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Journal of Elasticity

, Volume 84, Issue 3, pp 281–299 | Cite as

Twisted Elastic Rings and the Rediscoveries of Michell's Instability

  • Alain Goriely
Original Paper

Abstract

Elastic rings become unstable when sufficiently twisted. This fundamental instability plays an important role in the modeling of DNA mechanics and in cable engineering. In 1962, Zajac computed the value of the critical twist for the instability. This critical value was rediscovered in 1979 by Benham and independently by Le Bret in elastic models for DNA; unstable rings have since become an important example of elastic instabilities in rods both for the development of new methods and in applications. The purpose of this note is to show that the problem had been completely solved by John Henry Michell in 1889 in a rather elegant manner and to reflect on its history and modern developments.

Key words

Kirchhoff rods elastic rings twist instability 

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References

  1. 1.
    Michell, J.H.: On the stability of a bent and twisted wire. Messenger of Math. 11, 181–184, (1889–1990)Google Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, Berlin Heidelberg New York (1995)MATHGoogle Scholar
  3. 3.
    Benham, C.J.: An elastic model of the large structure of duplex DNA. Bioploymers 18, 609–623 (1979)CrossRefGoogle Scholar
  4. 4.
    Benham, C.J.: Geometry and mechanics of DNA superhelicity. Biopolymers 22, 2477–2495 (1983)CrossRefGoogle Scholar
  5. 5.
    Vologodskii, A.: Topology and Physics of Circular DNA. CRC Press, Boca Raton (1992)Google Scholar
  6. 6.
    Manning, R.S., Maddocks, J.H., Kahn, J.D.: A continuum rod model of sequence-dependent DNA structure. J. Chem. Phys. 105, 5626–5646 (1996)CrossRefADSGoogle Scholar
  7. 7.
    Silk, W.K.: On the curving and twining of stems. Environmental Exp. Bot. 29, 95–109 (1989)CrossRefGoogle Scholar
  8. 8.
    Goriely, A., Tabor, M.: Spontaneous helix-hand reversal and tendril perversion in climbing plants. Phys. Rev. Lett. 80, 1564–1567 (1998)CrossRefADSGoogle Scholar
  9. 9.
    Goldstein, R.E., Goriely, A., Hubber, G., Wolgemuth, C.: Bistable helices. Phys. Rev. Lett. 84, 1631–1634 (2000)CrossRefADSGoogle Scholar
  10. 10.
    Maddocks, J.H.: Bifurcation theory, symmetry breaking and homogenization in continuum mechanics descriptions of DNA. In: Givoli, M.J., Grote, D., Papanicolaou, G. (eds.) A Celebration of Mathematical Modeling: The Joseph B. Keller Anniversary Volume, pp. 113–136. Kluwer (2004)Google Scholar
  11. 11.
    Thomson, W.T., Tait, P.G.: Treatise on Natural Philosophy. Cambridge (1867)Google Scholar
  12. 12.
    Michell, J.H.: The small deformation of curves and surfaces with applications to the vibration of a helix and a circular ring. Messenger of Math. 19, 68–82 (1889–90)Google Scholar
  13. 13.
    Cherry, T.M.: J.H. Michell. Australian dictionary of biography 19, 494–495 (1986) (http://gutenberg.net.au/dictbiog/0-dict-biogMa-Mo.html)
  14. 14.
    Tuck, E.O.: The wave resistance formula of J.H. Michell (1898) and its significance to recent research in ship hydrodynamics. J. Austral. Math. Soc. Ser. B 30, 365–377 (1989)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Michell, A.G.M.: John Henry Michell Obituary Notices of Fellows of the Royal Society 3, 363–382 (1941)Google Scholar
  16. 16.
    Michell, J.H., Michell, A.G.M., Niedenfuhr, F.W., Radok, J.R.M.: The Collected Mathematical Works of J.H. and A.G.M. Michell. Noordhoff, Groningen, Netherlands (1964)Google Scholar
  17. 17.
    Basset, A.B.: On the deformation of thin elastic wires. Amer. J. Math. 17, 281–317 (1895)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1892)MATHGoogle Scholar
  19. 19.
