Advertisement

Journal of Elasticity

, Volume 116, Issue 1, pp 75–100 | Cite as

Stability and Bifurcation of a Soap Film Spanning a Flexible Loop

  • Yi-chao Chen
  • Eliot Fried
Article

Abstract

The Euler–Plateau problem, proposed by Giomi and Mahadevan in Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012), concerns a soap film spanning a flexible loop. The shapes of the film and the loop are determined by the interactions between the two components. In the present work, the Euler–Plateau problem is reformulated to yield a boundary-value problem for a vector field that parameterizes both the spanning surface and the bounding loop. Using the first and second variations of the relevant free-energy functional, detailed bifurcation and stability analyses are performed. For a spanning surface with energy density σ and a bounding loop with length 2πR and flexural rigidity a, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at R 3 σ/a=3, confirming a result obtained previously via an energy comparison. All other bifurcation solution branches emanating from the flat circular solution branch, including those to nonplanar solution branches, are found to be unstable.

Keywords

Surface tension Flexural rigidity Inextensibility Euler–Lagrange equations Second variation condition Plateau’s problem Thread problem Closed-curve problem 

Mathematics Subject Classification (2010)

49Q10 53A04 53A05 53A10 53A25 53C80 53Z05 

References

  1. 1.
    Dierkes, U., Hildebrandt, S., Tromba, A.J.: Regularity of Minimal Surfaces, 2nd edn. Springer, Berlin (2010) MATHGoogle Scholar
  2. 2.
    Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces, 2nd edn. Springer, Berlin (2010) CrossRefGoogle Scholar
  3. 3.
    Bernatzki, F., Ye, R.: Minimal surfaces with an elastic boundary. Ann. Glob. Anal. Geom. 19, 1–9 (2001) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Giomi, L., Mahadevan, L.: Minimal surfaces bounded by elastic lines. Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012) ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bernatzki, F.: Mass-minimizing currents with an elastic boundary. Manuscr. Math. 93, 1–20 (1997) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bernatzki, F.: On the existence and regularity of mass-minimizing currents with an elastic boundary. Ann. Glob. Anal. Geom. 15, 379–399 (1997) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Plateau, J.A.F.: Recherches expérimentales et théorique sur les figures d’équilibre d’une masse liquide sans pesanteur. Mém. Acad. R. Sci. Lett. Beaux-Arts Belg. 23, 1–151 (1849) Google Scholar
  8. 8.
    Singer, D.A.: Lectures on elastic curves and rods. In: Garay, O.J., García-Río, E., Vázquez-Lorenzo, R. (eds.) Curvature and Variational Modeling in Physics and Biophysics. Conference Proceedings of the American Institute of Physics, vol. 1002, pp. 3–32 (2008) Google Scholar
  9. 9.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976) MATHGoogle Scholar
  10. 10.
    Efimov, N.V.: Some problems in the theory of space curves. Usp. Mat. Nauk 2, 193–194 (1947) MathSciNetGoogle Scholar
  11. 11.
    Fenchel, W.: On the differential geometry of closed space curves. Bull. Am. Math. Soc. 57, 44–54 (1951) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. I. Springer, Berlin (1985) CrossRefMATHGoogle Scholar
  13. 13.
    Chen, Y.-C.: Singularity theory and nonlinear bifurcation analysis. In: Fu, Y.B., Ogden, R.W. (eds.) Nonlinear Elasticity: Theory and Applications. Cambridge University Press, Cambridge (2001) Google Scholar
  14. 14.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975) MATHMathSciNetGoogle Scholar
  15. 15.
    Julicher, F.: Supercoiling transitions of closed DNA. Phys. Rev. E 49, 2429–2435 (1994) ADSCrossRefGoogle Scholar
  16. 16.
    Dichmann, D.J., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: Mesirov, J.P., Schulten, K., Sumners, D.W. (eds.) Mathematical Approaches to Biomolecular Structure and Dynamics, pp. 71–113. Springer, Berlin (1996) CrossRefGoogle Scholar
  17. 17.
    Coleman, B.D., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elast. 60, 173–221 (2000) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Ericksen, J.L.: The thermo-kinetic view of elastic stability theory. Int. J. Solids Struct. 2, 573–580 (1966) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of HoustonHoustonUSA
  2. 2.Department of Mechanical EngineeringMcGill UniversityMontréalCanada

Personalised recommendations