Take an elastic filament who has no preferred curvature (naturally 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 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. The phenomenon is quite striking as the ring suddenly ring suddenly buckles and loops back on itself by forming an eight-shape.

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).

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 but also plants and microbial filaments. 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

History of the problem

The stability of twisted rings was first discussed by Thomson and Tait in their classical Treatise on Natural Philosophy-1867 . 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".

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".

His work on twisted elastic rings was almost completely forgotten (with the notable exception of one citation by Stu Antman) and rediscovered independently at least three times, first by Ed. Zajac in 1962 (motivated by the dynamics of oceanic cables), then by Craig Benham and Marc LeBret around 1978-79. I was recently told by Bernard Coleman that he first heard of thw ork of Zajac by one of Zajac's colleagues in a Engineering conference in the 1980's. This is why this instability is sometimes referred to as Zajac's instability (Zajac was not aware of all this work until I tild him recently--it turns out he lives also in Tucson and was until recently a Professor at the University of Arizona).

You can find more about this problems in our original Journal of Elasticity article and on these sites

• A bibliography of J. H. Michell on Wikipedia

• A biography of J. H. Michell in the Australian Dictionary of Biography (Onlined Ed)