Acta Mechanica Solida Sinica

, Volume 22, Issue 1, pp 18–26 | Cite as

A numerical study of indentation with small spherical indenters

Article

Abstract

The finite element method for the conventional theory of mechanism-based strain gradient plasticity is used to study the indentation size effect. For small indenters (e.g., radii on the order of 10 µm), the maximum allowable geometrically necessary dislocation (GND) density is introduced to cap the GND density such that the latter does not become unrealistically high. The numerical results agree well with the indentation hardness data of iridium. The GND density is much larger than the density of statistically stored dislocations (SSD) underneath the indenter, but this trend reverses away from the indenter. As the indentation depth (or equivalently, contact radius) increases, the GND density decreases but the SSD density increases.

Key words

indentation size effect spherical indenters geometrically necessary dislocations maximum density 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Tymiak, 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 Materialia, 2001, 49: 1021–1034.CrossRefGoogle Scholar
  2. 2.
    Swadener, J.G., Taljat, B. and Pharr, G.M., Measurement of residual stress by load and depth sensing indentation with spherical indenters. Journal of Materials Research, 2001, 16: 2091–2102.CrossRefGoogle Scholar
  3. 3.
    Swadener, J.G., George, E.P. and Pharr, G.M., The correlation of the indentation size effect measured with indenters of various shapes. Journal of the Mechanics and Physics of Solids, 2002, 50: 681–694.CrossRefGoogle Scholar
  4. 4.
    Nye, J.F, Some geometrical relations in dislocated crystals. Acta Metallurgica et Materialia, 1953, 1: 153–162.CrossRefGoogle Scholar
  5. 5.
    Ashby, M.F., The deformation of plastically non-homogeneous alloys. Philosophical Magazine, 1970, 21: 399–424.CrossRefGoogle Scholar
  6. 6.
    Gao, H. and Huang, Y., Geometrically necessary dislocation and size-dependent plasticity. Scripta Materialia, 2003, 48: 113–118.CrossRefGoogle Scholar
  7. 7.
    Xue, Z., Huang, Y., Hwang, K.C. and Li, M., The influence of indenter tip radius on the micro-indentation hardness. Journal of Engineering Materials and Technology, 2002, 124: 371–379.CrossRefGoogle Scholar
  8. 8.
    Qu, S., Huang, Y., Pharr, G.M. and Hwang, K.C., The indentation size effect in the spherical indentation of iridium, A study via the conventional theory of mechanism-based strain gradient plasticity. International Journal of Plasticity, 2006, 22: 1265–1286.CrossRefGoogle Scholar
  9. 9.
    Arsenlis, A. and Parks, D.M., Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Materialia, 1999, 47: 1597–1611.CrossRefGoogle Scholar
  10. 10.
    Shi, M., Huang, Y. and Gao, H., The J-integral and geometrically necessary dislocations in nonuniform plastic deformation. International Journal of Plasticity, 2004, 20: 1739–1762.CrossRefGoogle Scholar
  11. 11.
    Taylor, G.I., The mechanism of plastic deformation of crystals. Part I — theoretical. Proceedings of the Royal Society of London A, 1934, 145: 362–387.CrossRefGoogle Scholar
  12. 12.
    Taylor, G.I., Plastic strain in metals. Journal of the Institute of Metals, 1938, 62: 307–324.Google Scholar
  13. 13.
    Tabor, D., Hardness of Metals. Oxford: Clarendon Press, 1951.Google Scholar
  14. 14.
    Feng, G. and Nix, W.D., Indentation size effect in MgO. Scripta Materialia, 2004, 51: 599–603.CrossRefGoogle Scholar
  15. 15.
    Huang, Y., Zhang, F., Hwang, K.C., Nix, W.D., Pharr, G.M. and Feng, G., A model of size effects in nanoindentation. Journal of the Mechanics and Physics of Solids, 2006, 54: 1668–1686.CrossRefGoogle Scholar
  16. 16.
    Huang, Y., Feng X., Pharr, G.M. and Hwang, K.C., A nano-indentation model for spherical indenters. Modelling and Simulation in Materials Science and Engineering, 2007, 15: S255–S262.CrossRefGoogle Scholar
  17. 17.
    Huang, Y., Qu, S., Hwang, K.C., Li, M. and Gao, H., A conventional theory of mechanism based strain gradient plasticity. International Journal of Plasticity, 2004, 20: 753–782.CrossRefGoogle Scholar
  18. 18.
    Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W., Mechanism-based strain gradient plasticity — I. Theory. Journal of the Mechanics and Physics of Solids, 1999, 47: 1239–1263.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Huang, Y., Gao, H., Nix, W.D. and Hutchinson, J.W., Mechanism-based strain gradient plasticity — II. Analysis. Journal of the Mechanics and Physics of Solids, 2000, 48: 99–128.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wiedersich, H., Hardening mechanisms and the theory of deformation. Journal of Metals, 1964, 16: 425–430.Google Scholar
