Effective extensional–torsional elasticity and dynamics of helical filaments under distributed loads

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.