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.
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References
C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill (1978).
J. Carr, Applications of centre manifold theory. Applied Math. Sci. 35 (1981).
H. Cohen and R.G. Muncaster, The theory of pseudo-rigid bodies. Springer Tracts in Natural Philosophy 33 (1988). Springer-Verlag.
P.H. Coullet and E.A. Spiegel, Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43 (1983) 776–821.
S.M. Cox and A.J. Roberts, Centre manifolds of forced dynamical systems. J. Austral. Math. Soc. B 32 (1991) 401–436.
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.
Y.C. Fung, Foundations of Solid Mechanics. Prentice-Hall (1965).
Y.C. Fung, A First Course in Continuum Mechanics (2nd edition). Prentice-Hall (1977).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag (1983). Appl. Math. Sci. 42.
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.
G.N. Mercer and A.J. Roberts, A complete model of shear dispersion in pipes. Preprint (1992).
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.
A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, with Applications to Elliptic Variational Problems, Springer-Verlag (1991). Lect. Notes in Mathematics 1489.
R.G. Muncaster, Invariant manifolds in mechanics I: The general construction of coarse theories from fine theories. Arch. Rat. Mech. 84 (1983) 353–373.
R.G. Muncaster, Saint-Venant's problem for slender bodies. Utilitas Mathematica 234 (1983), 75–101.
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.
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.
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.
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.
A.J. Roberts, Boundary conditions for approximate differential equations. J. Austral. Math. Soc. B 34 (1992) 54–80.
J. Sijbrand, Properties of center manifolds. Trans. Amer. Math. Soc. 289 (1985) 431–469.
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Roberts, A.J. The invariant manifold of beam deformations. J Elasticity 30, 1–54 (1993). https://doi.org/10.1007/BF00041769
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DOI: https://doi.org/10.1007/BF00041769