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A ppl i e d Ma th e ma t i c s a n d Me c ha n ic s(English Edit ion, Vol .6 , No.4, Apr. 1985)

P ub l ishe d by S U T,Shanghai , China

O N T H E C O R R E C T I O N B Y M C C L IN T O C K A N D I R W I N *

W a n g M a o - h u a ( ~ )

(Be i j ing Ins t i tu te o f Aeronau t ics and Ast ronau t ics , B e i j ing)

(Receive d M arch 5, 1984)

A b s t r a c t

In the case o f quas i -b r i tt le f rac tu re , a t the c rack t ip the re i s a smal l p las t ic reg ion

whose a ffec t ion cannot be neg lec ted .There fore the I~.. ar e las tic asympo to t ic f ie lds mu s t be

correc ted . In 1965 F. A . M cC l in to ck a nd G . R . I rw in p resen ted a ca r rec tion which s ince

then has been adopted ex ten s ive ly. H ere in th is paper, i t mas t be po in ted ou t tha t suchcorrection is wrong. A correc t result is given.

I . M c C l i n t o c k a n d I r w i n s C o r r e c t i o n

In [ I ] , there areR = 2 r y

2a:O'~

( 1 . 2 ) T

( 1 . 8 ) t ~

w her e cyo is the tensile yield stress;d = a + r v .

Subs t i tu t ing eq . ( l .2 )min to eq .( 1 . 8 ) t ~ , i t t u r n s o u t

r u - - 2 m z _ 1 (1 , 1 5 ) ~ ~

r

w he r e r n = c r o / c r . Fig. 1Subs t i tu t ing eq . (1 .1 5 ) t~J in to r TM the correc t ive s tress intensi ty fac tor K'~ ca n be

f ound .S ince then in a la rge num ber o f l it e ra tures th i s cor rec t ion has been ado pted and i t was rega rded

tha t K I can be exac t ly foun d by the i te ra tion me thod . This ite ra tion program is a s fo l lows:

( ) Subs t i tu t ing a in to K l = c l d n a , the r e su l t K I i s taken a s K p ~ .(2) Sub st i tut ing K ~ ~ into eq .( 1 .8 ) t~J', the result ry is taken as r~~ .(3) Subst i tut ing a + r ~ o in to eq , ( 1 .2) t~J , the r e su lt K '~ i s taken a s K ~ ~(4) S ubs t i tu t ing K~ ~ in to eq . (1 .8 ) t~ , the r e su l t ru i s taken a s r~"

* Com municated by C hien Wei-zang.

387

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3 88 W a n g M a o - h u a

(5 )T h e c a l c u l a t i o n i s c a r r i e d o u t r e p e a t e d l y, u n ti l t h e d i f fe r e n c e o fs a t i s f i e d w i t h a c e r t a i n r e q u i r e m e n t .

H . T h e C o r r e c t R e s u l t s

T he w e l l - kn ow n l i n e a r e l a s t i c f i e ld s a r e a s f o l l ow s :

K t 0 I- 0 . 30 -I

K I 0 . 0 3 0

or , ----v (o r, + or, ) (p lan e s t r a in )

w he re Kt- - - -c r~/ z ra ~ v i s Po isso n ' s r a t io .

T h e r e f o r e , a t O f 0 , t h e r e a r e

K t K t

a n d

a t

i .e.

K ~ ~-*~ a nd K [ > is

K t 0 [ ' 1 + s i n ~ - s i n - ~ ]o = cos 2-

----0zz TUz

c~ffi= 0 (p lan e s t r e ss )

or3 = 0 ( p l a ne s t re s s ) ( 2 . 1 )

K s K I~ , ~ s = v ( ~ t + ~ 2 ) ( pla n e s tr ain ) ( 2 . 2 )

Vo n M i s e s 's c r i t e r io n i s

( o i - - ~ 2 ) 2 + ( a ~ - - o ' s) 2 + ( o ' s - - ~ , ) ~ ffi 2 o'~

F r om e qs . ( 2 . 1 ) a nd ( 2 .3 ) , i t l e a ds t o

F r om e qs . ( 2 . 2 ) a nd ( 2 .3 ) , w e ob ta in

G ~ l . t h u s t h e re a r e

a l - - - - - G ~

a ~( 7 , . f f i ( 1 - - 2 v ) '

f ~ O' le ~ O'e

u , ~ o ' t , ~ - O ' ~ 2 v )

( p l a ne s t r a in )

