(i)

ix 1 1

1.1 1 1.2 4 1.3 5 1.4 6

1.4.1 6 1.4.2 6 1.4.2.1 6 1.4.2.2 8 1.4.3 8 1.4.4 8 1.4.5 10

1.5 12 1.6 12

2 15 2.1 16 2.2 18 2.3 20 2.3.1 20 2.3.2 23 2.3.2.1 23 2.3.2.2 24 2.3.2.3 26 2.3.3 Griffith 27 2.3.4 Griffith 29 2.4 35 2.4.1 35

(ii)

2.4.2 38 2.4.3 42 2.5 K G 44 2.6 49 2.7 51 2.8 56 2.8.1 56 2.8.1.1 Westergaard 56 2.8.1.2 Muskhelishvili 60 2.8.1.3 Williams 60 2.8.2 61 2.8.3 64 2.8.4 66 2.8.4.1 70 2.8.4.2 K 75 2.8.5 79 2.8.5.1 K 80 2.8.5.2 K 81 2.8.6 86 2.8.6.1 K 86 2.8.6.2 K 87 2.8.7 88 2.8.8 90 2.9 91 2.9.1 92 2.9.2 93 2.9.3 94 2.9.3.1 Irwin 94 2.9.3.2 100

(iii)

2.10 102 2.11 104 2.11.1 CTOD Irwin 105 2.11.2 CTOD 106 2.12 109 2.13 110 : Williams 113

3 - 125 3.1 125 3.2 J- 128 3.2.1 128 3.2.2 J 131 3.2.3 J 133 3.3 J - CTOD 134 3.4 J 137 3.4.1 J 137 3.4.2 J 148 3.4.2.1 148 3.4.2.2 151 3.4.2.3 164 3.4.3 EPRI 166 3.4.4 170 3.5 J 175 3.5.1 J- 175 3.5.2 J 178 3.6 181 3.7 182

: Jpl EPRI 185 : J-integral estimation for a semi-elliptical surface crack in a round bar under tension 193

(iv)

4 197 4.1 197 4.2 199 4.3 201 4.3.1 201 4.3.2 202 4.3.2 202 4.4 206 4.5 214 4.5.1 214 4.5.2 214 4.5.3 217 4.5.4 219 4.5.5 220 4.5.6 221 4.6 KIc 223 4.7 KR 234 4.8 JIc 244 4.8.1 245 4.8.2 255 4.9 JR 262 4.10 JR 267 4.10.1 EPRI 268 4.10.2 268 4.11 CTOD 272

4.12 277 4.13 278 4.14 278 : 281

(v)

5 295 5.1 295 5.2 297 5.2.1 299 5.2.2 299 5.2.3 299 5.3 301 5.3.1 da/dN 302 5.3.2 da/dN incremental polynomial 302 5.3.3 da/dN modified difference 303 5.3.4 da/dN central difference 304 5.4 308 5.4.1 309 5.4.2 309 5.5 311 5.5.1 311 5.5.1.1 311 5.5.1.2 313 5.5.1.3 313 5.5.2 314 5.5.3 317 5.6 318 5.6.1 318 5.6.2 320 5.7 321 5.7.1 323 5.7.2 323 5.8 327 5.8.1 327 5.8.2 328

(vi)

5.8.3 Simplified Rainflow 334 5.8.4 339 5.8.4.1 340 5.8.4.2 Wheeler 341 5.8.4.3 Wheeler 343 5.8.4.4 Willenborg 343 5.8.4.5 346 5.8.4.6 346 5.9 347 5.10 - 349 5.11 353 5.12 357 5.13 357 : 363 : 375

6 381 6.1 381

6.1.1 381 6.1.2 SCC 383 6.1.2.1 384 6.1.2.2 384 6.1.3 387

6.2 389 6.2.1 389 6.2.2 391 6.3 395

6.3.1 C* 396 6.3.1.1 C* 397 6.3.1.2 C* EPRI 399 6.3.1.3 C* 401

(vii)

6.3.1.4 C* 402 6.3.2 403

6.3.3 407 6.4 410

6.5.1 412 6.5.2 412

6.5 414 6.6 415 : 419

7 425 7.1 425 7.2 425 7.3 427 7.3.1 427 7.3.2 428 7.3.3 431 7.3.4 435 7.4 CTOD 438 7.5 441 7.5.1 441 7.5.2 J- 445 7.6 R6 448 7.6.1 449 7.6.2 451 7.6.2.1 451 7.6.2.2 451 7.6.2.3 452 7.6.2.4 453 7.6.2.5 454 7.6.2.6 Lr 455

(viii)

7.6.2.7 Kr 455 7.6.2.8 458 7.7 472 : WES 2805-1997 473 479

(ix)

(simulation) 7 1 2 3 - 4 5 6 7 R6 Emeritus Prof. Dr. Yasuhide ASADA (), Assoc. Prof. Dr. Toshiya NAKAMURA

(x)

Prof. Shinsuke SAKAI TJTTP-OECF .. Jirapong.K@Chula.ac.th

2553

(xi)

stress intensity factor +++ singularity dominated-zone + + fracture toughness

1

1.1 1 (overload) (fatigue) (creep) (corrosion) (wear) (buckling) (failure criteria) ( ) ( ) () () ( ) (failure analysis) 2 ( 1) ( )f ( )g 1 2 failure damage corrosion damage corrosion failure

2

()( ),...f ( ),...g >b () 2a (6)

a

A 2 (7)

(7) ( = 0) (infinitely sharp) !

20

() [7] ( )0> (7)

2.3. .. 1921 A.A. Griffith (global analysis)

2.3.1 (conservation of energy theorem) (external work) W (strain energy) U UW = (8)

W P (load-line displacement) LL

LL

PdW = (9)

4 LLPW 2

1= (8) LLPU 2

1= 5 (strain energy density) Ud 2

1 E

2

21

dxdydzE

U2

21 = (10)

21

W

P

LL

LL

P

4 -

U

5 -

(central through-crack plate) 2a P 6 P LL (stiffness)

2a

P

P

LL

LL

1

6

22

W U Ws dA

dA

dWdAdU

dAdW s+=

( )dA

dWUW

dAd s= (11)

(11) (fracture criteria) (total potential energy)

WU = (12) (11)

dAdW

dAd s= (13)

..1956 Irwin (13) (11) (energy release rate, G 2) (crack driving force) (crack growth resistance, R)

dAd

dAdU

dAdWG == (14)

dA

dWR s= (15) RG = (16)

(16) G R ( dAdG dAdR )

2 Griffith

23

dAdR

dAdG > (17)

dAdR

dAdG < (17)

dAdR

dAdG = (17)

(17) (17) ()

2.3.2 1) (displacement-controlled condition) 2) (load-controlled condition)

2.3.2.1 a P1 ( LL 1) a a+da ( P1 1 dPP +1 1+d ) a a+da ( 7) O C D da C D dW ACDE dU ODE OCA [ (12)] ( )OCAODEACDEd = ( )[ ]OCAABDEOBAACDEd += ( ) ( )ABDEACDEOBAOCAd += OCDd =

- da ()

24

a

a+da

P1

1 LL d+1

-

O

C

D

A E

dB

dPP +1

7 a a+da

2.3.2.2 8 ( B) P1 1 OA P1 AB a+da dPP +1

11P

1 LL

A

B

1P

dPP +1

a daa +P

O

da daa +

C

a a

() () ()

8 () () da ()

25

(dW = 0) (14)

dAdUG = (18)

8 U a OAC 1121 P

a+da OBC ( ) 1121 dPP +

dU dPdU 12

1 = (19)

dAdP

dAdU

LL21= (20)

PCLLLL = (21)

CLL (compliance) P-LL

LLLLLL CdAdPP

dAdC

dAd += (22)

0=dA

d LL

dA

dCCP

dAdP LL

LL

= (23)

(20) dA

dCCP

dAdU LL

LLLL2

1=

(21) dA

dCPdAdU LL2

21=

(18) dA

dCPG LL221= (24)

B dA = Bda (24)

da

dCBPG LL

2

21= (25)

26

2.3.2.3 9 P1 1 OA P1 AB a+da 1+d

dAdU

dAdWG = (14)

9 a a+da 1121 P

( ) dP +1121

dPdU 121=

dPdW 1= dPdPdUdW 11 2

1=

dUdPdUdW == 121

(14)

dAdUG = (26)

1P

P

1P

1 d+1 LL

a daa +

O

A B

D

d

12

a daa +

1PC

E

a

9

27

dA

dPdAdU

dAdW LL

21= (27)

(22)

PdA

dCdA

d LLLL = (28) (27)

dAdCPG LL2

21= (29)

B dA = Bda (29)

da

dCBPG LL

2

21= (30)

(24) (29) (25) (30) G (18) (26) U G G (17)

2.3.3 Griffith Griffith (infinite plate) 2a [ 10()] y

E

2

21 [ 10()]

[ 10()]

( )( )E

BaaU2

22=

28

E

BaU22= (31) 3

= 0

21 222

=

EBaVol

E

Vol

EaB

dad 22=

GE

adAd =

22 (32)

Ws

( )( )ss aBW 22=

2aB s2 () ss aBW 4=

4a

2a

() ()

10

3 Griffith Griffith

29

ss BdadW 4=

RdA

dWs

s = 4 (33)

(32) (33) (16)

sf BEaB 42

2

= f

aE s

f 2= (34)

(34) (ideally brittle solids) (34)

2.3.4 Griffith Irwin Orowan Griffith (plastic work) p ( )

aE ps

f += 2 (35)

p s s

(crack meandering) 11() (crack branching) 11() Griffith

a

EWff

2= (36)

fW (fracture energy)

30

() () 11 G

() ()

1 [3] 3 .

, a (.)

, Pc ()

() , c (.)

30.0 4,000 0.40 40.0 3,500 0.50 50.0 3,120 0.63 60.0 2,800 0.78 70.0 2,620 0.94 77.5 2,560 1.09

() Gc CLL - 1) a a+da - a a+da ( E1)

( )( ) 1,1,,1,1,,,,1 212121 +++++ ++= icicicicicicicicii PPPPAOA (E1) i = 1, 2, 3, 4, 5 2.3.2.1

( )iiii

ic aaBAOAG = +

+1

1, (E2)

31

0

2000

4000

0.2 0.4 0.6 0.8 1.0 1.2

(

)

(.)

a= 3

0 .

a = 40

.a =

50

.

a = 60 .

a = 70 .

a = 77.5 .

