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Rank-one convexity implies polyconvexity for isotropic,objective and isochoric elastic energies in the two-dimensional case |
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| Authors: | Robert J Martin Patrizio Neff Ionel-Dumitrel Ghiba |
| Institution: | 1. University of Duisburg-Essen, Faculty of Mathematics, Thea-Leymann-Straße 9, 45127 Essen, Germany;2. University of Duisburg-Essen, Faculty of Mathematics, Thea-Leymann-Straße 9, 45127 Essen, Germany
Alexandru Ioan Cuza University of Ia?i, Department of Mathematics, Blvd. Carol I, no. 11, 700506 Ia?i, Romania Octav Mayer Institute of Mathematics of the Romanian Academy, Ia?i Branch, 700505 Ia?i |
| Abstract: | We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on GL+(2) is already polyconvex on 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. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) |
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