Geometrical Effect Explains Graphene Membrane Stiffening at Finite Vacancy Concentrations

The presence of defects such as vacancies in solids has prominent effects on their mechanical properties. It not only modifies the stiffness and strength of materials, but also changes their morphologies. The latter effect is extremely significant for low- dimensional materials such as graphene. We show in this work that graphene swells while point defects such as vacancies are created at finite concentrations. The distorted geometry resulted from this areal expansion, in combination with the in-plane softening effect, predicts an unusual defect concentration dependence of stiffness measured for supported graphene membrane in nanoindentation tests, which explains the defect- induced stiffening phenomenon reported recently. The mechanism is elucidated through an analytical membrane model as well as numerical simulations at atomistic and continuum levels. In addition to elucidate the counter-intuitive observations in experiments and computer simulations, our findings also highlight the role of defect- modulated morphology engineering that can be quite effective in designing nanoscale material and structural applications.