考虑挠曲电效应的压电纳米薄板力-电-热耦合特性研究

摘要：挠曲电效应是由应变梯度引起的,与尺度相关的力电耦合效应。基于Kirchhoff板假设和挠曲电理论,本文推导了温度和电压作用下的压电薄板力-电-热耦合微分控制方程,定量分析了微分控制方程中非线性项的影响,并针对四周固支压电薄板采用Ritz法求解,数值计算了压电薄板的弯曲和振动行为。在研究温度和挠曲电效应对薄板耦合特性和力学行为的影响时,本文分别考虑了材料系数不随温度变化和随温度线性变化两种情况。以PZT-5H为例,我们讨论了挠曲电和温度对压电薄板的横向位移和固有频率的影响。研究结果表明挠曲电效应对压电纳米薄板的力学行为影响很大,且具有明显的尺寸效应。此外,薄板对温度变化非常敏感。因此,可通过挠曲电效应和温度来调控压电纳米薄板的多场耦合特性和力学行为,进而优化基于压电薄板的NEMS/MEMS中传感器、作动器等电子器件的性能。     

基于广义应变梯度理论的纳米梁挠曲电效应研究

挠曲电效应是应变梯度与电极化的耦合,它存在于所有的电介质材料中。在纳米电介质结构的挠曲电效应研究中,应变梯度弹性对挠曲电响应的影响一直以来被低估甚至被忽略了。根据广义应变梯度理论,应变梯度弹性中独立的尺度参数只有三个,而文献中所采用的一个或两个尺度参数的应变梯度理论只是它的简化形式。基于该理论,论文建立了考虑广义应变梯度弹性的三维电介质结构的理论模型,并以一维纳米梁为例研究了其弯曲问题的挠曲电响应及其能量俘获特性。结果表明,纳米梁的挠曲电响应存在尺寸效应,并且弹性应变梯度会影响结构挠曲电的尺寸效应,特别是当结构的特征尺寸低于尺度参数时。论文的工作为更进一步理解纳米尺度下的挠曲电机理和能量俘获特性提供理论基础和设计依据。     

电学开路挠曲电悬臂梁自振频率分析

挠曲电效应是一种新兴的机电耦合效应,在微纳米尺度的传感器、致动器和俘能器方面有广阔的应用前景．本文基于挠曲电材料的变分原理和电吉布斯自由能,推导了表面覆盖电极的挠曲电悬臂梁在电学开路条件下的机电耦合动力学控制方程和相应的力电边界条件．进一步获得了求解电学开路条件下挠曲电悬臂梁自振频率的超越方程．以聚偏氟乙烯(PVDF)材料为算例,讨论了挠曲电系数、末端质量块和梁尺寸对结构自振频率和电学开路/短路条件下结构自振频率有效频移的影响．计算结果表明,挠曲电系数的增大会提高梁的自振频率;末端质量的增大可以降低梁的自振频率,并且末端质量块的转动效应对悬臂梁自振频率的影响很小;悬臂梁结构的有效频移随着结构尺寸减小而增加,并在某一厚度尺寸趋于饱和值．     

考虑电场梯度的挠曲电纳米板弯曲性能分析

具有尺度依赖的挠曲电效应在器件的设计中扮演着越来越关键的角色, 研究人员在微纳米尺度多物理场分析中进行了大量工作. 基于考虑挠曲电和电场梯度效应的弹性介电材料非经典理论, 以二维纳米板为例, 通过理论建模, 分析纳米板在弯曲问题中的力?电耦合行为. 根据Mindlin假设给出板的位移场和电势场的一阶截断, 选取板的材料为立方晶体(m3m点群), 将广义三维本构方程代入到高阶应力、高阶偶应力、高阶电位移和高阶电四极矩的表达式中得到相应的二维本构方程, 利用弹性电介质变分原理得到板的控制方程和边界上的线积分等式, 分别将二维本构方程和边界上外法线的方向余弦代入, 得到板的高阶弯曲方程、高阶电势方程以及对应的四边简支边界条件. 利用四边简支矩形板的高阶弯曲方程、高阶电势方程和相应的边界条件, 根据Navier解理论, 求解纳米板的电势场, 重点分析电场梯度对板内一阶电势的影响. 数值计算结果表明: 电场梯度对纳米板中由挠曲电效应产生的一阶电势有削弱作用, 且材料参数g₁₁越大, 一阶电势受到的削弱越大; 同时电场梯度的存在消除了纳米板在受横向集中载荷作用时一阶电势的奇异性. 本文是对具有挠曲电效应和电场梯度效应的纳米板结构分析理论的一个扩展, 为微纳米尺度器件的结构设计提供参考.      

