Abstract
The paper presents the basic ideas of the guidelines for the determination of the uncertainty which are presently being introduced into the ISO standards on the conventional hardness measuring methods Rockwell, Vickers, Brinell and Knoop. They are based on general guidelines for the evaluation of the uncertainty of measurements published by ISO (Guide to the Expression of Uncertainty in Measurement - GUM) and EA.

Introduction

In the past ten years, the efforts of industry to enhance the quality of products have been increasingly supported on the one hand by the development of high-precision measuring methods and measuring devices and, on the other hand, by the international standardization of the basis for quality assurance. In the field of hardness testing, this tendency can be seen for example by the development of the instrumented indentation test and the automatic image processing for the Vickers and Brinell measuring methods. During the standardization of the basis for quality assurance, apart from ISO 9001 [1], which describes a model for the quality assurance in the manufacturing process, several basic standards have been developed, which have sustainably influenced industrial measurements.
Among these are in particular the ISO "Guide to the Expression of Uncertainty in Measurement", which is abbreviated to "GUM", and the ISO 17025 standard on general requirements for the competence of testing and calibration laboratories [3]. The GUM was the first document to lay down a general, uniform guideline on how the uncertainty of measurement isto be determined. Basically, the GUM prescribes the following steps :

Quantify the essential influence quantities of the measurement
Determine the measurement deviations and the uncertainties of each influence quantity
Correct for the measurement deviations
Determine the sensitivity coefficients in order to transform the input quantities into the target quantity
Summarize the uncertainties of the influence quantities into the combined uncertainty via the sum of their variances
Calculate an expanded uncertainty for a given confidence level

The internationally harmonized ISO 17025 standard formulates clearly the requirements in particular for the traceability of measurements to national standards and with regard to the necessity of measurement uncertainty considerations when indicating the measurement results. In Germany we already have more than 20 DKD laboratories (DKD = German Calibration Service) in the field of the measurement quantity hardness which are accredited for the calibration of hardness reference blocks, indenters, hardness testing machines and rubber hardness testers. As for an accreditation, evidence of the measurement uncertainty is required, these laboratories performed important work in determining the uncertainty in the calibration of the hardness measuring means represented by them according to GUM and to the EA guideline for the estimation of the measurement uncertainty EA/4-02 derived from it [4].
Regardless of this remarkable progress one has to admit that the concept of measurement uncertainty is relatively new in industry because one used to operate here above all with the observance of tolerances.
After the publication of the GUM (which is kept quite general) it soon became clear that for the determination of the measurement uncertainty it will be necessary to elaborate guidelines specifically with regard to the measurement quantity. The requirement that for each measurement result its measurement uncertainty must be stated, can thus be realized best. Furthermore, it is necessary to apply uniform methods for the determination of the measurement uncertainty so that the values of the measurement uncertainty can be compared with each other. An important step in this direction was achieved through the elaboration of the EA 10-16 guideline on the estimation of the uncertainty of hardness measurements which was adopted within the framework of the EA (European Cooperation for Accreditation) organization [5]. On this basis, and by using the results of EU project "UNCERT" [6], it was decided during the periodic and just due revision of the ISO standards in the field of the hardness measuring methods Brinell, Vickers, Rockwell and Knoop (namely ISO 6506, ISO 6507, ISO 6508, ISO 4545) to supplement these standards by guidelines for the determination of the measurement uncertainty.
As these four ISO hardness standards lay down the requirements for the test method, the testing machines and the calibration of hardness reference blocks in three parts each, a total of twelve guidelines for the determination of the measurement uncertainty had to be elaborated. These guidelines are based as far as possible on uniform principles and take, at the same time, the specific features of the individual hardness measuring methods into account.
In the following, the basic principles of these guidelines for the determination of the measurement uncertainty will be presented.

Uncertainty of the Hardness Measuring Methods According to Rockwell, Vickers, Brinell and Knoop

The determination of the uncertainty of the hardness measuring methods relates to Part 1 of the ISO 6506, ISO 6507, ISO 6508 and ISO 4545 standards. To get a better understanding of how the uncertainty of measured hardness values comes about, the metrological chain for the definition and transmission of the hardness scales is shown in Fig. 1.

