi ðtÞ is defined as: YSi ðtÞ ¼ max Y0 , supt ðYi þ bYj Þ i ¼ 1, 2 i 6¼ j ð5Þ where b is an additional material parameter accounting for coupling effects between the two principal damage directions. It is further assumed that the function f is equal to: f YS ðtÞ ¼ a YSi Y0 ð6Þ where a and Y0 are two additional material parameters. In summary, the model uses a total of five material parameters: three parameters (a, b, Y0) govern the damage evolution laws and two parameters (, ) govern the permanent strain effects. To identify these parameters and validate the model, a series of specific tests have to be conducted on the material. The mechanical behavior of short fiber reinforced composites, especially sheet molding compound (SMC) materials, has been studied experimentally by several researchers [2–6]. Denton [2] characterized the mechanical properties of an SMC-R50 composite, a structural grade SMC with a 50% fiber content in weight, as a function of temperature under static and fatigue loading. Wang and Chim [3] studied fatigue degradation in a random SMC composite and identified various forms of damage mechanisms. Hour and Sehitoglu [4] studied damage evolution in SMC specimens by determining the damage volumetric strain. More recently, Berthaud et al. [5] analyzed the degradation mechanisms of a short fiber reinforced vinyl ester composite from tensile tests and microscopic observations. The material sensitivity to notches or holes was assessed. The effect of biaxial loading was also investigated. Although the mechanical behavior of short fiber composites has been studied for two decades, few attempts have been made to characterize the behavior of short fiber composites, once damage has occurred. Perreux and Siqueira [6] analyzed the influence of damage on a virgin SMC material. A specimen (SMC sheet) was tested in tension to induce predamage. The predamaged sheets were then sampled in the 90 and 45 directions with respect to the loading axis. Tensile tests were performed on the specimens to study the behavior of the damaged material. However, no attempt was made to measure the apparent elastic properties of the damaged material and determine the components of the compliance tensor. This article reports the results of an experimental program carried out to determine the mechanical behavior of a short glass fiber reinforced composite. Specifically, tests were conducted to (1) characterize the material behavior under tensile, compressive, and shear stresses, (2) study Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 82 M.-L. DANO ET AL. the evolution of damage, and (3) measure the apparent elastic properties of the damaged material to determine its compliance tensor components. Specific objectives of the study were also to identify the five material parameters of the theoretical model developed in [1] and to validate its predictions. In the sections to follow, a description of the material investigated is first presented. Then, the results of the tests performed to study the material mechanical behavior are discussed. In particular, the results of a progressive repeated tensile loading test are presented and used to identify the model parameters. Finally, the experiments to evaluate the apparent elastic properties of the damaged material are described and the results presented. MATERIAL DESCRIPTION The material investigated in this study is a short fiber reinforced composite fabricated using a spray-up open-mold technology. Chopped E-glass fiber strands and catalyzed polyester resin are sprayed on the mold surface (with a chopper/spray gun). Manual rollers are then used to remove the entrapped air. This process presents the advantages of being both economical and well adapted to high volume production. The end product contains 30% fiber in weight. The mean diameter and length of the fibers are about 10 mm and 30 mm, respectively. Given that each strand contains about 300 fibers, the length-to-diameter ratio of a strand is 175. Microscopic observations of the composite [7] still revealed the presence of a significant number of entrapped air bubbles, and confirmed the random orientation of the fibers through the thickness. Therefore, the material was expected to initially exhibit an in-plane isotropic behavior. MATERIAL MECHANICAL BEHAVIOR The material was tested in tension, compression, and shear for mechanical behavior characterization. The stress–strain curves shown in Figures 1, 2, 4, and 5 summarize the experimental results. Tensile Behavior The specimen configuration used for the tensile tests is shown in Figure 1(a). A total of ten specimens were cut from 60 cm by 60 cm sheet panels provided by the material supplier. To assess the isotropic behavior of the composite, specimens were cut along two perpendicular Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 