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7/23/2019 Strength analysis of massive reinforced concrete
http://slidepdf.com/reader/full/strength-analysis-of-massive-reinforced-concrete 1/19
A c ta Me c h a n ic a 9 9 , 7 5 -9 3 (1 9 9 3 )
C T M E C H N I C
9 Springer-Verlag 1993
E la s t i c p la s t i c a n a ly s i s o f cra ck ed p la t e s in p la n e st re s s:
a n e x p e r i m e n t a l s t u d y
P . S . T h e o c a r i s , A t h e n s , G r e e c e
(Received April 13, 1992)
S u mm a ry . A n e x p e r ime n ta l me th o d i s p re s e n te d fo r t h e c o m p le te s o lu t io n o f t h e e l a s t ic -p l a s ti c p l a n e s t r e s s
p ro b le m o f a n e d g e -c ra c k e d p la t e o b e y in g th e Mis e s y i e ld c r i te r io n a n d th e P ra n d t l -R e u s s in c re me n ta l
s t ress -s t ra in f low ru le . The m ate r ia l o f the p la te is a s sumed as a s t ra in -h arden ed one w ith d if fe ren t degrees o f
h a rd e n in g . T h e e l a s t i c a n d p la s t i c c o mp o n e n t s o f s t r a in w e re d e te rmin e d b y u s in g th e me th o d o f b i re f r in g en t
c o a t in g s c e me n te d o n th e s u r fa c e o f th e m e ta l li c s p ec ime n s ma d e o f th e m a te r i a l u n d e r s tu d y . N o rma l
inc idence o f c i rcu la r ly po la r ized l igh t y ie lded the isoc l in ics and isochrom atics o f the coa t ing w hich p rov ided
the p r inc ipa l e las t ic s t ra in d if fe rences and s t ra in -d irec t io ns a t the in te rface. Eva lua t io n o f the s t ress in tens i ty
fac to r a t th e c rack t ip , by us ing the G riff i th -Irwin de f in i t ion , gave the sum of p r inc ipa l s t re s ses a t the c rack t ip .
These da t a were suff icient to separa te the comp onents o f s t ra in a t the coa t ing -p la te in te rface by us ing the
c lass ica l shea r-d ifference m ethod .
T h e s t r e s s c o mp o n e n t s o n th e p a r t i a l ly p l a s t i c al ly d e fo rme d c ra c k e d p la t e w e re d e te rmin e d b y u s in g th e
Prand t!-Reu ss s t ress -s t ra in re la t ionsh ips in a s tep-by-s tep p rocess fo l lowing the whole h is to ry o f load ing o f
the p la te . Thus , a rad ia l d is t r ibu t ion law for the equ iva len t s t re s s ~ and s t r a in in a l l d i rec t ions o f the p la te was
e s t a b l is h e d w h ic h g a v e th e in s t a n ta n e o u s p o s i t i o n o f t h e e l a s ti c -p l a st i c b o u n d a ry a n d i t s e v o lu t io n d u r in g
load ing , a s we l l a s the d is t r ibu t ion o f e las t ic and p las t ic components o f s t res ses a l lover the p la te .
F o u r c a s e s w e re s o lv e d fo r v a r io u s a m o u n t s o f s t r a in -h a rd e n in g f ro m a q u a s i p e r fe c tly p la s t i c ma te r i a l t o
a n a lmo s t b r i t t l e s t r a in h a rd e n e d o n e . T h e v a lu e s o f th e c h a ra c te r i s t i c p a ra me te r s d e f in in g ea c h ty p e o f
mate r ia l were es tab l ished .
The resu l ts de r ived comp are exce l len t ly with ex is t ing ones based e i the r on exp erime nta l o r numerica l
s o lu t io n s a n d s in c e th e y a re b a s e d o n b o th th e th e o ry o f e l a st i c i ty a n d th e in c re me n ta l t h e o ry o f p l a s t i c i ty
th e y c o n s t i tu t e a s o u n d b a s i s fo r c o mp a r i s o n . Mo re o v e r , t h e a lg o r i th m b a s e d o n th i s h y b r id me th o d i s f a s t
a n d s t a b le r e q u i r in g a m in imu m c o mp u te r t ime , me mo ry a n d d a ta p re p a ra t io n .
Introduct ion
F a i l u r e b e h a v i o r o f c r a c k e d m a t e r i a l s i s s i g n i f i c an t l y i n f l u e n c e d b y c r a c k - t i p p l a s t i c i t y w h i c h
d i r e c t l y m o d u l a t e s t h e n e a r s t r e ss - f i el d a n d , c o n s e q u e n t l y , a f fe c ts t h e r e s u l t s o f t h e v a r i o u s
f r a c tu r e c r i te r i a. I n t h e p a s t , it w a s c o m m o n l y a s s u m e d t h a t f r a c t u r e i s a p h e n o m e n o n
i n d e p e n d e n t o f t h e g e n e r a l f ai l u r e a n d s o l e l y c o n n e c t e d w i t h b r i t t l e m a t e r i a l s . T h i s m a y b e t r u e i n
e x t r e m e c a s e s o f h i g h b r i t t l e n e s s , b u t m o s t o f e v e r y d a y m a t e r i a l s d o n o t b e h a v e s o . T h e y f a i l b y
b r i t t l e f r a c tu r e a f t e r t h e d e v e l o p m e n t o f p l a s t i c a l l y d e f o r m e d e n c l a v e s , w h i c h c o u l d b e v e r y s m a l l
a n d i n s i g n i f i c a n t i n h i g h l y b r i t t l e m a t e r i a l s .
T h e n , t h e c o r r e c t a p p l i c a t i o n o f a n y f r a c t u r e c r i t e r i o n r e q u i r e s t h e a - p r i o r i c o r r e c t e v a l u a t i o n
o f th e e l a s t i c - p l a s t i c st r e s s fi e l d i n t h e v i c i n i t y o f t h e c r a c k - t i p . T h i s h a s a l r e a d y b e e n a p p l i e d f o r
v a r i o u s e l a s t i c- p l a s ti c m a t e r i a l s a n d m o d e s o f l o a d i n g b y u s i ng v a r i o u s a n a l y t i c a l a n d n u m e r i c a l
m e t h o d s [ 1] - [ 10 ]. F i n a l l y , t h e u s e o f J - i n t e g r a l a s a f r a c t u r e c r i t e r i o n w a s d i s c u s s e d a n d p r o v e d
i n [ 11 ] - [ 1 3 ] .
7/23/2019 Strength analysis of massive reinforced concrete
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76 R S. Theo caris
O n t h e o t h e r h a n d i t i s g e n e r a l l y a c c e p t e d t h a t o n l y e x p e r i m e n t a l m e t h o d s c a n a c c u r a -
t e l y so l v e th e p r o b l e m o f t h e s p r e a d i n g o f t h e p l a s t ic e n c l a v e s a r o u n d d i s c o n t i n u i ti e s o f t h e
s t re s s fi e ld a n d e v a l u a t e i n a s a t i s f a c t o r y m a n n e r t h e d i s t r i b u t i o n o f e l a s ti c a n d p l a s t i c
s t re s s - a n d s t r a i n - c o m p o n e n t s i n s i d e t h e f ie ld a s w e l l a s t h e y d e f in e t h e e x a c t i n s t a n t a n e o u s
p o s i t i o n o f t h e e l a st i c -p l a s t i c b o u n d a r y a n d i t s e v o l u t i o n d u r i n g l o a d i n g . I t w a s s h o w n t h a t
t h e m e t h o d o f p h o t o e l a s t i c co a t in g s c o m b i n e d w i t h s o m e au x i l ia r y m e t h o d d e t e r m i n i n g a n
a d d i t i o n a l q u a n t i t y n e c e s s a r y f o r t h e s e p a r a t i o n o f t h e e l a st i c a n d p l a s t i c c o m p o n e n t s o f
s t re s s e s a n d s t r a i n s a l l o v e r t h e fi el d c a n d e t e r m i n e t h e p r o g r e s s i v e e x t e n s i o n o f th e p l a s t i c
e n c l a v e s w h i c h n u c l e a t e a t r e g i o n s o f t h e h i g h e s t e la s t ic s t re s s c o n c e n t r a t i o n a n d s u b s e -
que n t ly sp r ead co m p le te ly in the s t r e s s f ie ld . S ince the ax ia l s t r e s s and s t r a in d i s t r ibu t io n in
t h e p l a s t ic r a n g e o f m e t a l s a n d s i m i l a r m a t e r i a l s d e p e n d s o n t h e p a r t i c u l a r m e c h a n i c a l
p r o p e r t i e s o f t h e m a t e r i a l t h e u s e o f t h e s a m e m a t e r i a l a s a t e s t i n g p ie c e in t h e l a b o r a t o r y
b e c o m e s o b l i g a t o r y [ 1 4 ] - [ 1 6 ] . T h e m e t h o d w i t h i t s o w n v a r i a t i o n s w a s s u c c e s sf u l ly u s e d i n
a s e r ie s o f st u d i e s f o r e l a st i c - p e rf e c t ly p l a s t ic m a t e r i a l s o r s t r a i n - h a r d e n i n g o n e s a n d f o r
d i f f e r en t types o f d i s con t inu i t i e s am on g which a r e c la s s if i ed ho les no tc hes o f d i f f e r en t
r e - en t r an t ang les e tc . [ 1 7] - [ 20].
A n o t h e r s t u d y w a s a l s o u n d e r t a k e n t o e v a l u a t e t h e p o s s i b i l it i e s o f u s i n g t h e w e l l k n o w n
D u g d a l e - B a r e b l a t t m o d e l [ 2 1] i n a p p l i c a t i o n s w h i c h s i m p l i fi e s c o n s i d e r a b l y t h e p l a s t ic a n a l y s i s
o f s t r u c t u r e s [ 2 1 ] - [ 2 4 ] . A l t h o u g h a m o d i f i e d v e r s i o n o f t h i s m o d e l w a s i n t r o d u c e d i n th e s e
r e fe r e n c es w h i c h y i e ld s a m u c h h i g h e r a p p r o x i m a t i o n i t w a s s h o w n t h a t t h i s m o d e l a l s o i s
i n c a p a b l e t o g i v e a c c u r a t e d i s t r i b u t i o n s o f s t re s s e s a n d s t r a i n s in p r o b l e m s o f c o n t a i n e d
p l a s t ic i t y w h e r e t h e e l a st i c c o m p o n e n t s o f s t re s s e s a r e c o m p a r a b l e i n m a g n i t u d e w i t h t h e i r
c o u n t e r p a r t s o f th e p l a s t i c c o m p o n e n t s .
