考虑表面效应和非局部效应的磁-电-弹纳米板壳结构二维简化理论及在电声器件中的应用

On the basis of the nonlocal theory and surface effect theory of magneto-electro-elasticity and high-order plate theory (Mindlin plate and Lee plate theory), referring to Lee plate theory, all mechanical displacements, electric potential, and magnetic potential of the nanoplates and nanoshells are expanded into the trigonometric series of thickness under orthogonal curvilinear coordinate system, a novel two-dimensional high-order equations of magneto-electro-elastic nanoplates and nanoshells are derived by utilizing Hamilton’s principle. The established theory is effective for the analysis of high frequency vibration in electro-acoustic devices due to the shear deformation and rotary effect are all taken into account, which may make up the deficiency of the classical thin plate theory. It is suitable for the analysis of basic vibration modes, such as stretching, bending, thickness-extension, thickness-shear as well as the multi vibration modes coupling problem. Based the established theory, the following structure design and performance optimization of nanodevices will be carried out: electro-mechanical coupling properties of piezoelectric nanoresonators considering surface and nonlocal effects; magneto-electric coupling effect of multiferroic laminated nanoplates and nanoshells considering surface and nonlocal effects; structure design and mechanical behavior analysis of multiferroic nano energy harvesters. The outcomes are of good academic significance and potential practical application value for the design and optimization of the size-dependent piezoelectric nanoresonators and multiferroic nano energy harvesters.

从三维磁-电-弹理论出发，基于Hamilton原理、表面效应理论、非局部理论和高阶板理论(Mindlin板理论和Lee板理论)的基本思想，在正交曲线坐标系中，通过将板壳的位移、电势和磁势沿板壳的厚度方向作三角级数展开，建立考虑表面效应和非局部效应的磁-电-弹多场耦合的二维简化板壳理论。由于考虑了剪切变形和转动效应，该理论可以用来求解电声器件的高频振动问题，弥补了经典薄板理论的不足。也适用于分析板器件的伸缩、弯曲、厚度拉伸、厚度剪切等基本振动模式以及多种振动模态耦合的问题。基于该理论，将针对尺寸依赖的纳米器件的结构设计和性能优化开展以下工作：考虑表面效应和非局部效应的压电纳米谐振器力电耦合特性研究；考虑表面效应和非局部效应的多铁性板壳结构的磁电耦合效应研究；多铁性纳米俘能器的结构设计和力学行为分析。研究结果对尺寸依赖的压电纳米谐振器和多铁性纳米俘能器的设计和优化有重要的学术意义和实用价值。

从三维磁-电-弹-热理论出发，基于哈密尔顿原理、表面效应理论、非局部理论和高阶板理论（Mindlin板理论和Lee板理论）的基本思想，通过将板壳结构的位移、电势和磁势沿板、壳的厚度方向做三角级数展开，建立了考虑表面效应和非局部效应的磁-电-热-弹多场耦合二维简化板壳理论。该理论不仅可以用来求解电声器件的高频振动问题，弥补了经典薄板理论的不足，也适用于分析板器件的延伸、弯曲、厚度拉伸、厚度剪切等基本振动模式以及振动模态耦合问题。基于所建立理论，针对尺寸依赖的纳米器件的结构设计和性能优化开展了以下有典型代表性的研究工作：建立了纳米尺度上能够刻画俘能器工作性能的热-力-电耦合模型，定量给出了一个区别于宏观力学性能表现的表面效应影响的梁的临界厚度值；小于该临界厚度值，尺寸依赖特性就必须建议考虑而不能被忽略;开展了考虑表面效应和非局部效应的压电纳米谐振器力电耦合特性研究；多铁性纳米俘能器的结构设计和力学行为分析等。研究结果对于尺寸依赖的压电纳米谐振器和多铁性纳米俘能器的设计和优化有重要的学术意义和潜在的工程实用价值。