Computational Methods Based on an Energy Integral in Dynamic

8/3/2019 Computational Methods Based on an Energy Integral in Dynamic

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I n t e r n a t i o n a l J o u r n a l o f F r a c t u r e 2 7 ( 1 98 5 ) 2 2 9 - 2 4 3 .O 1985 M a r t i nus N i jho f f Pub l i she r s , D ord rech t . P r i n t ed i n The N e the r l an ds .

Com putat i ona l m ethods based on an energy i n tegra l i n dynam i cfractureT . N A K A M U R A , C .F . S H I H a nd L .B . F R E U N DDivision o f Engineering, Brown University, Providence, R1 02912, USA(R ece ived Sep t em ber 1 , 1984)

A b s t r a c t

D e v e l o p m e n t s i n t h e u s e o f t h e c r a c k t i p e n e r g y f l u x i n te g r a l in c o m p u t a t i o n a l d y n a m i c f r a c tu r e m e c h a n i c s o v e rthe pas t f ew yea r s a r e r ev i ew ed . A n exp re s s ion fo r t he c r ack t i p ene rgy f l ux i n t e rm s o f ne a r t i p m echan i ca l f i e l d sw h ich i s va l i d fo r gene ra l m a t e r i a l r e sponse i s de r ived . I t i s t hen de m o ns t r a t e d t ha t ce r t a in u se fu l ene rgy i n t eg ra l sm a y b e e x t r a c te d f r o m t h e g e n e r a l r e su l t b y i n v o k i n g t h e a p p r o p r i a t e c h a r a c t e r i z a t io n o f m a t e r i a l r e s p o n se .Seve ra l a l t e rna t i ve rep re sen t a t i ons o f ene rgy f lux i n t he fo rm o f i n t eg ra l s ove r som e f i n i t e r eg ion a rou nd t hec rack t i p a r e p r e sen t ed an d com pared w i th a v i ew tow ard im p lem en ta t i on i n f in i t e e l em en t s im u la t i on s t ud i e s .

1 . I n t r o d u c t i o n

The f i r s t c r ack t ip con tour in t eg ra l express ion fo r e l as todynamic energy r e l ease r a t e wasp r o p o s e d b y A t k i n s o n a n d E s h e l b y [1 ] , w h o a r g u e d t h a t t h e f o r m f o r d y n a m i c g r o w t hs h o u l d b e t h e s a m e a s f o r q u a s i - s t a ti c g r o w t h w i t h t h e e l a s ti c e n e rg y d e n s i t y r e p l a c e d b ythe to t a l m echan ica l energy dens i ty , t ha t i s , t he e l as t i c energy p lus the k ine t i c energy . Theequ iva len t i n t eg ra l express ion fo r dynamic energy r e l ease r a t e in t e rms o f c r ack t ip s t r essa n d d e f o r m a t i o n f i e l d s w a s s u b s e q u e n t l y d e r i v e d d i r e c t l y f r o m t h e f i e l d e q u a t i o n s o fe l a s t o d y n a m i c s b y K o s t r o v a n d N i k i t i n [ 2] a n d b y F r e u n d [ 3] . T h e y e n f o r c e d a n i n s t a n t a -n e o u s e n e r g y r a t e b a l a n c e f o r t h e t i m e - d e p e n d e n t v o l u m e o f m a t e r i a l b o u n d e d b y t h eo u t e r b o u n d a r y o f t h e s o l i d , t h e c r a c k f a c e s , a n d s m a l l l o o p s s u r r o u n d i n g e a c h m o v i n gcrack t ip and t r ans l a t ing wi th i t. By app l i ca t ion o f Reyn o ld ' s t r anspo r t t he orem [4 ] andthe d ivergence theorem an express ion fo r c r ack t ip energy f lux in the fo rm of an in t eg ra lo f f i e ld quan t i t i e s a long the c r ack t ip loop was ob ta ined . A r esu l t wh ich i s app l i cab le to am u c h b r o a d e r r a n g e o f m a t e r i a l r e s p o n s e a n d w h i c h c o n t a i n s t h e e a r l i e r e l a s t o d y n a m i cresu l t a s a spec ia l case i s de r ived here , wi th an observa t ion made by Eshe lby [5 ] on thefo rm of the equa t ions o f mot ion as a po in t o f depar tu r e . A s imi l a r r esu l t has a l so beenob ta ined by Wi l l i s [ 6 ] .

I n t h e a n a ly s i s o f d y n a m i c c r a c k p h e n o m e n a b y m e a n s o f c o m p u t a t i o n a l m e t h o d s , s u c has the f in i t e e l emen t method , a fund am enta l d i f f i cu l ty i s enco un te r ed in e f fo r t s t oc o m p u t e v a l u e s o f t h e c r a c k t ip e n e r g y f l u x v e rs u s t im e o r a m o u n t o f c r a c k g r o w t h . T h ed i f f i cu l ty a r i ses f rom the f ac t t ha t , on the o ne hand , t he c r ack t ip energy f lux i s de f ined int e rms o f the va lues o f f i e ld quan t i t i e s fo r po in t s a rb i t r a r ily c lose to the c r ac k t ip whi l e , onthe o ther hand , i t i s p r ec i se ly fo r po in t s near the c r ack t ip fo r which the accura teca lcu la t ion o f f i e ld quan t i t i e s i s mos t d i f f i cu l t . I n an e f fo r t t o c i r cumven t th i s d i f f i cu l ty ,f i n i t e d o m a i n i n t e g r a l s h a v e b e e n i n t r o d u c e d f o r t h e c o m p u t a t i o n o f d y n a m i c e n e r g yr e le a s e ra t e b y K i s h i m o t o , A o k i a n d S a k a t a [ 7 , 8 ], b y A t l u ri [ 9 ] , a n d b y N i s h i o k a a n dAt lu r i [ 10 -12] .

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2 30 T. Nakam ura, CF . S hih and LB . FreundT h e p u r p o s e h e r e i s to r e v i ew d e v e l o p m e n t s in t h is a r e a o v e r t h e p a s t f e w y e a rs . F i r s t ,

a n e x p r e s s i o n f o r c r a c k t i p e n e r g y f l u x w h i c h i s v a l i d f o r g e n e r a l m a t e r i a l r e s p o n s e i sd e r i v e d i n a d i r e c t m a n n e r . W h i l e a v a r i e t y o f s e e m i n g l y d i f f e r e n t p a t h - i n d e p e n d e n te n e r g y i n t eg r a l s h a v e b e e n p r o p o s e d i n r e c e n t y e a rs , e a c h m a y b e e x t r a c t e d f r o m t h i sg e n e r a l r e s u l t b y i n v o k i n g t h e a p p r o p r i a t e r e s t r i c t i o n s o n m a t e r i a l r e s p o n s e a n d c r a c k t i pm o t i o n . S e v e r a l a l t e r n a t e e x p r e s s i o n s f o r t h e e n e r g y f l u x i n t e g r a l a r e o b t a i n e d f r o m t h eg e n e r a l r e s u l t b y a p p l i c a t i o n o f t h e d i v e r g e n c e t h e o r e m o v e r s o m e p a r t o f t h e f r a c t u r i n gs o l id a n d t h e s e e x p r e s s io n s a r e c o m p a r e d , w i t h a v ie w t o w a r d i m p l e m e n t a t i o n i n f i n i tee l e m e n t s im u l a t i o n s . F o r t h e i ll u s t r a ti o n s b a s e d o n e l a s t o d y n a m i c c r a c k g r o w t h , e x a c ts o l u t i o n s a r e a v a i l a b l e t o t e s t t h e c o m p u t e d q u a n t i t i e s . I n t h e c a s e o f e l a s t i c - p l a s t i cr e s p o n s e , o n t h e o t h e r h a n d , n o e x a c t s o l u t i o n s e x i s t t o p r o v i d e a b a s i s f o r c o m p a r i s o n .2 . T h e c r a c k t i p e n e r g y f l u x i n t e g r a lI n t h e a b s e n ce o f b o d y f o r c es , th e e q u a t i o n o f m o t i o n i n t e r m s o f c a r t e si a n c o m p o n e n t s o fs t r e s s o~j a n d d i sp l a c e m e nt u i i s

~ ( l i j 0 2 U ib x j P - ' ~ - t = 0 , ( 2 . 1 )w h e r e p i s m a s s d e n s i t y . I f t h e i n n e r p r o d u c t o f (2 .1 ) w i t h t h e p a r t i c l e v e l o c i ty O u J O t isf o r m e d , t h e n t h e r e s u l t i n g e q u a t i o n m a y b e w r i t t e n i n t h e f o r m

o x j t - f f o , j ) + r ) = 0 , ( 2 . 2 )w h e r e U i s t h e s tr e s s w o r k d e n s i t y a n d T i s t h e k i n e t ic e n e r g y d e n s i ty , n a m e l y , a t a n ym a t e r i a l p o i n t ,

I ' t O Z u i , OUi OUi (2 . 3 )U = J o O i j ~ d t T = PO t O t"E q u a t i o n ( 2 .2 ) i s a d i f f e r e n t i a l f o r m o f t h e w o r k b a l a n c e p r i n c i p l e w h i c h i s v a li d f o r a n ym a t e r i a l r e s p o n s e .

