alternating method applied to analyse mode-iii fracture problems with multiple cracks in an infinite...

7/27/2019 Alternating Method Applied to Analyse Mode-III Fracture Problems With Multiple Cracks in an Infinite Domain

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E L S E V I E R Nuclear Engineering and Design 152 (1994) 135-145

N u c l e a rE n g ! n . n ga l D e m g n

A l t e r n a t i n g m e t h o d a p p l i e d t o a n a l y s e M o d e - I l l f r a c t u r ep r o b l e m s w i t h m u l t i p l e c r a c k s i n a n i n f i n i t e d o m a i n

K u en T i n g a, Ch en q -Sh y o u n g Ch an g b, K w e i -K u o Ch an g ~aDepartment of System Engineering, Chung Cheng Institute of Technology, Ta-Shi 33509, Taiwan, China

bChung Shan Institute of Science and Technology, Lung-Tan 32526, Taiwan, China(Received 20 October 1993; revised version 22 March 1994

Abstract

An efficient alternating method was developed in the present study for the analysis of Mode-III fracture problemswith multiple cracks in the infinite domain. To achieve this, an analytical solution for a-single crack in an infinitedomain subjected to arbitrary longitudinal shear loading across the crack surface is developed. This analyticalsolution correlates with a successive iterative superposition process capable of satisfying the prescribed boundarycondition for each crack of the problem. Several Mode-III fracture problems were analysed to confirm the validityof this method. An excellent correlation between the computed results and available referenced solutions have shownthe accuracy and efficiencyof this research effort.

1. In troduct ion

The development of various methods for calcu-lating the stress intensity factor for the Mode-IIIfracture problem has received much attention inmany research efforts (Zhang, 1987, 1989; Ma,1988, 1991; Liu, 1992). A series of analyticalsolutions of multiple cracks in an infinite domainhas been developed previously by Sih et al. (Sih,1973). However, their solution was concerned witha specific arrangement of cracks and simple load-ing conditions. Although both the finite elementmethod and the boundary element method canbe used to deal with the complexities of thegeometries, boundary conditions and multiplecracks, the discretization of the mesh makes

these problems complicated. Hence, a Schwarz-Neumann alternating method was successfullydeveloped to solve the fracture problems withmultiple cracks. Nevertheless, this efficient proce-dure was confined to the three-dimensional Mode-I(Vijayakumar , 1981; Nishioka, 1983; O'Donohue,1984, 1985) and mixed-mode (Mode-1 and Mode-II) problems, (Chen, 1989, 1990). Mode-III multi-ple fracture problems have received little attentionwith regard to this issue.In the process of the alternating method, ananalytical solution of the single crack in the infi-nite domain, subjected to arbitrary longitudinalshear loading over the crack surface, is used forthe successive iterative superposition procedureswhich satisfy the prescribed bounda ry conditions.

0029-5493/94/$07.00 1994 Elsevier Science S.A. All rights reservedSSDI 0029-5493(94)00727-G


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1 3 6 K. Ting e t a l . ] Nuclear Engineer ing and Des ign 152 (1994) 1 35 -14 5

R e f e r e n c e d t o t h e s o l u t i o n s o b t a i n e d b y S n e d d o na n d E l l io t t ( S n e d d o n , 1 9 5 1) f o r th e p r o b l e m w i t ht h e c r a c k s u r f a c e u n d e r s y m m e t r i c n o r m a l p r e s -s u r e , t h e g e n e r a l a n a l y t i c a l s o l u t i o n f o r M o d e - I I If r a c t u r e i n t h e i n f i n i t e d o m a i n i s f i r s t d e r i v e d . T h ed i s t r i b u t i o n o f t h e s h e a r f o r c e s a c r o s s t h e f i ct i-t i o u s c r a c k w i ll b e f i tt e d b y t h e a p p r o p r i a t e p o l y -n o m i a l s t h r o u g h t h e l e a s t- s q u a r e s m e t h o d . T h e n ,t h e s i n g u l ar b e h a v i o r o f th e s h e a r s t re s s n e a r t h ec r a c k t i p i s p r e c i s e l y d e s c r i b e d .

S e v e r a l M o d e - I I I f r a c t u r e p r o b l e m s w i t h m u l t i -p l e c r a c k s i n t h e i n f i n i t e d o m a i n a r e a n a l y s e d t ov e r i fy t h e v a l i d i ty o f th e p r e s e n t w o r k . T h e r e s u lt sa r e c o m p a r e d w i t h o t h e r r e f e r e n c e s o l u t i o n s , a n dt h e m e t h o d p u t f o r t h i n t h i s w o r k i s s h o w n t o b eq u i t e e f f i c ie n t a n d a c c u r a t e .

2. An alyt ical so lution for s ingle cra ck witharbitrary crack surface shear loading

A n i n fi n it e d o m a i n w i t h a c r a c k a t y = 0 a n d- a < x < a ( w h e r e a i s h a l f t h e c r a c k l e n g t h ) i ss h o w n i n F i g . 1 . T h e c r a c k s u r f a c e s a r e s u b j e c t e dt o a r b i t r a r y l o n g i t u d i n a l s h e a r lo a d i n g , w h i c h c a nb e r e p re s e n te d b y a p o l y n o m i a l o f a n y o r d e r . T h eg o v e r n i n g e q u a t i o n i s g i v e n as :VZw = 0 (1)w h e r e w i s a d i s p l a c e m e n t i n t h e z - d i r e c t i o n , a n dt h e b o u n d a r y c o n d i t i o n i s r e p r e s e n t e d a s f o ll o w s :

l O O>X

F i g . 1 . I n f i n i t e p l a t e w i t h a c r a c k u n d e r a r b i t r a r y l o n g i t u d i n a ll o a d i n g a t c r a c k s u r f a c e .

