# Junction of elastic plates and beams

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

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 3, page 419-457
- ISSN: 1292-8119

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topGaudiello, Antonio, et al. "Junction of elastic plates and beams." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 419-457. <http://eudml.org/doc/249992>.

@article{Gaudiello2007,

abstract = {
We consider the linearized elasticity system in a multidomain of $\{\bf R\}^3$. 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 $r^\{\varepsilon\}$. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and $r^\{\varepsilon\}$ tend to zero simultaneously, with $r^\{\varepsilon\}\gg \varepsilon^2$, we identify the limit problem. This limit problem involves six junction conditions.
},

author = {Gaudiello, Antonio, Monneau, Régis, Mossino, Jacqueline, Murat, François, Sili, Ali},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Junctions; thin structures; plates; beams; linear elasticity; asymptotic analysis; multidomain},

language = {eng},

month = {7},

number = {3},

pages = {419-457},

publisher = {EDP Sciences},

title = {Junction of elastic plates and beams},

url = {http://eudml.org/doc/249992},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Gaudiello, Antonio

AU - Monneau, Régis

AU - Mossino, Jacqueline

AU - Murat, François

AU - Sili, Ali

TI - Junction of elastic plates and beams

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/7//

PB - EDP Sciences

VL - 13

IS - 3

SP - 419

EP - 457

AB -
We consider the linearized elasticity system in a multidomain of ${\bf R}^3$. 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 $r^{\varepsilon}$. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and $r^{\varepsilon}$ tend to zero simultaneously, with $r^{\varepsilon}\gg \varepsilon^2$, we identify the limit problem. This limit problem involves six junction conditions.

LA - eng

KW - Junctions; thin structures; plates; beams; linear elasticity; asymptotic analysis; multidomain

UR - http://eudml.org/doc/249992

ER -

## References

top- E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string. J. Elasticity25 (1991) 137–148.
- D.R. Adams and L.I. Hedberg, Fonctions Spaces and Potential Theory. Springer Verlag, Berlin (1996).
- 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.
- D. Caillerie, Thin elastic and periodic plates. Math. Methods Appl. Sci.6 (1984) 159–191.
- P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis. Masson, Paris (1990).
- P.G. Ciarlet, Mathematical Elasticity, Volume II: Theory of Plates. North-Holland, Amsterdam (1997).
- P.G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model. J. Mécanique18 (1979) 315–344.
- 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. Elasticity19 (1988) 111–161.
- D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999).
- 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.
- 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.
- 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.
- 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.
- 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.
- 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. I335 (2002) 717–722.
- 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.
- I. Gruais, Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée. RAIRO: Modél. Math. Anal. Numér.27 (1993) 77–105.
- I. Gruais, Modeling of the junction between a plate and a rod in nonlinear elasticity. Asymptotic Anal.7 (1993) 179–194.
- 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.
- H. Le Dret, Problèmes Variationnels dans les Multi-domaines: Modélisation des Jonctions et Applications. Masson, Paris (1991).
- 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.
- 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.
- H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci.6 (1996) 59–84.
- R. Monneau, F. Murat and A. Sili, Error estimate for the transition 3d-1d in anisotropic heterogeneous linearized elasticity. To appear.
- 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.
- 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 Linéaire21 (2004) 271–293.
- 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. I328 (1999) 179–184.
- F. Murat and A. Sili, Anisotropic, heterogeneous, linearized elasticity problems in thin cylinders. To appear.
- O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992).
- D. Percivale, Thin elastic beams: the variational approach to St. Venant's problem. Asymptot. Anal.20 (1999) 39–60.
- L. Trabucho and J.M. Viano, Mathematical Modelling of Rods, Handbook of Numerical Analysis4. North-Holland, Amsterdam (1996).

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