Nonlinear computational models for composite materials using homogenization.

Abstract

The primary objective of this dissertation is to develop computational models that describe the overall mechanical response as well as characterize the microstructure behaviour of composite materials. The models presented are restricted to two different classes: the small deformation of linear elastic composites, and the large deformation of elasto-plastic composites. It is further assumed that the composite materials can be represented by a periodic repetition of a microstructural unit. Based on this assumption the homogenization theory is used to derive models characterizing both the macroscopic and the microscopic mechanical behavior of the composite, and the finite element method used for their computational implementation. For the case of small deformation of linear elastic composites, the accuracy of the finite element method is studied. A priori error estimations are derived and the convergence properties are discussed. After asserting the convergence properties of the method, the ideas of material pre-processor and material post-processor are introduced and examples for local stress analysis are presented. An adaptive finite element method based on the a priori error estimations is also implemented to improve the accuracy of the approximations. For the case of large deformation elasto-plasticity of composites, an incremental method is developed by applying the homogenization technique to the rate form of equilibrium equations. The constitutive equation for the components of the composite assumes time independent plasticity, additive decomposition of the velocity gradient, isotropic and kinematic hardening, and a normal flow rule. The particular case of J$\sb2$ flow theory of plasticity with isotropic and kinematic hardening is considered. The method is applied to local stress and plasticity analysis of general two dimensional structures, as well as to investigate the macroscopic properties of the composites. The numerical results show that the macroscopic behavior depends significantly on the geometry of the microstructure and is rather different from that of the constituents of the composite material.