T h i s a n a l y s i s w a s s u c c e s sf u l ly u s e d i n e x p e r i m e n t a l w o r k [ 17 ] - [ 20 ] w h e r e t h e i n c r e m e n t s o f
s t r a i n s w e r e d i r e c t ly e x p e r i m e n t a l l y m e a s u r e d w h e r e a s i n t h is p a p e r t h e i n c r e m e n t s o f s t r a i n s
i n s i d e a n d o u t s i d e t h e e l a s t i c - p l a s t i c b o u n d a r i e s a r e d e f i n e d b y u s i n g p h o t o e l a s t i c d a t a f r o m
b i r e f r i n g e n t c o a t i n g a n d b y i n t r o d u c i n g a
compli nce technique
f o r t h e e l a s t i c c o m p o n e n t s o f
s t r e sses . Thu s these quan t i t i e s t ake the i r e la s t i c va lues a t the f i r st load in g- s tep wh er e the fi r s t
p l a s t i c n u c l e i a r e d e f i n e d a c c o r d i n g t o t h e G r i f f i t h - I r w i n a s s u m p t i o n [ 1 ] a n d t h e e l a s t i c
c o m p o n e n t s o f s t re s s . T h e n s t re s s e s o b e y t h e H o o k e a n d t h e P r a n d t l - R e u s s l a w s fo r t h e i r e l a s ti c
a n d p l a s t i c p a r t s r e s p e c t i v e ly i n a m a n n e r s i m i l a r t o t h a t d e s c r i b e d i n [2 8 ] w i t h d i s t r i b u t i o n s o f
e l a s ti c s tr e s se s f o ll o w i n g p o w e r l a w s w i t h d e c r e a s i n g n e g a t i v e e x p o n e n t s v e r s u s d i s t a n c e f r o m t h e
c r a c k t i p i n c o m p l e t e a n a l o g y w i t h t h e e x i s t in g e x a c t s o l u t i o n o f a n e l a s t ic p l a t e c o n t a i n i n g a n
in te r na l o r edge c r ack [ 29] .
D e s c r i p t i on o f t h e m e t h o d
T h e s o l u t i o n i s b a s e d o n t h e f o l lo w i n g p ri n c i p le s w h i c h a r e in a g r e e m e n t w i t h e it h e r th e H o o k e
o r th e P r a n d t l - R e u s s l a w s c o n c e r n i n g t h e e l a s t ic o r p l a s t i c b e h a v i o r o f t h e m a t e r i a l . A l s o t h e y
p o s s e s s a u s u a l ly o b v i o u s p h y s i c a l r e a s o n i n g . T h e s e p r i n c ip l e s w e r e a p p l i e d t o a t h in p l a t e
c o n t a i n i n g a c e n t r a l c r a c k o f l e n g t h 2 a a n d l o a d e d u n i a x i a l l y b y a s t re s s a ~ a t a d i r e c t i o n
p e r p e n d i c u l a r t o t h e c r a c k - a x i s u n d e r g e n e r a l i z e d p l a n e - s t r e s s c o n d i t i o n s . T h e m a t e r i a l o f t h e
p l a t e o b e y s th e e q u i v a l e n t s t r e s s - st r a i n c u r v e s h o w n i n F i g . l b w h i c h f o r s i m p l i c i ty i s a s s u m e d
bi - l inea r . F or a low eno ug h a pp l i ed s t r e s s a t in f in i ty a~o the r e su l t ing in i t i a l e la s t i c - p la s t i c
b o u n d a r y c o r r e s p o n d i n g t o t h e o n s e t o f y i el d in g i s s o s m a ll as t o b e th e s a m e w h e n c o m p u t e d b y
m e a n s o f p u r e l y e l a s ti c o r e l a s t ic - p l a s ti c c o n s i d e r a t i o n s [1 ].
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Elast ic-plast ic analysis of crack ed plates 77
F o r t h i s i n i ti a l s t ep , t h e s t r e s s- f ie l d i s d e s c r i b e d b y t h e w e l l - k n o w n c o m p l e x s t r es s -
f u n c t i o n s
~ e Z )
an d 7~e(z) [29]:
Tm Z Too
~ e Z t =
2
Z 2 - -
a 2 ) 1 / 2 4
1 )
Too Z
Too
~ Y e Z ) - - - ~ - - .
2 (z 2 - - a 2 ) 1/ 2 4
T h e s e c o m p l e x s t r e s s f u n c t i o n s d e s c r i b e t h e e l a s t i c s t r e s s d i s t r i b u t i o n f o r a n i n f i n i t e p l a t e
c o n t a i n i n g a c e n t r a l tr a n s v e r s e c r a c k s u b j e c t ed t o s i m p l e t e n si o n . I n o r d e r t o h a v e t h e s o l u t i o n
f o r a n e d g e c r a c k , w h o s e s o l u t i o n i n a c l o s e d f o r m d o e s n o t e x i s t , i t i s c u s t o m a r y t o u s e t h e
s o l u t i o n f o r a n i n t e r n a l c r a c k a n d t o m u l t i p l y t h e v a l u e s o f t h e s t r e ss i n t e n s i t y f a c t o r a t th e c r a c k
t ip , o r t h e d i s t r i b u t i o n o f st r e ss i n s i d e t h e s t r e s s fi el d , b y a c o n v e n i e n t m u l t i p l i c a t i o n f a c t o r ,
d e f i n e d e i t h e r e x p e r i m e n t a l l y o r n u m e r i c a l l y [ 30 ], [ 31 ], w h i c h t a k e s c a r e o f t h e i n f l u e n c e o f t h e
e x a c t c r a c k o p e n i n g d i s p l a c e m e n t a t t h e i n t e r s e c ti o n o f t h e c r a c k f l a n k s w i t h t h e f r ee l o n g i t u d i n a l
b o u n d a r y a t w h i c h t h e e d g e c r a c k t e r m i n a t e s .
T h e e x p r e s s i o n s f o r t h e s tr e s s f u n c t i o n s
~ e Z )
a n d ~Pe(Z) i n r e l a t i on s ( 1 ) i n t r od uc e d i n t o t he
e x p r e s s i o n s f o r t h e c o m p o n e n t s o f s t re s s e s y ie l d
T x = T o o [ F 1 r , O ) - F 2 r , 0)] ,
c r y = T o o [ F l r , O ) + F 2 r ,
0)] ,
(2)
Z x y = a ~ F 3 r , 0 ) ,
w h e r e t h e f u n c t i o n s
F i g
= 1 , 2 , 3 ) a r e r e a l f u n c t i o n s o f t h e p o l a r c o o r d i n a t e s r e f e r r e d t o
a C a r t e s i a n f r a m e w i t h o r i g i n a t t h e c r a c k t ip a n d O x - a x i s th e s y m m e t r y a x is o f t h e c r a c k .
T h e F i - f u n c t i o n s a r e g i v e n b y :
1 ) 1
F l r , O ) = G l r , O ,
1/2) =
r l r 2 - 1 1 2
c o s Pl -- ~ ,~ 2
F 2 r , O ) = G 2 r , O ,
1/2) =
r l r 2 - 1 / 2
s i n O [ - s i n ( q h / 2 ) -
r 1 2 r 2 - 1
sin (2~01 - 1.5qh)] + ~ ,
(3)
F 3 r , O ) = G 3 r , O ,
1/2) =
r l r 2 - l I e
s in O [- c o s (q~2/2) - t- r12r2 -1 cos (2~01 - 1 .5q02)] ,
w h e r e t h e p o l a r r a d i i r l a n d r 2 a n d t h e p o l a r a n g l e s q h a n d (/)2 a r e e x p r e s s e d b y :
r I = [ (a + r cos O ) 2 + r 2 sin 2 O]a/2,
r e = [(r 2 Cos 2 0 + 2 a r cos O)a + r 2 s i n 20 + 2 a t s i n
0 ) 2 ] 1 / 2 ,
r s in 0
qol = a r c t a n 0 ( 4 )
a + r c os
@2 = a r c t a n
r e s in 2 0 + 2 a r s i n 0
r 2 c os 20 + 2 a r c o s O
a n d t h e p o l a r r a d i i r a n d a n g l e s q~ a r e m e a s u r e d f r o m t h e c r a c k t i p a n d i n a n a n t i - c l o c k w i s e s e n s e
f r o m t h e O x - a x i s .
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78 E S . Theo ca r i s
b
O
O
._>
V
E : I 'anlO /
r _ _ _ _ _ ~ ~ B d t D
~ ~ .-d~ -,-
0 A' _ B'
i+2
0'i+.3 6
p : plastic cur'ves b e P i - - - ~ i5 - i+2
d ~:=(cl~_.g+d[:p
O(cr.ack ip) dis tan ce x
Fig . 1 . a The a s sum ed rad ia l d i s t r ibu t ion o f fo r success ive inc rem en ts o f g b Th e p iecewise l inea r
idea l i za t ion o f the equ iva len t s t r e s s - s t r a in cu rve
I n o r d e r t o d e f in e t h e a c t u a l p o s i t i o n o f t h e e l a s ti c - p l a s ti c b o u n d a r y t h e M i s e s y i e ld c r i te r i o n
w a s u s e d w h i c h s t a t e s t h a t t h e e q u i v a l e n t o r e f f e c t i v e s t r e s s , ~ , i s e x p r e s s e d b y :
( ~ 2 = 0 . x 2 _ ~ 0 .y 2 _ _
0.x0.y F
3 7 7 x Z y = 0 0 2 ,
5 )
w h e r e 0 .o i s t h e y i e l d s t r e s s o f t h e m a t e r i a l i n s i m p l e t e n s i o n a t i n fi n i t y . I n t r o d u c i n g r e l a t i o n s ( 2 )
i n t o t h e y i e l d c o n d i t i o n (5 ) w e d e r i v e :
0 .oo = 0 .o {F x2 r , O ) + 3 F 2 2 ( r , 0 ) + 3 F 3 2 ( r , 0 ) } - 1 / 2 .
