Thermo-mechanical vibration of double-layer graphene nanosheets in elastic medium considering surface effects; developing a nonlocal third order shear deformation theory

Due to the superior mechanical properties of double-layer graphene sheets (DLGSs) for applications in nano devises, understanding the mechanical response and stability of such structural elements is essential and inevitable. In this paper a modified nonlocal continuum model is developed to consider the surface energy effects and to evaluate the size-dependent mechanical response of nanoscale structures. Moreover, an efficient nonlocal third order shear deformation plate theory (TSDT) is employed to analyze the vibrational behavior of DLGSs embedded in an elastic medium. An analytical method is introduced for determining the natural frequencies of DLGS and an accurate exact solution could be found for the problem which is validated via comparing the results for some simpler problems found in the literature. Based on the results, the developed theory predicts the fundamental natural frequencies more accurately compared to the other common theories. Moreover, comprehensive numerical results are given for the obtained size-dependent natural frequencies of the DLGSs. The influences of surface effects, small scale coefficient, stiffness of the internal springs, external foundation parameters, temperature changes and different dimensions of nanosheet on the frequencies in both in-phase and anti phase vibrational modes are studied.