Size-dependent static and dynamic responses of embedded double-layered graphene sheets under longitudinal magnetic field with arbitrary boundary conditions

This paper deals with the static and dynamic response of orthotropic double-layered graphene sheets (DLGSs) resting on visco-Pasternak foundation subjected to longitudinal magnetic field as well as mechanical transverse load with arbitrary boundary conditions based on first order shear deformation theory (FSDT) in the framework of Eringen's differential constitutive model. In order to obtain more accurate results in dynamic response, the properties of each single layer graphene sheet (SLGS) are assumed to be viscoelastic. The governing equations of motion are obtained via energy method and Hamilton's principle. Two solution procedures are proposed to solve the governing equations of motion. In the first method, the equations are solved analytically by means of Navier's and Laplace inversion technique in the space and time domains only for simply supported boundary conditions. In the second method, the governing equations are discretized numerically on the space and time domains via Ritz method and Newmark scheme, respectively, which are suitable for arbitrary boundary conditions. Employing the second solution procedure as the main approach, parametric studies are carried out to explore the effects of the magnetic parameter, nonlocal parameter, boundary conditions, structural damping, stiffness and damping coefficient of the foundation, aspect ratio, length to thickness ratio and van der Waals (vdW) interaction. Results indicate that with increasing the value of vdW interaction, the deflection of upper layer decreases while it increases for lower layer. Moreover, it is observed that when one edge of the nanoplate changes from free to simply supported or from simply supported to clamped, the deflection amplitude and time of response decrease and therefore the nanoplate reaches the equilibrium state much faster. The presented results can be used in practical design and manufacturing of graphene reinforced nanocomposites.