[Dérivation de la théorie non-linéaire des plaques avec la contrainte de l'incompressibilité]
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.
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.
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@article{CRMATH_2007__344_8_541_0,
author = {Conti, Sergio and Dolzmann, Georg},
title = {Derivation of a plate theory for incompressible materials},
journal = {Comptes Rendus. Math\'ematique},
pages = {541--544},
publisher = {Elsevier},
volume = {344},
number = {8},
year = {2007},
doi = {10.1016/j.crma.2007.03.013},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.03.013/}
}
TY - JOUR AU - Conti, Sergio AU - Dolzmann, Georg TI - Derivation of a plate theory for incompressible materials JO - Comptes Rendus. Mathématique PY - 2007 SP - 541 EP - 544 VL - 344 IS - 8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.03.013/ DO - 10.1016/j.crma.2007.03.013 LA - en ID - CRMATH_2007__344_8_541_0 ER -
%0 Journal Article %A Conti, Sergio %A Dolzmann, Georg %T Derivation of a plate theory for incompressible materials %J Comptes Rendus. Mathématique %D 2007 %P 541-544 %V 344 %N 8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.03.013/ %R 10.1016/j.crma.2007.03.013 %G en %F CRMATH_2007__344_8_541_0
Conti, Sergio; Dolzmann, Georg. Derivation of a plate theory for incompressible materials. Comptes Rendus. Mathématique, Tome 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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