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Wrinkles and folds in a fluid-supported sheet of finite size

Oz Oshri1,*, Fabian Brau2,†, and Haim Diamant3,‡

  • 1Raymond & Beverly Sackler School of Physics & Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
  • 2Nonlinear Physical Chemistry Unit, Université libre de Bruxelles (ULB), CP231, B-1050 Brussels, Belgium
  • 3Raymond & Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel

  • *ozzoshri@tau.ac.il
  • †fabian.brau@ulb.ac.be
  • ‡hdiamant@tau.ac.il
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Phys. Rev. E 91, 052408 – Published 29 May, 2015

DOI: https://doi.org/10.1103/PhysRevE.91.052408

Abstract

A laterally confined thin elastic sheet lying on a liquid substrate displays regular undulations, called wrinkles, characterized by a spatially extended energy distribution and a well-defined wavelength λ. As the confinement increases, the deformation energy is progressively localized into a single narrow fold. An exact solution for the deformation of an infinite sheet was previously found, indicating that wrinkles in an infinite sheet are unstable against localization for arbitrarily small confinement. We present an extension of the theory to sheets of finite length L, accounting for the experimentally observed wrinkle-to-fold transition. We derive an exact solution for the periodic deformation in the wrinkled state, and an approximate solution for the localized, folded state. We find that a second-order transition between these two states occurs at a critical confinement ΔF=λ2/L.

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References (58)

