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Delayed-elastic model for initiation and accumulation of creep
cavitation at high temperatures
Sinha, N. K.
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Ser
Trn
N21d
,,
1266
1
$
National Research
Conseil national
c.
2
Council Canada
de recherches Canada
'BEm
-
DELAYED-ELASTIC MODEL FOR INITIATION AND
7
ACCUMULATION OF CREEP CAVITATION AT HIGH
TEMPERATURES
4
by N.K. Sinha
ANALYZED
Reprinted from
Advances in Fracture Research
*---.-.
3Proceedings of the 6th. International
-
a0.-
!Conference on Fracture (ICF6)
I
-5s.
v
*New Delhi, India, 4
-
10 December 1984
D.
2295
-
2302
DBR Paper No. 1266
Division of Building Research
Price $1.00
*
Un modsle quantitatif a St6 mis au point pour d6terminer l'influence de la contrainte sur le temps d'incubation n6cessaire 2 la formation de fissures intergranulaires sous une pression modCr6e due au fluage A haute tempCrature, la contrainte minimum ngcessaire, et l'accumulation des dommages r6sultant des fissures. Ce modsle est bas6 sur une combinaison
de th5ories et d'observations : les fissures se forment 5 partir d'une d6formation Clastique difE6rSe critique qui correspond 5 un glissement entre les grains, et la cavit6 due au fluage augmente exponentiellement selon la d6formation 61astique diffEr6e qui dgpasse cette vnleur critique.
ADVANCES
IN
FRACTURE RESEARCH
Proceedings of the 6th International
Conference on Fracture (ICF6)
New Delhi, India, 4-10 December 1984
Editors
S. R. VALLURI,
D. M.
R. TAPLIN
P. RAMA RAO,
J. F.
KNOTT, R. D,UBEY
PERGAMON PRESS
1:
DELAYED-ELASTIC MODEL FOR INITIATION
AND ACCUMULATION OF CREEP CAVITATION
AT HIGH TEMPERATURES
N.
K. Sinha
Division of Building Research, National Research Council of Canada, Ottawa KIA OR6. Canada
ABSTRACT
A q u a n t i t a t i v e model h a s been developed f o r s t r e s s dependency o f t h e I n c u b a t i o n t i m e r e q u i r e d f o r i n i t i a t i o n of i n t e r g r a n u l a r c r a c k s u n d e r moderate s t r e s s d u r i n g high-temperature c r e e p , t h e minimum s t r e s s r e q u i r e d ,
and the subsequent accumulation o f damage from c r a c k i n g a c t i v i t y . It i s based on r comhination of t h e o r y and o b s e r v a t i o n : c r a c k s Porn a t a c r i t i c a l d e l a y e d - e l a s t i c s t r a i n c o r r e s p o n d i n g t o a c r i t i c a l grain-boundary s l i d i n g displacement, and f u r t h e r c r e e p c a v i t a t i o n i n c r e a s e s e x p o n e n t i a l l y w i t h d e l a y e d - e l a s t i c s t r a i n i n e x c e s s o f t h i s c r i t i c a l value.
KEYWORDS
High temperature c r e e p ; i n t e r g r a n u l a r f r a c t u r e ; c a v i t y n u c l e a t i o n ; i n c u b a t i o n time; c r e e p damage; damage accumulation; d e l a y e d e l a s t i c i t y ; p o l y c r y s t a l s ; grain-boundary s l i d i n g ; g r a i n s i z e ; i c e .