I f t h e t i m e c o o r d i n a t e i s n o w g i v e n t h e s a m e s t a t u s a s a s p a t i a l c o o r d i n a t e , t h e n ( 2 . 2) i st h e r e q u i r e m e n t t h a t a c e r t a i n v e c t o r e x p r e s s i o n i s d i v e r g e n c e f r e e i n a p a r t i c u l a r" v o l u m e " o f s p a c e - ti m e , t h a t is , i t r e p r es e n t s a m e c h a n i c a l c o n s e r v a t i o n l a w . T h u s , i f t h ed i v e r g e n c e t h e o r e m i s a p p l i e d o v e r t h i s v o l u m e , t h e r e s u l t i s t h a t t h e i n t e g r a l o v e r t h eb o u n d i n g s u r f a c e o f t h i s v o l u m e o f th e i n n e r p r o d u c t o f t h e d i v e r g e n c e fr e e v e c t o r w i t h t h es u r f a c e u n i t n o r m a l v e c t o r v a n is h e s . F o r d e f i n i t e n e s s , c o n s i d e r a t w o - d i m e n s i o n a l b o d yw h i c h c o n t a in s a n e x t e n d i n g c r a c k a n d b o u n d e d b y a c u rv e C ( a t ta c h e d t o a r b i t r a r ym a t e r i a l pa r t i c le s ) . The p l a ne o f t he bo dy i s t he x 1 , x2-p l a ne , t he c ra c k p l a ne i s x z = 0 a n dc r a c k g r o w t h i s a l o n g t h e x l - a x i s . T h e i n s t a n t a n e o u s c r a c k t ip s p e e d i s v . A s m a l l c o n t o u rF b e g i n s o n o n e t r a c t i o n f r e e f a c e o f t h e c r a c k , s u r r o u n d s t h e t i p , a n d e n d s o n t h eo p p o s i t e t r a c t i o n f r e e f a c e . T h i s c r a c k t i p c o n t o u r i s f i x e d i n s i z e a n d o r i e n t a t i o n w i t hr e s p e c t t o t h e c r a c k t i p c o o r d i n a t e s a s t h e c r a c k g r o w s . C o n s i d e r t h e v o l u m e i n t h e x 1, x 2 ,t - spa c e t h a t i s bou nd e d by t he p l a ne s t = t I a nd t = t 2 , wh e re t 1 a n d t 2 a re a rb i t r a ry t i me s ,b y t h e r i g h t c y l i n d e r s w e p t o u t b y C b e t w e e n t h e t w o t i m e s , b y t h e p l a n e s s w e p t o u t b yt h e c r a c k f a c e s b e t w e e n t h e t w o t i m e s , a n d b y t h e t u b u l a r s u r f a c e s w e p t o u t b y t h e s m a l lc o n t o u r F a s t h e c r a c k a d v a n c e s i n t h e x a - d i r e c t i o n w i t h i n c r e a s i n g t i m e . F o r t h i sp a r t i c u l a r c h o i c e o f s u r f a c e , t h e s u r f a c e i n t e g r a l w h i c h r e s u l t s f r o m a p p l i c a t i o n o f t h e


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C o m p u t a t i o n a l m e t h o d s b a s e d o n a n e n e r g y i n t e g r a l 23 1d i v e r g e n c e t h e o r e m a n d R e y n o l d ' s t r a n s p o r t t h e o r e m i s

t , J %n~dCd=c ( t ) ( U n c r ) d A + j , , j r l % n j - ~ - + ( u + T ) v n , d F d t ,t,

( 2 . 4 )w h e r e A ( t ) i s t h e c r o ss - s e ct i on o f t h e a b o v e - m e n t i o n e d v o l u m e o n a n y p l a n e t = c o n s t a n t ,n i is a u n i t v ec to r t h a t i s n o rm a l t o C o r I " i n an y p l an e t = co n s t an t an d t h a t p o in t s aw ayf r o m t h e c r a c k t ip , a n d v n 1 i s t h e i n s t a n t a n e o u s v e l oc i ty o f a n y p o i n t o n F i n t h e d i r e c t io nof n i a t th i s po in t .

Th e t e rm o n t h e l e f t s id e o f (2 .4 ) i s t h e t o t a l wo rk d o n e o n t h e b o d y i n t h e t im e i n t e rv a lo f i n t e re s t , an d t h e f i r s t t e rm o n t h e r i g h t s i d e i s th e i n c rease i n i n t e rn a l en e rg y d u r in g t h esame t ime . Co n seq u en t l y , t h e l a s t t e rm in (2 .4 ) i s t h e t o t a l en e rg y l o s t f ro m th e b o d y d u et o f l u x t h r o u g h F d u r i n g t h e i n t e r v a l t I < t < _ t z . Becau se t h i s t ime i n t e rv a l i s co mp le t e lya rb i t ra ry , i t fo l l o ws imm ed ia t e ly t h a t t h e c rack t i p l in e i n t eg ra l

r [ Oui )F ( r ) =jr~% ni-f f+(u+ r)vn , d r (2 .5 )i s t h e i n s t a n t a n e o u s r a t e o f e n e r g y f l o w o u t o f t h e b o d y t h r o u g h F . T h e i n t e r p r e t a t i o n o ft h e t e rms i n (2 .5 ) i s s t ra ig h t fo rward . Th e f i r s t t e rm i s t h e ra t e o f wo rk o f t h e ma te r i a lo u t s i d e o f F o n t h e m a te r i a l i n s id e o f F . I f t h e cu rv e were a ma te r i a l l in e , t h en t h i s wo u ldb e t h e o n l y c o n t r i b u t i o n t o t h e e n e r g y f l u x . T h e c u r v e i s m o v i n g t h r o u g h t h e m a t e r i a l ,h o wev e r , an d t h e seco n d t e rm in (2 .5 ) rep re sen t s t h e co n t r i b u t i o n t o t h e en e rg y f l u x d u e t oth e f l u x o f ma te r i a l .

T h e d y n a m i c e n e r g y r e le a s e r a te i f , t h a t i s, t h e e n e r g y r e le a s e d f r o m t h e b o d y p e r u n i tc r a c k a d v a n c e , i s d e f i n e d a s t h e l i m i t i n g v a l u e o f F ( F ) / v a s t h e c o n t o u r F i s s h r u n k o n t oth e c rack t i p . In o rd e r fo r t h is co n cep t t o h av e an y fu n d am en t a l s i g n if i can ce , i t i sn e c e s s a r y th a t t h e r e s u l t o f th e l i m i t in g p r o c e s s b e i n d e p e n d e n t o f t h e a c t u a l s h a p e o f F a si t s si ze b eco m es v an i sh in g ly sma l l . Su re ly F i t se l f i s n o t p a th - i n d e p en d en t i n g en e ra l , a sc a n b e s e e n b y m e a n s o f a d i r e c t c h e c k . C o n s i d e r t h e c l o s e d p a t h f o r m e d b y t w o c r a c k t i pc o n t o u r s F 1 a n d F z , a n d b y t h e s e g m e n t s o f t h e c r a c k f a c e s t h a t c o n n e c t t h e e n d s o f t h e s ec o n t o u r s . A p p l i c a t i o n o f t h e d i v e r g en c e t h e o r e m t o t h e e n e r g y f l u x in t e g r a l f o r t h i s e n ti r ec lo sed p a th l ead s t o t h e re su l t t h a t

