Uncertainties in the representative indentation stress and strain using spherical nanoindentation

In this study, we reviewed the definitions of indentation strain and stress proposed from earlier studies and then calculated indentation strain–stress curves according to 20 combinations of five indentation strain definitions and four indentation stress definitions. The finite element method was applied to predict the load–displacement curves and the force–displacement for spherical nanoindentation and tensile test, respectively. Thus, the load–displacement data were used to determine indentation strain–stress curves, which were compared with the stress–strain curve, obtained from the tensile test simulation. Comparing with the tensile stress–strain curve, three combinations of σX − ɛK, σO − ɛA, and σH − ɛM reveal better fitting curves than the other combinations, where the subscripts of X, K, O, A, H, M are denoted as Xu and Chen (J Mater Res 25:2297-2307, 2010), Kalidindi and Pathak (Acta Mater 56:3523-3532, 2008), Oliver and Pharr (J Mater Res 7:1564-1583, 1992, J Mater Res 19:3-20, 2004), Ahn and Kwon (J Mater Res 16:3170-3178, 2001), Hill et al. (P R Soc Lond 423:301-330, 1989), and Milman et al. (Acta Metal Mater 41:2523-2532, 1993), respectively. The limitation of the indentation strain is at 0.06, 0.03, and 0.01 for combinations of σX − ɛK, σO − ɛA, and σH − ɛM. The stress constraint factor (called the ratio of mean contact pressure to tensile stress) for combinations of σX − ɛK, σO − ɛA, and σH − ɛM is 3.4, 3.4, and 2.6 to obtain the optimized fitting indentation stress–strain curves with the parameter n of the power law of 0.306, 0.248 and 0.173, which reveal the deviation of 43.0%, 15.9%, and 19.2%, respectively, in comparison with that of the tensile flow curve. Considering the best fitting curve by the combination of σO − ɛA, together with the stress constraint factor $$\psi$$ of 3.4, the fitting parameters of n and K are 0.248 and 588.56 corresponding to the deviation of 15.9% and 7.7% in terms of these parameters in the tensile flow curve.