Skip to main content
SpringerLink
Log in

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

  • Article
  • Published:
Journal of Materials Research Aims and scope Submit manuscript

Abstract

Nix and Gao established an important relation between microindentation hardness and 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W.D. Nix: Mechanical properties of thin films. Mater. Trans. 20A, 2217 (1989).

    CAS  Google Scholar 

  2. W.C. Oliver and G.M. Pharr: An improved technique for determining hardness and elastic modulus using loading and displacement sensing indentation. J. Mater. Res. 7, 1564 (1992).

    Article  CAS  Google Scholar 

  3. M.S. de Guzman, G. Neubauer, P.A. Filnn, and W.D. Nix: The role of indentation depth on the measured hardness of materials, in Thin Films: Stresses and Mechanical Properties IV, edited by P.H. Townsend, T.P. Weihs, J.E. Sanchez, Jr., and P. Borgensen (Mater. Res. Soc. Symp. Proc. 308, Pittsburgh, PA, 1993) p. 613.

    Google Scholar 

  4. N.A. Stelmashenko, A.G. Walls, L.M. Brown, and Y.V. Milman: Microindentation on W and Mo oriented single crystals: An STM study. Acta Metall. Mater. 41, 2855 (1993).

    Article  CAS  Google Scholar 

  5. M. Atkinson: Further analysis of the size effect in indentation hardness tests of some metals. J. Mater. Res. 10, 2908 (1995).

    Article  CAS  Google Scholar 

  6. Q. Ma and D.R. Clarke: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853 (1995).

    Article  CAS  Google Scholar 

  7. W.J. Poole, M.F. Ashby, and N.A. Fleck: Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Mater. 34, 559 (1996).

    Article  CAS  Google Scholar 

  8. K.W. McElhaney, J.J. Vlasssak, and W.D. Nix: Determination of indenter tip geometry and indentation contact area for depthsensing indentation experiments. J. Mater. Res. 13, 1300 (1998).

    Article  CAS  Google Scholar 

  9. S. Suresh, T.G. Nieh, and B.W. Choi: Nano-indentation of copper thin films on silicon substrates. Scripta Mater. 41, 951 (1999).

    Article  CAS  Google Scholar 

  10. A.V. Zagrebelny, E.T. Lilleodden, W.W. Gerberich, and C.B. Carter: Indentation of silicate-glass films on Al2O3 substrates. J. Am. Ceram. Soc. 82, 1803 (1999).

    Article  CAS  Google Scholar 

  11. G.I. Taylor: The mechanism of plastic deformation of crystals. Part I.–theoretical. Proc. R. Soc. London A. 145, 362 (1934).

    Article  CAS  Google Scholar 

  12. G.I. Taylor: Plastic strain in metals. J. Inst. Metals 62, 307 (1938).

    Google Scholar 

  13. W.D. Nix and H. Gao: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411 (1998).

    Article  CAS  Google Scholar 

  14. Y.Y. Lim and M.M. Chaudhri: 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).

    Article  Google Scholar 

  15. J.G. Swadener, E.P. George, and G.M. Pharr: The correlation of the indentation size effect measured with indenters of various shapes. J. Mech. Phys. Solids 50, 681 (2002).

    Article  Google Scholar 

  16. G. Feng and W.D. Nix: Indentation size effect in MgO. Scripta Mater. 51, 599 (2004).

    Article  CAS  Google Scholar 

  17. A.A. Elmustafa and D.S. Stone: Nanoindentation and the indentation size effect: Kinetics of deformation and strain gradient plasticity. J. Mech. Phys. Solids 51, 357 (2003).

    Article  CAS  Google Scholar 

  18. Y.Y. Lim, A.J. Bushby, and M.M. Chaudhri: Nano and macro indentation studies of polycrystalline copper using spherical indenters, in Fundamentals of Nanoindentation and Nanotribology, edited by N.R. Moody, W.W. Gerberich, N. Burnham, and S.P. Baker (Mater. Res. Soc. Symp. Proc. 522, Warrendale, PA, 1998) p. 145.

    Google Scholar 

  19. N.I. Tymiak, D.E. Kramer, D.F. Bahr, T.J. Wyrobek, and W.W. Gerberich: Plastic strain and strain gradients at very small indentation depths. Acta Mater. 49, 1021 (2001).

