Advertisement

Roadmap to the Morphological Instabilities of a Stretched Twisted Ribbon

  • Julien Chopin
  • Vincent Démery
  • Benny DavidovitchEmail author
Chapter
  • 701 Downloads

Abstract

We address the mechanics of an elastic ribbon subjected to twist and tensile load. Motivated by the classical work of Green (Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 154(882):430, 1936; 161(905):197, 1937) and a recent experiment (Chopin and Kudrolli in Phys. Rev. Lett. 111(17):174302, 2013) that discovered a plethora of morphological instabilities, we introduce a comprehensive theoretical framework through which we construct a 4D phase diagram of this basic system, spanned by the exerted twist and tension, as well as the thickness and length of the ribbon. Different types of instabilities appear in various “corners” of this 4D parameter space, and are addressed through distinct types of asymptotic methods. Our theory employs three instruments, whose concerted implementation is necessary to provide an exhaustive study of the various parameter regimes: (i) a covariant form of the Föppl–von Kármán (cFvK) equations to the helicoidal state—necessary to account for the large deflection of the highly-symmetric helicoidal shape from planarity, and the buckling instability of the ribbon in the transverse direction; (ii) a far from threshold (FT) analysis—which describes a state in which a longitudinally-wrinkled zone expands throughout the ribbon and allows it to retain a helicoidal shape with negligible compression; (iii) finally, we introduce an asymptotic isometry equation that characterizes the energetic competition between various types of states through which a twisted ribbon becomes strainless in the singular limit of zero thickness and no tension.

Keywords

Buckling and wrinkling Far from threshold Isometry Helicoid 
ssFvK equations

“small-slope” (standard) Föppl–von Kármán equations

cFvK equations

covariant Föppl–von Kármán equations

t, W, L

thickness, width and length of the ribbon (non-italicized quantities are dimensional)

\(t\), \(W=1\), \(L\)

thickness, width and length normalized by the width

\(\nu\)

Poisson ratio

\(\mathrm{E}, \mathrm {Y},\mathrm {B}=\frac{\mathrm {Y}\mathrm {t}^{2}}{12(1-\nu^{2})}\)

Young, stretching and bending modulus

\(Y=1\), \(B=\frac{t^{2}}{12(1-\nu^{2})}\)

stretching and bending modulus, normalized by the stretching modulus

\(T=\mathrm {T}/\mathrm {Y}\)

tensile strain (tensile load normalized by stretching modulus)

\(\theta\), \(\eta=\theta/L\)

twist angle and normalized twist

\((\hat{\boldsymbol {x}},\hat{\boldsymbol {y}},\hat{\boldsymbol {z}})\)

Cartesian basis

\(s\), \(r\)

material coordinates (longitudinal and transverse)

\(z(s,r)\)

out of plane displacement (of the helicoid) in the small-slope approximation

\(\boldsymbol {X}(s,r)\)

surface vector

\(\hat{\boldsymbol {n}}\)

unit normal to the surface

\(\sigma^{\alpha \beta}\)

stress tensor

\(\varepsilon_{\alpha \beta}\)

strain tensor

\(g_{\alpha \beta}\)

metric tensor

\(c_{\alpha \beta}\)

curvature tensor

\(\mathcal{A}^{\alpha\beta\gamma\delta}\)

elastic tensor

\(\partial_{\alpha}\), \(D_{\alpha}\)

partial and covariant derivatives

\(H\), \(K\)

mean and Gaussian curvatures

\(\zeta\)

infinitesimal amplitude of the perturbation in linear stability analysis

\(z_{1}(s,r)\)

normal component of an infinitesimal perturbation to the helicoidal shape

\(\eta_{\mathrm{lon}}\), \(\lambda_{\mathrm{lon}}\)

longitudinal instability threshold and wavelength

\(\eta_{\mathrm{tr}}\), \(\lambda_{\mathrm{tr}}\)

transverse instability threshold and wavelength

\(\alpha=\eta^{2}/T\)

confinement parameter

\(\alpha_{\mathrm{lon}}\)

threshold confinement for the longitudinal instability

\(r_{\mathrm{wr}}\)

