Instrumented Indentation Test in the Nano-range: Performances Comparison of Testing Machines Calibration Methods
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Abstract
In the modern manufacturing industry, dealing with innovative productions and advanced materials, technological surface characterisation is becoming crucial to qualify components and optimise processes. Instrumented indentation test is an effective method for characterising mechanical behaviour of materials through the analysis of the force–displacement curve obtained during the implementation of a predefined loading/unloading cycle. Instrumented indentation test allows for hardness test to be performed at different force ranges, thus enabling bulk to local material characterisation. To guarantee the characterisation accuracy, rigorous procedures for the calibration of testing machines are defined in ISO 14577-2. In particular, calibration of frame compliance and indenter area function may be addressed according to methods which do not require the indenter area function to be known a priori, thus avoiding the need of high-resolution microscopes. The present work aims at comparing performances and compatibility of these methodologies by considering tests performed in the nano-range.
Keywords
Instrumented indentation test Frame compliance Indenter area1 Introduction
Instrumented indentation test (IIT) is a depth sensing technique which was introduced to assess hardness of material at nanoscales where, due to lateral resolution, traditional optical instruments are ineffective. It was early developed in the mid-1970s in the former Soviet Union [1, 2], even though, because of contingencies, it was not until the late 1980s and early 1990s; thanks to the works of Doerner and Nix [3] and Oliver and Pharr [4], it was capable to arouse actual interest in the research and industrial community.
When facing material characterisation, accuracy of results is core; therefore, testing machine has to be carefully calibrated according to ISO 14577-2 [12] to guarantee traceability and to establish uncertainty contribution to final results.
Recently, Barbato et al. [13] demonstrated that major contributions to measurement uncertainty of indentation modulus are the C_{f} and the parameters of A(h). In particular, Annex D of ISO 14577-2 [12] introduces five methods for their calibration. Method nos. 1, 3 and 5 require the use of a metrological atomic force microscope (AFM) to calibrate the area shape function parameters, whilst the remaining methods (i.e. method nos. 2 and 4) outline iterative procedures to achieve calibration of both frame compliance and A(h) parameters by exploiting relationships that can be drawn from IC. It is clear that adoption of metrological AFM yields lower measurement uncertainty [14]. However, considering that the availability of such a scanning force microscope (SFM) entails high cost and longer calibration time, which are critical for industrial users, often method nos. 2 and 4 are adopted. Despite this, ISO 14577-2 does not suggest a good practice to perform such calibrations and literature [14, 15] and practices of research laboratories or testing machine manufacturers show quite a variety of solution, whose compliancy is not reported.
This work aims at comparing results of C_{f} and parameters of A(h) calibration when method nos. 2 and 4 of ISO 14577-2 are adopted. Also, it will investigate effect of different load ranges to perform calibration, in order to establish route towards good practice in calibrating testing machine. The paper is structured as follows. Section 2 describes calibration methods and experimental set-up, Sect. 3 discusses results, and Sect. 4 eventually concludes the findings.
2 Methodology
This section discusses the two methods outlined in ISO 14577-2 which will be investigated in the present work. They both rely upon some common methodology based on relationships amongst the parameters to be calibrated and on general considerations about the indentation system.
ISO 14577-2 requires a set of indentations to be performed over a load range which is representative for the application field of the instrument and suggests frame compliance initialisation to be performed exploiting data from the indentations at the two higher loads.
2.1 Method No. 2 of ISO 14577-2
Method no. 2 (M2) describes calibration to be performed according to workflow discussed in the former section by indenting a single sample. Considering that, to achieve calibration of A(h) parameters, even at shallow depth, a relatively soft material is required, e.g. fused silica can be used.
2.2 Method No. 4 of ISO 14577-2
Method no. 4 (M4) describes calibration to be performed according to workflow discussed in the former section by indenting two samples of different material. A stiffer material, e.g. tungsten, shall be considered to calibrate C_{f}, whilst a softer material, e.g. fused silica, enables the calibration of A(h) parameters. Therefore, steps 1–4, and consequently 7, have to be performed considering data from tungsten indentations, whilst steps 5 and 6, which calibrate shape function parameters, require data from fused silica indentations. This method, by coupling calibration and material, guarantees faster convergence [14].