(6)
B y p u t t i n g i n t o E q . (6 ) a r e a s o n a b l y s m a l l v a l u e o f r , d e r i v e d b y a p p l y i n g I r w i n ' s p r o p o s i t i o n [1 ],
f o r 0 = 0 ~ w e c a n c o m p u t e a fi r st v a l u e o f 0 .~ . F o r 0 0 ~ a n d f o r c o n s t a n t 0% w e c a n c o m p u t e
n u m e r i c a l l y t h e c o r r e s p o n d i n g v a l u e s o f t h e i n it i a l r a d i i o f t h e e l a s t ic - p l a s t i c b o u n d a r y f o r 0 =t 0 .
R e f e r r i n g t o F i g . 1 a l l t h e p o i n t s o f th e e l a s t ic - p l a s t ic b o u n d a r y a r e r e p r e s e n t e d b y t h e p o i n t A o f
t h e e q u i v a l e n t s t r e s s - s t r a i n c u r v e .
A n i n c r e a s e o f t h e e q u i v a l e n t s t r e ss , c a u s e d b y a n in f i n i te s i m a l r e s p e c t i v e i n c r e m e n t o f
0 . ~ b e y o n d i t s i n i t i a l v a l u e r e s u l t s i n a n i n c r e m e n t
d g
o f t h e e q u i v a l e n t s t r a i n g c o n s i s t i n g o f a n
e l a s t i c p a r t d g , a n d a p l a s t i c o n e d g v [ 3 2 ] . T h e c o m p o n e n t s d g ~ a n d d g p o f t h e s t r a i n i n c r e m e n t o b e y
e i t h e r t h e H o o k e , o r th e P r a n d t l - R e u s s l a w s r e s p e c ti v e l y , t h e s a m e b e i n g t r u e fo r e a c h o n e o f t h e
s t re s s- a n d s t r a i n - c o m p o n e n t s .
F o l l o w i n g t h e a n a ly s i s d e v e l o p e d i n [ 1 7 ] - [ 2 0 ] f o r t h e h o m o g e n e o u s i s o t ro p i c e l a st ic -
p l a s t i c a n d p l a s t i c a l ly i n c o m p r e s s i b l e m a t e r i a l o b e y i n g t h e P r a n d t l - R e u s s f l o w r u l e u n d e r
p l a n e - s t r e s s c o n d i t i o n s , w e w r i t e d i r e c tl y t h e e x p r e s s i o n s f o r t h e r e s p e c t i v e i n c r e m e n t s o f
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Elas t i c -p la s t i c ana lys i s o f c racke d p la te s 79
s t r e s s :
31
~-1 + v dcr~ = de~ + ~ (dex + dey + de~) 2cr~ 3 cry d 2 ,
1 + v v 2cry -- a=~
E
dcry = dey + ~ (de~ + dey + de ~ ) ~ d2 ,
(7)
l + v
1
dz~y = ~ dTx y Cxyd 2 ,
w h e r e t h e n o n - n e g a t i v e s c a l a r f a c t o r o f p r o p o r t i o n a l i t y , d 2 , i s g i v e n f o r a s t r a i n - h a r d e n i n g
m a t e r i a l b y [ 3 2 ] :
d~p
d : ~ - 2 ~ 8 )
T h e s t r e s s - s t r a i n r e l a t i o n s h i p h y p o t h e s i s f o r t h e e la s t ic c o m p o n e n t s o f t h e s t ra i n i n c r e m e n t s
t a k e s t h e f o r m [ 3 2 ] :
de x: _ dey : _ dTxy, , _ dg~
(9)
T h e p l a s t i c c o m p o n e n t s o f t h e s t r ai n i n c r e m e n t s o b e y t h e P r a n d t l - R e u s s l a w [ 32 ]:
dex,p _ d e y p dyxy,p
crx cry gxy
3 a ~
d 2
2
(10)
w h e r e a x ', c ry ' a r e t h e d e v i a t o r i c c o m p o n e n t s o f st r es s e s. F r o m F i g . 1 i t i s v a l i d t h a t :
dge = ~ dg , dgp = 1 - dg .
(11 )
T a k i n g i n t o c o n s i d e r a t i o n t h e p l a s t i c i n c o m p r e s s i b i l i t y a s s u m p t i o n
d~x,e + dey, , + dez ,e -
1 - 2 v
E
( d a x + d a y ) , ( 1 2 )
E q s . ( 7 ) t a k e t h e f i n a l f o r m :
E
d c r x m
V 2
da y - 1 ~ 122 dg: , y v & x + 3[(v - 1) ay + (1 - 2 v) a~] ,
1 3 )
d z x y -
2 ( 1 + v )
d y ~ y - ~ z xy .
I n th i s s e t o f e q u a t i o n s a x , c ry , a n d y a r e k n o w n f r o m t h e p r e v i o u s l o a d i n g - s t e p a n d dex, d ~ ; y d T x y
a r e o b t a i n e d b y t h e r e s p e c t i v e s u m m i n g o f E q s . (9 ) a n d (1 0), w h i c h a r e f u n c t i o n s o f
dG , dgp ,
a n d
w h i c h i n t u r n a r e a l s o f u n c t i o n s o f t h e i n d e p e n d e n t v a r i a b l e
dg.
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8 0 R S . Th e o c a r i s
I f o n e r e s t r i c t s h i m s e l f to a s t e p s t a r t i n g f r o m t h e C r o -y ie ld s t r e s s b y p u t t i n g 6 = cro ( p o i n t A i n
F i g . lb ) , E q s ( 13 ) t a k e t h e s i m p l e f o r m :
H d g H d g H
d g
d a x
= ~- a~ --'Co d~ x
= ~
O'y --,~30
d c x y
= ~ Zxy ~-'eo (14)
T h e n o n - n e g a t i v e v a l u e o f d 2 in E q . ( 8) i m p l i e s t h a t a n y i n c r e a s e o f
dgp,
a n d t h e r e f o r e o f
dg,
r e s u l t s
i n h i g h e r v a l u e s o f ~. I t m e a n s t h a t , a f t e r t h e f ir s t i n c r e m e n t d g , t h e p o i n t ( r , 0 ) ( m a r k e d a s rg i n
F i g . l a ) l i e s i n s i d e t h e e l a s t i c - p l a s t i c r e g i o n w h i c h , n o w , e x t e n d s u p t o t h e p o i n t r~ + 1 i n t h e s a m e
f i gu r e . T h e n e w v a l u e i + ~ o f t h e e q u i v a l e n t s t r e s s a t t h e p o i n t r ~, a f t e r th e a d d i t i o n o f dg, is
c o m p u t e d f r o m t h e e x p r e s s i o n o f t h e M i s e s y i e ld c o n d i t i o n g i v e n in E q . ( 3) b y a d d i n g t h e s tr e s s
i n c r e m e n t s ( E q . 1 4 ), t o t h e p r e v i o u s v a l u e s o f t h e s t r e s se s . T h e n e w v a l u e f o r t h e r a d i u s r~ + 1 o f th e
e l a s t ic - p l a s t ic b o u n d a r y a t a g iv e n 0 - d i r e c t io n w i ll b e c o m p u t e d a s f o l lo w s .
I t h a s b e e n w e l l e s ta b l i s h e d t h a t , w h i l e t h e s t r e s s d i s t r i b u t i o n i n s id e a n e l a s ti c f ie l d p r e s e n t s
m a x i m u m o r m i n i m u m c o m p o n e n t s o f s t re s s es o n l y a l o n g t h e b o u n d a r i e s o f t h e st re s s f ie ld , t h e
s a m e i s n o t v a l i d f o r th e p l a s t i c o r t h e e l a s t o - p l a s t i c s t r es s f ie ld . I t h a s b e e n p r o v e d e x p e r i m e n t a l l y
a n d n u m e r i c a l l y [ 3 3] t h a t i n s i d e t h e p l a s ti c e n c la v e s t h e o n e C a r t e s i a n c o m p o n e n t o f s t re s s
p a r a l l e l t o t h e l o a d i n g d i r e c t i o n o f t h e p l a t e , e v e n f o r a n e l a s t i c - p e r fe c t l y p l a s t ic m a t e r i a l , p r e s e n t s
a m a x i m u m l o c a t ed s o m e w h e r e i n s id e t h e p l a st i c e n c la v e a n d a t a d i s ta n c e f r o m t h e f re e
b o u n d a r y e q u a l t o a p p r o x i m a t e l y o n e - t h i r d o f t h e r e s p ec t iv e r a d i u s o f th e e l a st ic p l a s ti c
b o u n d a r y [ 17 ] - [ 20 ]. T h u s , t h e s u r f a c e o f t h e i n t e n s i t y d i s t r i b u t i o n o f t h i s s t r e ss in s i d e t h e e n c l a v e
h a s t h e s h a p e o f a h i ll in s t e a d o f a s u r f a c e o f a c o n t i n u o u s l y d i m i n i s h i n g o r i n c r e a s i n g s l o p e f r o m
t h e b o u n d a r i e s t o t h e i n t e r i o r o f t h e s tr e s s f ie l d w i t h o u t p r e s e n t i n g a n y m a x i m u m i n s i d e i t.
A t y p i c a l e x a m p l e o f s u c h a v a r i a t i o n o f s t r e s s e s i s g i v e n , b e s i d e s [ 17 ] - [ 20 ], i n [3 4] , w h e r e t h i s
m a x i m u m o f t h e o - r- st re s s d i s t r i b u t i o n a p p e a r s i n s i d e th e p l a s t i c e n c la v e .
