THERMOELASTIC DAMPING IN FUNCTIONALLY GRADED MINDLIN RECTANGULAR MICRO PLATES
-
摘要: 功能梯度材料微板谐振器热弹性阻尼的建模和预测是此类新型谐振器热−弹耦合振动响应的新课题. 本文采用数学分析方法研究了四边简支功能梯度材料中厚度矩形微板的热弹性阻尼. 基于明德林中厚板理论和单向耦合热传导理论建立了材料性质沿着厚度连续变化的功能梯度微板热弹性自由振动控制微分方程. 在上下表面绝热边界条件下采用分层均匀化方法求解变系数热传导方程, 获得了用变形几何量表示的变温场的解析解. 从而将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的偏微分方程. 然后,利用特征值问题在数学上的相似性,求得了四边简支条件下功能梯度材料明德林矩形微板的复频率解析解, 进而利用复频率法获得了反映谐振器热弹性阻尼水平的逆品质因子. 最后, 给出了材料性质沿板厚按幂函数变化的陶瓷−金属组分功能梯度矩形微板的热弹性阻尼数值结果. 定量地分析了横向剪切变形、材料梯度变化以及几何参数对热弹性阻尼的影响规律. 结果表明, 采用明德林板理论预测的热弹性阻尼值小于基尔霍夫板理论的预测结果, 而且两者的差别随着相对厚度的增大而变得显著.Abstract: Accurately modelling and evaluating of thernoelastic damping (TED) in functionally graded material (FGM) micro plates are challenging novel topics in the study on the responses of thermoelastic coupled vibration of this kind of new type micro resonators. In this paper, TED in a simply supported FGM rectangular micro plate with moderate thickness is investigated by means of mathematical analysis. Based on the Mindlin plate theory and the one-way coupled heat conduction theory, differential equations governing the thermal-elastic free vibration of the FGM micro plates with the material properties varying continuously along with the thickness direction are established. Under the adiabatic boundary conditions at the top and the bottom surfaces, analytical solution of the temperature field expressed by the kinematic parameters is obtained by using layer-wise homogenization approach. As a result, the structural vibration equation including the thermal membrane force and moment is transformed into a partial differential equation only in terms of the amplitude of the deflection. Then, by using the mathematical similarity between the eigenvalue problems an analytical solution of the complex frequency for an FGM Mindlin micro plate with the four edges simply supported is arrived at, from which the inverse quality factor representing the TED is extracted. Finally, numerical results of TED for the FGM rectangular micro plate made of ceramic-metal constituents with the material properties varying in the thickness as power functions are presented. Effects of the transverse shear deformation, the gradient of the material property and the geometric parameters on the TED are quantitatively investigated in detail. The numerical results show that the TED evaluated by the Mindlin plate theory is smaller than that by the Kirchhoff plate theory and that the difference in the values predicted by the two plate theories becomes significant along with the increase of the thickness-to-side length ratio.
-
引 言
谐振器作为微/纳电机系统中的重要器件被广泛用作传感器、驱动器、陀螺仪、能量收集器、频率控制器、逻辑开关等. 而大多数谐振器的力学模型可以简化为弹性微梁或微板结构. 高性能的微/纳电机系统需要谐振器在工作时具有低耗能或高品质因子的优点. 然而, 谐振器在工作中不可避免地存在各种耗能机制. 例如热弹性阻尼或内摩擦[1]就是一种主要的内耗能机制. 热弹性阻尼是结构在一个振动周期内材料内部的热弹性耦合变形引起的内部能耗, 它不能通过外部条件的改善而消除. 因此, 热弹性阻尼有时将会成为微电机系统的主要能耗形式. 因此, 精确地分析和预测热弹性阻尼对高品质微/纳谐振器的研究和设计具有重要意义.
Zener[1]首先采用能量方法和单向耦合的热传导方程给出了细长矩形截面微梁谐振器的逆品质因子解析解. Bishop 和 Kinra [2-3]首先把Zener[1]关于均匀材料微梁谐振器热弹性阻尼分析的能量法推广到了具有非完善界面复合材料层合微纳梁/板谐振器的热弹性阻尼研究, 并通过计算一个振动周期内由不可逆传热产生的总热量与弹性总弹性势能之比给出了微层合结构的逆品质因子的解析解. 与上述能量法不同, Lifshitz 和Roukes [4] 通过联立求解单向耦合的结构振动方程和热传导方程获得了细长微梁热弹耦合自由振动的复频率, 首次采用复频率法获得了微梁谐振器热弹性阻尼的解析解. 此后, 能量法[1-3]和复频率法[4]则被广泛地应用于微/纳谐振器热弹性阻尼的理论分析和求解中. 研究者们采用不同的结构变形理论、不同热传导理论和不同的求解方法开展了不同结构形式的微/纳谐振器热弹性阻尼的理论研究. 其中, 许多工作是关于微板谐振器的热弹性耦合振动及其热弹性阻尼的理论研究[5-26].
在已有关于微板谐振器热弹性阻尼的理论研究中, 结构振动方程几乎都是基于经典板理论, 即基尔霍夫(Kirchhoff) 薄板理论建立的[5-12, 4-23]. 考虑材料性质为均匀、各向同性的情况, 采用复频率法, 文献[5-9]分别求得了矩形和圆(环)形微板谐振器热弹性阻尼的解析解, 分析了静电载荷和残余应力、边界条件、几何尺寸以及非线性几何变形对热弹性阻尼的影响规律. Li等[10]、Fang等[11-12]分别基于一维、二维和三维热传导理论建立了微板单向耦合的热传导数学模型, 并采用能量法求得了矩形和圆形薄板谐振器的热弹性阻尼. 最近, 基于考虑横向剪切变形的明德林(Mindlin)板理论, 马航空等[13]首次给出了四边简支中厚度矩形微板热弹性阻尼的解析解. 研究发现随着微板厚度的增大, 明德林微板的热弹性阻尼最大值单调减小, 而基尔霍夫微板的热弹性阻尼峰值却保持不变.
考虑热传导过程中温度与热流运动间的延滞效应(或波动效应), 部分作者[14-16]还利用广义热传导理论研究了微板谐振器的热弹性耦合振动响应. 例如, 文献[14-15]基于Lord-Shulman热传导理论求得了微板谐振器热弹性阻尼解析解, 分析了延滞时间参数和孔隙率变化对热弹性阻尼的影响. Guo等[16]采用双向延滞广义热传导模型研究了圆板谐振器的热弹性阻尼. Grover[17]采用Kelvin–Voigt 材料模型下的广义热黏弹性理论研究了均匀微圆板谐振器的热弹性阻尼. 最新的研究成果还包括: Ghugh和Partap[18]在微板的热弹性耦合振动模型建立中同时考虑了热松弛和微拉伸 (micro stretch) 效应, 研究结果表明微拉伸增强了微板热弹性阻尼的临界值; Wang 和 Li[19]综合考虑微板的尺度效应、热流矢量的延滞效应、非局部效应以及记忆效应, 分别采用复频率法和能量法求得了基尔霍夫板理论热弹性阻尼的解析解. 通过大量的数值结果详细地分析了微结构的尺度参数、热松弛时间参数以及边界条件对热弹性阻尼的影响规律.
针对材料性质沿单轴方向的阶梯式变化, 部分作者研究了由不同均匀材料分层组成的层合微板的热弹性阻尼特性. Sun等[20]采用复频率法研究了对称铺设的三层微圆板横向轴对称振动下的热弹性阻尼. 采用文献[3]中的方法, 忽略温度梯度在面内的变化, Zuo等[21-22]分别研究了双层和三层微板的热弹性阻尼, 发现随频率的变化热弹性阻尼会有多个峰值出现. Liu等[23-24]采用格林函数法分别求解了二维和三维热传导方程, 利用能量法获得了双层微圆板和非对称铺设三层矩形微板热弹性阻尼的级数形式解析解. 然而, 为了消去结构振动方程中的拉−弯耦合项, 文献[20-24]都不约而同地采用了物理中面法将几何中面的面内位移用挠度来表示, 从而大大简化了在数学上的求解难度. 但是, 在材料性质变化关于几何中面不对称的情况下, 热−弹耦合振动不但会产生热弯矩, 还会产生热薄膜力. 而他们在简化中完全忽略了热薄膜力对热弹性阻尼的贡献.
考虑材料性质沿着厚度连续变化, 最近亦有少数作者研究了功能梯度材料微板的热弹性阻尼[25-27]. 基于一阶剪切变形理论和应变梯度理论, 文献[25]研究了材料性质沿横向幂函数变化的功能梯度微板的热弹性阻尼. 应用泰勒级数展开方法, 获得了变系数热传导方程的级数形式的解析解, 进而获得了四边简支功能梯度中厚度矩形板的热弹性阻尼. 基于基尔霍夫经典板理论和准一维热传导理论, Li等采用解析方法研究了功能梯度圆形微板[26]和矩形微板[27]的热−弹耦合自由振动响应, 并成功地将分层均匀化方法应用于复杂变系数热传导方程的求解. 获得了材料性质沿厚度按幂函数变化的功能梯度微板的热弹性阻尼解析解. 首次定量地分析了热薄膜力对热弹性阻尼的影响程度.