( i ) s t re s ses Ty=a n d Z x= , d i s p l a c e m e n t w a p p r o a c h e sz e r o a t t h e in f in i te b o u n d a r i e s , i . e. x 2 + y 2 ~ o o

( i i ) s t r e s s Z yz r e p r e s e n t e d a s7 ~ y z = - f ( x ) = ( 2 )

n

a t y = 0 a n d Ix l < a( i ii ) d i s p l a c e m e n t w = 0 a t y = 0 a n d I xl > a

w h e r e N i s a p o s i ti v e i n te g e r a n d d . i s a p o l y n o -m i a l c o e f f i c i e n t .

T h e r e l a t i o n s h i p s b e t w e e n t h e d i s p l a c e m e n t wa n d r e l a t e d v a r i a b l e s a r e

0 wTx= = # - ~ x ( 3 )0 wZy: = # ~ y ( 4 )

w h e r e Zxz a n d Tyz a r e s h e a r l o a d i n g a n d # i s t h es h e a r m o d u l u s .

T o s o l v e E q . ( 1 ) , t h e v a r i a b l e ( w , y ) c a n b et r a n s f o r m e d i n t o t h e v a r i a b l e ( if , ~ ) b y t h eF o u r i e r t r a n s f o r m t e c h n i q u e ( S n e d d o n , 1 9 5 1) a s

w h e r ew (~' Y) = J ~-~ow ( x , y ) e x p ( + i ~ x ) d x ( 6)S i m i l a r l y , E q s . ( 3 ) a n d ( 4 ) c a n b e r e p r e s e n t e d a s : = - i # f f ( 7 )

O ff+,==. ( 8 )T h e s o l u t i o n o f E q . ( 5 ) is o b t a i n e d a s

f f( ~, y ) = A e x p ( - l ~ l y ) + B e x p ( + l C L v ) ( 9 )T h e c o e f f i c ie n t s B = 0 a n d A a r e d e t e r -m i n e d b y b o u n d a r y c o n d i t i o n s ( i ) a n d ( i i) . S u b s t i -t u t i n g c o e f f i c i e n t A i n t o E q . ( 9 ) a n d b y r e a r r a n g i n gE q s . ( 7 ) a n d ( 8 ) , t h e t e r m s k ( ~ , y ) , Cx~(~,Y) a n dZyz(~,Y ) c a n b e e x p r e s s e d a sf(oi f ( ~ , y) = ] - ~ exp ( - -. I~ L ) ( i 0 )Z yz (~ , y ) = - f ( ~ e x p ( - I ~ L ) ( 1 1 )xz(~,Y) = ~ f ( ~ ) exp ( - I ~L ) (12)


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K . T i n g e t a L / N u c l ea r E n g in e e r in g a n d D e s ig n 1 5 2 ( 1 9 9 4 ) 1 3 5 - 1 4 5 1 3 7

~ ( x , y ) =z ~ ( x , y ) =w h e r e

T a k i n g t h e i n v e r s e F o u r i e r t r a n s f o r m o f E q s .( 1 0 ) a n d ( 1 2 ) , t h e f o r m s o f w(x,y), Zyz(X,y),zx~(x, y ) ca n b e o b ta in ed a s1 ( ' ~ f ( O e x "w ( x ,y ) = ~ J .- o o -~ P ~ - I I Y ) e x p ( - - i ~ x ) d e

(13 )' Ly e (x , y ) = ~ f ( O ex p ( - I ~ lY ) e x p ( - i C x ) d e(14 )

- i f _ ~ -~( x , y) = ~ -n ~ ] -~ f (O exp ( - -1~ L )x e x p ( - i ~ x ) d e ( 15 )

I f f ( O is a n e v e n f u n c t io n , E q s . ( 1 3 ) - ( 1 5 ) c a nb e w r i t t e n a sw(x, y) = ~ J o - ~ - exp ( - ~ y ) cos ~ x d~ (16 )ry~(x,y) = - 2 f ~( ) e x p ( - { y ) c o s ~ x d { ( 17 )7 [

- 2 f o f ~( O e x p ( - { y ) s i n g x d ~ ( 1 8 )=(x, y) = -g -w h e r e

f f Z (o ( O = x ) c o s ~ x a x .I f f ( O is a n o d d f u n c ti o n , E q s . ( 1 3 ) - ( 1 5 ) c a nb e w r i t t e n a s

2 f ~ f o ( O e x p ( - ~ y ) s in ~ x d ~ ( 19 )w(x , y ) =-~ Jo - - ~--27[ o~fO (O e x p ( - - ~ y ) s i n ~ x d ( 20 )

yo- 2 f o ( O e x p ( - - ~ y ) c o s ~ x d ~ ( 2 1)7 [fo ( O = jo fo (X ) s in cx dx .

