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

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

3 Potential energy function

4 Equilibrium paths

5 Stable or unstable?

6 Critical points

7 Imperfection-sensitivity

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

9 Limit points u  0

10 Limit points u  0  0

11 Limit points u  > 0  0  0

12 Asymmetric point of bifurcation

13 Unstable-symmetric point of b.

14 Stable-symmetric point of b.

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

16 Monoclinal point of bifurcation

17 Homeoclinal point of bifurcation

18 Anticlinal point of bifurcation

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

20 Thom’s theorem II. Cuspoid catastrophes:

21 Two active variables

22 Thom’s theorem III. Umbilic catastrophes

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

24 Subclasses of folds limit point asymmetric point of bifurcation

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

26 Unstable-X point of bifurcation

27 Transition from standard to dual Butterfly catastrophe

28 Transition between umbilics

29 Summary of equilibrium paths

30 Double cusp scale of x scale of scale of y

31 Classification +1 C B C B

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

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

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

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

36 Imperfections

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

38 Point-like instability x y x y

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

40 Equilibrium surface perfect imperfect imperfection-sensitivity

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

42 Class 14

43 Class 8

44 Various classes

45 Class 6 x y

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