(half the) width of the longitudinally wrinkled zone

\(\Delta \alpha=\alpha-24\)

distance to the threshold confinement

\(f(r)\)

amplitude of the longitudinal wrinkles

\(U_{\mathrm{hel}}\), \(U_{\mathrm{FT}}\)

elastic energies (per length) of the helicoid and the far from threshold longitudinally wrinkled state

\(U_{\mathrm{dom}}\), \(U_{\mathrm{sub}}\)

dominant and subdominant (with respect to \(t\)) parts of \(U_{\mathrm{FT}}\)

\({\boldsymbol {X}_{\mathrm{cl}}}(s)\)

ribbon centerline

\(\hat{t} = d\boldsymbol {X}_{\mathrm{cl}}(s)/ds\)

tangent vector in the ribbon midplane

\(\hat{\boldsymbol {r}}(s)\)

normal to the tangent vector

\(\hat{\boldsymbol{b}}(s)\)

Frenet binormal to the curve \(\boldsymbol {X}_{\mathrm{cl}}(s)\)

\(\tau(s), \kappa(s)\)

torsion and curvature of \(\boldsymbol {X}_{\mathrm{cl}}(s)\)

Mathematics Subject Classification (2010)

74K20 53Z05 35Q74 74K35 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Green, A.E.: Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 154(882), 430 (1936) CrossRefzbMATHGoogle Scholar
  2. 2.
    Green, A.E.: Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 161(905), 197 (1937). http://www.jstor.org/stable/96910 CrossRefzbMATHGoogle Scholar
  3. 3.
    Chopin, J., Kudrolli, A.: Phys. Rev. Lett. 111(17), 174302 (2013). doi: 10.1103/PhysRevLett.111.174302 CrossRefGoogle Scholar
  4. 4.
    Mockensturm, E.M.: J. Appl. Mech. 68(4), 561 (2000). doi: 10.1115/1.1357517 CrossRefGoogle Scholar
  5. 5.
    Coman, C.D., Bassom, A.P.: Acta Mechanica 200(1–2), 59 (2008). doi: 10.1007/s00707-007-0572-3 CrossRefzbMATHGoogle Scholar
  6. 6.
    Sadowsky, M.: Teil II. Verhandl. des 3. Intern. Kongr. f. Techn. Mechanik, 444–451 (1930) Google Scholar
  7. 7.
    Korte, A.P., Starostin, E.L., van der Heijden, G.H.M.: Proc. R. Soc. A, Math. Phys. Eng. Sci. 467(2125), 285 (2011). doi: 10.1098/rspa.2010.0200 CrossRefzbMATHGoogle Scholar
  8. 8.
    Cerda, E., Mahadevan, L.: Phys. Rev. Lett. 90(7), 074302 (2003). doi: 10.1103/PhysRevLett.90.074302 CrossRefGoogle Scholar
  9. 9.
    Goriely, A., Nizette, M., Tabor, M.: J. Nonlinear Sci. 11(1), 3 (2001). doi: 10.1007/s003320010009 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Champneys, A.R., Thompson, J.M.T.: Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 452(1954), 2467 (1996). doi: 10.1098/rspa.1996.0132 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    van der Heijden, G., Thompson, J.: Phys. D: Nonlinear Phenom. 112(1–2), 201 (1998). Proceedings of the Workshop on Time-Reversal Symmetry in Dynamical Systems. doi: 10.1016/S0167-2789(97)00211-X CrossRefGoogle Scholar
  12. 12.
    van der Heijden, G.H.M., Thompson, J.M.T.: Nonlinear Dyn. 21(1), 71 (2000). doi: 10.1023/A:1008310425967 CrossRefzbMATHGoogle Scholar
  13. 13.
    Santangelo, C.: Geometric frustration in twisted strips. J. Club Condens. Matter Phys. (2014). http://www.condmatjournalclub.org/?p=2330
  14. 14.
    Davidovitch, B., Schroll, R.D., Vella, D., Adda-Bedia, M., Cerda, E.A.: Proc. Natl. Acad. Sci. 108(45), 18227 (2011). doi: 10.1073/pnas.1108553108 CrossRefGoogle Scholar
  15. 15.
