Some observations of grain boundary ledges and ledges as dislocation sources in metals and alloys
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T H E r o l e of grain b o u n d a r i e s in engineering m e t a l s and alloys has been t r a d i t i o n a l l y v i s u a l i z e d as one involving the p i l e - u p of d i s l o c a t i o n s of like sign g e n e r ated f r o m F r a n k - R e a d s o u r c e s within the g r a i n s ; i m p l i c i t in the o r i g i n a l H a l l - P e t c h r e l a t i o n between the yield or flow s t r e s s , a, and the grain size, l (Hall, ~ Petch, 2-4 Codd and Petch,5), a = 0 o + kl-l/2;
a = o o + otpb(8rn/Tr)l/Zl-l/z,
[1]
where k is t r a d i t i o n a l l y r e f e r r e d to as the Petch slope. Yielding o r flow takes place when the p i l e d - u p d i s l o c a tions e x e r t sufficient s t r e s s at the grain boundaries so that p l a s t i c d e f o r m a t i o n can be propagated f r o m one grain to another. Li, 8 on the other hand, has d e r i v e d the H a l l - P e r c h r e lation in Eq. [1] f r o m a c o n s i d e r a t i o n of grain boundary s o u r c e s of d i s l o c a t i o n s without the need of dislocation p i l e - u p s . The b a s i s of L i ' s derivation involves i n t r i n s i c grain boundary s t r u c t u r e ; in p a r t i c u l a r dislocation ledges (LIT). Dislocation ledges a r e p r e s u m e d to be dislocation s o u r c e s in the g r a i n boundary, having a d e n s i t y m p e r unit length or p e r unit a r e a , i . e . , the number of ledges p e r unit length of boundary having unit a r e a , or the number of ledges p e r unit a r e a of boundary plane. The e x t e r n a l s t r e s s r e q u i r e d to gene r a t e d i s l o c a t i o n s f r o m these ledge s o u r c e s is p r e sumed to be s m a l l when the m i s f i t angle (or m i s o r i e n tation, O) is l a r g e (LIT). The flow s t r e s s at the t i m e of yielding will be the s t r e s s needed to move d i s l o c a t i o n s in the f o r e s t f o r m e d by the d i s l o c a t i o n s g e n e r a t e d by the g r a i n - b o u n d a r y ledges, and if this f o r e s t is v i s u a l ized as a volume extending around the g r a i n boundary, the dislocation density in the f o r e s t will be p r o p o r t i o n a l to the ledge density. As a consequence, the flow s t r e s s becomes 7 L. E. MURR is Professor and Head, Department of Metallurgical and Materials Engineering, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801. Manuscript submitted April 8, 1974. METALLURGICAL TRANSACTIONS A
[2]
where ~ is of the o r d e r of 0.4 and depends somewhat on the d i s l o c a t i o n a r r a n g e m e n t (Bailey and HirschS), /~ is the s h e a r modulus, and b is the dislocation B u r g e r s vector. If the strength of the grain boundary is due to g r a i n boundary ledge density, the s t r e s s r e q u i r e d by a pileup of d i s l o c a t i o n s to p e n e t r a t e the boundary is e s s e n t i a l l y the same as the s t r e s s n e c e s s a r y to initiate p l a s t i c d e f o r m a t i o n by generating d i s l o c a t i o n s f r o m g r a i n boundc r y ledges to f o r m a f o r e
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