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A new approach of the Oliver and Pharr model to fit the unloading curve from instrumented indentation testing

Published online by Cambridge University Press:  17 April 2017

Stephania Kossman*
Affiliation:
Univ. Lille, FRE 3723—LML—Laboratoire de Mécanique de Lille, Lille F-59000, France; and Arts et Métiers ParisTech, MSMP, Lille 59800, France
Thierry Coorevits
Affiliation:
Arts et Métiers ParisTech, MSMP, Lille 59800, France
Alain Iost
Affiliation:
Arts et Métiers ParisTech, MSMP, Lille 59800, France
Didier Chicot
Affiliation:
Univ. Lille, FRE 3723—LML—Laboratoire de Mécanique de Lille, Lille F-59000France
*
a)Address all correspondence to this author. e-mail: [email protected]
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Abstract

The unloading part of a load–displacement curve from instrumented indentation tests is usually approximated by a power law (Oliver and Pharr model), where the load is the dependent variable. This approach generally fits well the data. Nevertheless, the convergence is occasionally quite questionable. In this regard, we propose a different approach for the Oliver and Pharr model, called the inverted approach, since it assigns the displacement as the dependent variable. Both models were used to fit the unloading curves from nanoindentation tests on fused silica and aluminum, applying a general least squares procedure. Generally, the inverted methodology leads to similar results for the fitting parameters and the elastic modulus (E) when convergence is achieved. Nevertheless, this approach facilitates the convergence, because it is a better conditioned problem. Additionally, by Monte Carlo simulations we found that robustness is improved using the inverted approach, since the estimation of E is more accurate, especially for aluminum.

