Model for the nanoindentation hardness-depth relationships of ion-irradiated bicrystals with grain boundary effect

• A mechanistic model with GB effect is proposed for the depth-dependent hardness of ion-irradiated bicrystals. • The ISE caused by GNDs, irradiation hardening determined by inhomogeneously distributed irradiation defects, and the contribution of SSDs are addressed in the model for the first time. • The hardness-depth relation of ion-irradiated bicrystals is divided into four different stages by three critical indentation depths. • This model possessed a predictive capacity, and the predicted results were in good agreement with the experimental results. This study investigates the hardening mechanism of bicrystalline materials under ion irradiation by integrating experimental measurements with theoretical analysis. The experimental process involved He + irradiation and nanoindentation, which revealed stage-dependent hardening behaviors. To explain the relevant experimental phenomena, a mechanistic model was developed to characterize the depth-dependent hardness evolution in ion-irradiated bicrystals. The three dominant hardening mechanisms throughout the entire nanoindentation process have been systematically analyzed for the first time, including the indentation size effect (ISE) caused by geometrically necessary dislocations (GNDs), irradiation hardening determined by inhomogeneously distributed irradiation defects, and the contribution of statistically stored dislocations (SSDs). The former two factors, influenced by the average density of dislocations and irradiation defect within the plastic zone, are affected by the s boundary (GB) and indentation depth. Considering the influence of GB on both the geometrical configuration and expansion capacity of the plastic zone, the relationship between hardness and indentation depth in ion-irradiated bicrystals was explicitly derived across four different stages. Based on this model, the evolution of associated microstructures can be quantitatively assessed, encompassing the plastic zone volume, the average density of irradiation defect and GNDs. The validity and accuracy of the proposed model were validated by comparing the theoretical results with the experimental data obtained from nanoindentation tests on double-layer copper samples. Furthermore, the model demonstrates predictive capability, with predictions showing strong agreement with experimental results.