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Activation energies for superplastic materials
Vol. I0, pp. 433 -436, 1976 Printed in the United States
Scripta METALLURGICA
ACTIVATION
Pergamon Press, Inc.
ENEliGIES FOli S U P E I I P L A S T I C M . A T E R / A L S
K.C. GIFKINS Metallurgy Department, University of Melbourne, Parkville 305Z Australia. (Received February 10, 1976)
This note discusses two recent analyses of results for superplastic materials (1,2) i n w h i c h a c t i v a t i o n e n e r g i e s QI a n d QII f o r t h e l o w - s t r e s s ( r e g i m e I) a n d s u p e r p l a s t i c ( r e g i m e II) r e g i o n s o f b e h a v i o u r w e r e o b t a i n e d . C a h o o n ( 1 ) u s e d s e v e r a l t y p e s of a n a l y s i s a n d o b t a i n e d w i d e l y d i f f e r i n g v a l u e s , o s t e n s i b l y f o r QII, w h i l s t M o h a m e d e t a l . c l e a r l y i d e n t i f i e d v a l u e s o f QI a n d QII a n d , r a t h e r s u r p r i s i n g l y , f o u n d QI -~ Q l ( l a t t i c e s e l £ - d i f f u s i o n ) , which does not agree with previous determinations and which further gives no support to any t h e o r y f o r r e g i m e I. T h e p r o c e d u r e s u s e d to o b t a i n b o t h s e t s of r e s u l t s a r e e x a m i n e d h e r e a n d a s u g g e s t i o n f o r a n a l t e r n a t i v e a p p r o a c h i s m a d e ; t h i s l e a d s to a n i n t e r p r e t a t i o n o f t h e e x p e r i m e n t a l r e s u l t s w h i c h y i e l d s QI < Q1" 1)
Cahoon(1) bases one of his analyses
on the general
~ , L.~ exp (-Q/lIT)
equation
..............
(I)
the subscripts ~ , L inferring measurements at constant stress and grain size. It is also i m p l i e d f o r E q u a t i o n 1, a n d m a d e m o r e e v i d e n t i n a l a t e r e q u a t i o n , t h a t t h e m e a s u r e m e n t s are also at constant stress exponent n( = l/m), as in
~ ,L
= ( A G b / k T ) (~/G)n exp (-Q/RT)
w h i c h i s t h e f o r m o f r a t e e q u a t i o n u s e d a l s o i n r e f . Z. m o d u l u s ).
..............
(Z)
(A i s a c o n s t a n t a n d G i s t h e s h e a r
Clearly, if n varies at constant • and L, the stress term makes a variable c o n t r i b u t i o n w h i c h t h e s i m p l e A r r h e n i u s t y p e p l o t ( E q u a t i o n 1) a s s u m e s to h a v e a r i s e n f r o m temperature alone. Z) The sigmoidal form of the logarithmic rate vs. stress curve and its displacement with temperature ( C a h o o n ' s F i g s . 3 a n d 4, w h i c h h a v e t h e i r c a p t i o n s i n t e r c h a n g e d i n the text), mean that n does vary at constant stress. F o r e x a m p l e , i n C a h o o n ' s F i g . 6 t h e v a l u e s o f n c o r r e s p o n d i n g to h i s p o i n t s a t 8 N / r n r n z h a v e n e q u a l t o 7 . 7 , 3 . 8 a n d 3. Z f o r 410, 460 a n d 5 1 4 ° C r e s p e c t i v e l y . This p r o b a b l y a c c o u n t s f o r t h e v e r y h i g h v a l u e of a c t i v a t i o n e n e r g y t h a t h e o b t a i n s f o r a n a l u m i n i u m - 17% c o p p e r a l l o y (163 k J / m o l e ) . S i m i l a r s p r e a d s of n w e r e p r e s e n t i n t h e a n a l y s e s f o r a c t i v a t i o n e n e r g y i n r e f e r e n c e s 2 a n d 14 c i t e d b y C a h o o n , a n d w h i c h g a v e h i g h v a l u e s of a c t i v a t i o n e n e r g y . There is, however, a further difficulty, in that the points at constant stress s t r a d d l e t h e r e g i m e s o f s u p e r p l a s t i c b e h a v i o u r I t o III a n d t h i s t a c i t l y a s s u m e s t h a t t h e r e i s no change in mechanism through this range. The general trend of opinion, echoed in Cahoon's Discussion, is to the contrary. 