The effect of magnetic field on buckling and nonlinear vibrations of Graphene nanosheets based on nonlocal elasticity theory

In this paper, the buckling behavior and nonlinear vibrations of graphene nanosheets in the magnetic field are studied analytically. By considering mechanical and magnetic interactions, new relationships have been proposed for the forces exerted by the magnetic field. The nonlinear governing equation is derived using Kirchhoff's thin plate theory in conjunction with the nonlocal strain gradient theory of elasticity and von Karman's nonlinear strain-displacement relation. The nonlinear governing equation is discretized using the Galerkin method. According to the method of multiple scales, the approximate analytical solutions are extracted. For the three considered boundary conditions, nonlinear natural frequencies and amplitude-frequency curves are computed for different values ​​of magnetic field and nonlocal parameters. The results show that increasing the nonlocal parameter and applying a magnetic field reduces the flexural stiffness and increases the in-plate compressive force which results in reducing the natural frequency. In addition, excessive magnification of the magnetic field causes static buckling. The value of the critical magnetic field is highly dependent on the type of boundary conditions.