How does the shape of an inclusion near a bi-material interface evolve to maintain uniform internal stress: the anti-plane shear case

In the theory of two-dimensional linear elasticity, an elliptical inclusion is known to attain a constant stress field when perfectly buried in an infinite homogeneous matrix if a uniform eigenstrain is applied to it. The focus of this paper falls on the question: when the initially elliptical inclusion verges on a bi-material interface, what would happen to its configuration if it is required to retain the internal constant stress? Specifically, we explore the anti-plane shear version of this question (the version of plane deformations or three-dimensional deformations seems, however, insoluble at this stage), in which an inclusion undergoing a uniform (anti-plane shear) eigenstrain is embedded in a bi-material structure composed of two infinite elastic half-planes whose interface is straight and perfectly bonded, and the shape of the inclusion is to be determined such that the eigenstrain-induced stress inside the inclusion appears to be a constant. Unlike most optimization methods-driven solution procedures for finding the shape of the inclusion approximately in which huge computation is required, we derive by a rigorous theoretical analysis an exact integral equation with respect to the boundary curve of the inclusion that is sufficiently and necessarily related to the existence of a constant stress inside the inclusion. We solve this integral equation via the use of some analytic techniques and present in several illustrative examples a variety of shapes of the inclusion achieving constant stresses. We discover some interesting phenomena for the evolution of the shape of the uniformly stressed inclusion relative to the stiffness of the nearby interface.