THE AUTOMATIC TESTERS IN MICROHARDNESS MEASUREMENT AND ISE EFFECT

The measurement of micro-hardness with the applied loads 0.09807 N, 0.24518 N, 0.49035 N and 0.9807 N has been carried out by three automatic and one manually-operated microhardness testers. The certified reference material (CRM) was the tested sample. Each operator obtained the readings of the tester, which she/he normally operates. The measurement was repeated after thirty months. The influence of the testers and their stability, as well as applied load on the measured values of the micro-hardness and the indentation size phenomenon (ISE), were evaluated. The parametric and non-parametric tests, Analysis of Variance (ANOVA), Zscore and Total Dispersion Zone were used for the evaluation of the statistical significance of obtained factors. The ISE was evaluated using Meyer’s and Proportional Specimen Resistance model and also by Hays – Kendall approach. The variability of measured values of the microhardness and parameters of ISE is high despite the use of automatic hardness testers with practically excluding the impact of the operator. The results are affected by operators, used testers and by applied loads. The measurement system can’t be considered to be stable.


Introduction
Measurement of micro-hardness can be carried out in a similar manner to the Vickers macroindentation tests with the diamond pyramid.However, the most important and intractable problem associated with low loads (the deep of indentation is less than 10 μm as a rule) is that concerned with a change in indentation size [1,2].The micro-hardness of solids depends on the applied load.The study of the relationship between micro-hardness and load has been carried out not only for metallic materials but also for semiconductors, glass, slag, ceramics, sintered materials and organic crystals [3][4][5][6][7][8].The dependence of measured values of the micro-hardness of solids on the applied load is known as the indentation size effect (ISE).It increases the uncertainty of the micro-hardness and may result in unreliable conclusions, particularly at low loads.They are required for the purpose of the measuring of small samples, coatings, thin layers or phases in metallography [9].The measured value of the micro-hardness is usually high if a low load is used; it decreases with an increase in applied load.Such phenomenon is called as "normal" ISE.It may be caused by DOI 10.12776 the testing equipment [10,11] or by intrinsic structural factors of the material: work hardening during indentation, load to initiate plastic deformation, elastic resistance and mixed elastic/plastic deformation, response of material [9,10,12], the effect of indenter/specimen friction resistance, the residual stress as the effect of the machining [10,[9][10][11][12][13].In the literature, there are many examples, which reveal that the "normal" ISE occurs in brittle materials including glass [10].In contrast to "normal" ISE, a reverse (inverse) ISE (RISE), where the apparent micro-hardness increases with increasing load, is also known.The reverse ISE essentially takes place in materials in which plastic deformation is predominant.
The repeatability is the condition of measurement, out of a set of conditions that includes the same measurement procedure, same operators, same measuring system, same operating conditions and same location, and replicate measurements on the same or similar objects over a short period of time [14].
An important aspect of maintaining the capability of the equipment to produce traceable and reliable measurement results is a determination of the maximum period that should be permitted between successive calibrations [15] -the interval of the calibration.Its length is related to the stability of the tester.The stability (or drift) is the total variation in the measurements obtained with a measurement system on the same standard or parts when measuring a single characteristic over an extended period.The measurement of stability is the change in bias over time.Knowledge of the equipment and measurement conditions help identify special causes when the system is unstable.Stability of measuring equipment is its property, whereby its metrological properties remain constant in time.It may be quantified by the duration of a time interval in which a metrological property or a quality is changed in a defined range.Stability is the absence of special causes of variation; the property of being in statistical control.It refers to both statistical stability of measurement process and measurement stability over time.Both are vital for a measurement system to be adequate for its intended purpose.Statistical stability implies a predictable, underlying measurement process operating within common cause variation (in-control).
Measurement stability addresses the necessary conformance to the measurement standard or reference over the operating life (time) of the measurement system [14,16].The validity of the three assumptions is analyzed in the present article: 1.The tester has no statistically significant effect on the value of the measured microhardness.2. The applied test load, in the relation to the Kick's law, has no statistically significant effect on the measured value of the micro-hardness.Otherwise, the presumption of the validity of Meyer's law is accepted.3. The differences between the values of the micro-hardness, repeatedly measured over a long period (the period between trial No. 1 and trial No. 2 was 30 months), under the same conditions are not statistically significant.In this case, the measurement system can be considered stable.To confirm or refute these assumptions were used the following statistical methods: Parametric and Non-parametric tests, Analysis of Variance (ANOVA), Z-score and Total Dispersion Zone.Existence, nature and size of the above mentioned ISE (statistically significant effect of the load on the micro-hardness) were evaluated by Meyer's, Proportional Specimen Resistance Model, and also by Hays -Kendall approach.DOI  The applied loads P were 0.09807 N, 0.24518 N, 0.49035 N and 0.9807 N.Each operator obtained readings of the tester, which she/he usually operates.An operator performed five indentations at each load.The result was the file of 20 indentations.The load duration time was 15 seconds, and the ambient temperature and the loading rate of automatic testers were in agreement with the standards [17,18].Some data concerning the trial No. 1 were presented in the paper [19].
To determine the stability of the equipment, the trial was repeated thirty months after trial No. 1 and was named as trial No. 2. The trial No. 2 was carried out by the same equipment, by the same operators on the same sample, and by the same conditions.The statistical significance of the applied load on the micro-hardness was tested by one way Analysis of Variance (ANOVA).Namely, if the value of p (p 2 in Tab. 2) is less than 0.05, the influence is significant.

