On end rotation for open rods undergoing large deformations
Authors:
G. H. M. van der Heijden, M. A. Peletier and R. Planqué
Journal:
Quart. Appl. Math. 65 (2007), 385-402
MSC (2000):
Primary 49N99, 51H99, 51N99, 74B20, 74K10
DOI:
https://doi.org/10.1090/S0033-569X-07-01049-X
Published electronically:
April 10, 2007
MathSciNet review:
2330564
Full-text PDF Free Access
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Additional Information
Abstract: We give a careful discussion of end rotation in elastic rods, focusing on ambiguities that arise if arbitrarily large deformations are allowed. By introducing a closure and restricting to a class of deformations we show that a rigorous treatment of end rotation can be obtained. The results underpin various non-rigorous discussions in the literature and serve to promote the variational analysis of boundary-value problems for rods undergoing large deformations. As an example we discuss the application to the model of a rod lying on the surface of a cylinder.
References
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- R. Planqué. Constraints in Applied Mathematics: Rods, Membranes, and Cuckoos. Ph.D. thesis, Technische Universiteit Eindhoven, 2005.
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References
- J. Aldinger, I. Klapper, and M. Tabor. Formulae for the calculation and estimation of writhe. J. Knot Theory and its Ramifications, 4(3):343–372, 1995. MR 1347359 (97d:57003)
- J.C. Alexander and S.S. Antman. The ambiguous twist of Love. Quarterly of Applied Math., 40:83–92, 1982. MR 652052 (84c:73052)
- M. V. Berry. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A, 392:45–57, 1984. MR 738926 (85i:81022)
- G. Călugăreanu. Sur les classes d’isotopie des nœuds tridimensionnels et leurs invariants. Czechoslovak Math. J., 11:588–625, 1961. MR 0149378 (26:6868)
- J. Cantarella. On comparing the writhe of a smooth curve to the writhe of an inscribed polygon. SIAM J. Numer. Anal., 42:1846–1861, 2005. MR 2139226 (2006e:53005)
- F.B. Fuller. The writhing number of a space curve. Proc. Nat. Acad. Sciences USA, 68(4):815–819, 1971. MR 0278197 (43:3928)
- F.B. Fuller. Decomposition of the linking number of a closed ribbon: A problem from molecular biology. Proc. Nat. Acad. Sciences USA, 75:3557–3561, 1978. MR 0490004 (58:9367)
- K. Klenin and J. Langowski. Computation of writhe in modeling of supercoiled DNA. Biopolymers, 54:307–317, 2000.
- A.E.H. Love. A Treatise on the Mathematical Theory of Elasticity. Dover Publications, 4th edition, 1944. MR 0010851 (6:79e)
- A.C. Maggs. Twist and writhe dynamics of stiff polymers. Phys. Rev. Lett., 85:5472–5475, 2000.
- J.F. Marko. Supercoiled and braided DNA under tension. Phys. Rev. E, 55(2):1758–1772, 1997.
- S. Neukirch and G.H.M. van der Heijden. Geometry and mechanics of uniform $n$-plies: from engineering ropes to biological filaments. J. Elasticity, 69:41–72, 2002. MR 2020510 (2004i:74070)
- R. Planqué. Constraints in Applied Mathematics: Rods, Membranes, and Cuckoos. Ph.D. thesis, Technische Universiteit Eindhoven, 2005.
- V. Rossetto and A. C. Maggs. Writhing geometry of open DNA. J. Chem. Phys., 118:9864–9874, 2003.
- E.L. Starostin. On the writhe of non-closed curves. arXiv:physics/0212095, 2002.
- T.R. Strick, J.F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette. The elasticity of a single supercoiled DNA molecule. Science, 271, 1835–1837 1996.
- 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, 457:695–715, 2001. MR 1841581 (2002d:74040)
- G.H.M. van der Heijden, M.A. Peletier, and R. Planqué. Self-contact for rods on cylinders. Arch. Rat. Mech. Anal., 182:471–511, 2006.
- A.V. Vologodskii and J.F. Marko. Extension of torsionally stressed DNA by external force. Biophys. J., 73:123–132, 1997.
- J.H. White. Self-linking and the Gauss integral in higher dimensions. Amer. J. Math., 91:693–728, 1969. MR 0253264 (40:6479)
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Additional Information
G. H. M. van der Heijden
Affiliation:
Centre for Nonlinear Dynamics and its Applications, University College London, Gower Street, London WC1E 6BT
M. A. Peletier
Affiliation:
Technische Universiteit Eindhoven, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven
R. Planqué
Affiliation:
Department of Mathematics, Faculty of Sciences, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam
Email:
rplanque@few.vu.nl
Keywords:
Rod theory,
end rotation,
twist,
writhe,
large deformations.
Received by editor(s):
August 28, 2006
Published electronically:
April 10, 2007
Additional Notes:
This research was partly funded by the EPSRC under contract GR/R51698/01. The first author also thanks The Royal Society for continuing support. The authors are grateful to Jason Cantarella for many interesting discussions, and to Rubber Import Amsterdam BV for supplying us with ample experimental material.
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.