Understanding nanoindentation unloading curves

Experiments have shown that nanoindentation unloading curves obtained with Berkovich triangular pyramidal indenters are usually well-described by the power-law relation P = alpha(h - h(f))(m), where h(f) is the final depth after complete unloading and alpha and m are material constants. However, the power-law exponent is not fixed at an integral value, as would be the case for elastic contact by a conical indenter (m = 2) or a flat circular punch (m = 1), but varies from material to material in the range m = 1.2-1.6. A simple model is developed based on observations from finite element simulations of indentation of elastic-plastic materials by a rigid cone that provides a physical explanation for the behavior. The model, which is based on the concept of an indenter with an "effective shape" whose geometry is determined by the shape of the plastic hardness impression formed during indentation, provides a means by which the material constants in the power law relation can be related to more fundamental material properties such as the elastic modulus and hardness. Simple arguments are presented from which the effective indenter shape can be derived from the pressure distribution under the indenter.