- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Collapse and folding of pressurized rings in two dimensions

Eleni Katifori1,2,*, Silas Alben3,4, and David R. Nelson1,†

  • 1Department of Physics, Harvard University, Cambridge, Massachussetts 02138, USA
  • 2Center for Studies in Physics and Biology, Rockefeller University, New York, New York 10065, USA
  • 3SEAS, Harvard University, Cambridge, Massachussetts 02138, USA
  • 4School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

  • *ekatifori@mail.rockefeller.edu
  • †nelson@physics.harvard.edu
PDF

Phys. Rev. E 79, 056604 – Published 8 May, 2009

DOI: https://doi.org/10.1103/PhysRevE.79.056604

Abstract

Hydrostatically pressurized circular rings confined to two dimensions (or cylinders constrained to have only z-independent deformations) undergo Euler-type buckling when the outside pressure exceeds a critical value. We perform a stability analysis of rings with arclength-dependent bending moduli and determine how weakened bending modulus segments affect the buckling critical pressure. Rings with a fourfold symmetric modulation are particularly susceptible to collapse. In addition we study the initial postbuckling stages of the pressurized rings to determine possible ring folding patterns.

View figure in article

Article Text

References (34)

  1. M. Levy, J. Math. Pures Appl. 10, 5 (1884).
  2. G. F. Carrier, J. Math. Phys. 26, 94 (1947).
  3. I. Tadjbakhsh and F. Odeh, J. Math. Anal. Appl. 18, 59 (1967).
  4. J. Chaskalovic and S. Naili, Z. Angew. Math. Phys. 46, 149 (1995).
  5. T. M. Atanackovic, Acta Mech. 127, 121 (1998).
  6. C. Y. Wang, J. Eng. Mech. 111, 1423 (1985).
  7. C. Y. Wang, Z. Angew. Math. Mech. 73, 301 (1993).
  8. J. Zang, A. Treibergs, Y. Han, and F. Liu, Phys. Rev. Lett. 92, 105501 (2004).
  9. T.-D. Park and S. Kyriakides, Int. J. Mech. Sci. 38, 557 (1996).
  10. R. Lo Frano and G. Forasassi, Sci. Technol. Nucl. Instal. 2008, 685805 (2008).
  11. O. Aksogan, and A. H. Sofiyev, J. Sound Vib. 254, 693 (2002).
  12. J. Joseph-Silverstein and W. Cohen, J. Cell Biol. 98, 2118 (1984).
  13. R. Waugh and G. Erwin, J. Cell Biol. 108, 1711 (1989).
  14. J. Italiano, E. Joseph, W. Bergmeier, S. Tiwari, H. Falet, J. H. Hartwig, K. M. Hoffmeister, P. Andre, D. D. Wagner, and R. A. Shivdasani, Blood 101, 4789 (2003).
  15. W. D. Cohen, Y. Sorokina, and I. Sanchez, Cell Motil. Cytoskeleton 40, 238 (1998).
  16. S. Leibler, in Statistical Mechanics of Membranes and Surfaces, edited by D. R. Nelson, T. Piran, and S. Weinberg (World Scientific, Singapore, 1989).
  17. S. Leibler, R. R. P. Singh, and M. E. Fisher, Phys. Rev. Lett. 59, 1989 (1987).
  18. E. Levinson, Phys. Rev. A 45, 3629 (1992).
  19. A. H. Romero, J. Phys. I 2, 15 (1992).
  20. U. M. B. Marconi and A. Maritan, Phys. Rev. E 47, 3795 (1993).
  21. G. Gaspari, J. Rudnick, and A. Beldjenna, J. Phys. A 26, 1 (1993).
  22. E. Haleva and H. Diamant, Eur. Phys. J. E 19, 461 (2006).
  23. M. K. Mitra, G. I. Menon, and R. Rajesh, Phys. Rev. E 77, 041802 (2008).
  24. R. Morikawa and Y. Saito, J. Phys. Soc. Jpn. 64, 3562 (1995).
  25. The thermal behavior of inhomogeneously elastic rings will be discussed in a future paper [E. Katifori and D. R. Nelson (unpublished)].
  26. For a treatment of the buckling transition of extensible rings the interested reader can refer to [5].
  27. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, New York, 1970).
  28. Consider, for example, a rod of thickness h and length L with compressive forces applied at the ends, sufficient to produce a strain of order unity. If Y is the three-dimensional Young’s modulus of the rod, the energy of compression is of order Yh2L. However, if the rod buckles out of the axis of compression (thus trading compression energy for bending energy), the energy is now of order Yh4/L. Bending deformations will be preferred whenever Eh4/L⪡Eh2L, i.e., (h/L)2⪡1. More generally, we can consider rings or rods inextensional whenever their perimeter or length greatly exceed their thickness.
  29. A. E. H. Love, Treatise on the Mathematical Theory of Elasticity (Dover, New York, 2002).
  30. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).
  31. S. Panyukov and Y. Rabin, Phys. Rev. E 64, 011909 (2001).
  32. Note that shift in area is always negative, confirming that a perfect circle is indeed the maximum area for a given perimeter.
  33. As discussed in Refs. [2,3], as the pressure keeps increasing above the critical pressure, secondary local minima (of energy higher than the ellipsoidal mode global minimum) appear in the energy landscape. The ring configurations that correspond to these minima have rotational symmetry with period higher than the n=2 ellipsoidal deformation.
  34. The energy E[δθ(s)] can be analytically evaluated to infinite order in δθ2 modes: E[δθ2]=π(8κ0|δθ2|2+psin(4|δθ2|)4|δθ2|).

Outline

Information

Phys. Rev. E 79, 056604– Published 8 May, 2009
  • Received 12 December 2008

© 2009 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation