Author(s): Konstantin Z Markov
The introductory and more elementary ideas and results of micromechanics of heterogeneous media are collected in the survey. The central problem under discussion is “homogenization.” It replaces such media by homogeneous ones, which behave macroscopically in the same way and possess certain gross effective properties. These properties are related in a complicated manner to the prescribed internal structure of the medium and their evaluation, in general, represents a profound challenge in any specific situation. A brief historical survey is given, underlying the reappearance of essentially the same “homogenization” quest in numerous guises and contexts over the last two centuries. Within the framework of the volume-averaging approach the basic notions are introduced and some of the central, now classical, results are then derived and discussed such as perturbation expansions, Hashin-Shtrikman’s bounds, variational estimates and Levin’s cross-property relation. A general “one-particle” scheme for approximate evaluation of the effective properties (in the static case) is detailed in its various implementations like selfconsistency, iterated limits and effective field. Illustrations concern conductivity, elasticity, and absorption phenomena in heterogeneous particulate media, as well as a simple self-consistent model for polycrystals’ homogenization.