2.3 Experimental Set-Up
Calibrated material characteristics mean and expanded uncertainty
Material | Calibration laboratory | E (GPa) | υ |
---|---|---|---|
SiO_{2} | NPL | 73.3 ± 0.6 | 0.161 ± 0.003 |
W | NPL | 413.0 ± 2.8 | 0.281 ± 0.003 |
Metrological characteristics of three-plate force–displacement transducer
Maximum force (mN) | 10 |
Load resolution (nN) | 1 |
Load noise floor (nN) | 100 |
Maximum displacement (µm) | 5 |
Displacement resolution (nm) | 0.04 |
Displacement noise floor (nm) | 0.2 |
Depending on the electronic circuit input, the cycle can be performed in force or displacement control. With reference to Fig. 6 and to a force-controlled cycle, the two fixed (drive) plates (violet and blue) are the electrodes that are driven by two AC voltage signals with same amplitude and a phase shift of 180°. This design makes zero the electric field potential at the mid (floating)-plate (red), which is connected to springs (green) for mechanical guide and to the output electrode. Force is applied electrostatically by means of a DC voltage bias at the lower plate. The three-plate design generates a linear electric field voltage, and because the input impedance is significantly larger than the output’s, the floating electrode electric potential is the same of the electric field at its location. Thus, by continuously recording the input voltages, e.g. both the AC and the DC, at the drive plates and the resulting output voltage at the floating plate, its resulting displacement can be retrieved by the known, by design, electric field.
2.4 Experimental Plan
The present work investigates the effect of method and load, in terms of steps within a given range, on the results of calibration of frame compliance and area shape function parameters as average values and related uncertainties.
Considered conditions for M2
Case | Material | Load range (mN) | Replications per each load |
---|---|---|---|
M2_1 | SiO_{2} | 0.5–1–5–10 | 10 × |
M2_1_bis | 0.5–1–5–10 | 5 × | |
M2_1_ter | 0.5:0.25:10 | 1 × |
Considered conditions for M4
Case | Material | Load range (mN) | Replications per each load |
---|---|---|---|
M4_1 | W | 0.5–1–5–10 | 10 × |
SiO_{2} | |||
M4_1_bis | W | 0.5–1–5–10 | 5 × |
SiO_{2} | |||
M4_1_ter | W | 0.5:0.25:10 | 1 × |
SiO_{2} |
Custom script was implemented on MATLAB R2018b, convergence was achieved as soon as the variation of mean values of calibrated parameter between cycle j and j − 1 was less than 0.1%, and it was furtherly checked on root-mean-square error stabilisation. To bound least-squared linear regression to the physics of the problem, additional constraints were set to force the intercept of the linear model (step 4 in Fig. 3) and the evaluated contact area (step 7 in Fig. 3) to be positive.
Combined uncertainty, u_{c}, is calculated, and when multiplied by a coverage factor k, the expanded uncertainty, U, which is half width of the confidence interval, is evaluated. k depends on the confidence level at which U is computed and on the probability distribution of y; typically, k = 2 corresponds to a confidence level of about 95% [16]. However, iterative computation hinders from writing explicit and independent relationships for the four calibrated parameters, which are necessary to apply Eq. 8. Therefore, according to JCGM 100 (GUM) [16] and to JCGM 101 [17], a Monte Carlo simulation was set up to provide an assessment of the expanded uncertainty, with a confidence level of 95%, of the calibrated parameters.
3 Results and Discussion
This section discusses calibration results obtained by applying the standard methods M2 and M4 to the cases listed in Tables 3 and 4.
Results are discussed in terms of relative consistency amongst the methods, expanded uncertainty, method computational speed and correctness of the estimation. In particular, the method speed is evaluated in terms of iteration to achieve convergence, as this impacts on the calibration cost and computational effort. Estimate correctness is addressed for C_{f} and a_{2} and a_{0}. The first parameter is expected, from experience, to have order of magnitude at most of 10^{−3} mm·N^{−1}; the second, due to deviation from ideal geometry, is expected to be in the neighbourhood of its theoretical value, i.e. 23.97; the third, provided that the indenter mounted on the machine was not brand new, should cater for the faster increase in contact area at small penetration depths, and hence it is expected to be slightly positive [6, 11, 12, 13, 14, 15]. Even though wear and tip blunting affect the whole area shape function, so that a_{0}, a_{1} and a_{2} deviations from theoretical values are intertwined, simple forecast can be only made for a_{0} and a_{2}.