E q s . ( 1 6 ) - ( 2 1 ) h a v e s a t i s f i e d b o u n d a r y c o n d i -t i o n ( i ) f o r a p p r o p r i a t e l y c o r r e s p o n d i n g c a s e s .O n c e b o u n d a r y c o n d i t i o n s ( i i) . a n d ( i ii ) h a v e b e e ns a ti sf ie d , f u n c t i o n s j ~ ( ) a n d f o ( O m u s t b e d e t e r -m i n e d . T w o s et s o f d u a l i n t e g r a l s c a n b e f o u n d

f r o m E q s . ( 1 7 ) a n d ( 1 6 ), a n d E q s . ( 2 0 ) a n d ( 1 9 ) a sfo l lows:2 ~ ( ~ ) c o s c x d ~ = f e ( x ) 0 < x < a ( 2 2 )7 [

- ~ cos Cx d~ = 0 x > a (23 )a n d2 I ~ f o ( O s i n ~ x d ~ = f o ( X ) 0 < x < a ( 2 4 )7 [ 3 0fo~f (O s in ~ x de = 0 x > a (25 )-U

B a s e d o n t h e s o l u t i o n o f d u a l i n t e g r a l s d e r iv e db y S n e d d o n ( 1 9 5 7 ) , E q s . ( 2 2 ) - ( 2 5 ) c a n b e m a n i p -u l a t e d , a n d s o l u t io n s f ~ ( ) a n d f o ( O o f t h e a b o v ed u a l i n t e g r a l s c a n b e o b t a i n e d a s

7 [ 1 / 2f ~ ( ) = 4 a2 ~ F ( n / 2 + 1 / 2 ), =0 ,2 . .. F ( n / 2 + 2 ) A,{Jo(a)+ a L I x n +2J l ( a~ x ) dx} (26 )7[ /2 N ~ I F (n /2 + 1 ) A n. . . . .altaQ7 a 2 n = 1 ,3 . ,. I ~ ( ~ ~ t~ 5 7 ~ )+ a~ jo x"+2Jz(a~x)dx} (27 )

w h e r e F ( n ) d e n o t e s t h e G a m m a f u n c t i o n .S u b s t i t u ti n g J ~( ) a n d f o ( O i n t o E q s . ( 1 7 ) a n d( 2 0 ), a n d f o l lo w i n g s o m e m a n i p u l a t i o n s t h e c l o s e dfo rm s o lu t ions o f zyz a r e th us fo un d as fo l low s .( i ) F o r even o rde r n , t he s t re s s Zyz can beexp res s ed a s1 ~ F ( n / 2 + 1 / 2 )

Zyz = 27[ 1/2 ~ F (n /2 + 2)n = 0 , 2 . . . .X ~ n ( - - a 2 r ( r l r 2 ) - 3 /2 C O S {O 3 +~ ( 0 , G ) }+ 3 _~ k~ __ 0 ~ k/ 2 ) 1 l ) ] _ ~ _ 3 [ ( r l r 2 ) ( 2 k _ 3 ) / 2x r n - Z k + 3 c o s { ( k - 1 ) 7 [ + ~ - -~ -- ~ (01 + 0 2 )+(n - 2k + 3)O}-r 'ncos{(k -1 )rc + n 0 } l )

( 2 8 )


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138 K. Ting et aL / Nuclear Engineering and Design 152 (1994) 135 -14 5( i i ) F o r o d d o r d e r n , t h e s t r e s s Zyz c a n b ee x p r e s s e d a s

1 N -I F ( n / 2 + l )zy = ,~ ,q / 2 ~ F ( n / 2 + 5 / 2 ) Anz ., s~ n = 1 ,3 , . .

3 /2 f3

+a-~3 ("+ ' ) /2+ ' ( ~ k lk = O ~ 2 k 1_ 3x [ ( r , r 2 ) ( 2 k - 3 ) / 2 r n - 2 k + 3 s i n { - ( 2 k 2 3 ) n+ - - (0~ + 02) + (n - 2 k- r n s i n { ~ - ~ ~ + n 0 } l ) (2 9)

w h e r e ( k ) i s a b i n o m i a l c o e f fi c ie n t a n d t h e re la -t i ons h i p s o f r , r~ , r2 , 0 , 0~ , and 02 a re s hown i nF i g . 2 . T h e d o s e d f o r m s o l u t i o n s o f Z~z a n d w a r ea l s o d e r i v e d s i m i l a r ly .

3. Com putation of M o ~ -m s tre ss intens i ty fac torA c r a c k i n a s o l i d c a n b e s t r e s s e d i n t h r e ed i f f e r e n t m o d e s , i.e . o p e n i n g m o d e ( M o d e - l ) , s l i d-i n g m o d e ( M o d e - I I ) a n d t e a r i n g m o d e ( M o d e -I I I ) . T h e s t a b i li t y o f a c r a c k i n a b r i t t le m a t e r i a l i s

d e s c r i b e d b y t h e s t re s s i n t e n s i t y f a c t o r s ( K ~ , K n ,K i n ) . F r o m t h e d e f i n i t io n b y E r d o g a n ( 1 9 8 3 ), K mc a n b e e x p r e s s e d a s

l ( x . y )

/ 0 2 / o > x( - , . . o ) ( o . o ) ( , , . o )F i g . 2 . R e l a t i o n s h i p b e t w e e n r , r l , r 2 , 0 , 0 1 a n d 0 2 a n d x a n dy , w h e r e a i s h a l f t h e c r a c k l e n g t h .

K il I = t i m { 2n (x - - a)}l /2Zez(X, 0) (30 )x - - -~a

S u b s t i t u t i n g t h e a b o v e a n a l y t i c a l s o l u t i o n o fzyz (x , 0 ) i n t o E q . ( 3 0 ) , K i l l i s o b t a i n e d a s f o l l o w s( t h e d e t a i le d m a n i p u l a t i o n s a r e g i v e n i n A p p e n d i xA ) . ( i) F o r e v e n o r d e r n , t h e s tr e s s i n t e n s i t yf a c t o r K TI~ c a n b e e x p r e s s e d a s

a l/ 2 N F ( n / 2 + 1 /2 ) A , ( n + 2 ) (31 )K~II = 2 ~ F ( n / 2 + 2 )n = 0 ,2 . . .( i i ) F o r o d d o r d e r n , t h e s t r e s s i n t e n s i t y f a c t o r

K ~ n c a n t h e n b e e x p r e s s e d a sal/2 N -1 F ( n / 2 + 1)K~ n = 2 ~ F (n / 2 + 5 / 2 ) A n (n + 3) (32)n = 1,3, . . .

