1-the fracture process.pdf

8/11/2019 1-the fracture process.pdf

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T he racture Process

2 .1 In t roduc t ion

Frac tu re i s o f ten cons idered as a p rocess in wh ich increased load ing sudden ly causes

acce le ra ted g row th o f a p re -ex is t ing c rack . A c loser s tudy , how ever , revea ls th ree d is-

t inc t phases , 1 ) load ing wi thou t c r ack g rowth , 2 )

stable crack growth

and 3 )

unstable

crack growth.

Stab le c rack g rowth may , in p r inc ip le , be con t ro l led wi th the load ing

dev ice , so tha t , fo r in s tance , a p rescr ibed s low crack g rowth may be ob ta ined . Th is i s

no t poss ib le fo r uns tab le c r ack g rowth , wh ich occu r s spon taneous ly .

Four d is t inc t ly d i f f e ren t r eg ions may be r ecogn ized in a c r ack edge v ic in i ty . Neares t

the edge i s the

process region

d i s c u s s e d i n t h e p r e v io u s c h a p t e r . W h e n th e c r a c k

edge advances , a

wake of the process region

i s l e f t beh ind . Ou ts ide the p rocess r eg ion

there i s genera l ly a plastic region. W h e n th e c r a c k e d g e a d v a n c e s a wake of the plastic

region

i s l e ft beh ind . In th i s wake the m ate r ia l i s aga in d efo rm ing e las tica l ly , because o f

u n lo a d in g , b u t r e v e rs e d p l a s t i c f lo w m a y e v e n tu a l l y o c c u r , s o t h a t a secondary plastic

region

a p p e a r s b e h in d t h e w a k e o f t h e

primary plas t ic region.

The p rocess r eg ion

a n d th e p r im a r y a n d s e c o n d a r y p l a s t i c r e g io n s a r e t h e

dissipative regions.

O u t s i d e

the m is the e las t ic r eg ion . In F ig . 2 .1 .1 the th ree phases o f the f r ac tu re p rocess a r e

i l l u s t ra t e d , t o g e th e r w i th t h e p r o c e s s r e g ion a n d t h e p r im a r y ) p l a s t ic r e g io n w i th i t s

wake.

T h e p r e s e n t c h a p t e r w i l l m a in ly d e a l w i th t h e o p e n in g m o d e , a n d s o m e f e a tu r e s

d iscussed may be r ecogn ized as be ing spec i f ic to th i s mode . However , severa l f ea tu res

a lso ca r ry over to the shear ing modes .

2.2 Pre ex is t ing cracks

P r e - e x i s t i n g c r a c k s a r e v e r y c o m m o n a n d v i r t u a l l y i m p o s s i b l e t o a v o i d i n l a r g e s t r u c -

t u r e s . I n s o m e s o l i d s , f o r i n s t a n c e g l a s s , t i n y s u r f a c e c r a c k s a p p e a r s p o n t a n e o u s l y a s

a r e s u l t o f c h e m i c a l a g e n t s , e v e n i n s e e m i n g l y n e u t r a l e n v i r o n m e n t s , s u c h a s a i r w i t h

n o r m a l h u m i d i t y . I n o t h e r c a s e s , c r a c k s a r e o p e n e d a s a r e s u l t o f t h e r m a l s t r e s s e s ,

c r e a t e d , f o r i n s t a n c e , a f t e r h e a t t r e a t m e n t h a r d e n i n g ) o r w e l d i n g . C r a c k s a r e a l s o

f r e q u e n t l y f o r m e d d u r i n g m a n u f a c t u r i n g o f t h e m a t e r i a l o r a s a r e s u l t o f m e c h a n i c a l

p r o c e s s e s d u r i n g m a n u f a c t u r i n g o r j o i n i n g s t r u c t u r a l p a r t s .

A p r e - e x i s t i n g c r a c k i s g e n e r a l l y n o t s i m p l y a s h a r p s l o t i n a v i r g i n m a t e r i a l . S u c h a

s l o t w o u l d b e o n l y a f e w i n t e r - a t o m i c d i s t a n c e s w i d e , b u t p r e - e x i s t i n g c r a c k s i n s t e e l s ,

27


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28 2 THE FRACTUR E PROCESS

Fi g. 2.1.1 The three phases of the fracture process are: 1. Loading witho ut crack growth

but with increasing damage near the crack edge. 2. Stable crack growth. 3. Unstable crack

growth. The four different regions near the crack edge are: 1. The process region. 2. The

plastic region. 3. The wake of the plastic region. 4. The virgin elastic region.

fo r i ns t ance ma y show open ings o f severa l hun dred in t e r -a tomic d i s t ances and some-

t imes much more . The mater i a l s t a t e i n t he c rack edge v i c in i ty var i es cons iderab ly

dep end ing on the h i s to ry o f c rack fo rmat ion . Th i s v ar i e ty ca ll s fo r a ph i losoph y o f han-

d l ing f r ac tu re p ro b lems tha t does no t need to cons ider the p rev ious h i s to ry which

moreover is genera l ly poor ly known. On the o ther hand i t is o f course des i r ab l e t o

know wh ethe r t he me chan i sm tha t ca used crack ing is s t il l p resen t . I t is fo r exam -

p le im po r t a n t t o recogn ize the ex i s t ence o f r es idua l s tr esses t ha t have cause d cracks

dur ing o r a f t e r weld ing .

M o s t o f t en a p r e - ex is t in g c r ack d o es n o t co n s t i t u t e a g r ea t e r h aza r d t h an a sh a r p

s lo t o f t he sa me s i ze . Therefo re t he s impl i fy ing ass um pt ion o f a sharp s lo t is gener -

a l ly app rop r i a t e fo r es t ima tes on the l i ke lihood o f c rack g row th and f rac tu re and i t

c e r t a i n l y m ak es a m a t h em a t i ca l an a l y s i s m u ch m o r e t r ac t ab l e .


3/18

2 .3 LOADING BEFORE CRACK GROW TH 29

F i g . 2 . 3 . 1 D u r i n g l o a d i n g o f a n o r i g i n al l y s h a r p c r ac k , p l a s ti c fl ow n e a r t h e c r a c k e d g e c a u s e s

b l u n t i n g . T h e b l u n t e d r e g i o n is s o m e t i m e s r e f e r r e d t o a s t h e s t r e t c h e d z o n e .

2 3 Loadingbefore cr ck growth

Su pp os e that a crack is so oriented that the amb ien t stresses tend to op en it. Ev en a

small load causes a separation of the crack faces, an d a strain concentr ation appe ars

at the crack edge s). In mo st materials, plastic flow follows, an d dur ing further load ing

the strains or stresses be co me sufficiently high to initiate micro- separa tions: a proces s

region develops. Cont inu ed increase of the load causes gr owt h of both the process

region an d the plastic region. Eventually, coalescences occur b et we en micro-sep arations

an d the ma in crack: the crack starts growing.

Th e seq uenc e of events is illustrated in Fig. 2.3.1. Du ri ng loading wit hou t crack

growth , the plastic flow near the crack edge an d the height increase of the process

region causes considerable blunting of the edge, fo rmin g the so-called stretched zone

In so me materials, for instance mild steel, the blunting ma y be visible by the nak ed

eye. A ma ch in e pr od uc ed pre-existing crack ma y be substantially blunted; this ma y

lead to considerab ly increased resistance to onset of crack gro wth.

