ON THE MAGNETICALLY INDUCED ELECTROMECHANICAL COUPLING OF CENTROSYMMETRIC FLEXOELECTRIC SANDWICH PLATE
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摘要: 现代工业的发展对材料性能和结构尺寸提出更高的要求, 机电器件的设计越来越偏向于小型化、高频化和智能化. 最新研究成果表明, 磁电耦合复合材料不仅能够以较强的磁电转换效率实现磁能、机械能和电能之间的相互转换, 还可以避免结构与机械驱动源的直接接触, 实现非接触调控, 这对于制备多功能微纳米器件具有重要意义. 文章基于Mindlin所发展的多物理场结构理论分析方法, 结合宏观压磁理论和偶应力挠曲电理论, 研究由单个挠曲电电介质层和两个对称压磁层构成的三明治型夹层板在外部横向磁场驱动下的动态力电耦合响应, 其中通过引入曲率将经典力电耦合理论拓宽到中心对称材料. 夹层板在正弦型全局磁场和均布局部磁场驱动下的动态数值算例表明: 位移和电势具有一定的频率依赖性, 当激振频率达到固有频率时, 振幅达到最大值; 此外, 对称式驱动压磁层分布方式趋于提高多层复合板的力电耦合性能. 文章理论模型和研究结果可为磁控机电器件的优化设计提供新的改进思路.Abstract: The development of modern industry inspires higher requirements for material properties and structural dimensions. The design of electromechanical devices is increasingly biased towards miniaturization, high frequency and intelligence. The most recent studies demonstrate that composite materials with magnetoelectric coupling can not only achieve mutual conversion of magnetic, mechanical, and electrical energy with high magnetoelectric conversion efficiencies, but can also avoid direct contact between the structure and the mechanical driving source to achieve non-contact control, which is crucial for the creation of multifunctional micro and nanoscale devices. Based on the multi-physics structural analysis framework developed by Mindlin, this paper studies the dynamic electromechanical coupling response of a sandwich plate composed of a flexoelectric dielectric layer and two symmetric piezomagnetic layers induced by external transverse magnetic fields. The macroscopic piezomagnetic and curvature-induced flexoelectric theories are employed and the classical electromechanical coupling theory is extended to centrosymmetric materials. The dynamic numerical examples of the sandwich plate driven by a sinusoidal global magnetic field and a uniformly distributed local magnetic field show that the magnitudes of displacement and potential are frequency dependent. When the excitation frequency reaches the natural frequency, the amplitude reaches the maximum. In addition, the distribution of symmetrical piezomagnetic layer tends to improve the electromechanical coupling performance of multilayer composite plates. Both the theoretical model and numerical results provide new ideas for the optimization design of magnetic-controlled electromechanical devices.
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引 言
力电耦合效应是广泛存在于各类电介质材料中的多场耦合效应之一, 常见的形式有压电效应、电致伸缩效应、铁电效应等[1-5]. 一些线性力电耦合效应, 如压电效应和挠曲电效应, 由于可以实现电能和机械能之间的相互转化而被广泛应用于多种机电器件和设备中, 例如谐振器、传感器、发电机和滤波器等[6-10].
挠曲电效应是电介质中应变梯度(非均匀应变)与电极化之间的耦合(正挠曲电效应), 或者电极化梯度与应力的耦合关系(逆挠曲电效应). 正挠曲电效应将机械能转化为电能, 可用于制造传感元件或发电元件. 逆挠曲电效应能够将电能转化为机械能, 可应用于制造驱动元件. 在传统压电材料中, 均匀应变(如单轴拉压)能够通过压电效应激发电极化, 其中, 非中心对称晶体是材料表现压电性的必要条件; 而挠曲电效应即使在中心对称晶体中, 也可通过非均匀应变(如弯曲变形)局部破坏材料内部正负电荷的对称性并诱发挠曲电极化. 近年来, 挠曲电效应逐渐引起了研究者们的广泛关注. 相比于压电效应, 挠曲电效应具有明显的优势. 一方面, 压电效应仅存在于20种非中心对称的介电材料中, 这极大限制了压电材料的进一步发展与应用. 而非均匀应变能够在所有介电材料中产生电极化, 即挠曲电材料的来源比压电材料更广泛. 另一方面, 由于压电材料在高于居里温度时会发生相变, 导致材料对称性增高, 压电性消失, 因而压电效应很难应用于高温场合, 而挠曲电效应没有这个限制, 故挠曲电材料的工作温度范围更广. 除此之外, 挠曲电效应会随着尺度的降低而急剧增强, 当材料特征尺寸达到微纳米量级时, 挠曲电效应会变得十分显著, 因此在微纳米材料中挠曲电效应变得更加不可忽视[11-12].
在考虑挠曲电效应的连续介质力学模型中, 机械载荷引起的弯曲变形和曲率引起的挠曲电极化是大多数研究人员关心的问题[13-20]. 除此之外, 利用热弹性、热电性或磁致伸缩效应引起的热或磁致变形和电极化在多场耦合领域也已得到广泛应用[21-24]. 特别地, 与直接施加机械载荷相比, 利用磁电耦合材料进行磁调控是一种非接触式的智能驱动方法, 通常出现在以压磁性材料作为驱动器的复合结构中, 能够实现磁−力−电的多场耦合. 磁电耦合材料具有很多重要的应用, 包括磁传感器、数据存储、医学中的远程药物传递和无线能量采集器等[25-27]. 虽然自然界中的磁电耦合单相材料确实存在, 如BiFeO3, Cr2O3和YMnO3, 但是, 它们具有较低的居里温度并且在室温下的磁电耦合性能很弱, 这无疑限制了磁电耦合材料的应用[25]. 为此, 人们转而采用压电和磁致伸缩复合结构, 用以规避单一构型磁电材料的缺点, 如BaTiO3和CoFe2O4磁电复合材料[28]、PZT压电陶瓷与稀土合金Terfenol-D磁电复合材料等[29]. 近期, Li等[30]针对一维非压电悬臂梁的研究表明, 通过磁场也可以激发挠曲电极化, 这进而拓宽了挠曲电器件的应用范围.
