Analysis of Cracks

8/8/2019 Analysis of Cracks

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Engineering Analysis w ith Boundary Elements 21 (1988) 169-178 1998 Elsevier Science Ltd. All rights reserved

Printed in Grea t BritainPII: S09 55 -7 997 (9 8)0 003 3 -7 0955-7997/98/$19.00+0.00

R e s e a r c h N o t eT h e m e t h o d o f a n a l y s i s o f c r a c k s i nt h r e e - d i m e n s i o n a l t r a n s v e r s e l y i s o t r o p i c m e d i a :b o u n d a r y i n t e g r a l e q u a t i o n a p p r o a c h

M . H . Z h a o a '* , Y . P . S h e n a , Y . J . L i u b & G . N . L i u baxi 'an Jiaotong Universi ty, Xi 'an 710049, Peop le's Republic of ChinabZhengzhou Research Inst i tute o f Mechan ical Engineering, Zhengzhou 450052, P eople's Repub lic of China

(Received 27 September 1996; accepted 24 February 1997)

The fundamental solut ions are obta ined for a uni t -concentra ted displacementdiscontinuity in a three-dimensional transversely isotropic m edium . The meth od ofsolut ion involves reducing the problem to a sys tem of hyper-s ingular in tegralequa tions by application of these fund ame ntal solutions. T he near crack border elasticdisplac eme nt and stress are obtained. Stress intensity factors can be expressed in termsof the displacement d iscont inui ty on the crack surface. An alogy is establi shed betweenthe bound ary integral equations for arbitrarily shape d crack s in a transversely isotropicand elastic medium such that once the stress intensity factors in the transverselyisot ropic medium can be determined di rect ly f rom that of the i sot ropic e las t icmedium. Resul ts fo r the penny-shaped crack are obta ined as an example . 1998Elsevier Science Ltd . A l l r ights reservedKey words: Fundamental solution, crack, transversely isotropic med ium, displac eme ntdiscontinuity, hyp er-sing ular integral equation.

I I N T R O D U C T I O NT h e r e s e a r c h e r s o f p r o b l e m s w i t h t r an s v e r s e l y i s o t r o p icp r o p e r t i e s h a v e r e c e i v e d a t t e n t i o n s i n c e 1 9 4 8 . I n t h i s fi e l dt h e p o t e n t i a l f u n c t i o n m e t h o d s 1 -4 a r e w e l l k n o w n a n dw i d e l y u s e d . W i t h t h e s e m e t h o d s m a n y i m p o r t a n t r e s u l t sf o r c r a c k p r o b l e m s h a v e b e e n o b t a i n e d 5 -8 . W h a t h a s n o tb e e n a c c o m p l i s h e d i s t h e m e t h o d o f s o l u t io n f o r a n a r b i-t r a r i l y s h a p e d p l a n e c r a c k .

T h e b o u n d a r y i n t eg r a l e q u a t i o n - b o u n d a r y e l e m e n tm e t h o d ( B I E M - B E M ) h a s m a n y ad v a n t a g es o v e r d o m a i nm e t h o d s t o b e u s e d f o r t h e p r o b l e m s o f c o n c e n t r a ti o n a n ds t r e ss s i n g u l a r i t y . 9 T h e d i s p l a c e m e n t d i s c o n t i n u i t y ( D D )B I E M - B E M i s m o r e e f f e ct iv e i n f r ac t ur e m e c h a n i cs ,w h i c h d e s c r i b e s t h e b a s i c c h a r a c t e r o f a c r a c k t h a t t h ed i s p l a c e m e n t a c r o s s i ts s u r f a c e s i s d i s c o n t i n u o u s . 1 M a n ya p p l i ca t i o n s h a v e b e e n m a d e t o s o l v e t w o - d i m e n s i o n a l , t *Corresponding author. Z hengzh ou Research Ins ti tu te o f Mech-a n i c a l E n g i n e e r i n g , Z h o n g y u a n R o a d , Z h e n g z h o u , H e n a n ,450052, P eople ' s Repu bl ic of China . Fax: (+86) 0371 7449148.

an t i p l ane , l 1 t o r si on ,~ 2 p l a t e -bend ing~ 3 and t h r ee -d im en s iona lp r o b l e m s . 1 4- 16 S o m e b o o k s o n t h e s u b j e c t c a m e o u t i np r i n t ] 7 I n t h is s t u d y , e f f o r t s a re f o c u s e d o n t h e D D B I E Mo f a t h r e e - d i m e n s i o n a l t r a n s v e r s e l y i s o t r o p i c e l a s ti c s o l id .

169

2 P R E L I M I N A R Y C O N S I D E R A T I O NI n t h e a b s e n c e o f b o d y f o r c e s , th e b a s i c e q u a t i o n s c a n b eo u t l i n e d a s f o l l o w s : 5

o 0 j = 0 ( 1 )g 6 = Cu k l e k l (2 )CO= (Uj, + U i,j ) (3 )

w h e r e i ,j ,k = 1 , 2 ,3 . Th e q uan t i t i e s ag~ egj, ug a r e t he c om pon en t so f t he s t r e s s , s t ra i n , and d i sp l ac em en t r e spec t i ve ly , an d C gjk/appea r a s t he e l a s t i c s t i f f nes s . Fo r a t r ansve r se ly i so t rop i ce l a s ti c m e d i u m , t h e r e a r e f i v e e l a s ti c c o n s t a n t s . I n w h a t f o l -l o w s o n l y t h e t r a n s v e r s e l y i s o tr o p i c m a t e r i a l w i l l b e e x a m i n e d .