T h e c o m p l e t e s o l u t i o n o f s tr e ss i n t e n s it y f a c t o rK m i s t h u sKi n = K~ i i + K ~ I I (33 )

O n c e t h e l o a d i n g c o e f f i c ie n t A n i s f o u n d , t h es t r e s s d i s t r i b u t i o n a n d d i s p l a c e m e n t a t t h e c r a c kt ip s c a n b e c o m p u t e d f r o m t h e d o s e d f o r m s o l u-t i o n w h i c h h a d b e e n d e r i v e d p r e v i o u s l y , a n d t h es t r e s s i n t e n s i t y f a c t o r s c a n b e o b t a i n e d f r o m E q s .( 3 1 ) - ( 3 3 ) .

4. Ap plication of alternat ing m ethod for solvingm ultiple crack sT h e a l t e r n a t i n g m e t h o d w a s p r e v i o u s l y d e -s c r i b e d b y K a n t o r o v i c h a n d K r y l o v ( 1 9 6 4 ) , w h oo b t a i n e d t h e s o l u t i o n f o r p o t e n t i a l p r o b l e m s b ys u c c e s s i v e , i t e r a t i v e s u p e r p o s i t i o n o f s e q u e n c e s o fa n a l y t i c a l s o l u t i o n s . T h e d e t a i l e d p r o c e d u r e s a p -p l y i n g th e a l t e rn a t i n g m e t h o d f o r s o l v in g m u l t ip l ec r a c k p r o b l e m s i n t h i s w o r k c a n b e s t a t e d a s

fo l l ows .( 1 ) T h e r e l a t i o n s h i p o f t h e g e o m e t r i c a l a r -r a n g e m e n t o f t h e c r a c k s i s i n p u t a n d t h e i n i t i a la r b i t r a r y s t re s s d i s t r i b u t i o n % z o n t h e c r a c k s u r -f a c e i s r e p r e s e n t e d b y a p o l y n o m i a l o f a n a p p r o -p r i a t e o r d e r a s

w here Zk i s t he s h ea r s t r e s s Zyz a l o ng t he c ra cks u r f a c e ,


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K. Ting et a l./ Nuclear Engineering and Design 152 (1994) 135-145 139

( l " lLIa= l,--,...,a k \ a k / Jand

i i i " i{ A } ~ , = { A o , A , , A 2 , A ' 3 , " " , A . }. . i s a( = 0 , l , 2 .) i s t h e p o l y n o m i a l o r d e r , { A } ~

c o ef f ic i en t , i is t h e i t e r a t i v c s t e p s , x k i s t h e c o o r d i -n a t e o f c r a c k k a n d a k i s h a l f t h e c r a c k l e n g t h o fcrack k.(2) Using the analytical solution obtained inthe previous section. The interactive residualstress Ztk induced by crack k at a fictitious crack 1is computed. Because the crack surfaces are trac-tion free, the value of the residual stresses Z~kalong the crack surfaces must be reversed andmust be fitted by a polynomial of appropriateorder, i.e. Ztk = [L]rAtk , where [L] is the vectorwhich is composed of the terms ( X /a k ) " ( n =0, 1, 2 . . . ) and A rk is the interaction coefficientvector of crack l induced by crack k; A rk can bedetermined by the least-squares method. The de-tailed statement can be described as follows.

(i) Three cracks are assumed to be present inthe infinite domain. Crack 1 is assumed only toappear in the domain and longitudinal shear load-ing {A }~ is applied toward this crack surface. Theinteractive residual stresses zik induced by crack 1at fictitious cracks 2 and 3 are computed (z2~ andz31). The interactive coefficients A2~ and A3~ areobtained using the least-squares method.(ii) The residual stresses T21 and z31 in step (i)must be reversed to dispose of crack 2 in thedomain. As well as the original longitudinal shearloading {A }21on crack surface, the addi tion of theresidual coefficient vector -A21 to {A}~ is re-quired, along wi th the upda ted coefficient {A }~ ={A }~- A21 being computed. The same procedurein step (i) is used to compute the residual co-efficient vector A k2 (k = 1, 3). In the same mannerof disposing of crack 3, the updated coefficient{A}~={A}t3-A31--A33 and the residual co-effcient vector A k3 (k = 1, 2) can be obtained. I f Kcracks are present in an infinite domain, the up-dated coefficient can be represented as

k - - I{ A = { A - E A k , k = 1 , 2 . . . . K

r = l

(iii) The residual stress of crack k induced bythe rest of the cracks should be released for thenext iterative process and can be computed by thecoefficient vector {A }~+ l, where

K

E Ak,r = k + l

No residual stresses are present when k equals toK , i.e. {A }~+~ = 0.Steps (i)-(iii) are shown in Fig. 3.

(3) The stress and stress intensity factor ofcrack k for the current iteration can be bothcalculated and updated by substituting the updatecoefficient vectors {A }~ into the analytical solu-tion which were derived in the previous section.(4) Two criteria of convergence in this work areas follows:

(i) the criterion of stress intensity factor , i.e.the ratio of the stress intensity factor incrementfor the current iteration (Ak) to the stress inten-sity factor for the previous iteration for eachcrack is smaller than a tolerance value 7 ( = 0.005).

(ii) the criterion of residual stress, i.e. the ratioof the residual stress to be released to the per-missible stress for each crack is smaller than atolerance value ~ (=0.005).

The iteration is completed once these two crite-ria are satisfied, otherwise, execute the next step.(5) The coefficient vectors {A }~ are replaced by{A }~+ l, and these factors are allowed to becomethe new applied loading on the crack surface. Allthe iterative procedures for steps (2)-(4) arerepeated until the criteria for each crack aresatisfied.

