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, 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.
Highlights
In isotropic linear elasticity, it is plain to see that a homogeneous stress is always accompanied by homogeneous strain, provided that the usual positive-definiteness assumptions on the elastic energy are required
We discuss whether homogeneous Cauchy stress implies homogeneous strain in isotropic nonlinear elasticity
It is plain to see that a homogeneous stress is always accompanied by homogeneous strain, provided that the usual positive-definiteness assumptions on the elastic energy are required
Summary
It is plain to see that a homogeneous stress is always accompanied by homogeneous strain, provided that the usual positive-definiteness assumptions on the elastic energy are required. The similar question of whether constant stress implies constant strain is considerably more involved. For a homogeneous isotropic hyperelastic body under finite strain deformation, the Cauchy stress tensor can be represented as follows [6,7,16,17]:. Where R ∈ SO(3) is an arbitrary constant rotation, b ∈ 3 is an arbitrary constant translation, and V is the left principal stretch tensor satisfying V2 = B, and which is uniquely determined from the given stress σ = T.
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