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17 0 M . H . Z h a o e t a l .2 . 1 P o t e n t i a l f u n c t i o n s ~ - s 2 . 2 D i s p l a c e m e n t s a n d s t r e s sLet the car tesian coordinate system ( x , y , z ) be co inc iden twi th t he i so t rop i c p l ane o f t he med ium. Eqns (1 ) - (3 ) canbe cast in to component forms. Eqn (3) can be subst i tu tedinto eqn (2) . Fur ther subst i tu t ion into eqn (1) yie lds thegove rn ing equa t ions :

32Ux 1 32Ux 32UxCll 7 X 4" 7 (cl l -- CI 2) C~ '2- 4" C44--~-Z-2-Z21 3 2 u v 3 2 u .4 , X ( C ll 4 "C 1 2) 7~ :- ,~ 4 " ( C l 3 4 " C 4 4 ) . . ' = 0z o x o y " o x o z

1 OZu., oeu v O2uvX cl l -- C12)_~-~.2 4"C11 a~-.2 4"c44 - 2"z a x o y o z (4 )1 32u~ 32U-4" 7 (C II 4" CI2 ) O ~V 4" (C13 4" C44 Oyez = 0#/32#A . 32btz"~ 02btz

4 4 t 0 - ~ 4 " 3 y 2 ) 4 " C 3 3 0322IZ O2u~ = 0 .4 " ( C ' 3 4 " 4 4 ) t o ~ O - Z 4 " 3 y O z )

The d i sp l acement s m ay be expre ssed i n t e rms o f t he po ten -t ia l funct ions ~ and X

3@ 3X 3@ 3X k O@u < - 3 x - 3 y ' = ~ + 3 x u : = o ~ ( 5 )

Put t ing eqn (5) in to eqn (4) gives2 3 X1 - c , 2 ) ~3"Y4.32X\ 2+ c44 _ZSY_2 0 (6 )c l l OX ~ y 2 ) "Z

/ 32@ 3 2 @ "~ . . . . 02 @c l l t 0-~5- 4" 7 ) 4" [c44 4 - tcl3 + c 4 4) k ~ = 0

, . f 0 2 @ 3 2 @ ' k 0 2@I, 13 +c 44 +c 44 k) t -~ f fx2 + -~ --y2 +c 33k ~-z 2 = O(7 )

I f eqn (7) has a non-t r ivia l solut ion, the fol low ing condi t ionmust be satisfied:

C44 4" (C l3 4" c 4 4 ) k - - c 3 3 k -~- )k (8 )Cll C13 + c44 + caak

thus the quadra t ic in k i sC 4 4 ( C 1 3 4 " C 4 4 ) k 2 4 " [ ( C 1 3 4 " C 4 4 ) 2 4 " C 2 4 - - C 1 1 C 3 3 ] k (9 )

4" C44 (Cl3 4" C44) = 0wi th t he two roo t s deno ted by k j and k2. Cor re spond ing t othe two rea l or conjugate kl and k2, there are two rea l orconjugate va lues of h expressed by hi and X2. Thereforeeqn (7) becomes

c] @ 32 @ i c ') " i"ax---T-t- O-~y2--4.X,--OTy-z2 0 ( i = 1 ,2 ) (1 0)

L e t X ( x , y , z ) = @ 3 ( x , y , z ) and def ine2c44X 3 - - -CII -- CI2

such t ha t( 1 1 )

3 2 @ 3 0 2 @ 3 . , 3 2@ 3374- T 2+ ^ 3 - ~ - ? = 0 ( 12 )I t fo l lows t ha t t he d i sp l acement s may be expre ssed i n t e rmsof the three po tent ia l funct ions @ ~, @> @3. Referred to acyl indr ica l polar coordinate system, the resul t s a re

0 1 3@3U r = ~ ( ' I ' ~ + @ 2 ) r a 01 3 .@ 3@ 3" 0 = 7 ~ ( I 4 " @ 2 ) 4 " 0 ~ -

O @ l , . 0 @ 2b/z = k I ---~-g I-K2 ~-g

(13)

The st resses are( 0 2 1 3 5 0 2 )o r r = c l , f f- r 4" C l 2 r- O - r 4" C i 2 r2 O O 2 j ( @ i 4-@2)

02"i'j , , 3 / 1 3 @ 3 \+ c , , k , v - c < , ,-c ,21 ( ;[" 3 2 1 3 1 3 ~ \ .000 = ~ ,2 ~ + ~,, ;~ + c ,, 7 ~ ) (@' + @ 2 )

. 3 2 @ , , . a f l O @ A+ + c o , , - c , 2 )/ / 02 1 3 1 32 \ , ,

az z C 13 t ~rrZ-r2 . 7 5 7 4 - 70--0-~)(@ 1 4. " i'2)+ c33k, 02@,- - 3 z 2

1 3 2 @ j , 3 2 @ 3

0 2 ~ j 1 0 2 @ 3O z r ( c 4 4 4 , c 4 4 k j ) - - - - c 4 4 - "3 r 3 z r 3 0 3 z( 1 02 1_ 32~ (,i,1 4- `i'2 )oro ( , , - c , 2) \ ; 37 o r230

4 . ~ ( C 1 1 _ _ C 1 2 ) ( 0 2 @ 3 1 0 2 @ 3 1 0 @ 3 "O r2 r 2 0 0 2 r ~ r ] "(14)

a r epea t ed i ndex deno t ing summat ion .Eqns (10) and (12) in cyl indr ica l polar coordinates

b e c o m e0 2@ i 1 O @ 1 0 2 @ i 0 2 @ i0 7 + ; G r + - y -~ + - ~ g = 0 i = 1 ,2 ,3 ( 15 )


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T h e m e t h o d o f a n a l y s i s o f c r a c k s i n 3 D t r a n s v e r s e l y i s o t ro p i c m e d i a 17 1w h e r e

z i = s iz and s = - - 1 ( 16)boun dar y condi t ions a long the p lace z - --- 0 a r e

1Uz(r , O,Z)[z=0 = ~5(r ) arz lz= o = 0 OzOIz= = 0 (21)

3 F U N D A M E N T A L S O L U T I O N S F O R T H E U N I T -C O N C E N T R A T E D D I S P L A C E M E N TD I S C O N T I N U I T YC o n s i d e r a p e n n y - s h a p e d c r a c k o f ra d i u s a w i t h t h e u p p e rsur f ace S + and low e r sur f ace S - show n in F ig . 1 .

