Journal of Elasticity

, Volume 30, Issue 1, pp 1–54 | Cite as

The invariant manifold of beam deformations

Part 1: the simple circular rod
  • A. J. Roberts
Article

Abstract

The subcentre invariant manifold of elasticity in a thin rod may be used to give a rigorous and appealing approach to deriving one-dimensional beam theories. Here I investigate the analytically simple case of the deformations of a perfectly uniform circular rod. Many, traditionally separate, conventional approximations are derived from within this one approach. Furthermore, I show that beam theories are convergent, at least for the circular rod, and obtain an accurate estimate of the limit of their validity. The approximate evolution equations derived by this invariant manifold approach are complete with appropriate initial conditions, forcing and, in at least one case, boundary conditions.

Key words

elastic beam theory subcentre invariant manifolds dynamical systems 

AMS (MOS) Classification

73C02 73C10 35A35 58G40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill (1978).Google Scholar
  2. 2.
    J. Carr, Applications of centre manifold theory. Applied Math. Sci. 35 (1981).Google Scholar
  3. 3.
    H. Cohen and R.G. Muncaster, The theory of pseudo-rigid bodies. Springer Tracts in Natural Philosophy 33 (1988). Springer-Verlag.Google Scholar
  4. 4.
    P.H. Coullet and E.A. Spiegel, Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43 (1983) 776–821.Google Scholar
  5. 5.
    S.M. Cox and A.J. Roberts, Centre manifolds of forced dynamical systems. J. Austral. Math. Soc. B 32 (1991) 401–436.Google Scholar
  6. 6.
    C. Domb and M.F. Sykes, On the susceptibility of a ferromagnetic above the Curie point. Proc. Roy. Soc. Lond. A 240 (1957) 214–228.Google Scholar
  7. 7.
    Y.C. Fung, Foundations of Solid Mechanics. Prentice-Hall (1965).Google Scholar
  8. 8.
    Y.C. Fung, A First Course in Continuum Mechanics (2nd edition). Prentice-Hall (1977).Google Scholar
  9. 9.
    J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag (1983). Appl. Math. Sci. 42.Google Scholar
  10. 10.
    G.N. Mercer and A.J. Roberts, The application of centre manifold theory to the dispersion of contaminant in channels with varying flow properties. SIAM J. Appl. Math. 50 (1990) 1547–1565.Google Scholar
  11. 11.
    G.N. Mercer and A.J. Roberts, A complete model of shear dispersion in pipes. Preprint (1992).Google Scholar
  12. 12.
    A. Mielke, On Saint-Venant's problem and Saint-Venant's principle in nonlinear elasticity. In J.F. Besseling and W. Eckhaus (eds), Trends in Appl. of Maths. to Mech. (1988), pp. 252–260.Google Scholar
  13. 13.
    A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, with Applications to Elliptic Variational Problems, Springer-Verlag (1991). Lect. Notes in Mathematics 1489.Google Scholar
  14. 14.
    R.G. Muncaster, Invariant manifolds in mechanics I: The general construction of coarse theories from fine theories. Arch. Rat. Mech. 84 (1983) 353–373.Google Scholar
  15. 15.
    R.G. Muncaster, Saint-Venant's problem for slender bodies. Utilitas Mathematica 234 (1983), 75–101.Google Scholar
  16. 16.
    A.J. Roberts, Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations. J. Austral. Math. Soc. Ser. B 27 (1985) 48–65.Google Scholar
  17. 17.
    A.J. Roberts, The application of centre manifold theory to the evolution of systems which vary slowly in space, J. Austral. Math. Soc. Ser. B 29 (1988) 480–500.Google Scholar
  18. 18.
    A.J. Roberts, Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems. J. Austral. Math. Soc. Ser. B 31 (1989) 48–75.Google Scholar
  19. 19.
    A.J. Roberts, The utility of an invariant manifold description of the evolution of dynamical systems. SIAM J. of Math. Anal. 20 (1989) 1447–1458.Google Scholar
  20. 20.
    A.J. Roberts, Boundary conditions for approximate differential equations. J. Austral. Math. Soc. B 34 (1992) 54–80.Google Scholar
  21. 21.
    J. Sijbrand, Properties of center manifolds. Trans. Amer. Math. Soc. 289 (1985) 431–469.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • A. J. Roberts
    • 1
  1. 1.Department of Applied MathematicsThe University of AdelaideAdelaideSouth Australia

Personalised recommendations