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Instrumented Indentation Test in the Nano-range: Performances Comparison of Testing Machines Calibration Methods

  • Maurizio GalettoEmail author
  • Giacomo Maculotti
  • Gianfranco Genta
  • Giulio Barbato
  • Raffaello Levi
Original Articles
  • 23 Downloads

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 area 

1 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.

IIT requires performing a loading–unloading cycle, throughout which applied force and indenter displacement are measured by means of devoted transducers, to indent material. With reference to Fig. 1, cycle typically consists of a loading phase, a holding at a maximum load, to compensate for creep phenomena, and an unloading. From the analysis of the loading–unloading cycle, mechanical properties can be retrieved. This mechanical characterisation requires limited sample preparation, and it can be often considered a non-destructive test. Therefore, it is attractive for online quality control and rapid set-up of manufacturing processes. In fact, it can effectively provide throughout mechanical characterisation in terms of hardness, estimate of Young’s modulus, creep and relaxation behaviour of the material [4, 5, 6]. Furthermore, it proved to be capable of characterising microstructure of metallic materials by distinguishing amongst different phases and precipitates [7] and by estimating characteristic dimension of microstructure for both mono- and polycrystalline materials [8, 9, 10]. Therefore, given the widespread appeal of this technique, it was standardised in 2002 by means of the ISO 14577, which consists of four parts, and was reissued in 2015.
Fig. 1

Example of indentation curve: (a) loading curve, (b) holding at maximum load necessary for creep compensation, (c) unloading curve and the residual indentation hp

Mechanical characterisation can be achieved by processing the loading–unloading cycle when expressed in terms of applied force as a function of indenter displacement, i.e. F(h), the so-called indentation curve (IC), an example of which is shown in Fig. 1. However, data continuously measured during the indentation cycle require to be corrected to account for some measurement errors. In particular, h is affected by a zero error, h0, by the elastic deformation of the sample reference surface depending on the indenter shape, ε F/S (where S is the contact stiffness, i.e. the sample stiffness, and ε caters for indenter geometry), and by the elastic deformation of the indentation testing machine, FCf (where Cf is the frame compliance) [11]. Eventually, corrected displacement, hc, is calculated, according to Eq. 1.
$$h_{\text{c}} = h - h_{0} - \varepsilon \frac{F}{S} - C_{\text{f}} F.$$
(1)
Furthermore, mechanical characterisation needs to estimate the contact area surface, which, being IIT a depth sensing technique, can be retrieved directly from continuous measurement of h if the area shape function, A(h), is known. This relationship describes the lateral surface of the indenter as a function of the distance from its apex; for the most typical case of an ideal regular Berkovich indenter, i.e. a tetrahedron with tip dihedral angle of 130.06° (Fig. 2), it is A(h) = 23.97h2. However, due to deviation from ideal geometry, which is mostly due to blunting, offset of the tip and wear, it results in a more general quadratic function, which can be written in terms of both raw (h) and corrected (hc) displacement:
Fig. 2

Left: Berkovich indenter geometry (θ is half tip dihedral angle). Right: image of Berkovich indenter obtained with an atomic force microscope (AFM)

$$A\left( h \right) = a_{2} h^{2} + a_{1} h + a_{0} .$$
(2)

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 Cf 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 Cf 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.

