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

Existing indentation models (both analytical models and numerical analysis) show a linear relationship between delta(r)/delta(m) and H/E(r), where delta(r) and delta(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 delta(r)/delta(m) and H/E, 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 delta(r), which is sensible given the self-similarity of the indentations at different scales, and that the ratio of R(p)/delta(r) is nearly constant for materials with different E/H, which contradicts the conventional view.