Journal of Elasticity

, Volume 109, Issue 1, pp 75–93 | Cite as

Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-dimensional Study

  • Apala Majumdar
  • Christopher Prior
  • Alain Goriely


The stability of an inextensible unshearable elastic rod with quadratic strain energy density subject to end loads is considered. We study the second variation of the corresponding rod-energy, making a distinction between in-plane and out-of-plane perturbations and isotropic and anisotropic cross-sections, respectively. In all cases, we demonstrate that the naturally straight state is a local energy minimizer in parameter regimes specified by material constants. These stability results are also accompanied by instability results in parameter regimes defined in terms of material constants.


Cosserat rods Stability of straight axis solution Variational criterion 

Mathematics Subject Classification (2000)

74B20 49K40 



This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). AG is a Wolfson Royal Society Merit Holder. AM is also supported by an EPSRC Career Acceleration Fellowship EP/J001686/1. We also would like to thank John Maddocks for comments on an earlier version of this manuscript, and bringing to our attention important references.


  1. 1.
    Antman, S.S., Kenney, C.S.: Large buckled states of nonlinearly elastic rods under torsion, thrust and gravity. Arch. Ration. Mech. Anal. 84, 289–338 (1981) MathSciNetGoogle Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems in Elasticity, 2nd edn. Springer, New York (2005) Google Scholar
  3. 3.
    Ball, J., Marsden, J.E.: Quasiconvexity at the boundary, positivity of the second variation, and elastic stability. Arch. Ration. Mech. Anal. 86, 251–277 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bazant, Z.P., Cedolin, L.: Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories. Dover, New York (2003) Google Scholar
  5. 5.
    Beck, M.: Die knicklast des einseitig eigenspanntan tangential gedrückten. Zeitschrift für angewandte Mathematik und Mechanik (1952) Google Scholar
  6. 6.
    Browne, R.C.: Dynamic stability of one-dimensional nonlinearly viscoelastic bodies. Arch. Ration. Mech. Anal. 68(3), 257–282 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Born, M.: Untersuchungen über die stabilität der elastischen Linie in Ebene und Raum: Unter verschiedenen Grenzbedingungen. Dieterich, Göttingen (1906) Google Scholar
  8. 8.
    Chouaieb, N., Goriely, A., Maddocks, J.H.: Helices. Proc. Natl. Acad. Sci. USA 103, 9398–9403 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Chouaieb, N., Maddocks, J.: Kirchhoff’s problem of helical equilibria of uniform rods. J. Elast. 77, 221–247 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti. apud Marcum-Michaelem Bousquet & socios (1744) Google Scholar
  11. 11.
    Euler, L.: Sur la force des colonnes. Mem. Acad. Berlin 13 (1759) Google Scholar
  12. 12.
    Evans, L.: Partial Differential Equations. American Mathematical Society, Providence (1998) zbMATHGoogle Scholar
  13. 13.
    Fain, B., Rudnick, J., Ostlund, S.: Conformations of linear DNA. Phys. Rev. E 55(6), 7364–7367 (1997) ADSCrossRefGoogle Scholar
  14. 14.
    Gore, J., Bryant, Z., Nöllmann, M., Le, M.U., Cozzarelli, N.R., Bustamante, C.: DNA overwinds when stretched. Nature 442(7104), 836–839 (2006) ADSCrossRefGoogle Scholar
  15. 15.
    Goriely, A.: Twisted elastic rings and the rediscoveries of Michell’s instability. J. Elast. 84(3), 281–299 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Goriely, A., Nizette, M., Tabor, M.: On the dynamics of elastic strips. J. Nonlinear Sci. 11(1), 3–45 (2001) MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Goriely, A., Tabor, M.: Nonlinear dynamics of filaments I: Dynamical instabilities. Physica D 105, 20–44 (1997) MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Greenhill, A.G.: On the strength of shafting when exposed both to torsion and to end thrust. ARCHIVE: Proceedings of the Institution of Mechanical Engineers 1847–1982 (vols. 1–196). 34(1883), 182–225 (1883) Google Scholar
  19. 19.
    Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley, New York (1966) zbMATHGoogle Scholar
  20. 20.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover, New York (1944) zbMATHGoogle Scholar
  21. 21.
    Maddocks, J.H.: Stability of nonlinearly elastic rods. Arch. Ration. Mech. Anal. 85(4), 180–198 (1984) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Manning, R.S., Rogers, K.A., Maddocks, J.H.: Isoperimetric conjugate points with application to the stability of DNA minicircles. Proc. R. Soc. Lond. A, Math. Phys. Eng. Sci. 454, 3047 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Michell, J.H.: On the stability of a bent and twisted wire. Messag. Math. 11 (1889) Google Scholar
  24. 24.
    Nizette, M., Goriely, A.: Toward a classification of Euler-Kirchhoff filaments. J. Math. Phys. 40, 2830–2866 (1999) MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    O’Reilly, O.M.: The dual Euler basis: constraints, potentials, and Lagrange’s equations in rigid body dynamics. J. Appl. Mech. 74, 256 (2007) zbMATHCrossRefGoogle Scholar
  26. 26.
    O’Reilly, O.M., Peters, D.M.: On stability analyses of three classical buckling problems for the elastic strut. J. Elast. 105, 117–136 (2011) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1961) Google Scholar
  28. 28.
    Virga, E.G.: An extended Wirtinger inequality. J. Phys. A, Math. Gen. 34, 1507–1511 (2001) MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Ziegler, H.: Principles of Structural Stability. Blaisdell, Waltham (1968) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Apala Majumdar
    • 1
  • Christopher Prior
    • 1
  • Alain Goriely
    • 1
  1. 1.OCCAM, Mathematical InstituteOxford UniversityOxfordEngland

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