The Föppl–von Kármán plate theory as a low energy <i>Γ</i>-limit of nonlinear elasticity

Gero Friesecke; Richard D. James; Stefan Müller

doi:10.1016/S1631-073X(02)02388-9

The Föppl–von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity
[La théorie Föppl–von Kármán des plaques comme Γ-limite de l'élasticité non linéaire]

Gero Friesecke ¹ ; Richard D. James ² ; Stefan Müller ³

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
² Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA
³ Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany

Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 201-206.

Abstract (VO)
Résumé

We show that the Föppl–von Kármán theory arises as a low energy Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [5] that for maps $v : {(0, 1)}^{3} \to ℝ^{3}$ , the L² distance of ∇v from a single rotation is bounded by a multiple of the L² distance from the set SO(3) of all rotations.

Nous montrons que la théorie Föppl–von Kármán des plaques émerge comme Γ-limite de la théorie de l'élasticité tridimensionnelle. La démonstration repose sur une generalisation aux derivées d'ordre supérieur de notre résultat de rigidité [5] que pour des fonctions $v : {(0, 1)}^{3} \to ℝ^{3}$ , la distance L² de ∇v à une rotation est bornée par un multiple de la distance L² à l'ensemble SO(3) des rotations.

Reçu le : 2002-03-20
Accepté le : 2002-03-28
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02388-9

Affiliations des auteurs :

Gero Friesecke ¹ ; Richard D. James ² ; Stefan Müller ³

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
² Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA
³ Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany

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     author = {Gero Friesecke and Richard~D. James and Stefan M\"uller},
     title = {The {F\"oppl{\textendash}von} {K\'arm\'an} plate theory as a low energy {\protect\emph{\ensuremath{\Gamma}}-limit} of nonlinear elasticity},
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Gero Friesecke; Richard D. James; Stefan Müller. The Föppl–von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. Comptes Rendus. Mathématique, Volume 335 (2002) no. 2, pp. 201-206. doi : 10.1016/S1631-073X(02)02388-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02388-9/

Bibliographie
Cité par

[1] S.S. Antman Nonlinear Problems of Elasticity, Springer, New York, 1995

[2] G. Anzelotti; S. Baldo; D. Percivale Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity, Asymptotic Anal., Volume 9 (1994), pp. 61-100

[3] P.G. Ciarlet Mathematical Elasticity II – Theory of Plates, Elsevier, Amsterdam, 1997

[4] D.D. Fox; A. Raoult; J.C. Simo A justification of nonlinear properly invariant plate theories, Arch. Rational Mech. Anal., Volume 124 (1993), pp. 157-199

[5] G. Friesecke; R.D. James; S. Müller Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, Série I, Volume 334 (2002), pp. 173-178

[6] G. Friesecke, R.D. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., to appear

[7] H. LeDret; A. Raoult Le modéle de membrane non linéaire comme limite variationelle de l'élasticité non linéaire tridimensionelle, C. R. Acad. Sci. Paris, Série I, Volume 317 (1993), pp. 221-226

[8] H. LeDret; A. Raoult The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., Volume 73 (1995), pp. 549-578

[9] A.E.H. Love A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, 1927

[10] J.J. Marigo; H. Ghidouche; Z. Sedkaoui Des poutres flexibles aux fils extensibles : une hiérachie de modèles asymptotiques, C. R. Acad. Sci. Paris, Série IIb, Volume 326 (1998), pp. 79-84

[11] R. Monneau, Justification of nonlinear Kirchhoff–Love theory of plates as the application of a new singular inverse method, Preprint, 2001

[12] O. Pantz Une justification partielle du modèle de plaque en flexion par Γ-convergence, C. R. Acad. Sci. Paris, Série I, Volume 332 (2001), pp. 587-592

[13] A. Raoult, Personal communication

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Holm Altenbach; Victor A. Eremeyev Cosserat-Type Shells, Generalized Continua from the Theory to Engineering Applications, Volume 541 (2013), p. 131 | DOI:10.1007/978-3-7091-1371-4_3
L.E. Perotti; A. Bompadre; M. Ortiz Automatically inf − sup compliant diamond‐mixed finite elements for Kirchhoff plates, International Journal for Numerical Methods in Engineering, Volume 96 (2013) no. 7, p. 405 | DOI:10.1002/nme.4555
Sören Bartels Finite element approximation of large bending isometries, Numerische Mathematik, Volume 124 (2013) no. 3, p. 415 | DOI:10.1007/s00211-013-0519-7
Igor Velčić Nonlinear Weakly Curved Rod by Γ-Convergence, Journal of Elasticity, Volume 108 (2012) no. 2, p. 125 | DOI:10.1007/s10659-011-9358-x
Igor Velčić Shallow-shell models by Γ-convergence, Mathematics and Mechanics of Solids, Volume 17 (2012) no. 8, p. 781 | DOI:10.1177/1081286511429889
Johannes Altenbach; Holm Altenbach; Victor A. Eremeyev On generalized Cosserat-type theories of plates and shells: a short review and bibliography, Archive of Applied Mechanics, Volume 80 (2010) no. 1, p. 73 | DOI:10.1007/s00419-009-0365-3
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Patrizio Neff Γ-convergene e for a geometrically exact Cosserat shell-model of defective elastic crystals, Poly-, Quasi- and Rank-One Convexity in Applied Mechanics, Volume 516 (2010), p. 301 | DOI:10.1007/978-3-7091-0174-2_9
Nabil Kerdid; Hervé Le Dret; Abdelkader Saı¨di Numerical approximation for a non-linear membrane problem, International Journal of Non-Linear Mechanics, Volume 43 (2008) no. 9, p. 908 | DOI:10.1016/j.ijnonlinmec.2008.06.007
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Erick Pruchnicki NonLinearly Elastic Membrane Model For Heterogeneous Shells by Using a New Double Scale Variational Formulation: A Formal Asymptotic Approach, Journal of Elasticity, Volume 84 (2006) no. 3, p. 245 | DOI:10.1007/s10659-006-9066-0
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