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耿亚南, 蔡宗熙. 内压载荷作用下薄膜椭球的稳定性分析[J]. 力学学报, 2016, 48(6): 1343-1352. DOI: 10.6052/0459-1879-16-142
引用本文: 耿亚南, 蔡宗熙. 内压载荷作用下薄膜椭球的稳定性分析[J]. 力学学报, 2016, 48(6): 1343-1352. DOI: 10.6052/0459-1879-16-142
Geng Yanan, Cai Zongxi. STABILITY OF A PRESSURIZED ELLIPSOIDAL BALLOON[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1343-1352. DOI: 10.6052/0459-1879-16-142
Citation: Geng Yanan, Cai Zongxi. STABILITY OF A PRESSURIZED ELLIPSOIDAL BALLOON[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1343-1352. DOI: 10.6052/0459-1879-16-142

内压载荷作用下薄膜椭球的稳定性分析

STABILITY OF A PRESSURIZED ELLIPSOIDAL BALLOON

  • 摘要: 超弹性橄榄状和南瓜状薄膜椭球在内压载荷作用下存在不同的分岔形式.对橄榄状薄膜椭球来说,细长比大于某一临界值时,在一定内压作用下会发生梨形分岔;小于该临界值时,薄膜椭球的分岔行为与圆管的局部起鼓现象相类似.对南瓜状薄膜椭球,无论圆扁,当内压达到某载荷值时都会发生梨形分岔.本文采用能量判据,分析了在压强控制和质量控制两种加载方式作用下,不同形状的薄膜椭球的均匀解及分岔解的稳定性.通过计算要考察的平衡状态及施加小扰动之后状态的能量差来判断当前状态是否稳定,结果表明,在压强控制下,P-V曲线下降段的均匀解和分岔解均为不稳定解.但在质量控制下,在P-V曲线下降段中只有均匀解出现时,均匀解为稳定解;而在均匀解和分岔解共存的区间内,均匀解为不稳定解,分岔解为稳定解.另外,P-V曲线两个上升段的均匀解则均为稳定解.

     

    Abstract: A pressurized ellipsoidal balloon may bifurcate into different shapes depending on its precise shape. For a rugby-shaped balloon, there exists a threshold ratio of the axes in the Z- and R- directions, above which the balloon tends to bifurcate into a pear shape. Otherwise, the pear shape is impossible and when the balloon is slender enough, it may bulge out locally in a symmetric manner more like a tube. However, for a pumpkin-shaped balloon, bifurcation into a pear shape is always possible. In this paper, by using an energy criterion, we determine the stability properties of the primary and bifurcated solutions under pressure control and volume control,respectively. The total energy of the equilibrium state and its disturbed state are calculated, and the di erence between these two states is used to evaluate the stability of current state. Our analyses indicate that under pressure control, both primary and bifurcated solutions that exist on the descending branch of the pressure versus volume curve are unstable, but under volume control, the bifurcated solution is always stable whenever it appears while the primary solution is only stable when there does not exist any bifurcated solution. However, the primary solutions that exist on the two ascending branches are always stable.

     

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