The effective material properties of a composite elastic half-space

Abstract

A classical problem in applied mathematics is the determination of the effective properties of a composite material by looking at its reflection and transmission properties. This model discussed here is an elastic half-space containing randomly distributed voids—to obtain its “average” material properties (i.e., the effective density and elastic moduli) we consider elastic waves incident from a homogeneous half-space onto the inhomogeneous material. We restrict attention to dilute dispersions of inclusions, and therefore, results are obtained under the assumption of small volume fraction. We look at several aspects of this problem, as times allows. First, we discuss how predictions derived from the non-isotropic Foldy or the Waterman-Truell multiple-scattering theories (MSTs) in the low-frequency limit are equivalent to results found by an asymptotic integral equation (homogenization) method developed by the authors [1]. Second, the effect of nonlinear pre-stress on each void, and hence on the averaged material properties, is considered [2]. And third, some comments are offered regarding higher order effects and the closure assumption (e.g., the quasi-crystalline approximation). [1] W. J. Parnell, I. D. Abrahams, and P. R. Brazier-Smith, Quart. J. Mech. Appl. Math. 63, 145–175, 2010. [2] T. Shearer, W. J. Parnell, and I. D. Abrahams, 471, 20150450, 2015.