Nonlinear electrical conductivity in heterogeneous media

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.