Comptes Rendus
Mathematical Problems in Mechanics/Calculus of Variations
Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence
[Dérivation de la théorie non linéaire des coques en flexion à partir de l'élasticité non linéaire tridimensionelle par Gamma-convergence]
Comptes Rendus. Mathématique, Volume 336 (2003) no. 8, pp. 697-702.

We show that the nonlinear bending theory of shells arises as a Γ-limit of three-dimensional nonlinear elasticity.

Nous montrons que la théorie non linéaire des coques en flexion émerge comme Γ-limite de la théorie de l'élasticité tridimensionelle.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00028-1

Gero Friesecke 1 ; Richard D. James 2 ; Maria Giovanna Mora 3 ; 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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Gero Friesecke; Richard D. James; Maria Giovanna Mora; Stefan Müller. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. Comptes Rendus. Mathématique, Volume 336 (2003) no. 8, pp. 697-702. doi : 10.1016/S1631-073X(03)00028-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00028-1/

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  • Shun Li; Peng-Fei Yao, Proceedings of the 10th World Congress on Intelligent Control and Automation (2012), p. 1585 | DOI:10.1109/wcica.2012.6358131
  • Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Archive for Rational Mechanics and Analysis, Volume 200 (2011) no. 3, pp. 1023-1050 | DOI:10.1007/s00205-010-0387-6 | Zbl:1291.74130
  • Philippe G. Ciarlet; Radu Gogu; Cristinel Mardare A notion of polyconvex function on a surface suggested by nonlinear shell theory, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 349 (2011) no. 21-22, pp. 1207-1211 | DOI:10.1016/j.crma.2011.10.002 | Zbl:1385.74010
  • Marta Lewicka; Mohammad Reza Pakzad Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1158-1173 | DOI:10.1051/cocv/2010039 | Zbl:1300.74028
  • Marta Lewicka Reduced theories in nonlinear elasticity, Nonlinear conservation laws and applications. Proceedings of the summer program, IMA, Minneapolis, MN, USA, July 13–31, 2009, New York, NY: Springer, 2011, pp. 393-403 | DOI:10.1007/978-1-4419-9554-4_22 | Zbl:1403.74051
  • Marta Lewicka Metric-induced Morphogenesis and Non-Euclidean Elasticity: Scaling Laws and Thin Film Models, Parabolic Problems, Volume 80 (2011), p. 433 | DOI:10.1007/978-3-0348-0075-4_22
  • 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, pp. 73-92 | DOI:10.1007/s00419-009-0365-3 | Zbl:1184.74042
  • Dominique Blanchard; Georges Griso Justification of a simplified model for shells in nonlinear elasticity, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 348 (2010) no. 7-8, pp. 461-465 | DOI:10.1016/j.crma.2010.02.015 | Zbl:1258.74136
  • Dominique Blanchard; Georges Griso Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant's strain tensor, Journal of Elasticity, Volume 101 (2010) no. 2, pp. 179-205 | DOI:10.1007/s10659-010-9255-8 | Zbl:1258.74137
  • PATRIZIO NEFF; KWON-IL HONG; JENA JEONG THE REISSNER–MINDLIN PLATE IS THE Γ-LIMIT OF COSSERAT ELASTICITY, Mathematical Models and Methods in Applied Sciences, Volume 20 (2010) no. 09, p. 1553 | DOI:10.1142/s0218202510004763
  • 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
  • Dominique Blanchard; Georges Griso Decomposition of shell deformations - asymptotic behavior of the Green-St Venant strain tensor, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 347 (2009) no. 17-18, pp. 1099-1103 | DOI:10.1016/j.crma.2009.06.018 | Zbl:1170.74031
  • Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad A nonlinear theory for shells with slowly varying thickness, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 347 (2009) no. 3-4, pp. 211-216 | DOI:10.1016/j.crma.2008.12.017 | Zbl:1168.74036
  • Pavel Bělík; Bob Jennings; Mikhail M. Shvartsman; Cristina U. Thomas Modeling the behavior of heat-shrinkable thin films, Journal of Elasticity, Volume 95 (2009) no. 1-2, pp. 57-77 | DOI:10.1007/s10659-009-9194-4 | Zbl:1160.74032
