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基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析

陈卫 方耀楚 孙冰 彭林欣

陈卫, 方耀楚, 孙冰, 彭林欣. 基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析. 力学学报, 2023, 55(6): 1-16 doi: 10.6052/0459-1879-23-040
引用本文: 陈卫, 方耀楚, 孙冰, 彭林欣. 基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析. 力学学报, 2023, 55(6): 1-16 doi: 10.6052/0459-1879-23-040
Chen Wei, Fang Yaochu, Sun Bing, Peng LinXin. Meshless analysis of linear bending and free vibration of functionally graded carbon nanotube-reinforced composite plate on elastic foundation based on improved reddy type third-order shear deformation theory. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1-16 doi: 10.6052/0459-1879-23-040
Citation: Chen Wei, Fang Yaochu, Sun Bing, Peng LinXin. Meshless analysis of linear bending and free vibration of functionally graded carbon nanotube-reinforced composite plate on elastic foundation based on improved reddy type third-order shear deformation theory. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1-16 doi: 10.6052/0459-1879-23-040

基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析

doi: 10.6052/0459-1879-23-040
基金项目: 国家自然科学基金(11562001, 12162004)和南华大学博士科研启动基金(Y00043-13)资助项目
详细信息
    通讯作者:

    彭林欣, 教授, 主要研究方向: 计算复合板壳力学中的无网格法. E-mail: penglx@gxu.edu.cn

  • 中图分类号: TU339

MESHLESS ANALYSIS OF LINEAR BENDING AND FREE VIBRATION OF FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE PLATE ON ELASTIC FOUNDATION BASED ON IMPROVED REDDY TYPE THIRD-ORDER SHEAR DEFORMATION THEORY

  • 摘要: 基于改进Reddy型3阶剪切变形理论(third-order shear deformation theory, TSDT)假设, 考虑碳纳米管(carbon nanotubes, CNTs)转向及功能梯度材料的不均匀性, 建立弹性地基上功能梯度碳纳米管增强复合材料(functionally graded carbon nanotube-reinforced composite, FG-CNTRC)板的线性弯曲和自由振动无网格分析模型. 利用改进Reddy型TSDT推导FG-CNTRC板的势能和动能, 给出弹性地基势能的表达式, 再将其分别进行叠加, 通过最小势能原理及Hamilton原理推导出弹性地基上FG-CNTRC板的线性弯曲和自由振动控制方程. 采用稳定移动克里金插值(stabilized moving Kriging interpolation, SMKI)对问题域内的节点进行离散, 该近似形函数的构造方法满足克罗内克条件, 可以直接施加边界条件. 文中首先给出基于三阶剪切变形理论的弹性地基FG-CNTRC板线性弯曲与自由振动无网格离散模型. 随后通过基准算例, 研究本文方法的有效性及精度问题. 最后数值分析了CNTs的分布形式、转向角、体积分数、地基系数、宽厚比和边界条件等对FG-CNTRC板的线性弯曲及自振频率的影响. 研究表明: 采用本文方法计算FG-CNTRC薄板、中厚板、甚至厚板的线性弯曲和自振频率均具有较高的计算精度; 随着CNTs体积分数和地基系数的增加, FG-CNTRC板结构刚度逐渐增大; FG-CTRC板结构刚度与宽厚比成正相关, 厚度增加的剪切效应会让CNTs转向角对结构刚度的影响逐渐降低.

