Abstract
Methods for analyzing nanoindentation load-displacement data to determine hardness and elastic modulus are based on analytical solutions for the indentation of an elastic half-space by rigid axisymmetric indenters. Careful examination of Sneddon’s solution for indentation by a rigid cone reveals several largely ignored features that have important implications for nanoindentation property measurement. Finite element and analytical results are presented that show corrections to Sneddon’s equations are needed if accurate results are to be obtained. Without the corrections, the equations underestimate the load and contact stiffness in a manner that leads to errors in the measured hardness and modulus, with the magnitudes of the errors depending on the angle of the indenter and Poisson’s ratio of the half-space. First order corrections are derived, and general implications for the interpretation of nanoindentation data are discussed.
Similar content being viewed by others
References
D. Newey, M. A. Wilkins, and H. M. Pollock, J. Phys. E: Sci. Instrum. 15, 119–122 (1982).
S. I. Bulychev, V. P. Alekhin, M. Kh. Shorshorov, A. P. Ternov-skii, and G. D. Shnyrev, Zavod. Lab. 41, 1137 (1975).
M. Kh. Shorshorov, S. I. Bulychev, and V. P. Alekhin, Sov. Phys. Dokl. 26, 769 (1981).
S. I. Bulychev and V. P. Alekhin, Zavod. Lab. 53, 76 (1987).
J. L. Loubet, J. M. Georges, O. Marchesini, and G. Meille, J. Tribology 106, 43 (1984).
F. Frolich, P. Grau, and W. Grellmann, Phys. Status Solidi 42, 79 (1977).
J. B. Pethica, R. Hutchings, and W. C. Oliver, Philos. Mag. A 48, 593 (1983).
D. Stone, W. R. LaFontaine, P. Alexopoulos, T-W. Wu, and C-Y. Li, J. Mater. Res. 3, 141 (1988).
W. C. Oliver and G. M. Pharr, J. Mater. Res. 7, 1564 (1992).
M. F. Doerner and W. D. Nix, J. Mater. Res. 1, 601 (1986).
J. S. Field and M. V. Swain, J. Mater. Res. 8, 297 (1993).
J. S. Field and M. V. Swain, J. Mater. Res. 10, 101 (1995).
G. M. Pharr, W. C. Oliver, and F. R. Brotzen, J. Mater. Res. 7, 613 (1992).
A. E. H. Love, Quart. J. Math. 10, 161 (1939).
A. E. H. Love, Philos. Trans. A 228, 377 (1929).
J. W. Harding and I. N. Sneddon, Proc. Cambridge Philos. Soc.41, 16 (1945).
I. N. Sneddon, Int. J. Eng. Sci. 3, 47 (1965).
I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951), pp. 450–467.
S. I. Bulychev, V. P. Alekhin, M. Kh. Shorshorov, A. P. Ternov-skii, and G. D. Shnyrev, Zavod. Lab. 41, 1137 (1975).
S. I. Bulychev, V. P. Alekhin, M. Kh. Shorshorov, and A. P. Ternovskii, Prob. Prochn. 9, 79 (1976).
C-M. Cheng and Y-T. Cheng, Appl. Phys. Lett. 71, 2623 (1997).
A. Bolshakov and G. M. Pharr, J. Mater. Res. 13, 1049 (1998).
A. Bolshakov, W. C. Oliver, and G. M. Pharr, in Thin Films: Stresses and Mechanical Properties VI, edited by W. W. Ger-berich, H. Gao, J-E. Sundgren, and S. P. Baker (Mater. Res. Soc. Symp. Proc. 436, Pittsburgh, PA, 1997), p. 141.
J. E. Ritter, T. J. Lardner, D. T. Madsen, and R. J. Gionazzo, in Materials in Microelectronic and Optoelectronic Packaging (The American Ceramic Society, Westerville, OH), p. 379.
H. Gao, C-H. Chiu, and J. Lee, Int. J. Solids Struct. 29, 2471 (1992).
Y-T. Cheng and C-M. Cheng, J. Appl. Phys. 84, 1284 (1998).
T. A. Larsen and J. C. Simo, J. Mater. Res. 7, 618 (1992).
A. Bolshakov and G. M. Pharr, in Thin Films: Stresses and Mechanical Properties VI, edited by W. W. Gerberich, H. Gao, J-E. Sundgren, and S. P. Baker (Mater. Res. Soc. Symp. Proc. 436, Pittsburgh, PA, 1997), p. 189.
J. Boussinesq, Applications des Potentiels a l’etude de equilibre dt du mouvement des solides elastiques (Gauthier-Villars, Paris, 1885).
E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon Press, Oxford, 1937).
I. W. Busbridge, Proc. London Math. Soc. 44, 115 (1938).
A. Bolshakov, Ph.D. Thesis, Rice University, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hay, J.C., Bolshakov, A. & Pharr, G.M. A critical examination of the fundamental relations used in the analysis of nanoindentation data. Journal of Materials Research 14, 2296–2305 (1999). https://doi.org/10.1557/JMR.1999.0306
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1557/JMR.1999.0306