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。     

多孔介质的热-电-化-力学耦合理论及应用

本文针对自然界中不同种类的多孔介质,评述了包含化学效应的多场耦合力学问题的国外内研究现状.介绍了研究多场耦合问题的理论架构,并基于唯象理论和热力学理论,建立了一般性的热-电-化-力学多场耦合理论,在此基础上简化为化学-力学耦合理论,利用相应的控制方程和本构关系, 给出了线性耦合系统的变分原理,并证明了化学-力学耦合理论架构的封闭性. 基于所提出的化-力耦合模型,通过数值算例解释了多孔介质中的化学-力学耦合现象.最后对多孔介质,特别是对活体生物软组织中的多场耦合研究中存在的问题进行了讨论,并展望了本领域的未来发展趋势.     

具有叉指式电极的1-3型压电复合材料层合板力电耦合行为的三维精确分析

基于三维弹性理论和压电理论,导出了含有1-3型压电复合材料层的有限长矩形层合简支板的静力平衡方程和边界条件,给出了该层合板在叉指式电极和外力共同作用下力电耦合特性的三维精确解.数值算例的计算结果与有限元解进行了对比,取得了很好的一致性.研究了压电矩阵各向异性和刚度矩阵各向异性以及电势等因素对其挠曲面扭率最大值的影响.数值结果表明层合板扭率最大值的绝对值随压电矩阵各向异性系数Rd的增大而增大并随刚度矩阵各向异性系数Rc的减小而增加.     

悬臂梁挠曲电俘能器的力电耦合模型及性能分析

挠曲电效应指应变梯度在电介质中引起的电极化现象,是一种普遍存在的力电耦合行为。应变梯度与材料的尺寸成反比,因此挠曲电效应有望在纳米尺度主导材料的物理性质,尤其是力电耦合性能。本文建立了悬臂梁挠曲电俘能器的理论模型,基于哈密顿原理得到了悬臂梁挠曲电俘能器的控制方程和相应的边界条件;进一步,得到了悬臂梁挠曲电俘能器的输出电压频率响应和功率密度频率响应随悬臂梁的振动频率、外电路阻抗、挠曲电层厚度以及弹性层模量的变化规律。聚偏氟乙烯和环氧树脂层合挠曲电悬臂梁俘能器模型的数值结果表明输出电压频率响应和功率密度频率响应在共振频率点取得最大值,且随着各阶模态对应的共振频率的增加悬臂梁挠曲电俘能器的输出电压和功率密度均增加。此外,计算结果还表明悬臂梁俘能器存在最佳匹配阻抗,在匹配阻抗附近悬臂梁俘能器的输出功率密度随挠曲电层厚度的减小而增大,表现出明显的尺寸效应。本文工作提供了一种基于挠曲电效应的悬臂梁俘能器的理论模型,为悬臂梁俘能器的设计提供了理论依据。     

智能软材料热""电-化-力学耦合问题的研究进展

正智能软材料是指具有较低模量,在外场(如热、电、磁、化、光等)作用下能够产生较大力学响应的材料。本文评述了聚合物胶体、水凝胶以及关节软骨等典型智能软材料的多场耦合力学问题的国内外研究现状,重点讨论了这类智能软材料以化学扩散为特征的热-电-化-力学耦合的基本理论和研宄方法。详细介绍了具有代表性的国内外主要研究团体的研究进展。阐述了基于热力学理论和哈密顿原理所建立的一般性热-电-化-力学多场耦合理论框架。针对等温过程的化学-力学耦合的本构关系和控制方程,证明了化学-力学耦合     

悬臂梁挠曲电俘能器的力电耦合模型及性能分析

挠曲电效应指应变梯度在电介质中引起的电极化现象，是一种普遍存在的力电耦合行为。应变梯度与材料的尺寸成反比，因此挠曲电效应有望在纳米尺度主导材料的物理性质，尤其是力电耦合性能。本文建立了悬臂梁挠曲电俘能器的理论模型，基于哈密顿原理得到了悬臂梁挠曲电俘能器的控制方程和相应的边界条件；进一步，得到了悬臂梁挠曲电俘能器的输出电压频率响应和功率密度频率响应随悬臂梁的振动频率、外电路阻抗、挠曲电层厚度以及弹性层模量的变化规律。聚偏氟乙烯和环氧树脂层合挠曲电悬臂梁俘能器模型的数值结果表明输出电压频率响应和功率密度频率响应在共振频率点取得最大值，且随着各阶模态对应的共振频率的增加悬臂梁挠曲电俘能器的输出电压和功率密度均增加。此外，计算结果还表明悬臂梁俘能器存在最佳匹配阻抗，在匹配阻抗附近悬臂梁俘能器的输出功率密度随挠曲电层厚度的减小而增大，表现出明显的尺寸效应。本文工作提供了一种基于挠曲电效应的悬臂梁俘能器的理论模型，为悬臂梁俘能器的设计提供了理论依据。     