A1A2

A3A4 A5 A6

() -

E1 -

(E1) (E2) Gc 10000, 10750, 11160, 9807, 9987 J/m2 Gc ( 9,807 J/m2) (10,340 J/m2)

2) -

ic

iciLL P

C,

,,

= ; i = 1, 2, 3, 4, 5, 6 (E2) (E2)

, a (.) 30 40 50 60 70 77.5 , CLL (/) 1.0x10-7 1.4x10-7 2.0x10-7 2.8x10-7 3.6x10-7 4.3x10-7 4

76243343 1075.11076.91006.31037.21053.7 ++= aaaaCLL

32

2 [5] D a P ( E1)

( )( )DadEDCd

Da

Da

Daa

GE LL41

21

18

=

E

a

D

P

P

AA

A-A

E1

CLLED (normalized compliance) (24) (29)

dAdCPG LL2

21= (24)

P 24DP

= 2

4DP = (E1)

E2

( )aDaaDDy =

=22

24 (E2)

= aDy2

arctan2

33

(E2) ( )

=

aDaDa

2arctan2 (E3)

A ( ) sin22

1 2

= DA (E4) dA/da

dad

ddA

dadA

= (E5)

(E4) ( ) cos1421 2 = D

ddA (E6)

(E3) ( )aDadad

=2 (E7)

(E6) (E7) (E5)

( )( ) daaDaDdA

= cos14

2

(E8)

(E3)

( )daaDadA = 2 (E9)

(E1) (E9) (24)

aD/2

Ay

E2 ( A-A)

34

( ) dadC

aDa

DG LL

=2

1621

422

( )( )

( )DadEDCd

aDaaD

aEG LL

=2

2 64

( )[ ]( )

( )21

418

=

DadEDCd

aDaa

Da

EG LL

( )( )21

41

21

18

=

DadEDCd

Da

Da

Daa

EG LL

3 [6] B 2h 2a P E1 h

35

33

2EBhPa

LL = E1 3

32EBh

aP

C LLLL == (E1)

(E1) (25)

= 3

22 321

EBha

BPG (E2)

() (E2)

3222

43

hEBaPG =

P

2a

hLL

E2

2.4 2.4.1 3

1) (opening mode) 1 [ 12()]

2) (in-plane shear mode) (sliding mode) 2 [ 12()]

3) (out-of-plane shear mode) (tearing mode) 3 [ 12()]

13 17 13 P M P M

36

() ( 1) () ( 2) () ( 3)

12 1 14

(circumferential crack) (twisting moment) T 3 15 45 1 (biaxial) 16 (neutral axis) 2 17 y x 1 2

+

A

AMM

PP

13 P M

37

T T

A

A

x+

14 T

T T

A

A

45o

15 T

16

1) 2)

38

x

yxy

xx yy

yx

17

2.4.2 (isotropic) 4 ij ( 18) [7]

( ) ( ) ( ) ( )=

++

=

1,

20,0,

mmij

m

mijijij grAgAfrkr (37)

r k A

ijf ijg

(37) 3 21r r0 () r

21r (dominant term) r ( P ) r ( P ) 21r (37)

4

39

( ) ( ) ( ) 0,0, ijijij gAfrkr +

= (38)

r (38) - (T-stress) [7] 5 - [8,9] (38) 6

( ) ijij frk

= (39)

r (singular point) (39) (37) 10 [10] (singularity dominated zone) 19 (infinite body) (cylindrical coordinate, zr ) [11] 2.1

x

y

z

r

P

xx

yy

zzzxzy

yz yx

xy

xz

P()

18

5 1 ( o0= ) - rr 124C (biaxial stress state) 6 ( 2.8.6)

40

(37 ) (39 )

ij

r -1/2

x

19 (37)

2.1 :

a2

x

y

r

= 2

3cos2

cos54

21 rArr

+= 2

3cos2

cos34

21 rA

+= 2

3sin2

sin4

21 rAr

( )

=2

3cos2

cos124

21 r

Aur

( )

++=2

3sin2

sin124

21 r

Au

a2x

yr

+= 2

3sin32

sin54

21 rBrr

= 2

3sin32

sin34

21 rB

+= 2

3cos32

cos4

21 rBr

( )

+=2

3sin32

sin124

21 r

Bur

( )

++=2

3cos32

cos124

21 r

Bu

a2

x

yr

+ + +

2sin

221 = rCrz

2cos

221 = rCz

2sin2

1 Cruz =

(shear modulus) 43 ( ) ( ) + 13

41

1 2 xyz - 2.1 [4]

cossin2sincos 22 rrrxx += (40) cossin2cossin 22 rrryy ++= (40) ( ) ( ) 22 sincoscossin += rrrxy (40)

20 3 xyz

( ) ( ) sincos dAdAdA rzzyz += ( ) ( ) sincos dAdAdA zrzxz = sincos rzzyz += (41) cossin rzzxz += (41) 2.2

++

rzz

dA+

+xz

yz

dA x

y

() () 20 3 () () xyz

2.2 ( xyz)

1

= 2

3sin2

sin12

cos21 Arxx

+= 2

3sin2

sin12

cos21 Aryy

23cos

2sin

2cos2

1 = Arxy

+=2

sin212

cos2

221

Aru

+=2

cos212

sin2

221

Arv

(shear modulus) 43 ( ) ( ) + 13

42

2.2 ()

2

+= 2

3cos2

cos22

sin21 Brxx

23cos

2cos

2sin2

1 = Bryy

= 2

3sin2

sin12

cos21 Brxy

++=2

cos212

sin2

221

Bru

=2

sin212

cos2

221

Brv

3 2sin

221 = rCxz

2cos

221 = rCyz

2sin

21

Crw =

2.4.3 2.1 2.2 1, 2 3 ( ), ( )Iij ijAr fr = (42) ( ) ( ) IIijij fr

Br =, (42) ( ) ( ) IIIijij fr

Cr2

, = (42)

(42) ( )( )

Iij

ij

frr

A,=

( o0= ) ( )( )00,

Iij

ij

frr

A=

2.2 ij yy ( ) 10 =oIyyf

( )0,rrA yy= 2

AA= ( )0,2 rrA yy=

Irwin [12] A r (stress intensity factor) 1 KI

( )( )oyyrI rrK 0,2lim0 = (43)

43

2 3 2BB

= 2CC

= Irwin ( )( )oxyrII rrK 0,2lim0 = (43) ( )( )oyzrIII rrK 0,2lim0 = (43)

KII KIII 2 3

2.2 2.3 (singularity stress amplitude)

2.3 ( xyz)

1

=2

3sin2

sin12

cos2

r

K Ixx

+=2

3sin2

sin12

cos2

r

K Iyy

23cos

2sin

2cos

2

rK I

xy =

( )yyxxzz += 0== yzxz

+=2

sin212

cos22

2 rKu I

+=2

cos212

sin22

2 rKv I

0=w

2

+=2

3cos2

cos22

sin2

r

K IIxx

23cos

2cos

2sin

2

rK II

yy =

=2

3sin2

sin12

cos2

r

K IIxy

( )yyxxzz += 0== yzxz

++=2

cos212

sin22

2 rKu II

=2

sin212

cos22

2 rKv II

0=w

3 2sin

2

rK III

xz =

2cos

2

rK III

yz = 2

sin2

rKw III=

44

(local parameter) K 21 P p K K K

2.5 K G .. 1957 Irwin G K K ( ) 1 a 1 xy [ 22 ()] y 00= xr 2.3 1 ( )

xKyx Iyy 20, == (44)

a+a [ 22()] 0=x ax = y

xx

yyxy

p

P

aa2

( )K,, PafK = ( )K,, pagK =

21 K

45

y

x

a

a a

( )x

Kx Iyy 2=

y

x

y

x

( ) ( )+ = xvxv

y

x( )

22

1 xKxv I+= +

()

()

()

()

22 (closure stress)

yx 0180= xr 1 2.3 ( )

22

1 xKxv I+= + (45)

( ) ( )+ = xvxv

( )+xv y () ( )xv -y ()

46

(44) (45) xy axx =+ xax = (45) ( )

22

1 xaKxv I+=+ (46)

a a [ 22()] (closure work) Wc 7 ( ) ( )

( )

( )dxxBdvyxW

a xv

xvyyc

==+

00,

( ) ( )( ) dxxdvyxBW a xv yyc

==+

0 00,2

( ) ( ) dxxvyxBW a yyc

==0

0,212

(44) (46) dxxaKx

KBWa

II

c

+=

0 221

2

dxx

xaBKWa

Ic +=0

2

41

2sinax = ( ) ++=

2

0

2 2cos14

1 daBKW Ic

aBKW Ic += 281

(14) G Ba aBKaGB I += 28

1

7

47

28

1IKG

+= (47)

( ) += 12E

+=

13 43

(47)

EKG I=

2

(48)

EE = 21 =EE

2.8.6 () BdxxaK

xKxaK

xKxaK

xKaGB

aIIIIII

IIII

II

++++=

0 22

2221

2221

2

++++=a

IIIIII dxx

xaKKKBaGB0

222 14

14

++++=

2

18

18

222IIIIII KKKaBaGB

2

222IIIIII K

EK

EKG ++= (49)

IIIIII GGGG ++= (49)

EKG II =

2

, EKG IIII =

2

22III

IIIKG =

4 [13] E1

P

L L

hh

a B

E1

48

( 2) (49)

E

KG II2

= (E1)

G ( 3) P (26) G () ( E2)

( ) xPxM21

= 0 < x < L ( ) xPPLxM

22= L < x < 2L

ax 0 h a 2h 2L a U

( )[ ] dxEIxMU

L

=2

0

2

2

( ) ( )

+

+

=

L

L

L

a

a

dxhBE

xPPLdx

hBE

xP

dxBhE

xP

U2

3

2

3

2

0 3

2

21212

2

21212

2

1212

221

2

( ) ( )3 3323323 332332 16 3216164 EBh aLPEBhLPEBhaLPEBhaPU +=++= (E2)

B (26) dadU

BG 1=

2LLax

M1(x) M2(x)

E2

49

(E2) 32

22

169

hEBaPG = (E3)

(E3) (E1) 234

3BhPaKII =

KII 1. KII 2. P KII a 3. KII ( E)

2.6 2.4.3 K () K K (effect of finite size) K 2a [ 23()] (finite width plate) [ 23()] (force flow line) W x y x (free edge) (gradient) K K K (geometry correction factor) K 23() aKI = 23() ( )WafaK I =

( )Waf

50

W2

a2

W2

a2

() () 23 K 2.4 23()

2.4 2W 2a

[6]

( )Waf Irwin

Wa

aW

2tan2 5% 5.0Wa

Brown ( ) ( ) ( )32 525.1288.0128.01 WaWaWa ++ 0.5% 7.0Wa Feddersen W

a2

sec 0.3% 7.0Wa 1% 8.0=Wa

Koiter ( ) ( )( )WaWaWa

+

1326.05.01 2 1% Wa

Tada ( ) ( ) ( )( )WaWaWaWa

+

1044.0370.05.01 32 0.3% Wa

Tada ( ) ( )[ ]WaWaWa

2sec06.0025.01 42 + 0.1% Wa

51

2.7 (superposition principle) K K K () K K K K ( [ (49)] ) 24 K 4321 ,,, TTTT 43214321 TITITITITTTTI KKKKK +++=+++ (50) 43214321 TIITIITIITIITTTTII KKKKK +++=+++ (50) 43214321 TIIITIIITIIITIIITTTTIII KKKKK +++=+++ (50)

4321 TTTTjK +++ K j ),,( IIIIIIj = 4321 TTTT +++ iTjK K j ),,( IIIIIIj = iT (i = 1, 2, 3, 4)

1T

2T

4T

3T

4321 TTTTjK

+++

1T

1TjK

2T

2TjK

3T

3TjK

4T

4TjK

+

+ +

=

24 K

52

25 P M ( 1) K P M totalIK K tensionIK K bendingIK

bendingItensionItotalI KKK +=

+

=

Waf

WBWM

Waf

WBPK bt

totalI

tf bf

K 26 26 () P K 1 26() 1) P p 2) P p KI 8 KI 26() KI 26()

pIPI KK ,, = (51)

