全应变梯度挠曲电纳米梁有限单元法研究

挠曲电效应是一种存在于所有电介质材料中的特殊的力电耦合效应，本质上是应变梯度与电极化之间的线性耦合。然而，应变梯度会引入位移的高阶偏量，常给挠曲电问题的理论求解带来困难。且已有研究表明应变梯度弹性项会影响纳米结构中的力电耦合响应，但是现有的挠曲电研究大多忽略了应变梯度弹性的影响。因此，本文提出了一种既考虑应变梯度弹性，又考虑挠曲电效应的有效数值方法。基于全应变梯度弹性理论，建立了包含3个独立材料尺度参数的纳米欧拉梁的理论模型和有限元模型，提出了满足C₂弱连续的两节点六自由度单元。基于本文的有限单元法，以简支欧拉梁为例，通过分析讨论挠度、电势和能量效率，得到了挠曲电效应和应变梯度弹性项对梁的力电响应的影响。结果表明，挠曲电效应存在尺寸依赖性，且应变梯度弹性项在纳米电介质结构的挠曲电研究中的影响不可忽略。

Research on finite element method of nanobeam considering flexoelectricity based on general strain gradient elasticity theory

Flexoelectricity is a special electromechanical coupling that exists in all dielectrics.It is essentially a linear coupling between strain gradient and polarization.However,the strain gradient introduces high-order partial derivate of displacements,which often make it difficult to solve the flexoelectric problem theoretically.Moreover,studies have shown that the strain gradient elastic term affects the electromechanical coupling responses in nanostructures,but the effect of strain gradient elasticity was mostly ignored in the existing research.Therefore,this paper proposed an effective numerical method that considered both strain gradient elasticity and flexoelectricity.Based on the general strain gradient elasticity theory,this paper established a theoretical model and a finite element model of nano-Euler beams.Three independent material length scale parameters were included in the two models.And an element with two nodes and six degrees of freedom was proposed,which satisfied C₂ weak continuity.Based on the finite element method proposed in this study,a simply supported Euler beam was studied.By analyzing the deflection,electrical potential and energy efficiency,the effects of flexoelectricity and strain gradient elasticity on the electromechanical coupling responses were studied.The results show that the flexoelectricity is size-dependent,and the influence of the strain gradient elasticity on the flexoelectric response of nano-dielectric structures cannot be ignored.