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.
Similar content being viewed by others
References
Antman S.S.: Nonlinear problems of elasticity. Springer-Verlag, 1995
Blom J.G., Peletier M.A. (2004). A continuum model of lipid bilayers. European J. Appl. Math. 15: 487–508
Le Bret M. (1984). Twist and writhing in short circular DNAs according to first-order elasticity. Biopolymers 23: 1835–1867
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
Cantarella J., Kusner R.B., Sullivan J.M. (2002). On the minimum rope length of knots and links. Invent. Math. 150: 257–286
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
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
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
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
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, 1977
Gonzalez O., Maddocks J.H. (1999). Global curvature, thickness and the ideal shape of knots. Proc. Natl. Acad. Sci. USA 96: 4769–4773 1999
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
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
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
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
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
Jülicher F. (1994). Supercoiling transitions of closed DNA. Phys. Rev. E 49(3): 2429–2436
Maddocks J.H. (1987). Stability and folds. Arch. Ration. Mech. Anal. 99: 301–328
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
Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Prentice-Hall, 1967
Schuricht F., von der Mosel F. (2004). Characterization of ideal knots. Calc. Var. partial Differential Equations 19: 281–315
Schuricht F., von der Mosel H. (2003). Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Ration. Mech. Anal. 168: 35–82
Starostin E.L. (2003). A constructive approach to modelling the tight shapes of some linked structures. Forma 18: 263–293
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
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
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
Tobias I., Swigon D., Coleman B.D. (2000). Elastic stability of DNA configurations I General theory. Phys. Rev. E(3) 61: 747–758
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Otto
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-006-0011-y