Finite element analysis of strain gradient middle thick plate model on the vibration of graphene sheets

The dynamics equation of the Mindlin middle thick plate model based on strain gradient theory is formulated to study the vibration of single-layered graphene sheets (SLGSs). Analytical solution of the natural frequency for free vibration of Mindlin plate with all edges simply-supported is derived. A 4-node 36-degree-of-freedom (DOF) Mindlin plate element is proposed to build the nonlocal finite element (FE) plate model with second order gradient of strain taken into consideration. This FE method is used to study the influences of the size, vibration mode and nonlocal parameters on the scale effect of vibration behaviors of SLGSs, which validates the reliability of the FE model for predicting the scale effect on the vibrational SLGSs with complex boundary conditions. The natural frequencies obtained by the strain gradient Mindlin plate are lower than that obtained by classical Mindlin plate model. The natural frequencies of SLGSs obtained by Mindlin plate model with first-order shear deformation taken into account are lower than that obtained by Kirchhoff plate model for both strain gradient model and classical case. The small scale effect increases with the increase of the mode order and the decrease of the size of SLGSs. ©, 2015, Editorial Office of Chinese Journal of Theoretical and Applied Mechanics. All right reserved.