Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity

A nonlocal elastic plate model accounting for the small scale effects is developed to investigate the vibrational behavior of multi-layered graphene sheets under various boundary conditions. Based upon the constitutive equations of nonlocal elasticity, derived are the Reissner–Mindlin-type field equations which include the interaction of van der Waals forces between adjacent and non-adjacent layers and the reaction from the surrounding media. The set of coupled governing equations of motion for the multi-layered graphene sheets are then numerically solved by the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions in a multi-layered graphene sheet. Based on exact solution, explicit expressions for the nonlocal frequencies of a double-layered graphene sheet with all edges simply supported are also obtained. The results from the present numerical solution, where possible, are indicated to be in excellent agreement with the existing data from the literature.