Functional series and Hashin–Shtrikman type bounds on the effective conductivity of random media

Abstract

A general method of placing variational bounds on the effective scalar conductivity of random heterogeneous solids is proposed. The method utilizes the Hashin-Shtrikman variational principle and trial fields which are obtained upon truncating the functional series expansion for the polarization field in the medium. It is shown how the earlier variational procedures due to Hashin and Shtrikman, Milton and Phan-Thien and Willis find natural places in the proposed general scheme. The new results reported include, for example, the fact of nonexistence of certain Gaussian-like random media and, for dispersions of spheres with volume fraction c, an exact O(c2) relation for the effective conductivity which contains absolutely convergent integrals only.