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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

*-

--.-.

3

Proceedings of the 6th. International

-

a

0.-

!

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

*

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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.

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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

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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 a

c 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).

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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 gbs

a 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.

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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 10

Fig. 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.

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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 e

r 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

ari

D 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 o

e 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 t

a 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 s

Thus, 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

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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

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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 r

c 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; = 1

m.n-z,

T = 263 €v1

Grain-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

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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 ] .

-

Thus

For 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.

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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

.,

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.,

6,

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.,

1,

140.

Mukherjee, A.K., J.E. Bird, and J.E. Dorn (1969).

Trans.,

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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SAVE

1

'

g5.00

US$120.00

!

BY

ORDERING

BEFORE

I

30

JUNE

1985

An Indispensable Reference Source for Materials Scientists,

Mechanical, Structural, and Design Engineers-

#

1 1

ADVANCES

(FRACTURE RESEARCH

I

*

Proceedings of the

In 6 Volumes

6th International

Over 4000 pages

Conference on Fracture

binding

New Delhi, 4 -10

~a tional

A

eronautical ~aborato&, Bangalore, India

Containing nearly 400 research and review

papers, this major six-volume work provides

the most comprehensive account of current

knowledge presently available on fracture

processes and the engineering application of

this knowledge to the design, fabrication and

operation of materials and structures.

An essential reference designed to enable

both engineers and scientists with a central

interest in fracture and its prevention to

define, examine and help solve current

problems associated with fracture failure

analysis and safe design. Of interest to

materials scientists, mechanical, structural,

aeronautical and design engineers.

Major Areas Covered:

Mechanics and mechanisms offracture,

fatigue and creep

Environmental effects

Fatigue and fracture of non-metallic

materials and composites

Dynamic fracture

Test techniques and fracture analysis

Engineering applications of fracture

mechanics

I I

i

4100 p p approx

1000 illus approx

1500 lit refs approx

June

1985

Regular price

0 08 029309 3 Hardcover

E375.00 US$600.00

,

Special pre-publication offer expires 30 June 1985

0 08 029310

7

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di300.00 US$480.00

@)

PERGAMON PRESS

(14)

T h i s p a p e r ,

w h i l e b e i n g d i s t r i b u t e d i n

r e p r i n t form by t h e D i v i s i o n of B u i l d i n g

R e s e a r c h ,

remains t h e c o p y r i g h t of

t h e

o r i g i n a l p u b l i s h e r .

It

s h o u l d n o t be

r e p r o d u c e d i n whole o r i n p a r t w i t h o u t t h e

p e r m i s s i o n of t h e p u b l i s h e r .

A

l i s t of a l l p u b l i c a t i o n s a v a i l a b l ' e from

t h e D i v i s i o n may be o b t a i n e d by w r i t i n g t o

t h e P u b l i c a t i o n s S e c t i o n , D i v i s i o n of

B u i l d i n g R e s e a r c h ,

N a t i o n a l R e s e a r c h

C o u n c i l

of

C a n a d a ,

O t t a w a ,

O n t a r i o ,

KIA OR6.

Figure

Fig.  2.  Time  dependence  of  crack  density  f o r   compressive  s t r e s s   (Gold  1972a)
TABLE  1   Creep  Parameters  f o r   I c e   Obtained  from  E a r l i e r   Creep  Experiments  (Sinha
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

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