First of all, the system can be thought as a series of springs modelling the testing machine, with compliance Cf, and the sample, with stiffness S. The resulting stiffness is actually measured from raw data, according to its definition as in Eq. 3. Moreover, considering the definition of reduced modulus Er, we can write Eq. 4, where E is Young’s modulus, ν is Poisson’s ratio and subscripts s and i stand, respectively, for sample and indenter. Substituting the right-hand side of Eq. 4 in Eq. 3, the linear relationship of Eq. 5 between 1/Sm and \({1 \mathord{\left/ {\vphantom {1 {\sqrt {A\left( {h_{{{\text{c}},{ \hbox{max} }}} } \right)} }}} \right. \kern-0pt} {\sqrt {A\left( {h_{{{\text{c}},{ \hbox{max} }}} } \right)} }}\) is obtained, whose intercept is the Cf and from which Eq. 6 follows.
$${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {S_{m} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${S_{m} }$}} = \left( {\left. {\frac{\partial F}{\partial h}} \right|_{{h_{ \hbox{max} } }} } \right)^{ - 1} = C_{\text{f}} + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 S}}\right.\kern-0pt} \!\lower0.7ex\hbox{$S$}}$$
(3)
$${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {E_{r} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${E_{r} }$}} = \left( {\frac{{1 - \nu_{s}^{2} }}{{E_{s} }} + \frac{{1 - \nu_{i}^{2} }}{{E_{i} }}} \right)^{ - 1} = \frac{{2\sqrt {A\left( {h_{{{\text{c}},{ \hbox{max} }}} } \right)} }}{S\sqrt \pi }$$
(4)
$${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {S_{m} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${S_{m} }$}} = C_{\text{tot}} = C_{\text{f}} + \frac{\sqrt \pi }{{2E_{r} \sqrt {A\left( {h_{{{\text{c}},{ \hbox{max} }}} } \right)} }}$$
(5)
$$A\left( {h_{{{\text{c}},{ \hbox{max} }}} } \right) = \frac{\pi }{{4E_{r}^{2} \left( {C_{\text{tot}} - C_{\text{f}} } \right)^{2} }}.$$
(6)
Thus, an iterative procedure, whose workflow is shown in Fig. 3, can be outlined to calibrate parameters and achieve convergence of the values obtained. Initialisation of the problem is performed in steps 1 and 2 assuming ideal conditions, i.e. infinitely stiff testing machine and ideal indenter geometry.
Fig. 3

Workflow of calibration iterative procedure

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 Cf, 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

The present work investigates performances of calibration methods for testing equipment to perform nano-indentation. A Triboindenter TI 950 by Hysitron (shown in Fig. 4), hosted in the facilities of the Istituto Italiano di Tecnologia and equipped with a regular diamond Berkovich indenter (elastic modulus 1140 GPa and Poisson’s modulus 0.07), shown in Fig. 2, was calibrated on reference samples, whose characteristics are summarised in Table 1.
Fig. 4

Hysitron TI 950 calibrated in this work

Table 1

Calibrated material characteristics mean and expanded uncertainty

Material

Calibration laboratory

E (GPa)

υ

SiO2

NPL

73.3 ± 0.6

0.161 ± 0.003

W

NPL

413.0 ± 2.8

0.281 ± 0.003

Oliver and Pharr [4] have outlined the working principle of the testing equipment and cycle, which enables the measurement of the IC. It consists of a rigid frame of which the indentation head is made up of a force and displacement transducers driven by an AC source and a controller, whose scheme is shown in Fig. 5. In particular, the TI 950 features a Hysitron-patented three-plate transducer, whose design (the scheme is shown in Fig. 6) aims at providing high sensitivity, a large dynamic range, a linear output signal and, by a low mass of the transducer mid-plate, small sensitivity to vibration. Table 2 summarises main metrological characteristics of the transducer.
Fig. 5

A schematic representation of the testing equipment: (A) sample; (B) indenter; (C) load application coil; (D) indentation column guide springs; (E) capacitive displacement sensor [4]

Fig. 6

Three-plate transducer actuation diagram.

(Courtesy of Hysitron)

Table 2

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.

According to the literature [12, 14, 15], the experimental plan, summarised in Tables 3 and 4, was outlined to properly cater for different degrees of freedom. The full force range of the commercial indentation platform Hysitron TI 950, i.e. from 0.5 to 10 mN, was considered. To cater for replicated measurements effect, the range was sampled according to a logarithmic scale; at these levels, ten and five replicated measurements were performed (respectively case “1” and “1_bis”). Furthermore, the force range was sampled at 0.25 mN, as trade-off between data set size and measurement time (case “1_ter”), to investigate effect of high-density individual measurements. Calibrated samples made out of fused silica and tungsten, due to their homogeneity and difference in mechanical properties (as reported in Table 1), were selected as calibration materials.
Table 3

Considered conditions for M2

Case

Material

Load range (mN)