F ( r : ) _ F ( I , ) = L , 2 o u v O -~ x ~+ w - ~ - l , o t : + v ~ t - ~ x d A , ( 2 .6 )w h e r e AI2 i s t h e a rea wi th in t h e c lo sed p a th . An eq u iv a l en t ex p ress io n fo r l i n ea r l y e l a s t i cre sp o n se ap p ea rs a s (14 ) i n [3 ]. Fo r p o in t s v e ry c lo se t o t h e c rack t i p , i t i s ev id en t t h a t Fw i l l i n d e e d b e p a t h - i n d e p e n d e n t ( p r o v i d e d t h a t t h e s i n g u l ar i ti e s i n f i e l d q u a n t i t ie s a r e s u c hth a t t h e i n t eg ra l d e f i n in g F ex i s t s n ea r t h e c rack t i p ) i n t h e l imi t a s F sh r i n k s o n to t h e t i pi f u ~ i s smo o th en o u g h fo r i t s seco n d d e r iv a t i v es t o ex i s t an d i f

O t t - v = o a s r - ~ 0 + , ( 2 . 7)wh ere f i s an y f i e ld v a r i ab l e an d r i s t h e rad i a l d i s t an ce f ro m th e m o v in g c rack t i p . I t i sn o t ed t h a t t h e r i g h t s i d e o f (2 .6 ) i s ze ro b y d e f in i t i o n fo r s t ead y -s t a t e c rack g ro wthp r o b l e m f o r m u l a t i o n s , s o t h a t ( 2 .5 ) p r o v i d e s a p a t h - i n d e p e n d e n t i n t e g r a l f o r t h is c l as s o fp r o b l e m s f o r a n y t y p e o f m a t e r i a l r e s p o ns e .

I f (2 .7 ) i s a s s u m e d t o h o l d f o r a r b i t r a r y c r a c k t i p m o t i o n u n d e r d y n a m i c c o n d i ti o n s ,t h e n O u J O t m a y h e r e p l a c e d b y - v O u J O x ~ i n ( 2.5 ) so t h a t

~ = r ~ 0r ~ l i m( ( U + T ) n l - o i j n y - ~ x l ) d F . (2 .8 )


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232 T . N a k a m u r a , C F . S h i h a n d L . B . F r e u n dAny difference between O u i / O t a n d - v O u J O x I which is less singular than O u J O x I cannotcontribute to the value of ~. The general result (2.8) underlies virtually all crack tipenergy integrals that have been defined and applied in fracture mechanics, in the sensethat each may be derived from (2.8) by invoking the appropriate restrictions on materialresponse (through U) and on crack tip motion. Some illustrations and applications arecited in the following discussion, with particular emphasis on the computation of crack tipenergy quantities.3 . E l a s t o d y n a m i c s3 . 1 . L i n e a r l y e l a s t i c m a t e r i a l sFor linearly elastic material response, U = o i j O u i / O x and the stress components have thecharacteristic inverse square root dependence on radial distance from the crack tip. Thedynamic energy release rate ~ is proportional to the square of the coefficient of thissquare root singularity, the so-called elastic stress intensity factor. For a growing crack,the particle velocity also has the inverse square root dependence, whereas the particlevelocity vanishes as the square root of the distance from the crack tip for a stationarycrack under dynamic loading. The case of dynamic crack growth is considered first.

Consider an arbitrary material area A o around the crack tip bounded by the curve Fo.The curve Fo begins on one crack face and ends on the opposite face, as shown in Fig. 1.To be somewhat more precise, A o is the area enclosed by Fo, F and the crack faces in thelimit as F is shrunk onto the crack tip. Application of the divergence theorem to (2.8)leads immediately to the expression,

~ = U n 1 - o i j n j ~ 1 + l p v 2 O x 1 O x I n I d F

O 2 U i O U O U 0 2 U i+ f A [ J r 2 O X1 D l ) 2 ~ X 1 ~ x ~ ) d A . ( 3 . 1 a )At first sight, it might appear that the area integral in (3.1a) is divergent because O u J O x 1and O E u i / O t 2 are singular as r-1/2 and r -3/2, respectively. This is not so, however, due tothe particular combination of terms in the integrand. The displacement gradient near thecrack tip has the form O u i / O x 1 = r - 1 / E f i ( O ) in crack tip polar coordinates, so that the

I " o

F i g u r e 1 . D o m a i n A o i n x ~ - x 2 p l a n e i s e n c l o s e d b y F , I " a n d c r a c k f a c e s . A t i s c r a c k t i p r e g i o n e n c l o s e d b y Fa n d c r a c k f a c e s .


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C o m p u t a t i o n a l m e t h o d s b a s e d o n a n e n e rg y i n te g r a l 233second der iva t ive has the fo rm ~ Z u i / O x ~ =r-3/2gi (O)=--r-3 /2[ lL(o COS 0+f [ (0 ) s in 0 ] . For any angular var i a t ion f~ (0) , i t i s eas i ly shown tha t

L " J ~ ( O ) g i ( O ) d O = O (3 .2 )prov id ed tha t f , (+ , r ) i s bou nde d . Thu s , the non in tegrab le s ingu lar ity in the a rea in tegra lin (3 .1a) i s on ly apparen t and the in tegra l does indeed have a f in i t e va lue .

While the representat ion for f f in (3.1a) is cer tainly correct , i t has a feature which is thebas i s fo r some concern in cons ider ing implementa t ion in f in i t e e l ement p rocedures . Thesecond t e rm of the in tegrand in the a rea in tegra l con ta ins the der iva t ives o f d i sp lacementgrad ien t s . To ob ta in non-van i sh ing s econd der iva t ives o f the d i sp lacements , the e lementsmus t have quadra t i c o r h igher o rder in te rpo la t ion func t ions fo r the d i sp lacements . Thus ,cer t a in c las ses o f e l ements , inc lud ing the commonly employed l inear d i sp lacement e l e -ments , a r e ru led ou t fo r such ca lcu la t ions . Fur therm ore , we no te tha t the p rodu ct s o fs t r es s and d i sp lacement g rad ien t s appear na tu ra l ly in the v i r tua l work s t a tement o fe q u i l i b r i u m a n d s u c h p r o d u c t s c a n b e c a l c u l a t e d a c c u r a t e l y a t t h e G a u s s i n t e g r a t i o npoin t s . H ow ever , th i s accuracy i s no t p reserved fo r the h igher der iva t ives. Hen ce , r epresen-ta t ion (3 .1a) may no t be idea l fo r comput ing the energy r e lease r a te .

Other r epresen ta t ions fo r energy r e lease r a te may be ob ta ined as fo l lows ,f= f ( Un l Ou, , Ou, Ou,F ,A - 'J n j- ~ xl + 2 P - ~ - - ~ n ' ) d r

f A ( a 2 u 'u ' O u , 2 u , )O ~ t 2 ~ x 1 O 3 t ~ x lO t d A ( 3 . 1 b )i f= l im [ p v 2 O U i ~ u i f r , , ( ~Ui~F - . o 0x---7 x ---7 ' d r + U n , - ) d r

32ui 3ui+ - A- to 0t 0x~ dA . (3.1c)

o

In (3 .1c) the f i r s t in tegra l i s eva lua ted a t the c rac k t ip an d the r emain ing two in tegra l s a r eeva lua ted over the f in i t e r eg ion . As no ted ear l i e r , an accura te eva lua t ion o f the in tegra l a tthe c rack t ip i s v i r tua l ly imposs ib le to ach ieve . However , i f the con t r ibu t ion f rom the f i r s tin tegra l i s smal l r e l a t ive to con t r ibu t ions f rom the o ther in tegra l s , then (3 .1c) cou ld be ausefu l r epresen ta t ion . To pur sue th i s po in t , no te tha t fo r l inear ly e las t i c f i e lds , the f i r s tin tegra l on the r igh t in (3 .10 can be eva lua ted us ing the known crack t ip f i e ld quan t i t i es[13] to y ie ld a r esu l t which i s p rop or t iona l to the s t r es s intens i~y f ac to r sq uared and whichd e p e n d s o n t h e i n st a n t a n e o u s c r a c k t ip s p e e d . I f e a c h t e r m i n ( 3 .1 c) i s d i v i d e d b y ~ , w h i c hi s a ls o p r o p o r t i o n a l t o t h e s t re s s in t e n s it y f a c t o r s q u a r e d a n d w h i c h d e p e n d s o n t h e c r a c kt ip speed , then the f i r s t t e rm on the r igh t in the r esu l t ing equa t ion i s a func t ion o f c r ackt ip s p e e d o n l y. I f t h is s p e e d d e p e n d e n t r a t i o ( d e n o t e d b y R b e l o w ) is m u c h s m a l le r t h a nuni ty , then the f i r s t t e rm can be neg lec ted and a r e la t ive ly s imple expres s ion fo r energyre lease r a te i s ob ta ine d .