    King, H., Schroll, R.D., Davidovitch, B., Menon, N.: Proc. Natl. Acad. Sci. 109(25), 9716 (2012). doi: 10.1073/pnas.1201201109 CrossRefGoogle Scholar
  16. 16.
    Grason, G.M., Davidovitch, B.: Proc. Natl. Acad. Sci. 110(32), 12893 (2013). doi: 10.1073/pnas.1301695110 CrossRefGoogle Scholar
  17. 17.
    Landau, L.D., Lifchitz, E.M., Kosevitch, A.M., Pitaevski, L.P., Sykes, J.B., Reid, W.: Course of Theoretical Physics: Theory of Elasticity. Butterworth-Heinemann, Stoneham (1986) Google Scholar
  18. 18.
    Cranford, S., Buehler, M.J.: Model. Simul. Mater. Sci. Eng. 19(5), 054003 (2011). http://stacks.iop.org/0965-0393/19/i=5/a=054003 CrossRefGoogle Scholar
  19. 19.
    Kit, O.O., Tallinen, T., Mahadevan, L., Timonen, J., Koskinen, P.: Phys. Rev. B 85(8), 085428 (2012). doi: 10.1103/PhysRevB.85.085428 CrossRefGoogle Scholar
  20. 20.
    Ogden, R.W.: Non-linear Elastic Deformations. Courier Dover Publications, New York (1997) Google Scholar
  21. 21.
    Efrati, E., Sharon, E., Kupferman, R.: J. Mech. Phys. Solids 57(4), 762 (2009). doi: 10.1016/j.jmps.2008.12.004 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dias, M.A., Hanna, J.A., Santangelo, C.D.: Phys. Rev. E 84(3), 036603 (2011). doi: 10.1103/PhysRevE.84.036603 CrossRefGoogle Scholar
  23. 23.
    Hohlfeld, E., Davidovitch, B.: (2014, submitted) Google Scholar
  24. 24.
    Stein, M., Hedgepeth, J.M.: Analysis of Partly Wrinkled Membranes. National Aeronautics and Space Administration, Washington (1961) Google Scholar
  25. 25.
    Pipkin, A.C.: IMA J. Appl. Math. 36(1), 85 (1986). doi: 10.1093/imamat/36.1.85 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mansfield, E.H.: The Bending and Stretching of Plates. Cambridge University Press, Cambridge (2005) Google Scholar
  27. 27.
    Davidovitch, B., Schroll, R.D., Cerda, E.: Phys. Rev. E 85(6), 066115 (2012). doi: 10.1103/PhysRevE.85.066115 CrossRefGoogle Scholar
  28. 28.
    Bella, P., Kohn, R.V.: Commun. Pure Appl. Math. (2013) Google Scholar
  29. 29.
    Audoly, B., Pomeau, Y.: Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells. Oxford University Press, Oxford (2010) Google Scholar
  30. 30.
    Huang, J., Davidovitch, B., Santangelo, C.D., Russell, T.P., Menon, N.: Phys. Rev. Lett. 105(3), 038302 (2010). doi: 10.1103/PhysRevLett.105.038302 CrossRefGoogle Scholar
  31. 31.
    Vandeparre, H., Piñeirua, M., Brau, F., Roman, B., Bico, J., Gay, C., Bao, W., Lau, C.N., Reis, P.M., Damman, P.: Phys. Rev. Lett. 106, 224301 (2011). doi: 10.1103/PhysRevLett.106.224301 CrossRefGoogle Scholar
  32. 32.
    Schroll, R.D., Adda-Bedia, M., Cerda, E., Huang, J., Menon, N., Russell, T.P., Toga, K.B., Vella, D., Davidovitch, B.: Phys. Rev. Lett. 111(1), 014301 (2013). doi: 10.1103/PhysRevLett.111.014301 CrossRefGoogle Scholar
  33. 33.
    Pogorelov, A.: Extrinsic Geometry of Convex Surfaces Google Scholar
  34. 34.
    Witten, T.A.: Rev. Mod. Phys. 79, 643 (2007). doi: 10.1103/RevModPhys.79.643 CrossRefzbMATHGoogle Scholar
  35. 35.
    Sharon, E., Roman, B., Marder, M., Shin, G.S., Swinney, H.L.: Nature 419(6907), 579 (2002). doi: 10.1038/419579a CrossRefGoogle Scholar
  36. 36.
