Constitutive modelling of composite laminates under uniaxial compression

Hsuan-Teh HU, Lung-Shen KE

Department of Civil Engineering, ‘‘National” Cheng Kung University, Tainan 701, China

KEYWORDS Composite laminates;Constitutive model;Finite element analysis;Nonlinearity;Uniaxial compression

Abstract A nonlinear constitutive model for a single lamina is proposed for the failure analysis of composite laminates. In the material model, both fiber and matrix are assumed to behave elasticplastically and the in-plane shear is assumed to behave nonlinearly with a variable shear parameter.The damage onset for individual lamina is detected by a mixed failure criterion, composed of the Tsai-Wu criterion and the maximum stress criterion. After damage takes place within the lamina,the fiber and in-plane shear are assumed to exhibit brittle behavior, and the matrix is assumed to exhibit degrading behavior. The proposed nonlinear material model is tested against experimental data of composite laminates subjected to uniaxial compressive loads, and good agreement is obtained.

1. Introduction

Due to lightweight and high strength, the use of composite laminate materials in aerospace industry has increased rapidly in recent years. Numerous cases involving the design of composite structures show that there is a need for more refined analysis that takes into account phenomena such as progressive failure and inelastic or nonlinear deformation of composite materials.1-12It is well known that unidirectional f ibrous composites exhibit severe nonlinearity in their in-plane shear stress-strain relations.13In addition, deviation from linearity is also observed with in-plane transverse loading but the degree of nonlinearity is not comparable to that observed with the inplane shear.14,15Therefore, appropriate modeling of the nonlinear behavior of fiber-reinforced composite materials becomes crucial.

A signif icant number of macro-mechanical models have been proposed to represent the constitutive relation of fiberreinforced composite materials such as nonlinear elasticity models,13,14,16plasticity models,17-19or damage theory coupled with elasticity.20In addition, various failure criteria have also been proposed to predict the onset of damage in single layers within fiber-reinforced composites, i.e., limit theories,21polynomial theories,22,23and direct mode-determining theories.24-27As for the post-damage process of individual lamina,two idealized types of failure modes have been defined in a previous study;19namely, brittle mode and ductile mode. In the case of the brittle mode,the material gives up its entire stiffness and strength in the dominant stress direction as the damage is reached.For the ductile mode,the material retains its strength but loses its overall stiffness in the direction of damage.

Obviously, a rational analysis of the individual layers within the laminate under loading must include three parts,i.e., pre-damage analysis, damage onset determination, and post-damage analysis. In the pre-damage analysis, the proper constitutive model of lamina is a key tool to describe the real behavior of each layer within the laminate under loading. In previous studies, it is assumed that the fiber and matrix perform with elastic-plastic behavior,19and the in-plane shear behaves nonlinearly with a constant shear parameter.25In this study, it is proposed that the in-plane shear behaves nonlinearly with a variable shear parameter. The difference between these two distinct types of shear parameters is investigated in this study. Up to now, the Tsai-Wu failure criterion8,23is the most common criterion used to determine the damage onset of individual layer. However, Hu et al.28-30point out that the Tsai-Wu failure criterion would cause overestimated failure stresses in the fiber direction of the composite lamina. To eliminate this unreasonable phenomenon, an extra limitation should be added into the Tsai-Wu failure criterion to obtain a more accurate and reasonable stresses in the composite lamina. As the results, Zhu and Sankar31and Hu et al.28-30suggested that the combination of both the Tsai-Wu criterion and the maximum stress criterion, which is called the mixed criterion, was a much better criterion for determining the damage to lamina. Thus, in this paper, a mixed criterion is employed to determine the damage onset of individual layers. For the post-damage analysis,a degrading mode for matrix and brittle modes for fiber and in-plane shear are proposed to simulate the post damage behavior of individual lamina.

In this paper, the proposed nonlinear analysis model is developed first.Then,various failure criteria and post damage modes are reviewed,and a mixed failure criterion and the postdamage modes are proposed. Finally, the ABAQUS finite element program32is used to verify the proposed constitutive model against experimental data of Petit and Waddoups14and the conclusions obtained from the numerical analysis are given.

