Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T18:52:19.515Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86(1984), 125–45.CrossRefGoogle Scholar
2Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
3Ball, J. M.. Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51 (1984), 699–728.CrossRefGoogle Scholar
4Ball, J. M. and Murat, F.. W 1.p quasiconvexity and variational problems for multiple integrals. J. Fund. Anal. 58 (1984), 225–53.CrossRefGoogle Scholar
5Bousselsal, M. and Brighi, B.. On the rank-one convexity domain of the Saint Venant–Kirchhoff stored energy function (to appear).Google Scholar
6Ciarlet, P. G.. Introduction to Numerical Linear Algebra and Optimization (Edinburgh: Cambridge University Press, 1987).Google Scholar
7Ciarlet, P. G.. Mathematical Elasticity, Volume I: Three-Dimensional Elasticity (Amsterdam: North-Holland, 1988).Google Scholar
8Dacorogna, B.. Quasiconvexity and relaxation of non convex variational problems. J. Fund. Anal. 46 (1982), 102–18.CrossRefGoogle Scholar
9Dacorogna, B.. Direct Methods in the Calculus of Variations. Applied Mathematical Sciences 78 (Berlin: Springer, 1989).CrossRefGoogle Scholar
10Kohn, R. V. and Strang, G.. Explicit relaxation of a variational problem in optimal design. Bull. Amer. Math. Soc. 9 (1983), 211–14.CrossRefGoogle Scholar
11Kohn, R. V. and Strang, G.. Optimal design and relaxation of variational problems, I. II and III. Comm. Pure Appl. Math. 39 (1986), 113-37; 139-82; 353–77.CrossRefGoogle Scholar
12Dret, H. Le. Sur les fonctions de matrices convexes et isotropes. C.R. Acad. Sci. Paris Ser. I 310 (1990), 617–20.Google Scholar
13Dret, H. Le and Raoult, A.. Le modele de membrane non lineaire comme limite variationnelle de l'élasticité non lineaire tridimensionnelle. C.R. Acad. Sci. Paris Sér. I 317 (1993), 221–6.Google Scholar
14Dret, H. Le and Raoult, A.. Enveloppe quasi-convexe de la densite d'energie de Saint Venant–Kirchhoff. C.R. Acad. Sci. Paris Ser. I 38 (1994), 93–8.Google Scholar
15Dret, H. Le and Raoult, A.. The nonlinear membrane model as variational limit of three-dimensional nonlinear elasticity. J. Math. Pures Appl. Comm. Appl. Nonlinear Anal. 1 (1994), 85–96.Google Scholar
16Dret, H. Le and Raoult, A.. Remarks on the quasiconvex envelope of stored energy functions in nonlinear elasticity Comm. Appl. Nonlinear Anal. 1 (1994), 85–96.Google Scholar
17Morrey, C. B. JrQuasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.CrossRefGoogle Scholar
18Raoult, A.. Non-polyconvexity of the stored energy function of a Saint Venant–Kirchhoff material. Aplikace Matematiky 31 (1986), 417–19.Google Scholar
19Thompson, R. C. and Freede, L. J.. Eigenvalues of sums of Hermitian matrices III. J. Research Nat. Bur. Standards B 75 (1971), 115–20.CrossRefGoogle Scholar
20Truesdell, C. and Noll, W.. The Nonlinear Field Theories of Mechanics. In Handbuch der Physik, Vol. III/3 (Berlin: Springer, 1965).Google Scholar