3) A n a l t e r n a t i v e a p p r o a c h a l s o u s e d b y C a h o o n ( 1 ) i s to f o l l o w A s h b y a n d V e r r a l l (3) a n d s u p p o s e t h a t r e g i m e I r e s u l t s f r o m t h e p r e s e n c e o f a t h r e s h o l d s t r e s s ~o • C a h o o n d e t e r m i n e d ~o b y e x t r a p o l a t i o n f r o m r e s u l t s a t e a c h t e m p e r a t u r e , f i n d i n g t h e s e v a l u e s to b e m u c h higher than the theory would predict. Activation energies were then extracted at constant values of
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( c r - v o ) , a n d f o u n d to b e = Q g b a t l o w s t r e s s e s , r i s i n g to 1 3 5 k J / m o l e (=Q1) a t ~ - V o = 1 2 N / r a m Z, b u t t h e p l o t l e a d i n g to t h e l a t t e r i s n o n - l i n e a r . For this reason, Cahoon r e v e r t e d i n h i s F i g . 10 to u s e a f o r m o f E q u a t i o n 2, b u t a g a i n a s s u m e d a c o n s t a n t v a l u e o f n = 4 . 3 , w h e r e a s h i s F i g . 4 s h o w s i t to t a k e v a l u e s o f 5 . 9 , 4 . 3 a n d 3 . 7 a t 1 4 N / m m Z. Once again, this must bias the value of the activation energy. I t s h o u l d a l s o b e n o t e d t h a t if o t h e r v a l u e s o f ( v - ~o) a r e t a k e n a n d p l o t t e d o n h i s F i g . 9, t h e v a l u e s o f t h e a c t i v a t i o n e n e r g i e s i n c r e a s e t h r o u g h o u t t h e r a n g e (~ - Vo) = 1 to 12 N / r a m Z, b u t r e a c h a p p r o x i m a t e l y 1 3 5 k J / m o l e a t 3 N / r a m 2, t h a t i s , a s s o o n a s ( v - cro) r i s e s s i g n i f i c a n t l y a b o v e Vo. I t m i g h t t h e r e f o r e b e c o n c l u d e d t h a t t h i s m o d e o f a n a l y s i s g i v e s QI = Q g b a n d Q1 f o r QII a n d QIII • In the light of the above remarks, i t i s n o t s u r p r i s i n g to f i n d t h a t u s i n g t h e e x p e r i m e n t a l ( e x t r a p o l a t e d ) v a l u e o f Vo a n d t h e v a l u e o f QI b a s e d o n t h i s , i n C a h o o n ' s F i g . l l , Ashby and Verrall's equation only fits the experimental results exactly at the lowest stress and increasingly diverges from them at higher stresses. It h a s b e e n f o r c e d to f i t a t t h e lowest stress. It i s n o t , t h e r e f o r e , a t r u e t e s t o f t h e v a l i d i t y o f t h e m o d e l o n w h i c h A s h b y and Verrall have based their equation. 4) M o h a m e d , S h e i a n d L a n g d o n (Z) h a v e a l s o d e t e r m i n e d v a l u e s f o r QI a n d QII, f o r a z i n c - Z2% a l u m i n i u m a l l o y . T h e y b a s e d t h e i r a n a l y s i s o n E q u a t i o n 2 a n d o b t a i n e d QI =Q1 a n d QII = Q g b • T h e y t o o k n = 4 . 5 a s c o n s t a n t t h r o u g h o u t r e g i m e I a t a l l t e m p e r a t u r e s and, i n d e e d , t h e i r F i g . 1 s h o w s g o o d f i t f o r s t r a i g h t l i n e s o f s l o p e c o r r e s p o n d i n g to t h i s . A v a l u e of QI = Q1, t h e y p o i n t o u t , i s n o t i n a g r e e m e n t w i t h A s h b y a n d V e r r a l l ' s t h e o r y f o r r e g i m e I a n d i s a l s o m u c h l a r g e r t h a n p r e v i o u s v a l u e s f o r t h i s m a t e r i a l ( s u m m a r i z e d i n t h e i r T a b l e 1). 