Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations

- In isotropic linear elasticity, homogeneous stress is induced by homogeneous strain.
- We show that, by contrast, different large strains can produce equal Cauchy stresses.
- A proof is carried out by exhibiting compatible deformations on a specific geometry.
- An example of strain energy function is provided for which this situation occurs.
- The example function is derived from a non rank-one convex elastic energy.

We discuss whether homogeneous Cauchy stress implies homogeneous strain in isotropic nonlinear elasticity. While for linear elasticity the positive answer is clear, we exhibit, through detailed calculations, an example with inhomogeneous continuous deformation but constant Cauchy stress. The example is derived from a non rank-one convex elastic energy.