Nonlinear electrical conductivity in heterogeneous media

Abstract
A new method is proposed for estimating the effective behaviour of strongly nonlinear composite materials. The central idea is to write down second–order Taylor expansions for the constituent phase energy functions, about appropriately chosen reference fields. This allows the restatement of the problem for the nonlinear composite in terms of a corresponding problem for a fictitious linear composite with anisotropic constituents, subjected to a given distribution of polarization fields. The procedure thus permits the straightforward conversion of well–known estimates for linear composites, such as the Maxwell–Garnett and effective–medium approximations, into corresponding estimates for nonlinear composites. The resulting nonlinear Maxwell–Garnett and effective-medium estimates are exact to second order in the contrast.