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 bifu rcations of the solutions to the Kirchhoff equation describing the evolutio n of thin elastic filaments. The control parameter is taken to be the initi al twist density in a straight rod. The first bifurcation occurs when the t wisted straight rod deforms into a helix. This helix is an exact solution o f the Kirchhoff equations, whose stability can be studied. The secondary bi furcation is reached when the helix itself becomes unstable and the localiz ation of the post-bifurcation modes is demonstrated for these solutions. Fi nally, 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 dyn amical character of the phenomena by studying the dispersion relation and d eriving amplitude equations for the different configurations.