D isp lace men t d i scon t inu i ty ~u~] ac r oss the sur f aces ma yb e e x p r e s s e d a s

{ ~ u x ~ :U x ( X ,Y , O + ) - u x ( x , y , O - )~ U y ] = U y ( x ,y , O + ) - U y ( X , y , O - ) ( x , y )I [u z ]] = u z( x,y ,O + ) - U z ( X , y , O - )

E S (17)

T h e f u n d a m e n t a l s o l u t i o n s c o r r e s p o n d i n g t o a u n i t - c o n c e n -t r a ted d i sp lacemen t d i scont inu i ty should sa t i s f y the gover n -ing equa t ions of t r ansver se ly i so t r op ic mate r ia l s and thef o l l o w i n g c o n d i t io n s :

l im f {~u~] ,~uy] ,~Uz]} d s = { 1 , 0 , 0 }a--,0 ( 18)

lira ~ { ~ ux] , ~ Uy] , ~ Uz]} d s = { 0 , 1 , 0 }a--,O (19)

l im ~ s [ ~ u x ] , [u y ~ , ~ U z ] } d s = { 0 , 0 , 11a---.O ( 20)Sa t i s f ac t ion of eqn ( 18) and eqn ( 19) r e f e r to condi t ions ofs k e w - s y m m e t r y a n d e q n ( 2 0 ) t o a x i s - s y m m e t r y . T h e f u n d a -menta l so lu t ion w i th ax is - symmetr y w i l l be de r ived .3 .1 Fundamenta l so lut ions sa t i s fy ing eqn (20)O w ing to the symmetr y , cons ider on ly the pa r t z - > 0 . The

Wh en v / r 2 + z 2 ---* ~ , the f in i te c ondi t ion s r equi r e tha t

O r r : 4700 : l Tz z ~ l Yz r ~ ( Tz O : {Tr O - ~ - 0 ( 22)A l l t h e c o m p o n e n t s d e p e n d o n l y o n ( r , z ) a n d i n d e p e n d e n tof 0 ow ing to ax is - sym metr y . H en ce , eqn ( 15) s impl i f ie s tothe f or m

0 2 ~ i 1 0 ~ i 0 2 ~ iO r2 + r- -~ -r + --~-zi2= 0 ( i = 1 ,2 ) ( 23)and ~ 3 = 0 . Thus , az0 i s au tomat ica l ly equa l to ze r o .

A pply the ze r o- or der H anke l t r ans f or m to eqn ( 23) , andobser v e eqn ( 22) . A su i tab le so lu t ion i s

~ i ( ~ , z i ) = A i ( ~ ) e - ~ z l ( i = 1 , 2 ) ( 24)w h e r e ~l i (~ ,Zi ) i s the ze r o- or der H anke l t r ans f or m ofq I i ( r , z i ) .Subs t i tu t ing eqn ( 24) in to the r esu l t s ob ta ined by ze r o-or der H anke l t r ans f or m of eqn ( 13) , the r esu l t i s

2u z = - Z A j ( ~ ) k j s j e - ~Z J ( 25)j = l

T h e i n v e r s e H a n k e l t r a n s f o r m c a n b e a p p l i e d t o e q n ( 2 4 ) t oy ie ld

f ~ A i ( ~ ) ~ el Z ( r , Z i ) ~ - - ~Zi o ( ~r ) d~ (26)Subs t i tu t ing eqn ( 26) in to eqn ( 14) , az r i s ob ta ined as

(7zr = 0 4 2 ( c 4 4 - ~ c 4 4 k j ) A j ( ~ ) s j e - ~ z ) J l ( ~ r ) d r

(27)T h e b o u n d a r y c o n d i t i o n s i n e q n ( 2 1 ) a r e t h e t r a n s f o r m e dva lues and can be appl ied us ing eqns ( 25) and ( 27) . Thisg ives

Z

Fig. 1. Circular crack.

Y

1s i A l + s 2 A 2 - 4 r ~1k l s l A 1 + k 2 s 2 A 2 - - 4 7 r~

w h i c h c a n b e s o l v e d t o o b ta i nm l m 2A I = - f f - A 2 = T

The cons tan ts mi a r em l m 1 + k 24 rs l ( k l -k2) m 2 -.~

l + k l4 r s 2 ( k l - k 2 )

( 28)

(29)

(30)


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1 7 2 M . H . Z h a o e t a l.W i t h t h e a i d o f e q n ( 2 9) , X I ' t i i s f o u n d f r o m e q n ( 2 6 ) a s

m i ( 3 1 )q l i ( r , Z ,) - - V / ~ - }- Z 2S u b s t i t u t i n g e q n ( 3 1 ) i n t o e q n s ( 1 3 ) a n d ( 1 4 ) , t h e s t r es s a n dd i s p l a c e m e n t f i e l d s a r e o b t a i n e d :

2u r = - r y ) 3 ,2j = l ( r 2 q - "

2 m j k j z jU ' = ( r 2 q - g 2 ) 3 /22

O r r " -~ - ~ _ . m j [ c , l ( 2 r 2 - Z 2 ) - - C I 2 ( F 2 - ] - Z )j = l+ c , 3 k j ( 2 z f - r 2 ) ]/ (r 2 + 2 ) s /2

20 0 0 = ~ . m j [ c l 2 ( 2 r 2 - z 2 ) - C l , ( r 2 q - z 2 )j = l

+ c ~ 3 k i ( 2 z ~ - r 2 ) ] / ( r 2 + z ~ ) 5 / 22 (2Z __ r 2)

z z = Z a j m j q_z2)5/2j = I ( r22a z ~ = 3 r Z ( c 4 4 + c 4 4 k j ) ( s j z j m j )j : 1 ( r2 q- Z2)5/2'

( 3 2 )

w h e r ea j = - c13 q - c 3 3 k j s 2 ( 3 3 )