2. Nonlinear analysis model

2.1. Proposed stress-strain curves and post damage models

Fig.1 shows the material,element and structure coordinates of fiber reinforced plastics. In Fig. 1, (1, 2, 3)means the material coordinate,where 1 indicates the fiber direction,2 indicates the orthogonal direction to the fiber direction in the plane of the lamina, and 3 refers the transverse direction to the plane of the lamina. In addition, (x, y, z) represents the element local coordinate system, and (X, Y, Z) the structural global coordinate system.

Fig. 1 Material, element and structure coordinates of fiber reinforced plastics.

For a single lamina subjected to tensile loading, the stressstrain curves of the nonlinear analysis model are shown in Figs. 2(a) and (b). It is assumed that the material response can be represented by elastic-plastic stress-strain curves in the principal material directions, i.e., the 1 direction (fiber direction)and the 2 direction(transverse direction),of the lamina. Let Xytand Xutbe the yield strength and the ultimate strength of the lamina for tension in the 1 direction, Yytand Yutbe the yield strength and the ultimate strength of the lamina for tension in the 2 direction. For the elastic regions, i.e.,σ1≤Xytand σ2≤Yyt, the elastic moduli are denoted by Eiie(i=1,2). For the plastic regions, i.e., Xyt≤σ1≤Xutand Yyt≤σ2≤Yut, the elastic moduli are denoted by Eiipt(i=1,2). For a lamina subjected to compressive loading, the stress-strain curves are shown in Figs. 2(c) and (d). It is obvious that Xycand Xucare the yield strength and the ultimate strength of the lamina for compression in the 1 direction and that Yycand Yucare the yield strength and the ultimate strength of the lamina for compression in the 2 direction.For the plastic regions, i.e., Xyc≤σ1≤Xucand Yyc≤σ2≤Yuc, the elastic moduli are denoted by Eiipc(i=1,2). Let S be the ultimate in-plane shear strength. It is assumed that the in-plane shear in the 1-2 direction can be modeled by a nonlinear stress-strain curve as shown in Fig. 2(e).

Fig. 2 Stress-strain curves of the proposed nonlinear failure model.

For the post-damage region,the strengths of the lamina are dropped to zero (brittle modes) in the 1 direction (Figs. 2(a)and (c)) and the 1-2 direction (Fig. 2(e)). This is because the failures of the lamina in 1 and 1-2 directions are dominated by fiber and its failure mode is sudden and brittle.It is known that the failure of the lamina in the 2 direction is dominated by matrix. After the failure of the matrix occurs, its strength would drop to zero gradually. Hence, in the proposed model the elastic stiffness of the lamina is assumed to have a negative modulus E22fin the 2 direction(degrading mode)and the damaged lamina unloads in the transverse direction until no stress remains in the lamina (Figs. 2(b) and (d)).

2.2. Nonlinear constitutive model of the lamina

For fiber-composite laminate materials, each lamina can be considered to be an orthotropic layer in a plane stress condition. Taking into account the elastic-plastic behavior in the 1 and 2 directions and the nonlinear behavior on the 1-2 plane within the lamina,the strain-stress relations for an orthotropic lamina in the material coordinates (1,2) can be written as13

where ε1, ε2, and γ12represent the strains in the 1 direction, 2 direction and the 1-2 plane, respectively. The σ1, σ2and τ12denote the stresses in the 1 direction, 2 direction and the 1-2 plane, respectively. The υ12and υ21are Poisson’s ratios, and E11and E22are the elastic moduli in the 1 and 2 directions.If the lamina is in the elastic stage in the 1 or 2 direction,then E11=E11eor E22=E22e.If the lamina is in the plastic stage in the 1 or 2 direction,then E11=E11por E22=E22p.The G12is the shear modulus,and S66is a shear parameter to account for the in-plane shear nonlinearity.The S66shear parameter in Eq.(1) was derived by Hahn and Tsai13using the complementary elastic energy density and has been verif ied against the experimental data on off-axis tests.For a specific composite lamina,the value of S66can be determined by a curve f it to pure shear test data.