PM

W2

B

a2

25 P M

8

53

P

p

= +

() () ()

P

p

=p

26

K 26() [ 27()] p [ 27()] K () p K K 2.8.3 2.8.4 K (concentrated load per unit thickness) P K p 2.8.3

P

p

P

p

P

() () ()

27

54

5 E1 P AP = ; A 2 a P K D

P/2

P/2

E1 P

E1 E2 ( A) B A B ( C) P - ( D) DICIBIAI KKKK +=+ A B ( )DICIAI KKK += 21

2.0Da [18] () E3 C D a2 aDa 22 += K C D

55

WaaK CI 2

sec=

Wa

Wa

Wa

Wa

aPK DI

+

=

1

16.0957.05.0132

P

+ = +

P

P

Pa aD

A B C D E2

P

P

) D) C

a aD

a aD

P

P

2W 2W2W 2W

a2a2

E3

56

2.8 K K K 4

1) (analytical approach) (stress function) (Greens function) (weight function) (stress concentration) (body force) (Laurent s series)

2) (numerical approach) (finite element) (boundary element) (boundary collocation)

3) (experimental approach) (compliance) (photoelasticity) (strain gage)

4) (bounding) (compounding)

K ( ) K K K 6, 14, 15 K

2.8.1

K (43) Westergaard Muskhelishvili William

2.8.1.1 Westergaard Westergaard

{ } { }ZyZ ImRe += (52)

= ZdzZ , dzZZ = iyxz += , 1=i

57

[ (52)] [3]

{ } { }ZyZxx = ImRe (53) { } { }ZyZyy += ImRe (53) { }Zyxy = Re (53)

dzdZZ =

[3] { } { } = ZyZu ImRe2 121 (54) { } { } += ZyZv ReIm2 121 (54) K [3]

=

III

II

I

zz

III

II

I

ZZZ

zzKKK

00

lim2 (55)

III ZZ , IIIZ Westergaard 1, 2 3 0z

6 [3] Westergaard 2a P b ( E1)

( ) 2222

azba

bzPZI

=

KI

x

y

a

P

Pa

b

E1 2a P

58

ax = az = az += ( )[ ] ( ) 22

22

aaba

baPZI +

+=

(55) KI ax =

( ) ( )[ ] ( )

+

+== 2222

0lim2

aaba

baPaxK I (E1)

0/0

( ) 221

aa + (E1) ( ) K+=+ 2322 24

1211

aaaa

(E1) ( ) ( )[ ]

++

== K 2322

0 24

121lim2

aababaPaxK I

( ) ( )[ ] ( )[ ]

++

+

+== K

23

22

0

22

0 24

1lim2

lim2aba

baPaba

baPaxK I

( ) ( )[ ] 021lim2

22

0+

+

== ababaPaxK I

( )baba

aPaxKI

+==

KI ax = ax = P bx =

( )baba

aPaxKI +

==

7 [3] Westergaard 2a W ( E1)

=

Wa

Wz

Wz

ZI

22 sinsin

sin

59

KI

x

y

a2

W W

E1 2a

ax = az = az +=

+

+=

Wa

Wa

W

Wa

WZI

22 sinsin

sin

(55) 0/0

+W

aW

aW

22 sinsin1

+W

aW

aW

22 sinsin1

Wa

Wa

W cossin2

1

K+

Wa

Wa

Wa

Wa

W

Wa

Wa

W

cossincossin24

sincos 22

(E1)

(E1)

60

=

=

Wa

WWa

Wa

W

Wa

Z I

tan

2cossin2

sin

(55) KI ax =

= Wa

W

K I

tan2

lim20

=W

aWKI tan

=W

aa

WaKI

tan

2.8.1.2 Muskhelishvili Muskhelishvili [16]

( ) ( )[ ]zzz += Re (56)

( ){ } ( ){ } ( ){ }zzyzx ReImRe ++= (56)

( )z ( )z z ( iyxz += )

Muskhelishvili [16]

( ) ( ) ( ){ }zzzi yyxx =+=+ Re422 (57) ( ) ( )zzzi xyxxyy +=+ 222 (57) ( ) ( ) ( )[ ]zzzzivu =+ 21 (57) K ( )( )zzziKK

zzIII = 00lim22 (58)

2.8.1.3 Williams William ( 1 2)

( ) fr 1+= (59)

61

William (59) 2

( ) ( ) ( ) ( )[ ] 1sin1cos1sin1cos 43211 +++++= + CCCCr (60)

321 ,, CCC 4C

(60) 0= 0= r =

K 1 2

2.8.2 ( 28) 1

K+

+

= 23sin

2sin1

2cos

223cos

22

rK

rrK II

xx (61) K+

++

= 2

3sin2

sin12

cos22

3cos22

r

Krr

K IIyy (61)

K++

= 23cos

2cos

2sin

223sin

22

rK

rrK II

xy (61)

2=r 0=

( ) IIIyyoyy KKK 20,2 max =+=

KI

( )2

lim max0

yyIK = (62)

+

/2

r

x

y

28

62

( 28) K 2 3

( ) max0lim xyIIK = (62) ( ) max0lim yzIIIK = (62)

8 [17] E1 P KI y ( )

aP

yy 2

max=

x

y

b2

a2

P

P

E1 P (62) KI

2

2

lim0

= a

P

KI

a

PKI =

9 [17] E1 2b P KI

( )

bbb

bb

bP

yy

+

+

+

=arctan1

12

2max

63

x

y

P

P

b2

E1 P

(62)

2

arctan1

12

2

lim0

+

+

+

=

bbb

bb

bP

KI

bbb

b

bPKI

+

+

+

= arctan1

1lim

2 0

+

= 1arctan1

lim2

2

0

bbb

b

bPKI

bbbb

bb

PKIarctanarctan

lim2 0 ++

+=

( ) ( ) bP

bb

bPKI

=

+++=

22000

2

64

2.8.3 K p(x) (concentrated load) K

( )axGa

PK ,= (63)

P () x G(x,a) (, )

p(x) K

( ) ( )= dxaxGxpaK ,1 (64)

P K

( )1 ( , )K P x G x a dxa

= (64)

b ( )P x bx P bx = (64)

( )abGa

PK ,=

K (64) (63) G(x,a)

10 K 2a P, Q T x ( E1) K A [6]

xaxa

TQP

aKKK

AIII

II

I

+

=

1

K B [6]

xaxa

TQP

aKKK

BIII

II

I

+

=

1

P

P

A B x

yx

2a

Q

Q

T

T

E1

2a

65

K (63) A B ( )

xaxaaxGA +

=,

( )xaxaaxGB

+=,

11 KI x E1 KI ( W = 100 . a = 50 .)

) p0 ) p(x) p0 x = 0 x = a ) P (x = 0)

KI ( E1) [6]

( )( )

( )( )

( )( ) ( ) ( )( )[ ]

+

+

=

Waaxaxax

ax

Wa

ax

Wa

axa

PK I

1176.183.01

30.030.1

1

28.535.4

1

152.32

2

23

21

23

W

y

x

a

P

Px

E1

66

KI (63)

( ) ( )( )

( )( )

( )( ) ( ) ( )( )[ ]

+

+

=

Waaxaxax

ax

Wa

ax

Wa

axaxG

1176.183.01

30.030.1

1

28.535.4

1

152.32,

2

23

21

23

) p0 : KI ( )=

a

I dxaxGpaK

00 ,

1

mpKI 0979.0= p0 Pa

) p0 : KI ( ) =

a

I dxaxGaxp

aK

00 ,1

1

mpKI 0509.0= p0 Pa

) P : KI ( ) ( )=

a

I dxaxGPaK

0

,01

( )a

aPGKI ,0=

m

PKI1571.24=

P N/m

2.8.4 2.7 ( 26) K K yy(x),

67

yx(x), yz(x) 1, 2 3 ( 29) K

( ) ( )dxaxmxK Ia

yyI ,= (65) ( ) ( )dxaxmxK II

ayxII ,= (65)

( ) ( )dxaxmxK IIIa

yzIII ,= (65)

mI(x), mII(x) mIII(x) 1, 2 3 ( ) ( )

aaxv

KEaxm

II

= ,2

, (66) ( ) ( )

aaxu

KEaxm

IIII

= ,2

, (66) ( ) ( )

aaxw

Kaxm

IIIIII

= ,22, (66)

u(x,a), v(x,a) w(x,a) x, y z

Bueckner [19] Paris [20] K 30 KI KI v(x,a) p0 (66) KI p(x) p(x) (65)

x

y

a

yy(x) yx(x)

+ + + +

+

yz(x)

29

68

p0

mI(x,a)

KI v(x,a)

a

p(x)

a

KI

( ) ( )dxaxmxpK Ia

I ,0=

30 KI

12 [16] K 2a P ( E1)

x

y

a

P

Pa

E1

( E2) 26 E2() [ E2()] KI v(x,a) aK I = (E1) ( ) 222, xa

Eaxv = (E2)

(E1) (E2) (66)

( ) ( ) ( )222222, xa axaE aaEaxmI =

=

(E3)

KI E1 E3 KI

( ) =d

dI dxxa

apK 22 (E4)

69

x

y

aa

x

y

aa

() ()

E2 () ()

=adapKI arcsin2

P p

( )dpP 2= KI E1

aP

ada

dPK

dI =

= arcsin22lim0

Dirac delta function (E4) ( ) ( )2 2

a

Ia

aK P x dxa x

=

( ) aPa aPK =

= 22 0

x

y

a

p

ad d

E3

70

(66) v(x,a) 2 1) [21-23] 2) K [19,24,25] K 2.8.4.2

2.8.4.1 K KI,ref (66) ( ) ( )

aaxv

KEaxm

refII

= ,2

,,

(65) K

( )dxa

axvKEK

a refIrefI = ,2 ,,

KI,ref x

( )dxa

axvEKa

refI = ,22,

(profile) [21-23] 2 - y [ 31()]

( ) 0,0

=

=xxaxv (67)

- [ 31()] ( ) 0,

02

2

=

=xxaxv (67)

Petroski Achenbach [21] 1

71

( ) ( )=

+ =0

21

21

,i

ii

i xaaCaxv (68)

Ci

v(x,a)

( ) ( ) ( )232112121

0, xaaCxaaCaxv += (69)

y

x

y

xP P

31

() )

13 2a Petroski Achenbach

( E2 12) KI

aK I = (E1) (69) ( ) ( ) ( )23211212

1

0, xaaCxaaCaxv += (E2)

a

( )

+

+

=

23

21

1

21

210 12

1123

2

1

12

1,ax

axCa

x

ax

Ca

axv

72

ax

( ) ( ) ( ) ( )2312110210 12123

21

2,

++=

CCCCa

axv (E3)

(E1) (E3) (66)

( ) ( ) ( ) ( )

++= 231211021

0 12

123

21

22,

CCCCa

EaxmI (E4)

