Homogenization of random structures in probabilistic terms

Viktor Berdichevsky, vberd@eng.wayne.edu
Homogenization of random structures in probabilistic terms
Studying of materials with evolving random microstructures requires the knowledge of probabilistic characteristics of local fields because the path of the microstructure evolution is determined by the local fields. The mathematical problem for determining the probabilistic characteristic was formulated as a minimization problem for
energy with the respect to probabilistic measure long ago (Berdichevsky, 1987). This variational problem contains an infinite chain of constraints for probability distributions. It remained unclear how to properly truncate
this infinite chain. In this talk a way of truncation is suggested. The corresponding probability distributions are
found for two-phase composites. As a by-product, one gets also the dependencies of the effective coefficients
on the parameters of microstructures. Other issues that will be discussed are: formulation of Hashin–Shtrikman
results in probabilistic terms, a universal extension of the Hashin–Shtrikman bounds, the law of decrease of
microstructure entropy.