Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case

Abstract

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on ${\mathrm{GL}}^{+}(2)$ is already polyconvex on ${\mathrm{GL}}^{+}(2)$. Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. Our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.

References in corpus (4)

• Geometry of logarithmic strain measures in solid mechanics
• The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers
• The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity
• Ahlfors-Beurling operator on radial functions

Cited by in corpus (12)

• Weak lower semicontinuity of integral functionals and applications
• Characterizations of symmetric polyconvexity
• A non-ellipticity result, or the impossible taming of the logarithmic strain measure
• A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity
• Two-by-two upper triangular matrices and Morrey's conjecture
• Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture
• The exponentiated Hencky strain energy in modelling tire derived material for moderately large deformations
• Remarks on Analytic Solutions in Nonlinear Elasticity and Anti-Plane Shear Problem
• Rank-one convexity vs. ellipticity for isotropic functions
• Morrey's conjecture for the planar volumetric-isochoric split. Part I: least convex energy functions
• Automatic quasiconvexity of homogeneous isotropic rank-one convex integrands
• A rank-one convex, non-polyconvex isotropic function on ${\mathrm{GL}}^{+}(2)$ with compact connected sublevel sets