Non-uniqueness and Non-existence in the Statistical Theory of Elastic Composites

The Correlational Approximation

Abstract

I give the boundary conditions corresponding to the phenomenological model of a statistically homogeneous and isotropic elastic body, in the correlational approximation. I show that the involved strain energy density is not positive definite. By a simple ewample I show that a correctly formulated traction boundary value problem can have a unique solution, can have an infinity of solutions, or can have no solution, according to the relation existing between the geometrical dimension of the body and the correlation radius of the considered material.