Uniform stresses inside a non-elliptical inhomogeneity and a nearby half-plane with locally wavy interface

We develop a two-parameter conformal mapping function for a doubly connected domain to solve the inverse problem in anti-plane and plane elasticity associated with a non-elliptical inhomogeneity with internal uniform stresses embedded in a half-plane bonded to another half-plane also with internal uniform stresses via a locally wavy interface. The internal uniform stresses are found to be independent of the specific shapes of the locally wavy and non-elliptical interfaces. The permissible range of the two parameters in the mapping function for a one-to-one mapping is obtained. Typical geometries corresponding to the three-phase composite are illustrated.