Dynamics of helical strips

The dynamics of inertial elastic helical thin rods with noncircular cross sections and arbitrary intrinsic curvature, torsion, and twist is studied. The classical Kirchhoff equations are used together with a perturbation scheme at the level of the director basis, and the dispersion relation for helical strips is derived and analyzed. It is shown that all naturally straight helical strips are unstable whereas free-standing helices are always stable. There exists a one-parameter family of stationary helical solutions depending on the ratio of curvature to torsion. A bifurcation analysis with respect to this parameter is performed, and bifurcation curves in the space of elastic parameters are identified. The different modes of instabilities are analyzed.