A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity. (March 2020)

• Record Type: Journal Article
• Authors: Schweickert, Eva; Mihai, L. Angela; Martin, Robert J.; Neff, Patrizio
• Is Part Of: International journal of non-linear mechanics. Volume 119(2020)
• Publication Date: 2020-03
• Subjects: 74B20; Nonlinear elasticity -- Ellipticity -- Conformal mappings -- Möbius transformations; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2019.103282
• Languages: English
• ISSNs: 0020-7462

Abstract

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. Highlights: We consider a strictly rank-one convex hyperelastic energy potential. Previous results show that laminates inducing a homogeneous Cauchy stress are affine. We construct a non-affine deformation which induces a constant Cauchy stress field.