Blun ting ma y be studied exp erime ntally in different ways. It can be obse rved opti-

cally in a cut nor ma l to the crack edge. Ano th er wa y is to pou r a mo ul d into the crack,

whi ch is wit hdr awn after the mo uld has solidified and the crack has been opened. In

a less direct way, a CM OD- mea sur em ent Crack Mou th Op eni ng Displacement), the

chang e in crack openi ng during loading is dete rmin ed at the crack mou th, generally by

me an s of a clip-gauge; see Fig. 2.3.2. This me th od , whi ch is extensively used in frac-

ture mechanics tests, is know n as COD- dete rmin ati on Crack Opening Displacement).

It is based on so me estimated relation bet wee n crack blunting an d mo uth opening,

ass umi ng that the crack has not grown.


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30 2. THE FRACTU RE PROCESS

i ' I

i l i n i l 9

;

STRAIN

GAUGES

MACHINED CRACK

S OT

Fi g. 2 .3 .2 Cl ip-gauge measurem ent of crack m outh opening.

2 4 O n s e t o f c r a c k g r o w t h

I t i s u s u a l l y v e r y d i f f i c u l t t o d e t e c t w h e n a c r a c k s t a r t s g r o w i n g . E v e n t h e v e r y c o n -

cep t o f inc ip ien t c rack g ro wth i s d i f fi cu lt to de fine . Crack g ro wth o ccurs w hen m ic ro-

s e p a r a t i o n s i n fr o n t o f t h e c r a c k e d g e c o a le s ce w i t h t h e m a i n c r a c k , b u t t h e m i c r o -

s e p a r a t i o n s a r e u n e v e n l y s p a c e d a n d o f d i ff e re n t s iz es . T h e y m a y e v e n b e o f d i f fe r e n t

t y p e s a l o n g t h e c r a c k e d ge . S o m e c o a le s c en c e s w i t h t h e m a i n c r a c k m a y t h e r e f o r e o c c u r

l o n g b e f o re c o a l e sc e n c e s a lo n g t h e m a j o r p a r t o f t h e e d g e . E v e n f r o m a m a c r o s c o p i c

p o i n t o f v i ew , c r a c k g r o w t h m a y o c c u r e a r l y a l o n g s o m e p a r t o f t h e c r a c k e d g e a n d

l a t e r a t o t h e r p a r t s , a s a t th e t h u m b n a i l e ff e ct .

T h e d i f f i c u l t y i n i d e n t i f y i n g i n c i p i e n t c r a c k g r o w t h i s s i m i l a r t o t h a t e n c o u n t e r e d i n

t h e d e t e r m i n a t i o n o f t h e e l a st ic l i m i t f r o m a t e n s il e t e s t . I n t h a t c a s e t h e d i f fi c u lt y

i s re so lved by a con ven t ion , the idea o f which i s to de f ine the y ie ld s t re s s a s the

s t r e ss w h e n a sm a l l , b u t y e t s a fe l y d e t e c t a b l e p e r m a n e n t e l o n g a t i o n ( u s u a l l y 0. 2 % ) h a s

o c c u r r e d . T h e s a m e i d e a , a p p l i e d t o c r a c k g r o w t h , l e a d s to d e f i n i ti o n o f i n c i p i e n t c r a c k

g r o w t h a s th e s t a t e w h e n a s m a l l, b u t y e t s af e ly d e t e c t a b l e a m o u n t o f c r a c k g r o w t h h a s

o c c u r r e d . E x t r a o r d i n a r y a n d s o p h i s t i c a t e d t e c h n i q u e s s h o u l d n o t b e n e e d e d . M e t h o d s

t o d e t e r m i n e t h e a m o u n t o f s t a b l e c r a c k g r o w t h w i ll b e d i s c u ss e d i n th e n e x t s e c ti o n .

A c o n v e n t i o n t h a t a ll ow s u n a m b i g u o u s d e t e r m i n a t i o n o f i n c i p ie n t c r a c k g r o w t h , l ik e

t h e o n e d i s c u s s e d , m a y b e n e e d e d f o r t e s t i n g p u r p o s e s . I n t h e o r e t i c a l t r e a t m e n t s , i t

i s n e v e r t h e l e s s u s u a l l y a s s u m e d t h a t o n s e t o f c r a c k g r o w t h o c c u r s s m o o t h l y a n d s i-

m u l t a n e o u s l y a l o n g t h e w h o l e c r a c k e d g e . S u c h i d e a l i z a t i o n s a r e c o m m o n i n a p p l i e d

mec han ics : in e la s t i c -p la s t i c theor ie s , fo r ins tance , th e s t a t e i s a s su m ed to be com-

p le te ly e la s t i c un t i l the y ie ld cond i t ion i s reached , and homogeneous p la s t i c f low i s

a s s u m e d t o o c c u r i m m e d i a t e l y a f t e r w a r d s .

T h e o n s e t o f c r a c k g r o w t h d e p e n d s o n s e v e ra l f a c to r s : m a t e r i a l p r o p e r t i e s , b o d y

g e o m e t r y ( i n c l u d i n g c r a c k g e o m e t r y ) , l o a d d i s t r i b u t i o n , l o a d m a g n i t u d e a n d e n v i r o n -

m ent a l co nd i t ions . T im e e f fec t s o f ten p lay a pa r t a s a re su l t o f v i s cop las t i c f low in the

p r o c e s s r e g i o n a n d i t s v i ci n it y . I n o t h e r c a s es d if fu s io n o f i m p u r i t y a t o m s t o w a r d s t h e

p r o c e s s r e g i o n m a y c a u s e d e l a y e d o n s e t o f c r a c k g r o w t h a f t e r l o a d a p p l i c a ti o n . T i m e

e f f e c t s w i l l b e m o r e i m p o r t a n t t h e f u r t h e r t h e c r a c k g r o w t h p r o c e s s p r o c e e d s t o w a r d

u n s t a b l e c r a c k g r o w t h .

O n s e t o f s t a b l e c r a c k g r o w t h i s g o v e r n e d b y a

l oca l cond i t ion

d e s c r i b i n g w h e n t h e


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2.5 STABLE CRACK GROWTH 31

) A a

Fig . 2 .5 .1 Rela t ion between s table crack growth, Aa, and outer load, P .

p r o c e s s r e g i o n r e a c h e s a c e r t a i n c r i t i c a l s t a t e . W i t h e x p e r i e n c e f r o m t h e d e v e l o p m e n t

o f t h e p r o c e s s r e g i o n u p t o o n s e t o f c r a c k g r o w t h i n a c e r t a i n m a t e r i a l , a n i m a g i n e d

o b s e r v e r w h o c o u l d o v e r l o o k t h e w h o l e p r o c e s s r e g i o n a n d i t s i m m e d i a t e v i c i n i t y , b u t

n o t n e c e s s a r i l y o t h e r p a r t s o f t h e b o d y , w o u l d b e a b l e t o t e l l w h e n c r a c k g r o w t h

i s ab ou t to occur . In mos t cases o f eng in ee r ing in te re s t , the d eve lo pm ent o f s ize ,

s h a p e a n d d e f o r m a t i o n s o f t h e p r o c e s s r e g i o n w i ll al w a y s b e t h e s a m e i n t h e s a m e

mate r ia l : the obse rve r wi l l no t be ab le to s ee any d i f fe rences , excep t those re la ted to the

n o n r e g u l a r d i s t r i b u t i o n o f m i c r o - s e p a ra t io n s . T h i s i n d e p e n d e n c e o n b o d y a n d l o a d i n g

g e o m e t r y i s w h a t B a r e n b l a t t 1 9 5 9c ) c a l le d autonomy a c o n c e p t t h a t w i l l b e u s e d

f r e q u e n t l y i n t h e p r e s e n t w o r k . I t h a s p l a y e d a d o m i n a t i n g r o l e i n f r a c t u r e m e c h a n i c s ,

a l t h o u g h o f t e n i n t u i t i v e l y t a k e n f o r g r a n t e d r a t h e r t h a n e x p l i c i tl y re c o g n i ze d . I t s h o u ld ,

h o w e v e r , b e r e m a r k e d a l r e a d y h e r e t h a t t h e r e a r e s e v e r a l e x c e p t i o n s t o a u t o n o m y ; t h u s ,

t h e l o a d i n g s t u a t i o n i n p a r t i c u l a r w h e t h e r i t p r o d u c e s c r a c k f ac e o p e n i n g o r s li d in g )

a n d e n v i r o n m e n t a l c o n d i t i o n s h a v e t o b e s p e c i f i e d .

