Abstract
We study the mechanics of uniform n-plies, correcting and extending previous work in the literature. An n-ply is the structure formed when n pretwisted strands coil around one another in helical fashion. Such structures are encountered widely in engineering (mooring ropes, power lines) and biology (DNA, proteins). We first show that the well-known lock-up phenomenon for n=2, described by a pitchfork bifurcation, gets unfolded for higher n. Geometrically, n-plies with n>2 are all found to behave qualitatively the same. Next, using elastic rod theory, we consider the mechanics of n-plies, allowing for axial end forces and end moments while ignoring friction. An exact expression for the interstrand pressure force is derived, which is used to investigate the onset of strand separation in plied structures. After defining suitable displacements we also give an alternative variational formulation and derive (nonlinear) constitutive relationships for torsion and extension (including their coupling) of the overall ply. For a realistic loading problem in which the ends are not free to rotate one needs to consider the topological conservation law, and we show how the concepts of link and writhe can be extended to n-plies.
Similar content being viewed by others
References
J.F. Allemand, D. Bensimon, R. Lavery and V. Croquette, Stretched and overwound DNA forms a Pauling-like structure with exposed bases. Proc. National Acad. Sci. USA 95 (1998) 14152–14157.
S.S. Antman, Nonlinear Problems of Elasticity. Springer, New York (1995).
U. Bockelmann, B. Essevaz-Roulet and F. Heslot, DNA strand separation studied by single molecule force measurements. Phys. Rev. E 58 (1998) 2386–2394.
G. Călugăreanu, Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants. Czechoslovak Math. J. 11 (1961) 588–625.
A. Cardou and C. Jolicoeur, Mechanical models of helical strands. ASME Appl. Mech. Rev. 50 (1997) 1–14.
C.R. Chaplin, Torsional failure of a wire rope mooring line during installation in deep water. Eng. Failure Anal. 6 (1998) 67–82.
B.D. Coleman and D. Swigon, Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elasticity 60 (2000) 173–221.
G.A. Costello, Theory of Wire Rope, 2nd edn. Springer, New York (1997).
E.H. Dill, Kirchhoff's theory of rods. Arch. Hist. Exact Sci. 44 (1992) 1–23.
B. Essevaz-Roulet, U. Bockelmann and F. Heslot,Mechanical separation of the complementary strands of DNA. Proc. National Acad. Sci. USA 94 (1997) 11935–11940.
B. Fain, J. Rudnick and S. Östlund, Conformations of linear DNA. Phys. Rev. E 55 (1997) 7364–7368.
W.B. Fraser and D.M. Stump, The equilibrium of the convergence point in two-strand yarn plying. Internat. J. Solids Struct. 35 (1998) 285–298.
W.B. Fraser and D.M. Stump, Twist in balanced-ply structures. J. Text. Inst. 89 (1998) 485–497.
F.B. Fuller, The writhing number of a space curve. Proc. National Acad. Sci. USA 68 (1971) 815–819.
F.B. Fuller, Decomposition of the linking of a closed ribbon: a problem from molecular biology. Proc. National Acad. Sci. USA 75 (1978) 3557–3561.
D.E. Gilbert and J. Feigon, Multistranded DNA structures. Curr. Opin. Struct. Biol. 9 (1999) 305–314.
O. Gonzalez and J.H. Maddocks, Global curvature, thickness, and the ideal shapes of knots. Proc. National Acad. Sci. USA 96 (1999) 4769–4773.
O. Gonzalez, J.H. Maddocks and J. Smutny, Curves, circles, and spheres. Contemporary Math. 304 (2002) 195–215.
J.W.S. Hearle and A.E. Yegin, The snarling of highly twisted monofilaments. Part I: The loadelongation behaviour with normal snarling. J. Text. Inst. 63 (1972) 477–489; Part II: Cylindrical snarling. J. Text. Inst. 63 (1972) 490-501.
W. Jiang, A general formulation of the theory of wire ropes. ASME J. Appl. Mech. 62 (1995) 747–755.
W.G. Jiang, J.L. Henshall and J.M. Walton, A concise finite element model for three-layered straight wire rope strand. Internat. J. Mech. Sci. 42 (2000) 63–86.
C. Kang, X. Zhang, R. Ratliff, R. Moyzis and A. Rich, Crystal structure of four-stranded Oxytricha telomeric DNA. Nature 356 (1992) 126–131.
