Metrological Comparison of Indirect Calibration Methods for Nanoindentation: A Bootstrap-Based Approach

Abstract

Area shape function and frame compliance are the most critical parameters in nanoindentation, as they control measurement accuracy and represent the largest contributions to measurement uncertainty. Despite the availability of direct calibration methods, indirect calibrations are the most practical and fast. Thus, the indirect calibration methods proposed in ISO 14577-2 are most typically applied in academic and industrial research, as well as in quality controls. Previous research has highlighted some criticalities, but a holistic metrological framework was missing. This work aims to compare the performances of indirect calibration methods for area shape function and frame compliance in the nano-range, considering different alternatives suggested in the standard and most recent literature. The comparison will be based on uncertainty estimation using bootstrap estimation, which will innovatively highlight and introduce the effect of the nanoindentation dataset in the uncertainty estimation. The results show that the optimization of accuracy and uncertainty in mechanical characterization is achieved by indenting pairs of certified reference materials, resulting in a more robust approach to calibration experimental conditions than methods that require a single sample to be indented.

Keywords: bootstrap; calibration; frame compliance; indenter calibration; instrumented indentation test; nanoindentation; uncertainty.

Figure A1

Empirical probability density function (pdf) of the 10,000 calibrated parameters for the 10 calibration methods for the frame compliance (C_f).

Figure A2

(a) Relative accuracy and (b) relative uncertainty for the E_IT of SiO₂ (FS blue star) and W (red circle). Not all values are represented to improve visualization (extreme values are excluded).

Figure 1

Indentation curve (IC): applied force (F) as a function of the penetration depth (h), with highlighted parts showing the corrected depth (h_c), the residual penetration depth (h_p), and the measured contact stiffness (S_m).

Figure 2

The iterative process for the application of the indirect calibration of area shape function parameters and frame compliance as per ISO 14577-2 methods #2 and #4. The M4 requires carrying out step 4 on the stiffer material (W) and step 6 on the more elastic material (SiO₂). Step 7 shows one of the several possible models for the A_p. Adapted from [50].

Figure 3

Empirical probability density function (pdf) of the 10,000 calibrated parameters for the 10 calibration methods. (a) Frame compliance (C_f): notice the calibration method dependency of the dispersion, and the difference in the average estimates due to different calibration methods. (b) a₂: theoretical value is 24.49; notice the significant dependence of both average and dispersion on the calibration method for all area shape function parameters, i.e., also for (c) a₁ and (d) a₀. Due to the scale, the histogram of C_f for the calibration method M2-FS8 is not reported. Appendix A reports individual histograms for C_f (see Figure A1).

Figure 4

Error bar plots of the calibrated parameters as a function of the calibration method. (a) frame compliance, (b) a₂, (c) a₁, and (d) a₀. Error bars represent the expanded uncertainty with a coverage factor of 2 (confidence level of 95%).

Figure 5

SiO₂ E_IT for the different calibration methods. E_IT as a function of maximum characterization force. Error bars represent expanded uncertainty (coverage factor of 2). Red dashed lines are the confidence interval for the independently calibrated value of the Young’s modulus of the sample.

Figure 6

W E_IT for the different calibration methods. E_IT as a function of maximum characterization force. Error bars represent expanded uncertainty (coverage factor of 2). Red dashed lines are the confidence interval for the independently calibrated value of the Young’s modulus of the sample.

Figure 7

(a) Relative accuracy and (b) relative uncertainty for the E_IT of SiO₂ (FS blue star) and W (red circle). Improved visualization with large values excluded is available in Figure A2.