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

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.
Citation: Juan Casado-Díaz, Manuel Luna-Laynez, Francois Murat. The behavior of a beam fixed on small sets of one of its extremities. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1017/S0308210500003620. Google Scholar

[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. doi: 10.1016/j.crma.2003.10.033. Google Scholar

[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, 338 (2004), 975. doi: 10.1016/j.crma.2004.02.020. Google Scholar

[4]

J. Casado-Díaz, M. Luna-Laynez and F. Murat, The diffusion equation in a notched beam,, Calc. Var., 31 (2008), 297. doi: 10.1007/s00526-006-0073-6. Google Scholar

[5]

P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity,, Studies in Math. and its Appl., (1988). Google Scholar

[6]

P. G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates,, Studies in Math. and its Appl., (1988). Google Scholar

[7]

D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures,, Applied Mathematical Sciences Series, (1999). doi: 10.1007/978-1-4612-2158-6. Google Scholar

[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. doi: 10.1016/S1631-073X(02)02543-8. Google Scholar

[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. doi: 10.1051/cocv:2007036. Google Scholar

[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), (1987), 118. Google Scholar

[11]

H. Le Dret, Problèmes Variationnels dans les Multi-Domaines: Modélisation des Jonctions et Applications,, Masson, (1991). Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0764-4442(99)80159-1. Google Scholar

[14]

F. Murat and A. Sili, Asymptotic behavior of solutions of linearized anisotropic heterogeneous elasticity system in thin cylinders,, to appear., (). Google Scholar

[15]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization,, North-Holland, (1992). Google Scholar

[16]

L. Trabucho and J. M. Viaño, Mathematical Modelling of Rods,, Handbook of Numerical Analysis, (1996). Google Scholar

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. doi: 10.1017/S0308210500003620. Google Scholar

[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. doi: 10.1016/j.crma.2003.10.033. Google Scholar

[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, 338 (2004), 975. doi: 10.1016/j.crma.2004.02.020. Google Scholar

[4]

J. Casado-Díaz, M. Luna-Laynez and F. Murat, The diffusion equation in a notched beam,, Calc. Var., 31 (2008), 297. doi: 10.1007/s00526-006-0073-6. Google Scholar

[5]

P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity,, Studies in Math. and its Appl., (1988). Google Scholar

[6]

P. G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates,, Studies in Math. and its Appl., (1988). Google Scholar

[7]

D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures,, Applied Mathematical Sciences Series, (1999). doi: 10.1007/978-1-4612-2158-6. Google Scholar

[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. doi: 10.1016/S1631-073X(02)02543-8. Google Scholar

[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. doi: 10.1051/cocv:2007036. Google Scholar

[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), (1987), 118. Google Scholar

[11]

H. Le Dret, Problèmes Variationnels dans les Multi-Domaines: Modélisation des Jonctions et Applications,, Masson, (1991). Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0764-4442(99)80159-1. Google Scholar

[14]

F. Murat and A. Sili, Asymptotic behavior of solutions of linearized anisotropic heterogeneous elasticity system in thin cylinders,, to appear., (). Google Scholar

[15]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization,, North-Holland, (1992). Google Scholar

[16]

L. Trabucho and J. M. Viaño, Mathematical Modelling of Rods,, Handbook of Numerical Analysis, (1996). Google Scholar

[1]

S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577

[2]

Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107

[3]

Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237

[4]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

[5]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181

[6]

Daniele Boffi, Franco Brezzi, Michel Fortin. Reduced symmetry elements in linear elasticity. Communications on Pure & Applied Analysis, 2009, 8 (1) : 95-121. doi: 10.3934/cpaa.2009.8.95

[7]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178

[8]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114

[9]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[10]

Bernd Schmidt. On the derivation of linear elasticity from atomistic models. Networks & Heterogeneous Media, 2009, 4 (4) : 789-812. doi: 10.3934/nhm.2009.4.789

[11]

Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619

[12]

Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573

[13]

Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040

[14]

Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113

[15]

Bogdan Sasu, A. L. Sasu. Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. Communications on Pure & Applied Analysis, 2006, 5 (3) : 551-569. doi: 10.3934/cpaa.2006.5.551

[16]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[17]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[18]

Ahmed El Kaimbillah, Oussama Bourihane, Bouazza Braikat, Mohammad Jamal, Foudil Mohri, Noureddine Damil. Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1685-1708. doi: 10.3934/dcdss.2019113

[19]

Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721

[20]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

[Back to Top]