[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.
Accepté le :
Publié le :
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
[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
- A nonlinear bending theory for nematic LCE plates, M
AS. Mathematical Models Methods in Applied Sciences, Volume 33 (2023) no. 7, pp. 1437-1516 | DOI:10.1142/s0218202523500331 | Zbl:1519.74044 - Bending, buckling and free vibration analysis of incompressible functionally graded plates using higher order shear and normal deformable plate theory, Applied Mathematical Modelling, Volume 69 (2019), pp. 47-62 | DOI:10.1016/j.apm.2018.11.047 | Zbl:1461.74043
- Plates with incompatible prestrain of high order, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 34 (2017) no. 7, pp. 1883-1912 | DOI:10.1016/j.anihpc.2017.01.003 | Zbl:1457.74121
- Convergence of equilibria for incompressible elastic plates in the von Kármán regime, Communications on Pure and Applied Analysis, Volume 14 (2015) no. 1, pp. 143-166 | DOI:10.3934/cpaa.2015.14.143 | Zbl:1317.74058
- The von Kármán theory for incompressible elastic shells, Calculus of Variations and Partial Differential Equations, Volume 48 (2013) no. 1-2, pp. 185-209 | DOI:10.1007/s00526-012-0549-5 | Zbl:1314.74043
- The Kirchhoff theory for elastic pre-strained shells, Nonlinear Analysis: Theory, Methods Applications, Volume 78 (2013), p. 1 | DOI:10.1016/j.na.2012.07.035
-
-convergence for incompressible elastic plates, Calculus of Variations and Partial Differential Equations, Volume 34 (2009) no. 4, pp. 531-551 | DOI:10.1007/s00526-008-0194-1 | Zbl:1161.74038 - Approximating
isometric immersions, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 346 (2008) no. 3-4, pp. 189-192 | DOI:10.1016/j.crma.2008.01.001 | Zbl:1149.46030 - Derivation of a plate theory for incompressible materials, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 344 (2007) no. 8, pp. 541-544 | DOI:10.1016/j.crma.2007.03.013 | Zbl:1112.74037
Cité par 9 documents. Sources : Crossref, zbMATH
Commentaires - Politique
Vous devez vous connecter pour continuer.
S'authentifier