Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. II. Bounds for the effective permittivity of statistically anisotropic materials

Upper and lower bounds are derived for the effective permittivity of random heterogeneous materials that are statistically homogeneous but not necessarily statistically isotropic. As trial functions in standard variational principles, we use modified perturbation expansions for the electric field and the electric displacement. The bounds are expressed in terms of the many-point correlation functions of the spatial variation of permittivity. For a wide class of random multiphase materials called cell materials, explicit calculation is performed taking account of the three-point correlation effects.