83 Experimental Characterization of Damage in Composites 100 50 mm 200 mm 60 mm 18 mm Stress σ1 (MPa) 90 80 x1 70 60 50 40 30 20 10 0 -0.010 -0.005 x2 24 mm Thickness = 4±0.4 mm 0.000 0.005 0.010 Strain (ε) ε2 (a) 0.015 0.020 0.025 ε1 (b) Figure 1. Monotonic tensile behavior: (a) specimen configuration and (b) stress–strain relation. σ1 90 Stress σ1 (MPa) 80 70 60 50 d E1 d E1 40 x1 d 30 ν12 =−(ε1/ε2) 20 x2 10 0 -0.010 -0.005 ε2 0.000 0.005 Strain (ε) 0.010 ε1 0.015 0.020 σ1 Figure 2. Repeated tensile loading test. directions. The rather important specimen thickness variation (0.4 mm) is due to the manufacturing process, which does not allow for tighter tolerances. The specimens were tested in tension at a testing speed of 2 mm/min using a universal testing machine. The longitudinal (x1-direction) and transverse (x2-direction) strains were recorded using an axial and a transverse extensometer. The stress–strain curves obtained for the ten specimens are illustrated in Figure 1(b). No significant differences were visible between the results obtained in the two perpendicular directions. This confirms that the material initially has an in-plane isotropic behavior. The stress–strain curve Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 84 M.-L. DANO ET AL. is quite linear at the beginning. But, as the applied stress reaches a value of about 30 MPa, the curve becomes nonlinear. The tensile modulus was determined from the slope of the initial linear portion of the stress–strain curve, and, accordingly, the Poisson’s ratio was evaluated from the slope of the initial portion of the transverse versus longitudinal strain curve. The average results for elastic modulus, Poisson’s ratio, and tensile strength were 7.08 GPa, 0.35, and 91.6 MPa, respectively. During testing, before the curve becomes nonlinear, noise signaling crack initiation is clearly heard. As the load was increasing, the number of cracks kept increasing at an accelerating pace up to failure. From these observations, it seems that damage initiates as a critical stress level is reached and that it increases gradually until failure occurs. Repeated tensile loading tests with progressively increasing maximum stress were performed on a series of ten specimens to study damage initiation and evolution. The specimens were instrumented with an axial and a transverse extensometer. A typical stress–strain curve is presented in Figure 2. Up to a stress of approximately 30 MPa, the material shows a linear elastic behavior. Beyond this critical stress, the elastic modulus decreases progressively. After unloading, residual strains are observed, though they remain rather small. Such a behavior strongly suggests the apparition and development of damage in the material. Similar observations were reported by other investigators [5,6]. Based on these results, the material behavior can be considered as damageable elastic. For each cycle, the elastic modulus E d1 , Poisson’s ratio d12 , and the residual strains "p11 and "p22 were determined. From the elastic modulus measurements, D1, the damage variable in the direction of loading (x1-direction) could be quantified using Equation (1): E d1 ¼ Eð1 D1 Þ2 ) D1 ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffi E d1 =E ð7Þ In theory, D2, the damage variable in the direction perpendicular to loading (x2-direction) can be evaluated from Poisson’s ratio measurements since from Equation (1): d12 ¼ 1 D1 ) D2 ¼ 1 d ð1 D1 Þ 1 D2 12 ð8Þ However, it turned out that Poisson’s ratio measurements were not precise enough to correctly evaluate D2. Table 1 presents all the experimental data obtained from the repeated tensile test. As can be observed, as the applied stress increases, elastic Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 85 Experimental Characterization of Damage in Composites Table 1. Experimental data from the repeated tensile test. Maximum stress level (MPa) 24.50 39.43 51.41 61.34 70.27 77.30 Elastic modulus Ed1 (GPa) Poisson’s ratio d12 Permanent axial strain ep11 (km/m) Permanent transverse strain ep22 (mm/m) Damage in the direction of loading D1 7.016 6.843 6.591 6.281 5.947 5.643 0.332 0.333 0.331 0.325 0.315 0.299 72.24 240.8 337.1 550.7 722.4 987.2 126.7 267.6 352.1 450.6 549.2 647.8 0.0134649 0.025708 0.043811 0.066579 0.091765 0.115231 Table 2. Numerical values of the unknown parameters (from [1]). Y0 (MPa) a (MPa1) b 0.06819 0.07884 0.5878 0.010071 0.008371 modulus progressively decreases while the permanent strains increase. From the elastic modulus, variable D1 could be evaluated using Equation (7). Poisson’s ratios are also indicated in Table 1. However, the measurements could not be used to accurately evaluate damage variable D2. At the exception of the Poisson’s ratios, the test data of Table 1 were used to identify the five unknown parameters of the model