Calculus of Variations
Derivation of a plate theory for incompressible materials
Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 541-544.

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.

Nous dérivons un modèle bidimensionnel pour les plaques élastiques comme Γ-limite de la théorie de l'élasticité tridimensionnelle avec contrainte d'incompressiiblité. La densité d'énergie du problème réduit est déterminée à partir des modules d'élastiques de la densité d'énergie tridimensionnelle à l'identité. Sans contrainte d'incompressibilité, Friesecke, James et Müller sont les premiers à avoir rigoureusement justifié le modèle de plaque en flexion par Γ-convergence. La difficulté principale de l'extension de ce résultat au cas incompressible réside dans la construction, afin d'établir l'inégalité de Γ-limsup, d'une suite de déformations satisfaisant la contrainte non-linéaire d'incompressibilité.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2007.03.013
Conti, Sergio 1; Dolzmann, Georg 2

1 Fachbereich Mathematik, Universität Duisburg-Essen, Lotharstr. 65, 47057 Duisburg, Germany
2 NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany
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Conti, Sergio; Dolzmann, Georg. Derivation of a plate theory for incompressible materials. Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 541-544. doi : 10.1016/j.crma.2007.03.013. http://www.numdam.org/articles/10.1016/j.crma.2007.03.013/

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