Eshelby's problem for two–dimensional piezoelectric inclusions of arbitrary shape

Eshelby's problem for piezoelectric inclusions of arbitrarily shaped cross–section remains a challenging topic. In this paper, a simple method is presented to obtain an analytic solution for Eshelby's problem of a two–dimensional inclusion of any shape in a piezoelectric plane or half–plane. The exact solutions are derived in terms of some auxiliary functions. A general approach is given to construct these auxiliary functions based on the conformal mappings which map the exterior of some closed curves onto the exterior of the unit circle. The problem is studied in the physical plane rather than in the image plane. The conformal mappings are used to construct auxiliary functions with which the technique of analytic continuation can be applied to the inclusion of any shape. The solution obtained is exact provided that the expansions of all conformal mappings include only a finite number of terms. On the other hand, if an exact conformal mapping includes infinite terms, a truncated polynomial mapping function should be used and then the method gives an approximate solution. One remarkable feature of the method is that simple elementary expressions can be obtained for the internal elastic and electrical fields within the inclusion in a piezoelectric entire plane. Elliptical and polygonal inclusions are used to illustrate the construction of the auxiliary functions and the details of the method.