Stability of anisotropic, naturally straight, helical elastic thin rods

Abstract

There has been a differential-geometric interpretation which associates each elastic thin rod with a curve on the three-dimensional unit sphere equipped with a Riemannian metric related to the bending and twisting stiffnesses of the rod. In this paper, we exploit this interpretation to study the stability of anisotropic, naturally straight, helical equilibrium rods with clamped ends. Such a rod is called geodesic here if its associated curve is a geodesic, or equivalently, its twisting stiffness equals one of the bending stiffnesses. We establish criteria for geodesic equilibria to be stable and possibly unstable separately, and develop a scheme predicting unstable non-geodesic equilibria. We also present an example to emphasize the necessity of examining whether a helical equilibrium rod is geodesic or not when one is concerned with its stability.