本文拟在前期工作[13, 27]的基础上进一步研究横向剪切变形对功能梯度中厚度微板热弹性阻尼的影响规律. 基于明德林板理论和单向耦合的准一维热传导理论建立材料性质沿厚度连续变化的功能梯度矩形微板热−弹耦合自由振动数学模型. 采用分层均匀化方法求解复杂变系数热传导微分方程, 利用结构振动特征值问题在数学上的相似性求得用基尔霍夫均匀微板的无阻尼固有频率表示的四边简支功能梯度材料微板的复频率, 进而采用复频率法提取表征热弹性阻尼的逆品质因子. 最后, 以材料性质按照幂函数变化的两种陶瓷−金属 (Al-SiC 和 Ni-Si3N4) 功能梯度微板为例, 给出热弹性阻尼的数值结果, 定量分析剪切变形、梯度指数以及几何尺寸对热弹性阻尼的影响规律.
1. 问题的控制方程
![]()
图 1 微板的几何尺寸和坐标示意图Figure 1. Schematic illustration of micro plate and the coordinate system1.1 运动方程
图1为微板的几何尺寸和坐标示意图, 基于明德林板理论[13, 25, 28], 可给出下列功能梯度中厚度板的位移场
$$ u\left( {x,y,z,t} \right) = {u_0}\left( {x,y,t} \right) + z{\varphi _x}\left( {x,y,t} \right) \tag{1a}$$ $$ v\left( {x,y,z,t} \right) = {v_0}\left( {x,y,t} \right) + z{\varphi _y}\left( {x,y,t} \right) \tag{1b}$$ $$ w\left( {x,y,z,t} \right) = {w_0}\left( {x,y,t} \right) \tag{1c}$$ 其中, t 为时间;
$ {u_0} $ ,$ {v_0} $ 和$ {w_0} $ 分别为几何中面上一点分别在$ x $ ,$ y $ 和$ z $ 方向的位移分量;$ {\varphi _x} $ 和$ {\varphi _y} $ 为中面法线分别绕$ y $ 轴和$ x $ 轴的转角.根据弹性力学的几何方程可得微板的应变场
$$ {\varepsilon _x} = \frac{{\partial {u_0}}}{{\partial x}} + z\frac{{\partial {\varphi _x}}}{{\partial x}} , \;\; {\varepsilon _y} = \frac{{\partial {v_0}}}{{\partial y}} + z\frac{{\partial {\varphi _y}}}{{\partial y}} \tag{2a}$$ $$ {\gamma _{xy}} = \frac{{\partial {u_0}}}{{\partial y}} + \frac{{\partial {v_0}}}{{\partial x}} + z\left( {\frac{{\partial {\varphi _x}}}{{\partial y}} + \frac{{\partial {\varphi _y}}}{{\partial x}}} \right) \tag{2b}$$ $$ {\gamma _{xz}} = {\varphi _x} + \frac{{\partial {w_0}}}{{\partial x}}, \;\; {\gamma _{yz}} = {\varphi _y} + \frac{{\partial {w_0}}}{{\partial y}} \tag{2c}$$ 对于线弹性材料由胡克定律可得到应力分量
$$ {\sigma _x} = \frac{E}{{1 - {v^2}}}\left[ {{\varepsilon _x} + v{\varepsilon _y} - \left( {1 + v} \right)\alpha \theta } \right] \tag{3a}$$ $$ {\sigma _y} = \frac{E}{{1 - {v^2}}}\left[ {{\varepsilon _y} + v{\varepsilon _x} - \left( {1 + v} \right)\alpha \theta } \right] \tag{3b}$$ $$ ({\tau _{xy}},{\tau _{xz}},{\tau _{yz}}) = \frac{E}{{2\left( {1 + \nu } \right)}}({\gamma _{xy}},{\gamma _{xz}},{\gamma _{yz}}) \tag{3c}$$ 其中,
$ \theta (x,y,z,t) = T(x,y,z,t) - {T_0} $ 为变温场, 是由热−弹耦合振动产生的;$ T $ 为瞬态温度,$ {T_0} $ 为初始温度;$ E $ ,$ \nu $ 和$ \alpha $ 分别为弹性模量、泊松比和热膨胀系数, 它们都是坐标z的函数. 由于功能梯度板的泊松比通常相比其他材料参数变化很小, 为了后面方便计算, 将其看作常数(取组分材料的平均值).$$ \frac{{\partial {N_x}}}{{\partial x}} + \frac{{\partial {N_{xy}}}}{{\partial y}} = {I_0}\frac{{{\partial ^2}{u_0}}}{{\partial {t^2}}} + {I_1}\frac{{{\partial ^2}{\varphi _x}}}{{\partial {t^2}}} $$ (4) $$ \frac{{\partial {N_{xy}}}}{{\partial x}} + \frac{{\partial {N_y}}}{{\partial y}} = {I_0}\frac{{{\partial ^2}{v_0}}}{{\partial {t^2}}} + {I_1}\frac{{{\partial ^2}{\varphi _y}}}{{\partial {t^2}}} $$ (5) $$ \frac{{\partial {M_x}}}{{\partial x}} + \frac{{\partial {M_{xy}}}}{{\partial y}} - {Q_x} = {I_1}\frac{{{\partial ^2}{u_0}}}{{\partial {t^2}}} + {I_2}\frac{{{\partial ^2}{\varphi _x}}}{{\partial {t^2}}} $$ (6) $$ \frac{{\partial {M_{xy}}}}{{\partial x}} + \frac{{\partial {M_y}}}{{\partial y}} - {Q_y} = {I_1}\frac{{{\partial ^2}{v_0}}}{{\partial {t^2}}} + {I_2}\frac{{{\partial ^2}{\varphi _y}}}{{\partial {t^2}}} $$ (7) $$ \frac{{\partial {Q_x}}}{{\partial x}} + \frac{{\partial {Q_y}}}{{\partial y}} = {I_0}\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}} $$ (8) 其中
$ {I_i} $ 为等效惯性参数, 具体表示为$$ {I_i} = \int_{ - h/2}^{h/2} {\rho {z^i}{\text{d}}z} \;\; (i = 0,1,2) $$ (9) $ \rho $ 为板的质量密度. 式(4)~式(8)中的等效内力和内力矩分别定义为$$ \left( {{N_x},{N_y},{N_{xy}}} \right) = \int_{ - {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}^{{h \mathord{\left/ {\vphantom {h 2}} \right. } 2}} {\left( {{\sigma _x},{\sigma _y},{\tau _{xy}}} \right)} {\text{d}}z \tag{10a}$$ $$ \left( {{M_x},{M_y},{M_{xy}}} \right) = \int_{ - {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}^{{h \mathord{\left/ {\vphantom {h 2}} \right. } 2}} {\left( {{\sigma _x},{\sigma _y},{\tau _{xy}}} \right)} z{\text{d}}z \tag{10b}$$ $$ \left( {{Q_x},{Q_y}} \right) = {k_s}\int_{ - {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}^{{h \mathord{\left/ {\vphantom {h 2}} \right. } 2}} {\left( {{\tau _{xz}},{\tau _{yz}}} \right)} {\text{d}}z \tag{10c}$$ 其中
$ {k_s} = {5 \mathord{\left/ {\vphantom {5 6}} \right. } 6} $ 为剪切修正系数.将式(2)、式(3)代入式(10), 可得用位移分量表示的等效内力和弯矩
$$ {N_x} = {S {_0}}\left( {\frac{{\partial {u_0}}}{{\partial x}} + \nu \frac{{\partial {v_0}}}{{\partial y}}} \right) + {S {_1}}\left( {\frac{{\partial {\varphi _x}}}{{\partial x}} + \nu \frac{{\partial {\varphi _y}}}{{\partial y}}} \right) - {N_T} \tag{11a}$$ $$ {N_y} = {{S {_0}}}\left( {\nu \frac{{\partial {u_0}}}{{\partial x}} + \frac{{\partial {v_0}}}{{\partial y}}} \right) + {{S {_1}}}\left( {\nu \frac{{\partial {\varphi _x}}}{{\partial x}} + \frac{{\partial {\varphi _y}}}{{\partial y}}} \right) - {N_T} \tag{11b}$$ $$ {N_{xy}} = {S {_{r0}}}\left( {\frac{{\partial {v_0}}}{{\partial x}} + \frac{{\partial {u_0}}}{{\partial y}}} \right) + {S {_{r1}}}\left( {\frac{{\partial {\varphi _y}}}{{\partial x}} + \frac{{\partial {\varphi _x}}}{{\partial y}}} \right) \tag{11c}$$ $$ {M_x} = {{S {_1}}}\left( {\frac{{\partial {u_0}}}{{\partial x}} + \nu \frac{{\partial {v_0}}}{{\partial y}}} \right) + {{S {_2}}}\left( {\frac{{\partial {\varphi _x}}}{{\partial x}} + \nu \frac{{\partial {\varphi _y}}}{{\partial y}}} \right) - {M_T} \tag{11d}$$ $$ {M_y} = {{S {_1}}}\left( {\nu \frac{{\partial {u_0}}}{{\partial x}} + \frac{{\partial {v_0}}}{{\partial y}}} \right) + {{S {_2}}}\left( {\nu \frac{{\partial {\varphi _x}}}{{\partial x}} + \frac{{\partial {\varphi _y}}}{{\partial y}}} \right) - {M_T} \tag{11e}$$ $$ {M_{xy}} = {S {_{r1}}}\left( {\frac{{\partial {v_0}}}{{\partial x}} + \frac{{\partial {u_0}}}{{\partial y}}} \right) + {S {_{r2}}}\left( {\frac{{\partial {\varphi _y}}}{{\partial x}} + \frac{{\partial {\varphi _x}}}{{\partial y}}} \right) \tag{11f}$$ $$ {Q_x} = {k_s}{S {_{r0}}}\left( {{\varphi _x} + \frac{{\partial {w_0}}}{{\partial x}}} \right) \tag{11g}$$ $$ {Q_y} = {k_s}{S {_{r0}}}\left( {{\varphi _y} + \frac{{\partial {w_0}}}{{\partial y}}} \right) \tag{11h}$$ 其中
