A DYNAMIC FRACTURE ANALYSIS OF CRACK CURVING AND BRANCHING

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A DYNAMIC FRACTURE ANALYSIS OF CRACK CURVING AND BRANCHING

A. Kobayashi, M. Ramulu

To cite this version:

A. Kobayashi, M. Ramulu. A DYNAMIC FRACTURE ANALYSIS OF CRACK CURV- ING AND BRANCHING. Journal de Physique Colloques, 1985, 46 (C5), pp.C5-197-C5-206.

�10.1051/jphyscol:1985525�. �jpa-00224755�

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Colloque C5, supplkment au n08, Tome 46, aoQt 1985 page C5-197

A D Y N A M I C F R A C T U R E A N A L Y S I S O F C R A C K C U R V I N G A N D B R A N C H I N G

A . S . Kobayashi and M. Ramulu

University of Washington, Department of MechanicaZ Engineering, SeattZe, Washington 98195, U.S.A.

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6 - On presente un c r i t e r e de courbure de f i s s u r e fond& s u r sa s t a b i l i t i ! d i r e c t i o n n e l l e a i n s i qu'un c r i te r e de r a m i f i c a t i o n necessi t a n t un f a c t e u r dy- namique d ' i n t e n s i t@ de c o n t r a i n t e de r a m i f i c a t i o n e t l e c r i te r e de courbure.

Ces c r i t e r e s sont u t i l i s e s pour p r e d i r e l e s courbures e t l e s r a m i f i c a t i o n s de f i s s u r e s dans des e c h a n t i l l o n s de Homalite-100, de polycarbonate, d ' a c i e r e t de tubes d'aluminium.

Abstract

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D e s p i t e t h e many papers and a number o f comprehensive review a r t i c l e s on dynamic f r a c t u r e * papers on elastodynamic analyses o f crack c u r v i n g and branching a r e few.

Among these are an approximate t h e o r e t i c a l s o l u t i o n f o r t h e c u r v i n g o f a s t a t i c crack w i t h nonvanishing mode I1 s t r e s s i n t e n s i t y f a c t o r s /1,2/, and an elasto- s t a t i c s s o l u t i o n s w i t h v a n i s h i n g mode I1 s t r e s s i n t e n s i t y f a c t o r f o r p r e d i c t i n g t h e i n i t i a l angle o f crack branching / 3 r 4 / . A c r i t i c a l review o f s t a t i c crack branching a n a l y s i s i s provided by Lo / 5 / .

Experimental observations o f crack c u r v i n g i n b r i t t l e m a t e r i a l s /6-12/ show t h a t t h e curved crack path appears t o d i f f e r w i t h specimens and l o a d i n g conditions.

For example, crack path i n a bend specimen under impact l o a d i n g may have a charac- t e r i s t i c S-shape as t h e crack approaches t h e compression s i d e o f t h e specimen / 6 * 7 / ; a double c a n t i l e v e r beam specimen under wedge l o a d i n g o f t e n y i e l d s a gradual c u r v i n g crack path a f t e r f r a c t u r e i n i t i a t i o n /&11/; s i n g l e d edge notch specimens under u n i a x i a l l o a d i n g may have a s l i g h t l y d e f l e c t e d crack path /12/.

Long . b e f o r e o t h e r e l a s t o s t a t i c s o l u t i o n s t o crack branching problems became a v a i l a b l e , Y o f f e l s pre-branching a n a l y s i s /13/ showed t h a t t h e maximum circum- f e r e n t i a l s t r e s s * oget e x h i b i t s two symmetrical maximum along t h e crack a x i s a t a crack v e l o c i t y o f about C/C1 = 0.38# where Cl i s t h e d i l a t a t i o n a l wave v e l o c i t y . T h i s c r i t i c a l v e l o c i t y , which s h i f t e d t h e maximum Gee away from t h e d i r e c t i o n o f s e l f - s i m i l a r crack propagation* was deemed t h e c o n d i t i o n f o r crack branching.