5) A suggestion which can bring these and other results in the literature into a m o r e c o n s i s t e n t p a t t e r n i s to b a s e t h e a n a l y s i s f o r r e g i m e I o n a d i f f e r e n t e q u a t i o n ( a n d model). T h i s s u p p o s e s r e g i m e I to r e p r e s e n t t h e e m e r g e n c e o f GBS a s r a t e - c o n t r o l l i n g (4), w h e r e a s i n r e g i m e s II a n d HI GBS i s c o n t r o l l e d b y a c c o m m o d a t i n g m e c h a n i s m s a t t h e t r i p l e e d g e s . A f u l l d e s c r i p t i o n o f t h i s i n t e r l o c k i n g s e r i e s of m e c h a n i s m s h a s b e e n s u b m i t t e d e l s e w h e r e (5). T h e p o i n t to n o t e h e r e i s t h a t if GBS i n v o l v e s d i s l o c a t i o n m o t i o n (in t h i s c a s e of g r a i n - b o u n d a r y d i s l o c a t i o n s G B D ' s ) , w e c a n f o l l o w N i c h o l s (6) a n d e x p e c t t h e l o g a r i t h m i c p l o t o f ~ v s • to b e n d a r o u n d to g i v e n = 1 a t l o w s t r e s s e s , s e e F i g . 1, c u r v e W. W e do n o t n e e d , a t t h i s s t a g e , to s p e c i f y t h e m e c h a n i s m i n g r e a t e r d e t a i l . T h e e f f e c t o f r e d u c i n g t e m p e r a t u r e i s to d i s p l a c e t h i s c u r v e s o m e w h a t a s s h o w n ( c u r v e s X, Y, Z ) , s u c h t h a t t h e v a l u e o f n d e c r e a s e s w i t h d e c r e a s e d t e m p e r a t u r e a t c o n s t a n t stress. O n c e a g a i n , i t c a n b e s e e n t h a t a s s u m i n g n i s c o n s t a n t m a y b i a s t h e v a l u e of QI, making it appear too large.
j
FIG. 1
c
Log Stress.
Log strain rate vs. log stress curves (schematic) for temperatures T1 - T4 (TI>Tz>T3>T4), based on amodel involving dislocation motion through barriers of large activation area.
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It i s q u i t e e a s y to d r a w c u r v e s o f t h e t y p e s h o w n i n F i g . 1 t h r o u g h t h e p o i n t s i n F i g . 1 of r e f . 2 and o b t a i n as good a fit as f r o m the s t r a i g h t l i n e s w h i c h give, in e f f e c t , t h e a v e r a g e slope of the curve. 6) The s u g g e s t i o n m a d e in the p r e v i o u s s e c t i o n can be g i v e n p h y s i c a l s i g n i f i c a n c e and some support from other experimental results. An apparent actzvatlon A area may be found from A * = ( k T / b ) n (2/o-) .............. (3) a n e x p r e s s i o n p u t f o r w a r d b y L i (7). A t y p i c a l v a l u e o f A ~ a t t h e t r a n s i t i o n f r o m r e g i m e I to II b a s e d o n t h e w o r k o f M o h a m e d a n d L a n g d o n (8), t h e i r F i g . 3, i s 1 . 6 x 1 0 - 1 Z c m 2 w h i c h i s e q u i v a l e n t to ~ 1000b Z ; o t h e r m a t e r i a l s g i v e a s i m i l a r v a l u e (5). F a s t GBS i n b i c r y s t a l s o f z i n c h a s b e e n f o u n d b y T u r n e r to o c c u r a t t h e v e r y e a r l y s t a g e s o f e x t r e m e l y l o w - s t r e s s d e f o r m a t i o n (9, 10) a n d h i s r e s u l t s a l s o g i v e a n a c t i v a t i o n a r e a o f - 1000b Z. T h i s i s a l a r g e v a l u e f o r an a c t i v a t i o n a r e a , but in t e r m s of the m o v e m e n t of G B D ' s s u g g e s t s b a r r i e r s a s s o c i a t e d w i t h t h e l e d g e s a n d e d g e s b e t w e e n m i c r o f a c e t s o f " g o o d f i t " (11). T w o i n v e s t i g a t i o n s u s i n g f i e l d - i o n m i c r o s c o p y (12, 13) h a v e d e m o n s t r a t e d s u c h f e a t u r e s i n r a n d o m h i g h a n g l e g r a i n b o u n d a r i e s a n d s e t t h e i r m i n i m u m a r e a a t 2 0 - 5 0 a t o m s b y Z-5 a t o m s = 100b Z . F o r p r o t r u s i o n s (1Z) t h e b a r r i e r w o u l d h a v e p h y s i c a l d i m e n s i o n s o f Z to 3 t i m e s t h i s , t h e G B D ' s h a v i n g to c r o s s s e v e r a l e d g e s o f a m i c r o f a c e t t o a d v a n c e a l o o p a c r o s s t h e f a c e o f t h e p r o t r u s i o n o r m i c r o f a c e t ( s e e F i g . 2).
