Dynamic Buckling of Functionally Graded Cylindrical Shells under Axial Loading
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摘要: 基于Donnell壳体理论和经典板壳理论，利用Hamilton变分原理得到轴向荷载作用下材料属性呈幂律分布的功能梯度材料圆柱壳的动力屈曲控制方程。根据圆柱壳周向连续性设出径向位移的函数表达，利用分离变量法求解得到功能梯度材料圆柱壳在轴向荷载作用下的动力屈曲临界荷载的解析表达式和屈曲解。利用MATLAB软件编程计算，讨论了径厚比、梯度指数、环向模态数、轴向模态数等对功能梯度材料圆柱壳动力屈曲临界荷载的影响。结果表明：圆柱壳的临界荷载随临界长度的增加而减小；冲击端为夹支的临界荷载比冲击端为简支的临界荷载大，说明约束条件对临界荷载有较大影响；圆柱壳的临界荷载随着模态数的增加而增大，说明临界荷载越大，高阶模态越易被激发；屈曲模态图随着模态数的增加而复杂化。
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关键词:
- 功能梯度材料 /
- 屈曲临界荷载 /
- 圆柱壳 /
- Hamilton原理 /
- 模态数
Abstract: Based on the Donnell shell theory and classical shell theory, we established the dynamic buckling governing equation of functionally graded cylindrical shell under axial load using Hamilton principle.According to the expression of the radial displacement based on the circumferential continuity of cylindrical shell, we also obtained the dynamic buckling critical load expression and the buckling solution of functionally graded cylindrical shell under axial loading using the separation variable method.Using MATLAB, we performed the numerical analysis of functionally graded cylindrical shells, and discussed the influence of the diameter-thickness ratio, the gradient index, the number of circumferential mode and axial mode on the critical load of dynamic buckling.The results show that the critical load of cylindrical shells decreases with the increase of the critical length.The constraint conditions have effects on the critical load, and the critical load of the clamped edges is higher than that of the simple support.Moreover, the critical load of cylindrical shells grows as the modal number increases, indicating that the higher the critical load is, the easier the high-stage mode excites.The dynamic buckling modal diagram becomes more complicated as the modal number increases. -
图 3 不同材料组成下临界荷载与临界长度的关系
Figure 3. Critical load vs.critical length under different material composition conditions
图 4 冲击端为夹支时不同径厚比下临界荷载与临界长度的关系
Figure 4. Critical load vs.critical length under clamped edge and different diameter-thickness ratios conditions
图 5 冲击端为简支时不同径厚比下临界荷载与临界长度的关系
Figure 5. Critical load vs.critical length under simple support and different diameter-thickness ratios conditions
图 6 冲击端为夹支时不同梯度指数下临界荷载与临界长度的关系
Figure 6. Critical load vs.critical length under clamped edge and different gradient indexes conditions
图 7 冲击端为简支时不同梯度指数下临界荷载与临界长度的关系
Figure 7. Critical load vs.critical length under simple support and different gradient indexes conditions
图 8 冲击端为夹支时不同环向模态数下临界荷载与临界长度的关系
Figure 8. Critical load vs.critical length under clamped edge and different circumferential modal number conditions
图 9 冲击端为简支时不同环向模态数下临界荷载与临界长度的关系
Figure 9. Critical load vs.critical length under simple support and different circumferential modal number conditions
图 10 冲击端为夹支时不同轴向模态数下临界荷载与临界长度的关系
Figure 10. Critical load vs.critical length under clamped edge and different axial modal number conditions
图 11 冲击端为简支时不同轴向模态数下临界荷载与临界长度的关系
Figure 11. Critical load vs.critical length under simple support and different axial modal number conditions
图 12 不同环向屈曲模态图(n=1;m=1, 2, 3, 4, 5, 6)
Figure 12. Different circumferential buckling modes (n=1;m=1, 2, 3, 4, 5, 6)
图 13 不同轴向屈曲模态图(m=2;n=1, 2, 3, 4, 5, 6)
Figure 13. Different axial buckling modes (m=2;n=1, 2, 3, 4, 5, 6)
表 1 材料参数
Table 1. Material parameters
Material E/GPa ρ/(g·cm-3) μ Ceramic 385 3.96 0.230 Ti 109 4.54 0.410 Fe 155 7.86 0.291 Cu 119 8.96 0.326 
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