Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff–von Kármán–Love plate theory

Philippe G. Ciarlet; Giuseppe Geymonat; Françoise Krasucki

doi:10.1016/j.crma.2013.05.012

Mathematical Problems in Mechanics

Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff–von Kármán–Love plate theory

Presented by: Philippe G. Ciarlet

Philippe G. Ciarlet ¹; Giuseppe Geymonat ²; Françoise Krasucki ³

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire de mécanique des solides, UMR 7649-0176, École polytechnique, 91128 Palaiseau cedex, France
³ I3M, UMR–CNRS 5149, université de Montpellier-2, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 405-409.

Abstract
Résumé

Let ω be a simply-connected domain in $R^{2}$ and let $(E_{α β})$ and $(F_{α β})$ be two symmetric $2 \times 2$ matrix fields with components in $L^{2} (ω)$ . In this Note, we identify nonlinear compatibility conditions “of Donati type” that the components $E_{α β}$ and $F_{α β}$ must satisfy in order that there exists a vector field $(η_{1}, η_{2}, w) \in H_{0}^{1} (ω) \times H_{0}^{1} (ω) \times H_{0}^{2} (ω)$ such that:

\frac{1}{2} (\partial_{α} η_{β} + \partial_{β} η_{α} + \partial_{α} w \partial_{β} w) = E_{α β} and \partial_{α β} w = F_{α β} in ω .

The left-hand sides of these relations are the components of tensors found in the Kirchhoff–von Kármán–Love theory of nonlinearly elastic plates.

Soit ω un domaine simplement connexe de $R^{2}$ et soient $(E_{α β})$ et $(F_{α β})$ deux champs de matrices $2 \times 2$ symétriques dont les composantes sont dans $L^{2} (ω)$ . Dans cette Note, on identifie et justifie des conditions non linéaires de compatibilité « de type Donati » que doivent satisfaire les composantes $E_{α β}$ et $F_{α β}$ afin quʼil existe un champ de vecteurs $(η_{1}, η_{2}, w) \in H_{0}^{1} (ω) \times H_{0}^{1} (ω) \times H_{0}^{2} (ω)$ tel que :

\frac{1}{2} (\partial_{α} η_{β} + \partial_{β} η_{α} + \partial_{α} w \partial_{β} w) = E_{α β} et \partial_{α β} w = F_{α β} dans ω .

Les membres de gauche de ces relations sont les composantes de tenseurs trouvés dans la théorie de Kirchhoff–von Kármán–Love des plaques non linéairement élastiques.

Received: 2013-05-24
Published online: 2013-05-01

DOI: 10.1016/j.crma.2013.05.012

Author's affiliations:

Philippe G. Ciarlet ¹; Giuseppe Geymonat ²; Françoise Krasucki ³

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire de mécanique des solides, UMR 7649-0176, École polytechnique, 91128 Palaiseau cedex, France
³ I3M, UMR–CNRS 5149, université de Montpellier-2, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

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     author = {Philippe G. Ciarlet and Giuseppe Geymonat and Fran\c{c}oise Krasucki},
     title = {Nonlinear {Donati} compatibility conditions for the nonlinear {Kirchhoff{\textendash}von} {K\'arm\'an{\textendash}Love} plate theory},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {405--409},
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     year = {2013},
     doi = {10.1016/j.crma.2013.05.012},
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Philippe G. Ciarlet; Giuseppe Geymonat; Françoise Krasucki. Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff–von Kármán–Love plate theory. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 405-409. doi : 10.1016/j.crma.2013.05.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.05.012/

References
Cited by

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[2] P.G. Ciarlet Mathematical Elasticity, Volume II: Theory of Plates, North-Holland, Amsterdam, 1997

[3] P.G. Ciarlet Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013

[4] P.G. Ciarlet; P. Ciarlet Another approach to linearized elasticity and a new proof of Kornʼs inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271

[5] P.G. Ciarlet, G. Geymonat, F. Krasucki, Nonlinear Donati compatibility conditions and the intrinsic approach for nonlinearly elastic plates, in preparation.

[6] P.G. Ciarlet; S. Mardare Nonlinear Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates, Math. Models Methods Appl. Sci. (2013) (in press) | DOI

[7] G. Geymonat; F. Krasucki On the existence of the Airy function in Lipschitz domains. Application to the traces of $H^{2}$ , C. R. Acad. Sci. Paris, Ser. I, Volume 330 (2000), pp. 355-360

[8] G. Geymonat; F. Krasucki Some remarks on the compatibility conditions in elasticity, Accad. Naz. Sci. XL, Volume 123 (2005), pp. 175-182

[9] G. Geymonat; F. Krasucki Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains, Commun. Pure Appl. Anal., Volume 8 (2009), pp. 295-309

[10] S. Kesavan On Poincaréʼs and J.-L. Lionsʼ lemmas, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 27-30

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