2 5 Stable crack growth

eneral propert ies of stable cra ck growth

E v e n t h o u g h t h e t o t a l a m o u n t o f c r a ck g r o w t h d u r i n g t h e s t a b l e p h a s e o f t en i s m u c h

s m a l l e r t h a n t h e c r a c k l e n g t h , a n d t h e r e f o r e m i g h t b e d i f f i c u l t t o d e t e c t , t h e o u t e r

load gene ra l ly has to be inc reased cons ide rab ly , o f t en by a fac to r o f two or more . F ig .

2 .5 .1 s h o w s a t y p i c a l r e l a t i o n b e t w e e n t h e a m o u n t o f s t a b l e c r a c k g r o w t h a n d t h e o u t e r

load.

I n a r a t e - i n d e p e n d e n t m a t e r i a l , t h e r a t e o f g r o w t h i s d i r e c t l y r e l a t e d to t h e c u r r e n t

ra te o f load ing . I f the load ing dev ice , fo r ins tance the g r ips in a t ens i l e t e s t ing ma ch ine ,

c a n b e m a n i p u l a t e d , t h e n i t is p o s s ib l e t o s to p t h e g r o w t h o r c h a n g e t h e r a t e o f g r o w t h

as des ired . Th is pos s ib l i li ty , even i f i t on ly ex i s t s in p r inc ip le , m ay be t ak en to de f ine

t h e c o n c e p t o f s t a b l e c r a c k g r o w t h . I n a r a t e - d e p e n d e n t m a t e r i a l , t h e c o n c e p t o f s t a b l e

c rack g ro wth i s l e ss we l l de f ined , bu t i s o f t en used i f the con t r ibu t ion to c rack g ro wth

f r o m t h e r a t e - d e p e n d e n c e i s s m a l l .

C o m p a r a t i v e l y s m a l l c h a n g e s o f l o a d or g ri p c a u se s u b s t a n t i a l c r a c k g r o w t h t o w a r d s

the e nd o f the s t ab le phase , e spec ia l ly a t sm a l l sca le y ie ld ing . Viscop las t i c e f fec ts ma y

the n be s ign i f ican t , even i f the y hav e been neg l ig ib le a t ea r l i e r s t ages .


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3 2. THE FRACTURE PROCESS

Fig . 2 . 5 . 2 Exa mp le o f l oa d ing ge om e t r y supp r e s s ing uns t a b l e c ra c k g r owth , t hus c a pa b l e o f

p r oduc ing v i r t ua l l y un l im i t e d s t a b l e c r a c k g r owth .

d t t

F i g 2 5 3

Exa m ple o f l oa d ing ge ome t r y supp r e s s ing s t a b l e c r a c k g r owth , t hus c a pa b l e o f

p r oduc ing uns t a b l e c r a c k g r owth d i r e c t l y f r om a s t a t i ona r y c r a c k .

Th e total am ou nt of stable crack growth dep end s on several factors: material proper-

ties, envir onmen tal conditions, bo dy geo met ry and loading geometry. In fact, virtually

unlimited am ou nt s of stable crack gro wth ma y be obtained solely by choosing a suitable

loading geometry. We dg in g of a plate is an obvious example , althou gh conceptually

so mew hat ambiguou s, because the crack often grows through jumps. Ano ther exam ple

is sh ow n in Fig. 2.5.2, sho wi ng conical crack surfacest. On the other hand , bo dy an d

loading geome tries can b e design ed so as to suppre ss the stable pha se entirely. Fig.

2.5.3 sh ow s the idea of such designs.

In co mm on engineering cases the total am ou nt of stable crack gro wth dep end s very

mu ch on the em be dm en t of the process region in the plastic region. As a result it ma y

be either larger or smaller but sel dom mu ch smaller) than the forward extent of the

t Such conical surfaces were obtaine d in spectacular experim ents by Roesler 1956), using a cylindrical

indentor on a glass block.


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2.5 STABLE CRAC K GROWTH 33

p l a s t i c r e g i o n a t s m a l l s c a l e y i e l d i n g . A n i l l u s t r a t i v e e x a m p l e i s f o u n d f r o m n u m e r i c a l

c a l cu l a ti o n s b y T v e r g a a r d a n d H u t c h i n s o n ( 19 9 2) . A sh a ll o w e m b e d m e n t w a s o b t a in e d

b y c h o o s i n g a l o w r a t i o b e t w e e n c o h e s i v e s t r e n g t h a n d y i e l d s t r e n g t h ; t h i s r e s u l t e d i n

s t a b l e c r a c k g r o w t h a b o u t e q u a l t o t h e f o r w a r d s e x t e n s i o n o f t h e p l a s t i c r e g io n . O n

t h e o t h e r h a n d , a d e e p e m b e d m e n t , o b t a i n e d b y c h o o s in g a h i g h e r r a t io , r e s u l t e d i n

s t a b l e g r o w t h a m o u n t i n g t o s e v e r a l t i m e s t h e f o r w a r d s e x t e n s i o n o f t h e p l a s t i c r e g io n .

S t a b l e c r a c k g r o w t h i s g o v e r n e d b y l o c a l c o n d i t i o n s a t t h e p r o c e s s r e g i o n , b u t t h e

onse t o f uns tab le g ro wth fo llows a s a re su l t o f a g loba l ins tab i l i ty . Th i s wi l l be d i scus sed

i n S e c t i o n 2 . 6 . S t o p p i n g s t a b l e c r a c k g r o w t h a t s o m e i n s t a n t f o r e x a m i n a t i o n , r e v e a l s

t h a t t h e t h e b l u n t i n g d o e s n o t a c c o m p a n y t h e m o v i n g e d g e ; t h i s i s a g a i n s h a r p a p a r t

f rom the h e igh t ex tens ion o f the p roces s reg ion ; s ee F ig . 2 .1.1 . Th i s red uc t io n o f

b l u n t i n g s e e m s t o i m p l y t h a t t h e r a t i o b e t w e e n h y d r o s t a t i c a n d d e v i a t o r i c s t r a i n s a n d

s t re s s e s i n th e p r o c e s s r e g i o n v i c i n it y i n c r ea s e s a f t e r s o m e a m o u n t o f c r a c k g r o w t h ,

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

easurement of the amount of crack growth

M e a s u r i n g t h e a m o u n t o f s t a b l e c r a c k g r o w t h c a u s e s i n g e n e r a l g r e a t e x p e r i m e n t a l

p r o b l e m s . T h e r e a r e , h o w e v e r , e x c e p t i o n s . I n t h i n p l a t e s , s t a b l e c r a c k g r o w t h m a y

a m o u n t t o s e v e r a l t i m e s t h e p l a t e t h i c k n e s s , a n d i t m a y t h e n e a s i l y b e o b s e r v e d o n

t h e p l a t e s u r f a c e s ( s e e e . g . B r o c k 1 9 6 8 ) . S o m e u n c e r t a i n t y m a y p r e v a i l d u e t o t h e

p o s s i b il i ty o f a t h u m b n a i l e ff ec t. I t w a s o n c e d e s c r ib e d t o t h e a u t h o r t h a t a f ai le d

s h i p - p l a t e s h o w e d a c r a c k t h a t c o n t i n u e d i n si d e t h e p l a t e a l o n g a d i s t a n c e o f a b o u t

