Homogeneous Cauchy stress induced by non‐homogeneous deformations

Abstract

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

A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity

“…In particular, for a (planar) conformal mapping ϕ : Ω ⊂ R 2 → R 2 , the deformation gradients ∇ϕ(x 1 ), ∇ϕ(x 2 ) at x 1 , x 2 ∈ Ω cannot differ by a rank-one matrix. Figure 5: Structure of a rank-one laminate [16,15].…”