F u r t h e r m o r e , i t i s o b v i o u s l y s t i p u l a t e d t h a t o u t s i d e t h e p l a s ti c e n c l a v e t h e m a t e r i a l b e h a v e s
e l a s ti c a l ly a n d t h e r e i s a s m o o t h c o n t i n u a t i o n o f s t r e s s - c o m p o n e n t s o n b o t h s i d es o f t h e
e l a s t i c - p l a s t i c b o u n d a r y .
T h e a b o v e r e a s o n i n g i s a l g e b r a i c a l l y e x p r e s s e d b y t h e f o l l o w i n g c o n d i t i o n s :
a ~ r )
i ) ~ . . . . . . = 0 , 1 5 )
i i) ( r ) . . . (_ ) = c~(r ) . . . (+) = i+1 ,
(16)
w h e r e c ri+ 1 i s d e f i n e d i n F i g . l b ,
i i i ) d 6 r ) _ d 6 r ) ,
d r
I t=r , ( - )
d r
r=r , (+)
1 7 )
w h e r e r m .x i s t h e p o l a r r a d i u s o f t h e p o s i t i o n o f t h e m a x i m u m 6 ( r )- s tr e s s a n d
r~(O)
s t h e r e s p e c t i v e
r a d i u s o f t h e e l a st i c - p la s t i c b o u n d a r y a t th e s a m e d i r e ct i o n . H e r e f o r a t r a n s v e r s e l y c r a c k e d p l a t e
s u b j e c t e d t o s i m p l e t e n s i o n a t a d i r e c t i o n n o r m a l t o t h e c r a c k - a x i s i t is v a l i d t h a t 0 = 0 ~
T o m a t e r i a l i z e t h e a b o v e - s t a t e d c o n d i t i o n s f o r 6 (r ), w e a s s u m e t h a t t h e f u n c t i o n 6 ( r ) t a k e s t h e
f o r m :
c l r 2 - c 2 r + c 3
fo r r < r /
~(r) = ~ o .oo(G12(r 0 , n) + 3 G 22(r , 0 , n) + 3G 32(r , 0 , n)) 1 /2 fo r r > r l
(18.1)
(18 .2 )
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Elas t i c -p la s t i c ana lys i s o f c rack ed p la te s 81
w h e r e c i, i = 1 , 2 , 3, a r e c o n s t a n t s a n d t h e f o r m o f # ( r ) f o r r >
r i
i s s i m i l a r t o t h a t o f E q . ( 5) , w h e r e
t h e r e a l f u n c t i o n s G i ( r , 0 , n ), i = 1 , 2 , 3 , a r e d e t e r m i n e d i n a s i m i l a r m a n n e r t o t h a t f o r t h e e l a s ti c
c a s e. R e a l l y , w e s t i p u l a t e t h e f o l l o w i n g e x p r e s s i o n s f o r th e c o m p l e x s t r e s s - f u n c t i o n s
ebp(Z)
a n d
Up ( Z) f o r r > r~, i n a n a l o g y w i t h t h e e l a s t ic c o m p l e x s t r e ss f u n c t i o n s ~)e(Z) a n d % ( z ) g iv e n i n
r e l a t i o n s ( 1 ) :
~oo z2 n O'c~
( z ) -
2 (r 2 - ae) 4 '
~o~ z2 n 0 0o
T i P ( Z ) - 2 ( r a - a 2 ) + ~ '
(19 )
w h e r e t h e u n k n o w n e x p o n e n t n is a p o s i t i v e n u m b e r a n d i t t a k e s t h e v a l u e n = 1 / 2 f o r t h e
r e s p ec t iv e e l a st ic p r o b l e m . E q u a t i o n s (1 9) ar e a n a l y t ic a n d s a ti s fy t he C a u c h y - R i e m a n n
c o n d i t i o n s , a s E q s . ( 1) d o . E x p r e s s i o n s ( 19 ) y i e l d t h e f o l l o w i n g e x p r e s s i o n s f o r t h e c o m p o n e n t s o f
s t r e s s e s :
o-~ = cr~o[G l(r,
O , n ) - G z ( r , O ,
n) ] ,
c ry = a ~ [ G a ( r , O , n ) + G 2 ( r , O ,
n) ] , (20 )
Zx y = a ~ o G3 ( r , O , n ) ,
w h e r e G a ,
G2
G 3 a r e r e a l f u n c t i o n s d e f i n e d i n c o m p l e t e a n a l o g y w i t h f u n c t i o n s F ~ (i = 1 , 2 , 3 )
g i v e n b y r e l a t i o n s ( 3 ) . T h e s e f u n c t i o n s a r e g i v e n b y :
G a ( r , O , n ) = r l Z r 2 -
co s (2ncpl - ngo2) - -
2 '
G z ( r , O , n ) = 2 n r r ~ 2 - l r 2 -n
s in 0{s in [ (2n - 1 )
D1 --
nrg2]
- r l Z r 2 - 1
sin [(2n + 1) go1 - (n + 1) ~o2]} + 2 ' (21)
G 3 ( r , O , n ) = 2 n r r x 2 - i r 2 -
s i n 0 { - c o s [ ( 2 n - 1 ) g oa - n g oz ]
+ r l e r 2 a co s [(2n + 1) rpl - (n + 1) go2]}.
T h e r a d i a l b e h a v i o r c o n d i t i o n s , t h e n , r e s u l t in :
3 D D r D
c a = 4 r r ' c z
= ~ - , c 3 = # i + 1 - ~ - , ( 22 )
w h e r e D i s t h e r a d i a l d e r i v a t i v e o f
or(r)
f o r r = r i , g i v e n b y :
0 G a 0 G 2 ~ G 3
~ # ( r ) G a ~ + 3 G a ~ - r + 3 G 3 a ~ -
D - ~?r - c ry~ (G i2 + 3G2 2 + 3G 32) l / e (23 )
a n d G a ,
G 2 , G are
e x p r e s s e d b y r e l a t i o n s (2 1) w h i l e r l i s t h e r e s p e c t i v e p o l a r r a d i u s o f t h e
e l a s t i c - p l a s t i c b o u n d a r y .
T h e e x p o n e n t n i n t h e a b o v e e q u a t i o n s i s g e n e r a l l y a f u n c t i o n o f t h e a n g l e 0 , a l o n g w h i c h t h e
c o m p u t a t i o n s a r e p e r f o r m e d a t t h e e f fe c t iv e l o a d i n g s t e p , o r , e q u i v a l e n t l y , t h e r e s p e c t i v e v a l u e
r l o f t h e r a d i u s o f th e p l a s t i c e n c l a v e .
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8 2 R S . Th e o c a r i s
I n o r d e r t o e v a l u a t e t h i s q u a n t i t y w e p r o c e e d b y i n c r e a s i n g t h e to t a l s t r a i n b y de o n t h e
e l a s t ic - p l a s t ic b o u n d a r y w h i c h c o r r e s p o n d s t o a n e w v a l u e f o r g i v e n b y :
8 i+1 = ~ i - H dg .
(24)
T h e s t r e s s i + 1 c a u s e s a s h i f t o f t h e c u r v e e i t o t h e c u r v e e ~ + , f o r t h e e l a s t i c m a t e r i a l a n d
a c o r r e s p o n d i n g s h i ft o f t h e c u r v e P l t o p ~+ 1 f o r t h e s t r a i n - h a r d e n i n g m a t e r i a l , a s i t is s h o w n i n
F i g . l b . I n b o t h c a s e s, t h e i n c r e a s e i n s t r a i n - e n e r g y d e n s i t y i s r e p r e s e n t e d b y t h e a r e a w e o f t h e
s t r ip b e t w e e n e~ a n d e ~ + l i n t h e e l a s ti c m a t e r i a l a n d t h e a r e a w p b e t w e e n p~ a n d P ~ + I i n t h e
e l a s t ic - p l a s ti c m a t e r i a l . T h e s e t w o q u a n t i t i e s w e a n d w , a r e r e l a t e d , a s it i s s h o w n i n F i g. l b , b y :
O max
W e A r e a ( O F ' F ' ) O -o2 - + H -
-
- 2 , 25)
w p A r e a ( O A F F ' ) E { o - 0 ( l _ l ) + ~ }
w h e r e O 'm a i s t h e m a x i m u m v a l u e o f i n s i d e t h e p l a s t i c a l ly d e f o r m e d e n c l a v e .
O n t h e o t h e r h a n d , c o n c e r n i n g w e a n d
wp,
i t i s v a l i d t h a t :
w j = ~ [~{ +,(r) - ,J (r)] dr , j = e , p , (26)
0
w h e r e i + l ( r) a n d i(r) a r e g i v e n b y E q s . (1 8 ) w i t h t h e s u p e r s c r i p t j = p f o r t h e e l a s t i c - p l a s t i c
m a t e r i a l a n d b y E q . ( 5) f o r th e e l a s ti c m a t e r i a l , w i t h t h e s u p e r s c r i p t e . T a k i n g i n t o c o n s i d e r a t i o n
E q s . ( 5 ), ( 1 8) a n d ( 26 ), r e l a t i o n ( 2 5) h a s t w o u n k n o w n q u a n t i t i e s , O -oo a n d n .
T h e v a l u e f o r a~o m a y b e f o u n d b y a v e r a g i n g t h e % - s t re s s d i s t ri b u t i o n a l o n g t h e m i n i m u m
s e c t i o n o f t h e c r a c k e d p l a t e . T h u s t h e m e a n v a l u e o f t h e O -y -s tre ss c o m p o n e n t a l o n g t h e m i n i m u m
s e c t i o n y i e l d s t h e v a l u e a ~ . I n t r o d u c i n g t h i s v a l u e i n t o E q . (2 3 ) f o r r > r~ w e d e f i n e t h e v a l u e f o r
t h e e x p o n e n t n . I n t r o d u c t i o n o f t h e c o m p u t e d v a l u e s o f O-ooa n d n i n t o E q . ( 1 8 .2 ) g iv e s a n e w v a l u e
r i + 1 f o r t h e r a d i u s o f t h e e l a s t i c - p l a s t i c b o u n d a r y p o i n t ( r i+ 1 , a ~+ ~ = O -o ) n F i g . l b . T h i s p o i n t
r e p l a c e s t h e p r e v i o u s o n e ( rl , ~ = O-o )a n d l ie s o n t h e n e w e l a s t i c - p l a s t i c b o u n d a r y , i .e . i t i s a g a i n
r e p r e s e n t e d b y t h e p o i n t A i n F i g . l a .
