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Archive for Rational Mechanics and Analysis

, Volume 85, Issue 4, pp 311–354 | Cite as

Stability of nonlinearly elastic rods

  • J. H. Maddocks
Article

Abstract

This paper describes previously unknown stabilities and instabilities of planar equilibrium configurations of a nonlinearly elastic rod that is buckled under the action of a dead-load. The governing equations are derived from variational principles, including ones of isoperimetric type. Properties of stability are accordingly determined by study of the second variation. Stabilities to deformations both in the plane and out of the plane are considered.

Among the newly discovered properties are: secondary bifurcation from the first buckled mode, marked differences between stability to two-dimensional and to three-dimensional variations, and the stabilizing influence of resistance to twist. In the isoperimetric examples, the analysis makes crucial use of a novel device to account for the dependence of the second variation on constraints.

Keywords

Neural Network Complex System Governing Equation Nonlinear Dynamics Variational Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. J. C. Alexander & S. S. Antman (1982), The ambiguous twist of Love, Q. Appl. Math., XL, p. 83.Google Scholar
  2. S. S. Antman & E. R. Carbone (1977), Shear and Necking Instabilities in Nonlinear Elasticity, J. of Elasticity, 7, pp. 125–151.Google Scholar
  3. S. S. Antman & C. S. Kenney (1981), Large, buckled states of nonlinearly elastic rods under torsion, thrust, and gravity, Arch. Rational Mech. and Anal., 76, pp. 289–338.Google Scholar
  4. S. S. Antman & G. Rosenfeld (1978), Global behavior of buckled states of nonlinearly elastic rods, SIAM Review 20, p. 513, Corrigenda (1980), 22, p. 186.Google Scholar
  5. M. Born (1906), Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen, Dieterichsche Univ.-Buchdruckerei, Göttingen.Google Scholar
  6. R. C. Browne (1979), Dynamic Stability of One-dimensional Viscoelastic Bodies, Arch. Rational Mech. Anal., 68, pp. 257–282.Google Scholar
  7. R. E. Caflisch & J. H. Maddocks (1984), Nonlinear dynamical theory of the elastica, submitted to Proc. Roy. Soc. Edinburgh A.Google Scholar
  8. C. V. Coffman (1976), The nonhomogeneous classical elastica, Technical Report, Department of Mathematics, Carnegie-Mellon University.Google Scholar
  9. L. Euler (1744), Methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes, Opera Omnia I, Vol. 24, Füssli, Zurich 1960, pp. 231–297.Google Scholar
  10. R. L. Fosdick & R. D. James (1981), The elastica and the problem of the pure bending for a non-convex stored energy function, J. Elasticity 11, pp. 165–186.Google Scholar
  11. I. M. Gelfand & S. V. Fomin (1963), Calculus of Variations, Prentice-Hall, Inc., New Jersey.Google Scholar
  12. H. Goldstein (1980), Classical Mechanics, Addison-Wesley, Reading, Mass., Second Edition.Google Scholar
  13. J. Gregory (1980), Quadratic Form Theory and Differential Equations, Academic Press, New York.Google Scholar
  14. M. R. Hestenes (1951), Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific Journal of Mathematics, 1, pp. 525–581.Google Scholar
  15. M. R. Hestenes (1966), Calculus of Variations and Optimal Control Theory, John Wiley, New York.Google Scholar
  16. E. L. Ince (1927), Ordinary Differential Equations, Longmans, Green & Co; London.Google Scholar
  17. R. D. James (1981), The equilibrium and post-buckling behaviour of an elastic curve governed by a non-convex energy, J. Elasticity 11, pp. 239–269.Google Scholar
  18. G. Kirchhoff (1859), Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. reine angew. Math. (Crelle), 56.Google Scholar
  19. K. Kovári (1969), Räumliche Verzweigungsprobleme des dünnen elastischen Stabes mit endlichen Verformungen, Ing.-Arch., 37, pp. 393–416.Google Scholar
  20. A. E. H. Love (1927), A Treatise on the Mathematical Theory of Elasticity, Fourth Edition, Cambridge University Press.Google Scholar
  21. J. H. Maddocks (1981), Analysis of nonlinear differential equations governing the equilibria of an elastic rod and their stability, Thesis, University of Oxford.Google Scholar
  22. J. H. Maddocks (1984), Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles, SIAM J. of Math. Anal., in press.Google Scholar
  23. J. Pierce & A. P. Whitman (1980), Topological properties of the Manifolds of configurations of Several Simple Deformable Bodies, Arch. for Rational Mech. & Anal., 74, p. 101.Google Scholar
  24. E. L. Reiss (1969), Column buckling—An elementary example of bifurcation, pp. 1–16 of Bifurcation Theory and Nonlinear Eigenvalue Problems, Eds. J. B. Keller & S. S. Antman, Benjamin, New York.Google Scholar
  25. H. F. Weinberger (1973), Variational Methods for Eigenvalue Approximation, C. B. M. S. Conference Series # 15, SIAM.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1984

Authors and Affiliations

  • J. H. Maddocks
    • 1
  1. 1.Mathematical InstituteOxford UniversityUK

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