A two-dimensional Eshelby problem for two bonded piezoelectric half-planes

The present paper concerns a two-dimensional Eshelby problem for an inclusi on of arbitrary shape embedded within one of two bonded dissimilar piezoele ctric half-planes. The elastic and piezoelectric constants of the inclusion and its surrounding half-plane are assumed to be the same. A simple explic it solution is derived in terms of some auxiliary functions which can be de termined using several related conformal mappings of the inclusion shape. T he obtained solution is exact provided that the expansions of all conformal mappings include only a finite number of terms. On the other hand, if an e xact conformal mapping includes infinite terms, a truncated polynomial mapp ing function should be used and then the method gives an approximate soluti on. The existing solutions obtained in an earlier work for a homogeneous pi ezoelectric plane or half-plane can be derived from the present solution as special cases. In particular, the closed-form solutions are given for the Eshelby problem of an arbitrarily shaped inclusion in a piezoelectric half- plane with various mixed surface conditions, such as rigid insulating surfa ce or traction-free conducting surface. These solutions are used to study t he effects of various surface electrical conditions on an electro-elastic f ield in a piezoelectric half-plane.