Replications per each load

M2_1

SiO2

0.5–1–5–10

10 ×

M2_1_bis

0.5–1–5–10

5 ×

M2_1_ter

0.5:0.25:10

1 ×

Table 4

Considered conditions for M4

Case

Material

Load range (mN)

Replications per each load

M4_1

W

0.5–1–5–10

10 ×

 

SiO2

  

M4_1_bis

W

0.5–1–5–10

5 ×

 

SiO2

  

M4_1_ter

W

0.5:0.25:10

1 ×

 

SiO2

  
The measured contact stiffness is evaluated according to the power law method, described in ISO 14577-1 [6]. Sneddon’s general solution of Boussinesq’s problem states that the unloading curve (Fig. 1) yields a power law (PL) relationship between force and displacement, as shown by Eq. 7, in which the exponent m depends on the indenter geometry (for Berkovich indenter theoretical value is m = 2), whose equation can be obtained by fitting the experimental raw data. This analytical model is then differentiated to obtain Sm according to its definition in Eq. 3.
$$F = \alpha h^{m} .$$
(7)

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.

Uncertainty evaluation of the four calibrated parameters (a0, a1, a2 and Cf) is addressed to cater for the several influencing factors, i.e. calibrated sample and indenter material properties uncertainty, testing equipment force and displacement transducers accuracy and the regressions within the iterative procedure itself. Ishikawa diagram in Fig. 7 summarises influencing factors.
Fig. 7

Ishikawa diagram for influencing factors of standard uncertainty. SE stands for standard error

Individual contributions, which are represented by influencing factors covariances, can be combined according to Eq. 8 [16]: f is a generic function of the inputs x, i.e. the influencing factors, and u(xi, xj) is the covariance associated with xi and xj; partial derivatives are the sensitivity coefficients, which cater for sensitivity of the output y on the inputs x.
$$u_{\text{c}}^{2} = \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} \frac{\partial f}{{\partial x_{i} }}\frac{\partial f}{{\partial x_{j} }}u\left( {x_{i} ,x_{j} } \right).$$
(8)

Combined uncertainty, uc, 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 Cf and a2 and a0. 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 a0, a1 and a2 deviations from theoretical values are intertwined, simple forecast can be only made for a0 and a2.

Figures 8, 9, 10 and 11 show results for the calibrated parameters, respectively, a0, a1, a2 and Cf. Error bars of the figures represent expanded uncertainty with coverage factor equal to 2: black error bars graph overall variability, whilst green bars the variability component due to regression.
Fig. 8

Comparison of calibration methods. Parameter a0. Error bars represent expanded uncertainty with coverage factor 2: black—overall variability, green—regression variability

Fig. 9

Comparison of calibration methods. Parameter a1. Error bars represent expanded uncertainty with coverage factor 2: black—overall variability, green—regression variability

Fig. 10

Comparison of calibration methods. Parameter a2. Error bars represent expanded uncertainty with coverage factor 2: black—overall variability, green—regression variability. Red dash-dotted line is theoretical value

Fig. 11

Comparison of calibration methods. Parameter Cf. Error bars represent expanded uncertainty with coverage factor 2: black—overall variability, green—regression variability

The different methods show that the mean estimate of a2 and Cf quite oscillates about the expected value, which in some cases may result in bias, e.g. M2_1 and M2_1_bis provide relatively underestimated means. This notwithstanding, they all, but for the couple M2_1 and M4_1, provide results compatible with each other, even though this is mostly thanks to the high expanded uncertainty, as shown in Table 5, which summarises relative expanded uncertainties at a confidence level of 95%. For reference, average estimates of parameters are reported in Table 6.
Table 5

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

Table 6

Average value of calibrated parameters

Method

Cf (µm·N−1)

a0 (nm2)

a1 (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

Cf is expected to be at most 1 µm·N−1, a0 to be positive and a2 about 23.97

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 a0. 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 a2 is concerned, M2_1 and M2_1_bis tend to relatively underestimate the mean value.

Area shape function parameter a0 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, Cf and a2 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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Copyright information

© International Society for Nanomanufacturing and Tianjin University and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.DIGEPPolitecnico di TorinoTurinItaly

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