基于应变梯度有限元的单层石墨烯振动研究

建立了单层石墨烯等效非局部薄板的一种新的有限元模型，并运用有限元法分析不同边界条件下单层石墨烯振动的小尺度效应。给出了基于弹性应变梯度理论下Kirchhoff板的振动方程。发展了一种4节点24自由度的板单元，用于离散化求解考虑微纳结构尺度效应的高阶微分方程。在研究四边简支板振动时，考虑应变梯度的非局部弹性有限元数值计算结果与理论分析结果相一致。用有限元方法研究了不同尺寸、振动波长、振动模态阶数、边界条件类型以及非局部参数的单层石墨烯振动。

Finite Element Analysis of Strain Gradient on the Vibration of Single-layered Graphene Sheets

The vibration of single-layered graphene sheets (SLGS) is investigated using a new finite element method based on nonlocal continuum plate model with second order strain gradient taken into consideration. The paper starts with dynamic equation of Kirchhoff plate of strain gradient elasticity. Then a 4-node 24-degree of freedom (DOF) plate element is developed to discretize the higher order partial differential equations with the small scale effect taken into consideration. The finite element results are in good agreement with theoretical results in analyzing simply-supported SLGS, which validated the reliability of the finite element method. Finally, the finite element method is used to study the vibration of graphene sheets with different size,wave length，boundary conditions and the nonlocal parameter which show the small scale effects.