Homogenization in probabilistic terms: The variational principle and some approximate solutions

Studying of materials with evolving random microstructures requires the knowledge of probabilistic characteristics of local fields because the path of the microstructure evolution is controlled by the local fields. The probabilistic characteristics of local fields are determined by the probabilistic characteristics of material properties. In this paper it is considered the problem of finding the probabilistic characteristic of local fields, if the probabilistic characteristics of material properties are given. The probabilistic characteristics of local fields are sought from the variational principle for probabilistic measure. Minimizers of this variational problem provide all statistical information of local fields as well as the effective coefficients. Approximate solutions are obtained for electric current in composites for two cases: multi-phase isotropic composites with lognormal distribution of conductivities and two-phase isotropic composites. The solutions contain a lot of statistical information that has not been available previously by analytical treatments.