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E L S E V I E R 0955-7997(95 )00012-7

Engineering Analysis w ith Boundary Elem ents15 (1995) 137-149Copyright 1995 Elsevier Science LimitedPrinted in Great Britain. All rights reserved0955-7997/95/$09.50

C o m puta t i o n o f energ y re l ea s e ra te us i ng m a ter i a ld i f ferent ia t ion o f e las t i c BIE for 3 -D e las t i cfractureM a r c B o n n e t & H a i h o n g X i a o

L a b o r a t o i r e d e MOc a ni qu e d e s So l i d es , U R A C N R S 3 1 7 , c e n t r e c o mm u n P o l y t e c h n i q u e - M i n e s - P o n t s e t Ch a us s~ e s,E c o l e P o l y t e c h n i q u e , 9 1 1 2 8 P a l a i s e a u C e d e x , F r a n c e

Th i s p ap er d ea l s wi t h a n o v e l ap p ro ach , b ased o n mat e r i a l d i f f e r en t i a t i o n o fe l as ti c BIE fo rmu l a t i o n s , f o r t h e n u mer i ca l co m p u t a t i o n o f t h e en e rg y r el ease r a t eG a l o n g a c r ack f ro n t i n 3 -D e l as t i c f r ac t u re p ro b l ems . I t i s b ased u p o n t h edef in i t ion of G in terms of the mater ia l der ivat ive of the po ten t ia l energy atequ i l ib r ium W w i th respect to a l l possib le regu lar v i r tual crack ex tensions . W canbe formulated in terms of surface in tegrals ; th i s fact in tu rn al lows for ab o u n d ary -o n l y fo rm u l a t i o n o f i ts m a t e r i a l d e r iv a t i v e wit h r e sp ec t t o v i r t u a l c r ackextensions . The necessary s tep of comput ing the shape sensi t iv i t ies o f theb o u n d ary e l a s ti c v a r iab l es i s d o n e b y m ean s o f a d e r i v a ti v e BIE . Th e l a t t e r r e su l tsf rom a mater ia l d i f feren t ia t ion of the pr imary elas t ic BIE, which i s performedon a regu lar ized (weakly s ingular) d isp lacement BIE so that the p rocess i sma t h emat i ca l l y so u n d . Th e u n k n o wn s o f b o t h p r i mary an d d e r i v a t i v e BIEs a r eg o v ern ed b y t h e same i n t eg ra l o p e ra t o r , w i t h o b v i o u s co mp u t a t i o n a l ad v an t ag es .The presen t approach thus does no t resor t to any f in i te-d i f ference evaluat ion ofder ivat ives wi th respect to crack f ron t per tu rbat ions .The implementat ion of the presen t method , includ ing the key technical s tep ofco n s t ru c t i n g ap p ro p r i a t e v ec t o r i n t e rp o l a t i o n fu n c t i o n s fo r t h e t r an s fo rmat i o nveloci ty associated w i th a v i r tual cra ck ex tension , i s d iscussed . F inal ly , in o rder tod emo n s t r a t e t h e p o t en t i a l o f t h e p ro p o sed ap p ro ach , n u mer i ca l r e su l t s a r epresen ted for two mode I examples where reference resu l t s are avai lab le fo rcompar ison : the round bar wi th a c i rcu lar in ternal crack and the semiel l ip t icalsurface crack , in bo th cases under un i form tension .Key words. Regular ized elas t ic BIE, mater ia l d i f feren t ia t ion , var iab le domains ,l i nea r f r ac t u re mech an i cs , v i rt u a l c r ack ex t en si o n , b o u n d ary e l emen t me t h o d

1 I N T R O D U C T I O NO n e o f t h e b a s i c q u a n t i t i e s i n v o l v e d i n e l a s t i c f r a c t u r em e c h a n i c s i s t h e e n e r g y r e l e a s e r a t e G(s), a f u n c t i o n o ft h e a r c l e n g t h s a l o n g t h e f r o n t O F o f a c r a c k F :

I G ( s )6 l (s )d s = - 6 W (1 )OFw h e r e 6 W i s t h e p e r t u r b a t i o n o f th e p o t e n t i a l e n e r g ya t e q u il i b ri u m W i n d u c e d b y a c r a c k f r o n t n o r m a le x t e n s i o n 61 a n d i n t h e a b s e n c e o f l o a d v a r i a t i o n . F o rt w o - d i m e n s i o n a l p r o b l e m s , G i s t h e v a l u e o f th e w e l l-k n o w n p a t h - i n d e p e n d e n t J i n t e g r a l . I n l i n e a r f r a c -t u r e m e c h a n i c s , G i s l i n k e d t o t h e c r a c k t i p s i n g u l a r i t yo f th e s t r e s s fi el d t h r o u g h t h e w e l l - k n o w n I r w i n

137

f o r m u l a : 1= 1 + v 2 - ~G(s) [ K Z ( s ) + K Z ( s ) ] + _ _ Kin(s) (2)

w h e r e KI(S) , KH(S) , K i l t ( S ) a re t h e s t r e s s i n t e n s i t yf a c t o r s ( S I F s ) , # a n d v b e i n g , r e s p e c t i v e l y , t h e s h e a rm o d u l u s a n d P o i s s o n r a t i o . F u r t h e r m o r e , G h a s ac l e ar t h e r m o d y n a m i c a l m e a n i n g 2 a n d p l a y s a c e n t r a lr o l e f o r th e p r e d i c t i o n o f c r a c k e x t e n s i o n a c c o r d i n g t oG r i f f i t h - t y p e c r i t e r i a .T h u s , t h e c o n s i d e ra t i o n o f p e r t u rb a t i o n s o f W u n d e rf i c t i t i o u s b o d y c h a n g e s a s s o c i a t e d w i t h v i r t u a l c r a c ke x t e n s i o n s p r o v i d e s a c o m p u t a t i o n a l t o o l f o r e l a s t i cc r a c k a n a l y s i s : t h i s i s t h e v i r t u a l c r a c k e x t e n s i o na p p r o a c h . I n t h e fi r st n u m e r i c a l a p p l i c a t i o n s , d e r i v a t i v e s

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138 M. Bonne t , H . Xiaoo f W a r e e v a l u a t e d u s i n g s m a l l fi n it e c r a c k p e r t u r b a t i o n sand f i n i te d i f fe rences .3 '4 In l a t e r w orks 5 -9 t he con cep t o fm a t e r i a l d i f f e r e n t i a t i o n is a p p l i e d t o W ; t h i s l e a d s t or i g o r o u s f o r m u l a t i o n s f o r G , s t a r t i n g f r o m v a r i a t i o n a lf o r m u l a t i o n s o f e l a s t i c i t y p r o b l e m s . T h i s a p p r o a c h ,s o m e t i m e s k n o w n a s t h e ' 0 - m e t h o d ' ( 0 r e f e r s t o t h en o t a t i o n u s e d i n R e f s 6 a n d 7 a n d h e r e i n f o r t h e t r a n s -f o r m a t i o n v e l o c it y ) , h a s l e d t o F E M i m p l e m e n t a t i o n s . 10T h e p r e s e n t p a p e r a i m s a t f o r m u l a t i n g a B I E v e r s i on o ft h e 0 - m e t h o d .A c o m m o n f a c t o r s h a r e d b y t h e p r e se n t i n v e s ti g a t io ns u b j e c t a n d o t h e r s i t u a t i o n s w h e r e t h e g e o m e t r i c a ld o m a i n i s a p r i m a r y v a r i a b l e , l i k e s h a p e o p t i m i z a t i o n ,g e o m e t r i c a l i n v e r s e p r o b l e m s o r f r e e - b o u n d a r y p r o b -l e m s , i s t h e p r e s e n c e o f i n t e g r a l f u n c t i o n a l s b e i n gb o t h d i r e c t l y ( t h r o u g h t h e g e o m e t r i c a l s u p p o r t o f t h ei n t e g r a l ) a n d i n d i r e c t l y ( t h r o u g h m e c h a n i c a l f i e l d sw h i c h s o l v e , e . g . e l a s t i c b o u n d a r y - v a l u e p r o b l e m s )s h a p e d e p e n d e n t , w h o s e d e r i v a t i v e s w i t h r e s p e c t t os h a p e p a r a m e t e r s , o r m o r e a b s t r a c t l y t h e i r d o m a i nd e r i v a t i v e s , a r e o f p r a c t i c a l i n t e r e s t . I t i s k n o w n t h a tf i n it e - d if f e re n c e e v a l u a t i o n s o f g r a d i e n t s a r e b o t h c o m -p u t a t i o n a l l y e x p e n s i v e a n d p r o n e t o i n a c c u r a c i e s d u et o t h e k n o w n m a t h e m a t i c a l l y i l l - p o s e d n a t u r e o fth i s ope ra t i on . I I Henc e , i t i s na tu r a l t o r eve r t t oa n a l y t i c a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o a v a r i a b l ed o m a i n . T h i s c o n c e p t h a s b e e n d i s c u s s e d b y s e v e r a la u t h o r s ( s e e , e . g . R e f s 1 2 a n d 1 3 , i n F E M - o r i e n t e dc o n t e x t s ) . S i n c e i n s u c h p r o b l e m s t h e d o m a i n ( a n dh e n c e i t s b o u n d a r y ) i s a p r i m a r y v a r i a b l e , i t i s n a t u r a lt o c o n s i d e r t h e u s e o f b o u n d a r y i n t e g r a l f o r m u l a t i o n s ,w h i c h a l l o w f o r t h e ' m i n i m a l ' m o d e l l i n g . T h e f o r m u l a -t i o n o f s h a p e s e n s i t i v i t i e s i n a B I E c o n t e x t m a y r e s u l tf rom e i the r t he ad jo in t va r i ab l e a pp roa ch 14 17 o r t hed i rec t d i f fe ren t i a t i on app roa ch . 18 21I n t h e p r e s e n t p a p e r , t h e v i r t u a l c r a c k e x t e n s i o na p p r o a c h i s f o r m u l a t e d f o r B I E a n a l y si s o f c r a c k p r o b -l e m s , u s i n g t h e b a s i c c o n c e p t s d e v e l o p e d i n R e f s 6 a n d 7 .F i r s t , s o m e b a s i c d e f i n i t i o n s a n d r e s u l t s a b o u t m a t e r i a ld i f f e r e n t i a t i o n a r e r e c a l l e d . T h e n , t h e d e f i n i t i o n o f G ,f o r a t h r e e - d i m e n s i o n a l ( 3 - D ) c r a c k e d s o l i d , i s g i v e n ab o u n d a r y - o n l y r e f o r m u l a t i o n ; a s a r e s u l t , G s o l v e s av a r i a t i o n a l e q u a t i o n w h o s e r i g h t - h a n d s i d e d e p e n d sl i n e a r l y o n t h e t r a n s f o r m a t i o n v e l o c i t y t h r o u g h t h eb o u n d a r y s h a p e s e n s i t i v i t i e s o f e l a s t i c v a r i a b l e s . I no r d e r t o u s e o n l y c o n v e n t i o n a l d i s p l a c e m e n t c o l l o c a -t i o n B I E f o r m u l a t i o n s , t h e f r a m e w o r k o f m u t t i r e g i o nm o d e l l i n g i s u s e d . T h e f o r m u l a t i o n i s g e n e r a l i n t e r m so f t h e f r a c t u r e m o d e s i n v o l v e d . N e x t , t h e g o v e r n i n gB I E f o r m u l a t i o n f o r f i r s t - o r d e r e l a s t i c s h a p e s e n s i t i v -i ti e s, w h o s e s o l u t i o n i s a n e c e s s a r y s t ep f o r t h e n u m e r i -c a l e v a l u a t i o n o f G i n t h e p r e s e n t w o r k , i s g i v e n .T h e n , t h e n u m e r i c a l i m p l e m e n t a t i o n o f t h e m e t h o d i sd e s c r i b e d , a n d s p e c i a l a t t e n t i o n i s d e v o t e d t o t h e k e ys t e p o f b u i l d i n g a p p r o p r i a t e ( v e c t o r ) i n t e r p o l a t i o n f u n c -t i o n s f o r t h e t r a n s f o r m a t i o n v e l o c i t y i n t h e v i c i n i t y o ft h e c r a c k f r o n t . F i n a l l y , n u m e r i c a l r e s u l t s a r e p r e s e n t e d