现有文献对于挠曲电效应的结构理论研究大多只针对弯曲和屈曲等静态问题, 缺乏对于较为复杂的受迫振动等动态问题的研究, 尤其是将磁场作为驱动源时的动态振动机制尚不明晰. 有鉴于中心对称挠曲电介电材料在微纳米尺度下显著的力电耦合效应和磁场的非接触驱动特性, 本文以具有单个挠曲电层和两个对称压磁层组成的夹层板为例, 通过多物理场结构理论建模, 研究夹层板在全局和局部磁场驱动下的动态力电耦合响应, 以期为磁控机电器件的优化设计提供新的改进思路.
1. 结构理论框架
从连续介质力学的角度看, 对于特定的模型结构, 可以将三维框架简化为一维的梁、杆或二维的板模型, 再通过解析法对这些模型进行求解. 因此, 本节将基于Mindlin发展的多物理场结构理论分析方法建立挠曲电夹层板的二维理论分析框架.
1.1 单层挠曲电板
如图1所示的单层挠曲电薄板, 长度和宽度分别为a和b, 厚度为2h. 采用直角坐标系$({x_1},{x_2},{x_3})$, 在外部横向力$q({x_1},{x_2})$的作用下发生变形. 根据Mindlin结构理论假设, 板内任意质点的位移可沿板的厚度方向按级数形式展开, 其一阶截断的表达式为[31-32]
$$ \left.\begin{split} & {u_1}({\boldsymbol{x}},t) = u_1^{(0)}({x_1},{x_2},t) + {x_3}u_1^{(1)}({x_1},{x_2},t) \\ & {u_2}({\boldsymbol{x}},t) = u_2^{(0)}({x_1},{x_2},t) + {x_3}u_2^{(1)}({x_1},{x_2},t) \\ & {u_3}({\boldsymbol{x}},t) = u_3^{(0)}({x_1},{x_2},t) \end{split} \right\} $$ (1) 其中, ${u_1}$, ${u_2}$和${u_3}$分别表示点$({x_1},{x_2},{x_3})$在$t$时刻沿${x_1}$, ${x_2}$和${x_3}$方向的位移, $u_1^{(0)}$和$u_2^{(0)}$表示板中面上一点$({x_1},{x_2},0)$沿面内两个方向的位移, $u_3^{(0)}$表示板中面上一点的挠度, $u_1^{(1)}$和$u_1^{(2)}$代表薄板直法线在两个竖直坐标面内的转角. 与此相应, 板内的电势也可以表达为
$$ \varphi ({\boldsymbol{x}},t) = {\varphi ^{(0)}}({x_1},{x_2},t) + {x_3}{\varphi ^{(1)}}({x_1},{x_2},t) $$ (2) 其中, ${\varphi ^{(0)}}$和${\varphi ^{(1)}}$表示沿板面的零阶电势和沿板横向的一阶电势.
根据经典偶应力理论[33-36], 无穷小应变张量定义为${\boldsymbol{\varepsilon}} = (\nabla {\boldsymbol{u}} + {\boldsymbol{u}}\nabla )/2$; 曲率张量表达式为${ {\textit{χ}}} = {\boldsymbol{\theta}} \otimes \nabla$, 其中${\boldsymbol{\theta }}$($= (\nabla \times {\boldsymbol{u}})/2$)为旋转矢量. 结合式(1)可得非0应变和曲率分量分别为
$$ \left.\begin{split} & {\varepsilon _{11}} = u_{1,1}^{(0)} + u_{1,1}^{(1)}{x_3} \\ & {\varepsilon _{12}} = {\varepsilon _{21}} = \frac{1}{2}(u_{1,2}^{(0)} + u_{2,1}^{(0)}) + \frac{1}{2}(u_{1,2}^{(1)} + u_{2,1}^{(1)}){x_3} \\ & {\varepsilon _{13}} = {\varepsilon _{31}} = \frac{1}{2}(u_1^{(1)} + u_{3,1}^{(0)}) \\ & {\varepsilon _{22}} = u_{2,2}^{(0)} + u_{2,2}^{(1)}{x_3}{\text{ }} \\ & {\varepsilon _{23}} = {\varepsilon _{32}} = \frac{1}{2}(u_2^{(1)} + u_{3,2}^{(0)}) \end{split} \right\} $$ (3) $$ \left. \begin{split} & {\chi _{11}} = \frac{1}{2}(u_{3,12}^{(0)} - u_{2,1}^{(1)}),\;\;\;{\kern 1pt} {\chi _{12}} = \frac{1}{2}(u_{3,22}^{(0)} - u_{2,2}^{(1)}) \\ & {\chi _{21}} = \frac{1}{2}(u_{1,1}^{(1)} - u_{3,11}^{(0)}),\;\;\;{\kern 1pt} {\chi _{22}} = \frac{1}{2}(u_{1,2}^{(1)} - u_{3,12}^{(0)}) \\ & {\chi _{31}} = \frac{1}{2}[(u_{2,11}^{(0)} - u_{1,12}^{(0)}) + (u_{2,11}^{(1)} - u_{1,12}^{(1)}){x_3}] \\ & {\chi _{32}} = \frac{1}{2}[(u_{2,12}^{(0)} - u_{1,22}^{(0)}) + (u_{2,12}^{(1)} - u_{1,22}^{(1)}){x_3}] \\ & {\chi _{33}} = \frac{1}{2}(u_{2,1}^{(1)} - u_{1,2}^{(1)}) \end{split} \right\} $$ (4) 此外, 静电场强度可由电势的负梯度计算得到, 即${\boldsymbol{E}} = - \nabla \varphi$. 结合式(2)可得其分量形式为
$$ {E_1} = - \varphi _{,1}^{(0)} - \varphi _{,1}^{(1)}{x_3},{\text{ }}{E_2} = - \varphi _{,2}^{(0)} - \varphi _{,2}^{(1)}{x_3},{\text{ }}{E_3} = - {\varphi ^{(1)}} $$ (5) 根据偶应力挠曲电理论[36-37], 单层挠曲电板的Gibbs自由能(G)在$[0,T]$时间内的一阶变分为
$$ \delta \int_0^T {G{\text{d}}t} = \int_0^T {\int_V {(\sigma _{ij}^S\delta {\varepsilon _{ij}} + {m_{ij}}\delta {\chi _{ij}} - {D_i}\delta {E_i}){\text{d}}V{\text{d}}t} } $$ (6) 其中, $ \sigma _{ij}^S $和$ {m_{ij}} $分别表示与应变共轭的对称应力和与曲率共轭的偶应力, $ {D_i} $为电位移, $V$代表板的体积, ${\text{d}}V$为板内任意微分单元, ${\text{d}}t$表示时间微分增量.
