Hostname: page-component-77f85d65b8-grvzd Total loading time: 0 Render date: 2026-03-28T08:55:35.921Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  31 January 2011

Baoxing Xu  and
Xi Chen
Baoxing Xu
Affiliation:
Columbia Nanomechanics Research Center, Department of Earth and Environmental Engineering, Columbia University, New York, New York 10027
Xi Chen*
Affiliation:
Columbia Nanomechanics Research Center, Department of Earth and Environmental Engineering, Columbia University, New York, New York 10027; Department of Civil & Environmental Engineering, Hanyang University, Seoul 133-791, Korea; and School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China
*
a)Address all correspondence to this author. e-mail: xichen@columbia.edu
Get access

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.

Keywords

NanoindentationStress/Strain Relationship

Information

Type
Articles
Information
Journal of Materials Research , Volume 25 , Issue 12 , December 2010 , pp. 2297 - 2307
Copyright
Copyright © Materials Research Society 2010

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

1.Pharr, G.M.: Measurement of mechanical properties by ultra-low-load indentation. Mater. Sci. Eng., A 253, 151 (1998)CrossRefGoogle Scholar
2.Gouldstone, A., Chollacoop, N., Dao, M., Li, J., Minor, A.M., Shen, Y-L.: Indentation across size scales and disciplines: Recent developments in experimentation and modeling. Acta Mater. 55, 4015 (2007)CrossRefGoogle Scholar
3.Zhao, M., Chen, X., Xiang, Y., Vlassak, J.J., Lee, D., Ogasawara, N., Chiba, N., Gan, Y.X.: Measuring elastoplastic properties of thin films on an elastic substrate using sharp indentation. Acta Mater. 55, 6260 (2007)CrossRefGoogle Scholar
4.Greer, J.R., Oliver, W.C., Nix, W.D.: Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater. 53, 1821 (2005)CrossRefGoogle Scholar
5.Dao, M., Chollacoop, N., VanVliet, K.J., Venkatesh, T.A., Suresh, S.: Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Mater. 49, 3899 (2001)CrossRefGoogle Scholar
6.Cheng, Y.T., Cheng, C.M.: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng., R 44, 91 (2004)CrossRefGoogle Scholar
7.Zhao, M., Ogasawara, N., Chiba, N., Chen, X.: A new approach to measure the elastic–plastic properties of bulk materials using spherical indentation. Acta Mater. 54, 23 (2006)CrossRefGoogle Scholar
8.Lee, H., Lee, J.H., Pharr, G.M.: A numerical approach to spherical indentation techniques for material property evaluation. J. Mech. Phys. Solids 53, 2037 (2005)CrossRefGoogle Scholar
9.Huber, N., Tsakmakis, C.: 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)CrossRefGoogle Scholar
10.Cao, Y.P., Lu, J.: A new method to extract the plastic properties of metal materials from an instrumented spherical indentation loading curve. Acta Mater. 52, 4023 (2004)CrossRefGoogle Scholar
11.Cheng, Y.T., Cheng, C.M.: Scaling relationships in conical indentation of elastic-perfectly plastic solids. Int. J. Solids Struct. 36, 1231 (1999)CrossRefGoogle Scholar
12.Chollacoop, N., Dao, M., Suresh, S.: Depth-sensing instrumented indentation with dual sharp indenters. Acta Mater. 51, 3713 (2003)CrossRefGoogle Scholar
13.Cheng, Y.T., Cheng, C.M.: Can stress–strain relationships be obtained from indentation curves using conical and pyramidal indenters? J. Mater. Res. 14, 3493 (1999)CrossRefGoogle Scholar
14.Chen, X., Ogasawara, N., Zhao, M., Chiba, N.: On the uniqueness of measuring elastoplastic properties from indentation: The indistinguishable mystical materials. J. Mech. Phys. Solids 55, 1618 (2007)CrossRefGoogle Scholar
15.Tabor, D.: A simple theory of static and dynamic hardness. Proc. R. Soc. London, Ser. A 192, 247 (1948)Google Scholar
16.Herbert, E.G., Pharr, G.M., Oliver, W.C., Lucas, B.N., Hay, J.L.: On the measurement of stress–strain curves by spherical indentation. Thin Solid Films 398, 331 (2001)CrossRefGoogle Scholar
17.Kalidindi, S.R., Pathak, S.: Determination of the effective zero-point and the extraction of spherical nanoindentation stress–strain curves. Acta Mater. 56, 3523 (2008)CrossRefGoogle Scholar
18.Beghini, M., Bertini, L., Fontanari, V.: Evaluation of the stress–strain curve of metallic materials by spherical indentation. Int. J. Solids Struct. 43, 2441 (2006)CrossRefGoogle Scholar