INTRODUCTION
D e f o n n a t l m as a r e s u l t of grain-boundary s l i d i n g ( s h e a r i n g ) p l a y 8 a dominant r o l e i n c r e e p p r o c e s s e s a t t e m p e r a t u r e s above a b o u t 0.4
T,,
where T, i s m e l t i n g p o i n t i n Kelvin. T h i s grain-boundary s l i d i n g could r e s u l t i n s t r e s a c o n c e n t r a t i o n s a t t r i p l e p o i n t s o r a t i r r e g u l a r i t i e s o r ledges i n t h e g r a i n boundaries s u f f i c i e n t t o n u c l e a t e c r a c k s (Zener, 1948; G i f k i n s , 1956). S i n h a (1979, 1982a) r e l a t e d g r a i l r b o u n d a r y s l i d i n g t o d e l a y e & e l a s t i c e f f e c t*
and showed t h a t c r a c k f o r m a t i o n d u r i n g constant-load c r e e p is i n i t i a t e d a t ac r i t i c a l d e l a y e & e l a s t i c s t r a i n c o r r e s p o n d i n g t o a c r i t i c a l grain-boundary s l i d i n g displacement. The p r e s e n t p a p e r e x t e n d s this h y p o t h e s i s and e s t a b l i s h e s a n i n t e r d e p e n d e n c e between damage a c c u m u l a t i o n d u r i n g c r e e p and t h e d e l a y e d - e l a s t i c s t r a i n under c o n d i t i o n s where t h e r a t e - c o n t r o l l i n g mechanisms a r e b o t h g r a i r r b o u n d a r y s l i d i n g and d i s l o c a t i o n c r e e p (Gandhf a n d Ashby, 1979; Mukherjee, Bird and Dorn, 1969).
EXPERIMENT
Gold (1967, 1972a) i n v e s t i g a t e d c r a c k i n g a c t i v i t y i n t r a n s v e r s e l y i s o t r o p i c , columnar-grained S-2 i c e s u b j e c t e d t o c o n s t a n t compressive l o a d a p p l i e d p e r p e n d i c u l a r t o t h e long d i r e c t i o n o f t h e g r a i n s a t 0.96 T,. The c r a c k s were c l e a r l y v i s i b l e ; they were l o n g and narrow, w i t h t h e l r l o n g d i r e c t i o n
i n t h e l o n g d i r e c t i o n of t h e g r a i n s , and t h e i r p l a n e tended t o b e p a r a l l e l t o t h e d i r e c t i o n of compressive s t r e s s . Formation of f i r s t c r a c k s f o r s t r e s s e s i n t h e r a n g e o f a = 5 x 1 0 ' ~ t o 2 x 1 0 ' ~ E (where F, i s Young's modulus) was, however, r e p o r t e d t o be a r e a s o n a b l y w e l l - d e f i n e d e v e n t i n p r e v i o u s l y undeformed specimens (Fig. 1). The f i r s t t h r e e c r a c k s w e r e a n a l y s e d f o r s t a t i s t i c a l s i g n i f i c a n c e . S t r e s s ( o ) dependence o f t h e t i m e of f o r m a t i o n of t h e f i r s t c r a c k ( t f c ) was s i m i l a r t o t h a t of c r e e p f a i l u r e time on a p p l i e d load f o r o t h e r m a t e r i a l s (Bartenev and Zuyev, 1968;
Zhurkov, 1965)
where t o and a a r e c o n s t a n t s , k is t h e Roltzmann c o n s t a n t , and Qo i s the a p p a r e n t a c t i v a t i o n energy a t z e r o s t r e s s . The p h y s i c a l a b s u r d i t y i n t h i s r e l a t i o n i s t h e p r e d i c t i o n t h a t c r a c k s would d e v e l o p e v e n n e a r z e r o s t r e s s ( a
-
.)'0 Gold (1967, 1972a) d i d not observe c r a c k i n g a c t i v i t y i n i c e w i t h i n t h e e x p e r i m e n t a l t i m e f o r stresses less t h a n about 0.6 M N T ~ - ~( - 6 x E l .
Gold (1972a, 1972b) a l s o s t u d i e d t h e accumulation of damage d u r i n g c r e e p i n S-2 i c e . The dependence of c r a c k d e n s i t y on time a t 0.96 Tm(-10°C) i s shown i n Fig. 2, where e a c h curve r e p r e s e n t s t h e a v e r a g e of s i x t e s t s . Crack d e n s i t y i s g i v e n a s t h e number of c r a c k s p e r u n i t a r e a because of t h e two- dimensional n a t u r e of t h e deformation and c r a c k formation.