Grafik:

Figure 1:Metrological chain for the definition and transmission of the hardness scales

The chain starts on the international level with the definitions of different hardness scales in the ISO standards on the basis of which international comparisons of different hardness scales are carried out. With a series of hardness standardizing machines on the national level, primary hardness reference blocks are calibrated for the calibration in the world national metrology institutes exist which have set up hardness standardizing machines. In Germany, such hardness standards are operated at PTB. The accuracy of the hardness standardizing machines is achieved by direct calibrations on the highest level and verified by international comparisons. The largest part of hardness reference blocks is calibrated by the calibration laboratories of DKD. For this, hardness reference standard machines are used in the DKD laboratories which are traced back to the national standards by direct and indirect calibrations. The hardness reference blocks calibrated by them serve to monitor the hardness testing machines at the premises of the users, above all in industry. Besides this indirect calibration with hardness reference blocks, also direct calibrations of the essential influence quantities in hardness testing machines like test force, length measurement, indenter geometry and test cycle are carried out. When laying down the method for estimating the uncertainty we started from the assumption that data as easily available as possible will be used for this purpose. Also, the calculation should not be unnecessarily complicated. Therefore, the uncertainty of the hardness values measured with hardness testing machines is obtained from the measurement with hardness reference blocks and a test specimen. The influences are considered in detail in Table 1.

Influence of the uncertainty	Source of their determination
Uncertainty of the hardness refeence block u_CRM Standard uncertainty of the hardness testing maschine when measuring the hardness reference block u_H Standard uncertainty when measuring a test specimen u _x Standard uncertainty due to the resolution of the length measuring system u_ms	Calibration certificate of the hardness reference block Measurement result of the hardness reference block on the hardness testing machine Measurement result of a test specimen on the hardness testing machine Specification of the hardness testing machine

Table 1: Influences and sources of their dertermination taken into account in the uncertainty of the hardness measuring method

At this point, the method divides into two branches:

If the user of the hardness testing machine does not want to use the hardness measurement deviation
b = H - H_CRM
which can be determined with the hardness reference block, the permissible limit deviation u_E according to ISO 6508-2 (in the case of a Rockwell hardness testing machine) must be added to the influence quantities mentioned above. This yields Method 1.
If the hardness measurement deviation b is taken into account in the measurement result, the uncertainty of the hardness measurement deviation u_b has to be included instead of limit deviation u_E.This yields Method 2.

As u_E is a general maximum value which has to be taken from the ISO standard, the determination of the measurement uncertainty is simplified, but in general the measurement uncertainty according to Method 1 is larger than that according to Method 2.
The calculation formulas for Methods 1 (eq.1) and 2 (eq. 2) are as follows:

(1)
(2)

The root represents according to GUM the combined uncertainty which follows from the sum of the variances of the above discussed influence quantities. The expanded uncertainty U is then the combined uncertainty multiplied by the expansion factor 2. By this it is achieved that the hardness measuring value generally lies with a probability of approximately 95% in the interval given by the expanded uncertainty.
Method 2 can be recommended not only when the smallest possible measurement uncertainty has to be proven but also when a quality control card, e.g. with the statistical quantities mean value X and the range R, is conducted. The calculation of the measurement uncertainty is demonstrated by the example as given in Table 2:

Tabelle 2

Table 2: Calculation (in excerpts) of the uncertainty of the Rockwell hardness measurement

Uncertainty of the Calibration of Hardness Testing Machines

The calibration of hardness testing machines encompasses both the direct calibration of test force, length measurement, indenter geometry and test cycle and the indirect calibration of the overall function with hardness reference blocks.
The uncertainties for the four mentioned measurement quantities calculated on the basis of direct calibration are compared with the corresponding tolerances in ISO 6508-2 and ISO 4545-2. The uncertainty of the indirect calibration is compared with the permissible limit deviation of the hardness testing machine

Uncertainty of the Direct Calibration of Hardness Testing Machines
The uncertainty of the direct calibration will be demonstrated here by means of an example, i.e. the calibration of the test force.
The combined relative standard uncertainty of the test force calibration is calculated according to the following equation:
Formel Unsicherheit direkte Kalibrierung von Härteprüfmaschinen (3)
where:
u_FRS - relative measurement uncertainty of the force transducer (from the calibration certificate)
u_FHTM - relative standard uncertainty of the test force generated by the hardness testing machine
The results of the test force calibration:
Formel relative Prüfkraft und Formel relative Standardunsicherheit (4), (5)
where:
t = 1.32 for n = 3 and a = 68.3%
The calculation of the uncertainty of the test force is given in Table 4.