对应于上式的本构方程为[38]
$$ \sigma _{ij}^S = {C_{ijkl}}{\varepsilon _{kl}},\quad {m_{ij}} = - {f_{kij}}{E_k},\quad {D_i} = {s_{ij}}{E_j} + {f_{ikl}}{\chi _{kl}} $$ (7) 其中, $ {C_{ijkl}} $为挠曲电板的4阶弹性刚度张量、$ {s_{ij}} $为介电系数张量、$ {f_{ikl}} $为3阶挠曲电系数张量, 反映电极化和曲率之间的力电耦合效应.
1.2 夹层板的三维本构关系
考虑图2所示的三明治式压磁驱动挠曲电夹层板结构, 该模型建立在直角坐标系$({x_1},{x_2},{x_3})$之中, 由中间的挠曲电层和沿着挠曲电层的中面呈上下对称分布的两个压磁层组成. 其中, 挠曲电层选用具有中心对称的立方晶体结构(m3m点群)材料, 具有挠曲电性. 压磁层选用具有横观各向同性的六方晶体结构(6mm点群)材料, 不含压电性和磁电耦合效应, 仅考虑压磁性能. 两侧六方晶体压磁层的c轴虽均沿${x_3}$方向, 但是二者极化方向相反. 在横向磁场${H_3}$作用下, 当其中一个压磁层在面内由磁致伸缩效应产生拉伸变形时, 另一个压磁层则会产生压缩变形, 夹层板结构整体的拉压变形得以相互抵消, 反之亦然.
对于立方晶体结构材料, 挠曲电层的三维本构关系为[38]
$$ \left. \begin{split} & \sigma _{11}^S = {{\bar C}_{11}}{\varepsilon _{11}} + {{\bar C}_{12}}{\varepsilon _{22}},{\text{ }}\sigma _{22}^S = {{\bar C}_{12}}{\varepsilon _{11}} + {{\bar C}_{11}}{\varepsilon _{22}} \\ & \sigma _{13}^S = 2{C_{44}}{\varepsilon _{13}},{\text{ }}\sigma _{23}^S = 2{C_{44}}{\varepsilon _{23}},{\text{ }}\sigma _{12}^S = 2{C_{44}}{\varepsilon _{12}}{\text{ }} \\ & {m_{13}} = - {m_{31}} = - {f_{16}}{E_2},\;\;\;{\kern 1pt} {m_{23}} = - {m_{32}} = {f_{16}}{E_1} \\ & {m_{12}} = - {m_{21}} = {f_{16}}{E_3},\;\;\;{\kern 1pt} {D_1} = {s_{11}}{E_1} + {f_{16}}{\chi _{32}}{\text{ }} \\ & {D_2} = {s_{11}}{E_2} - {f_{16}}{\chi _{31}},\;\;\;{\kern 1pt} {D_3} = {s_{11}}{E_3} + {f_{16}}({\chi _{21}} - {\chi _{12}}) \end{split} \right\} $$ (8) 其中, $ {\bar C_{11}} = {C_{11}} - C_{12}^2/{C_{11}} $, $ {\bar C_{12}} = {C_{12}} - C_{12}^2/{C_{11}} $为薄板的等效弹性刚度系数, 二者通过令$ \sigma _{33}^S \approx 0 $求得, 基于该等效方法(应力松弛处理)能够有效解释沿板厚度方向的泊松效应.
对于横观各向同性结构材料, 上部压磁层的三维本构关系为[39]
$$ \left. \begin{split} & \sigma _{11}^S = {{\hat C}_{11}}{\varepsilon _{11}} + {{\hat C}_{12}}{\varepsilon _{22}} - {{\hat h}_{31}}{H_3} \\ & \sigma _{22}^S = {{\hat C}_{12}}{\varepsilon _{11}} + {{\hat C}_{11}}{\varepsilon _{22}} - {{\hat h}_{31}}{H_3} \\ & \sigma _{13}^S = 2{C'_{44}}{\varepsilon _{13}},{\text{ }}\sigma _{23}^S = 2{C'_{44}}{\varepsilon _{23}},{\text{ }}\sigma _{12}^S = 2{C'_{66}}{\varepsilon _{12}} \\ & {D_1} = {s'_{11}}{E_1},{\text{ }}{D_2} = {s'_{11}}{E_2},{\text{ }}{D_3} = {s'_{33}}{E_3} \end{split} \right\} $$ (9) 其中, ${\hat C_{11}} = {C{'_{11}}} - C{'_{1{\text{3}}}}^2/{C{'_{33}}}$, ${\hat C_{12}} = {C{'_{12}}} - C{'_{1{\text{3}}}}^2/{C{'_{33}}}$为压磁层的等效弹性刚度系数, $ {\hat h_{31}} $($ = {h'_{31}} - {C'_{13}}{h'_{33}}/{C'_{33}} $)对压磁系数$ {h'_{31}} $进行修正. 式(9)中的上标 “' ”表示该材料参数仅适用于压磁层.