FIG. 2 GBS i n r e g i m e I, g e n e r a t e d b y t h e m o t i o n o f g r a i n - b o u n d a r y d i s l o c a t i o n s ( n u m b e r e d 1-6) h e l d up a t s t r u c t u r a l f e a t u r e s o f grain boundary as shown.
It i s t h i s l a r g e v a l u e o f A ~ t h a t g i v e s r i s e t o r e l a t i v e l y h i g h v a l u e s o f n a t c o m p a r a t i v e l y l o w s t r e s s e s i n r e g i m e I ( u s u a l l y n -- 5 a t ~ 1 M P a ) , t h i s a l s o m e a n s t h a t n r a p i d l y d r o p s to ~ 1 a t s t i l l l o w e r s t r e s s e s , i n a r e g i o n w h i c h h a s s e l d o m b e e n e x a m i n e d experimentally. A case where this may have been seen has been reported by Geckinli and B a r r e t t (14) w h o s h o w a s p u r w i t h s l o p e n = 1 o n t h e u s u a l s i g m o i d a l c u r v e a t t h e l o w e s t stresses. T h e y a t t r i b u t e t h i s to d i f f u s i o n c r e e p , b u t c a l c u l a t i o n u s i n g t h e f o r m u l a o f Kaj a n d A s h b y (15) s h o w s d i f f u s i o n c r e e p to b e m u c h t o o f a s t to a c c o u n t f o r t h i s s p u r (5). 7) T h e p r e s e n t m o d e l c a n b e d e v e l o p e d (5) t o y i e l d a n e q u a t i o n f o r s t r a i n r a t e i n v o l v i n g ~ n w h e r e n i s r e l a t e d to A ~ t h r o u g h E q u a t i o n 3, a n d t h i s t h e n g i v e s Q I < QI" A n u m b e r o f r e s u l t s i n t h e l i t e r a t u r e a r e b e s t f i t t e d w i t h QI = 0. Z Q1, w h i c h i s a l s o t h e v a l u e found by Turner bicrystals.
(9) f o r t h e a c t i v a t i o n e n e r g y f o r t h e i n i t i a l GBS i n h i s l i g h t l y l o a d e d
8) The c o m p l e x i t y of b e h a v i o u r in s u p e r p l a s t i c i t y and r e l a t e d f i e l d s of c r e e p i n v o l v i n g GBS b r i n g s a b o u t t h e n e e d f o r s p e c i a l c a r e i n e x t r a c t i n g v a l u e s f o r p a r a m e t e r s such as activation energy. Apart from the obvious possibility that a single activation energy d o e s n o t a p p l y , i t i s a l s o n e c e s s a r y to r e c o g n i z e t h a t t h e a p p a r e n t v a l u e o f Q d e p e n d s u p o n t h e m o d e l u s e d f o r t h e i n t e r a c t i o n o f s t r e s s a n d t e m p e r a t u r e i n o v e r c o m i n g b a r r i e r s to dislocation motion. This in turn depends on the assumed nature of these barriers. This u n d e r l i n e s t h e n e c e s s i t y to c h o o s e m o d e l s b y r e f e r e n c e to o t h e r c r i t e r i a , s u c h a s s t r u c t u r a l
changes.
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References 1.
JoR. Cahoon, M e t a l Sci. _9, 346 (1975).
Z.
F . M o h a m e d , S - A Shei and T . G . Langdon, A c t a M e t . , in p r e s s .
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M . F . A s h b y and R . A . V e r r a l l , ibid, Z_!I, 149 (1973).
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1%, C. G i f k i n s , S c r i p t a M e t . , 7, Z7 (1973).
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F . Ao N i c h o l s , M a t e r . S c i . E n g . , 8, 108 (1971).
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J . C . M . Li, " D i s l o c a t i o n D y n a m i c s " (Ed. A. 1%. R o s e n f i e l d et a l . ) 1969, New Y o r k ( M c G r a w Hill) p. 87,
8.
ToG. L a n g d o n and F . M o h a m e d , A c t a M e t . , Z__~3, 117 (1975).