o n t w o m o d e I e x a m p l e s w h e r e r e f e r e n c e r e s u l t s a r ea v a i l a b l e f o r c o m p a r i s o n : t h e r o u n d b a r w i t h a c i r c u l a rin t e rna l c rack and the semi -e l l i p t i ca l su r face c rack , i nb o t h c a se s u n d e r u n i f o r m t e n s i o n .

2 F I R S T - O R D E R M A T E R I A L D E R I V A T I V E O F AS U R F A C E I N T E G R A LL e t u s c o n s i d e r , i n t h e 3 - D E u c l i d e a n s p a c e ~ 3 ,e q u i p p e d w i t h a C a r t e s i a n o r t h o n o r m a l b a s i s ( e l , c 2, e 3) ,a body ~2p whose shape depends on a t ime- l i kep a r a m e t e r p t h r o u g h a c o n t i n u u m k i n e m a t i c s - t y p eL a g r a n g i a n d e s c r i p t i o n , w i t h t h e i n i t i a l c o n f i g u r a t i o nf~o conven t iona l ly a ssoc i a t ed wi th p = 0 :

Y E f~o --~ Y = P(Y ;P) e f~pw h e r e(VY E f~0) O (Y ; 0) = Y (3)T h r o u g h o u t t h i s p a p e r , l o w e r - c a s e b o l d f a c e l e t t e r s x , yd e n o t e g e o m e t r i c a l p o i n t s o n t h e c u r r e n t c o n f i g u r a t i o nt ip . T h e d i f fe o m o r p h i s m @ ( - ; p ) , o r geometrical trans-.formation, m u s t p o s s e s s a s t r i c t l y p o s i t i v e J a c o b i a ne v e r y w h e r e a n d f o r a n y p _> 0 . A g i v e n d o m a i ne v o l u t i o n c o n s id e r e d a s a w h o l e a d m i t s m a n y d i f fe r e n tr e p r e s e n t a t i o n s , e q n ( 3 ) , w i t h d i f f e r e n t t r a n s f o r m a t i o n s

2 . 1 M a te r ia l d e r iv a t iv e o f s c a la r o r t e n s o r f i e ld sT h e transformation velocity 0(y ; p ) , de f ined b y

0 ( y ; p ) = ~ p ( Y ; p ) f o r y = ~ ( Y ; p ) ( 4)i s t h e ( E u l e r i a n r e p r e s e n t a t i o n f o r t h e ) ' v e l o c i t y ' o f t h e' m a t e r i a l ' p o i n t w h i c h c o i n c i d e s w i t h t h e g e o m e t r i c a lp o i n t y a t ' t i m e ' p .N e x t , l e t f ( y ; p ) d e n o t e a s c a la r , v e c to r o r t e n s o rf ie ld . T h e m a t e r ia l d e r i v a ti v e f ( y ; p ) i n t h e d o m a i nt r a n s f o r m a t i o n y = ~ ( Y ; p ) i s d e f i n e d (s e e, e .g . R e f .22) as. 1f ( y ; p ) = l i m ~ [ . f ( ~ ( Y ; p + h ) , p + h ) - f ( ~ ( V ; p ) , p ) ]

= f p ( y ; p ) + V f ( y ; p ) . 0 (y ; p ) (5 )w h e r e V d e n o t e s t h e g r a d i e n t w i t h r e s p e c t t o E u l e r i a nc o o r d i n a t e s ( V f = ( f y ) e i) . T h e m a t e r i a l d e r i v a t i v eo f t h e g r a d ie n t V f is t h u s g i ve n b y

( V f ) * = V f - V f . V 0 (6 )I t s h o u l d b e s t re s s e d th a t t h e t r a n s f o r m a t i o n ~ i s u s e dt o r e p r e s e n t a c o n t i n u o u s c h a n g e o f d o m a i n , e a c h f ~ pb e i ng t h e g e o m e t r ic a l s u p p o r t o f a b o u n d a r y v a l u e p r o b -l e m , a s o p p o s e d t o a m a t e r i a l d e f o r m a t i o n o f a g i v e np h y s i c a l b o d y ; t h e ' m a t e r i a l ' q u a l i f i c a t i o n i s t h u s o n l ya c o n v e n i e n t l a n g u a g e a b u s e .

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Energy release rate fo r elastic fractu re 1392 . 2 M a t e r i a l d e r i v a ti v e o f s u r f a c e in tegralsA s s h o w n i n A p p e n d i x 1 , t h e m a t e r i a l d e r i v a t iv e s o f t h eu n i t n o r m a l n a n d t h e s u r f a c e di f fe r e n ti a l e l e m e n t d S o na m a t e r i a l s u r f a c e S p = O ( S ; p ) a r e g i v e n b y

d S = d i v s 0 dS = Dr Or dS, ( 7 )h = - n . V s O = - n r D j O r e ji n t e rm s o f t h e s u r fa c e g r a d i e n t V s a n d t h e s u r f a c ed ivergence d ivs :

V s f = V f - ( V f . n )n = ( f i - n i f n ) e i -- ( D i f ) e ,(8)

d i v s u = d i v u - ( V u - n ) . n = Diu i (9)Then , fo r a gener ic su r f ace in tegra l J (p ) :

= J f ( Y P )(P) sj , dSpone has , us ing eqn (7 ) :d J = j = J { f d S + f ( d S ) * }dp Sp

= [ { f + f d i v s O } d S (10)J s~Indee d J cou ld be expressed in severa l o ther w ays , 13 bu tt h e a b o v e f o r m u l a s e r v e s t h e p u r p o s e o f t h e p r e s e n tp a p e r .3 F O R M U L A T I O N O F T H E E N E R G Y R E L EA S ER A T E I N A B E M C O N T E X TL e t t h e b o d y f~ b e e l a s ti c ( s h e a r m o d u l u s # , P o i s s o nra t io u ) , i t s ex terna l boundary be ing sp l i t in to Sr (onwhich the t r ac t ion vec to r t is g iven: t = t ) and S ,, (onw h i c h t h e d i s p l a c e m e n t o is gi ve n: u = u ) . N o b o d yforces a r e p resen t . The po ten t ia l energy o f fZ a t e las t i ce q u i l i b r i u m t a k e s t h e v a l u e W :