下部压磁层的c轴方向为负时, 压磁系数的符号发生改变, 因而其面内正应力本构方程变为
$$ \left. \begin{split} & \sigma _{11}^S = {{\hat C}_{11}}{\varepsilon _{11}} + {{\hat C}_{12}}{\varepsilon _{22}} + {{\hat h}_{31}}{H_3} \\ & \sigma _{22}^S = {{\hat C}_{12}}{\varepsilon _{11}} + {{\hat C}_{11}}{\varepsilon _{22}} + {{\hat h}_{31}}{H_3} \end{split} \right\} $$ (10) 1.3 夹层板的控制方程和边界条件
根据Mindlin的电介质变分原理可同时推导出夹层板的控制方程和边界条件[38], 而已有研究表明[38,40], 一阶电势由电极化与曲率(由弯曲变形引起)的耦合直接产生, 零阶电势再与一阶电势耦合. 由于结构沿厚度方向具有对称性, 横向磁场作用下三明治夹层板的零阶电势${\varphi ^{(0)}}$与一阶电势${\varphi ^{(1)}}$得以解耦, 面内位移$u_1^{(0)}$和$u_2^{(0)}$相互抵消并与挠度$u_3^{(0)}$和转角$u_1^{(1)}$和$u_2^{(1)}$发生解耦. 因此, 下文将着重关注横向磁场驱动引起的挠度、转角和横向电势分布, 夹层板的控制方程为[38]
$$ \left. \begin{split} & \sigma _{13,1}^{S(0)} + \sigma _{23,2}^{S(0)} - \frac{1}{2}m_{11,12}^{(0)} - \frac{1}{2}m_{12,22}^{(0)} + \frac{1}{2}m_{21,11}^{(0)}+ \\ & \qquad \frac{1}{2}m_{22,12}^{(0)} + f_3^{(0)} = {m^{(0)}}\ddot u_3^{(0)} \\ & - \sigma _{13}^{S(0)} + \sigma _{11,1}^{S(1)} + \sigma _{12,2}^{S(1)} + \frac{1}{2}m_{21,1}^{(0)} + \frac{1}{2}m_{22,2}^{(0)} - \frac{1}{2}m_{33,2}^{(0)}+ \\ & \qquad \frac{1}{2}m_{31,12}^{(1)} + \frac{1}{2}m_{32,22}^{(1)} + f_1^{(1)} = {m^{(2)}}\ddot u_1^{(1)} \\ & - \sigma _{23}^{S(0)} + \sigma _{12,1}^{S(1)} + \sigma _{22,2}^{S(1)} - \frac{1}{2}m_{11,1}^{(0)} - \frac{1}{2}m_{12,2}^{(0)} + \frac{1}{2}m_{33,1}^{(0)}- \\ & \qquad \frac{1}{2}m_{31,11}^{(1)} - \frac{1}{2}m_{32,12}^{(1)} + f_2^{(1)} = {m^{(2)}}\ddot u_2^{(1)} \\ & D_{1,1}^{(1)} + D_{2,2}^{(1)} - D_3^{(0)} = 0\end{split} \right\} $$ (11) 对应的边界条件为
$$ \left.\begin{split} &{\sigma }_{n3}^{S(0)} + \frac{1}{2}{m}_{ss,s}^{(0)} + \frac{1}{2}{m}_{sn,n}^{(0)}-\frac{1}{2}{m}_{nn,s}^{(0)} = 0\text{ }{\rm{or}}\text{ }{u}_{3}^{(0)} = {{\bar u}}_{3}^{(0)}\\ &{\sigma }_{nn}^{S(1)} + \frac{1}{2}{m}_{3n,s}^{(1)} + \frac{1}{2}{m}_{sn}^{(0)} = 0\text{ }{\rm{or}}\text{ }{u}_{n}^{(1)} = {{\bar u}}_{n}^{(1)}\\ &{\sigma }_{ns}^{S(1)} + \frac{1}{2}{m}_{33}^{(0)}-\frac{1}{2}{m}_{3n,n}^{(1)}-\frac{1}{2}{m}_{3s,s}^{(1)}-\frac{1}{2}{m}_{nn}^{(0)} = 0\text{ } {\rm{or}}\\ &{u}_{s}^{(1)} = {{\bar u}}_{s}^{(1)},{m}_{sn}^{(0)} = 0\text{ }{\rm{or}}\text{ }{u}_{3,n}^{(0)} = {{\bar u}}_{3,n}^{(0)}\\ &{m}_{3n}^{(1)} = 0\text{ }{\rm{or}}\text{ }{u}_{s,n}^{(1)} = {{\bar u}}_{s,n}^{(1)}\\ &{D}_{n}^{(1)} = 0\text{ }{\rm{or}}\text{ }{\varphi }^{(1)} = {{\bar \varphi }}^{(1)}\end{split}\right\} $$ (12) 其中, 顶标“··”表示物理量对时间的二阶导数, “–”表示给定值. $\sigma _{ij}^{S(n)}$为n阶应力, $m_{ij}^{(n)}$为n阶偶应力, $D_i^{(n)}$为n阶电位移, $f_i^{(n)}$为n阶外力, ${m^{(n)}}$表示n阶转动惯量, 它们被定义为
$$ \begin{split} & [\sigma _{ij}^{S(0)},\sigma _{ij}^{S(1)},m_{ij}^{(0)},m_{ij}^{(1)},D_i^{(0)},D_i^{(1)},{m^{(0)}},{m^{(2)}}]= \\ &\qquad \int_{ - h}^h {[\sigma _{ij}^S,{x_3}\sigma _{ij}^S,{m_{ij}},{x_3}{m_{ij}},{D_i},{x_3}{D_i},\rho ,x_3^2\rho ]{\text{d}}{x_3}} \end{split} $$ (13) 其中, $\rho $表示质量密度. 式(13)中的物理量建立在($n,s,{x_3}$)下, 由坐标系(${x_1},{x_2},{x_3}$)转换得到[41].
