Paper Review

Unidirectional fiber composite materials exhibit nonlinear rate dependent behavior under off-axis loading. There are two points of view to discuss this physical phenomenon, i.e. macromechanical and micromechanical mechanics, and all published literatures originated from either of the two perspectives. Based on the

viewpoint of macromechanics, Sun and Chen [1] developed a single parameter yield function under plane stress assumption and brought it into the flow rule with a power law curve fitting effective stress – effective plastic strain relation to describe the nonlinearity of fiber composites. This single parameter in the yield function was chosen suitably so that all off-axis experimental data collapse into a single master curve in the effective stress versus effective plastic strain domain. It is a fact that the single parameter model has good agreements with experiments. Because of rate independence in this model, some improvements were carried out. Gates and Sun [2]

combined the over stress model [3] with the single parameter model to predict the rate dependent behavior of composites under loading (the over stress is positive) and unloading (the over stress is zero) conditions. In order to use the over stress model, a quasistatic stress - strain relation was set as a reference state, and when the strain rate is higher, the corresponding relative effective stress was calculated by subtracting the quasistatic effective stress from the current effective stress associated with the same strain level. The relative effective stress and effective plastic strain rate relations obtained from the over stress model were then employed together with the flow rule for characterizing the plastic deformation of composites subjected off-axis loading. Yoon and Sun [4] used the same way as Gates and Sun [2] to investigate the effects of variant strain rates on a monotonic tension process under off-axis loading. The results were also compared with a modified Bodner and Partom’s model [5]. Weeks and Sun [6] modeled off-axis composites using a mathematical form similar to Johnson-Cook model [7] in conjunction with the single parameter model. A quasistatic state was chosen as a reference state to get the corresponding reference effective plastic strain rate and effective stress while the Johnson-Cook model was working. Then, the effective stress and effective plastic strain rate relation at high strain rate analyses could be obtained via this model and applied into

flow rule to get corresponding plastic responses like the over stress model.

Thiruppukuzhi and Sun [8] directly introduced a rate dependent term into the effective stress – effective plastic strain power law relation and proposed a three parameters model for modeling the nonlinear rate dependent behavior of unidirectional fiber composites. Since the power law equation is a convenient form to use, in this study, the three parameters model was adopted as the viscoplasticity model to describe the rate dependent nonlinearity of the matrix phase.

In order to investigate the nonlinear effect of matrix on the mechanical behavior of fiber composites, a micromechanical approach is proposed by modeling the composites as heterogeneous solids consisting of fiber and matrix phases.

Through the characteristics of repetition, a Representative Volume Element (RVE) was selected to represent the whole composite materials. By analyzing the mechanical behavior of the RVE, the overall material responses of composites could be determined. There are several micromechanical models available for describing the mechanical behaviors of composites, i.e., Eshelby model [9], Mori-Tanaka model [10,11], square fiber model [12] and generalized method of cells [13-15]. Eshelby [9]

introduced Eshelby’s tensor together with the equivalent principal concept to model a homogeneous inclusion embedded in an infinite matrix. Basically, Eshelby model is a dilute model because only one inclusion is considered. Mori and Tanaka [10]

extended Eshelby’s approach to establish a non-dilute model in which the stress and strain states of the inclusion and the matrix were considered in an average sense.

Benveniste [11] gave alternative explanations of Eshelby model and Mori-Tanaka model by introducing the strain concentration concept and obtained succinct formulas for these two models. Commonly, the Eshelby model and Mori-Tanaka model were mainly applied to characterize the stiffness of short fiber composites. However, they could be extended to characterize the long fiber composites if the aspect ratio of the

inclusion was assumed to be infinity [16] and the nonlinear behavior of composites can be described if an incremental Mori-Tanaka mean field approach was adopted [17]. Sun and Chen [12] proposed a “Square Fiber Model” constructed by a RVE composed of one square fiber and two pure matrix regions. A 2-D plane stress plastic potential modified from von Mises J2 function was applied in conjunction with the associated flow rule to describe the plastic strain of the matrix material, while the fiber was regarded as an orthotropic elastic material. The entire stiffness matrix of the composite was derived from some suitable constant stress and constant strain assumptions between each subcell in the RVE. Therefore, we can obtain the total strain increments due to a given stress history by using this model. Similar to this way, Goldberg and Stouffer [18] suggested a four regions model with one square fiber and three matrix regions. Not a plane stress condition but both two transverse directions have to be applied constant stress and strain assumptions in all subregions to obtain the overall constitutive equation. The matrix phase was described using the Bodner and Partom’s model [5] and the corresponding deformation was solved by using the Runge-Kutta method. Away from the forgoing theories, Aboudi [13, 14]

derived a four regions micro-mechanical model called “Method of Cells”, which is very efficient in modeling the elastic and inelastic behavior of fiber-reinforced unidirectional composites. Based on the displacement and traction continuity at the interfaces of all subcells as well as the periodicity at the RVE, a stress - strain relation was described in a matrix form to predict mechanical behavior of composite materials.

By extending the method of cells, Paley and Aboudi [15] proposed a scheme called Generalized Method of Cells (GMC) which can deal with an undetermined numbers of subcells. The weak point in GMC is that the more subcells you have, the more CPU time is required. To enhance computational efficiency of GMC, Orozco [19]

took advantage of the sparse features of the strain concentration matrix. It is

basically an improvement in the numerical processing. The sparse implementation of GMC made it possible to solve the problems with complex micro-structures and tiny refinements. Pindera and Bednarcyk [20] adopted a different manner to enhance the computational efficiency of GMC. They expressed the displacement continuity between the subcells in terms of stresses and then derived a modified formulation of GMC. This formulation is regarded as the most efficient way in the employment of GMC until now. The feature of the GMC is that which cells were fibers or matrices were not indicated in advance. In other words, we can assign the cells with either fibers or matrix after forward when the final constitutive equation was established.

In applications of the GMC, Orozco and Pindera [21] combined the GMC with an available tangent plasticity matrix to analyze transverse mechanical behavior of composites under different fiber arrangements and fiber shapes. A large number of subcells were constructed in their study to model the complex microstructures. It showed that different fiber arrangements and fiber shapes lead to distinct constitutive behavior. Ogihara et al. [22] characterized the nonlinear behavior of carbon/epoxy unidirectional and angle-ply laminates. The GMC was applied first to obtain the property of unidirectional fiber composites under off-axis loading. Together with the laminate plate theory, the angle-ply laminates were calculated from the unidirectional composites. Kawai et al. [23] investigated the AS4/PEEK composites under loading and unloading conditions on the off-axis response at strain rate up to 0.01/min. The PEEK matrix was described by Chaboche model and the composite was predicted using GMC. The results showed good agreements with the experimental results for AS4/PEEK composites. However, the strain rates in their investigation were not high enough for engineering applications. Using finite element analysis, Zhu and Sun [24] investigated the nonlinear behaviors of fiber composites by applying suitable boundary conditions on a RVE selected properly with three different fiber

arrangements. It was shown that the square diagonal packing array provides the best prediction on the experimental results for all samples with various off-axis angles