using a constrained optimization technique [1]. The results are reported in Table 2. Note that parameter b is nonzero, which means that applying stresses in the x1-direction has induced damage not only in the x1-direction, but also in the x2-direction. Figure 3 shows typical damage in the material after being tested in direct tension. The micrographs were obtained from scanning electron microscopic observations of a polished sample cut from a specimen tested up to failure. As illustrated in Figure 3(a), large matrix cracks form primarily in the direction perpendicular to the tensile stresses. Their development undergoes different stages. First, cracks initiate at fiber ends and around air bubbles entrapped in the material [7]. As the stress is further increased, the microcracks grow in the material. As illustrated in Figure 3(b), cracks propagate from the bulk matrix towards the matrix–fiber interface, and finally across the fibers. From these microscopic observations, it is expected that damage will induce anisotropy in the material. The elastic properties of the damaged material should be more affected in the direction parallel to the applied load than in the transverse direction. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 86 M.-L. DANO ET AL. σ1 x1 x2 σ1 (a) (b) Figure 3. Scanning electron micrographs of a tension specimen: (a) overall view and (b) close-up view. 200 Back-to-back gages 180 20 mm x1 x2 Stress σ1 (MPa) 160 140 120 100 80 60 40 20 12.7 mm Thickness=5.7 mm (a) 0 -0.024 -0.021 -0.018 -0.015 -0.012 -0.009 -0.006 -0.003 0.000 Strain (ε1) (b) Figure 4. Monotonic compressive behavior: (a) specimen configuration and (b) stress–strain relationship. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 87 Experimental Characterization of Damage in Composites Compressive Behavior The specimen configuration used for the compressive tests is shown in Figure 4(a). Two back-to-back strain gages were mounted on the specimen surface. A total of five specimens were tested in compression at a testing speed of 1 mm/min. Typical compressive test results are presented in Figure 4(b). The stress– strain curve is clearly linear until the specimen failure, without any stiffness loss nor residual strain. The compressive modulus was found to be equal to 8.33 GPa, which is slightly higher than in tension. The compressive strength is 153 MPa, which is 70% larger than the tensile strength. These results are consistent with previously reported data [3,4] showing that short fiber SMC composites are stronger in compression than in tension. Examination of fractured specimens showed that failure was due to matrix shear failure. From the test results, it is clear that the material has different mechanical behaviors in tension and in compression. The material exhibits a damageable-elastic behavior in tension and a brittle linear elastic behavior in compression. Therefore, the model developed to predict the material response [1] has to take into account this so-called ‘unilateral’ behavior. In-plane Shear Behavior The in-plane shear mechanical behavior of the material was determined by applying edgewise shear loads using a three-rail device. As shown in Figure 5(a), specimens are rectangular plates with (0 /90 ) strain gage (0°/90°) Strain gage 152 mm Stress τ (MPa) 80 60 40 20 0 0.000 0.010 0.020 137 mm Strain (γ) (a) (b) 0.030 0.040 Figure 5. Monotonic shear behavior: (a) specimen configuration and (b) stress–strain relation. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 88 M.-L. DANO ET AL. 70 Stress τ (MPa) 60 50 40 30 20 10 0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 Strain (γ ) Figure 6. Repeated shear loading test. rosettes, oriented at 45 to the main axis, mounted in the center of both test sections. The specimens were tested to failure at a testing speed of 1.5 mm/min. A typical stress–strain curve obtained in monotonic shear is shown in Figure 5(b). The material has an average shear strength of 68.8 MPa and an average shear modulus of 2.63 GPa. It is interesting to note that the elastic properties of the intact material satisfy the isotropic condition: G ¼ E=2ð1 þ Þ ¼ 7:08=2ð1 þ 0:35Þ ¼ 2:62 GPa ð9Þ Therefore, the test results confirm that the intact material is isotropic. Repeated shear loading tests with progressively increasing maximum load were performed to study the material damage response. As shown in Figure 6, the shear modulus decreases progressively as the stress increases and a small permanent strain develops. Therefore, the in-plane shear behavior may also be considered as damageable-elastic. Samples cut from specimens tested to failure were examined using a scanning electron microscope to study shear damage. As revealed by the micrographs in Figure 7, matrix microcracks develop at a 45 angle with respect to the direction of the applied shear load. As a matter of fact, the microcracks are oriented perpendicular to the principal tensile stress direction. