Hostname: page-component-f554764f5-nt87m Total loading time: 0 Render date: 2025-04-12T23:42:22.885Z Has data issue: false hasContentIssue false
  • English
  • Français

Indenter tip radius effect on the Nix–Gao relation in micro- and nanoindentation hardness experiments

Published online by Cambridge University Press:  01 November 2004

S. Qu ,
Y. Huang ,
W.D. Nix ,
H. Jiang ,
F. Zhang  and
K.C. Hwang
S. Qu
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois 61801
Y. Huang*
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois 61801
W.D. Nix
Affiliation:
Department of Material Science and Engineering, Stanford University, Stanford, California 94305
H. Jiang
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois 61801
F. Zhang
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
K.C. Hwang
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
*
a) Address all correspondence to this author. e-mail: [email protected]
Get access

Abstract

Nix and Gao established an important relation between microindentation hardnessand indentation depth. Such a relation has been verified by many microindentation experiments (indentation depths in the micrometer range), but it does not always hold in nanoindentation experiments (indentation depths approaching the nanometer range). We have developed a unified computational model for both micro- and nanoindentation in an effort to understand the breakdown of the Nix–Gao relation at indentation depths approaching the nanometer scale. The unified computational model for indentation accounts for various indenter shapes, including a sharp, conical indenter, a spherical indenter, and a conical indenter with a spherical tip. It is based on the conventional theory of mechanism-based strain gradient plasticity established from the Taylor dislocation model to account for the effect of geometrically necessary dislocations. The unified computational model for indentation indeed shows that the Nix–Gao relation holds in microindentation with a sharp indenter, but it does not hold in nanoindentation due to the indenter tip radius effect.

Keywords

HardnessMicroindentationNanoindentation
Type
Articles
Information
Journal of Materials Research , Volume 19 , Issue 11 , November 2004 , pp. 3423 - 3434
Copyright
Copyright © Materials Research Society 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

