Skip to main content
SpringerLink
Log in

Determining engineering stress-strain curve directly from the load-depth curve of spherical indentation test

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

Abstract

The engineering stress-strain curve is one of the most convenient characterizations of the constitutive behavior of materials that can be obtained directly from uniaxial experiments. We propose that the engineering stress-strain curve may also be directly converted from the load-depth curve of a deep spherical indentation test via new phenomenological formulations of the effective indentation strain and stress. From extensive forward analyses, explicit relationships are established between the indentation constraint factors and material elastoplastic parameters, and verified numerically by a large set of engineering materials as well as experimentally by parallel laboratory tests and data available in the literature. An iterative reverse analysis procedure is proposed such that the uniaxial engineering stress-strain curve of an unknown material (assuming that its elastic modulus is obtained in advance via a separate shallow spherical indentation test or other established methods) can be deduced phenomenologically and approximately from the load-displacement curve of a deep spherical indentation test.

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. G.M. Pharr: Measurement of mechanical properties by ultra-low-load indentation. Mater. Sci. Eng., A 253, 151 (1998).

    Article  Google Scholar 

  2. A. Gouldstone, N. Chollacoop, M. Dao, J. Li, A.M. Minor, and Y-L. Shen: Indentation across size scales and disciplines: Recent developments in experimentation and modeling. Acta Mater. 55, 4015 (2007).

    Article  CAS  Google Scholar 

  3. M. Zhao, X. Chen, Y. Xiang, J.J. Vlassak, D. Lee, N. Ogasawara, N. Chiba, and Y.X. Gan: Measuring elastoplastic properties of thin films on an elastic substrate using sharp indentation. Acta Mater. 55, 6260 (2007).

    Article  CAS  Google Scholar 

  4. J.R. Greer, W.C. Oliver, and W.D. Nix: Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater. 53, 1821 (2005).

    Article  CAS  Google Scholar 

  5. M. Dao, N. Chollacoop, K.J. VanVliet, T.A. Venkatesh, and S. Suresh: Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Mater. 49, 3899 (2001).

    Article  CAS  Google Scholar 

  6. Y.T. Cheng and C.M. Cheng: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng., R 44, 91 (2004).

    Article  Google Scholar 

  7. M. Zhao, N. Ogasawara, N. Chiba, and X. Chen: A new approach to measure the elastic-plastic properties of bulk materials using spherical indentation. Acta Mater. 54, 23 (2006).

    Article  CAS  Google Scholar 

  8. H. Lee, J.H. Lee, and G.M. Pharr: A numerical approach to spherical indentation techniques for material property evaluation. J. Mech. Phys. Solids 53, 2037 (2005).

    Article  CAS  Google Scholar 

  9. N. Huber and C. Tsakmakis: Determination of constitutive properties from spherical indentation data using neural networks. Part I: The case of pure kinematic hardening in plasticity laws. J. Mech. Phys. Solids 47, 1569 (1999).

    Article  Google Scholar 

  10. Y.P. Cao and J. Lu: A new method to extract the plastic properties of metal materials from an instrumented spherical indentation loading curve. Acta Mater. 52, 4023 (2004).

    Article  CAS  Google Scholar 

  11. Y.T. Cheng and C.M. Cheng: Scaling relationships in conical indentation of elastic-perfectly plastic solids. Int. J. Solids Struct. 36, 1231 (1999).

    Article  Google Scholar 

  12. N. Chollacoop, M. Dao, and S. Suresh: Depth-sensing instrumented indentation with dual sharp indenters. Acta Mater. 51, 3713 (2003).

    Article  CAS  Google Scholar 

  13. Y.T. Cheng and C.M. Cheng: Can stress-strain relationships be obtained from indentation curves using conical and pyramidal indenters? J. Mater. Res. 14, 3493 (1999).

    Article  CAS  Google Scholar 

  14. X. Chen, N. Ogasawara, M. Zhao, and N. Chiba: On the uniqueness of measuring elastoplastic properties from indentation: The indistinguishable mystical materials. J. Mech. Phys. Solids 55, 1618 (2007).

    Article  Google Scholar 

  15. D. Tabor: A simple theory of static and dynamic hardness. Proc. R. Soc. London. Ser. A 192, 247 (1948).

    Article  Google Scholar 

  16. E.G. Herbert, G.M. Pharr, W.C. Oliver, B.N. Lucas, and J.L. Hay: On the measurement of stress-strain curves by spherical indentation. Thin Solid Films 398, 331 (2001).

    Article  Google Scholar 

  17. S.R. Kalidindi and S. Pathak: Determination of the effective zero-point and the extraction of spherical nanoindentation stress-strain curves. Acta Mater. 56, 3523 (2008).

    Article  CAS  Google Scholar 

  18. M. Beghini, L. Bertini, and V. Fontanari: Evaluation of the stress-strain curve of metallic materials by spherical indentation. Int. J. Solids Struct. 43, 2441 (2006).

    Article  CAS  Google Scholar 

  19. S. Jayaraman, G.T. Hahn, W.C. Oliver, C.A. Rubin, and P.C. Bastias: Determination of monotonic stress-strain curve of hard materials from ultra-low-load indentation tests. Int. J. Solids Struct. 35, 365 (1998).

    Article  Google Scholar 

  20. B. Taljat, T. Zacharia, and F. Kosel: New analytical procedure to determine stress-strain curve from spherical indentation data. Int. J. Solids Struct. 35, 4411 (1998).

