Self-Contact for Rods on Cylinders

  • G. H. M. van der Heijden
  • M. A. Peletier
  • R. Planqué
Article

Abstract

We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality.

Using techniques from ordinary differential equation theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods.

References

  1. 1.
    Antman S.S.: Nonlinear problems of elasticity. Springer-Verlag, 1995Google Scholar
  2. 2.
    Blom J.G., Peletier M.A. (2004). A continuum model of lipid bilayers. European J. Appl. Math. 15: 487–508CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Le Bret M. (1984). Twist and writhing in short circular DNAs according to first-order elasticity. Biopolymers 23: 1835–1867CrossRefGoogle Scholar
  4. 4.
    Cantarella, J., Fu, J.H.G., Kusner, R., Sullivan, J.M., Wrinkle ,N.C.: Criticality for the Gehring link problem. arXiv: math.DG/0402212, 2004Google Scholar
  5. 5.
    Cantarella J., Kusner R.B., Sullivan J.M. (2002). On the minimum rope length of knots and links. Invent. Math. 150: 257–286CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Coleman B.D., Swigon D. (2000). Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elasticity 60: 173–221CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Coleman B.D., Swigon D., Tobias I. (2000). Elastic stability of DNA configurations II. Supercoiled plasmids with self-contact. Phys. Rev. E 61(3): 759–770CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Sandstede, B., Wang, X.: Auto97: Continuation and bifurcation software for ordinary differential equations; available by ftp from ftp.cs.concordia.ca in directory pub/doedel/autoGoogle Scholar
  9. 9.
    Fraser W.B., Stump D.M. (1998). The equilibrium of the convergence point in two-strand yarn plying. Internat. J. Solids Structures 35(3–4): 285–298CrossRefMATHGoogle Scholar
  10. 10.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, 1977Google Scholar
  11. 11.
    Gonzalez O., Maddocks J.H. (1999). Global curvature, thickness and the ideal shape of knots. Proc. Natl. Acad. Sci. USA 96: 4769–4773 1999CrossRefADSMathSciNetMATHGoogle Scholar
  12. 12.
    Gonzalez O., Maddocks J.H., Schuricht F., von der Mosel H. (2002). Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differential Equations 14: 29–68CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    van der Heijden G.H.M. (2001). The static deformation of a twisted elastic rod constrained to lie on a cylinder. Proc. Soc. Lond sec. A math. phys. Eng. Sci 457: 695–715ADSMATHGoogle Scholar
  14. 14.
    van der Heijden G.H.M., Neukirch S., Goss V.G.A., Thompson J.M.T. (2003). Instability and self-contact phenomena in the writhing of clamped rods. Int. J. Mech. Sci. 45: 161–196CrossRefMATHGoogle Scholar
  15. 15.
    van der Heijden, G.H.M., Peletier, M.A., Planqué, R.: On end rotations for open rods undergoing large deformations. submitted to Arch. Ration. Mech. Anal. arXiv: math-ph/0310057, 2005Google Scholar
  16. 16.
    van der Heijden G.H.M. and Thompson J.M.T. (1998). Lock-on to tape-like behaviour in the torsional buckling of anisotropic rods. Phys D 112: 201–224CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Jülicher F. (1994). Supercoiling transitions of closed DNA. Phys. Rev. E 49(3): 2429–2436CrossRefADSGoogle Scholar
  18. 18.
    Maddocks J.H. (1987). Stability and folds. Arch. Ration. Mech. Anal. 99: 301–328CrossRefMathSciNetGoogle Scholar
  19. 19.
    Neukirch S., van der Heijden G.H.M. (2002). Geometry and mechanics of uniform n-plies: from engineering ropes to biological filaments. J. Elasticity 69: 41–72CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Prentice-Hall, 1967Google Scholar
  21. 21.
    Schuricht F., von der Mosel F. (2004). Characterization of ideal knots. Calc. Var. partial Differential Equations 19: 281–315CrossRefMathSciNetGoogle Scholar
  22. 22.
    Schuricht F., von der Mosel H. (2003). Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Ration. Mech. Anal. 168: 35–82CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Starostin E.L. (2003). A constructive approach to modelling the tight shapes of some linked structures. Forma 18: 263–293MathSciNetGoogle Scholar
  24. 24.
    Starostin E.L. (2004). Symmetric equilibria of a thin elastic rod with self-contacts. Phil. Trans. R. Soc. Lond. Philos. Ser. A Math. Phys. Eng. Sci. 362: 1317–1334CrossRefADSMathSciNetMATHGoogle Scholar
  25. 25.
    Stump D.M., van der Heijden G.H.M. (2001). Birdcaging and the collapse of rods and cables in fixed-grip compression. Internat. J. Solids and Structures 38: 4265–4278CrossRefMATHGoogle Scholar
  26. 26.
    Thompson J.M.T., van der Heijden G.H.M., Neukirch S. (2002). Supercoiling of DNA plasmids: mechanics of the generalized ply. Proc. R. Soc. Lond Ser. A Math. Phys. Eng. Sci. 458: 959–985ADSCrossRefMATHGoogle Scholar
  27. 27.
    Tobias I., Swigon D., Coleman B.D. (2000). Elastic stability of DNA configurations I General theory. Phys. Rev. E(3) 61: 747–758CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • G. H. M. van der Heijden
    • 1
  • M. A. Peletier
    • 2
  • R. Planqué
    • 3
  1. 1.Centre for Nonlinear DynamicsUniversity College LondonLondonEngland
  2. 2.Dept. of Mathematics and Computing ScienceTechnical University EindhovenEindhovenThe Netherlands
  3. 3.Department of Computer ScienceUniversity of BristolBristolEngland

Personalised recommendations