Linear reduction of stiffness and vibration frequencies in defected circular monolayer graphene

Mechanical stiffness of monolayer graphene with randomly distributed vacancies is studied using molecular-dynamics simulation and elasticity theory. Nanoindentation is used to obtain Young's modulus and the effective spring constant which decrease linearly with the percentage of vacancies. The load and unload curves are the same and the breaking force and breaking points depend on the percentage of vacancies. Fracture may appear near the boundaries. We introduce a simple method to make the system vibrate by pulling up the atomic force microscopy tip from the center of the clamped circular monolayer graphene which then starts to vibrate.