5 . Resu l ts and di scuss ion

Several examples of an infinite domain wi th twoor multiple cracks subjected to unifo rm longitudi-nal shear loading at infinity are considered toverify the validity of this work. These solutionscan be superposed by the solution of an infinitedomain with cracks subjected to negative pre-scribed longitudinal shear load ing ~ over the cracksurfaces, and uniform shear stress distribution ofthe infinite domain without cracks.


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1 4 0 K. Ting et al. / Nuclear Engineeringand D esign 152 (1994) 135-14 5

+ +

+ - A , r A ~ 3 - A ~ {o1

X

6 )F i g . 4. T w o u n e q u a l c r a c k s u n d e r l o n g i t u d i n a l s h e a r l o a d in gi n a n i n f i n it e p l a t e .

stress intensity factors for the inner crack tips ofthe cracks can be written asF i n A ( a ~ 2 a ] - f - a 2 ) - K I I I A' d Z o ( n a ) i/zF n lB ( a 2 , a l d a 2 ) K n l B\ a , - Zo(n a) i/2

The functions FHIA and FnIB are shown graphi-cally for various geometric parameters in Fig. 5.The results listed by Sih (1973) are also providedfor comparison sake. A good agreement is seenbetween the analytical solution and reference

2 .22 .1 , a t : T h e h e l l l e n 4 t th o f c r a c k 1 02 . 0 , a 2 : T h e h a l f l e n 4 t t h o f c r a c k 2

1 a I .

&" 7F n I A 1 . 6 O " 1 " ~ # /1 . 4 - - : S l h ( Y o k o b o r i T . e t a l ) ~ / ~1.2,: I ' - -- o r 'o .o o .1 o .2 o . s o .4 o .5 o .6 0 .7 o . s o .9 1 .o

( a l + a z ) / dF i g . 5 . N o r m a l i z e d s t r e s s i n t e n s i t y f a c t o r F m f o r t w o u n e q u a lc o l l i n e a r c r a c k s i n a n i n f i n i t e p l a t e .


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K. l ing e t a l . / Nuclear Engineer ing and Design 152 (1994) 13 5-14 5 141

0 3 C21

(a) al = a2 = 0 .5; (al q - a 2 ) / d = 0.8

0 ,% ~ - - - - - - ~ ~

( b ) a l = 3 , a 2 = 1 ; ( a l +a2)/d = 0 . 9Fig. 6. Norm alized stress contour for two equal and unequa l collinear cracks in an infinite plate.

d a t a . T h e s h e a r t r a c t i o n c o n t o u r s a r e p r e s e n t e d inF i g . 6 f o r b o t h e q u a l a n d u n e q u a l c r a c k s , s o a s t od i s p l a y t h e i n t e r a c t i v e e f fe c t s b e t w e e n m u l t i p l ec r a c k s . T h e s t r e s s c o n c e n t r a t i o n s a t i n n e r c r a c kt i p s a r e o b s e r v e d i n t h e s e f i g u r e s t o b e l a r g e r t h a nt h o s e a t t h e o u t e r c r a c k t i p s . I n F i g s . 5 a n d 6 , t h ev a l u e s o f t h e s h e a r s t r e s s a r e n o r m a l i z e d a s z / % ,where z = ( 'Cyz -~ "Cxz ) 1/2.5 .2 . P a i r o f ec c e n t ri c p a r a l l e l c r a c k s

T h i s e x a m p l e t o b e s o l v e d i s a n i n f i n i t e s h e e tw i t h t w o e c c e n t ri c p ar a l le l c r a c k s u n d e r u n i f o r ml o n g i t u d i n a l s h e a r s t re s s ~o o n t h e e d g e o f t h ei n fi n it e d o m a i n . T h e g e o m e t r y a n d d i s t r i b u t io n o ft h e c r a c k s a r e s h o w n i n F i g . 7 . T h e n o r m a l i z e ds t re s s i n t e n s i ty f a c t o r f o r t h e i n n e r c r a c k t i p A c a nb e w r i t t e n a s

FmA ' b Zo(n a) , /2w h e r e r = ( e 2 + b 2 ) ~ /2, a n d t h e f u n c t i o n F m A i si n d i c at e d g r a p h i c a l l y f o r v a r i o u s g e o m e t r i cp a r a m e t e r s i n F i g . 8 . C o m p a r a t i v e d a t a l i s t e d b yS i h ( 1 9 7 3 ) a r e a l s o p r o v i d e d . E x c e l l e n t c o r r e l a -t i o n s b e t w e e n t h e c o m p u t e d r e s u l t s a n d r e f e r e n c es o l u t i o n s a r e o b s e r v e d .F i g . 9 d is p l a y s t h e v a r i a t i o n o f t h e n o r m a l i z e dM o d e - I I I s t r e s s i n t e n s i t y f a c t o r F m A v s . t [ a f o rt h e c a s e s o f b / 2 a = 0 . 0 1 , 0 . 1 , 0 .5 a n d 1 . W h e n tb e c o m e s e q u a l to z e r o a n d t w o c r a c k t i p s A a n d Ba r e c l o s e t o e a c h o t h e r ( i . e . b / 2 a = 0 . 0 1 ) , t h e c r a c kt i p A i s o b s e r v e d t o h a v e a l a r g e r s t r e s s i n t e n s i t yf a c t o r ; t h i s p e a k v a l u e i n d i c a t e s t h a t t w o p a r a l l e lc r a c k s h a v e s t r o n g l y i n t e r a c t i v e e f f e c t s . W h e n t w oc r a c k s b e c o m e s e p a r a t e d b y a n a p p r o p r i a t e d i s -


