Representative indentation elastic modulus evaluated by unloading of nanoindentation made with a point sharp indenter

Abstract

The conventional method to extract elastic modulus from the nanoindentation on isotropic linearly elastic solids is based on Sneddon’s solution (1965). However, it is known that the solution is valid only for incompressive elastic solids with the Poisson’s ratio ν of 0.5. This paper first proposes the modification of the solution in a wide range of ν from 0 to 0.5 through the numerical analysis on the unloading behavior of a simulated conical nanoindentation with a finite element method. As a result of the modification, the coefficient of linearity between the indentation elastic parameter ke and Young’s modulus E is empirically given as a function of ν and the inclined face angle of the indenter, β, where ke is defined as ke≡P/h2 with the indentation load P and penetration depth of the indenter h. According to the linear relationship between ke and E, it is found that elastic rebound during unloading of a nanoindentation is uniquely characterized by a representative indentation elastic modulus E∗ defined in terms of E, ν and β, and that the value of E∗ can be evaluated from the P–h relationship with ke and β. For an isotropic elastoplastic solid, the indentation unloading parameter k2 defined as k2≡P/(h–hr)2 for a residual depth hr is different from ke even though a linearly elastic solid with ke and elastoplastic solid with k2 have a common E∗. In order to evaluate E∗ of an elastoplastic solid, the corresponding ke is estimated from k2 with an empirical equation as a function of the relative residual depth ξ defined as ξ≡hr/hmax for the maximum penetration depth hmax. A nanoindentation experiment confirmed the validity of the numerical analysis for evaluating the elastic modulus.