Type
Articles
Copyright
Copyright © Materials Research Society 2017 

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Footnotes

Contributing Editor: George M. Pharr

References

REFERENCES

VanLandingham, M.R.: Review of instrumented indentation. J. Res. Natl. Inst. Stand. Technol. 108, 249 (2003).Google Scholar
Hay, J.: Introduction to instrumented indentation testing. Exp. Tech. 33, 66 (2009).Google Scholar
Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A., and Suresh, S.: Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Mater. 49, 3899 (2001).CrossRefGoogle Scholar
Antunes, J.M., Fernandes, J.V., Menezes, L.F., and Chaparro, B.M.: A new approach for reverse analyses in depth-sensing indentation using numerical simulation. Acta Mater. 55, 69 (2007).Google Scholar
Mata, M., Anglada, M., and Alcalá, J.: Contact deformation regimes around sharp indentations and the concept of the characteristic strain. J. Mater. Res. 17, 964 (2002).CrossRefGoogle Scholar
Anstis, G.R., Chantikul, P., Lawn, B.R., and Marshall, D.B.: A critical evaluation of indentation techniques for measuring fracture toughness: I, direct crack measurements. J. Am. Ceram. Soc. 64, 533 (1981).Google Scholar
Field, J.S., Swain, M.V., and Dukino, R.D.: Determination of fracture toughness from the extra penetration produced by indentation-induced pop-in. J. Mater. Res. 18, 1412 (2003).Google Scholar
Cao, Y.P. and 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).Google Scholar
Attaf, M.T.: Connection between the loading curve models in elastoplastic indentation. Mater. Lett. 58, 3491 (2004).CrossRefGoogle Scholar
Jha, K.K., Suksawang, N., and Agarwal, A.: Analytical method for the determination of indenter constants used in the analysis of nanoindentation loading curves. Scr. Mater. 63, 281 (2010).Google Scholar
Chicot, D. and Mercier, D.: Improvement in depth-sensing indentation to calculate the universal hardness on the entire loading curve. Mech. Mater. 40, 171 (2008).Google Scholar
Chicot, D., Gil, L., Silva, K., Roudet, F., Puchi-Cabrera, E.S., Staia, M.H., and Teer, D.G.: Thin film hardness determination using indentation loading curve modelling. Thin Solid Films 518, 5565 (2010).Google Scholar
Zeng, K. and Chiu, C-h.: An analysis of load–penetration curves from instrumented indentation. Acta Mater. 49, 3539 (2001).Google Scholar
Gong, J., Miao, H., and Peng, Z.: A new function for the description of the nanoindentation unloading data. Scr. Mater. 49, 93 (2003).Google Scholar
Oliver, W.C. and 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).Google Scholar
Doerner, M.F. and Nix, W.D.: A method for interpreting the data from depth-sensing indentation instruments. J. Mater. Res. 1, 601 (1986).CrossRefGoogle Scholar
Gubicza, J., Juhász, A., Tasnádi, P., Arató, P., and Vörös, G.: Determination of the hardness and elastic modulus from continuous Vickers indentation testing. J. Mater. Sci. 31, 3109 (1996).Google Scholar
Yetna N’Jock, M., Roudet, F., Idriss, M., Bartier, O., and Chicot, D.: Work-of-indentation coupled to contact stiffness for calculating elastic modulus by instrumented indentation. Mech. Mater. 94, 170 (2016).CrossRefGoogle Scholar
Jha, K.K., Suksawang, N., and Agarwal, A.: A new insight into the work-of-indentation approach used in the evaluation of material’s hardness from nanoindentation measurement with Berkovich indenter. Comput. Mater. Sci. 85, 32 (2014).Google Scholar
Oliver, W.C. and 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).Google Scholar
Pharr, G.M. and Bolshakov, A.: Understanding nanoindentation unloading curves. J. Mater. Res. 17, 2660 (2002).CrossRefGoogle Scholar
Loubet, J.L., Bauer, M., Tonck, A., Bec, S., and Gauthier-Manuel, B.: Nanoindentation with a Surface Force Apparatus, Nastasi, M. et al., eds. (Springer, Dordrecht, 1993); p. 429.Google Scholar
Hochstetter, G., Jimenez, A., and Loubet, J.L.: Strain-rate effects on hardness of glassy polymers in the nanoscale range. Comparison between quasi-static and continuous stiffness measurements. J. Macromol. Sci., Part B: Phys. 38, 681 (1999).Google Scholar
Bevington, P.R. and Robinson, D.K.: Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 2003).Google Scholar
Brown, L.C. and Mac Berthouex, P.: Statistics for Environmental Engineers (CRC Press, Boca Raton, 2002).Google Scholar
Smyth, K.: Nonlinear Regression in Encyclopedia of Environmetrics, El-Shaarawi, A.H. and Piegorsch, W.W., eds. (Wiley, New York, 2002).Google Scholar
4.1.4.2. Nonlinear Least Squares Regression. [Online]. Available at: http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd142.htm (accessed: 13-Aug-2016).Google Scholar
Madsen, K., Nielsen, H.B., and Tingleff, O.: Methods for non-linear least squares problems. In Informatics and Mathematical Modeling (Technical University of Denmark, Kongens Lyngby, 2004).Google Scholar
Gratton, S., Lawless, A.S., and Nichols, N.K.: Approximate Gauss-Newton methods for nonlinear least squares problems. SIAM J. Optim. 18, 106 (2007).Google Scholar
Whitehouse, D.J.: Handbook of Surface and Nanometrology, 2nd ed. (CRC Press, Boca Raton, 2010).Google Scholar
Fischer-Cripps, A.C.: Critical review of analysis and interpretation of nanoindentation test data. Surf. Coat. Technol. 200, 4153 (2006).Google Scholar
Menčík, J. and Swain, M.V.: Errors associated with depth-sensing microindentation tests. J. Mater. Res. 10, 1491 (1995).Google Scholar
Fischer-Cripps, A.: A review of analysis methods for sub-micron indentation testing. Vacuum 58, 569 (2000).CrossRefGoogle Scholar
Peters, C.A.: Statistics for analysis of experimental data. In Environ. Eng. Process. Lab. Man. (S. E. Powers, Champaign, 2001); p. 125.Google Scholar
Trefethen, L.N. and Bau, D. III: Numerical Linear Algebra (SIAM, Philadelphia, 1997).Google Scholar
Erhel, J., Nassif, N., and Bernard, P.: Calcul matriciel et systèmes linéaires (INSA Rennes, Rennes, 2012).Google Scholar
Sofroniou, M. and Spaletta, G.: Precise numerical computation. J. Log. Algebr. Program. 64, 113 (2005).Google Scholar
“JCGM 101:2008-Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’-Propagation of distributions using a Monte Carlo method.” [Online]. Available: http://www.bipm.org/en/publications/guides/gum.html (accessed: 27-Feb-2017).Google Scholar
“ISO 14577-1:2015-Metallic materials—Instrumented indentation test for hardness and materials parameters—Part 1: Test method,” ISO. [Online]. Available: http://www.iso.org/iso/home/store/catalogue_ics/catalogue_detail_ics.htm?csnumber=56626 (accessed: 04-Sep-2016).Google Scholar
Iost, A., Guillemot, G., Rudermann, Y., and Bigerelle, M.: A comparison of models for predicting the true hardness of thin films. Thin Solid Films 524, 229 (2012).Google Scholar
Isselin, J., Iost, A., Golek, J., Najjar, D., and Bigerelle, M.: Assessment of the constitutive law by inverse methodology: Small punch test and hardness. J. Nucl. Mater. 352, 97 (2006).Google Scholar