We study slender, helical elastic rods subject to distributed forces and moments. Focussing on the case when the helix axis remains straight, we employ the method of multiple scales to systematically derive an ‘equivalent-rod’ theory from the Kirchhoff rod equations: the helical filament is described as a naturally-straight rod (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Importantly, our analysis is asymptotically exact in the limit of a ‘highly-coiled’ filament (i.e., when the helical wavelength is much smaller than the characteristic lengthscale over which the filament bends due to external loading) and is able to account for large, unsteady displacements. In addition, our analysis yields explicit conditions on the external loading that must be satisfied for a straight helix axis. In the small-deformation limit, we exactly recover the coupled wave equations used to describe the free vibrations of helical coil springs, thereby justifying previous equivalent-rod approximations in which linearised stiffness coefficients are assumed to apply locally and dynamically. We then illustrate our theory with two loading scenarios: (I) a heavy helical rod deforming under its own weight; and (II) the dynamics of axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. In both scenarios, we demonstrate excellent agreement with solutions of the full Kirchhoff rod equations, even beyond the formal limit of validity of the ‘highly-coiled’ assumption. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, and yields analytical insight into their elastic instabilities. In particular, our analysis indicates that tensile instabilities are a generic phenomenon when helical rods are subject to a combination of distributed forces and moments.

中文翻译：

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分布式载荷下螺旋丝的有效拉伸-扭转弹性和动力学


我们研究了受分布力和力矩作用的细长螺旋弹性杆。着眼于螺旋轴保持笔直的情况，我们采用多尺度的方法从基尔霍夫杆方程中系统地推导出“等效杆”理论：螺旋丝被描述为自然笔直的杆（与螺旋轴对齐），拉伸和扭转变形耦合。重要的是，我们的分析在“高度卷曲”细丝的极限内（即，当螺旋波长远小于细丝因外部载荷而弯曲的特征长度尺度时）是渐近精确的，并且能够解释大的、不稳定的位移。此外，我们的分析还得出了直螺旋轴必须满足的外部载荷的明确条件。在小变形极限中，我们精确地恢复了用于描述螺旋螺旋弹簧自由振动的耦合波动方程，从而证明了以前的等效杆近似是合理的，其中线性化刚度系数假设局部和动态应用。然后，我们用两种载荷场景来说明我们的理论：（I） 一根沉重的螺旋杆在自身重量下变形;（II） 粘性流体中轴向旋转（旋转）的动力学，可以认为是细菌鞭毛丝的简单模型。在这两种情况下，我们都证明了与完整 Kirchhoff 杆方程的解非常吻合，甚至超出了“高度卷曲”假设的正式有效性极限。更广泛地说，我们的分析提供了一个框架，可以在各种物理和生物环境中开发螺旋杆的简化模型，并产生对其弹性不稳定性的分析见解。 特别是，我们的分析表明，当螺旋杆受到分布力和力矩的组合时，拉伸不稳定性是一种普遍现象。