Fernand Ellyin, Zihui Xia and Yunfa Zhang

Department of Mechanical Engineering, University of Alberta
Edmonton, Alberta, Canada T6G 2G8
fernand. ellyin@ualberta. ca


The micro/meso-mechanical approach for composites is commonly based on the analysis of a representative volume element or a so-called repeated unit cell (RUC). Through analysis of the RUC model one can predict not only macroscopic mechanical properties but also microscopic damage initiation and its propagation in composites. In this paper, we present an overview of our contributions in the following three essential areas in the micro/meso-mechanical analyses: (1) a unified form of periodic boundary conditions for the RUC modelling; (2) a nonlinear viscoelastic constitutive model for polymer matrix materials; and (3) a post-damage constitutive model based on the concept of smeared crack. Application examples combining the above three topics are presented, in which three types of glass/epoxy laminates are analyzed using finite element method. The predicted results are compared with experimental data and they are in good agreement.


Composite materials are widely used in advanced structures in aeronautics, astronautics, automotives, marine, petrochemical and many other industries due to their superior properties over conventional industrial materials. In the past couple of decades many researchers have devoted considerable effort to evaluate macro-mechanical properties of composites by using micro/meso-mechanical modelling methods. The latter method provides overall behaviour of the composites from known properties of the reinforcing phase (particles, fibre or fibre yarns) and the matrix phase (polymers, metals) through analysis of a representative volume element (RVE) or a repeated unit cell (RUC) model, see Aboudi (1991), Nemat-Nasser and Hori (1993). Furthermore, damage in composites generally occurs in a microscopic scale and its effect is manifested progressively into meso – and macro-scales. Micro/meso – mechanical modelling can also be used to study damage initiation and growth in composites.

There are three essential prerequisites for successful micro/meso-mechanical analyses of composites, viz: (1) An appropriate RUC model must be selected and correct periodical boundary conditions should be applied to the RUC model. (2) The constitutive models must accurately describe the mechanical behaviour of the constituents, especially of the matrix material. (3) Appropriate damage criteria for different damage mechanisms and post-damage constitutive model should be introduced to simulate damage initiation and propagation in composites.

For many composite materials, such as fibrous laminates and textile composites, the microstructure can be envisioned as a periodic array of RUC. Therefore in the micro/meso-analysis based on a RUC, the equilibrium equation should be solved under appropriate periodic boundary conditions. Valid periodic boundary conditions should ensure the compatibility of the neighbouring RUCs, i. e. both displacement and traction continuity conditions should be met. However, there still exist ambiguities in application of correct periodic boundary conditions. For example, in cases of shear loading or multiaxial loading in which shear stress is involved, homogeneous boundary conditions (plane – remains-plane or uniform boundary tractions) were applied to the RUC in some previous publications. Many researchers, e. g. Sun and Vaidya (1996), Yuan et al. (1997), Xia et al. (2003a), have pointed out


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 505-516.

© 2006 Springer. Printed in the Netherlands.

that the homogeneous boundary conditions will over – or under-estimate the effective modulus, since by applying homogeneous boundary conditions, the displacement and traction continuity conditions can not always be satisfied at the same time.

Since the micro/meso-mechanical analyses are based on properties of individual constituents of the composite, an accurate constitutive model of each constituent becomes a prerequisite in attempting to predict the composite behaviour. Although the reinforcing phases, such as fibres, ceramic particles behave elastically for most of their stress-strain range, the polymer matrices are viscoelastic or viscoelastic-viscoplastic materials. The analysis of Hashin (1966) demonstrated that the viscoelastic effect in a unidirectional fibre composite is significant for axial shear, transverse shear and transverse uniaxial stress, for which the influence of matrix is dominant. Comprehensive experimental studies on an epoxy polymer (Hu et al. 2003, Shen et al. 2004) have indicated that these materials exhibit complex time – and loading-history-dependent properties. The viscoelastic effects become even more pronounced under conditions of high temperature, sustained loading and/or high stress level. Therefore, it is necessary to develop an accurate constitutive model for the polymer matrix material.

Due to co-existence of multi-phase materials with quite different mechanical properties, damage mechanisms of composites become more complicated. The initiation and evolution of the damages are essentially in the microscopic scale. Therefore, micro/meso-mechanical approaches are desirable to capture these damage mechanisms. The polymer matrices usually have much lower strength and stiffness than the reinforcing phases. Some damage modes (matrix micro-cracking, fiber/matrix interfacial debonding) could occur at relatively low applied loads or even during the manufacturing process. However, certain degree of damages may be tolerated in composites before the structural failure modes (delamination, fiber fracture) occur. Therefore, it is essential for damage analysis of composites to be able to identify different damage modes, to predict damage evolution within each mode and between different modes. To this end, appropriate damage criteria and post-damage constitutive relations are required.

In this paper, our recent contributions in the aforementioned three areas for polymeric composites will be elucidated. A unified form of periodic boundary conditions for any multiaxial loading conditions will be presented first, and uniqueness of the solution by applying the unified periodic boundary conditions on the RUCs will be proved. For an epoxy polymer matrix a nonlinear viscoelastic constitutive model in differential form has been developed based on comprehensive multiaxial experimental test data. The nonlinear viscoelasticity is described through introduction of a modulus function and a time-scale factor. A post-damage constitutive model based on the concept of smeared crack has also been introduced to simulate the behaviour after damage initiation. It permits crack description in terms of stress-strain relations and stiffness reduction in particular orientations instead of the common element-death method in FEM codes. Application examples combining the above three essential topics will be presented, in which three types of glass/epoxy laminates ( 45° unidirectional, cross-ply, and ±45° angle-ply) will be analyzed using finite element method. The predicted results are compared with experimental data and it is shown that they are in good agreement.