General Interface Integral Equations in Elasticity of Random Structure Composites

Abstract

One considers linearly elastic composite material (CM), which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. nonellipsoidal) shape. The new general integral equations (GIE) connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals that makes it possible to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of ellipsoidal symmetry. In particular, we used a meshfree method based on fundamental solutions basis functions for a transmission problem in linear elasticity. As a particular problem allowing us to get an exact result, we consider a linear elastic CM, which consists of a homogeneous matrix containing a statistically homogeneous random set of non-canonical inclusions. The elastic properties of the matrix and the inclusions are the same, but the stress-free strains are different. Increasing of volume fraction of inclusions can lead to change of a sign of local residual stresses estimated by either the new approach or the classical one. The main properties of the method are analyzed and illustrated with several numerical simulations in 2D infinite domains containing statistically homogeneous random field of inclusions.