W = ~ I I n a ( u ) : e (u ) d V - I s r t ' u d S (1 1)where u , ~ , t r a r e the d i sp lacement , s t r a in and s t r essf ie lds so lu t ion to the e las t i c equ i l ib r ium prob lem wi thb o u n d a r y d a t a u , t .3 . 1 M u l t i r e g i o n a p p r o a c h f o r a c r a c k e d e l a s t i c s o i ldN o w s u p p o s e t h a t a c r a c k F , w i t h u p p e r a n d l o w e rt r ac t ion- f r ee f aces F (F ig . 1 ) and un i t norm al nd i r e c t e d f r o m F - t o P + , is e m b e d d e d i n f ~. I n o r d e r t ou s e o n l y u s u a l d i s p l a c e m e n t B I E f o r m u l a t i o n s , t h e s o -ca l led 'm ul t i r eg ion app roac h z3 i s cons idered : f~ i s sp l iti n t o t w o s u b d o m a i n s f ~ + , f ~ - s e p a r a t e d b y a s u r f a c e Scon ta in ing the c r ack F (F ig . 1 ) . The cr acked so l ide q u i l i b r i u m i s t h e n f o r m u l a t e d i n t e r m s o f t w o e l a s t i c

u~

t ,

- ~ ~ f~ =f2 +n f~t o

Fig. 1. M ultiregion modelling of cracked solids: notation.p r o b l e m s ( o n e f o r e a c h s u b d o m a i n ) w i t h t h e f o l l o w i n gc o n d i t i o n s o n t h e e x t e r n a l b o u n d a r y :

u = u on Sut = t on Sf - fo r each sub dom ain f~ (12)t = 0 o n F

a n d t h e f o l l o w i n g c o u p li n g c o n d i t i o n s ( p e r f e c t b o n d i n gbetween f~+ and f~- ou ts ide the c r ack P) :u = u - S F ( 1 3 )nt + = - t -

I n t h e p a r t i c u l a r c a s e o f p r a c ti c a l i m p o r t a n c e w h e r e f ~h a s a s y m m e t r y p l a n e I I c o n t a i n i n g a p l a n e c r a c k Fand the ex terna l loa d ing t i s symme tr ic w i th r espe c tto I I , i t i s su f f ic ien t to cons ider the subprob lem overf~+ (say) , the coupling relat ions, eqn (13) , beingr e p l a c e d b y

u + .n = 0t + - ( t + . n ) n 0 o n ( I I n 0 f V ) - F (14)3 . 2 D e f i n i t i o n o f t h e e n e r g y r e le a s e r a t eT h e e n e r g y r e l e a s e r a t e G a s s o c i a t e d w i t h t h e c r a c k e dso l id f~ and the load ing u , t i s def ined by eqn (1 ),o r , e q u i v a l e n t l y b yI G ( 0 . v ) d s = - # v 0 ~ e (15)OFw h e r e v i s t h e u n i t n o r m a l t o t h e c r a c k f r o n t O F e x t e r i o rt o F a n d t a n g e n t t o F . A l s o , O d e n o t e s t h e s e t o f v i r tu a lcr ack ex tens ions , tha t i s , those t r ans fo rmat ion ve loc i t i es0 a s s o c i a t e d w i t h g e o m e t r i c a l t r a n s f o r m a t i o n s O ( . ; p )which descr ibe a c r ack ex tens ion : one has

0 . n = 0 o n F , 0 = 0 on Su, ST ( 1 6 )Thus , on ly r egu lar v i r tua l c r ack ex tens ions ( i .e .w i t h o u t k i n k i n g ) a r e c o n s i d e r e d . M o r e o v e r , i n e q n( 1 5 ) t h e v a r i a t i o n W o f W i s t a k e n f o r c o n s t a n tload ing (u , t ) s o t h a t o n e h a s , w i t h i n t h e m u l t ir e g i o n

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140 M. Bonnet, H . Xiaof r a m e w o r k ,

ti = 0 o n S ~0 o n S ~ f o r e a c h s u b d o m a i n f~ 0 o n r

with the coupling conditions{ ~+ =,i~+=_{(_ onS-r17)

18)3 . 3 B o u n d a r y - o n l y f o r m u l a t i o n o f t h e e n e r g y r e l e a s e ra t eN o w t h e v a l u e W o f th e p o t e n t i a l e n e r g y a t e q u i l i b r iu m ,e q n ( 11 ), c a n b e r e f o r m u l a t e d u s i n g b o u n d a r y i n t e g r a ls :a p p l y i n g t h e d i v e r g e n c e - f l u x t h e o r e m t o t h e d o m a i ni n t e g r a l i n e q n ( 1 1) a n d t a k i n g i n t o a c c o u n t t h e e q u i l ib -r i u m e q u a t i o n d i v c r = 0 g iv e s t h e w e l l - k n o w n a l t e r n a t i v eexpre ss ion :

J t . u D d S _ ~ . u d SV = ~ s, - . s ~T h e v a r i a t i o n W o f W i n a c r a c k e x t e n s i o n t h u s s t e m sf r o m a p p l i c a t i o n o f e q n ( 1 0 ) t o t h e a b o v e e q u a t i o n .S i n c e 0 E (9 , e q n ( 1 1) , a n d a c c o u n t i n g f o r t h e b o u n d a r ycon di t i o ns (eqn (17)) , I~ i s f i na l l y expre ssed in t e rm s o ft h e m a t e r i a l d e r i v a t i v e o f t h e b o u n d a r y e l a s ti c v a r ia b l e sa s fo l l ows:

J * 1 1 t D . 6 d S (19)t . u D d S _ 5T h e d e r i v a t i v e s ( ~ , t ) d e p e n d l i n e a r l y o n 0 , t h r o u g h ade r iva t i ve BIE to be d i scussed in sec t i on 4 , which hasto be so lved fo r a l l 0 i n ( a f i n i t e -d imens iona l subspaceof) 6) .T h u s , i n c o r p o r a t i o n o f t h e v a r i a n t e x p r e s s i o n ( 1 9 )f o r W i n t o t h e v a r i a t i o n a l e q u a t i o n ( 1 5 ) l e a d s t o ab o u n d a r y - o n l y a p p r o a c h t o t h e c o m p u t a t i o n o f G ( s ) .Note t ha t i t i s known ( see , e .g . Re f . 7 ) t ha t t he r i gh t -h a n d s id e - I ~ o f eq n ( 15 ) d e p e n d s u l ti m a t e l y o n l y o nt h e n o r m a l c r a c k f r o n t v i r t u a l e x t e n s i o n ( 0 - u ) l o r , n o to n a n y p a r t i c u l a r c o n t i n u a t i o n o f 0 o u t s i d e OILa l t h o u g h t h i s f a c t i s n o t a p p a r e n t T h i s i s a n i m p o r t a n tc o n s i d e r a t i o n b e c a u s e t h e p r e s e n t u se o f L a g r a n g i a n -t y p e m a t e r i a l d i f f e r e n t i a t i o n n e c e s s a r i l y r e q u i r e s s u c hc o n t i n u a t i o n o f 0 .

4 F I R S T - O R D E R E L A S T I C S H A P E S E N S I T I V I T YF O R M U L A T I O NT h e i n t e r m e d i a r y s t e p o f d e t e r m i n i n g t h e m a t e r i a lde r iva t i ve s ( f i , t ) fo r eve ry 0 E 6 ) i s now d i scussed fo rt h e s a k e o f c o m p l e t e n e s s . G o v e r n i n g B I E s f o r t h e f ir s t-and second-orde r e l a s t i c sens i t i v i t i e s a re i nves t i ga t ed i nmore de t a i l i n Re f . 19 .

4 . 1 G o v e r n i n g r e g u l a r i z e d e l a s t i c B I EA n y e l a s to s t a t i c s t a te o n ~ w i t h z e r o b o d y f o r c e s c a n b ec h a r a c te r i z ed b y t h e b o u n d a r y v a r i a bl e s g o v e r n e d b y t h efo l low ing re gu la r i zed d i sp l a cem ent BIE : 24'25

k[u/(y) (x, y) dS,,i(x)]nj(y) Z0~ ) . i/

t i ( y ) U [ ( x , y ) d S v = 0 (20)0~o r , i n t r o d u c i n g f o r l a t e r c o n v e n i e n c e a n a b b r e v i a t e dn o t a t i o n :

l l ( x , u ) : / 2 ( x , t ) : 0 ( 2 1 )u s i n g t h e K e l v i n i n f i n i t e - s p a c e f u n d a m e n t a l d i s p l a c e -m e n t a n d e l a s t i c s t r e s s t e n s o r

U )(x , y) - 1 [ (3 - 4u)6ik + r.ir, k] (22)16w#(1 -- u)rx 1

/ ~ / ( x , y ) 8 7 r ( 1 - u ) r 2 [(1 - 2u)(bikr/ + 6k r,; - ;~ur, k)+ 3r ir / r . k] (23)

c r e a t e d a t y E ~ 3 b y a u n i t p o i n t f o r c e a p p l i e d a t t h ec o l l o c a t i o n p o i n t x a l o n g t h e e k - d i r e c ti o n . A l s o , r =[y - x[ i s t he E uc l id i an d i s t ance be tw een y , x , an d ( ).~deno te s a pa r t i a l de r iva t i ve wi th re spec t t o Yi .The BIE , eqn (20) , ho lds fo r any c o l l o c a t i o n p o i n tx ~ .~3 . The d i sp l acement u i s r equ i red t o be H61de r -c o n t i n u o u s a t x w h e n x i s t a k e n o n 0 f l : 26

3C > 0 , 3a E [0 , 1 ] such tha tl u ( y ) - u ( x ) [ _< Cly - x l (~ (24 )

f o r t h e r e g u l a r i z a t i o n p r o v i d e d b y t h e p r e s e n c e o f t h eterm [u (y) - u (x)] in eqn (20) to b e effec t ive4 . 2 F i r s t - o r d e r s e n s i t i v i t y f o r m u l a t i o nA s m a l l p e r t u r b a t i o n o f t h e d o m a i n ~ = ~ p a s s o c i a t e dw i t h a s m a l l i n c r e m e n t d p i n d u c e s a p e r t u r b a t i o n o ft h e e l a s t o s t a t i c s t a t e ( a , t ) , w h i c h m a y b e e x p r e s s e d i nt e rms o f t he f i r s t -o rde r ma te r i a l de r iva t i ve s ( f i , ) :

