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
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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.
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关键词:
- 改进Reddy型三阶剪切变形理论 /
- 功能梯度碳纳米管增强复合材料板 /
- 弹性地基 /
- 稳定移动克里金插值 /
- 线性弯曲 /
- 自由振动
Abstract: Based on the improved Reddy type third-order shear deformation theory (TSDT), and considering the orientation of carbon nanotubes (CNTs) and the inhomogeneity of functionally gradient materials, a meshless analysis model for the linear bending and free vibration of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates on elastic foundation is established. The potential energy and kinetic energy of FG-CNTRC plate are derived by the improved Reddy type TSDT, and then the expression of the potential energy of elastic foundation is given, and then they are respectively superposed. The linear bending and free vibration control equations of FG-CNTRC plate on elastic foundation are derived by the principle of minimum potential energy and Hamilton principle. Stable moving Kriging interpolation (SMKI) is used to discretize the nodes in the problem domain. The construction method of the approximate shape function satisfies the Kronecker condition and can directly apply the boundary conditions. In this paper, a meshless discrete model of linear bending and free vibration of FG-CNTRC plate on elastic foundation based on the third-order shear deformation theory is presented. Then, the effectiveness and accuracy of the proposed method are studied by a benchmark example. Finally, the effects of CNTs distribution, orientation angle, volume fraction, foundation coefficient, width thickness ratio and boundary conditions on the linear bending and natural frequency of FG-CNTRC plate are numerically analyzed. The results show that the proposed method has a good accuracy in calculating the linear bending and natural frequencies of FG-CNTRC thin, medium-thick, and even thick plates. As the volume fraction of CNTs and the foundation coefficient increase, the stiffness of the FG-CNTRC plate structure gradually increases. The stiffness of the FG-CTRC plate structure is positively correlated with the width-thickness ratio, and the shear effect of increasing thickness gradually reduces the influence of the CNTs orientation angle on the plate stiffness. -
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θ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θ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 modes 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
Parameters Matrix CNTs Poisson’s ratio νm = 0.34 $ {\nu }_{\text{12}}^{\text{CNT}} $ = 0.175 density/(kg·m−3) ρm = 1150 ρCNT = 1400 Young’s modulus/GPa Em = 2.1 ${{E} }_{\text{11} }^{\text{CNT} }$ = 5646.6,
${{E} }_{\text{22} }^{\text{CNT} }$ = 7080shear modulus/GPa — ${{G} }_{\text{12} }^{\text{CNT} }$ = 1944.5 
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表 2 CNTs的效能参数
Table 2. The efficiency parameters of CNTs
${{V} }_{\text{CNT} }^{\text{*} }$ η1 η2 η3 0.11 0.149 0.934 0.934 0.14 0.150 0.941 0.941 0.17 0.149 1.381 1.381
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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/h CNT type FEM-FSDT[8] IGA-TSDT[16] Present 10 UD 3.739 × 10−3 3.717 × 10−3 3.716 × 10−3 FG-V 4.466 × 10−3 4.427 × 10−3 4.447 × 10−3 FG-O 5.230 × 10−3 5.438 × 10−3 5.430 × 10−3 FG-X 3.177 × 10−3 3.102 × 10−3 3.140 × 10−3 20 UD 3.628 × 10−3 3.624 × 10−3 3.629 × 10−3 FG-V 4.879 × 10−3 4.877 × 10−3 4.880 × 10−3 FG-O 6.155 × 10−3 6.248 × 10−3 6.243 × 10−3 FG-X 2.701 × 10−3 2.685 × 10−3 2.697 × 10−3 50 UD 1.155 1.156 1.156 FG-V 1.653 1.654 1.652 FG-O 2.157 2.163 2.158 FG-X 0.790 0.791 0.792
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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/h CNT
typeFEM-
FSDT[8]FEM-
TSDT[32]SMKI-
FSDT (error)Present (error) 5 UD — 8.832 8.627 (2.321) 8.753 (0.894) FG-V — 8.407 8.551 (1.713) 8.504 (1.154) FG-O — 8.029 8.133 (1.295) 7.946 (1.034) FG-X — 9.122 8.877 (2.686) 9.040 (0.899) 10 UD 13.532 13.601 13.519 (0.603) 13.553 (0.353) FG-V 12.452 12.352 12.423 (0.575) 12.458 (0.858) FG-O 11.550 11.371 11.544 (1.521) 11.322 (0.431) FG-X 14.616 14.727 14.601 (0.856) 14.677 (0.340) 20 UD 17.355 17.354 17.333 (0.121) 17.320 (0.196) FG-V 15.110 15.038 15.050 (0.080) 15.081 (0.286) FG-O 13.532 13.432 13.526 (0.700) 13.405 (0.201) FG-X 19.939 19.945 19.905 (0.201) 19.914 (0.155) 50 UD 19.223 19.181 19.268 (0.454) 19.199 (0.094) FG-V 16.252 16.218 16.261 (0.265) 16.252 (0.210) FG-O 14.302 14.275 14.410 (0.946) 14.302 (0.189) FG-X 22.984 22.930 22.993 (0.275) 22.935 (0.022)
