Nonlocal vibration and buckling analysis of single and multi-layered graphene sheets using finite strip method including van der Waals effects

Detailed studies on the nanoscale vibration and buckling characteristics of rectangular single and multi-layered graphene sheets (SLGSs and MLGSs) are carried out using semi-analytical finite strip method (FSM), based on the classical plate theory (CPT). The displacement functions of the sheets are evaluated using continuous harmonic function series which satisfy the boundary conditions in one direction and a piecewise interpolation polynomial in the other direction. Nonlocal continuum mechanics is employed to derive the differential equation of the system. The weighted residual method is employed to obtain stiffness, stability and mass matrices of the graphene sheets. The effects of van der Waals (vdW) forces which are present as bonding forces between the layers are considered in the stiffness matrix of the system. The analysis of MLGSs is much more complex due to the influence of vdW forces. The mechanical properties of the graphene sheet are assumed in two ways as orthotropic or isotropic materials. A matrix eigenvalue problem is solved to find the natural frequency and critical stress of GSs subjected to different types of in-plane loadings including uniform and non-uniform uniaxial loadings. The accuracy of the proposed model is validated by comparing the results with those reported by the available references. Furthermore, a comprehensive parametric study is performed to investigate the effects of various parameters such as boundary conditions, nonlocal parameter, aspect ratio and the type of loading on the results.