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Indentation load–displacement curve, plastic deformation, and energy

  • J. Malzbender (a1) and G. de With (a1)


Various methods to access indentation data are considered on the basis of the load P–displacement h curve, its derivative, or its integral. This paper discusses and extends the various analytical models to estimate the indentation P–h curve, the slope, and the dissipated energy to aid the development of a concise methodology to analyze indentation data. Special consideration is given to the effect of pile-up and sink-in. Relationships for sharp and spherical indenters are presented and in addition for sharp indenters with a rounded tip. An overview over analytic expressions for the P–h curve is given and compared to finite element simulations and experimental data. An expression derived for the representative strain at the onset of yield under sharp and spherical indenters compares well with literature results. The effect of a rounded tip on the yielding under a sharp indenter is discussed. The ratio of loading to unloading slope and the ratio of the plastically dissipated energy to the total energy is related to hardness and elastic modulus. In combination these ratios can be used to determine the strain-hardening coefficient.


Corresponding author

a) Address all correspondence to this author. Current address: Institute for Materials and Processes in Energy Systems, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany. e-mail:


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1.Pharr, G.M. and Oliver, W.C., MRS Bull. 7, 28 (1992).
2.Oliver, W.C. and Pharr, G.M., J. Mater. Res. 7, 1564 (1992).
3.Chechenin, N.G., Boettiger, J., and Krog, J.P., Thin Solid Films 261, 219 (1995).
4.Chechenin, N.G., Boettiger, J., and Krog, J.P., Thin Solid Films 261, 228 (1995).
5.Chechenin, N.G., Thin Solid Films 304, 78 (1997).
6.Chen, X. and Vlassak, J.J., J. Mater. Res. 16, 2974 (2001).
7.Malzbender, J., With, G. de, and Toonder, J.M.J. den, Thin Solid Films 372, 134 (2000).
8.Spence, D.A., J. Elast. 5, 297 (1975).
9.Fischer-Cripps, A.C. and Collins, R.E., J. Mater. Sci. 29, 2216 (1994).
10.Field, J.S. and Swain, M.V., J. Mater. Res. 8, 297 (1993).
11.Korunsky, A.M., J. Strain Anal. 34, 391 (2001).
12.Hainsworth, S.V., Chandler, H.W., and Page, T.F., J. Mater. Res. 11, 1987 (1995).
13.Larsson, P-L., Giannakopoulos, A.E., Soederlund, E., Rowcliffe, D.J., and Vestergaard, R., Int. J. Solids Struct. 33, 221 (1996).
14.Cheng, Y-T. and Cheng, C-M., J. Mater. Res. 14, 3493 (1999).
15.Storakers, B. and Larsson, P.L., J. Mech. Phys. Solids 42, 307 (1994).
16.Hill, R., Storakers, B., and Zdunek, A.B., Proc. Roy. Soc. London, A 423, 301 (1989).
17.Bower, A.F., Fleck, N.A., Needleman, A., and Ogbonna, N., Proc. R. Soc. London, A 41, 97 (1993).
18.Dao, M., Chollacoop, N., Vliet, K.J. van, Venkatesh, T.A., and Suresh, S., Acta Mater. 49, 3899 (2001).
19.Malzbender, J., Toonder, J.M.J. den, and With, G. de, J. Mater. Res. 5, 1209 (2000).
20.Malzbender, J. and With, G. de, Surf. Coat. Technol. 127, 266 (2000).
21.Malzbender, J. and With, G. de, Surf. Coat. Technol. 137, 72 (2001).
22.Hertz, H., Miscellaneous Papers (MacMillan, London, United Kingdom, 1896).
23.Fischer-Cripps, A.C., Introduction to Contact Mechanics (Springer, Berlin, Germany, 2000).
24.Chaudhri, M.M., J. Mater. Res. 16, 336 (2001).
25.Lo, R.Y. and Bogy, D.B., J. Mater. Res. 14, 2276 (1999).
26.Fischer-Cripps, A.C., J. Mater. Res. 16, 3050 (1999).
27.Hay, J.C., Bolshakov, A., and Pharr, G.M., J. Mater. Res. 14, 2296 (1999).
28.Hay, J.L. and Wolff, P.J., J. Mater. Res. 16, 1280 (2001).
29.Cheng, Y-T. and Cheng, C-M., Surf. Coat. Technol. 417, 133134 (2000).
30.Alcala, J., Baronr, A.C., and Anglada, M., Acta Mater. 48, 3451 (2000).
31.Malzbender, J. and With, G. de, Surf. Coat. Technol. 135, 60 (2000).
32.Taljat, B. and Zacharia, T., Int. J. Solids Struct. 35, 4411 (1998).
33.Johnson, K.L., Contact Mechanics (Cambridge University Press, Cambridge, United Kingdom, 1985).
34.Fischer-Cripps, A.C., J. Mater. Sci. 34, 129 (1999).
35.Marsh, D.M., Proc. Roy. Soc. A279, 420 (1964).
36.Hirst, W. and Howse, M.G.J.W., Proc. Roy. Soc. A311, 429 (1969).
37.Giannakopoulos, A.E., Larsson, P-L., and Vestergaard, R., Int. J. Solids Struct. 31, 2679 (1994).
38.Cheng, Y-T. and Cheng, C-M., Philos. Mag. Lett. 77, 39 (1998).
39.Shimamoto, A., Tanaka, K., Akiyama, Y., and Yoshizake, H., Philos. Mag. A 74, 1097 (1996).
40.Murakami, Y., Tanaka, K., Itokatu, M., and Shimamoto, A., Philos. Mag. A 69, 1131 (1994).
41.Sneddon, I.N., Int. J. Eng. Sci. 3, 47 (1965).
42.Cheng, Y-T. and Cheng, C-M., J. Mater. Res. 13, 1059 (1998).
43.Larsson, P-L., Int. J. Mech. Sci. 43, 895 (2001).
44.Tabor, D., The Hardness of Metals (Oxford, New York, 1951).
45.Shorshorov, M.K., Bulychev, S.I., and Alekin, V.P., Sov. Phys. Dokl. 26, 769 (1981).
46.Cheng, Y-T. and Cheng, C-M., Appl. Phys. Lett. 73, 614 (1998).
47.Cheng, Y-T., Li, Z., and Cheng, C-M., in Fundamentals of Nanoindentation and Nanotribology II, edited by Baker, S.P., Cook, R.F., Corcoran, S.G., and Moody, N.R. (MRS Proc. 649, Warrendale, PA, 2001), p. Q1.1.
48.Milman, Y.V., Galanov, B.A., and Chugunova, S.I., Acta Metall. Mater. 41, 2523 (1993).
49.Marx, V. and Balke, H., Acta Mater. 45, 3791 (1997).
50.Venkatesh, T.A., Vleit, K.J. van, Ginnakopoulos, A.E., and Suresh, S., Scr. Mater. 42, 833 (2000).
51.Bulychev, S.I., Tech. Phys. 44, 775 (1999).
52.Sakai, M., Acta Metall. Mater. 41, 1751 (1993).
53.Tuck, J.R., Korunsky, A.M., Bull, S.J., and Davidson, R.I., Surf. Coat. Technol. 137, 217 (2001).
54.Suresh, S. and Giannakopoulos, A.E., Acta Mater. 46, 5755 (1998).
55.Swadener, J.G., Taljat, B., and Pharr, G.M., J. Mater. Res. 16, 2091 (2001).
56.Carlsson, S. and Larsson, P.L., Acta Mater. 49, 2179 (2001).


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