The Mechanics of Ribbons and Möbius Bands pp 293-320 | Cite as
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.
Keywords
Elastic plates Elastic rods Energy minimizationMathematics Subject Classification
74K20 74K10 74G65Preview
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References
- 1.Antman, S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, Berlin (2005) zbMATHGoogle Scholar
- 2.Dias, M.A., Audoly, B.: A non-linear rod model for folded elastic strips. J. Mech. Phys. Solids 62, 57–80 (2014) MathSciNetCrossRefGoogle Scholar
- 3.Dias, M.A., Audoly, B.: “Wunderlich, meet Kirchhoff”: a general and unified description of elastic ribbons and thin rods. J. Elast. (2014) Google Scholar
- 4.Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.J.: AUTO-07p: continuation and bifurcation software for ordinary differential equations (2007). http://indy.cs.concordia.ca/auto/
- 5.Goriely, A.: Twisted elastic rings and the rediscoveries of Michell’s instability. J. Elast. 84, 281–299 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Goriely, A., Shipman, P.: Dynamics of helical strips. Phys. Rev. E 61(4), 4508–4517 (2000) MathSciNetCrossRefGoogle Scholar
- 7.Haijun, Z., Zhong-can, O.Y.: Spontaneous curvature-induced dynamical instability of Kirchhoff filaments: application to DNA kink deformations. J. Chem. Phys. 110, 1247 (1999) CrossRefGoogle Scholar
- 8.Hibbitt, D., Karlsson, B., Sorensen, P.: Abaqus manual version 6.11. Tech. rep., Dassault Systèmes (2011) Google Scholar
- 9.Hinz, D.F., Fried, E.: Translation and interpretation of Michael Sadowsky’s paper “Theory of elastically bendable inextensible bands with applications to the Möbius band”. J. Elast., 1–11 (2014) Google Scholar
- 10.Hinz, D.F., Fried, E.: Translation of Michael Sadowsky’s paper “An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem”. J. Elast. (2014) Google Scholar
- 11.Hinz, D.F., Fried, E.: Translation of Michael Sadowsky’s paper “The differential equations of the Möbius band”. J. Elast., 1–4 (2014) Google Scholar
- 12.Hoffman, K.A.: Methods for determining stability in continuum elastic-rod models of DNA. Philos. Trans. R. Soc. Lond. A 362, 1301–1315 (2004) CrossRefzbMATHGoogle Scholar
- 13.Hoffman, K.A., Manning, R.S., Maddocks, J.H.: Link, twist, energy and the stability of DNA minicircles. Biopolymers 70(2), 145–157 (2003) CrossRefGoogle Scholar
- 14.Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, P.H. (ed.) Applications of Bifurcation Theory, pp. 359–384. Academic Press, San Diego (1977) Google Scholar
- 15.Manning, R.S., Hoffman, K.A.: Stability of \(n\)-covered circles for elastic rods with constant planar intrinsic curvature. J. Elast. 62(1), 1–23 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Michell, J.H.: On the stability of a bent and twisted wire. Messenger Math. 11, 181–184 (1889–1890) Google Scholar
- 17.Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes, 3rd edn. Cambridge University Press, Cambridge (2007) Google Scholar
- 18.Sadowsky, M.: Die Differentialgleichungen des Möbiusschen Bandes. In: Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 39 (2. Abt. Heft 5/8), pp. 49–51 (1929) Google Scholar
- 19.Sadowsky, M.: Ein elementarer Beweis für die Existenz eines abwickelbaren Möbiusschen Bandes und die Zurückführung des geometrischen Problems auf ein Variationsproblem. In: Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 17. Juli 1930, pp. 412–415 (1930). Mitteilung vom 26. Juni Google Scholar
- 20.Sadowsky, M.: Theorie der elastisch biegsamen undehnbaren Bänder mit Anwendungen auf das Möbiussche Band. In: Proceedings of the 3rd International Congress of Applied Mechanics, Stockholm, vol. 2, pp. 444–451 (1930) Google Scholar
- 21.Seffen, K.A., Pellegrino, S.: Deployment dynamics of tape springs. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455(1983), 1003–1048 (1999) MathSciNetCrossRefGoogle Scholar
- 22.Starostin, E.L., van der Heijden, G.H.M.: The shape of a Möbius strip. Nat. Mater. 6(8), 563–567 (2007) CrossRefzbMATHGoogle Scholar
- 23.Starostin, E.L., van der Heijden, G.H.M.: Tension-induced multistability in inextensible helical ribbons. Phys. Rev. Lett. 101, 084301 (2008) CrossRefzbMATHGoogle Scholar
- 24.Thompson, J.M.T., Champneys, A.R.: From helix to localized writhing in the torsional post-buckling of elastic rods. Proc. R. Soc. A, Math. Phys. Eng. Sci. 452(1944), 117–138 (1996) MathSciNetCrossRefGoogle Scholar
- 25.Todres, R.E.: Translation of W. Wunderlich’s “on a developable Möbius band”. J. Elast., 1–12 (2014) Google Scholar
- 26.Wunderlich, W.: Über ein abwickelbares Möbiusband. Monatshefte Math. 66(3), 276–289 (1962) MathSciNetCrossRefzbMATHGoogle Scholar