DELAYED ELASTICITY AND CRACK INITIATION
The a u t h o r h a s a l r e a d y d i s c u s s e d t h e h y p o t h e s i s t h a t s h e a r o r s l i d i n g i n t h e grain-boundary r e g i o n s g i v e s r i s e t o d e l a y e d e l a s t i c e f f e c t ( S i n h a , 1979) and developed f o r m u l a t i o n s f o r s t r e s s ( o ) , time ( t ) , t e m p e r a t u r e (T), and g r a i n s i z e ( d ) dependence of t h e d e l a y e d - e l a s t i c s t r a i n ( d e s ) , ~ d , u n d e r u n i a x i a l l o a d i n g c o n d i t i o n s
where E i s Young's modulus and c l is a c o n s t a n t c o r r e s p o n d i n g t o t h e u n i t o r r e f e r e n c e g r a i n s i z e , d l ; b and s a r e c o n s t a n t s and l/aT i s t h e temperature- dependent r e l a x a t i o n time. The primary assumptions were
and
where E is t h e s t r a i n induced by grain-boundary s l i d i n g ( g b s ) ,
x
i s t h e gbsa v e r a g e grain-boundary d i s p l a c e m e n t , and K i s a c o n s t a n t n e a r l y e q u a l t o 1 ( G i f k i n s , 1956). The v a l u e s of b o t h s and K a r e g i v e n a s 1 i n Table 1 f o r t h e p r e s e n t a n a l y s i s f o r i c e . I n g e n e r a l i z i n g , however, b o t h c o n s t a n t s a r e r e t a i n e d i n t h i s paper.
T = 263 K I - 1 0 " C I
9 F I R S T C R A C K S O N L Y . 6 6 TESTS
A T H I R D C R A C K
'000.4 0 . 6 0 . 8 1 0 1 . 2 1 . 4 1 6 1 . 8 2.0 2 . 2
F i g . 1. S t r e s s dependence of average times t o formation of f i r s t three cracks i n S-2 i c e under compressive s t r e s s (Gold 1967, 1972a). The s o l i d l i n e i s based on present theory.
' y ' I 1 " ' I " ' I " ' I 1 ' ' I " ' l " ' , T = 2 6 3 K I - 1 0 " C I - o 0 . 7 8 M N . ~ . ~ - o 0 . 9 8 M N . ~ - ~
-
A 1 . 1 8 M N . ~ . '1
:: A*"C.
1 . 5 7 PAN.m" - :',.
A 1 . 9 6 M N . m - ' --
E X P E R I M E N T A L...
T H E O R Y...
__A,....
-
.... .
I: 1 6 8 i n 1:) i n 10Fig. 2. Time dependence of crack density f o r compressive s t r e s s (Gold 1972a). Broken l i n e s a r e based on present theory.
C r a c k s c a n d e v e l o p a t t r i p l e p o i n t s o r i r r e g u l a r i t i e s i n t h e g r a i n
b o u n d a r i e s d u r i n g c r e e p i f t h e stress c o n c e n t r a t i o n s prodoced by g r a i n -
boundarv s b p a r t n p : a r e n o t r e l a x e d by t h e p r o c e s s e s of i n t e r n a l accomntodnttan
(Sinha, 1984). A c r i t i c a l grain-boundary d i s p l a c e m e n t ,
2,
might b er e q u i r e d ~ ~ ~ t o c r:r lcks a r e i n i t i a t e d . Equations ( 3 ) and ( 4 ) g i v e t ! ~ e
c r i t i c a l g b s , E : ~ ~ , and t h e c r i t i c a l d e s ,
€2
ariD e l a y e d - e l a s t i c s t r a i n s c a l c u l a t e d f o r a l l o b s e r v a t i o n s ( o , t f c p a i r s ) i n
Fig. 1 a r e p r e s e n t e d i n Fig. 3. C a l c u l a t i o n s were made on t h e b a s i s of
e q u a t i o n ( 2 ) f o r a g r a i n s i z e of 4.5 mm and t h e v a l u e s o f t h e m a t e r i a l
c o n s t a n t s i n T a b l e 1. The f i r s t c r a c k s seem t o form, i r r e s p e c t i v e of s t r e s s
l e v e l , f o r
€2
= 1.04 x 10-4 ( w i t h a s c a t t e r of 10%). According t oe q u a t i o n (5) t h i s g i v e s Fc = 0.47 vm f o r K = 1 and d = 4.5 mm.