$ {N_T} $ 和$ {M_T} $ 分别为热薄膜力和热弯矩, 其定义为$$ \left( {{N_T},{M_T}} \right) = \int_{ - {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}^{{h \mathord{\left/ {\vphantom {h 2}} \right. } 2}} {\frac{{\alpha E\theta }}{{1 - \nu }}\left( {1,z} \right)} {\text{d}}z $$ (12) 方程(11)中的刚度系数由下式给出
$$ \left.\begin{array} {l} {S_i} = \displaystyle\int_{ - {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}^{{h \mathord{\left/ {\vphantom {h 2}} \right. } 2}} {\dfrac{{E\left( z \right)}}{{1 - {\nu ^2}}}} {z^i}{\text{d}}z\\ {S_{ri}} = \dfrac{{(1 - \nu )}}{2}{S_i}\;\;(i = 0,1,2) \end{array}\right\} $$ (13) 将式(11)代入式(4)~式(8)可得用位移表示的运动方程. 这是包含5个基本未知函数的相互耦合的偏微分方程组, 其中还包含由温度场来确定的热薄膜力和热弯矩. 因此, 联立求解这些微分方程在数学上是很困难的. 本文通过消去面内位移和转角, 最终将5个方程转化为只用挠度表示的运动方程. 首先将式(11a) ~ (11c) 分别代入式(4)和式(5), 并分别对两式关于坐标
$ x $ 和$ y $ 求偏导数后相加, 利用式(13)可得$$ {{S {_0}}}{\nabla ^2}{\varepsilon _0} + {{S {_1}}}{\nabla ^2}{\kappa _0} - {\nabla ^2}{N_T} = \frac{{{\partial ^2}}}{{\partial {t^2}}}({I_0}{\varepsilon _0} + {I_1}{\kappa _0}) $$ (14) 其中
$$\left.\begin{array} {l} {\varepsilon _0} = \dfrac{{\partial {u_0}}}{{\partial x}} + \dfrac{{\partial {v_0}}}{{\partial y}} \\ {\kappa _0} = \dfrac{{\partial {\varphi _x}}}{{\partial x}} + \dfrac{{\partial {\varphi _y}}}{{\partial y}} \\ \;{\nabla ^2} = \dfrac{{{\partial ^2}}}{{\partial {x^2}}} + \dfrac{{{\partial ^2}}}{{\partial {y^2}}} \end{array} \right\}$$ (15) 类似地将式(11d) ~ (11f)代入式(6)和式(7), 利用式(8)可得
$$ {{S {_1}}}{\nabla ^2}{\varepsilon _0} + {S {_2}}{\nabla ^2}{\kappa _0} - {\nabla ^2}{M_T} = \frac{{{\partial ^2}}}{{\partial {t^2}}}\left( {{I_0}w + {I_1}{\varepsilon _0} + {I_2}{\kappa _0}} \right) $$ (16) 利用式(11g)和(11 h)可得方程(8)的位移形式
$$ {\kappa _0} = - {\nabla ^2}{w_0} + \frac{{{I_0}}}{{{k_s}{S_{r0}}}}\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}} $$ (17) 最后, 将式(14)代入式(16), 忽略面内应变
$ {\varepsilon _0} $ 的惯性项, 并利用式(17)可得只用横向位移$ w $ 表示的运动方程$$ \begin{split} & {{\bar S}_2}{\nabla ^4}{w_0} + {\nabla ^2}({M_T} - {z_0}{N_T}) = \hfill \\ &\qquad - \frac{{{\partial ^2}}}{{\partial {t^2}}}\left[ {{I_0}{w_0} - \left( {{{\bar I}_2} + \frac{{{{\bar S}_2}{I_0}}}{{{k_s}{S_{r0}}}}} \right){\nabla ^2}{w_0} + \frac{{{I_2}{I_0}}}{{{k_s}{S_{r0}}}}\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}}} \right] \end{split} $$ (18) 其中
$$\left.\begin{array} {l} {\bar S_2} = {S_2} - {z_0}{S_1} \\ {\bar I_2} = {I_2} - {z_0}{I_1} \\ {z_0} = {S_1}/{S_0} \end{array}\right\} $$ (19) 上式中热弯矩和热薄膜力由热−弹耦合产生的变温场确定.
1.2 热传导方程
根据单向耦合的热弹性动力学理论, 可给出功能梯度微板的热传导方程[27]
$$ k{\nabla ^2}\theta + \frac{\partial }{{\partial z}}\left( {k\frac{{\partial \theta }}{{\partial z}}} \right) = \rho C\frac{{\partial \theta }}{{\partial t}} + \frac{{\alpha E{T_0}}}{{1 - 2\nu }}\frac{{\partial e}}{{\partial t}} $$ (20) 其中,
$ k $ ,$ C $ 分别是热传导系数和比热, 它们都是坐标 z 的已知函数;$ e $ 是体积应变. 板的体积应变可表示为$$ e = {\varepsilon _x} + {\varepsilon _y} + {\varepsilon _z} = \frac{{1 - 2\nu }}{{1 - \nu }}\left( {{\varepsilon _0} + z{\kappa _0}} \right) + \frac{{1 + \nu }}{{1 - \nu }}\alpha \theta $$ (21) 将上式代入式(20), 并考虑到板内温度改变主要是由横向振动引起的, 因此, 为了方便求解可忽略温度场在面内梯度变化, 可得单相耦合的准一维热传导方程[13, 27]
$$ \frac{\partial }{{\partial z}}\left( {k\frac{{\partial \theta }}{{\partial z}}} \right) = \rho C\frac{{\partial \theta }}{{\partial t}} + \frac{{\alpha E{T_0}}}{{1 - \nu }}\frac{\partial }{{\partial t}}\left( {{\varepsilon _0} + z{\kappa _0}} \right) $$ (22) 2. 热弹性阻尼的求解
2.1 自由振动响应
假设位移场和温度场的动态响应同步, 则系统的热−弹耦合自由振动响应可表示为下列调和形式
$$ ({u_0},{v_0},{w_0}) = [\bar u(x,y),\bar v(x,y),\bar w(x,y)]{{\rm{e}}^{{\text{i}}\omega t}} \tag{23a}$$ $$ ({\varphi _x},{\varphi _y},\theta ) = [{\bar \varphi _x}(x,y),{\bar \varphi _y}(x,y),\bar \theta (x,y,z)]{{\rm{e}}^{{\text{i}}\omega t}} \tag{23b}$$ 其中,
$ \omega $ 为固有频率;$ \bar u $ ,$ \bar v $ ,$ \bar w $ ,$ {\bar \varphi _x} $ ,$ {\bar \varphi _y} $ 和$ \bar \theta $ 分别为对应位移物理量的振幅;$ {\text{i}} = \sqrt { - 1} $ . 将式(23)分别代入式(18)和式(22)消去时间变量可得$$ {\bar S_2}{\nabla ^4}\bar w + {\mu _2}{\nabla ^2}\bar w + {\mu _0}\bar w + {\nabla ^2}\left( {{{\bar M}_T} - {z_0}{{\bar N}_T}} \right) = 0 $$ (24) $$ \frac{\partial }{{\partial z}}\left( {k\frac{{\partial \bar \theta }}{{\partial z}}} \right) = {\text{i}}\omega \left[ {\rho C\bar \theta + \frac{{\alpha E{T_0}}}{{1 - \nu }}({{\bar \varepsilon }_{\text{0}}} + z{{\bar \kappa }_{\text{0}}})} \right] $$ (25) 其中
$$\left. \begin{array} {l} {\mu _{\text{0}}} = {\omega ^2}{I_0}\left( {\dfrac{{{\omega ^2}{I_2}}}{{{k_s}{S_{r0}}}} - 1} \right)\\ {\mu _{\text{2}}} = {\omega ^2}\left( {{{\bar I}_2} + \dfrac{{{I_0}{{\bar S}_2}}}{{{k_s}{S_{r0}}}}} \right) \end{array} \right\}$$ (26) $$ \left.\begin{array} {l} \left( {{{\bar N}_T},{{\bar M}_T}} \right) = \int_{ - {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}^{{h \mathord{\left/ {\vphantom {h 2}} \right. } 2}} {\dfrac{{\alpha E\bar \theta }}{{1 - \nu }}\left( {1,z} \right)} {\text{d}}z \\ {\bar \varepsilon _0} = \dfrac{{\partial \bar u}}{{\partial x}} + \dfrac{{\partial \bar v}}{{\partial y}}\\ {\bar \kappa _0} = \dfrac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \dfrac{{\partial {{\bar \varphi }_y}}}{{\partial y}} \end{array}\right\}$$ (27) 若谐振器由均匀材料组成, 则有
$ {z_0} = 0 $ ,$ {\bar \varepsilon _0} = 0 $ , 方程(24)和(25)可简化为[13]$$ {S_2}{\nabla ^4}\bar w + {\mu _2}{\nabla ^2}\bar w + {\mu _0}\bar w + {\nabla ^2}{\bar M_T} = 0 $$ (28) $$ k\frac{{{\partial ^{\text{2}}}\bar \theta }}{{\partial {z^{\text{2}}}}} = {\text{i}}\omega \left( {\rho C\bar \theta + \frac{{\alpha E{T_0}}}{{1 - \nu }}z{{\bar \kappa }_{\text{0}}}} \right) $$ (29) 这时振动方程(28)中没有热薄膜力, 热传导方程(29)中的系数是常数, 于是可容易得到其解析通解[13].
