Comptes Rendus
The Föppl–von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity
[La théorie Föppl–von Kármán des plaques comme Γ-limite de l'élasticité non linéaire]
Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 201-206.

Nous montrons que la théorie Föppl–von Kármán des plaques émerge comme Γ-limite de la théorie de l'élasticité tridimensionnelle. La démonstration repose sur une generalisation aux derivées d'ordre supérieur de notre résultat de rigidité [5] que pour des fonctions v:(0,1) 3 3 , la distance L2 de ∇v à une rotation est bornée par un multiple de la distance L2 à l'ensemble SO(3) des rotations.

We show that the Föppl–von Kármán theory arises as a low energy Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [5] that for maps v:(0,1) 3 3 , the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.

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Accepté le :
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DOI : 10.1016/S1631-073X(02)02388-9

Gero Friesecke 1 ; Richard D. James 2 ; Stefan Müller 3

1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
2 Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA
3 Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany
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Gero Friesecke; Richard D. James; Stefan Müller. The Föppl–von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 201-206. doi : 10.1016/S1631-073X(02)02388-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02388-9/

[1] S.S. Antman Nonlinear Problems of Elasticity, Springer, New York, 1995

[2] G. Anzelotti; S. Baldo; D. Percivale Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity, Asymptotic Anal., Volume 9 (1994), pp. 61-100

[3] P.G. Ciarlet Mathematical Elasticity II – Theory of Plates, Elsevier, Amsterdam, 1997

[4] D.D. Fox; A. Raoult; J.C. Simo A justification of nonlinear properly invariant plate theories, Arch. Rational Mech. Anal., Volume 124 (1993), pp. 157-199

[5] G. Friesecke; R.D. James; S. Müller Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, Série I, Volume 334 (2002), pp. 173-178

[6] G. Friesecke, R.D. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., to appear

[7] H. LeDret; A. Raoult Le modéle de membrane non linéaire comme limite variationelle de l'élasticité non linéaire tridimensionelle, C. R. Acad. Sci. Paris, Série I, Volume 317 (1993), pp. 221-226

[8] H. LeDret; A. Raoult The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., Volume 73 (1995), pp. 549-578

[9] A.E.H. Love A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, 1927

[10] J.J. Marigo; H. Ghidouche; Z. Sedkaoui Des poutres flexibles aux fils extensibles : une hiérachie de modèles asymptotiques, C. R. Acad. Sci. Paris, Série IIb, Volume 326 (1998), pp. 79-84

[11] R. Monneau, Justification of nonlinear Kirchhoff–Love theory of plates as the application of a new singular inverse method, Preprint, 2001

[12] O. Pantz Une justification partielle du modèle de plaque en flexion par Γ-convergence, C. R. Acad. Sci. Paris, Série I, Volume 332 (2001), pp. 587-592

[13] A. Raoult, Personal communication

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