We present the prediction of nonlinear in-plane shear stress strain curves based on complementary strain energy density function. When the [θo] unidirectional laminates are subjected to a combination of shear stresses and axial stresses, the predicted shear responses of laminates are compared with measured experimental data for verifying this accuracy of this model. In this process, the compliance constants are determined to match the measured test results in terms of the influence of on-axis axial stress. Finally, it is shown that the predicted shear stress strain behavior of E-glass/vinylester concurs with experimental curves for all off-axis angles, while shear behavior predictions of boron/epoxy and T800H/3633 laminates agree with experimental results for 30° 45° 60° off-axis angles laminates except 15° off-axis angle laminates. This is because the elastic longitudinal modulus is still significantly higher than the other modulus even for a small off axis angle.

본 논문에서는 보완된 변형 에너지 밀도 함수를 기반으로 하여 평면내 비선형 응력-변형률 곡선 모델을 제시한다. 다방향의 적측판에 전단방향과 축방향의 응력이 조합되어 작용할 때의 전단방향의 응답을 예측하였으며, 실험값과 비교하여 제시 모델의 정확성을 검증하였다. 이 과정에서 compliance 상수는 기존 축방향의 응력에 대한 측정값과 일치하도록 결정되었다. 마지막으로 제시된 모델을 이용한 E-glass/vinylester의 전단 응력-변형률 거동은 모든 축외 각도에 대하여 실험 곡선과 일치하며, boron/epoxy와 T800H/3633 적층판의 전단 거동은 15도를 제외한 30도, 45도, 60도의 축외 각도(off-axis angle)에 대하여 실험값과 일치한다. 이는 작은 적층 각도의 변화에도 종방향의 탄성계수는 다른 탄성계수에 비하여 매우 높기 때문이다.