( p l a ne s t r a in )

0 ~ 0

1 ) P l a n e s t r e s s

F r o m e q s . ( 2. 1) a n d ( 2 .6 ) w e m a y k n o w

g to '~ 0 , ~ / 2z r r,

_ K ;

A c c o r d i n g t o t h e s t r e s s r e l a x a t i o n e q u i l i b r i u m , t h e r e s h o u l d b e ( S e e F i g . 2 )

2.3)

2 . a )

2 . 5 )

2 . 6 )

2 . 7 )

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O n t h e C o r r e c t i o n b y M c C l i n t o c k a n d I rw i n 3 8 9

Rat ,=I~ (a t )8-odr

I n c o n s i d e r a t i o n o f e q s. ( 2 .1 ) a n d

( 2 .6 ) , t h e a b o v e e q u a t i o n b e c o m e sr, K t r

R a ' = I o

F r o m e q ( 2. 7) , i t t u r n s o u t

1 K t z _ _ a 2 r,2 . 8 )

Let a= a + r v bet h e e f f e c t i v e h a l f c r a c k l e n g t h , t h e r e i s

_ _ ID F Lx

Fig. 2

/C t = c r d ~r(a+rg)As o r i g i n o f c o o r d i n a t e s i s l o c a t e d a t t h e c r a c k t i p O , t h e r e is

( 2 . 9 )

e . = K 2 . 1 0 ), , f 2 ~ F

I n o r d e r t o k e e p u p t h e s t r e s s r e l a x a t i o n e q u i l i b r i u m i .e . t h e sh a p e o f c u r v e AB C '~ H , f r o m e q s

(2 .6 ) and (2 .10) , i t l e ads to

a , = / 2 r r ( R _ r , ) ( 2 . 1 1 )

I n c o n s i d e r a t i o n o f e q s. ( 2 . 8 ), ( 2 .9 ) a n d ( 2 .1 ) , we c a n f i n d

r~ = 2m z + 1 ( 2 . 1 2 )

i . e .

2 ) P l a n e s t r a i nF r o m c q s . ( 2 . 2 ) a n d ( 2 . 6 ) , t h e r e i s

a s K t~O 1 s~

( 1 - - 2 v )

/ e l

A c c o r d i n g t o t h c s t r es s r e l a x a t i o n c q u l i b r i u m t h e re s h o u l d b e

Rau,= (ar )o .odr

I n c o n s i d e r a t i o n o f eq s . (2 .2 ) a n d ( 2 .6 ), t h e a b o v e e q u a t i o n b e c o m e s

ao ~re g tR ( 1 - - 2 v ) = 0~,/----~-~-dr

There fore , f rom eq , (2 .7 ) , i t l e ads to

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3 90 W a n g M a o - h u a

K ~ ( l _ 2 v ) z = _ . ~ _ ( l _ 2 v ) ~ = 2 r ,R = ,~ c r . 2 .14)

I n o r d e r t o k e e p u p t h e s t r e s s r e l a x a t io n e q u i l ib r i u m , i .e . t h e sh a p e o f c u r v eA B C H , f r o m e q s .(2 .6 ) and (2 .10) , t he re shou ld be

( 1 - - 2 v ) - -, ~ / 2 ~ ( R - - r y ) ( 2 . 1 5 )

F ro m eqs . (2 .9 ) , (2 .14) and (2 .15) , we can f ind

m zr = 1 2 . 1 o )

In eqs . (2.12)and (2.16 l, r ~ is t he cor rec t ive va lue o f o r ig ina l ha l f c rac k l en g th a . S ub s t i tu t in gry

i n t o e q . (2 .9 ) t h e c o rr e c t iv e st re s s i n t e n s i t y f a c t o r ~ ' t c a n b e f o u n d .I n t h e c a se o f m o d e I I I, o n l y w h e n t h e f o l lo w i n g tr a n s f o r m a t i o n

~ - > r , , ( ~ 8 ~ r , , K t - - > K 3 ( 2 . 1 7 )

is c a r r i e d o u t , r v a n d ~ '~ c a n b e f o u n d .