The incremental stress-strain relations for a nonlinear orthotropic lamina can be given as follows:

where

The terms α1and α2are the shear correction factors and are taken to be 0.83 in this study.33It is assumed that the transverse shear stresses always behave linearly and do not affect the nonlinear in-plane behavior of individual lamina.

3. Failure criterion, degradation of stiffness and laminate governing equations

3.1. Review of failure criteria

As previously mentioned, failure criteria fall into three basic categories: (A) limit theories,21(B) polynomial theories,22,23and (C) direct mode determining theories.24-27Among them,the most popular criteria, i.e., the maximum stress criterion21and the Tsai-Wu failure criterion,23are selected to be brief ly reviewed.

3.1.1. Maximum stress criterion

The maximum stress criterion is the dominant member of the limit failure theory category.21For the plane stress condition,the maximum stress criterion for an orthotropic material can be expressed as follows:

3.1.2. Tsai-Wu failure criterion

The Tsai-Wu failure criterion has a general nature, because it contains almost all other polynomial theories as special cases.Under the plane stress condition,the Tsai-Wu failure criterion has the following form:23

where

The stress interaction term F12in Eq. (9a) is difficult to determine. In lieu of additional experimental data on other materials and loading conditions, a numerical experiment was performed by Narayanaswami and Adelman34to estimate the errors for ten different composite systems under six different loadings. The maximum error in predicted failure loads among all cases was below 10 percent. These results suggest that the Tsai-Wu failure criterion with F12=0 can predict failure of practical composite materials under general biaxial loading with suff icient accuracy for engineering applications.Thus, the use of Tsai-Wu failure criterion with F12=0 is recommended as a preferred alternative to the experimental determination of F12for orthotropic laminae.As the result,F12=0 is used in this study.

3.2. Mixed failure criterion

Although the Tsai-Wu failure criterion is widely used in determining the damage onset of a lamina, there are some drawbacks with it.Among them is the fact that the failure stress of fiber in a lamina may exceed the strength of the material in the case of symmetric angle-ply laminates with a small fiber angle(say 0°＜θ ＜20°)subjected to off-axis tension.28,29In order to eliminate this unreasonable phenomenon,the limitation of the maximum stress of the lamina in the fiber direction is added into the Tsai-Wu failure criterion to obtain a mixed failure criterion, which has the following formulations:

and

3.3. Normalized failure stresses and failure contribution

The Tsai-Wu failure criterion and the mixed failure criterion consider the coupling effect of in-plane stresses, σ1, σ2and τ12, in the lamina when the collapse occurs. In order to determine the individual stress ratio in the lamina, the normalized failure stresses are defined, which represent the stress ratios(failure stresses/corresponding strengths)in the lamina for various stresses at the onset of collapse. The expressions for the normalized failure stresses are described as follows:

where σ11f（ ）n, σ22f（ ）nand τ12f（ ）ndenote the normalized failure stresses in the 1 direction, 2 direction, and the 1-2 plane of the lamina, respectively. The σ11f, σ22fand τ12fare the stresses of the lamina in the 1 direction,2 direction and the 1-2 plane at the onset of failure.

3.4. Proposed degradation models

Upon damage within the lamina occurring,the material properties begin to degrade. Material degradation within the damaged area is evaluated by the mode of failure predicted by the failure criterion. Therefore, the residual stiffness of composite strongly depends on the failure mode in each layer.According to the literature,the degradation models for each layer are categorized into three types of failure modes, i.e., the brittle mode, the ductile mode19and the degrading mode.28-30For the brittle mode, the material loses its entire stiffness and strength in the dominant stress direction.For the ductile mode,the material retains its strength but loses all of its stiffness in the failure direction. For the degrading mode, the material loses its stiffness and strength in the failure direction gradually until the stress in that direction is reduced to zero.

In this investigation, it is proposed that the post damage modes are the brittle behavior for σ1and τ12, and the degrading behavior for σ2(Fig.2).The following three rules are used to determine whether the ply failure is caused by matrix fracture, shear failure, or as a result of fiber breakage or buckling35,36:

(1) If a ply fails in the condition of Xuc＜σ1＜Xut, and-S ＜τ12＜S, the damage is assumed to be matrix induced. Consequently, the degradation of transverse stiffness occurs. Due to the interlock action with the neighboring plies, the damaged ply gradually loses its capability to support transverse stress,until the fracture in shear or the breakage or buckling in fiber on the same plyoccurs. However, the lamina remains able to carry the longitudinal and shear stresses.In this case,the constitutive matrix of the lamina becomes

where E22fis a negative tangent modulus in the transverse direction of the lamina after matrix damage. In the proposed model, the shear parameter S66has a variable value.