3 . 2 U n i t - c o n c e n t r a t e d t a n g e n t i a l (y - d i r e c ti o n )d i s p l a c e m e n t d i s c o n t i n u i t yT h i s f u n d a m e n t a l s o l u t i o n s a t i s fy i n g e q n ( 1 9 ) w i l l b eo b t a i n e d u s i n g t h e b o u n d a r y c o n d i t i o n s f o r z > 0 :

o z l z = o = 0( 3 4 )1 1u~ = 2-6(r) sin 0 U o = ~ 6 ( r ) c o s 0

w h e r e 6 ( 0 i s t h e D i r a c d e l t a f u n c t i o n . B a s e d o n e q n ( 1 5 )a n d t h e e x p r e s s i o n o f oz z i n e q n ( 1 4 ) , t h e p o t e n t i a l f u n c t i o n ss a t i s f y i n g t h e f ir s t c o n d i t i o n i n e q n ( 3 4 ) a r e g i v e n b y

q l l (r , O , Z l ) = H l f l ( r , O , z l )q/t2(r , 0, Z2) = H 2fl (r , 0, z2) (35 )

q/3( r , 0 , Z3) = f2 ( r , 0 , z3)w h e r e

s u c h t h a t f l a n d f 2 m u s t s a t i s f y32J ) . 1 32 f 1 . I 02 f l 02 f ,j + r - g -r + + o - 7 = 0 ( 3 7 )& 2 l & 2 1 0 21 2 & 25 7 + ; - g - r + + O z- =

w h e r e j = 1 ,2 . T o s a t i s f y e q n s ( 3 7 ) a n d ( 2 2 ) , f l a n d f 2 c a nb e t a k e n a s 53 ~c

f l ( r , 0 , z i) = 7 o : f o ( P lm ( ( ) e - ~Zi J m( ( r ) d ( - s i n m O

f 2 ( r , O , z3) = Z ~ P 2 m ( ~ ) e - ~ Z ~ J m ( ~ r d ~ - c o s m Om = 0 ~( 3 8 )

S u b s t i tu t i n g e q n ( 3 8 ) i n t o e q n ( 3 5 ) a n d m a k i n g u s e o f e q n( 1 3 ) , t h e n o n - v a n i s h i n g d i s p l a c e m e n t s b e c o m e

b ~ r = 2 7 0 s i am 0 I o ~ 2 { [ J m - l ( ~ r ) - J m + l ( ~ r ) ] [ H , e - ~ ' ' ' q - n 2 e - ~ Z 2 ] P l m ( ~ ) - l - [ J m _ l ( ~ r )+ J m + l ( ~ r ) ] p 2 , , ( ~ ) e - ~ z ~ }d ~

U o = ~m~_ 0c s m 0 f 2 ~ . 2 { [ J m - I ( ( r ) + J , n + | ( ] ~ r ) ]

+ [ J ,, _ ~ ( ( r ) - J m + , ( ( r ) ] P 2 m ( ( ) e - ~z3 } d ( .( 3 9 )

T h e d i s p l a c e m e n t c o n d i t i o n s i n e q n ( 3 4 ) m a y b e e n f o r c e do n e q n ( 3 9 ) . I t c a n b e c o n c l u d e d t h a t o n l y t h e c a s e o f m - --- 1n e e d s t o b e r e t a i n e d wh i l e p In a n d P2 m a r e a l l z e r o f o r m =~ 1.I t f o l l o w s t h a t

f ~ 2 { [ J o ( ~ r ) - J 2 ( F ~ r ) ] H l 2 P l l ( f ; )+ [ J o ( ( r ) + J 2 ( ~ r) ]p 2 1 ( ( ) } d ~ = 6 ( r ) ( 4 0 )

J o ~ 2 { [ d ( ~ r ) q - J 2 ( ~ r ) l H l 2 P l ' ( ~ )+ [ J 0 ( ~ r ) - - d 2 ( ~ r ) ] P 2 1 ~ ) } d ~ = 6 ( r )

w h e r e H I 2 = H l q - H 2. F r o m e q n ( 4 0 ) , P l l ( ~ ) a n d P 21 (~ ) a reo b t a i n e d :

1 1

1 1P 21( ~) - - 27 r H12

( 4 1 )

a lH l = 1 H e - - ( 3 6 )a2E q n ( 4 1 ) m a y b e i n s e r t e d i n t o e q n ( 3 8 ) . F u r t h e r a p p l i c a t i o no f i n v e r s e H a n k e l t r a n s f o r m r e n d e r s


5/10

T h e m e t h o d o f a n a l y s i s o f c r a c k s i n 3 D t r a n s v e r s e l y i s o tr o p i c m e d i a 173

{ ~ i( r , O, zi) = H-'----L~I1 - z i ( r 2 - t - Z2) - 112] sin 027rH1z r( i = 1, 2)

1 1 1/2]x l 1 3 (r ,O , 2 3 )= ~ - ~ r [ 1 - - z 3 ( r 2 4 - Z~ ) - c o s O/ 4 j d ) - 1 12 ]l i( r , O, Zi = ~ 1 [ 1 -- z i ( r 2 4 - c o s o27rH12

i = 1 , 21 1 [1 - z 3 ( r 2 4- z~ ) - l /Z] s in 0q l 3 ( r ' O ' z 3 ) = 2 r r

( 42 ) ( 4 5 )T h e d i s p l a c e m e n t s f o l l o w f r o m e q n ( 1 4 ) :

U r = _ ~ _ _ . ~ { j ~ = lH J r l Z J 2 r 2 4 - 2 2 ) ]H , 2 LZ 3 1- t - ( l - ( r Z 4 - z ] ) m ) } - f i s i n O

uo= ~ w i l l -- [1 23(2r2 4- z2)1V z e l } o , o

2r s m 0 5 - = ~ )-3 12u z = 2 7 r H j 2 j ~ = ~ j s j H j ( r 2 4 -

( 43 )

T h e s t r e s s e s c a n b e o b t a i n e d b y p u t t i n g e q n ( 4 2 ) i n t o e q n( 14 ) . I t i s f ound t ha t

3r s in 0 ~- - , . . 2 , 2 , - 5 / 2a z z = - - - - ; 7 , - 7 . n j a j z j t r t z j )2 r r t l 2 j = 1

