The behavior of a  beam fixed on small sets of one of its extremities

Juan Casado-Díaz; Manuel Luna-Laynez; Francois Murat

doi:10.3934/dcds.2014.34.4039

Article Contents

2014, Volume 34, Issue 10: 4039-4070. Doi: 10.3934/dcds.2014.34.4039

This issue Previous Article Asymptotic behaviour of a logistic lattice system Next Article The transition point in the zero noise limit for a 1D Peano example

The behavior of a beam fixed on small sets of one of its extremities

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Facultad de Matemáticas, c/ Tarfia s/n, 41012 Sevilla
2.
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris VI), boîte courrier 187, 75252 Paris cedex 05

Received: March 2013

Revised: October 2013

Published: October 2014

Abstract / Introduction Related Papers Cited by

Abstract

In this paper we study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity system in a thin cylinder (a beam). The beam is fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities but only on several small fixing sets on the other extremity; on the remainder of the boundary the Neumann boundary condition holds. As far as the boundary conditions are concerned, the result depends on the size and on the arrangement of the small fixing sets. In particular, we show that it is equivalent to fix the beam at one of its extremities on 3 unaligned small fixing sets or on 1 or 2 fixing set(s) of bigger size.

Keywords:

Mathematics Subject Classification: 74K10, 35Q74.

Citation:

Related Papers

Cited by

References

[1]	J. Casado-Díaz and M. Luna-Laynez, Homogenization of the anisotropic heterogeneous linearized elasticity system in thin reticulated structures, Proc. Roy. Soc. Edinburgh A, 134 (2004), 1041-1083.doi: 10.1017/S0308210500003620.
[2]	J. Casado-Díaz, M. Luna-Laynez and F. Murat, Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths, C.R. Acad. Sci. Paris Ser. I, 338 (2004), 133-138.doi: 10.1016/j.crma.2003.10.033.
[3]	J. Casado-Díaz, M. Luna-Laynez and F. Murat, Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities, C. R. Acad. Sci. Paris, C. R. Acad. Sci. Paris Ser. I, 338 (2004), 975-980.doi: 10.1016/j.crma.2004.02.020.
[4]	J. Casado-Díaz, M. Luna-Laynez and F. Murat, The diffusion equation in a notched beam, Calc. Var., 31 (2008), 297-323.doi: 10.1007/s00526-006-0073-6.
[5]	P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, Studies in Math. and its Appl., 20, North-Holland, Amsterdam 1988.
[6]	P. G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates, Studies in Math. and its Appl., 27, North-Holland, Amsterdam, 1988.
[7]	D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Sciences Series, 136, Springer-Verlag, Berlin 1999.doi: 10.1007/978-1-4612-2158-6.
[8]	A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili, On the junction of elastic plates and beams, C. R. Acad. Sci. Paris Ser. I, 335 (2002), 717-722.doi: 10.1016/S1631-073X(02)02543-8.
[9]	A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili, Junction of elastic plates and beams, ESAIM Control Optim. Calc. Var., 13 (2007), 419-457.doi: 10.1051/cocv:2007036.
[10]	G. Geymonat, F. Krasucki and J. J. Marigo, Stress distribution in anisotropic elastic composite beams, in Applications of Multiple Scalings in Mechanics (eds. P. G. Ciarlet and E. Sanchez Palencia), Masson, Paris, 1987, 118-133.
[11]	H. Le Dret, Problèmes Variationnels dans les Multi-Domaines: Modélisation des Jonctions et Applications, Masson, Paris, 1991.
[12]	H. Le Dret, Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asympt. Anal., 10 (1995), 367-402.
[13]	F. Murat and A. Sili, Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces, C. R. Acad. Sci. Paris Ser. I, 328 (1999), 179-184.doi: 10.1016/S0764-4442(99)80159-1.
[14]	F. Murat and A. Sili, Asymptotic behavior of solutions of linearized anisotropic heterogeneous elasticity system in thin cylinders, to appear.
[15]	O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.
[16]	L. Trabucho and J. M. Viaño, Mathematical Modelling of Rods, Handbook of Numerical Analysis, IV, North-Holland, Amsterdam, 1996.