Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane

Analytical solution for Eshelby's problem of an anisotropic non-elliptical inclusion remains a challenging problem. In this paper, a simple method is presented to obtain an analytical solution for Eshelby's problem of an inclusion of arbitrary shape within an anisotropic plane or half-plane of the same elastic constants. The method is based on an observation that the interface conditions for arbitrary inclusion-shape can be written in a decoupled form in which three unknown Stroh's functions are decoupled from each other. The solution is given in terms of three auxiliary functions constructed by three conformal mappings which map the exteriors of three image curves of the inclusion boundary, defined by three Stroh's variables, onto the exterior of the unit circle. With aid of these auxiliary functions, techniques of analytical continuation can be applied to the inclusion of any shape. The solution is given in the physical plane rather than in the image plane, and is exact provided that the expansion of every mapping function includes only a finite number of terms. On the other hand, if at least one of the mapping functions includes infinite terms, a truncated polynomial mapping should be used, and thus the method gives an approximate solution. A remarkable feature of the present method is that it gives elementary expressions for the internal stress field within an inclusion in an anisotropic entire plane. Elliptical and polygonal inclusions are used to illustrate the construction of the auxiliary functions and the details of the method.