Indentation of a Viscoelastic Half-Space

  • Ivan ArgatovEmail author
  • Gennady Mishuris
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 91)


In this chapter, we consider frictionless indentation problems for a viscoelastic half-space. We note that, throughout the chapter, the indenter displacement is denoted by w(t). Previous notation has been altered to avoid confusion with Dirac’s delta function \(\delta (t)\), as well as with the loss angle \(\delta (\omega )\). As is well known, a viscoelastic material is characterized by a number of time-dependent moduli, and a certain combination of them, called the indentation relaxation modulus and denoted by \(M_3(t)\), will determine the material’s response to indentation. Depending on the type of loading protocol, different information can be gathered from the indentation test about the indentation relaxation modulus and its inverse counterpart \(C_3(t)\), called the indentation creep compliance.


  1. 1.
    Appleyard, R.C., Swain, M.V., Khanna, S., Murrell, G.A.C.: The accuracy and reliability of a novel handheld dynamic indentation probe for analysing articular cartilage. Phys. Med. Biol. 46, 541–550 (2001)CrossRefGoogle Scholar
  2. 2.
    Argatov, I.: Depth-sensing indentation of a transversely isotropic elastic layer: second-order asymptotic models for canonical indenters. Int. J. Solids Struct. 48, 3444–3452 (2011)CrossRefGoogle Scholar
  3. 3.
    Argatov, I.: Sinusoidally-driven flat-ended indentation of time-dependent materials: asymptotic models for low and high rate loading. Mech. Mater. 48, 56–70 (2012)CrossRefGoogle Scholar
  4. 4.
    Argatov, I.: An analytical solution of the rebound indentation problem for an isotropic linear viscoelastic layer loaded with a spherical punch. Acta Mech. 223, 1441–1453 (2012)CrossRefGoogle Scholar
  5. 5.
    Argatov, I.I.: Mathematical modeling of linear viscoelastic impact: application to drop impact testing of articular cartilage. Trib. Int. 63, 213–225 (2013)CrossRefGoogle Scholar
  6. 6.
    Argatov, I., Daniels, A.U., Mishuris, G., Ronken, S., Wirz, D.: Accounting for the thickness effect in dynamic spherical indentation of a viscoelastic layer: application to non-destructive testing of articular cartilage. Eur. J. Mech. A/Solids 37, 304–317 (2013)CrossRefGoogle Scholar
  7. 7.
    Argatov, I., Iantchenko, A., Kocherbitov, V.: How to define the storage and loss moduli for a rheologically nonlinear material? Continuum Mech. Therm. 29, 1375–1387 (2017)CrossRefGoogle Scholar
  8. 8.
    Argatov, I., Mishuris, G.: An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: evaluation of the rebound indentation test with application for assessing viability of articular cartilage. Mech. Res. Commun. 38, 565–568 (2011)CrossRefGoogle Scholar
  9. 9.
    Argatov, I.I., Popov, V.L.: Rebound indentation problem for a viscoelastic half-space and axisymmetric indenter. Z. Angew. Math. Mech. 96, 956–967 (2016)CrossRefGoogle Scholar
  10. 10.
    Brown, C.P., Crawford, R.W., Oloyede, A.: An alternative mechanical parameter for assessing the viability of articular cartilage. Proc. Inst. Mech. Eng., Part H 223, 53–62 (2009)Google Scholar
  11. 11.
    Cao, Y., Ma, D., Raabe, D.: The use of flat punch indentation to determine the viscoelastic properties in the time and frequency domains of a soft layer bonded to a rigid substrate. Acta Biomater. 5, 240–248 (2009)CrossRefGoogle Scholar
  12. 12.
    Christensen, R.M.: Theory of Viscoelasticity. Academic Press, New York (1982)Google Scholar
  13. 13.
    Galin, L.A.: Spatial contact problems of the theory of elasticity for punches of circular shape in planar projection. J. Appl. Math. Mech. (PMM) 10, 425–448 (1946). (in Russian)Google Scholar
  14. 14.
    Giannakopoulos, A.E.: Elastic and viscoelastic indentation of flat surfaces by pyramid indentors. J. Mech. Phys. Solids 54, 1305–1332 (2006)CrossRefGoogle Scholar
  15. 15.
    Golden, J.M., Graham, G.A.C.: The steady-state plane normal viscoelastic contact problem. Int. J. Eng. Sci. 25, 277–291 (1987)CrossRefGoogle Scholar
  16. 16.
    Golden, M.J., Graham, G.A.C.: Boundary Value Problems in Linear Viscoelasticy. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  17. 17.
