Bounds for the non-local effective properties of random media

The paper deals with a random medium subjected to a static scalar field with inhomogeneous mean values. Then, effective linear material parameters show dispersion, i.e. they depend on the “wave vector” k of the mean field. The variational methods of P.H. Dederichs and R. Zeller (1973) are generalized to derive upper and lower bounds for scalar effective material parameters as functions of k. In the limit k → 0 (homogeneous mean fields), bounds of the Hashin-Shtrikman type are reproduced. For k → ∞, the bounds coincide with the exact result. In the general case, a two-point moment of the stochastic material parameter is involved. Especially, composites with cell structure and binary mixtures are considered. Detailed calculations are carried out for effective dielectricity, relating mean electric displacement to the mean electric field (which is mathematically equivalent to electrical and thermal conductivities and other scalar parameters), of a binary system composed of nearly spherical grains of equal size.