2.2 热传导方程的求解
利用文献[26-27, 29]中的分层均匀化方法将功能梯度材料微板沿板厚平均划分为N层子区域, 当分层数足够大时每层的材料性质可近似看作均匀的. 这样, 就可以将变系数微分方程(25)转化为分别定义在各分层上的一系列常系数的微分方程组
$$ \begin{split} & \frac{{{\partial ^2}{{\bar \theta }_j}}}{{\partial {z^2}}} = \frac{{{\text{i}}\omega }}{{{k_j}}}\left[ {{\rho _j}{C_j}{{\bar \theta }_j} + \frac{{{\alpha _j}{E_j}{T_0}}}{{1 - \nu }}({{\bar \varepsilon }_{\text{0}}} + z{{\bar \kappa }_{\text{0}}})} \right]\;\; \hfill \\ &\qquad ({z_j} < z < {z_{j + 1}};j = 1,2,3, \cdots ,N) \end{split} $$ (30) 其中, 具有下标为j的物性参数是在中点
${\bar z_j} = ({z_j} + {z_{j + 1}})/2$ 处取值;$ {\bar \theta _j}(z) $ 为第j层温度场.依据微分方程理论, 方程(30)的通解可表示为
$$ \begin{split} & {{\bar \theta }_j} = {A_j}\sin ({p_j}z) + {B_j}\cos ({p_j}z) - {q_j}({{\bar \varepsilon }_0} + z{{\bar \kappa }_0}) \hfill \\ & \qquad ({z_j} < z < {z_{j + 1}},j = 1,2,3, \cdots ,N) \end{split} $$ (31) 其中
$$ \left.\begin{array} {l} {p_j} = ({\text{i}} - 1)\sqrt {\dfrac{{\omega {\rho _j}{C_j}}}{{2{k_j}}}} \\ {q_j} = \dfrac{{{E_j}{\alpha _j}{T_0}}}{{(1 - \nu ){\rho _j}{C_j}}} \end{array}\right\} $$ (32) $ {A_j} $ 和$ {B_j} $ 为与坐标z无关而与几何中面的变形量$ {\bar \varepsilon _0} $ 和曲率参数$ {\bar \kappa _0} $ 有关的系数.考虑上下表面为绝热的功能梯度微板, 温度边界条件以及层间界面处连续性条件可分别表示为
$$ \frac{{\partial {{\bar \theta }_1}}}{{\partial z}}\left| {_{z = - {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}} \right. = 0 \tag{33a}$$ $$ {\bar \theta _j}(x,y,{z_j}) = {\bar \theta _{j + 1}}(x,y,{z_{j + 1}}) \tag{33b}$$ $$ {k_j}\frac{{\partial {{\bar \theta }_j}}}{{\partial z}}\left| {_{z = {z_j}}} \right. = {k_{j + 1}}\frac{{\partial {{\bar \theta }_{j + 1}}}}{{\partial z}}\left| {_{z = {z_{j + 1}}}} \right. \tag{33c}$$ $$ \frac{{\partial {{\bar \theta }_N}}}{{\partial z}}\left| {_{z = {h \mathord{\left/ {\vphantom {h 2}} \right. } 2}}} \right. = 0 \tag{33d}$$ 分析式(31)和齐次边界及连续性条件式(33), 系数
$ {A_j} $ 和$ {B_j} $ 可表示为$$ \left.\begin{array}{l} {A_j} = {\bar A_{1j}}{\bar \varepsilon _0} + {\bar A_{2j}}{\bar \kappa _{\text{0}}}\\ {B_j} = {\bar B_{1j}}{\bar \varepsilon _0} + {\bar B_{2j}}{\bar \kappa _{\text{0}}} \end{array}\right\} $$ (34) 其中,
$ {\bar A_{1 j}} $ ,$ {\bar B_{1 j}} $ ,$ {\bar A_{2 j}} $ 和$ {\bar B_{2 j}} $ 为常复数. 将式(34)代入式(31), 进而代入式(33), 比较$ {\bar \varepsilon _0} $ 和$ {\bar \kappa _0} $ 的系数可以获得关于这四个系数的代数方程组, 以确定这4N个常数. 最终可得各层的温度场$ {\bar \theta _j} = {\bar A_j}{\bar \varepsilon _0} + {\bar B_j}{\bar \kappa _{\text{0}}} $ ($ {z_j} < z < {z_{j + 1}} $ ;$j = 1,2,\cdots,N$ ) (35)其中
$$ {\bar A_j} = {\bar A_{1j}}\sin ({p_j}z) + {\bar B_{1j}}\cos ({p_j}z) - {q_j} \tag{36a} $$ $$ {\bar B_j} = {\bar A_{2j}}\sin ({p_j}z) + {\bar B_{2j}}\cos ({p_j}z) - {q_j}z \tag{36b} $$ 2.3 热弹性阻尼
热弹性阻尼要从振动方程(24)在给定边界条件下的复频率中提取. 而要求解结构自由振动响应, 就需要将方程(24)中的热薄膜力和热弯矩用挠度表示. 为此, 先将式(23)代入式(14), 忽略关于中面应变
$ {\varepsilon _0} $ 的惯性力可得$$ {\nabla ^2}{\bar \varepsilon _0} = - {z_0}{\nabla ^2}{\bar \kappa _0} + \frac{1}{{{S_0}}}{\nabla ^2}{\bar N_T} - \frac{{{I_1}{\omega ^2}}}{{{S_0}}}{\bar \kappa _0} $$ (37) 再将式(35)代入式(27)积分可得
$$ {\bar N_T} = {\bar \beta _A}{\bar \varepsilon _0} + {\bar \beta _B}{\bar \kappa _0} ,\;\;\; {\bar M_T} = {\bar \gamma _A}{\bar \varepsilon _0} + {\bar \gamma _B}{\bar \kappa _0} $$ (38) 其中系数
$ {\bar \beta _A} $ ,$ {\bar \beta _B} $ ,$ {\bar \gamma _A} $ 和$ {\bar \gamma _B} $ 的计算公式为$$ ({\bar \beta _A},{\bar \beta _A},{\bar \gamma _A},{\bar \gamma _B}) = \sum\limits_{j = 1}^N {\frac{{{\alpha _j}{E_j}}}{{1 - \nu }}\int_{{z_j}}^{{z_{j + 1}}} {({{\bar A}_j},} \;{{\bar B}_j},z{{\bar A}_j},z{{\bar B}_j}){\text{d}}z} $$ (39) 于是由式(37)和式(38)可以得到
$$ {\nabla ^2}{\bar \varepsilon _0} = {a_0}{\omega ^2}{\bar \kappa _0} + {a_2}{\nabla ^2}{\bar \kappa _0} \tag{40a}$$ $$ {\nabla ^2}{\bar N_T} = {b_0}{\omega ^2}{\bar \kappa _0} + {b_2}{\nabla ^2}{\bar \kappa _0} \tag{40b}$$ $$ {\nabla ^2}{\bar M_T} = {c_0}{\omega ^2}{\bar \kappa _0} + {c_2}{\nabla ^2}{\bar \kappa _0} \tag{40c}$$ 其中
