The parabolic umbilic catastrophe and its application in the theory of elastic stability
Authors:
David Hui and Jorn S. Hansen
Journal:
Quart. Appl. Math. 39 (1981), 201-220
MSC:
Primary 73H05; Secondary 58C28
DOI:
https://doi.org/10.1090/qam/625469
MathSciNet review:
625469
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Abstract: The implications of the parabolic umbilic catastrophe in the theory of elastic stability are investigated. In particular, the influence of terms in the potential energy which are deemed necessary for a complete analysis and the isolation of primary critical surfaces are considered. The results are demonstrated for the example of the buckling and initial post-buckling of a spherical shell under the influence of a constant as well as a spatially variable pressure.
- René Thom, Stabilité structurelle et morphogénèse, W. A. Benjamin, Inc., Reading, Mass., 1972 (French). Essai d’une théorie générale des modèles; Mathematical Physics Monograph Series. MR 0488155
- John M. T. Thompson and Giles W. Hunt, Towards a unified bifurcation theory, Z. Angew. Math. Phys. 26 (1975), no. 5, 581–603 (English, with German summary). MR 388441, DOI https://doi.org/10.1007/BF01594031
- M. J. Sewell, Some mechanical examples of catastrophe theory, Bull. Inst. Math. Appl. 12 (1976), no. 6, 163–172. MR 649268
W. T. Koiter, On the stability of elastic equilibrium, Dissertation, Polytechnic Institute, Delft, Holland, 1945
- J. M. T. Thompson and G. W. Hunt, A general theory of elastic stability, Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1973. MR 0400868
- Jorn S. Hansen, Some two-mode buckling problems and their relation to catastrophe theory, AIAA J. 15 (1977), no. 11, 1638–1644. MR 461560, DOI https://doi.org/10.2514/3.7463
J. W. Hutchinson, Imperfection sensitivity of externally pressurized spherical shells, J. Appl. Mech. 34, 49–55 (1967)
A. N. Godwin, Three-dimensional pictures for Thom’s parabolic umbilic, Math. Pbl. I.H.E.S. 40, 117–138 (1971)
- Th. Bröcker, Differentiable germs and catastrophes, Cambridge University Press, Cambridge-New York-Melbourne, 1975. Translated from the German, last chapter and bibliography by L. Lander; London Mathematical Society Lecture Note Series, No. 17. MR 0494220
- D. Hui and J. S. Hansen, Two-mode buckling of an elastically supported plate and its relation to catastrophe theory, Trans. ASME Ser. E. J. Appl. Mech. 47 (1980), no. 3, 607–612. MR 586373
R. Thom, Structural stability and morphogenesis, Benjamin, Reading, Mass., 1975
J. M. T. Thompson and G. W. Hunt, Towards a unified bifurcation theory, J. Appl. Math. Phys. 26, 581–604 (1975)
M. J. Sewell, Some mechanical examples of catastrophe theory, Inst. Math. Applics., 12, 163–172 (1976)
W. T. Koiter, On the stability of elastic equilibrium, Dissertation, Polytechnic Institute, Delft, Holland, 1945
J. M. T. Thompson and G. W. Hunt, A general theory of elastic stability, Wiley, New York, 1973
J. S. Hansen, Some two-mode buckling problems and their relation to catastrophe theory, AIAA J. 15, 1638–1644 (1977)
J. W. Hutchinson, Imperfection sensitivity of externally pressurized spherical shells, J. Appl. Mech. 34, 49–55 (1967)
A. N. Godwin, Three-dimensional pictures for Thom’s parabolic umbilic, Math. Pbl. I.H.E.S. 40, 117–138 (1971)
Th. Bröcker and L. Lander, Differential germs and catastrophes, London Math. Society, Lecture Notes 17, 1975
D. Hui and J. S. Hansen, Two-mode buckling of an elastically supported plate and its relation to catastrophe theory, J. Appl. Mech. 47, 607–612 (1980)
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Article copyright:
© Copyright 1981
American Mathematical Society