     

  • 图  1  弹性地基上FG-CNTRC板的等效模型

    Figure  1.  Equivalent model of FG-CNTRC plate on elastic foundation

    图  2  四边简支FG-CNTRC板中点挠度收敛性分析(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    Figure  2.  Convergence analysis of central deflection of simply supported FG-CNTRC plate(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    图  3  不同边界条件下FG-CNTRC板中点无量纲轴向应力${\bar \sigma }_{xx}$(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.17, b/h = 50)

    Figure  3.  Normalized axial stress ${{\bar \sigma }}_{xx}$ of central point of FG-CNTRC plate under different boundary condition(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.17, b/h = 50)

    图  4  本文方法与SMKI-FSDT之间的计算效率比较(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    Figure  4.  Comparison of computational efficiency between the present method and SMKI-FSDT(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    图  5  不同宽厚比下四边固支UD板中点归一化挠度随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    Figure  5.  Normalized central deflection versus CNT orientation angle θ for the clamped UD plate with different width-thickness ratio(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    图  6  弹性地基上四边固支FG-CNTRC板的无量纲中点挠度随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  6.  Dimensionless central deflection versus CNT orientation angle θ for the clamped FG-CNTRC plate on elastic foundation (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    图  7  FG-CNTRC板的无量纲基础频率随边界条件的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  7.  Dimensional fundamental frequency versus boundary condition for the FG-CNTRC square plate(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    图  8  不同长宽比下四边固支UD板的无量纲基础频率随转向角的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    Figure  8.  Dimensional fundamental frequency versus CNT orientation angle θ for the clamped UD plate with different length-width ratio (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    图  9  不同宽厚比下四边固支UD板的归一化基础频率随转向角的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    Figure  9.  Normalized fundamental frequency versus CNT orientation angle θ for the clamped UD plate with different width-thickness ratio(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    图  10  弹性地基上四边固支FG-CNTRC板的无量纲基础频率随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  10.  Dimensionless fundamental frequencies versus CNT orientation angle θ for the clamped FG-CNTRC plate on elastic foundation (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    图  11  四边固支UD板前5阶自振模态(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  11.  The first five natural vibration of clamped UD plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    表  1  材料参数

    Table  1.   The properties of material

    ParametersMatrixCNTs
    Poisson’s ratioνm = 0.34$ {\nu }_{\text{12}}^{\text{CNT}} $ = 0.175
    density/(kg·m−3)ρm = 1150ρCNT = 1400
    Young’s modulus/GPaEm = 2.1$ {\text{E}}_{\text{11}}^{\text{CNT}} $ = 5646.6,$ {\text{E}}_{\text{22}}^{\text{CNT}} $ = 7080
    shear modulus/GPa$ {\text{G}}_{\text{12}}^{\text{CNT}} $ = 1944.5
    下载: 导出CSV

    表  2  CNTs的效能参数

    Table  2.   The efficiency parameters of CNTs

    $ {\text{V}}_{\text{CNT}}^{\text{*}} $η1η2η3
    0.110.1490.9340.934
    0.140.1500.9410.941
    0.170.1491.3811.381
    下载: 导出CSV

    表  3  四边简支FG-CNTRC板中点无量纲挠度(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)

    Table  3.   Dimensionless central deflection of simply supported FG-CNTRC plate(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    b/hCNT typeFEM-FSDT[8]IGA-TSDT[16]Present
    10UD3.739 × 10−33.717 × 10−33.716 × 10−3
    FG-V4.466 × 10−34.427 × 10−34.447 × 10−3
    FG-O5.230 × 10−35.438 × 10−35.430 × 10−3
    FG-X3.177 × 10−33.102 × 10−33.140 × 10−3
    20UD3.628 × 10−33.624 × 10−33.629 × 10−3
    FG-V4.879 × 10−34.877 × 10−34.880 × 10−3
    FG-O6.155 × 10−36.248 × 10−36.243 × 10−3
    FG-X2.701 × 10−32.685 × 10−32.697 × 10−3
    50UD1.1551.1561.156
    FG-V1.6531.6541.652
    FG-O2.1572.1632.158
    FG-X0.7900.7910.792
    下载: 导出CSV

    表  4  四边简支FG-CNTRC板无量纲基础频率(${\boldsymbol{V}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)

