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

Part of: Equilibrium (steady-state) problems Functions of several variables Elastic materials

Published online by Cambridge University Press:  18 May 2017

Robert J. Martin ,
Ionel-Dumitrel Ghiba  and
Patrizio Neff
Robert J. Martin
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Strasse 9, 45127 Essen, Germany (robert.martin@uni-due.de)
Ionel-Dumitrel Ghiba
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Strasse 9, 45127 Essen, Germany, Department of Mathematics, Alexandru Ioan Cuza University of Iaşi, Blvd Carol I, no. 11, 700506 Iaşi, Romania and Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi Branch, 700505 Iaşi, Romania (dumitrel.ghiba@uni-due.de; dumitrel.ghiba@uaic.ro)
Patrizio Neff
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Strasse 9, 45127 Essen, Germany (patrizio.neff@uni-due.de)
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We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL+(2) is already polyconvex on GL+(2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. 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.

Keywords

rank-one convexitypolyconvexityquasi-convexityMorrey's conjectureisochoric energiesnonlinear elasticity

MSC classification

Secondary: 74B20: Nonlinear elasticity 74G65: Energy minimization 26B25: Convexity, generalizations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 3 , June 2017 , pp. 571 - 597
Copyright
Copyright © Royal Society of Edinburgh 2017