Nonlinear Dynamics of Filaments IV: Spontaneous Looping of Twisted Elastic Rods

Everyday experience shows that twisted elastic filaments spontaneously form loops. We model the dynamics of this looping process as a sequence of bifurcations of the solutions to the Kirchhoff equation describing the evolution of thin elastic filament. The control parameter is taken to be the initial twist density in a straight rod. The first bifurcation occurs when the twisted straight rod deforms into a helix. This helix is an exact solution of the Kirchhoff equations whose stability can be studied. The secondary bifurcation is reached when the helix itself becomes unstable and the localization of the post-bifurcation modes is demonstrated for these solutions. Finally, the tertiary bifurcation takes place when a loop forms at the middle of the rod and the looping becomes ineluctable. Emphasis is put on the dynamical character of the phenomena by studying the dispersion relation and deriving amplitude equations for the different configurations.