1Nix, W.D.: Mechanical properties of thin films. Mater. Trans. 20A, 2217 (1989).Google Scholar
2Oliver, W.C. and Pharr, G.M.: An improved technique for determining hardness and elastic modulus using loading and displacement sensing indentation. J. Mater. Res. 7, 1564 (1992).CrossRefGoogle Scholar
3de Guzman, M.S., Neubauer, G., Filnn, P.A. and Nix, W.D.: The role of indentation depth on the measured hardness of materials, in Thin Films: Stresses and Mechanical Properties IV, edited by Townsend, P.H., Weihs, T.P., Sanchez, J.E. Jr., and Borgensen, P. (Mater. Res. Soc. Symp. Proc. 308, Pittsburgh, PA, 1993) p. 613.Google Scholar
4Stelmashenko, N.A., Walls, A.G., Brown, L.M. and Milman, Y.V.: Microindentation on W and Mo oriented single crystals: An STM study. Acta Metall. Mater. 41, 2855 (1993).Google Scholar
5Atkinson, M.: Further analysis of the size effect in indentation hardness tests of some metals. J. Mater. Res. 10, 2908 (1995).Google Scholar
6Ma, Q. and Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853 (1995).CrossRefGoogle Scholar
7Poole, W.J., Ashby, M.F. and Fleck, N.A.: Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Mater. 34, 559 (1996).CrossRefGoogle Scholar
8McElhaney, K.W., Vlasssak, J.J. and Nix, W.D.: Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments. J. Mater. Res. 13, 1300 (1998).Google Scholar
9Suresh, S., Nieh, T.G. and Choi, B.W.: Nano-indentation of copper thin films on silicon substrates. Scripta Mater. 41, 951 (1999).Google Scholar
10Zagrebelny, A.V., Lilleodden, E.T., Gerberich, W.W. and Carter, C.B.: Indentation of silicate-glass films on Al2O3 substrates. J. Am. Ceram. Soc. 82, 1803 (1999).CrossRefGoogle Scholar
11Taylor, G.I.: The mechanism of plastic deformation of crystals. Part I.–theoretical. Proc. R. Soc. London A. 145, 362 (1934).Google Scholar
12Taylor, G.I.: Plastic strain in metals. J. Inst. Metals 62, 307 (1938).Google Scholar
13Nix, W.D. and Gao, H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411 (1998).Google Scholar
14Lim, Y.Y. and Chaudhri, M.M.: The effect of the indenter load on the nanohardness of ductile metals: An experimental study on polycrystalline work-hardened and annealed oxygen-free copper. Philos. Mag. A 79, 2879 (1999).Google Scholar
15Swadener, J.G., George, E.P. and Pharr, G.M.: The correlation of the indentation size effect measured with indenters of various shapes. J. Mech. Phys. Solids 50, 681 (2002).CrossRefGoogle Scholar
16Feng, G. and Nix, W.D.: Indentation size effect in MgO. Scripta Mater. 51, 599 (2004).Google Scholar
17Elmustafa, A.A. and Stone, D.S.: Nanoindentation and the indentation size effect: Kinetics of deformation and strain gradient plasticity. J. Mech. Phys. Solids 51, 357 (2003).Google Scholar
18Lim, Y.Y., Bushby, A.J. and Chaudhri, M.M.: Nano and macro indentation studies of polycrystalline copper using spherical indenters, in Fundamentals of Nanoindentation and Nanotribology, edited by Moody, N.R., Gerberich, W.W., Burnham, N., and Baker, S.P. (Mater. Res. Soc. Symp. Proc. 522, Warrendale, PA, 1998) p. 145.Google Scholar
19Tymiak, N.I., Kramer, D.E., Bahr, D.F., Wyrobek, T.J. and Gerberich, W.W.: Plastic strain and strain gradients at very small indentation depths. Acta Mater. 49, 1021 (2001).CrossRefGoogle Scholar
20Iwashita, N. and Swain, M.V.: Elasto-plastic deformation of silica glass and glassy carbons with different indenters. Philos. Mag. A 82, 2199 (2002).CrossRefGoogle Scholar
21Kucheyev, S.O., Bradby, J.E., Williams, J.S. and Jagadish, C.: Mechanical deformation of single-crystal ZnO. Appl. Phys. Lett. 80, 956 (2002).Google Scholar
22Huang, Y., Qu, S., Hwang, K.C., Li, M. and Gao, H.: A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plast. 20, 753 (2004).CrossRefGoogle Scholar
23Bailey, J.E. and Hirsch, P.B.: The dislocation distribution, flow stress, and stored energy in cold-worked polycrystalline silver. Philos. Mag. 5, 485 (1960).CrossRefGoogle Scholar
24Ashby, M.F.: The deformation of plastically non-homogeneous alloys. Philos. Mag. 21, 399 (1970).CrossRefGoogle Scholar
25Nye, J.F.: Some geometrical relations in dislocated crystals. Acta Metall. Mater. 1, 153 (1953).CrossRefGoogle Scholar
26Cottrell, A.H.: The Mechanical Properties of Materials (J. Willey, New York, 1964), p. 277.Google Scholar
27Arsenlis, A. and Parks, D.M.: Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater. 47, 1597 (1999).Google Scholar
28Shi, M., Huang, Y. and Gao, H.: The J-integral and geometrically necessary dislocations in nonuniform plastic deformation. Int. J. Plast. 20, 1739 (2004).CrossRefGoogle Scholar
