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1 Imperfection-sensitivity and catastrophe theory Zs. Gáspár BME Dept. of Structural Mechanics

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2 Contents Concepts Early results Thom’s theorem Most important cases Double cusp catastrophes – Classification –Equilibrium paths –Imperfection-sensitivities

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3 Potential energy function

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4 Equilibrium paths

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5 Stable or unstable?

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6 Critical points

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7 Imperfection-sensitivity

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8 Koiter (1945, 1965) Limit point Asymmetric point of bifurcation Unstable-symmetric point of b. Stable-symmetric point of b.

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9 Limit points u 0

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10 Limit points u 0 0

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11 Limit points u > 0 0 0

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12 Asymmetric point of bifurcation

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13 Unstable-symmetric point of b.

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14 Stable-symmetric point of b.

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15 Thompson & Hunt (1971) Monoclinal point of bifurcation Homeoclinal point of bifurcation Anticlinal point of bifurcation

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16 Monoclinal point of bifurcation

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17 Homeoclinal point of bifurcation

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18 Anticlinal point of bifurcation

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19 Thom’s theorem I. Typically a smooth, (r<6) is: - structurally stable, - equivalent around any point to one of the forms: 1. 2.

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20 Thom’s theorem II. Cuspoid catastrophes:

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21 Two active variables

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22 Thom’s theorem III. Umbilic catastrophes

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23 Typical catastrophes Time: fold Symmetry:cusp Optimization: elliptic and hyperbolic umbilic Symmetry + optimizations: double cusp

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24 Subclasses of folds limit point asymmetric point of bifurcation

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25 New subclasses of cusps Unstable-symmetric p. of b. Unstable-X point of bifurcation

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26 Unstable-X point of bifurcation

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27 Transition from standard to dual Butterfly catastrophe

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28 Transition between umbilics

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29 Summary of equilibrium paths

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30 Double cusp scale of x scale of scale of y

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31 Classification +1 C B C B

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32 Equilibrium paths x y4 or 2(B-2)x 2 8x28x2 2x22x2 0x3 2(B-2C)y 2 8Cy 2 2Cy 2 y Sj2Sj2 Sj1Sj1 yxj

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33 Subclasses (-) +1 C B 1 6 2a 4a4a 4e4e 8a8a 11 13a 15 B=2 B=2C 2b 4b4b 2c 2e 2d 8b8b 13b 4d4d 4c4c 4c4c 8c8c

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34 Equilibrium paths in some cases x y 1 x y 2c x y 12b

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35 Projections of the equilibrium paths 12b – – – – + + – 3a3b + – – + 3c 12a 3d – + 3e + – 0 – – 0 + – + 0 7a7a7b7b + + – + 0 7c7c + – – 2a 2b + – 0 – + – 2c2d – + + – 2e up down horizontal

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36 Imperfections

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37 Horizontal paths x y 6a perfect x y x y

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38 Point-like instability x y x y

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39 Asymmetric point of bifurcation x y 4a x y perfect imperfect

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40 Equilibrium surface perfect imperfect imperfection-sensitivity

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41 Determinacy Classes 5, 6, 8 and 9 Classes 10, 11, 12 and 13 Classes 14, 15

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42 Class 14

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43 Class 8

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44 Various classes

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45 Class 6 x y

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

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