Sébastien Neukirch
Institut Jean le Rond d'Alembert
Centre National de la Recherche Scientifique
Sorbonne Université, Campus Pierre et Marie Curie
Paris, Francetel: +33 1 44 27 72 61
e-mail: sebastien.neukirch (-atat-) upmc.fr
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Stable elastic knots with no self-contact
D. Moulton, P. Grandgeorge, and S. Neukirch
Journal of the Mechanics and Physics of Solids, vol. 116 (2018) 33–53
Abstract : We study an elastic rod bent into an open trefoil knot and clamped at both ends. The question we consider is whether there are stable configurations for which there are no points of self-contact. This idea can be fairly easily replicated with a thin strip of paper, but is more difficult or even impossible with a flexible wire. We search for such configurations within the space of three tuning parameters related to the degrees of freedom in a simple experiment. Mathematically, we show, both within standard Kirchhoff theory as well within an elastic strip theory, that stable and contact-free knotted configurations can be found, and we classify the corresponding parametric regions. Numerical results are complemented with an asymptotic analysis that demonstrates the presence of knots near the doubly-covered ring. In the case of the strip model, quantitative experiments of the region of good knots are also provided to validate the theory.
DOI: 10.1016/j.jmps.2018.03.019
download the journal version : PDF
Please note a typo in Eq. 4.3, first line, and a typo in Eq. 2.8 which are corrected in the following PDF.
Submitted (Nov 8th, 2017)
Reports (received Jan 17th, 2018)
Reviewer 1In this manuscript knotted configurations of elastic beams without self contact are investigated. The main finding is that configurations of this type are possible provided the beams have finite length and the clamps are misaligned. The analysis appears to be sound and the style of the manuscript is good. However I am not convinced that the question addressed by the authors is very significant, which is a particular concern for a journal like JMPS. The authors currently do not elaborate as to why it is worth investing time and energy to solve this particular question, and why it could be of interest beyond a narrow circle of specialists in beam theory.
On the positive side a nice experimental validation is provided, as well as a detailed comparison between the beam and strip models.
Minor remarks follow in no particular order. I am not comfortable with the statement that 'The walled-knotted configurations are visually most satisfying knots… and form our primary focus': if aesthetics is really driving the analysis, this should be mentioned up front and not hidden in the middle of the paper. There is no point in spelling out the equations in the coordinates basis, as it makes them difficult to read: they should be kept in compact and readable form, and the reader should be referred to the literature or to an appendix for the implementation details. The word 'jumps' on page 3 should be clarified: it is unclear why the loss of self-contact entails a discontinuous jump in configurations. At the bottom of p.5, what is meant by 'in simple cross-sectional geometries' is unclear as well: please clarify how/whether K=3 could be achieved in an experiment.
Overall, I have the feeling that the question addressed by this manuscript is too focused and that the work is too limited in scope for JMPS and I recommend rejection.
Reviewer 2
In this manuscript, the authors discuss the problem of stability of a trefoil knotted structure without self-contact. The knot is made out of a thin (thickness h) and narrow (width w) elastic material, which makes it appropriate to treat this system as a 1-dimensional variational problem. Hence, the authors perform their theoretical analysis using both Kirchhoff rod and elastic strip models to validate their experiments. Stable and contact-free knots are found and their stability classified in space of three parameters (which determines one set of boundary conditions), namely end-rotation, end-to-end displacement, and transverse end-displacement (the other end of the rod/strip remains fixed). Additionally, asymptotic analysis is carried out. The authors show relatively good agreement with their experimental results and offer a reasonable rational for when the results fail to match. I think this manuscript is very well written and scientifically relevant. Therefore, I do recommend their manuscript for publication in JMPS.
I do, however, have a few questions and perhaps a few points to be addressed.
1) Page 3: “However, in modeling a ribbon, for instance a strip of paper as in Fig 1(b) […]” -> Are you referring to Fig 1, without subfigure b?
2) Page 5: In the last couple of sentences, the conditions of the elastic constant leads to unusual material properties, such as the Poisson’s ratio. Perhaps the discussion around this point is a little terse. Extending this discussion could be useful.
3) Page 8: $\bar{n}_1(0)^2+ \bar{n}_2(0)^2 +\bar{n}_3(0)^2+\bar{m}_1(0)^2+ \bar{m}_2(0)^2 +\bar{m}_3(0)^2= 1$. I guess you meant two separate conditions, $\bar{n}_1(0)^2+ \bar{n}_2(0)^2 +\bar{n}_3(0)^2=1$ and $\bar{m}_1(0)^2+ \bar{m}_2(0)^2 +\bar{m}_3(0)^2= 1$.
4) Page 14: I’m a bit confused with the following sentence: “This situation is analogous to the constraint of inextensibility where the extensional strain in the rod is zero while the axial stress is not.” If I understand correctly this says that the curvature of the center-line with respect to the plane of the ribbon remains zero (enforced by the constraint). Perhaps this is what the authors mentioned, but I was just wondering for my own clarification. The authors may feel free to extend the discussion for clarification of the readers.
5) Page 16: It would be instructive to have seen experiments also done for rod-like structures.
6) Page17: Capitalizing W and H at this point, to distinguish from experimental parameters, may not be necessary. But that’s just a matter of taste.
7) Pages 17 and 18: The discussion about the validity for the strip model. This is done for fixed H. Do you think that playing with the thickness of the strip would change the range of validity for the ratio W/H, thus suggesting W/H isn’t the right dimensionless parameter to think about in this case?