Quantifying uncertainty in load–hardness relationships

Abstract

This article presents methodology for constructing a probability space that quantifies the load–hardness relationship in ceramics. Aspects of this space are indicative of uncertainties introduced by variations in material microstructure, instrument repeatability, and technician skill. The developed method is general in nature, and can be made specific to particular types of hardness measurements or equations used to describe the load–hardness relationship such as Meyer’s law, the modified proportional specimen resistance model, and others. Construction of the probability space is accomplished by applying Bayesian hypothesis testing to determine the likelihood function of critical parameters of the chosen load–hardness equation. A demonstration of this methodology is presented for Vickers hardness measurements made at four applied loads on tungsten carbide. The utility of the technique in quantifying microstructural uncertainty is shown using Knoop hardness datasets for aluminum oxynitride and two types of silicon carbide. Analysis of the normality of hardness values was shown to provide an objective criterion for determining when enough measurements have been made to adequately describe material behavior. The probability spaces constructed for each material were used to quantify uncertainty in the load–hardness curve that would extend to predictions regarding microstructural features or performance based on this relationship.