Frictionless Contact of Thin Viscoelastic Layers

  • Ivan ArgatovEmail author
  • Gennady Mishuris
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 50)


The chapter begins with an introduction to linear viscoelastic theory, and then proceeds to a generalization of the elastic leading-order asymptotic models for the viscoelastic case, based on the correspondence principle. In Sect. 4.2, we consider the main features of the analytical technique for solving unilateral contact problems for a viscoelastic foundation. The axisymmetric contact problem for a thin bonded incompressible viscoelastic layer is analyzed in Sect. 4.3 and in the refined formulation accounting for tangential displacements in Sect. 4.4. Finally, in Sect. 4.5 we solve the problem of frictionless contact for thin incompressible viscoelastic layers bonded to rigid substrates shaped like elliptic paraboloids.


  1. 1.
    Ateshian, G.A., Lai, W.M., Zhu, W.B., Mow, V.C.: An asymptotic solution for the contact of two biphasic cartilage layers. J. Biomech. 27, 1347–1360 (1994)CrossRefGoogle Scholar
  2. 2.
    Argatov, I.I.: Asymptotic modeling of the impact of a spherical indenter on an elastic half-space. Int. J. Solids Struct. 45, 5035–5048 (2008)zbMATHCrossRefGoogle Scholar
  3. 3.
    Argatov, I.: A general solution of the axisymmetric contact problem for biphasic cartilage layers. Mech. Res. Comm. 38, 29–33 (2011)zbMATHCrossRefGoogle Scholar
  4. 4.
    Argatov, I.I.: Development of an asymptotic modeling methodology for tibio-femoral contact in multibody dynamic simulations of the human knee joint. Multibody Syst. Dyn. 28, 3–20 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Argatov, I.I., Mishuris, G.S.: Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface. Eur. J. Mech. A/Solids 29, 1051–1064 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Argatov, I., Mishuris, G.: Elliptical contact of thin biphasic cartilage layers: exact solution for monotonic loading. J. Biomech. 44, 759–761 (2011)CrossRefGoogle Scholar
  7. 7.
    Argatov, I., Mishuris, G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl. Math. Model. 35, 3201–3212 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Barber, J.R.: Contact problems for the thin elastic layer. Int. J. Mech. Sci. 32, 129–132 (1990)zbMATHCrossRefGoogle Scholar
  9. 9.
    Belokon’, A.V., Vorovich, I.I.: Contact problems of the linear theory of viscoelasticity ignoring friction and cohesion forces [in Russian]. Izv. Akad. Nauk SSSR. MTT [Mechanics of Solids], 6, 63–73 (1973)Google Scholar
  10. 10.
    Chadwick, R.S.: Axisymmetric indentation of a thin incompressible elastic layer. SIAM J. Appl. Math. 62, 1520–1530 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Christensen, R.M.: Theory of Viscoelasticity. Academic Press, New York (1982)Google Scholar
  12. 12.
    Graham, G.A.C.: The contact problem in the linear theory of viscoelasticity. Int. J. Eng. Sci. 3, 27–46 (1965)zbMATHCrossRefGoogle Scholar
  13. 13.
    Hoang, S.K., Abousleiman, Y.N.: Poroviscoelasticity of transversely isotropic cylinders under laboratory loading conditions. Mech. Res. Commun. 37, 298–306 (2010)zbMATHCrossRefGoogle Scholar
  14. 14.
    Jaffar, M.J.: Asymptotic behaviour of thin elastic layers bonded and unbonded to a rigid foundation. Int. J. Mech. Sci. 31, 229–235 (1989)CrossRefGoogle Scholar
  15. 15.
    Korhonen, R.K., Laasanen, M.S., Töyräs, J., Rieppo, J., Hirvonen, J., Helminen, H.J., Jurvelin, J.S.: Comparison of the equilibrium response of articular cartilage in unconfined compression, confined compression and indentation. J. Biomech. 35, 903–909 (2002)Google Scholar
  16. 16.
    Kravchuk, A.S.: On the Hertz problem for linearly and nonlinearly elastic bodies of finite dimensions. J. Appl. Math. Mech. 41, 320–328 (1977)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lakes, R.S.: Viscoelastic Materials. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  18. 18.
    Lee, E.H., Radok, J.R.M.: The contact problem for viscoelastic bodies. J. Appl. Mech. 27, 438–444 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Mishuris, G., Argatov, I.: Exact solution to a refined contact problem for biphasic cartilage layers. In: Nithiarasu, P., Löhner, R. (eds.) Proceedings of the 1st International Conference on Mathematical and Computational Biomedical Engineering—CMBE2009, pp. 151–154. Swanseak, UK, June 29–July 1 2009Google Scholar
  20. 20.
    Naghieh, G.R., Jin, Z.M., Rahnejat, H.: Contact characteristics of viscoelastic bonded layers. Appl. Math. Model. 22, 569–581 (1998)CrossRefGoogle Scholar
  21. 21.
    Naghieh, G.R., Rahnejat, H., Jin, Z.M.: Characteristics of frictionless contact of bonded elastic and viscoelastic layered solids. Wear 232, 243–249 (1999)CrossRefGoogle Scholar
  22. 22.
    Pipkin, A.C.: Lectures on Viscoelastic Theory. Springer, New York (1986)CrossRefGoogle Scholar
  23. 23.
    Ting, T.C.T.: Contact problems in the linear theory of viscoelasticity. J. Appl. Mech. 35, 248–254 (1968)zbMATHCrossRefGoogle Scholar
  24. 24.
    Tschoegl, N.W.: Time dependence in material properties: an overview. Mech. Time-Depend. Mat. 1, 3–31 (1997)CrossRefGoogle Scholar
  25. 25.
    Wu, J.Z., Herzog, W., Epstein, M.: An improved solution for the contact of two biphasic cartilage layers. J. Biomech. 30, 371–375 (1997)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsAberystwyth UniversityAberystwythUK

Personalised recommendations