General investigation of elastic thin rods as subject to a terminal twist

The recent development of DNA structure, brought by the elastic rod model, revives the study of the so-called Michell-Zajac instability for isotropic naturally straight elastic rings. The instability states that when subjected to a terminal twist, a manipulation which cuts, rotates, and then seals closed rods, an elastic ring does not writhe until the amount of rotation exceeds a rod-dependent threshold. From the data generated by a finite element method, Bauer, Lund, and White [Proc. Natl. Acad. Sci. USA. 90, 833 (1993)] concluded that the instability becomes extreme for isotropic naturally singly bent, doubly bent, and O -ring elastic rings since they writhe immediately as subject to a terminal twist. This paper continues their study for other closed rods. In order to understand DNA structure in DNA-protein interactions, this paper also extends the study to open rods with clamped ends; for such rods, a terminal twist is a manipulation which releases, rotates, and then reclamps one end of the rods. Moreover, the rods under consideration need not be isotropic or may violate Kirchhoff-Clebsch conservation law of total energy. By linearizing the Euler-Lagrange equations which govern equilibrium rods and analyzing the linearized equations, this paper establishes an inequality such that if the initial values of the bending curvatures, their first derivatives, and the twisting density of an equilibrium rod satisfy the inequality, the rod axis deforms immediately as the rod is subject to a terminal twist. Since the initial data dissatisfying the inequality form a hypersurface in the five-dimensional Euclidean space, this paper asserts that a terminal twist makes the axis deformed instantly for almost every equilibrium rod.