Cho os ing F to be a c i rcu la r pa th an d us ing the p lane s t r a in r e la t ionsh ip be tw een theenergy r e lease r a te ~ and s t res s in tens i ty f ac to r K g iven in [13] , the r a t io R i s g iven by

f ~ t ov 2 ~ u i ~ u~Ox-----~Ox-----( l rd OR ( o ) = , (3 .3)

wh ere v is Po i s son ' s r a tio , E i s the e las ti c modulus , K i s the e las ti c st r es s in tens i ty f ac to r ,


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234 T. Nakam ura , C .F . "Sh ih and L .B . Freunda n d t h e d i m e n s i o n l e s s f u n c t i o n o f s p e e d A ( o ) i s g iv en i n [1 3 ] . Va lu es o f t h i s ra t i o h av eb e e n c o m p u t e d u s i n g t h e e x p l ic i t e x p r e s s io n s f o r a n g u l a r v a r i a t i o n o f c r a c k t i p f i e l d s g i v e nin [1 3 ] . F r o m th e re su l t s it is c lea r t h a t fo r c rack t i p sp eed s sma l l e r t h an h a l f t h e sh ea rwav e sp eed , t h e i n t eg ra l co n t r i b u t e s l e s s t h an 1 0 p e rcen t t o t h e en e rg y re l ea se ra t e ,h o w e v e r , t h e c o n t r i b u t i o n b e c o m e s s i g n i f i c a n t a t h i g h e r c r a c k t i p s p e e d s.

W i t h r e g a r d t o t h e o b s e r v a t i o n m a d e a b o v e , a f e a t u r e o f ( 3 .1 b ) w h i c h i s o f i m p o r t a n c ef r o m t h e p o i n t o f v i e w o f c o m p u t a t i o n i s n o t e d . T h e t h i r d a n d f i f t h t e r m s i n ( 3 . 1 b ) a r et y p i c a l l y c o m p a r a b l e i n m a g n i t u d e b u t a r e o f t h e o p p o s i t e s ig n s , a n d t h e i r n e t c o n t r i b u -t i o n t o t h e en e rg y re lea se ra t e ( i.e . t h e su m o f t h e tw o t e rms) i s sma l l. In f ac t t h e se t e rmsa re eq u iv a l en t t o t h e f i rs t t e rm in (3.1c ) w h ich h as b een sh o w n to co n t r i b u t e n eg l i g ib ly t oth e en e rg y re lea se ra t e a t l o wer c rack t i p sp eed . In o th e r wo rd s , a sm a l l co n t r i b u t i o n t o t h ee n e r g y r e l e a s e r a t e i s c o m p u t e d f r o m t h e d i f f e r e n c e o f t h e t w o t e r m s o f c o m p a r a b l em a g n i t u d e . T h i s i s a p o t e n t i a l s o u r c e o f c o m p u t a t i o n a l e r r o r . W e c o n c l u d e t h a t t h ed i f f e r e n c e b e t w e e n ( 3 . 1 b) a n d ( 3 .1 c ) is b o u n d e d a n d s m a l l f o r v / c 2 < 0 .5 . The resu l t (3 .2)a l so su g g es t s t h a t t h e re l a t iv e co n t r i b u t i o n f r o m th e v i c in i t y o f t h e c rack t i p t o t h e a reai n t e g r a l in ( 3 . 1c ) i s s m a l l f o r u n i f o r m l y r i s in g l o a d s . O u r n u m e r i c a l c a l c u l a ti o n s c o n f i r m e dth i s b eh av io r . Hen ce ex p ress io n (3 .1 c ) ap p ea rs t o b e su i t ed fo r ev a lu a t i n g t h e en e rg yr e l ea s e r a t e f r o m f i n i te e l e m e n t f i e l d q u a n t i ti e s f o r c r a c k s p e e d s v / c 2 < 0.5.

F in a l l y , i t i s n o t ed t h a t t h e ex p ress io n (3 .1 c) w i th v - - - , 0 i s ex ac t l y t h e en e rg y re l ea ser a t e e x p r e s s i o n f o r t h e c a s e o f a s t a t i o n a r y c r a c k s u b j e c t e d t o d y n a m i c l o ad s . I n t h is c a se ,t h e q u a n t i t y ~ m u s t b e i n t e r p r e t e d a s a v i r tu a l e n e r g y r e l ea s e r a t e, t h a t is , ~ c o m p u t e df ro m (3 .1 c) wi th v = 0 i s t h e en e rg y ch an g e t h a t wo u ld b e re l ea sed i f t h e mech a n i ca l f i e l d sw e r e f r o z e n a t a n y i n s t a n t a n d t h e c r a c k t i p w e r e g i v e n a n i n c r e m e n t o f e x t e n s i o n i n t h ex l - d i r e c t io n . O f c o u r se , I r w i n ' s r e la t i o n s h i p b e t w e e n t h e s t r es s i n t e n s i t y f a c t o r a n dh o l d s i n t h is c as e , a n d t h e i n t e r p r e t a t i o n o f ~ a s a m e a s u r e o f c r a c k t ip f i e l d i n t e n s i t yr a t h e r t h a n a s a n e n e r g y r e le a s e r a te i s m o r e a t t r a c ti v e i n s o m e r e s p ec t s.3.2. Nonlinearly elastic materialsA l l o f t h e e n e r g y r e l e a s e r a t e e x p r e s s i o n s d e r i v e d i n t h e p r e c e d i n g s e c t i o n a p p l y f o rn o n l i n ea r ly e l a s t i c ma te r i a l r e sp o n se , a s we l l a s l i n ea r l y e l a s t i c re sp o n se . Ho wev er , t h e rea r e t o o f e w t h e o r e t i c a l r e s u l t s a v a i l a b l e f o r d y n a m i c f r a c t u r e o f n o n l i n e a r l y e l a s t i cm a t e r i a l s a t t h e p r e s e n t t i m e t o p r o v i d e a b a s i s f o r a g e n e r a l d i s c u s si o n . F o r a s t a t i o n a r yc r a c k i n a m a t e r i a l w i t h a p o w e r l a w s t r e s s -s t r a in r e l a t i o n s h ip , i t w o u l d s e e m t h a t t h e n e a rt ip f i e l d s a r e a d e q u a t e l y d e s c ri b e d b y t h e H R R a s y m p t o t i c s o l u ti o n [1 4,1 5] p r o v i d e d t h a tt h e i r t i m e v a r i a t i o n i s n o t t o o r a p i d . A n a l y s i s o f s u c h p r o b l e m s w o u l d p r o c e e d a s i n t h ea n a l y s i s o f c e r t a in p l a s t i c i t y p r o b l e m s a s c o n s i d e r e d b e l o w i n S e c t i o n 4.

