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Elasticity of Knots

a recent pre-print concerning the elasticity of a knotted rod (submitted to Journal of the Mechanics and Physics of Solids)

http://arxiv.org/abs/0812.2881

Authors:  N. Clauvelin, B. Audoly, S. Neukirch

Abstract:

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The problem is formulated as a non-linear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium.

Comments

In my opinion, you have done a very interesting topic that should be of broad interest not only for mechanicians, physical scientists but also for life scientists, which is the reason why your relevant work can be published in PRL. Well done! I am very interested in your work.

Thanks for your comment !
In fact we have already published  a PRL paper in 2007. It was less detailled about the calcultations, but some other experimental results were presented.
Elastic Knots (PRL): http://dx.doi.org/10.1103/PhysRevLett.99.164301

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Nicolas Clauvelin
UPMC University Paris 06
Institut de Mécanique d'Alembert
http://www.lmm.jussieu.fr/~clauvelin/

Mechanical study of knots is important in not only mechanics but also biophysics. Recently, a knot has been found in a certain protein domain. In a recent study by Cieplak and coworkers (Sulkowska et al., PNAS, 105, 19714 (2008)), it is shown that knot enhances the stability of protein domain, and that knot affects the protein unfolding mechanics. I believe that mechanics of knots will enrich our understanding in biophysics as well.

Thanks for your comment and the reference.
I am totally agree with you that a better comprehension of the mechanics of a knotted rod will help to undertand proteins unfolding processes.
On this point our paper presents some results concerning an instability when torsional loads are apply to a knotted rod. Such an instability can make easier to unknot the rod and might be of some interest in the case of proteins.
--
Nicolas Clauvelin
UPMC University Paris 06
Institut de Mécanique d'Alembert
http://www.lmm.jussieu.fr/~clauvelin/

Arash_Yavari's picture

Dear Nicolas:

Very interesting work. Just a minor comment. Right after Eq.(1) you mention that the tangent vector has unit length because the rod is inextensible. This is not the real reason: as soon as you parametrize any curve with arc length, the tangent vector will be of unit length. Of course in your theory, the deformed and underofmed curves are both parametrized by the same arc length and so this holds for both of them.

Regards,
Arash

Thanks a lot for your interest in my work.

In fact, in the case of the one-dimensional elastic rod theory (Kirchhoff equations) it is common to write the equations on the deformed rod. However the arc-length definition is always related to the undeformed configuration (for which the length is known)and then saying that the rod is inextensible leads to obtain a unitary tangent on the deformed configuration.

Anyway I think it's more a question of considering the problem written on the deformed configuration or on the undeformed one.
--
Nicolas Clauvelin
UPMC University Paris 06
Institut de Mécanique d'Alembert
http://www.lmm.jussieu.fr/~clauvelin/

Arash_Yavari's picture

Dear Nicolas:

Tangent vector of a curve being of unit length, in general, has nothing to do with any rod theory. As I said earlier, as soon as you parametrize any given curve by its arc length the tangent vector becomes of unit length. Again, this has nothing to do with formulating a rod theory referentially or spatially. In what you're doing assuming inextensibility you can parametrize your rod by the same arc length in both the reference and current configurations. You may look at "Differential Geometry of Curves and Surfaces" by Do Carmo or "Differential Geometry and Its Applications" by by John Oprea for discussions on parametrization of curves by arc length.

Note that your (inextensible) rod need not be parametrized by arc length and if not in the new parametrization the tangent vector is not of unit length anymore even when the rod is inextensible.

Regards,
Arash

Dear Arash,
thanks for your comment and your pertinent remarks.

I am completely agree with you about the fact the unitarity of the tangent is directly related to the arc-length parametrization of a curve (a kind of "canonical" parametrization). However in a one-dimensional elastic rod theory some explanations are needed.

In Kirchhoff elastic rod theory the rod is parametrized by a spatial curve and a continuum of material frames (different from the Frénet-Serret frame). It is convenient to write the equilibrium equations on the deformed configuration in term of the arc-length of the underformed configuration (unitary tangent for the undeformed configuration). Then the tangent of the deformed configuration, expressed with the arc-length of the underformed one, is not unitary. Nevertheless it is possible under certain assumptions, one is the inextensibility of the rod, to show that this tangent expressed in term of the arc-length of the undeformed configuration is unitary. In the paper such considerations are not fully explained as it is common in the use of the Kirchhoff elastic rod theory (see for examples [1] and [2] — these references do not directly deal with this question but contain some remarks). I agree with you that the remark we write in our paper about the unitarity of the tangent can mislead readers who are not familiar with our approach of the elastic rod theory.

Sincerely.

References:
[1] A. Goriely. Twisted Elastic Rings and the Rediscoveries of Michell’s Instability. Journal of Elasticity, 84(3):281–299, 2006.
[2] D. M. Stump, W. B. Fraser, K. E. Gates. The writhing of circular cross-section rods: undersea cables to DNA supercoils. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 454(1976):2123–2156, 1998.

--
Nicolas Clauvelin
UPMC University Paris 06
Institut de Mécanique d'Alembert
http://www.lmm.jussieu.fr/~clauvelin/

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