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 引用本文: 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.
 Citation: Li Shirong. Thermoelastic damping in functionally graded Mindlin rectangular micro plates. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1601-1612.

## 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.

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