    Golden, M.J., Graham, G.A.C.: A proposed method of measuring the complex modulus of a thick viscoelastic layer. Rheol. Acta 28, 414–416 (1989)CrossRefGoogle Scholar
  18. 18.
    Golden, J.M., Graham, G.A.C., Lan, Q.: Three-dimensional steady-state indentation problem for a general viscoelastic material. Quart. Appl. Math. 52, 449–468 (1994)CrossRefGoogle Scholar
  19. 19.
    Graham, G.A.C.: The contact problem in the linear theory of viscoelasticity. Int. J. Eng. Sci. 3, 27–46 (1965)CrossRefGoogle Scholar
  20. 20.
    Graham, G.A.C.: The contact problem in the linear theory of viscoelasticity when the time dependent contact area has any number of maxima and minima. Int. J. Eng. Sci. 5, 495–514 (1967)CrossRefGoogle Scholar
  21. 21.
    Graham, G.A.C., Golden, J.M.: The three-dimensional steady-state viscoelastic indentation problem. Int. J. Eng. Sci. 26, 121–126 (1988)CrossRefGoogle Scholar
  22. 22.
    Graham, G.A.C., Golden, J.M.: The generalized partial correspondence principle in linear viscoelasticity. Quart. Appl. Math. 46, 527–538 (1988)CrossRefGoogle Scholar
  23. 23.
    Greenwood, J.A.: Contact between an axisymmetric indenter and a viscoelastic half-space. Int. J. Mech. Sci. 52, 829–835 (2010)CrossRefGoogle Scholar
  24. 24.
    Hertz, H.: Hertz’s Miscellaneous Papers; Chap. 5 and 6. Macmillan, London, UK (1896)Google Scholar
  25. 25.
    Hori, R.Y., Mockros, L.F.: Indentation tests of human articular cartilage. J. Biomech. 9, 259–268 (1976)CrossRefGoogle Scholar
  26. 26.
    Hunter, S.C.: The Hertz problem for a rigid spherical indenter and a viscoelastic half-space. J. Mech. Phys. Solids 8, 219–234 (1960)CrossRefGoogle Scholar
  27. 27.
    Lakes, R.S.: Viscoelastic Materials. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  28. 28.
    Lee, E.H., Radok, J.R.M.: The contact problem for viscoelastic bodies. J. Appl. Mech. 27, 438–444 (1960)CrossRefGoogle Scholar
  29. 29.
    Lee, S., Knauss, W.G.: A note on the determination of relaxation and creep data from ramp tests. Mech. Time-Depend. Mat. 4, 1–7 (2000)CrossRefGoogle Scholar
  30. 30.
    Love, A.E.H.: Boussinesq’s problem for a rigid cone. Quart. J. Math. Oxford Ser. 10, 161–175 (1939)Google Scholar
  31. 31.
    Lu, H., Wang, B., Ma, J., Huang, G., Viswanathan, H.: Measurement of creep compliance of solid polymers by nanoindentation. Mech. Time-Depend. Mat. 7, 189–207 (2003)CrossRefGoogle Scholar
  32. 32.
    Oyen, M.L.: Spherical indentation creep following ramp loading. J. Mater. Res. 20, 2094–2100 (2005)CrossRefGoogle Scholar
  33. 33.
    Park, S.W.: Analytical modeling of viscoelastic dampers for structural and vibration control. Int. J. Solids Struct. 38, 8065–8092 (2001)CrossRefGoogle Scholar
  34. 34.
    Pipkin, A.C.: Lectures on Viscoelastic Theory. Springer, New York (1986)CrossRefGoogle Scholar
  35. 35.
    Popov, V.L., Heß, M.: Method of Dimensionality Reductions in Contact Mechanics and Friction. Springer, Heidelberg (2014)Google Scholar
  36. 36.
    Radok, J.R.M.: Visco-elastic stress analysis. Quart. Appl. Math. 15, 198–202 (1957)CrossRefGoogle Scholar
  37. 37.
    Segedin, C.M.: The relation between load and penetration for a spherical punch. Mathematika 4, 156–161 (1957)CrossRefGoogle Scholar
  38. 38.
    Ting, T.C.T.: The contact stresses between a rigid indenter and a viscoelastic half-space. J. Appl. Mech. 33, 845–854 (1966)CrossRefGoogle Scholar
  39. 39.
    Ting, T.C.T.: Contact problems in the linear theory of viscoelasticity. J. Appl. Mech. 35, 248–254 (1968)CrossRefGoogle Scholar
  40. 40.
    Tschoegl, N.W.: The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  41. 41.
    Tschoegl, N.W.: Time dependence in material properties: an overview. Mech. Time-Depend. Mat. 1, 3–31 (1997)CrossRefGoogle Scholar
  42. 42.
    Vandamme, M., Ulm, F.-J.: Viscoelastic solutions for conical indentation. Int. J. Solids Struct. 43, 3142–3165 (2006)CrossRefGoogle Scholar
  43. 43.
    Yao, H., Gao, H.: Optimal shapes for adhesive binding between two elastic bodies. J. Colloid Interface Sci. 298, 564–572 (2006)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

Personalised recommendations