9.
P o A . T u r n e r , P h . D . T h e s i s , I m p e r i a l C o l l e g e , London (1963).
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R . L . B e l l and T . G . Langdon, " I n t e r f a c e s C o n f e r e n c e " (Ed. 1%. Co Gifkins) 1970, Sydney ( B u t t e r w o r t h ' s).
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R . C . G i f k i n s , M a t e r . S c i . E n g , , _Z, 181 (1967).
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R y a n and J.Wo S u i t e r , P h i l . M a g . ,
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Bolin, R . J . B a y u z i c k and R . N . R a n g a n a t h a n , ibid, 3__ZZ,895 (1975).
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A . E . G e c k i n l i and C. 1%. B a r r e t t , S c r i p t a M e t . , 8, 115 (1974).
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1%. Raj and M . F . A s h b y , M e t . T r a n s . ,
1_~0, 7Z7 (1964).
Z, 1113 (1971).
Scripta METALLURGICA
ACTIVATION
Pergamon Press, Inc.
ENEliGIES FOli S U P E I I P L A S T I C M . A T E R / A L S
K.C. GIFKINS Metallurgy Department, University of Melbourne, Parkville 305Z Australia. (Received February 10, 1976)
This note discusses two recent analyses of results for superplastic materials (1,2) i n w h i c h a c t i v a t i o n e n e r g i e s QI a n d QII f o r t h e l o w - s t r e s s ( r e g i m e I) a n d s u p e r p l a s t i c ( r e g i m e II) r e g i o n s o f b e h a v i o u r w e r e o b t a i n e d . C a h o o n ( 1 ) u s e d s e v e r a l t y p e s of a n a l y s i s a n d o b t a i n e d w i d e l y d i f f e r i n g v a l u e s , o s t e n s i b l y f o r QII, w h i l s t M o h a m e d e t a l . c l e a r l y i d e n t i f i e d v a l u e s o f QI a n d QII a n d , r a t h e r s u r p r i s i n g l y , f o u n d QI -~ Q l ( l a t t i c e s e l £ - d i f f u s i o n ) , which does not agree with previous determinations and which further gives no support to any t h e o r y f o r r e g i m e I. T h e p r o c e d u r e s u s e d to o b t a i n b o t h s e t s of r e s u l t s a r e e x a m i n e d h e r e a n d a s u g g e s t i o n f o r a n a l t e r n a t i v e a p p r o a c h i s m a d e ; t h i s l e a d s to a n i n t e r p r e t a t i o n o f t h e e x p e r i m e n t a l r e s u l t s w h i c h y i e l d s QI < Q1" 1)
Cahoon(1) bases one of his analyses
on the general
~ , L.~ exp (-Q/lIT)
equation
..............
(I)
the subscripts ~ , L inferring measurements at constant stress and grain size. It is also i m p l i e d f o r E q u a t i o n 1, a n d m a d e m o r e e v i d e n t i n a l a t e r e q u a t i o n , t h a t t h e m e a s u r e m e n t s are also at constant stress exponent n( = l/m), as in
~ ,L
= ( A G b / k T ) (~/G)n exp (-Q/RT)
w h i c h i s t h e f o r m o f r a t e e q u a t i o n u s e d a l s o i n r e f . Z. m o d u l u s ).
..............
