Shape of a single inclusion enclosing uniform stress for a non-uniform far-field loading in anti-plane shear

Abstract

The problem as to whether an inclusion when embedded in an elastic infinite matrix with a perfectly bonded interface could achieve uniform internal stress has been studied extensively in the literature. It has been shown that in almost all cases ellipsoidal shape (including elliptical shape for two-dimensional deformation) is the only inclusion shape ensuring the uniformity of the stress inside the inclusion if the external loading exerted remotely on the edge of the matrix is uniform. From a practical point of view, this result remains valid for a non-uniform remote loading imposed on the matrix provided the product of the inclusion's size and the gradient of the non-uniform loading is small enough (as compared with the overall level of the stress field induced by the non-uniform loading at the inclusion's position). In this paper, we explore the possibility of designing an inclusion enclosing uniform internal stress in an elastic infinite plane under anti-plane shear deformation for a non-uniform remote loading with arbitrary gradients. We derive the equations with respect to the parameters describing the configuration of the inclusion leading to a uniform internal stress inside the inclusion for a general polynomial remote loading, and establish the analytic and numerical solutions for the shape of the inclusion which are illustrated via several examples.