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Comptes Rendus Mathématique
Volume 344, n° 8
pages 541-544 (avril 2007)
Doi : 10.1016/j.crma.2007.03.013
Received : 9 December 2005 ;  accepted : 14 Mars 2007
Derivation of a plate theory for incompressible materials
Dérivation de la théorie non-linéaire des plaques avec la contrainte de lʼincompressibilité
 

Sergio Conti a , Georg Dolzmann b
a Fachbereich Mathematik, Universität Duisburg-Essen, Lotharstr. 65, 47057 Duisburg, Germany 
b NWF I - Mathematik, Universität Regensburg, 93040 Regensburg, Germany 

Abstract

We derive a two-dimensional model for elastic plates as a Γ -limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The energy density of the reduced problem describes plate bending, and is determined from the elastic moduli at the identity of the energy density of the three-dimensional problem. Without the constraint of incompressibility, Γ -convergence to a plate theory was first derived by Friesecke, James and Müller. The main difficulty in the present result is the construction of a recovery sequence which satisfies pointwise the nonlinear constraint of incompressibility. To cite this article: S. Conti, G. Dolzmann, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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

Nous dérivons un modèle bidimensionnel pour les plaques élastiques comme Γ -limite de la théorie de lʼélasticité tridimensionnelle avec contrainte dʼincompressiiblité. La densité dʼénergie du problème réduit est déterminée à partir des modules dʼélastiques de la densité dʼénergie tridimensionnelle à lʼidentité. Sans contrainte dʼincompressibilité, Friesecke, James et Müller sont les premiers à avoir rigoureusement justifié le modèle de plaque en flexion par Γ -convergence. La difficulté principale de lʼextension de ce résultat au cas incompressible réside dans la construction, afin dʼétablir lʼinégalité de Γ -limsup, dʼune suite de déformations satisfaisant la contrainte non-linéaire dʼincompressibilité. Pour citer cet article : S. Conti, G. Dolzmann, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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


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