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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.

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The axial and torsional wave propagation in a double-walled carbon nanotube (DWCNT) embedded on elastic foundations are investigated using nonlocal continuum shell theory. The effects of the surrounding elastic medium are considered using the spring constant of the Winkler-type and the shear constant of the Pasternak-type. The van der Waals (vdW) forces between the inner and the outer nanotubes are taken into account. The dynamic response of the carbon nanotube is formulated on the basis of nonlocal elasticity shell theory. The cut-off frequencies are obtained and it has been concluded that the cut-off frequencies are independent of small scale coefficient and shear modulus of the elastic medium. It has been found that the phase velocity sharply decreases by increasing the axial half wave number and approaches a constant value. It has also been concluded that the maximum phase velocity predicted by nonlocal theory is located between 5 and 10 nanometers while for local theories the phase velocity sharply decreases in this interval and approaches a constant value. Results show that the effect of Pasternak-type on phase velocity is significant but the effect of Winkler-type is not really considerable.