Theory of high temperature intercrystalline fracture under static or fatigue loads, with or without irradiation damage
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p l a s t i c deformation or by i r r a d i a t i o n damage. (Although wedge type voids a r e c o n s i d e r e d to f o r m by grain bounda r y sliding p r o c e s s e s t h e r e is no r e a s o n that part, even a m a j o r part, of t h e i r growth may not occur by vacancy condensation. Heald and Williams 2 have suggested such a vacancy growth m e c h a n i s m for w-type voids.) Skelton 3 f u r t h e r developed the H u l l - R i m m e r t h e o r y in an attempt to make it account for high t e m p e r a t u r e (push-pull) fatigue f r a c t u r e . In this paper we also gene r a l i z e Skelton's a n a l y s i s . In the a n a l y s i s to follow it is to be understood that the voids a r e in an e a r l y stage of t h e i r growth and consequently t h e i r r a d i u s a is s m a l l c o m p a r e d with the s e p a r a t i o n distance 2L between voids. (See the Appendix for meaning of symbols.) It is a s s u m e d that a grain boundary is a p e r f e c t sink or s o u r c e for v a c a n c i e s . Thus the v a c a n c y concentration is the equilibrium concentration a p p r o p r i a t e for the local value of the n o r m a l s t r e s s that a c t s a c r o s s the boundary. (The vacancy conc e n t r a t i o n c at a grain boundary i n t e r f a c e is an i l l - d e fined concept. So is the diffusion c o e f f i c i e n t Dgb for grain boundary v a c a n c i e s . However the product CDgb, which is p r o p o r t i o n a l to the grain boundary self diffusion coefficient, is a well defined quantity that can be m e a s u r e d r e a s o n a b l y well in an experiment. In the equations that a p p e a r in this paper the quantities c and Dgb always a p p e a r s as the product CDgb and there is no difficulty in considering c and Dgb to be p e r f e c t l y well defined quantities. There also is no r e a l difficulty in c o n s i d e r i n g that the s t r e s s changes only the value of VOLUME 5, AUGUST 1974-1743
c and not of D g b although it could change the v a l u e of both t e r m s . Our a n a l y s i s is c a r r i e d out in t e r m s of the m o r e e a s i l y v i s u a l i z e d c o n c e n t r a t i o n g r a d i e n t s r a t h e r than the c h e m i c a l p o t e n t i a l g r a d i e n t s . ) We a l s o a s s u m e that the r a d i u s a i s of such a v a l u e that the i n e q u a l i t y a < WDgb/D b is s a t i s f i e d . H e r e w is the e f f e c t i v e width (w ~ 50 nm) of a g r a i n b o u n d a r y and D b is the v o l u m e d i f f u s i o n c o e f f i c i e n t of l a t t i c e v a c a n c i e s . S i n c e D g b / D b ~ 1 0 2 at t e m p e r a t u r e s equal to about half the mel~ing t e m p e r a t u r e a n d D K b / D b ~ 1 0 4 at the m e l t ing point, v o i d s have to b e r a t h e r l a r g e b e f o r e t h i s i n e q u a l i t y is not s a t i s f i e d . When the i n e q u a l i t y i s s a t i s f i e d v a c a n c i e s can b e t r a n s p o r t e d f r o m the m o r e d i s t a n t p a r t s of the g r a i n b o u n d a r y to the p a r t s c l o s e r to a v o i
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