Tabelle 3

Table 3: Results of the force calibration

Tabelle 4

Table 4: Calculation of the measurment uncertainty of the test force

Finally, the maximum relative deviation of the test force ΔF_max including the measurement uncertainty of the used standard is calculated.

ΔF_max= ΔF_rel + U_F (6)

The result of the example means that the deviation of the test force including the measurement uncertainty of the used standard observes the tolerance in paragraph 4.2 of DIN EN ISO 6508-2.

Tabelle 5

Table 5: Calculation of the maximum relative deviation of the test force including the measurment uncertainty

Uncertainty of the Indirect Calibration of the Hardness Testing Machine

The measurement uncertainty of the indirect calibration of the hardness testing machine is calculated as follows:
Formel für Messunsicherheit der indirekten Kalibrierung der Härteprüfmaschine (7)
where:
u_H - Standard uncertainty of the hardness testing machine when measuring a hardness reference block
u_CRM - Calibration uncertainty of the hardness reference block
u_CRM-D- Hardness change of the hardness reference block since the last calibration due to drift (negligible when using the hardness reference block according to the standar)
u_ms - Uncertainty due to the resolution of the measurement system of the hardness testing machine

According to the calibration certificate for k = 1, the following example is considered :
Hardness reference block H_CRM = 45.4 HRC
Uncertainty of the hardness reference block u_CRM = ±0.5 HRC
Resolution of the measurement system of the testing hardness testing machine ms = 0.1 µm (= 0.05 HRC).

Using the hardness reference block on the hardness testing machine, the measuring results were obtained and are given in Table 6:

Table 6: Result of the indirect calibration

Formel 8

(8)

The measurement uncertainty according to eq. (8) is calculated and is given in Table 7.

(9)

Measurement quantity X_i	Estimated value x_i	Standard measurement uncertainty u(x_i)	Distribution type	Sensitivity coefficient c_i	Contribution to uncertainty u_i(H)
u_H	45.4 HRC	0.37 HRC	Normal	1.0	0 HRC
u_CRM	45.4 HRC	0.25 HRC	Normal	1.0	0 HRC
u_ms	0 HRC	0.058 HRC	Rectangle	1.0	0 HRC
u_CRM-D	0 HRC	0 HRC	Triangle	1.0	0 HRC
Combined uncertainty Combined uncertainty u_HTM					0 HRC
Expanded measurement uncertainty U_HTM (k = 2)					0 HRC

Table 7: Measurement uncertainty of the indirect calibration

Hardness H, measured on the hardness testing machine	45.1 HRC
Expanded measurement uncertainty U_HTM,HRC	0.9
Deviation of hardness testing machine when calibrated with hardness refence block II, HRC	0.5
Maximum deviation of hardness testing machine including measurement uncertainty Δ H_HTMmax, HRC	1.4

Table 8: Maximum deviation of hardness testing machine including measurement uncertainty

The result (Table 8) means that the permissible limit deviation of the hardness testing machine, which - according to paragraph 5 of ISO 6508-2 - amounts to ± 1.5 HRC, is observed.

Uncertainty of the Calibration of Hardness Reference Blocks

During the indirect calibration with primary hardness reference blocks, the overall function of the hardness reference standard machine is checked. Thereby, the repeatability and the deviation of the hardness reference standard machine from the actual hardness measuring value are determined. The measurement uncertainty of the indirect calibration of the hardness reference standard machine is determined with the following equation.

Formel für Messunsicherheit der indirekten Kalibrierung der Härte-Bezugsnormalmesseinrichtung (10)
where :
u_CRM-1- Calibration uncertainty of the primary hardness reference block according to the calibration certificate for k = 1
u_xCRM-1 - Standard uncertainty of the hardness reference standard machine due to its repeatability
u_CRM-D- Hardness change of the primary hardness reference block since its last calibration due to drift
u_ms - Uncertainty due to the resolution of the measurement system of the hardness reference standard machine

Eq. (10) corresponds in its structure to eq. (7) according to which the measurement uncertainty is determined during the indirect calibration of the hardness testing machine.The measurement uncertainty of the hardness reference block is then calculated as follows:
Formel für Messunsicherheit der Härtevergleichsplatte (11)

where:
u_CRM - Calibration uncertainty of the hardness reference block
u_xCRM-1 - Standard uncertainty due to the inhomogeneity of the hardness distribution on the hardness reference block