  21. 21.
    Basinski, S.J. and Basinski, Z.S., Plastic deformation and work hardening. In: Nabarro, F.R.N. (ed.), Dislocations in Solids, North-Holland, 1979, 4: 261–362.Google Scholar
  22. 22.
    Gil Sevillano, J., In: Cahn, R.W., Hassen, P., Kramer, E.J. (eds.), Materials Science and Technology: A comprehensive treatment. VCH, Weinheim, 1993, 6: 39.Google Scholar
  23. 23.
    Kocks, U.F. and Mecking, H., Physics and phenomenology of strain hardening: the FCC case. Progress in Materials Science, 2003, 48: 171–273.CrossRefGoogle Scholar
  24. 24.
    Cottrell, A.H., The Mechanical Properties of Materials, New York: J. Willey, 1964.MATHGoogle Scholar
  25. 25.
    Nix, W.D. and Gao, H., Indentation size effects in crystalline materials: A law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 1998, 46: 411–425.CrossRefGoogle Scholar
  26. 26.
    Bishop, J.F.W. and Hill, R., A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philosophical Magazine, 1951, 42: 414–427.MathSciNetMATHGoogle Scholar
  27. 27.
    Bishop, J.F.W. and Hill, R., A theoretical derivation of the plastic properties of a polycrystalline face-centered metal. Philosophical Magazine, 1951, 42: 1298–1307.MATHGoogle Scholar
  28. 28.
    Kocks, U.F., The relation between polycrystal deformation and single crystal deformation. Metallurgical and Materials Transactions, 1970, 1: 1121–1144.CrossRefGoogle Scholar
  29. 29.
    Kok, S., Beaudoin, A.J. and Tortorelli, D.A., A polycrystal plasticity model based on the mechanical threshold. International Journal of Plasticity, 2002, 18: 715–741.CrossRefGoogle Scholar
  30. 30.
    Kok, S., Beaudoin, A.J. and Tortorelli, D.A., On the development of stage IV hardening using a model based on the mechanical threshold. Acta Materialia, 2002, 50: 1653–1667.CrossRefGoogle Scholar
  31. 31.
    Kok, S., Beaudoin, A.J., Tortorelli, D.A. and Lebyodkin, M., A finite element model for the Portevin-Le Chatelier effect based on polycrystal plasticity. Modelling and Simulation in Materials Science and Engineering, 2002, 10: 745–763.CrossRefGoogle Scholar
  32. 32.
    Zhang, F., Huang, Y., Hwang, K.C., Qu, S. and Liu, C., A three-dimensional strain gradient plasticity analysis of particle size effect in composite materials. Materials and Manufacturing Processes, 2007, 22: 140–148.CrossRefGoogle Scholar
  33. 33.
    Acharya, A. and Bassani, J.L., Lattice incompatibility and a gradient theory of crystal plasticity. Journal of the Mechanics and Physics of Solids, 2000, 48: 1565–1595.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Huang, Y., Xue, Z., Gao, H., Nix, W.D. and Xia, Z.C., A study of microindentation hardness tests by mechanism-based strain gradient plasticity. Journal of Materials Research, 2000, 15: 1786–1796.CrossRefGoogle Scholar
  35. 35.
    Begley, M.R. and Hutchinson, J.W., The mechanics of size-dependent indentation. Journal of the Mechanics and Physics of Solids, 1998, 46: 2049–2068.CrossRefGoogle Scholar
  36. 36.
    Huang, Y., Xue, Z., Gao, H., Nix, W.D. and Xia, Z.C., A study of microindentation hardness tests by mechanism-based strain gradient plasticity. Journal of Materials Research, 2000, 15: 1786–1796.CrossRefGoogle Scholar
  37. 37.
    Saha, R., Xue, Z., Huang, Y. and Nix, W.D., Indentation of a soft metal film on a hard substrate: strain gradient hardening effects. Journal of the Mechanics and Physics of Solids, 2001, 49: 1997–2014.CrossRefGoogle Scholar
  38. 38.
    Qu, S., Huang, Y., Nix, W.D., Jiang, H., Zhang, F. and Hwang, K.C., Indenter tip radius effect on the Nix-Gao relation in micro- and nanoindentation hardness experiments. Journal of Materials Research, 2004, 19: 3423–3434.CrossRefGoogle Scholar
  39. [39]
    Hibbitt, Karlsson & Sorenson, Inc., ABAQUS/Standard User’s Manual Version 6.2., 2001.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2009

Authors and Affiliations

  1. 1.Department of Engineering MechanicsTsinghua UniversityBeijingChina
  2. 2.Department of MechanicsZhejiang UniversityHangzhouChina
  3. 3.Department of Civil and Environmental EngineeringNorthwestern UniversityEvanstonUSA
  4. 4.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA

Personalised recommendations