Nonlinear vibration analysis of graphene sheets resting on Winkler–Pasternak elastic foundation using an atomistic-continuum multiscale model

Based on the higher-order Cauchy–Born (HCB) rule, an atomistic-continuum multiscale model is proposed to address the large-amplitude vibration problem of graphene sheets (GSs) embedded in an elastic medium under various kinds of boundary conditions. By HCB, a linkage is established between the deformation of the atomic structure and macroscopical deformation gradients without any parameter fitting. The elastic foundation is formulated according to the Winkler–Pasternak model which considers both normal pressure and transverse shear stress effects. The weak form of nonlinear governing equations is derived via a variational approach, namely based on the variational differential quadrature (VDQ) method and Hamilton's principle. In order to solve the obtained equations, a numerical scheme is adopted in which the generalized differential quadrature (GDQ) method together with a numerical Galerkin technique is utilized for discretization in the space domain, and the time-periodic discretization method is used to discretize in the time domain. The effects of the arrangement of atoms, the Winkler and Pasternak coefficients of the elastic foundation, and boundary conditions on the frequency–response curves of GSs are illustrated. It is revealed that the nonlinear effects on the response of GSs with larger size in armchair direction are less important.