V. Katritch, W.K. Olson, P. Pierański, J. Dubochet and A. Stasiak, Properties of ideal composite knots. Nature 388 (1997) 148–151.
T. Kunoh and C.M. Leech, Curvature effects on contact position of wire strands. Internat. J. Mech. Sci. 27 (1985) 465–470.
A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge Univ. Press, Cambridge (1927).
A. Maritan, C. Micheletti, A. Trovato and J.R. Banavar, Optimal shapes of compact strings. Nature 406 (2000) 287–290.
M. Moakher and J.H. Maddocks, A double-strand elastic rod theory. Preprint Institut Bernoulli, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland (2002).
D.A.D. Parry and J.M. Squire (eds), Fibrous Proteins (Special issue). J. Struct. Biol. 122 (1998).
M. Peyrard and A.R. Bishop, Statistical mechanics of a nonlinear model for DNA denaturation. Phys. Rev. Lett. 62 (1989) 2755–2758.
P. Pierański, In search of ideal knots. In: [35], pp. 20–41.
S. Przybyl and P. Pierański, Helical close packings of ideal ropes. European Phys. J. E 4 (2001) 445–449.
I. Rouzina and V.A. Bloomfield, Force-induced melting of the DNA double helix 1: Thermodynamic analysis. Biophys. J. 80 (2001) 882–893; 2: Effect of solution conditions. Biophys. J. 80 (2001) 894-900.
A. Sarkar, J.-F. Léger, D. Chatenay and J.F. Marko, Structural transitions in DNA driven by external force and torque. Phys. Rev. E 63 (2001) 051903.
T. Simonsson, G-quadruplex DNA structures - variations on a theme. Biol. Chem. 382 (2001) 621–628.
A. Stasiak, V. Katritch and L.H. Kauffman (eds), Ideal Knots, Series on Knots and Everything, Vol. 19. World Scientific, Singapore (1998).
A. Stasiak and J.H. Maddocks, Best packing in proteins and DNA. Nature 406 (2000) 251–253.
E.L. Starostin, On the writhe of non-closed curves. Preprint, Institut Bernoulli, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland (2002); available at arXiv: physics/0212095.
T.R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon and V. Croquette, The elasticity of a single supercoiled DNA molecule. Science 271 (1996) 1835–1837.
D.M. Stump, W.B. Fraser and K.E. Gates, The writhing of circular cross-section rods: undersea cables to DNA supercoils. Proc. Roy. Soc. London A 454 (1998) 2123–2156.
J.M.T. Thompson, G.H.M. van der Heijden and S. Neukirch, Super-coiling of DNA plasmids: The mechanics of the generalised ply. Proc. Roy. Soc. London A 458 (2002) 959–985.
L.R.G. Treloar, The geometry of multy-ply yarns. J. Text. Inst. 47 (1956) T348–T368.
P.N.T. Unwin and P.D. Ennis, Two configurations of a channel-forming membrane protein. Nature 307 (1984) 609–613.
G.H.M. van der Heijden, The static deformation of a twisted elastic rod constrained to lie on a cylinder. Proc. Roy. Soc. London A 457 (2001) 695–715.
G.H.M. van der Heijden and J.M.T. Thompson, Helical and localised buckling in twisted rods: a uniform analysis of the symmetric case. Nonlinear Dynamics 21 (2000) 71–99.
G.H.M. van der Heijden, J.M.T. Thompson and S. Neukirch, A variational approach to loaded ply structures. J. Vibration Control 9 (2003) 175–185.
K.M. Vasquez, L. Narayanan and P.M. Glazer, Specific mutations induced by triplex-forming oligonucleotides in mice. Science 290 (2000) 530–533.
J.H. White, Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91 (1969) 693–728.
M. Yeager and B.J. Nicholson, Structure of gap junction intercellular channels. Curr. Opin. Struct. Biol. 6 (1996) 183–192.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Neukirch, S., van der Heijden, G. Geometry and Mechanics of Uniform n-Plies: from Engineering Ropes to Biological Filaments. Journal of Elasticity 69, 41–72 (2002). https://doi.org/10.1023/A:1027390700610
Issue Date:
DOI: https://doi.org/10.1023/A:1027390700610