It s h o u l d be menttoned (and can b e shown by s u b s t i t u t i n g Ed i n e q u a t i o n ( 2 )
by t h e r i g h t s i d e of e q u a t l o n (5)) t h a t t h e c a l c u l a t e d v a l u e of
xC
was n o ta f f e c t e d by t h e somewhat a r b i t r a r y c h o i c e o f g r a l n s i z e . T h i s c h o i c e was
made because of t h e e x t e n s i v e g r a i n d i a m e t e r d e t e r m i n a t i o n s c a r r i e d o u t d u r l n g s t r e n g t h t e s t s ( S i n h a , 1981, 1982b) on i c e produced, essentially, by
t h e method used by Gold (1972a).
S u b s t i t u t i n g
€2
f o r E~ and t f c f o r t i n e q u a t i o n ( 2 ) and r e a r r a n g i n g g i v e sThus, on s u b s t i t u t i o n of
€2
from e q u a t i o n ( 5 ) ,Thus tfc is i n d e p e n d e n t of g r a i n s i z e . C a l c u l a t i o n s b a s e d on T a b l e 1 and
e q u a t i o n (6), w i t h E$ = 1.04 x 10-4 and d = 4.5 mm ( o r e q u a t i o n ( 7 ) , w i t h
iic = 0.47 urn) a r e compared w i t h t h e e x p e r i m e n t a l r e s u l t s i n Fig. 1. The
r a p i d i n c r e a s e i n t f c w i t h d e c r e a s e i n o, p a r t i c u l a r l y a t t h e lower end of
t h e s t r e s s . Ls now r e p r e s e n t e d more r e a l i s t i c a l l y t h a n hy t h e Zhurkov t y p e
e q w t i o n (1). S u b s t i t u t i o n of t f c = w i n e q u a t i o n s ( 6 ) and ( 7 ) g i v e s t h e
minimum s t r e s s , urnin, for c r a c k i n g as
T h i s i s independent of g r a i n s i z e and g i v e s omin = 0.5 M N * ~ - ~ f o r
-
xC= 0.47 um and o t h e r c o n s t a n t s i n T a b l e 1. It a g r e e s w e l l w i t h
G o l d ' s (1967, 1972a) o b s e r v a t i o n of minimum s t r e s s of 0.6 MN -m-2 f o r
c r a c k i n g .