~ u = f i d p + o ( d p ) , 6 t = t d p + o ( d p )T h i s i d e a i s c o n s i s t e n t w i t h t h e p r e s e n t B I E f r a m e w o r k :t h e b o u n d a r y Otlp o f a m a t e r i a l d o m a i n ~ p i s i t s e l fm a t e n a l , h e n c e ( u , t ) 1 0 ~ , a r e c o m p l e t e l y d e t e r m i n e d b ythe knowledge o f (u , t ) I0a , ,+d , , fo r t he ne ighbour ingp e r t u r b e d b o u n d a r y c o n f i g u r a t i o n s . I n o t h e r w o r d s ,

, . * *t h e m a t e r i a l d e n v a U v e s ( u , t) 1 0 e P a r e t a k e n w h i l es t a y i n g o n t h e m o v i n g b o u n d a r y T h e B I E , e q n ( 2 1 ) ,g o v e r n s a n y e l a s t o s t a t i c s t a t e d e f i n e d o n t l p f o r a n yp _> 0 s u c h t h a t O ( . ; p ) i s de f in e d T h u s , t a k i n g t h em a t e r i a l d e r i v a t i v e u s i n g e q n ( 1 0 ) y i e l d s t h e g o v e r n i n g

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142 M . B o n n e t , H . X i a ot r a n s f o r m a t i o n v e l o c i t y 0 . T h e d e r i v a t i v e B I E ( e q n( 3 2 ) ) i s w e a k l y s i n g u l a r u n d e r t h e r e q u i r e m e n t s o f e q n s(24) and (29) and th i s i s a l so t rue fo r a l l i n t e rmedia t ec a l c u l a t i o n s u s e d f o r i t s d e r i v a t io n .4 . 4 Com m ents ab out the der ivat ive BI EF i r s t l y , i t i s a p p a r e n t f r o m e q n s ( 2 1 ) a n d ( 3 2 ) t h a t t h es a m e i n t e g r a l o p e r a t o r g o v e r n t h e p r i m a r y v a r i a b l e s( n , t ) a n d t h e i r m a t e r i a l d e r i v a t i v e s . A l s o , f o r a g i v e nt r a n s f o r m a t i o n v e l o c i t y O , t h e d e r i v a t i v e B I E , e q n( 3 2 ) , h a s m a n y s o l u t i o n s ( f i , t ) . I n o r d e r t o e n s u r eu n i q u e n e s s o f ( f i , ) , o n e h a s i n a d d i t i o n t o s p e c i f y h o wt h e b o u n d a r y c o n d i t i o n s a s s o c i a t e d w i t h t h e e l a s t i cp r o b l e m e v o l v e w i t h ~ p . I t i s s i m p l e s t t o a s s u m e t h a tt h e t r a n s f o r m a t i o n O ( . ; p ) , w h i c h d e s c r i b e s a g i v e nc h a n g e o f d o m a i n , i s c h o s e n s o t h a t t h e D i r i c h l e t a n dN e u m a n n p a r t s S~,p, ST,p o f O f 2p a re r e spec t ive ly t r ans-f o r m e d i n t o t h e D i r i c h l e t a n d N e u m a n n p a r t s S u , p + d p ,ST,p+dpo f O ~ p + d p , SO t h a t ( u ,f i ) a n d ( t , t ) a r e u n k n o w n( a n d k n o w n o v e r t h e s a m e p o r t i o n s o f t h e b o u n d a r y .T h e n , ( fi, t ) a r e l i n e a r f o r m s o v e r 0 p r o v i d e d t h a t t h e i rp r e s c r i b e d p a r t s a r e t h e m s e l v e s l i n e a r f o r m s o v e r 0 ;t h u s , t h e u n k n o w n p a r t s o f ( u , t ) a n d o f ( u , t ) s h a r et h e s a m e i n t e g r a l g o v e r n i n g o p e r a t o r . T h i s r e m a r kc o n s t i t u t e s a n i m p o r t a n t c o m p u t a t i o n a l a d v a n t a g e : t h ed i s c r e t i z e d i n t e g r a l o p e r a t o r i s b u i l t a n d f a c t o r e d o n l yo n c e , i n t h e c o u r s e o f a b o u n d a r y e l e m e n t s o l u t i o n t oe q n ( 21 ), t h e n l a t e r r e p e a t e d l y r e u s e d f o r t h e n u m e r i c a ls o l u t i o n o f t h e d e r i v a t i v e B I E ( e q n ( 3 2 )) .5 B E M F O R M U L A T I O NT h e s u r f a c e s 0 f ~ + a n d 0 ~ a r e a p p r o x i m a t e d b y c la s si -c a l b o u n d a r y e l e m e n t s . E a c h e l e m e n t E e i s m a p p e d o n apa r en t e l em ent Ae , wh ich i s e i t he r t he sq ua re ( ( l , ~2) E[-1, 1J2 or the tria ng le (1 -> 0, ~2 _> 0, 0 < ~1 + ( 2 < 1,i n te r m s o f N U s h a p e f u n c t i o n s N k a n d n o d e s y k ( ing l o b a l n u m b e r i n g ) ; i s o p a r a m e t r i c i n t e r p o l a t i o n o f t h ee l a s t i c va r i ab l e s (u , t ) i s cons ide red , so t ha t{ y }(y )t (y ) { y }Z Nk(~) n k ,k= l t k

(33)W e deno te r e spec t ive ly by a . (~ ) (c~ = 1 ,2 ) and n (~ ) t hen a t u r a l b a s is o f th e t a n g e n t p l a n e a n d t h e u n i t n o r m a l a tY(():

N U (c~ = 1,2 )(34)

a~(~) = Z Nk, ~(()ykk = l1n(y) = ~ a A b)

w i t hJ : g l l g 2 2 - g ~ 2 : [ ]a l A a z H 2 (35)g ,~ = as " a;~ (c~,/3 = 1,2)

T h e t h r e e b a s i c s t e p s i n v o l v e d i n t h e c o m p u t a t i o n o fG u s i n g t h e p r e s e n t a p p r o a c h a r e n o w d e s c r i b e d .N i n e - n o d e d q u a d r i l a t e r a l e l e m e n t s w e r e u s e d f o r t h en u m e r i c a l e x a m p l e s p r e s e n t e d i n t h i s p a p e r .5 . 1 S o lut ion o f the pr im ary BI EThe BIE (eqn (20) ) i s d i sc re t i zed a long the l i ne s ou t l i neda b o v e . T h e c o u p l e d e l a s t o s t a t ic p r o b l e m s o n f~ =f ~ + u f~ a r e n u m e r i c a ll y so l v ed w i t h b o u n d a r y c o n -d i t i ons (12) and con t inu i ty cond i t i ons (13) . I f f~ i ss y m m e t r i c w i t h r e s p e c t t o a p l a n e H a n d i f F C I I ,o n l y o n e b o u n d a r y v a l u e p r o b l e m i s t o b e s o l v e d , t h ec o n t i n u i t y c o n d i t i o n s b e c o m i n g a s e q n ( 1 4 ) . T h i s s t e pi n v o l v e s t h e b u i l d i n g o f t h e u s u a l B E M d i s c r e t e l i n e a re q u a t i o n :

[A]{u} + [B]{t} = {0 }A f t e r a p p r o p r i a t e c o l u m n s w i t c h e s , o n e o b t a i n s t h eg o v e r n i n g l i n e a r s y s t e m s o f e q u a t i o n s o n t h e v e c t o r{ v } o f e l a s t o s t a t i c u n k n o w n s :

[K]{v} = {gO} (36)5 . 2 S o lut ion o f the der ivat ive BI ET h i s s t e p i n v o l v e s t h e c o n s t r u c t i o n o f a d i s c r e t e s e t o fa d m i s s i b l e t r a n s f o r m a t i o n v e l o c i ty fi e ld s 0 C O . D e n o t eb y E ( 0 F ) t h e se t o f b o u n d a r y e l e m e n t s a d j ac e n t t o t hec r a ck f r o n t O F a n d l et A l . . . . , A N c b e t h e N C m e s hn o d e s l o c a t e d o n O F ( F i g . 2 ) . T h e l o c a l n u m b e r i n g o fn o d e s o n e a c h e l e m e n t E E E ( 0 F ) i s a r r a n g e d s o t h a tt he curve (~ 2 = -1 ) , a ssoc i a t ed wi th t he nodes 1 , 2 , 3 , i sl o c a t e d o n O F . I n o r d e r t o t a k e i n t o a c c o u n t t h ek n o w n f a c t t h a t W u l t i m a t e l y d e p e n d s o n l y o n t h en o r m a l e x t e n s io n v e lo c i ty ( 0 . u ) o f th e c r a c k f r o n t ,t r a n s f o r m a t i o n v e l o c i t i e s o f t h e f o l l o w i n g f o r m a r ei n t r o d u c e d :

N C0(y ) = Z 0kB k(~) ' y = y(~) (37)k - 1i n t e r m s o f N C s c a l a r n o d a l v a l u e s 0 h = ( 0 . u ) ( A k ) o f

. . -" aF

F- C r a c k f ro n t n o d e so O t h e r n o d e s

Fig. 2. Crack surface (shaded area: E(0F)).

