Effective conductivity of nonlinear composites

Abstract

The perturbation expansion method is used to compute the effective conductivity of nonlinear composite media. We develop perturbation expansions to solve nonlinear partial differential equations pertaining to the electrostatic boundary value problems. As an example in two dimensions, we apply the method to deal with a cylindrical inclusion in a host, both of either linear or nonlinear current-voltage characteristics, and derive the zeroth-, first-, and second-order series in the nonlinear conductivity coefficient. We also consider a composite of cylindrical inclusions embedded in a host. For low concentrations of inclusions, we derive the exact formulas of the effective conductivity to first, third, and fifth order. We show that the approximate results of Zeng et al. (to third order) are indeed exact.