Experimentally observed c r a c k v e l o c i t i e s a t crack branching, unfortunate1 y, a r e much s m a l l e r than t h e t h e o r e t i c a l l y p r e d i c t e d c r i t i c a l crack v e l o c i t i e s f o r branching. Bowdeds e t al. /14/ experimental observations showed t h a t a crack propagated we1 l be1 ow t h e t e r m i n a l v e l o c i t y b e f o r e branching* b u t accelerated up t o t h e i n s t a n t of branching, and i s accompanied by a drop o f 5 t o 10 percent i n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985525

C5-198 JOURNAL DE PHYSIQUE

t h e crack v e l o c i t y a f t e r branching. The p r e c i s e u l t r a s o n i c r i p p l e marking t e c h n i - ques used by ~ o l l /15/ and by K e r k h o f f /16/ showed t h a t t h e crack v e l o c i t y . de- creased about 10 p e r c e n t i n g l a s s immediately a f t e r branchingr whereas Schardin /17/ and Acloque /18/ observed no change i n crack v e l o c i t i e s d u r i n g branching i n p l a t e g l a s s and a 6 percent change i n pre-stressed glass, r e s p e c t i v e l y . Crack branching v e l o c i t i e s i n v a r i o u s s t e e l s r e p o r t e d by I r w i n /19/. Hahn e t al. /20/#

Congelton e t al. /21,22,23/, and Weimer and Rogers /24,25/ and i n glassy polymer specimens by Kobayashi e t al. /26-28/, D a l l y e t al. /29,30/, and Ravi-Chandar and Knauss /31,32/ showed t h a t crack branching occurred c o n s i s t e n t 1 y a t v e l o c i t i e s w e l l below t h e crack v e l o c i t i e s o f C/C1 = 0.25. S i m i l a r observations were made by Paxson e t al. /33/ and Doyle /34/. Table 1 shows t h e t h e o r e t i c a l and exper- imental c r i t i c a l crack v e l o c i t i e s where t h e Experimentally observed low branching v e l o c i t i e s , which h a r d l y decreased a f t e r branching c o u l d n o t be a p r e r e q u i s i t e t o c r a c k branching i n these materials.

Tab1 e 1. T h e o r e t i c a l and Experimental C r i t i c a l V e l o c i t i e s

Source Mater i a1 Cb/Co*

c a l P r e d i a Y o f f e /13/

Fxper Measurements

Anthony e t al. /22/

Bow den e t al. / 14/

Congl eton /23/

Do1 l 1151 Hahn e t al. /20/

I r w i n e t al. /19/

Kobayashi e t al. /27r28/

Paxson e t al. /33/

Schardin /17/

G1 ass G1 ass Tool s t e e l P1 a t e G1 ass FK-52 G1 ass A533B Steel H m a l i te-100 H m a l i te-100 P1 e x i g l ass G1 ass

*Cb i s t h e crack v e l o c i t y a t t h e onset o f branching CO i s t h e bar wave v e l o c i t y

Since crack branching was observed a t lower v e l o c i t i e s , a branching c r i t e r i o n which i s v a l i d f o r a crack v e l o c i t y l e s s than t h e c r i t i c a l v e l o c i t y , o f C = 0.38Cl, must be considered. For example, C l a r k and I r w i n /35/ concluded t k a t crack branching occurs by advanced cracking, which r e q u i r e s a c r i t i c a l s t r e s s i n t e n s i t y f a c t o r , K f o r i t s generation. Independently, Congleton and Petch /21/ proposed a cr&>al s t r e s s i n t e n s i t y f a c t o r c r i t e r i o n f o r c r a c k branching based on advanced cracking. Table 2 shows t h e c a l c u l a t e d r a t i o s o f t h e c r i t i c a l Table 2. C r i t i c a l S t r e s s I n t e n s i t y Factors a t Onset o f Branching ( f r a

Experimental Study

Source M a t e r i a l K ~ b / K ~ ~ *

Congleton e t al. /22/

D a l l y e t al. /29,30/

Do1 l / 15/

I r w i n e t al. /35/

Hahn e t al. /20/

K i rchner & K i r c h n e r /36/

K i r c h n e r e t al. /37/

Kobayashi e t al. /27,28/

Weimer and Rogers /24,25/

Tool Steel H m a l i t e - 1 0 0 G1 ass

Homal ite-100 A5338 Steel F1 i n t Glass Ti-ZrO H m a l ize-100 HF-Steel FS-01 Steel

*KIb i s branching s t r e s s i n t e n s i t y f a c t o r KI

c i s f r a c t u r e toughness

s t r e s s i n t e n s i t y f a c t o r s K /K a t t h e onset o f branching f o r v a r i o u s m a t e r i a l s where KI i s t h e f r a c t u r e %ugk%ess. The experimental data v a r i e d from m a t e r i a l t o materyal and a l s o appeared t o be a f u n c t i o n o f t h e t e s t method. D a l l y e t al.