(E1) (E4) (65)

( ) ( ) ( ) ( ) ( )

++= 11212321222

0

1

23

121

102

10 daCCCC

aEa

ECC =+ 10 5

434 (E5)

(E2) (67)

03 10 =+ CC (E6) (E5) (E6)

EC

1615

0 = EC

165

1 = (E7)

(E7) (E4)

( ) ( ) ( )

+= 2321 1

6451

64151, aaxmI

12 ax

( ) ( )21212 111,

=a

axmEx

= ( ) ( )( ) %100,,,

12

12 axm

axmaxm

Ex

Ex (E8)

E1 3

73

( ) aaxmI ,

0 0.2 0.4 0.6 0.8 1.0

1.00.75

1.5

2.0

%

13

12

(E8)

0.5

0

1.0

1.5

2.0

E1 ( 12) ( 13)

14 2 E1 KI Bowie

1, 3

2

I BowieFK a FaF

r

= + +

F1 = 0.6865, F2 = 0.2772 F3 = 0.9439 KI 2.3

2r aa

E1 2

74

2.3 y

+=2

cos212

sin22

2 rKv I (E1)

E1 o180= xar = ( x-y ) ( ) += 12E ( ) ( ) += 13

24xa

EKv I =

KI Bowie (66) ( )

= 242,

,

,

xaE

KaK

Eaxm BowieIBowieI

( ) ( ) ( )( )

rraF

FK

xaaxa

xaa

axmBowieI

2

2

1

,

22221,

++=

KI (66) =

a

II dxaxmxpK0

),()(2

++

++=

42

322

)(xr

rxr

rxp x ()

KI ( ) a E2 Bowie 2=ra 12 1) K 2) K ( 1 ) Petroski Achenbach

75

aKI

raa

K BowieI

,

0 0.5 1 1.5 21

1.5

2

2.5

3 KI

E2 KI Bowie

2.8.4.2 K Glinka et al. [19, 24, 25] 1

( ) ( )

+

+

+=23

3221

1 111122,

axM

axM

axM

xaaxmI (70)

M1, M2, M3

(semi-elliptical surface crack) (corner crack) 32 ( A)

( ) ( )

+

+

+=23

3221

1, 111122,

axM

axM

axM

xaaxm AAAAI (71)

M1A, M2A, M3A ( B)

76

( )

+

+

+= 23

3221

1, 12,

axM

axM

axM

xaxm BBBBI (71)

M1B, M2B, M3B (70) (71) 3 K 3 Shen et al.[24, 25] [ (67)] 2 (66)

( ) ( )

=

a

axvxK

Ex

axm

I

I ,2

,

( )

=

xaxv

aKE

I

,2

67() ( ) 0,

0

=

=xI

xaxm (72)

67()

( ) 0,0

2

2

=

=x

I

xaxm (72)

2c aB

B Ax

y

z

c aB A

x

y

z

t t

() ()

32

77

[22] K [24] (closed form) K 9 (residual stress) ryy (butt weld) 33 [26]

2

22

41

= l

xryy el

xA

A l 4~6 [26]

x

ryy

y

33

15 [24] K 2a 2 1) 0 [ E1()]

aK I 0= 2) [ E1()] ( ) 210 = aK I

9 Numerical Methods for Engineers Chapra, S.C., Canale, R.P McGraw-Hill

78

x

y

x

y0 0

() ()

E1 K E1() E1()

( ) 01 =xp (E1) ( )

=

ax

xp 102 (E2)

(70) KI (65)

( ) dxaxM

axM

axM

xaa

a

+

+

+= 23

3221

10

00 111122

(E3)

( ) dxaxM

axM

axM

xaax

aa

+

+

+

=

2

3

3221

10

00 111122121 (E4)

(72)

( ) 0111122

0

23

3221

1 =

+

+

+

=xaxM

axM

axM

xax (E5)

(E3) (E4) (E5)

221

322 321

=+++ MMM

2

231

52

21

32

321=+++ MMM

021 32 = MM

79

M1, M2, M3

1685.01 =M 4730.02 =M 7365.03 =M

(70) 2a

2.8.5 K ANSYSTM (routine) K K (node) K K K [27]

1) (displacement extrapolation method) 2) G 3) J- ( 3)

() K (extrapolation) K ( 0=r ) a aa + (18) G (49) K (closed path) J - J- K K 3.2.1 [28,29] (KI KII)

80

2.8.5.1 K 2.3 = y 1 x 2

22

1 rKv I+== (73)

221 rKu II

+== (73) (73) K

+= =

vr

KrI

21

2lim0

(74)

+= =

ur

KrII

21

2lim0

(74) (74) K [30] sin(x)/x x 34 10 x 5x10-6

1.0000000002

1.0000000000

0.9999999998

0.9999999996

0.999999999510-7 10-6 10-5 10-4

x

xxsin

1sinlim0

= xx

x

34 sin(x)/x

10 MathCadTM

81

2.8.5.2 K (74) K 21r 3

1 y [28]

( ) ( )( )

++

++

++=

2sin

23sin

3121

sin312

3sin2

sin1224

1

23

2

1

rE

A

rE

ArE

Kv I (75)

= (75) y

( )( ) ( )( ) 232 11322

211 r

EAr

EKv I

++++==

(76)

( )( ) rAvrEK I 2

*

322

11

+++= =

( ) rAvrK I 2* 322

12

++= = (77)

KI* KI r

KI* r (77) r KI (77) ( 35)

=;r

r

*IK

( )aKI

a

35 KI

82

K [31] K ( 4 ) (crack element) (crack-tip element) () 36 Byskov [32] Barsoum [33] 8 37() - 1/4 37() - (singularity isoparametric element) - 37 8 38() ( 1, 8 7) 37() 1, 8 7 r-1 (perfectly plastic) 6 38() K K 39 A B y A B r = L/4 r = L (76)

( )( ) ( )( ) 232 11121211

41 L

EAL

EKv IA

++++=

( )( ) ( )( ) 232 1132211

21 L

EAL

EKv IB

++++=

A2 KI

( )( ) ( )BAI vvLEK ++= 8

2113

(78)

83

1

2

3

4

36 Byskov

4LL

1 2 3

4

567

8

L

1 2 3

4

567

8

() 8 () 8

37

4L

L

1,8,7

2

3

4

5

6

4L

L

12

3

4

5

6

() 8 () 6

38

84

=

=

AB

CD

0.25LL

39 6

3 x KII

( )( ) ( )BAII uuLEK ++= 8

2113

(78)

uA uB x A B

( = ) ( = ) K 1 2 [34] ( )( ) ( ) ( )

++= DBCAI vvvvL

EK2142

113

(79)

( )( ) ( ) ( )

++= DBCAII uuuuLEK

2142

113

(79)

16 [29] KI (single edge notch tension, SENT) E1 E1 E = 210 GPa, = 0.3

E1 (1)

(.)

(.) A 0.04 6.91410-3 B 0.16 1.38810-2

(1) 39

85

MPa200=

h = 50 .

W = 50 .

a = 25 .

E1

== 43 1.8 L = 0.16 . (78)

( )( ) ( )5639

10388.110914.681016.0

218.13.013

10210 ++

= IK mMPaKI 89.157=

K H W 11

23

1

,

=

Wa

WH

WaF

aKI

1=WH 5.0=Wa 9989.0=F mMPaK I 36.158= 0.3

K

11 Fett, T. Stress intensity factors for edge-cracked plates under arbitrary loading. Fatigue and Fracture of Engineering Material and Structures, Vol. 22, p.301-305. Boundary collocation K

86

2.8.6 (bounding method) [35,36] (upper bound) (lower bound) K (integrand) (63) K p(x)

2.8.6.1 K 40() p(x) 2 ( 40()) pmean p(x) pmean () (63) ( ) ( )=

a

dxaxGxpa

K0

,1 (63)

( ) ( )( ) ( ) ( )( ) ( )

++= a

bmean

b

mean

a

mean dxaxGpxpadxaxGpxp

adxaxGp

aK ,1,1,1

00

b pmean p(b)

( )dxxpa

pa

mean =0

1 (80)

tippx

y

( )xp

x

y

a

meanp2

1b

a

( )xp

) )

40 K

87

40() ( 2) p(x) ( 1) K KUpper ( )=

amean

Upper dxaxGapK

0

, (81) 2.8.6.2 K 41() p(x) 2 [ 41()] ptip p(a) p(x) ptip (63) ( ) ( )( ) ( ) += a tipa tip dxaxGpxpadxaxGpaK 00 ,

1,1

( )( ) ( ) ( )( ) ( ) > a tipa tip dxaGpxpadxaxGpxpa 00 ,0

1,1 G(0,a) x ( )( ) ( ) ( ) ( )

=

a

tip

aa

tip dxpdxxpaaGdxaGpxp

a 000

,0,01

( ) ( )

=

a

tipapdxxpaaG

0

,0

tippx

y

a

( )xp

x

y

3

a

tipp

( )xp

) )

41 K

88

K KLower

( ) ( ) ( )

+= apdxxpa

aGdxaxGpa

K tipaa

tipLower00

,0,1 (82)

(81) (82)

( ) ( ) lpBBKapBpB meanmeantip 2121 +

89

17 K 2a 2W ( 23())

E1 A ( B)

aK =0

2a

2W

2a

2a

A B

2a

K0Kreq

+ +

K1 K2

A B

W-a W-a

A B A B

E1 2 A [6]

+

+

= 0003.10366.04782.07089.055.0

234

Wa

Wa

Wa

WaaK A

0003.10366.04782.07089.0550.0234

10

+

+

=Wa

Wa

Wa

WaQ

KK A (E1)

3 A K0 [6]

004.12803.0568.22279.50317.4234

20

+

+

=Wa

Wa

Wa

WaQ

KK A (E2)

(E1) (E2) (85)

+

+

= 004.1317.0046.3937.5582.4

234

Wa

Wa

Wa

WaaK req

90

Fedderson ( 2.4) E2 ( 10 a/W = 0.7)

1.0

1.1

1.2

1.3

1.4

1.5

0.90 0.2 0.4 0.6 0.8 W

a

aK

Fedderson

Compounding

E2 K Fedderson

2.8.8 K CLL G [ (24) (29)] G K 1 P K CLL

da

dCB

EPK LLI 2= (86)

EE = ( )21 = EE CLL 1) 2) P

LL

91

3) P-LL () 42()

4) - 42() 5) (86) KI

(86) KI 0= 3.0= KI 5 KI

2.9 2.4.2 - (yield strength) (plastic zone) (yield zone) K () 21r 21r K , P

, LL

1a 2a

3a

4a

LLC1

1

, CLL

, a1a 2a 3a 4a

() ()

42

92

K ( 4) ( 5 6) Irwin K K a K aa + a ( ) a LEFM (small scale yielding, SSY) 21r K ( 3)

2.9.1 B 43 - zz z 43 ( z yy ) (zz ) [38] ( 44 )

2

5.2,,,

Y

IKhbaB (87) B a b h ( b )

93

z

xy

B

zz

xxyy

zz

Bz

z

0.50

43 zz z [7]

a b

Bh

44

2.9.2 Von-Mises

( ) ( ) ( ) ( ) 2222222 26 Yxzyzxyxxzzzzyyyyxx =+++++ (88)

