An epitrochoidal thermal inclusion in a coated circular domain

We employ Muskhelishvili’s complex variable method to derive a closed-form solution to the two-dimensional Eshelby’s problem associated with an epitrochoidal thermal inclusion undergoing uniform in-plane dilatational eigenstrains concentrically embedded in a coated circular domain with a rigidly clamped or traction-free boundary. In general, the mean stress within the epitrochoidal inclusion and its supplement to the circular domain is non-uniform. However, when the ratio of the shear modulus of the coating to that of the circular domain is analytically determined for given geometric and material parameters, the mean stress within the epitrochoidal inclusion and its supplement to the circular domain can indeed remain uniform. In order to accomplish this uniformity property, a thin coating with a rigidly clamped boundary can be replaced by a spring-type interface, and a thin coating with a traction-free boundary can be replaced by a membrane-type interface.