Eshelby tensor of a polygonal inclusion and its special properties

Nozaki and Taya (ASME J. Appl. Mech. 64 (1997) 495-502) analyzed the elastic field in a convex polygonal inclusion in an infinite body. By numerical analysis, they found that, when the shape of the inclusion is a regular polygon, the strain at the center of inclusion and the strain energy per unit volume of inclusion have strange and remarkable properties: these values are the same as those of a circular inclusion and are invariant for inclusion's orientation if the shape of the inclusion is not a square. In this paper, we first derive a simple, exact expression of the Eshelby tensor for an arbitrary polygonal inclusion. Using the expression, we then show a mathematical explanation why these special properties appear.