The elliptic homoeoid inclusion in plane elasticity

The transformation problem of an elliptical homoeioid inclusion with a uniform eigenstrain embedded in an unbounded homogeneous isotropic medium is studied in the context of plane elasticity. The term homoeoid is used to name a region of a plane medium bounded by two concentric, similar and similarly-oriented elliptic contours. The solution to the problem is achieved by solving first an auxiliary problem corresponding to the case in which the region of the medium (core) surrounded by the inclusion is replaced by a hole. A particular feature of the elastic field of the auxiliary problem is the unmoving of the hole boundary. This result suggests that the solution to the auxiliary problem is, also, the solution to the problem under consideration; additionally, it is the solution whatever is the mechanical property of the core and its bonding conditions with the inclusion. The solution to the problem is obtained in closed form, in terms of the complex potentials of the inclusion and its surrounding (matrix). Based on the complex potentials obtained, a simple expression for the total elastic energy stored in the unbounded medium is derived. It is shown that the total area change of the unbounded medium is that of the inclusion, which is determined in a simple form.