Analysis of Thermal Post-buckling Behavior of Graded Porous Circular Plate with Variable Thickness
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摘要: 为分析多孔梯度材料圆板在非均匀温度场中的热后屈曲响应，基于经典板理论和物理中面概念建立了梯度多孔材料圆板在热载荷作用下的控制微分方程，其中假设厚度变化沿半径为二次抛物线型且板在其厚度上具有对称和非对称的非均匀孔隙率分布。采用打靶法数值求解了问题的屈曲和后屈曲响应，给出了均匀升温和热传导下的梯度多孔非线性变厚度圆板后屈曲平衡路径。结果显示：变厚度系数、孔隙率系数、孔隙分布方式以及温度场对板的临界载荷和后屈曲平衡路径均有影响；在不同温度场中孔隙率系数越大，屈曲时的临界载荷越小；孔隙率对称分布下的临界载荷大于非对称情况下的。Abstract: In order to analyze the thermal post-buckling response of the graded porous material circular plate in the non-uniform temperature field, the governing differential equation of the graded porous material circular plate under thermal load was established on the basis of classical plate theory and the concept of physical neutral plane. It is assumed that the thickness change is quadratic parabolic along the radius and the plate has symmetric and asymmetric non-uniform porosity distribution across their thickness. Using shooting method technique, the buckling and post-buckling responses of the problem was numerically solved, and the post-buckling equilibrium paths of the graded porous nonlinear variable thickness circular plate under different types of thermal loads as heat conduction and uniform temperature rise were obtained. The results show that the variable thickness coefficient, porosity coefficient, pore distribution and temperature field have the effects on the critical load and post buckling equilibrium path of the plate; under different temperature fields, the larger the porosity coefficient, the smaller the critical load during buckling; the critical load under symmetrical porosity distribution is greater than that under asymmetric porosity distribution.
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Key words:
- graded porous materials /
- physical neutral plane /
- circular plate /
- post-buckling /
- numerical solution
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图 1 变厚度多孔圆板的示意图
Figure 1. Schematic diagram of variable thickness porous circular plate
图 2 孔隙率沿截面厚度分布模式图
Figure 2. Porosity distribution pattern along the thickness of the section
图 3 厚度变化系数下
$ {W_{\max }} - \tau $ 的关系曲线Figure 3. Relationship curves of
$ \tau $ versus${W_{\max }} $ for different η
图 4 不同孔隙率下
$ {W_{\max }} - \tau $ 的关系曲线（$ \eta {\text{ = 0}}{\text{.5}} $ ）Figure 4.
$\tau $ versus${W_{\max }} $ for different porosity e0(η = 0.5)
图 5 非均匀升温对
$ {W_{\max }} - \tau $ 的影响（$ {e_0} = 0.2,\eta {\text{ = 0}}{\text{.5}} $ ）Figure 5. The influence of non-uniform rise on
${W_{\max }} - \tau $ 
图 6 不同孔隙率下
$ {W_{\max }} - \tau $ 的关系曲线（$ {T_{\text{r}}} = 1.5,\eta {\text{ = 0}}{\text{.5}} $ ）Figure 6.
$\tau $ versus${W_{\max }} $ for different porosity e0 (Tr = 1.5, η = 0.5)
图 7 不同
$ \eta $ 下Mode I板的屈曲构形图（$ \tau {\text{ = 15}} $ ）Figure 7. Buckling configuration of Mode I plate under different η (τ = 15)
图 8 不同
$ \eta $ 下Mode II板的屈曲构形图（$ \tau {\text{ = 15}} $ ）Figure 8. Buckling configuration of Mode II plate under different η (τ = 15)
表 1 梯度多孔圆板的临界温度（
$\eta = 0.{\text{1}}$ ）Table 1. The critical temperature of
$\Delta T_{\rm{cr}} $ porous plate (η=0.1)孔隙率系数
$ {e_0} $均匀升温/K 非均匀升温/K Mode I Mode II Mode I Mode II 0.1 162.59 157.42 132.35 111.42 0.2 140.15 128.35 122.48 101.36 0.3 133.57 122.30 113.35 92.35 0.4 121.38 107.54 104.87 84.24 0.5 110.41 100.46 97.28 76.90 
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表 2 不同变厚度系数下多孔圆板的临界温度（
${e_0} = 0.{\text{3}}$ ）Table 2. Critical temperature of porous circular plate with variable thickness coefficient（e0=0.3）
变厚度系数
$ \eta $均匀升温/K 非均匀升温/K Mode I Mode II Mode I Mode II 0.0 127.14 116.41 107.89 80.486 0.2 147.53 139.66 118.93 88.721 0.5 160.45 145.53 136.16 101.57 1.0 195.63 179.12 166.01 123.84 2.0 269.64 246.89 228.81 170.69
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