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 Experimental Characterization of Damage in Composites 89 Figure 7. Scanning electron micrographs of a shear specimen showing 45 matrix microcracks: (a) overall view and (b) close-up view. ELASTIC PROPERTIES OF THE DAMAGED MATERIAL Test Description A series of tests were performed to determine the elastic properties of the damaged material. Material sheets were preloaded in uniaxial tension to produce various degrees of damage. Afterwards, specimens cut from the damaged sheets were tested to determine their residual engineering Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 90 M.-L. DANO ET AL. elastic properties, i.e., the effective Young’s moduli, Poisson’s ratios, and shear moduli. Similar tests were performed in [8] to evaluate the degradation of the elastic properties of steel. Methodology The geometry of the material sheets that were pre-loaded in tension is presented in Figure 8. The specimens were 4 mm-thick, 900 mm-long and 200 mm-wide with tabs at both ends. They were loaded in tension up to four levels of stress (30, 50, 60, and 70 MPa) at a testing speed of 1.5 mm/min. For each stress level, two plates had to be tested to provide a sufficient amount of tensile and shear specimens. Therefore, a total of eight rectangular sheets were tested. Figure 8 shows how the specimens were cut from the sheets. Dog bone shaped specimens (12) were cut along three directions (0, 90, and 45 with respect to the initial loading direction). Rectangular specimens (2) were Plate P1 Figure 8. Sheet dimensions and specimen configurations. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 Plate P2 Experimental Characterization of Damage in Composites 91 cut along the longitudinal and transverse directions. Tensile tests were performed on the dog bone specimens to determine the effective Young’s modulus and Poisson’s ratio in the 0, 90, and 45 directions, referred to respectively, as E d1 and d12 , E d2 and d21 , and E d(45) and d(45). Shear tests were carried out on the rectangular specimens to measure the effective shear moduli Gd12 and Gd21 . Due to symmetry of the elastic tensor, it could be expected to obtain identical results for the two shear moduli. The strains were recorded using the same equipment as for the characterization tests. A longitudinal and a transverse extensometer were used for the tensile specimens and two (0 /90 ) strain gage rosettes were mounted on the shear specimens. Experimental Results Figures 9–11 show the evolution of the various elastic properties as a function of the pre-stress applied to the material. For each pre-stress level, several data points are plotted. Each point was obtained from the test result of a single specimen. Data scatter is important for most properties except for shear modulus. The experimental results reveal that all three effective Young’s moduli, E d1 , E d2 , and E d(45) decrease as the pre-stress increases (Figure 9). Modulus E d1 decreases at the highest rate, which was expected since it is measured in the pre-stress direction. Modulus E d2 exhibits a trend similar to the one of modulus E d(45). The reduction of elastic modulus E d2 shows that damage develops also in the direction transverse to the load, but to a lesser degree than in the loading direction. These results are consistent with scanning electron microscopic observations made on the damaged material. An interesting comparison can be made with somewhat similar tests performed by Perreux and Siqueira [6] on SMC specimens. The investigated material was a random short fiber reinforced sheet molding compound with 26 wt% 25 mm long E-glass fibers in a polyester resin containing calcium carbonate filler. It was also found that the elastic modulus of the material is affected in the 45 -direction. However, tensile tests performed in the 90 -direction showed that the elastic modulus of the material was unchanged. A micrographic study revealed that only small matrix microcracks perpendicular to the load develop under uniaxial tension. Therefore, although the SMC material investigated by Perreux and Siqueira had the same resin type and the same reinforcement, the two materials did not undergo the same damage mechanisms. A deeper analysis would be required to explain the different results, but it is beyond the scope of this study. Figure 10 shows that the effective Poisson’s ratios d12 , d21 , and d(45) are not affected significantly by the pre-stress, irrespective of its level. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 92 M.