Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers
Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers
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外力作用下具有边缘的轴对称膜：屈曲、最小表面和系绳
DOI：
10.1039/d1sm00827g
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发表时间：
2021
期刊：
影响因子：
3.4
通讯作者：
Powers, Thomas R.
中科院分区：
文献类型：
--
作者：
Jia, Leroy L.;Pei, Steven;Pelcovits, Robert A.;Powers, Thomas R.
我们用理论和数值计算来确定一个轴对称的流体膜的形状与抗弯曲和恒定面积。该膜以经典几何形状连接两个环，在肥皂膜中产生悬链形状。在我们的问题中，我们发现无穷多个分支的解决方案的形状和外力的功能分离的环，类似于本征模式的无限家庭的欧拉屈曲的细长杆。特别注意的悬链，出现的形状时，最大允许分离的面积小于一个临界面积等于平面面积包围的两个环。微扰理论的论点直接涉及到的张力的悬链膜的稳定性的悬链肥皂膜在这一制度。当膜面积大于临界面积，我们发现额外的圆柱形系绳解决方案的形状方程在大环分离，任意大的环分离是可能的。这些结果适用于消失的高斯曲率模量的情况下，当高斯曲率模量是非零的，该地区是低于临界面积，力和膜张力发散的环分离接近其最大值。我们还研究了我们的形状的稳定性和分析表明，悬链膜有显着不同的稳定性比他们的肥皂膜同行。
We use theory and numerical computation to determine the shape of an axisymmetric fluid membrane with a resistance to bending and constant area. The membrane connects two rings in the classic geometry that produces a catenoidal shape in a soap film. In our problem, we find infinitely many branches of solutions for the shape and external force as functions of the separation of the rings, analogous to the infinite family of eigenmodes for the Euler buckling of a slender rod. Special attention is paid to the catenoid, which emerges as the shape of maximal allowable separation when the area is less than a critical area equal to the planar area enclosed by the two rings. A perturbation theory argument directly relates the tension of catenoidal membranes to the stability of catenoidal soap films in this regime. When the membrane area is larger than the critical area, we find additional cylindrical tether solutions to the shape equations at large ring separation, and that arbitrarily large ring separations are possible. These results apply for the case of vanishing Gaussian curvature modulus; when the Gaussian curvature modulus is nonzero and the area is below the critical area, the force and the membrane tension diverge as the ring separation approaches its maximum value. We also examine the stability of our shapes and analytically show that catenoidal membranes have markedly different stability properties than their soap film counterparts.