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 k e and Young’s modulus E is empirically given as a function of ν and the inclined face angle of the indenter, β , where k e is defined as k e  ≡  P / h 2 with the indentation load P and penetration depth of the indenter h . According to the linear relationship between k e 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 k e and β . For an isotropic elastoplastic solid, the indentation unloading parameter k 2 defined as k 2  ≡  P /( h – h r ) 2 for a residual depth h r is different from k e even though a linearly elastic solid with k e and elastoplastic solid with k 2 have a common E ∗ . In order to evaluate E ∗ of an elastoplastic solid, the corresponding k e is estimated from k 2 with an empirical equation as a function of the relative residual depth ξ defined as ξ  ≡  h r / h max for the maximum penetration depth h max . A nanoindentation experiment confirmed the validity of the numerical analysis for evaluating the elastic modulus.