Nonlocal Theory of the Effective Linear Constitutive Laws in Heterogeneous Materials.

In many cases, the macroscopic behavior of heterogeneous materials (multiphase mixtures, composite materials, polycrystals) can be described by linear effective constitutive laws connecting mean fields which are defined by ensemble averaging. These laws contain such effective properties as effective dielectricity, effective conductivity, effective Hooke's tensor. In general, the effective laws turn out to be nonlocal, in other words, the Fourier transforms of the effective properties depend on the wave vector. By means of stochastic Green's function, a general expression for the effective constitutive laws is obtained without any reference to perturbation methods. Except simple cases, the nonlocality of the effective properties cannot be determined unambiguously. The considerations are carried out mainly on the example of elastostatics (e. g. effective Hooke's tensor).