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

Abstract

The present paper concerns a two–dimensional Eshelby problem for an inclusion of arbitrary shape embedded within one of two bonded dissimilar piezoelectric halfplanes. The elastic and piezoelectric constants of the inclusion and its surrounding half–plane are assumed to be the same. A simple explicit solution is derived in terms of some auxiliary functions which can be determined using several related conformal mappings of the inclusion shape. The 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 exact conformal mapping includes infinite terms, a truncated polynomial mapping function should be used and then the method gives an approximate solution. The existing solutions obtained in an earlier work for a homogeneous piezoelectric 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 surface or traction–free conducting surface. These solutions are used to study the effects of various surface electrical conditions on an electro–elastic field in a piezoelectric half–plane.