29Bishop, J.F.W. and Hill, R.: A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philos. Mag. 42, 414 (1951).Google Scholar
30Bishop, J.F.W. and Hill, R.: A theoretical derivation of the plastic properties of a polycrystalline face-centered metal. Philos. Mag. 42, 1298 (1951).Google Scholar
31Kocks, U.F.: The relation between polycrystal deformation and single crystal deformation. Metall. Mater. Trans. 1, 1121 (1970).Google Scholar
32Hutchinson, J.W.: Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. R. Soc. London A 348, 101 (1976).Google Scholar
33Canova, G.R. and Kocks, U.F.: In Seventh International Conference on Textures of Materials, edited by Brakman, C.M., Jongenburger, P., and Mittemeijer, E.J. (Soc. Mater. Sci., The Netherlands, 1984), p. 573.Google Scholar
34Asaro, R.J. and Needleman, A.: Texture development and strain hardening in rate dependent polycrystals. Acta Metal. Mater. 33, 923 (1985).Google Scholar
35Kok, S., Beaudoin, A.J. and Tortorelli, D.A.: A polycrystal plasticity model based on the mechanical threshold. Int. J. Plast. 18, 715 (2002).Google Scholar
36Kok, S., Beaudoin, A.J. and Tortorelli, D.A.: On the development of stage IV hardening using a model based on the mechanical threshold. Acta Mater. 50, 1653 (2002).Google Scholar
37Kok, S., Beaudoin, A.J., Tortorelli, D.A. and Lebyodkin, M.: A finite element model for the Portevin-Le Chatelier effect based on polycrystal plasticity. Modell. Simul. Mater. Sci. Eng. 10, 745 (2002).CrossRefGoogle Scholar
38Acharya, A. and Bassani, J.L.: Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys. Solids 48, 1565 (2000).CrossRefGoogle Scholar
39Acharya, A. and Beaudoin, A.J.: Grain-size effect in viscoplastic polycrystals at moderate strains. J. Mech. Phys. Solids 48, 2213 (2000).CrossRefGoogle Scholar
40Evers, L.P., Parks, D.M., Brekelmans, W.A.M. and Geers, M.G.D.: Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. J. Mech. Phys. Solids 50, 2403 (2002).CrossRefGoogle Scholar
41Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W.: Mechanism-based strain gradient plasticity–I. Theory. J. Mech. Phys. Solids 47, 1239 (1999).CrossRefGoogle Scholar
42Huang, Y., Gao, H., Nix, W.D. and Hutchinson, J.W.: Mechanism-based strain gradient plasticity–II. Analysis. J. Mech. Phys. Solids 48, 99 (2000).CrossRefGoogle Scholar
43Huang, Y., Xue, Z., Gao, H., Nix, W.D. and Xia, Z.C.: A study of microindentation hardness tests by mechanism-based strain gradient plasticity. J. Mater. Res. 15, 1786 (2000).Google Scholar
44Shi, M., Huang, Y., Jiang, H., Hwang, K.C. and Li, M.: The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity. Int. J. Fracture 112, 23 (2001).Google Scholar
45Fleck, N.A. and Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825 (1993).Google Scholar
46Fleck, N.A. and Hutchinson, J.W. in Advances in Applied Mechanics 33, edited by Hutchinson, J.W. and Wu, T.Y. (Academic Press, New York, 1997), p. 295.CrossRefGoogle Scholar
47Fleck, N.A. and Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245 (2001).Google Scholar
48Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50 5 (2002).CrossRefGoogle Scholar
49Hibbitt, Sorenson, Karlsson &Inc., ABAQUS/Standard user's manual version 6.2. (2001).Google Scholar
50Qu, S., Huang, Y., Jiang, H., Liu, C., Wu, P.D. and Hwang, K.C.: Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity. Int. J. Fracture 2004 (in press).CrossRefGoogle Scholar
51Begley, M.R. and Hutchinson, J.W.: The mechanics of size-dependent indentation. J. Mech. Phys. Solids. 46, 2049 (1998).CrossRefGoogle Scholar
52Xue, Z., Huang, Y., Hwang, K.C. and Li, M.: The influence of indenter tip radius on the micro-indentation hardness. J. Eng. Mater. Technol. 124, 371 (2002).Google Scholar
53Wei, Y. and Hutchinson, J.W.: Hardness trends in micron scale indentation. J. Mech. Phys. Solids 51, 2037 (2003).Google Scholar
54Shu, J.Y. and Fleck, N.A.: Strain gradient crystal plasticity: Size-dependent deformation of bicrystals. J. Mech. Phys. Solids 47, 297 (1999).CrossRefGoogle Scholar
55McLean, D.: Mechanical Properties of Metals (John Wiley & Sons, New York, 1962).Google Scholar
56Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W.: Strain gradient plasticity: Theory and experiments. Acta Metall. Mater. 42, 475 (1994).CrossRefGoogle Scholar
57Qiu, X., Huang, Y., Wei, Y., Gao, H. and Hwang, K.C.: The flow theory of mechanism-based strain gradient plasticity. Mech. Mater. 35, 245 (2003).Google Scholar