1 1 2 2 2 -- 3/Zz0 = - = [ . - = Y +zj )Z ~ L / - - /1 2 j = 1C44S3(Z~ _ 2r2 )(r 2 + Z ) - 5/2] COS 0

O z r = 2 7 r ~ ~ " ( c4 4 4 - c 4 4 k j ) s j a j ( z 2 - 2 r 2 )1 2 j = 1

X ( r 2 4 - 2 2 2 ) - 5 /2 4 - c 4 4 s 3 ( r 2 4 - z 2 ) - 3/21 sin 0 .-I (44)

T h e d i s p l a c e m e n t i s o b t a i n e dUr-- - - - ~ - ~ { j ~ H ~ z I Zj(2r2+zf)]-(fi~]

4 - ( 1 ( r 2 ~ - ~ 3 2 ) ,1 2 . ) ~ -~ 2 o s 0u 0 = - - n , 2 _ l - - _ ( r 2 + ( 4 6 )

- [ I z 3 ( 2 r Z + z b ] 1z ~ 1 } ;~ s in 0r c o s 0 2= ;~ - f z j ) z ~ - ~ l- @ ~ = l k ) S jH j ( r 2 " 2 " - 3 1 2

The s t r e s s i s3 r c o s 0 ~ H ja jz j ( r 2 + z2 ) _ 512z z -= 27rH12 j = 1

1azO = (c44 + c44 k j )s jH j(r 2 + z~) - 312

+ c44s3(z ] _ 2r2) ( r 2 + z32) - 5 /2] s in 0

1 ~ , 0 Z ( c 4 4 + c 4 4 k j ) sj H j ( z2 - 2 r 2 )O zr ~ 27 r 1 2 j = 1- I - 2 Z 2 ~ - 5 / 2 - I - C S t r 2 4 - Z 2~ - 3 / 2 ] ( r 2 j ) 44 3 t 3~ | cos 0 .

.. I

( 47 )T h i s c o m p l e t e s t h e f o u n d a t i o n f o r t h e p r o b l e m o f u n it -c o n c e n t r a t e d d i s p l a c e m e n t d i s c o n t i n u i t y f u n d a m e n t a ls o l u t i ons .

3 . 3 U n i t - c o n c e n t r a t e d t a n g e n t i a l ( x - d i r e c t i o n )d i s p l a c e m e n t d i s c o n t i n u i t yT h e f u n d a m e n t a l s o l u t io n f o r a u n i t- c o n c e n t r a te d d i s p l a c e-m e n t d i s c o n t i n u i t y i n t h e x - d ir e c t io n c a n b e o b t a i n e d i n t h es a m e w a y a s t h a t i n t h e y - d i r e c t i o n d i s c u s s e d e a r l i e r . T h epo t e n t i a l f unc t i ons a r e

4 B O U N D A R Y I N T E G R A L E Q U A T I O NL e t a c r a c k o f a rb i t ra r y s h a p e i n c o i n c i d e n c e w i t h t h e p l a n eo f i s o t r o p y t h a t l i e s i n t h e x y - d o m a i n a s s h o w n i n F i g . 2 .D e n o t e d b y S + a n d S - a r e, r e s p e c t i v e l y , t h e u p p e r s u r f a c ewi t h ou t e r no r m a l d i r e c t i ona l c o s i ne {0 , 0 , - 1 } a nd l ow e rc r a c k s u r f a c e w i t h { 0 ,0 ,1 } . Th e r e c i p r oc a l r e l a t i ons i n t hea b s e n c e o f b o d y f o r c e s a r e g iv e n b y


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17 4 M . H . Z h a o e t a l .Y

Fig. 2. Planar crack.

- - X

f s I " ( 1 ) ( 2 ) ' ~ f / (1) ( 2 ) ~~ P i U i ) d s + J s - ~ P i I 'l i ) d s ( 48 )I { ( 2 ) . ( 1 ) , ~ / ( 2 ) ( I ) ' ~~ - s + ~ P i t~ i ) d s - ~ - s - ~ P i u i ) d s

whe re ul l ), @ ) and ul2) are two se ts of solut ions for thebod y wh i le p( l) = a ( O m .

Con sider u} ) , a l l )' as th e rea l solut ion of the p roblem andul ~) , el 2) as t he un i t - concen t ra t ed d i sp l acement d i scon t inu i tyfundamenta l so lu t ions i n Sec t i on 3 . I t c an be ea s i l y shown tha t

p12) l,+ = -p lZ ) l~- (49)The t rac t ions on the crack surfaces sa t i sfy the condi t ions

p l l ) ls + = - - p l l ) l s - ( 5 0 )t ha t a re a ssumed th roughou t t h i s work . Inse r t i ng t he se con-di t ions into eqn (48) gives the boundary integra l equat ionsfor a plane crack of arbi t rary shape:

[ f s + L z z l l U z U ~ d s = p z ( x , Y )I s + { [ L z r c o s 2 0 - t - L z o3 sin 20]~ux~

+ [ L z r3 - Lz o3] sin 0 cos O ~ u y ] } ~ ds((, ~7)= p x ( x , y )

I s { [ t z r 3 - - g z 0 3 ] s i n O c o s O ~ U x ~d s

+ [ L z r3 s in 2 0 + L z o 3 cos 20]~uy]} - 7 = p y ( x , y ) .

where2

L z z I = ~ . a i m ii = 1t z r 3=~ - 2 , = ~-~-12(l+kj)+s3 c44

1 [ 1 ~-L ]L z o 3 = w - I , - r -- 2 _ s j l 4 j ( 1 + k j ) - 2 s 3 c 4 4 .zTr Lrt l2j= 1

(51)

(52)

The foreg oing constants are related to the material properties.The d i sp l acement d i scon t inu i t ie s a re expre ssed a s

{ ~ u ~ ] = u x ( x , y , 0 + ) - u x ( x , y , O - )~ U y ~ = U y ( X , y ,O + ) U y ( X , y , O - ) ( x , y ) E S (53)~ uz U = u z ( x , y , 0 + ) - u z ( x , y , O - )

I t shou ld be no t ed t ha tr 2 = ( x - - ~ ) 2 + ( y _ ~ / )2 ( 5 4 )

The ke rne l func t i ons i n eqn (51) have t he same o rde r o fsingular i ty as those in the displacement discont inui tybounda ry i n t eg ra l equa t ions fo r t he co r re spond ing e l a s t ic i t yp rob l em. Hen ce eqn (51) can a l so be r ega rded a s t he hype r -singular in tegra l equat ions.