CREEP D m G E ACCUMULATION
It i s p o s s i b l e t h a t more and more c r a c k s w i l l d e v e l o p f o r o > oa, and
t > t f c i f t h e number of s i t e s of s t r e s s c o n c e n t r a t i o n become c t r t i c a l owing
t o i n c r e a s e d s t r a i n ( i n c r e a s e i n grain-boundary s l i d i n g ) . T h i s p o s s i b i l i t y
F I R S T C R A C K 1 6 6 T E S T S 1 0 F I R S T C R A C K i a T H I R D C R A C K Fig. 3. Computed d e l a y e d - e l a s t i c s t r a i n v e r s u s s t r e s s f o r a l l e x p e r i m e n t a l p o i n t s i n F i g u r e 1. Fig. 4 . Dependence of e x p e r i m e n t a l l y observed c r a c k d e n s i t y on comptited d e l a y e d - e l a s t i c s t r a i n f o r experiments i n F i g u r e 2. e x p e r i m e n t a l r e s u l t s i n Fig. 2 and comparing t h e dependence o f c r a c k d e n s i t y w i t h computed v a l u e s f o r e a c h stress l e v e l . R e s u l t s shown i n Fig. 4, c a l c u l a t e d f o r d = 4.5 mm and i n f o r m a t i o n i n T a b l e 1, i n d i c a t e a strong dependence between c r a c k i n g a c t i v i t y a n d d e l a y e d - e l a s t i c s t r a i n ( o r g r a i n - boundary d i s p l a c e m e n t ) i r r e s p e c t i v e of s t r e s s l e v e l . T h e r e i s d e v i a t i o n from t h i s dependence a f t e r l o n g e r p e r i o d s , depend in^ nn P h n s t r e q s l e v e l . This 1s t o hr expected because t h e p u c e n t i a l s i t e s f o r c r a c k
n u c l e a t i o n would d e c r e a s e w i t h time and, moreover, t h e
accumulated damage should have a
profound i n f l u e n c e on t h e p r o c e s s e s of f u r t h e r c a v i t a t ioq dependency. The damage
a c c ~ u n u l a t i o n d u r i n g c r e e p may be e x p r e s s e d by
or N = N, exp
[ $
( Z
-
Z c ) ] ( 9 b )where 5 is a c o n s t a n t and J, = 5 K/d, and N c i s t h e c r i t i c a l c r a c k d e n s i t y
corresponding t o t h e c r i t i c a l v a l u e of B: o r
F.
Regression a n a l y s i s of t h e r e s u l t s i n Fig. 4 f o r e q u a t i o n ( g a l and
p r e v i o u s l y o b t a i n e d zC = 1.04 x lod4 (ZC 3 0.47 LIIU) and d = 4.5 wm gave
N c = 0.055 and 5 = 6.8 x 104 ($I = 1.33 x l o 7 m-l), w i t h a c o r r e l a t i o n c o e f f i c i e n t of 0.97, g i v i n g
N
-
0.055 exp [ 6 x l o 4 ( E-
~ 1.04 x 10'"}
( l O a )o r N = 0.055 exp t1.33 x
l o 7
(z
-
0.47 x ( l o b )T h i s i s shown i n Fig. 4 by t h e s o l i d l i n e .
The dependence of c r a c k i n g a c t i v i t y on s t r e s s and time c a n be o b t a i n e d by e l i m i n a t i n g cd i n e q u a t i o n ( g a l , u s i n g e q u a t i o n ( 2 )
c l d l b
N = N~ exp
[ E
(
-
(+)
-
exp[-
(aT t )Ij-~i
)I
d ( 1 l a )
Equation ( l l a ) w i t h t h e above v a l u e s o f Nc, 5,
€2
and t h e v a l u e s of o t h e rc o n s t a n t s from Table 1 is compared i n Fig. 2 w i t h t h e e x p e r i m e n t a l r e s u l t s .
TABLE 1 Creep Parameters f o r I c e Obtained from
E a r l i e r Creep Experiments (Sinha. 1979).
E = 9.5 G N * ~ - ~ ; Q = 67 kJ/mol (16 k c a l f m o l ) ;
c1 = 9 ; dl = 1 mm;
-
s = 1 ; K = 1; n = 3;263 K) = 2.5 x
lom4
s-1;?