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~xi

( 0 . v ) a n d v e c t o r i n te r p o l a t io n f u n c t io n s B k . M a n ychoices a re poss ib le for the Bk; he re , they a re de f ineds o t ha t 0 ( y ) = 0 ou t s i de E ( 0 F ) ( F ig . 3 ) a nd , on a n y e le -m e n t Ee E E ( 0 F ) :Bk (~l, 1) = 0, ~l e [--1, 1] (38 )Bk(? /~ , -1) = 6ktw(rl~,--1), 1 = 1 ,2 ,3

w he r e r / t E A e i s t he a n t e c e de n t o f t he c r a c k f r on t nodeA t E Ee C E ( 0 F ) a nd t he l oc a l num be r i ng k = l , 2 , 3 i su s e d . D e f i n i ng a c on t i nua t i o n v ( ) o f t he un i t no r m a lto OF as fo l lows :1 g12 gllv = ] -~ (a l A n) = ~ a 1 gv~ ._] a2 (39)

the Bk(~) ( in loca l number ing) a re de f ined (F ig . 4) a sa k ( ) = f ( ~ 2 ) Sk (~1)v() (40)

w he r e S l , $2 , $3 a r e t he c l a s s i c a l one - d i m e ns i ona lqua d r a t i c s ha pe f unc t i ons :S l ( ~ l ) = ~ 1 ~ 1 - - 1)/ 2, S2(~ l) = 1 - ~ , S3(~1 )(41)

a n d f Is a c o n t i n u o u s f u n c t i o n s u c h t h a t f ( 1 ) = 0 ,f ( - 1 ) = t . I t i s s ugge s t e d t o u s e f ( ~ ) = ( 3 - 2~ - ~ 2 ) / 4w i t h qua r t e r - node e l e m e n t s ( t h i s a l l ow s f o r a l i ne a rva r i a t i on o f t he f a c t o r f i n t he phys i c al s pa c e ) o rf ( ~ ) = (1 - ~ ) / 2 w i t h o r d i n a r y e l em e n t s . T h e d e f in i ti o n(40) s a t i s f i e s the cons t ra in t s (38) ; moreover , t het r a ns f o r m a t i on ve l oc i t i e s 0 , e qn ( 37 ) , a r e c on t i nuousove r 0 f ~ a n d s a t is f y t he r e q u i r e m e n t s ( 16 ) . T he i n t e r -po l a t i on o f ( 0 . ~ ,) on F t a ke s t he f o r m o f a s t a nda r done - d i m e ns i ona l i n t e r po l a t i on :

NC( 0 . v ) ( y ) = E OgSk(~) (42)k=l

T he de f i n it i on , e qns ( 37 ) - ( 40 ) , i s the n s ubs t i t u t e d i n t o

Fig. 3.

Energy release rate for elastic fracture

x2F

Geometrical support and nodal values for thediscretized crack extension velocity.

, i(~2=+1) I I/ : (~2=-1)/ '

(Crack front)Fig. 4. The vector interpolation function B 2.

t he de r i va t i ve B I E ( e qn ( 32 ) ) . D ue t o t he l i ne a r i t y o ft he r i gh t - ha nd s i de o f e qn ( 32 ) w i t h r e s pe c t t o 0 , i ti s suf f i c ient to co mp ute th e so lu t ions (Uk ,T k) to eqn (32)for the pa r t i cu la r cho ices 0 = B k , so tha t the pa i r (~ , t )as soc ia t ed wi th 0 de f ined by eqn (37) i s g iven by

NC NC~1 = E Ok(Jk ' t= E Ok~k (43)k =l k = l* k * kThe pa i r (U ,T ) s a t i s f i e s the mat r ix re l a t ion:

[A]{ I3 k} + [B]{ 'F k} = { f t ( u , t ;B k) }w he r e t he r i gh t - ha nd s i de { f l ( u , t ; B k ) } c om e s f r om t hed i s c r e t i z a t i o n o f t h e r i g h t - h a n d s i d e J 2 ( x , t ; 0 ) -J l ( X , U ; 0 ) o f e qn ( 32 ) w i t h O = B k . T h e a b o v e e q u a -t i o n , t o g e t h e r w i t h t h e h o m o g e n e o u s b o u n d a r y c o n d i -t i ons ( 17 ) , l e a ds t o t he gove r n i ng l i ne a r s y s t e m s o fe qua t i ons f o r t he ve c t o r { ~ } o f unkn ow n de r i va t ive s :

[K ] {~ } = { f l (u, t ; Bk) } ( 44 )The same mat r ix [K] appears in eqns (36) and (44) ,be c a us e t he p r e s e n t c ons t r uc t i on o f 0 i s s uc h t ha t t heD i r i c h l e t a nd N e um a nn pa r t s S ~ , S ~ : r e m a i n f i xe da nd t hus a r e m a t e r i a l s u r f a c e s .5.3 Solut ion of the governing variat ional equation for GThe energy re l ease ra t e i s in t e rpola ted , s imi la r ly to(0 . v)10r , a s

NCG(s ) = E GkS k((2) (45)k =l

G be i ng t he noda l va l ue s G ( A k) . T he n ( u , t ) , e qn . ( 43 ) ,a re subs t i tu t ed in to the expres s ion (eqn (19) ) for W, sot ha t t he d i s c r e t iz e d f o r m o f t he va r i a ti ona l e q ua t i on(15) reads:f i nd Gk, V 0 m ( 1< m < N C ){ j 1I*oo, , Gk Sk(S) Sm(s)ds + Tin dSF -2 S,,

2 Sr

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144 M . B o n n e t , H . X i a ow h i c h , e q u a t i n g t o z e r o t h e c o e f f i ci e n t o f e a c h Om, l e adst o a l i n e a r (s y m m e t r i c a n d b a n d e d ) m a t r i x e q u a t i o n f o rt h e N C u n k n o w n s G k .5 . 4 E v a lu a t io n o f ta n g e n t ia l d i f fe r e n t ia l o p e r a to r sT h e d e r i v a t i v e B I E ( e q n ( 3 2 ) ) m a k e s u s e o f t a n g e n t i a ld i f f e r e n t i a l o p e r a t o r s , f o r w h i c h B E i m p l e m e n t a t i o n -o r i e n t e d e x p r e s s i o n s a r e n o w g i v e n . F i r s t , f r o m c l a s s i c ald i f f e r e n t i a l g e o m e t r y , t h e s u r f a c e g r a d i e n t V s f o f as c a l a r f u n c t i o n e x p r e s s e d i n t e r m s o f t h e v a r i a b l eE A~ i s g iven by

V s f = ( D ~ f ) e~ w i t h D r , ) = f ~ g ' ~ i ~ ( a ~ . e r ) (47)w h e r e g ~3 a r e c o n t r a v a r i a n t c o m p o n e n t s o f g : g'~)gn:~ =tSff , see eqn (35) . E qu a t io n (47) a l so h o lds f o r C a r t e s i anc o m p o n e n t s o f v e c t o r o r t e n s o r f i e l d s , s o t h a t

DrOr = O~,,g~i~(a~ er) (48)Dr j u i d S y = e~jq[Ui.l( a2-eq) - ui,2(al %)] d( (49)

5 . 5 N u m e r i c a l e v a l u a t i o n o f s i n g u la r i n t eg r a l sSing u la r i n t egra l s ove r an e l em ent E e occur i f x c E~ .F o l l o w i n g a c o m m o n p r a c t i c e i n B E M , 27 s et ~ j =p c o s ~ , ~ 2 = p s i n ~ , w h e re r t = 0 h , z / 2 ) d e n ot e th ea n t e c e d e n t o f x o n A e . T h e n ,

d ~ = p d p d ~ (50)M o r e o v e r , f o r a n y s h a p e f u n c t i o n N ( ~ ) , o n e c a ni n t r o d u c e m o d i f i e d s h a p e f u n c t i o n s N ( p , ~ ; r /) , r e g u l a ra t p = 0 a n d s u c h t h a t

U ~ ) - N O ) = p / c q p , ~; rt)s o t h a t o n e h a s f r o m e q n ( 3 3 ) :

P ~ r/)Yk= Ix-yl = l V k ( p , ~ ; = p~(p, @;r/)k = l

a n d # ( p , ~ , ;r / ) ~ O . C o n s e q u e n t l y , s i nc e U ~ ( x , y ) ,~ k ( x , y ) b e h a v e li k e r - l , r -2, respect ive ly:U~ (x ,y ) =_1 8 /k (p ,~? ; r / )

Pk ( 5 1 )~ ki ~ / ( x , y ) =

~ kwh ere U~(p , ~ ; 7 /) and ~ i . j (P , ~; r /) a re reg ular a t p = 0.F i n a l l y , u s i n g t h e m o d i f i e d i n t e r p o l a t i o n f u n c t i o n / V k ,13k a s s o c i a t e d w i t h t h e i n t e r p o l a t i o n s ( 3 3 ) o f u a n d(37) o f 0 ( see Appendix) , one can wr i t e :Nu(y ) - u(x ) -- p Z uk/Vk(p' ~; ~/) (52)

k - IB k( y) _ B~,-(x) --_ p B k ( p , ~; r/) (53)

From eqns (50) t o (53) , i t i s e a sy t o see t ha t a l l s i ngu la rin t egra l s a re r ecas t i n to regu la r i n t egra l s expre ssed in

I PIli

Fig. 5. Example 1: geometrical notations.t e r m s o f ( p , ~ ) . T h i s m a k e s f u l l u s e o f t h e r e g u l a r -i z a t i o n . T h e s u b s e q u e n t n u m e r i c a l i n t e g r a t i o n s c a n b ep e r f o r m e d w i t h s t a n d a r d p r o d u c t G a u s s i a n q u a d r a t u r ef o r m u l a e , u s i n g a l a s t c o o r d i n a t e c h a n g e ( p , ~ )( v l , v 2) i n o r d e r t o r e c o v e r a n i n t e g r a l o v e r t h e s q u a r e[ - 1 , 1 ] 2

6 N U M E R I C A L E X A M P L E SE x a m p l e 1 - - R o u n d b a r w i t h a p e n n y - sh a p e d a x i a l c r a c kA n i n t e r n a l p e n n y - s h a p e d p l a n e c r a c k o f r a d i u s R li s s i t ua t ed i n a cy l indr i ca l ba r ( l eng th 2H, ex t e rna lr a d i u s R > R l ) ; t h e c r a c k p l a n e i s t h e p l a n e o f s y m -m e t r y o r t h o g o n a l t o t h e a x i s o f r o t a t i o n a l s y m m e t r y ,a s s h o w n i n F i g . 5 . T h e b a r i s s u b j e c t e d t o a u n i f o r mt e n s i o n p , a l o n g t h e d i r e c t i o n o r t h o g o n a l t o t h e c r a c kp l a n e ( m o d e I ) . A n a p p r o x i m a t e s o l u t i o n f o r K 1 ,o b t a i n e d u s i n g s e m i - a n a l y t i c a l m e t h o d s , i s k n o w n f o r

Y

i~ ~ i i ~: ii

Fig. 6. Example 1: boundary element subdivision of crackplane (coarse mesh).

