Effective extensional-torsional elasticity and dynamics of helical filaments under distributed loads
arxiv(2024)
摘要
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 'effective-column'
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 the small-deformation limit, we exactly
recover the coupled wave equations used to describe the free vibrations of
helical coil springs, thereby justifying previous effective-column
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. 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.
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