A strain-gradient viscoplastic model with dislocation and grain-boundary mechanisms: from Hall-Petch strengthening to inverse Hall-Petch softening effects

Hall-Petch (HP) strengthening and inverse Hall-Petch (IHP) softening are two competing grain-size-dependent phenomena in crystalline materials. However, integrating these two effects into a predictive, cross-scale theoretical framework remains a challenge. We analyze the plastic deformation mechanisms in crystalline materials, discussing the competitive relationship between grain boundary (GB) sliding/diffusion and dislocation motion. We then quantify their respective viscous (rate-dependent) contributions and derive the critical grain size (Formula presented) and yield strength, both of which are in excellent agreement with experimental results. Different microstructures exhibit distinct viscous responses, enabling viscosity to serve as a descriptor of microstructure. By treating viscosity as a microstructure-sensitive bridge variable, the grain-size dependence of the competing dislocation-mediated and grain-boundary-mediated mechanisms is transferred into the constitutive length scale of the strain-gradient viscoplastic framework. We propose a strain-gradient viscoplastic constitutive model capable of capturing both HP strengthening and IHP softening effects. The model is validated through three illustrative cases—compression/shear of axisymmetric Cu thin films, torsion of Au microwires, and bending of Ni microbeams—accurately predicting mechanical responses across varying geometries and grain sizes. This study provides actionable guidance for grain size control and the design of advanced materials.