A critical examination of the relationship between plastic deformation zone size and Young's modulus to hardness ratio in indentation testing

Abstract

Existing indentation models (both analytical models and numerical analysis) show a linear relationship between δ_r/δ_m and H/E_r, where δ_r and δ_m are the residual and maximum indentation depth, and E_r and H are the reduced Young's modulus and hardness of the test material. Based on the analysis of Oliver and Pharr, a new relationship between δ_r/δ_m and H/E_r has been derived in a different way without any additional assumptions, which is nonlinear, and this has been verified by finite element analysis for a range of bulk materials. Furthermore, this new relationship for residual depth is used to derive an analytical relationship for the radius of the plastic deformation zone R_p in terms of the residual depth, Young’s modulus, and hardness, which has also been verified by finite element simulations for elastic perfectly plastic materials with different work hardening behavior. The analytical model and finite element simulation confirms that the conventional relationship used to determine R_p developed by Lawn et al. overestimates the plastic deformation, especially for those materials with high E/H ratio. The model and finite element analysis demonstrate that R_p scales with δ_r, which is sensible given the self-similarity of the indentations at different scales, and that the ratio of R_p/δ_r is nearly constant for materials with different E/H, which contradicts the conventional view.