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

Abstract

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, ρ (the ratio of bending stiffness out of the plane of curvature to bending stiffness in the plane of curvature), and γ (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 ρ ≥ 1, and the n-covered circle (n>1) is stable if and only if γ>1, ρ>1, and

$$\frac{{n - 1}}{n} \leqslant \sqrt {\frac{{\gamma - 1}}{\gamma } \cdot \frac{{\rho - 1}}{\rho }} .$$

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 ${\hat u}$ 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.