Improved Nix-Gao-Feng model considering the strain-rate effect on indentation size effect

The indentation size effect (ISE) and strain-rate effect are two fundamental phenomena in nanomechanical characterization; however, a model that can predict the influence of the strain-rate effect on hardness measurements across different ISE categories remains lacking. Here, we develop an improved Nix-Gao-Feng model considering the strain-rate effect on ISE, termed the INGF-SS model, by systematically identifying and formulating the strain-rate dependence of three key parameters: the lattice friction contribution, the dislocation hardening contribution, and the plastic-zone evolution parameter. The model is established by first modifying the relationship between the plastic zone size ratio and indentation depth, and then determining mathematical descriptions for the rate-dependent evolution of these key parameters. The descriptive capability of INGF-SS model is validated against experimental data covering three ISE categories, normal ISE (N-ISE), normal-to-reverse ISE (NR-ISE), and complex ISE (C-ISE), across different materials under various strain-rates. The validation shows that the INGF-SS model can describe the rate-dependent of three ISE categories. Compared with a simplified INGF‑SS model accounting only for rate effects on lattice friction, the INGF‑SS model yields a higher mean determination coefficient (DC) of 0.9630 and a lower mean root mean square error (RMSE) of 0.4648 GPa. Furthermore, the predictive capability of the model is evaluated via two extrapolation strategies: the strategy using both low and high strain-rate data for parameters calibration achieves acceptable medium-strain-rate predictions for most materials, while direct extrapolation to the highest strain rate shows larger deviations at large indentation depths. This work thus provides a semi-empirical theoretical model for describing coupled size and rate effects in different materials under dynamic indentation.