$$ {a_0} = \frac{{ - {I_1}}}{{1 - {{\bar \beta }_A}}} ,\;\;\; {a_2} = \frac{{{{\bar \beta }_B} - {z_0}}}{{1 - {{\bar \beta }_A}}} \tag{41a}$$ $$ {b_0} = {a_0}{\bar \beta _A} ,\;\;\; {b_2} = {a_2}{\bar \beta _A} + {\bar \beta _B} \tag{41b}$$ $$ {c_0} = {a_0}{\bar \gamma _A} ,\;\;\; {c_2} = {a_2}{\bar \gamma _A} + {\bar \gamma _B} \tag{41c}$$ 将式(23)代入式(17)可得
$$ {\bar \kappa _0} = - {\nabla ^2}\bar w - \frac{{{I_0}}}{{{k_s}{S_{r0}}}}{\omega ^2}\bar w $$ (42) 最后, 将式(40b)、式(40c)和式(42)代入方程(24)可得以挠度的振幅表示的功能梯度明德林微板的结构振动方程
$$ {\bar a_4}{\nabla ^4}\bar w + {\bar a_2}{\omega ^2}{\nabla ^2}\bar w + ({\bar a_{01}} + {\bar a_{02}}{\omega ^2}){\omega ^2}\bar w = 0 $$ (43) 其中
$$ {\bar a_{01}} = - {I_0}/{\bar S_2} , \; {\bar a_{02}} = ({f_0} + {\bar c_2})/{K_s} \tag{44a}$$ $$ {\bar a_2} = (1 + {f_2})/{K_s} + {f_0} + {\bar c_2}\; , \;\;\; {\bar a_4} = 1 + {f_2} \tag{44b}$$ $$ {f_0} = ({z_0}{b_0} - {c_0})/{\bar S_2} \text{, }\;\;\; {f_2} = ({z_0}{b_2} - {c_2})/{\bar S_2} \tag{44c}$$ $$ {K_s} = {k_s}{S_{r0}}/{I_0} , \;\;\; {\bar c_2} = {\bar I_2}/{\bar S_2} \tag{44d}$$ 微分方程(43)可以写成标准形式
$$ ({\nabla ^2} + {\varLambda _1})({\nabla ^2} + {\varLambda _2})\bar w = 0 $$ (45) 其中
$$ {\varLambda _1},{\varLambda _2} = \frac{{ - {{\bar a}_2}{\omega ^2} \pm \omega \sqrt {{{\bar a}_2}^2{\omega ^2} - 4{{\bar a}_4}({{\bar a}_{01}} + {{\bar a}_{02}}{\omega ^2})} }}{{2{{\bar a}_4}}} $$ (46) 等温的均匀基尔霍夫微板的自由振动方程可以表示为[27]
$$ ({\nabla ^2} + \lambda _0^*)({\nabla ^2} - \lambda _0^*)\bar w_0^* = 0 $$ (47) 其中,
$ \lambda _0^* = \omega _0^*\sqrt {I_0^*{\text{/}}S_2^*} $ ,$ \omega _0^* $ 是基尔霍夫均匀板无阻尼自由振动的固有频率,$ {w_0}^* $ 是相应振幅;$ I_0^* = h{\rho _0} $ ,$ S_2^* = {E_0}{h^3}/[12(1 - {\nu _0}^2)] $ ;$ {\rho _0} $ ,$ {E_0} $ 和$ {\nu _0} $ 是参考均匀板的质量密度、弹性模量和泊松比.考虑边界条件为周边简支, 则方程(47)的边界条件可表示为
$$ \bar w_0^* = {\text{0}},\;\;{\nabla ^{\text{2}}}\bar w_0^* = {\text{0}}$$ (48) 可以证明(见附录A), 方程(45)在四边简支约束下的边界条件也可表示为
$$ \bar w = {\text{0}},\;\;{\nabla ^{\text{2}}}\bar w = {\text{0}}$$ (49) 从而, 由微分方程边值问题式(45)、式(49)与式(47)、式(48)在数学上的相似性, 可得它们特征值之间的关系
$$ {\varLambda _{\text{1}}} = \lambda _0^* \text{, }\;\;\; {\varLambda _{\text{2}}} = - \lambda _0^* $$ (50) 将式(46)代入式(50), 可得关于固有频率代数方程
$$ {\bar a_{02}}{\omega ^4} + ({\bar a_{01}} + {\bar a_2}\lambda _0^*){\omega ^2} + {\bar a_4}\lambda {_0^{*2}} = 0 $$ (51) 由参数
$ {f_0} $ 和$ {f_2} $ 的定义可知, 代数方程(51)的系数是关于复频率$ \omega $ 的超越函数, 因此, 很难直接求得解析解. 根据文献中普遍采用的简化方法[4-9,13-15,26-27,29], 可做如下近似, 即令$ {f_0}(\omega ) = {f_0}({\omega _0}) $ ,$ {f_2}(\omega ) = {f_2}({\omega _0}) $ , 其中$ {\omega _0} $ 为不考虑热弹性阻尼的功能梯度明德林微板的固有频率. 显然, 只要在式(51)中令$ {f_0} = {f_2} = 0 $ , 即可得到频率$ {\omega _0} $ , 然后可计算出包含阻尼参数的函数$ {f_1}({\omega _0}) $ 和$ {f_2}({\omega _0}) $ , 进而代入式(44)求得方程(51)的系数, 即可获得功能梯度明德林微板的复频率$$ \omega = {\left[ {\frac{{ - ({{\bar a}_{01}} + {{\bar a}_2}\lambda _0^*) \pm \sqrt {{{({{\bar a}_{01}} + {{\bar a}_2}\lambda _0^*)}^2} - 4{{\bar a}_{02}}{{\bar a}_4}\lambda {{_0^*}^2}} }}{{2{{\bar a}_{02}}}}} \right]^{1/2}} $$ (52) 最后, 采用复频率法可得用逆品质因子表示的微板的热弹性阻尼
$$ {Q^{ - 1}} = 2\left| {\frac{{{{\rm{Im}}} (\omega )}}{{{{\rm{Re}}} (\omega )}}} \right| $$ (53) 其中
$ {{\rm{Re}}} (\omega ) $ 和$ {\text{Im}}(\omega ) $ 分别是复频率的实部和虚部.3. 数值结果与讨论
数值计算时分别选取Ni-Si3N4 和Al-SiC两种由金属和陶瓷组分相复合而成的功能梯度微板. 在表1中列出了上述四种均匀组分材料在室温
${T_0} = 300\;{\text{K}}$ 下的物性参数. 假设材料体积分数在厚度方向按幂函数连续变化, 则等效材料性质参数可由混合律给出[26-29]Materials E/GPa $ \rho $
/(kg·m−3)$ \kappa $
/(W·m·K−1)$ C $
/(J·kg·K−1)$ \alpha $/
(10−6·K−1)$ \nu $ SiC (Pc) 427 3100 65 670 4.3 0.17 Al (Pm) 70 2707 233 896 23.4 0.3 Si3N4 (Pc) 250 3200 8 937.5 3.0 0.27 Ni (Pm) 210 8900 92 438.2 13.0 0.3 $$ P = {P_{\text{m}}} + ({P_{\text{c}}} - {P_{\text{m}}}){\left( {\frac{1}{2} + \frac{z}{h}} \right)^n} $$ (54) 其中,
$ n \geqslant 0 $ 表示材料性质梯度变化指数;$ {P_{\text{c}}} $ 和$ {P_{\text{m}}} $ 分别表示纯陶瓷和纯金属材料的物性参数. 在计算中选择金属板为参考均匀板. 将式(54)分别代入式(9)、式(13)可计算出刚度系数和惯性参数. 将式(54)代入式(30)可得各分层内的常系数热传导方程.关于分层均匀化方法的收敛性已在前期研究中[26-27,29]给以了检验和讨论. 在后面的数值计算中取分层数N = 500, 这已达到非常高的精度.