T h e a b o v e d e s c r i b e d p r o c e d u r e c a n b e r e p e a t e d f o r t h e n e w i n c r e m e n t dg, a s w e l l a s f o r a n y
o t h e r 0 - d i r e c t i o n . T h i s i t e r a t i v e p r o c e d u r e c o n t i n u e s u p t o t h e d e s i r e d l o a d i n g s t e p o r , a n y h o w ,
u p t o a v a l u e o f ~o o n o t e x c e e d i n g O-o-
F i n a l l y , b y c o n n e c t i n g p o i n t s r b e l o n g i n g a t t h e s a m e l o a d - s t e p O -oo o r - rc < 0 < ~ , w e p l o t
s u c c e s s iv e e l a s t ic - p l a s t ic b o u n d a r i e s . T h e s a m e p r o c e d u r e c a n b e a p p l i e d e i th e r a l o n g t h e
e l a s t i c - p l a s t i c b o u n d a r y o r a l o n g c i r c l e s c e n t e r e d o n t h e c r a c k - t i p , i n o r d e r t o e v a l u a t e t h e
s t r e s s - c o m p o n e n t s . I t r e m a i n s n o w t o d e f in e t h e f o r m o f s t r es s d i s t r i b u t i o n i n s i d e th e p l a s t i c
e n c l a v e a n d e v a l u a t e t h e c o e f f i c i e n t s c ~ ( i = 1 , 2 , 3 ) i n r e l a t i o n ( 1 8. 1 ).
3 T h e p h o t o e l a s t i c e v a l u a t i o n o f t h e s t r e s se s a l o n g t h e m i n i m u m s e c t i o n
I t r e m a i n s t o e v a l u a t e t h e e x a c t s t r e s s d i s t r ib u t i o n i n s id e t h e e l a s t ic f ie l d a n d t h e p l a s t i c e n c l a v e s
a t le a s t a l o n g t h e m i n i m u m s e c t i o n o f t h e p l a t e i n o r d e r t o d e f i n e a c c u r a t e l y t h e f a c t o r s
c l ( i = 1 , 2 , 3 ) d e t e r m i n i n g E q . ( 1 8. 1) y i e l d i n g t h e d i s t r i b u t i o n o f t h e e f f e c t i v e s t r e s s ( r ) i n s i d e t h e
s t r es s f ie ld . F o r t h i s p u r p o s e u s e w a s m a d e o f t h e m e t h o d o f p h o t o e l a s t i c c o a t i n g s w h i c h i s
a p o w e r f u l e x p e r i m e n t a l m e t h o d f o r d e f in i n g t h e st r e ss d i s t r i b u t i o n i n s i d e a c o n t a i n e d p l a s t i c it y
p r o b l e m [ ) 4 ] - [1 61 .
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Elastic-plastic analysis of cracked plates 83
The m etho d consists of cementing a thin layer of a polymeric substance presenting a high
coeffic ient of birefringence on the polished surface of the specimen m ade of the same materia l
which is unde r study. Linear and circular polarizat ion of a white or mon och rom atic l ight bundle ,
impinging on and reflected from the surface of the load ed m atal l ic specimen, yie lds the isol inic
and the isochro matic ensembles created because of the stra ining of the coating, which fol lows the
deform ations of the specimen, which const i tutes the substra te of the canap6 . Then, while the
metal l ic specimen m ay d eform elastically or plast ically, the birefringent coating, which is a r a ther
bri t t le materia l an d yields insignificantly, is deform ed only elast ical ly. Then the co mp onen ts of
stra in in the coating may be readily determined by solving the e last ic problem. These stra ins,
being equal to the stra ins of the substra te , yie ld the means to evaluate the e lastic and plast ic
components of the plast ical ly deformed specimen.
Variat ions of the m etho d have been used with success for various e lastoplast ic problems
[17] - [20] and they are considered as reason ably accurate for obviou s reasons, but a lso since they
were used as standards to com pare the accuracies of other experim ental meth ods a nd especial ly
the num erical m etho ds [35],[36], [41]. The refor e, the m eth od will not be descr ibed her e in detail,
s ince i t will be applied in i ts simplest form to yield the values of the stra in co mp onen ts a lon g o nly
Fig. 2. The isochromatic patterns for a symmetrically edge-cracked plate at four different loading steps
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84 R S. Theoc aris
t h e m i n i m u m s e c t io n s o f th e s p e c i m e n s , w h i c h a r e s u f f ic i e nt f o r t h e d e t e r m i n a t i o n o f t h e
coeffic ien ts c i .
F o r t h i s p u r p o s e t h e s h e a r d i f fe r e n c e m e t h o d , a s it h a s b e e n i n t r o d u c e d a n d a p p l i e d
e x t e n s i v e ly in [ 3 7] , w a s u s e d f o r t h e s e p a r a t i o n o f th e C a r t e s i a n c o m p o n e n t s o f st r e s se s a l o n g t h e
m i n i m u m s e c t i o n s o f t h e t e s t e d c r a c k e d p l a t e s. S i n c e t h e s h e a r d i f f e r en c e m e t h o d i s b a s e d
e x c l u s iv e l y o n t h e e q u i l i b r i u m e q u a t i o n s f o r s t re s se s , i t s v a l i d i t y is n o t r e s t r i c te d o n l y t o p u r e l y
e l a s ti c p r o b l e m s a n d t h e r e f o r e i t s u se i s a l l o w e d a l l o v e r a n y i n t e r s e c t i o n o f th e e l a s t i c - p l a s ti c
stress field.
F o r t h e s e p a r a t i o n o f t h e C a r t e s i a n c o m p o n e n t s o f st r e ss e s it is n e c e s s a r y to d i s p o s e , b e s id e s
t h e e n s e m b l e o f is o c h r o m a t i c s o r t r a j e c t o r i e s o f t h e p r i n c i p a l s t r e s s d if f er e n c es , t h e n e t w o r k o f
i s o c li n ic s . F o r t h e m i n i m u m s e c t io n s , w h i c h c o n s t i t u t e s e c t io n s o f s y m m e t r y o f t h e p l a t e , i t is w e l l
k n o w n t h a t t h e c o m p o n e n t s o f st re s se s p a ra l le l a n d n o r m a l t o t h e m i n i m u m s e c ti o n a r e p r i n c ip a l
s t re s s es . T h e r e f o r e t h e s e a x e s b e l o n g t o a z e r o o r d e r i s o c li n i c a n d t h e s h e a r s t r e s s a t t h e a b o v e
s y s t e m i s e q u a l t o z e r o . T h i s s i m p l i fi e s c o n s i d e r a b l y t h e e v a l u a t i o n o f in d i v i d u a l s t r es s e s a l o n g
these s ec t ions .
F i g u r e 2 ( a - d ) p r e s e n ts t h e n e t w o r k o f t h e i so c h r o m a t i c s o f a s te e l p l a te , s y m m e t r i c a l ly
e d g e - c r a c k e d , s u b j e c t e d t o s i m p l e t e n s i o n , n o r m a l t o t h e c r a c k - a x i s . T h e m a t e r i a l o f th e p l a t e i s
a h i g h - y i e l d s t r e n g t h a l l o y s t e e l u n d e r t h e t r a d e n a m e U S S - T 1 , q u e n c h e d a n d t e m p e r e d , w h i c h
p r e s e n t s a l i n e a r e l a s ti c s t r e s s -s t r a i n r e l a t i o n s h i p u p t o t h e y i e l d p o i n t a n d a n a l m o s t f l a t y i e l d
c h a r a c t e r i s t i c c o r r e s p o n d i n g t o a v a l u e H E = 0 .0 5 . T h e m e c h a n i c a l p r o p e r t i e s o f th i s m a t e r i a l
are given in [38].
T h e f a m i l ie s o f i s o c h r o m a t i c s p r e s e n t e d i n F i g . 2 c o r r e s p o n d t o f o u r s u c c e s si v e s t e p s o f
l o a d i n g . W h i l e F i g . 2 a p r e s e n t s t h e l o a d i n g s t e p f o r i n c ip i e n t p l a s t i c d e f o r m a t i o n , F i g . 2 b
c o r r e s p o n d s t o a s i g n if i c a n t a d v a n c e o f t h e p l a s t i c e n c l a v e s a r o u n d t h e c r a c k t i p s o f t h e p l a t e .
F i g u r e s 2 c a n d 2 d i n d i c a t e c a s e s o f p r o g r e s s i v e l o a d i n g o f t h e p l a t e w h e r e t h e b r i t t le b i r e f r i n g e n t
/ I
- ~ . 0 - 3 . 0 - 2 .0 - I i0 0 A 1 . 0 2 . 0 3 . 0
~ccr ack-tip (cm)
Fig. 3. The network of isoclinics arou nd the crack tip o f an edge cracked plate u nder plane stress for a low
strain-hardening material H E = 0.05
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Elastic-plastic analysis of cracked plates 85
c o a t i n g , i n c a p a b l e t o f o l lo w t h e p l a s t ic d e f o r m a t i o n s o f th e e x t e n d e d p l a s t i c e n c l a v e s p r ese n t s
a pa r t i a l d i s loca t ion wi th the me ta l l i c subs t ra t e . The d i s t inc t ive ly de f ined d i s loca t ion bou nda r i e s
i n b o t h f ig u r e s c o r r e sp o n d e x a c t l y t o t h e r e sp e c ti v e e la s t ic - p la s t ic b o u n d a r i e s o f t h e m e t a l l ic p l a t e
u n d e r p l a n e - st r e s s c o n d i t i o n s .
