The role of dislocations in the flow stress grain size relationships

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F O L L O W I N G the work of Hall 1 and P e t c h , z m a n y exp e r i m e n t a l and t h e o r e t i c a l s t u d i e s have b e e n conducted to u n d e r s t a n d the l i n e a r r e l a t i o n s h i p between the flow s t r e s s and the r e c i p r o c a l s q u a r e root of g r a i n s i z e , see Fig. 1. The o r i g i n a l m e c h a n i s m proposed by Hall 1 i n volves a pileup of d i s l o c a t i o n s a g a i n s t the g r a i n b o u n d a r y as shown in Fig. 2. The flow s t r e s s i s the e x t e r n a l s t r e s s which, with the help of the pileup, c r e a t e s a c r i t i c a l s t r e s s c o n c e n t r a t i o n at a c e r t a i n d i s t a n c e ahead of the pileup. His model was modified l a t e r by C o t t r e l l 3 who suggested that the c r i t i c a l s t r e s s conc e n t r a t i o n i s that which can unpin a d i s l o c a t i o n s o u r c e n e a r the g r a i n b o u n d a r y . A s i m i l a r m e c h a n i s m was p r o p o s e d by Petch ~ except that the s t r e s s c o n c e n t r a tion is at the g r a i n b o u n d a r y . Yielding is to take place when such s t r e s s c o n c e n t r a t i o n r e a c h e s the s t r e n g t h of the g r a i n b o u n d a r y . Since all these e a r l y m e c h a n i s m s involve pileups of d i s l o c a t i o n s , m a n y t h e o r e t i c a l s t u d i e s have b e e n conducted to e x a m i n e the p r o p e r t i e s of v a r i o u s k i n d s of p i l e u p s . These include s i n g l e - l a y e r s i n g l e - e n d e d p i l e ups in homogeneous, h e t e r o g e n e o u s , and a n i s o t r o p i c m e d i a , s i n g l e - l a y e r d o u b l e - e n d e d pileups, c i r c u l a r p i l e u p s , and m u l t i p l e - l a y e r pileups. No a t t e m p t will be made to r e v i e w all these s t u d i e s . Some r e l e v a n t ones will be s e l e c t e d and t h e i r effects on the H a l l - P e t c h r e l a t i o n examined. More r e c e n t l y o b s e r v a t i o n s have b e e n r e p o r t e d which indicate the need to s e a r c h for m e c h a n i s m s which do not involve pileups. T h e s e o b s e r v a t i o n s include the lack of d i r e c t evidence of pileups in pure m e t a l s , d i s l o c a t i o n g e n e r a t i o n in the p r e y i e l d m i c r o s t r a i n r e g i o n , an i n c r e a s e of H a l l - P e t c h slope in a l l o y s where p i l e ups a r e o b s e r v e d , and the effect or s e g r e g a t i o n of i m p u r i t i e s to g r a i n b o u n d a r i e s . Some of these nonpileup m e c h a n i s m s will be r e v i e w e d and c r i t i c a l l y a s s e s s e d .

tion p i l e u p s , the d i s c r e t e a p p r o a c h and the continuum approach. In the f i r s t a p p r o a c h , d i s c r e t e d i s l o c a t i o n s of n o r m a l B u r g e r s v e c t o r s a r e a r r a n g e d to a s s u m e e q u i l i b r i u m p o s i t i o n s . It is a p r o b l e m of solving s i m u l t a n e o u s n o n l i n e a r a l g e b r a i c e q u a t i o n s . In the second a p p r o a