将式(3) ~ 式(5)、式(8) ~ 式(10)代入式(13), 再将所得二维本构代入式(11), 可得到以位移$u_3^{(0)}$, $u_1^{(1)}$, $u_2^{(1)}$, 电势${\varphi ^{(1)}}$以及磁场强度${H_3}$表示的夹层板控制方程, 即
$$ \left. \begin{split} & C_{44}^{(0)}(u_{3,11}^{(0)} + u_{3,22}^{(0)} + u_{1,1}^{(1)} + u_{2,2}^{(1)}) + \frac{1}{2}f_{16}^{(0)}(\varphi _{,11}^{(1)} + \varphi _{,22}^{(1)})+ \\ & \qquad f_3^{(0)} = {m^{(0)}}\ddot u_3^{(0)} \\ & - C_{44}^{(0)}u_{3,1}^{(0)} - C_{44}^{(0)}u_1^{(1)} + C_{11}^{(2)}u_{1,11}^{(1)} + (C_{12}^{(2)} + C_{66}^{(2)})u_{2,12}^{(1)}+ \\ & \qquad C_{66}^{(2)}u_{1,22}^{(1)} + \frac{1}{2}f_{16}^{(0)}\varphi _{,1}^{(1)} + f_1^{(1)} = {m^{(2)}}\ddot u_1^{(1)} + h_{31}^{(1)}{H_{3,1}} \\ & - C_{44}^{(0)}u_{3,2}^{(0)} + (C_{12}^{(2)} + C_{66}^{(2)})u_{1,12}^{(1)} - C_{44}^{(0)}u_2^{(1)} + C_{66}^{(2)}u_{2,11}^{(1)}+ \\ &\qquad C_{11}^{(2)}u_{2,22}^{(1)} + \frac{1}{2}f_{16}^{(0)}\varphi _{,2}^{(1)} + f_2^{(1)} = {m^{(2)}}\ddot u_2^{(1)} + h_{31}^{(1)}{H_{3,2}} \\ & - s_{11}^{(2)}\varphi _{,11}^{(1)} - s_{11}^{(2)}\varphi _{,22}^{(1)} + s_{33}^{(0)}{\varphi ^{(1)}} + \frac{1}{2}f_{16}^{(0)}(u_{3,11}^{(0)} + u_{3,22}^{(0)}- \\ & \qquad u_{1,1}^{(1)} - u_{2,2}^{(1)}) = 0 \end{split} \right\} $$ (14) 其中
$$ \left. \begin{split} & f_{16}^{(0)} = {f_{16}}(2h - c),\;C_{44}^{(0)} = {{C'_{44}}}c + {C_{44}}(2h - c) \\ & s_{33}^{(0)} = {{s'_{33}}}c + {s_{11}}(2h - c),\;h_{31}^{(1)} = {{\hat h}_{31}}\left[{h^2} - {\left(h - \frac{c}{2}\right)^2}\right] \\ & C_{11}^{(2)} = \frac{2}{3}{{\hat C}_{11}}\left[{\left( - h + \frac{c}{2}\right)^3} + {h^3}\right] + \frac{2}{3}{{\bar C}_{11}}{\left(h - \frac{c}{2}\right)^3} \\ & C_{12}^{(2)} = \frac{2}{3}{{\hat C}_{12}}\left[{\left( - h + \frac{c}{2}\right)^3} + {h^3}\right] + \frac{2}{3}{{\bar C}_{12}}{\left(h - \frac{c}{2}\right)^3} \\ & C_{66}^{(2)} = \frac{2}{3}{{C'_{66}}}\left[{\left( - h + \frac{c}{2}\right)^3} + {h^3}\right] + \frac{2}{3}{C_{44}}{\left(h - \frac{c}{2}\right)^3} \\ & s_{11}^{(2)} = \frac{2}{3}{{s'_{11}}}\left[{\left( - h + \frac{c}{2}\right)^3} + {h^3}\right] + \frac{2}{3}{s_{11}}{\left(h - \frac{c}{2}\right)^3} \end{split} \right\} $$ (15) 2. 受迫振动分析
为了论证上节建立的挠曲电夹层板二维理论模型, 本节将分别对其在全局和局部横向磁场作用下的动态力电耦合响应展开研究. 以简支板闭合电路状态为例(边界配置如图3所示), 边界条件可根据式(12)得到, 即
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图 3 横向磁场驱动下夹层板的边界配置Figure 3. Boundary configuration of sandwich plate under a transverse magnetic field$$ \left. \begin{split} & u_3^{(0)} = 0,\;\;\;{\kern 1pt} \sigma _{nn}^{S(1)} + \frac{1}{2}m_{3n,s}^{(1)} + \frac{1}{2}m_{sn}^{(0)} = 0 \\ & u_s^{(1)} = 0,\;\;\;{\kern 1pt} m_{sn}^{(0)} = 0,\;\;\;m_{3n}^{(1)} = 0,\;\;\;{\kern 1pt} {\varphi ^{(1)}} = 0 \end{split} \right\} $$ (16) 对于矩形板, 沿${x_1} = 0$和${x_1} = a$, 边界条件为