1
Y:;iix
2-$Ts-l; = 1m.n-z,
T = 263 €v1Grain-boundary s h e a r i n g i n p o l y c r y s t a l s i s a complex p r o c e s s depending o n e x t e r n a l c o n d i t i o n s of s t r e s s and t e m p e r a t u r e and i n t e r n a l c o n d i t t o n s s u c h a s c r y s t a l l i n e s t r u c t u r e of t h e m a t r i x , type of d e f e c t , t e x t u r e and f a b r i c of t h e m a t e r i a l , g r a i n s i z e and i t s d i s t r i b u t i o n , i m p u r i t i e s i n t h e
n a t e r i a l and i n c l u s i o n s a t t h e g r a i n boundaries. The a n a l y s i s , however
h y p o t h e t i c a l , r e s u l t e d i n a meaningful way of h a n d l i n g e x p e r i m e n t a l o b s e r v a t i o n s , p a r t i c u l a r l y t h e g t r e s s and time dependence of t h e o n s a t of
c r a c k f o r n a t i o n and sirbsequent c r e e p damage. Although t h e p r e s e n t a n a l y s i s
i s based mainly on s p e c u l a t i o n , i t i s not w i t h o u t d i r e c t e x p e r i m e n t a l
evidence. The dependence of c a v i t y f o r m a t i o n on anount of g r a i r r b o u n d a r y
s l L d i n g was r e p o t t e d f i r s t by I n t r a t e r and Machlin (1959) i n copper b i c r y s t a l s . S i m i l a r a b s e r v a t i o n s have been r e p o r t e d by F l e c k , T a p l i n and
Reevers (1975) i n a copper a l l o y . That c r e e p f r a c t u r e t a k e s p l a c e when t h e
h a s a l s o been observed d i r e c t l y from experiments on copper b i c r y s t a l s (Watanabe, 1983). B i c r y s t a l r e s u l t s cannot, however, be d i r e c t l y a p p l i e d t o p o l y c r y s t a l l i n e m a t e r i a l s .
A f u r t h e r s i g n i f i c a n t t e s t of t h e model developed i n t h i s paper i s t o u s e
i t t o p r e d i c t t h e s t r a i n dependence of c r a c k i n g a c t i v i t y . Creep s t r a i n , E,
was d e s c r i b e d (Sinha, 1979) a s composed of t h r e e components
where E~ i s pure e l a s t i c s t r a i n ( a a/E), E~ i s t h e des d e s c r i b e d by
-
e q u a t i o n ( 2 ) , and E, i s v i s c o u s o r permanent deformation[=
\
t ( a/$ In,
1 where i s t h e v i s c o u s s t r a i n r a t e f o r u n i t o r r e f e r e n c e s t r e s s ,
v1
a1 = 1 M N * ~ - ~ , and n i s a c o n s t a n t ] .
-
ThusFor a given stress, temperature, and g r a i n s i z e , equation (13) g i v e s t o t a l s t r a i n a s a f u n c t i o n of time. A s E depends on g r a i n s i z e and s t r e s s , i t
can r e a d i l y be shown t h a t s t r a i n a t f i r s t c r a c k s w i l l depend on t h e s e q u a n t i t i e s . Equation (11) g i v e s , f o r t h e same imposed c o n d i t i o n s , t h e dependence of crack d e n s i t y on time. Equations (11) and (13) can t h e r e f o r e be used t o examine t h e dependence of c r a c k i n g a c t i v i t y on s t r a i n . A s e t of c a l c u l a t e d r e s u l t s is shown i n Fig. 5 t h a t compares w e l l w i t h t h e
experimental o b s e r v a t i o n s of Gold (1972b).
-
T = 263 K I - 1 0 " C I G R A I N SIZE = 4 . 5 m m
-
Fig. 5. T h e o r e t i c a l c r a c k d e n s i t y versus s t r a i n a t -lO°C f o r g r a i n s i z e of 4.5 mm.