Keywords: nonlinearity, in-plane shear, off-axis angles, GFRP, CFRP

As composite laminates are the widely used in practical engineering, the accurate analyses of mechanical properties are becoming important, so it is necessary to confirm the correct stress strain relationship in the failure analysis of composite laminates. There are actually numerous experimental results and theoretical studies on the stress strain relationship. In this work, mechanical responses of E-glass/vinylester, boron/epoxy and carbon/epoxy laminates are compared with theoretical predictions. The following three paragraphs are a review of three different laminates experimented by other researchers, and the remaining paragraphs in the section are different kinds of approaches to describe the nonlinear behavior.
The paragraph is intended to describe the mechanical behavior of E-glass/epoxy laminates. Aboudi¹ measured shear stress strain responses of unidirectional laminates, and also measured axial stress stain curves of angle-ply laminates (±15°, ±30°, ±35°, ±41°, ±45°). Haj-Ali² performed experiments to achieve the transverse compressive and shear responses of unidirectional laminates. Paepegem³ investigated the in-plane shear response of [±45°] and off-axis [10°] stacking sequences lam-inates under tensile tests. Soden⁴ listed experimental data of inplane shear and transverse compressive behavior of E-glass/epoxy unidirectional laminates. The axial stress strain relationship of UD laminates seems to be linear while the hoop stress strain behavior shows the elastic – plastic performance, however, the axial stress strain response could be nonlinear when off-axis angles of UD laminates are nonzero. Comparing with the in plane shear stress train response, although it can be easily observed that the nonlinearity of hoop stress strain response exists at a high strain level, the nonlinearity of shear direction is so more significantly pronounced, no matter what the stacking sequence is UD or [±45°].
The mechanical properties of boron/epoxy laminates are investigated considering axial and in-plane shear directions. P. H. Petit⁵ presented the stress strain responses in the longitudinal, transverse and in-plane shear directions, respectively. These laminates are classified as 0° unidirectional lamina, balanced and symmetric angle-ply laminates: [±20°], [±30°], [±60°], balanced and symmetric cross-ply laminates: [0°/90°] and quasi-isotropic laminates [0°/±45°]. Aboudi1 presented axial stress versus axial strain and axial stress versus hoop strain curves of off-axis laminates (θ=15°, 30°, 45°, 60°) under uniaxial loading, and also presented the axial stress strain curves of angle-ply laminates (±20°, ±30°, ±45°, ±60°). The uniaxial stress strain relation of UD laminates is almost linear when the lamina is subjected to the axial loading, while the inplane shear stress show significant nonlinearity. The axial stress versus axial strain and axial stress versus hoop strain curves are nonlinear for off-axis laminates, and the nonlinearity is more obvious as the off-axis angles are 15°, 30° and 45°. The uniaxial stress strain response of 60° off-axis laminates shows slightly nonlinear. When the stacking sequences are [±30°], [±45°], and [±60°], the nonlinear responses of uniaxial stress strain are easily to be observed, while response of [±20°] laminates is nearly to be linear.
As for studies of mechanical behavior of carbon/epoxy laminates composed of different classes of fiber (AS4, T300, T700), Soden4 presented experimental data and figures for the properties of the two kinds of AS4/3501 and T300/BSL914C laminates. The stress strain curves are slightly nonlinear at a high strain levels under the conditions of longitudinal tensile loading and transverse compressive loading, respectively. And in-plane shear stress strain curves show very significantly nonlinear properties. Ogihara⁶ investigated the mechanical behavior of T700S/2500 unidirectional and angle-ply laminates under uniaxial tensile loading. For unidirectional laminates except [0o] laminates, all axial stress strain curves show obvious nonlinearity, and the stress strain curves for angle-ply laminates except [±15°] also show nonlinearity, in particular for [±45°] laminates.
In order to get an accurate stress strain relationship of composite laminates, both macromechanical models and micromechanical models are proposed to predict the linear slope at
a low strain level and nonlinear curve at a high strain level. Ramberg-Osgood method is widely used for describing elastic-plastic stress strain relationship, moreover, the model can imply material plastic response at a low strain level. In order to derive this function, a yield offset value is introduced so as to satisfy the elastic stress strain relation at a reference stress σ0. There are indeed numerous studies and tests performed on this formula, nevertheless, many expressions of the formula are provided by that strain are derived from stress with initial shear modulus and compliance constants consisted of three adaptations of material parameters,^2,7 while the Bogetti⁸ and Richard⁹ proposed another Ramberg-Osgood model that the stress is expressed by the strain with initial shear modulus, asymptotic stress, and a shape parameter. Comparing with the previous formula, the advantage of this formula is that the prediction from the formula can match the experimental data well when the stress almost rarely change with increasing strain. The reason why the differences exist between the two kinds of expressions is that the formula expressed by stress cannot be derived for the interval in which stress almost keeps the same with the increasing strain, based on mathematical explanation.
The above model is based on physical phenomenon, which is derived from the experimental curves. The following discussions are related to the mechanical behavior of laminates considering that failure of laminates is controlled by transverse matrix crack or inter-fiber failure.
Sun¹⁰ and Majid¹¹ described the shear stress strain relation by using the determination of effective shear modulus. In the failure analysis of laminates, they took first initial failure and subsequent progression of matrix cracking into account, hence, the stiffness degradation is needed to be verified. In order to achieve the effective stiffness, initial modulus and crack density as well as a curve fitting parameter are necessary.
Puck¹² thought that hoop stress has an influence on the risk of fracture, hence, the stress exposure fE is recommended. In his research, a computational method is proposed by means of Puck’s failure criteria with the consideration of an influence of the normal stress on the shear strain. The author assumed that the nonlinear behavior occurred until the stress exposure fE exceeded a threshold value fEthr. According to the comparisons of stress strain relationship between measured experiments and calculations, the author recommend these values of the three numerical adaptation of parameters (fEthr, n, C), and there is still two parameter need to be verified by using failure criteria. In the model, the stress exposure fE is decisive parameter.
There is also a model based on the definition of complementary strain energy density. Researchers discussed the applications of the model by means of comparing the theoretical predictions with experiments. Hahn and Tsai¹³ discussed the nonlinear shear stress strain relationship of material boron/epoxy without coupling uniaxial stress and in plane shear stress under on-axis coordinates and off-axis coordinates. It is observed that longitudinal tensile modulus changes with tensile stress in study of Ishikawa et al..¹⁴ The fitting curves match the test for carbon composites when the laminates are under longitudinal tensile loading. Luo¹⁵ employed the complementary energy method for flexible composites composed of continuous fibers in an elastomeric matrix. Axial stress strain responses of materials tirecord/rubber, Kevlar-49/silicone are compared with the experiments for off-axis specimen. The correlation between predicted response and experiment is good. Kroupa¹⁶ performed tensile tests of carbon/epoxy with various fiber orientations and identified the material parameters.
The above are macromechanical modes that identified parameters by fitting the experimental data. Certainly, there are also micromechanical approaches analysed by a generalized cells model. The cell model is based on a rectangular cell constituted four square sub-cells (fiber and matrix) along with the cell geometry. The strains are the same in the sub-cells and the sum stress of the sub-cells equals the stress of the cell, as a result, it is proposed that mechanical behavior of laminates is denoted by the overall average values. The procedure of derivation and verification of this model are reported by Haj-Ali,² Aboudi1 and Paley.¹⁷