Abstract
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.
Similar content being viewed by others
References
Achenbach, J.: Wave Propagation in Elastic Solids. Elsevier, Amsterdam (2012)
Aghalovyan, L.: Asymptotic Theory of Anisotropic Plates and Shells. World Scientific, Singapore (2015)
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)
Alexandrov, V.M.: Contact problems on soft and rigid coatings of an elastic half-plane. Mech. Solids 45(1), 34–40 (2010)
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)
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)
Biot, M.A.: Bending of an infinite beam on an elastic foundation. Z. Angew. Math. Phys. 2(3), 165–184 (1922)
Cai, Z., Fu, Y.: Exact and asymptotic stability analyses of a coated elastic half-space. Int. J. Solids Struct. 37(22), 3101–3119 (2000)
Cai, Z., Fu, Y.: On the imperfection sensitivity of a coated elastic half-space. Proc. R. Soc. A 455(1989), 3285–3309 (1999)
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)
Den Hartog, J.P.: Advanced Strength of Materials. McGraw-Hill, New York (1952)
Destrade, M., Fu, Y., Nobili, A.: Edge wrinkling in elastically supported pre-stressed incompressible isotropic plates. Proc. R. Soc. A 472(2193), 20160410 (2016)
Erbaş, B., Yusufoğlu, E., Kaplunov, J.: A plane contact problem for an elastic orthotropic strip. J. Eng. Math. 70(4), 399–409 (2011)
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)
Gei, M., Ogden, R.W.: Vibration of a surface-coated elastic block subject to bending. Math. Mech. Solids 7(6), 607–628 (2002)
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)
Gorbunov-Posadov, M.I.: Beams and Plates on Elastic Foundation. Gosstroiizdat, Moscow (1949). (in Russian)
Gorbunov-Posadov, M.I.: Calculation of Constructions on Elastic Foundation. Gosstroiizdat, Moscow (1953). (in Russian)
Gorbunov-Posadov, M.I.: Tables for the Computation of Thin Plates on Elastic Foundations. Gosstroiizdat, Moscow (1959). (in Russian)
Hetenyi, M.: Beams on Elastic Foundation. The University of Michigan Press, Ann Arbor (1958)
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)
Kaplunov, J.D.: Long-wave vibrations of a thinwalled body with fixed faces. Q. J. Mech. Appl. Math. 48(3), 311–327 (1995)
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)
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)
Kaplunov, J., Prikazchikov, D., Sultanova, L.: Justification and refinement of Winkler–Fuss hypothesis. Z. Angew. Math. Phys. 69(3), 80 (2018)
Kuznetsov, V.I.: Elastic Foundations. Gosstroiizdat, Moscow (1952)
Popov, GYa.: Plates on a linearly elastic foundation (a survey). Sov. Appl. Mech. 8(3), 231–242 (1972)
Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453(1959), 853–877 (1997)
Tiersten, H.F.: Elastic surface waves guided by thin films. J. Appl. Phys. 40(2), 770–789 (1969)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kaplunov, J., Prikazchikov, D. & Sultanova, L. Elastic contact of a stiff thin layer and a half-space. Z. Angew. Math. Phys. 70, 22 (2019). https://doi.org/10.1007/s00033-018-1068-9
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-018-1068-9