CONCLUSION
A s i m p l e model h a s been developed t h a t a g r e e s w e l l w i t h o b s e r v a t i o n s o f c r a c k f o r m a t i o n i n p o l y c r y s t a l l i n e i c e d u r i n g c r e e p a t e l e v a t e d
temperatures. It d e s c r i b e s t h e s t r e s s , t i m e , and s t r a i n dependence o f c r a c k i n g a c t i v i t y a t a c o n s t a n t temperature. It i n d i c a t e s t h a t c r a c k s a r e i n i t i a t e d on a t t a i n i n g a c r i t i c a l grain-boundary s l i d i n g displacement t h a t
i s n o t dependent on g r a i n s i z e o r s t r e s s ; t h a t t h e c o r r e s p o n d i n g c r i t i c a l delayed e l a s t i c s t r a i n depends on g r a i n s i z e b u t not on s t r e s s ; and t h a t t h e c o r r e s p o n d i n g t o t a l s t r a i n depends o n b o t h g r a i n s i z e and s t r e s s . The model p r e d i c t s t h e s t r e s s dependence of o n s e t of c r a c k i n g a c t i v i t y b e t t e r t h a n t h e u s u a l Zhurkov t y p e r e l a t i o n . It a l s o p r e d i c t s t h a t c r a c k s d o n o t develop below a minimum s t r e s s , i r r e s p e c t i v e of g r a i n s i z e . A s t h e a n a l y s i s
i s v e r y g e n e r a l , t h e approach s h o u l d f i n d a p p l i c a t i o n t o h i g h t e m p e r a t u r e e n g i n e e r i n g problems i n v o l v i n g m e t a l s and o t h e r m a t e r i a l s .
ACKNOWLEDGMENT
The a u t h o r is i n d e b t e d t o L.W. Gold f o r v a l u a b l e d i s c u s s i o n and t o
R. Jerome f o r h i s a s s i s t a n c e i n p r e p a r i n g t h e g r a p h i c a l p r e s e n t a t i o n . T h i s paper i s a c o n t r i b u t i o n from t h e Divison of B u i l d i n g Research, N a t i o n a l Research C o u n c i l Canada, and i s p u b l i s h e d w i t h t h e a p p r o v a l o f t h e D i r e c t o r of t h e Division.
REFERENCES
Rartenev, G.M. and Y.S. Zuyev (1968). S t r e n g t h and F a i l u r e of V i s c o e l a s t i c
Materials,
Perganon P r e s s , 164.F l e c k , R.G., D.M.R. T a p l i n , and C.J. Reevers (1975). Acta Met
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2,
415. Gandhi, C. and M.F. Ashby (1979). Acta Met.,27,
1565.G i f k i n s , R.C. (1956). Acta Met.,
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98.Gold, L.W. (1967). Time t o f o r m a t i o n of f i r s t c r a c k s i n i c e . I n H. Oura (Ed.), P h y s i c s of Snow and I c e I n s t i t u t e of Low Temperature S c i e n c e , ~okkaid-370.
Gold, L.W. (1972a). The f a i l u r e p r o c e s s i n c o l ~ ~ m n a r g r a i n e d i c e . N a t i o n a l Research Council of Canada. D i v i s i o n of B u i l d i n g Research, NRCC 12637. Gold, L.W. (1972b). P h i l . Mag.,
26,
311. -I n t r a t e r , J. and E.S. Machlin (1959). Acta Met.,
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140.Mukherjee, A.K., J.E. Bird, and J.E. Dorn (1969).
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Am. Soc. M e t a l s ,3
155.Sinha, N.K. (1979).
Phil.
Mag.,, 825. Sinha, N.K. (1981). Exper. Mech.,21,
209.S i n h a , N.K. (1982a). Delayed e l a s t i c s t r a i n c r i t e r i o n f o r f i r s t c r a c k s i n i c e . IUTAM Symposium on Deformation and F a i l u r e of Granular M a t e r i a l s , D e l f t . A.A. Ralkema P u b l i s h e r . 323-330.
Sinha,
N.K.
(1982b). J. Mats. ~ c i . ,17,
785-802.S i n h a , N.K. (1984). ~ n t e r c r ~ s t a l l i n e c r a c k i n g , grain-boundary s l i d i n g , and d e l a y e q e l a s t i c i t y a t h i g h t e m p e r a t u r e s . J. Mats. S c i .
Watanabe, T. (1983). Met. Trans. A.
*,
531-545.Zener, C. (1948). ~ra-al~ Amer. Soc. Metals, Cleveland, Ohio, U.S.A.. 3. 3-31.
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