5 C R A C K B O R D E R S I N G U L A R B E H A V I O U R

Choose an arbi t rary point o on the crack border I" foranalysing the singular behaviour . The border I" of thecrack i s smooth a t point o . Without loss in genera l i ty theca r t es i an coord ina t e sys t em o x y z i s p laced such tha t the y-axis and x-axis are tangent and normal to I ' respect ive ly ,wh ereas the z-axis i s normal to the crack plane S as shown inFig. 3 . Le t e den ote the radius o f a smal l c i rc le r~ conta in edin S. The hype r-singu lar in tegra l in eqn (51) sho uld be f ini tein E and can be wri t ten as

f 1r L z z I ~ U z ]~ ds(~, r/) = R z ( x , y )I ~ [ ( L z r3 0 + L z o3 0 ) ~ u x ]o s 2 sin 2

1q - ( L z r3 - t z o 3 ) si n 0.cosO~uy]] ds(~, 7) = R ~ ( x , y )I E [ (L z r 3 - - L z 0 3 ) sin 0-cos O~ux]]

1+ ( Lz r 3 si n2 0 + L zo 3 cos 20)~uy~] ~ ds(~, ~/)= R y ( x , y ) .

(55)z

Fig. 3. Crack-tip analysis.

Y


7/10

T h e m e t h o d o f a n a l y s i s o f c r a c k s i n 3 D t r a n s v e r s e l y i s o tr o p i c m e d i a 17 5w h e r e R x( , y ) , R y( , y ) an d R z( , y ) are all f ini te functions for( x , y ) E E .5 . I D i s c o n t i n u i t y

A z ( o ) c o t Otz.Lzz1 = 0[2Lzr3 + Lzo3]Ax(o co t rC~x = 0[L zr3 + 2 L z o 3 ] A y ( o ) c o t r O t y = 0

(58)

T h e d i s p l a c e m e n t n e a r p o i n t o c a n b e o b t a i n e d b ysuper pos ing p lane s t r a in and the an t ip lane d i sp lacementsin the o x z plane . / s The d i sp lacement d i scont inu i t i e s a t then e i g h b o u r h o o d o f p o i n t o a r e g i v e n b y 15

{ ~ U x ]= A x ( o ) ~ x[ U y ] = A y ( o ) ~ '~ U z ]= A z ( o )~ ~

( 56)

w h e r e A x, A y a n d A z d e p e n d o n t h e l o c a t i o n o f p o i n t o . T h eindices ax , ay and a z r e f e r t o t h e u n k n o w n s o l u t i o n s a n dthe i r va lues a r e be tw een [ 0 ,1] .

I nse r t ing eqn ( 56) in to eqn ( 55) , in the H adamar d-pr inc ip le va lue sense , 15 the h yper - s ingu la r in tegr a l s can beeva lua ted w i th accur ac y w hen ~ i s suf f ic ien t ly smal l and x isinf ini tes imal, such as

f L ~ _ d~ d T = x a z - l A z ( o ) f ~ z d ~ f~ -o ~d ~ [ (1 - - ~)2 + (Y -- ~)2 ] 3/2

= -- 2x c~= tAz(o )Tr cot 7r~z~ r ~ c o s 2 O d ~ d 7 = ~ r ~u~ ]](x -~)2 r5

4 a~= - -~x - tAx(o)rr c o t t a x~ ~ U x~ d s ( L 7 ) = 0in 0 0o sE r 3

d~ d7

f ~Ux]s in2 0 d ~ d 7 = ~ ~ u x ]( Y -- 7 ) 2 d ~r. r 3 z r 5 ~ d72 a= - -~ x " - l A x ( o ) r c o t r o tx

( 57)

w her e the in tegr a l st (1 ~- 0 2 dt = - lr co t 7rc

f ~ d t 2oc (t 2 + X2)3/2 = X2

f ~ d t 4- ~ ( t 2 + x 2 ) 5 /2 - - 3 X4have been used in de r iv ing eqn ( 57) .

Inser t ing e qn (57) in to eqn (55) a nd let t ing x ---* 0, thenresults in

5 . 2 S i n g u l a r b e h a v i o u rT h e v a l u e s o f A x , A y an d A z a r e not ze r o in gener a l .Mor eover , the coef f ic ien ts in eqn ( 58) a r e ce r ta in ly no tze r o . H ence , non- t r iv ia l so lu t ions of eqn ( 58) ex i s t on lyw hen the f o l low ing condi t ions a r e sa t i s f ied :

cot 7rczx= c o t 7 1 " O / y = c o t rc a = 0 (59)Reca l l tha t the va lues of ax, a r and a z are w ith in the in te r va l[ 0 ,1] . Eqn ( 59) r equi r es tha t

ax = ay = a z = ( 60)This r esu l t r evea ls tha t the d i sp lacement near the c r ack t ipbeh aves as r 1/2, wh ich is the same as fo r an elas t ic so l id.