Relative expanded uncertainties (at a confidence level of 95%) of calibrated parameters
Method | \(U_{{{\text{rel}}\,C_{\text{f}} }}\) (%) | \(U_{{{\text{rel}}\,a_{0} }}\) (%) | \(U_{{{\text{rel}}\,a_{1} }}\) (%) | \(U_{{{\text{rel}}\,a_{2} }}\) (%) |
---|---|---|---|---|
M2_1 | > 100 | > 100 | 38 | 9 |
M2_1_bis | > 100 | > 100 | 62 | 12 |
M2_1_ter | > 100 | > 100 | > 100 | 18 |
M4_1 | 20 | 78 | 71 | 6 |
M4_1_bis | 55 | > 100 | 74 | 8 |
M4_1_ter | 51 | > 100 | > 100 | 12 |
Average value of calibrated parameters
Method | C_{f} (µm·N^{−1}) | a_{0} (nm^{2}) | a_{1} (nm) | a _{2} |
---|---|---|---|---|
M2_1 | 0.043 | 226 | 886 | 22.59 |
M2_1_bis | 0.041 | 2336 | 758 | 23.38 |
M2_1_ter | 0.368 | 21,124 | 322 | 26.87 |
M4_1 | 0.297 | 13,597 | 415 | 26.45 |
M4_1_bis | 0.159 | 9434 | 510 | 25.27 |
M4_1_ter | 0.268 | 20,054 | 386 | 25.95 |
As expected, cases with five load replications provided higher uncertainty to the measurement than ten replication cases. Even greater variability occurs when continuously increasing loads are adopted in the calibration procedure, though they should provide higher data density improving the fitting. In fact, when continuously increasing loads are adopted, Monte Carlo simulation shows the larger variability associated with the method definition since extracting single values from several distributions yields larger uncertainty than extracting multiple values from a smaller number of distributions.
In general, the use of two materials, i.e. method no. 4, yields lower uncertainty, thanks to the slight decoupling of frame compliance and area shape function parameters evaluation that is achieved by means of this procedure; in particular, method no. 2 is associated with poor performances generating relative uncertainties larger than 100%, which makes this method questionable for metrological purposes. However, high relative uncertainties are always expected when dealing with small numbers, i.e. mainly in the case of a_{0}. Furthermore, regression variability is the main uncertainty contribution and M4_1 produces the smallest expanded uncertainties.
Consistently with Herrmann et al.’s [14] results, M4 provides faster convergence with less than ten iterations for all the cases, whilst M2 requires, depending on the case, about thirty iterations for M2_1 and M2_1_bis and sixty for M2_1_ter.
Parameters are correctly estimated with respect to their theoretical value in all the cases, though high expanded uncertainty deeply affects these results; however, M4_1 represents an exception in this case as will be discussed in the following. As far as a_{2} is concerned, M2_1 and M2_1_bis tend to relatively underestimate the mean value.
Area shape function parameter a_{0} results to be slightly positive, as expected, and to include the nominal zero value in all the cases but for M4_1. Frame compliance calibration shows that a small but non-negligible correction is required to cater properly for testing equipment stiffness.
To conclude, C_{f} and a_{2} calibration provides an interesting validation of the results: provided the relationship established between these two parameters in Eq. 5, a proportionality is expected between their mean estimates, which is in fact shown comparing Figs. 10 and 11.
4 Conclusions
The present work discussed iterative methods for calibrating area shape function parameters and frame compliance for instrumented indentation testing equipment. These parameters were demonstrated to be major source of uncertainty in mechanical characterisation by means of instrumented indentation test, and the calibration methods considered are broadly adopted in practice because they do not require scanning force microscopes. Calibration methods are described in the related standard ISO 14577-2, and the present work investigates some degrees of freedom of the calibration procedure in the nano-range, which have been defined according to laboratories practice and literature references of the standards.
Methods show high expanded uncertainty resulting in poor precision which disguises a general lack of robustness and is enhanced when only one material for calibration is adopted, i.e. method no. 2. Method no. 4 yields more precise results with less computational effort, thanks to the different materials adopted for the calibration of the frame compliance and area shape function parameters. Even if method M4_1 is the most precise, its accuracy might be questioned.
Monte Carlo simulation showed that regression generates the larger contribution to uncertainty. Moreover, uncertainty is strongly dependent on the data set size and, in particular, it benefits from replicated data.
The standard appears to be little prescriptive since expanded uncertainty and accuracy of the results significantly depend on the choice of the method and its implementation. The authors highlight that further analysis shall be conducted to improve traceability of the instrumented indentation test, and they are working in that direction.
Notes
Acknowledgements
The authors would like to thank Dr Massimo Lorusso of Istituto Italiano di Tecnologia for his availability and expertise in operating the testing equipment and for having performed the measurements.
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