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October  2014, 34(10): 4039-4070. doi: 10.3934/dcds.2014.34.4039

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

Juan Casado-Díaz 1, , Manuel Luna-Laynez 1, and  Francois Murat 2,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Facultad de Matemáticas, c/ Tarfia s/n, 41012 Sevilla, Spain, Spain
2. 

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris VI), boîte courrier 187, 75252 Paris cedex 05, France

Received  March 2013 Revised  October 2013 Published  April 2014

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: linear elasticity, small fixing areas., correctors, Asymptotic behavior, thin beams.
Mathematics Subject Classification: 74K10, 35Q7.
Citation: Juan Casado-Díaz, Manuel Luna-Laynez, Francois Murat. The behavior of a beam fixed on small sets of one of its extremities. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4039-4070. doi: 10.3934/dcds.2014.34.4039
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.

show all references

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.

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