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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A one-dimensional model for elastic ribbons: A little stretching makes a big difference
Basile Audoly and Sébastien Neukirch
Journal of the Mechanics and Physics of Solids, 153 (2021) 104457
Abstract : Starting from the theory of elastic plates, we derive a non-linear one-dimensional model for elastic ribbons with thickness t, width a and length L, assuming t << a << L. It takes the form of a rod model with a specific non-linear constitutive law accounting for both the stretching and the bending of the ribbon mid-surface. The model is asymptotically correct and can handle finite rotations. Two popular theories can be recovered as limiting cases, namely Kirchhoff's rod model for small bending and twisting strains, |k_i| << t/a^2, and Sadowsky's inextensible ribbon model for |k_i| >> t/a^2; we point out that Sadowsky's inextensible model may be a poor approximation even for ribbons having a very thin cross-section (say, with t/a as small as 0.02). By way of illustration, the one-dimensional model is applied (i) to the lateral-torsional instability of a ribbon, showing good agreement with both experiments and finite-element shell simulations, and (ii) to the stability of a twisted ribbon subjected to a tensile force. The non-convexity of the one-dimensional model is discussed; it is addressed by a convexification argument.
DOI: 10.1016/j.jmps.2021.104457
pre-print version: hal-03132668
journal version: PDF
Received 3 February 2021
Received in revised form 15 April 2021
Accepted 16 April 2021
Reviewer 1
In this paper the authors provide a new non-linear model for elastic ribbons. The model is derived from the theory of elastic plates by means of an asymptotic analysis based on a particular scaling of the “macroscopic” deformation measures. The asymptotic analysis is performed in a quite interesting way by means of the following steps:1. the deformation measures of the one-dimensional (unknown) limit problem are fixed. Hereafter, we call them macroscopic deformation measures;
2. from the macroscopic deformation measures the directors and the de- formation of the “mid-line” are determined;
3. with the quantities derived in step 2) the displacement of the plate is defined by introducing undetermined “microscopic displacements”;
4. the measures of deformation associated to the displacement of the plate are computed and then only the terms consistent with the scaling adopted are retained;
5. assuming the macroscopic deformation measures constant and the mi- croscopic displacements to depend only on the variable in the direction of the width of the ribbon, the microscopic displacements are deter- mind in such a way to make the energy stationary;
6. the solutions found are substituted into the energy to obtain an ex- pression depending only on the macroscopic deformation measures;
7. the energy obtained is further simplified by using again the scalings adopted. This step introduces constraints in the model and these, in turn, force a convexification of the energy.
Besides the new ribbon model, the paper contains two stability analysis and also several comparisons with other theories. The quality and wealth of the results contained in the manuscript are quite remarkable. It is therefore natural for me to strongly advice publication.
Just a remark on the presentation style. The paper is carefully and clearly written and it is easy to follow if one accepts the line of thought of the authors. I always have to read the details before moving on to the next step and therefore I was unable to accept the path drawn by the authors. Hence, in reading the manuscript I had to read part of Section 2, to jump to Section 6, then move to Appendix A, and then maybe, now I do not remember, go back and continue with Section 3. It has been a continuous rollercoaster. I understand that this is a matter of style, so I am not asking to make changes, but I wanted to point this out.
Reviewer 2
The authors derive an equivalent rod model (i.e., a 1D elastic model) for elastic ribbons (having thickness << width << length), starting from nonlinear plate theory that takes account of both stretching and bending of the midsurface.By a careful scaling analysis and a separation of macroscopic and microscopic quantities the model is made to capture the two classical 1D elastic models available: the Kirchhoff (flat) rod model and the Sadowsky developable strip model. These two models are quite different and the precise relationship between them has so far remained unclear. The newly proposed model has a wider domain of validity and interpolates between these two limiting models.
The derivation of the model is not rigorous and therefore needs validation, which is provided by the detailed discussion of two buckling problems. For both, the new model is found to give good results. The application to lateral-torsional buckling is particularly impressive as the new ribbon model satisfactorily captures the singular limit in which both width (a) and thickness (t) go to zero and the Kirchhoff rod model and the Sadowsky strip model predict significantly different buckling loads. It is found that if t/a is very small the ribbon buckles initially as a rod but then shows a quick transition to the strip post-buckling curve (a phenomenon not observed before as far as I know), in good agreement with both experiments and finite-element simulations.
The new model is a welcome addition to the literature. The results give new insight into the mechanical behaviour of thin and narrow elastic structures. The paper is generally carefully written and clearly explained.
Detailed comments:
- The authors' reference to the literature is less than generous. It is stated in the Introduction (p. 2) that "DA15 ... have shown that Sadowsky's inextensible ribbon model can be formulated in a way that fits the nonlinear theory of elastic rods: an inextensible ribbon is effectively a rod endowed with a nonlinear constitutive law". This modelling of a ribbon as an effective rod goes back to Starostin & Van der Heijden, Nature Materials 6 (2007) where the equivalent rod equations (for more general problems) were first given. Should it be thought that the authors of this 2007 paper were not aware that the equations effectively describe a rod with nonlinear constitutive relations then reference may be made to Starostin & Van der Heijden, Physical Review E79, 066602 (2009). The authors should give a fair discussion of the literature.
- In Section 7.3 the 'corrected' Sadowsky strain energy of Freddi et al. (FHMP15) is recovered. But it seems to me that the argument is not quite completed. The condition (7.3) was ASSUMED in Section 7.1, so it still needs to be shown that this is consistent with the Sadowsky limit.
- On p. 11 results are presented of an application of Koiter's expansion method. No details are provided, but can at least a reference for the method be given?
- On p.1 it is written that "the Kirchhoff rod model ... is based on the classical theory of rods". I would say it IS the classical theory of rods.
- A little further down it is written: "The ribbon model of [Wunderlich] extends that of Sadowsky to include the effect of the longitudinal gradient of twisting strain". This is a curious statement. The former extends the latter to include the effect of finite width. And this does not just give rise to a gradient of twisting strain, but also a gradient of bending strain.
- Section 6.1: what does it mean for the basis d_i to be "direct"?
- The accents on Von Karman are frequently wrong.