Advertisement

Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 4, pp 1969–1986 | Cite as

A contact problem for a thermoelastic Timoshenko beam

  • Alessia Berti
  • Jaime E. Muñoz Rivera
  • Maria Grazia NasoEmail author
Article

Abstract

In this paper, a dynamic contact problem between a Timoshenko beam and two rigid obstacles is considered. Thermal effects are also taken into account and the contact is modeled using the classical Signorini condition. The global existence in time of solutions is found by considering related penalized problems, proving some a priori estimates and passing to the limit. An exponential decay property is also showed.

Keywords

Timoshenko thermoelastic beam Signorini condition Contact Asymptotic behavior 

Mathematics Subject Classification

74H40 74M15 35B40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alabau-Boussouira, F.: Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control. NoDEA Nonlinear Differ. Equ. Appl. 14(5–6), 643–669 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, K.T., Shillor, M., Wright, S.: On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle. J. Elast. 42(1), 1–30 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antes, H., Panagiotopoulos, P.D.: The boundary integral approach to static and dynamic contact problems. In: International Series of Numerical Mathematics, Equality and Inequality Methods, vol. 108. Birkhäuser, Basel (1992)Google Scholar
  4. 4.
    Arantes, S., Muñoz Rivera, J.E.: Exponential decay for a thermoelastic beam between two stops. J. Therm. Stress. 31(6), 537–556 (2008)CrossRefGoogle Scholar
  5. 5.
    Araruna, F.D., Feitosa, A.J.R., Oliveira, M.L.: A boundary obstacle problem for the Mindlin–Timoshenko system. Math. Methods Appl. Sci. 32(6), 738–756 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arnold, D.N., Madureira, A.L., Zhang, S.: On the range of applicability of the Reissner–Mindlin and Kirchhoff–Love plate bending models. J. Elast. 67(3), 171–185 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berti, A., Copetti, M.I.M., Fernández, J.R., Naso, M.G.: A dynamic thermoviscoelastic contact problem with the second sound effect. J. Math. Anal. Appl. 421(2), 1163–1195 (2015)Google Scholar
  8. 8.
    Berti, A., Copetti, M.I.M., Fernández, J.R., Naso, M.G.: Analysis of dynamic nonlinear thermoviscoelastic beam problems. Nonlinear Anal. 95, 774–795 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berti, A., Naso, M.G.: Unilateral dynamic contact of two viscoelastic beams. Q. Appl. Math. 69(3), 477–507 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Berti, A., Naso, M.G.: Vibrations of a damped extensible beam between two stops. Evol. Equ. Control Theory (EECT) 2(1), 35–54 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bonfanti, G., Fabrizio, M., Muñoz Rivera, J.E., Naso, M.G.: On the energy decay for a thermoelastic contact problem involving heat transfer. J. Therm. Stres. 33(11), 1049–1065 (2010)CrossRefGoogle Scholar
  12. 12.
    Bonfanti, G., Muñoz Rivera, J.E., Naso, M.G.: Global existence and exponential stability for a contact problem between two thermoelastic beams. J. Math. Anal. Appl. 345(1), 186–202 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bonfanti, G., Naso, M.G.: A dynamic contact problem between two thermoelastic beams. In: Applied and Industrial Mathematics in Italy III, Ser. Adv. Math. Appl. Sci., vol. 82, pp. 123–132. World Sci. Publ., Hackensack, NJ (2009)Google Scholar
  14. 14.
    Copetti, M.I.M., Fernández, J.R.: A dynamic contact problem involving a Timoshenko beam model. Appl. Numer. Math. 63, 117–128 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Copetti, M.I.M.: Numerical approximation of dynamic deformations of a thermoviscoelastic rod against an elastic obstacle. M2AN Math. Model. Numer. Anal. 38(4), 691–706 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    de Pater, A.D., Kalker, J.J. (eds.): The Mechanics of the Contact Between Deformable Bodies. Delft University Press, Delft (1975)Google Scholar
  17. 17.
    Djebabla, A., Tatar, N.-E.: Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping. J. Dyn. Control Syst. 16(2), 189–210 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Djebabla, A., Tatar, N.-E.: Stabilization of the Timoshenko beam by thermal effect. Mediterr. J. Math. 7(3), 373–385 (2010)Google Scholar
  19. 19.
    Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  20. 20.
    Elliott, C.M., Tang, Q.: A dynamic contact problem in thermoelasticity. Nonlinear Anal. 23(7), 883–898 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fernández Sare, H.D., Racke, R.: On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Arch. Ration. Mech. Anal. 194(1), 221–251 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Frémond, M.: Contact with Adhesion, Topics in Nonsmooth Mechanics, pp. 157–185. Birkhäuser, Basel (1988)Google Scholar