首先, 在
$ n = 0 $ 的特殊情况下, 表2中给出了纯陶瓷(Si3N4)明德林正方形微板在不同振动模态和边厚比(a/h)下的热弹性阻尼值. 并与相应基尔霍夫微板的结果进行了比较. 由此可见, 随着板厚和模态阶数的增加, 两种板理论预测的热弹性阻尼之间的相对误差变大. 在$ a/h = {\text{10}} $ 时, 最大误差达到了10%. 进一步在表3中给出了具有不同梯度变化指数和不同边厚比的功能梯度(Ni-Si3N4)正方形微板($h = 1\;{\text{μm}}$ )在基频振动时的热弹性阻尼值. 从两种板理论的预测结果比较可见,随着金属组分的增加, 两种理论预测的值相对误差随之增大. 上述数值结果定量地表明, 随着板厚的增大剪切变形对热弹性阻尼的影响变得显著.表 2 具有不同边厚比的纯陶瓷(Si3N4)正方形微板在不同振动模态下的热弹性阻尼($ {Q^{ - 1}} \times {10^5} $ )($h = 1\;{\text{μm}}$ )Table 2. TED ($ {Q^{ - 1}} \times {10^5} $ ) in a full ceramic (Si3N4) square micro plate for different values of a/h in different modes ($h = 1\;{\text{μm}}$ )Modes $ a/h = 30 $ $ a/h = 20 $ $ a/h = {\text{10}} $ Kirchhoff Mindlin Err/% Kirchhoff Mindlin Err/% Kirchhoff Mindlin Err/% (1,1) 14.613 14.558 0.38 7.6773 7.6312 0.60 2.1528 2.1110 1.95 (1,2) 6.9934 6.9477 0.65 3.3417 3.2980 1.31 0.8967 0.8593 4.16 (2,2) 4.5733 4.5287 0.97 2.1528 2.1110 1.95 0.5684 0.5344 5.97 (1,3) 3.7261 3.6821 1.18 1.7427 1.7016 2.36 0.4573 0.4252 7.02 (2,3) 2.9215 2.8782 1.48 1.3568 1.3170 2.94 0.3538 0.3240 8.41 (3,3) 2.1528 2.1110 1.95 0.9926 0.9546 3.83 0.2571 0.2304 10.4 ${ {Err} } = (Q_{ {\text{Kirchhoff} } }^{ - 1} - Q_{ {\text{Mindlin} } }^{ - 1})/Q_{ {\text{Kirchhoff} } }^{ - 1}\times100 {\text{%} }$ 表 3 两种板理论下功能梯度正方形微板的热弹性阻尼$ ({Q^{ - 1}} \times {10^4}) $ 比较 ($h = 1\;{\text{μm}}$ , 一阶模态)Table 3. Comparison of TED$ ({Q^{ - 1}} \times {10^4}) $ in an FGM (Ni-Si3N4) square plate based on the two plate theories ($h = 1\;{\text{μm}}$ , 1st mode)n Theories a/h 5 10 20 30 50 100 0.5 Mindlin 1.2395 3.3630 6.3841 5.4018 2.4535 0.6379 Kirchhoff 1.2944 3.4649 6.4609 5.4514 2.4618 0.6384 Err/% 4.24 2.94 1.18 0.91 0.34 0.07 1 Mindlin 2.4114 6.3177 8.6528 5.4878 2.1874 0.5562 Kirchhoff 2.6271 6.5289 8.7886 5.5367 2.1947 0.5566 Err/% 8.21 3.23 1.55 0.88 0.33 0.07 3 Mindlin 4.4540 11.729 9.5238 4.8808 1.8236 0.4591 Kirchhoff 4.8390 12.196 9.7117 4.9278 1.8298 0.4595 Err/% 7.95 4.33 3.83 0.95 0.33 0.09 10 Mindlin 6.3314 16.558 10.650 5.2113 1.9246 0.4838 Kirchhoff 6.8618 17.308 10.874 5.2632 1.9316 0.4842 Err/% 7.73 4.33 2.06 0.98 0.36 0.08 为了更进一步考查剪切变形对热弹性阻尼的影响, 在图2中给出了不同边厚比对应的功能梯度正方形微板(Ni-Si3N4)的热弹性阻尼随着板厚连续变化的特性曲线. 图中结果再次表明剪切变形对中厚板和厚板的影响显著. 但是, 在
$ a = 30 h $ 时两种板理论的预测值已经非常接近. 这说明对于薄板, 基尔霍夫板理论已给出十分满意的解答.![]()
图 2 具有不同边厚比$ a/h $ 的功能梯度(Ni-Si3N4)微板的热弹性阻尼与板厚$ h $ 的关系曲线(一阶模态)Figure 2. Curves of TED versus the plate thickness$ h $ of FGM (Ni-Si3N4) micro plate with different side-to-thickness ratio a/h (1st mode)图3中给出了边长为
$a = 100\;{\text{μm}}$ 的功能梯度(Al-SiC)正方形微板的热弹性阻尼随着材料性质梯度变化指数和板厚尺寸的变化规律. 在厚度$h < 6\;{\text{μm}}$ 时, 热弹性阻尼随着指数的增大(金属组分增加)先增大后减小, 有峰值出现; 当$h > 6\;{\text{μm}}$ 后热弹性阻尼则为单调递增的. 这种热弹性阻尼随指数增大在一定范围内发生的非单调性是由于材料性质非均匀性在这部分区间的急剧变化而引起的复杂热−弹耦合振动引起的. 从图中可见, 极大值发生在材料性质梯度变化最剧烈的区间内. 这一结果也和所选取的两种组分材料的性质有关. 对于Ni-Si3N4组分材料, 由文献[27]中的结果可知在板很薄时热弹性阻尼随梯度指数增加会出现极小值. 由此可见, 材料性质的非均匀性以及组份材料的物性参数不同会导致这种复杂的热−弹耦合振动效应. 由此可见非均匀材料谐振器热−弹耦合振动响应的复杂性.![]()
图 3 具有不同厚度的功能梯度 (Al-SiC) 正方形微板的热弹性阻$ {Q^{ - 1}} $ 尼与梯度指数$ n $ 间的变化曲线($a = b = 100\;{\text{μm}}$ )Figure 3. Curves of TED$ {Q^{ - 1}} $ versus the gradient index$ n $ of an FGM (Al-SiC) micro square plate with different values of the thickness ($a = b = 100\;{\text{μm}}$ )图4中绘出了具有不同材料梯度指数的正方形微板(
$a = b = 100\;{\text{μm}}$ )的热弹性阻尼随着板厚的连续变化的特性曲线. 结果表明, 热弹性阻尼随着厚度的增大先单调增加, 达到极大值后单调减小. 但是, 对于两种不同组分材料的微板, 热弹性阻尼极大值的变化规律并不相同. Ni-Si3N4微板的热弹性阻尼最大值随梯度指数增大单调增加, 而Al-AiC微板的热弹性阻尼的最大值的变化却并非单调. 为了更加清晰地显示这种变化规律, 图5给出两种微板的最大热弹性阻尼$ Q_{\max }^{ - 1} $ 与梯度指数$ n $ 的连续变化曲线. 图中可见, 同等条件下Al-SiC微板的$ Q_{\max }^{ - 1} $ $- h $ 曲线是局部非单调的.![]()
图 4 具有不同梯度指数的功能梯度正方形微板的热弹性阻尼与板厚之间的关系曲线($a = b = 100\;{\text{μm}}$ )Figure 4. Curves of TED versus the plate thickness of an FGM square micro plate with different values of the gradient index ($a = b = 100\;{\text{μm}}$ )![]()
图 5 正方形功能梯度微板的最大热弹性阻尼值随材料梯度指数变化的特性曲线Figure 5. Characteristic curves of the maximum TED versus the material gradient index of FGM rectangular micro plate图6中给出了对应不同长宽比 (a/b) 的功能梯度 (Ai-SiC) 矩形微板在两种板理论下的热弹性阻尼随板厚的变化曲线. 结果表明, 长度固定后随着宽度的减小, 基尔霍夫微板的热弹性阻尼的峰值保持不变, 而明德林微板的峰值有所减小. 然而, 使得热弹性阻尼取得峰值的临界厚度
$ {h_{{\text{cr}}}} $ 随着长宽比的增大而减小.![]()
图 6 不同长宽比的FGM (Al-SiC)矩形微板的TED随板厚变化的特性曲线($n = 0.5,\;a = 100\;{\text{μm}}$ )Figure 6. Curves of the TED versus the plate thickness of an FGM (Al-SiC) rectangular micro plate with different length-to-wideness ratio ($n = 0.5,\;a = 100\;{\text{μm}}$ )为了反映振动模态对热弹性阻尼的影响规律, 图7给出了梯度指数
$ n = 0.5 $ 的功能梯度 (Al-SiC) 正方形微板分别以四前阶模态振动时的热弹性阻尼与厚度关系曲线. 其中的变化特性与图6类似. 结果表明, 微板抗弯刚度的增大(长度给定, 宽度减小; 振动模态阶数增大) 对热弹性阻尼的峰值影响甚微, 但是对临界厚度的影响显著.![]()
图 7 对应不同振动模态的FGM (Al-SiC)正方形微板的热弹性阻尼随板厚变化的特性曲线($n = 0.5,\;a = b = 100\;{\text{μm}}$ )Figure 7. TED versus of the plate thickness of an FGM (Al-SiC) square micro plate corresponding to different vibration modes ($n = 0.5,\;a = b = 100\;{\text{μm}}$ )4. 结论
基于明德林一阶剪切变形板理论和准一维单向耦合的热传导理论建立了材料性质沿厚度按连续变化的功能梯度矩形微板热−弹性耦合自由振动控制微分方程. 采用分层均匀化方法求得了复杂变系数热传导方程半解析解. 在四边简支边界条件, 利用结构振动微分方程边值问题的相似性, 获得了用相同尺寸和边界条件的参考均匀无阻尼基尔霍夫微板的固有频率表示的功能梯度明德林微板的复频率, 从而由复频率法求得了代表热弹性阻尼的逆品质因子半解析解. 通过数值算例定量地分析了材料梯度变化指数、组份材料物性参数、几何尺寸以及振动模态对功能梯度微板热弹性阻尼的影响规律. 通过与基尔霍夫薄板理论的预测结果进比较, 定量地分析了剪切变形对热弹性阻尼的影响程度. 结果表明, 基尔霍夫板理论预测的热弹性阻尼值大于明德林板理论的预测值. 而且随着板厚的增大二者的差别逐渐变得显著. 两种理论预测的热弹性阻尼随梯度指数和板厚的变化趋势是相同的.
