  A classical problem in elasticity theory involves an inhomogeneity embedded
in a material of given stress and shear moduli. The inhomogeneity is a region
of arbitrary shape whose stress and shear moduli differ from those of the
surrounding medium. In this paper we present a new, semi-analytic method for
finding the stress tensor for an infinite plate with such an inhomogeneity. The
solution involves two conformal maps, one from the inside and the second from
the outside of the unit circle to the inside, and respectively outside, of the
inhomogeneity. The method provides a solution by matching the conformal maps on
the boundary between the inhomogeneity and the surrounding material. This
matching converges well only for relatively mild distortions of the unit circle
due to reasons which will be discussed in the article. We provide a comparison
of the present result to known previous results.
