Advertisement

Journal of Elasticity

, Volume 68, Issue 1–3, pp 95–121 | Cite as

Classification of the Spatial Equilibria of the Clamped Elastica: Symmetries and Zoology of Solutions

  • Sébastien Neukirch
  • Michael E. Henderson
Article

Abstract

We investigate the configurations of twisted elastic rods under applied end loads and clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. We show that all solutions of the clamped boundary value problem exhibit a π-flip symmetry. The Kirchhoff equations which describe the equilibria of these rods are integrated in a formal way which enable us to describe the boundary conditions in terms of 2 closed form equations involving 4 free parameters. We show that the flip symmetry property is equivalent to a reversibility property of the solutions of the Kirchhoff differential equations. We sort these solutions according to their period in the phase plane. We show how planar untwisted configurations as well as circularly closed configurations play an important role in the classification.

classification of boundary value problem solutions for elastic rods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S.S. Antman, Nonlinear Problems of Elasticity. Springer, New York (1995).MATHGoogle Scholar
  2. 2.
    A.R. Champneys and J.M.T. Thompson, A multiplicity of localised buckling modes for twisted rod equations. Proc. Roy. Soc. London A 452 (1996) 2467–2491.MATHMathSciNetADSCrossRefGoogle Scholar
  3. 3.
    B.D. Coleman and D. Swigon, Theory of supercioiled elastic rings with self contact and its application to DNA plasmids. J. Elasticity 60 (2000) 173–221.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    R.L. Devaney, Reversible diffeomorphisms and flows. Trans. Amer. Math. Soc. 218 (1976) 89–113.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    D.J. Dichmann, Y. Li and J.H. Maddocks, Hamiltonian formulations and symmetries in rod mechanics. In: J.P. Mesirov, K. Schulten and D.W. Sumners (eds), Mathematical Approaches to Biomolecular Structure and Dynamics, The IMA Volumes in Mathematics and Its Applications, Vol. 82 (1996) pp. 71–113.Google Scholar
  6. 6.
    G. Domokos, Global description of elastic bars. Z. angew. Math. Mech. 74 (1994) T289–T291.Google Scholar
  7. 7.
    G. Domokos, A group-theoretic approach to the geometry of elastic rings. J. Nonlinear Sci. 5 (1995) 453–478.MATHMathSciNetCrossRefADSGoogle Scholar
  8. 8.
    G. Domokos and T. Healey, Hidden symmetry of global solutions in twisted elastic rings. J. Nonlinear Sci. 11 (2001) 47–67.MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    L. Euler, Methodus inveniendi lineas curvas maximi minimivi propreitate gaudentes. Opera Omnia I 24(1744) 231–297, Füssli, Zürich (1960).Google Scholar
  10. 10.
    M.E. Henderson and S. Neukirch, Classification of the spatial clamped elastica, Numerical continuation of the solution set. Int. J. Bif. Chaos (2004) in press.Google Scholar
  11. 11.
    K.A. Hoffman, R.S. Manning and R.C. Paffenroth, Calculation of the stability index in parameter-dependent calculus of variations problems: Buckling of a twisted elastic strut. SIAM J. Appl. Dyn. Systems 1(1) (2002) 115–145.MATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    A.A. Ilyukhin, Spatial Problems of Nonlinear Theory of Elastic Rods. Naukova Dumka, Kiev (1979) (in Russian).Google Scholar
  13. 13.
    S. Kehrbaum, Hamiltonian formulations of the equilibrium conditions governing elastic rods: Qualitative analysis and effective properties. PhD Thesis, University ofMaryland, College Park (1997).Google Scholar
  14. 14.
    S. Kehrbaum and J.H. Maddocks, Elastic rods, rigid bodies, quaternions and the last quadrature. Phil. Trans. Roy. Soc. London A 355 (1997) 2117–2136.MATHMathSciNetADSGoogle Scholar
  15. 15.
    L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Course of Theoretical Physics, Vol. 7, 3 edn. Pergamon Press, Oxford (1986).Google Scholar
  16. 16.
    J. Langer and D.A. Singer, Lagrangian aspects of the Kirchhoff elastic rod. SIAM Rev. 38(4) (1996) 605–618.MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Y. Li and J.H. Maddocks, On the computation of equilibria of elastic rods, part I: Integrals, symmetry and a Hamiltonian formulation. J. Comput. Phys. Preprint (1994).Google Scholar
  18. 18.
    A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover, New York (1944).MATHGoogle Scholar
  19. 19.
    J.H. Maddocks, Stability of nonlinearly elastic rods. Arch. Rational Mech. Anal. 85(4) (1984) 180–198.MathSciNetCrossRefGoogle Scholar
  20. 20.
    S. Neukirch, G.H.M van der Heijden and J.M.T. Thompson, Writhing instabilities of twisted rods: From infinite to finite length. J. Mech. Phys. Solids 50 (2002) 1175–1191.MATHMathSciNetCrossRefADSGoogle Scholar
  21. 21.
    M. Nizette and A. Goriely, Toward a classification of Euler-Kirchhoff filaments. J. Math. Phys. 40 (1999) 2830–2866.MATHMathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Y. Shi and J.E. Hearst, The Kirchhoff elastic rod, the nonlinear Schrödinger equation and DNA supercoiling. J. Chem. Phys. 101 (1994) 5186–5200.CrossRefADSGoogle Scholar
  23. 23.
    E.L. Starostin, Three-dimensional shapes of looped DNA. Meccanica 31 (1996) 235–271.MATHCrossRefGoogle Scholar
  24. 24.
    D. Swigon, Configurations with self contact in the theory of the elastic rod model for DNA. PhD Thesis, Rutgers State University of New Jersey, U.S.A. (1999).Google Scholar
  25. 25.
    H. Tsuru, Equilibrium shapes and vibrations of thin elastic rod. Journal of the Physical Society of Japan 56(7) (1987) 2309–2324.MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    G.H.M. van der Heijden and J.M.T. Thompson, Helical and localised buckling in twisted rods: A unified analysis of the symmetric case. Nonlinear Dynamics 21 (2000) 71–79.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Sébastien Neukirch
    • 1
  • Michael E. Henderson
    • 2
  1. 1.Bernoulli Institute, School of Basic Sciences, École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.T.J. Watson Research Center I.B.M.Yorktown HeightsU.S.A.

Personalised recommendations