(Z)
(A i s a c o n s t a n t a n d G i s t h e s h e a r
Clearly, if n varies at constant • and L, the stress term makes a variable c o n t r i b u t i o n w h i c h t h e s i m p l e A r r h e n i u s t y p e p l o t ( E q u a t i o n 1) a s s u m e s to h a v e a r i s e n f r o m temperature alone. Z) The sigmoidal form of the logarithmic rate vs. stress curve and its displacement with temperature ( C a h o o n ' s F i g s . 3 a n d 4, w h i c h h a v e t h e i r c a p t i o n s i n t e r c h a n g e d i n the text), mean that n does vary at constant stress. F o r e x a m p l e , i n C a h o o n ' s F i g . 6 t h e v a l u e s o f n c o r r e s p o n d i n g to h i s p o i n t s a t 8 N / r n r n z h a v e n e q u a l t o 7 . 7 , 3 . 8 a n d 3. Z f o r 410, 460 a n d 5 1 4 ° C r e s p e c t i v e l y . This p r o b a b l y a c c o u n t s f o r t h e v e r y h i g h v a l u e of a c t i v a t i o n e n e r g y t h a t h e o b t a i n s f o r a n a l u m i n i u m - 17% c o p p e r a l l o y (163 k J / m o l e ) . S i m i l a r s p r e a d s of n w e r e p r e s e n t i n t h e a n a l y s e s f o r a c t i v a t i o n e n e r g y i n r e f e r e n c e s 2 a n d 14 c i t e d b y C a h o o n , a n d w h i c h g a v e h i g h v a l u e s of a c t i v a t i o n e n e r g y . There is, however, a further difficulty, in that the points at constant stress s t r a d d l e t h e r e g i m e s o f s u p e r p l a s t i c b e h a v i o u r I t o III a n d t h i s t a c i t l y a s s u m e s t h a t t h e r e i s no change in mechanism through this range. The general trend of opinion, echoed in Cahoon's Discussion, is to the contrary. 3) A n a l t e r n a t i v e a p p r o a c h a l s o u s e d b y C a h o o n ( 1 ) i s to f o l l o w A s h b y a n d V e r r a l l (3) a n d s u p p o s e t h a t r e g i m e I r e s u l t s f r o m t h e p r e s e n c e o f a t h r e s h o l d s t r e s s ~o • C a h o o n d e t e r m i n e d ~o b y e x t r a p o l a t i o n f r o m r e s u l t s a t e a c h t e m p e r a t u r e , f i n d i n g t h e s e v a l u e s to b e m u c h higher than the theory would predict. Activation energies were then extracted at constant values of
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( c r - v o ) , a n d f o u n d to b e = Q g b a t l o w s t r e s s e s , r i s i n g to 1 3 5 k J / m o l e (=Q1) a t ~ - V o = 1 2 N / r a m Z, b u t t h e p l o t l e a d i n g to t h e l a t t e r i s n o n - l i n e a r . For this reason, Cahoon r e v e r t e d i n h i s F i g . 10 to u s e a f o r m o f E q u a t i o n 2, b u t a g a i n a s s u m e d a c o n s t a n t v a l u e o f n = 4 . 3 , w h e r e a s h i s F i g . 4 s h o w s i t to t a k e v a l u e s o f 5 . 9 , 4 . 3 a n d 3 . 7 a t 1 4 N / m m Z. Once again, this must bias the value of the activation energy. I t s h o u l d a l s o b e n o t e d t h a t if o t h e r v a l u e s o f ( v - ~o) a r e t a k e n a n d p l o t t e d o n h i s F i g . 9, t h e v a l u e s o f t h e a c t i v a t i o n e n e r g i e s i n c r e a s e t h r o u g h o u t t h e r a n g e (~ - Vo) = 1 to 12 N / r a m Z, b u t r e a c h a p p r o x i m a t e l y 1 3 5 k J / m o l e a t 3 N / r a m 2, t h a t i s , a s s o o n a s ( v - cro) r i s e s s i g n i f i c a n t l y a b o v e Vo. I t m i g h t t h e r e f o r e b e c o n c l u d e d t h a t t h i s m o d e o f a n a l y s i s g i v e s QI = Q g b a n d Q1 f o r QII a n d QIII • In the light of the above remarks, i t i s n o t s u r p r i s i n g to f i n d t h a t u s i n g t h e e x p e r i m e n t a l ( e x t r a p o l a t e d ) v a l u e o f Vo a n d t h e v a l u e o f QI b a s e d o n t h i s , i n C a h o o n ' s F i g . l l , Ashby and Verrall's equation only fits the experimental results exactly at the lowest stress and increasingly diverges from them at higher stresses. It h a s b e e n f o r c e d to f i t a t t h e lowest stress. It i s n o t , t h e r e f o r e , a t r u e t e s t o f t h e v a l i d i t y o f t h e m o d e l o n w h i c h A s h b y and Verrall have based their equation. 4) M o h a m e d , S h e i a n d L a n g d o n (Z) h a v e a l s o d e t e r m i n e d v a l u e s f o r QI a n d QII, f o r a z i n c - Z2% a l u m i n i u m a l l o y . T h e y b a s e d t h e i r a n a l y s i s o n E q u a t i o n 2 a n d o b t a i n e d QI =Q1 a n d QII = Q g b • T h e y t o o k n = 4 . 5 a s c o n s t a n t t h r o u g h o u t r e g i m e I a t a l l t e m p e r a t u r e s and, i n d e e d , t h e i r F i g . 