$$ u_2^{(1)} = 0,\quad u_3^{(0)} = 0,\quad {\varphi ^{(1)}} = 0 $$ (17) $$ \sigma _{11}^{S(1)} + \frac{1}{2}m_{31,2}^{(1)} + \frac{1}{2}m_{21}^{(0)} = 0,\quad m_{21}^{(0)} = 0,\quad m_{31}^{(1)} = 0 $$ (18) 将二维本构代入式(18)可进一步得到
$$ C_{11}^{(2)}u_{1,1}^{(1)} + C_{12}^{(2)}u_{2,2}^{(1)} - \frac{1}{2}f_{16}^{(2)}\varphi _{,22}^{(1)} + \frac{1}{2}f_{16}^{(0)}{\varphi ^{(1)}} - h_{31}^{(1)}{H_3} = 0 $$ (19) 沿${x_2} = 0$和${x_2} = b$, 边界条件为
$$ u_1^{(1)} = 0,\quad u_3^{(0)} = 0,\quad {\varphi ^{(1)}} = 0 $$ (20) $$ \sigma _{22}^{S(1)} - \frac{1}{2}m_{32,1}^{(1)} - \frac{1}{2}m_{12}^{(0)} = 0,\quad m_{12}^{(0)} = 0,\quad m_{32}^{(1)} = 0 $$ (21) 将二维本构代入式(21)可进一步得到
$$ C_{12}^{(2)}u_{1,1}^{(1)} + C_{11}^{(2)}u_{2,2}^{(1)} - \frac{1}{2}f_{16}^{(2)}\varphi _{,11}^{(1)} + \frac{1}{2}f_{16}^{(0)}{\varphi ^{(1)}} - h_{31}^{(1)}{H_3} = 0 $$ (22) 本节主要分析外部磁场对模型的驱动作用, 不考虑体力项, 即令$ f_3^{(0)} = f_1^{(1)} = f_2^{(1)} = 0 $. 夹层板受到时谐磁场的作用, 磁场强度表达式为
$$ {H_3} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {{{\hat H}_{mn}}\sin ({\xi _m}{x_1})\sin ({\zeta _n}{x_2}){{\rm{e}}^{{\rm{i}}\omega t}}} } $$ (23) 其中, ${\xi _m} = m\text{π} /a$, ${\zeta _n} = n\text{π} /b,$ ω是板的激振频率, i是虚数单位并满足i2 = −1. $ {\hat H_{mn}} $代表磁场强度正弦级数展开式的系数, 可通过积分计算得到, 即
$$ {\hat H_{mn}} = \frac{4}{{ab}}\int_0^a {\int_0^b {{H_3}\sin ({\xi _m}{x_1})\sin ({\zeta _n}{x_2}){\text{d}}{x_1}{\text{d}}{x_2}} } $$ (24) 取决于外加驱动磁场的形式, 位移场和电势场也必然具有相应的时空分布特征, 即
$$ \left. \begin{split} & u_3^{(0)} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {W_{mn}^{V(0)}\sin ({\xi _m}{x_1})\sin ({\zeta _n}{x_2}){{\rm{e}}^{{\rm{i}}\omega t}}} } \\ & u_1^{(1)} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {U_{mn}^{V(1)}\cos ({\xi _m}{x_1})\sin ({\zeta _n}{x_2}){{\rm{e}}^{{\rm{i}}\omega t}}} } \\ & u_2^{(1)} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {V_{mn}^{V(1)}\sin ({\xi _m}{x_1})\cos ({\zeta _n}{x_2}){{\rm{e}}^{{\rm{i}}\omega t}}} } \\ & {\varphi ^{(1)}} = \sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {\varPhi _{mn}^{V(1)}\sin ({\xi _m}{x_1})\sin ({\zeta _n}{x_2}){{\rm{e}}^{{\rm{i}}\omega t}}} } \end{split} \right\} $$ (25) 其中, $ W_{mn}^{V(0)} $, $ U_{mn}^{V(1)} $, $ V_{mn}^{V(1)} $和$\varPhi _{mn}^{V(1)}$是与m, n有关的傅里叶级数待定系数. 对于任意$ {\hat H_{mn}} $, $ W_{mn}^{V(0)} $, $ U_{mn}^{V(1)} $, $ V_{mn}^{V(1)} $和$\varPhi _{mn}^{V(1)}$, 由式(23)和式(25) 给出的展开式均能满足边界条件(17)和式(19) ~ 式(21).