    Article  Google Scholar 

  21. S.D. Mesarovic and N.A. Fleck: Spherical indentation of elastic-plastic solids. Proc. R. Soc. London. Ser. A 455, 2707 (1999).

    Article  Google Scholar 

  22. S. Qu, Y. Huang, G.M. Pharr, and K.C. Hwang: The indentation size effect in the spherical indentation of iridium: A study via the conventional theory of mechanism-based strain gradient plasticity. Int. J. Plast. 22, 1265 (2006).

    Article  CAS  Google Scholar 

  23. G. Feng, S. Qu, Y. Huang, and W.D. Nix: An analytical expression for the stress field around an elastoplastic indentation/contact. Acta Mater. 55, 2929 (2007).

    Article  CAS  Google Scholar 

  24. A.C. Fischer-Cripps: Use of combined elastic modulus in the analysis of depth-sensing indentation data. J. Mater. Res. 16, 3050 (2001).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  26. B. Taljat and G.M. Pharr: Development of pile-up during spherical indentation of elastic-plastic solids. Int. J. Solids Struct. 41, 3891 (2004).

    Article  Google Scholar 

  27. J.R. Matthews: Indentation hardness and hot pressing. Acta Metall. 28, 311 (1980)

    Article  CAS  Google Scholar 

  28. R. Hill, B. Storakers, and A.B. Zdunek: A theoretical study of the Brinell hardness test. Proc. R. Soc. London, Ser. A 436, 301 (1989).

    Google Scholar 

  29. Y. Tirupataiah: On the constraint factor associated with the indentation of work-hardening materials with a spherical ball. Metall. Trans. A 11, 2375 (1991).

    Article  Google Scholar 

  30. J.S. Field and M.V. Swain: Determining the mechanical properties of small volumes of material from submicrometer spherical indentations. J. Mater. Res. 10, 101 (1995).

    Article  CAS  Google Scholar 

  31. I.N. Sneddon: The relationship between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. En?. Sci. 3, 47 (1965).

    Article  Google Scholar 

  32. D. Tabor: Hardness of Metals (Oxford University Press, Oxford, UK, 1951).

    Google Scholar 

  33. H.A. Francis: Phenomenological analysis of plastic spherical indentation. J. Em. Mater. Technol. Trans. ASME 98, 272 (1976).

    Article  Google Scholar 

  34. H. Habbab, B.G. Mellor, and S. Syngellakis: Post-yield characterisation of metals with significant pile-up through spherical indentations. Acta Mater. 54, 1965 (2006).

    Article  CAS  Google Scholar 

  35. F.M. Haggag, R.K. Nanstad, J.T. Hutton, D.L. Thomas, and R.L. Swain: Use of automated ball indentation testing to measure flow properties and estimate fracture toughness in metallic materials, in Applications of Automation Technology to Fatigue and Fracture Testing, edited by A.A. Braun, N.E. Ashbaugh, and F.M. Smith (ASTM, Philadelphia, PA, 1990), pp. 188–208.

    Chapter  Google Scholar 

  36. K.L. Johnson: The correlation of indentation experiments. J. Mech. Phys. Solids 18, 115 (1970).

    Article  Google Scholar 

  37. J.H. Ahn and D. Kwon: Derivation of plastic stress-strain relationship from ball indentations: Examination of strain definition and pileup effect. J. Mater. Res. 16, 3170 (2001).

    Article  CAS  Google Scholar 

  38. F.P. Bowden and D. Tabor: The Friction and Lubrications of Solids (Oxford University Press, Oxford, UK, 1950).

    Google Scholar 

  39. Y.P. Cao, X. Qian, and N. Huber: Spherical indentation into elastoplastic materials: Indentation-response based definitions of the representative strain. Mater. Sci. Eng..A 454-455, 1 (2007).

    Article  CAS  Google Scholar 

  40. J.H. Lee, T. Kim, and H. Lee: A study on robust indentation techniques to evaluate elastic-plastic properties of metals. Int. J. Solids Struct. 47, 647 (2010).

    Google Scholar 

  41. K.S. Zhang and Z.H. Li: Numerical analysis of the stress-strain curve and fracture initiation for ductile material. Ens. Fract. Mech. 49, 235 (1994).

    Article  Google Scholar 

  42. W.C. Oliver and G.M. Pharr: Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. J. Mater. Res. 19, 3 (2004).

    Article  CAS  Google Scholar 

  43. A.L. Norbury and T. Samuel: The recovery and sinking-in or piling-up of material in the brinell test, and the effects of these factors on the correlation of the brinell with certain other hardness tests. J. Iron Steel Inst. 17, 673 (1928).

    Google Scholar 

  44. M.F. Ashby: Materials Selection in Mechanical Design, 2nd ed. (Butterworth-Heinemann, Woburn, MA, 1999).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Columbia Nanomechanics Research Center, Department of Earth and Environmental Engineering, Columbia University, New York, New York, 10027, USA

    Baoxing Xu & Xi Chen

  2. Department of Civil & Environmental Engineering, Hanyang University, Seoul, 133-791, Korea

    Xi Chen

  3. School of Aerospace, Xi’ an Jiaotong University, Xi’an, 710049, China

    Xi Chen

Authors
  1. Baoxing Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xi Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, B., Chen, X. Determining engineering stress-strain curve directly from the load-depth curve of spherical indentation test. Journal of Materials Research 25, 2297–2307 (2010). https://doi.org/10.1557/jmr.2010.0310

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1557/jmr.2010.0310

Search

Navigation