On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation


Results of Sneddon’s analysis for the elastic contact between a rigid, axisymmetric punch and an elastic half space are used to show that a simple relationship exists among the contact stiffness, the contact area, and the elastic modulus that is not dependent on the geometry of the punch. The generality of the relationship has important implications for the measurement of mechanical properties using load and depth sensing indentation techniques and in the measurement of small contact areas such as those encountered in atomic force microscopy.

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  1. 1.

    J. B. Pethica, R. Hutchings, and W. C. Oliver, Philos. Mag. A 48, 593 (1983).

    CAS  Article  Google Scholar 

  2. 2.

    W. C. Oliver, R. Hutchings, and J. B. Pethica, in ASTM STP 889, edited by P. J. Blau and B. R. Lawn (American Society for Testing and Materials, Philadelphia, PA, 1986), pp. 90–108.

  3. 3.

    M. F. Doerner and W. D. Nix, J. Mater. Res. 1, 601 (1986).

    Article  Google Scholar 

  4. 4.

    J. L. Loubet, J. M. Georges, O. Marchesini, and G. Meille, J. Tribology 106, 43 (1984).

    CAS  Article  Google Scholar 

  5. 5.

    S. I. Bulychev, V. P. Alekhin, M. Kh. Shorshorov, A. P. Ternovskii, and G. D. Shnyrev, Zavod. Lab. 41, 1137 (1975).

    CAS  Google Scholar 

  6. 6.

    D. Newey, M. A. Wilkens, and H. M. Pollock, J. Phys. E: Sci. Instrum. 15, 119 (1982).

    CAS  Article  Google Scholar 

  7. 7.

    D. Stone, W. R. LaFontaine, P. Alexopoulos, T-W. Wu, and Che-Yu Li, J. Mater. Res. 3, 141 (1988).

    CAS  Article  Google Scholar 

  8. 8.

    M. Kh. Shorshorov, S. I. Bulychev, and V. P. Alekhin, Sov. Phys. Sokl. 26, 769 (1982).

    Google Scholar 

  9. 9.

    N. Gane and J. M. Cox, Philos. Mag. 22, 881 (1970).

    CAS  Article  Google Scholar 

  10. 10.

    I. N. Sneddon, Int. J. Engng. Sci. 3, 47 (1965).

    Article  Google Scholar 

  11. 11.

    N. A. Stillwell and D. Tabor, Proc. Phys. Soc. Lond. 78, 169 (1961).

    Article  Google Scholar 

  12. 12.

    W. C. Oliver and G. M. Pharr, in preparation.

  13. 13.

    R. B. King, Int. J. Solids Structures 23, 1657 (1987).

    Article  Google Scholar 

  14. 14.

    D. Tabor, Proc. Roy. Soc. A 192, 247 (1948).

    Google Scholar 

  15. 15.

    K. L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1985).

    Google Scholar 

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Pharr, G.M., Oliver, W.C. & Brotzen, F.R. On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. Journal of Materials Research 7, 61 (1992). https://doi.org/10.1557/JMR.1992.0613

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