BENDING PERFORMANCE ANALYSIS OF FLEXOELECTRIC NANOPLATE CONSIDERING ELECTRIC FIELD GRADIENTS
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摘要: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 g 11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.Abstract:The size-dependent flexoelectric effect plays an increasingly critical role in the design of smart devices. Researchers have done much work in multi-physics field analysis at the micro- or nano-scale. The electromechanical coupling behavior of nanoplate in the bending problem is analyzed based on the non-classical theory of elastic dielectric materials considering the flexoelectric effect and electric field gradient effect, using two-dimensional nanoplate as an example. The Mindlin assumption is used to obtain the first-order truncation of the displacement field and electric potential field of the Mindlin plate, the material of plate is assumed to be a cubic crystal in m3m class, the two-dimensional constitutive equations are obtained by substituting the three-dimensional constitutive equations into the expressions of higher-order stress, higher-order couple stress, higher-order electric displacement and higher-order quadrupole, the governing equations of the plate and the line integral equation on the boundary are simultaneously derived through the elastic dielectric variational principle, hence the higher-order bending equations, the higher-order electric potential equation, and the corresponding simply-supported boundary conditions of the rectangular plate are obtained by substituting the two-dimensional constitutive equations and the directional cosine on the boundary into the governing equations of the plate and the line integral equation on the boundary, respectively. According to the higher-order bending equations, higher-order electric potential equation, the corresponding simply-supported boundary conditions of rectangular plate, and the Navier solution theory, the electric potential field of nanoplate are analytically solved, with a focus on the influence of electric field gradient effect on the electric potential in the plate. The numerical results show that the electric field gradient weakens the first-order electric potential generated by the flexoelectric effect in the nanoplate, and the greater the material parameter g 11, the greater the weakening of the first-order electric potential. In addition, the existence of the electric field gradient eliminates the singularities of the first-order electric potential of nanoplate under transverse concentrated loading. Present work can be seen as an extension of the structural analysis theory of nanoplate with flexoelectric effect and electric field gradient effect, which provides a reference for the structural design of micro- or nano-scale devices.
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图 1(a) Mindlin板中面上的直角坐标系和(b) 变形前和变形后板在x2= 0平面上的几何形状
Figure 1.(a) Rectangular coordinate system on the midplane of Mindlin plate and (b) geometry of the plate edge on thex2= 0 plane before and after deformation
图 3四边简支矩形板受集中载荷P作用
Figure 3.Four-sided simply supported rectangular plate subjected to concentrated loadingP
图 4四边简支矩形板的一阶电势φ(1)分布
Figure 4.First-order electric potentialφ(1)of the four-sided simply supported Mindlin plate
图 5不同的材料参数g11对四边简支矩形板在x2=b/2处的一阶电势的影响
Figure 5.Influences of different material parametersg11on the first-order electric potential of four-sided simply supported plate atx2=b/2
6四边简支矩形板的电四极矩分量云图
6.Higher-order electric quadrupole of four-sided simply supported Mindlin plate
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[1] Deng Q, Kammoun M, Erturk A, et al. Nanoscale flexoelectric energy harvesting.International Journal of Solids and Structures, 2014, 51: 3218-3225doi:10.1016/j.ijsolstr.2014.05.018 [2] 陈春林, 李肇奇, 梁旭等. 悬臂梁挠曲电俘能器的力电耦合模型及性能分析. 固体力学学报, 2020, 41(2): 159-169 (Chen Chunlin, Li Zhaoqi, Liang Xu, et al. Electromechanical coupling model and performance analysis of the unimorph cantilever beam-based flexoelectric energy harvester.Chinese Journal of Solid Mechanics, 2020, 41(2): 159-169 (in Chinese) [3] 李鹏. 压电复合结构中弹性波传播特性分析及其在高性能声波器件中的应用研究. [博士论文]. 西安: 西安交通大学, 2017Li Peng. Investigation of elastic waves propagation properties in piezoelectric composite structures and applications for acoustic wave devices with high performance. [PhD Thesis]. Xi’an: Xi’an Jiaotong University, 2017 (in Chinese) [4] Pan E. Exact solution for simply supported and multilayered magneto-electro-elastic plates.Journal of Applied Mechanics, 2001, 68: 608-618doi:10.1115/1.1380385 [5] Ho CM, Tai YC. Micro-electro-mechanical-systems (MEMS) and fluid flows.Annual Review of Fluid Mechanics, 1998, 30: 579-612doi:10.1146/annurev.fluid.30.1.579 [6] 张乐乐, 刘响林, 刘金喜. 压电纳米板中SH型导波的传播特性. 力学学报, 2019, 51(2): 503-511 (Zhang Lele, Liu Xianglin, Liu Jinxi. Propagation characteristics of SH guided waves in a piezoelectric nanoplate.Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(2): 503-511 (in Chinese) [7] 孟莹, 丁虎, 陈立群. 带附加质量块的压电圆板能量采集器振动分析. 力学学报, 2021, 53(11): 2950-2960 (Meng Ying, Ding Hu, Chen Liqun. Vibration analysis of a piezoelectric circular plate energy harvester considering a proof mass.Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(11): 2950-2960 (in Chinese) [8] 李世荣. 功能梯度材料明德林矩形微板的热弹性阻尼. 力学学报, 2022, 54(6): 1601-1612 (Li Shirong. Thermoelastic damping in functionally graded Mindlin rectangular micro plates.Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1601-1612 (in Chinese) [9] Yang J. An Introduction to the Theory of Piezoelectricity. New York: Springer, 2005 [10] Wang B, Gu Y, Zhang S, et al. Flexoelectricity in solids: progress, challenges, and perspectives.Progress in Materials Science, 2019, 106: 100570doi:10.1016/j.pmatsci.2019.05.003 [11] Bhaskar UK, Banerjee N, Abdollahi A, et al. A flexoelectric microelectromechanical system on silicon.Nature Nanotechnology, 2016, 11(3): 263-266doi:10.1038/nnano.2015.260 [12] Deng Q, Lü S, Li Z, et al. The impact of flexoelectricity on materials, devices, and physics.Journal of Applied Physics, 2020, 128: 080902doi:10.1063/5.0015987 [13] Qu YL, Jin F, Yang JS. Effects of mechanical fields on mobile charges in a composite beam of flexoelectric dielectrics and semiconductors.Journal of Applied Physics, 2020, 127: 194502doi:10.1063/5.0005124 [14] Quang HL, He QC. The number and types of all possible rotational symmetries for flexoelectric tensors.Proceedings of the Royal Society A:Mathematical,Physical and Engineering Sciences, 2011, 467(2132): 2369-2386doi:10.1098/rspa.2010.0521 [15] Shen S, Hu S. A theory of flexoelectricity with surface effect for elastic dielectrics.Journal of the Mechanics and Physics of Solids, 2010, 58: 665-677doi:10.1016/j.jmps.2010.03.001 [16] Shu L, Wei X, Pang T, et al. Symmetry of flexoelectric coefficients in crystalline medium.Journal of Applied Physics, 2011, 110(10): 104106doi:10.1063/1.3662196 [17] Maranganti R, Sharma ND, Sharma P. Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions.Physical Review B, 2006, 74: 014110doi:10.1103/PhysRevB.74.014110 [18] El Dhaba AR, Gabr ME. Flexoelectric effect induced in an anisotropic bar with cubic symmetry under torsion.Mathematics and Mechanics of Solids, 2020, 25(3): 820-837doi:10.1177/1081286519895569 [19] Qu YL, Jin F, Yang JS. Magnetically induced charge redistribution in the bending of a composite beam with flexoelectric semiconductor and piezomagnetic dielectric layers.Journal of Applied Physics, 2021, 129: 064503doi:10.1063/5.0039686 [20] Qu YL, Jin F, Yang JS. Flexoelectric effects in second-order extension of rods.Mechanics Research Communications, 2021, 111: 103625doi:10.1016/j.mechrescom.2020.103625 [21] Qu Y, Jin F, Yang J. Torsion of a flexoelectric semiconductor rod with a rectangular cross section.Archive of Applied Mechanics, 2021, 91(5): 2027-2038doi:10.1007/s00419-020-01867-0 [22] Li A, Zhou S, Qi L, et al. A flexoelectric theory with rotation gradient effects for elastic dielectrics.Modelling and Simulation in Materials Science and Engineering, 2016, 24: 015009doi:10.1088/0965-0393/24/1/015009 [23] Qu YL, Zhang GY, Fan YM, et al. A non-classical theory of elastic dielectrics incorporating couple stress and quadrupole effects: part I–reconsideration of curvature-based flexoelectricity theory.Mathematics and Mechanics of Solids, 2021, 26(11): 1647-1659doi:10.1177/10812865211001533 [24] Gao XL, Zhang GY. A non-classical Mindlin plate model incorporating microstructure, surface energy and foundation effects.Proceedings of the Royal Society A:Mathematical,Physical and Engineering Sciences, 2016, 472: 20160275doi:10.1098/rspa.2016.0275 [25] Li YS, Pan E. Static bending and free vibration of a functionally graded piezoelectric microplate based on the modified couple-stress theory.International Journal of Engineering Science, 2015, 97: 40-59doi:10.1016/j.ijengsci.2015.08.009 [26] Yang J. The Mechanics of Piezoelectric Structures. Singapore: World Scientific, 2006 [27] Mindlin RD. An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. Singapore: World Scientific, 2006 [28] Yang J. Mechanics of Piezoelectric Structures. Singapore: World Scientific, 2020 [29] Koiter WT. Couple-stresses in the theory of elasticity.Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen B, 1964, 67: 17-44 [30] Mindlin RD. Influence of couple-stresses on stress concentrations.Experimental Mechanics, 1963, 3(1): 1-7doi:10.1007/BF02327219 [31] Zhang GY, Gao XL, Guo ZY. A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium.Acta Mechanica, 2017, 228: 3811-3825doi:10.1007/s00707-017-1906-4 [32] Mindlin RD. High frequency vibrations of piezoelectric crystal plates.International Journal of Solids and Structures, 1972, 8: 895-906doi:10.1016/0020-7683(72)90004-2 [33] Mindlin RD. Polarization gradient in elastic dielectrics.International Journal of Solids and Structures, 1968, 4: 637-642doi:10.1016/0020-7683(68)90079-6 [34] 胡淑玲, 申胜平. 具有挠曲电效应的纳米电介质变分原理及控制方程. 中国科学: G辑, 2009, 39(12): 1762-1769 (Hu Shuling, Shen Shengping. Variational principles and governing equations in nano-dielectrics with the flexoelectric effect.Science China Physics,Mechanics&Astronomy, 2009, 39(12): 1762-1769 (in Chinese) [35] Gao XL, Mall S. Variational solution for a cracked mosaic model of woven fabric composites.International Journal of Solids and Structures, 2001, 38: 855-874doi:10.1016/S0020-7683(00)00047-0 [36] Wang L, Liu S, Feng X, et al. Flexoelectronics of centrosymmetric semiconductors.Nature Nanotechnology, 2020, 15: 661-667doi:10.1038/s41565-020-0700-y [37] Yang XM, Hu YT, Yang J. Electric field gradient effects in anti-plane problems of polarized ceramics.International Journal of Solids and Structures, 2004, 41(24-25): 6801-6811doi:10.1016/j.ijsolstr.2004.05.018 -
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