Modelling of nanomechanical nanoindentation measurements using an AFM or nanoindenter for compliant layers on stiffer substrates

Finite element analysis (FEA) is used to model the nanoindentation process for a rigid, spherically shaped indenter acting on an elastic two-phase system of an elastic layer that is more compliant than the underlying elastic substrate. A review of current analytical equations to model this process is made and compared to FEA. The FEA results may be expressed analytically by a simple function that describes the reduced modulus value obtained with Oliver and Pharr's method for any modulus value, thickness of layer or radius of the indenter tip. This function is used to investigate Buckle's rule, that to measure the properties of a layer, the indentation depth should be 10% or less of the total layer thickness. The results show that Buckle's rule is invalid for layer thicknesses below 5 mu m and a new rule is developed which depends on the layer thickness, the indenter radius and the ratio of the reduced moduli of the substrate and overlayer. This rule is based on FEA data. We present a guide to the analysis of the maximum depth that may be indented in order to keep the uncertainty in the reduced modulus for the layer to better than 10%.