    Table  4.   Dimensionless fundamental frequencies of simply supported FG-CNTRC plate(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    b/hCNT typeFEM- FSDT[8]FEM- TSDT[32]SMKI- FSDT (error)Present (error)
    5UD8.8328.627 (2.321 )8.753 (0.894 )
    FG-V8.4078.551 (1.713 )8.504 (1.154 )
    FG-O8.0298.133 (1.295 )7.946 (1.034 )
    FG-X9.1228.877 (2.686 )9.040 (0.899 )
    10UD13.53213.60113.519 (0.603 )13.553 (0.353 )
    FG-V12.45212.35212.423 (0.575 )12.458 (0.858 )
    FG-O11.55011.37111.544 (1.521 )11.322 (0.431 )
    FG-X14.61614.72714.601 (0.856 )14.677 (0.340 )
    20UD17.35517.35417.333 (0.121 )17.320 (0.196 )
    FG-V15.11015.03815.050 (0.080 )15.081 (0.286 )
    FG-O13.53213.43213.526 (0.700 )13.405 (0.201 )
    FG-X19.93919.94519.905 (0.201 )19.914 (0.155 )
    50UD19.22319.18119.268 (0.454 )19.199 (0.094 )
    FG-V16.25216.21816.261 (0.265 )16.252 (0.210 )
    FG-O14.30214.27514.410 (0.946 )14.302 (0.189 )
    FG-X22.98422.93022.993 (0.275 )22.935 (0.022 )
    下载: 导出CSV

    表  5  不同体积分数及地基系数下四边简支FG-CNTRC板中点无量纲挠度${{\tilde w}}$

    Table  5.   Dimensionless central deflections ${\tilde w}$ of simply supported FG-CNTRC plates with different volume fraction and foundation coefficient

    (kw, ks)Theory$\text{}{{V} }_{\text{CNT} }^{\text{*} }$ = 0.11$\text{}{{V} }_{\text{CNT} }^{\text{*} }$ = 0.17
    UDFG-VFG-OFG-XUDFG-VFG-OFG-X
    uniform load
    (0, 0)TSDT0.73560.88061.07050.62060.47120.56630.68440.4012
    SSDT0.73400.87921.07430.61790.47020.56550.68620.4001
    Present0.73520.87971.07410.62140.47100.56560.68540.4013
    (100, 0)TSDT0.69830.82860.99550.93500.45560.54400.65300.3897
    SSDT0.69690.82740.99880.59100.45480.54370.65460.3887
    Present0.69800.82780.99870.59420.45540.54370.65400.3898
    (100, 50)TSDT0.47730.53460.59910.42620.35000.40000.45570.3098
    SSTD0.47670.53410.60030.42500.34950.39970.45650.3092
    Present0.47740.53460.60080.42660.35000.39990.45650.3100
    sinusoidal load
    (100, 0)TSDT0.49640.58690.70810.42270.31770.37690.45260.2723
    SSDT0.49530.58590.71040.42080.31700.37630.45370.2715
    Present0.49630.58650.71020.42350.31760.37650.45320.2726
    (100, 0)TSDT0.47290.55440.66120.40560.30790.36320.43300.2651
    SSDT0.47190.55340.66330.40380.30720.36260.43400.2644
    Present0.47280.55400.66310.40630.30780.36280.43350.2653
    (100, 50)TSDT0.32400.35830.40010.28960.23610.26740.30340.2101
    SSDT0.32190.35790.40090.28880.23570.26710.30390.2097
    Present0.32260.35840.40110.29020.23620.26730.30380.2104
    Notes: The data TSDT and SSDT in the table are from the analytical solution of the literature[28]
    下载: 导出CSV

    表  6  不同宽厚比及地基系数下四边固支FG-CNTRC板中点无量纲挠度(${{V}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)

