On the uniformity of stresses inside an inhomogeneity of arbitrary shape

We consider an inhomogeneity of ‘arbitrary’ shape embedded within an infinite isotropic elastic medium (matrix) subjected to antiplane shear deformations under the assumption of uniform remote loading. The inhomogeneity–matrix interface is assumed to be imperfect, characterized by a single interface function. Under these assumptions, we present a novel method leading to the solution of the problem concerned with identifying the shape of the inhomogeneity and the form of the corresponding interface function which leads to a uniform interior stress field. The analysis is based on complex variable methods. Specific solutions are derived in closed form and verified by comparison with existing solutions. As a consequence of our analysis, we present an interesting result on the uniqueness (within a certain class of smooth curves) of the circular inhomogeneity as the only inhomogeneity which, under the given conditions, leads to a uniform interior stress field when the interface function is constant (homogeneously imperfect interface).