I m , w i t h o u t p e n e t r a t i n g t h e s u r fa c e s. T h i s p a r t o f t h e c r a c k w a s d e t e c t e d b e c a u s e t h e

n e c k i n g o f t h e r e m a i n i n g l i g a m e n t s w a s v is i bl e o n t h e p l a t e s u r fa c e s. T h i s p a r t i c u l a r

c a s e w a s c e r t a i n l y a s s o c i a t e d w i t h u n s t a b l e a n d r a t h e r f a s t c r a c k g r o w t h , b u t t h e

o b s e r v a t i o n s h o w s t h a t c o n c l u si o n s d r a w n f r o m c r a c k s v i si b le o n t h e s u r f a c e o f a b o d y

m a y b e m i s l e a d i n g .

C o n t i n u o u s r e c o r d i n g o f c r a c k g r o w t h m a y b e m a d e b y m e a n s o f a c o u s t i c a l o r e le c-

t r i c a l i m p e d a n c e m e t h o d s . S u c h m e t h o d s d o n o t a l w a y s g i v e a s a t i s f a c t o r y a c c u r a c y ,

b u t t h e r e h a s b e e n s u b s t a n t i a l h o p e t h a t r e c o r d i n g o f

a c o u s t i c e m i s s i o n

on eng inee r -

i n g s t r u c t u r e s c o u l d g i v e a w a r n i n g a b o u t c r a c k g r o w t h t h a t m i g h t e l u d e i n s p e c t i o n s .

S i m i la r ly , t r e m o r s i n t h e e a r t h s c r u s t m a y g iv e a w a r n i n g a b o u t a n i m p e n d i n g e a r t h -

quake .

I n t e r m i t t e n t m e a s u r e m e n t s m a y b e c a r r ie d o u t i n a n u m b e r o f d i ff e re n t w a y s on l ab -

o r a t o r y t e s t s p e c i m e n s . O n e s i m p l e w a y is t o m a k e c o m p l i a n c e m e a s u r e m e n t s the stiff-

n e s s o f t h e s p e c i m e n d e c r e a s e s w i t h i n c r e a s i n g c r a c k l e n g t h . T h e s p e c i m e n i s u n l o a d e d ,

whereby the e la s t i c s t i f fnes s , o r i t s inve rse , the compl iance , i s recorded . Seve ra l de te r -

m i n a t i o n s , b y c y cl e s o f u n l o a d i n g , r e l o a d i n g a n d f u r t h e r l o a d in g , m a y b e p e r f o r m e d

f o r t h i s p u r p o s e d u r i n g a t e s t w h i c h o t h e r w i s e i s d e s i g n e d f o r m o n o t o n e l o a d i n g ; s e e

F ig . 2 .5 .4 . Th i s me thod i s f requen t ly used and i s fa i r ly accura te fo r deep ly c racked

s p e c i m e n s . I t is n o t a l w a y s r e m e m b e r e d , h o w e v e r, t h a t e v e n a s m a l l n u m b e r o f l o a d

cyc le s migh t s ign i f i can t ly in f luence the g rowth p roces s by low-cyc le fa t igue ac t ion ; c f .

p a g e s 6 1 7 f f . A c a l i b r a t i o n p r o c e d u r e m a y u s e a m a c h i n e d s l o t , r a t h e r t h a n t h e s h a r p

c r a c k u s u a l l y r e q u i r e d f o r t e s t s p e c i m e n s . I n t h i s w a y , c r ac k g r o w t h i s s u p p r e s s e d , a n d

b y u s i n g s l o ts o f d i ff e r en t l e n g t h s c a l i b r a t i o n c u r v e s c a n b e p r o d u c e d .

A v e r y a c c u r a t e m e t h o d o f d e t e r m i n i n g c r a ck g r o w t h u n d e r l a b o r a t o r y c o n d it io n s ,


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34 2. THE FRACTUR E PROCESS

Fig . 2 .5 .4 Load-displacement P-5) record, obta ined by frequent unloading and re loading.

Th e chan ge of s lope in elastic regions g ives a me asure of the com pliance changes du ring

loading.

ST BLE

CRACK ~

GROWTHL

i ' i i . , i : i l i ,i i i i i :i : ; i

FIN L FR CT UR E

~ D Y E M A R K S

F TIGUE

C R C K

M C H I N E D S LO T

Fig . 2 .5 .5 A dye may be used during labora tory tes t ing to mark the pos i t ion of the crack

edge at different s tages of the stable phase.

c o n s i s ts o f i m m e r s i n g t h e c r a c k w i t h a d y e r e p e a t e d l y d u r i n g l o a d i n g B r o b e r g 1 9 7 5,

T r y d i n g 1 9 9 1 ) . T h e s p e c i m e n s h o u l d b e o r i e n t e d w i t h t h e c r a c k m o u t h p o i n t i n g u p -

w a r d s . T h e l o a d i n g i s k e p t c o n s t a n t f o r a w h i l e u n t i l t h e d y e h a s d r i e d . B e c a u s e o f

g r a v i ty , t h e d y e c o n c e n t r a t e s n e a r t h e c r a c k ed g e , a n d t h e a m o u n t o f c r a c k g r o w t h c a n

b e s e e n o n t h e f r a c t u r e s u r f a c e s a f t e r c o m p l e t e d t e s t f r o m t h e d y e m a r k i n g s ; s e e F i g .

2 .5 .5 . A n a d v a n t a g e w i t h t h e m e t h o d i s t h a t n o t o n l y t h e p o s i t i o n b u t a ls o t h e s h a p e o f

t h e c r a c k e d g e a t i n te r v a ls , m a y b e d e t e r m i n e d . A n a l t e r n a t i v e , b u t s o m e w h a t b r u t a l

m e t h o d , i s t o o x i d i z e t h e c r a c k s u r f a c e s b y h e a t i n g t h e c r a c k r e g i o n , f o r i n s t a n c e b y

m e a n s o f a n a c e t y l e n e f l a m e .


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2 6 ONSET OF UNSTABLE CRACK GROWTH 35

Fig . 2 .6 .1 Load-disp lacement record , showing sudden, bu t ra ther sm al l crack advances.

2 6 On set of unstable crack growth

A c e n t r a l q u e s t i o n i n f r a c t u r e m e c h a n i c s c o n c e r n s t h e c i r c u m s t a n c e s l e a d i n g t o u n -

s t a b l e c r a c k g r o w t h , w h i c h is u s u a l l y c o n s i d e r e d a s s y n o n y m o u s w i t h f r a c t u r e . I t is

t h e r e s u l t o f g l o b l i n s t b i l i t y B r o b e r g 1 9 7 3 a) . T h i s i m p l i e s i n v o l v e m e n t o f t h e w h o l e

b o d y , i n c l u d i n g t h e l o a d i n g d e v i c e . T h i s s i t u a t i o n i s w e l l - k n o w n f r o m t h e f i n a l f r a c -

t u r i n g o f a t e n s il e t e s t p i e c e , w h i c h o c c u r s s o o n e r i n a s o f t t h a n i n a s ti f f m a c h i n e , a n d

a l so sooner fo r a l ong t es tp i ece than fo r a sho r t . Fo r a c rack in a l a rge p l a t e , uns t ab le

crack g ro w th oc cu rs a t l a rge sca l e y i e ld ing i .e ., fo r a r e l a t ive ly sm al l c rack ) befo re the

P - A a cu rve , F ig . 2.5 .1 , i s l eve l li ng ou t , bu t a t smal l sca le y i e ld ing w h en the cu rve

is level l ing out .