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E n e r g y r e l ea s e r a t e f o r e l a s ti c f r a c t u r e 145Y

Fig. 7. Example 1: boundary element subdivision of crackplane (fine mesh).t h i s p r o b l e m 32 in t h e f o r m :

lK , 1 - ( R 2 / R 2 ) P ~ V / - ~ I F ( R I / R )a n d v a l u e s f o r F a r e p r o v id e d b y a p p r o x i m a t e f o r m u l a e .T w o d i f f e r e n t m e s h e s , a s s h o w n i n F i g s 6 a n d 7 , w e r eu s e d f o r o n e - e i g h t h o f t h e s t r u c t u r e . T h e n u m e r i c a lv a l u e s o b t a i n e d f o r G a t t h e c r a c k f r o n t n o d e s a r ec o m p a r e d w i t h t h e a b o v e r e f e r e n c e s o l u t i o n i n F i g s 8a n d 9 , w h i c h s h o w t h e g o o d a g r e e m e n t b e t w e e n t h e m .E x a m p l e 2 - - S e m i - e l li p t i c a l s u r f a c e c r a c kA s a s e c o n d i l l u s t r a t i v e e x a m p l e , t h e c a s e o f a s e m i -e l li p t ic a l s u r fa c e c r a c k s i t u a t e d i n a s y m m e t r y p l a n e o f ar e c t a n g u l a r p a r a l l e p i p e d i s c o n s i d e r e d . T h e g e o m e t r i c a ln o t a t i o n s a r e g i v e n i n F i g . 1 0 . T h e a s p e c t r a t i o b / a ist h e e l l i p t ic i t y o f t h e c r a c k ( b = a f o r t h e s e m i c i r c u l a rc r a c k ) . F o r c o m p a r i s o n w i t h o t h e r a v a i l a b l e s o l u t i o n s ,t h e p a r a U e p i p e d i s s u b j e c t e d t o a u n i f o r m t e n s i o n p ,a l o n g t h e d i r e c t i o n o r t h o g o n a l t o t h e c r a c k p l a n e

0 . 8 0

0 . 7 0

0 .60 '0 . 5 0

0 40

0 . 3 0

0 . 2 0

0 . 1 00 . 0 0 0 . 0

. . . . . . X 7 . .. . " V . . . . . V - . . . . .

. . . . . ~ . . . ~ . . . . ~ - . . . .

. . . . . . . . r . . . . . . . . ~r . . . . . . . .o . . . . . . . . .p O C C, i

, i45 .0 80 .08 ( d ~ r e e s )

a / b . 0 . 2 5 r e f e r e n c e )O a / b = 0 . 2 5 ( p r e s e n t )

. . . . . . ~ = 0 . 3 7 5 r e f e r e n c e )[ 3 a / b = 0 . 3 7 5 ( p r e s e n t )

. . . . a f o = 0 . 5 r e f e r e n c e )0 a / o = 0 . 5 p r e s e n t )

- - - - - a / b = 0 . 6 2 5 r e f e r e n c e ), ~ a / b . 0 . 6 2 5 ( p r e s e n t)

- - - - - a / b = 0 . 7 5 r e f e r e n c e )V a / b = 0 . 7 5 ( p r e s e n t )

Fig. 8. Example 1: comparison between numerical results andreference solu tion (coarse mesh).

0 .800 . 7 0

0 . 6 0

0 .50G 0 . 4 0

0 3 0

0 ,200 10

0 .000 . 0

. - , - - , - - , - , - - , . - , - , . - - .

. - - . & - - . - . ~ - - . ~ - - - . . ~ - - - ~ - . - . . . . ~ - - . 4 ~ - - -

, - - . - - . - - . - . - - . - - . - - . - .

i----t~---~---~---~---Q----a---~r---ii C C O C 0 0 C i

4 5 .0 9 0 .00 (degntes)

aJb = 0.25 reference)O af o 0 25 present)

a/b = 0 375 reference)0.375 (present)~ . 0.5 reference)0.5 (present)0 625 reference)A a~o, 0.625 (present)a/b 0.75 reference)

1-'- -v ~o" 075 (p ... . )

Fig. 9. Example 1: comparison between numerical results andreference solution (fine mesh).( m o d e I ) . O w i n g t o g e o m e t r i c a l s y m m e t r y , o n l y o n e -q u a r t e r o f t h e i n i t i a l p a r a ll e l i p i p e d i s d is c r e t i z ed i n t o at o t a l o f 1 3 6 n i n e - n o d e d b o u n d a r y e l e m e n t s ; t h e c r a c kf r o n t i t s e l f i s m a d e o f s i x e le m e n t e d g e s , so t h a t Ga n d 0 a r e i n t e r p o l a t e d u s i n g 1 3 n o d a l v a l u e s . T w ov a r i a n t m e s h e s M 1 a n d M 2 w e r e u s e d , b o t h h a v i n gs ix b o u n d a r y e l e m e n t s a l o n g t h e c r a c k f r o n t , s o t h a t t h ea n g u l a r s p a c i n g b e t w e e n c r a c k f r o n t n o d e s ( u s i n gn o t a t i o n s o f F i g . 1 0) i s u n i f o r m ( A 0 = 7 r/ 24 , m e s hM 1 ; F i g . l l ) o r n o n - u n i f o r m ( A 0 = 7 r/ 32 ( r e s p e c t i v e ly7r/16) f or 0 E [0, 7r/4] (resp ecti ve ly 0 E [7r/4, 7r/2]), m es hM 2 ; F i g . 1 2 ) , 0 = 0 b e i n g t h e a n g u l a r l o c a t i o n o f t h ec r a c k e d g e .N u m e r i c a l v a l u e s o f t h e n o n d i m e n s i o n a l S I F K 7 - - -K I / K [ w e r e o b t a i n e d f r o m t h e v a l u e s o f G c o m p u t e dw i t h t h e p r e s e n t m e t h o d , u s i n g

K , = 1 - -- ~ G K [ = ~t h e s h a p e f a c t o r Q b e i n g d e f i n e d a s

I ?= E ( k ) = V / 1 - k 2 sin2 0 dOk2 = ~ l - (b /a ) 2 (b _ a lT h e s e a r e c o m p a r e d w i t h o t h e r n u m e r i c a l r e s u l t s f r o m

i i : : : : : : : : :: : : : : : : : :: : : : : : : : : :: : : : : ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i l i i i i i i i i i ii i i i i i i i i_ [ iiiiill l i i : : i i i i i i i i : : _- i a I LfFig. I0. Example 2: geometrical notation for the crack plane.

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iI

146 M . B o n n e t , H . X i a o

c r a c k f r ,

t YI

Fig. 11 . Exam ple 2 : bou nda ry elem ent subdiv ision of crackplane (mesh M 1, b = a) .

c r a c k

Fig . 12 . Example 2 : boundary element subdiv is ion of crackplane (mesh M2, b = 0-6a) .1.20

. . . ~[ r -7,~ " '%i . ~ o ..-*' ' :~,~ -:e-- ~. . . g . . -

1.00 2 ~

. . . ~ . E ) '

0 . 8 0 . - ~ ; . ,~ i F L , _ , ~ _ , - -/ [ ~ - - - O p r . a n t (K41) ] :0.70 " [G- - -~)present (M2)[~ - - - - Ra j u & Newman Ji I - - ~ Extrapolat ion (M2)I

0.60 . . . . . . . ~ . . . . . ~ , , ,0.0 30.0 60.0 90.00 (degrees)

Fig . 13 . Example 2 : compar ison between resu l t s (shal lowsemiell iptical surface crack, b = 0.4a).N e w m a n a n d R a j u 28 ( F ig s 1 3 - 1 6 ) a n d f r o m T a n a k a a n dI t o h ~ ( F i g s 1 4 a n d 1 5) . T h e l a t t e r o n e s w e r e o b t a i n e du s i n g a s o p h i s t i c a t e d s p e c i a l c r a c k - f r o n t e l e m e n t ,w h i c h a l l o w s f o r th e m o d e l l i n g o f b o t h t h e s q u a r e - r o o tc r a c k f r o n t s i n g u l a r i t y a n d t h e c r a c k e d g e s i n g u l a r i t y( w h o s e e x p o n e n t d i f fe r s f r o m - 1 / 2 e x c e p t f o r v = 0 ),a n d a r e t h u s e x p e c t e d t o p r o v i d e a b e t t e r r e f e r e n c e

1 . 1 o ~ . . . . . - . . : : ' ~r 1 , - ' ~ : 5 _ _ - . ~

1 . o 5 ~ , ,,.,.~.,,;,'.:..~,' , . . " . . ~ . : ~ . . "t~1 . o o t i . ' " : . ~ : " ,~ , I , . f , J

~ ; ' ~ . . . . I . . . . T a n a k a i lo , I J0.90; ~ j~ - - - '~Prosent (M1) lO"'O Present M2) M~2 ii l , - - , E : l r a ~ l a t , o o0 . 8 5 0.0 30.0 60.0 90.00 (degrees)

Fig . 14 . Example 2 : compar ison between resu l t s (shal lowsemiell iptical surface crack, b = 0 '6a).

