On the junction of elastic plates and beams

Gaudiello, Antonio; Monneau, Régis; Mossino, Jacqueline; Murat, François; Sili, Ali

Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 717-722.

On considère le système linéarisé de l'élasticité, dans un multidomaine de ${\mathrm{R}}^3$ 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 r^ε. La frontière latérale de la plaque et le haut de la poutre sont supposés encastrés. Nous identifions le problème limite quand ε et r^ε tendent simultanément vers zéro, avec r^ε⪢ε². Ce problème limite fait intervenir six conditions de jonction.

We consider the linearized elasticity system in a multidomain of ${\mathrm{R}}^3$. 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 r^ε. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and r^ε tend to zero simultaneously, with r^ε⪢ε², we identify the limit problem. This limit problem involves six junction conditions.

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

[2] Anzellotti, G.; Baldo, S.; Percivale, D. 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] Caillerie, D. Thin elastic and periodic plates, Math. Methods Appl. Sci., Volume 6 (1984), pp. 159-191

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

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

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

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

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

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

[10] Gaudiello, A.; Gustafsson, B.; Lefter, C.; Mossino, J. 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] Gaudiello, A.; Gustafsson, B.; Lefter, C.; Mossino, J. 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] Gruais, I. 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] Kozlov, V.A.; Ma'zya, V.G.; Movchan, A.B. Asymptotic representation of elastic fields in a multi-structure, Asymptotic Anal., Volume 11 (1995), pp. 343-415

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

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

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

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

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

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