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1 4 2 K. Ting et al . / Nuclear Engineering and Design 152 (1994) 135-145

Y

T

O

> X

F i g . 7. T w o e c c e n t r i c p a r a l l e l c r a c k s u n d e r l o n g i t u d i n a l s h e a rl o a d i n g i n a n i n f in i t e p l a t e .

t a n c e ( i . e . b/2a = 1 ) , t h e c r a c k t i p s A a n d B a r eo b s e r v e d t o h a v e m u c h t h e s a m e v a l u e; t h i s p h e -n o m e n o n i n d i c a t e s t h a t t w o p a r a l l e l c r a c k s h a v ew e a k l y i n t e r a c t i v e e f fe c t s. I n a d d i t i o n , t h e i n t e r a c -

1 . 8 I1 .5 r _ _ : S t h ( I s i d a )

1 . 41 . 3

F D I a 1 . 21 . 11 , 0 10 . 9 ' ,o . o o . ~ o l e o ' . a o ~ 4 o 1 6 o ' .e o 1 7 o l e o : . t . o

2 a / rF i g . 8 . N o r m a l i z e d s t r e s s in t e n s i t y f a c t o r F m f o r t w o e c c e n t r i cp a r a l l e l c r a c k s w i t h v a r i o u s r a t i o s e /b i n a n i n f i n i t e p l a t e .

t i v e e f f e c t s b e t w e e n t w o c r a c k s c a n b e s e e n f r o mF i g . 1 0 f o r t h e s e c a s e s .5.3. Three collinear cracks

L e t u s c o n s i d e r a n i n f i n i t e e l a s t i c s h e e t c o n t a i n -i n g t h r e e c o l l i n e a r c r a c k s s u b j e c t e d t o l o n g i t u d i n a ls h e a r s t r es s T , a s s h o w n i n F i g . 1 1. T h e r e s u l ts o f

4 .08 ,88 .02 . 6

~ I R J . O ,1.8 ,1. 003 50.0

~ I U m : m -

. ~ a - - O . O 1

- u - i ~ s - i . o - d , s o ~ o o ~ t ~ o 2 o s~ s~o s~s 4.0t / a(,.)2.0.2 . 0 .

1 . 8 .

1 . 0

0 .0

0 .0

~ l , t _ _1PIs f l rB : ~ - -

. - - - - " ' ~ a ' O ' l

; : ," : : ; ; ," f I- a . 0 - - 1 ,2 1 - - 1 . 0 - -0 . 0 0 . 0 0 . 8 1 . 0 1 . 0 2 . 0 2 . 8 3 . 0 $ . 0 4 . 0t / , ,

1.8.

1.0.

0.8.

f f ~

1 e r a: - - _~ a - - O 5

0.0 ; ; ; . . . . . . 0 - 1 . o - . .0 - . o , , o~ o 0 : , :o l : s . : o , , : 6 ~ o ~ . ,t / a( c )

1. 1

I..0

0 .o 1 er a : - - -

0.0 : : ; : ; ; : ; : : :- J ~ - l ~ - l ~ - O ~ O ~ 0 ,8 1 . 0 1 ,1 1 2 . 0 2 . 6 ~ 2 . 0 4 , 0t / .( d )

F i g . 9 . N o r m a l i Z e d s t r e s s i n t e n s i t y f a c t o r F H , f o r t w o e c c e n t r i c p a r a l l e l c r a c k s w i t h v a r i o u s r a t i o s b/2a: ( a ) 0 . 0 1 , ( b ) 0 . 1 ; ( c ) 0 . 5 ;(d) I.


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K. Tin g et a l./ Nuclear Engineering and Design 152 (1994) 135-145 14 3

F i g . 1 0 . N o r m a l i z e d s t r e ss c o n t o u r f o r t w o e c c e n t r i c p a r a l l e lc r a c k s ( a = 0 . 5 ; b /e = 1; 2air = 0 . 8 ) i n a n i n f i n i t e p l a t e .

1 . 71 .6 A ~ A : F m ' t )1 .5 O - - O : F r o , t P r e s e n t1 .4 o ~ 0 : F m c ) / /

Fm 1.a - - - ( F = , t ) : S i h ~ o / / /

1 . 1 ' ~ ~ . ~ O ~ . . O ~ O1.0: - -= ~ T I t J I0 . 0 0 . 1 0 . 2 0 ~ .8 0 . 4 0 . 6 O . fl 0 . 7 0 . 8 o . g

P - a / dF i g . 1 2. N o r m a l i z e d s t r e s s i n t e n s it y f a c t o r F m f o r t h r e ec o l l i n e a r c r a c k s i n a n i n f i ni t e l a t e .

T a b l e 1C o m p a r i s o n o f o r d e r N o f t h e p o l y n o m i a l f it te d

YT_ ~ O " r o o

OF i g . 1 1 . T h r e e c o l l i n e a r c r a c k s u n d e r l o n g i t u d i n a l s h e a r l o a d -i n g i n a n i n f in i t e p l a t e . C r a c k t i p s A , B a n d C a r e s h o w n i n t h ef igure .

the normalized stress intensity factor Fin for cracktips A, B and C for various values of 2aid arepresented in Fig. 12. The results of FmA arecompared with the solution obtained by Sih(1973). Good agreement is observed.Table 1 presents a comparison of the polyno-mial order N taken for fitting the longitudinalshear force on crack surfaces. It is show n thatN = 3 is sufficient for these cases. Since the solu-

