Uniformity of stresses inside a hypotrochoidal inhomogeneity

We prove the existence of uniform stresses inside a hypotrochoidal inhomogeneity in the context of both anti-plane elasticity and plane elasticity. Specifically, we consider an Eshelby inclusion of hypotrochoidal shape into which is entirely embedded a hypotrochoidal elastic inhomogeneity. The inclusion is surrounded by an infinite elastic matrix. Additionally, both the inclusion and the inhomogeneity undergo the same corresponding uniform anti-plane eigenstrains or in-plane volumetric eigenstrains. When the single mismatch parameter for anti-plane elasticity or plane elasticity is judiciously chosen in accordance with the four given geometric parameters of the composite, the internal stresses inside the hypotrochoidal inhomogeneity are found to be uniform. In addition, the internal uniform stresses are hydrostatic in the case of plane elasticity.