1 3 5 71.30 ~ F,----T RaN & Newman| . . . . Tanaka & Itoh ~.

i I~-- ~ Present (U l ) lIO- - -O P resent (M2)1.25 I~ -- ~ Extrapolat ion (M2)II5 1 2 0 f , , , , ,

~_~ .. 41.15

1.1o - - - - - - - . . . . . , _ . . . . . . . . . . .

1.000.0 30.0 60.0 90.00 (degrees)

Fig . 15 . Example 2 : compar ison between resu l t s (semici rcu larsurface crack , b = a) .o . 9 o F , , : . . . . . . . . r . . . . . . . , . . . . . .

0.50 O" --present MI)O ~Oprasent (M2)V - - - - , R a ju & N e w m a n- - .~ Ext rapo ar ian (M2)0.40 [ -0.0 30.0 . . . . . 60,0

9 (degrees)Fig . 16 . Example 2 : compar ison between resu l t ssemiell iptical surface crack, b = 2a).900

(deep

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E n e r g y r e l ea s e r a t e f o r e l a s ti c f r a c t u r e 147s o l u t i o n n e a r t h e c r a c k e d g e . O n e s ee s i n F i g s 1 4 a n d 1 5t h a t o u r r e s u l t s s h o w a b e t t e r t e n d e n c y t o r e p r o d u c et h e s m a l l p e a k n e a r t h e c r a c k e d g e w h e n m e s h M 2 ,w h i c h i s f i n e r t h a n M 1 n e a r t h e c r a c k e d g e , i s u s e d .G e n e r a l l y s p e a k i n g , o u r r e s u l t s a g r e e r e a s o n a b l y w e l lw i t h t h e r e f e r e n c e o n e s . F o r c o m p a r i s o n ' s s a k e , w eh a v e a l s o d e p i c t e d i n F ig s 1 3 - 1 6 t h e v a l u e s o f K 7o b t a i n e d b y e x t r a p o l a t i o n o f t h e c r a c k o p e n i n g d i s p la c e -m e n t . T h e y t e n d t o b e s o m e w h a t l e s s g o o d t h a n t h o s eo b t a i n e d u s i n g t h e p r e s e n t a p p r o a c h , d e s p i t e t h e u s e o fq u a r t e r - n o d e e l e m e n t s a l o n g t h e c r a c k f r o n t .7 C O N C L U S I O NI n t h i s p a p e r , a n o v e l a p p r o a c h t o t h e c o m p u t a t i o n o fe n e r g y r e le a s e r a t e G ( s ) h a s b e e n p r e s e n t e d f o r 3 - D e l as -t i c f r a c t u r e . I t i s b a s e d o n a b o u n d a r y - o n l y a p p r o a c hf o r t h e f o r m u l a t i o n o f p e r t u r b a t i o n s o f th e e l a s ti cp o t e n t i a l e n e r g y a t e q u i l ib r i u m i n d u c e d b y c r a c k f r o n tv i r t u a l a d v a n c e s . A s s u c h , t h i s w o r k p a r a l l e l s o t h e ri n v e s ti g a ti o n s d e v o t e d t o t h e F E M - b a s e d 0 - m e t h o d .A k e y i n g r e d i e n t i n t h e p r e s e n t a p p r o a c h i s th e u s e o ft h e d e r i v a t i v e B I E , w h i c h g o v e r n s t h e e l a s t i c s e n s i t i v i t i e so n t h e b o u n d a r y . T h e d e r i v a t i v e B I E i t s e l f o f c o u r s e i sa l s o a p p l i c a b l e t o o t h e r k i n d s o f s i tu a t i o n s , l i k e s h a p eo p t i m i z a t i o n o r i n v e r s e p r o b l e m s .T h e e x t r a c o m p u t a t i o n a l c o s t a s s o c i a t e d w i t h t h eb u i l d i n g a n d s o l u t i o n o f th i s d e r i v a t i v e B I E i s r e a s o n -a b l e t h a n k s t o t h e f a c t t h a t t h e p r i m a r y a n d d e r i v a t i v eB I E s s h a r e t h e s a m e g o v e r n i n g o p e r a t o r , w h i c h i s t h u sb u i l t a n d f a c t o r e d o n l y o n c e . T h e n u m e r i c a l e x a m p l e sp r e s e n t e d s h o w t h e p o t e n t i a l o f t h e m e t h o d . I n p a r t i c u -l a r , t h e y t e n d t o b e m o r e a c c u r a t e t h a n t h e e v a l u a t i o n sb a s e d o n d i s p l a c e m e n t e x t r a p o l a t i o n p r o d u c e d b y t h es a m e r u n , a l t h o u g h q u a r t e r - n o d e e l e m e n t s w e r e u s e da l o n g t h e c r a c k f r o n t .F u r t h e r d e v e l o p m e n t s s h o u l d i n c l u d e t h e s t u d y o fa c t u a l c r a c k g r o w t h ( i n c l u d i n g s t a b i l i t y a n d n o n -b i f u r c a t i o n c o n s i d e r a t i o n s ) u n d e r t h e G r i f f i t h c r i t e r i o n .T h i s r e q u i r e s t h a t t h e e l a s t i c p o t e n t i a l e n e r g y b e d i f -f e r e n t i a t e d u p t o t h e s e c o n d o r d e r w i t h r e s p e c t t oc r a c k f r o n t a d v a n c e s .R E F E R E N C E S

1. Bui, H. D. M~canique de la Rupture F ragile. Masson, Paris,1978.2. Ng uyen , Q. S . Bifurcat ion an d s tabi l i ty in diss ipat ivemedia (plasticity, fr iction, fracture) . Appl. Mech. Rev.,1994, 37, 1-31.3. Hel len, T . K. On the method of vir tual crack extens ion.Int . J . Num. Meth. in Engng. 1975, 9, 187-207.4. Parks , D . M . A s t if fness der ivat ive f ini te e lement techniqu efor determ inat ion o f crack t ip s t ress intens i ty factors . Int .J . Fract., 1974, 10, 487-501.5. Delorenzi , H. G. On the energy re lease ra te and the J -integral for 3-D crack conf igurat ions . Int. J . Fract., 1982,19, 183-94.

6. Destu ynde r , P . , Djao ua, M. & Lescure , S . Quelquesremarques sur la mrcanique de la rupture 61as t ique. J .M~can. Th~or. Appl., 1983, 2, 113-35.7 . M ia lon , P . Ca lcu l de l a d& ivr e d ' une g r andeur pa r r appo r t~i un fond de f issure par la m& hod e 0. Bullet in ED F/D ER,Sdr. C, 1987, 3.8. Ohtsuka, A . General ized J - integral and three-dim ensionalfracture mechanics. Hiroshima Math. J . , 1981 , 11 , 21-52.9 . S uo , X. Z . & Com bescur e , A. S ur une f o r m ula t ionmathrmatique de la dr r iv~e de l ' rnergie potent ie l le enthro r ie de la ruptu re f ragile . C.R . Acad. Sci. Paris, Sgr. II ,1989, 308 , 1119-22 .10. Wadie r , Y . & M alak , O. The the ta m e thod app l i ed to theanalys is of 3D elas t ic-plas t ic cracked bodies . S M I R T , L o sAngeles, 1989.11. Tikhonov, A. N. & Arsenin, V. Y. Solutions to Ill-posedProblems. Wins ton- Wi ley , New Yor k , 1977 .12. Haug, Choi , J . O. , Komkov. Design Sensitivity Analysis ofStructural Systems. Academic Press, 1986.13. Petryk, H. & Mroz, Z. Time der ivat ives of integrals and

f unc t iona l s de f ined on va r y ing vo lum e and sur f acedomains . Arch. Mech. , 1986, 38, 6 94-724.14. Aitha l, R. & Saigal, S. Shape sensitivity analysis in the rma lpr ob lem s us ing BEM. Engng Anal. with Bound. Elem.,present issue.15. Bonnet , M. BIE an d ma ter ia l different ia t ion appl ied to theform ulat ion o f obs tacle inverse problems. Eng. Anal. withBoundary Elem., 1995, 15, 121-36.16. Choi , J . O. & Kwak, B. M. Boundary integral equat ionme thod for shape opt im izat ion of e las tic s t ructures . Int . J .Num. Meth. in Engng, 1988, 26, 1579-95.17. Mer ic , R. A . D if ferent ia l and integral sensi t ivi ty formu la-t ions and shape op t im iza t ion by BEM. Engng Anal. withBoundary Elem., present issue.18. Baron e, M. R . & Yang, R. J . A bo und ary e lementapproach for recovery of shape sens i t ivi t ies in three-dim ensio nal elastic solids. Comput . Meth. in Appl. Mech. &Engng, 1989, 74, 6 9-82.19. Bonnet , M. Regular ized BIE formulat ions for f i r s t - andsecond-order shape sensitivity of elastic f ields. Computersand Structures, (submitted).20 . Mukher jee , S . & C handr a , A. A boun dar y e lem entformulation for design sensitivities in problems involvingbo th geom etr ic and mater ia l nonl ineari ties. Math. Comput .Modelling, 1991, 15, 245 -55.21. Z hang, Q. & Mu kher jee , S . Second-ord er des ign sens it ivi tyanalys is for l inear e las t ic problems by the der ivat iveboundar y e lem ent m e thod . Comp. Meth. in Appl . Mech.& Engng, 1991, 86, 32 1-35.22. Salenqon, J. Mdcanique des Milieux Continus. Presse del 'Ecole Polytechnique , 1992.23. Cruse, T. A. Boundary Element Analysis in ComputationalFracture Mechanics. Kluw er Academic Publishers , 1988.24. Rizzo, F . J . & Skippy, D. J . An advanc ed bou nda ryintegral equat ion method for three-dimensional e las t ic i ty.Int . J . Num. Meth. in Engng, 1977, 11, 1753-68.25. Bui , H. D. , Loret , B. & Bonnet , M. Rrgular isa t ion des6quations intrgrales de l '61astostatique et de l '61asto-dynam ique . C.R. Acad. Sci. Paris, S(r. I I , 1985, 300,633- 6 .26. Kupradze, V. D. (ed. ) Three-dimensional Problems of theMa them atica l Theory of Elasticity an d Thermoelasticity.Nor th Hol land , Am s te r dam , 1979 .27. Rizzo, F . J . , Skippy, D. J . & Rezayat , M. A boundaryin tegr a l equa t ion m e thod f o r r ad ia t ion and s ca t t e r ing o felastic waves in three-dimensions. Int . J . Num. Meth. inEngng, 1986, 23, 425 -36.