O n t h e o t h e r h a n d , i t c a n b e r e a d i l y s h o w n t h a t a s i m i l a r s e p a r ab l e c r a c k t i p a s y m p t o t i cf i e ld c a n n o t b e c o n s t r u c t e d f o r d y n a m i c g r o w t h o f a c r a c k i n t h e s a m e m a t e r i a l . A ni n d i c a t i o n t h a t a n a l y s i s o f d y n a m i c c r a c k g r o w t h i n n o n l i n e a r l y e l a s t i c m a t e r i a l s m a yp r o v i d e s o m e s u r p r i s e s i s c o n t a i n e d i n t h e r e c e n t w o r k r e p o r t e d b y A c h e n b a c h a n dN i s h i m u r a [ 1 6 ] . T h e y c o n s i d e r e d t h e e f f e c t o f i n e r t i a o n m o d e I I I c r a c k g r o w t h i n an o n l i n e a r l y e l a s ti c m a t e r i a l f o r w h i c h t h e e f f e c ti v e s t re s s h a s a c o n s t a n t m a g n i t u d e f o r a lle f fec t i v e s t ra in s g rea t e r t h an a p a r t i cu l a r l ev e l. A p r in c ip a l re su l t i s t h a t t h e t o t a l s t ra i nr e m a i n s b o u n d e d a s t h e c r a c k ti p i s a p p r o a c h e d f r o m a d i r e c t io n d i r e c t ly a h e a d o f t h e ti p .T h i s r e s u l t p o s es a d i l e m m a , f o r i t s u g g e s ts t h a t t h e r e m a y b e n o e n e r g y r e le a s e d a t t h ec r a c k t i p a l t h o u g h e n e r g y c o n t i n u e s t o f l o w in t o t h e c r a c k t i p r e g io n f r o m r e m o t e p o i n t s .T h e r e s o l u t i o n o f t h i s d i l e m m a m a y b e f o u n d t h r o u g h t h e o b s e r v a t i o n t h a t t h e m a t e r i a lc o n s i d e r e d a d m i t s l in e s a l o n g w h i c h d i s p l a c e m e n t g r a d i e n t s a r e d i s c o n t i n u o u s [ 17 ] , a n dt h e e n e r g y f l o w i n g i n t o t h e c r a c k t i p r e g i o n m a y b e d i s s i p a te d t h r o u g h p r o p a g a t i o n o fs u c h l in e s o f d i s c o n t i n u i t y .


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C o m p u t a t i o n a l m e t h o d s b a s e d on a n e n e r g y i n t e g ra l 23 54 . Elas t i c -plas t i c materia l responseF o r a s t a t i o n a r y c r a c k i n a d y n a m i c a l l y lo a d e d b o d y o f i n c r e m e n t a l e l a s ti c - p la s t icma te r i a l , t h e i n t eg ra l q u an t i t y i n (2 .8 ) can s t i l l b e u sed t o so me ad v an t ag e . Of co u rse , t h ein t e rp re t a t i o n o f t h e i n t eg ra l a s an en e rg y re l ea se ra t e i s l o s t fo r t h i s ca se . No n e th e l e s s , i th a s b e e n d e m o n s t r a t e d t h a t i f th e l o a d i n g a p p l i e d t o t h e c r a c k t i p r e g io n is m o n o t o n i c a l l yin c reas in g an d i f t h e s t re s s h i s t o r i e s fo r ma te r i a l p a r t i c l e s n ea r t h e c rack t i p a re n ea r lyp r o p o r t i o n a l , t h e n t h e v a l u e o f t h e i n te g r a l p r o v id e s a m e a s u r e o f t h e i n t e n s i t y o f t h e n e a rt i p d e fo rma t io n . I t i s t h i s ro l e t h a t i s p e r t i n en t t o e l a s t i c -p l a s t i c f rac tu re mech an i c s an dc o n s i d e r a b l e a n a l y t i c a l , n u m e r i c a l a n d e x p e r i m e n t a l i n v e s t i g a t i o n s h a v e s h o w n t h a t t h ev a lu es o f t h i s i n t eg ra l i s a re l ev an t ch a rac t e r i z in g p a rame te r o f t h e n ea r t i p f i e l d s o fs t a t i o n a r y c r a c k s w h e n c e r t a i n si ze r e q u ir e m e n t s a r e m e t . F o r d y n a m i c l o a d i n g c o n d i t i o n sa n d a s t a t i o n a r y c r a c k ti p , t he k i n e t ic e n e r g y te r m T m a k e s n o c o n t r i b u t i o n t o t h e i n t e g r a lin (2 .8) so tha t

f C = J = l i m f ( Ou, ro i j n j - ~ x l J d , (4 .1 )F ~ 0 J F \w h e r e U i s a s t r e ss w o r k d e n s i t y a p p r o p r i a t e f o r p r o p o r t i o n a l l o a d i n g o f t h e m a t e r i a l . F o rq u a s i s t a t i c l o a d i n g c o n d i t i o n s a n d p r o p o r t i o n a l s t r e s s h i s t o r y a t e a c h m a t e r i a l p o i n t , t h ei n t e g r a l in ( 4. 1) i s p a t h i n d e p e n d e n t a n d i ts v a l u e d o e s n o t d e p e n d o n a l i m i t i n g p r o c e s s i nwh ich t h e c rack t i p co n to u r F i s sh ru n k o n to t h e c rack t i p . In t h i s ca se , t h e i n t eg ra l i sp rec i se ly R ice ' s J - i n t eg ra l [1 8 ], a s an t i c i p a t ed i n t h e n o t a t i o n i n (4.1) . Th e b ack g ro u n d fo rt h i s ap p ro ach t o e l a s t i c -p l a s t i c f rac tu re mech an i c s i s o u t l i n ed i n recen t rev i ew a r t i c l e s b yR i c e [ 1 9 ] a n d b y H u t c h i n s o n [ 2 0 ]. I n t h e c a s e o f d y n a m i c n e a r t i p f i e ld s , o n t h e o t h e rh an d , i n c lu s io n o f t h e l imi t i n t h e d e f i n i t i o n o f J i s e s sen t i a l, t h a t i s , t h e J - i n t eg ra l i s n o tp a t h i n d e p e n d e n t u n d e r d y n a m i c l o a d in g c o n d it io n s .

J u s t a s i n t h e c a s e o f a n a l y s i s o f e l a s t i c c r a c k p r o b l e m s b y m e a n s o f c o m p u t a t i o n a lt e c h n i q u e s , t h e a c c u r a t e d e t e r m i n a t i o n o f f i e l d q u a n t i t i e s v e r y n e a r t h e c r a c k t i p i sd i f f i c u l t a n d ( 4. 1) c a n n o t b e e v a l u a t e d d i r e c tl y . A p p l i c a t i o n o f t h e d i v e r g e n c e t h e o r e m ,h o w e v e r , l e ad s t o t h e a l t e r n a t e f o r m

J = f ( U n , -r o , ' J n j - U ' d F + f A_ o 0 2 u i- ~ X 1 d A , (4 .2 )w h e r e th e p a t h Fo a n d t h e a r e a A o a re a s sh o w n in F ig . 1 . A n i l l u s t ra t i o n o f t h e fo rm (4 .2 )w i l l b e p r e s e n t e d i n t h e f o l l o w i n g s e c ti o n . E v e n i n t h e c a s e o f d y n a m i c g r o w t h o f a c r a c kin an i n c remen ta l e l a s t i c -p l a s t i c ma te r i a l , t h e i n t eg ra l (2 .8 ) can b e ap p l i ed fo r ana p p r o p r i a t e d e f i n i t i o n o f t h e t e r m U . I t h a s n o t b e e n f o u n d t o b e p a r t i c u l a r l y u s ef u l ,h o w e v e r , b e c a u s e t h e n e a r t i p f i e l d s f o r g r o w i n g c r a c k s t e n d t o b e l e s s s i n g u l a r t h a n t h ec o r r e s p o n d i n g f i e l d s f o r s t a t i o n a r y c r a c k s a n d t h e v a l u e o f f f i s t h e n z e r o . T h e p h y s i c a li n t e r p r e t a t i o n o f t h i s r e s u lt i s th a t a l l o f t h e e n e r g y t h a t f l o w s in t o t h e c r a c k t i p r e g i o n i se i t h e r d i s s ip a t e d t h r o u g h p l a s t ic f l o w o r i s t r a p p e d a s r e s i d u a l e la s t ic e n e r g y i n t h e w a k e o ft h e c rack t i p p l a s t i c zo n e .

