Elastic contact of a stiff thin layer and a half-space

  • J. KaplunovEmail author
  • D. Prikazchikov
  • L. Sultanova


The 3D problem in linear elasticity for a layer lying on a half-space is subject to a two-parametric asymptotic treatment using the small parameters corresponding to the relative thickness of the layer and stiffness of the foundation. General scaling for the displacements and stresses is inspired by the analysis of the exact solution of the toy plane strain problem for a vertical sinusoidal load. The direct asymptotic procedure widely used in mechanics of thin structures is adapted for the layer. It is demonstrated that the Kirchhoff theory for thin plates is only applicable for sufficiently high contrast of the coating and half-space stiffnesses. In the scenario, in which the Kirchhoff theory fails, alternative approximate formulations are introduced, reducing the original problem for a coated solid to problems for a homogeneous half-space with Neumann, mixed or effective boundary conditions along its surface.


Stiff thin coating Asymptotic Kirchhoff plate Contrast Substrate 

Mathematics Subject Classification

74G10 74K35 



  1. 1.
    Achenbach, J.: Wave Propagation in Elastic Solids. Elsevier, Amsterdam (2012)zbMATHGoogle Scholar
  2. 2.
    Aghalovyan, L.: Asymptotic Theory of Anisotropic Plates and Shells. World Scientific, Singapore (2015)CrossRefGoogle Scholar
  3. 3.
    Aleksandrova, G.P.: Contact problems in bending of a slab lying on an elastic foundation. Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela (1), 97–116 (1973)Google Scholar
  4. 4.
    Alexandrov, V.M.: Contact problems on soft and rigid coatings of an elastic half-plane. Mech. Solids 45(1), 34–40 (2010)CrossRefGoogle Scholar
  5. 5.
    Bigoni, D., Gei, M., Movchan, A.B.: Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics. J. Mech. Phys. Solids 56(7), 2494–2520 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bigoni, D., Ortiz, M., Needleman, A.: Effect of interfacial compliance on bifurcation of a layer bonded to a substrate. Int. J. Solids Struct. 34(33–34), 4305–4326 (1997)CrossRefGoogle Scholar
  7. 7.
    Biot, M.A.: Bending of an infinite beam on an elastic foundation. Z. Angew. Math. Phys. 2(3), 165–184 (1922)CrossRefGoogle Scholar
  8. 8.
    Cai, Z., Fu, Y.: Exact and asymptotic stability analyses of a coated elastic half-space. Int. J. Solids Struct. 37(22), 3101–3119 (2000)CrossRefGoogle Scholar
  9. 9.
    Cai, Z., Fu, Y.: On the imperfection sensitivity of a coated elastic half-space. Proc. R. Soc. A 455(1989), 3285–3309 (1999)CrossRefGoogle Scholar
  10. 10.
    Dai, H.-H., Kaplunov, J., Prikazchikov, D.A.: A long-wave model for the surface elastic wave in a coated half-space. Proc. R. Soc. A 466(2122), 3097–3116 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Den Hartog, J.P.: Advanced Strength of Materials. McGraw-Hill, New York (1952)Google Scholar
  12. 12.
    Destrade, M., Fu, Y., Nobili, A.: Edge wrinkling in elastically supported pre-stressed incompressible isotropic plates. Proc. R. Soc. A 472(2193), 20160410 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Erbaş, B., Yusufoğlu, E., Kaplunov, J.: A plane contact problem for an elastic orthotropic strip. J. Eng. Math. 70(4), 399–409 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fu, Y.B., Cai, Z.X.: An asymptotic analysis of the period-doubling secondary bifurcation in a film/substrate bilayer. SIAM J. Appl. Math. 75(6), 2381–2395 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gei, M., Ogden, R.W.: Vibration of a surface-coated elastic block subject to bending. Math. Mech. Solids 7(6), 607–628 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Goldenveizer, A.L., Kaplunov, J.D., Nolde, E.V.: On Timoshenko–Reissner type theories of plates and shells. Int. J. Solids Struct. 30(5), 675–694 (1993)CrossRefGoogle Scholar
  17. 17.
    Gorbunov-Posadov, M.I.: Beams and Plates on Elastic Foundation. Gosstroiizdat, Moscow (1949). (in Russian)Google Scholar
  18. 18.
    Gorbunov-Posadov, M.I.: Calculation of Constructions on Elastic Foundation. Gosstroiizdat, Moscow (1953). (in Russian)Google Scholar
  19. 19.
    Gorbunov-Posadov, M.I.: Tables for the Computation of Thin Plates on Elastic Foundations. Gosstroiizdat, Moscow (1959). (in Russian)Google Scholar
  20. 20.
    Hetenyi, M.: Beams on Elastic Foundation. The University of Michigan Press, Ann Arbor (1958)zbMATHGoogle Scholar
  21. 21.
    Höller, R., Aminbaghai, M., Eberhardsteiner, L., Eberhardsteiner, J., Blab, R., Pichler, B., Hellmich, C.: Rigorous amendment of Vlasov’s theory for thin elastic plates on elastic Winkler foundations, based on the principle of virtual power. Eur. J. Mech. A Solids 73, 449–482 (2019)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kaplunov, J.D.: Long-wave vibrations of a thinwalled body with fixed faces. Q. J. Mech. Appl. Math. 48(3), 311–327 (1995)CrossRefGoogle Scholar
  23. 23.
    Kaplunov, J.D., Markushevich, D.G.: Plane vibrations and radiation of an elastic layer lying on a liquid half-space. Wave Motion 17(3), 199–211 (1993)CrossRefGoogle Scholar
  24. 24.
    Kaplunov, J.D., Nolde, E.V.: Long-wave vibrations of a nearly incompressible isotropic plate with fixed faces. Q. J. Mech. Appl. Math. 55(3), 345–356 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kaplunov, J., Prikazchikov, D., Sultanova, L.: Justification and refinement of Winkler–Fuss hypothesis. Z. Angew. Math. Phys. 69(3), 80 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kuznetsov, V.I.: Elastic Foundations. Gosstroiizdat, Moscow (1952)Google Scholar
  27. 27.
    Popov, GYa.: Plates on a linearly elastic foundation (a survey). Sov. Appl. Mech. 8(3), 231–242 (1972)CrossRefGoogle Scholar
  28. 28.
    Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453(1959), 853–877 (1997)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tiersten, H.F.: Elastic surface waves guided by thin films. J. Appl. Phys. 40(2), 770–789 (1969)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Computing and MathematicsKeele UniversityKeeleUK

Personalised recommendations