T h e c o n t r i b u t i o n t o s t a b l e c r a c k g r o w t h b y v i s c o p l a s t i c f l o w m a y b e c o m e s i g n i f i c a n t

t o w a r d s t h e e n d o f t h e s t a b l e p h a s e , a s o ft e n e x p e r i e n c e d f o r t h i n s h e e t s , f o r i n s t a n c e

a lu m in iu m o r ce l lu lo id shee t s . I t imp l i es a con cep tua l d i f f icu l ty , bec ause o nse t o f uns t a -

b l e c r a c k g r o w t h i s n o t c o n t r o l l e d s o l e l y b y t h e c u r r e n t l o a d i n g s i t u a t i o n , b u t a l s o b y

v i s c o p l a s ti c i ty . F r o m a p r a c t i c a l p o i n t o f v i ew , u n s t a b l e c r a c k g r o w t h i s a s u n w a n t e d

i f caus ed b y v i scop las t i c f low as i f by load increase . In t es t s , am p le t im e sh ou ld b e

a l lo w e d f o r v i s c o p l a s t i c flo w. I n s p e c t i o n o f t h e l o a d - d i s p l a c e m e n t r e l a t i o n d u r i n g t h e

t e s t w i l l g e n e r a l l y g iv e i n f o r m a t i o n a b o u t w h e t h e r t h e r a t e o f l o a d i n g i s s u f fi c ie n t ly

lOW

A s w el l as v i scop las t i c f low , o ther t ime-ef fec t s may be p resen t , fo r i n s t ance d i f fu s ion

o f i m p u r i t y a t o m s t o w a r d t h e c r a c k e d g e. S u c h e ff e ct s m i g h t b e v e r y s t r o n g . A n e x c e s s

o f h y d r o g e n i n s te e l m a y c a u s e f r a c t u r e a t c o m p a r a t i v e l y s m a l l lo a d s ; s ee e. g. D i e t e r

1 9 8 8 ) . W a t e r v a p o u r m a y c a u s e f r a c t u r e i n g l a s s a f t e r a f e w w e e k s a t a l o a d w h i c h i s

o n l y h a l f t h e l o a d n e e d e d t o c a u s e f r a c t u r e a f t e r s o m e m i n u t e s ; s e e e .g . S h a n d 1 9 5 8) .

S e v e r a l o t h e r m a t e r i a l c o m b i n a t i o n s l e a d t o e n v i r o n m e n t a l l y a s s i s t e d c r a c k i n g ; c f . p a g e

9.

G e n e r a l l y , a s t r u c t u r a l p a r t o r t e s t p i e ce i s r a p i d l y f u l l y b r o k e n b y a f a st r u n n i n g f ra c -

t u r e , a c c o m p a n i e d b y a s h a r p s o u n d i m m e d i a t e l y a f t e r o n s e t o f u n s t a b l e c r a c k g ro w t h .

S o m e t i m e s , h o w e v e r , s o m e d o u b t m a y a r i s e a s t o w h e t h e r u n s t a b l e c r a c k g r o w t h h a s

o c c u r r e d o r n o t . I n t h e s e c a s e s , a s u d d e n , t h o u g h v e r y s m a l l , c r a c k g r o w t h o c c u r s ,

w h e r e u p o n s t a b l e c r a c k g r o w t h c o n t i n u e s d u r i n g f u r t h e r l o a d in g , u n t il a n u n d i s p u t a b l e

f r a c t u r e o c c u r s . S u c h e v e n t s a p p e a r a s s t e p s i n a l o a d v e r s u s d i s p l a c e m e n t r e c o r d ; s e e

Fig. 2.6.1.

T h e c a u s e o f s u c h s u d d e n c r a c k g r o w t h s t e p s m a y b e o f d i f fe r e n t k i n d s . I n s o m e c a s es ,


10/18

6 2 T H E F R A C T U R E P R O C E S S

t h e o r i g i n c a n b e t r a c e d t o i n h o m o g e n e o u s m a t e r i a l . O n e e x a m p l e i s w e l d e d p l a t e s ,

w h e r e d i f f e r e n t r e g i o n s p o s s e s s d i f f e r e n t p r o p e r t i e s . U n s t a b l e c r a c k g r o w t h m a y b e

i n i t i a t e d i n t h e h e a t a f f e c t e d z o n e b u t b e a r r e s t e d i n t h e b a s e m a t e r i a l . I n o t h e r c a s e s

t h e b o d y s h a p e , o r t h e l o a d d i s t r i b u t i o n , i s s u c h t h a t a c r a c k i s s o o n a r r e s t e d a f t e r o n s e t

o f u n s t a b l e g r o w t h . O n e e x a m p l e i s c r a c k g r o w t h d u e t o t h e r m a l s t r e s s e s i n a p l a t e

o r a h o s e : s u c h s t r e s s e s m i g h t v a r y f r o m t e n s i l e t o c o m p r e s s i v e a l o n g t h e p r o s p e c t i v e

c r a c k p a t h . F r e q u e n t l y t h e e x p l a n a t i o n t o a p e r c e i v e d s u d d e n c r a c k g r o w t h s t e p i s d u e

t o a s o - c a l l e d

p o p i n

p h e n o m e n o n , o c c u r r i n g o n l y i n t h e p l a t e i n t e r i o r . T h u s , i t i s a

t h u m b n a i l e f f e c t . T h e n e c k i n g r e g i o n s n e a r t h e p l a t e s u r f a c e s p r o v i d e h i g h e r r e s i s t a n c e

t o c r a c k g r o w t h , s o t h a t f u r t h e r i n c r e a s e o f t h e l o a d i s n e e d e d t o p r o d u c e u n s t a b l e

g r o w t h o v e r t h e w h o l e c r o s s - s e c t i o n o f t h e p l a t e .

2 7 Un s tab le c rack grow th

Gen eral considerations

A f t e r o n s e t o f u n s t a b l e c r a c k g r o w t h , t h e c r a c k e d g e g e n e r a l l y a c c e l e r a t e s t o a v e r y h i g h

v e l o c i t y , o f t e n s e v e r a l h u n d r e d m e t e r s p e r s e c o n d , a n d s o m e t i m e s t o a f e w t h o u s a n d

m e t e r s p e r s e c o n d . T h e e n e r g y r e q u i r e d f o r c o n v e r s i o n f r o m a s t a t i c t o a d y n a m i c s t a t e

o f t h e s t r u c t u r e i s p r o v i d e d b y s t r e s s - s t r a i n e n e r g y t r e l e a s e f r o m t h e b o d y , s o m e t i m e s

a s s i s t e d b y e n e r g y s u p p l y f r o m t h e l o a d i n g d e v i c e . A t g r i p c o n t r o l , t h e s o u r c e i s s t r e s s -

s t r a i n e n e r g y f r o m t h e b o d y , o n l y . A t l o a d c o n t r o l , u n s t a b l e c r a c k g r o w t h o c c u r s w h e n

t h e m a x i m u m l o a d i s r e a c h e d , i . e . a s s o o n a s t h e p r e c e d i n g s t a b l e g r o w t h t e n d s t o

o c c u r u n d e r c o n s t a n t l o a d . T h e s t r e s s - s t r a i n e n e r g y r e l e a s e d , t o g e t h e r w i t h t h e e n e r g y

s u p p l i e d f r o m t h e o u t e r l o a d g o e s t o k i n e t i c e n e r g y a n d t o w h a t i s r e q u i r e d b y t h e

d i s s i p a t i v e r e g i o n a t t h e c r a c k e d g e t o s u s t a i n c r a c k g r o w t h .