将式(23)和式(25)代入控制方程(14)得线性方程组
$$ \{ {{\boldsymbol{K}}_{ij}} + {\omega ^2}{{\boldsymbol{M}}_{ij}}\} \left[ {\begin{array}{*{20}{c}} {W_{mn}^{V(0)}} \\ {U_{mn}^{V(1)}} \\ {V_{mn}^{V(1)}} \\ {\varPhi _{mn}^{V(1)}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ {h_{31}^{(1)}{\xi _m}\hat H_{mn}^V} \\ {h_{31}^{(1)}{\zeta _n}\hat H_{mn}^V} \\ 0 \end{array}} \right] $$ (26) 其中, ${{\boldsymbol{K}}_{ij}}$和${{\boldsymbol{M}}_{ij}}$分别为4 × 4的刚度矩阵和质量矩阵, 二者的非零分量为
$$ \left. \begin{split} & {K_{11}} = - C_{44}^{(0)}(\xi _m^2 + \zeta _n^2) \\ & {K_{12}} = {K_{21}} = - C_{44}^{(0)}{\xi _m} \\ & {K_{13}} = {K_{31}} = - C_{44}^{(0)}{\zeta _n} \\ & {K_{14}} = {K_{41}} = - f_{16}^{(0)}(\xi _m^2 + \zeta _n^2)/2 \\ & {K_{22}} = - C_{44}^{(0)} - C_{11}^{(2)}\xi _m^2 - C_{66}^{(2)}\zeta _n^2 \\ & {K_{23}} = {K_{32}} = - (C_{12}^{(2)} + C_{66}^{(2)}){\xi _m}{\zeta _n} \\ & {K_{24}} = {K_{42}} = f_{16}^{(0)}{\xi _m}/2 \\ & {K_{33}} = - C_{44}^{(0)} - C_{66}^{(2)}\xi _m^2 - C_{11}^{(2)}\zeta _n^2 \\ & {K_{34}} = {K_{43}} = f_{16}^{(0)}{\zeta _n}/2 \\ & {\kern 1pt} {K_{44}} = s_{11}^{(2)}\xi _m^2 + s_{11}^{(2)}\zeta _n^2 + s_{33}^{(0)} \\ & {M_{11}} = {m^{(0)}},{\text{ }}{M_{22}} = {M_{33}} = {m^{(2)}} \end{split} \right\} $$ (27) 通过对方程组(26)中的激振频率进行扫频分析, 可以求出满足条件的$ W_{mn}^{V(0)} $, $ U_{mn}^{V(1)} $, $ V_{mn}^{V(1)} $和$\varPhi _{mn}^{V(1)}$, 再将这些求出的傅里叶系数代回式(25), 进而可计算出$u_3^{(0)}$, $u_1^{(1)}$, $u_2^{(1)}$和${\varphi ^{(1)}}$, 并完成求解.
为了求解数值结果, 本节选取的板尺寸参数为: h = 10 nm, c = 0.4h, a = b = 40h. 压磁层和挠曲电层材料分别选取CoFe2O4和Si, 二者的物理参数如表1所示[40,42]. 值得注意的是, 由于CoFe2O4的挠曲电系数无法在现有文献中得出, 本文忽略了压磁层的挠曲电效应[43].
此外, 为了避免数值计算的奇异性, 本节在数值计算中采用复刚度法, 通过引入复数弹性刚度, 以考虑机械阻尼的影响. 即令$\bar C_{ij}^{(\alpha )} = (1 - {\rm{i}}/Q)C_{ij}^{(\alpha )}$ (α = 0, 2), 并以$\bar C_{ij}^{(\alpha )}$代替刚度矩阵${{\boldsymbol{K}}_{ij}}$中的$C_{ij}^{(\alpha )}$进行以上数值计算. 这里Q为材料品质因数, 本文取为Q = 100[44].
表 1 CoFe2O4和Si的材料参数Table 1. Material parameters of CoFe2O4 and SiProperty/Unit CoFe2O4 Si elastic constants/GPa ${ {C'_{11}} } = 286 \\$ ${ {C'_{1{\text{2} }} } } = {\text{173} }$ $ {C_{11}} = 165.7 $ ${ {C'_{13}} } = 170 \\$ $ {C_{12}} = 63.9 \\ $ ${ {C'_{33} }} = 269.5 \\$ $ {C_{44}} = 79.56 $ ${ {C'_{44}} } = 45.3$ dielectric constants/(C2·(N·m2)−1)×10−9 ${ {s'_{11}} } = 0.{\text{08} } \\$ ${s_{11} } = 0.1035$ ${ {s'_{33}} } = 0.{\text{093} }$ piezomagnetic constants/(N·(A·m)−1) ${ {h'_{31}} } = 580.3$ — ${ {h'_{33} }} = 699.7$ flexoelectric coefficient/(nC·m−1) — $ {f_{16}} = 0.4 $ mass density/(kg·m−3) $\rho ' = 5300$ $\rho = 2332$ 2.1 全局磁场驱动
本小节研究全局磁场驱动下复合板的受迫振动问题, 考虑幅值为h0 = 1.0 × 106 A/m的正弦型磁场强度分布形式
$$ {\hat{H}}_{11}^{V} = {h}_{0},{\hat{H}}_{mn}^{V} = 0\text{ (}m\ne 1\text{ }{\rm{or}}\text{ }n\ne 1\text{)} $$ (28) 图4展示了单侧压磁层驱动挠曲电双层板(蓝色线条)(参见文献[40])与当前夹层板模型(红色线条)在正弦型全局磁场驱动下的一阶电势幅频关系对比, 两种模型的材料组分占比和几何尺寸均保持一致. 从图中可以看出, 一阶电势表现出明显的频率依赖性, 当激振频率远离固有频率时, 一阶电势的振动幅值几乎为零, 而当激振频率靠近固有频率时, 一阶电势的振动幅度急剧增加, 在固有频率处达到最大值. 此外, 相较于“双层复合板”、“当前夹层板”的一阶电势明显增大, 峰值提高了10%以上, 这表明对称式驱动压磁层分布方式趋于提高多层复合板的力电耦合性能.