(2) If the ply fails in the condition of Xuc＜σ1＜Xut, and τ12≥S or τ12≤-S,the damage is assumed to be shear induced. Consequently, the damaged lamina loses its capability to support transverse and shear stresses, but remains able to carry longitudinal stress. In this case,the constitutive matrix of the lamina becomes

(3) If the ply fails with σ1≥Xut, or σ1≥Xuc, the ply failure is caused by fiber breakage or buckling and a total ply rupture is assumed. Thus, the constitutive matrix of the lamina becomes

3.5. Laminate governing equations

The forgoing nonlinear failure analysis model for fiberreinforced composite lamina can be combined with classical lamination theory to form the following incremental laminate force-strain relations: in-plane

4. Numerical analysis

4.1. Numerical simulations and material properties

The aforementioned nonlinear constitutive model combined with various failure criteria and various post damage modes for composite materials are written in a FORTRAN subroutine, i.e., the user-defined material model, UMAT. Then this UMAT subroutine is linked to the ABAQUS finite element program32for the numerical analyses. The subroutine will be called at all material calculation points of elements for which the material def inition includes a user-defined material behavior. The ABAQUS program will pass the previous strains and the current incremental strains to the UMAT subroutine. The subroutine must update the stresses and solution-dependent state variables to their values at the end of the increment for which it is called. In addition, it must provide the material Jacobian matrixto the ABAQUS main program.The analyzed laminate plate is simply supported around all edges and subjected to uniaxial compressive loads in the longitudinal direction only (Fig. 3). The length of the plate L is equal to 12 cm and the width of the plate W is equal to 2 cm. The laminate plate contains 4 plies with the thickness t of each ply equal to 0.1016 mm. The laminae are assumed to be perfectly bounded and no slipping occurs within the laminate. Due to the boundary conditions, the laminated plate is free to expand or contract in both x and y directions. Since the plate has no geometric imperfection in z direction, lateral def lection of the plate in z direction will not occur. Thus, the failure of the plate is due to material and not due to buckling.The stresses are uniformly distributed throughout the entire laminate plate.Hence,only one eight-node isoparametric shell elements with six degrees of freedom per node (three displacements and three rotations) is used to model the plate. The reduced integration rule together with hourglass stiffness control is employed to formulate the element stiffness matrix.32

In the ABAQUS program, stresses and strains in material coordinates (1, 2, 3) are calculated at each incremental step,and are evaluated by the failure criteria to determine both the occurrence of failure and the mode of failure. Mechanical properties of each lamina in the damaged area are reduced,according to proper degradation models. Stresses and strains are then recalculated to determine any additional damage as a result of stress redistribution at the same load. This procedure continues until no additional damage is found. Then,the next increment of load is pursued. The final collapse load is determined when the composite plates cannot sustain any additional load.

Fig. 3 Geometry and boundary conditions of composite laminates.

In order to verify the proposed nonlinear failure analysis model, numerical results generated from the model are compared with the test data of Boron/Epoxy composites.14The material properties of Boron/Epoxy composites used in the analysis are E11e=207 GPa, E11pt=E11pc=180 GPa,Yuc=-262 MPa, S=128.6 MPa. It should be noted that the shear parameter S66has two types, a variable type and a constant type. The variable shear parameter is obtained by curve f itting from the pure shear test data.15

4.2. Verif ication of the proposed nonlinear constitutive model

It is necessary to assure that the proposed constitutive model can correctly simulate the stress-strain relations in the principal directions and in a pure shear of a lamina before it is utilized to predict the mechanical behavior and failure stresses of entire composite laminates under various loadings. Fig. 4 shows the numerical results for a single lamina subjected to pure shear loading against the experimental data.14It is obvious that the shear stress-strain curve simulated by the proposed constitutive model with the variable S66model agrees with the test data well and is much better than those with the linear shear model(S66=0)and the nonlinear shear model with S66being constant.

