Effective conductivity of nonlinear composites of spherical particles: A perturbation approach

The perturbation expansion method is employed to compute the effective nonlinear conductivity of a random composite material characterized by a weakly nonlinear relation between the current density J and the electric field E of the form J=σE+χ‖E ${\mathrm{\Vert }}^2$ E, where σ and χ take on different values in the inclusion and in the host. We develop perturbation expansions to obtain analytic formulas of the potential to arbitrary order in χ. As an example, we apply the method to deal with a spherical inclusion in a host, both of either linear or nonlinear J-E relations, and obtain the potential to second order in χ. For low inclusion concentrations, we derive the effective conductivity to the first, the third, and the fifth order.