/29,30/, showed t h a t t h e existence o f a p l a t e a u i n t h e C-KID c u r v e as a necessary c o n d i t i o n f o r branching where KDI i s t h e dynamic f r a c t u r e toughness.

Theories used t o p r e d i c t crack branching angles under s t a t i c l o a d i n g c o n d i t i o n s a r e t h e maximum c i rcumferenti a1 s t r e s s theory /38/, t h e minimum s t r a i n energy d e n s i t y f a c t o r theory /39/, and t h e maximum s t r a i n energy r e l e a s e r a t e theory /40-42/. Kal t h o f f /4/ anal yzed symmetrically branched edged cracks subjected t o u n i a x i a l t e n s i o n by a s t a t i c , p o s t branching s t r e s s analysis. The s i g n o f t h e mixed mode s t r e s s i n t e n s i t y f a c t o r s r a t i o K /K changed and became z e r o f o r a c h a r a c t e r i s t i c k i n k i ng angle 'ac which was de%rmfned by using t h e maximum s t r e s s c r i t e r i o n , 8,.

K a l t h o f f found t h a t a crack branch w i t h a small branching angle tended t o repel one another when K /K > 0. Also, a branch w i t h a l a r g e i n i t i a l branching angle tended t o a t t r a c t l L a c i o t h e r when K /K < 0. S e l f - s i m i l a r crack propagation occurred when KI1 = 0. Kitagawa e t a\! h3,44/ and V i t e k /45/ a r r i v e d a t a sim- i l a r conclusion and t h a t t h e crack kinked w i t h a k i n k i n g angle o f 18 degrees a t zero KII. T h e o r e t i c a l l y p r e d i c t e d and e x p e r i m e n t a l l y observed branching angles a r e given i n Table 3.

Table 3. Crack Branching Angles

Branching Angle*

Source i n Dearees

D e o r e t i c a l R e s m Achenback /46/

Anderson /40/

Hussain e t al. /41/

Kitagawa e t al. /43,44/

V i t e k /45/

Y o f f e /13/

i c a l Results P a r l e t u n /3/

K a l t h o f f /4/

k ~ e r i m e n t a l Re- Bowden e t al. /14/

Bul l e n e t al. /47/

C h r i s t i e /48/

C l a r k and I r w i n /35/

Congleton e t al. /22,23/

Kal t h o f f /4/

K i r c h n e r e t al. /37/

Kobayashi e t al. /27,28/

Nakasa and Takei /49/

B u l l o c k and Kaae /SO/

"Branching angles measured between t h e two c r a c k t i p s

The h y p o t h e t i c a l crack c u r v i n g and crack branching mechanisms can be explained by c r a c k - t i p micro-cracking. F i r s t t h e micro-cracks n u c l e a t e from t h e i n c l usions and v o i d s i n t h e v i c i n i t y o f t h e crack t i p and grow as a r e s u l t o f t h e imposed crack- t i p s t r e s s f i e l d . Continual crack growth r e s u l t s i n coalescence o f t h e micro- cracks i n t h e c r a c k - t i p r e g i o n and reduce t h e s t r e s s i n t e n s i t y f a c t o r w i t h i n - crease i n compl iance o f t h e c r a c k - t i p c o r e region. Simultaneously, t h e remote s t r e s s componentr Uoxr which a c t s p a r a l l e l t o t h e crack and which i s t h e second o r d e r term i n t h e c r a c k - t i p s t r e s s f i e l d , increases i n magnitude /51/. The i n - creased magnitude i n p a r a l l e l s t r e s s i n t u r n a c t i v a t e s micro-cracks away from t h e

