The problem is simple to state: "If a wire of isotropic section and naturally straight be twisted, and the ends joined so as to form a continuous curve, the circle will be a stable form of equilibrium for less than a certain amount of twist." In other words, consider an isotropic elastic rod (the rod has no preferred bending direction) that is stress-free when held straight. Now, paint a straight line on the straight rod and shape the rod so that the rod centerline is a circle. At the junction, the tangent from the two ends agree but the cross-sections can be rotated so that the line painted on the straight unstressed shape twists around the central curve. The twist is the total angular rotation of the line with respect to the central curve. The line will close on itself at the junction if the twist is an integral multiple of 2π. The rod is glued at this point and released. For small values of the twist, the twisted ring is stable. For sufficiently high twist, the elastic ring will become unstable and will start writhing out of the plane. The phenomenon is quite striking as the instability appears to be subcritical (in the sense that no stable equilibrium shape exists close to the unstable ring). The ring suddenly buckles and loops back on itself by forming an eight-shape where self-contact plays a particularly important role. For reasons that will soon become apparent, we shall refer to the twisted elastic ring instability as Michell's instability and the problem addressed here is to identify the value of the critical twist at which the instability sets in.
This phenomenon raises many interesting questions. Qualitatively, one may understand the instability as a balance between torsional and bending energy. The torsional energy of the ring increases as the square of the twist and is eventually relieved by a change of shape that increases the bending energy (proportional to the square of the curvature). Quantitatively, the first natural question is to determine the value of the critical twist that makes the ring unstable in terms of its geometric (ring and cross-section radii) and elastic parameters (torsional and flexural rigidity). Once the instability threshold has been determined, many other questions can be approached such as determining the static and dynamic behavior of the ring after the instability sets in and how this phenomenon can be generalized to more complicated systems. The instability of the twisted elastic ring is a fundamental instability of elastic materials akin to the Euler instability describing the buckling of loaded beams. Beyond its obvious importance as a natural philosophy question and its application in engineering problems, the problem of twisted elastic rings and related instability of elastic materials has gained some renewed interest in science largely due to the realization that Kirchhoff models for elastic rods are suitable models for the study of macromolecules such as DNA molecules [C J Benham, An elastic model of the large structure of duplex DNA (1979); C J Benham, Geometry and mechanics of DNA superhelicity (1983); A Vologodskii, Topology and Physics of Circular DNA (1992); R S Manning, J H Maddocks and J D Kahn, A continuum rod model of sequence-dependent DNA structure (1996)] but also plants [W K Silk, On the curving and twining of stems (1989); A Goriely and M Tabor, Spontaneous helix-hand reversal and tendril perversion in climbing plants (1998)] and microbial filaments [R E Goldstein, A Goriely, G Hubber and C Wolgemuth, Bistable helices (2000)]. In particular, the analysis of mini-DNA rings made out of a few hundred bases offers a unique perspective to characterize physical properties of DNA and twisted elastic rings are the natural theory to understand and extract these properties [Bifurcation theory, symmetry breaking and homogenization in continuum mechanics descriptions of DNA (2004)].
The stability of twisted rings was first discussed by Thomson and Tait in their classical Treatise on Natural Philosophy. In paragraph 123, they discussed the problem of the respective stability of the circle versus the eight form and reached the conclusion that "the circular form, which is always a figure of equilibrium, may be stable or unstable, according as the ratio of torsional to flexural rigidity is more or less than a certain value depending on the actual degree of twist." Motivated by this assertion, John Henry Michell wrote a four-page paper where he determined the critical twist as a function of the ratio a = M/L of torsional to flexural rigidity. His original paper is given in Appendix A and a modern proof based on his analysis is given in the next Section. His analysis rests on an application of a general theory of vibration of rods around an equilibrium shape [J H Michell, The small deformation of curves and surfaces with applications to the vibration of a helix and a circular ring (1890)] (his first published work at age 26). Apparently, Michell realized that when these frequencies become imaginary the equilibrium shape loses its stability and he applied this idea to derive a simple criterion for the instability of a twisted elastic ring.
