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

In this paper, a general method is presented for the analytic solution of E shelby'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 genera l 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 bo undary value problem is studied in the physical plane lather than in the im age plane. The solution obtained is exact provided that the expansion of th e mapping function reduces to only a finite number of terms. When the numbe r of terms in the expansion is infinite, a truncated polynomial mapping fun ction can be used to obtain an approximate solution. Explicit expressions f or the general solution of the governing equations are derived in terms of the auxiliary function. It is shown that existing solutions for an inclusio n 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 inclusio n. Several examples are used to illustrate the construction of the auxiliar y function.