Fig. 4 Pure shear stress-strain curves for Boron/Epoxy lamina.

Fig. 5 Uniaxial tensile stress-strain curves for [±45]s Boron/Epoxy laminate.

Fig. 6 Longitudinal compressive stress-strain curves for Boron/Epoxy lamina.

The results simulated by the variable S66model and the constant S66model for a[±45]slaminate subjected to uniaxial tension loading against the experimental data14are shown in Fig.5.It can be seen that the result simulated by the proposed nonlinear shear model with the variable S66exhibits better f it with the test data than that simulated by the nonlinear shear model with the constant S66.

Figs. 6 and 7 illustrate the numerical results for a single lamina subjected to uniaxial longitudinal compressive loading and uniaxial transverse compressive loading against the experimental data.14It can be seen that the proposed elastic-plastic behavior in the longitudinal direction and transverse direction of the lamina exhibit quite good correlation with the experimental data. As a result, the proposed material model with the variable S66is proved to model the nonlinear behavior of composite laminates adequately.

4.3. Comparisons among various matrix post failure modes

The load-deformation behavior of a composite laminate is greatly affected by the stress-strain behavior of individual layers within the laminate,and the ultimate strength of a composite laminate is greatly controlled by the post-damage mode of damaged lamina within the laminate. In order to verify that the proposed post-damage mode in the transverse direction of the lamina is suitable, three idealized post failure modes,brittle, ductile and degrading modes, are taken into account.Fig. 8 shows the uniaxial compressive stress-strain curves predicted by various matrix post failure modes for a[±20]sangleply laminate against the experimental data.14The ultimate loads predicted by the brittle mode and the degrading mode are exactly the same and equal to 0.82 GPa, which is close to the experimental data 0.75 GPa.For the ductile mode,the predicted ultimate load is signif icantly overestimated and is misleading.

Fig. 7 Transverse compressive stress-strain curves for Boron/Epoxy lamina.

Fig. 9 shows the uniaxial compressive stress-strain curves predicted by various matrix post failure modes for a [±30]sangleply laminate against the experimental data.14The ultimate load predicted by the degrading mode is 0.36 GPa,which is close to the experimental data 0.30 GPa. In the case of the brittle mode, instability exists in the fiber direction immediately after the failure of the lamina occurs. As the result, the compressive strain in the laminate is suddenly and signif icantly increased, which is contradictory to the experimental data.When the stress redistribution in the laminate is completed,the stress in the laminate starts to increase again, and the predicted ultimate load is 0.44 GPa, which is overestimated. For the ductile mode, the predicted ultimate load is again signif icantly overestimated. Hence, it is appropriate and justif ied to use the degrading mode to model the post failure of a lamina.

Fig. 8 Uniaxial compressive stress-strain curves predicted by various matrix post failure modes for [±20]s laminate.

Fig. 9 Uniaxial compressive stress-strain curves predicted by various matrix post failure modes for [±30]s laminate.

Fig. 10 Uniaxial compressive stress-strain curves predicted by various matrix post failure modes for [0/90]s laminate.

Fig. 10 shows the uniaxial compressive stress-strain curves predicted by various matrix post failure modes for a [0/90]scross ply laminate against the experimental data.14While the ultimate load predicted by the brittle mode is 1.50 GPa, the ultimate loads predicted by the ductile mode and by the degrading mode are the same and equal to 1.54 GPa. It can be seen that all these predicted ultimate loads are close to the experimental data 1.74 GPa.

Figs. 11-13 show the uniaxial compressive stress-strain curves predicted by various matrix post failure modes for[15/75]s, [30/60]sand [±45]scross ply laminates. Since there are no experimental data to compare, we can only see the trends. It can be observed that the predicted stress-strain curves as well as the ultimate loads with different post failure modes are exactly the same for all these three cross ply laminates. In the case of the laminates with a [θ/θ-90]scross ply layup, as the fiber angles are more deviated from 0° and 90°,the stress-strain curves of the laminates exhibit more nonlinear behavior. This is due to the nonlinear in-plane shear effect. In addition, as the fiber angles are more deviated from 0° and 90°, the ultimate loads of the laminates become lower.

