Classical and differential hardness: aspects of quantifying the deformation response in indentation experiments

Abstract

In depth-sensing nanoindentation, the load–depth curve F(h) is acquired, from which a single value for hardness H and a second for the indentation modulus E _ind are inferred. This is a very poor outcome since F(h) is a source of much more information. This paper describes a technique to extract the hardness H(h) as a continuous depth-dependent function from the load–depth-curve. This was accomplished by assigning each depth h a corresponding contact depth h _C = h _C(h) that can be calculated using an iteration algorithm. The hardness is then simply H(h) = F(h)/A _C(h _C(h)). For very simple area functions A _C, an analytical solution h _C(h; F) can even be found. Furthermore, the differential hardness H _d is introduced as an additional hardness quantity, which is obtained when dividing the load increment ΔF by the resulting increase in contact area ΔA _C. It turns out that H and H _d are identical quantities for a material of constant hardness only. When the hardness is depth- and, therefore, size-dependent, H _d differs from H in a definite way, which depends on the hardness evolution with depth, i.e. on the indentation size effect of the material under investigation. The differential hardness proves particularly useful for inhomogeneous samples and situations where the hardness is time-dependent.

Acknowledgments

The experimental results used in this paper were obtained during stays by the author at Technische Universität Dresden, Technische Fachhochschule Wildau (Laboratory of Surface Physics, Professor Asta Richter) and Universität des Saarlandes, Saarbrücken. The author would like to express his gratitude to the staff of these institutions for very fruitful cooperation.