    Article  CAS  Google Scholar 

  20. N. Iwashita and M.V. Swain: Elasto-plastic deformation of silica glass and glassy carbons with different indenters. Philos. Mag. A 82, 2199 (2002).

    Article  CAS  Google Scholar 

  21. S.O. Kucheyev, J.E. Bradby, J.S. Williams, and C. Jagadish: Mechanical deformation of single-crystal ZnO. Appl. Phys. Lett. 80, 956 (2002).

    Article  CAS  Google Scholar 

  22. Y. Huang, S. Qu, K.C. Hwang, M. Li, and H. Gao: A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plast. 20, 753 (2004).

    Article  Google Scholar 

  23. J.E. Bailey and P.B. Hirsch: The dislocation distribution, flow stress, and stored energy in cold-worked polycrystalline silver. Philos. Mag. 5, 485 (1960).

    Article  CAS  Google Scholar 

  24. M.F. Ashby: The deformation of plastically non-homogeneous alloys. Philos. Mag. 21, 399 (1970).

    Article  CAS  Google Scholar 

  25. J.F. Nye: Some geometrical relations in dislocated crystals. Acta Metall. Mater. 1, 153 (1953).

    Article  CAS  Google Scholar 

  26. A.H. Cottrell: The Mechanical Properties of Materials (J. Willey, New York, 1964), p. 277.

    Google Scholar 

  27. A. Arsenlis and D.M. Parks: Crystallographic aspects of geometrically- necessary and statistically-stored dislocation density. Acta Mater. 47, 1597 (1999).

    Article  CAS  Google Scholar 

  28. M. Shi, Y. Huang, and H. Gao: The J-integral and geometrically necessary dislocations in nonuniform plastic deformation. Int. J. Plast. 20, 1739 (2004).

    Article  Google Scholar 

  29. J.F.W. Bishop and R. Hill: A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philos. Mag. 42, 414 (1951).

    Article  CAS  Google Scholar 

  30. J.F.W. Bishop and R. Hill: A theoretical derivation of the plastic properties of a polycrystalline face-centered metal. Philos. Mag. 42, 1298 (1951).

    Article  CAS  Google Scholar 

  31. U.F. Kocks: The relation between polycrystal deformation and single crystal deformation. Metall. Mater. Trans. 1, 1121 (1970).

    Article  Google Scholar 

  32. J.W. Hutchinson: Bounds and self-consistent estimates for creep of polycrystalline materials. Proc. R. Soc. London A 348, 101 (1976).

    Article  CAS  Google Scholar 

  33. G.R. Canova and U.F. Kocks: In Seventh International Conference on Textures of Materials, edited by C.M. Brakman, P. Jongenburger, and E.J. Mittemeijer (Soc. Mater. Sci., The Netherlands, 1984), p. 573.

  34. R.J. Asaro and A. Needleman: Texture development and strain hardening in rate dependent polycrystals. Acta Metal. Mater. 33, 923 (1985).

    Article  CAS  Google Scholar 

  35. S. Kok, A.J. Beaudoin, and D.A. Tortorelli: A polycrystal plasticity model based on the mechanical threshold. Int. J. Plast. 18, 715 (2002).

    Article  Google Scholar 

  36. S. Kok, A.J. Beaudoin, and D.A. Tortorelli: On the development of stage IV hardening using a model based on the mechanical threshold. Acta Mater. 50, 1653 (2002).

    Article  CAS  Google Scholar 

  37. S. Kok, A.J. Beaudoin, D.A. Tortorelli, and M. Lebyodkin: A finite element model for the Portevin-Le Chatelier effect based on polycrystal plasticity. Modell. Simul. Mater. Sci. Eng. 10, 745 (2002).

    Article  CAS  Google Scholar 

  38. A. Acharya and J.L. Bassani: Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys. Solids 48, 1565 (2000).

    Article  Google Scholar 

  39. A. Acharya and A.J. Beaudoin: Grain-size effect in viscoplastic polycrystals at moderate strains. J. Mech. Phys. Solids 48, 2213 (2000).

