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Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

Published online by Cambridge University Press:  11 June 2019

Robert J. Martin
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany ([email protected]; [email protected]) ([email protected])
Jendrik Voss
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany ([email protected]; [email protected]) ([email protected])
Patrizio Neff
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg–Essen, Campus Essen, Thea-Leymann-Straße 9, 45127Essen, Germany ([email protected]; [email protected]) ([email protected])
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, ([email protected])
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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.

Type
Research Article
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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