  1. P. J. Hudleston and S. H. Treagus, Information from folds: A. review, J. Struct. Geol. 32, 2042 (2010).
  2. S. T. Milner, J.-F. Joanny, and P. Pincus, Buckling of Langmuir monolayers, Europhys. Lett. 9, 495 (1989).
  3. V. M. Kaganer, H. Möhwald, and P. Dutta, Structure and phase transitions in Langmuir monolayers, Rev. Mod. Phys. 71, 779 (1999).
  4. W. Lu, C. M. Knobler, R. F. Bruinsma, M. Twardos, and M. Dennin, Folding Langmuir Monolayers, Phys. Rev. Lett. 89, 146107 (2002).
  5. A. Gopal, V. Belyi, H. Diamant, T. A. Witten, and K. Y. C. Lee, Microscopic folds and macroscopic jerks in compressed lipid monolayers, J. Phys. Chem. B 110, 10220 (2006).
  6. Q. Zhang and T. A. Witten, Microscopic wrinkles on supported surfactant monolayers, Phys. Rev. E 76, 041608 (2007).
  7. K. Y. C. Lee, Collapse mechanisms of Langmuir monolayers, Annu. Rev. Phys. Chem. 59, 771 (2008).
  8. M. Kücken and A. C. Newell, A model for fingerprint formation, Europhys. Lett. 68, 141 (2004).
  9. M. Kücken and A. C. Newell, Fingerprint formation, J. Theor. Biol. 235, 71 (2005).
  10. E. Cerda and L. Mahadevan, Geometry and physics of wrinkling, Phys. Rev. Lett. 90, 074302 (2003).
  11. E. Cerda, L. Mahadevan, and J. Pasini, The elements of draping, Proc. Natl. Acad. Sci. USA 101, 1806 (2004).
  12. B. Davidovitch, Period fissioning and other instabilities of stressed elastic membranes, Phys. Rev. E 80, 025202 (2009).
  13. P. M. Reis, F. Corson, A. Boudaoud, and B. Roman, Localization through surface folding in solid foams under compression, Phys. Rev. Lett. 103, 045501 (2009).
  14. D. P. Holmes and A. J. Crosby, Draping films: A wrinkle to fold transition, Phys. Rev. Lett. 105, 038303 (2010).
  15. Y. Ebata, A. B. Croll, and A. J. Crosby, Wrinkling and strain localization in polymer thin films, Soft Matter 8, 9086 (2012).
  16. B. Li, Y.-P. Cao, X.-Q. Feng, and H. Gao, Mechanics of morphological instabilities and surface wrinkling in soft materials: A review, Soft Matter 8, 5728 (2012).
  17. C. C. Donato, M. A. F. Gomes, and R. E. de Souza, Crumpled wires in two dimensions, Phys. Rev. E 66, 015102(R) (2002).
  18. N. Stoop, F. K. Wittel, and H. J. Herrmann, Morphological phases of crumpled wire, Phys. Rev. Lett. 101, 094101 (2008).
  19. E. Bayart, S. Deboeuf, F. Corson, A. Boudaoud, and M. Adda-Bedia, Measuring order in the isotropic packing of elastic rods, Europhys. Lett. 95, 34002 (2011).
  20. G. W. Hunt, M. K. Wadee, and N. Shiacolas, Localized elasticae for the strut on the linear foundation, J. Appl. Mech. 60, 1033 (1993).
  21. S. H. Lee and A. M. Waas, Initial post-buckling behavior of a finite beam on an elastic foundation, Int. J. Non-Linear Mech. 31, 313 (1996).
  22. L. Boué, M. Adda-Bedia, A. Boudaoud, D. Cassani, Y. Couder, A. Eddi, and M. Trejo, Spiral patterns in the packing of flexible structures, Phys. Rev. Lett. 97, 166104 (2006).
  23. S. Deboeuf, M. Adda-Bedia, and A. Boudaoud, Energy distributions and effective temperatures in the packing of elastic sheets, Europhys. Lett. 85, 24002 (2009).
  24. H. Aharoni and E. Sharon, Direct observation of the temporal and spatial dynamics during crumpling, Nat. Mater. 9, 993 (2010).
  25. J. Huang, B. Davidovitch, C. D. Santangelo, T. P. Russell, and N. Menon, Smooth cascade of wrinkles at the edge of a floating elastic film, Phys. Rev. Lett. 105, 038302 (2010).
  26. F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud, and P. Damman, Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators, Nat. Phys. 7, 56 (2011).
  27. A. D. Cambou and N. Menon, Three-dimensional structure of a sheet crumpled into a ball, Proc. Natl. Acad. Sci. USA 108, 14741 (2011).
  28. R. D. Schroll, E. Katifori, and B. Davidovitch, Elastic building blocks for confined sheets, Phys. Rev. Lett. 106, 074301 (2011).
  29. H. Vandeparre, M. Pineirua, F. Brau, B. Roman, J. Bico, C. Gay, W. Bao, C. N. Lau, P. M. Reis, and P. Damman, Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains, Phys. Rev. Lett. 106, 224301 (2011).
  30. M. R. Semler, J. M. Harris, A. B. Croll, and E. K. Hobbie, Localization and length-scale doubling films on soft substrates, Phys. Rev. E 88, 032409 (2013).
  31. L. Zhuo and Y. Zhang, The mode-coupling of a stiff film/compliant substrate system in the post-buckling range, Int. J. Solids Struct. 53, 28 (2015).