Evaluation of the influence of the load on the micro-hardness
The parameters n and A ln are determined from a straight line graph of ln d (mm) versus ln P (N).Meyer's index n (work hardening coefficient) is calculated using the slope of the line.The value of A ln is the y-intercept of the straight line, Tab. 3. If n = 2, the micro-hardness is independent of the applied load and is given by Kick's Law.However, n<2 indicated "normal" ISE behavior.
If n >2, there is the reverse ISE behavior.
Several authors [9,11] have proposed that the Eq. ( 2) may describe ISE behavior: Li and Bradt pointed out that the parameters a 1 (N mm -1 ) and a 2 (N mm -2 ) of Eq. ( 2) are related to the elastic and plastic properties of the material, respectively [6].Eq. ( 2) may be rearranged in the form: ) The parameters a 1 , and a 2 of Eq. ( 3) may be obtained from the plot of P/d (N mm -1 ) against d (mm).Measured values of a 1 and a 2 are given in Tab. 3. The values of constants c 0 (N), c 1 ≈ a 1 (N mm -1 ) and c 2 ≈ a 2 (N mm -2 ) of Eq. ( 4), obtained from the quadratic polynomial regressions of P/d (N mm -1 ) against d (mm) are given in Tab. 4. The parameter c 1 characterizes the load dependence of micro-hardness (elastic properties).It consists of the elastic resistance of the test specimen, and the friction resistance developed at the indenter facet/specimen interface [6].The parameter c 2 is the measure of the load-independent microhardness (plastic properties).The ratio a 1 /a 2 or c 1 /c 2 may be treated approximately as a measure of the residual stress due to machining and polishing [11,20].Hays and Kendall proposed that there exists a minimum load W (N) necessary to initiate plastic deformation and below which only elastic deformation occurs.Then the load dependence of hardness is expressed by Eq. ( 5), where A 1 (N mm -2 ) is a constant independent of load.
The values of W and A 1 may be obtained from the regressions of P (N) against d 2 (mm) [10], and their measured values are given in Tab. 4. The constant A 1 (and also a 2 and, if necessary not very reliable c 2 ) can be used for calculation of "true hardness" H PSRA1 = 0.1891 A 1 .The load independent "true hardness" is free of the ISE effect.It is corresponding to the hardness obtained for an "infinite" load and characteristic length d* = a 1 /a 2 (or d*= c 1 /c 2 ) [21].Two-way ANOVA without replication was used for evaluation of the statistical significance of the effect of the tester and the trials on Meyer's index n.The effect of the tester is statistically significant (p = 0.0481) and the trials are not statistically significant (p = 0.25134).

