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

Abstract

It has recently been shown that for a Cauchy stress response induced by a strictly rank-one convex hyperelastic energy potential, a homogeneous Cauchy stress tensor field cannot correspond to a non-homogeneous deformation if the deformation gradient has discrete values, i.e. if the deformation is piecewise affine linear and satisfies the Hadamard jump condition. In this note, we expand upon these results and show that they do not hold for arbitrary deformations by explicitly giving an example of a strictly rank-one convex energy and a non-homogeneous deformation such that the induced Cauchy stress tensor is constant. In the planar case, our example is related to another previous result concerning criteria for generalized convexity properties of conformally invariant energy functions, which we extend to the case of strict rank-one convexity.

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• Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case
• Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations
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• Hyperelastic bodies under homogeneous Cauchy stress induced by three-dimensional non-homogeneous deformations

Cited by in corpus (1)

• Rank-one convexity vs. ellipticity for isotropic functions