附录
$ \;$ 考虑简支边界条件, 在x = 0, a处有
$$ \bar w = 0 , \;\;{\bar N_x} = 0 ,\;\; {\bar M_x} = 0 \tag{A1}$$ 由于微板周边面内自由, 可以认为薄膜力在边界处为零, 利用式(11a)、式(11b)和式(23)可得
$$ \frac{{\partial {{\bar u}_0}}}{{\partial x}} + \nu \frac{{\partial {{\bar v}_0}}}{{\partial y}} = - {z_0}\left( {\frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \nu \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right) + \frac{{{{\bar N}_T}}}{{{S_0}}} \tag{A2}$$ $$ \nu \frac{{\partial {{\bar u}_0}}}{{\partial x}} + \frac{{\partial {{\bar v}_0}}}{{\partial y}} = - {z_0}\left( {\nu \frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right) + \frac{{{{\bar N}_T}}}{{{S_0}}} \tag{A3}$$ 将式(23)代入式(11d)和(11e), 并利用式(A1)和式(A2)得到
$$ {\bar M_x} = {\bar S_2}\left( {\frac{{\partial {\varphi _x}}}{{\partial x}} + \nu \frac{{\partial {\varphi _y}}}{{\partial y}}} \right) - {\bar M_T} + {z_0}{\bar N_T} \tag{A4}$$ $$ {\bar M_y} = {\bar S_2}\left( {\nu \frac{{\partial {\varphi _x}}}{{\partial x}} + \frac{{\partial {\varphi _y}}}{{\partial y}}} \right) - {\bar M_T} + {z_0}{\bar N_T} \tag{A5}$$ 将式(A2)和式(A3)相加可得
$$ {\bar \varepsilon _0} = - {z_0}{\bar \kappa _0} + \frac{{2{{\bar N}_T}}}{{{S_0}(1 + \nu )}} \tag{A6}$$ 进一步将式(38)代入式(A4)和式(A5)可得
$$ \begin{split} & {{\bar M}_x} = {{\bar S}_2}\left( {\frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \nu \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right)- \hfill \\ &\qquad ({{\bar \gamma }_A} + {z_0}{{\bar \beta }_A}){{\bar \varepsilon }_0} + ({{\bar \gamma }_B} + {z_0}{{\bar \beta }_B}){{\bar \kappa }_0} \end{split} \tag{A7}$$ $$ \begin{split} & {{\bar M}_y} = {{\bar S}_2}\left( {\nu \frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right)- \hfill \\ & \qquad ({{\bar \gamma }_A} + {z_0}{{\bar \beta }_A}){{\bar \varepsilon }_0} + ({{\bar \gamma }_B} + {z_0}{{\bar \beta }_B}){{\bar \kappa }_0} \end{split} \tag{A8}$$ 由式(A6)式和式(38)可得
$$ {\bar \varepsilon _0} = \bar \beta {\bar \kappa _0} \text{, }\;\; \bar \beta = \frac{{(1 + \nu ){S_1} + 2{{\bar \beta }_B}}}{{(1 + \nu ){S_0} - 2{{\bar \beta }_A}}} \tag{A9}$$ 将式(A9)代入式(A7)和式(A8)得到
$$ \begin{split} & {{\bar M}_x} = {{\bar S}_2}\left( {\frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \nu \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right) + [{{\bar \gamma }_B} + {z_0}{{\bar \beta }_B} - \hfill \\ & \qquad ({{\bar \gamma }_A} + {z_0}{{\bar \beta }_A})\bar \beta ]\left( {\frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right) \end{split} \tag{A10}$$ $$ \begin{split} & {{\bar M}_y} = {{\bar S}_2}\left( {\nu \frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right) + [{{\bar \gamma }_B} + {z_0}{{\bar \beta }_B} - \hfill \\ &\qquad ({{\bar \gamma }_A} + {z_0}{{\bar \beta }_A})\bar \beta ]\left( {\frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}} \right) \end{split} \tag{A11}$$ 在简支边x = 0, a处, 还有条件
$$ {\bar \varphi _y} = 0 , \;\; \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}}{\text{ = }}0 \tag{A12}$$ 将式(A12)代入式(A10), 并利用式(A1)中第三式可得
$$ \frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} = 0 \tag{A13}$$ 于是, 由式(A12)和式(A13)知
$$ {\bar \kappa _0} = \frac{{\partial {{\bar \varphi }_x}}}{{\partial x}} + \frac{{\partial {{\bar \varphi }_y}}}{{\partial y}} = 0 \tag{A14}$$ 最后利用式(A1)第一式和式(42), 可将x = 0, a处的边界条件表示为
$$ \bar w = 0 ,\;\;\; {\nabla ^2}\bar w = 0 \tag{A15}$$ 类似地, 可证明式(A15)在y = 0, b处也成立. 进一步可以预测, 对于直边多边形的功能梯度简支微板式(A15)也会成立.
-
![]()
图 1 微板的几何尺寸和坐标示意图
Figure 1. Schematic illustration of micro plate and the coordinate system
![]()
图 2 具有不同边厚比
$ a/h $ 的功能梯度(Ni-Si3N4)微板的热弹性阻尼与板厚$ h $ 的关系曲线(一阶模态)Figure 2. Curves of TED versus the plate thickness
$ h $ of FGM (Ni-Si3N4) micro plate with different side-to-thickness ratio a/h (1st mode)![]()
图 3 具有不同厚度的功能梯度 (Al-SiC) 正方形微板的热弹性阻
$ {Q^{ - 1}} $ 尼与梯度指数$ n $ 间的变化曲线($a = b = 100\;{\text{μm}}$ )Figure 3. Curves of TED
$ {Q^{ - 1}} $ versus the gradient index$ n $ of an FGM (Al-SiC) micro square plate with different values of the thickness ($a = b = 100\;{\text{μm}}$ )![]()
图 4 具有不同梯度指数的功能梯度正方形微板的热弹性阻尼与板厚之间的关系曲线(
$a = b = 100\;{\text{μm}}$ )Figure 4. Curves of TED versus the plate thickness of an FGM square micro plate with different values of the gradient index (
$a = b = 100\;{\text{μm}}$ )![]()
图 5 正方形功能梯度微板的最大热弹性阻尼值随材料梯度指数变化的特性曲线
Figure 5. Characteristic curves of the maximum TED versus the material gradient index of FGM rectangular micro plate
![]()
图 6 不同长宽比的FGM (Al-SiC)矩形微板的TED随板厚变化的特性曲线(
$n = 0.5,\;a = 100\;{\text{μm}}$ )Figure 6. Curves of the TED versus the plate thickness of an FGM (Al-SiC) rectangular micro plate with different length-to-wideness ratio (
$n = 0.5,\;a = 100\;{\text{μm}}$ )![]()
图 7 对应不同振动模态的FGM (Al-SiC)正方形微板的热弹性阻尼随板厚变化的特性曲线(
$n = 0.5,\;a = b = 100\;{\text{μm}}$ )Figure 7. TED versus of the plate thickness of an FGM (Al-SiC) square micro plate corresponding to different vibration modes (
$n = 0.5,\;a = b = 100\;{\text{μm}}$ )表 1 功能梯度微板组分材料的物性参数值 (
${T_0} = 300\;{\text{K}}$ )[24, 26]Table 1 Values of the parameters of the material properties of the constituents of FGM micro plate (
${T_0} = 300\;{\text{K}}$ ) [24, 26]Materials E/GPa $ \rho $
/(kg·m−3)$ \kappa $
/(W·m·K−1)$ C $
/(J·kg·K−1)$ \alpha $/
(10−6·K−1)$ \nu $ SiC (Pc) 427 3100 65 670 4.3 0.17 Al (Pm) 70 2707 233 896 23.4 0.3 Si3N4 (Pc) 250 3200 8 937.5 3.0 0.27 Ni (Pm) 210 8900 92 438.2 13.0 0.3 
下载: 导出CSV
表 2 具有不同边厚比的纯陶瓷(Si3N4)正方形微板在不同振动模态下的热弹性阻尼(
$ {Q^{ - 1}} \times {10^5} $ )($h = 1\;{\text{μm}}$ )Table 2 TED (
$ {Q^{ - 1}} \times {10^5} $ ) in a full ceramic (Si3N4) square micro plate for different values of a/h in different modes ($h = 1\;{\text{μm}}$ )Modes $ a/h = 30 $ $ a/h = 20 $ $ a/h = {\text{10}} $ Kirchhoff Mindlin Err/% Kirchhoff Mindlin Err/% Kirchhoff Mindlin Err/% (1,1) 14.613 14.558 0.38 7.6773 7.6312 0.60 2.1528 2.1110 1.95 (1,2) 6.9934 6.9477 0.65 3.3417 3.2980 1.31 0.8967 0.8593 4.16 (2,2) 4.5733 4.5287 0.97 2.1528 2.1110 1.95 0.5684 0.5344 5.97 (1,3) 3.7261 3.6821 1.18 1.7427 1.7016 2.36 0.4573 0.4252 7.02 (2,3) 2.9215 2.8782 1.48 1.3568 1.3170 2.94 0.3538 0.3240 8.41 (3,3) 2.1528 2.1110 1.95 0.9926 0.9546 3.83 0.2571 0.2304 10.4 ${ {Err} } = (Q_{ {\text{Kirchhoff} } }^{ - 1} - Q_{ {\text{Mindlin} } }^{ - 1})/Q_{ {\text{Kirchhoff} } }^{ - 1}\times100 {\text{%} }$
下载: 导出CSV
表 3 两种板理论下功能梯度正方形微板的热弹性阻尼
$ ({Q^{ - 1}} \times {10^4}) $ 比较 ($h = 1\;{\text{μm}}$ , 一阶模态)Table 3 Comparison of TED
