Advertisement

“Wunderlich, Meet Kirchhoff”: A General and Unified Description of Elastic Ribbons and Thin Rods

  • Marcelo A. Dias
  • Basile AudolyEmail author

Abstract

The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-of-plane curvature, and a variable width. Both the case of a finite width (Wunderlich’s model) and the limit of small width (Sadowksky’s model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (Nat. Mater. 6(8):563–567, 2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Frenet–Serret frame, the present work unifies the description of thin ribbons and thin rods.

Keywords

Elastic plates Elastic rods Energy minimization 

Mathematics Subject Classification

74K20 74K10 74G65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Armon, S., Efrati, E., Kupferman, R., Sharon, E.: Geometry and mechanics in the opening of chiral seed pods. Science (New York) 333(6050), 1726–1730 (2011). http://www.ncbi.nlm.nih.gov/pubmed/21940888. doi: 10.1126/science.1203874 CrossRefGoogle Scholar
  2. 2.
    Audoly, B., Pomeau, Y.: Elasticity and Geometry: From Hair Curls to the Nonlinear Response of Shells. Oxford University Press, London (2010) Google Scholar
  3. 3.
    Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. ACM Trans. Graph. 27(3), 63:1–63:12 (2008) CrossRefGoogle Scholar
  4. 4.
    Cheng-Chung, H.: A Differential-Geometric Criterion for a Space Curve to be Closed. Proceedings of the American Mathematical Society 83(2), 357–361 (1981). http://www.jstor.org/stable/2043528. doi: 10.2307/2043528 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chopin, J., Kudrolli, A.: Helicoids, wrinkles, and loops in twisted ribbons. Phys. Rev. Lett. 111(17), 174302 (2013) CrossRefGoogle Scholar
  6. 6.
    Chouaïeb, N.: Kirchhoff’s problem of helical solutions of uniform rods and stability properties. Ph.D. thesis, École polytechnique fédérale de Lausanne, Lausanne, Switzerland (2003) Google Scholar
  7. 7.
    Cohen, H.: A non-linear theory of elastic directed curves. International Journal of Engineering Science 4(5), 511–524 (1966). http://www.sciencedirect.com/science/article/pii/0020722566900139. doi: 10.1016/0020-7225(66)90013-9 CrossRefGoogle Scholar
  8. 8.
    Coleman, B., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. Journal of Elasticity 60(3), 173–221 (2000). doi: 10.1023/A:1010911113919 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cosserat, E., Cosserat, F.: Théorie des Corps déformables. A. Hermann et Fils, Paris (1909) Google Scholar
  10. 10.
    Dias, M.A., Audoly, B.: A non-linear rod model for folded elastic strips. J. Mech. Phys. Solids 62, 57–80 (2014). http://linkinghub.elsevier.com/retrieve/pii/S0022509613001658. doi: 10.1016/j.jmps.2013.08.012 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dias, M.A., Dudte, L.H., Mahadevan, L., Santangelo, C.D.: Geometric Mechanics of Curved Crease Origami. Phys. Rev. Lett. 109(11), 1–5 (2012). http://link.aps.org/doi/10.1103/PhysRevLett.109.114301. doi: 10.1103/PhysRevLett.109.114301 CrossRefGoogle Scholar
  12. 12.
    Efimov, N.V.: Some problems in the theory of space curves. Uspekhi Mat. Nauk 2(3), 193–194 (1947). http://mi.mathnet.ru/umn6961 Google Scholar
  13. 13.
    Ericksen, J.L.: Simpler static problems in nonlinear theories of rods. International Journal of Solids and Structures 6(3), 371–377 (1970). http://www.sciencedirect.com/science/article/pii/0020768370900454. doi: 10.1016/0020-7683(70)90045-4 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Frenchel, W.: On the differential geometry of closed space curves. Bulletin of the American Mathematical Society 57(1), 44–54 (1951). http://projecteuclid.org/euclid.bams/1183515801 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Giomi, L., Mahadevan, L.: Statistical mechanics of developable ribbons. Phys. Rev. Lett. 104, 238104 (2010). http://link.aps.org/doi/10.1103/PhysRevLett.104.238104. doi: 10.1103/PhysRevLett.104.238104 CrossRefGoogle Scholar
  16. 16.
    Green, A.E.: The elastic stability of a thin twisted strip. II. Proc. R. Soc. Lond. A 161, 197–220 (1937) CrossRefzbMATHGoogle Scholar
  17. 17.
    Korte, A.P., Starostin, E.L., van der Heijden, G.H.M.: Triangular buckling patterns of twisted inextensible strips. Proc. R. Soc. A, Math. Phys. Eng. Sci. 467(2125), 285–303 (2010). http://rspa.royalsocietypublishing.org/cgi/doi/10.1098/rspa.2010.0200. doi: 10.1098/rspa.2010.0200 CrossRefGoogle Scholar
  18. 18.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944) zbMATHGoogle Scholar
  19. 19.
    Mahadevan, L., Keller, J.B.: The shape of a Möbius band. Proc. R. Soc. A, Math. Phys. Eng. Sci. 440, 149–162 (1993) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mockensturm, E.M.: The elastic stability of twisted plates. J. Appl. Mech. 68(4), 561–567 (2001) CrossRefzbMATHGoogle Scholar
  21. 21.
    Sadowsky, M.: Ein elementarer Beweis für die Existenz eines abwickelbares Möbiusschen Bandes und Zurückfürung des geometrischen Problems auf ein Variationsproblem. Sitzungsber. Preuss. Akad. Wiss. 22, 412–415 (1930) Google Scholar
  22. 22.
    Seffen, K.A., Audoly, B.: Buckling of a closed, naturally curved ribbon (2014, in preparation) Google Scholar
  23. 23.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 3, 3rd edn. Publish or Perish, Inc., Houston (1999) zbMATHGoogle Scholar
  24. 24.
    Starostin, E., van der Heijden, G.: Tension-induced multistability in inextensible helical ribbons. Phys. Rev. Lett. 101(8), 084301 (2008). http://link.aps.org/doi/10.1103/PhysRevLett.101.084301. doi: 10.1103/PhysRevLett.101.084301 CrossRefGoogle Scholar
  25. 25.
    Starostin, E.L., van der Heijden, G.H.M.: The shape of a Möbius strip. Nat. Mater. 6(8), 563–567 (2007). http://www.ncbi.nlm.nih.gov/pubmed/17632519. doi: 10.1038/nmat1929 CrossRefGoogle Scholar
  26. 26.
    Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast. 33(1), 1–26 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wu, Z.L., Moshe, M., Greener, J., Therien-Aubin, H., Nie, Z., Sharon, E., Kumacheva, E.: Three-dimensional shape transformations of hydrogel sheets induced by small-scale modulation of internal stresses. Nat. Commun. 4, 1586 (2013). http://www.ncbi.nlm.nih.gov/pubmed/23481394. doi: 10.1038/ncomms2549 CrossRefGoogle Scholar
  28. 28.
    Wunderlich, W.: Über ein abwickelbares Möbiusband. Monatshefte Math. 66(3), 276–289 (1962). http://link.springer.com/10.1007/BF01299052. doi: 10.1007/BF01299052 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yang, Y., Tobias, I., Olson, W.K.: Finite element analysis of DNA supercoiling. J. Chem. Phys. 98(2), 1673–1686 (1993) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of EngineeringBrown UniversityProvidenceUSA
  2. 2.Sorbonne Universités, UPMC Univ Paris 06, CNRSUMR 7190 Institut Jean Le Rond d’AlembertParisFrance

Personalised recommendations