Contact Mechanics

  • Anthony C. Fischer-Cripps
Part of the Mechanical Engineering Series book series (MES)


There has been considerable recent interest in the mechanical characterisation of thin film systems and small volumes of material using depth-sensing indentation tests with either spherical or pyramidal indenters. Usually, the principal goal of such testing is to extract elastic modulus and hardness of the specimen material from experimental readings of indenter load and depth of penetration. These readings give an indirect measure of the area of contact at full load, from which the mean contact pressure, and thus hardness, may be estimated. The test procedure, for both spheres and pyramidal indenters, usually involves an elastic—plastic loading sequence followed by an unloading. The validity of the results for hardness and modulus depends largely upon the analysis procedure used to process the raw data. Such procedures are concerned not only with the extraction of modulus and hardness, but also with correcting the raw data for various systematic errors that have been identified for this type of testing. The forces involved are usually in the millinewton (10−3 N) range and are measured with a resolution of a few nanonewtons (10−9 N). The depths of penetration are on the order of microns with a resolution of less than a nanometre (10−9 m). In this chapter, the general principles of elastic and elastic—plastic contact and how these relate to indentations at the nanometre scale are considered.


Contact Pressure Plastic Zone Specimen Material Spherical Indenter Indenter Load 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Hertz, “On the contact of elastic solids,” J. Reine Angew. Math. 92, 1881, pp. 156–171. Translated and reprinted in English in Hertz’s Miscellaneous Papers, Macmillan & Co., London, 1896, Ch. 5.Google Scholar
  2. 2.
    H. Hertz, “On hardness,” Verh. Ver. Beförderung Gewerbe Fleisses 61, 1882, p. 410. Translated and reprinted in English in Hertz’s Miscellaneous Papers,Macmillan & Co, London, 1896, Ch. 6.Google Scholar
  3. 3.
    S. Timoshenko and J.N. Goodier, Theory of Elasticity, 2nd Ed. McGraw-Hill, N.Y. 1951.Google Scholar
  4. 4.
    A.C. Fischer-Cripps, “The use of combined elastic modulus in the analysis of depth sensing indentation data,” J. Mater. Res. 16 11, 2001, pp. 3050–3052.CrossRefGoogle Scholar
  5. 5.
    M.M. Chaudhri, “A note on a common mistake in the analysis of nanoindentation test data”, J. Mater. Res. 16 2, 2001, pp. 336–339.CrossRefGoogle Scholar
  6. 6.
    A.C. Fischer-Cripps, “The use of combined elastic modulus in depth-sensing indentation with a conical indenter,” J. Mat. Res. 18 5, 2003, pp. 1043–1045.Google Scholar
  7. 7.
    I.N. Sneddon, “Boussinesq’s problem for a rigid cone,” Proc. Cambridge Philos. Soc. 44, 1948, pp. 492–507.Google Scholar
  8. 8.
    J.R. Barber and D.A. Billings, “An approximate solution for the contact area and elastic compliance of a smooth punch of arbitrary shape,” Int. J. Mech. Sci. 32 12, 1990, pp. 991–997.CrossRefGoogle Scholar
  9. 9.
    G.G. Bilodeau, “Regular pyramid punch problem,” J. App. Mech. 59, 1992,’pp. 519–523.Google Scholar
  10. 10.
    P.-L. Larsson, A.E. Giannakopolous, E. Soderlund, D.J. Rowcliffe and R. Vestergaard, “Analysis of Berkovich Indentation,” Int. J. Structures, 33 2, 1996, pp. 221–248.CrossRefGoogle Scholar
  11. 11.
    A.C. Fischer-Cripps, Introduction to Contact Mechanics, Springer-Verlag, New York, 2000.Google Scholar
  12. 12.
    D. Tabor, The Hardness of Metals, Clarendon Press, Oxford, 1951.Google Scholar
  13. 13.
    F. Auerbach, “Absolute hardness,” Ann. Phys. Chem. (Leipzig) 43, 1891, pp.61100. Translated by C. Barns, Annual Report of the Board of Regents of the Smithsonian Institution, July 1, 1890 — June 30 1891, reproduced in “Miscellaneous documents of the House of Representatives for the First Session of the Fifty-Second Congress,” Government Printing Office, Washington, D.C., 43, 1891–1892, pp.207236.Google Scholar
  14. 14.
