Graphene mechanics: II. Atomic stress distribution during indentation until rupture†
Previous Atomic Force Microscopy (AFM) experiments found single layers of defect-free graphene to rupture at unexpectedly high loads in the micronewton range. Using molecular dynamics simulations, we modeled an AFM spherical tip pressing on a circular graphene sheet and studied the stress distribution during the indentation process until rupture. We found the graphene rupture force to have no dependency on the sheet size and a very weak dependency on the indenter velocity, allowing a direct comparison to experiment. The deformation showed a non-linear elastic behavior, with a two-dimensional elastic modulus in good agreement with previous experimental and computational studies. In line with theoretical predictions for linearly elastic sheets, rupture forces of non-linearly elastic graphene are proportional to the tip radius. However, as a deviation from the theory, the atomic stress concentrates under the indenter tip more strongly than predicted and causes a high probability of bond breaking only in this area. In turn, stress levels decrease rapidly towards the edge of the sheet, most of which thus only serves the role of mechanical support for the region under the indenter. As a consequence, the high ratio between graphene sheets and sphere radii, hitherto supposed to be necessary for reliable deformation and rupture studies, could be reduced to a factor of only 5–10 without affecting the outcome. Our study suggests time-resolved analysis of forces at the atomic level as a valuable tool to predict and interpret the nano-scale response of stressed materials beyond graphene.