zbMATH — the first resource for mathematics

The Föppl-von Kármán plate theory as a low energy \(\Gamma\)-limit of nonlinear elasticity. (English. Abridged French version) Zbl 1041.74043
Summary: We show that the Föppl-von Kármán theory arises as a low energy \(\Gamma\)-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [the authors, ibid. 334, No. 2, 173–178 (2002; Zbl 1012.74043)] that for maps \(v:(0, 1)^3\to\mathbb{R}^3\), the \(L^2\) distance of \(\nabla v\) from a single rotation is bounded by a multiple of the \(L^2\) distance from the set SO(3) of all rotations.

74K20 Plates
74B20 Nonlinear elasticity
Full Text: DOI
[1] Antman, S.S., Nonlinear problems of elasticity, (1995), Springer New York · Zbl 0820.73002
[2] Anzelotti, G.; Baldo, S.; Percivale, D., Dimension reduction in variational problems, asymptotic development in γ-convergence and thin structures in elasticity, Asymptotic anal., 9, 61-100, (1994) · Zbl 0811.49020
[3] Ciarlet, P.G., Mathematical elasticity II - theory of plates, (1997), Elsevier Amsterdam · Zbl 0888.73001
[4] Fox, D.D.; Raoult, A.; Simo, J.C., A justification of nonlinear properly invariant plate theories, Arch. rational mech. anal., 124, 157-199, (1993) · Zbl 0789.73039
[5] Friesecke, G.; James, R.D.; Müller, S., Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. acad. sci. Paris, Série I, 334, 173-178, (2002) · Zbl 1012.74043
[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 · Zbl 1021.74024
[7] LeDret, H.; Raoult, A., Le modéle de membrane non linéaire comme limite variationelle de l’élasticité non linéaire tridimensionelle, C. R. acad. sci. Paris, Série I, 317, 221-226, (1993) · Zbl 0781.73037
[8] LeDret, H.; Raoult, A., The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. math. pures appl., 73, 549-578, (1995) · Zbl 0847.73025
[9] Love, A.E.H., A treatise on the mathematical theory of elasticity, (1927), Cambridge University Press Cambridge
[10] Marigo, J.J.; Ghidouche, H.; Sedkaoui, Z., Des poutres flexibles aux fils extensibles : une hiérachie de modèles asymptotiques, C. R. acad. sci. Paris, Série iib, 326, 79-84, (1998) · Zbl 0924.73109
[11] R. Monneau, Justification of nonlinear Kirchhoff-Love theory of plates as the application of a new singular inverse method, Preprint, 2001 · Zbl 1030.74030
[12] Pantz, O., Une justification partielle du modèle de plaque en flexion par γ-convergence, C. R. acad. sci. Paris, Série I, 332, 587-592, (2001) · Zbl 1033.74028
[13] A. Raoult, Personal communication
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.