Junction of elastic plates and beams

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

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 419-457.

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 $\epsilon$, and of a vertical beam with fixed height and small cross section of radius $r^{\epsilon }$. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When $ϵ$ and $r^{\epsilon }$ tend to zero simultaneously, with $r^{\epsilon }\gg {ϵ}^2$, we identify the limit problem. This limit problem involves six junction conditions.

Classification : 35B40, 74B05, 74K30
Mots clés : junctions, thin structures, plates, beams, linear elasticity, asymptotic analysis

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

[2] D.R. Adams and L.I. Hedberg, Fonctions Spaces and Potential Theory. Springer Verlag, Berlin (1996). | MR | Zbl

[3] G. Anzellotti, S. Baldo and D. Percivale, Dimension reduction in variational problems, asymptotic development in $\Gamma$-convergence and thin structures in elasticity. Asymptot. Anal. 9 (1994) 61-100. | Zbl

[4] D. Caillerie, Thin elastic and periodic plates. Math. Methods Appl. Sci. 6 (1984) 159-191. | Zbl

[5] P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis. Masson, Paris (1990). | MR | Zbl

[6] P.G. Ciarlet, Mathematical Elasticity, Volume II: Theory of Plates. North-Holland, Amsterdam (1997). | MR | Zbl

[7] P.G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model. J. Mécanique 18 (1979) 315-344. | Zbl

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

[9] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999). | MR | Zbl

[10] M. Dauge and I. Gruais, Asymptotics of arbitrary order for a thin elastic clamped plate, I: Optimal error estimates. Asymptot. Anal. 13 (1996) 167-197. | Zbl

[11] G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461-1506. | Zbl

[12] G. Friesecke, R.D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Rat. Mech. Anal. 180 (2006) 183-236. | Zbl

[13] A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino, Asymptotic analysis of a class of minimization problems in a thin multidomain. Calc. Var. Part. Diff. Eq. 15 (2002) 181-201. | Zbl

[14] A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino, Asymptotic analysis for monotone quasilinear problems in thin multidomains. Diff. Int. Eq. 15 (2002) 623-640. | Zbl

[15] A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili, On the junction of elastic plates and beams. C.R. Acad. Sci. Paris Sér. I 335 (2002) 717-722. | Zbl

[16] A. Gaudiello and E. Zappale, Junction in a thin multidomain for a fourth order problem. M3AS: Math. Models Methods Appl. Sci. 16 (2006) 1887-1918. | Zbl

[17] I. Gruais, Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée. RAIRO: Modél. Math. Anal. Numér. 27 (1993) 77-105. | Numdam | Zbl

[18] I. Gruais, Modeling of the junction between a plate and a rod in nonlinear elasticity. Asymptotic Anal. 7 (1993) 179-194. | Zbl

[19] V.A. Kozlov, V.G. Ma'Zya and A.B. Movchan, Asymptotic representation of elastic fields in a multi-structure. Asymptot. Anal. 11 (1995) 343-415. | Zbl

[20] H. Le Dret, Problèmes Variationnels dans les Multi-domaines: Modélisation des Jonctions et Applications. Masson, Paris (1991). | MR | Zbl

[21] H. Le Dret, Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero. Asymptot. Anal. 10 (1995) 367-402. | Zbl

[22] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. | Zbl

[23] H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6 (1996) 59-84. | Zbl

[24] R. Monneau, F. Murat and A. Sili, Error estimate for the transition 3d-1d in anisotropic heterogeneous linearized elasticity. To appear.

[25] M.G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by $\Gamma$-convergence. Calc. Var. Part. Diff. Eq. 18 (2003) 287-305. | Zbl

[26] M.G. Mora and S. Müller, A nonlinear model for inextensible rods as a low energy $\Gamma$-limit of three-dimensional nonlinear elasticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 271-293. | Numdam | Zbl

[27] F. Murat and A. Sili, Comportement asymptotique des solutions du sytème de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces. C.R. Acad. Sci. Paris Sér. I 328 (1999) 179-184. | Zbl

[28] F. Murat and A. Sili, Anisotropic, heterogeneous, linearized elasticity problems in thin cylinders. To appear.

[29] O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992). | MR | Zbl

[30] D. Percivale, Thin elastic beams: the variational approach to St. Venant's problem. Asymptot. Anal. 20 (1999) 39-60. | Zbl

[31] L. Trabucho and J.M. Viano, Mathematical Modelling of Rods, Handbook of Numerical Analysis 4. North-Holland, Amsterdam (1996). | MR | Zbl