Stability and bifurcation of circular Kirchhoff elastic rods

Abstract

Imagine a thin elastic rod like a piano wire. We assume its unstressed state is straight and it can be bent and twisted but inextensible. Certain particular case of the Kirchhoff elastic rod is one of the mathematical models of such an elastic rod. We consider the equilibrium states of the elastic rod when it is bent and twisted and both ends are welded together to form a smooth loop. Then, the simplest equilibrium state is a circular state with uniformly distributed torsion. The purpose of this paper is to investigate the stability and the bifurcation of such circular Kirchhoff elastic rods in equilibrium. The theory of elastic rods has been studied since the age of Leonhard Euler and Daniel Bernoulli. They initiated the theory of elastica. An elastica is a mathematical model of the equilibrium states of an elastic rod when it is assumed to be subjected to bending only. It is characterized by the critical curve of the bending energy or the total squared curvature functional. One of the mathematical models of an elastic rod with bending and twisting was considered by G. Kirchhoff. In this paper, we are mainly interested in closed objects. In the case of closed elasticae, Langer and Singer completely classified the closed elasticae and determined their knot types ([6]). Now, the integrability of the Euler-Lagrange equation of the stored energy of bending and twisting was essentially known by Kirchhoff ([4]). In recent years, Y. Shi and J. Hearst ([9]) have given the explicit expressions of the solutions of the Euler-Lagrange equation in cylindrical coordinates and investigated certain closed solutions to study supercoiled DNA. We use the model known as the uniform and symmetric case of Kirchhoff elastic rods, which is characterized by the following energy functional. We consider the totality of curves with unit normal{7, M}, that is, 7 = 7(5) : [si, s2] ->• R > is a unit-speed curve, and M(s) is a unit normal vector field along 7(5), and we define the torsional elastic energy T on it by