-L. DANO ET AL. 8 Young's modulus (GPa) (a) 7 6 5 4 Data from Plate P1 3 Data from Plate P2 2 Initial modulus 1 0 0 20 40 60 80 Pre-stress (MPa) 8 Young's modulus (GPa) (b) 7 6 5 4 3 2 Data from Plate P1 1 Data from Plate P2 0 0 20 40 60 80 Pre-stress (MPa) 8 Young's modulus (GPa) (c) 7 6 5 4 3 Data from Plate P1 2 1 0 0 20 40 60 80 Pre-stress (MPa) Figure 9. Degradation of effective Young’s modulus caused by initial pre-stress: (a) effective Young’s modulus E d1 vs pre-stress; (b) effective Young’s modulus E d2 vs pre-stress; and (c) effective Young’s modulus E d(45) vs pre-stress. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 93 Experimental Characterization of Damage in Composites 0.40 (a) Poisson's ratio 0.35 0.30 0.25 0.20 Data from Plate P1 Data from Plate P2 Initial value 0.15 0.10 0.05 0.00 0 20 40 60 80 Pre-stress (MPa) (b) 0.40 Poisson's ratio 0.35 0.30 0.25 0.20 0.15 Data from Plate P1 Data from Plate P2 0.10 0.05 0.00 0 20 40 60 80 Pre-stress (MPa) (c) 0.40 Poisson's ratio 0.35 0.30 0.25 0.20 0.15 Data from plate P1 0.10 0.05 0.00 0 20 40 60 80 Pre-stress (MPa) Figure 10. Degradation of effective Poisson’s ratios caused by initial pre-stress: (a) effective Poisson’s ratio d12 vs pre-stress; (b) effective Poisson’s ratio d21 vs pre-stress; and (c) effective Poisson’s ratio d(45) vs pre-stress. Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 94 M.-L. DANO ET AL. Shear modulus (GPa) 3 2 d G 12 1 d G 21 Initial modulus 0 0 20 40 60 80 Pre-stress (MPa) Figure 11. Effective shear modulus vs initial pre-stress. Finally, Figure 11 presents the results obtained for the shear tests. For each level of pre-stress, the average effective shear moduli Gd12 and Gd21 obtained from the two specimens are very close. As for the tensile modulus, the shear modulus decreases slightly as damage develops. The experimental results presented in Figures 9–11 were used to validate the choice for the elastic tensor of the damaged material and verify the reliability of the identified set of parameters. The reduction of the effective Young’s modulus E d2 (Figure 9(b)) revealed that damage was induced in the direction transverse to the load, which is consistent with the value found for parameter b in the identification process. The evolution of shear modulus Gd12 and Poisson’s ratios d12 and d21 were used to verify the expressions derived for the elastic properties of the damaged material. Finally, the results for the tensile tests along the 45 -direction allowed a further validation of the model. Comparisons between the experimental and simulated results can be found in [1]. Overall, the predictions agree quite satisfactorily with the test data. CONCLUSIONS An experimental program was conducted to characterize the mechanical behavior of a sprayed random short fiber reinforced material. The results of tensile, compressive, and shear tests showed that the material is initially characterized by an in-plane isotropic behavior, a damageable-elastic behavior in tension, and a brittle elastic behavior in compression. The residual elastic properties of the damaged material were also determined. The experimental results revealed that damage causes the elastic Downloaded from jtc.sagepub.com at Bibliotheque de l'Universite Laval on March 26, 2015 Experimental Characterization of Damage in Composites 95 stiffness of the material to decrease. However, damage does not develop in an isotropic fashion. The Young’s modulus in the loading direction is more affected than in the direction perpendicular to the applied load. Part of the experimental program was used to identify the parameters of a continuum damage mechanics constitutive model. A total of five parameters defining the constitutive laws for damage and plasticity were evaluated. Other results obtained from the experimental program were used to partially validate the predictions of the model. Additional experiments, in particular biaxial tests, will be required to complete the model validation. NOMENCLATURE E ¼ Young’s modulus of the undamaged material E d1 ¼ Young’s modulus of the damaged material in the x1-direction E d2 ¼ Young’s modulus of the damaged material in the x2-direction d E (45) ¼ Young’s modulus of the damaged material in the 45 -direction G ¼ shear modulus of the undamaged material Gd12 ¼ shear modulus of the damaged material in the 1–2 plane Gd21 ¼ shear modulus of the damaged material in the 2–1 plane ¼ Poisson’s ratio of the undamaged material d12 ¼ Poisson’s ratio of the damaged material for uniaxial stress in the x1-direction d21 ¼ Poisson’s ratio of the damaged material for uniaxial stress in the x2-direction d(45) ¼ Poisson’s ratio of the damaged material for uniaxial stress in the 45 -direction 1 ¼ uniaxial stress applied in the x1-direction ¼ shear stress applied in the 1–2 plane ACKNOWLEDGMENTS The authors would like to thank NSERC, ADS Groupe Composites, Bombardier and Prevost Car for their financial support. REFERENCES 1. 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