Internal residual stresses in heterogeneous solids—A statistical theory for particulate composites

Abstract

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a homogeneous and statistically uniform random set of ellipsoidal inclusions. Because of the differential thermal expansion, a microstructural residual stress state arises. By means of the “multiparticle effective field” method we first derive the functional relation between the stored elastic energy and the thermoelastic constants of the components. Using this result an exact estimation of all components of the statistical second moment tensor of the stress fields is given. Furthermore, an expression for the second moment of the stress in the matrix in the vicinity of the ellipsoidal inclusion and a correlation function of internal stresses is obtained. The application of the theory is demonstrated by some numerical results for a WC-Co composite.