Y ( = 0o) 1

94

r

KIxx 2= , xxyy = , 0=zz , 0=== xzyzxy (89)

rKI

xx 2= , xxyy = , ( )yyxxzz += , 0=== xzyzxy (89) (89) (89) (88) yy y Yyy = (90) Yyy 2441

1

+=

= 0.33 Yyy 3 (90) (90) (90) ( y) (90) (plastic constraint factor, PCF) PCF xx (89) PCF 3 [39] 68.122 = 3

2.9.3 2 Irwin (strip yield model) 2.9.3.1 Irwin Irwin 12 ( o0= ) 1 ry ( A 45)

Yy

I

rK =2

ry 2

21

=

Y

Iy

Kr (91)

12 3

95

(91) - Irwin AA ( 46) rp 46 ==

yy rI

r

yypY BdrrKBdrBr

00 2

B yy

r

rKI

yy 2= ( )

yr

Y A

45 ()

yy

r

pr

Y AA

yr

-

46

96

rp [ (91)] y

Y

Ip r

Kr 212

=

= (92)

PCF 3

2

31

=

Y

Ip

Kr (92) 2 3 (shear yield stress) Y Y 2

21

=

Y

IIp

Kr (93)

2

31

=

Y

IIp

Kr (93)

3 2

1

=

Y

IIIp

Kr (94)

2

31

=

Y

IIIp

Kr (94)

- a a ry ( 47) (effective crack length) aeff

yeff raa += (95)

(effective stress intensity factor) Keff

( )effeff aKK = (96)

ry K Keff ry Keff (iteration)

1. K a Kold 2. ry (91) (92)

3. aeff (95) 4. Keff (96) Knew

97

5. Knew Kold - Knew Kold Keff = Knew - Kold Knew 2-5

yy

r

Y AA

yr

-

O O

a+ry

yy

47 Irwin

18 Keff 2a aK =

(96) ( )yeff raK += (91)

+=

2

21

Y

effeff

KaK

Keff 2211

=

Y

effaK

98

19 Keff 2a Irwin ( 2.4 ) 2a 2 . 40 . 100 250 MPa W 150 . MPaY 420=

2.4

=W

aa

WaK tan

+

+

+=

2

2

2

21tan

212

1

Y

eff

Y

effY

effeff

Ka

WKa

WKaK

i j

+

+

+=+

2

,2

,

2

,1, 2

1tan

212

1

Y

jieffi

Y

jieffi

Y

jieffijieff

Ka

WKa

WKaK

=Wa

aWaK i

iiieff

tan0,

) 100 MPa K 4 25 E1()

2a () 1 2 3 4 5

2 5.606 5.685 5.687 5.687 5.687 6 9.723 9.861 9.865 9.865 9.865 14 14.951 15.172 15.179 15.179 15.179 22 18.974 19.277 19.286 19.287 19.287 30 22.573 22.975 22.990 22.990 22.990 40 26.954 27.541 27.567 27.568 27.568

99

) 250 MPa 60 E1()

2a () 1 2 3 4 5 6 7 8

2 14.015 15.207 15.409 15.444 15.451 15.452 15.452 15.452 6 24.306 26.393 26.750 26.813 26.825 26.827 26.827 26.827 14 37.377 40.735 41.342 41.457 41.479 41.483 41.484 41.484 22 47.436 52.074 53.003 53.200 53.241 53.250 53.252 53.253 30 56.432 62.702 64.178 64.548 64.643 64.667 64.674 64.675 40 67.386 76.844 79.875 80.959 81.360 81.511 81.567 81.589

5

10

25

20

15

10 20 30 400 2a (.)

( )mMPaKeff

0

20

80

60

40

10 20 30 400

( )mMPaKeff

2a (.)

() 100 MPa () 250 MPa

E1 K

100

2.9.3.2 Dugdale [7] ( 48) (strip yield) () Dugdale (strip yield model) Dugdale 2 1) 2(a+) 49() 2) 2(a+) (closure stress) Y 49() () K 49() K 49() K 49() ( ) += aK (97)

K 47() K P x ( 50) [7]

( )( )( )

( )( )

+++++

+++= xa

xaxaxa

aPKP

(98)

( )+a2a2

48 Dugdale

101

= +( )+a2

a2( )+a2 ( )+a2

Y

() () ()

49 Dugdale - () 2a () 2(a+) () () 2(a+) ()

x

PP

PP

( )+a2

50

x

dxP Y= 49() ( ) dxxa

xaxaxa

aK

a

a

Yclosure

+

+++++

+++

=

++=

a

aaK Yclosure arccos2 (99)

102

0=+ closureKK

(97) (99)

=+ Ya

a

2cos (100)

= 1

2sec

Ya

(101)

(100)

...2!6

12!4

12!2

11642

+

+

=+ YYYa

a

2

2

22

88

==

Y

I

Y

Ka

(101)

Y

103

SSY LEFM Hauf [8] LEFM SSY 3 (middle tension plate, MT) (single-edge notch tension, SENT) (double edge notch tension, DENT) 52 a/W 0.2, 0.35 0.5 LL 1 LEFM SSY 2.5 (limit load)15 a/W () SSY LEFM LEFM SSY LEFM [38] ( )

24,,

Y

IKhaba (102)

(102) 44

-

LL

LEFM

SSY

51 - LEFM, SSY, -

15

104

2a

2W

a

W 2W

a a

() MT () SENT () DENT

52

2.5 LEFM SSY a/W

0.2 0.35 0.5

LEFM SSY LEFM SSY LEFM SSY MT 60(1) 65 40 50 30 40 SENT 75 85 40 50 20 35 DENT 70 85 50 65 35 50

(1) 8

2.11 (crack tip opening displacement, CTOD) CTOD (large-scale yielding) LEFM SSY K CTOD - 3 ( ) 53 CTOD 16 CTOD Irwin 16 1 2 (crack tip sliding distance, CTSD)

105

Dugdale CTOD

2.11.1 CTOD Irwin y ( 2.3)

+=2

cos212

sin22

2 rKv I

54 yrr = =

221 y

Ir

Kv += CTOD 54

2

12 yIr

KvCTOD +== (103)

a

CTOD

53 CTOD

vCTOD

y

x

a yr

54 CTOD Irwin

106

(91) (103)

E

KY

I

24= (104)

(48)

Y

IG

4= (105)

2.11.2 CTOD Burdekin Stone [40] Westergaard 55() 55() 55() CTOD

22 +a 55()

( )22

+=

az

zZ (106)

22 +a Y ax = += ax 55()

P x 22 +a ( 50) ax = += ax

( )( ) ( )dxxzazxaz

Za

a

Y+

++=

2222

222

( )+a2

Y

a2

( )+a2

+

( )+a2

Y

a2

=

() () ()

55

107

( )

( )( )

++

++=

22

22

22cotarccos2

aaaz

zaarc

aa

az

zZ Y

(107)

(106) (107) 55()

( ) ( )( )

( )

++

+++

+=

22

22

2222cotarccos2

aaaz

zaarc

aa

az

z

az

zZ Ytotal

(108) (53) ()

{ } { }ZyZxx = ImRe { } { }ZyZyy += ImRe { }Zyxy = Re

( 0=y ) { }Zyyxx Re== 0=xy += ax (108)

( ) ( )

++=

+

aa

az

z

az

z Y arccos22222

+=

aa

Yarccos2

=+ Ya

a

2cos

+ aak

( )

+= 2

22

1cot2

kaz

zkarcZZ Ytotal

(109)

y 54()

{ } ( ) { }[ ]ZyZE

v Re1Im21 += = ZdzZ

108

( 0=y ) ZE

v Im2= (110)

(109) [ ]212 azZ Y = (111)

+=

11

1cot

2

2

1

k

za

arc

( )( )

++=

22

22

2 cot aaazarc

(111) (110)

( ) ( )

+

+

+=

2

221

2

221

1coth

11coth4

kza

zkz

kza

aa

Ev Y

CTOD 56

== = kEavCTOD Yyaz

1ln82lim 0

+

+

= ...

2121

2218

42

YY

Y

Ea

+

+= ...

2611

22

YY

I

EK

(112)

CLCLa

CTOD

56 CTOD

109

SSY Y (112)

Y

I

Y

I GE

K ==

2

(113)

(104) (113) CTOD Irwin Dugdale

Em

KY

I

= 2

(114)

m 2.12

( 2.2) Griffith ( 2.3) K ( 2.4) Irwin ( 2.5) Irwin K K ( 4)

K ; ; K ( 2.6) ( 2.7) 2 K K ( (50))

110

( (51)) K ( 2.8.1 - 2.8.4) ( 2.8.5) ( 2.8.6 - 2.8.7) ( 2.8.8)

2 Irwin Dugdale ( (104) (113)) - () Irwin K LEFM LEFM SSY (2.9.3) LEFM SSY ( (102) 2.10)

CTOD ( 2.11) CTOD K 2.13 [1] Timoshenko, S.P. History of strength of materials, McGraw-Hill, 1953. [2] Parton V.Z. Fracture mechanics: From theory to practice, Gordon and Breach Science

Publisher, 1992. [3] Gdoutos, E.E. Fracture mechanics : An introduction, Kluwer Academic Publishers, 1993. [4] Timoshenko, S.P., and Goodier, J.N. Theory of elasticity, 3rded, McGraw-Hill International

Edition, 1970. [5] Daoud, O.E.K., Cartwright, D.J., and Carney, M. Strain energy release rate for a single edge-

cracked circular bar in tension. J. of Strain Analysis, Vol. 13 No. 2 1978 p.83-89. [6] Tada, H., Paris, P.C., and Irwin, G.R. The stress analysis of cracks handbook. Del Research

Corporation, 1973. [7] Anderson,T.L. Fracture mechanics: Fundamental and application, 2nd ed., CRC Press 1995.

111

[8] Hauf, D.E. Modified effective crack-length formulation in elastic-plastic fracture mechanics. BSc.Thesis. MIT, 1992.

[9] Wang, X. Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions. Eng. Frac. Mech., Vol. 69, 2002, pp.1339-1352.

[10] Suresh, S. Fatigue of materials. Cambridge University Press, UK, 1991. [11] Hellan, K. Introduction to fracture mechanics. McGraw-Hill, 1984. [12] Irwin, G.R. Analysis of stresses and strains near the end of a crack transversing a plate. Trans

ASME J. of App. Mech., Vol. 24, No. 3, 1957, pp.361-364. [13] Wang, C.H. Introduction to fracture mechanics. DSTO-GD-0103, 1996. [14] Rooke, D.P., and Cartwright, D.J. Compendium of stress intensity factors, Her Majestys

Stationery Office, London, 1974. [15] Murakami, Y. Stress intensity factor handbook. Pergamon Press, New York, 1987. [16] Parker, A.P. The mechanics of fracture and fatigue. E.&F.N. Spon, USA, 1981. [17] Okamura, H. Senkei Hakai Riki Gaku Nu Mon. (In Japanese), 1995. [18] Broek,D. Practical use of fracture mechanics. Kluwer academic publisher, Netherlands,1989. [19] Boyd, K.L., Krishnan, S., Litvinov, A., Elsner, J.H., Harter, J.A., Ratwani, M.M. and Glinka, G.

