s. goyal, n.c. perkins and c.l. lee- nonlinear dynamics and loop formation in kirchhoff rods with...

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Nonlinear dynamics and loop formation in Kirchhorods with implications to the mechanics of DNA and cables S. Goyal a , N.C. Perkins a, * , C.L. Lee b a Mechan ical Engineeri ng, 2250 G. G. Brown Laborator ies, Universi ty of Michi gan, 2350 Haywar d Stree t, Ann Arbor, MI 48109-2125, United States b Lawrence Livermore National Laboratory, New Technologies Engineering Division, 7000 East Avenue, Liver more, CA 94550, United States Received 6 August 2004; received in revised form 2 March 2005; accepted 9 March 2005 Available online 23 May 2005 Abstract The paper contributes a general dynamical formulation and numerical solution procedure for studying nonlinear and three-dimension al dynamic s of Kirch horods. Targe t applic ations include the dynami c format ion of DNA loops and supercoils as well as loops (hockles) in marine cables. The formulation accommodates non-homogeneous and non- isotropic inextensible rods both with and without coupling of tension and torsion. The utility of this formulation is illus- trated by studying the dynamics and quasi-static response of a clamped–clamped rod subject to compression and/or twist. For slow loading rates, the computed quasi-static responses converge to published equilibrium solutions for a benchmark problem. As loading rates increase, new behaviors are observed including hysteresis in the neighborhood of equilibrium bifurcations. The addition of chirality in the form of tension–torsion coupling has a pronounced inu- ence on the computed looped geometr ies. This nding has implicati ons for DNA loops forme d by DNA–pr otein bind- ing as well as loop formation in helically wound wire and synthetic cables. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Kirchhorod; Dynamics; Supercoiling; Looping; Cable mechanics; DNA mechanics 1. Introdu ction Marine cables under torsion and low tension may form loops and tangles as shown in Fig. 1. In this con- text, the loops are sometimes referred to as ‘‘hockles’’ and these can lead to mechanical damage or to signal attenuation in ber optic cables [1]. The resulting large deformations are dominated by bending and torsion 0021-9991/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2005.03.027 * Corresponding author. Tel.: +1 734 936 0403; fax: +1 734 615 6647. E-mail address: [email protected] (N.C. Perkins). Journal of Computational Physics 209 (2005) 371–389 www.elsevier.com/locate/jcp

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Page 1: S. Goyal, N.C. Perkins and C.L. Lee- Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables

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