Flexural Wave Propagation of Double-Layered Graphene Sheets Based on the Hamiltonian System

Double-layered graphene sheets (DLGSs) as a new type of nanocomponents, with special mechanical, electrical and chemical properties, have the potential of being applied in the nanoelectro-mechanical systems (NEMS) and nanoopto-mechanical systems (NOMS). In DLGSs structure, the two graphene sheets are connected by van der Waals (vdW) interaction. Thus, it can exhibit two vibration modes during the propagation of the flexural wave, i.e., in-phase mode and anti-phase mode. Based on the Kirchhoff plate theory and the nonlocal elasticity theory, Hamiltonian equations of the DLGSs are established by introducing the symplectic dual variables. By solving the Hamiltonian equation, the dispersion relation of the flexural wave propagation of the DLGSs is obtained. The numerical calculation indicates that the bending frequency, phase velocity and group velocity of the in-phase mode and anti-phase mode for the DLGSs are closely related to the nonlocal parameters, the foundation moduli and the vdW forces. The research results will provide theoretical basis for the dynamic design of DLGSs in micro-nanofunctional devices.