Multiparticle Effective Field and Related Methods in Micromechanics of Composite Materials

Abstract

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.

On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles

“…The second moment of the strain field is obtained by derivation of the effective energy of the composite with respect to the elastic moduli L (r) of the individual phases (see for instance Ponte Castañeda and Suquet, 1998, Buryachenko, 2001or Ponte Castañeda, 2002 :…”

An incremental-secant mean-field homogenization method with second statistical moments for elasto-plastic composite materials

“…Following the argumentation in [48,49], the elastic predictor can be obtained in each phase from the second statistical moment estimation of the total strain, as…”

Effective elastic moduli of nanocomposites with prescribed random orientation of nanofibers

“…Estimation of effective elastic moduli of nanocomposites was performed by the means of effective field method (MEF, see for references and details [1]) based on a so-called effective field hypothesis: each fiber is embedded in the homogeneous effective field depending on prescribed random orientation of this fiber [in contrary to the Mori-Tanaka method (MTM) where the effective field coincides with the average stresses in the matrix]. It may be noted the certain drawbacks of the extension of MoriTanaka method to multiphase composites: Mori-Tanaka moduli may violate the Hashin-Shtrikman bounds and are non-symmetric for the general diphase composites.…”

“…For example, many studies have shown that both tensile ductility and fracture properties of multi-phase composite materials are strongly affected by the spatial heterogeneity of the reinforcing phases (see for references e.g. [1]). Even after many years of comprehensive study by extremely laborious direct measurements and empirical relations, the effect of the structure of microinhomogeneous materials and their influence on the mechanical properties is not completely clear.…”