4.4. Comparisons among various failure criteria

The Tsai-Wu failure criterion23is the most popular failure criterion for composite laminate and has been extensively used in the literature on this topic. As mentioned previously, with the Tsai-Wu failure criterion,the failure stress of fiber in a lamina may exceed the strength of the material in the case of symmetric angle-ply laminates with a small fiber angle subjected to offaxis tension.28,29Hence, it is replaced by the mixed failure criterion, i.e., the Tsai-Wu failure criterion combined with the maximum stress criterion.In this section,the mixed failure criterion is compared with other popular criteria such as the Rotem criterion27, the Edge criterion26and the Chang criterion25against the experimental data.14

Fig. 11 Uniaxial compressive stress-strain curves predicted by various matrix post failure modes for [15/-75]s laminate.

Fig. 12 Uniaxial compressive stress-strain curves predicted by various matrix post failure modes for [30/-60]s laminate.

Fig. 13 Uniaxial compressive stress-strain curves predicted by various matrix post failure modes for [±45]s laminate.

Fig. 14 shows the uniaxial compressive stress-strain curves predicted by various failure criteria for a[±20]sangleply laminate against the experimental data.14The predicted failure loads are 818 MPa for the Rotem criterion, 775 MPa for the Edge criterion, 810 MPa for the Chang criterion and 818 GPa for the mixed criterion and the experimental failure load is 755 MPa (see Table 1). All the failure loads and the stiffnesses of the laminate predicted by the four criteria are reasonably close to the experimental data.

Fig. 14 Uniaxial compressive stress-strain curves predicted by various failure criterion for [±20]s laminate.

Fig. 15 shows the uniaxial compressive stress-strain curves predicted by various failure criteria for a [±30]sangleply laminate against the experimental data.14The predicted failure loads are 406 MPa for the Rotem criterion,355 MPa for the Edge criterion, 370 MPa for the Chang criterion and 362 MPa for the mixed criterion, and the experimental failure load is 300 MPa(see Table 1). It can be seen that the Rotem criterion overestimates the stiffness and the failure load of the laminate as compared to the other three criteria. Although, the failure loads of the laminate predicted by the other three criteria are also higher than the experimental data, the stiffnesses of the laminate predicted by the other three criteria are acceptable.

Fig. 16 shows the uniaxial compressive stress-strain curves predicted by various failure criteria for a[±60]sangleply laminate against the experimental data14. The predicted failure loads are 226 MPa for the Rotem criterion, 229 MPa for the Edge criterion, 249 MPa for the Chang criterion and 249 MPa for the mixed criterion, and the experimental failure load is 389 GPa (see Table 1). Although, the failure loads of the laminate predicted by all criteria are signif icantly underestimated as compared to the experimental data,the stress-strain curve predicted by the Rotem criterion still deviates from those predicted by the other three criteria.

Fig. 17 shows the uniaxial compressive stress-strain curves predicted by various failure criteria for a[0/90]scross ply laminate against the experimental data.14The predicted failure loads are 1503 MPa for the Rotem criterion, 1503 MPa for the Edge criterion, 1535 MPa for the Chang criterion and 1535 MPa for the mixed criterion,and the experimental failure load is 1740 MPa(see Table 1).Again all the failure loads and the stiffnesses of the laminate predicted by the four criteria are reasonably close to the experimental data.

Fig. 15 Uniaxial compressive stress-strain curves predicted by various failure criteria for [±30]s laminate.

Fig. 16 Uniaxial compressive stress-strain curves predicted by various failure criteria for [±60]s laminate.

Fig. 18 shows the uniaxial compressive failure stresses σxfpredicted by various failure criteria for [±θ]sangleply laminates against the experimental data.14Fig. 19 shows the normalized material failure stresses predicted by the various failure criteria for [±θ]sangleply laminates subjected to uniaxial compressive loading. It can be observed that the failure stress σxfand the normalized failure stresses predicted by the Tsai-Wu failure criterion and the mixed failure criterion are almost the same with only a small discrepancy in the neighborhood of the angle θ=45°. It should be noted that around the region θ=45°,the predicted normalized failure stress(τ12f)nof the Tsai-Wu failure criterion exceeds the value of 1(see Fig.19(a)), which is unreasonable and unlikely to occur. This is thereason why the authors propose a mixed failure criterion for the cases of laminates subjected to uniaxial tensile loads.28,29

Table 1 Ultimate loads predicted by the various model versus the experimental data.

