材料梯度、几何非线性和高阶剪切效应对压电纳米梁力电特性的耦合影响

Coupling Effects of Material Gradient,Geometric Nonlinearity and High-order Shear Effect on the Electromechanical Properties of Piezoelectric Nanobeam

目的压电材料由于其优越的力电性能在MEMS/NEMS得到广泛应用。针对目前对压电纳米结构力电响应计算忽视了微/纳米尺度下压电材料的挠曲电效应以及剪切效应问题,提出了囊括挠曲电效应和压电效应的功能梯度压电(Functionally Graded Piezoelectric,FGP)纳米梁数学模型。纳米梁由压电层和功能梯度层组成,其中功能梯度层材料遵循幂律指数分布。方法首先,基于Reddy三阶剪切变形理论、非局部应变梯度理论(NGST)和哈密顿原理,并考虑了Von Kármán几何非线性,获得了梯度梁的非线性力电耦合控制方程及相应的边界条件;然后结合Runge-Kutta方法和Galerkin方法得到了简支梁的线性和非线性固有频率以及均方根(RMS)输出电压。结果提出的模型与已有文献结果对比十分吻合。此外,数值结果表明挠曲电常数、压电常数、应变梯度尺度参数、非局部参数、幂律指数和几何尺寸对非线性固有频率和均方根电压有影响。结论相较于Euler梁理论和Timoshenko梁理论,采用Reddy三阶剪切变形理论得到的梯度梁在相同质量下具有更高的RMS电压,同时会降低非线性固有频率。

Objective Piezoelectric materials are widely used in MEMS/NEMS due to their excellent electromechanical properties.A mathematical model of functionally graded piezoelectric(FGP)nanobeams including the flexoelectric effect and the piezoelectric effect was proposed to solve the problem that the flexoelectric effect and shear effect of piezoelectric materials at the microscale/nanoscale are neglected in the current calculation of the electromechanical response of piezoelectric nanostructures.The nanobeam consists of a piezoelectric layer and a functionally graded layer,while the materials of the functionally graded layer follow the power-law exponential distribution.Methods Firstly,based on Reddy's third-order shear deformation theory,nonlocal strain gradient theory(NSGT),Hamilton's principle,and Von Kármán geometric nonlinearity,the nonlinear electromechanical coupling governing equations and corresponding boundary conditions of the gradient beam were obtained.Then,the linear frequency,nonlinear natural frequency,and the root mean square(RMS)output voltage of the simply supported beam were obtained by combining the Runge-Kutta method and the Galerkin method.Results The presented model in this study was verified by existing literature results.In addition,the numerical results showed that flexoelectric constants,piezoelectric constants,strain gradient parameters,nonlocal parameters,power-law index,and geometric dimensions have an impact on the nonlinear natural frequency and RMS voltage.Conclusion Compared with the Euler beam theory and Timoshenko beam theory,the gradient beam with the same mass has higher RMS voltage and lower nonlinear natural frequency when using Reddy's third-order shear deformation theory.