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A buckling model for the set of umbilic catastrophes

Published online by Cambridge University Press:  24 October 2008

J. M. T. Thompson  and
Z. Gaspar
J. M. T. Thompson
Affiliation:
University College London and the Technical University of Budapest
Z. Gaspar
Affiliation:
University College London and the Technical University of Budapest
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Abstract

Zeeman has drawn attention to the sequences in which catastrophes, or modes of instability, can be linked, and it is a common observation that sequences of catastrophes of low order are always found in the environment of catastrophes of higher order. In this paper, a simple buckling model is presented that generates in an elegant manner a complete sequence of the umbilic catastrophes as represented by the semi-symmetric branching points. The scan of a single fundamental parameter of this model is shown to trace a route through all regimes of the umbilic bracelet, giving in turn the hyperbolic, symbolic, elliptic, parabolic and hyperbolic umbilic catastrophes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 3 , November 1977 , pp. 497 - 507
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Thom, R.Structural stability and morphogenesis (translated from the French edition by Fowler, D. H.) (Reading, Benjamin, 1975).Google Scholar
(2)Zeeman, E. C.Catastrophe theory. Scientific American 234 (4) (1976), 65–83.CrossRefGoogle Scholar
(3)Thompson, J. M. T. Catastrophe theory and its role in applied mechanics, Sectional Lecture at the 14th International Congress of Theoretical and Applied Mechanics, Delft, August, 1976. (Theoretical and applied mechanics, vol. 2, page 451, ed. Koiter, W. T. (Amsterdam, North Holland, 1977).)Google Scholar
(4)Thompson, J. M. T. and Hunt, G. W.A general theory of elastic stability (London, Wiley, 1973).Google Scholar
(5)Thompson, J. M. T. and Hunt, G. W.Towards a unified bifurcation theory. J. Appl. Math. Phys. (Z.A.M.P.), 26 (5) (1975), 581–604.Google Scholar
(6)Hansen, J. S. Buckling of imperfection-sensitive structures: A probabilistic approach. Thesis, University of Waterloo (1973).Google Scholar
(7)Thompson, J. M. T.Experiments in catastrophe. Nature 254 (5499) (1975), 392–395.CrossRefGoogle Scholar
(8)Thompson, J. M. T. and Shorrock, P. A.Hyperbolic ombilic catastrophe in crystal fracture. Nature 260 (5552) (1976), 598–599.CrossRefGoogle Scholar
(9)Thompson, J. M. T. and Supple, W. J.Erosion of optimum designs by compound branching phenomena. J. Mech. Phys. Solids 21 (3) (1973), 135–144.CrossRefGoogle Scholar
(10)Zeeman, E. C. The ombilic bracelet and the double-cusp catastrophe. In Structural stability, the theory of catastrophes, and applications in the sciences, ed. Hilton, P. (Lecture Notes in Mathematics 525 (Berlin, Springer, 1976)).Google Scholar
(11)Poston, T. and Stewart, I. N.Taylor expansions and catastrophes (Research Notes in Mathematics 7 (London, Pitman, 1976)).Google Scholar
(12)Koiter, W. T. On the stability of elastic equilibrium. Dissertation. Delft, Holland, 1945 (English translation: NASA, Tech. Trans., F10, 1967).Google Scholar
(13)Budiansky, B. Theory of buckling and post-buckling behaviour of elastic structures. In Advances in Applied Mechanics, vol. 14 (New York, Academic Press, 1974).Google Scholar
(14)Huseyin, K.Nonlinear theory of elastic stability (Leyden, Noordhoff, 1975).Google Scholar
(15)Zeeman, E. C. The classification of elementary catastrophes of codimension ≤ 5. In Structural stability, the theory of catastrophes, and applications in the sciences, ed. Hilton, P. (Lecture Notes in Mathematics 525 (Berlin, Springer, 1976)).Google Scholar
(16)Porteous, I. R.The normal singularities of a submanifold. J. Differential Geometry 5 (1971), 543–564.CrossRefGoogle Scholar
(17)Thompson, J. M. T. and Hunt, G. W.The instability of evolving systems. Interdisciplinary Science Reviews 2 (3) (1977), 240–262.CrossRefGoogle Scholar
(18)Hunt, G. W.Imperfection-sensitivity of semi-symmetric branching (to appear in Proc. Roy. Soc.).Google Scholar
(19)Roorda, J.Stability of structures with small imperfections. J. Eng. Mech. Div. Am. Soc. Civ. Eng. 91 (1965), 87–106.Google Scholar
(20)Chillingworth, D. The catastrophe of a buckling beam. In Dynamical systems – Warwick 1974, ed. Manning, A. (Lecture Notes in Mathematics 468 (Berlin, Springer, 1975)).Google Scholar
(21)Zeeman, E. C. Euler buckling. In Structural stability, the theory of catastrophes, and applications in the sciences, ed. Hilton, P. (Lecture Notes in Mathematics 525 (Berlin, Springer, 1976)).Google Scholar