Effective properties of composite materials with periodic microstructure: a computational approach

Abstract

This study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents. The theoretical background of the method is recalled first. Two different families of numerical methods are considered to solve the problem. The first is based on the Finite Element Method. The concept of ‘macroscopic degrees of freedom’ is presented. The implementation of periodicity conditions is discussed. A general framework permitting either a strain or stress control is proposed. The second numerical method is based on Fast Fourier Transforms. It considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain. The solution of this problem is based on the explicit form of the periodic Green's function of the reference medium. The relative merits of the two methods are compared and several examples are discussed. Both methods give very comparable results on test examples and their domains of applications appear to be complementary.