Nonlinear dynamics of filaments. III - Instabilities of helical rods

The time-dependent Kirchhoff equations for thin elastic rods are used to study the linear stability of twisted helical rods with intrinsic curvature and twist. Using a newly developed perturbation scheme, we derive the general dispersion relations governing the stability of various helical configurations. We show that helices with no terminal forces are always dynamically stable. We also compute the most stable helical shape against twist perturbations and show that different unstable modes can be excited in different regions of the parameter space and can sometimes coexist. The linearly unstable modes are computed and explicit forms are given.