Critical analysis of generalized Maxwell homogenization schemes and related prospective problems

Abstract

A linear composite medium consisting of a homogeneous matrix containing either the periodic or random set of heterogeneities is considered. Both the original and generalized Maxwell schemes are analyzed by reducing the problem to calculation of elastic fields in a finite volume of the composite embedded in the infinite homogeneous matrix medium and subjected to a constant external stress or strain field at infinity. We investigate (qualitatively and quantitatively) a correspondence between the Maxwell schemes (the original and generalized versions) and other different basic assumptions, conceptions, and methods of analytical micromechanics such as effective field method (EFM) and Mori–Tanaka method (MTM). Some deficiencies and inconsistencies of generalized Maxwell schemes in particular examples are detected and explained how the mentioned difficulties can be easily overcome by the classical micromechanical methods EFM and MTM. For the modeling of both periodic and random structure composites, the generalized Maxwell schemes taking into account the inclusion interactions inside the cluster are analyzed. The incorrectness of using of some generalized Maxwell schemes for the modeling of random structure composite materials (CMs) is shown and the directions for its improvements in the framework of the second background of analytical micromechanics are proposed.