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Justification and refinement of Winkler–Fuss hypothesis

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Abstract

Two-parametric asymptotic analysis of the equilibrium of an elastic half-space coated by a thin soft layer is developed. The initial scaling is motivated by the exact solution of the plane problem for a vertical harmonic load. It is established that the Winkler–Fuss hypothesis is valid only for a sufficiently high contrast in the stiffnesses of the layer and the half-space. As an alternative, a uniformly valid non-local approximation is proposed. Higher-order corrections to the Winkler–Fuss formulation, such as the Pasternak model, are also studied.

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References

  1. Achenbach, J.: Wave Propagation in Elastic Solids. Elsevier, Amsterdam (2012)

    MATH  Google Scholar 

  2. Aghalovyan, L.: Asymptotic Theory of Anisotropic Plates and Shells. World Scientific, Singapore (2015)

    Book  MATH  Google Scholar 

  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)

  4. Alexandrov, V.M.: Contact problems on soft and rigid coatings of an elastic half-plane. Mech. Solids 45(1), 34–40 (2010)

    Article  MathSciNet  Google Scholar 

  5. Borodich, F.M.: The Hertz-type and adhesive contact problems for depth-sensing indentation. Adv. Appl. Mech. 47, 225–366 (2014)

    Article  Google Scholar 

  6. Challamel, N., Meftah, S.A., Bernard, F.: Buckling of elastic beams on non-local foundation: a revisiting of Reissner model. Mech. Res. Commun. 37(5), 472–475 (2010)

    Article  MATH  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Di Paola, M., Marino, F., Zingales, M.: A generalized model of elastic foundation based on long-range interactions: integral and fractional model. Int. J. Solids Struct. 45(17), 3124–3137 (2009)

    Article  MATH  Google Scholar 

  9. Dutta, S.C., Roy, R.: A critical review on idealization and modeling for interaction among soil-foundation-structure system. Comput. Struct. 80(20–21), 1579–1594 (2002)

    Article  Google Scholar 

  10. Epshtein, S.A., Borodich, F.M., Bull, S.J.: Evaluation of elastic modulus and hardness of highly inhomogeneous materials by nanoindentation. Appl. Phys. A 119(1), 325–335 (2015)

    Article  Google Scholar 

  11. Erbaş, B., Yusufoğlu, E., Kaplunov, J.: A plane contact problem for an elastic orthotropic strip. J. Eng. Math. 70(4), 399–409 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fischer-Cripps, A.C.: Nanoindentation Testing. Nanoindentation, pp. 21–37. Springer, Berlin (2011)

    Google Scholar 

  13. Frýba, L.: History of Winkler foundation. Veh. Syst. Dyn. 24(1), 7–12 (1995)

    Article  Google Scholar 

  14. Gazetas, G., Dobry, R.: Horizontal response of piles in layered soils. J. Geotech. Eng. 110(1), 20–40 (1984)

    Article  Google Scholar 

  15. 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)

    Article  MATH  Google Scholar 

  16. Hetenyi, M.: Beams on Elastic Foundation. The University of Michigan Press, Ann Arbor (1958)

    MATH  Google Scholar 

  17. Kaplunov, J., Nobili, A.: The edge waves on a Kirchhoff plate bilaterally supported by a two-parameter elastic foundation. J. Vib. Control 23(12), 2014–2022 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kerr, A.D.: Elastic and viscoelastic foundation models. J. Appl. Mech. 31(3), 491–498 (1964)

    Article  MATH  Google Scholar 

  19. Kovalenko, E.V., Buyanovskii, I.A.: On the durability prediction for an elastic cylinder-elastic layer pair reinforced by thin coating according to the wear criterion. J. Mach. Manuf. Reliab. 44(4), 357–362 (2015)

    Article  Google Scholar 

  20. Kuznetsov, V.I.: Elastic Foundations. Gosstroiizdat, Moscow (1952)

    Google Scholar 

  21. Liyanapathirana, D.S., Poulos, H.G.: Pseudostatic approach for seismic analysis of piles in liquefying soil. J. Geotech. Geoenviron. Eng. 131(12), 1480–1487 (2005)

    Article  Google Scholar 

  22. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  23. Martynyak, R.M., Prokopyshyn, I.A., Prokopyshyn, I.I.: Contact of elastic bodies with nonlinear Winkler surface layers. J. Math. Sci. 205(4), 535–553 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Morozov, N.F., Tovstik, P.E.: On modes of buckling for a plate on an elastic foundation. Mech. Solids 45(4), 519–528 (2010)

    Article  Google Scholar 

  25. Nikolaou, S., Mylonakis, G., Gazetas, G., Tazoh, T.: Kinematic pile bending during earthquakes: analysis and field measurements. Geotechnique 51(5), 425–440 (2001)

    Article  Google Scholar 

  26. Nobili, A.: Superposition principle for the tensionless contact of a beam resting on a Winkler or a Pasternak foundation. J. Eng. Mech. 139(10), 1470–1478 (2012)

    Article  Google Scholar 

  27. Pasternak, P.L.: On a new method of an elastic foundation by means of two foundation constants. Gos. Izd. Lit. po Stroit. i Arkhitekt., Moscow (1954)

  28. Wang, Y.H., Tham, L.G., Cheung, Y.K.: Beams and plates on elastic foundations: a review. Prog. Struct. Eng. Mater. 7(4), 174–182 (2005)

    Article  Google Scholar 

  29. Winkler, E.: Die Lehre von der Elastizit at und Festigkeit. Domimicus, Prague (1867)

    Google Scholar 

  30. Yim, C.-S., Chopra, A.K.: Earthquake response of structures with partial uplift on Winkler foundation. Earthq. Eng. Struct. Dyn. 12(2), 263–281 (1984)

    Article  Google Scholar 

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  1. School of Computing and Mathematics, Keele University, Keele, Staffordshire, ST5 5BG, UK

    J. Kaplunov, D. Prikazchikov & L. Sultanova

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  1. J. Kaplunov
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  2. D. Prikazchikov
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  3. L. Sultanova
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Correspondence to J. Kaplunov.

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Kaplunov, J., Prikazchikov, D. & Sultanova, L. Justification and refinement of Winkler–Fuss hypothesis. Z. Angew. Math. Phys. 69, 80 (2018). https://doi.org/10.1007/s00033-018-0974-1

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  • DOI: https://doi.org/10.1007/s00033-018-0974-1

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