Determination of sensitivity coefficients and their uncertainties in Rockwell hardness measurement: a Monte Carlo method for multiple linear regression

Abstract

In the last few decades, many researchers have been studying how hardness measurements can be affected by possible influence variables (i.e. velocity of the indenter, dwell times, temperature, etc). This interest is particularly motivated by the newly adopted international definitions for the realization of Rockwell superficial hardness scales (HR45N, HR30N and HR15N) provided by the Consultative Committee for Mass and Related Quantities of Bureau International des Poids et Mesures, which deal with all the above-mentioned parameters. In this paper, the effect of two such parameters, namely the velocity of the final load application and the time interval of the force variation from the preliminary force value to the total force value, on superficial Rockwell hardness scales at different levels is studied and the related sensitivity coefficients are determined. The coefficients obtained are in the order of 10−3 HR s μm−1 and 10−2 HR s−1, respectively, in agreement with other National Metrology Institutes (NMIs), i.e. NIST and NPL. However, the uncertainties associated by the other NMIs are usually underestimated since they are simply given as the standard deviation calculated from the ordinary least squares method for the multiple linear regression, or, in other cases, not reported. For this reason, we propose a methodology for calculating the uncertainties of the sensitivity coefficients via a Monte Carlo method applied to multiple linear regression in order to consider the variability of both input and output quantities: with this method, the squared uncertainties are given as the squared sum of the standard deviation calculated from the ordinary least squares method and the uncertainty contribution due to the repeatability obtained via the proposed Monte Carlo method. The proposed method yields uncertainties of about 10−2 HR, while the uncertainties reported in other related published papers are in the order of 10−3 HR.