Effective conductivity of composites containing aligned spheroidal inclusions of finite conductivity

Abstract

The effective conductivity of composites containing aligned spheroids of finite conductivity is determined with the pair interaction rigorously taken into account. The pair interaction is evaluated by solving a boundary‐value problem involving two aligned spheroids with a boundary collocation scheme. Our results are in the form of virial expansion in the inclusion volume fraction f, truncated at the O(f²) term. Effective conductivities obtained from this study agree well with Willis’s [J. Mech. Phys. Solids 25, 185 (1977)] bounds and Kim and Torquato’s [J. Appl. Phys. 74, 1844 (1993)] first‐passage‐time simulation results. An expression for the effective conductivity tensor based on an equivalent inclusion method originated by Hasselman and Johnson [J. Compos. Mater. 21, 508 (1987)] is also derived, from which a criterion for testing the virial expansion validity is constructed. The relevant series ratio is found to be a good measure of intensity of the system thermal interaction. The smaller the value of the series ratio, the better agreement of the virial expansion with the simulation results. Explicit expressions for the second‐order virial expansion for the parallel and perpendicular components of the effective conductivity tensor are tabulated for inclusion aspect ratios 1, 10/9, 2, 5, 10 and phase conductivity ratios σ=0, 0.1, 2, 10, 100, 10 000, and ∞.