A n e x c e p t i o n t o t h is s i t u a t i o n a r is e s f o r r a t e - d e p e n d e n t p l a s t ic r e s p o n s e w h e n t h e r a t es e n s i ti v i ty is s u f f i c i e n t ly p r o n o u n c e d . A p r o b l e m o f a c r a c k g r o w i n g a t h i g h s p e e d i n a ne l a s t i c / r a t e - d e p e n d e n t p l a s t i c s o l i d u n d e r s m a l l s c a l e y i e l d i n g c o n d i t i o n s w a s c o n s i d e r e db y F r e u n d a n d H u t c h i n s o n [ 2 1] . A t h i g h s t re s s le v el s, s u c h a s t h o s e t h a t a r i s e n e a r t h e t i pa t h ig h c rack sp eed , i t was a s su m ed t h a t t h e p l a s t i c s t ra in ra t e i n c reases l i n ea r l y wi ths t re s s. In t h i s ca se , t h e n ea r t i p f i e l d s a re su f f i c i en t l y s i n g u l a r t o re su l t i n a f i n i t e v a lu e fo r

i n (2.8) , a n d a g en e ra l ex p ress io n i s d e r i v ed i n [2 1 ] fo r t h e n ea r t i p en e rg y re l ea se ra t e i nt e r m s o f t h e o v e r a l l r e l e a s e r a t e f o r s t e a d y g r o w t h . I n f a c t , f o r s t e a d y s t a t e g r o w t h , t h ein t eg ra l i n (2 .8 ) i s p a th i n d ep en d en t fo r g en e ra l ma t e r i a l r e sp o n se [6 ,2 1 ] .


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23 6 T . N a k a m u r a , C . F S h i h a n d L . B . F r e u n d5 . Finite e lement analyses o f stat ionary and propagat ing cracks5 . 1 . E l a s t o d yn a mi c a n a l ys i s

Computat ional aspects of the domain integral representat ions for the energy re lease ra teas given in Sect ion 3 are discussed in th is sect ion. To emphasize the interpreta t ion of theseenerg y integrals (3.1) as a mea sure of the crack tip field intensity, the sy mb ol J will beemplo yed in place of ~ hereaf ter . In (3.1) the domain A o is enclosed by I" , F and thecrack faces in the limit as F is shrunk onto the crack tip. From result (3.2), i t is clear thatthe contr ibut ions f rom the area integrals in (3 .1) taken over the domain A, ( indicated inFig. 1) as F is shrunk onto the crack t ip must vanish. Hence in the context of numericalintegrat ion A o will be taken to be the ent i re area enclosed by Fo and the crack faces . Theenergy integrals have the form given in (3.1), of course.

The n umerical s tudies were carr ied ou t with the f in i te e lement program AB AQ US [22].The Hi lb e r -H ugh es -T ay lo r a lgor i thm [23] i s employed to in tegra te the semi-di scre teequat ion of motion. A small amou nt of dampin g is invoked by set t ing the a value to be- 0.05. Both 4-noded bi l inear isoparametr ic e lement and 8-noded biq uadrat ic isopa ramet-t ic e lement are used in the calculat ions . Unless otherwise s ta ted, e lem ent s t i f fness matr icesare obtained by 2 2 Gaussian integrat ions .

To indicate the implementat ion of these energy integrals in our f in i te e lement calcula-tions, the discrete form of (3.1b) is given as follows

2[ ( aU 1 aU 2J = E E wi U-Oalax---1 -0 21 ax---1all segments on Fo i = 1[ au~ au2/ ]

4 r[ a2Ul aUl+ X Xw,[ o t ax 'all elements in A o i = 1

-I'- l p aUl aUl 1 aU2 aU2 )o--7 a--7 +

a2uau2 aUlO , au 2u2) ]+ P Ot 2 ax 1 P a t ax la t p a t ax la t D .

(5.1)Here P1 and P2 are the path factors given in [24] , w are the appropria te Gauss weightsand al l quant i t ies within [ ] i are evaluated a t the respect ive Gauss points . Also D is thedeterminant of the Jacobian matr ix of the isoparametr ic t ransformation. Similar discreteexpressions can be written for (3.1a) and (3.1c). A more detailed discussion of a discreterepresen ta t ion fo r a domain in tegra l r ep resen ta t ion fo r the dynamic J can be found in[25].The dy nam ic J- integrals (3 .1) are independe nt of the s ize of the dom ain A o enclosed bythe con tour F o as indicated in Fig . 1 . This invar iant fe ature of the d om ain integrals canserve as a useful check on the self -consis tency of the nu merical calculat ions . As note dpreviously , accurate f in i te e lement solut ions for points near the crack t ip are dif f icul t toobtain . H ence the do main s of in tegrat ions typical ly extend over ten or mo re e lements .

To evaluate the computat ional aspects of the numerical a lgor i thm and of the energyintegrals , a s t ress wave problem with an exact solut ion is employed as the benchmarkproblem. Freu nd [26] considered a semi- inf ini te crack in an inf ini te plane subjected to atensile stress wave normal to the crack plane. At t = 0, the stress wave arrives at the crackplane and at a la ter t ime, t = % the crack s tar ts to propagate . The analyt ical solut ion forthe dy nam ic stress intensi ty facto r in the interval 0 < t < (i.e. , the crack is stationa ry) is

K ( t ) = Boo cv~,t. (5.2)


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C o m p u t a t io n a l m e t h o d s b a s e d o n a n e n e r g y i n t e g ra l 237H e r e c ] is t h e l o n g i t u d i n a l w a v e s p e e d f o r t h e m e d i u m , % i s t h e m a g n i t u d e o f t h e te n s i lep u l s e w i t h s te p f u n c t i o n t i m e d e p e n d e n c e a n d B i s a f u n c t i o n o f P o i s s o n 's r a t i o g i ve n i n[26 ]. Fo r t im es t > ~ ', t h e d y n am ic s tre s s i n t en s i t y fac to r fo r t h e p ro p ag a t i n g c rack i s

K ( t , o ) = k ( o ) B % c ~ /T t, ( 5 . 3 )wh ere o i s t h e c rack t i p sp eed an d k ( v ) i s d e f i n e d i n [ 2 7 ] a s a f u n c t i o n w h i c h d e c r e a s e sm o n o t o n i c a l l y f r o m o n e a t o = 0 t o z e r o w h e n o i s t h e R a y l e i g h w a v e s pe e d .

F o r t h e p u r p o s e o f c o m p a r i s o n w i t h n u m e r i c a l re s u l ts p r e s e n t e d s u b s e q u e n t l y , th e w e l lk n o w n r e l a t io n b e t w e e n t h e d y n a m i c J a n d t h e d y n a m i c st r es s i n t e n s i t y f a c t o r u n d e rp l a n e s t r a in c o n d i t i o n i s g iv e n b y

J ( t , t J ) (1 ~ 2 ) A ( o ) [ K ( t , o)] 2. (5.4 )H e r e A ( o ) i s a u n iv e rsa l fu n c t i o n o f c rack t i p sp eed [1 3 ]. Fo r a s t a t i o n a ry c rack (0 = 0 ) , Ai s u n i t y , an d .4 i n c reases mo n o to n i ca l l y wi th i n c reas in g v . In t h e su b seq u en t an a ly ses , ~, i st ak en t o b e 0 .3 .5 .2 . E l a s t i c a n a l ys i s o f a s t a t i o n a r y c r a ck s u b j ec t t o a s i n g l e s t ep p u l s eA s t e p p u l s e i s a p p l i e d a t t = - H / c I a t t h e e d g e x 2 = H o f t h e m o d e l s h o w n b y i n s e t i nF ig . 2 , an d t h e p u l se a r r iv es a t t h e c rack p l an e a t t = 0 . S in ce th e p ro b l em h as n o re f l ec t i v es y m m e t r y w i t h r e s p e c t to t h e c r a c k p l a n e ( x 2 = 0 p l an e ) , t h e d o m a i n d o w n s t r e a m f r o m t h ec r a c k p l a n e i s i n c l u d e d i n t h e m o d e l . T o s i m u l a t e t h e i n f i n i t e p l a n e p r o b l e m f o r w h i c h t h es o l u t i o n ( 5 . 2 ) i s a p p l i c a b l e , t h e d i m e n s i o n H m u s t b e l a r g e e n o u g h s o t h a t w i t h i n t h ep e r i o d o f i n te r e s t , r e f le c t e d w a v e s f r o m t h e b o u n d a r i e s d o n o t r e a c h t h e v i c i n i ty o f t h ec r a c k t ip . I n a d d i t i o n t h e l e n g t h o f t h e m o d e l ( i n X l - d i re c t io n ) m u s t b e s u f f i c i e n tl y l a r g ec o m p a r e d t o H s o t h a t w i t h i n t h e i n t e rv a l o f in t e r e s t, t h e c r a c k t ip r e g i o n i s u n a f f e c t e d b yt h e u n l o a d i n g w a v e s e m a n a t i n g f r o m t h e r e m o t e e d g e s . T h e s e f a c t o r s w e r e t a k e n i n t oc o n s i d e r a t i o n i n s e l e c t in g t h e d i m e n s i o n s o f t h e f i n i te e l e m e n t m o d e l . T h e f u l l p l a n e m o d e lc o n s t r u c t e d w i t h 8 - n o d e d d e m e n t s h a s 2 00 d e m e n t s ( 1 0 b y 2 0 a r r a n g em e n t ) a n d 6 81