    Audoly, B., Boudaoud, A.: Phys. Rev. Lett. 91, 086105 (2003). doi: 10.1103/PhysRevLett.91.086105 CrossRefGoogle Scholar
  37. 37.
    Klein, Y., Venkataramani, S., Sharon, E.: Phys. Rev. Lett. 106, 118303 (2011). doi: 10.1103/PhysRevLett.106.118303 CrossRefGoogle Scholar
  38. 38.
    Gemmer, J.A., Venkataramani, S.C.: Nonlinearity 25(12), 3553 (2012). http://stacks.iop.org/0951-7715/25/i=12/a=3553 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Giomi, L., Mahadevan, L.: Phys. Rev. Lett. 104, 238104 (2010). doi: 10.1103/PhysRevLett.104.238104 CrossRefGoogle Scholar
  40. 40.
    Kohn, R.V., Nguyen, H.M.: J. Nonlinear Sci. 23(3), 343 (2013). doi: 10.1007/s00332-012-9154-1 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Audoly, B., Boudaoud, A.: J. Mech. Phys. Solids 56(7), 2444 (2008). doi: 10.1016/j.jmps.2008.03.001 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Michell, J.: Messenger of Math. 11, 181 (1889–1890) Google Scholar
  43. 43.
    Goriely, A.: J. Elast. 84(3), 281 (2006). doi: 10.1007/s10659-006-9055-3 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Majumdar, A., Prior, C., Goriely, A.: J. Elast. 109(1), 75 (2012). doi: 10.1007/s10659-012-9371-8 MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (2013) zbMATHGoogle Scholar
  46. 46.
    Ashwell, D.G.: Q. J. Mech. Appl. Math. 15(1), 91 (1962). doi: 10.1093/qjmam/15.1.91 MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Nayyar, V., Ravi-Chandar, K., Huang, R.: Int. J. Solids Struct. 48(25–26), 3471 (2011). doi: 10.1016/j.ijsolstr.2011.09.004 CrossRefGoogle Scholar
  48. 48.
    Healey, T., Li, Q., Cheng, R.B.: J. Nonlinear Sci. 23(5), 777 (2013). doi: 10.1007/s00332-013-9168-3 MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Kim, T.Y., Puntel, E., Fried, E.: Int. J. Solids Struct. 49(5), 771 (2012). doi: 10.1016/j.ijsolstr.2011.11.018 CrossRefGoogle Scholar
  50. 50.
    Lee, C., Wei, X., Kysar, J.W., Hone, J.: Science 321(5887), 385 (2008) CrossRefGoogle Scholar
  51. 51.
    Novoselov, K.S., Fal[prime]ko, V.I., Colombo, L., Gellert, P.R., Schwab, M.G., Kim, K.: Nature 490(7419), 192 (2012). doi: 10.1038/nature11458 CrossRefGoogle Scholar
  52. 52.
    Mahadevan, L., Vaziri, A., Das, M.: Europhys. Lett. 77(4), 40003 (2007). http://stacks.iop.org/0295-5075/77/i=4/a=40003 CrossRefGoogle Scholar
  53. 53.
    Schroll, R.D., Katifori, E., Davidovitch, B.: Phys. Rev. Lett. 106, 074301 (2011). doi: 10.1103/PhysRevLett.106.074301 CrossRefGoogle Scholar
  54. 54.
    Klein, Y., Efrati, E., Sharon, E.: Science 315(5815), 1116 (2007). doi: 10.1126/science.1135994 MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Kim, J., Hanna, J.A., Byun, M., Santangelo, C.D., Hayward, R.C.: Science 335(6073), 1201 (2012). doi: 10.1126/science.1215309 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Julien Chopin
    • 1
    • 2
  • Vincent Démery
    • 3
  • Benny Davidovitch
    • 3
    Email author
  1. 1.Civil Engineering Department, COPPEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Department of PhysicsClark UniversityWorcesterUSA
  3. 3.Physics DepartmentUniversity of MassachusettsAmherstUSA

Personalised recommendations

Cite chapter

Cite chapter

Buy options