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 Naso


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.


Timoshenko thermoelastic beam Signorini condition Contact Asymptotic behavior 

Mathematics Subject Classification

74H40 74M15 35B40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Berti, A., Naso, M.G.: Unilateral dynamic contact of two viscoelastic beams. Q. Appl. Math. 69(3), 477–507 (2011)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  20. 20.
    Elliott, C.M., Tang, Q.: A dynamic contact problem in thermoelasticity. Nonlinear Anal. 23(7), 883–898 (1994)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kuttler, K.L., Shillor, M.: A dynamic contact problem in one-dimensional thermoviscoelasticity. Nonlinear World 2(3), 355–385 (1995)MathSciNetMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Muñoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Discret. Contin. Dyn. Syst. 9(6), 1625–1639 (2003)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51(2), 105–126 (1998)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Said-Houari, B., Kasimov, A.: Decay property of Timoshenko system in thermoelasticity. Math. Methods Appl. Sci. 35(3), 314–333 (2012)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Timoshenko, H.: Vibration Problems in Engineering. D van Nostrand Company Inc., New-York (1937)MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Alessia Berti
    • 1
  • Jaime E. Muñoz Rivera
    • 2
  • Maria Grazia Naso
    • 3
  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