An application of the asymptotic theory to a threshold model for the estimate of Martens Hardness

Abstract

Hardness measurements have a significant role in mechanical metrology, as they are frequently used to characterise materials properties relevant to industrial processes. A recently introduced method, called Martens Hardness, is based on force and indentation records obtained during a test cycle; the Force/Depth Curve, which describes the indetation pattern, is typically formed by two parts having a zero-point in common. A segmented regression model is proposed in this paper, based on the introduction of a threshold parameter in order to estimate the unknown zero-point. The problem is not trivial, since the relationship between observed force and indentation depth is structural and, moreover, the number of nuisance parameters grows with the number of measured data. The asymptotic likelihood theory leads to an estimate of the unknown parameters of the model. Monte Carlo simulations are resorted to in order to analyse the properties of estimators under different hypotheses about measurement errors, and to etablish the applicability conditions of the method proposed.