9 - Homogenization of elastoplastic composites

Summary

The object of the chapter This chapter provides a short introduction to the notion of homogenization (i.e., determining the parameters of a unique fictitious material that ‘best’ represents the real heterogeneous material or composite) and then, at some length, its application to the case where all or some of the constitutive components have an elastoplastic behaviour. The essential notions are those of representative volume element, procedure of localization, and the representation of some microscopic effects by means of internal variables. Composites with unidirectional fibres, polycrystals and cracked media provide examples of application.

Notion of homogenization

Homogenization is the modelling of a heterogeneous medium by means of a unique continuous medium. A heterogeneous medium is a medium of which material properties (e.g., elasticity coefficients) vary pointwise in a continuous or discontinuous manner, in a periodic or nonperiodic way, deterministically or randomly. While, obviously, homogenization is a modelling technique that applies to all fields of macroscopic physics governed by nice partial differential equations, we focus more particularly on the mechanics of deformable bodies with a special emphasis on composite materials (as used in aeronautics) and polycrystals (representing many alloys.) Most of the composite materials developed during the past three decades present a brittle, rather than ductile behaviour. As emphasized in Chapter 7, the elastic behaviour then prevails and there is no need to consider the homogenization of an dastoplastic behaviour.