Development of structural integrity analysis technologies for aging aircraft structures : Bonded composite patch repair and weight function methods. Wright Laboratory, WL-TR-97-3105, July, 1997.

[20] Paris, P.C., McMeeking, R.M. and Tada, H. The weight function method for determining stress intensity factors. Cracks and Fracture, ASTM STP 601. American Society for Testing and Materials, 1976, pp.471-489.

[21] Petroski, H.J. and Achenbach. Computation of the weight function from a stress intensity factor, Eng. Frac. Mech., Vol. 10, 1978, pp. 257-266.

[22] Varfolomeyev, I.V. and Hodulak, L. Improved weight functions for infinitely long axial and circumferential cracks in a cylinders. Int. J. Pres & Piping, Vol. 70, 1997, pp. 103-109.

[23] Fett, T. and Mattheck, C. On the Calculation of crack opening displacement from the stress intensity factor. Eng. Frac. Mech., Vol. 27, No.6, 1987, pp. 697-715.

[24] Shen, G. And Glinka, G. Determination of weight functions from reference stress intensity factors. Theoretical and Applied Fracture Mechanics, Vol. 15, 1991, pp. 237-245.

112

[25] Shen, G.and Glinka, G. Weight functions for a surface semi-elliptical crack in a finite thickness plate. Theoretical and Applied Fracture Mechanics, Vol. 15, 1991, pp. 247-255.

[26] Bergman, M., Brickstad, B., Dahlberg, L., Nilsson, F. and Sattari-Far, I. A procedure for safety assessment of components with crack handbook. SA/FoU-Report 91/01, 1991.

[27] Shiratori, M., Miyoshi, T., and Matsushita, H. SuuChi Hakai Riki Gaku. (In Japanese), 1996. [28] Guinea, G.V., Planas, J. and Elices, M. KI evaluation by the displacement extrapolation

technique. Eng. Frac. Mech., Vol. 66, 2000, pp.243-255. [29] Zhu, W.X. and Smith, D.J. On the use of displacement extrapolation to obtain crack tip singular

stresses and stress intensity factors. Eng. Frac. Mech., Vol. 51, No.3, 1995, pp.391-400. [30] Sanford R.J. Principles of fracture mechanics. Pearson Education, 2003. [31] Chan, S.K., Tuba, I.S., and Wilson, W.K. On the finite element method in linear fracture

mechanics. Eng. Frac. Mech., Vol.2, 1970, pp.1-17. [32] Byskov, E. The calculation of stress intensity factors using finite element method with cracked

elements. Int. J. of Frac. Mech. Vol. 6, No. 2, 1970, pp. 159-167. [33] Barsoum, R.S. On the use of isoparametric finite elements in linear fracture mechanics. Int. J.

of Num Methd. Eng, Vol.1, No. 1, 1976, pp.25-37. [34]

. 2544. [35] Smith, E. Simple approximate methods for determining the stress intensification at the tip of a

crack. Int. J. of Fracture, Vol. 13, No. 4, 1977, pp.515-518. [36] Cartwright, D.J. Bounding functions for stress intensity factors. Int. J. of Fracture, Vol. 24.

1984, pp. 35-44. [37] Rooke, D.P., Baratta, F.I., and Cartwright, D.J. Simple methods of determining stress intensity

factors. Eng. Frac. Mech., Vol. 14, 1981, pp.379-426. [38] Dowling, N.E. Mechanical behavior of engineering materials : Engineering methods for

deformation, fracture and fatigue. Prentice Hall Internaltional Inc, New Jersey, 1993. [39] Broek,D. Elementary engineering fracture mechanics, 4th eds., Martinus Nijhoff Publish-

ers,1986. [40] Burdekin, F.M. and Stone, D.E.W. The crack opening approach to fracture mechanics in

yielding materials. J. of Strain Analysis., Vol. 1, No.2, 1966, pp.145-153.

113

William

1. 1 2 1.1 William 1 [11]

( ) fr 1+= (1)

2 04 = (2) (2)

011112

22

2

2

2

22

2

=

+

+

+

+ rrrrrrrr

( ) ( ) 0112 2222244 =+++ fd fdd fd (3)

(3)

( ) ( ) ( ) ( ) 1sin1cos1sin1cos 4321 +++++= CCCCf (4)

x

yrr

r

1

114

( )( ) ( ) frr 122

1 +== (5)

22

211

+=

rrrrr

( ) ( ) ( ) frr

frr

++= +12111 ( ) ( ) ( )[ ] ffr ++= 11 (6)

=

rrr1

( )

= + fr

rr11

( ) fr = 1 (7)

1 = 0== r (5) (7) 0= 0=f 0= r 0=d

df

1 : = , 0=f

( ) ( ) ( ) ( ) 1sin1cos1sin1cos0 4321 +++++= CCCC (8)

2 : = , 0=f

( ) ( ) ( ) ( ) 1sin1cos1sin1cos0 4321 +++= CCCC (9)

3 : = , 0=ddf

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

1cos11sin11cos11sin10

4

321

++++++=

CCCC (10)

4 : = , 0=ddf

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

1cos11sin11cos11sin10

4

321

+++++++=

CCCC (11)

(8) (9)

( ) ( ) 01cos1cos 31 =++ CC (12)

115

(8) (9)

( ) ( ) 01sin1sin 42 =++ CC (13)

(10) (11)

( ) ( ) ( ) ( ) 01cos11cos1 42 =+++ CC (14)

(10) (11)

( ) ( ) ( ) ( ) 01sin11sin1 31 =+++ CC (15)

(12) (15)

( ) ( )( ) ( ) ( ) ( )

=

++

+00

1sin11sin11cos1cos

3

1

CC

(16)

(13) (14)

( ) ( )( ) ( ) ( ) ( )

=

++

+00

1cos11cos11sin1sin

4

2

CC

(17)

(16) (17) (non-trivial solution) (16)

( )[ ]( ) ( ) ( )[ ]( ) ( ) 01sin11cos1sin11cos =+++

( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ] 01sin1cos1sin1cos1sin1cos1sin1cos

=++++++

( )( ) ( )[ ] 011sin11sin =++++

02sin2sin =+ (18)

(17)

( )[ ]( ) ( ) ( )[ ]( ) ( ) 01cos11sin1cos11sin =+++

( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ] 01cos1sin1cos1sin1cos1sin1cos1sin

=++++++

( ) ( )[ ] ( ) ( )[ ] 011sin11sin =++++

( ) 02sin2sin =+

02sin2sin =+ (19)

116

= = (18) (19)

(18) : ( ) ( )[ ] 02sin2sin =+ 02sin =

(19) : ( ) ( )[ ] 02sin2sin =+ 02sin =

( ) 02sin =

2n= n n = ,...3,2,1n

C1 C2 C3 C4 ( (12) (15)) 2n= (16) (17) (16)

( ) ( )

( ) ( )

=

+

+

+

00

12

sin12

12

sin12

12

cos12

cos

3

1

n

n

CC

nnnn

nn

( ) ( ) nn CnCn 31 12cos12cos

+=

( )( ) nn

Cn

n

C 131

2cos

12

cos

+

= (20)

( ) ( ) nn CnnCnn 31 12sin1212sin12

+

+=

( )( ) nn

Cnn

nn

C 131

2sin1

2

12

sin12

+

+

= (20)

117

(20) (20) n

,...6,4,2=n (20) nn CC 13 = (21) ,...5,3,1=n (20) nn Cn

nC 13 22

+= (21)

(17)

( ) ( )

( ) ( )

=

+

+

+

00

12

cos12

12

cos12

12

sin12

sin

4

2

n

n

CC

nnnn

nn

( ) ( ) nn CnCn 42 12sin12sin

+=

( )( ) nn

Cn

n

C 241

2sin

12

sin

+

= (22)

( ) ( ) nn CnnCnn 42 12cos1212cos12

+

+=

( )( ) nn

Cnn

nn

C 241

2cos1

2

12

cos12

+

+

= (22)

(22) (22) n

,...5,3,1=n (22) nn CC 24 = (23) ,...6,4,2=n (22) nn Cn

nC 24 22

+= (23)

(21) (23) f

+

++

+

= 1

2sin1

2cos

221

2sin1

2cos 2121

nCnCnnnCnCf nnnnn ; ,...5,3,1=n

++

+

+

= 1

2sin

221

2cos1

2sin1

2cos 2121

nCnnnCnCnCf nnnnn ; ,...6,4,2=n

118

=

=

++

+

+

+

+

+

++

=

evennnn

oddnnnn

nnnnCnnC

nnCnnnnCf

12

sin221

2sin1

2cos1

2cos

12

sin12

sin12

cos221

2cos

21

21

William

++

+

+

+

+

+

++

=

=

+

=

+

12

sin221

2sin1

2cos1

2cos

12

sin12

sin12

cos221

2cos

21

12

21

12

nnnnCnnCr

nnCnnnnCr

nnevenn

n

nnoddn

n

(24)

1.2 (24) (5)

=

=

++

+

+

++

+

+

++

+=

evennn

nn

oddnn

nn

nnnnC

nnCrnn

nnC

nnnnC

rnn

12

sin221

2sin

12

cos12

cos

21

2

12

sin12

sin

12

cos221

2cos

21

2

2

11

2

2

11

2

n = 1 n = 2

( )

2cos12

23sin

2sin

23cos

31

2cos

21

23

12

211121

+

+

+

=

C

CCr

( )

2cos12

23sin

2sin

43

23cos

31

2cos

43

12

2121

1121

+

+

+=

C

CrCr (25)

119

rr (24) (6)

=

+

=

+

=

=

+

++

+

+

+

++

+

++

+

+

++

+

+

++

+

+

++

+

+

++

+=

evenn

n

nn

oddn

n

nn

evennn

nn

oddnn

nn

rr

nnnnnnC

nnnnC

rr

nnnnC

nnnnnnC

rr

nnnnC

nnCrn

r

nnC

nnnnC

rnr

12

sin122

212

sin12

12

cos12

12

cos121

12

sin12

12

sin12

12

cos122

212

cos121

12

sin221

2sin

12

cos12

cos1

21

12

sin12

sin

12

cos221

2cos

12

1

22

2

22

11

22

22

2

22

11

22

2

1

2

2

1

2

n = 1 n = 2

( )

2cos122

3sin49

2sin

41

23cos

43

2cos

41

23sin

2sin

23cos

31

2cos

23

12

211121

211121

++

++

+

+

+=

C

CCr

CCrrr

( )

2cos12

23sin

43

2sin

45

23cos

41

2cos

45

12

2121

1121

++

++

=

C

CrCrrr (26)

120

r (24) (7)

