Imperfectionsensitivity and catastrophe theory

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Imperfection-sensitivity and catastrophe theory

• Zs. Gáspár
• BME Dept. of Structural Mechanics

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Contents

• Concepts
• Early results
• Thoms theorem
• Most important cases
• Double cusp catastrophes
• Classification
• Equilibrium paths
• Imperfection-sensitivities

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Potential energy function
4
Equilibrium paths
5
Stable or unstable?
6
Critical points
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Imperfection-sensitivity
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Koiter (1945, 1965)

• Limit point
• Asymmetric point of bifurcation
• Unstable-symmetric point of b.
• Stable-symmetric point of b.

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Limit points
l
u
e
0
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Limit points
l
u
e
0
e lt
0
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Limit points
l
u
e
gt 0
e
0
e lt
0
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Asymmetric point of bifurcation
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Unstable-symmetric point of b.
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Stable-symmetric point of b.
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Thompson Hunt (1971)

• Monoclinal point of bifurcation
• Homeoclinal point of bifurcation
• Anticlinal point of bifurcation

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Monoclinal point of bifurcation
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Homeoclinal point of bifurcation
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Anticlinal point of bifurcation
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Thoms theorem I.
, (rlt6) is
Typically a smooth
- structurally stable,
- equivalent around any point to one of the forms
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Thoms theorem II.
Cuspoid catastrophes
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Two active variables
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Thoms theorem III.
Umbilic catastrophes
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Typical catastrophes
Time fold
Symmetry cusp
Optimization elliptic and hyperbolic
umbilic
Symmetry optimizations double cusp
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Subclasses of folds
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New subclasses of cusps
Unstable-symmetric p. of b.
Unstable-X point of bifurcation
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Unstable-X point of bifurcation
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Transition from standard to dual
Butterfly catastrophe
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Transition between umbilics
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Summary of equilibrium paths
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Double cusp
scale of x

scale of
scale of y
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Classification

-
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Equilibrium paths
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Subclasses (-)
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Equilibrium paths in some cases
?
?
x
x
y
y
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Projections of the equilibrium paths
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Imperfections
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Horizontal paths
perfect
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Point-like instability
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Asymmetric point of bifurcation
imperfect
perfect
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Equilibrium surface
perfect
imperfection-sensitivity
imperfect
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Determinacy
Classes 5, 6, 8 and 9

Classes 10, 11, 12 and 13
Classes 14, 15
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Class 14
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Class 8
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Various classes
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Class 6
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Conclusions

• New subclasses for cusps
• Transitions
• 36 subclasses (4th degree) for double cusps
• Determinacy
• Imperfection-sensitivity surfaces