    E. Meyer, “Untersuchungen über Harteprufung und Harte,” Phys. Z. 9, 1908, pp. 66–74.Google Scholar
  15. 15.
    S.L. Hoyt, “The ball indentation hardness test,” Trans. Am. Soc. Steel Treat. 6, 1924, pp. 396–420.Google Scholar
  16. 16.
    M.C. Shaw, “The fundamental basis of the hardness test,” in The Science of Hardness Testing and its Research Applications, J.H. Westbrook and H. Conrad, Eds. American Society for Metals, Cleveland, OH, 1973, pp. 1–15.Google Scholar
  17. 17.
    M.V. Swain and J.T. Hagan, “Indentation plasticity and the ensuing fracture of glass,” J. Phys. D: Appl. Phys. 9, 1976, pp. 2201–2214.Google Scholar
  18. 18.
    M.T. Huber, “Contact of solid elastic bodies,” Ann. D. Physik, 14 1, 1904, pp. 153163.Google Scholar
  19. 19.
    A.C. Fischer-Cripps, “Elastic-plastic response of materials loaded with a spherical indenter,” J. Mater. Sci. 32 3, 1997, pp. 727–736.CrossRefGoogle Scholar
  20. 20.
    S.Dj. Mesarovic and N. A. Fleck, “Spherical indentation of elastic-plastic solids,” Proc. Roy. Soc. A455, 1999, pp. 2707–2728.Google Scholar
  21. 21.
    R. Hill, E.H. Lee and S.J. Tupper, “Theory of wedge-indentation of ductile metals,” Proc. Roy. Soc. A188, 1947, pp. 273–289.Google Scholar
  22. 22.
    R. Hill, The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950.Google Scholar
  23. 23.
    D.M. Marsh, “Plastic flow in glass,” Proc. Roy. Soc. A279, 1964, pp. 420–435.Google Scholar
  24. 24.
    L.E. Samuels and T.O. Mulheam, “An experimental investigation of the deformed zone associated with indentation hardness impressions,” J. Mech. Phys. Solids, 5, 1957, pp. 125–134.Google Scholar
  25. 25.
    T.O. Mulheam, “The deformation of metals by Vickers-type pyramidal indenters,” J. Mech. Phys. Solids, 7, 1959, pp. 85–96.Google Scholar
  26. 26.
    K.L. Johnson, “The correlation of indentation experiments,” J. Mech. Phys. Sol. 18, 1970, pp. 115–126.Google Scholar
  27. 27.
    M.C. Shaw and D.J. DeSalvo, “A new approach to plasticity and its application to blunt two dimension indenters,” J. Eng. hid. Trans. ASME, 92, 1970, pp. 469–479.Google Scholar
  28. 28.
    M.C. Shaw and D.J. DeSalvo, “On the plastic flow beneath a blunt axisymmetric indenter,” J. Eng. hid. Trans. ASME 92, 1970, pp. 480–494.Google Scholar
  29. 29.
    C. Hardy, C.N. Baronet, and G.V. Tordion, “The elastic-plastic indentation of a half-space by a rigid sphere,” Int. J. Numer. Methods Eng. 3, 1971, pp. 451–462.Google Scholar
  30. 30.
    C.M. Perrott, “Elastic-plastic indentation: Hardness and fracture,” Wear 45, 1977, pp. 293–309.Google Scholar
  31. 31.
    S.S. Chiang, D.B. Marshall, and A.G. Evans, “The response of solids to elastic/plastic indentation. 1. Stresses and residual stresses,” J. Appl. Phys. 53 1, 1982, pp. 298–311.CrossRefGoogle Scholar
  32. 32.
    S.S. Chiang, D.B. Marshall, and A.G. Evans, “The response of solids to elastic/plastic indentation. 2. Fracture initiation,” J. Appl. Phys. 53 1, 1982, pp. 312317.Google Scholar
  33. 33.
    K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985.Google Scholar
  34. 34.
    J.H. Ahn and D. Kwon, “Derivation of plastic stress-strain relationship from ball indentations: Examination of strain definition and pileup effect,” J. Mater. Res. 16 11, 2001, pp. 3170–3178.CrossRefGoogle Scholar
  35. 35.
