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.
Accepted:
Published online:
Sergio Conti 1; Georg Dolzmann 2
@article{CRMATH_2007__344_8_541_0,
author = {Sergio Conti and Georg Dolzmann},
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},
}
Sergio Conti; Georg Dolzmann. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.013/
[1] Γ-Convergence for Beginners, Oxford University Press, Oxford, 2002
[2] Theory of Plates, Mathematical Elasticity, II, Elsevier, Amsterdam, 1997
[3] Semi-soft elasticity and director reorientation in stretched sheets of nematic elastomers, Phys. Rev. E, Volume 66 (2002), p. 061710
[4] S. Conti, G. Dolzmann, A two-dimensional plate theory as a Gamma-limit of three-dimensional incompressible elasticity, in preparation
[5] Derivation of elastic theories for thin sheets and the constraint of incompressibility (A. Mielke, ed.), Analysis, Modeling and Simulation of Multiscale Problems, Springer, 2006, pp. 225-247
[6] An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993
[7] Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. (8) (1975), pp. 842-850
[8] Macroscopic response of nematic elastomers via relaxation of a class of -invariant energies, Arch. Ration. Mech. Anal., Volume 161 (2002), pp. 181-204
[9] A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math, Volume 55 (2002), pp. 1461-1506
[10] A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Ration. Mech. Anal., Volume 180 (2006), pp. 183-236
[11] Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 173-178
[12] Le modèle de membrane nonlinéaire comme limite variationelle de l'élasticité non linéaire tridimensionelle, C. R. Acad. Sci. Paris, Volume 317 (1993), pp. 221-226
[13] The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., Volume 73 (1995), pp. 549-578
[14] On the Sobolev space of isometric immersions, J. Differential Geom., Volume 66 (2004), pp. 47-69
[15] Une justification partielle du modèle de plaque en flexion par Γ-convergence, C. R. Acad. Sci. Paris, Ser. I, Volume 332 (2001), pp. 587-592
[16] On the justification of the nonlinear inextensional plate model, Arch. Ration. Mech. Anal., Volume 167 (2003), pp. 179-209
[17] Incompressible nonlinearly elastic thin membranes, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 75-80
[18] Modeling of a nonlinear membrane plate for incompressible materials via Gamma-convergence, Anal. Appl. (Singap.), Volume 4 (2006), pp. 31-60
[19] Nematic elastomers – a new state of matter?, Prog. Polym. Sci., Volume 21 (1996), pp. 853-891
[20] Liquid Crystal Elastomers, Oxford Univ. Press, 2003
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