no. 2
Rigorous derivation of nonlinear plate theory and geometric rigidity
[Dérivation rigoureuse de la théorie non linéaire des plaques et rigidité géométrique]
Friesecke, Gero1 ; Müller, Stefan2 ; James, Richard D.3
1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
2 Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
3 Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA
Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 173-178.

Nous montrons que la théorie non linéaire des plaques émerge comme Γ-limite de la théorie de l'élasticité tridimensionnelle. La démonstration repose sur un résultat de rigidité pour des fonctions v:(0,1) 3 →ℝ 3 . Nous montrons que la distance L2 de ∇v d'une rotation est bornée par un multiple de la distance L2 à l'ensemble SO(3) des rotations.

We show that nonlinear plate theory arises as a Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a sharp rigidity estimate for maps v:(0,1) 3 →ℝ 3 . We show that 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02133-7
Friesecke, Gero 1 ; Müller, Stefan 2 ; James, Richard D. 3

1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
2 Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
3 Department of Aerospace Engineering and Mechanics, 107 Akerman Hall, University of Minnesota, Minneapolis, MN 55455, USA 
@article{CRMATH_2002__334_2_173_0,
     author = {Friesecke, Gero and M\"uller, Stefan and James, Richard~D.},
     title = {Rigorous derivation of nonlinear plate theory and geometric rigidity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {173--178},
     publisher = {Elsevier},
     volume = {334},
     number = {2},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02133-7},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02133-7/}
}
TY  - JOUR
AU  - Friesecke, Gero
AU  - Müller, Stefan
AU  - James, Richard D.
TI  - Rigorous derivation of nonlinear plate theory and geometric rigidity
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 173
EP  - 178
VL  - 334
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(02)02133-7/
DO  - 10.1016/S1631-073X(02)02133-7
LA  - en
ID  - CRMATH_2002__334_2_173_0
ER  - 
%0 Journal Article
%A Friesecke, Gero
%A Müller, Stefan
%A James, Richard D.
%T Rigorous derivation of nonlinear plate theory and geometric rigidity
%J Comptes Rendus. Mathématique
%D 2002
%P 173-178
%V 334
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(02)02133-7/
%R 10.1016/S1631-073X(02)02133-7
%G en
%F CRMATH_2002__334_2_173_0
Friesecke, Gero; Müller, Stefan; James, Richard D. Rigorous derivation of nonlinear plate theory and geometric rigidity. Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 173-178. doi : 10.1016/S1631-073X(02)02133-7. http://www.numdam.org/articles/10.1016/S1631-073X(02)02133-7/

[1] Ben Belgacem, H.; Conti, S.; DeSimone, A.; Müller, S. Rigorous bounds for the Föppl–von Kármán theory of isotropically compressed plates, J. Nonlinear Sci., Volume 10 (2000), pp. 661-685

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

[3] DalMaso, G. An Introduction to Γ-Convergence, Birkhäuser, 1993

[4] DeGiorgi, E.; Franzoni, T. Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., Volume 58 (1975), pp. 842-850

[5] Evans, L.C.; Gariepy, R.F. Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992

[6] Jin, W.; Sternberg, P. Energy estimates for the von Kármán model in thin-film blistering, J. Math. Phys., Volume 42 (2001), pp. 192-199

[7] John, F. Rotation and strain, Commun. Pure Appl. Math., Volume 14 (1961), pp. 391-413

[8] John, F. Bounds for deformations in terms of average strains (Shisha, O., ed.), Inequalities III, 1972, pp. 129-144

[9] Kirchhoff, G. Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. Reine Angew. Math., Volume 40 (1850), pp. 51-88

[10] Kohn, R.V. New integral estimates for deformations in terms of their nonlinear strains, Arch. Rat. Mech. Anal., Volume 78 (1982), pp. 131-172

[11] Kufner, A. Weighted Sobolev Spaces, Wiley, New York, 1985

[12] LeDret, H.; Raoult, A. Le modèle 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

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

[14] LeDret, H.; Raoult, A. The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci., Volume 6 (1996), pp. 59-84

[15] Liu, F.-C. A Lusin property of Sobolev functions, Indiana Univ. Math. J., Volume 26 (1977), pp. 645-651

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

[17] Nečas, J. Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa, Volume 16 (1962), pp. 305-326

[18] Pantz, O. 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

[19] Reshetnyak, Y.G. Liousville's theory on conformal mappings under minimal regularity assumptions, Sibirskii Math., Volume 8 (1967), pp. 69-85

[20] Willmore, T.J. Total Curvature in Riemannian Geometry, John Wiley & Sons, New York, 1982

[21] Ziemer, W. Weakly Differentiable Functions, Springer-Verlag, 1989

Cité par Sources :