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c r a c k - t i p plane. I f t h e micro-cracks are s u f f i c i e n t l y c l o s e t o t h e primary crack t i p , they w i l l tend t o d i v e r t t h e crack away from i t s o r i g i n a l plane. A s i n g l e c r a c k - t i p d i v e r s i o n r e s u l t s i n crack c u r v i n g and mu1 t i p l e c r a c k - t i p d i v e r s i o n r e s u l t s i n crack branching. Also, w i t h i n c r e a s i n g s t r e s s i n t e n s i t y factor, t h e micro-cracks grow s u f f i c i e n t l y l a r g e t o e n g u l f t h e main crack t i p which now d e p i c t s a b l u n t crack /52/. The l a r g e r energy release r a t e associated w l t h t h e apparent b l u n t crack t i p w i l l i n t u r n r e q u i r e t h e generation o f brancned cracks f o r l a r g e r energy d i s s i p a t i o n . As a r e s u l t , t h e primary crack w i l l extend through t h i s damaged r e g i o n and l e a v e i n i t s wake a s e r i e s o f d i s t i n c t attempted branches /53/. At a s u f f i c i e n t l y high a p p l i e d s t r e s s i n t e n s i t y f a c t o r o f K , t h e branched cracks w i l l continue t o propagate and thus complete successful crack branches. I n t h e foregoing cases considered, t h e necessary p r e r e q u i s i t e f o r crack c u r v i n g and branching i s t h e growth o f micro-cracks i n t o macro-cracks w i t h eventual hookup w i t h t h e main crack. As shown i n previous papers /51,52,54/, t h e remote s t r e s s component, which i s a c t i n g p a r a l l e l t o t h e d i r e c t i o n o f crack propagation, con- t r o l s t h e l o c a t i o n and o r i e n t a t i o n o f micro-cracking and hence governs t h e d i r e c - t i o n a l s t a b i l i t y o f an extending crack.

I11

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With t h e foregoing assumption t h a t t h e advanced o f f - a x i s micro-cracks d i c t a t e t h e d i r e c t i o n o f crack propagation, e i t h e r o f t h e two dynamic crack c u r v i n g c r i t e r i a /54/ advanced by t h e authors, namely t h e maximum c i r c u m f e r e n t i a l s t r e s s o r t h e minimum s t r a i n energy density c r i t e r i a , can be used t o p r e d i c t crack curving.

Both c r i t e r i a p r e d i c t nearly i d e n t i c a l crack c u r v i n g angles i n t h e crack v e l o c i t y range and i n t h e presence o f smaller KII. For b r e v i t y , only t h e crack c u r v i n g c r i t e r i o n based on t h e maximum c i r c u m f e r e n t i a l s t r e s s thus w i l l be discussed.

The maximum c i rcumferenti a1 s t r e s s c r i t e r i o n , as modi f i e d by Ramul U and Kobayashi /54/, assumes t h a t t h e crack w i l l extend towards t h e maximum c i rc u m f e r e n t i a l stress, 000, a t a distance, ro, away f r w n t h e r a p i d l y propagating crack t i p . The c o n d i t i o n f o r a s e l f - s i m i l a r propagation o f a s t r a i g h t crack can be obtained i n terms o f ro, which i s a c h a r a c t e r i s t i c distance associated w i t h t h e c u r r e n t dy- namic s t a t e o f stress. For a dynamically propagating crack, it i s always l e s s than t h e corresponding ro o f a s t a t i o n a r y crack.

The d i r e c t i o n o f microcracks depends both on t h e m a t e r i a l m i c r o s t r u c t u r e and t h e s t r e s s f i e l d . These two parameters a r e l i n k e d together i n a c r i t e r i o n i n v o l v i n g t h e d i r e c t i o n a l s t a b i l i t y o f an extending crack. The onset o f crack c u r v i n g o f a r a p i d l y propagating crack i s governed by t h e d i r e c t i o n a l s t a b i l i t y o f t h e propa- g a t i n g s t r a i g h t crack and i s assumed t o occur when r <rc. T h i s rc i s a charac- t e r i s t i c distance derived from a d i r e c t i o n a l s t a b i l E y c r i t e r i o n i n v o l v i n g t h e c r i t i c a l crack t i p s t a t e o f s t r e s s which suddenly s h i f t s o f f t h e s e l f - s i m i l a r crack path. The corresponding angle, rc, f o r a maximum age, under a pure mode I dynamic crack extension can be determined through a transcendental r e l a t i o n i n v o l - v i n g t h e s t a t e of s t r e s s and t h e c r i t i c a l values o f r

.

