Vibration analysis of graphene sheets resting on Winkler/Pasternak foundation: A multiscale approach

In the present article, an atomistic-continuum multiscale model is developed to study the free-vibration response of single-layered graphene sheets (SLGSs) embedded in an elastic medium based upon the higher-order Cauchy-Born (HCB) rule. In order to take both transverse shear stress and normal pressure into account, the elastic foundation is considered to be of Winkler-Pasternak type. The governing equations are derived within a variational formulation using a newly proposed method called Variational Differential Quadrature (VDQ). Using the VDQ approach together with the Generalized Differential Quadrature (GDQ) technique, the variational form of the governing equation is discretized in a computationally efficient manner. Finally, a generalized eigenvalue problem is solved to calculate the frequencies of SLGSs. The convergence and correctness of the presented numerical solutions are examined firstly. Then, a number of numerical examples are given to study the effects of boundary conditions, elastic medium and arrangement of atoms on the vibrational response of SLGSs. The present model does not involve any additional phenomenological input, and it considers size effect and material nonlinearity due to atomic interactions.