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图 4 全局磁场驱动下的一阶电势幅频关系对比Figure 4. Comparison of amplitude-frequency curve of first-order potential under the global magnetic field2.2 局部磁场驱动
本小节采用${H_3} = {h_0}\exp ({\rm{i}}\omega t)$的均布局部磁场驱动复合板, 磁场的驱动范围是一个长度为2d (d < a/2), 宽度为2e (e < b/2)的矩形区域, 如图5所示的灰色区域, 关于矩形板的中心(x0, y0)对称. 对于该均布局部磁场, 式(24)中的傅里叶系数可计算为
$$ \begin{split} & {{\hat H}_{mn}} = \frac{{4{h_0}}}{{mn{\text{π} ^2}}}\left\{\cos \left[{\xi _m}({x_0} - d)\right] - \cos \left[{\xi _m}({x_0} + d)\right]\right\} \cdot \\ &\qquad \left\{\cos \left[{\zeta _n}({y_0} - e)\right] - \cos \left[{\zeta _n}({y_0} + e)\right]\right\}\sin ({\xi _m}{x_1})\sin ({\zeta _n}{x_2}) \end{split} $$ (29) 局部磁场的具体驱动范围可通过改变d和e进行调整, 为了便于进行数值计算, 本小节取d = e = 25 nm.
图6展示了局部磁场驱动下复合板的挠度振动幅值与激振频率关系, 其中, h0 = 1.0 × 106 A/m. 在式(23)中共保留了30项, 此时的计算结果与保留60项时在小数点后第3位依然一致, 表明在驱动磁场的傅里叶级数中取30项可以满足收敛性要求. 从图6中可以看出, 在0 ~ 5.0 × 1010 rad/s的激振频率范围内, 一共出现了3阶固有频率, 相应产生了3个共振峰, 并且, 随着固有频率的增加, 共振峰值越来越低. 受到复合板振动阻尼的影响, 固有频率不再表现为单一的频率值, 而是表现为一小段固有频率区间.
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图 6 局部磁场驱动下夹层板的挠度幅频关系Figure 6. Deflection amplitude-frequency curve of the sandwich plate under local magnetic field与图6的前两阶固有频率相对应, 图7给出了夹层板的挠度、转角和横向电势振型. 虽然这些振型实际上是复值, 但由于它们的虚部远小于实部, 因此可略去虚部. 由于矩形板几何尺寸和磁场作用范围的对称性, 夹层板的转角$u_2^{(1)}$与$u_1^{(1)}$的分布类似, 出于简化的目的, $u_2^{(1)}$的分布未做展示. 从图7中可以看出, 这些振型具有很强的频率依赖性, 低阶振型具有更大的振动幅值, 而高阶振型沿复合板长度和宽度方向具有更短的周期. 特别地, 从图7(b)和图7(f)中还可以明显看出, 由于磁场的局部驱动特征, 在作用区域中心附近产生挠度、转角和电势峰值. 图7所展示的结果对板型磁驱动器(磁能转换为机械能)和板型磁传感器(磁能转换为电能)的优化设计具有重要的参考价值.
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图 7 局部磁场驱动下夹层板的前两阶挠度、转角和一阶电势振型Figure 7. The first two order deflection, rotation angle and first-order potential mode of the sandwich plate under the local magnetic field![]()
7 局部磁场驱动下夹层板的前两阶挠度、转角和一阶电势振型 (续)7. The first two order deflection, rotation angle and first-order potential mode of the sandwich plate under the local magnetic field (continued)3. 结 论
本文以探索新型磁控纳米机电器件为研究背景, 提出了一种新的三明治式压磁驱动挠曲电夹层板模型, 构建了夹层板的二维控制方程和边界条件, 并分析了矩形简支板在全局和局部磁场激励下的动态力电耦合响应. 基于本文的数值研究结果可以获得以下两点重要结论.
(1) 在外部时谐磁场驱动下, 夹层板的位移和电势振动幅值依赖于激振频率. 随着固有频率的增加, 振型幅值迅速衰减, 振型表现出周期变化特征.
(2) 驱动压磁层的分布方式对复合板结构的力电耦合响应具有重要影响, 本文设计的对称式驱动压磁层分布方式趋于提高复合板的力电耦合性能.
本文的理论模型和研究结果可为磁控机电器件的优化设计提供新的改进思路.
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图 3 横向磁场驱动下夹层板的边界配置
Figure 3. Boundary configuration of sandwich plate under a transverse magnetic field
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图 4 全局磁场驱动下的一阶电势幅频关系对比
Figure 4. Comparison of amplitude-frequency curve of first-order potential under the global magnetic field
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图 6 局部磁场驱动下夹层板的挠度幅频关系
Figure 6. Deflection amplitude-frequency curve of the sandwich plate under local magnetic field
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图 7 局部磁场驱动下夹层板的前两阶挠度、转角和一阶电势振型
Figure 7. The first two order deflection, rotation angle and first-order potential mode of the sandwich plate under the local magnetic field
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7 局部磁场驱动下夹层板的前两阶挠度、转角和一阶电势振型 (续)
7. The first two order deflection, rotation angle and first-order potential mode of the sandwich plate under the local magnetic field (continued)
表 1 CoFe2O4和Si的材料参数
Table 1 Material parameters of CoFe2O4 and Si
Property/Unit CoFe2O4 Si elastic constants/GPa ${ {C'_{11}} } = 286 \\$ ${ {C'_{1{\text{2} }} } } = {\text{173} }$ $ {C_{11}} = 165.7 $ ${ {C'_{13}} } = 170 \\$ $ {C_{12}} = 63.9 \\ $ ${ {C'_{33} }} = 269.5 \\$ $ {C_{44}} = 79.56 $ ${ {C'_{44}} } = 45.3$ dielectric constants/(C2·(N·m2)−1)×10−9 ${ {s'_{11}} } = 0.{\text{08} } \\$ ${s_{11} } = 0.1035$ ${ {s'_{33}} } = 0.{\text{093} }$ piezomagnetic constants/(N·(A·m)−1) ${ {h'_{31}} } = 580.3$ — ${ {h'_{33} }} = 699.7$ flexoelectric coefficient/(nC·m−1) — $ {f_{16}} = 0.4 $ mass density/(kg·m−3) $\rho ' = 5300$ $\rho = 2332$ 
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