I n d e e d , it is r e a so n a b l e a n d o b v i o u s t o a c c e p t t h a t a s so o n a s t h e c o m p o n e n t s o f s tr a in s ,
and e spec ia l ly the equiv a len t o r e f fec t ive s t ra in g ove rpasse s the e l a s t i c -p la s t i c bo un da ry a nd
s t a r t s t o inc rease rap id ly i t fo rces the s t il l e la s t i ca l ly de form ing br i t t l e coa t ing to d i s loca te f rom
the me ta l l i c subs t ra tum , thus c rea t ing th i s sp l i t ti ng or de l a m ina t ion e ffec t. I t i s wor th wh i l e
ind ica t ing th a t i t i s the f i r s t and on ly d i rec t expe r im enta l de f in i t i on of t he e l a s t i c -p l a s t i c
b o u n d a r y i n p l a s ti c a ll y d e f o r m e d m a t e r i a l s a n d t h e p i c t u r e c o i n c i d es e x a c t l y w i th r e sp e c t i v e
f igures o f such bou nda r i e s d e f ined e i the r expe r im enta l ly [17 ] - [20] o r nu mer i ca l ly [35], [36] and
[41].
F i g u r e 3 p r e sen t s t h e e n se m b l e o f th e i so cl in i cs f o r t h e s a m e p r o b l e m a n d f o r t h e i m p e n d i n g
plas t i c i ty load ing s t ep . How ever , t he ne two rk of i soe lin i c s i n the success ive load in g s t eps , wh en
the p l a s t i c enc laves a re spread ing ins ide the s t re ss fi eld, does no t ch ange s ign i f i can tly e spec ia l ly in
t h e z o n e d e l i n e a t e d b y a n e ll ip se h a v i n g a s m a j o r a x i s t h e m i n i m u m se c t io n o f t h e sp e ci m e n . T h e
o n l y v a r i a t i o n i s t h a t i t s m i n o r a x is i s r e d u c e d a s t h e l o a d i n g i s p r o c e s s in g a n d t h e o r d e r s o f th e
i so c h r o m a t i c s m o v e t o w a r d s t h e a x i s o f sy m m e t r y o f th e p l a t e . T h i s a g a i n f a c i li ta t e s t h e
c a l c u l a ti o n s e sp e c i a ll y i n t h i s i n t e r i o r z o n e , w h i c h i s t h e m o s t i m p o r t a n t o n e .
Th e sh ea r d i f fe rence ana lys i s , y i e ld ing the va lues of t he ind iv idua l s t re sses ins ide the s t re ss
f ie ld, a l low s the eva lua t ion of the coeff ic ients c i ( i = 1, 2, 3) and, thu s, Eq. (18.1) i s com ple te ly
de f ined . The n , app ly ing the com ple t e ly de f ined re l a t ionsh ips (18), (23) and (24) we can d e te rm ine
poin t s o f t he s t re ss f ie ld cor re spo ndin g to the same lo ad l eve l ~% e i the r a long the e l a s t i c -p l a s t i c
b o u n d a r y o r a l o n g a n y p o l a r d i s t a n c e i n s id e e i t h e r t h e e l a st ic o r t h e p l a s t i c r e g i o n o f t h e s t re s s
field.
F igure 4 presen t s t he e quiv a len t o r e f fec t ive s t re ss d i s t r ibu t ion # , norm a l i zed to th e y i e ld
stress ~ro in s imple t ens ion o f t he re spec t ive p l a in spec imen, a long the m in im um sec t ion , fo r
1.5
HIE =0.3
, \ . '- , ~ .35
. 0N
H =O ~ - - . . . . .
0 e las t i c )
0 0.1 0.2 0.3 O.t~ 0.5
xl
Fig. 4. Th e radial distribution of the reduced value of equivalent stress c?/Cro or 0 = 0 ~ versus the reduced
ratios r / a x / a from the crack tip for a material with H I E = 0.3
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86 R S. Theocaris
p a r a m e t r i c v a l u e s o f t h e r a t i o c % / a o , t h a t i s a s t h e e x t e r n a l l y a p p l i e d t e n s i le l o a d i s in c r e a s e d
c o n t i n u o u s l y .
I t i s w o r t h w h i l e n o t i n g t h a t t h e r e s u l t s g i v e n f o r t h e m a t e r i a l w i t h
H E
= 0 . 3 0 a l o n g t h e
m i n i m u m s e c t i o n r e s e m b l e t h e r e s p e c ti v e re s u l t s f o r t h e r e m a i n i n g t h r e e o t h e r t y p e s o f m a t e r i a l s
s t u d ie d . T h e o n l y d if f er e n c e, a p a r t f o r t h e v a l u e s o f a ~ / a o a t t h e c r a c k t i p a n d t h e m i d d l e p o i n t s o f
t h e m i n i m u m s e c t i o n s , i s t h a t , a s t h e m a t e r i a l s t r a i n - h a r d e n s t h e i n s i d e , t h e p l a s t i c e n c l a v e s
m a x i m a o f t h e a y - s t re s s es a r e r e d u c e d r e l a t iv e l y t o t h e i r r e s p e c t i v e v a l u e s a t t h e c r a c k t i ps .
H o w e v e r , t h e t r e n d s o f t h e c u r v e s a r e s im i l a r , t h u s j u s ti f y i n g t h e r e p r e s e n t a t i o n o f t h e v a r i a t i o n o f
t h e ( r ) -c u r v e s b y t h e r e l a t i o n (1 8 .1 ). O u t s i d e t h e a r e a a r o u n d t h e m i n i m u m s e c t io n s o f t h e p l a t e s
t h e i n t e r i o r m a x i m a o f t h e ( r ) - d i s tr i b u t i o n s p r o g r e s s i v e l y d i s a p p e a r a s t h e p o l a r a n g l e s
0 i n c r e a s e f r o m 0 = 0 ~ a t t h e m i n i m u m s e c t i o n t o 0 - -, _+ ~ z/2 a l o n g t h e A y - a x i s. T h i s m e a n s t h a t
the coe f f i c ien t s c~ in re la t ion (18 .1) a re s l igh t ly chang ing wi th the po la r ang le 0 .
Appl icat ion of the method and resul ts
T h e a b o v e - d e s c r i b e d m e t h o d w a s a p p l i e d i n f o u r t y p e s o f d i f f er e n t e n g i n e e r i n g m a t e r i a l s
p r e s e n t i n g d i f f e r en t a m o u n t s o f st r a i n h a r d e n i n g a n d i ts re s u l ts w e r e c o m p a r e d t o a l r e a d y
e x i s t i n g r e s u l t s d e r i v e d b y o t h e r m e t h o d s . T o i n i t i a t e t h e c o m p u t a t i o n s , t h e f o l l o w i n g m a t e r i a l
p r o p e r t i e s a n d c o n s t a n t s a r e r e q u i r e d : t h e m o d u l u s o f e l a st i c it y , E , P o i s s o n ' s r a t io , v , t h e p l a s t i c
t a n g e n t i a l m o d u l u s , H , a n d t h e y i e l d s tr e s s i n s i m p l e t e n s i o n a o o f t h e u n c r a c k e d m a t e r i a l . A l s o ,
Y
g / o " o =
0 7 5
8 e d g
H / E : O . 0 5
0.72
0.7
Fig. 5. Po lar distribution of the r~-
radii of the elastic-plastic bo und ary
around the crack-tip for various
levels of the applied load for the
materials with H E = 0.3
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Elast ic-plast ic analysis of crack ed plates 87
crack
HIE:0 3
Fig. 6. Po lar d istr ibution of the r l-radii of the elastic-plast ic
boundary around the crack- t ip for var ious l evel s of the
applied load fo r the materials with HIE = 0.5
t h e m a g n i t u d e o f t h e i n c r e m e n t d g o f t h e e q u i v a l e n t s t ra i n , g , w a s d e f i n e d a c c o r d i n g t o t h e d e t a il s
o f t h e s o l u t i o n o f e a c h p r o b l e m a n d t h e f ir st v a l u e o f th e r a t io r/a o f th e r a d i u s o f t h e i n i t i a l p l a s t i c
e n c l a v e f o r 0 = 0 ~ t o t h e c r a c k l e n g t h , a , a s e v a l u a t e d a c c o r d i n g t o t h e I r w i n f o r m u l a [ 1 ]. F o r a l l
t h e m a t e r i a l s i t w a s t a k e n c ro = 0 . 7 7 M P a , E = 2 1 0 M P a , v = 0 . 3 4 , dg = 0.001, r/a = 0 . 0 1 a n d
a = 7 .5 x 1 0 . 3 m . I t m u s t b e n o t e d t h a t t h e v a l u e o f d g a f fe c ts o n l y t h e n u m b e r o f i t e r a ti o n s
l e a v i n g u n a l t e r e d t h e r e s u l t s f o r s m a l l d d s .
U s i n g t h e d a t a d e r i v e d b y t h e s o l u t i o n o f t h e s y s t e m o f e q u a t i o n s a l o n g d i f f er e n t p o l a r a n g l e s
0 0 < < 0 < + ~ / 2 ) w e h a v e p l o t t e d i n F i g s. 5 a n d 6 t h e e la s t i c - p la s t i c b o u n d a r i e s f o r t h e t y p e s o f
m a t e r i a l s w i t h I I /E = 0 . 0 5 a n d H /E = 0 . 3 0 r e s p e c t iv e l y , a s t h e y h a v e e v o l u t e d w i t h t h e i n c r e a s e
o f t h e e x t e r n a l l y a p p l i e d l o a d cr~ ~ / ~ o = 0 . 3 8 - 0 . 75 f o r HIE = 0 . 0 5 a n d a~/C~o= 0.173 - 0 .70
f o r H /E = 0.30).
F i g u r e 7 p r e s e n t s t h e e v o l u t i o n o f t h e e l a st i c p l a s t i c b o u n d a r y i n a n in f i n it e p l a t e c o n t a i n i n g
a n i n t e r n a l c r a c k o f l e n g t h 2 a a n d s u b j e c t e d t o a t e n s il e l o a d cr~ a t i n f in i t y a n d , t h e r e f o r e , l o a d e d
i n p l a n e - st r e ss c o n d i t i o n s . T h e s e b o u n d a r i e s w e r e d e r iv e d b y a n u m e r i c a l s o l u t i o n o f t h e p r o b l e m
u s i n g t h e P A P S T p r o g r a m m e w i t h f i n i t e e l e m e n t s [ 3 9 ] , [ 4 0 ] .
