Elastic surface deformation due to indenters with arbitrary symmetry of revolution

More than four years ago Pharr proposed the concept of the virtual rigid indenter to describe the elastic stress part of indentation problems beyond the linear elastic limit (Pharr 1999 Mechanical Properties of Films, Coatings and Interfacial Materials (IL Chiocco, Italy June 27-July 2, 1999). He suggested substituting the problem of a well defined indenter acting on a predeformed surface with the problem of an effectively shaped indenter and a flat surface. This way he was able to explain the unloading curves of quite a variety of different materials even with sharp indenters like Vickers or Berkovich indenters. The method works well even when for the sake of simplicity the assumed effective indenter is chosen to have symmetry of revolution, which a real indenter does not (e.g. Vickers and Berkovich indenters are four and three sided pyramids, respectively). To make this concept applicable, the resulting pressure distributions of a wide range of different indenter shapes are needed. The author presents solutions for indenters with symmetry of revolution with a radius (r) dependent shape described as a series of powers of r, up to power 8. In addition, the resulting potential functions are calculated in order to be able to construct the resulting elastic field completely analytically and in closed form. The effect of the different indenter shape on the resulting elastic indentation depth is discussed, and some practical examples are considered.