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Buckling of Naturally Curved Elastic Strips: The Ribbon Model Makes a Difference

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Abstract

We analyze the stability of naturally curved, inextensible elastic ribbons. In experiments, we first show that a loop formed using a metallic strip can become unstable if its radius is larger than its natural radius of curvature (undercurved case): the loop then folds onto itself into a smaller, multiply-covered loop. Conversely, a multi-covered, overcurved metallic strip can unfold dynamically into a circular configuration having a lower covering index. We analyze these instabilities using a one-dimensional mechanical model for an elastic ribbon introduced recently (Dias and Audoly in J. Elast., 2014), which extends Sadowsky’s developable elastic ribbon model in the presence of natural curvature. Combining linear stability analyses and numerical computations of the post-buckled configurations, we classify the equilibria of the ribbon as a function of the ratio of its natural curvature to its actual curvature. Our ribbon model is formulated in close analogy with classical rod models; this allows us to adapt classical stability methods for rods to the case of a ribbon. The stability of a ribbon is found to differ significantly from that of an anisotropic rod: we attribute this difference to the fact that the tangent twisting modulus of a ribbon can be negative, in contrast to what is possible in the well-studied case of linearly elastic rods. The specific stability properties predicted by the curved ribbon model are confirmed by a finite element analysis of cylindrical shells having a small height-to-radius ratio.

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Notes

  1. An inextensible surface is a surface that preserves its metric, hence the length of all curves drawn onto it, upon deformation. We do not consider the case of surfaces that are inextensible along a given set of material directions only, such as surfaces made up of woven networks of inextensible fibres.

  2. By contrast, Kirchhoff’s kinematical hypothesis, which underpins rod models, valid when w and h are comparable, but it does not capture developable configurations well in the limit h≪w.

  3. To show these equalities, note that the left-hand side of the first equation is the derivative of \(\hat{\mathbf{M}}\) in the moving basis following the virtual motion. The second equation is a classical formula for calculating the bending and twisting strain increments; see for instance [2].

  4. We just found that the stability boundaries for the ribbon model are either rigid-body modes, or out-of-plane modes (b=0) but never in-plane modes (c=0 has no root for g=∞). This confirms the assumption made earlier in Sect. 5.3 that the stability of the ribbon model can be analyzed assuming that the perturbations associated with in-plane modes are zero.

  5. We have ignored the rotational inertia in Eq. (16c) for the balance of angular momentum and, as a result, pure twist modes are associated with zero inertia and an infinite frequency: this is why c=0 makes \((\overline{\varOmega}^{2})_{2}\) infinite in Eq. (26a).

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Authors and Affiliations

  1. UMR 7190 Institut Jean Le Rond d’Alembert, Sorbonne Universités, UPMC Univ Paris 06, CNRS, 75005, Paris, France

    Basile Audoly

  2. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK

    Keith A. Seffen

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  1. Basile Audoly
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  2. Keith A. Seffen
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Correspondence to Basile Audoly.

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Audoly, B., Seffen, K.A. Buckling of Naturally Curved Elastic Strips: The Ribbon Model Makes a Difference. J Elast 119, 293–320 (2015). https://doi.org/10.1007/s10659-015-9520-y

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