Mak ing use of the un i t d i sp lacements so lu t ions in Sec t ion3 , the s t r es ses near the c r ack t ip ou ts ide of the c r ack a r eobta ined:

azz(X, y, O )= - ~ [Lzzll[Uz] ~ d~ d 7azx(X, y, O) = Ig [(Lzr3 cOs2 0 + Lz o3 s in 2 O)~Ux]

1+ (Lzr3 - Lz03) s in 0 cos O~uy[ -~ d~ d 7azy(X, y, O) = I~ [(Lzr3 - Lz3) s in 0 cos O~ux]

1+ ( L z r 3 s i n 2 0 + L zo 3 c o s 2 0 ) [ u y ~ ] ~-~ d ~ d 7( 61)

The r esu l t s~ ~ - d~ dT = Tr az (o ) /v / Of [[Ux]] d~ d 7 =c o s 2 0 2r 3 gAx()Trlv/-O (62)

~ ~uy]] d~ d 7 = Osi n 0 0OSr r ~IlUx] sin 2 0 7rd~ d7 = :~ax(o) lv / -OF. r 3

w he n appl ied to eqn ( 61) g ives the s t r ess near the c r ack t ipa t po in t ( - p , y ) as

azz(X, y, O) = -- [LzzlAz(o)]Tr/ v ~1Ozx( X , y , O) = - [ ~Lzr3 + ~Lzo3]A x( o) Tr /V / - O2a z y ( X , y , O ) ~ - [ I L z r 3 + ~ L z o 3 ] a y ( o ) ' t r [ V / O

( 63)


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17 6 M . H . Z h a o e t a l .Table 1 . Constants of mater ia ls

Mate r i a l Ela s t i c cons t r a in t s ( 10 3 N m- -~)CII C33 C44 C13 CI2

M agnes ium 41.193 42.573 11.316 14.973 18.078Zin c 111.021 42.09 0 26.427 34.569 23.115Bar ium t i t ana te 115.920 130.41 37.674 48.990 53.820/3-Quartz 80.454 76.17 6 24.90 9 22.63 2 11.523

S t r e s s s i n g u l a r i t y o f t h e o r d e r o f r - j / 2 i s o b t a i n e d n e a r t h ec r a c k t i p , w h i c h c o r r e s p o n d s t o t h a t f o r t h e i s o t r o p i c e l a s t icm a t e r i a l .

5 . 3 I n t e n s i t y f a c t o r sT h e s t r es s i n t e n s i t y f a c t o r sg i v e n b y

K I = lin~ V / ~ r c r z z ( x , y , O )K , , = l i r a V ~ r o ~ ( x , y , O )K I u = l i r a v / ~ r O z y ( X , y , O )

I t f o l l o w s f r o m e q n ( 6 3 ) t h a t

n e a r t h e c r a c k t i p a r e

( 6 4 )

K ~- - - - - V / ~ T r [ L z z l A z ( o ) ]( 6 5 )

S i n c e A x , A t, a n d A z i n e q n ( 6 5 ) a r e k n o w n , i t c a n b e f u r t h e rs h o w n t h a t

K i = - ~ / ~ . r ! i m o [ L z z , ~ u z ] ] / V / ~2 1g l l = - V / ~ . T r l i m [ ~ L z r 3 - I - ~ L z o ] [[ ux ]] /-~ - ' ~ k o o j v r1 2

r - O L 5 - ,5 - j v / r

( 6 6 )

6 M E T H O D O F A N A L Y S I S6 .1 S y m m e t r i c p r o b l e m sC o n s i d e r a p l a n a r c r a c k o f a r b i t r a r y s h a p e w i t h t h e s u r f a c esp a r a l l e l to t h e i s o t r o p i c p l a n e o f a t h r e e - d i m e n s i o n a l i n f i n it et r a n s v e r s e l y i s o t ro p i c m e d i u m . T h e c r a c k s u r f a c e i s l o a d e db y p r e s s u r e p ( x , y ) s u c h t h a t b o u n d a r y c o n d i t i o n i s

O z ( X , y , 0 + ) = - p ( 6 7 )T h e b o u n d a r y i n t e g r a l e q u a t i o n i s g i v e n b y

I ~ [ L z z . [ l u z ] ] ]~ d ~ d r l = p ( x , y ) ( 6 8 )L i s t e d i n T a b l e 1 a r e c o n s t r ai n t s f o r s e v e r a l t r a n s v e r s e l yi s o t r o p i c m a t e r i a l s ) T h e y c a n b e u s e d t o e v a l u a t e t h ev a l u e s o f t h o s e i n T a b l e 2 .

T h e d i s p l a c e m e n t d i s c o n t in u i t y b o u n d a r y i n t eg r a le q u a t i o n f o r i s o t r o p i c s o l id t a k e s t h e f o r m

E ~ M d ~ d ~ l = _ t z ( x , y ) ( 6 9 )871.( i ~_ p2) sI t h a s t h e s a m e f o r m a s e q n ( 6 8 ) . I n e q n ( 6 9 ) , E a n d v a r et h e Y o u n g ' s m o d u l e a n d P o i s s o n ' s r at io r e s p e c ti v e l y . T h ep r e s s u r e o n t h e c r a c k s u r f a c e i s t z ( x , y ) .

N o w , l e t t z = - p , w h i c h c o r r e s p o n d s t o [ [W p ] a s g i v e n b ye q n ( 6 9 ) . T h e m o d e I s tr e s s i n t e n s i t y f a c t o r K p a l o n g t h ec r a c k b o r d e r i s

E l im V ~ p ~ w p BK[' - 8 ( 1 - v 2 ) p - 0 ( 7 0 )I t f o l l o w s t h a t t h e s o l u t i o n t o e q n ( 6 8 ) i s

EL z z l ~ U z ]= 8 r ( 1 - . 2 )U , ,A ' " ( 7 1 )w h i l e ~ u .~ i s o b t a i n e d a s

Table 2 . Values of the constants re lated to the mater ia lsMa gnes ium Zinc Ba r ium t i t ana te 13-Quart z

SI$2L z z l (1 0 3 )L~3(103)L z r 3 ( l O )P

0.69840 1 .08769 - 0 .6641 0i 0 .96472 - 0 .11013 i 0 .772771.40844 1 .08769 + 0 .6641 0i 0 .96472 + 0 .11013 i 1 .32989- 1.37954 - 1.86112 - 4.26593 - 2.715 20- 0 .92624 - 4 .80316 - 2 .84293 - 3 .74574- 3.60787 - 6.66638 - 10.64441 - 6.498 320.32936 0 .10274 0 .32329 0 .16441