Measuring quantity X_i	Estmated value x_i	Standard uncertainty u(x_i)	Distribution type	Sensitivity coefficient c_i	Contribution to uncertainty u_i(H)
u_CRM-1	45.5 HRC	0.25 HRC	Normal	1.0	0.25 HRC
u_xCRM-1	0 HRC	0.11 HRC	Normal	1.0	0.11 HRC
u_ms	0 HRC	0.05 HRC	Rectangle	1.0	0.029 HRC
u_CRM-D	0 HRC	0 HRC	Trianggle	1.0	0 HRC
Combined uncertainty u_CM					0.27 HRC

Table 9: calculation of the uncertainty of a Rockwell hardness reference standard machine

In the example given in Table 9, first the uncertainty of a Rockwell hardness reference standard machine uCM is calculated with eq. (10). After that, the measurement uncertainty of the hardness reference block is determined with eq. (11).

Tabelle 10

Table 10: Determination of the inhomogeneity of the hardness reference block

The standard deviation (Table 10) of the hardness reference block due to the inhomogeneity is obtained from:

(12)

with t = 1.14 and n = 5 one obtains :

u_{xCRM_2} = 0.12 HRC

Finally, the expanded calibration uncertainty of the hardness reference block is calculated (Table 11).

Hardness of the hardness reference block H_CRM,HRC	45.4
Inhomogeneity of the hardness reference u_xCRM-2,HRC	0.12
Measurement uncertainty of the hardness reference standard machine u_CM, HRC	0.27
Expanded calibration uncertainty of the hardness reference U_CRM,HRC	0.59

According to the example given, an expanded uncertainty of the hardness reference block U = 0.6 HRC is thus obtained.

Summary

Within the framework of the revision of the standards ISO 6506, ISO 6507, ISO 6508 and ISO 4545 for the hardness measuring methods according to Brinell, Vickers, Rockwell and Knoop, guidelines for the determination of the measurement uncertainty for the hardness measuring method, the hardness testing machine and the hardness reference blocks were elaborated. These guidelines, which have been presented by means of calculation examples, show that the calculation of the measurement uncertainty can be carried out without difficulties. The introduction of the guidelines for the determination of the measurement uncertainty creates the preconditions for a further improvement of quality assurance in connection with products on which the hardness has to be measured. In particular, one requirement given in the basic standards of quality assurance can be fulfilled now: that in practical hardness measurement it must be possible to indicate for each hardness measuring value its uncertainty. If it is necessary to frequently carry out uncertainty calculations it is recommended to convert the guidelines into a software program. [8]

Acknowledgement

The guidelines for the determination of the measurement uncertainty here presented have been developed within the scope of a working group of DIN NMP 141 "Hardness testing of metals". Special thanks are owed to the members of this working group - Dr. Wehrstedt, Dr. Polzin, Mr. Patkovszky and Dr. Ullner - for their constructive contributions and discussions.

References
[1] DIN EN ISO 9001: [1] DIN EN ISO 9001: Quality management and quality assurance standards - Part 1 : Guidelines for selection and use
[2] BIPM, IEC; IFCC, ISO, IUPAC, IUPAP, OIML: Guide to the Expression of Uncertainty in Measurement, 1. Edition 1993, corrected and reprinted, Genf 1995
[3] DIN EN ISO/IEC 17025: [3] DIN EN ISO/IEC 17025: General requirements for the competence of testing and calibration laboratories
[4] EA/4-02: Expression of the Uncertainty of Measurement in Calibration, December 1999
[5] EA 10-16, Guidelines on the Estimation of Uncertainty in Hardness Measurements, 2001
[6] EA/LC(04)36 Draft: Guideline to the evaluation of the uncertainty of the Brinell and the Vickers measuring method, 2004
[7] Gabauer W., Manual of Codes of Practice for the Determination of Uncertainties in Mechanical Tests on Metallic Materials, The Estimation of Uncertainties in Hardness Measurements, Project, No. SMT4-CT97-2165, UNCERT COP 14: 2000

[8] T. Polzin, D. Schwenk.: Estimation of uncertainty of hardness testing; PC file for the
determination, Materialprüfung, 2002, 44(3), 64 - 71