Lamellar inhomogeneities in a uniform stress field

Abstract

Solutions for lamellar inhomogeneities in an infinitely extended isotropic solid subjected to a uniform stress field at infinity are obtained by using Eshelby's equivalent inclusion method. First, two limiting cases are studied: cracks (i.e. inhomogeneities with elastic moduli identically zero), and anticracks (i.e. inhomogeneities with infinitely large elastic moduli). Solutions are obtained for different modes of uniform loading in plane strain (biaxial load and in-plane shear), and anti-plane strain. It is observed that the stress field of an anticrack under biaxial load has an inverse square root singularity at the tip of the anticrack. The stress field arises from the contribution of a planar dislocation distribution and a dislocation dipole distribution. Finally the most general case of inhomogeneities with finite non-zero elastic moduli, which here are called quasicracks, is considered and new solutions are provided. Quasicracks are useful to model fibers in fiber-reinforced materials. Analytical solutions are provided for the stress field in the matrix, including the interface shear stress.