Some results on the indentation of an elastic half space

Abstract

This paper presents some results on the indentation of an elastic half space. In the first part, an approximate analytical expression is proposed for the stress distribution on a half space under a flat punch with a cross-section in the form of a regular polygon. Accurate values for the β parameter are computed for triangular and square cross-sections. These values, corresponding, for example, to Berkovich and Vickers indenters, are, respectively, 1.06142 and 1.02121. In the second part, the influence of radial displacements on the β parameter is examined according to a previous idea of Bolshakov and Pharr [Mater. Res. Symp. Proc. 521 189 (1997).]. An alternative expression for this parameter is proposed giving lower values than those proposed by these authors. Since in most elastoplastic materials, the equivalent indenters used during unloading are in the form of paraboloids rather than cones, as in the well-known method of Oliver and Pharr [J. Mater. Res. 7 1564 (1992).], this case is examined in details. A simple formula is also proposed for the radial displacements in the case of indenters described by polynomials or obeying power laws with non-integer exponents. The conclusions concerning equivalent indenters in the form of paraboloids are then compared with preliminary experimental results obtained with a new indenter made of a cube-corner terminated by a Berkovich part. This indenter and another prismatic one allow a direct determination of the contact area, eliminating uncertainties due to the presence of pile-up or sinking-in.