Elastic foundation effect on nonlinear thermo-vibration of embedded double-layered orthotropic graphene sheets using differential quadrature method

In this article, transverse nonlinear vibration of orthotropic double-layered graphene sheets embedded in an elastic medium (spring and shear constants of the Winkler and Pasternak models) under thermal gradient is studied using nonlocal elasticity orthotropic plate theory. The equations of motion are derived based on application of Hamilton's principles. These are coupled, two-dimensional and time-dependent equations, which cannot be solved analytically due to their nonlinear terms. Hence, differential quadrature method is employed to solve the governing differential equations for the two boundary conditions of simply and clamped support in all four sides. The plots for the ratio of nonlinear to linear frequencies versus maximum transverse amplitude for armchair and zigzag graphene sheet structures are presented to investigate the effects of nonlocal parameters, Winkler and Pasternak effects, temperature, and various aspect ratios. The study also indicates that the nonlinear effect represented by nonlinear frequency ratio is considerable at lower Winkler and Pasternak constants, length aspect ratio and thickness aspect ratio while it might be neglected for higher values of these parameters. Regarding the influence of temperature difference on support type, with increased temperature difference, nonlinear frequency ratio increases when the graphene sheet is simply supported, but for clamped one, no specific change in nonlinear frequency ratio is observed.