Harmonic shapes in finite elasticity

We consider the design of harmonic shapes in a particular class of compressible hyperelastic materials of harmonic-type undergoing finite plane deformations. Harmonic shapes are characterized by a 'harmonicity condition' imposed on the final stress field. The 'harmonicity condition' used in this paper is a generalization of the original condition used in the corresponding problems of linear elasticity: specifically, that the first invariant of the stress tensor (i.e. the sum of the normal stresses) in the original stress field remains unchanged everywhere after the introduction of the harmonic hole or inclusion. Using complex variable techniques, we formulate the general equations for the identification of harmonic shapes in a material of harmonic-type subjected to plane deformations. Under the assumption of uniform biaxial loading, we identify shapes of harmonic rigid inclusions and harmonic free holes. Finally, comparisons are drawn to the analogous cases from linear elasticity.