Let a spatial domain \(V\) be divided into domains \(V_r\) \((r=1,2, \dots, N)\), and let \(f_r\) be the characteristic function of the set \(V_r\). The authors consider a composite material derived by the nonlinear equations \(\text{div} {\mathbf J} =0\), \({\mathbf J}= {\partial \over \partial {\mathbf E}} (\sum^N_{r=1} f_r({\mathbf x}) f_r ({\mathbf E}))\), with respect to \({\mathbf E}\) and \({\mathbf J}\). A new variational method is proposed for estimating the effective behavior of the composite material. The central idea is to use second-order Taylor expansions for the constituent phase energy functions about an appropriately chosen reference field.