Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a circular Eshelby inclusion in anti-plane shear

We use conformal mapping techniques to examine the uniformity of stresses inside a non-elliptical inhomogeneity interacting with a circular Eshelby inclusion in an elastic matrix subjected to remote uniform stresses in anti-plane shear. We show that for a prescribed set of two real loading and two complex geometric parameters, it is possible to determine the single unknown complex coefficient in the mapping function and the (unique) shape of the corresponding inhomogeneity enclosing internal uniform stresses. Our results indicate that the shape of the inhomogeneity depends on the circular Eshelby inclusion whereas the uniform stress field inside the inhomogeneity does not. Finally, we note that the influence of the circular Eshelby inclusion in the vicinity of the inhomogeneity allows for the possibility of a sharp corner on the boundary of the inhomogeneity.