19.Jayaraman, S., Hahn, G.T., Oliver, W.C., Rubin, C.A., Bastias, P.C.: Determination of monotonic stress–strain curve of hard materials from ultra-low-load indentation tests. Int. J. Solids Struct. 35, 365 (1998)CrossRefGoogle Scholar
20.Taljat, B., Zacharia, T., Kosel, F.: New analytical procedure to determine stress–strain curve from spherical indentation data. Int. J. Solids Struct. 35, 4411 (1998)CrossRefGoogle Scholar
21.Mesarovic, S.D., Fleck, N.A.: Spherical indentation of elastic–plastic solids. Proc. R. Soc. London, Ser. A 455, 2707 (1999)CrossRefGoogle Scholar
22.Qu, S., Huang, Y., Pharr, G.M., 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. Int. J. Plast. 22, 1265 (2006)CrossRefGoogle Scholar
23.Feng, G., Qu, S., Huang, Y., Nix, W.D.: An analytical expression for the stress field around an elastoplastic indentation/contact. Acta Mater. 55, 2929 (2007)CrossRefGoogle Scholar
24.Fischer-Cripps, A.C.: Use of combined elastic modulus in the analysis of depth-sensing indentation data. J. Mater. Res. 16, 3050 (2001)CrossRefGoogle Scholar
25.Oliver, W.C., Pharr, G.M.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564 (1992)CrossRefGoogle Scholar
26.Taljat, B., Pharr, G.M.: Development of pile-up during spherical indentation of elastic–plastic solids. Int. J. Solids Struct. 41, 3891 (2004)CrossRefGoogle Scholar
27.Matthews, J.R.: Indentation hardness and hot pressing. Acta Metall. 28, 311 (1980)CrossRefGoogle Scholar
28.Hill, R., Storakers, B., Zdunek, A.B.: A theoretical study of the Brinell hardness test. Proc. R. Soc. London, Ser. A 436, 301 (1989)Google Scholar
29.Tirupataiah, Y.: On the constraint factor associated with the indentation of work-hardening materials with a spherical ball. Metall. Trans. A 22, 2375 (1991)CrossRefGoogle Scholar
30.Field, J.S., Swain, M.V.: Determining the mechanical properties of small volumes of material from submicrometer spherical indentations. J. Mater. Res. 10, 101 (1995)CrossRefGoogle Scholar
31.Sneddon, I.N.: The relationship between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47 (1965)CrossRefGoogle Scholar
32.Tabor, D.: Hardness of Metals (Oxford University Press, Oxford, UK 1951)Google Scholar
33.Francis, H.A.: Phenomenological analysis of plastic spherical indentation. J. Eng. Mater. Technol. Trans. ASME 98, 272 (1976)CrossRefGoogle Scholar
34.Habbab, H., Mellor, B.G., Syngellakis, S.: Post-yield characterisation of metals with significant pile-up through spherical indentations. Acta Mater. 54, 1965 (2006)CrossRefGoogle Scholar
35.Haggag, F.M., Nanstad, R.K., Hutton, J.T., Thomas, D.L., Swain, R.L.: Use of automated ball indentation testing to measure flow properties and estimate fracture toughness in metallic materials, 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)188 208CrossRefGoogle Scholar
36.Johnson, K.L.: The correlation of indentation experiments. J. Mech. Phys. Solids 18, 115 (1970)CrossRefGoogle Scholar
37.Ahn, J.H., Kwon, D.: Derivation of plastic stress–strain relationship from ball indentations: Examination of strain definition and pileup effect. J. Mater. Res. 16, 3170 (2001)CrossRefGoogle Scholar
38.Bowden, F.P., Tabor, D.: The Friction and Lubrications of Solids (Oxford University Press, Oxford, UK 1950)Google Scholar
39.Cao, Y.P., Qian, X., Huber, N.: Spherical indentation into elastoplastic materials: Indentation-response based definitions of the representative strain. Mater. Sci. Eng., A 454–455, 1 (2007)CrossRefGoogle Scholar
40.Lee, J.H., Kim, T., Lee, H.: A study on robust indentation techniques to evaluate elastic–plastic properties of metals. Int. J. Solids Struct. 47, 647 (2010)CrossRefGoogle Scholar
41.Zhang, K.S., Li, Z.H.: Numerical analysis of the stress–strain curve and fracture initiation for ductile material. Eng. Fract. Mech. 49, 235 (1994)CrossRefGoogle Scholar
42.Oliver, W.C., Pharr, G.M.: Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. J. Mater. Res. 19, 3 (2004)CrossRefGoogle Scholar
43.Norbury, A.L., Samuel, T.: 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.Ashby, M.F.: Materials Selection in Mechanical Design 2nd ed (Butterworth-Heinemann, Woburn, MA 1999)Google Scholar