1 imperfection-sensitivity and catastrophe theory zs. gáspár bme dept. of structural mechanics
TRANSCRIPT
2
Contents
• Concepts
• Early results
• Thom’s theorem
• Most important cases
• Double cusp catastrophes– Classification– Equilibrium paths– Imperfection-sensitivities
8
Koiter (1945, 1965)
• Limit point
• Asymmetric point of bifurcation
• Unstable-symmetric point of b.
• Stable-symmetric point of b.
15
Thompson & Hunt (1971)
• Monoclinal point of bifurcation
• Homeoclinal point of bifurcation
• Anticlinal point of bifurcation
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
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
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
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
35
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
36
Imperfections yxyxCyyBxxyxVj 21
22422421
4 ,,,,
3/2 cr
sin ,cos , 32
31
2 ttεQt
sincos,,,, 3322242244 ytxtyxQtCyyBxxtQyxVj
321 ,, 0 CQtCytCxVV yx H
41
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
43
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