Closed-form solutions for the effective conductivity of two-phase periodic composites with spherical inclusions

In this paper, we use approximate solutions of Nemat-Nasser et al. to estimate the effective conductivity of two-phase periodic composites with non-overlapping spherical inclusions. Systems with different inclusion distributions are considered: cubic lattice distributions (simple cubic, body-centred cubic and face-centred cubic) and random distributions. The effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast Fourier transform-based methods. For systems containing randomly distributed spherical inclusions, the solutions are shown to be directly related to the static structure factor, and we obtain its analytical expression in the infinite-volume limit.