    Article  Google Scholar 

  40. L.P. Evers, D.M. Parks, W.A.M. Brekelmans, and M.G.D. Geers: Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. J. Mech. Phys. Solids 50, 2403 (2002).

    Article  CAS  Google Scholar 

  41. H. Gao, Y. Huang, W.D. Nix, and J.W. Hutchinson: Mechanismbased strain gradient plasticity–I. Theory. J. Mech. Phys. Solids 47, 1239 (1999).

    Article  Google Scholar 

  42. Y. Huang, H. Gao, W.D. Nix, and J.W. Hutchinson: Mechanismbased strain gradient plasticity–II. Analysis. J. Mech. Phys. Solids 48, 99 (2000).

    Article  Google Scholar 

  43. Y. Huang, Z. Xue, H. Gao, W.D. Nix, and Z.C. Xia: A study of microindentation hardness tests by mechanism-based strain gradient plasticity. J. Mater. Res. 15, 1786 (2000).

    Article  CAS  Google Scholar 

  44. M. Shi, Y. Huang, H. Jiang, K.C. Hwang, and M. Li: The boundary- layer effect on the crack tip field in mechanism-based strain gradient plasticity. Int. J. Fracture 112, 23 (2001).

    Article  Google Scholar 

  45. N.A. Fleck and J.W. Hutchinson: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825 (1993).

    Article  Google Scholar 

  46. N.A. Fleck and J.W. Hutchinson: In Advances in Applied Mechanics 33, edited by J.W. Hutchinson and T.Y. Wu (Academic Press, New York, 1997), p. 295.

  47. N.A. Fleck and J.W. Hutchinson: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245 (2001).

    Article  Google Scholar 

  48. M.E. Gurtin: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5 (2002).

    Article  Google Scholar 

  49. Hibbitt, Karlsson & Sorenson, Inc.: ABAQUS/Standard user’s manual version 6.2. (2001).

    Google Scholar 

  50. S. Qu, Y. Huang, H. Jiang, C. Liu, P.D. Wu, and K.C. Hwang: Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity. Int. J. Fracture (2004) (in press).

    Google Scholar 

  51. M.R. Begley and J.W. Hutchinson: The mechanics of sizedependent indentation. J. Mech. Phys. Solids. 46, 2049 (1998).

    Article  CAS  Google Scholar 

  52. Z. Xue, Y. Huang, K.C. Hwang, and M. Li: The influence of indenter tip radius on the micro-indentation hardness. J. Eng. Mater. Technol. 124, 371 (2002).

    Article  Google Scholar 

  53. Y. Wei and J.W. Hutchinson: Hardness trends in micron scale indentation. J. Mech. Phys. Solids 51, 2037 (2003).

    Article  Google Scholar 

  54. J.Y. Shu and N.A. Fleck: Strain gradient crystal plasticity: Sizedependent deformation of bicrystals. J. Mech. Phys. Solids 47, 297 (1999).

    Article  Google Scholar 

  55. D. McLean: Mechanical Properties of Metals (John Wiley & Sons, New York, 1962).

    Google Scholar 

  56. N.A. Fleck, G.M. Muller, M.F. Ashby, and J.W. Hutchinson: Strain gradient plasticity: Theory and experiments. Acta Metall. Mater. 42, 475 (1994).

    Article  CAS  Google Scholar 

  57. X. Qiu, Y. Huang, Y. Wei, H. Gao, and K.C. Hwang: The flow theory of mechanism-based strain gradient plasticity. Mech. Mater. 35, 245 (2003).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois, 61801, USA

    S. Qu, H. Jiang & Y. Huang

  2. Department of Material Science and Engineering, Stanford University, Stanford, California, 94305, USA

    W. D. Nix

  3. Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

    F. Zhang & K. C. Hwang

Authors
  1. S. Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. D. Nix
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. F. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. K. C. Hwang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Y. Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Qu, S., Nix, W.D., Jiang, H. et al. Indenter tip radius effect on the Nix–Gao relation in micro- and nanoindentation hardness experiments. Journal of Materials Research 19, 3423–3434 (2004). https://doi.org/10.1557/JMR.2004.0441

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1557/JMR.2004.0441

Search

Navigation