1 .5- o - - o - ( 3 .1 c ) A t = 1 .0 L / c I / o- -o -- -o - ( 3 . 1 c ) A t = 0 . 5 L / c I

-7- m _ _ e x ac t [ 26 ] S / , o", 7 ; ,~" , . .o- ~ 7:r"~ b . ~ - - ~

0 . 5

. .0 .0 - " e , , , I , , , , I . . . . I0 . 0 0 . 5 I . 0 1 .5

t c l / HF i g u r e 2 . V a r ia t i o n o f n o r m a l i z e d J w i t h n o r m a l i z e d t i m e a s d e t e r m i n e d f r o m t h e f u ll p l a n e m o d e l . R e s u l t so b t a i n e d f r o m t w o a n a l y s e s u s i n g d i f f e r e n t t i m e s t e p s a n d t h e e x a c t s o l u t i o n a r e s h o w n . S h o w n i n i n s e t is as c h e m a t i c o f t h e fu l l p l a n e m o d e l a n d t h e f i x e d r e f e r e n c e c o o r d i n a t e s x ] a n d x 2 .


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238 T . N a k a m u r a , C .F . S h i h a n d L . B . F r e u n dnodes. All dements are square shaped and are identical in size with the length (and width)being L. The time required for a longitudinal wave to traverse an element is then L / c ~ .Two calculations were carried out; the first employs uniform time steps of At = 1 . OL/c~and the second employs uniform time steps of At = 0 . 5 L / c ~ . At the end of each time step,the dynamic J is computed from the stress and deformation fields using the discreterepresentation of (3.1c). Typically the dynamic J is evaluated on ten integrat ion domains;the smallest domain has 4 elements and the largest domain has 200 elements. With theexception of the near-tip domain, which has four elements, the values of J evaluated onthe other domains are within one percent of its mean value which attest to the consistencyof the numerical solution. The dynamic J values determined from an alternate representa-tion (3.1b) are also in agreement with the values determined from (3.1c). This provides anaddit ional check on the consistency of the results. In subsequent discussions, the mean ofthe domain values will be reported as the dynamic J.

The dynamic J computed from (3.1c) is shown in Fig. 2 together with the exact solution(5.2) expressed in terms of J through (5.4). The computational solution clearly convergesto the exact solution for the smaller time step. At sufficiently large times, say t > 0 . 5 H / c 1,the finite element results for the smaller time step is within three percent of the exactsolution. At shorter times, the error is substantially larger. This is not unexpected since atshort times the refracted wave from the crack tip has traveled through relatively fewelements (i.e., small domain), and as previously discussed, an accurate evaluation of thedynamic J in this case is rather difficult.

Similar calculations were also carried out using a mesh where the crack tip region iscomprised of regular triangular and singular triangular elements [28]. Solutions from thesemodels and from a model comprised solely of 4-noded elements (the latter model has 800elements) are in substantial agreement with the numerical results shown in Fig. 2.Solutions from these models are also exhibit the same convergence characteristics de-scribed above. It is clear from this numerical study that (3.1c) and (3.1b) are suited for thedeterminat ion of the dynamic J using field quantities obtained from finite elementanalysis.5 .3 . E las t ic ana lys i s o f a s ta tionary a nd propag a t ing crack sub jec t to s ymm etr ica l ly app l ieds tep pu lsesA computational analysis of crack propagation requires a finite element model which issubstantially larger than the models employed in the preceding section for the stationarycrack analysis. To reduce the size of the computational problem we consider a crackedplate symmetrically loaded by tensile step pulses on the edges x 2 = +H. Since thisproblem possesses reflective symmetry with respect to the crack plane, only the upper halfplane needs to be considered. The solutions (5.2) and (5.3) are still applicable to thisproblem and by superposition, the magnitude of the dynamic K is twice that for the singlestep pulse.For this analysis the upper half plane is modelled by 4-noded isoparametric elements.This choice is guided by our numerical experimentations on dynamically propagatingcracks which indicate that the node release technique is better suited with 4-nodedelements. The upper half plane model has 1290 elements (15 by 86 arrangemen0 and 1392nodes. All elements are square shaped and identically sized. Homogeneous displacementboundary condit ions are prescribed on the symmetry plane x 2 = 0, x 1 > 0.To compare the convergence characteristics for the hal f plane model with the full planemodel of the preceding section, calculations similar to those reported in Section 5.2 werecarried out for a stationary crack. The convergence behavior with respect to element sizeand time steps for the present model is similar to that described in Section 5.2.


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Computat ional methods based on an energy in tegral 23 9Never the less the re su l t s f rom the fu l l p lane mode l a re in be t te r ag reemen t wi th the exac tso lu t ion when equ iva len t meshes and iden t ica l t ime s teps a re employed . The so lu t ionob ta ined f rom the ha l f p lane re f ined mesh mode l (a s desc r ibed in the p reced ing pa ra -g raph) wi th un i fo rm t ime s teps o f 0 .25L /c~ and the exact solut ion are shown in Fig . 3 .For t > 0.2 H / c l ; the dynam ic J agreed very well with the exact solut ion.

Befor e descr ib ing the main results , the a lgori th m for extendi ng the crack is br ief lyd iscussed . Cons ide r a c rack p ropaga t ing a t cons tan t speed v . The t ime requ i red fo r thec rack advance th rough one e lemen t leng th i s L / v a n d d e n o t e d b y t e. At the s ta r t o f thec rack advance , the ze ro d isp lacemen t boundary cond i t ion a t the c rack t ip i s rep laced byan equ iva len t concen t ra ted fo rce , Fo, requ i red to ma in ta in the p resc r ibed d isp lacemen t .The noda l fo rce is re laxed to ze ro in fo l lowing inc remen ts accord ing to a so -ca l led"constant energy re lease ra te" [29,30] re la t ion,

F ( t ) = F o ~ / ] - t / t e , (5 .5)

where t i s the t ime e lapsed f rom the s ta r t o f the node re lease and F ( t ) i s the noda l fo rcea t t ime t . The m er i t s o f thi s noda l fo rce re laxa t ion scheme and o the r node re laxa t ionschemes a re d i scussed in [30 ] . We exper imen ted wi th severa l noda l fo rce re laxa t ionschemes on severa l benchmark p rob lems and reached the conc lus ion tha t the schemeaccor ding to (5.5) gave dynam ic J resul ts which are consis tent with analyt ical solut ions .

The p ro b lem to be ana lyzed is a s fo l lows . At t ime t = -H /c I a s tep pulse is applied a tx 2 = H, and the lon gitudinal wave rea ches th e crack plan e a t t = 0 . The crack t ip iss ta t io nary fo r t < z , and i t prop agate s a t a cons tant speed v for t >_ r . Wit hin the in terval- H / c 1 < t < ~', the calcula t ions e mp loy uni form t ime s teps of At = 0 . 2 5 L / c l , w h e r e L / c 1is the t ime required for the longitudinal w ave to t ravel thro ugh an e lem ent. Fo r t >_ r thet ime s teps a re governed by the speed o f the p ropaga t ing c rack v and the e lemen t s ize L .Th e crack advance s across an e lem ent bou nd ary (on x 2 = 0) in 20 increme nts , a iad th isco r respond s to un i fo rm t ime s teps o f At = L / 2 O v . The ana lyses were ca r r ied ou t fo r c racktip speeds of 0 .4c 2 a n d 0.6C 2 respectively , whe re c 2 is the shear wave speed.