A l l t h e s e c u r v e s a r e s y m m e t r i c t o t h e c r a c k a x i s w h e r e t h e y p r e s e n t a s h a l l o w m i n i m u m .
F u r t h e r m o r e , t h e y p r e s e n t m a x i m a a t s y m m e t r i c p o s i t io n s o n b o t h s id e s o f t h e c r a c k a x is a n d a t
a n g l e s 0m d e p e n d i n g o n t h e r a t i o s H/E a s w e ll a s o n t h e a m o u n t o f t h e e x t e rn a l l o a d i n g o f t h e
p l at e s. F i g u r e 8 p r e s e n t s t h e v a r i a t i o n o f t h e p o s i t i o n o f t h e se m a x i m a a s t h e l o a d i n g o f th e p l a te s
i s i nc r e a s i ng .
I t i s i n t e r e s t i n g t o p o i n t o u t t h a t t h e s e m a x i m a f o r 0 a r e d i m i n i s h i n g a s t h e l o a d i n g i s
i n c r e a s in g a n d p l a st i c e n c la v e s i n it ia t e a n d e x p a n d . T h i s m e a n s t h a t i n t r o d u c t i o n o f p l a s ti c it y
r e s u lt s i n a d i s p la c e m e n t o f t h e m a x i m a o f t h e e l a st ic - p la s ti c b o u n d a r i e s t o w a r d s t h e a x is o f
s y m m e t r y o f t h e p la te . M o r e o v e r , t h is p h e n o m e n o n o f r a p i d a n g u l a r d i s p la c e m e n t o f th e m a x i m a
i n th e e l a s t i c -p l a s t i c b o u n d a r y i s m o r e i n t e n s e f o r d u c t i le m a t e r i a l s a n d i t i s p r o g r e s s i v e l y s l o w e d
d o w n a s th e s t r a i n - h a r d e n i n g o f t h e m a t e r i a l i s h i g h e r.
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8 8 R S . T h e o c a r i s
in[ernalc r a c k
H/E=0.05
t ip
A O A I
F i g . 7 . P o l a r d i s t r i b u t i o n o f t h e r l - r a d i i o f t h e e l a s t ic - p l a s t i c b o u n d a r y a r o u n d t h e c r a c k - t i p f o r v a r i o u s l e v e ls
o f th e a p p l i e d l o a d a s d e r i v e d b y a p p l y i n g a f in i te e l e m e n t a n al y s is w i t h t h e P A P S T p r o g r a m m e f o r an
i n t e r n a ll y c r a c k e d p l a t e m a d e o f a lo w s t r a i n - h a r d e n i n g m a t e r i a l w i t h HIE = 0.05
75
70~ ~ ~ 0.50
1 6 5
60 o
55~
0 0.25 0.50 0.75
F i g . 8 . V a r i a t i o n o f t h e d i r e c t i o n O o f t h e m a x i m u m r a d i u s o f t h e e l a s t ic - p I a s t i c b o u n d a r y v e r s u s a p p l i e d
t o a d f o r m a t e r i a l s w i t h d i f f e r e n t H/E ratios
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Elast ic-plast ic analysis of crack ed plates 89
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5
H
0 . 1 0 . 2
H / E = 0 . 3 0
/ E : o s o
0 . 3 0 . ~ -
x / a
Fig. 9. Variat ion of the ex pon ent n versus reduce d radius rp/a of the elastic-plast ic bo un da ry for four
materials with different H/E-ratios
F i g u r e 9 p re s e n t s t h e v a r i a t i o n o f t h e e x p o n e n t n v e rs u s t h e r e d u c e d r a d i u s rp/a o f t h e
e l a s t ic - p l a s t ic b o u n d a r y f o r 0 = 0 ~ r p = x ) f o r t h e f o u r s e l e c t e d m a t e r i a ls . T h i s e x p o n e n t i s
a l w a y s g r e a t e r t h a n 1 /2 . A s it c a n b e s e en , t h e m o r e d u c t i l e t h e b e h a v i o r o f a m a t e r i a l t h e h i g h e r
v a l u e s f o r n a r e v a li d . T h e b r i t tl e m a t e r i a l
H / E
= 0 . 3 0 s h o w s a l i m i t e d v a r i a t i o n i n t h e v a l u e s o f
t h i s e x p o n e n t a n d , f i n a ll y , t h e v a l u e s f o r t h e b r i t tl e s t o n e w i t h H / E = 0 . 5 0 a l m o s t c o i n c i d e w i t h
t h e e l a s ti c v a l u e f o r n = 1 / 2, a s i t s h o u l d b e e x p e c t e d . C o n c e r n i n g t h e p o l a r v a r i a t i o n o f t h i s
p a r a m e t e r , i t is c o n s t a n t a r o u n d t h e c r a c k - t i p w i t h in 2 p e r c e n t.
F i g u r e 1 0 s h o w s t h e v a r i a t i o n o f t h e r e d u c e d v a l u e
rp/a rp/a - x/a)
o f t h e r a d i u s o f th e
e l a s t ic - p l a s t ic b o u n d a r y a h e a d o f t h e c r a c k - a x i s , f o r 0 = 0 ~ v e r s u s t h e r e d u c e d v a l u e a~/ao o f t h e
s t re s s a t i n f i n it y f o r th e s a m e f o u r m a t e r ia l s . F o r c o m p a r i s o n , t h e e l a s ti c a l ly c o m p u t e d r a d i u s i s
a l s o p l o t t e d . A s f o r t h e e x p o n e n t n , t h e h a r d e r t h e m a t e r i a l t h e s m a l l e r i s t h e s i ze o f t h e p l a s t i c a l l y
d e f o r m e d z o n e , t e n d i n g t o t h e l o w e r e l as ti c ) l i m i t i n th e c a s e o f t h e b r i t t l e s t m a t e r i a l w i t h
H / E
= 0.50.
F i g u r e 1 1 gi v es a c o m p a r i s o n o f t h e r e s u lt s o f t h e p r e s e n t m e t h o d w i t h a l r e a d y e x i st in g
m e t h o d s f o r a m a t e r i a l s i m u l a t i n g p o l y c a r b o n a t e o f B i s p h e n o l - A P C B A ) w i t h E = 2 .8 M P a ,
a 0 = 0 .05 M Pa , v = 0 .34 , H / E = 0 . 15 . T h e r e s u l t s p l o t t e d i n t h is f i g u r e a r e t a k e n f r o m
e x p e r i m e n t a l d a t a g i v e n in [ 34 ] f o r s h a r p l y c r a c k e d P C B A - p l a t e s , a s w e l l a s f r o m t h e n u m e r i c a l
r e s u lt s u s i n g th e f i n it e -e l e m e n t m e t h o d p r o g r a m m e P A P S T [4 0] a n d f r o m [7 ] f r o m a s o l u t i o n f o r
s m a l l s c a l e y i e l d i n g . I t c a n b e c o n c l u d e d t h a t a l l t h e e x i s t i n g d a t a a g r e e s a t i s f a c t o r i l y w i t h t h e
p r e s e n t r e su l ts , t a k i n g i n t o c o n s i d e r a t i o n t h a t P C B A w a s s u p p o s e d b i - l in e a r a n d t h a t t h e r e s u lt s
o f [7 ] a r e f o r h a r d e n i n g e x p o n e n t n = 5 w h i c h i s n o t e x a c t ly t h e c a s e f o r P C B A .
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9 0 R S . T h e o c a r i s
0A
0.3
~ 0 2
x
ll
0.1
0.8
0 6
~0.~ .
t~
x
I I I
2
0 . 2
m
0.25 0 50
o Mo ~
F i g . 1 0 . V a r i a l i o n o f t h e r e d u c e d v a l u e r Ja
o f t h e r a d i u s o f t h e e l a s t i c - p l a s t ic b o u n d a r y
a h e a d t h e c r a c k t i p f o r 0 = 0 ~ v e r s u s t h e
r e d u c e d v a l u e a ~ - | o o f t h e s tr e s s a t i n f i n i t y
f o r t h e f o u r m a t e r i a l s w i t h d i f f e r e n t H/E
r a t i o s
0.75
x : P o p s f 6 i f f o r d a n d H i l .f o n [4 0 1
A : Th eoc c~r is [31.]
] : H i t i ' o n a n d H u t c h i n s o n [ 7 ]
,. : P r e s e n f m e t h o d
f
0 0.25
7 :
0.50 0.75
or=l
F i g . 11 . C o m p a r i s o n o f t h e v a l u es rp/av e r -
s u s c r~ /C r o o b t a i n e d f r o m t h e p r e s e n t m e -
t h o d a n d t h r e e e x i s t i n g o n e s f o r p o l y c a r b o -
n a t e
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Elastic-plastic analysis of cracked plates 91
1.50
1.00
t O . 5 0
I-
I 1 I
..
Hi t ton ond Hufch inson
[71
'+-, ~, - - : P r e s e n ~ m e f h o d
r
0.50
0 ~1~.