I V

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The aforementioned micro-mechanics i n v o l v e d i n crack c u r v i n g and crack branching i n d i c a t e t h a t t h e crack branching c r i t e r i o n r e q u i r e s both a c r i t i c a l s t r e s s i n - t e n s i t y f a c t o r t h a t i s accompanied by t h e c h a r a c t e r i s t i c distance r&rc. The crack branching c r i t e r i o n can thus be s t a t e d as

K~ ₂ K ~ b Necessary c o n d i t i o n

ro .Irc S u f f i c i e n t c o n d i t i o n

V

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The v a l i d i t y o f t h e above crack c u r v i n g and branching c r i t e r i o n was v e r i f i e d by dynamic p h o t o e l a s t i c i t y r e s u l t s o f Hanalite-100 /51,52,54/ and polycarbonate /55,56/ f r a c t u r e specimens. I n t h e following, t y p i c a l r e s u l t s from each o f t h e above s e r i e s o f experiments w i l l be discussed.

ECCENTRIC

L O A D I N G ( E L ) ~ !CENTRAL LOADING (CL)

28.6mm 2 2 . 2 m m

514- 810717 THICKNESS 3.2mm

2 8 . 6 m m 2 2 . 2 m m

S15 - 8 1 0 7 2 7

( 0 ) SPECIMEN DIMENSIONS l b ) CRACK PATHS

Fig. 1. Pol ycarbonate Double Edge Crack Tension Specimen

F i g u r e 1 shows specimen c o n f i g u r a t i o n s and c r a c k paths o f f i v e polycarbonate, double edge crack t e n s i o n specimens w i t h e i t h e r o f f s e t p a r a l l e l cracks, o f f s e t s l a n t e d cracks and symmetrically l o c a t e d t w i n cracks which were used i n t h i s dynamic photoel a s t i c study. The annealed t h i n p01 ycarbonate specimens w i t h b l u n t s t a r t e r cracks e x h i b i t e d b r i t t l e f r a c t u r e w i t h shear l i p s l e s s than 10 p e r c e n t o f t h e t h i c k n e s s and an apparent crack t i p y i e l d zone o f l e s s than 1.5mm. The ro associated w i t h crack c u r v i n g i s shown i n F i g u r e 2; t h e average rc v a l u e was 0.5mml and i s c o n s i s t e n t w i t h t h e rc estimated by Theocaris /57/.

10 20 30 4 0 SO 60 70 80

CRACK EXTENSION, mm

Fig. 2. ro o f Propagating Crack i n Polycarbonate Specimens

C5 -202 JOURNAL DE PHYSIQUE

The range o f f r a c t u r e parameters associated w i t h t h i s s e r i e s o f crack c u r v i n g ex- periments on Homal ite-100 and p01 ycarbonate f r a c t u r e specimens a r e summarized i n Table 4. The t h e o r e t i c a l l y p r e d i c t e d crack c u r v i n g angles, e i t h e r i n pure mode I o r mode I and mode I 1 l o a d i n g conditions, i n t h e presence o f ao,, were w i t h i n t h r e e degrees o f t h e corresponding measured values,

F i g u r e 3 shows t h e two t y p i c a l dynamic isochromatic p a t t e r n s associated w i t h crack branching i n a f r a c t u r i n g sing1 e edged notch p01 ycarbonate specimen. F i g u r e 4 shows t h e v a r i a t i o n s i n dynamic KI, ,IKI and a ox j u s t p r i o r t o and a f t e r c r a c k branching i n t h e f r a c t u r i n g polycarbona e specimen shown i n F i g u r e 3. By extra- p o l a t i n g t h e KI, a branching s t r e s s i n t e n s i t y f a c t o r o f KbI = 3.3 MPat'ii i s obtained. The r reached an estimated minimum value o f 0.7 mm a t branching. The r e s u l t s o f t h e o t h e r t h r e e experiments y i e l d e d an average ro o f 0.7 mm a t branch- i n g and a K = 3.3 MPafi. A t t h e onset o f crack branching, t h e c r a c k v e l o c i t i e s were foundlio be above 0.24 Cl. A summary o f crack branching r e s u l t s on Homalite-100 and polycarbonate f r a c t u r e specimens i s g i v e n i n Table 5.