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148 M . B o n n e t , H . X i a o28. Ta da, P aris & Irwin. The stress analysis of crack handbook.Del. Research Corp. , Hel ler town, PA, 1973.28. Newman, J. C. & Raju, I . S. An empirical stress-intensityfactor equ at ion for the surface crack. Engng Fract. Mech.,1981, 15, 185-92.29. Tanaka, M. & I toh, H. New crack e lements for boundary

eleme nt analysis of elastostatics consid ering arbitrar ystress singularities. Appl. Math. Modelling, 1987, 11,357-63.30. Nguyen, Q. S . , S tolz , C. & Debru yne, G. En ergy method sin fracture mechanics: stability, bifurcation and secondvar ia t ions . Eur. J . Mech . A/Sol ids, 1990 , 9, 157 73.31. Pradeilles-Duval, R. M. Ev olut ion de syst6mes avecsurfaces de discontinuit~ mobiles: application aud61aminage. Th~se de doctorat de l 'Ecole Polytechnique,1992.

A P P E N D I X 1. M A T E R I A L D E R I V A T IV E O Fd S A N D nF o r t h e s a k e o f c o m p l e t e n e s s , e q n ( 7 ) i s b r i e f l y e s t a b -l i s h e d . F i r s t , t h e m a t e r i a l d e r i v a t i v e o f a m a t e r i a lv e c t o r a a t t a c h e d t o t h e m o v i n g p o i n t y = O ( Y ; p ) i sg i v e n b y

= V 0 - a ( A I .1 )T h e n , l e t Sp b e a m a t e r i a l s u r f a c e , a n d d e n o t e b y ( a , b ) ap a i r o f m a t e r i a l v e c t o r s a t t a c h e d t o a m a t e r i a l p o i n ty = O ( Y ; p ) , c h o s e n s o a s t o b e u n i t a r y a n d o r t h o g o n a la t a f i x ed v a l u e P 0 o f p a n d t o b e l o n g t o t h e t a n g e n tp l a n e a t y t o S p f o r a l l p i n a n e i g h b o u r h o o d o f P 0.F o r a n y s u c h p , t h e u n i t n o r m a l t o S p a t y i s t h u sg i v e n b y

1n ( y ; p ) - ( a A b ) ( A 1 . 2 )l a A b lM o r e o v e r , t h e s u r f a c e d i f f e r e n t i a l e l e m e n t d S a t y i sp r o p o r t i o n a l t o ] a A b [, s o t h a t

d S = t a A b l * d S ( A 1.3 )T h e m a t e r i a l d e r i v a t iv e o f t h e v e c t o r p r o d u c t a A b i st h e n t a k e n f o r t h e p a r t i c u l a r v a l u e P0 o f p , u s i n g e q n( A I . 1 )

( a A b)* = ( V 0 . a ) A b + a A ( V 0 - b )= [ ( V O ) a a - -[ - ( V O ) b b ] n - - ( V O ) n a a - - ( V 0 ) n b b= ( d i v s 0 ) n - n . V s 0 ( A 1 . 4 )

w h e r e t h e f a c t t h a t ( a , b , n ) i s a n o r t h o n o r m a l v e c t o rf r a m e a t p = P 0 h a s b e e n u s e d . M o r e o v e r , o n e h a sa A b[a A b I* = - - . ( a A b )* = n - (a A b )* : d i v s 0[ a A b l

( A l S )E q u a t i o n ( 7 ) i s t h e n e a s i l y o b t a i n e d f r o m e q n s ( A 1 . 2 ) ,( A1. 3) , ( A1 . 4) , ( A1 . 5) .

A P P E N D I X 2 . M O D I F I E D SH A P E F U N C T I O N SF O R S I N G U L A R I N T E G R A T IO NS hape f unc t i ons f or n i ne - noded q uadr i l a t era l e l ement

T h e c l a s s i c a l L a g r a n g i a n s h a p e f u n c t i o n s f o r t h e n i n e -n o d e d q u a d r i l a t e r a l e l e m e n t a r eNI({) =S1({1)S1(~2)N 2 ( ~ ) = &( ~ ) &( ~ 2 )N3(~) :$3(~1)S1(~2)N4(~) = 33(~1)32(~2 )Ns(~) =$3(~1)$3(~2)

m 6( ~) = $2 ( ~1 ) $3 ( ~2 )N7(~) = S1(~1)$3(~2)N8(~) = S1(~1)$2(~2)Ng(~) =$2(~1)$2({2)

(A2.1)w i t h t h e S ~ g i v e n b y e q n ( 4 1 ) . T h e i r d e r i v a t i v e s N ~ a r et h u s g i v e n b y

NI ' (~) = S[ (~1)Sj(~2)N J ( ~ ) : S i ( ~ I ) S j t ( ~2 ) ( A 2 . 2 )

w i t h a p p r o p r i a t e l y c h o s e n i , j .M odi f i ed s hape f unc t i onsS i n g u l a r i n t e g r a t i o n s u s e m o d i f i e d s h a p e f u n c t i o n s A ?s u c h t h a t

N ( ~ ) - N ( r l ) = p N ( p , ~ ; r l)F i r st , n o t e t h a t , f o r a n y p a i r f ( ~ ) , g ( ~ ) o f r e g u l a rf u n c t i o n s :

jg (p , qo, ~1) = f ( P , qo; r l )g (~ ) + f ( r l )g , ( p , ~ ; r / ) ( A2 . 3)s o t h a t t h e N , N , ~ , e q n s ( A 2 . 1 ) a n d ( A 2 . 2 ) , c a n b er e a d i l y o b t a i n e d i n t e r m s o f t h e S i , S [ , w i t hS 1 ( / 9 , ~ ; / ] ~ ) = c~ [2r/~ - 1 + p c , ] / 2S2( p, ~; ~l,) = c~ [ - 2~1~ - pc~]S 3 ( p , ~ ; T I , ) = cc~[2r/o + 1 + p c ~ ] / 2

S~(p, ~; ~,~) = -2c,~SI(P, ~;'J~) = c, ,

(A2.4)I n t h e a b o v e f o r m u l a , C l , c 2 s t a n d f o r c o s % s i n ~ ,r e s p e c t i v e l y .

^M odi f i ed v ec t or s hape f unc t i on BymI t i s d e f i n e d b y

Bm(~) -- Bin(n) = PBm(P, ~; '1)F r o m t h e d e f i n i t i o n ( e q n ( 4 0 ) ) a n d u s i n g e q n ( A 2 . 3 ) , B mi s g i v e n b y

^mB ( p , ~ o , T ] ) : ( / ( p , C /9 ;T ] 2 ) S m ( ~ l )+ f ( r h ) S m (P , ~ ; r h ) ) v ( ~ )+ f ( T l 2 ) S m ( * h ) ~ ' m ( P , ~ ,* l ) ( A 2 . 5 )

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Energ y release rate for elas tic fractur e 149Nex t , eqn (39 ) fo r u and the ru le ( eqn (A2 .3 ) ) g ive

~m(p,~o , r l )_ g lc , ()f i , (p ,~o;n)g,/~7~ (1~2(p,~; n)( gl2 ~ A ( p , o ; . r /) a l ( .r / )+ \ ~ /

__ ( g l l ~ A P , ~; r / ) a 2 ( I r / )\ g , / ~ /w i t h NUfi~(p , ~o; rl) = Z ]Vk,c~(P,~o; rl) y kk=lF i n a l l y , o n e c a n s h o w t h a t

gla ~A(p ,~;r /) = 1~ .] ~'a (A~lgll + A~2g12 + A~2922)(c~ = 1,2) (A2 .6)

w i t h~, ~( p, p; r/) = fi,~(p, p; 0) " a3( ) + ac,(rt )~a (p, ~o; rl)

D,~ ( , r /) = ~/g ll (~ )J(~) ~/ g ll ( r / )J(n ) [gla() V/gll ( ' r /)J(r/)+ gla (r/) -v/gl 1 ~) J(~ ) ]

A~I (~, r /) = ~ (r /)g~1 2() - g ,m(r /)g l2(r /) [g l2()gn ( r /) + g l2(r /)gu ()]

A~2(~, r/) = g,~,~(rl)gu (r/)[g l2(~ )gu (r/)+ g12(r/)gll (~)]A~2(~, r/) = -~,~ (r/)g2 1 (~)

A P P E N D I X 3 . E X P R E S S I O N O F D r B ~One can show us ing c las s ica l d i f f e r en t ia l geomet ry tha t ,fo r any vec to r B such tha t B . n = 0 :

d i v s B d S y = ( x / J B ~ ) ,~ d ~

The n , f rom the de f in i t ion ( eqn (40 ) ) o f B m, one has :

+ - s ~ - ~ s ' g ~ ]J mvg,1 - J mT l, deT h e a b o v e f o r m u l a e u s e t h e s e c o n d d e r i v a t i v e s Nk, ~ ~ o fthe shape func t ions Ark .