Analytic Solution for Eshelby’s Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane

Abstract

Despite extensive study of the Eshelby problem for inclusions of simple shape, little effort has been made to inclusions of arbitrary shape. In this paper, with aid of the techniques of analytical continuation and conformal mapping, a novel method is presented to obtain analytic solution for the Eshelby’s problem of an inclusion of arbitrary shape in a plane or a half-plane. The boundary of the inclusion is characterized by a conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. However, the boundary value problem is studied in the physical plane rather than in the image plane. The conformal mapping is used to construct an auxiliary function with which the technique of analytic continuation can be applied to the inclusion of arbitrary shape. The solution obtained by the present method is exact, provided that the expansion of the mapping function includes only a finite number of terms. On the other hand, if the exact mapping function includes infinite terms, a truncated polynomial mapping function should be used and then the method gives an approximate solution. In particular, this method leads to simple elementary expressions for the internal stresses within the inclusion in an entire plane. Several examples of practical interest are discussed to illustrate the method and its efficiency. Compared to other existing approaches for the two-dimensional Eshelby’s problem, the present method is remarked by its elementary characters and applicability to inclusions of arbitrary shape in a plane or a half-plane.