The relation between mathematical expectations of stress and strain tensors in elastic microheterogeneous media. PMM vol.35, no.4, 1971, pp. 744-750

Abstract

Microheterogeneous media (composite materials, polycrystals and others) are examined for which the elastic moduli tensor cijmnis considered a homogeneous random function of coordinates. The question of the relation between mathematical expectations of stresses 〈σij〉 and strains 〈geij〉 in such media was studied by a number of authors [1-5] under the condition that the fields of stresses and strains are statistically homogeneous. The author of [6] examined the case of inhomogeneous fields and proposed a method of solution for the inhomogeneous stochastic problem. In this paper the program outlined in [6] is carried out. In Sect, 1 the initial stochastic inhomogeneous problem is reduced to an infinite sequence of homogeneous problems. This is achieved through the representation of the solution in the form of a series which satisfies the equilibrium equations for a volume element of the body, and the equations of compatibility of deformations. The coefficients of this series are homogeneous random tensor functions which are independent of body form and also independent of the determined external load acting on the body. These tensor functions depend only on the elastic properties of the body and are completely determined through the given random tensor cijmn. In Sect, 2 the coefficients of the above mentioned series are expressed in terms of the characteristics of the microstructure under the assumption that the medium is strongly isotropie. This assumption permits to avoid the limitation to the case of small inhomogeneities, and at the same time to avoid the need for giving multipoint correlation functions. It follows from the constructed solution that the dependence between 〈σij〉 and 〈ε{lunate}ij〉 analogous to the relation between stresses and strains in the multicouple stress theory of elasticity [7], The transition is made from the differential to the integral form of the realtion. The integral form is characteristic for nonlocal theory of elasticity ([8] and others). © 1971.