  32. D. Vella, P. Aussillous, and L. Mahadevan, Elasticity of an interfacial particle raft, Europhys. Lett. 68, 212 (2004).
  33. J. Huang, M. Juszkiewicz, W. H. de Jeu, E. Cerda, T. Emrick, N. Menon, and T. P. Russell, Capillary wrinkling of floating thin polymer films, Science 317, 650 (2007).
  34. B. D. Leahy, L. Pocivavsek, M. Meron, K. L. Lam, D. Salas, P. J. Viccaro, K. Y. C. Lee, and B. Lin, Geometry stability and elastic response of a supported nanoparticle film, Phys. Rev. Lett. 105, 058301 (2010).
  35. D. Vella, M. Adda-Bedia, and E. Cerda, Capillary wrinkling of elastic membranes, Soft Matter 6, 5778 (2010).
  36. T. J. Wagner and D. Vella, Floating carpets and the delamination of elastic sheets, Phys. Rev. Lett. 107, 044301 (2011).
  37. H. King, R. D. Schroll, B. Davidovitch, and N. Menon, Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities, Proc. Natl. Acad. Sci. USA 109, 9716 (2012).
  38. R. D. Schroll, M. Adda-Bedia, E. Cerda, J. Huang, N. Menon, T. P. Russell, K. B. Toga, D. Vella, and B. Davidovitch, Capillary deformations of bendable films, Phys. Rev. Lett. 111, 014301 (2013).
  39. M. Piñeirua, N. Tanaka, B. Roman, and J. Bico, Capillary buckling of a floating annulus, Soft Matter 9, 10985 (2013).
  40. M. Rivetti and A. Antkowiak, Elasto-capillary meniscus: pulling out a soft strip sticking to a liquid surface, Soft Matter 9, 6226 (2013).
  41. V. Démery, B. Davidovitch, and C. D. Santangelo, Mechanics of large folds in thin interfacial films, Phys. Rev. E 90, 042401 (2014).
  42. L. Pocivavsek, R. Dellsy, A. Kern, S. Johnson, B. Lin, K. Y. C. Lee, and E. Cerda, Stress and fold localization in thin elastic membranes, Science 320, 912 (2008).
  43. H. Diamant and T. A. Witten, Instability of infinitesimal wrinkles against folding, arXiv:1009.2487.
  44. B. Audoly, Localized buckling of a floating elastica, Phys. Rev. E 84, 011605 (2011).
  45. H. Diamant and T. A. Witten, Compression induced folding of a sheet: An integrable system, Phys. Rev. Lett. 107, 164302 (2011).
  46. F. Brau, P. Damman, H. Diamant, and T. A. Witten, Wrinkle to fold transition: Influence of the substrate response, Soft Matter 9, 8177 (2013).
  47. H. Diamant and T. A. Witten, Shape and symmetry of a fluid-supported elastic sheet, Phys. Rev. E 88, 012401 (2013).
  48. M. Rivetti, Non-symmetrical localized fold of a floating sheet, Comptes Rendus Mecanique 341, 333 (2013).
  49. M. Rivetti and S. Neukirch, The mode branching route to localization of the finite-length floating elastica, J. Mech. Phys. Solids 69, 143 (2014).
  50. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, New York, 2010); see also http://dlmf.nist.gov.
  51. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover Publications, New York, 1972), Chap. 16.
  52. F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions (Cambridge University Press, Cambridge, 2003).
  53. G. L. Lamb, Jr., Analytical descriptions of ultrashort optical pulse propagation in a resonant medium, Rev. Mod. Phys. 43, 99 (1971).
  54. R. M. DeLeonardis, S. E. Trullinger, and R. F. Wallis, Theory of boundary effects on sine-Gordon solitons, J. Appl. Phys. 51, 1211 (1980).
  55. G. Costabile, R. D. Parmentier, B. Savo, D. W. McLaughlin, and A. C. Scott, Exact solutions of the sine-Gordon equation describing oscillations in a long (but finite) Josephson junction, Appl. Phys. Lett. 32, 587 (1978).
  56. Note that there are two definitions for the modulus of the elliptic functions. In this section, we use the one given in Ref. [50]. The other one, m=k2, is used in Ref. [51] and by mathematica.
  57. The Jacobian function dc yields a solution identical to Eq. (43). The function cn describes a solution whose amplitude decreases with increasing displacement. The remaining nine functions either diverge with increasing L and/or are odd, thus reversing the symmetry of the solution.
  58. We use here the envelope at its lowest mode (no nodes). Higher modes, satisfying κεL=2nK(m) with n>1, are valid solutions but with higher energy.

Outline

Information

Phys. Rev. E 91, 052408– Published 29 May, 2015
  • Received 3 March 2015
  • Revised 12 May 2015

©2015 American Physical Society

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