Vibration investigation of circular graphene sheet with geometrical defect considering two-phase local/nonlocal theory exposed to the magnetic field

In this work, the mixed local/nonlocal elasticity theory is developed for the investigation of the vibration of a circular graphene sheet with a structural defect located in a magnetic field. When graphene is placed in a magnetic field, the Lorentz force is applied to it, which is calculated using Maxwell’s equations. The insufficiency of Eringen’s nonlocal theory (ENT) caused some authors to employ the two-phase theory (TPT) to study nanostructures. Geometric imperfections can happen in the manufacturing process of graphene sheets. Lots of these imperfections can be modeled as a hole. So, in this work, an imperfection is considered as the centric hole. Governing equations, in Newtonian formulation, are extracted in the integrodifferential form. The boundary conditions are selected as clamped at inner and outer edges. To discretize the equation of motion we employ Galerkin’s approach. The solution is validated using a comparison study between the presented results and those that exist in the literature, and the accuracy of the suggested approach is verified. The effectiveness of the mixture parameter, magnetic field, radius of imperfection, and nonlocal parameter is examined on the natural frequency. The results exhibit that an increase in the mixture parameter and magnetic field increases the natural frequency of the graphene sheet.