9/10

T h e m e t h o d o f a n a l y s i s o f c r a c k s i n 3 D t r a n s ve r s e ly i s o tr o p i c m e d i a 17 71 E

Lzz 87r(1 - v2 )~vvp]~" ( 72)With the a id o f eqn ( 66) , the s t r ess in tens i ty f ac tor s near thec r a c k b o r d e r a r e o b t a i n e d :

K~ = K Ip ( 73)Eqn ( 73) is indepen dent o f the shape , the d imens ion , theg e o m e t r y o f t h e c r a c k a n d t h e d i s tr i b u ti o n o f t h e m e c h a n i -ca l load p . H er e w e can con c lude tha t the s t r es s in tens i tyf ac tor s of a p lanar c r ack of a r b i t r a r y shape in t r ansver se lyi s o t r o p i c m e d i a c a n b e c a l c u l a t e d b y e q n ( 7 3 ) w i t h t h ecor r espo nding so lu t ions of e las t ic i ty . 5 The same holds f orthe pe nny - shape d or e l l ip t ica l c r acks . 56 . 2 A n t i - s y m m e t r i c a l p r o b l e m sT h e b o u n d a r y i n t e g ra l e q u a t i o n s o f t h e c r a c k s u b j e c t e d t oant i - symmetr ica l load ing a r e

f s [ L z r 3 0 + L z o 3 0 ) [ u x ]O S2 s in 21+ ( L z r 3 - Lz03) sin 0 co s 0[uy]] ~ ds(~, 7)

= p x ( x , Y ) ( 74)~ s [ ( L z r 3 - - L z3 ) s in 0 cos O[u~]

+ ( L z r 3 s in 2 0 + L z o 3 CO S2 0 ) ~ u y ] ] ~ d s ( ~ , )1 )= p y ( x , y ) .

w h e r e p , a n d p y a r e the t r ac t ions on the c r ac k sur f ace in thex- and y- d i r ec t ions r espec t ive ly . The equiva len t i so t r op ice las t ic cons tan ts E and v can be iden t i f i ed w i th the t r ans -ver se mate r ia l cons tan ts as

Lz__03_ 1 -- 2_______vL zr3 - Lzo3 3v ( 75)

ELzr3 - Lz3 - - 8r ( 1 - v2)3vTaking eqn ( 75) and eqn ( 74) in to account , i t i s f ound tha t

f ~ { [(1 - 2 v ) + 3 v c o s20]~ux]]+ 3v s in 0 cos O~ur] -~ ds (~ , ~t )

87r(1 -- v2)-~x,Y)t-'xE (76)

~ { [ 3v s in 0 cos O~u~]]1+ [(1 - 2v) + 3 sin 20]~Uy]} - '~ d s (~ , 7 )

8 r ( 1 - v2)- - E - p y ( x , y )

T h e y a r e h y p e r - s i n g u l a r b o u n d a r y i n t e g r a l e q u a t i o n s i nelas ticity . 14,15A s bef or e , the so lu t ion of a c r ack subjec ted to an t i -

symmetr ica l load ing in t r ansver se media can a l so beobta ine d f r om tha t f or an i so t r op ic e las t ic mate r ia l .

K n o w i n g t h a tE= 8 ( l - f f r r (77)

m8 ( 1 + l i ,- u U y~the s t r es s in tens i ty f ac tor s in t r ansver se ly i so t r op ic mate r ia lb e c o m e

[ 2 L zr 3 + L z 0 3 ] r - -. o W ~ ru x ]n = - ~ r [ - - ~ - - - - - J l imr ,.P [ L z r 3 + 2 L z 0 3 ] l im ~ / ~ u v ]^ h ' = - ~ ' [ ~ "i r -- -0 V r -

( 78)

The s t r ess in tens i ty f ac tor s a r e r e la ted to those in e las t ic i tya s

K ll I ~-~ K Pl ( 79)

S o l u t i o n s t o m a n y c r a c k p r o b l e m s c a n b e o b t a i n e d i n t h esame w ay. 5 Suppose tha t a pe nny- sh aped c r ack i s shear edin the d i r ec t ion o f the x- ax is , w i th r be ing the shear s t r es s.Th e s t r es s in tens i ty f ac tor s K H and K m in e las t ic so l id a r e : 5

p 4 r c o s 0K u -- ~-(2--~-~v/TraK}IIP= _ 4(1 -r(_~._v ) ( rin 0)X~ ( 80 )

Subs t i tu t ing eqn ( 80) in to eqn ( 79) , the s t r es s in tens i tyf ac tor s can be ob ta ined . T he c ons tan t ~, i s l i s ted in Table 2 .

7 C O N C L U S I O NThe H anke l t r ans f or m i s used to de r ive the f unda-menta l so lu t ions f or a un i t - concent r a ted d i sp lacementd iscont inu i ty .

T h e d i s p l a c e m e n t d i s c o n t i n u i ty b o u n d a r y i n t e g ra l e q u a -t ions a r e de r ived f or an a r b i t r a r i ly shaped c r ack in at r ansver se ly i so t r op ic so l id . They cor r espond to the hyper -s ingula r equa t ions . D isp lacement and s t r es s near the c r ackbor de r ar e ob ta ined . The f ami l ia r r - m s ingula r behavio ur isf ound f o r the s t r ess in f r on t o f the c r ac k t ip .

M or e spec ia l ly , the s t r es s in tens i ty f ac tor s a r e show n to beobta inable f r om the cor r espo nding so lu t ions f or an i so t r op icelas t ic mater ial .


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178 M. H. Zhao et al.

ACKNOWLEDGEMENTST h e w o r k w a s s u p p o r t e d b y t h e N a t io n a l N a t u r e S c i e n c e

F o u n d a t i o n o f t h e P e o p l e ' s R e p u b l i c o f C h i n a a n d t h e T e c h -n i c al D e v e l o p m e n t F o u n d a t i o n o f t he M a c h i n e r y B u i l d i n gI n d u s tr y o f t h e P e o p l e ' s R e p u b l i c o f C h i n a .

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