2a/d N N = 3 N = 4 N = 50 . 7 Fm A 1 . 196 1 . 195 1 . 195

FIIIB 1.165 1.164 1.164FIIIC 1.077 1.077 1.077

0.8 FIIIA 1.327 1.319 1.319F m a 1 . 288 1 . 281 1 . 280F m c 1 . 112 1 . 108 1 . 109

0 .9 FIIlA 1.650 1.587 1.577FIIIB 1.603 1.534 1.528Fll lC 1.174 1.155 1.155

F i g . 1 3 . N o r m a l i z e d s t r e ss c o n t o u r f o r t h r e e c o l l in e a r c r a c k s(a ~ = a 2 = a 3 = 0 . 5 ; d = 1 .2 5; i.e . 2 a / d = 0 . 8 ) i n an i n fi n it ep l a t e .

tion converges very fast, the number of iterationsis never greater than four. The shear tractioncontours are displayed in Fig. 13 to depict theinteractive effects between the three cracks. Thecrack tips A and B are observed in Fig. 13 to have


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1 4 4 K . : l in g e t a l . / N u c l e a r E n g i n e e r i n g a n d D e s i g n 1 5 2 ( 1 9 9 4 ) 1 3 5 - 1 4 5

l a rg e r s t r es s c o n c e n t r a t i o n s t h a n t h a t o f c r a c k t i pC (c f . F ig . 11) .

6. ConclusionsT h e S c h w a r z - N e u m a n n a l t er n at in g m e t h o d , in

c o n j u n c t i o n w i t h t h e a n a l y t i c a l s o l u t i o n d e r i v e df o r a s i n g l e c r a c k , w a s s u c c e s s f u l l y a p p l i e d i n t h ep r e s e n t w o r k t o a n a l y s e a r b i t r a r i l y d i s t r i b u t e dm u l t i p l e c r a c k s i n a n i n f i n i t e s h e e t u n d e r a r b i t r a r yl o a d i n g . T h i s m e t h o d , i n c o n j u n c t i o n w i t h th eb o u n d a r y e l em e n t m e t h o d , c a n b e d e v e lo p e d i n toa b o u n d a r y e l e m e n t a lt e rn a t in g m e t h o d w h i c h c anb e a p p l i e d f o r a n a l y z i n g m u l t i p l e c r a c k s d i s -t r i b u t e d i n a s h e e t w i t h a f i n i t e d o m a i n , a n d w i l lb e p r e s e n t e d i n t h e n e a r f u t u r e .

Appendix AF o r b r e v i t y , t h e e v e n o r d e r n i s c o n s i d e r e d o n l y .

T h e d e f i n it io n o f K ~n c a n b e e x p r e s s e d a sK ~ n = l i m { 2 n ( x - a)} I /2Zy~(X, 0 ) ( A 1 )

x - - - ~ a +

F r o m F i g . 2 , i f x > 0 , t h e n 0 = 01 = Oz = 0 . S u b -s t i t u t in g 0 , 0 1 a n d 0 2 i n t o E q . ( 2 8 ) , w e o b t a i n e dZy~(X, 0) as

- - 1 N F ( n / 2 + 1/2)Z y ~ ( X , O ) - 2 - ~ 3 2 n = o~,2 , . . . r ( n / 2 + 2 )[A n - - a 2 r ( r l r 2 ) - 3 1 2 + - ~ n ~ 1u k = O1x ~ { ( r l r 2 ) ( 2 x - 3 ) 1 2 r n - z ~ + 3 _ r n }

x c o s ( k - 1 ) Tt }] ( A 2 )w h e r e r l r 2 = ( x 2 - a 2 ) a n d r = x . E q . ( A 2 ) c a n b er e w r i t t e n a szyz(x , 0) - --__~1 ~ r ( n /2 4 - 1 t2)- - 2 ~ 1/2 F ( n / 2 + 2 )n = 0 ,2 . . ..

3 n / 2 + 1x A n _ a 2 x ( x 2 _ a 2 ) - -3 / 2 4 - a - n ~ E 0

( ~ 4 - 1k ) 2 k ~ { ( x2 - a z )( 2 k -3 ) /2x x n - z k + 3 - x n } c o s ( k - 1 ) ~ } [ ( A 3 )

-. IL e t u s s e t v a r i a b l e C . a s- 1 ~ F(n/24- 1/2)

C n - - 2 ~ 2 " -" r ( n / 2 + 2) A n (A4)n = 0,2 . . . .

R e a r r a n g i n g E q . ( A 3 ) , w e c a n o b t a i n ~ z ( X , o ) = C n I _ a 2 x ( x 2 a Z)_ 3/2 4-a -" \3 ~ 4 - 0 1 )

X ( ~ 3 1 ) { ( x 2 - - a 2 ) - - 3 / 2 x n + 3 - - x n } + 3 a ( ~ l l ) ( - - 1 ) { ( x Z - - a 2 ) - - ' / 2 x n + l - - x n }

3 n/2 + 1{~ ++ ~ - ~ k ~ 2 \ " k 1 ) ~ -~ 1- 3X { ( X 2 - - a 2 ) ( 2 k - 3 ) / 2 x n - 2 k + 3 _ X n }

x c o s ( k - 1 ) ~ / ( A 5 )d

n / ~ 1 _ _ _ _

E q . ( A 5 ) c a n b e e x p r e s s e d a s~(x n + 2) _ a , - - 2x ( 3 n / 2 + 3 ) x " + iZ y z( X , O ) = C n [ a - ~ 7 a - ~ - a n ( x z - a 2) i/2

+ ( 3 n / 2 + 2 ) x n ) D 3a n ~ + , ~- ~ { ( x 2 _ a 2 ) ~ , - 3 ) / 2 x . - 2 , + 3 _ x n }