Table 4. Experimental and T h e o r e t i c a l Crack Curving R e s u l t s

Homal fte-100 Polvc&onate

T o t a l Number o f Experiments Type o f F r a c t u r e Specimen Crack V e l o c i t y , C/C1 KI (MPCm)

K11IK1 ro (mm)

Measured Crack Curving Angle P r e d i c t e d Crack Curving Angle

9

SEN DCB, DTT About 0.21 0.50 t o 1.59 -0.22 t o 0.18 1.0 t o 1.5

-20 t o 26 degrees -20 t o -25 degrees

5

Double edge c r a c k spec About 0.22

1.5 t o 3.2 -0.33 t o 0.19 0.25 t o 0.75 -20 t o 3 degrees -19 t o 5 degrees

SIXTH FRAME, 59 p SECONDS THIRTEENTH FRAME, 162

p

Fig. 3. Dynamic I s o c h r a n a t i c s o f a Branched Crack i n a Polycarbonate Specimen It i s i n t e r e s t i n g t o observe t h a t t h e branching s t r e s s i n t e n s i t y f a c t o r appears t o be independent o f t h e t h i c k n e s s as w e l l as t h e i n i t i a l and branching crack lengths. Also n o t e t h a t t h e d e v i a t i o n between t h e estimated and measured c r a c k branching angles was 6 degrees and t h e average d e v i a t i o n was 3 degrees.

The abovementioned crack c u r v i n g as w e l l as t h e crack branching c r i t e r i a a r e a l s o appl i c a b l e t o q u a s i - s t a t i c problems. The crack branching c r i t e r i o n was t h u s used t o e v a l u a t e t h e branching data o f f i v e 76 X 152 X 9.5 mm t h i c k Hanalite-100, wedge loaded, r e c t a n g u l a r double c a n t i l e v e r beam specimens /51/. Since t h e cracks

-I I ^{I I ! , I I} I

0 2 0 4 0 e O m 1 0 0 1 2 0

CRACK 'EXTENSION, m m

Fig. 4. Dynamic K I, KII, and aox P r i o r t o Crack Branching

Table 5. Crack Branching Data i n Hcinalite-100 and Polycarbonate F r a c t u r e

Test Number

Homal ite-1OQ B8

B9

Specimen I n i t i a l Crack Thickness Crack Lenght C KIb

Length Branching ^r~

h ab MPafi mm

mm mm mm

3.18 5.6 66.0 2.08 1.2

3.18 4.3 177.0 2.03 1.3

3.58 5.8 139.7 2.03 1.4

9.53 5.1 52.6 2.00 1.4

9.53 13.5 19.1 2.08 1.2

9.53 13.5 28.7 2.09 1.3

Average 2.04 1.3

3.2 15 86 3.3 0.85

6.4 9 52 3.2 0.78

3.2 16 65 3.3 0.78

3.2 15 4 1 3.4 0.58

Average 3.3 0.75

Branching Angle Measured Estimated c

degrees

immediate1 y branched from t h e b l u n t s t a r t e r crack upon crack i n i t i a t i o n , t h e necessary and s u f f i c i e n t c o n d i t i o n s f o r branching were deduced by s t a t i c f i n i t e element analysis. LikewSse, crack c u r v i n g r e s u l t s were obtained i n t h r e e b i - a x i a1 l y loaded H m a l ite-100 p1 a t e s w i t h b l u n t c e n t r a l cracks /53/. Excel l e n t agreements between t h e p r e d i c t e d and measured branch angles was noted i n t h e WL-DCB as w e l l as t h e b i a x i a l l y loaded Hanal i t e - 1 0 0 f r a c t u r e specimens.

