Rigorous derivation of nonlinear plate theory and geometric rigidity

Gero Friesecke; Stefan Müller; Richard D. James

doi:10.1016/S1631-073X(02)02133-7

Rigorous derivation of nonlinear plate theory and geometric rigidity
[Dérivation rigoureuse de la théorie non linéaire des plaques et rigidité géométrique]

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

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
² Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
³ Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA

Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 173-178.

Résumé
Abstract

Nous montrons que la théorie non linéaire des plaques émerge comme Γ-limite de la théorie de l'élasticité tridimensionnelle. La démonstration repose sur un résultat de rigidité pour des fonctions $v : {(0, 1)}^{3} \to ℝ^{3}$ . Nous montrons que la distance L² de ∇v d'une rotation est bornée par un multiple de la distance L² à l'ensemble SO(3) des rotations.

We show that nonlinear plate theory arises as a Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a sharp rigidity estimate for maps $v : {(0, 1)}^{3} \to ℝ^{3}$ . We show that 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 : 2001-08-16
Accepté le : 2001-09-13
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02133-7

Affiliations des auteurs :

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

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
² Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
³ Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA

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Gero Friesecke; Stefan Müller; Richard D. James. Rigorous derivation of nonlinear plate theory and geometric rigidity. Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 173-178. doi : 10.1016/S1631-073X(02)02133-7. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02133-7/

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