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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Numerical modeling of inextensible elastic ribbons with curvature-based elements
Raphael Charrondiere, Florence Bertails-Descoubes, Sebastien Neukirch, Victor Romero
Computer methods in applied mechanics and engineering, vol. 364 (2020) 112922
Abstract : We propose a robust and efficient numerical model to compute stable equilibrium configurations of clamped elastic ribbons featuring arbitrarily curved natural shapes. Our spatial discretization scheme relies on elements characterized by a linear normal curvature and a quadratic geodesic torsion with respect to arc length. Such a high-order discretization allows for a great diversity of kinematic representations, while guaranteeing the surface of the ribbon to remain perfectly inextensible. Stable equilibria are calculated by minimizing the sum of the gravitational and elastic energies of the ribbon, under a developability constraint. Our algorithm compares favorably to standard shooting and collocation methods, as well as to experiments. Our model furthermore shows significant differences in behavior compared to a rod model, while yielding a substantial speed-up compared to a more general shell model. These results confirm the benefit of designing a special numerical model dedicated to ribbons.
DOI: 10.1016/j.cma.2020.112922
download the journal version : PDF
supplementary movie :
Submitted on july 26th, 2019
Reports received on jan. 7th, 2020
2nd version submitted on feb. 5th, 2020
Accepted on feb. 10th, 2020
Reviewer 1
I have read with interest this manuscript by Charrondiere et al. Overall, I think this is a good manuscript that the CMAME readership will find interesting and I recommend it for publication. In my opinion, the main contribution of the manuscript is to develop a computational framework to obtain stable equilibrium configurations of inextensible ribbons using material curvatures as primary variables. This computational framework is partially inspired by previous numerical methods for elastic rods.
I have the following minor comments:
1) In page 19, line 42, the authors say that continuation methods cannot easily account for the Wunderlich model. The authors make a similar claim in the caption of Table 2. The authors should either be more specific about why continuation methods cannot easily account for the Wunderlich model or just state that continuation methods have not been used for the Wunderlich model thus far.
2) The paper is well-written, but it has some small typos:
- In page 12, line 54, the sentence "In the following we report in details our experiments necessary to justify the choice for our final method" does not make sense.
- In page 13, line 26, the sentence "The convergence criterion used if often based ..." should be "The convergence criterion used is often based ...".
Reviewer 2
This manuscript presents a computational ribbon model based on a finite element discretization of the curvature. The manuscript is well written and the numerical examples are generally well chosen. The work is interesting and relevant. I recommend publication provided the following issues are addressed:
1. The concept of "developability" is not consistently introduced. According to the abstract it denotes a surface that cannot be stretched or sheared. The first aspect implies inextensibility, but then in Sec. 2.1 inextensibility is mentioned separately, indicating that developabaility does not contain inextensibility. The second aspect implies parallel rulings, which is not the case considered later. Perhaps the authors mean transverse shear instead of in-plane shear.
2. Sec. 2: The authors should cite the original source when mentioning the "ribbon model of Sadowsky". Likewise, the original works by Naghdi should be cited on line 506.
3. I'm surprised that the authors don't cite the non-linear beam and shell formulations of Simo (see J.C. Simo, "A finite strain beam formulation. The three-dimensional dynamic problem. Part I", Comput. Methods Appl. Mech. Eng. 49 (1985) 55-70; J.C. Simo, D.D. Fox, "On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parameterization", Comput. Methods Appl. Mech. Engrg. 72 (1989) 267-304; and following works). I would expect that the model proposed by the present authors falls in between these models if it is not actually contained already. Perhaps some aspect of the present work is new, but I'm convinced that many aspects are already contained in Simo's work. Thus the authors should discuss their work in the context of existing approaches by Simo and followers.
4. The authors claim that "A ribbon cannot be properly simulated using a rod model with a rectangular cross-section". I don't believe this statement is generally true. The ribbon model considered by the authors in eq. (17) assumes linear elasticity and does not account for the natural curvature of the structure. If a general non-linear rod model is used (even one based on Kirchhoff-Love kinematics), I would expect a much closer fit. For a fair comparison between the proposed ribbon model and a rod model, the same elastic energy should be used.
5. The numerical results in Fig. 5 (and 9) show very large in-plane shear deformations during which the length of the rulings extend substantially. Both the shear and the length change seem rather unphysical, esp. as inextensibility is assumed in axial direction. The authors should comment on this.
6. Fig. 11: How come there are multiple solutions from the ribbon simulation for a given Gamma?
7. The statement "Nevertheless, we can already conclude that, in the presence of torsion and natural curvature, the ribbon model captures the experimental behaviour better than an anisotropic rod model." is unfounded as the authors have only looked at a simplistic rod model. I'm convinced that a more general rod model, such as the one by Simo, would give a much more accurate picture of the situation.