1 s h o w s g o o d f i t f o r s t r a i g h t l i n e s o f s l o p e c o r r e s p o n d i n g to t h i s . A v a l u e of QI = Q1, t h e y p o i n t o u t , i s n o t i n a g r e e m e n t w i t h A s h b y a n d V e r r a l l ' s t h e o r y f o r r e g i m e I a n d i s a l s o m u c h l a r g e r t h a n p r e v i o u s v a l u e s f o r t h i s m a t e r i a l ( s u m m a r i z e d i n t h e i r T a b l e 1). 5) A suggestion which can bring these and other results in the literature into a m o r e c o n s i s t e n t p a t t e r n i s to b a s e t h e a n a l y s i s f o r r e g i m e I o n a d i f f e r e n t e q u a t i o n ( a n d model). T h i s s u p p o s e s r e g i m e I to r e p r e s e n t t h e e m e r g e n c e o f GBS a s r a t e - c o n t r o l l i n g (4), w h e r e a s i n r e g i m e s II a n d HI GBS i s c o n t r o l l e d b y a c c o m m o d a t i n g m e c h a n i s m s a t t h e t r i p l e e d g e s . A f u l l d e s c r i p t i o n o f t h i s i n t e r l o c k i n g s e r i e s of m e c h a n i s m s h a s b e e n s u b m i t t e d e l s e w h e r e (5). T h e p o i n t to n o t e h e r e i s t h a t if GBS i n v o l v e s d i s l o c a t i o n m o t i o n (in t h i s c a s e of g r a i n - b o u n d a r y d i s l o c a t i o n s G B D ' s ) , w e c a n f o l l o w N i c h o l s (6) a n d e x p e c t t h e l o g a r i t h m i c p l o t o f ~ v s • to b e n d a r o u n d to g i v e n = 1 a t l o w s t r e s s e s , s e e F i g . 1, c u r v e W. W e do n o t n e e d , a t t h i s s t a g e , to s p e c i f y t h e m e c h a n i s m i n g r e a t e r d e t a i l . T h e e f f e c t o f r e d u c i n g t e m p e r a t u r e i s to d i s p l a c e t h i s c u r v e s o m e w h a t a s s h o w n ( c u r v e s X, Y, Z ) , s u c h t h a t t h e v a l u e o f n d e c r e a s e s w i t h d e c r e a s e d t e m p e r a t u r e a t c o n s t a n t stress. O n c e a g a i n , i t c a n b e s e e n t h a t a s s u m i n g n i s c o n s t a n t m a y b i a s t h e v a l u e of QI, making it appear too large.
j
FIG. 1
c
Log Stress.
Log strain rate vs. log stress curves (schematic) for temperatures T1 - T4 (TI>Tz>T3>T4), based on amodel involving dislocation motion through barriers of large activation area.
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It i s q u i t e e a s y to d r a w c u r v e s o f t h e t y p e s h o w n i n F i g . 1 t h r o u g h t h e p o i n t s i n F i g . 1 of r e f . 2 and o b t a i n as good a fit as f r o m the s t r a i g h t l i n e s w h i c h give, in e f f e c t , t h e a v e r a g e slope of the curve. 6) The s u g g e s t i o n m a d e in the p r e v i o u s s e c t i o n can be g i v e n p h y s i c a l s i g n i f i c a n c e and some support from other experimental results. An apparent actzvatlon A area may be found from A * = ( k T / b ) n (2/o-) .............. (3) a n e x p r e s s i o n p u t f o r w a r d b y L i (7). A t y p i c a l v a l u e o f A ~ a t t h e t r a n s i t i o n f r o m r e g i m e I to II b a s e d o n t h e w o r k o f M o h a m e d a n d L a n g d o n (8), t h e i r F i g . 3, i s 1 . 6 x 1 0 - 1 Z c m 2 w h i c h i s e q u i v a l e n t to ~ 1000b Z ; o t h e r m a t e r i a l s g i v e a s i m i l a r v a l u e (5). F a s t GBS i n b i c r y s t a l s o f z i n c h a s b e e n f o u n d b y T u r n e r to o c c u r a t t h e v e r y e a r l y s t a g e s o f e x t r e m e l y l o w - s t r e s s d e f o r m a t i o n (9, 10) a n d h i s r e s u l t s a l s o g i v e a n a c t i v a t i o n a r e a o f - 1000b Z. T h i s i s a l a r g e v a l u e f o r an a c t i v a t i o n a r e a , but in t e r m s of the m o v e m e n t of G B D ' s s u g g e s t s b a r r i e r s a s s o c i a t e d w i t h t h e l e d g e s a n d e d g e s b e t w e e n m i c r o f a c e t s o f " g o o d f i t " (11). T w o i n v e s t i g a t i o n s u s i n g f i e l d - i o n m i c r o s c o p y (12, 13) h a v e d e m o n s t r a t e d s u c h f e a t u r e s i n r a n d o m h i g h a n g l e g r a i n b o u n d a r i e s a n d s e t t h e i r m i n i m u m a r e a a t 2 0 - 5 0 a t o m s b y Z-5 a t o m s = 100b Z . F o r p r o t r u s i o n s (1Z) t h e b a r r i e r w o u l d h a v e p h y s i c a l d i m e n s i o n s o f Z to 3 t i m e s t h i s , t h e G B D ' s h a v i n g to c r o s s s e v e r a l e d g e s o f a m i c r o f a c e t t o a d v a n c e a l o o p a c r o s s t h e f a c e o f t h e p r o t r u s i o n o r m i c r o f a c e t ( s e e F i g . 2).
