Study on the effect of an eccentric hole on the vibrational behavior of a graphene sheet using an analytical approach

Because of the production process and constraint conditions, a circular graphene sheet may be opposed to structural defect and pin hole, respectively. Some of the defects and pin hole on a circular graphene sheet can be considered as an eccentric hole. So, analyzing the behavior of a circular graphene sheet with an eccentric hole is important. Free vibration of an eccentric annular graphene sheet, as the basis of any dynamical analysis, is analytically studied in this paper. Nonlocal thin plate theory is used to model the problem. The translational addition theorem for cylindrical vector wave functions is employed to solve the equation of motion for various boundary conditions. Results are compared with the literature, and their accuracy is approved. Effects of boundary conditions, geometrical properties and nonlocal parameter changes on symmetric and antisymmetric vibrational modes are investigated. It is approved that the eccentricity has a significant effect on the natural frequencies. Also, symmetric and antisymmetric modes of an annular graphene sheet have different behavior when geometrical and nonlocal parameters change.