Strain Gradient Finite Element Analysis For Vibration Of Graphene Sheets

Graphene exhibits unique remarkable mechanical, electronic, thermotical and optical properties which have fascinated the scientific community. The vibrational characteristics of the graphene received extensive attention. Experiment, quantum mechanics, molecular dynamics(MD) and continuum mechanics are used to study the problem. In this thesis, we build a plate model for both the single-layered graphene sheets(SLGSs) and the double-layered graphene sheets(DLGSs) based on the strain gradient nonlocal elastic theory to study the vibration behaviors of graphene, which accounts for the small scale effect. The main progresses and contributions of the thesis are given as follows.The strain gradient theory is applied to build the dynamic equation of the SLGSs. An explicit formula is derived to predict the natural frequency of the SLGSs with all edges simple-supported. Then a 4-node 24-degree of freedom(DOF) Kirchhoff plate element is developed by the principle of virtual work to discretize the higher order partial differential equations. A good agreement between natural frequencies of the vibration of the simply-supported SLGSs predicted by finite element method(FEM) and that predicted by analytical solutions validates the reliability of the FEM. The small scale effect increases with the increase of the mode order and decrease of the size. Finally, this new FEM is used to study the influences of the effective wave length, nonlocal parameters and boundary conditions on the vibration behaviors of the SLGSs.An explicit formula is derived to predict the natural frequency of the DLGSs with all edges simple-supported based on the strain gradient theory, which considers van der Waals(vd W) interaction between layers. Then the 4-node 24 Kirchhoff plate element is applied to deal with the higher order strain gradient due to the consideration of small scale effect. It indicates the small scale effect increases with the increase of the mode order and nonlocal parameters. The vd W forces have significant effects on the low-order natural frequencies of DLGSs when the corresponding vd W coefficients decrease to a certain value.Analytical solutions for vibration of the equivalent Mindlin plate models of graphene are derived. Then a 4-node 36-DOF Mindlin plate element is proposed to investigate the vibrational characteristics of graphene and corresponding scale effect. FEM results indicate that the importance of small scale effect in the free vibration of the SLGSs is dependent on the geometric sizes, nonlocal parameters, vibration mode, and boundary conditions.