Asymptotic behavior of a structure made by a plate and a straight rod

Abstract

This paper is devoted to describing the asymptotic behavior of a structure made by a thin plate and a thin perpendicular rod in the framework of nonlinear elasticity. The authors scale the applied forces in such a way that the level of the total elastic energy leads to the Von-Kármán’s equations (or the linear model for smaller forces) in the plate and to a one-dimensional rod-model at the limit. The junction conditions include in particular the continuity of the bending in the plate and the stretching in the rod at the junction.

Keywords

Nonlinear elasticity / Junction / Straight rod / Plate

References

[1] Antman, S. S., The theory of rods, in Handbuch der Physik, Band VIa/2, S. Flügge & C. Truesdell eds., Springer-Verlag, Berlin, 1972, 641–703.

[2] Antman S S. Nonlinear Problems of Elasticity, 1995, New York: Springer-Verlag

[3] Aufranc M. Junctions between three-dimensional and two-dimensional non-linearly elastic structures. Asymptot. Anal., 1991, 4(4): 319-338

[4] Blanchard D, Gaudiello A, Griso G. Junction of a periodic family of elastic rods with a 3d plate. II. J. Math. Pures Appl. (9), 2007, 88(1): 149-190

[5] Blanchard D, Gaudiello A, Griso G. Junction of a periodic family of elastic rods with a thin plate. I. J. Math. Pures Appl. (9), 2007, 88(2): 1-33

[6] Blanchard D, Griso G. Microscopic effects in the homogenization of the junction of rods and a thin plate. Asymptot. Anal., 2008, 56(1): 1-36

[7] Blanchard D, Griso G. Decomposition of deformations of thin rods. Application to nonlinear elasticity. Anal. Appl., 2009, 7(1): 21-71

[8] Blanchard D, Griso G. Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant’s strain tensor. J. Elasticity, 2010, 101(2): 179-205

[9] Blanchard D, Griso G. Modeling of rod-structures in nonlinear elasticity. C. R. Math. Acad. Sci. Paris, 2010, 348(19–20): 1137-1141

[10] Blanchard D, Griso G. A simplified model for elastic thin shells. Asymptot. Analysis, 2012, 76(1): 1-33

[11] Blanchard D, Griso G. Asymptotic behavior of structures made of straight rods. J. Elasticity, 2012, 108(1): 85-118

[12] Blanchard, D. and Griso, G., Junction between a plate and a rod of comparable thickness in nonlinear elasticity, J. Elasticity, DOI: 10.1007/s10659-012-9401-6.

[13] Ciarlet P G. Mathematical Elasticity, Vol. I, 1988, Amsterdam: North-Holland

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

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

[16] Ciarlet P G, Le Dret H, Nzengwa R. Junctions between three dimensional and two dimensional linearly elastic structures. J. Math. Pures Appl., 1989, 68: 261-295

[17] Ciarlet P G, Rabier P. Les équations de Von Kármán, 1980, Berlin: Springer-Verlag

[18] Friesecke G, James R D, Müler S. A hierarchy of plates models derived from nonlinear elasticity by Γ-convergence. Arch. Rat. Mech. Anal., 2006, 180: 183-236

[19] Gaudiello A, Monneau R, Mossino J Junction of elastic plates and rods. ESAIM Control Optim. Calc. Var., 2007, 13(3): 419-457

[20] Griso G. Asymptotic behavior of rods by the unfolding method. Math. Meth. Appl. Sci., 2004, 27: 2081-2110

[21] Griso G. Asymptotic behavior of structures made of plates. Anal. Appl., 2005, 3(4): 325-356

[22] Griso G. Asymptotic behavior of structures made of curved rods. Anal. Appl., 2008, 6(1): 11-22

[23] Griso G. Decomposition of displacements of thin structures. J. Math. Pures Appl. (9), 89, 2008 199-233

[24] Gruais I. Modeling of the junction between a plate and a rod in nonlinear elasticity. Asymptotic Anal., 1993, 7: 179-194

[25] Gruais I. Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée. RAIRO: Modè. Math. Anal. Numér., 1993, 27: 77-105

[26] Le Dret H. Modeling of the junction between two rods. J. Math. Pures Appl., 1989, 68: 365-397

[27] Le Dret H. Modeling of a folded plate. Comput. Mech., 1990, 5: 401-416

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

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