Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom.
Phys Rev E. 2017 May;95(5-1):053106. doi: 10.1103/PhysRevE.95.053106. Epub 2017 May 11.
A long elastic cylinder, with radius a and shear-modulus μ, becomes unstable given sufficient surface tension γ. We show this instability can be simply understood by considering the energy, E(λ), of such a cylinder subject to a homogenous longitudinal stretch λ. Although E(λ) has a unique minimum, if surface tension is sufficient [Γ≡γ/(aμ)>sqrt[32]] it loses convexity in a finite region. We use a Maxwell construction to show that, if stretched into this region, the cylinder will phase-separate into two segments with different stretches λ_{1} and λ_{2}. Our model thus explains why the instability has infinite wavelength and allows us to calculate the instability's subcritical hysteresis loop (as a function of imposed stretch), showing that instability proceeds with constant amplitude and at constant (positive) tension as the cylinder is stretched between λ_{1} and λ_{2}. We use full nonlinear finite-element calculations to verify these predictions and to characterize the interface between the two phases. Near Γ=sqrt[32] the length of such an interface diverges, introducing a new length scale and allowing us to construct a one-dimensional effective theory. This treatment yields an analytic expression for the interface itself, revealing that its characteristic length grows as l_{wall}∼a/sqrt[Γ-sqrt[32]].
一个长的弹性圆柱，半径为 a，切变模量为 μ，在有足够的表面张力 γ 的情况下会变得不稳定。我们通过考虑这样一个圆柱在均匀纵向拉伸 λ 下的能量 E(λ)，可以简单地理解这种不稳定性。尽管 E(λ)有一个唯一的最小值，但如果表面张力足够大 [Γ≡γ/(aμ)>sqrt[32]]，它在有限的区域内会失去凸性。我们使用麦克斯韦构造来表明，如果拉伸到这个区域，圆柱将分成两段，两段的拉伸比分别为 λ_{1}和 λ_{2}。我们的模型因此解释了为什么不稳定性具有无限的波长，并允许我们计算不稳定性的亚临界滞后环（作为施加拉伸的函数），表明随着圆柱在 λ_{1}和 λ_{2}之间拉伸，不稳定性以恒定的幅度和恒定的（正）张力进行。我们使用全非线性有限元计算来验证这些预测，并对两相之间的界面进行了特征化。在 Γ=sqrt[32]附近，这样的界面的长度发散，引入了一个新的长度尺度，并允许我们构建一个一维有效理论。这种处理方法为界面本身提供了一个解析表达式，揭示了其特征长度的增长规律为 l_{wall}∼a/sqrt[Γ-sqrt[32]]。
