THERMOELASTIC DAMPING IN FUNCTIONALLY GRADED MINDLIN RECTANGULAR MICRO PLATES
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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.
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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 
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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 ${ {Err} } = (Q_{ {\text{Kirchhoff} } }^{ - 1} - Q_{ {\text{Mindlin} } }^{ - 1})/Q_{ {\text{Kirchhoff} } }^{ - 1}\times100 {\text{%} }$
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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
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[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): 1383-1393Ma 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): 871-877Li 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): 308-316Xu 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) -
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