Morphological stability of inclusions: two-dimensional analysis with a superelliptic approximation

The two-dimensional inclusion problem of elasticity has been extended to treat general non-elliptic shapes described as ( x 1 2 ) p / 2 + ( x 2 2 / α 2 ) p / 2 ≤ 1 , p ≥ 2. By using the Green function, the average Eshelby tensors are evaluated and elastic strain energy of the superelliptic inclusion is calculated for a material with cubic elastic anisotropy. In addition, by introducing the isotropic interface energy, equilibrium inclusion shapes to minimize the sum of elastic strain energy and interface energy are discussed as a function of the size and shape of the two-dimensional inclusion. It is found that an intermediate shape between a circle and a square or between an ellipse and a rectangle can be a minimum-energy shape under certain sets of given conditions.