Solid solution strengthening in Ti-Al alloys
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1 dv
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where v is u n i t c e l l v o l u m e and c is solute c o n c e n t r a t i o n . In r e l a t e d work N o r d b u r y showed that the i n d e n t a t i o n h a r d n e s s of copper can v a r y l i n e a r l y with solute c o n c e n t r a t i o n . Subsequent work has a m p l y c o n f i r m e d these e a r l y r e s u l t s . 4 - a Mott and N a b a r r o 9 l a t e r u s e d the m i s f i t p a r a m e t e r in a m o r e g e n e r a l s e n s e and p r o p o s e d a s i m p l e p r e c i p i t a t i o n h a r d e n i n g model w h e r e b y the c r i t i c a l s h e a r s t r e s s i n c r e a s e s l i n e a r l y with t h e v o l u m e f r a c t i o n of p r e c i p i t a t e . T h e i r r e l a t i o n is given by
w h e r e A r is the s h e a r s t r e s s i n c r e a s e to be a s c r i b e d to p r e c i p i t a t i o n , /x is the s h e a r m o d u l u s , f is the v o l u m e f r a c t i o n of p r e c i p i t a t e , and E' in this c a s e , is the m i s f i t b e t w e e n s o l v e n t and p r e c i p i t a t e . N a b a r r o 4 extended this r e s u l t to include dilute s u b s t i t u t i o n a l solid s o l u tions. A difficulty with the M o t t - N a b a r r o model is that d i s location s e g m e n t s having s t r a i g h t e l e m e n t s over d i s t a n c e s l a r g e c o m p a r e d to the m e a n f r e e path between solute a t o m s should not i n t e r a c t s t r o n g l y with such solute a t o m s . Mott and N a b a r r o l a t e r a s s u m e d that d i s l o c a t i o n s bow between p i n n i n g points and found s t r e n g t h e n i n g for that case to be n o n l i n e a r in both c and E' .~o Following S e e g e r ' s 'I s u g g e s t i o n that the flow s t r e s s m u s t be divided into its t h e r m a l and a t h e r m a l c o m p o n e n t s , both F r i e d e l ~2 and Suzuki ~3 have c o n s i d e r e d s t r e n g t h e n i n g models in which both c o m p o n e n t s of flow s t r e s s i n c r e a s e l i n e a r l y with solute above a c r i t i c a l c o n c e n t r a t i o n and below the c r i t i c a l t e m p e r a t u r e . The c r i t i c a l t e m p e r a t u r e , Tc, is here defined as that t e m p e r a t u r e above which the flow s t r e s s dependence on t e m p e r a t u r e r e f l e c t s only that of the m o d u l u s . The F r i e d e l - S u z u k i t h e o r i e s find q u a l i t a t i v e s u p p o r t in the w o r k s of K o s t o r z 5 and Koss.6 H o w e v e r , n e i t h e r theory p r e d i c t s the o b s e r v e d r a t i o s between the t h e r m a l H. W. ROSENBERG is Supervisor, MetallurgicalResearch, TIMET Divisionof Titanium Metals Corp. of America, Henderson, Nev. 89015. W. D. NIX is Professor, Materials Science Department, Stanford Univeristy, Stanford, Calif. 94305. Manuscript submitted June 22, 1972. METALLURGICALTRANSACTIONS
and a t h e r m a l c o m p o n e n t s of flow s t r e s s in the Nb-W s y s t e m or the r e s p e c t i v e solute c o n c e n t r a t i o n s above which l i n e a r h a r d e n i n
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