V 1

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The crack branching c r i t e r i o n was a l s o used t o p r e d i c t t h e crack branching angle i n a f r a c t u r i n g t h i n m i l d s t e e l tube which b u r s t a t -120' C /23,58/. The p i p e was 12.7 mm t h i c k , 152 mm diameter, w i t h a t o t a l l e n g t h o f 457.2 mm and a c e n t r a l through crack o f 2a = 57.2 mm. Two-dimensional dynamic f i n i t e element a n a l y s i s was used i n i t s generation mode, and determined n u m e r i c a l l y t h e dynamic KDI and t h e Go,. D e t a i l s o f t h i s a n a l y s i s are found i n Reference /52/. The c h a r a c t e r i s t i c rc = l mm f o r t h i s s t e e l was estimated by measuring t h e l e n g t h s o f t h e secondary cracks i n t h e photanicrograph. The n u m e r i c a l l y obtained KbI = 124 MPa& was w i t h - i n 5 percent o f Congletonfs estimated v a l u e based on s t a t i c analysis. The p r e - d i c t e d brancing angles agreed w e l l w i t h t h e measured values.

JOURNAL DE PHYSIQUE

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High-speed photoel a s t i c i n v e s t i g a t i o n s /28*54,55/ o f crack c u r v i n g reveal t h a t crack c u r v i n g i s o f t e n associated w i t h t h e backward tilt o f crack t i p i s o c h r o - matics. T h i s i n d i c a t e s p o s i t i v e values o f remote s t r e s s as observed i n dynamic t e a r t e s t (DTT) specimens under impact loading, and DCB under wedge loading. The proposed crack c u r v i n g c r i t e r i a p r e d i c t s t h e c u r v i n g angle w e l l and t h e angle o r i e n t a t i o n appears t o be c o n t r o l l e d by t h e p o s i t i o n o f t h e advance cracks, which causes t h e crack t o curve e i t h e r t o t h e upper o r l o w e r h a l f o f t h e crack plane, l o c a t e d ahead o f t h e crack t i p . Stress wave i n t e r a c t i o n and t h e r e s u l t i n g mixed mode e f f e c t a l s o c o n t r o l t h e crack d i r e c t i o n .

The d i r e c t i o n o f maximum age a t a c r i t i c a l r a d i a l d i s t a n c e i s always a t 8 = 0 f o r a < 0 and G/C1 0.32. However, crack branching occurred a t an average i n c l u d e d c%ck branching angle o f 28 = 28 degrees when C/C1 < 0.32 and when oox < 0. The c o n t r a d i c t i o n between t h e t % e o r e t i c a l and experimental r e s u l t s can be r e s o l v e d by examining t h e instantaneous c i r c u m f e r e n t i a l s t r e s s associated w i t h t h e onset o f crack branching a t t h e c r i t i c a l r a d i a1 distance. The r e d u c t i o n i n c i rc u m f e r e n t i a1 s t r e s s a t 0 = 14 degrees i s l e s s than 3 percent f o r t h e observed crack v e l o c i t y . Presumably, t h e propagating crack can extend i n any d i r e c t i o n w i t h i n 8 = 14 degrees i n o r d e r t o b i f u r c a t e and double i t s energy r e l e a s e rate. The averaged v a l u e o f t h e estimated one-ha1 f branching angle, BC = 15.4 degrees, i s i n good agreement w i t h t h e measured one-ha1 f branching angle. T h i s demonstrates t h a t t h e crack branching c r i t e r i o n remains v a l i d d e s p i t e t h e p r e d i c t e d s e l f - s i m i l a r c r a c k extension f o r K KIb. The c a l c u l a t e d one-ha1 f branching angle w i t h a < 0 i s t h e best estimaied angle under a pure mode I condition. The dynamic s??ess i n - t e n s i t y f a c t o r a t t h e onset o f crack branching reached an average maximum value o f 3.3 MPa& i n polycarbonate and 2.0 MpaJk i n Hanalite-100 f r a c t u r e specimens. T h i s branching s t r e s s i n t e n s i t y f a c t o r was found t o be independent o f t h e t h i c k n e s s o f t h e specimen, as w e l l as t h e i n i t i a l and branching c r a c k lengths, as shown i n Table 5.

