Comptes Rendus
On the junction of elastic plates and beams
Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 717-722.

We consider the linearized elasticity system in a multidomain of R 3 . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius rε. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and rε tend to zero simultaneously, with rεε2, we identify the limit problem. This limit problem involves six junction conditions.

On considère le système linéarisé de l'élasticité, dans un multidomaine de R 3 constitué d'une plaque horizontale de section fixée et de faible épaisseur ε et d'une poutre verticale de hauteur fixée et de petite section dont le rayon est rε. La frontière latérale de la plaque et le haut de la poutre sont supposés encastrés. Nous identifions le problème limite quand ε et rε tendent simultanément vers zéro, avec rεε2. Ce problème limite fait intervenir six conditions de jonction.

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Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)02543-8
Antonio Gaudiello 1; Régis Monneau 2; Jacqueline Mossino 3; François Murat 4; Ali Sili 5

1 Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale, Università di Cassino, Via G. Di Biasio 43, 03043 Cassino (FR), Italy
2 CERMICS, École nationale des ponts et chaussées, 6 et 8, avenue Blaise Pascal, cité Descartes, 77455 Champs-sur-Marne cedex 2, France
3 CMLA, École normale supérieure de Cachan, 61, avenue du Président Wilson, 94235 Cachan cedex, France
4 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boı̂te courrier 187, 75252 Paris cedex 05, France
5 Département de mathématiques, Université de Toulon et du Var, BP 132, 83957 La Garde cedex, France
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Antonio Gaudiello; Régis Monneau; Jacqueline Mossino; François Murat; Ali Sili. On the junction of elastic plates and beams. Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 717-722. doi : 10.1016/S1631-073X(02)02543-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02543-8/

[1] E. Acerbi; G. Buttazzo; D. Percivale A variational definition of the strain energy for an elastic string, J. Elasticity, Volume 25 (1991), pp. 137-148

[2] G. Anzellotti; S. Baldo; D. Percivale Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity, Asymptotic Anal., Volume 9 (1994) no. 1, pp. 61-100

[3] D. Caillerie Thin elastic and periodic plates, Math. Methods Appl. Sci., Volume 6 (1984), pp. 159-191

[4] P.G. Ciarlet Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson, Paris, 1990

[5] P.G. Ciarlet Mathematical Elasticity, Vol. II: Theory of Plates, North-Holland, Amsterdam, 1997

[6] P.G. Ciarlet; P. Destuynder A justification of the two-dimensional linear plate model, J. Mécanique, Volume 18 (1979), pp. 315-344

[7] D. Cioranescu; J. Saint Jean Paulin Homogenization of Reticulated Structures, Appl. Math. Sci., 139, Springer-Verlag, New York, 1999

[8] M. Dauge; I. Gruais Asymptotics of arbitrary order for a thin elastic clamped plate, I: Optimal error estimates, Asymptotic Anal., Volume 13 (1996), pp. 167-197

[9] G. Friesecke; R.D. James; S. Müller Rigourous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, Série I, Volume 334 (2002), pp. 173-178

[10] A. Gaudiello; B. Gustafsson; C. Lefter; J. Mossino Asymptotic analysis of a class of minimization problems in a thin multidomain, Calc. Var. Partial Differential Equations, Volume 15 (2002) no. 2, pp. 181-201

[11] A. Gaudiello; B. Gustafsson; C. Lefter; J. Mossino Asymptotic analysis for monotone quasilinear problems in thin multidomains, Differential Integral Equations, Volume 15 (2002), pp. 623-640

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

[13] I. Gruais Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée, Modélisation Mathématique et Analyse Numérique, Volume 27 (1993), pp. 77-105

[14] V.A. Kozlov; V.G. Ma'zya; A.B. Movchan Asymptotic representation of elastic fields in a multi-structure, Asymptotic Anal., Volume 11 (1995), pp. 343-415

[15] H. Le Dret Problèmes Variationnels dans les Multi-domaines : Modélisation des Jonctions et Applications, Masson, Paris, 1991

[16] 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

[17] H. Le Dret; A. Raoult The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., Volume 74 (1995), pp. 549-578

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

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

[20] F. Murat, A. Sili, Anisotropic, heterogeneous, linearized elasticity problems in thin cylinders, to appear

[21] O.A. Oleinik; A.S. Shamaev; G.A. Yosifian Mathematical Problems in Elasticity and Homogenization, North-Holland, 1992

[22] D. Percivale Thin elastic beams: the variational approach to St. Venant's problem, Asymptotic Anal., Volume 20 (1999), pp. 39-60

[23] L. Trabucho; J.M. Viano Mathematical Modelling of Rods, Handbook of Numerical Analysis, 4, North-Holland, Amsterdam, 1996

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