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 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.

[1]

S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks and 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 and 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 and Continuous Dynamical Systems, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237

[4]

Samuel Amstutz, Nicolas Van Goethem. Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3769-3805. doi: 10.3934/dcdsb.2020240

[5]

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

[6]

Abdessatar Khelifi, Siwar Saidani. Asymptotic behavior of eigenvalues of the Maxwell system in the presence of small changes in the interface of an inclusion. Communications on Pure and Applied Analysis, 2022, 21 (9) : 2891-2909. doi: 10.3934/cpaa.2022080

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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.

[12]

Cornel Marius Murea, Dan Tiba. Topological optimization and minimal compliance in linear elasticity. Evolution Equations and Control Theory, 2020, 9 (4) : 1115-1131. doi: 10.3934/eect.2020043

[13]

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

[14]

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

[15]

Rita Ferreira, Elvira Zappale. Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1747-1793. doi: 10.3934/cpaa.2020072

[16]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete and 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 and Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (1)

[Back to Top]