The Föppl–von Kármán plate theory as a low energy <i>Γ</i>-limit of nonlinear elasticity

Gero Friesecke; Richard D. James; Stefan Müller

doi:10.1016/S1631-073X(02)02388-9

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]

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

Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 201-206.

Résumé
Abstract

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 $v : {(0, 1)}^{3} \to ℝ^{3}$ , 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 $v : {(0, 1)}^{3} \to ℝ^{3}$ , 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.

Reçu le : 2002-03-20
Accepté le : 2002-03-28
Publié le : 2002-01-01

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

Affiliations des auteurs :

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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     author = {Gero Friesecke and Richard~D. James and Stefan M\"uller},
     title = {The {F\"oppl{\textendash}von} {K\'arm\'an} plate theory as a low energy {\protect\emph{\ensuremath{\Gamma}}-limit} of nonlinear elasticity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {201--206},
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     doi = {10.1016/S1631-073X(02)02388-9},
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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/

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