Skip to main content
Log in

Self-Contact for Rods on Cylinders

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Antman S.S.: Nonlinear problems of elasticity. Springer-Verlag, 1995

  2. Blom J.G., Peletier M.A. (2004). A continuum model of lipid bilayers. European J. Appl. Math. 15: 487–508

    Article  MathSciNet  MATH  Google Scholar 

  3. Le Bret M. (1984). Twist and writhing in short circular DNAs according to first-order elasticity. Biopolymers 23: 1835–1867

    Article  Google Scholar 

  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, 2004

  5. Cantarella J., Kusner R.B., Sullivan J.M. (2002). On the minimum rope length of knots and links. Invent. Math. 150: 257–286

    Article  MathSciNet  MATH  Google Scholar 

  6. Coleman B.D., Swigon D. (2000). Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elasticity 60: 173–221

    Article  MathSciNet  MATH  Google Scholar 

  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): 759–770

    Article  ADS  MathSciNet  Google Scholar 

  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/auto

  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): 285–298

    Article  MATH  Google Scholar 

  10. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, 1977

  11. Gonzalez O., Maddocks J.H. (1999). Global curvature, thickness and the ideal shape of knots. Proc. Natl. Acad. Sci. USA 96: 4769–4773 1999

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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: 29–68

    Article  MathSciNet  MATH  Google Scholar 

  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: 695–715

    ADS  MATH  Google Scholar 

  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: 161–196

    Article  MATH  Google Scholar 

  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, 2005

  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: 201–224

    Article  ADS  MathSciNet  Google Scholar 

  17. Jülicher F. (1994). Supercoiling transitions of closed DNA. Phys. Rev. E 49(3): 2429–2436

    Article  ADS  Google Scholar 

  18. Maddocks J.H. (1987). Stability and folds. Arch. Ration. Mech. Anal. 99: 301–328

    Article  MathSciNet  Google Scholar 

  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: 41–72

    Article  MathSciNet  MATH  Google Scholar 

  20. Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Prentice-Hall, 1967

  21. Schuricht F., von der Mosel F. (2004). Characterization of ideal knots. Calc. Var. partial Differential Equations 19: 281–315

    Article  MathSciNet  Google Scholar 

  22. Schuricht F., von der Mosel H. (2003). Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Ration. Mech. Anal. 168: 35–82

    Article  MathSciNet  MATH  Google Scholar 

  23. Starostin E.L. (2003). A constructive approach to modelling the tight shapes of some linked structures. Forma 18: 263–293

    MathSciNet  Google Scholar 

  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: 1317–1334

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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: 4265–4278

    Article  MATH  Google Scholar 

  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: 959–985

    Article  ADS  MATH  Google Scholar 

  27. Tobias I., Swigon D., Coleman B.D. (2000). Elastic stability of DNA configurations I General theory. Phys. Rev. E(3) 61: 747–758

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Planqué.

Additional information

Communicated by F. Otto

Rights and permissions

Reprints and permissions

About this article

Cite this article

van der Heijden, G.H.M., Peletier, M.A. & Planqué, R. Self-Contact for Rods on Cylinders. Arch Rational Mech Anal 182, 471–511 (2006). https://doi.org/10.1007/s00205-006-0011-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-006-0011-y

Keywords

Navigation