Comptes Rendus
no. 12
Mathematical Problems in Mechanics
Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities
[Comportement asymptotique d'une poutre élastique fixée sur une petite partie de l'une de ses extrémités]

Présenté par : Philippe G. Ciarlet

Juan Casado-Díaz1 ; Manuel Luna-Laynez1 ; François Murat2
1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 338 (2004) no. 12, pp. 975-980.

Nous étudions le comportement asymptotique de la solution d'un problème d'élasticité linéaire anisotrope et hétérogène dans un cylindre dont le diamètre ε tend vers zéro. Le cylindre est fixé (condition de Dirichlet homogène) sur la totalité de l'une de ses extrémités, mais seulement sur une petite partie (de taille εrε) de l'autre base ; sur le reste de la frontière on a la condition de Neumann. Nous montrons que le résultat depend de rε, et qu'il existe 3 tailles critiques, à savoir r ϵ =ϵ 3 ,r ϵ =ϵ et rε=ε1/3, et au total 7 comportements différents. Nous donnons un résultat de correcteur pour tous les comportements de rε.

We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter ε tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities, but only on a small part (of size εrε) of the second one; the Neumann boundary condition is imposed on the remainder of the boundary. We show that the result depends on rε, and that there are 3 critical sizes, namely r ϵ =ϵ 3 ,r ϵ =ϵ, and rε=ε1/3, and in total 7 different regimes. We also prove a corrector result for each behavior of rε.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.02.020
Juan Casado-Díaz 1 ; Manuel Luna-Laynez 1 ; François Murat 2

1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris cedex 05, France 
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     pages = {975--980},
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Juan Casado-Díaz; Manuel Luna-Laynez; François Murat. Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities. Comptes Rendus. Mathématique, Volume 338 (2004) no. 12, pp. 975-980. doi : 10.1016/j.crma.2004.02.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.02.020/

[1] J. Casado-Dı́az, F. Murat, The diffusion equation in a notched beam, in preparation

[2] J. Casado-Dı́az; M. Luna-Laynez; F. Murat Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 133-138

[3] J. Casado-Dı́az, M. Luna-Laynez, F. Murat, Diffusion problems in a domain made of two thin cylinders of different diameters and lengths, in preparation

[4] J. Casado-Dı́az, M. Luna-Laynez, F. Murat, Elasticity problems in a beam fixed on only a small part of one of its extremities, in preparation

[5] P.G. Ciarlet Mathematical Elasticity, vol. I: Three-Dimensional Elasticity, North-Holland, 1988

[6] A. Cimetière; G. Geymonat; H. Le Dret; A. Raoult; Z. Tutek Asymptotic theory and analysis for displacements and stress distribution in nonlinear straight slender rods, J. Elasticity, Volume 19 (1988), pp. 111-161

[7] A. Gaudiello; R. Monneau; J. Mossino; F. Murat; A. Sili On the junction of elastic plates and beams, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 717-722

[8] A. Gaudiello, R. Monneau, J. Mossino, F. Murat, A. Sili, Junction of elastic plates and beams, in press

[9] H. Le Dret Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asymptotic Anal., Volume 10 (1995), pp. 367-402

[10] F. Murat; A. Sili Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces, C. R. Acad. Sci. Paris, Ser. I, Volume 328 (1999), pp. 179-184

[11] F. Murat, A. Sili, Anisotropic, heterogeneous, linearized elasticity in thin cylinders, in preparation

[12] L. Trabucho; J.M. Viaño Mathematical Modelling of Rods, Handbook of Numerical Analysis, vol. IV, North-Holland, 1996

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