Exact Contrast Expansions

Abstract

For two-phase media in which variations in the phase properties are small, formally exact perturbation series for both the effective conductivity (Beran 1968, Phan-Thien and Milton 1982) and effective elastic moduli (Beran 1968, Dederichs and Zeller 1973, Willis 1981) have been developed. Such weak-contrast expansions are found by first obtaining corresponding expansions of either the local electric field E(x) or the local strain field ε(x) via integral equations. For specificity, it is useful to state the weak-contrast form of the effective conductivity σ_e of a macroscopically isotropic medium, keeping in mind that analogous results exist for anisotropic media and for the effective elastic tensor.