Multiparticle Effective Field and Related Methods in Micromechanics of Composite Materials

The numerous approaches used in micromechanics can be classified into four broad categories: perturbation methods, self-consistent methods of truncation of a hierarchy, variational methods, and the model methods. In detail we will consider the self-consistent methods applied to linear elastic problems, based on some approximate and closing assumptions for truncating of an infinite system of integral equations involved and their approximate solution. We consider multiparticle effective field methods, effective medium methods, the Mori-Tanaka method, differential methods and some others. This review article tends to concentrate on methods and concepts, their possible generalizations, and connections of different methods, rather than explicit results. In the framework of a unique scheme, we undertake an attempt to analyze the wide class of statical and dynamical, local and nonlocal, linear and nonlinear micromechanical problems of composite materials with deterministic (periodic and non-periodic) and random (statistically homogeneous and inhomogeneous, so-called graded) structures, containing coated or uncoated inclusions of any shape and orientation and subjected to coupled or uncoupled, homogeneous or inhomogeneous, external fields of different physical natures. The last section contains a discussion of prospects for future work. The article includes 540 references.