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Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

Part of: Real functions Functions of several variables Elastic materials Existence theories

Published online by Cambridge University Press:  11 June 2019

Robert J. Martin ,
Jendrik Voss ,
Patrizio Neff  and
Ionel-Dumitrel Ghiba
Robert J. Martin
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany (robert.martin@uni-due.de; max.voss@uni-due.de) (patrizio.neff@uni-due.de)
Jendrik Voss
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany (robert.martin@uni-due.de; max.voss@uni-due.de) (patrizio.neff@uni-due.de)
Patrizio Neff
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany (robert.martin@uni-due.de; max.voss@uni-due.de) (patrizio.neff@uni-due.de)
Ionel-Dumitrel Ghiba
Affiliation:
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, 700505Iaşi, (dumitrel.ghiba@uaic.ro)
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Abstract

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In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

Keywords

quasiconvexityrank-one convexitypolyconvexityquasiconvex envelopesnonlinear elasticityincompressibilityhyperelasticityrelaxationmicrostructure

MSC classification

Primary: 26B25: Convexity, generalizations
Secondary: 26A51: Convexity, generalizations 49J45: Methods involving semicontinuity and convergence; relaxation 74B20: Nonlinear elasticity
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2620 - 2631
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Royal Society of Edinburgh 2019

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