    J. Thum, D.J. Morris, and R.F. Cook, “Depth-sensing indentation at macroscopic dimensions,” J. Mater. Res. 17 10, 2002, pp. 2679–2690.Google Scholar
  36. 36.
    F. Frölich, P. Grau, and W. Grellmann, “Performance and analysis of recording microhardness tests,” Phys. Stat. Sol. (a), 42 1977, pp. 79–89.Google Scholar
  37. 37.
    J.B. Pethica, “Microhardness tests with penetration depths less than ion implanted layer thickness in ion implantation into metals,” Third International Conference on Modification of Surface Properties of Metals by Ion-Implantation, Manchester, England, 23–26, 1981, V. Ashworth et al. eds., Pergammon Press, Oxford, 1982, pp. 147–157.Google Scholar
  38. 38.
    J.S. Field, “Understanding the penetration resistance of modified surface layers,” Surface and Coatings Technology, 36, 1988, pp. 817–827.Google Scholar
  39. 39.
    N.A. Stilwell and D. Tabor, “Elastic recovery of conical indentations,” Phys. Proc. Soc. 78 2, 1961, pp. 169–179.CrossRefGoogle Scholar
  40. 40.
    R.W. Armstrong and W.H. Robinson, “Combined elastic and plastic deformation behaviour from a continuous indentation hardness test,” New Zealand Journal of Science, 17, 1974, pp. 429–433.Google Scholar
  41. 41.
    B.R. Lawn and V.R. Howes, “Elastic recovery at hardness indentations,” J. Mat. Sci. 16, 1981, pp. 2745–2752.Google Scholar
  42. 42.
    S.I. Bulychev, V.P. Alekhin, M. Kh. Shorshorov, and A.P. Ternorskii, “Determining Young’s modulus from the indenter penetration diagram,” Zavod. Lab. 41 9, 1975, pp. 11137–11140.Google Scholar
  43. 43.
    J.L. Loubet, J.M. Georges, O. Marchesini, and G. Meille, “Vicker’s indentation of magnesium oxide,” J. Tribol. 106, 1984, pp. 43–48.Google Scholar
  44. 44.
    M.F. Doerner and W.D. Nix, “A method for interpreting the data from depth-sensing indentation instruments,” J. Mater. Res. 1 4, 1986, pp. 601–609.CrossRefGoogle Scholar
  45. 45.
    W.C. Oliver and G.M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mater. Res. 7 4, 1992, pp. 1564–1583.CrossRefGoogle Scholar
  46. 46.
    T.J. Bell, A. Bendeli, J.S. Field, M.V. Swain, and E.G. Thwaite, “The determination of surface plastic and elastic properties by ultra-micro indentation,” Metrologia, 28, 1991, pp. 463–469.Google Scholar
  47. 47.
    J.S. Field and M.V. Swain, “A simple predictive model for spherical indentation,” J. Mater. Res. 8 2, 1993, pp. 297–306.CrossRefGoogle Scholar
  48. 48.
    A.C. Fischer-Cripps, “Study of analysis methods for depth-sensing indentation test data for spherical indenters,” J. Mater. Res. 16 6, 2001, pp. 1579–1584.CrossRefGoogle Scholar
  49. 49.
    A.G. Atkins, “Topics in indentation hardness,” Metal Science, 16, 1982, pp. 127137.Google Scholar
  50. 50.
    H.M. Pollock, “Nanoindentation”, ASM Handbook, Friction, Lubrication, and Wear Technology, 18, 1992, pp. 419–429.Google Scholar
  51. 51.
    J.L. Hay and G.M. Pharr, “Instrumented indentation testing,” ASM Handbook, Materials Testing and Evaluation, 8, 2000, pp. 232–243.Google Scholar
  52. 52.
    S.A. Syed, K.J. Wahl, and R.J. Colton, “Quantitative study of nanoscale contact and pre-contact mechanics using force modulation,” Mat. Res. Soc. Symp. Proc. 594, 2000, pp. 471–476.Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Anthony C. Fischer-Cripps
    • 1
  1. 1.CSIROLindfieldAustralia

Personalised recommendations