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 inex tensible-unshearable elastic rods with constant planar intrinsic curvature (u) over cap and constant values for the twisting stiffness and two bending stiffnesses. A simple expression is derived for the index as a function of (u) over cap, rho (the ratio of bending stiffness out of the plane of curv ature to bending stiffness in the plane of curvature), and gamma (the ratio of twisting stiffness to bending stiffness in the plane of curvature). In particular, for intrinsically straight rods ((u) over cap = 0) we prove tha t the 1-covered circle is stable if and only if rho greater than or equal t o 1, and the n-covered circle (n >1) is stable if and only if gamma >1, rho >1, and n-1/n less than or equal to root gamma -1/gamma.rho -1/rho. The index is computed by framing the standard Euler-Lagrange equations of e quilibrium within a constrained variational principle with an isoperimetric constraint ensuring the ring closure. The fact that (u) over cap appears l inearly in the second variation allows the second variation to be diagonali zed using the eigenfunctions of an appropriate eigenvalue problem similar t o a Sturm-Liouville problem. This diagonalization allows the direct computa tion 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 const rained index in terms of this unconstrained index and a correction computab le 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.