John Henry Michell is an interesting, almost tragic, figure of applied mathematics at the turn of the 20th century. A bright Australian student, he went to Cambridge (UK) for his postgraduate study and then returned to the University of Melbourne where he was eventually appointed Professor of Mathematics and retired at age 65. His entire research publication records took place between 1889 and 1902 when he published 23 papers. His contributions are believed to be "the most important contributions ever made by an Australian mathematician" [E O Tuck, The wave resistance formula of J H Michell and its significance to recent research in ship hydrodynamics (1898)]. While Michell was very active in teaching and science in Australia after 1902, the reasons of his abandonment of research activity are unclear and may be due to his dedication to teaching and a lack of positive response from the scientific community. To date, his single most recognized work is the computation of wave resistance to a ship, that is, the energy loss into a wave pattern by a steadily moving ship known as the wave resistance formula, which was not fully appreciated before his death.
Michell's work on the stability of rods received some attention at the turn of the 20th century and eventually led him, with other contributions, to his election at the Royal Society in 1902. At this time his work in elasticity was well received as is evidenced by the discussion in Basset's paper [A B Basset, On the deformation of thin elastic wires (1895)] and in the second edition (1906) of Love's classic treatise [A E H Love, A Treatise on the Mathematical Theory of Elasticity]. However, his pioneering work in elasticity seems to have fallen in complete darkness as it is completely absent from the literature after 1945 (with the notable exception of Antman and Kenney [S S Antman and C S Kenney, Large buckled states of nonlinearly elastic rods under torsion, thrust and gravity (1981)]). At the turn of the 21st century, the name of John Henry Michell has resurfaced in Australia through the establishment of the J H Michell Medal, awarded yearly since 1999 by the Australian Mathematical Society to a young outstanding applied mathematician.
In 1962, Edward E Zajac, then at Bell Laboratory, published an article on the stability of twisted elastic rings and the stability of the clamped looped elastica [E E Zajac, Stability of two planar loop elasticas (1962)]. His work was motivated by the coiling and kinking of submarine cables but he argued that these problems "are of intrinsic interest in applied mechanics". He was apparently unaware of Michell's work as he states that contrary to what one might expect, neither of the foregoing problems is solved in Born's thesis on the stability of elastic line. In the first part of the paper, Zajac studies the stability of the twisted elastic ring based on Love's formulation in Euler angles and rederives Michell's criterion by linearizing the static equation of rod equilibrium. His paper is clear and concise and a good example of applied mechanics at his best, a well-formulated problem of interest solved elegantly by direct analysis of Kirchhoff equations.
The story of repeated discoveries does not end with Zajac. Zajac's paper was published in an engineering journal and aside from the submarine cable community it did not receive much attention. However, the stability of twisted elastic rings became of interest to the biophysics DNA community when it was first realized that geometric and topological characterizations of curves could be of importance to understand DNA configurations [F B Fuller, The writhing number of a space curve (1971); F B Fuller, Decomposition of the linking number of a closed ribbon: A problem from molecular biology (1978)]. Shortly after, Benham and LeBret independently proposed to model DNA as an elastic rod and both considered the stability of twisted elastic rings and essentially rederived Michell's criterion [M LeBret, Twist and writhing in short circular DNA according to first-order elasticity (1984); C J Benham, Onset of writhing in circular elastic polymers (1989)]. The connection with Zajac's work was only realized years later by Coleman, Tobias, Olson, and collaborators in a series of papers [B D Coleman, E H Dill, M Lembo, Z Lu and I Tobias, On the dynamics of rods in the theory of Kirchhoff and Clebsch (1993); I Tobias and W K Olson, The effect of intrinsic curvature on supercoiling - Predictions of elasticity theory (1993); Y Yang, I Tobias and W K Olson, Finite element analysis of DNA supercoiling (1993)]. Since then, Zajac's work has been considered as the original paper on the subject and the instability of the twisted rings has even been referred to as Zajac's instability.
The basic idea behind Michell's analysis is to study the linearized dynamics of the rings and to identify vibration frequencies. The instability threshold is reached when the frequencies become imaginary, that is when small perturbations are exponentially amplified. Therefore, the analysis should be based on the full dynamical equations. However, in Michell's analysis, the rotational acceleration of the cross-section, is not taken into account. This leads to a simpler formulation (independent of the spin vector w) that gives the wrong vibration frequencies, but since the stability threshold does not depend on the dynamical part of the equations but only on the static part, it does not change the basic computation for the critical value of the twist. Accordingly, to follow closely Michell's basic analysis while remaining mathematically consistent, we give a simpler and shorter proof by considering the static equations and looking for non-trivial periodic solutions of the linearized equation. This proof is actually very close to the derivation of Euler criterion for the instability of a beam and to the best of my knowledge the simplest self-contained proof available.