Mathematical Problems in Mechanics
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.

Let ω be a simply-connected domain in R2 and let (Eαβ) and (Fαβ) be two symmetric 2×2 matrix fields with components in L2(ω). In this Note, we identify nonlinear compatibility conditions “of Donati type” that the components Eαβ and Fαβ must satisfy in order that there exists a vector field (η1,η2,w)H01(ω)×H01(ω)×H02(ω) such that:

12(αηβ+βηα+αwβw)=Eαβandαβw=Fαβin 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 R2 et soient (Eαβ) et (Fαβ) deux champs de matrices 2×2 symétriques dont les composantes sont dans L2(ω). Dans cette Note, on identifie et justifie des conditions non linéaires de compatibilité « de type Donati » que doivent satisfaire les composantes Eαβ et Fαβ afin quʼil existe un champ de vecteurs (η1,η2,w)H01(ω)×H01(ω)×H02(ω) tel que :

12(αηβ+βηα+αwβw)=Eαβetαβw=Fαβdans 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:
Published online:
DOI: 10.1016/j.crma.2013.05.012
Ciarlet, Philippe G. 1; Geymonat, Giuseppe 2; Krasucki, Françoise 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
@article{CRMATH_2013__351_9-10_405_0,
     author = {Ciarlet, Philippe G. and Geymonat, Giuseppe and Krasucki, Fran\c{c}oise},
     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},
     publisher = {Elsevier},
     volume = {351},
     number = {9-10},
     year = {2013},
     doi = {10.1016/j.crma.2013.05.012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.05.012/}
}
TY  - JOUR
AU  - Ciarlet, Philippe G.
AU  - Geymonat, Giuseppe
AU  - Krasucki, Françoise
TI  - Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff–von Kármán–Love plate theory
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 405
EP  - 409
VL  - 351
IS  - 9-10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.05.012/
DO  - 10.1016/j.crma.2013.05.012
LA  - en
ID  - CRMATH_2013__351_9-10_405_0
ER  - 
%0 Journal Article
%A Ciarlet, Philippe G.
%A Geymonat, Giuseppe
%A Krasucki, Françoise
%T Nonlinear Donati compatibility conditions for the nonlinear Kirchhoff–von Kármán–Love plate theory
%J Comptes Rendus. Mathématique
%D 2013
%P 405-409
%V 351
%N 9-10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.05.012/
%R 10.1016/j.crma.2013.05.012
%G en
%F CRMATH_2013__351_9-10_405_0
Ciarlet, Philippe G.; Geymonat, Giuseppe; Krasucki, Françoise. 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. http://www.numdam.org/articles/10.1016/j.crma.2013.05.012/

[1] Amrouche, C.; Ciarlet, P.G.; Gratie, L.; Kesavan, S. On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., Volume 86 (2006), pp. 116-132

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

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

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

[7] Geymonat, G.; Krasucki, F. On the existence of the Airy function in Lipschitz domains. Application to the traces of H2, C. R. Acad. Sci. Paris, Ser. I, Volume 330 (2000), pp. 355-360

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

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

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

Cited by Sources: