@article{763415f6f90c43d9bb6650af3a5da3e8,
title = "La th{\'e}orie F{\"o}ppl-von K{\'a}rm{\'a}n des plaques comme Γ-limite de l'{\'e}lasticit{\'e} non lin{\'e}aire",
abstract = "We show that the F{\"o}ppl-von K{\'a}rm{\'a}n 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 → ℝ3, 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.",
author = "Gero Friesecke and James, \{Richard D.\} and Stefan M{\"u}ller",
note = "Funding Information: Acknowledgement. RDJ thanks AFOSR/MURI (F49620-98-1-0433) and NSF (DMS-0074043) for supporting his work. GF and SM were partially supported by the TMR network FMRX-CT98-0229. References [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. 9 (1994) 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. 124 (1993) 157–199. [5] G. Friesecke, R.D. James, S. M{\"u}ller, Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, S{\'e}rie I 334 (2002) 173–178. [6] G. Friesecke, R.D. James, S. M{\"u}ller, 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 mod{\'e}le de membrane non lin{\'e}aire comme limite variationelle de l{\textquoteright}{\'e}lasticit{\'e} non lin{\'e}aire tridimensionelle, C. R. Acad. Sci. Paris, S{\'e}rie I 317 (1993) 221–226. [8] H. LeDret, A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 73 (1995) 549–578. [9] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn., Cambridge University Press, Cambridge, 1927. [10] J.J. Marigo, H. Ghidouche, Z. Sedkaoui, Des poutres flexibles aux fils extensibles : une hi{\'e}rachie de mod{\`e}les asymptotiques, C. R. Acad. Sci. Paris, S{\'e}rie IIb 326 (1998) 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 mod{\`e}le de plaque en flexion par Γ-convergence, C. R. Acad. Sci. Paris, S{\'e}rie I 332 (2001) 587–592. [13] A. Raoult, Personal communication.",
year = "2002",
month = jul,
day = "15",
doi = "10.1016/S1631-073X(02)02388-9",
language = "French",
volume = "335",
pages = "201--206",
journal = "Comptes Rendus Mathematique",
issn = "1631-073X",
publisher = "Academie des sciences",
number = "2",
}