Bounds for the non-local effective properties of random media—II

The paper deals with a random medium subjected to a static 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. Starting from a variational method previously developed by the authors, upper and lower bounds for k-dependent scalar effective parameters are derived in terms of two-point and three-point correlation functions of the stochastic material parameters. Taking into consideration the three-point correlation function gives a substantial improvement of the generalized Hashin-Shtrikman bounds obtained previously. In particular, composites with cell structure and arbitrary binary systems are considered. In order to illustrate the general results, numerical evaluations are carried out for effective permittivity of a binary cell material composed of nearly spherical grains of equal size.