Elliptic rigid inclusion in soft materials of harmonic type

We investigate finite plane deformations of an elliptic rigid inclusion embedded in a soft matrix which is made of a particular class of harmonic-type hyperelastic materials. The inclusion is assumed to be perfectly bonded to the matrix which is subjected to a constant remote in-plane loading. Utilizing the Cauchy integral techniques associated with conformal mappings, we derive closed-form solutions for the full-field deformation, Piola stress and Cauchy stress in the entire matrix. Numerical examples are presented to illustrate the current solutions in comparison with those established from linear elasticity theory. We find that in terms of the Cauchy stress around the inclusion, the maximum normal stress component always appears at the endpoints of the major axis of the inclusion irrespective of the magnitude of the remote loading, while the maximum hoop stress component occurs not exactly at the above-mentioned endpoints when the remote loading exceeds a certain value. In particular, we identify an exact explicit formula for determining the relative rotation of the inclusion during deformation induced by a remote uniaxial loading of arbitrarily-given magnitude and direction.