$ ({Q^{ - 1}} \times {10^4}) $ in an FGM (Ni-Si3N4) square plate based on the two plate theories ($h = 1\;{\text{μm}}$ , 1st mode)n Theories a/h 5 10 20 30 50 100 0.5 Mindlin 1.2395 3.3630 6.3841 5.4018 2.4535 0.6379 Kirchhoff 1.2944 3.4649 6.4609 5.4514 2.4618 0.6384 Err/% 4.24 2.94 1.18 0.91 0.34 0.07 1 Mindlin 2.4114 6.3177 8.6528 5.4878 2.1874 0.5562 Kirchhoff 2.6271 6.5289 8.7886 5.5367 2.1947 0.5566 Err/% 8.21 3.23 1.55 0.88 0.33 0.07 3 Mindlin 4.4540 11.729 9.5238 4.8808 1.8236 0.4591 Kirchhoff 4.8390 12.196 9.7117 4.9278 1.8298 0.4595 Err/% 7.95 4.33 3.83 0.95 0.33 0.09 10 Mindlin 6.3314 16.558 10.650 5.2113 1.9246 0.4838 Kirchhoff 6.8618 17.308 10.874 5.2632 1.9316 0.4842 Err/% 7.73 4.33 2.06 0.98 0.36 0.08
下载: 导出CSV
-
[1] Zener C. Internal fraction in solids I. Theory of internal fraction in reeds. Physical Review, 1937, 53: 90-99
[2] Bishop GE, Kinra V. Equivalence of the mechanical and entropic description of elastothermodynamics in composite materials. Mechanics of Composite Materials and Structures, 1996, 3: 83-95
[3] Bishop GE, Kinra V. Elastothermaldynamic damping in laminated composites. International Journal of Solids and Structures, 1997, 34: 1075-1092
[4] Lifshitz R, Roukes ML. Thermoelastic damping in micro- and nano-mechanical systems. Phys. Rev. B, 2000, 61: 5600-5609
[5] Nayfeh AH, Younis MI. Modeling and simulations of thermoelastic damping in microplates. Journal of Micromechanics and Microengineering, 2024, 14: 1711-1717
[6] Sun YX, Tohmyoh H. Thermoelastic damping of the axisymmetric vibration of circular plate resonator. Journal of Sound and Vibration, 2009, 319: 392-405
[7] Sun YX, Saka M. Thermoelastic damping in micro-scale circular plate resonators. Journal of Sound and Vibration, 2010, 329: 328-337
[8] Ali NA, Mohammadi AK. Thermoelastic damping in clamped-clamped annular microplate. Applied Mechanics and Materials, 2012, 110-116: 1870-1878
[9] Salajeghe S, Khadem SE, Rasekh M. Nonlinear analysis of thermoelastic damping in axisymmetric vibration of micro circular thin-plate resonators. Applied Mathematical Modelling, 2012, 36: 5991-6000
[10] Li P, Fang YM, Hu RF. Thermoelastic damping in rectangular and circular microplate resonators. Journal of Sound and Vibration, 2012, 331: 721-733
[11] Fang YM, Li P, Wang ZL. Thermoelastic damping in the axisymmetric vibration of circular micro plate resonators with two-dimensional heat conduction. Journal of Thermal Stresses, 2013, 36: 830-850
[12] Fang YM, Li P, Zhou HY, et al. Thermoelastic damping in rectangular microplate resonators with three-dimensional heat conduction. International Journal of Mechanical Sciences, 2017, 133: 578-589
[13] 马航空, 周晨阳, 李世荣. Mindlin 矩形微板的热弹性阻尼解析解. 力学学报, 2020, 52(5): 1383-1393 Ma Hangkong Zhou Chenyang, Li Shirong. Analytical solution of thermoelastic damping in Mindlin rectangular plate. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1383-1393 (in Chinese)
[14] Sharma JN, Sharma R. Damping in micro-scale generalized thermoelastic circular plate resonators. Ultrasonics, 2011, 51: 352-358
[15] Sharma JN, Grover D. Thermoelastic vibration analysis of MEMS/NEMS plate resonators with voids. Acta Mechanica, 2012, 223: 167-187
[16] Guo FL, Song J, Wang GQ, et al. Analysis of thermoelastic dissipation in circular micro-plate resonators using the generalized thermoelasticity theory. Journal of Sound and Vibration, 2014, 333: 2465-2474
[17] Grover D. Damping in thin circular viscothermoelastic plate resonators. Canadian Journal of Physics, 2015, 93: 1597-1605
[18] Chugh N, Partap G. Study of thermoelastic damping in microstrecth thermoelastic thin circular plate. Journal of Vibration Engineering and Technologies, 2021, 9: 105-114
[19] Wang YW, Li XF. Synergistic effect of memory-size-microstructure on thermoelastic damping of a micro-plate. International Journal of Heat and Mass Transfer, 2021, 181: 122031 doi: 10.1016/j.ijheatmasstransfer.2021.122031
[20] Sun YX, Jiang Y, Yang JL. Thermoelastic damping of the axisymmetric vibration of laminated trilayered circular plate resonators. Canada Journal of Physics, 2014, 92: 1026-1032
[21] Zuo WL, Li P, Zhang JR, et al. Analytical modeling of thermoelastic damping in bilayered microplate resonators. International Journal of Mechanical Science, 2016, 106: 128-137
[22] Zuo WL, Li P, Du JK, et al. Thermoelastic damping in trilayered microplate resonators. International Journal of Mechanical Sciences, 2019, 151: 595-608
[23] Liu SB, Ma JX, Yang XF, et al. Theoretical analysis of thermoelastic damping in bilayered circular plate resonators with two-dimensional heat conduction. International Journal of Mechanical Sciences, 2018, 135: 114-123
[24] Liu SB, Ma JX, Yang XF, et al. Theoretical 3D model of thermoelastic damping in laminated rectangular plate resonators. International Journal of Structural Stability and Dynamics, 2018, 18: 1850158
[25] Emami AA, Alibeigloo A. Thermoelastic damping analysis of FG Mindlin microplates using strain gradient theory. Journal of Thermal Stresses, 2016, 39: 1499-1522
[26] Li SR, Chen S, Xiong P. Thermoelastic damping in functionally graded material circular micro plates. Journal of Thermal Stresses, 2018, 41: 1396-1413
[27] Li SR, Ma HK. Analysis of free vibration of functionally graded material micro-plates with thermoelastic damping. Archive Applied Mechanics, 2020, 90: 1285-1304
[28] 李世荣, 张靖华, 徐华. 功能梯度与均匀圆板弯曲解的线性转换关系, 力学学报, 2011, 43(5): 871-877 Li Shirong, Zhang Jinghua, Xu Hua. Linear transformation between the bending solutions of functionally graded and homogenous circular plates based on the first-order shear deformation theory, Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(5): 871-877 (in Chinese)
[29] 许新, 李世荣. 功能梯度材料微梁的热弹性阻尼研究, 力学学报, 2017, 49(2): 308-316 Xu Xin, Li Shirong. Analysis of elastic damping of functionally graded micro- beams. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(2): 308-316 (in Chinese)
-
期刊类型引用(10)
1. 于彦彦，丁海平，芮志良. 二维层状盆地地震动附加放大特征研究:SV波入射. 振动与冲击. 2024(04): 166-178 . 
百度学术
2. 邢浩洁，刘爱文，李小军，陈苏，傅磊. 多人工波速优化透射边界在谱元法地震波动模拟中的应用. 地震学报. 2022(01): 26-39 . 百度学术
3. 孟雪莉，刘少林，杨顶辉，汪文帅，徐锡伟，李小凡. 基于优化数值积分的谱元法模拟地震波传播. 石油地球物理勘探. 2022(03): 602-612+492-493 . 百度学术
4. 邢浩洁，李小军，刘爱文，李鸿晶，周正华，陈苏. 弹性波模拟中局部透射边界的反射特征及参数优化. 振动与冲击. 2022(12): 301-312 . 百度学术
5. 孔曦骏，邢浩洁，李鸿晶. 流固耦合地震波动问题的显式谱元模拟方法. 力学学报. 2022(09): 2513-2528 . 本站查看
6. 邢浩洁，李小军，刘爱文，李鸿晶，周正华，陈苏. 波动数值模拟中的外推型人工边界条件. 力学学报. 2021(05): 1480-1495 . 本站查看
7. 邢浩洁，李鸿晶，李小军. 一维波动有限元模拟中透射边界的时域稳定条件. 应用基础与工程科学学报. 2021(03): 617-632 . 百度学术
8. 孔曦骏，邢浩洁，李鸿晶，周正华. 多次透射公式飘移问题的控制方法. 力学学报. 2021(11): 3097-3109 . 本站查看
9. 陈少林，柯小飞，张洪翔. 海洋地震工程流固耦合问题统一计算框架. 力学学报. 2019(02): 594-606 . 本站查看
10. 陈少林，程书林，柯小飞. 海洋地震工程流固耦合问题统一计算框架——不规则界面情形. 力学学报. 2019(05): 1517-1529 . 本站查看
其他类型引用(4)
计量
- 文章访问数: 717
- HTML全文浏览量: 198
- PDF下载量: 128
- 被引次数: 14