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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 UD FG-V FG-O FG-X UD FG-V FG-O FG-X uniform load (0, 0) TSDT 0.7356 0.8806 1.0705 0.6206 0.4712 0.5663 0.6844 0.4012 SSDT 0.7340 0.8792 1.0743 0.6179 0.4702 0.5655 0.6862 0.4001 present 0.7352 0.8797 1.0741 0.6214 0.4710 0.5656 0.6854 0.4013 (100, 0) TSDT 0.6983 0.8286 0.9955 0.9350 0.4556 0.5440 0.6530 0.3897 SSDT 0.6969 0.8274 0.9988 0.5910 0.4548 0.5437 0.6546 0.3887 present 0.6980 0.8278 0.9987 0.5942 0.4554 0.5437 0.6540 0.3898 (100, 50) TSDT 0.4773 0.5346 0.5991 0.4262 0.3500 0.4000 0.4557 0.3098 SSTD 0.4767 0.5341 0.6003 0.4250 0.3495 0.3997 0.4565 0.3092 present 0.4774 0.5346 0.6008 0.4266 0.3500 0.3999 0.4565 0.3100 sinusoidal load (100, 0) TSDT 0.4964 0.5869 0.7081 0.4227 0.3177 0.3769 0.4526 0.2723 SSDT 0.4953 0.5859 0.7104 0.4208 0.3170 0.3763 0.4537 0.2715 present 0.4963 0.5865 0.7102 0.4235 0.3176 0.3765 0.4532 0.2726 (100, 0) TSDT 0.4729 0.5544 0.6612 0.4056 0.3079 0.3632 0.4330 0.2651 SSDT 0.4719 0.5534 0.6633 0.4038 0.3072 0.3626 0.4340 0.2644 present 0.4728 0.5540 0.6631 0.4063 0.3078 0.3628 0.4335 0.2653 (100, 50) TSDT 0.3240 0.3583 0.4001 0.2896 0.2361 0.2674 0.3034 0.2101 SSDT 0.3219 0.3579 0.4009 0.2888 0.2357 0.2671 0.3039 0.2097 present 0.3226 0.3584 0.4011 0.2902 0.2362 0.2673 0.3038 0.2104 Notes: The data TSDT and SSDT in the table are from the analytical solution of Ref. [28]
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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(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 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 type b/h 5 10 20 50 100 (0, 0) UD 3.833 × 10−4 2.119 × 10−3 1.315 × 10−2 2.627 × 10−1 3.632 FG-V 3.875 × 10−4 2.252 × 10−3 1.572 × 10−2 3.663 × 10−1 5.293 FG-O 4.251 × 10−4 2.604 × 10−3 1.934 × 10−2 4.797 × 10−1 7.040 FG-X 3.722 × 10−4 1.985 × 10−3 1.120 × 10−2 1.896 × 10−1 2.470 (100, 0) UD 3.792 × 10−4 2.056 × 10−3 1.195 × 10−2 1.434 × 10−1 0.521 FG-V 3.832 × 10−4 2.181 × 10−3 1.406 × 10−2 1.708 × 10−1 0.538 FG-O 4.201 × 10−4 2.511 × 10−3 1.693 × 10−2 1.926 × 10−1 0.544 FG-X 3.683 × 10−4 1.929 × 10−3 1.030 × 10−2 1.181 × 10−1 0.492 (100, 50) UD 3.428 × 10−4 1.592 × 10−3 6.580 × 10−3 2.900 × 10−2 0.064 FG-V 3.462 × 10−4 1.665 × 10−3 7.110 × 10−3 2.980 × 10−2 0.065 FG-O 3.759 × 10−4 1.847 × 10−3 7.730 × 10−3 3.030 × 10−2 0.065 FG-X 3.340 × 10−4 1.516 × 10−3 6.080 × 10−3 2.810 × 10−2 0.064
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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) TSDT SSDT Present TSDT SSDT Present TSDT SSDT Present 0.11 UD 13.55 13.57 13.55 13.88 13.90 13.89 16.82 16.83 16.81 FG-V 12.45 12.46 12.46 12.81 12.82 12.82 15.94 15.95 15.93 FG-O 11.34 11.20 11.32 11.73 11.72 11.72 15.09 15.07 15.06 FG-X 14.69 14.72 14.68 15.00 15.03 14.98 17.75 17.77 17.73 0.14 UD 14.36 14.38 14.36 14.67 14.69 14.67 17.46 17.47 17.44 FG-V 13.28 13.29 13.28 13.62 13.63 13.62 16.57 16.59 16.57 FG-O 12.13 21.10 12.12 12.50 12.48 12.48 15.67 15.66 15.65 FG-X 15.41 15.45 15.40 15.70 15.74 15.69 18.33 18.36 18.31 0.17 UD 16.83 16.85 16.84 17.10 17.12 17.10 19.53 19.54 19.52 FG-V 15.44 15.46 15.45 15.73 15.74 15.74 18.34 18.35 18.33 FG-O 14.09 14.08 14.08 14.41 14.39 14.40 17.21 17.20 17.20 FG-X 18.19 18.21 18.18 18.43 18.46 18.43 20.70 20.73 20.69 Notes: The data TSDT and SSDT in the table are from the analytical solution of Ref. [28]
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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(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 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) UD FG-V FG-O FG-X UD FG-V FG-O FG-X UD FG-V FG-O FG-X 5 10.61 10.57 10.12 10.76 10.66 10.62 10.17 10.81 11.20 11.16 10.74 11.34 10 18.04 17.56 16.40 18.59 18.29 17.82 16.68 18.84 20.77 20.37 19.38 21.25 20 28.57 26.39 23.95 30.74 29.82 27.74 25.43 31.91 40.82 39.39 37.83 42.32 50 39.53 33.98 29.95 46.01 52.08 48.00 45.24 57.15 121.31 119.69 118.51 123.44 100 42.56 35.81 31.33 50.98 104.94 102.39 100.91 108.62 323.58 322.49 321.73 325.06
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