+

++

+

+

++

+

+

+

+

+

+

+++

=

=

+

=

+

12

cos122

212

cos12

12

sin12

12

sin12

12

cos12

12

cos12

12

sin122

212

sin12

2

11

2

2

11

2

nnnnnnC

nnnnCr

nnnnC

nnnnnnC

r

n

n

evenn

n

n

n

oddn

n

+

++

+

+

++

+

+

+

+

+++

=

=

=

12

cos122

212

cos12

12

sin12

12

sin12

2

12

cos12

12

cos12

12

sin122

212

sin12

21

2

11

2

2

11

2

nnnnnnC

nnnnCrn

nnnnC

nnnnnnC

rnr

n

n

evenn

n

oddnn

nn

n = 1 n = 2

2sin2

23cos

23

2cos

21

21

23sin

21

2sin

21

21

12

2121

1121

C

CrCrr

=

2sin223cos

43

2cos

41

23sin

41

2sin

41

122121

1121

CCrCrr

++

+= (27)

1 2 ( 1)

( ) 2cos122

3cos2

cos54 12

21

11 ++

= CrCrr (28)

( ) 2cos1223cos

2cos3

4 1221

11 +

+= CrC (28)

2sin223sin

2sin

4 1221

11 CrCr

+= (28)

121

( 2)

+= 2

3sin32

sin54

21

21 rCrr (29)

= 2

3sin32

sin34

21

21 rC (29)

+= 2

3cos32

cos4

21

21 rCr (29) 1.3

1

[ ] = rrrr E1

=

23cos

2cos3

23cos

2cos5

41 2

111 rC

E (30)

rur

rr =

( ) ( )

+=

23cos1

2cos35

421

11 rE

Crur

( ) ( )

+

= 23cos1

2cos35

214

21

11 rE

Cur

( )

++=

23cos

2cos

13512

421

11 r

ECur

( ) ( )( )

++=

23cos

2cos

1126

421

11

r

Cur

( )

=2

3cos2

cos124

21

11 rCur (31)

+

=13

122

( )rrE =1

( ) ( )

++= 2

3cos12

cos534

1 21

11 rCE

++=

23cos

2cos

153

821

11

rC

( )

++=

23cos

2cos

13326

821

11

r

C

( )

+=

23cos

2cos32

8

21

11 rC (32)

ruru =

( ) ( )

+=

23cos2

2cos122

23cos

2cos32

821

11 rCu

( )

+=

23cos3

2cos12

821

11 rCu

( )

++=2

3sin22

sin1228

21

11 rCu

( )

++=2

3sin2

sin124

21

11 rCu (33)

(31) (33) ( )0=r

2

( )

+=2

3sin32

sin124

21

21 rCur (34)

( )

+=2

3cos32

cos124

21

21 rCu (35)

123

2. 3 2 0== vu 0==== xyzzyyxx

ruGG zrzrz == (36)

== zzz ur

GG (37)

( ) 0=+

zrzrr

0=

+

zz u

rG

ruGr

r

01 222

=

+

+

zzz urr

ururG (38)

02 = zu

0=z 0=zu = (39)

( ) gruz = (38) ( ) ( ) ( ) ( ) 01 112 =++ grgrgrr ( ) ( )[ ] 021 =+ ggr (40)

(39)

( ) 0= gr = ( ) 0= g = (41)

(40) sincos BAg += (42)

x

y r

rz

2

124

z (42)

sinAg = 0=

= ddg ( 41)

( ) 0cos =A

= 0cos =A

0= (trivial solution) 0cos = 2n= ,...3,2,1=n

==

,...5,3,1 2sin

nn

nDg

=

=,...5,3,1

2

2sin

n

n

nznrDu

2

sin21

1rDuz = (43)

ruz

rz =

2

sin2,...5,3,1

12 nrnD

n

n

nrz =

=

2

sin2

21

1 = rDrz (44)

= zz ur

=

=,...5,3,1

2

2cos

2n

n

nznnrD

r

2

cos22

cos2

21

121

1 =

= rDrDrz

(45)

3 -

- (medium strength steel) (stainless steel) r-1/2 K (LEFM) SSY 2 - (elastic-plastic fracture mechanics, EPFM) 1.2 EPFM LEFM 2 CTOD - J- J- J-

3.1 2.11 2 CTOD ( ) SSY K Burdekin Stone [1] (large scale yielding, LSY) c Burdekin Stone c - el (rigidly plastic) pl

plel += (1)

126

el (114) 2 pl (crack mount opening displacement, CMOD) plm pl plm (uncrack ligament) W-a (rigid body rotation) 1 2 2 3 (3-point bending specimen) C(T) (compact tension specimen) 1() P P m 1() m - plm elm elm 1() plm m elm O [ 1()] plm pl ()

( ) ( ) aaWraWr plpl

m

pl

pl

+=

( )( ) aaWraWr

pl

plmpl

pl += (2)

rpl (plastic rotational factor) 0.432 a/W = 0.8 0.451 a/W = 0.3 [2] (1) ( )( ) aaWr

aWrEm

K

pl

plmpl

Y

I

++=

2

(3)

C(T) 2 pl plm

( )( ) ZaaWraWr

pl

plmpl

pl ++= (4)

127

plm

P

m

pl

m

W

pla

( )aWrpl O

P

S

Wb

m

elm

m

1

m

1

() () 1 plm pl 3

P

P

a

W

plpl

m O

Z

( )aWrpl

2 plm pl C(T)

[2]

+

+

= 2174.011

2

aWW

aWW

aWWrpl (5)

Z

128

P ( LL ) JIc 4 plLL LL Z = 0 (4) ( 3) 45 [3] 3.2 J- 3.2.1 Rice[4] J- B ( 4)

= ds

xuTUdyJ ii (6)

U ( = ijij dU )

iT (surface force)

J- - (loading) (6) [3]

dad

BJ = 1 (7)

3

129

ds

Tva

x

y

A

4 J-

(7) G [ (18) 2] 5 J- (7) - a a+da U= (7)

dadU

BJ 1= (8)

5 U = LLPdU (9) (9) (8) = LLdaPBJ LL

1 (10)

WU = LLPW = 6 = U

U (complementary strain energy)

(7) a

UB

J =

*1 (11)

6 = dPU LL (12) (11) dP

aBJ

P

LL = 1 (13)

130

LLP a daa +

P

LL

5 ( 5) J-

(8) (11)

GJ (14)

P

LL

U

*U

6 U *U

131

3.2.2 J- J- [ (6)] 1) J- 2) J- J- (path-independent integral)

7 (direction cosines) ds

dsdynx = ds

dxny = jiji nT =

(6) = dsxunUnJ ijij1

( ) +=+ dAyfxfdsnfnf Ayx 2121 (15)

dAxu

yxu

xxUJ

A = 1211

x = x1 , y = x2 u = u1

dAxu

xxUJ

A

iij

j

= (16)

x

y

=

y

x

nn

nvdsdy

dx

7 xy

132

(16) x

UxU ij

ij

=

x

ijij

=

+

=

i

j

j

iij x

uxx

ux2

1

=

xu

xi

jij (17)

(16) j

ijii

jij

iij

j xxu

xu

xxu

x

+

=

0=

j

ij

x

=

xu

xxu

xi

ijj

i

jij (17)

(17) (17) (16) J-

= = 0dsxuTUdyJ ii (18)

( 8) 1 3 2 4 1 3 0

4321=+++ JJJJ (19)

2 4 ( )0=iT dy = 0 J2 = J4 = 0 (19)

21 = JJ

J- 1 2 J-

12 34

8 xy

133

3.2.3 J- Hutchinson [6] Rice Rosengren [7]

n

YYY

+=

(20)

Y Y (yield strain) EYY = n

(20)

Y

ijn

Y

eYijkkijij

sE

sE

1

23

3211

+++= (21)

ijs (deviatoric stress components) ijije ss2

3= (effective stress) (21)

( ) ,~1

1

nrI

Jij

n

nYYYij

+

= (22)

( ) ,~1

nrI

Jij

nn

nYYYij

+

= (23)

( ) ,~111

nurI

Ju innn

nYYYi

++

= (24)

nI n ij~ , ij~ iu~ n

134

[6]

32 001262.00404.04744.0568.6 nnnIn += (25)

[6]

342 1045816.00175.02827.0546.4 nnnIn += (25)

(22) (24) J- - ( K) ( )0=r n = 1 (22) (24) ( r-1/2) J- HRR (HRR singular zone) (Hutchinson, Rice Rosengren ) 3.3 J- CTOD K G J- SSY 9 Y ( A C) ( B) 0=dy (6) y

Yyi TT == J-

= dxxvJ Y (26)

dvJa

aY

+=

2 ( ) ( )[ ]avavY += 2

YJ = (27)

135

CLCL

a

A

B

C

x

y

X

Y

Y

9 J-

Rice [7] J n (22) 5 YJ 7.1= YmJ =

Shih [3] J HRR 3 10 A x y u v = rr =

x

y

v

*r

u

A

A'

10

136

( ) ,2

= rv (28)

( ) ( ) ,, *** rvrur += (29)

(24) (29) uu 1 vu 2

( ) ( ) ( )[ ]nY

nn

nY IJnvnur

11* ,~,~

++= (30)

(30) (24) y v

( ) ( ) ( )[ ] ( )nuI

JnvnuI

Jv in

nY

nn

nY

nn

nYYY ,

~,~,~1

1111

+++

+

=

( ) ( ) ( )[ ] ( )nuI

Jnvnuv inY

nnY ,~,~,~

11 += (31) (28) ( )( ) ( ) ( )[ ]

Yn

nnY JI

nvnunv

11

,~,~,~2 += (32)

Ynd

J 1= (33) dn ( )( ) ( ) ( )( )[ ]

n

nnYn I

nvnunvd11

,~,~,~2 += (34)

11 dn n1 1= dn n EY ( )=n ( 16()) dn 1 (27) Shih HRR 1) 2) ( 2

137

0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1.0

nd

n1

0.0080.0040.0020.001

0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1.0nd

n1

0.0080.0040.0020.001

EY EY

() ()

11 n EY dn [ (34)]

3.4 J- J- 2 1) J- [ (6)] 2) [ (9)] J- (reference stress method) K (limit load)

3.4.1 J- [11]

1 [4] E1 a+b H a v () J- b x = b

138

x

y

v

a b

H

E1 H

E2 0654321

=+++++ JJJJJJ 1 5 : 0=ij 0=

xui 0

1=J 05 =J

2 4 : 0=dy 0=

xui 0

2=J 04 =J

3 : 0=

xui ( ) ( )HbxUdybxUJ ====

3

3

6 7 : 0=dy 0=iT 06 =J

( )HbxUJJ === 3

( ) ( ) ( )bxbxbxU yyyy ==== 2

1 221

yyE=

x

y

1 2

3

45 67

E2 J-

139

211 = bx =

y Hv

yy =

HEvJ

2

21 =

J- (isoparametric element) (interpolation function) (mapping function) x-y 12 8 x-y Ni ( ) ,ii NN = (35) i 12 ( )( )( )1 1 1 1 14N = + + (36) ( )( )22 1 1 12N = (36) ( )( )( )3 1 1 1 14N = + (36)

u

12

v

3

4

56

7

8

1 2 3

4

567

8

1=1=

1=

1=

x

y

() x-y () ()

12 (map) x-y

140

( )( )24 1 1 12N = + (36) ( )( )( )5 1 1 1 14N = + + + (36) ( )( )26 1 1 12N = + (36) ( )( )( )7 1 1 1 14N = + + (36) ( )( )28 1 1 12N = (36) (x,y)

