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

c 1 c 2

l 2l 1

a

1 1

2222

21112

1lbclbcU cosLP

PUV 0

VV

4

Equilibrium paths

,V 0V

5

Stable or unstable? ,V 0grad V definite pos. H

6

Critical points0grad V 0Hdet

1

2

3

12

1

2

3

1

7

Imperfection-sensitivity

,,V

cr

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

c r C

c r

u> 0

0

0

12

Asymmetric point of bifurcation

u

> 0

0

0

> 0

0

c r C

c r

13

Unstable-symmetric point of b.

0

0 0

00

u c r -C

c r

14

Stable-symmetric point of b.

0

0

0

0

0

u

c r C

c r

15

Thompson & Hunt (1971)

• Monoclinal point of bifurcation

• Homeoclinal point of bifurcation

• Anticlinal point of bifurcation

16

Monoclinal point of bifurcation

xxx

y

17

Homeoclinal point of bifurcation

xxx

y

18

Anticlinal point of bifurcation

xxx

y

19

Thom’s theorem I.Typically a smooth RRRf rn : , (r<6) is:

- structurally stable,

- equivalent around any point to one of the forms:

1u1.

22

1

22

1 nii uuuu ni 02.

20

Thom’s theorem II.Cuspoid catastrophes:

Mututu 11

2

12

4

14.

Mutututu 11

2

12

3

13

5

15.

Mututututu 11

2

12

3

13

4

14

6

16.

Mutututututu 11

2

12

3

13

4

14

5

15

7

17.

Mutu 11

3

13.

22

1

22

2 nii uuuuM ni 1

21

Two active variables 22

321 nyyccxycxyf ,,,,~

0 00 0 2 2 2

H =

22

Thom’s theorem III.Umbilic catastrophes

ni 2 22

1

22

3 nii uuuuN

Nutututuuu 1122

2

13

3

22

2

18.

Nutututuuu 1122

2

13

3

22

2

19.

Nututututuuu 1122

2

13

2

24

4

22

2

110.

Nutututututuuu 1122

2

13

2

24

3

25

5

22

2

111.

Nutututututuuu 1122

2

13

2

24

3

25

5

22

2

112.

Nututuututuutuu 1122213

2

24

2

215

4

2

3

113.

23

Typical catastrophes

Time: fold

Symmetry: cusp

Optimization: elliptic and hyperbolic umbilic

Symmetry + optimizations: double cusp

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Subclasses of folds

+ - a = 0

a

limit point

asymmetric point of bifurcation

25

New subclasses of cusps

a

b

Unstable-symmetric p. of b. Unstable-X point of bifurcation

a

b

26

Unstable-X point of bifurcation 2224

8

1

12

1uuV ,,,

c r

c r - |C

3112 / cr

213 3

2/

> 0

0

0

> 0 0

> 0

0

27

Transition from standard to dual

A 2A 2

A 2

A 2

A 2A 2

A 2

A 3

A 3

A 2

A 2

A 2

A 2A 2

+

+

+

+

+

+

+

+

-

-

-

-

-

-

-

-

-

-

-

-

--

-

-

-

-

-

-

-

-

-

d

c a < 0

-

Butterfly catastrophe

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Transition between umbilics

H y p e rb o lic

E llip tic

P a ra b o lic

S y m b o lic

-1 3

1

2

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Summary of equilibrium paths

xxx

y

x

y

xxx

y

xxx

y

xxx

y

xxx

y

30

Double cusp 2242244 ,, byaxCyyBxAxyxVj

yx then C if A

2242244 ,, yxCyyBxxyxVj

scale of x 1A

scale of 1a

scale of y 1b

yx then 1C new if

31

Classification

2

42

4,3,2,1CBB

yx

-1

+1

C

B

1

52

3 3

7

10 12

14

4BC

2

4224 CyyBxxf

-1

+1

C

B

1

62

4 4

8

11 13

15

4BC

2

+ -

32

Equilibrium paths Vj4

224224 yxCyyBxxV

x

y4or

2(B-2)x28x22x20x3

2(B-2C)y28Cy22Cy2y02

001

Sj2Sj1yxj

2

2

B

CBy 2

2

2

4y

B

CB

2

2

2

4y

B

CB

2)2(4 yBC

CB

Bx

2

2

22

2

4x

CB

CB

2

2

2

44 x

CB

CB

224 xB

22

33

Subclasses (-)

-1

+1

C

B

1

62a

4a 4e

8a

11 13a

15

B=2

B=2C

2b

4b

2c

2e2d

8b

13b

4d

4c4c

4BC

2

8c

34

Equilibrium paths in some cases

x

y

1

x

y

2c

x

y

12b

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Projections of the equilibrium paths12b

+ 0

+ –

+ –

+ –+ –

1

+ 0

+ –

+ 0+ –

10

+ +

+ –

+ ++ –

3a 3b

+ –

+ 0

+ –

+ +

3c 12a 3d

+ 0

+ +

+ +

+ –

+ –+ +

3e

+ –

+ –

+ –0 –

5 14

+ –

0 0

+ –

+ 0

7a 7b

+ 0

+ 0

+ +

+ –

+ –+ 0

7c

+ –

+ –

+ –– –

2a 2b

+ –

0 –

+ –

+ –

2c 2d

+ 0

+ –

+ +

+ –

+ –+ –

2e

up

down

horizontal

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Imperfections yxyxCyyBxxyxVj 21

22422421

4 ,,,,

3/2 cr

sin ,cos , 32

31

2 ttεQt

sincos,,,, 3322242244 ytxtyxQtCyyBxxtQyxVj

321 ,, 0 CQtCytCxVV yx H

37

Horizontal paths

0

0

2

1

x

y

6a

perfect0

0

2

1

x

y

x

y

38

Point-like instability

0

x

y

12

3

4

5x

y

8.0

39

Asymmetric point of bifurcation

x

y

4a

x

y

perfect imperfect

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Equilibrium surface

x

y

u

1

1

cr

2

perfect imperfect imperfection-sensitivity

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Determinacy

642246224, GyyFxyExDxyBxxyxf

3

4u

Dux

32

224v

B

Fvu

B

EDvy

6224, GvvBuuvug

6642

224

4, GvHuv

BvBuuvug

6424, GvvFuuvug

Classes 5, 6, 8 and 9

Classes 10, 11, 12 and 13

Classes 14, 15

42

Class 14 yxyxGyyFxyExDxxV 21

226422464

3/212 30 cr

5/42

1 24150

GGcr

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Class 8

sincos24

1

8

1

16

1 222246 yxyxyxxyV

x

y

= 0

1 2

3

x

y

0 < < 0,56754

1 2

3 4

x

y

1 2

0,56754 < < /2

x

y

= /2

1

2

1

cr

2

0,0 /2 cr

–0,010

–0,020

–0,005

–0,015

/8 /8 /4

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Various classes

1

cr

2

0,0 /2

2

4

13

4

cr

–0,010

–0,020

–0,005

–0,015

/8 /8 /4

8

45

Class 6

/2

cr

x

y

12

3

4

5

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Conclusions

• New subclasses for cusps

• Transitions

• 36 subclasses (4th degree) for double cusps

• Determinacy

• Imperfection-sensitivity surfaces