T h e g e n e r a l t h e o r e t i c a l u p p e r l i m i t o f c r a c k p r o p a g a t i o n v e l o c i t y i s t h e R a y l e i g h

w a v e v e l o c i t y f o r m o d e I c r a c k s B a r e n b l a t t a n d C h e r e p a n o v 1 9 6 0 , B r o b e r g 1 9 6 0 ,

C r a g g s 1 9 6 0 ) , t h e P w a v e v e l o c i t y f o r m o d e I I c r a c k s F r e u n d 1 9 7 9 , B u r r i d g e

e t a l .

1 9 7 9 , B r o b e r g 1 9 8 9 a ) , a n d t h e S w a v e v e l o c i t y f o r m o d e I I I c r a c k s . F o r m o d e I I c r a c k

p r o p a g a t i o n , v e l o c i t i e s b e t w e e n t h e R a y l e i g h a n d t h e S w a v e v e l o c i t i e s a r e t h e o r e t i c a l l y

i m p o s s i b l e . T h e a r g u m e n t s f o r t h e o r e t i c a l l y p o s s i b l e a n d i m p o s s i b l e r e g i o n s w i l l b e

d i s c u s s e d i n S e c t i o n 6 . 2 . T h e g e n e r a l t h e o r e t i c a l u p p e r l i m i t s a r e h a r d l y r e a c h e d i n

r e a l i t y . F o r m o d e I c r a c k s , v e l o c i t i e s o f a b o u t 7 0 p e r c e n t o f t h e R a y l e i g h w a v e v e l o c i t y

h a v e b e e n o b s e r v e d S c h a r d i n 1 9 5 9 ) , b u t h a r d l y m o r e . F o r e a c h m a t e r i a l , a n u p p e r

l i m i t s e e m s t o e x i s t . T h e r e m i g h t b e s e v e r a l r e a s o n s f o r t h i s l i m i t a t i o n . O n e r e a s o n i s

t h e i n c r e a s i n g e n e r g y d e m a n d o f t h e p r o c e s s r e g i o n i n t h e h i g h v e l o c i t y r a n g e . A n o t h e r

i s o n s e t o f b r a n c h i n g . T h e s e p h e n o m e n a w i l l b e d i s c u s s e d i n S e c t i o n s 9 . 1 a n d 9 . 2 .

V e r y o f t e n , t h e m a x i m u m c r a c k e d g e v e l o c i t y r e a c h e d d e p e n d s o n l i m i t a t i o n s o f

t h e e n e r g y f l o w t o t h e c r a c k e d g e r e g i o n r a t h e r t h a n o n t h e m a t e r i a l . F o r t h e l o n g

s t r i p c o n f i g u r a t i o n s e e F i g . 2 . 7 . 1 ) u n d e r g r i p c o n t r o l l e d c o n d i t i o n s , t h e e n e r g y r e l e a s e

f r o m t h e s t r e s s - s t r a i n f i e l d p e r u n i t o f c r a c k g r o w t h i s l i m i t e d t o s o m e m a x i m u m v a l u e ,

g i v e n b y t h e i m p o s e d g r i p d i s p l a c e m e n t . T h i s m a y b e s h o w n b y m e a n s o f a v e r y s i m p l e

a r g u m e n t , c f . S e c t i o n 6 . 7 .

t T h i s e n e r g y i s a l s o c a l l e d s t r a i n e n e r g y o r s t o r e d e n e r g y .


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2.7 UNSTABLE CRACK GROWTH 37

51

~

w

Fig . 2.7.1 Crack growth in a long str ip under grip control . Because no energy disappears

through the str ip edges, and the only dissipat ive region is that at the crack edge, the energy

flow to the crack edge per unit of crack advance mu st equal the energy accum ulated in the

strip per unit of length before introduction of the crack.

In t he low ve loc i ty r eg ion , t he energy demand fo r t he d i ss ipa t ive r eg ions may de-

crease somewhat wi th i ncreas ing crack ve loc i ty i n some mater i a l s , bu t i n o ther mate-

r i a l s i t i ncreases , t hough a t a much lower r a t e t han a t h igh ve loc i t i es . The d i f f e rences

be tween d i f f e ren t mater i a l s i n t he l ow ve loc i ty r eg ion may be r e l a t ed to mater i a l be-

hav iour known f rom rap id t ens i l e t es t s : t he energy d i ss ipa t ion decreases i n t he l ow

veloc i ty r eg ion wi th i ncreas ing load ing ve loc i ty i n some mater i a l s bu t i ncreases i n

o ther (Wih lborg 1976) . Th i s wi l l be fu r ther d i scussed in Sec t ion 9 .1 .

A s imple cons idera t ion shows tha t v i scop las t i c f l ow qu i t e obv ious ly p l ays an impor -

t an t par t i n r ap id c rack g rowth . Suppose , fo r i n s t ance , t ha t t he l i near d imens ions o f

the p l as t i c r eg ion are o f t he o rder o f i mm and tha t t he c rack edge ve loc i ty is of t he

order o f I m / s . A ma ter i a l po in t c lose t o t he c rack p a th is t hen be ing swep t over by

the p l as t i c r eg ion dur ing on ly i ms . Due to de l ay o f t he onse t o f p l as t i c f low (Clark an d

W ood 1949) and to v i scop las ti c i ty , a t ime dur a t ion o f t h i s ma gn i tu de wi ll genera l ly

impede p l as t i c f l ow cons iderab ly .

rack growth direction

A crack in an eng ineer ing s t ruc tu re wi l l genera l ly g row in a p l ane normal t o t he

m ax i m u m p r i n c i p a l s t r e s s ( t ac i t l y u n d e r s t o o d a s t h e s t r e s s a t t h e c r ack s i t e i n t h e

absence o f t he c rack) . I f t he o r ig ina l c rack o r i en t a t ion does no t co inc ide wi th such

a p lane , a sm oo th a d ju s tm en t occurs a f t e r i n i t ia l k ink ing , i. e. a change o f d i r ec t ion ,

which i s ab rup t a t smal l sca le o f y i e ld ing ; see F ig . 2 .7.2. Th ere a re some exc ep t ions

to th i s genera l ru l e. C racks in an i so t rop ic m ater i a l s t end to fo llow p lanes o f ma ter i a l

sy m m et r y r a the r t h an load ing symm et ry . In a reg ion o f i n t ense p l as t ic f low, a c rack

ma y g row und er i n -p l ane shear as , fo r i n s t ance , i n a cup and cone f r ac tu re o f a

t ens i le t es t p i ece . In t he p rev ious ch ap ter , i t was men t ioned tha t a band o f neck ing in

a th in shee t may ou t l i ne t he pa th fo r a c rack , and th i s pa th i s o f t en no t no rmal t o t he

overa l l max imum pr inc ipa l s t r ess , c f . page 14 .

Even a t i so t ro py and smal l scal e y i e ld ing , t he c rack m ay g row a t app rox im ate ly 45 ~

ang le t o t he m ax im um pr inc ipa l s t ress , i .e ., i n a d i r ec t ion o f p r inc ipa l shea r s t r ess.

