Simple Algebraic Approximations for The Effective Elastic Moduli of a Cubic Array of Spheres

Abstract

The method of elastostatic resonances [1] is applied to the three-dimensional problem of nonoverlap** spherical inclusions arranged in a cubic array in order to calculate the effective elastic moduli. The leading order in this systematic perturbation expansion is related to the Clausius-Mossotti (CM) approximation of electrostatics. It takes into account the dipole-dipole interaction between strain fields of different inclusions, and makes use of the concept of the the local Lorentz field. The derived CM-type approximations are in the form of simple algebraic expressions [2]. They provide accurate results at low volume fractions of the inclusions and are good estimates at moderate volume fractions even when the contrast is high. The expression for the bulk modulus turns out to be identical to one of the Hashin-Shtrikman bounds.