    Antman, S.S., Kenney, C.S.: Large buckled states of nonlinearly elastic rods under torsion, thrust and gravity. Arch. Ration. Mech. Analysis 84, 289–338 (1981)ADSMathSciNetGoogle Scholar
  20. 20.
    Zajac, E.E.: Stability of two planar loop elasticas. ASME J. Applied Mech. 29, 136–142 (March 1962)MATHMathSciNetGoogle Scholar
  21. 21.
    Fuller, F.B.: The writhing number of a space curve. Proc. Nat. Acad. Sci. 68, 815–819 (1971)MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Fuller, F.B.: Decomposition of the linking number of a closed ribbon: A problem from molecular biology. Proc. Natl. Acad. Sci. 78, 3557–3561 (1978)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Benham, C.J.: Elastic model of supercoiling. Proc. Nat. Acad. Sci. USA 74, 2397–2401 (1977)CrossRefADSGoogle Scholar
  24. 24.
    LeBret, M.: Catastrophic variations of twist and writhing of circular DNA with constraint? Biopolymers 18, 1709–1725 (1979)CrossRefGoogle Scholar
  25. 25.
    LeBret, M.: Twist and writhing in short circular DNA according to first-order elasticity. Biopolymers 23, 1835–1867 (1984)CrossRefGoogle Scholar
  26. 26.
    Benham, C.J.: Onset of writhing in circular elastic polymers. Phys. Rev. A 39, 2582–2586 (1989)CrossRefADSGoogle Scholar
  27. 27.
    Coleman, B.D., Dill, E.H., Lembo, M., Lu, Z., Tobias, I.: On the dynamics of rods in the theory of Kirchhoff and Clebsch. Arch. Rational Mech. Anal. 121, 339–359 (1993)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Tobias, I., Olson, W.K.: The effect of intrinsic curvature on supercoiling – Predictions of elasticity theory. Biopolymers 33, 639–646 (1993)CrossRefGoogle Scholar
  29. 29.
    Yang, Y., Tobias, I., Olson, W.K.: Finite element analysis of DNA supercoiling. J. Chem. Phys. 98, 1673–1686 (1993)CrossRefADSGoogle Scholar
  30. 30.
    Klapper, I., Tabor, M.: A new twist in the kinematics and elastic dynamics of thin filaments and ribbons. J. Phys. A 27, 4919–4924 (1994)MATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Schlick, T.: Modeling superhelical DNA: Recent analytical and dynamical approaches. Curr. Opin. Struct. Biol. 5, 245–262 (1995)CrossRefGoogle Scholar
  32. 32.
    Aldinger, J., Klapper, I., Tabor, M.: Formulae for the calculation and estimation of writhe.J. Knot Theory 4, 243–372 (1995)MathSciNetGoogle Scholar
  33. 33.
    Goriely, A., Tabor, M.: Nonlinear dynamics of filaments I: Dynamical instabilities. Physica D 105, 20–44 (1997)MATHCrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Liu, G.H., Schlick, T., Olson, A.J., Olson, W.K.: Configurational transitions in Fourier series-represented DNA supercoils. Biophys. J. 73, 1742–1762 (1997)Google Scholar
  35. 35.
    Westcott, T.P., Tobias, I., Olson, W.K.: Modeling self-contact forces in the elastic theory of DNA supercoiling. J. Chem. Phys. 107, 3967–3980 (1997)CrossRefADSGoogle Scholar
  36. 36.
    Wiggins, C.H.: Biopolymer mechanics: Stability, dynamics, and statistics. Math. Methods Appl. Sci. 24, 1325–1335 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  37. 37.
    Goriely, A., Tabor, M.: Nonlinear dynamics of filaments. II. Nonlinear analysis. Physica D 105, 45–61 (1997)MATHCrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Goriely, A., Tabor, M.: Nonlinear dynamics of filaments. III. Instabilities of helical rods. Proc. Roy. Soc. London (A) 453, 2583–2601 (1997)MATHADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Goriely, A., Tabor, M.: Nonlinear dynamics of filaments. IV. The spontaneous looping of twisted elastic rods. Proc. Roy. Soc. London (A) 455, 3183–3202 (1998)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Manning, R.S., Hoffman, K.A.: Stability of \(n\)-covered circles for elastic rods with constant planar intrinsic curvature. J. Elasticity 62, 456–479 (2001)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Hoffman, K.A., Manning, R.S., Maddocks, J.H.: Link, twist, energy, and the stability of DNA minicircles. Biopolymers 70, 145–157 (2003)CrossRefGoogle Scholar