Total Dispersion Zone
The value of the Total Dispersion Zone S M evaluates the ability of the testers achieve the same values of the micro-hardness for an individual load.It is necessary to calculate the mean values HV D , HV S , HV L, HV HP , HV HB and to calculate their standard deviations s D, s S, s L, s HP , s HB of five repeated measurements of the individual tester at different load [22].Total scatter zone S M will be calculated by Eq. ( 6) and ( 7) as a relative value: Mean standard deviation of all values of micro-hardness under the same load was calculated by Eq. ( 8) and ( 9): s v is a standard deviation of 5 mean values HV D , HV S ..., HV HB measured under the same load.
The sign tolerance T = 39 HV in Eq. ( 7), the same for all test loads, was calculated under maximal permissible error (10 % of 195 HV 0.05) according to standard [17].We regard S M % as follows: 0 to 20 % good, 21 to 30 % limited usable and more than 30 % unacceptable.As can be seen in Fig. 2, the values of S M % are "good" for all four applied loads and both trials.The differences between the results of hardness obtained by the individual tester are not significant by this method.

Z-score
The graphical method Z-score, employed for the visualization of results is routinely applied in inter-laboratory comparisons (round-robin tests).[23,24].As can be seen in Fig. 3, unsatisfactory results were not observed.The results are better in the trial No. 2. The increasing of the load also improves the results.    ) and also the load (p = 5.30 E10 -5 ) have statistically significant influence on the micro-hardness.The tester has a significant effect (p = 1.55 E -7 ) in the trial No. 2, but the effect of the load is not significant (p = 0.70012) on the micro-hardness.The interaction between the tester and the load is statistically significant in the trial No. 1. Taking into account all measured values, both the trial (p = 0.00074) and the tester (p = 1.80 E -21 ) have statistically significant effect on the micro-hardness.

Discussion
The assumption No. 1 (about the tester) can be rejected.The tester has a significant effect on measured value of the micro-hardness.It seems that the Total Dispersion Zone is not a sufficiently sensitive method for the measurement of the micro-hardness.The effect of the tester may be connected to different and, in general, high level of their uncertainty.This observation supports the fact that 4/5 of the used testers, including automatic, do not meet the requirements of the standard.The uncertainty of micro-hardness affects the evaluation of the ISE; therefore it can be the basis for decisions relating to the assumption No. 2. Petrík and Palfy in [26] investigated the relationship between the ISE and the uncertainty of the micro-hardness tester.
The assumption No. 2 (about the negligibility of the load) can also be rejected.The load has statistically significant effect on the measured value.This finding corresponds to the conclusion presented almost in all works cited in this paper.Detected ISE is the result of this impact.The problem is its nature and size.The sample with defined properties (CRM!) was used, however, normal and reverse ISE were detected.The tester influences the effect of the load on the microhardness.The evidence of this fact is a statistically significant amount of interaction between the load and the hardness in the trial No. 1.The Meyer's index is close to 2 for testers L and HP and the load affect the micro-hardness only negligibly.On the contrary, the results of tester S show a significant normal ISE while the results of the tester HB show significant reverse ISE.The influence of the operator is more significant than the impact of the equipment when using the manually-operated tester H.
The assumption No. 3 (the stability of testers) is also rejected.The changes of the mechanical properties of the sample over time are unlikely.This fact is the result of the progressive decrement of the metrological characteristics of the testers.Therefore the measurement system cannot be considered to be stable.The influence of the operator on the result is marginal for the automatic function of testers.High variability of the measured values was observed despite the same sample and automatic DOI 10.12776/ams.v22i3.758p-ISSN 1335-1532 e-ISSN 1338-1156 measurement system.The sample was the CRM with high uniformity of the microstructure and chemical composition.Therefore, it is expected that its micro-hardness and the residual stress due to machining and polishing of its surface are also uniform.For that reason, the testing equipment apparently may cause the ISE.The experimental error resulting from the measurement of the indentation diagonals is due to the limitations of the objective lens resolution, inadequate measurement capability of small areas of indentations and the determination of the applied load.They can also affect the nature and the size of the ISE [10,11,12].A variability of nature (normal and reverse) of ISE was observed on the same block and was measured manually by Hanemann tester.The value of n varied between 1.874 and 2.360 [26].High variability of n was also observed in repeated measurements of more hard (up to 392 HV0.05) reference blocks, for example [27].

Conclusion
All three assumptions mentioned in the introduction can be rejected.The variability of the micro-hardness and ISE parameters are affected by applied load and by hardness tester.4/5 of the used testers, including automatic, do not meet the requirements of the standard.Despite the using of automatic hardness testers with practically excluding the impact of the researcher, this variability was not solved.Moreover, these parameters vary with the time.The differences between the values of the micro-hardness repeatedly measured over a long period under the same conditions show that the measurement system is not considered as stable.Therefore, the repeated calibration of the tester with the calculation of the uncertainty and the ISE, as well as, the determination of the appropriate calibration interval due to the expected drift are necessary for the control of the measurement process.

Fig. 1
Fig. 1 Mean values of the micro-hardness, measured in both trials

Fig. 2
Fig. 2 Total Dispersion Zone -the values of S M %

x 2 .
is the mean micro-hardness, measured at individual load by an individual tester in one trial, x is specified hardness H c = 195 HV0.05 and "s" is the standard deviation of all (n = 200) measured values, Tab.The results | i z | ≤ 2 are satisfactory and | i z | ≥ 3 are unsatisfactory Files of values measured by individual testers in the trials No. 1 and 2 have normal distributions.But the file containing all values (n = 200) has other, very likely 2-Parameter Logistic DOI 10.12776/ams.v22i3.758p-ISSN 1335-1532 e-ISSN 1338-1156 distribution.The same distribution also has the file of the results, measured in the trial No. 2 (Tab.2).If the input data are non-normally distributed, it is appropriate to use non-parametric (robust) tests.Hence, if parametric t-test or the Analysis of Variance (ANOVA) is used, the results are not reliable.

Table 1
The tester Hanemann: loading rate v 1 and the speed of the penetration of the indenter v 2 .
The penetration speed was calculated only for manually-operated Hanemann tester.The penetration speed of the indenter into the test piece v 2 was calculated by dividing the depth of indentation (1/7 of the mean length of the indentation diagonals) with the time elapsed between the initial application of the load and the full test load.As shown in Tab. 1 the loading rate v 1 (N s -1 ) and the speed of the penetration of the indenter v 2 (μm s -1 ) of both operators are different and moreover change with time.Mean values of the micro-hardness of individual files (HV) and their standard deviation (s) are in Tab. 2 (HV).The statistical outliers were detected by Grubbs' test (significance level α = 0.05).No outlier was found.The absence of outliers suggests that the measurement process has avoided the gross errors and is under statistical control.The normality (determined by Freeware Process Capability Calculator software (Anderson -Darling test, if p 0.05 the distribution is normal) was confirmed for all files, Tab. 2. All files  DOI 10.12776/ams.v22i3.758p-ISSN 1335-1532 e-ISSN 1338-1156 have a normal distribution.But the set of all measured values (n = 200) has other than the normal distribution.

Table 2
Mean values of the micro-hardness of individual files (HV), their standard deviation (s), P 1the test of the normality: p-value, the repeatability r rel , maximum error of the tester E rel , relative expanded uncertainty of the calibration U rel , and p 2 as the p-value indicating the statistical significance of the load by ANOVA

Table 3
The values of Meyer's index n and indices A ln , a 1 and a 2

Table 4
The values of c 0 , c 1 , c 2 , W, A 1 and "true hardness" H PSRA1 .

5 .
The difference between files is not statistically significant according to paired t-test (1) and Mann-Whitney test (2).Other differences are significant.According to Kruskal-Wallis test, it can be greater than 99.99 % confident that Medians are different for both trials.As far as Mood's median test, there are not enough observations that are greater than the median for trial No. 1.Therefore, the test could not be used.For the trial No. 2, all differences are statistically significant according to this test.

Table 5
(2) results hypothesis teststhe difference between files is not statistically significant according to paired t-test (1) and Mann-Whitney test(2)Two way ANOVA with replication was used for evaluation of the statistical significance of the load, tester and the trial on the measured value of the micro-hardness.In the trial No. 1 the tester (p = 4.63 E