Elastic properties of hybrid composites by the effective field approach

Abstract

The work is dedicated to the calculation of the overall elastic properties of matrix composite materials containing two different populations of inclusions (three phase hybrid composites). The application of the well known Mori–Tanaka method or self-consistent effective medium method to the solution of this problem gives overall elastic moduli tensors of such composites that do not have the necessary symmetry (the symmetry with respect to the first and second pairs of indices). In this work, a new version of the effective field method that takes into account specific features of the microstructure of three phase composites is developed. In this version, the field that acts on every inclusion in the composite is assumed to be different for inclusions of different populations. It is shown that the modified effective field method gives a correct symmetry of the overall elastic moduli tensors of three phase composites. The method allows us to describe the influence of the peculiarities in spatial distributions of inclusions on the overall elastic constants. The cases of media containing infinite cylindrical fibers and thin ellipsoidal disks or spherical pores are considered. Various boolean type probabilistic models of random sets of such inclusions are proposed and the elastic moduli tensors of the corresponding three phase composites are obtained and analyzed. It turns out that these tensors strongly depend on statistical properties of the random fields of inclusions. It is shown that for two phase composites, the Mori–Tanaka method is a particular case of the effective field method. In the case of three phase composites, the formulas of the Mori–Tanaka method follow from the equations of the effective field method if a general property of the symmetry of cross-correlation functions of different populations of inclusions is violated. As a result, the overall elastic moduli tensors obtained by Mori–Tanaka method lose their natural symmetry.