  • Bernd Schmidt Qualitative properties of a continuum theory for thin films, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 25 (2008) no. 1, pp. 43-75 | DOI:10.1016/j.anihpc.2006.09.001 | Zbl:1142.74026
  • Bernd Schmidt On the passage from atomic to continuum theory for thin films, Archive for Rational Mechanics and Analysis, Volume 190 (2008) no. 1, pp. 1-55 | DOI:10.1007/s00205-008-0138-0 | Zbl:1156.74028
  • Philippe G. Ciarlet A brief introduction to mathematical shell theory, Classical and Advanced Theories of Thin Structures, Volume 503 (2008), p. 111 | DOI:10.1007/978-3-211-85430-3_5
  • E. L. Starostin; G. H. M. van der Heijden Tension-Induced Multistability in Inextensible Helical Ribbons, Physical Review Letters, Volume 101 (2008) no. 8 | DOI:10.1103/physrevlett.101.084301
  • David J. Steigmann Asymptotic finite-strain thin-plate theory for elastic solids, Computers Mathematics with Applications, Volume 53 (2007) no. 2, pp. 287-295 | DOI:10.1016/j.camwa.2006.02.025 | Zbl:1129.74026
  • Bernd Schmidt Plate theory for stressed heterogeneous multilayers of finite bending energy, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 88 (2007) no. 1, pp. 107-122 | DOI:10.1016/j.matpur.2007.04.011 | Zbl:1116.74034
  • C.-M. Lee; V.N. Goverdovskiy; A.I. Temnikov Design of springs with “negative” stiffness to improve vehicle driver vibration isolation, Journal of Sound and Vibration, Volume 302 (2007) no. 4-5, p. 865 | DOI:10.1016/j.jsv.2006.12.024
  • Gero Friesecke; Richard D. James; Stefan Müller A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Archive for Rational Mechanics and Analysis, Volume 180 (2006) no. 2, pp. 183-236 | DOI:10.1007/s00205-005-0400-7 | Zbl:1100.74039
  • Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare A nonlinear Korn inequality on a surface, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 85 (2006) no. 1, pp. 2-16 | DOI:10.1016/j.matpur.2005.10.010 | Zbl:1094.53001
  • 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, pp. 245-280 | DOI:10.1007/s10659-006-9066-0 | Zbl:1100.74043
  • Jens Frehse; Moritz Kassmann Nonlinear partial differential equations of fourth order under mixed boundary conditions, Mathematische Zeitschrift, Volume 254 (2006) no. 1, pp. 33-54 | DOI:10.1007/s00209-005-0917-3 | Zbl:1220.35071
  • Bernd Schmidt A Derivation of Continuum Nonlinear Plate Theory from Atomistic Models, Multiscale Modeling Simulation, Volume 5 (2006) no. 2, p. 664 | DOI:10.1137/050646251
  • R. Monneau Some remarks on the asymptotic invertibility of the linearized operator of nonlinear elasticity in the context of the displacement approach, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, Volume 86 (2006) no. 5, pp. 400-409 | DOI:10.1002/zamm.200510249 | Zbl:1097.35011
  • Karim Trabelsi Nonlinearly elastic thin plate models for a class of Ogden materials. II: The bending model, Analysis and Applications (Singapore), Volume 3 (2005) no. 3, pp. 271-283 | DOI:10.1142/s0219530505000571 | Zbl:1236.74022
  • Xiao Zhong; Daniel Faraco Geometric rigidity of conformal matrices, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, Volume 4 (2005) no. 4, pp. 557-585 | Zbl:1170.30308
  • Philippe G. Ciarlet An introduction to differential geometry with applications to elasticity, Journal of Elasticity, Volume 78-79 (2005) no. 1-3, pp. 3-201 | DOI:10.1007/s10659-005-4738-8 | Zbl:1086.74001
  • Pedro Miguel Santos; Elvira Zappale Second-order analysis for thin structures, Nonlinear Analysis: Theory, Methods Applications, Volume 56 (2004) no. 5, p. 679 | DOI:10.1016/j.na.2003.10.007
  • Philippe G. Ciarlet The continuity of a surface as a function of its two fundamental forms., Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 82 (2003) no. 3, pp. 253-274 | DOI:10.1016/s0021-7824(03)00017-5 | Zbl:1042.53003
  • Gero Friesecke; Richard D. James; Stefan Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Communications on Pure and Applied Mathematics, Volume 55 (2002) no. 11, pp. 1461-1506 | DOI:10.1002/cpa.10048 | Zbl:1021.74024

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