Uncertainty

The determination of uncertainty associated with Rockwell hardness measurements is a relatively new concept for many users of Rockwell hardness as well as for laboratories engaged in hardness calibrations, such as test block standardizing agencies. Traditionally, the acceptance criterion for Rockwell hardness measurements has been through the use of acceptability tolerances. This has been true for most all aspects of the Rockwell method including measurements made as part of the direct and indirect verifications of hardness machines and the standardization of test blocks. Tolerance limits will continue to be used in Rockwell hardness. They provide general criteria for determining whether a Rockwell hardness machine is operating at an acceptable level of performance. What the acceptability tolerance limits do not indicate is the accuracy in the measurements made with the hardness machine. When it is important that the measurement accuracy be known, then the uncertainty in the measurements should be determined.

8.3.1 Uncertainty Limits

Uncertainty values are usually written as numerical limits bracketing the measurement value. Stating a measurement value in this way tells the user that the “true value" of the measurement would fall somewhere within these uncertainty bounds. As an example of uncertainty as it might apply to Rockwell hardness, consider a standardized hardness block that is certified with a value of 25.3 HRC ± 0.4 HRC. In this example, the 25.3 HRC value is the certified average hardness of the block, and the ± 0.4 HRC is the uncertainty in this certified value. This means that although the standardizing agency estimated the average hardness value of the test block to be 25.3 HRC, the “true value" would fall somewhere within 24.9 HRC to 25.7 HRC. For a complete understanding of this measurement and uncertainty, the values should be accompanied with a brief statement defining what the uncertainty interval represents. This statement should usually indicate the statistical process used to calculate the uncertainty and state the confidence level of the uncertainty interval.

It is important to understand the difference between uncertainty intervals, such as given in the example above, and acceptance tolerance limits traditionally provided with commercial test blocks. In the example above, the ± 0.4 HRC states that the standardizing agency can only estimate the “true" average hardness of the test block and that the “true value" falls somewhere within ± 0.4 HRC of 25.3 HRC. In contrast to these uncertainty limits, the certified value marked on commercial test blocks in the United States has included tolerance limits that reflect an ASTM acceptability requirement. This requirement states that when using the test block to conduct an indirect verification or daily check of a hardness machine, the machine’s measurement value must fall within these limits. As a rule, the acceptance tolerances have been stated in the same format as demonstrated above for uncertainty statements, for example 25.3 HRC ± 1.0 HRC. These are clearly two different concepts.

8.3.2 Highest Reference Standard

Before the level of uncertainty can be determined, the laboratory must choose a reference standard to which the measurement value will be compared. For example, the level of error in a Rockwell hardness machine might be determined by comparing the result of a measurement made on a standardized test block with the test block’s certified hardness value. It then follows that the certified value of the test block also includes a level of error with respect to another reference standard, typically the performance of the hardness machine used to standardize it. The performance of the standardizing hardness machine also includes a level of error with respect to a higher-level reference standard, and so it goes to the highest level of reference. The highest level of reference to which a measurement value is compared might be referred to as the “true value." The level of error in the measurement is then determined with respect to this “true value" taking into account the errors at each of the reference levels between the measurement value and the “true value". Ideally, the highest level of reference should be an internationally agreed upon standard. In some cases, international agreement does not exist; consequently national reference standards (i.e., NIST in the United States) are typically considered the highest reference level. At present, this is the case for Rockwell hardness.

8.3.3 Calculation of Rockwell Hardness Uncertainties

Over the past decade, there has been an increasing industry trend towards obtaining quality program accreditation, as well as obtaining accreditation for testing and calibration facilities. A common element of most of these programs is the requirement for reporting the uncertainty of measurement data. As a result, users of Rockwell hardness have struggled to develop procedures to determine the uncertainty of Rockwell measurements.

Currently, there are no generally agreed upon U.S. or international methods for calculating the measurement uncertainty of a Rockwell hardness machine or the uncertainty in the certified value of standardized test blocks. A reason for this may be that, until recently, there has been very little desire or need by industries that use Rockwell hardness to use uncertainty values. Also, the determination of Rockwell hardness uncertainty is not as straightforward as it would seem.

For example, suppose the uncertainty is to be calculated by combining all of the sources of error together. The errors associated with the hardness machine are typically not in hardness units, but they are in other units, such as force, length, and time. In order to determine an uncertainty in the hardness measurement, the relationships between how these errors affect the hardness value must be determined, often by experiment. Amplifying this problem is the fact that these relationships vary by Rockwell scale and hardness level and are often material dependent. In addition, the errors associated with a diamond indenter are difficult to identify and more difficult to relate to errors in hardness. Thus, it is clear that determining the hardness uncertainty by assessing the individual components of the hardness machine is extremely difficult to accomplish.

A different approach to determining Rockwell hardness uncertainty is to assume that by passing a direct and indirect verification, the errors in the individual operating components of the hardness machine are small enough that the indirect verification measurements are not the result of multiple large errors offsetting each other. Thus, the individual machine components can be considered to be operating together as a single component. The individual operating components include the force application system, depth measuring system, indenter, test cycle, and the remaining parts of the machine frame and test specimen support system. By considering the hardness machine as a single component, the uncertainties may be estimated with respect to the overall performance of the hardness machine without having to assess the uncertainty contributions for each of the separate machine components. When this approach is used, the most significant sources of error have been determined to be the following:

(1) Repeatability in the performance of the hardness machine. (2) Reproducibility in the day-to-day performance of the hardness machine, including operator influence. (3) Resolution of the measurement indicating display. (4) Uncertainty in the certified average hardness value of the reference test block. (5) Non-uniformity in hardness across the surface of the test block or test material. (6) Bias in the hardness machine measurement with respect to the reference standard to which traceability is claimed. (7) Determining the hardness machine measurement bias. (8) Correcting for the measurement biases. (9) The remaining bias in the hardness machine after a correction for bias is made.

As this guide is being written, there are efforts both internationally and within the United States to develop general procedures to assist Rockwell hardness standardization laboratories and users of Rockwell hardness in evaluating their measurement uncertainty. In the United States, the ASTM has initiated the development of such a procedure, and the ISO is to take up this issue at the next committee meeting of ISO TC164/SC 3 subcommittee on hardness testing in 2001.