Multi-coating an inclusion of arbitrary shape to achieve uniformity of interior stresses

The idea of a multi-coating has been used extensively in the design of invisibility cloaks in which the objective is to achieve near-invisibility of a particular structure. To this end, this paper is concerned with the internal stress field of an elastic inclusion of arbitrary shape bonded to the surrounding matrix through multiple coatings when the matrix is subjected to remote uniform anti-plane stresses. For any number of intermediate coatings, we identify non-elliptical shapes of the inclusion leading to an internal uniform stress field. The problem is solved analytically for an inclusion of arbitrary shape with two coatings, and iteratively for an inclusion of arbitrary shape with three or more coatings. Our results indicate that the shape of an inclusion permitting an internal uniform stress field can essentially be arbitrary when the number of the coatings is sufficiently large.