Analysing nanoindentation unloading curves using Pharr's concept of the effective indenter shape

In this paper, an extended Hertzian approach will be used in order to develop a tool for the analysis of nanoindentation unloading curves of sharp indenters. The approach is based upon the concept of the effectively shaped indenter introduced by Pharr and co-workers. However, in contrast to the Sneddon formulae used by Pharr, the extended Hertzian approach allows a complete evaluation of the elastic field produced by the effective indenter in closed and elementary form and thus a much more thorough discussion of the effects observed during the indentation experiment. As an example the new approach is used here to determine the critical yield stress of a variety of materials of ultra thin coatings directly from the unloading curves of Berkovich indentations. The results are compared with those of other methods and the deviations are discussed.