Elastic fields in two jointed half-planes with an inclusion of arbitrary shape

Abstract

In this paper, a general method is presented for the analytic solution of Eshelby's problem concerned with an inclusion of arbitrary shape within one of two jointed dissimilar elastic half-planes. The method, based on the use of an auxiliary function and analytic continuation, is sufficiently general to accommodate an inclusion of arbitrary shape. The auxiliary function is constructed using a simple approach, from the conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. The boundary value problem is studied in the physical plane rather than in the image plane. The solution obtained is exact provided that the expansion of the mapping function reduces to only a finite number of terms. When the number of terms in the expansion is infinite, a truncated polynomial mapping function can be used to obtain an approximate solution. Explicit expressions for the general solution of the governing equations are derived in terms of the auxiliary function. It is shown that existing solutions for an inclusion of arbitrary shape in a homogeneous plane or half-plane can be obtained, as special cases, from the present solution. In particular, the solution in this paper reduces to a very simple form in the case of a thermal inclusion. Several examples are used to illustrate the construction of the auxiliary function.