Th i s i s ex t remely se ldom exper i enced in eng ineer ing s t ruc tu res , excep t i n some cases

of h igh-cyc le f a ti gue , bu t i t app ear s as a ru l e in ear thquak es . Th e reas on seems to


12/18

38 2. THE FRACTURE PROCESS

Fi g. 2.7. 2 Cracks in engineering struc tures ten d to grow in a plane normal to the ma ximu m

principal stress. If the crack is not originally oriented so a smooth a djust ment towards this

direction occurs after initial kinking.

Fig. 2.7.3 Examples of branching geometries.

be that the open ing mo de is suppressed by a high confining pressure (Ne mat -Na sse r

and Horii 1982, Horii and Nemat -Na sse r 1985,1986, Melin 1986, Nem at- Nas ser 1995).

By simulating such loading conditions at laboratory tests, crack propagatio n in the

direction of in-plane shear ma y be prov oked even in PM MA (Brob erg 1987b).

ranching

Bra nch ing of a propag ating crack generally occurs only at high crack edge veloci-

ties, mo st often larger tha n one fifth of the Ray leig h wa ve velocity. In materials like

PM MA , branc hing generally does not occur until abou t 7 0 of the Rayleig h velocity is

reached (Pa xson and Lucas 1973). The p hen ome non appears to be associated mainly

with mo de I crack growth, and it seem s to be the maj or factor that sets an upp er limit

to the ma xi mu m attainable crack edge velocity in ma ny materials. In mos t cases, two

symm etri cal crack branche s are created, but som eti mes there is asym metr ical branch-

ing and also branc hing with three or mo re branches. Fig. 2.7.3 show s exa mpl es of

branc hing geometries. Ver y often, multiple branch ing occurs, i.e. each bra nch encoun -

ters branc hing and so on. This creates som eti mes very regular geometrical patterns;

see Fig. 2.7.4. Br an ch in g will be further d iscussed in Section 9.2.


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2.8 THE PLA STIC REGION AND ITS WAKE 39

Fig. 2.7.4 Multiple branching.

2 8 Th e plastic region and its wake

eneral considerations

In cont ras t to the process reg ion , the p las t ic reg ion ma y usua l ly be adequ a te ly an a lysed

by cont inuu m mechanics . Thu s , the concept s of s t ress and s t ra in ma y be used , bu t the

occur rence of ve ry l arge s t ra ins in the inn ermo s t p a r t s o f the p las t i c reg ion jus t ou t s ide

the process reg ion) might necess i t a t e l a rge s t ra in theor ies and a l so cons ide ra t ions of

void form at ion and growth , a s in the Gurso n mode l Gurso n 1977, Tv erg aard 1981,

1982b, Tvergaard and Needleman 1988) . I t might appear cont rad ic tory to cons ide r

void growth in a cont inu um theory , bu t th i s can be done i f the n um ber of vo ids is

l a r ge i n e a c h vo l ume s ub j e c t e d t o a pp r ox i ma t e l y un i f o r m bounda r y c ond i t i ons .

Th e mo s t s t ra ined pa r t s o f the p las t i c reg ion may som et imes be seen d i rec t ly wi tho ut

any spec ia l p recaut ions , a s a roughness on the body sur face or , in some t ransparent

mater ia ls , l ike cel luloid, as an opaque region. Certa in techniques , for ins tance e tching,

m ay be used to impro ve or develop the vis ibi l i ty of plas t ic f low. Plas t ic s t ra ins m ay

also be determined by us ing gr ids , as shown in Fig. 2 .8.1, or moir~ methods .

The plastic region before unstable crack growth

Th e size of the plastic region at onset of unstable crack growth me as ur ed for instance

as the largest linear exte nsio n fr om the crack edge increases durin g loa ding until

unstable crack grow th occurs. Fig. 2.8.2 shows schematically ho w the size of the

plastic region at the e dge of a crack in a large plate can var y at onset of unst able

crack growth de pe nd in g on the crack length. It is smallest at the largest crack lengths

an d largest at the smallest crack lengths.

As indicat ed in Fig. 2.8.2 not only the size of the plastic region but also its sh ap e

is ap pr ox im at el y the sa me for different crack lengths if these are larger th an a certain

mi ni mu m value. The n au to no my prevails not only for the process region but also for

the plastic region i.e. it holds for the w ho le dissipative region at the cr ack edge. T his

is a case of small scale yielding but the co nven tion al definition of small scale yielding

an AS TM condition to be discussed on page 581 is generous en oug h to include cases


14/18

4 2 THE FRAC TURE PROCESS

C R C K

Fi g. 2.8 .1 Plas t ic strains in a celluloid sheet , revealed by a grid, applied on the shee t surfaces

before loading. Sketched from Berg kvist and Ande rsson 1972)

I I

I l l

t l l

Fig . 2 .8 .2 Exam ples of how the s ize of the p las t ic reg ion just before onset of unstab le crack

growth can va ry wi th the length of a crack in a large p late .

f o r w h i c h t h e s iz e a n d s h a p e o f t h e p l a s t i c r e g io n m a y v a r y ta n g i b l y L a r s s o n a n d

Car l s son 1973) .

Cases in w h ich the p l as t i c r eg ion i s t oo l a rge to qua l i fy as smal l sca l e y i e ld ing , a re

r e f e r r e d t o a s large scale y ie lding For c l a r i ty , how ever , t he concep t ne t sec t ion p las t ic

f l o w w i l l b e u s e d w h e n a p l a s t i c r e g i o n t r a v e r s e s a l i g a m e n t f r o m t h e p r o c e s s r e g i o n

t o a n o u t e r b o u n d a r y o f t h e b o d y .

B e f o r e c r a c k g r o w t h , t h e s t r a i n s i n t h e p l a s t i c r e g i o n a t s m a l l s c a le y i e l d i n g d e c r e a s e

a p p r o x i m a t e l y a s t h e i n v e r s e d i s t a n c e t o t h e c r a c k e d g e , e x c e p t n e a r t h e p r o c e s s r e g i o n .


15/18


16/18

42 2 . THE FRA CTU RE PROCESS

NE KING

NE KING

F ig . 2 .8 .5 N eck in g f rom a c rack ed g e in a th in sh ee t . I f t h e l i g amen t b e tw een th e c rack edg e

and the sheet boundary i s shor t enough , sp l i t neck ing occurs .

b o u n d a r y o f t h e p l a s t i c re g i o n m a y b e s i t u a t e d i n t h e p l a n e s t r e ss r e g i o n t . T h e s h a p e

o f s u c h a r e g i o n a t s m a l l s c a l e y i e l d i n g i s r a t h e r r o u n d e d , a s a l so s h o w n i n F i g . 2 .8 . 1.

H o w e v e r , e v e n in t h i n s h e e t s , t h e p l a s t i c r e g io n m a y b e r o u n d e d . B e r g k v i s t a n d A n -

d e r s s o n ( 19 7 2) p e r f o r m e d e x p e r i m e n t s o n P V C a n d c e l lu lo i d. T e n s i le t e s t s r e v e a l e d

a l m o s t p e r f e c t l y p l a s t ic b e h a v i o u r f or b o t h m a t e r i a l s , b u t w i t h a c o n s i d e r a b le d i f fe r-

e n c e i n u l t i m a t e s t r a i n : 3. 5 f o r a P V C s h e e t a n d 2 5 f o r a c e l l u lo i d s h e e t . A t t h e

e d g e o f a c r a c k i n t h e P V C s h e e t a n e c k i n g r e g i o n d e v e l o p e d , b u t i n t h e c e l l u l o i d s h e e t

a r o u n d e d p l a s t i c r e g i o n w a s f o r m e d .

