Analytical investigation on free vibration of circular double-layer graphene sheets including geometrical defect and surface effects

Effects of a single vacancy defect or a pin hole on free vibration behavior of a double layer graphene sheet are investigated. Using the nonlocal continuum theory as well as the Gurtin–Murdoch theory, the nonlocality and surface effects are considered in equations of motion. Both of in-phase and anti-phase vibration modes are analytically analyzed. Employing the translational addition theorem for cylindrical vector wave functions, the geometrical defect as a circular hole in arbitrary size and location is modeled. The van der Waals interaction between the upper and lower layers is included using the Lennard–Jones pair potential. The computational efficiency and accuracy of results are validated by literature. Effects of boundary conditions, geometrical properties, nonlocality and surface effect parameters on in-phase and anti-phase vibrational modes are investigated. Results reveal that the fundamental natural frequency of an annular double-layer graphene sheet with a free eccentric circular defect is less affected by the size and location of the defect. Moreover, the surface effect parameters have more significant effects on the in-phase vibration modes than the anti-phase ones.