  42. 42.
    Lembo, M.: On the stability of elastic annular rods. Int. J. Solids. Structures 40, 317–330 (2003)MATHCrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Hoffman, K.A.: Methods for determining stability in continuum elastic rod-models of DNA. Phil. Trans. R. Soc. Lond. A 362, 1301–1315 (2004)MATHCrossRefADSGoogle Scholar
  44. 44.
    Ivey, T.A., Singer, D.A.: Knot types, homotopies and stability of closed elastic rods. Proc. London Math. Soc. 3, 429–450 (1999)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Tobias, I., Coleman, B.D., Lembo, M.: A class of exact dynamical solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids. J. Chem. Phys. 105, 2517–2526 (1996)CrossRefADSGoogle Scholar
  46. 46.
    Qian, H., White, J.H.: Terminal twist induced continuous writhe of a circular rod with intrinsic curvature. J. Biomol. Struct. Dyn. 16, 663–669 (1998)Google Scholar
  47. 47.
    Haijun, Z., Zhong can, O.-Y.: Spontaneous curvature-induced dynamical instability of Kirchhoff filaments: Application to DNA kink deformations. J. Chem. Phys. 110, 1247–1251 (1999)CrossRefADSGoogle Scholar
  48. 48.
    Shipman, P., Goriely, A.: On the dynamics of helical strips. Phys. Rev. E 61, 4508–4517 (2000)CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Han, W., Lindsay, S.M., Dlakic, M., Harrington, R.E.: Kinked DNA. Nature 386, 563 (1997)CrossRefADSGoogle Scholar
  50. 50.
    Goriely, A., Nizette, M., Tabor, M.: On the dynamics of elastic strips. J. Nonlinear Sci. 11, 3–45 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Chouaieb, N., Maddocks, J.H.: Kirchhoff's problem of helical equilibria of uniform rods. J. Elast. 77, 221–247 (2005)CrossRefMathSciNetGoogle Scholar
  52. 52.
    Goriely, A., Tabor, M.: Nonlinear dynamics of filaments. Nonlinear Dyn. 21, 101–133 (2000)MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Tobias, I., Swigon, D., Coleman, B.D.: Elastic stability of DNA configurations. I. General theory. Phys. Rev. E 61, 747–758 (2000)CrossRefADSMathSciNetGoogle Scholar
  54. 54.
    Coleman, B.D., Swigon, S., Tobias, I.: Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact. Phys. Rev. E 61, 759–770 (2000)CrossRefADSMathSciNetGoogle Scholar
  55. 55.
    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)MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Starostin, E.L.: Equilibrium configurations of a thin elastic rod with self-contacts. PAMM, Proc. Appl. Math. Mech. 1, 137–138 (2002)CrossRefMATHGoogle Scholar
  57. 57.
    Wolgemuth, C.W., Goldstein, R.E., Powers, T.R.: Dynamic supercoiling bifurcation of growing elastic filaments. Phys. D 190, 266–289 (2004)MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Domokos, G., Healey, T.: Hidden symmetry of global solutions in twisted elastic rings. J. Nonlinear Sci. 11, 47–67 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  59. 59.
    Thompson, J.M.T., van der Heijden, G.H.M., Neukirch, S.: Super-coiling of DNA plasmids: Mechanics of the generalized ply. Proc. Roy. Soc. Lond. A 458, 959–985 (2001)Google Scholar
  60. 60.
    Panyukov, S., Rabin, Y.: Fluctuating elastic rings: Statics and dynamics. Phys. Rev. E. 64:#0011909 (2001)CrossRefADSGoogle Scholar
  61. 61.
    Tobias, I.: A theory of thermal fluctuations in DNA miniplasmids. Biophysical J. 74, 2545–2553 (1998)ADSGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Program in Applied Mathematics and Department of MathematicsUniversity of ArizonaTuczonUSA

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