V e r y p e c u l i a r d i s t r i b u t i o n s o f l a r g e s tr a i n s h a v e s o m e t i m e s b e e n o b s e r v e d i n p l a t e s

( B e r g k v i s t a n d A n d e r s s o n 1 97 2) ; se e F i g . 2 .8 .1 . T h e y s e e m t o b e c a u s e d b y i n t e r a c t i o n

b e t w e e n p l a n e s t r a i n a n d p l a n e s t r e s s p l a s t i c r e g i o n s i n c a s e s w h e n t h e i n n e r p a r t o f

t h e p l a s t i c r e g i o n i s s u b j e c t e d t o p l a n e s t r a i n a n d t h e o u t e r p a r t s t o p l a n e s t r e s s. T h e

i n t e r f e re n c e m a y c a u s e a w a v y p a t t e r n o f p l a s t ic f lo w ( B e r g k v i s t a n d A n d e r s s o n 1 97 2 ).

T h e p l a n e s t r a i n r e g i o n i s s t i f f e r t h a n t h e p l a n e s t r e s s r e g i o n , b e c a u s e d e f o r m a t i o n s

o c c u r u n d e r a p p r o x i m a t e l y e q u a l i n - p l a n e p r i n c i p a l s t r e s s e s . I t c a n t h e r e f o r e a c t a s

a h i n g e w h e n t h e c r a c k is b e i n g o p e n e d . I n f a c t, c o m p r e s s i v e i n - p l a n e s t r a i n s h a v e

b e e n o b s e r v e d n e a r a c r a c k e d g e ( B e r g k v i s t a n d A n d e r s s o n 1 9 7 2) ; s e e F i g . 2. 8 .1 . T h i s

i n d i c a t e s t h e p o s s i b l e e x i s t e n c e o f a s m a l l e l a s t ic e n c l a v e i n s i d e t h e p l a s t i c r e g i o n .

T h e d i r e c t i o n s o f p r i n c i p a l s t r a i n v a r y l a r g e l y a r o u n d t h e c r a c k e d g e . S o m e r o t a t i o n s

o c c u r d u r i n g l o a d in g , b u t t h e y a r e g e n e r a l l y n o t v e r y s ig n i fi c a n t b e f o re c r a c k g r o w t h ,

a n d t h e r e f o r e a p p r o x i m a t e p r o p o r t i o n a l l o a d i n g p r e v a i l s . T h i s i m p l i e s t h a t a t o t a l

s t r a i n t h e o r y m a y g iv e r e a s o n a b l e e s t i m a t e s , a n d i n a f ew c as e s a n e x a c t d e s c r i p t i o n ,

o f t h e s t r e s s - s t r a i n d i s t r i b u t i o n i n t h e p l a s t i c r e g i o n .

I n t h e i n - p l a n e s h e a r m o d e , t h e s h a p e o f t h e s m a l l s ca l e y i e l d i n g p l a s t i c r e g i o n

d e p e n d s o n t h e s t r a i n h a r d e n i n g a n d o n t h e e l a s t i c c o n s t a n t s . O n e p o s s i b l e e x a m p l e

i s s h o w n i n F i g . 2 .8 .6 . T h e r e a r e m a i n l y t e n s i l e s t r e s s e s o n o n e s i d e o f t h e c r a c k p l a n e

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

h a v e t h e s a m e s i z e a n d s h a p e , b e c a u s e t h e p l a s t i c f l o w p r o p e r t i e s m a y b e d i f f e r e n t f o r

c o m p r e s s i v e a n d t e n s i l e s t r e s s e s .

T h e p l a s t i c r e g i o n a t a n t i - p l a n e s h e a r m a y b e a n a l y t i c a l l y d e t e r m i n e d i n a n u m b e r

o f c a s e s . I n t h e s i m p l e s t c a s e , s m a l l s c a l e y i e l d i n g a n d p e r f e c t p l a s t i c i t y , t h e p l a s t i c

r e g i o n i s c i r c u l a r ( H u l t a n d M c C l i n t o c k 1 9 5 6 ) ; s e e F i g . 2 . 8 . 7 . I t i s a l s o c i r c u l a r a t s m a l l

s c a l e y i e l d i n g a n d p o w e r - l a w s t r a i n h a r d e n i n g ( R i c e 1 9 6 7 ) .

t Plane stress conditions dominate at distances from the crack edg e tha t are long er tha n ab out half

the plate thickness; see page 205.


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2.8 THE PLASTIC REGION AND ITS WAK E 43

Fi g . 2 .8 . 6 The s hape o f the sm al l sca le y ie ld ing p las t ic reg ion unde r in -p lane shear ing i s no t

n ecessa r i ly sy m m et r i c w i th r e sp ec t t o th e c rack p l an e . P lu s an d m in u s s ig n s in d ica t e t en s i l e

and compress ive s t ress s ta tes respect ively .

F ig . 2 .8 .7 Und er an t i -p lane s t ra in the smal l sca le y ie ld ing p las t ic reg ion is c i rcu lar . Arrows

p o in t in g to w ard th e r ead e r a re sy m b o l i zed b y Q an d a r ro w s p o in t in g aw ay a re sy m b o l i zed

by |

he plastic region during crack growth

During crack growth, rotations of the directions of principal strains occur in the plastic

region, necessitating an incremental strain theory. Dur ing the stable phase, the size

of the plastic region increases, due to increased loading of the body ; see Fig. 2.1.1.

Th e sha pe of a small scale yielding plastic region does not c han ge significantly during

this phase, except in cases when it gro ws into the plane stress region during stable

crack growth, after havi ng bee n develop ed inside the plane strain region. Th e strain

gradients are smaller than before crack growth; see Sections 5.6-5.7.

Th e shape and size of a small scale yielding plastic region chang e durin g unstable

crack growth. Th e situation is highly complex. Th e gro wth of the process region with

increasing crack velocity, discussed in the preceding chapter, tends to increase the size

of the plastic region. Th e opposite trend is associated with rate depende nt delayed

yielding and viscoplasticity Brob erg 1979a).


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44 2. THE FRAC TURE PROCESS

Fig. 2.8.8 Primary and secondary plast ic regions.

he wakes of the primary plastic region and the process region the

secondary plastic region

Du ring crack grow th a wake i s left beh ind the plast ic region; see Fig . 2 .1.1 . The w ake i s

c r ea t ed t h r o u g h u n l o ad i n g w h e r eb y d e f o r m a t i o n ch an g es ag ai n b eco m e p u r e l y e la s ti c

and the p l as t i c s t r a ins r emain con s t an t a t each mate r i a l po in t. Actua l ly a smal l par t

o f t he r ea r bou nda ry o f t he p l as t i c r eg ion s t a r t s m oving fo rwards a l r eady befo re c rack

growth because o f un load ing t r ansfe r red f rom the p rocess r eg ion as shown in F ig .

2.3.1.

The process region also leaves a wake behind as i t moves forward. This wake i s very

ins ign i f ican t and may be cons idered as cons i s t ing o f pass ive m ater i a l wi th ve ry smal l

stiffness.

In some cases two d i sconnected p l as t i c r eg ions develop : t o d i s t i ngu i sh them the f i r s t

one i s cal led the

primary plastic region

and the second one which develops in t he

wake of the p r im ar y plast ic region beca use of reversed plast ic f low is referred to as

t h e

secondary plastic region

Fig . 2 .8 .8 shows one example . The he igh t o f t he se conda ry

p las t i c r eg ion is cons iderab ly sm al l e r t han the he igh t o f t he p r ima ry one and ma y be

neg lec t ed in mos t con tex t s .

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