Effective thermal conductivities of heterogeneous media containing multiple imperfectly bonded inclusions

Abstract

In a recent paper [Duan et al., Phys. Rev. B 73, 174203 (2006)], we derived explicit expressions for the effective conductivities of heterogeneous media containing perfectly bonded ellipsoidal inclusions of diverse shapes, spatial distributions, and orientations. In this paper, we take into account the effect of three types of imperfect bonding between the inclusions and the matrix by replacing the imperfectly bonded ellipsoidal inclusions with equivalent perfectly bonded homogeneous inclusions using the average $t$-matrix approximation of the multiple-scattering approach. The explicit expressions remain unaltered in form but involve the parameters of the equivalent homogeneous inclusions. It is shown that our approximate scheme gives very accurate predictions of the effective conductivity of the heterogeneous materials, while retaining the simplicity of the explicit expressions. However, in contrast to the perfectly bonded inclusions, the effective conductivities of a heterogeneous medium containing imperfectly bonded inclusions depend upon the size of the inclusions. This size dependence is shown to be captured by simple scaling laws depending upon the type of bond imperfection.