Axisymmetric beading instabilities in soft, elongated cylinders have been observed in a plethora of scenarios, ranging from cellular nanotunnels and nerves in biology to swollen cylinders and electrospun fibers in polymer physics. One of the common geometrical features that can be seen in these systems is the finite wavelength of the emerging pattern. However, modelling studies often predict that the instability has an infinite wavelength, which can be associated with localized necking or bulging. In this paper, we consider a soft elastic cylinder with a thin coating that resists bending, as described by the Helfrich free energy functional. The bending stiffness and natural mean curvature of the coating are two novel features whose competition against bulk elasticity and capillarity is investigated. For intermediate values of the bending stiffness, a linear stability analysis reveals that the mismatch between the current and natural mean curvature of the coating can lead to patterns emerging with a finite wavelength. This analysis creates a continuous bridge between the classical solutions of the shape equation obtained from the Helfrich functional and a curvature-controlled zero-wavemode instability, similar to the one induced by the competition between bulk elasticity and capillarity. A weakly non-linear analysis predicts that the criticality of the bifurcation depends on the controlling parameter, with both supercritical and subcritical bifurcations possible. When capillarity is introduced, the criticality of the bifurcation changes in a non-trivial way.

中文翻译：

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曲率控制软涂层弹性圆柱体中的卷边：有限波模不稳定性和局部调制

在许多场景中都观察到了柔软细长圆柱体中的轴对称珠状不稳定性，从生物学中的细胞纳米隧道和神经到聚合物物理学中的膨胀圆柱体和电纺纤维。在这些系统中可以看到的常见几何特征之一是新兴图案的有限波长。然而，建模研究通常预测不稳定性具有无限波长，这可能与局部颈缩或凸出有关。在本文中，我们考虑了具有抗弯曲薄涂层的软弹性圆柱体，如赫尔弗里希自由能泛函所描述的那样。涂层的弯曲刚度和自然平均曲率是两个新颖的特征，研究了它们与体积弹性和毛细管现象的竞争。对于弯曲刚度的中间值，线性稳定性分析表明，涂层的当前平均曲率与自然平均曲率之间的不匹配可能导致出现具有有限波长的图案。该分析在赫尔弗里希泛函获得的形状方程的经典解和曲率控制的零波模式不稳定性之间建立了一座连续的桥梁，类似于体弹性和毛细管现象之间的竞争引起的不稳定性。弱非线性分析预测分岔的临界程度取决于控制参数，超临界和亚临界分岔都是可能的。当引入毛细管现象时，分叉的临界性会以一种不平凡的方式发生变化。