Due to surface tension, a beading instability takes place in a long enough fluid column that results in the breakup of the column and the formation of smaller packets with the same overall volume but a smaller surface area. Similarly, a soft elastic cylinder under axial stretching can develop an instability if the surface tension is large enough. This instability occurs when the axial force reaches a maximum with fixed surface tension or the surface tension reaches a maximum with fixed axial force. However, unlike the situation in fluids where the instability develops with a finite wavelength, for a hyperelastic solid cylinder that is subjected to the combined action of surface tension and axial stretching, a linear bifurcation analysis predicts that the critical wavelength is infinite. We show, both theoretically and numerically, that a localized solution can bifurcate sub-critically from the uniform solution, but the character of the resulting bifurcation depends on the loading path. For fixed axial stretch and variable surface tension, the localized solution corresponds to a bulge or a depression, beading or necking, depending on whether the axial stretch is greater than a certain threshold value that is dependent on the material model and is equal to $\sqrt[3]{2}$ when the material is neo-Hookean. At this single threshold value, localized solutions cease to exist and the bifurcation becomes exceptionally supercritical. For either fixed surface tension and variable axial force, or fixed axial force and variable surface tension, the localized solution corresponds to a depression or a bulge, respectively. We explain why the bifurcation diagrams in previous numerical and experimental studies look as if the bifurcation were supercritical although it was not meant to. Our analysis shows that beading in fluids and solids are fundamentally different. Fluid beading resulting from the Plateau–Rayleigh instability follows a supercritical linear instability whereas solid beading in general is a subcritical localized instability akin to phase transition.

由于表面张力，在足够长的流体柱中会发生串珠不稳定性，这会导致柱破裂，并形成具有相同总体积但表面积较小的较小包装。类似地，如果表面张力足够大，则在轴向拉伸下的软弹性圆柱体可能会不稳定。当轴向力在固定的表面张力下达到最大值或表面张力在固定的轴向力下达到最大值时，会发生这种不稳定性。但是，与流体在有限的波长下发生不稳定性的情况不同，对于承受表面张力和轴向拉伸的联合作用的超弹性实心圆柱，线性分叉分析预测临界波长是无限的。我们在理论上和数字上都显示 局部化的溶液可以从均匀溶液中亚临界地分叉，但是最终分叉的特性取决于加载路径。对于固定的轴向拉伸和可变的表面张力，取决于轴向拉伸是否大于某个阈值（取决于材料模型并等于），局部化解决方案对应于凸起或凹陷，成珠或颈缩$\sqrt[3]{2}$当材料是新Hookean。在此单一阈值下，局部解不再存在，并且分支变得异常超临界。对于固定的表面张力和可变的轴向力，或者对于固定的轴向力和可变的表面张力，局部解分别对应于凹陷或凸起。我们解释了为什么先前的数值和实验研究中的分叉图看起来好像分叉是超临界的，尽管这并不是故意的。我们的分析表明，流体和固体中的珠粒是根本不同的。高原-瑞利不稳定性产生的流体串珠遵循超临界线性不稳定性，而固体串珠通常是类似于相变的亚临界局部不稳定性。