Asymptotic Analysis of the Contact Problem for Two Bonded Elastic Layers

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


The first part of the chapter deals with the distributional asymptotic analysis of the contact problem of frictionless unilateral interaction of two bonded elastic layers. The case of incompressible layer materials is thoroughly treated in the second part of the chapter, beginning in Sect. 2.4.


  1. 1.
    Alblas, J.B., Kuipers, M.: On the two-dimensional problem of a cylindrical stamp pressed into a thin elastic layer. Acta Mech. 9, 292–311 (1970)zbMATHCrossRefGoogle Scholar
  2. 2.
    Aleksandrov, V.M.: The solution of a type of two-dimensional integral equations. J. Appl. Math. Mech. 28, 714–717 (1964)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aleksandrov, V.M.: Asymptotic solution of the contact problem for a thin elastic layer. J. Appl. Math. Mech. 33, 49–63 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Aleksandrov, V.M.: Asymptotic solution of the axisymmetric contact problem for an elastic layer of incompressible material. J. Appl. Math. Mech. 67, 589–593 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aleksandrov, V.M., Babeshko, V.A.: Contact problems for an elastic strip of small thickness. Izv. Akad. Nauk SSSR. Ser. Mekhanika. 2, 95–107 (1965). [in Russian]Google Scholar
  6. 6.
    Alexandrov, V.M., Pozharskii, D.A.: Three-Dimensional Contact Problems. Kluwer, Dordrecht (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Argatov, I.I.: Pressure of a punch in the form of an elliptic paraboloid on a thin elastic layer. Acta Mech. 180, 221–232 (2005)zbMATHCrossRefGoogle Scholar
  8. 8.
    Argatov, I.I.: Pressure of a paraboloidal die on a thin elastic layer. Dokl. Phys. 50, 524–528 (2005)CrossRefGoogle Scholar
  9. 9.
    Chadwick, R.S.: Axisymmetric indentation of a thin incompressible elastic layer. SIAM J. Appl. Math. 62, 1520–1530 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Estrada, R., Kanwal, R.P.: A distributional theory for asymptotic expansions. Proc. R. Soc. Lond. A 428, 399–430 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fabrikant, V.I.: Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation. Z. Angew. Math. Phys. 57, 464–490 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fabrikant, V.I.: Numerical methods of solution of contact problems in layered media. Int. J. Comput. Meth. Eng. Sci. Mech. 12, 84–95 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Garcia, J.J., Altiero, N.J., Haut, R.C.: An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load. J. Biomech. Eng. 120, 608–613 (1998)CrossRefGoogle Scholar
  14. 14.
    Gol’denveizer, A.L.: Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity. J. Appl. Math. Mech. 26, 1000–1025 (1962)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Itskov, M., Aksel, N.: Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials. Acta Mech. 157, 81–96 (2002)zbMATHCrossRefGoogle Scholar
  16. 16.
    Johnson, K.L.: Contact Mechanics. Cambridge Univ. Press, Cambridge, UK (1985)zbMATHCrossRefGoogle Scholar
  17. 17.
    Hlaváček, M.: A note on an asymptotic solution for the contact of two biphasic cartilage layers in a loaded synovial joint at rest. J. Biomech. 32, 987–991 (1999)CrossRefGoogle Scholar
  18. 18.
    Koiter, W.T.: Solutions of some elasticity problems by asymptotic methods. In: Applications of the theory of functions to continuum mechanics. Muskhelishvili, N.I. (ed.), pp. 15–31. International Symposium Tbilisi, USSR 1963. Moscow, Nauka (1965)Google Scholar
  19. 19.
    Leibenzon, L.S.: Course in the Theory of Elasticity. Gostekhizdat, Moscow-Leningrad (1947)Google Scholar
  20. 20.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)zbMATHGoogle Scholar
  21. 21.
    Mehrabadi, M.M., Cowin, S.C.: Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math. 43, 15–41 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Mishuris, G.: Imperfect transmission conditions for a thin weakly compressible interface. 2D problems. Arch. Mech. 56, 103–115 (2004)Google Scholar
  23. 23.
    Noble, B.: Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, New York (1958)zbMATHGoogle Scholar
  24. 24.
    Sneddon, I.N.: Fourier Transforms. Dover, New York (1995)Google Scholar
  25. 25.
    Vorovich, I.I., Aleksandrov, V.M., Babeshko, V.A.: Non-classical Mixed Problems of the Theory of Elasticity. Nauka, Moscow (1974). [in Russian]Google Scholar
  26. 26.
    Wong, R.: Distributional derivation of an asymptotic expansion. Proc. Amer. Math. Soc. 80, 266–270 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Wu, J.Z., Herzog, W.: On the pressure gradient boundary condition for the contact of two biphasic cartilage layers. J. Biomech. 33, 1331–1332 (2000)CrossRefGoogle Scholar
  28. 28.
    Yang, F.: Indentation of an incompressible elastic film. Mech. Mater. 30, 275–286 (1998)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsAberystwyth UniversityAberystwythUK

Personalised recommendations