Post-branching cracks i n a l l t e s t s always curved. K a l t h o f f /4/ observed t h a t t h e d i r e c t i o n o f two branched cracks (i.e. whether t h e y curve toward o r a p a r t from each o t h e r ) i s c o n t r o l l e d by K /KI. The p h o t o e l a s t i c p a t t e r n s o f running branched cracks i n i t i a l l y showed $&ong mixed mode and h i g h e r order term e f f e c t s . These p h o t o e l a s t i c r e s u l t s suggest t h a t t h e crack w i l l run p a r a l l e l t o t h e com- p r e s s i v e s t r e s s d i r e c t i o n and t h a t t h e mixed mode i s o c h r a n a t i c s a f t e r branching advance cracks. Therefore, post-branching crack propagation i s a l s o s t r o n g l y dependent on t h e KII/KI r a t i o as w e l l as on G.o, Post-branching crack c u r v i n g show t h a t t h e crack c u r v i n g angle g r a d u a l l y decreased i n magnitude along w i t h an increase i n n e g a t i v e Go, and continued t o propagate when t h e crack branching angle was 18 degrees. Although inconclusive^ i t appears from our unpublished exper- imental r e s u l t s * t h a t successful branching i s p o s s i b l e o n l y i f 28, > 20 degrees.

For s m a l l e r 20, < 20 degrees, however, branched cracks re-merge w i t h t h e main crack and propagate i n accordance w i t h K a l t h o f f c s a n a l y s i s /4/.

Work performed so f a r i n dynamic crack c u r v i n g and crack branching p r i m a r i l y considered two-dimensional behavior. I n many engineering m a t e r i a l s , crack growth, c u r v i n g * and branching i s associated w i t h advance nucl e a t i o n o f micro-cracks.

These microcracks a r e accompanied by i n t e r a c t i o n s between o f t h e s t r e s s waves and t h e crack t i p loading, as observed by t h e high-speed photomicrography experiments o f Ravi-Chandar and Knauss /32/, who a l s o demonstrated these three-dimensional e f f e c t s by observing t h e n u c l e a t i o n o f microcracks i n t h e i r r e a l - t i m e experiment.

V I I I - CONCLUSIONS

The review o f t h e present and o f t h e p r e v i o u s s t u d i e s on crack c u r v i n g and crack branching show t h a t t h e c r i t e r i a based on crack t i p micro-mechanics proposed by

t h e authors can s u c c e s s f u l l y p r e d i c t t h e c u r v i n g and branching i n b r i t t l e m a t e r i a1 S. Speci f i c a l l y:

1. A dynamic crack c u r v i n g c r i t e r i o n * which i s based on crack i n s t a b i l i t y and which i n v o l v e s t h e nonsingular s t r e s s component i n a d d i t i o n t o t h e s i n g u l a r stresses* i s presented.

2. A dynamic crack branching c r i t e r i o n * which r e q u i r e s as a necessary c o n d i t i o n a c r i t i c a l crack branching s t r e s s i n t e n s i t y f a c t o r and as a s u f f i c i e n t condi- t i o n t h e .above crack c u r v i n g criterion^ was developed.

3. The branching s t r e s s i n t e n s i t y f a c t o r i s found t o be a m a t e r i a l p r o p e r t y and i s independent o f t h i c k n e s s as we1 l as i n i t i a l and branching crack l e n g t h s o f t h e f r a c t u r e d specimens under u n i f o r m and l i n e a r l y v a r y i n g loads.

4. This dynamic crack branching c r i t e r i o n p r e d i c t e d t h e actual crack branching angle i n dynamically as w e l l as s t a t i c a l l y f r a c t u r i n g p h o t o e l a s t i c specimens.

I t a l s o p r e d i c t e d t h e crack branching angle i n a b u r s t i n g metal pipe.

ACKNCWLECG EMENT

The work r e p o r t e d here was obtained under ONR Contract NG0014-76-C-000 NR-064-478. The authors wish t o acknowledge t h e s u p p o r t and encouragement o f Dr.

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