An integral approach of indentation of Functionally Graded Materials

Determination of hardness profiles of Functionally Graded Materials could be currently performed by two principal methods. One method consists in the measurement of hardness on the cut cross-section of the sample, which is destructive, expensive and time consuming. Another method involves the measurement of hardness at the surface with increasing loads. This is an inverse problem that is usually solved in indentation by multilayer models. Our aim is to generalize the different multilayer models developed in recent years from geometrical considerations. This proposal is based on an integral approach of the centroid model proposed by Jonsson and Hogmark (JH) (1984). A brief calculation gives a definite integral relating the composite hardness obtained by normal indentation tests and the pressure exerted on a section of the indenter. The reasoning of JH model gives the cross-sectional hardness as a function of the distance from the top surface, i.e. the hardness profile, which is calculated thanks to the definition of the mechanical pressure exerted on a section of the indenter. Then, an inverse problem must be solved. Two gradient models are proposed and tested on data of previous works, the Gauss error function (erf function) that models diffusion phenomena, and a linear gradient model in a piecewise-defined function. In these models, the influences of the indentation size effect and of the residual stresses are not considered. An important application of this work is the prediction of the hardness profile of mechanical parts subjected to thermochemical treatments (carburizing, nitriding, carbonitriding, among others) or deposited coatings when indentation tests are carried out on the normal surface of the material. Hardness profile calculated from normal hardness measurements at the surface by using the proposed approach could provide an appropriated hardness description of the sample cross-section.