In the f i r s t ana lys i s , the c rack i s s ta t ionary be tween -H/c I < t < 0 .46 7H/ c 1 a n d t h e npresc r ibed to p rop aga te a t a un i fo rm speed o f 0 .4c 2. For the m ode l em ployed , the t imespan where the numer ica l so lu t ion pe r ta ins to the semi- in f in i te c rack in an in f in i te p lane

3"I -

N~ 2

00 .0

--o--o-o- (3.1c) ~ j

~ r ' '' J, , ~ I . ~ , I . . . . I , , , . I , . , , I

0 .2 0 .4 0 .6 0 .8 I0tcilH

Figure 3. Comparison of normalized J as determined from the half plane model with the exact solution.


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240 T. Nakamura, CF. Shih and LB . Freund

4

-o- --o- (5.1 b) ~ ' /-r 5 exact [26] ~ /"u.I

b 2- /- - , -c rack p ropa ga tes : v / c 2 = 0 .4

0 . . . . J . . . . I . . . . J0.0 0.5 1.0 1.5 2.0t c l / H

Figure 4. Comparison of normalized J wi th the exa ct solution. Crack propagates at sp ee d v/c 2= 0.4 att > 0.467H/c 1.

p r o b l e m i s -H/ c] < t < 2H/ c] . During th i s i n t e rva l t he c rack t i p advanced th rough fou rnodes . The dy nam ic J i s eva lua t ed a t eve ry t ime s t ep. F o r t < 1.667H/c 2, t h e d y n a m i c Jva lues eva lua t ed on va r ious domains u s ing (3 .1b ) a re w i th in seve ra l pe rcen t o f i t s meanva lue . Thus the numer i ca l re su l t s a re cons i s t en t w i th t he domain independence o f t hed y n a m i c J i n te g r al . A t l o n g e r ti m e s, t h e v a l u es b e c o m e d o m a i n d e p e n d e n t a n d w ea t t r i bu t e t h i s t o t he d i spe rs ion o f t he waves re f l ec t ed f rom the boundar i e s and numer i ca le r ro rs a ssoc i a t ed w i th t he node re l ease t echn ique . T he ave rag e o f t he dyna mic J a sso c i a t edwi th each un load ing p rocess i s shown in F ig . 4 . Wi th in t he i n t e rva l i nd i ca t ed , t henumer i ca l re su l t s ag reed reasonab ly we l l w i th t he exac t so lu t ion .

Num er i ca l re su l t s o f t he dyn ami c J fo r t he c rack p ropaga t ing a t v = 0 .6c 2 a re shown inFig. 5. He re ~- is set at 0 .467H/c I and the c rack t i p advanced th rough s ix nodes w i th in t he

"I"LIJo,J 0b-3

3.0

2.0

1.0

---~-o-- (3.1 b)

. . . . . . ,1.0 1.5 2.0

exact [26 ] / ~ ~

---.crack propagates: v/ c z =0 .6I , , , , I = , , I , ,0.5

t cjl HFigure 5. Comparison of normalized J wit h the ex act solution. Cr ack propagates at spe ed v/c 2= 0.6 att > 0.467H/c 1.


13/15


14/15

242 T. Nakamura , C.F. Shih and L.B. FreundHere n is the strain-hardening exponent, co and % are reference strain and stress relatedby % = Ec o. In this particular study n has a value of 10 which corresponds to a lightlyhardening material. The relation (5.6) is generalized to multiaxial stress states according toa J2 flow theory of plasticity.Elastic-plastic dynamic calculations were carried out for a mesh comprised of only4-noded elements and another mesh comprised of only 8-noded elements. The results fromboth calculations are very similar and we restrict the discussion to results obtained fromthe 8-noded element model. The half plane model has 456 elements (12 by 38 arrange-ment) and 1469 nodes. Elements near the crack tip are square shaped and identicallysized. Larger elements are employed at further distances from the crack tip. Symmetryconditions, that is, zero displacement boundary conditions are prescribed on the crackplane x 2 = 0 and x 1 >1 0. A tensile step load oa of magnitude 0.600 is applied on theboundary at x 2 = H. Uniform time steps of 0.5L/c I where L is the size of the squareelements in the vicinity of the crack tip is employed throughout the calculations. Thecalculation is terminated when the reflected waves from the boundaries reach the plasticzone emanating from the crack tip.The dynamic J is computed according to (4.2). In this case the integration of the stresswork density U follows the actual strain trajectory of a material point. (For a stationarycrack, (3.1c) is identical to (4.2) if U in (3.1c) is taken to be the stress work density.) Againwe observed that values evaluated on differently sized domains are within one percent ofthe main value throughout the entire analysis. This lends confidence to this particularelastic-plastic dynamic solution. The rise of dynamic J with respect to time is shown inFig. 6. To gauge the effect of plasticity, we have included in the figure the elastodynamic Jobtained from the same mesh using the identical time steps. The behavior of the dynamicJ with respect to the opening displacement of the node at a distance L behind the cracktip as determined from the elastic and elastic-plastic analyses are shown in Fig. 7. Thetrend of these solutions are in accord with quasi-static results. An analytical solution tothis problem is not available and more definite statements concerning the accuracy of theelastic-plastic numerical solution cannot be made. Nevertheless, results from anotherstudy [25] do provide information which is pertinent to the present discussion. In thatstudy, a deeply notched round bar was subjected to a tensile pulse with a wave lengthwhich is long compared to the bar diameter. The dynamic J obtained from (4.2) agreedwell with the values determined from a deep crack formula based on the remote load anddisplacement [31]. These earlier comparison taken together with the present resultsindicate that finite domain integrals can be employed to extract accurate values of thedynamic J from finite element solutions.

AcknowledgementsWe gratefully acknowledge the research support of the Naval Sea Systems Command(SEA 05R15) through NBS Grant NB82RA20004, the National Science Founda tionthrough Grant MEA82-10931, and the NSF Materials Research Laboratory at BrownUniversity through Grant DMR83-16893.The calculations described here were performed in the VAX-11/780 ComputationalMechanics Facility at Brown University. This facility was made possible by grants fromthe National Science Founda tion (Solid Mechanics Program), the General Electric Foun-dation, and the Digital Equipment Corporation.


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C o m p u t a t i o n a l m e t h o d s b a s e d o n a n e n e r g y i n te g r a l 243Reference s

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3 8 - 5 3 .[31] J .R . R ice , P .C . Par is and J .G . Merkle , in A S T M S T P 536 , A m er i can Soc i e ty fo r Tes t i ng and M a te r i a l s

(1973) 231-245 .

R ~ s u m ~

O n pas se en r evue l es d rve lopp em e n t s dans l ' u t i l i s a t i on de l ' i n t rg r a l e du f l ux d ' r ne rg i e h l ' ex t r rm i t6 d ' une f i s su redans l e ca l cu l de l a m rcan ique de l a r up tu re dynam ique e t c e , au cou r s des 5 de rn i~ re s ann~ es . O n e t a t i r e unee x p r e s s io n d u f lu x d ' r n e r g i e ~ l ' e x tr r m i t 6 d ' u n e f i s s u re e n f o n c t i o n d u c h a m p m r c a n i q u e a u v o i s in a g e d e c e t tee x t r r m i t r , e x p r e s s i o n a p p l i c a b l e a u x r r a c t i o n s g r n r r a l e s d ' u n m a t r r i a u . O n d ~ m o n t r e e n s u i t e q u e c e r t a i n e sin t rg r a l e s d ' r ne rg i e i n t r r e s san t e s peuven t ~ t r e ex t r a i t e s de s r r su l t a t s g rn r r aux en m e t t an t en exe rgue l e sc a r a c t r r i s a ti o n s a d r q u a t e s d e l a r ~ p o n se d u m a t r r i a u . D i v e rs e s r e p r r s e n t a t i o n s a l t e rn a t i v e s d u f lu x d ' r n e r g i e s o u sfo rm e d ' i n t rg r a l e couvran t ce r t a ines r rg ions f i n i e s au tou r de l ' ex t r rm i t6 de l a f i s su re son t p r r s en t r e s e tc o m p a r r e s , e n v u e d ' u n e i n s e r ti o n d a n s d e s 6 t u de s d e s i m u l a t i o n p e r d r m e n t s f in i s.

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