,
\ j
, , 2 , , I
\
~12 3r~IZ~ g
Fig. 12. Variation of the polar stress-com-
ponen ts ~r, a0, ~r0, n the up per half-plane of
the cracked plate according to the present
method and that of Hil ton and H utchinson
[ ]
F ina l ly , F ig . 12 p r ese n t s the d i s t r ib u t io n of the r edu ced to the y ie ld - s t r e s s , ~ ro , s t r e s ses , a t , or0
a n d rr0 v e r s u s t h e p o l a r a n g l e 0 a r o u n d t h e c r a c k t i p (0 =< 0 < 7 ) a c c o r d i n g t o t h e p r e s e n t m e t h o d
a n d a s t h e y a r e r e p o r t e d i n [ 7 ] f o r a m a t e r i a l w i t h H/E = 0 . 15 . T a k i n g i n t o c o n s i d e r a t i o n t h e
a l r e a d y m e n t i o n e d d i v e rg e n c e s b e t w e e n t h e r e s u l ts o f th e p r e s e n t m e t h o d a n d t h o s e g i v e n b y
H i l t o n a n d H u t c h i n s o n [7 ] c o n c e r n i n g t h e e l a st i c - p la s t i c b o u n d a r y , t h e o b s e r v e d q u a l i t a t i v e
a g r e e m e n t m a y b e c o n s i d e r e d a s s a t i s f a c t o r y . I n a n y c a s e t h e r e s u l t s d e r i v e d b y t h e m e t h o d
d e v e l o p e d i n t hi s p a p e r p r e s e n t a m o r e r e a s o n a b l e d i s t ri b u t i o n i n a g r e e m e n t w i t h b a s i c p h y s ic a l
laws.
5 C o n c l u s i o n s
T h e p a p e r p r e s e n t s a s i m p l e a n d f a s t m e t h o d f o r t h e c o m p u t a t i o n o f t h e e l a s t ic - p l a s ti c s tr e s s f i el d
a r o u n d a c r a c k - t ip , b a s e d o n t h e f u n d a m e n t a l l a w s o f t h e t h e o r y o f p l a s t ic i t y . T h i s w a s d o n e b y
m e a n s o f t w o s i m p l e h y p o t h e s e s c o n c e r n i n g t h e r e l a ti v e b e h a v i o r o f t h e e l a st i c a n d p l a s t i c
c o m p o n e n t s o f s t r a i n s a n d t h e r a d i a l d i s t r i b u t i o n o f t h e e q u i v a l e n t s t re s s. A n ' ~ c o n d i t i o n
w a s , a l s o , a d d e d t o i n i t i a t e t h e w h o l e p r o c e d u r e . T h e t w o m a i n h y p o t h e s e s w e r e s h o w n t o
c o m p l y , a c c o r d i n g t o t h e c a se , e it h e r w i t h t h e t h e o r y o f e l a s ti c i ty , o r w i t h t h e i n c r e m e n t a l t h e o r y
o f p l a s t ic i t y , a s it i s d e s c r i b e d b y t h e P r a n d t l - R e u s s e q u a t i o n s .
I n a d d i t i o n , i n s o m e a s p e c t s , t h e p r e s e n t m e t h o d s e e m s t o r e s u l t i n b e t t e r a p p r o x i m a -
t i o n s o f r e a l s i t u a t io n s . F o r e x a m p l e , i n F i g . 1 2, a c c o r d i n g t o t h e s o l u t i o n g i v e n i n [ 7] t h e
r a d i a l s t re s s es , at , ( a n d c o n s e q u e n t l y ) h a v e n o n - z e r o v a l u e s (h i g h l y c o m p r e s s i v e a t) a t t h e
c r a c k f l a n k s ( 0 = 1 8 0 ~ a l t h o u g h t h i s r e g i o n i s g e n e r a l l y a s s u m e d a s f r ee o f l o a d i n g . T h e
p r e s e n t m e t h o d g i v e s s t r e ss - f re e c r a c k - li p s . T o t h is a s p e c t t h e p r e s e n t m e t h o d s e e m s t o b e
super io r .
A l s o , t h e r e s u lt s o b t a i n e d b y t h e p r e s e n t m e t h o d a r e m o r e s e n s i ti v e t o t h e v a r i a t i o n s o f t h e
m e c h a n i c a l p r o p e r t i e s o f t h e m a t e r i a l s , as c o m p a r e d w i t h t h o s e o f t h e fi n i t e -e l e m e n t m e t h o d . A t
l e a st , t h e s p e c if i c p r o g r a m m e ( P A P S T ) u s e d f o r c o m p a r i s o n i s c o m p l e t e l y i n se n s i t iv e t o t h e
i n f lu e n c e o f th e r a t i o H/E wh ich on ly s l igh t ly a f f ec t s i t s r e su l t s.
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92 R S . Theoca r is
T h e s e n si t iv i t y o f t h e p r e s e n t m e t h o d d e p e n d s c o n s i d e r a b l y o n t h e a c c u r a t e e v a l u a t i o n
t h r o u g h p h o t o e l a s t i c i t y o f t h e s t r e ss d i s t r i b u t i o n i n s i d e t h e p l a s t i c e n c l a v e s a n d t h e e v a l u a t i o n o f
t h e e q u i v a l e n t s t r es s i n s id e t h e p l a s t i c a l l y d e f o r m e d z o n e s . H o w e v e r , t h e s p e c i f ic p o l y n o m i a l
s e l e c t e d i n ( 1 8. 1) t o e x p r e s s t h e s t r e ss d i s t r i b u t i o n i n s i d e t h e p l a s t i c e n c l a v e s h a s t h e a d v a n t a g e t o
s i m u l a t e t h e " l o c a l - h a r d e n i n g e f f e ct " e x p e r i m e n t a l l y v e r i fi e d f o r P C B A [3 4] .
S i m i l a r r e m a r k s c a n b e m a d e f o r t h e i n i t i a l ( e la s ti c ) v a l u e o f t h e r a d i u s o f th e e l a s t i c - p l a s t i c
b o u n d a r y a h e a d o f t h e c r a c k . S u c h a q u a n t i t y i s n e c e s s a r y t o d e s c r i b e th e o n s e t o f p l a s t i c i t y a n d
i t s v a l u e m u s t b e c o m p a r a b l e t o t h e d i a m e t e r o f a g r a i n in m e t a l s o r a s i m i l a r q u a n t i t y i n o t h e r
m a t e r i a l s . T h i s " s t r u c t u r a l p a r a m e t e r " o b v i o u s l y af f ec t s t h e p l a s t i c b e h a v i o r o f t h e m a t e r i a l
a l t h o u g h i t i s a b s e n t f r o m t h e p o i n t o f v i ew o f c o n t i n u u m m e c h a n i cs . H o w e v e r i t is a c o m m o n u s e
f o r a ll m e t h o d s s o l v i n g p l a s t i c i t y p r o b l e m s t o m a k e u s e o f t h e I r w i n a s s u m p t i o n [ 1] f o r t h e
d e f i n i t i o n o f t h e i n i t i a l r p .
E x p o n e n t n , i n t r o d u c e d i n E q s . (1 4) , m u s t n o t b e c o n f u s e d w i t h t h e h a r d e n i n g c o e f f i ci e n t. T h e
l a t t e r i s a c o n s t a n t n e c e s s a r y t o d e s c r i b e a l g e b r a i c a l l y t h e w h o l e s h a p e o f t h e c u r v e 6 =
f g ) .
O n
t h e c o n t r a r y , o u r e x p o n e n t n v a r ie s n o t o n l y w i t h t h e s p e ci f ic m a t e r i a l , b u t a l s o w i t h t h e
c o r r e s p o n d i n g l o a d - o r s t r a i n - le v e l , e n c o u n t e r i n g t h e p a t h - h i s t o r y o f t h e m a t e r i a l , i n a s e n se
m o r e s u i t a b l e t o t h e c o n s i d e r a t i o n s o f th e t h e o r y o f p la s t i c i ty .
F i n a l l y , t h e m a n - a n d c o m p u t e r - c o s t t o g e t t h e s a m e r e s u l ts f r o m t h e p r e s e n t m e t h o d i s m u c h
l o w e r t h a t t h a t fr o m t h e P A P S T n u m e r i c a l m e t h o d w h i c h w a s a l s o u s ed , a n d a l t h o u g h w e h a v e
n o t d a t a o n t h e c o s t m e t h o d s s i m i l a r t o t h e m e t h o d b y H u t c h i n s o n , i t s ee m s p r o b a b l e t h a t t h e i r
c o s t is c o m p a r a b l e r a t h e r w i t h t h e P A P S T - c o s t t h a n w i t h t h e c o s t o f t h e p r e s e n t m e t h o d .
cknowledgement
T h e a u th o r i s in d e b te d to h i s s ec re t a ry Mrs . A . Z o g ra fa k i fo r h e lp in g h im in ty p in g th e m a n u s c r ip t a n d
p o t t ing the f igures o f the pap er .
References
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8 4 -1 1 3 (1 9 6 5 ) .
[2] W il l is, J . R . : Crack p r opa ga t i on in v iscoe las t ic media . J . Mech . Phys . S o l ids 15, 229 -2 40 (1967).
[3] Rice , J. R . : Ma them atica l ana lys is in the mechan ics o f f rac tu re . In : Fra c tu re , I I , An a dvance d t rea t ise
(Liebowitz , H. , ed .) , pp . 1 91- 311 . New York : Acad emic Press 1968 .
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[5] Hutch inso n , J . W. : S ingu la r beha v ior a t the end of a tens i le c rack in a ha rd en ing m ate r ia l . J. Mech . Phys .
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[6] Hutch inso n , J . W. : P las t ic st res s and s t ra in f ie lds a t a c ra ck t ip . J. M ech . Phys . So l ids 16, 33 7 - 347 (1968).
[7 ] Hi l ton , R D. , Hutch inson , J . W. : P las t ic in tens i ty fac to rs fo r c racked p la tes . Eng . Frac t . Mech . 3 ,
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[9] Edm unds , T . M. , Wil l is , J . R . : M atched asym pto t ic expans ions in non l inea r f rac tu re mechan ics -II I . In
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[10] Rice J . R ., Sorensen , E . R: C ont inu in g c rack -t ip de forma t ion and frac tu re fo r p lane s t ra in c rack g row th
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E l a s t i c - p l a s t ic a n a l y s i s o f c r a c k e d p l a t e s 9 3
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e d .) T h e 1 9 71 N a t i o n a l S y m p o s i u m o f F r a c t u r e o f M e c h a n i c s , P a r t I I , A S T M - S T P 5 1 4, 1 - 2 0 1 97 2 ).
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2 3 , 1 6 7 - 1 8 3 1 97 5 ).
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1961).
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p l a s t i c i t y p r o b l e m s . E x p . M e c h . 3, 2 0 7 - 2 1 4 1 9 63 ).
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