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The quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function

Published online by Cambridge University Press:  14 November 2011

Hervé Le Dret  and
Annie Raoult
Hervé Le Dret
Affiliation:
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France
Annie Raoult
Affiliation:
Laboratoire de Modélisation et Calcul, Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
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Abstract

We give an explicit expression for the quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function in terms of the singular values. This envelope is also the convex, polyconvex and rank 1 convex envelope of the Saint Venant–Kirchhoff stored energy function. Moreover, it coincides with the Saint Venant–Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arranged in increasing order are located outside an ellipsoid. It vanishes on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant–Kirchhoff material can be compressed under zero external loading.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1179 - 1192
Copyright
Copyright © Royal Society of Edinburgh 1995

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