Stability of n-covered Circles for Elastic Rods with Constant Planar Intrinsic Curvature

A stability index is computed for the n-covered circular equilibria of inextensible-unshearable elastic rods with constant planar intrinsic curvature û and constant values for the twisting stiffness and two bending stiffnesses. A simple expression is derived for the index as a function of û, (the ratio of bending stiffness out of the plane of curvature to bending stiffness in the plane of curvature), and (the ratio of twisting stiffness to bending stiffness in the plane of curvature). In particular, for intrinsically straight rods (û = 0) we prove that the 1-covered circle is stable if and only if 1, and the n-covered circle (n>1) is stable if and only if >1, >1, and The index is computed by framing the standard Euler–Lagrange equations of equilibrium within a constrained variational principle with an isoperimetric constraint ensuring the ring closure. The fact that appears linearly in the second variation allows the second variation to be diagonalized using the eigenfunctions of an appropriate eigenvalue problem similar to a Sturm–Liouville problem. This diagonalization allows the direct computation of an unconstrained index (disregarding ring closure). We then apply a result of Maddocks (SIAM J. Math. Anal. 16 (1985) 47–68) to find the constrained index in terms of this unconstrained index and a correction computable from the linearized constraint.With numerical computations, we verify these analytic results on n-covered circles and determine the index of non-circular equilibria bifurcating from the branches of n-covered circles.