    Table  6.   Dimensionless central deflections of clamped FG-CNTRC plates with width-thick ratio and foundation coefficient (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    (kw, ks)CNT typeb/h
    5102050100
    (0, 0)UD3.833 × 10−42.119 × 10−31.315 × 10−22.627 × 10−13.632
    FG-V3.875 × 10−42.252 × 10−31.572 × 10−23.663 × 10−15.293
    FG-O4.251 × 10−42.604 × 10−31.934 × 10−24.797 × 10−17.040
    FG-X3.722 × 10−41.985 × 10−31.120 × 10−21.896 × 10−12.470
    (100, 0)UD3.792 × 10−42.056 × 10−31.195 × 10−21.434 × 10−10.521
    FG-V3.832 × 10−42.181 × 10−31.406 × 10−21.708 × 10−10.538
    FG-O4.201 × 10−42.511 × 10−31.693 × 10−21.926 × 10−10.544
    FG-X3.683 × 10−41.929 × 10−31.030 × 10−21.181 × 10−10.492
    (100, 50)UD3.428 × 10−41.592 × 10−30.658 × 10−20.290 × 10−10.064
    FG-V3.462 × 10−41.665 × 10−30.711 × 10−20.298 × 10−10.065
    FG-O3.759 × 10−41.847 × 10−30.773 × 10−20.303 × 10−10.065
    FG-X3.340 × 10−41.516 × 10−30.608 × 10−20.281 × 10−10.064
    下载: 导出CSV

    表  7  不同体积分数及地基系数下四边简支FG-CNTRC板的无量纲基础频率

    Table  7.   Dimensionless fundamental frequencies of simply supported FG-CNTRC plates with different volume fraction and foundation coefficient

    $\text{}{{V} }_{\text{CNT} }^{\text{*} }$CNT type(kw, ks) = (0, 0)(kw, ks) = (100, 0)(kw, ks) = (100, 50)
    TSDTSSDTPresentTSDTSSDTPresentTSDTSSDTPresent
    0.11UD13.5513.5713.5513.8813.9013.8916.8216.8316.81
    FG-V12.4512.4612.4612.8112.8212.8215.9415.9515.93
    FG-O11.3411.2011.3211.7311.7211.7215.0915.0715.06
    FG-X14.6914.7214.6815.0015.0314.9817.7517.7717.73
    0.14UD14.3614.3814.3614.6714.6914.6717.4617.4717.44
    FG-V13.2813.2913.2813.6213.6313.6216.5716.5916.57
    FG-O12.1321.1012.1212.5012.4812.4815.6715.6615.65
    FG-X15.4115.4515.4015.7015.7415.6918.3318.3618.31
    0.17UD16.8316.8516.8417.1017.1217.1019.5319.5419.52
    FG-V15.4415.4615.4515.7315.7415.7418.3418.3518.33
    FG-O14.0914.0814.0814.4114.3914.4017.2117.2017.20
    FG-X18.1918.2118.1818.4318.4618.4320.7020.7320.69
    Notes: The data TSDT and SSDT in the table are from the analytical solution of the literature[28].
    下载: 导出CSV

    表  8  不同宽厚比及地基系数下四边固支FG-CNTRC板的无量纲基础频率(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    Table  8.   Dimensionless fundamental frequencies of clamped FG-CNTRC plates with different width-thickness ratio and foundation coefficient (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    b/h(kw, ks) = (0, 0)(kw, ks) = (100, 0)(kw, ks) = (100, 50)
    UDFG-VFG-OFG-XUDFG-VFG-OFG-XUDFG-VFG-OFG-X
    510.6110.5710.1210.7610.6610.6210.1710.8111.2011.1610.7411.34
    1018.0417.5616.4018.5918.2917.8216.6818.8420.7720.3719.3821.25
    2028.5726.3923.9530.7429.8227.7425.4331.9140.8239.3937.8342.32
    5039.5333.9829.9546.0152.0848.0045.2457.15121.31119.69118.51123.44
    10042.5635.8131.3350.98104.94102.39100.91108.62323.58322.49321.73325.06
    下载: 导出CSV
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