Non-Euclidean Ribbons

Generalized Sadowsky Functional for Residually-Stressed Thin and Narrow Bodies
  • Efi EfratiEmail author


The classical theory of ribbons as developed by Sadowsky and Wunderlich has received much attention in recent years. It concerns the equilibrium conformations of thin and narrow ribbons whose intrinsic structure favors a rectangular and flat state. However, the intrinsic structure of naturally formed ribbons will often be more complicated; Spatial variations in the in-plane distance metric can give rise to both geodesic curvature and Gaussian curvature, curving the ribbon in and out of its plane. Moreover, metric variation across the thickness of the ribbon may result in nontrivial reference normal curvatures. The resulting geometric structure is likely to have no zero-energy (stress-free) realizations in Euclidean space.

This paper presents a generalization of the Sadowsky functional, which measures the bending energy of narrow ribbons, for the case of incompatible ribbons (having no stress-free configuration). Specific solutions to special cases where the reference normal curvatures vanish, and for a naturally curved developable ribbon are presented and the resulting twist-stretch relations are discussed.


Non-Euclidean plates Residual stress Ribbons Sadowsky 

Mathematics Subject Classification (2010)

74K20 74K25 53A05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dias, M.A., Audoly, B.: arXiv preprint (2014). arXiv:1403.2094. doi: 10.1007/s10659-014-9487-0 Google Scholar
  2. 2.
    Starostin, E.L., Van der Heijden, G.H.M.: Nat. Mater. 6(8), 563 (2007) CrossRefGoogle Scholar
  3. 3.
    Efrati, E., Sharon, E., Kupferman, R.: Phys. Rev. E 83(4), 046602 (2011) CrossRefGoogle Scholar
  4. 4.
    Wunderlich, W.: Monatshefte Math. 66(3), 276 (1962) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Efrati, E., Sharon, E., Kupferman, R.: J. Mech. Phys. Solids 57(4), 762 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Armon, S., Aharoni, H., Moshe, M., Sharon, E.: Soft Matter 10(16), 2733 (2014) CrossRefGoogle Scholar
  7. 7.
    Armon, S., Efrati, E., Kupferman, R., Sharon, E.: Science 333(6050), 1726 (2011) CrossRefGoogle Scholar
  8. 8.
    Chopin, J., Kudrolli, A.: Phys. Rev. Lett. 111(17), 174302 (2013) CrossRefGoogle Scholar
  9. 9.
    Efrati, E., Sharon, E., Kupferman, R.: Soft Matter 9(34)(8187), 00002 (2013) Google Scholar
  10. 10.
    Ciarlet, P.G.: An Introduction to Differential Geometry with Applications to Elasticity, 2005th edn. Springer, Berlin (2006) Google Scholar
  11. 11.
    Struik, D.J.: Lectures on Classical Differential Geometry, 2nd edn. Dover, New York (1988) zbMATHGoogle Scholar
  12. 12.
    Lewicka, M., Reza Pakzad, M.: ESAIM Control Optim. Calc. Var. 17(4), 1158 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Willmore, T.J.: An Introduction to Differential Geometry, 1st edn. Clarendon, Oxford (1959) zbMATHGoogle Scholar
  14. 14.
    Sadowsky, M.: Sitz.ber. Preuss. Akad. Wiss. Berl. Philos.-Hist. Kl. 22, 412–415 (1930) Google Scholar
  15. 15.
    Chopin, J., Démery, V., Davidovitch, B.: J. Elast., 1–53 (2014). doi: 10.1007/s10659-014-9498-x Google Scholar
  16. 16.
    Gore, J., Bryant, Z., Nöllmann, M., Le, M.U., Cozzarelli, N.R., Bustamante, C.: Nature 442(7104), 836 (2006) CrossRefGoogle Scholar
  17. 17.
    Ðuričković, B., Goriely, A., Maddocks, J.H.: Phys. Rev. Lett. 111, 108103 (2013) CrossRefGoogle Scholar
  18. 18.
    Efrati, E., Sharon, E., Kupferman, R.: Phys. Rev. E 80(1), 016602 (2009) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Plewa, J.S., Witten, T.A.: J. Chem. Phys. 112(22), 10042 (2000) CrossRefGoogle Scholar
  20. 20.
    Kamien, R.D., Lubensky, T.C., Nelson, P., O’Hern, C.S.: Europhys. Lett. 38(3), 237 (1997) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Moakher, M., Maddocks, J.H.: Arch. Ration. Mech. Anal. 177(1), 53 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bets, K.V., Yakobson, B.I.: Nano Res. 2(2), 161 (2009) CrossRefGoogle Scholar
  23. 23.
    Thomas, B.N., Lindemann, C.M., Corcoran, R.C., Cotant, C.L., Kirsch, J.E., Persichini, P.J.: J. Am. Chem. Soc. 124(7), 1227 (2002) CrossRefGoogle Scholar
  24. 24.
    Chung, D.S., Benedek, G.B., Konikoff, F.M., Donovan, J.M.: Proc. Natl. Acad. Sci. USA 90(23), 11341 (1993) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Physics of Complex SystemsWeizmann Institute of ScienceRehovotIsrael
  2. 2.James Franck InstituteThe University of ChicagoChicagoUSA

Personalised recommendations