I I I C o n c l u s i o n

(1 ) M c C l i n t o c k a n d I r w i n ' s c o r r e c ti v e r e su l t i.e . e q . ( 1 . 1 5 ) t 'J i s wr o n g , b e c a u se

(a ) In F ig . 2 r, is t ak en a s rv in F ig . 1 , wh ich l eads to the d i sc rep ance , i .e . acc ord ing to the

s t re ss r e l axa t ion equ i l ib r ium the re i s

R = 2 r s = 2 r u = @

b u t i n t h e o t h e r h a n d , t h e r e su l t i s e x p r e s se d b y e q . ( 1 . 1 5 ) t ~ J .I n f a c t , in o r d e r t o k e e p u p t h e s t re s s r e l a x a t i o n e q u i l i b r iu m t h e r e m u s tbe r~ :~ r v .Ot h e r wi se ,

w he n the ha l f c rack l eng th i s a , f ro m eq . (2 .7 ) t he re i s

ro = a / 2 m z

a n d wh e n t h e h a l f c r a c k l e n g t h i sa = o + r y , t h e r e sh o u l d b e

2rn 2 = ~ + 2m z

T h e r e f o r e , t h e f o l l o w i n g r e su l t

a _ D _ = R + rvr v + ~ . = r , + g ~ - - m + 2 m~ 2m 2

is tu r n e d o u t . I n o t h e r w o r d s , i n F i g . 1 p o i n t C m u s t b e sh i f t e d t o t h e r i g h t, wh i c h l e a d s to t h a t t h e

c o n d i t i o n o f R = 2 r ~ c a n n o t b e s a t i sf i ed .

I n o r d e r t o k e e p u p R = 2 r , , t h e r e m u s t b e r v ( r ~ , i .e . in F ig . 2 and the e ffec t ive c ra ck t ips h o u l d b e a t O , i n s t e a d o f0 .

(b ) In eq . (1 .2 ) ~ ' ~, h7 t i s i d en t i f i e d w i t h K I i n eq . ( ] . 8 ) lu , wh ich in fac t impl i e s tha t V8i s iden t i f i ed wi th rv . Th e re fo re , i n F ig . 1 po in t C m us t be sh i f t ed to the r igh t fu r the r, i. e.

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O n the Co r r e c t ion by Mc C l in toc k a nd I r w in 391

- a

A C = 2 r , = - - ~ - + m z = R + - - - ~ -

(2) In this pa per the iesu l ts a re sa t isf ied with the s tress re laxat ion equ il ibr ium. F ro m eq. (2 .7) ,w e k n o w

F , = a + r ~2m z

There fore , f rom the above equa t ion , (2 .8 ) and (2 .12) , i t tu rns ou t

a + ( Z m Z + l ) r y = - - ~ 2 r ~r ~ F , = , 2 m z .

(3 ) The re su l t s ob ta ined by i te ra t ion me thods a re same as ones by MeCl in tock and I rwin .

B e c a u se a ll o f t h em m a k e u s e o f r , f r t a n dK t = g t , a n d a r e b a s e d o n e q s . ( 1 . 2 ) t n a n d(1 .8 ) t t3 .

(4) L e t r y , a - a + r u a n d Kt=o ~/ na represent the correc t ive resul ts in this paper ;r ] , a * = a + r ] and ~7~_- -e ~r-~ 'g- M cCl in toc k an d I rwin ' s cor rec t ive r e su l ts .

( a ) F ro m eqs . (1 .1 5 ) t t ~ nd (2 .12) , we ob ta ina - a 2

r u = 2 m z - l

W hen m = 2 , the cor rec t ive va lue o f the ha l f c rack leng th is inc reased a bo ut 28 .6% by

McCl in tock and I rwin ' s cor rec t ion .(b) Fr om eqs. (1 .15) tu , (2 .12) and (2.9) w e obta in

R~-RR --K~

1 - % / 1 4 1~ / 1 + 2m--~--~_ 2m~------~

~ / 1 I 1 12 m l

W hen m --2 , the cor rec t ive va lue of s tr e ss in tens i ty f ac tor is inc reased abo ut 27 .6% by

McCl in tock and I rwin ' s cor rec t ion .F o r e xa mp le , w h e n a -- --10rnm, o ' ,= 1 4 0k g /m m z , c r =7 0k g / m m z , t h e s t re s s i nt e ns i ty

fac tors are l is ted in Table 1 .Tab le 1 . Un i t : kg /m m 3/'

U nc o r r e c t ion ] Mc C l in toc k a nd I r w in I n t hi s pa pe r392.35 I 419 .44 413.57

R e f e r e n c e s

[1 ] M cCl in tock , F. A . and G . R. I rwin , P la s t ic i ty a spec ts o ff rac tu r r mechanics ,

A S T M S T P ,381 (1965) , 84- 113.