  23. 23.
    Fung, Y.C.: Foundations of Solid Mechanics. Prentice-Hall Inc., Englewood Cliffs (1965)Google Scholar
  24. 24.
    Inman, D.J.: Engineering Vibration. Prentice-Hall Inc., Englewood Cliffs (1994)Google Scholar
  25. 25.
    Jarušek, J., Eck, C.: Dynamic contact problem with Coulomb friction for viscoelastic bodies, existence of solutions for a general body. In: Analysis, Numerics and Applications of Differential and Integral Equations (Stuttgart, 1996), Pitman Res. Notes Math. Ser., vol. 379, pp. 111–115. Longman, Harlow (1998)Google Scholar
  26. 26.
    Karasudhi, P.: Foundations of Solid Mechanics, Solid Mechanics and its Applications, vol. 3. Kluwer Academic Publishers Group, Dordrecht (1991)Google Scholar
  27. 27.
    Kikuchi, N., Oden, J.T.: Contact problems in elasticity: a study of variational inequalities and finite element methods. In: SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), vol. 8. Philadelphia, PA (1988)Google Scholar
  28. 28.
    Kim, J.U., Renardy, Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kuttler, K.L., Park, A., Shillor, M., Zhang, W.: Unilateral dynamic contact of two beams. Math. Comput. Model. 34(3–4), 365–384 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kuttler, K.L., Shillor, M.: A dynamic contact problem in one-dimensional thermoviscoelasticity. Nonlinear World 2(3), 355–385 (1995)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kuttler, K.L., Shillor, M.: Vibrations of a beam between two stops. Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Algorithms 8(1), 3–110 (2001)MathSciNetGoogle Scholar
  32. 32.
    Labuschagne, A., van Rensburg, N.F.J., van der Merwe, A.J.: Comparison of linear beam theories. Math. Comput. Model. 49(1–2), 20–30 (2009)CrossRefzbMATHGoogle Scholar
  33. 33.
    Lin, Y.-H.: Vibration analysis of Timoshenko beams traversed by moving loads. J. Mar. Sci. Technol. 2(1), 25–35 (1994)Google Scholar
  34. 34.
    Martins, J.A.C., Oden, J.T.: A numerical analysis of a class of problems in elastodynamics with friction. Comput. Methods Appl. Mech. Eng. 40(3), 327–360 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Messaoudi, S.A., Fareh, A.: Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds. Arab. J. Math. (Springer) 2(2), 199–207 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Messaoudi, S.A., Said-Houari, B.: Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl. 348(1), 298–307 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Muñoz Rivera, J.E., de Lacerda Oliveira, M.: Exponential stability for a contact problem in thermoelasticity. IMA J. Appl. Math. 58(1), 71–82 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Muñoz Rivera, J.E., Jiang, S.: The thermoelastic and viscoelastic contact of two rods. J. Math. Anal. Appl. 217(2), 423–458 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Muñoz Rivera, J.E., Racke, R.: Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 276, 248–278 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Muñoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Discret. Contin. Dyn. Syst. 9(6), 1625–1639 (2003)CrossRefzbMATHGoogle Scholar
  41. 41.
    Newland, D.E.: Mechanical Vibration Analysis and Computation. Longman Scientific & Technical, Harlow (1989)Google Scholar
  42. 42.
    Raposo, C.A., Ferreira, J., Santos, M.L., Castro, N.N.O.: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 18(5), 535–541 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51(2), 105–126 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Said-Houari, B., Kasimov, A.: Decay property of Timoshenko system in thermoelasticity. Math. Methods Appl. Sci. 35(3), 314–333 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Shi, P., Shillor, M.: Uniqueness and stability of the solution to a thermoelastic contact problem. Eur. J. Appl. Math. 1(4), 371–387 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Lect. Notes Phys., vol. 655. Springer, Berlin (2004)CrossRefGoogle Scholar
  47. 47.
    Soufyane, A.: Exponential stability of the linearized nonuniform Timoshenko beam. Nonlinear Anal. Real World Appl. 10(2), 1016–1020 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Timoshenko, H.: Vibration Problems in Engineering. D van Nostrand Company Inc., New-York (1937)zbMATHGoogle Scholar
  49. 49.
    Yan, Q.-X., Hou, S.-H., Feng, D.-X.: Asymptotic behavior of Timoshenko beam with dissipative boundary feedback. J. Math. Anal. Appl. 269(2), 556–577 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Alessia Berti
    • 1
  • Jaime E. Muñoz Rivera
    • 2
  • Maria Grazia Naso
    • 3
    Email author
  1. 1.Facoltà di IngegneriaUniversità e-CampusNovedrate (CO)Italy
  2. 2.National Laboratory for Scientific ComputationRio de JaneiroBrazil
  3. 3.Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di MatematicaUniversità degli Studi di BresciaBresciaItaly

Personalised recommendations