Frictional Indentation of an Elastic Half-Space

Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 91)

Abstract

In this chapter, we study the axisymmetric indentation problem for a transversely isotropic elastic half-space with finite friction. By treating the indentation problem incrementally, its general solution is reduced to that of the problem for a flat-ended cylindrical indenter with an unknown stick-slip radius. The solution to the latter problem in the transversely isotropic case is obtained via Turner’s equivalence principle Turner (Int J Solids Struct 16:409–419, 1980 [15]), from the analytical solution given by Spence (J Elast 5:297–319, 1975 [12]) in the case of isotropy. The generalization, due to Storåkers and Elaguine (J Mech Phys Solids 53:1422–1447, 2005 [14]), of the BASh relation for incremental indentation stiffness, and also accounting for the friction effects, is presented. The case of self-similar contact with friction is considered in more detail.

References

  1. 1.
    Borodich, F.M.: The hertz frictional contact between nonlinear elastic anisotropic bodies (the similarity approach). Int. J. Solids Struct. 30, 1513–1526 (1993)CrossRefGoogle Scholar
  2. 2.
    Borodich, F.M., Keer, L.M.: Contact problems and depth-sensing nanoindentation for frictionless and frictional boundary conditions. Int. J. Solids Struct. 41, 2479–2499 (2004)CrossRefGoogle Scholar
  3. 3.
    Bulychev, S.I., Alekhin, V.P., Shorshorov, MKh, Ternovskii, A.P., Shnyrev, G.D.: Determination of young’s modulus according to indentation diagram. Ind. Lab. 41, 1409–1412 (1975)Google Scholar
  4. 4.
    Fabrikant, V.I.: Four types of exact solution to the problem of an axisymmetric punch bonded to a transversely isotropic half-space. Int. J. Eng. Sci. 24, 785–801 (1986)CrossRefGoogle Scholar
  5. 5.
    Gauthier, A., Knight, P.A., McKee, S.: The Hertz contact problem, coupled volterra integral equations and a linear complementarity problem. J. Comput. Appl. Math. 206, 322–340 (2007)CrossRefGoogle Scholar
  6. 6.
    Hills, D.A., Nowell, D., Sackfield, A.: Mechanics Of Elastic Contacts. Butterworth-Heineman, Oxford (1993)Google Scholar
  7. 7.
    Hills, D.A., Sackfield, A.: The stress field induced by normal contact between dissimilar spheres. J. Appl. Mech. 54, 8–14 (1987)CrossRefGoogle Scholar
  8. 8.
    Linz, P., Noble, B.: A numerical method for treating indentation problems. J. Eng. Math. 5(3), 227–231 (1971)CrossRefGoogle Scholar
  9. 9.
    Mossakovskii, V.I.: Compression of elastic bodies under conditions of adhesion (axisymmetric case) [in Russian]. J. Appl. Math. Mech. 27, 630–643 (1963)CrossRefGoogle Scholar
  10. 10.
    Shayanfar, N., Hadizadeh, M.: \(\lambda \)-matrix formulation applied to the Hertz contact problem with finite friction. Comp. Math. Appl. 64, 2478–2483 (2012)CrossRefGoogle Scholar
  11. 11.
    Spence, D.A.: An eigenvalue problem for elastic contact with finite friction. Proc. Camb. Phil. Soc. 73, 249–268 (1973)CrossRefGoogle Scholar
  12. 12.
    Spence, D.A.: The hertz contact problem with finite friction. J. Elast. 5, 297–319 (1975)CrossRefGoogle Scholar
  13. 13.
    Spence, D.A.: Similarity considerations for contact between dissimilar elastic bodies. In: Proceedings of the IUTAM Symposium on Mechanics of Contact, pp. 76–89. Delft University Press, Delft (1975)Google Scholar
  14. 14.
    Storåkers, B., Elaguine, D.J.: Hertz contact at finite friction and arbitrary profiles. J. Mech. Phys. Solids 53, 1422–1447 (2005)Google Scholar
  15. 15.
    Turner, J.R.: Contact on a transversely isotropic half-space, or between two transversely isotropic bodies. Int. J. Solids Struct. 16, 409–419 (1980)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MechanicsTechnical University of BerlinBerlinGermany
  2. 2.Department of Mathematics, IMPACSAberystwyth UniversityAberystwythUK

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