Model for the onset of plasticity and hardness in bulk metallic glasses investigated by nanoindentation with a spherical indenter

Despite extensive research over the past three decades into how indentation depth affects the hardness (H) of both crystalline and non-crystalline materials, a mechanistic understanding of this phenomenon remains elusive. Here, we report that the depth dependence of H is also present in bulk metallic glasses. Importantly, indentation depth dependence is observed not only in hardness but also in the reduced elastic modulus E_r. We observed that H initially increases with increasing indentation depth h_t up to the yielding point. Beyond this point, however, it decreases with further increase of h_t, indicating the presence of an indentation depth dependence in the plastic regions. The evolution of E_r follows a similar trend. Based on our findings, firstly, we established the relationship between indentation hardness and the ratio of contact radius to indentation depth using classical Hertzian contact mechanics. Then, we developed a model based on the atomic-scale cooperative shear mechanism to interpret the indentation size effects in bulk metallic glasses. Furthermore, we observed that H correlates with the cube of the ratio of indentation elastic depth h_e to total depth h_t, or alternatively, with the ratio of indentation elastic work to total work. Our findings gave a scaling law, H = (h_e/h_t)³(H_e-H_p) + H_p, that uncovers an inherent relationship of hardness H with the mean pressure H_e at the onset of plasticity, flow hardness H_p, and the ratio h_e/h_t. The work underscores that the indentation depth effect stems from the interplay between elasticity and plasticity, rather than being solely influenced by factors like indentation depth, contact area, or indenter radius. This highlights its crucial role in comprehending and evaluating the plastic deformation of bulk metallic glasses at the submicron scale. © 2025 Elsevier Ltd.