Nanoscale vibration analysis of embedded multi-layered graphene sheets under various boundary conditions

The present article deals with the vibrational analysis of multi-layered graphene sheets with different boundary conditions amongst sheets. An elastic multiple-plate model is utilized in which the nested plates are coupled with each other through the van der Waals interlayer force. The interaction of van der Waals forces between adjacent and non-adjacent layers and the reaction from the surrounding media are included in the Reissner–Mindlin-type field equations on which the theoretical formulation is based. The set of coupled equations of motion for the multi-layered graphene sheets is then solved by the generalized differential quadrature method. The numerical analysis presented herein provides the possibility of considering various combinations of layerwise boundary conditions in a multi-layered graphene sheet. Based on exact solution, explicit formulas for the frequencies of a double-layered graphene sheet with all edges simply supported are also obtained. The results of the present numerical solution are shown to be in excellent agreement with those of exact solution for simply supported graphene sheets.