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功能梯度材料明德林矩形微板的热弹性阻尼

李世荣

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 doi: 10.6052/0459-1879-22-055
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 doi: 10.6052/0459-1879-22-055
Li Shirong. Thermoelastic damping in functionally graded Mindlin rectangular micro plates. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1601-1612 doi: 10.6052/0459-1879-22-055
Citation: Li Shirong. Thermoelastic damping in functionally graded Mindlin rectangular micro plates. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1601-1612 doi: 10.6052/0459-1879-22-055

功能梯度材料明德林矩形微板的热弹性阻尼

doi: 10.6052/0459-1879-22-055
基金项目: 国家自然科学基金资助项目(11672260)
详细信息
    作者简介:

    李世荣, 教授, 主要研究方向: 结构非线性分析及新型材料结构力学行为. E-mail: srli@yzu.edu.cn

  • 中图分类号: O343

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.

     

  • 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]

    MaterialsE/GPa$ \rho $
    /(kg·m−3)
    $ \kappa $
    /(W·m·K−1)
    $ C $
    /(J·kg·K−1)
    $ \alpha $/
    (10−6·K−1)
    $ \nu $
    SiC (Pc)4273100656704.30.17
    Al (Pm)70270723389623.40.3
    Si3N4 (Pc)25032008937.53.00.27
    Ni (Pm)210890092438.213.00.3
    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}} $
    KirchhoffMindlinErr/%KirchhoffMindlinErr/%KirchhoffMindlinErr/%
    (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)

    nTheoriesa/h
    510203050100
    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
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出版历程
  • 收稿日期:  2022-01-28
  • 录用日期:  2022-03-19
  • 网络出版日期:  2022-03-20
  • 刊出日期:  2022-06-18

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