FIG. 2 GBS i n r e g i m e I, g e n e r a t e d b y t h e m o t i o n o f g r a i n - b o u n d a r y d i s l o c a t i o n s ( n u m b e r e d 1-6) h e l d up a t s t r u c t u r a l f e a t u r e s o f grain boundary as shown.
It i s t h i s l a r g e v a l u e o f A ~ t h a t g i v e s r i s e t o r e l a t i v e l y h i g h v a l u e s o f n a t c o m p a r a t i v e l y l o w s t r e s s e s i n r e g i m e I ( u s u a l l y n -- 5 a t ~ 1 M P a ) , t h i s a l s o m e a n s t h a t n r a p i d l y d r o p s to ~ 1 a t s t i l l l o w e r s t r e s s e s , i n a r e g i o n w h i c h h a s s e l d o m b e e n e x a m i n e d experimentally. A case where this may have been seen has been reported by Geckinli and B a r r e t t (14) w h o s h o w a s p u r w i t h s l o p e n = 1 o n t h e u s u a l s i g m o i d a l c u r v e a t t h e l o w e s t stresses. T h e y a t t r i b u t e t h i s to d i f f u s i o n c r e e p , b u t c a l c u l a t i o n u s i n g t h e f o r m u l a o f Kaj a n d A s h b y (15) s h o w s d i f f u s i o n c r e e p to b e m u c h t o o f a s t to a c c o u n t f o r t h i s s p u r (5). 7) T h e p r e s e n t m o d e l c a n b e d e v e l o p e d (5) t o y i e l d a n e q u a t i o n f o r s t r a i n r a t e i n v o l v i n g ~ n w h e r e n i s r e l a t e d to A ~ t h r o u g h E q u a t i o n 3, a n d t h i s t h e n g i v e s Q I < QI" A n u m b e r o f r e s u l t s i n t h e l i t e r a t u r e a r e b e s t f i t t e d w i t h QI = 0. Z Q1, w h i c h i s a l s o t h e v a l u e found by Turner bicrystals.
(9) f o r t h e a c t i v a t i o n e n e r g y f o r t h e i n i t i a l GBS i n h i s l i g h t l y l o a d e d
8) The c o m p l e x i t y of b e h a v i o u r in s u p e r p l a s t i c i t y and r e l a t e d f i e l d s of c r e e p i n v o l v i n g GBS b r i n g s a b o u t t h e n e e d f o r s p e c i a l c a r e i n e x t r a c t i n g v a l u e s f o r p a r a m e t e r s such as activation energy. Apart from the obvious possibility that a single activation energy d o e s n o t a p p l y , i t i s a l s o n e c e s s a r y to r e c o g n i z e t h a t t h e a p p a r e n t v a l u e o f Q d e p e n d s u p o n t h e m o d e l u s e d f o r t h e i n t e r a c t i o n o f s t r e s s a n d t e m p e r a t u r e i n o v e r c o m i n g b a r r i e r s to dislocation motion. This in turn depends on the assumed nature of these barriers. This u n d e r l i n e s t h e n e c e s s i t y to c h o o s e m o d e l s b y r e f e r e n c e to o t h e r c r i t e r i a , s u c h a s s t r u c t u r a l
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