~ x ( x " + a 2 x n - 2 _ a 4 x n - 4 4 - . . . )=Cnt( 3 n / 2 +- - ~ - 2 - - a 2 - - - ~3 ) x+ ' ( 3 n / 2 _ ~ 2 ) x n }

+ D . 3 { ( x2 _ a2 ) ~2k - 3/2)Xn 2k +3 _ X n }H e r e t h e p o l y n o m i a l x n + a Z x n - 2 + a4x n-4 4-" h a v e ( n / 2 + 1 ) t e r m s . W e t h e n h a v e

L e t u s s e t v a r i a b l e D n a s


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K. Tin g et al . / N uclear Engineering and D esign 152 (1994) 135-145 14 5

r z (X , o ) 1 ( x + a ) 1 /2( x - - a ) 1/2 a n+ - ~ - + 3 x n ( x 2 - - a 2 ) 1/2 - - [- ( x .. b a )l / 2D n { ( x 2 - - a 2 ) k - l x n - 2 k + 3 - - ( x 2 - - a 2 ) l / 2 x n } ( A 6 )

Subs t i t u t ing Eq . (A6) in to Eq . (A1) , we found the s t r e ss i n t ens i ty f ac to r ( K ~ I I ) a s1{~ n = ( 2 r O l l2 C ~ a . ( 2 a )l l2 a (a ~ + a 2 a ~ - 2 + a 4 a ~ - 4 + . . .) - + 3 a ~+1: ~ a ) ~ - - g J a ~ + l a n - + 3 a n +l = - ( n a ) ' / z C n ( n + 2 ) ( A 7 )

Subs t i t u t ing Eq . (A4) in to Eq . (A7) , t he genera l f o rm o f t he s t r e ss i n t ens i ty f ac to r K~u can then beexpressed asa l l 2 N F ( n / 2 + 1 / 2 )

K~xi = 2 )-" F( n/ 2 + 2) A,, (n + 2)n = 0 , 2 , . . .

R e f e r e n c e sW. H. Che n a nd C . S . Cha ng , Ana l y s i s o f t wo d i me ns i on f r a c -t u r e p r ob l e ms wi t h mu l t i p l e c r a c ks unde r mi xe d bounda r y

c ond i t i ons , Eng . F r a c t . Me c h . 34 ( 1989 ) 921 - 934 .W. H . Ch e n a nd C . S . Cha ng , A na l y s i s o f mu l t i p l e c r a c ks i n a ninf in i te p la te under arb i t rary crack sur face t rac t ions , Ing .Arch. 60 (1990) 202-212.F. Erdogan, St ress in tens i ty fac tors , Trans . ASME, J . Appl .Mech. 50 (1983) 992-1002.

L . V . K a n t o r o v i c h a n d V . I. K r y l o v , A p p r o x im a t e M e t h o d s o fHi ghe r Ana l y s i s , Wi l ey , Ne w Yor k , 1964 .N . L i u a nd N . J . A l t i e r o , An i n t e g r a l e qua t i on me t hod a pp l i e d

t o mode I I I c r a c k p r ob l e ms , Eng . F r a c t . Me c h . 41 ( 1992 )587 - 596 .S.W. Ma, A cent ra l c rack in a rec tangular sheet where i t sbounda r y i s s ub j e c t e d t o a n a r b i t r a r y a n t i - p l a ne l oa d , Eng .Frac t . Mech. 30 (1988) 435-443.

S .W. M a a nd L . X . Zha ng , A ne w s o l u t i on o f a n e c c e n t r icc r a c k o f f t he c e n t e r l i ne o f a r e c t a ngu l a r s he e t f o r mod e - I I I ,Eng. Frac t . Mech. 40 (1991) 1-7 .T . N i s h i oka a nd S . N . A t l u r i , Ana l y t i c a l s o l u t i on f o r e mbe dde d

el l ip t ica l c racks , a nd f in i te c leme nt a l te rnat ing me tho d fo re l l ip t ica l sur face cracks , subjec ted to arbi t rary loadings ,Eng. Frac t . Mech. 17 (1983) 247-268.P . E . O ' Donoghue , T . N i s h i oka a nd S . N . A t l u r i , Mu l t i p l esurface cracks in pressure vesse ls , Eng. Frac t . Mech. 20(1984) 545-560.

G . C . S i h , Ha nd boo k o f S tr es s I n t e ns i ty Fa c t o r s , Noor dho f f ,Leyden, 1973.P . E . O ' Donoghue , T . N i s h i oka a nd S . N . A t l u r i , Mu l t i p l ecoplanar embedded e l l ip t ica l c racks in an inf in i te so l id sub-

jec ted to arbi t rary crack face t rac t ions , In t . J . Numer .Methods . Eng. 21 (1985) 437-449.I . N . Sne ddon , Fou r i e r T r a ns f o rm, M c Gr a w - Hi l l , Ne w Yor k ,1951.K . V i j a ya kuma r a nd S . N . A t l u r i , An e mbe dd e d e l l ip t i c al f la w ,in an inf in i te so l id , subjec t to arb i t rary crack-face t rac t ions ,T r a ns . ASME, J . App l . Me c h . 48 ( 1981 ) 88 - 96 .

X . S . Zha ng , The g e ne r a l s o lu t i on o f a c e n t r a l c r a c k o f f thec e n t e r l ine o f a r e c t a ngu l a r s he e t f o r mode - I I I , Eng . F r a c t .Mech. 28 (1987) 147-155.X . S . Zha ng , A t e a r mode c r a c k l oc a t e d a nywhe r e i n a f i n i t erec tangular sheet , Eng. Frac t . Mech. 33 (1989) 509-516.

alternating method applied to analyse mode-iii fracture problems with multiple cracks in an infinite...

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