Fig. 17 Uniaxial compressive stress-strain curves predicted by various failure criteria for [0/90]s laminate.

Fig. 18 Uniaxial compressive failure stresses predicted by various failure criteria for [±θ]s angleply laminates.

Comparing the Chang failure criterion with the mixed criterion, we can determine that the failure stress σxfand the normalized failure stresses predicted with these two criteria are about the same except in the region 0°≤θ ≤15°. This is because the Chang failure criterion does not consider the joint effect due to axial stress, transverse stress and in-plane shear stress acting simultaneously. When 0°≤θ ≤10°. we can see that the failure of the angleply laminate is due to axial failure stress(σ11f)nfor the Chang criterion(see Fig.19(a)).If the joint effect of axial stress,transverse stress and in-plane shear stress is considered, such as the Tsai-Wu failure criterion, the predicted failure stress σxfand the normalized failure stresses (in absolute values) will not be so high within the 0°≤θ ≤15°region.

In the case of the Edge criterion,it can be seen that its predicted failure stress σxfis usually lower than those predicted by other criteria (see Fig. 18). The reason for this is the fact that its predicted stresses σ1,σ2and τ12are smaller (in absolute value) than those predicted by other criteria, which can be clearly seen in Fig. 19(b).

Fig.19 Normalized material failure stresses predicted by various failure criteria for [±θ]s angleply laminates subjected to uniaxial compressive loading.

In the case of the Rotem criterion, the predicted failure stress σxfis usually higher than those predicted by other failure criteria when θ ≤45°and lower than those predicted by other failure criteria when θ ≥45°(see Fig.18).When 0°≤θ ≤15°,the Rotem criterion, like the Chang criterion, does not consider the joint effect due to axial stress, transverse stress and in-plane shear stress acting simultaneously. Hence, it has the similar trend as the Chang criterion. When 15°≤θ ≤80°,the predicted transverse stress σ2by the Rotem criterion is lower than that by mixed criterion, and the predicted in-plane shear stress τ12by the Rotem criterion is higher than that by mixed criterion(see Fig.19(b)).This is due to the Rotem model treating the in-plane shear as having linear behavior. Therefore, it overestimates the in-plane shear stress. After the failure criterion is employed, the transverse stress is then underestimated.

5. Conclusions

This paper proposes a material constitutive model for the analysis of composite laminates subjected to uniaxial compressive loads. The main outcomes of the model contain three parts:(A) nonlinear constitutive model, (B) mixed failure criterion,and (C) post-damage mode. In the nonlinear constitutive model, the fiber and matrix are simulated by elastic-plastic behavior and the in-plane shear is simulated by nonlinear behavior with a variable shear parameter. The mixed failure criterion is composed of the Tsai-Wu failure criterion and the maximum stress criterion,which can avoid the overestimation of stresses predicted by the Tsai-Wu failure criterion. In the post-damage regions, the fiber and the in-plane shear are simulated by a brittle mode, and the matrix is simulated by a degrading mode. The validity of the constitutive model has been verif ied against the experimental data14and reasonable accuracy has been achieved.

The proposed constitutive model has been compared with other popular failure criteria for[±θ]sangleply laminates subjected to uniaxial compressive loads. In the case of the Chang criterion, the predicted failure stress σxfis usually higher than the mixed criterion and the Tsai-Wu criterion when 0°≤θ ≤15°. In the case of the Edge criterion, the predicted failure stress σxfis usually lower than the mixed criterion and the Tsai-Wu criterion. In the case of the Rotem criterion,the predicted failure stress σxfis usually higher than those predicted by the mixed criterion and the Tsai-Wu criterion when θ ≤45°and lower than those predicted by the mixed criterion and the Tsai-Wu criterion when θ ≥45°.