The modified affine homogenisation scheme for nonlinear composites: Comparison with variational bounds and estimates

Abstract
The effective behaviour of heterogeneous materials exhibiting a random microstructure, such as composites and polycrystals, can be efficiently estimated or bounded with homogenisation methods. For materials with a nonlinear behaviour, use is classically made of a Linear Comparison Composite (LCC) presenting generally the same microstructure as the nonlinear material, but for which the local (linearised) compliance is uniform per phase. Homogenisation schemes, developed for linear behaviour, can thus be applied to the LCC. The way the LCC is defined has a great influence on the estimated effective behaviour and local mechanical fields, in particular for large nonlinearities and contrasts between the phases. For few years, it has become clear that not only the phase averages of strain and stress must be used for defining the LCC, but also their intraphase fluctuations. In that framework, the present authors have recently proposed a "modified affine" procedure [1] which presents the advantage over other formulations to be applicable to any type of behaviour, including nonlinear viscoelasticity. In this paper, we present a comparison, for simpler behaviour (nonlinear elasticity), between the proposed approach and several formulations previously proposed in the literature.