Relation Between Microscopic and Macroscopic Mechanical Properties in Random Mixtures of Elastic Media

Abstract

A material composed of a mixture of distinct homogeneous media can be considered as a homogeneous one at a sufficiently large observation scale. In this work, the problem of the elastic mixture characterization is solved in the case of linear random mixtures, that is, materials for which the various components are isotropic, linear, and mixed together as an ensemble of particles having completely random shapes and positions. The proposed solution of this problem has been obtained in terms of the elastic properties of each constituent and of the stoichiometric coefficients. In other words, we have explicitly given the features of the micro-macro transition for a random mixture of elastic material. This result, in a large number of limiting cases, reduces to various analytical expressions that appear in earlier literature. Moreover, some comparisons with the similar problem concerning the electric characterization of random mixtures have been drawn. The specific analysis of porous random materials has been performed and largely discussed. Such an analysis leads to the evaluation of the percolation threshold, to the determination of the convergence properties of Poisson’s ratio, and to good agreements with experimental data.