Universal scaling functions for continuous stiffness nanoindentation with sharp indenters

Abstract

This paper presents numerical and scaling analysis for continuous stiffness measurement (CSM) in nanoindentation tests. It shows numerically and experimentally that in CSM with sharp indenters the indentation displacement is proportional to stiffness and load is proportional to stiffness squared. The slopes of the two linear functions have been represented by two universal scaling functions obtained using self-similarity analysis. The scaled functions depend only on two governing parameters for materials with power law strain hardening behavior. One is the strain hardening exponent and the other is the nondimensional parameter ε ˜ Y ∗ = σ Y tan θ / E ∗ where E ∗ = E/(1 − ν 2), E is Young’s modulus, σ Y is yield stress, ν is Poisson’s ratio and θ is the equivalent half angle of the sharp indenter. The large deformation finite element method (FEM) has been used to simulate CSM nanoindentation and validate the universality of the functions over scaling parameters. To simulate small displacement oscillations applied on the indenter during CSM nanoindentation linear perturbation in the FEM computation has been used. Explicit equations for the universal scaling functions with and without friction have been obtained by fitting numerical results. The scaling functions asymptotically approach the elastic solution at low plasticity ( ε ˜ Y ∗ → ∞ ) and the full plastic solution at high plasticity ( ε ˜ Y ∗ → 0 ) and are valid in the elasto-plastic regime. Applicability of the scaling functions for nonpower-law stress–strain relations has been demonstrated with use of two representative strains.