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Comptes Rendus Mathématique
Volume 334, n° 7
pages 615-620 (2002)
Doi : S1631-073X(02)02310-5
Received : 5 November 2001 ;  accepted : 4 February 2002
Justification de la théorie non linéaire de Kirchhoff-Love, comme application d'une nouvelle méthode d'inversion singulière
Justification of the nonlinear Kirchhoff-Love theory, as the application of a new singular inverse method
 

Régis Monneau
CERMICS, École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, cite Descartes, Champs sur Marne, 774455 Marne la Valleé cedex 2, France 

Note présentée par Philippe G. Ciarlet

Résumé

Nous considérons dans le cadre de l'élasticité non linéaire une plaque tridimensionnelle isotrope homogène de St Venant-Kirchhoff, d'épaisseur 2ε et périodique dans les deux autres directions. Le problème se ramène par changement d'échelle à un problème non linéaire de perturbations singulières sur un ouvert fixe. Nous introduisons une nouvelle méthode d'inversion singulière. En appliquant cette méthode nous prouvons, pour des forces extérieures suffisamment petites et fixées, que la solution tridimensionnelle en déplacement converge vers la solution du modèle de Kirchhoff-Love non linéaire des plaques lorsque l'épaisseur 2ε tend vers zéro. Le modèle de plaque limite contient en particulier celui de von Kármán . Nous donnons aussi une estimation a priori de la vitesse de convergence. Pour citer cet article : R. Monneau, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 615-620.

The full text of this article is available in PDF format.
Abstract

In the framework of nonlinear elasticity, we consider a three-dimensional plate made of a St Venant-Kirchhoff isotropic and homogeneous material of thickness 2ε and periodic in the two other directions. By a change of scales, the problem can be mapped on a fixed open set, and seen as a nonlinear singular perturbation problem. We introduce a new singular inverse method. Applying this method, we prove that for fixed and small enough exterior forces, the three-dimensional displacement converges to the solution of the nonlinear Kirchhoff-Love theory of plate as the thickness 2ε tends to zero. The limit plate model contains in particular that of von Kármán . We also give a quantitative estimate of the convergence. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 615-620.

The full text of this article is available in PDF format.


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