==

8

1iii xNx

==

8

1iii yNy (37)

(chain rule) (37) f (Jacobian matrix)

[ ]

=

=

==

==8

1

8

1

8

1

8

1

2221

1211

ii

i

ii

i

ii

i

ii

i

yNxN

yNxN

yx

yx

JJJJ

J

(38)

[ ]

=1121

12221 1JJJJ

JJ (39)

[ ] 1J f

[ ]

=

yfxf

Jf

f

(40)

[ ]

=

f

f

J

yfxf

1 (40)

f u(x,y) v(x,y)

141

[ ]

=

u

u

J

yuxu

1 (41)

[ ] 1vv

x Jv vy

=

(41)

x-y ( )

==

r

iiiuNu

1

, (42)

( ) =

=r

iiivNv

1

, (42)

=

8

8

2

2

1

1

821

821

821

821

000

000

000

000

vu

vuvu

NNN

NNN

NNN

NNN

v

v

u

u

M

K

K

K

K

(43)

J- U

+

+

+=

yv

xv

yu

xuU yyxyxx 2

1 (44)

xuT ii

xvT

xuT

xuT yxii

+=

(45)

{ }T ij

{ }

=

y

x

yyxy

xyxx

y

x

nn

TT

T (46)

142

yxyxxxx nnT += (47) yyyxxyy nnT += (47)

nv 13

dsdynx = (48)

dsdxny = (48)

22 dydxds += (49)

(numerical integration) J- (6) (Gauss quadrature) ( ) ,f ( -1 1 ) -1 1 ) f (Gauss point) (Gauss points weight)

p [ 14()]

( ) ( )=

NG

ipiip fwdf

1

1

1

,, (50)

NG p iw i p

x

y

ds

dx

dy

=

y

x

nn

nv

13 (unit normal vector)

143

x

y

12

3

4

56

7

8

+

++

+

GP1

GP2

GP3

GP4

3

1=p

12

3

4

56

7

8

+

++

+

GP1

GP2

GP3

GP43

1=

3

1=p

31=

() ()

14 2 ) ) p [ 14()]

( ) ( )=

NG

iipip fwdf

1

1

1

,, (50)

NG p iw i p J- J-

=

=NE

e

eJJ1

(51)

NE eJ J- [( 14()]

ds

xuTyUJ ii

e

=1

1

(52)

(44), (47), (49) (50) (52)

144

( ) ( )

=

=

+

++

+

+

+

+=

NG

iyyyxxyyxyxxxi

yyxyxx

NG

ii

e

yxxvnn

xunnw

yyv

xv

yu

xuwJ

1

22

1 21

(52)

2

12211

12

22 JJ

J

ddy

ddx

ddy

dsd

ddynx +

=

+

==

2

122

11

11

JJ

Jny +=

[ 14()]

ds

xuTyUJ ii

e

=1

1

(53) (44), (47), (49) (50) (53)

( ) ( )

=

=

+

++

+

+

+

+=

NG

iyyyxxyyxyxxxi

yyxyxx

NG

ii

e

yxxvnn

xunnw

yyv

xv

yu

xuwJ

1

22

1 21

(53)

222

221

22

JJJnx +=

, 222

221

21

JJJny +

=

2 [25] 2W t 2a - 1/4 7, 13, 12 2 x, y E1-E4 E5 E6 J- 53 ( 7, 8 9)

145

2

12 13

7

3 4 5 13 14 1526 27

41 424350

554952

54

48

53

3433

32

2221

xy

a = 4W = 10

E1 1/4 E1 7

13 14 15 27 43 42 41 26 x 5.50 6.10 6.70 6.70 6.70 6.10 5.50 5.50 y 0.00 0.00 0.00 0.70 1.40 1.15 0.90 0.45

u -0.0321941 -0.0328437 -0.0339877 -0.03307040 -0.0307576 -0.0295602 -0.0286617 -0.03104010 v 0.0000000 0.0000000 0.0000000 0.00929602 0.0189894 0.0163396 0.0138731 0.00682375

E2 13 41 42 43 55 54 52 49 50

x 5.50 6.10 6.70 5.35 4.00 4.00 4.00 4.75 y 0.90 1.15 1.40 2.20 3.00 2.30 1.60 1.25

u -0.0286617 -0.0295602 -0.0307576 -0.0193732 -0.00890561 -0.0106854 -0.0137014 -0.0216570 v 0.0138731 0.0163396 0.0189894 0.0360462 0.06031980 0.0514632 0.0416548 0.0244707

E3 12 49 52 54 53 32 33 34 48

x 4.00 4.00 4.00 2.65 1.30 1.900 2.50 3.250 y 1.60 2.30 3.00 2.50 2.00 1.575 1.15 1.375

u -0.0137014 -0.0106854 -0.00890561 -0.00439876 -0.00312630 -0.00619434 -0.0109292 -0.0111176 v 0.0416548 0.0514632 0.06031980 0.07330100 0.08593660 0.07904330 0.0692646 0.0552660

146

E4 2 5 22 34 33 32 21 3 4

x 2.50 2.50 2.50 1.900 1.30 1.30 1.30 1.90 y 0.00 0.575 1.15 1.575 2.00 1.00 0.00 0.00

u -0.0275023 -0.0179369 -0.0109292 -0.00619434 -0.00312630 -0.00736483 -0.0143537 -0.0209016 v 0.0667081 0.0682252 0.0692646 0.07904330 0.08593660 0.08396450 0.0814629 0.0755574

E5 7 13

7 13 xx yy xy xx yy xy

1 0.37199E+2 0.15790E+3 0.51486E+1 0.90793E+0 0.16895E+3 0.18441E+2 2 0.28437E+2 0.15924E+3 0.93048E+1 -0.57147E+1 0.18323E+3 -0.58063E+1 3 0.13471E+2 0.15872E+3 0.13299E+2 -0.32772E+2 0.14831E+3 -0.42625E+2 4 0.27844E+2 0.14624E+3 0.34419E+1 -0.22689E+1 0.15892E+3 0.16829E+2 5 0.21711E+2 0.14860E+3 0.75206E+1 -0.69162E+1 0.16695E+3 -0.56363E+1 6 0.10260E+2 0.14936E+3 0.11409E+2 -0.26368E+2 0.13752E+3 -0.37561E+2 7 0.19479E+2 0.13825E+3 0.25777E+1 -0.51520E+1 0.15068E+3 0.15013E+2 8 0.15372E+2 0.14137E+3 0.66520E+1 -0.95707E+1 0.15356E+3 -0.69286E+1 9 0.65775E+1 0.14309E+3 0.10516E+2 -0.25042E+2 0.12627E+3 -0.36223E+2

E6 12 2 12 2

xx yy xy xx yy xy 1 -0.20803E+2 0.12518E+3 -0.61342E+2 -0.24893E+2 0.86811E+1 -0.25268E+2 2 -0.14749E+2 0.177971E+2 -0.52408E+2 -0.50804E+2 0.69207E+1 -0.11597E+2 3 -0.17390E+2 0.17148E+2 -0.36484E+2 -0.94998E+2 -0.32461E+0 0.90438E+0 4 -0.19258E+2 0.11564E+3 -0.53432E+2 -0.25198E+2 0.96199E+1 -0.22749E+2 5 -0.13851E+2 0.72075E+2 -0.44941E+2 -0.53174E+2 0.61809E+1 -0.11517E+2 6 -0.15274E+2 0.19199E+2 -0.30989E+2 -0.95832E+2 -0.16628E+1 -0.12293E+1 7 -0.16007E+2 0.10598E+3 -0.47863E+2 -0.21566E+2 0.11567E+2 -0.18048E+2 8 -0.11288E+2 0.63011E+2 -0.39582E+2 -0.52426E+2 0.65699E+1 -0.83908E+1 9 -0.12265E+2 0.13157E+2 -0.26756E+2 -0.95613E+2 -0.21253E+1 0.46707E+0

147

53 7, 8 9 E2 E7

E7

1 2 3 4 5 6 7 8 7 13 14 15 27 43 42 41 26 13 41 42 43 55 54 52 49 50 12 49 52 54 53 32 33 34 48 2 34 33 32 21 3 4 5 22

3 7, 8, 9 ( ) ( )0,53,53,53 ( )53,53 5/9, 8/9 5/9 53() 7

+

+

+

53,

53

53,

53

53,

53

53,

53

53,

53

21

95

2,27,

7,7,

Jyv

xv

yu

xu

GPyy

GPxyGPxx

2

12 13

7

3 4 5 13 14 15

262741

4243

50

55

49

52

54

48

53

34

33

32

2221

+++

+

++

++

+

+

++ 7

8

9

7

8

978

9

7

8

9

E2 ()

148

53() 7

+

+

+

+

53,

53

53,

53

53,

53

53,

53

53,

53

53,

53

53,

53

53,

53

95

21,2

22,27,7,

7,7,

JJxvnn

xunn

yGPyyxGPxy

yGPxyxGPxx

8 9 8/9 5/9 ( )0,53 ( )53,53 8 9

7 1.280 -0.057 1.337

13 1.772 -2.017 3.789 12 -0.449 -2.272 1.822 2 -0.416 -0.893 0.478

J- 2(1.337+3.789+1.822+0.478) = 14.852

3.4.2 J- 3.4.2.1 Begley Landes [11] J- ( 1)

1) 2) [ 15()] LL P LL

15() 3) ( U) LL

15() LL1 U a1 a4 LL 4) U LL [ 15()]

149

5) U-a ( aU ) (8) J- LL [ 15()]

LLP

U,

a,1a 2a 3a 4a

1LL2LL3LL4LL

JdadU

B 1

1a

2a

3a

4a

LL,

dUda

()

1a

2a

3a

4a

UU

P,

LL,1LL 2LL 3LL 4LL

()

() ()

1LL 2LL 3LL 4LL

a

15 J-

150

3 [11] E1 BU a 3 Ni-Cr-Mo-V 200 1 J- LL 0.1 , 0.2 0.3 0.4

0

50

100150

200

250

300

0 0.1 0.2 0.3 0.4, a ()

,

U/B

(-

/)

E1 BU (-/)

LL () a () 0.025 0.020 0.015 0.010

0.120 278 211 129 60 0.170 221 163 104 50 0.220 168 123 80 40 0.260 128 95 67 30 0.325 70 51 39 18 0.352 42 30 21 12

E1

23.42530.130637.632 2025.0

+== aaBU LL 89.33020.109333.693 2

020.0+== aaBU LL

38.18228.45797.18 2015.0

+== aaBU LL 537.8520.21284.9 2

010.0+== aaBU LL

(8) 30.130674.1264025.0

+== aJ -/2 20.109366.1386

020.0+== aJ -/2

28.45794.37015.0

+== aJ -/2 20.21268.19

010.0+== aJ -/2

151

J- E2 E2 J (-/)

, a () LL () 0.1 0.2 0.3 0.4

0.025 1179.83 1053.35 926.88 800.40 0.020 954.53 815.87 677.20 538.54 0.015 453.49 449.69 445.90 442.10 0.010 210.23 208.26 206.30 204.33

.

3.4.2.2 J- Rice[3], Zahoor [12,13] J- (10) (13) LL 2 elLL p