An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

In this paper, we analytically study vibration of functionally graded piezoelectric(FGP) nanoplates based on the nonlocal strain gradient theory. The top and bottom surfaces of the nanoplate are made of PZT-5 H and PZT-4, respectively. We employ Hamilton's principle and derive the governing differential equations. Then, we use Navier's solution to obtain the natural frequencies of the FGP nanoplate. In the first step, we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies. In the second step, we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded(FG) nanoplates. Finally, the effects of the FG power index, the nonlocal parameter, the aspect ratio, and the lengthto-thickness ratio, and the nanoplate shape on natural frequencies are investigated.     

Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales

This paper presents a nonlinear thickness-shear vibration model for onedimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity. The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle. The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode. Only the shear strain gradient through the thickness is considered in the present model. With geometric nonlinearity, the governing equations are converted into differential equations as the function of time by the Galerkin method. The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation. Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent, and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates. The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly. The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices.     

Buckling and post-buckling analyses of size-dependent piezoelectric nanoplates

This paper attempts to investigate the buckling and post-buckling behaviors of piezoelectric nanoplate based on the nonlocal Mindlin plate model and von Karman geometric nonlinearity. An external electric voltage and a uniform temperature rise are applied on the piezoelectric nanoplate. Both the uniaxial and biaxial mechanical compression forces will be considered in the buckling and post-buckling analysis. By substituting the energy functions into the equation of the minimum total potential energy principle,the governing equations are derived directly, and then discretized through the differential quadrature(DQ) method. The buckling and post-buckling responses of piezoelectric nanoplates are calculated by employing a direct iterative method under different boundary conditions. The numerical results are presented to show the influences of different factors including the nonlocal parameter, electric voltage,and temperature rise on the buckling and post-buckling responses.     

Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory

In this paper, the free vibration of magneto- electro-elastic （MEE） nanoplates is investigated based on the nonlocal theory and Kirchhoff plate theory. The MEE nanoplate is assumed as all edges simply supported rectan gular plate subjected to the biaxial force, external electric potential, external magnetic potential, and temperature rise. By using the Hamilton＇s principle, the governing equations and boundary conditions are derived and then solved analytically to obtain the natural frequencies of MEE nanoplates. A parametric study is presented to examine the effect of the nonlocal parameter, thermo-magneto-electro-mechanical loadings and aspect ratio on the vibration characteristics of MEE nanoplates. It is found that the natural frequency is quite sensitive to the mechanical loading, electric loading and magnetic loading, while it is insensitive to the thermal loading.     

四角点支撑纳米板稳态受迫振动解析解

本文基于非局部弹性理论及辛叠加方法,得到放置在黏弹性介质上四角点支撑矩形纳米板稳态受迫振动问题的解析解．将纳米板受迫振动问题导入哈密顿体系,得到哈密顿控制方程,在无需任何预设函数的情况下可直接对哈密顿控制方程进行求解,得到简支纳米板稳态受迫振动问题在辛空间展开形式的解析解．进而通过边界叠加,可求出四角点支撑纳米板稳态受迫振动的解析解．数值算例中验证了本文应用辛叠加方法得到解析解的准确性,并以石墨烯纳米板为例,分析了非局部参数和黏弹性介质参数对四角点支撑石墨烯纳米板稳态受迫振动的影响．结果表明,非局部参数和黏弹性介质参数的变化会影响石墨烯纳米板的共振频率及共振幅值．     

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials (FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated effective material properties, various homogenization schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less significant for a higher value of the material property gradient index. Furthermore, among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.     

Size-dependent shear buckling response of FGM skew nanoplates modeled via di?erent homogenization schemes

The size e?ects on the shear buckling behaviors of skew nanoplates made of functionally graded materials(FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated e?ective material properties, various homogenization schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the signi?cance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less signi?cant for a higher value of the material property gradient index. Furthermore,among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the di?erence between them reduces by increasing the aspect ratio of the skew nanoplate.