Mean field theory for weakly nonlinear composites

We discuss the nonlinear behavior of a random composite material characterized by a weakly nonlinear relation between the electric displacement of the form D = ϵE + χ|E|2E, where ϵ and χ are position dependent. A general expression for the effective nonlinear susceptibility to first order in the nonlinear susceptibility of the constituents in the composite is given. A general method of approximation is introduced which gives the effective nonlinear susceptibility in terms of the solution of the linear dielectric function of the random composite. Various applications of the proposed approximation are demonstrated.