The influence of punch blunting on the elastic indentation response

Elastic solutions of a family of axisymmetric problems concerning frictionless contact between a rigid punch and a semi-infinite substrate are considered using the method of Green and Zerna and of Collins. The analysis is relevant to the interpretation of experimental results in materials indentation testing, e.g. when substrate properties need to be determined from load-displacement traces, and precise information about the indenter tip shape is crucial. Commonly used solutions for ideal punch shapes, e.g. those having spherical or conical tips, may only be viewed as approximations since, in practice, indenter tips are neither perfectly round nor infinitely sharp. In order to illustrate the influence that small variations in punch shape may have on the contact behaviour, analytical solutions for a blunted Hertzian indenter and a rounded cone are obtained in parametric form, and their asymptotic behaviour at the extremes of low and high loads is investigated. A smooth punch is then considered of a general shape, given by a power series, and the resulting general solution is used as a basis for developing an inverse problem formulation of the tip shape calibration procedure. The method allows the best match between the measured and predicted load-displacement dependencies to be established. An example of the application of this procedure to the analysis of some nanoindentation data is presented.