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Tensile behaviour of alpha brasses at elevated temperature Orman, Leszek 1982

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TENSILE BEHAVIOUR OF ALPHA BRASSES AT ELEVATED TEMPERATURE by LESZEK ORMAN B . S c , ( M e t a l l u r g i c a l E n g i n e e r i n g ) , Academy o f M i n i n g and M e t a l l u r g y , C racow, P o l a n d , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Depar tment o f M e t a l l u r g i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1982 • '.-©Leszek Orman, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of M e t a l l u r g i c a l Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date December 14, 1982 (.3/81) i i ABSTRACT The t e n s i l e d e f o r m a t i o n b e h a v i o u r o f t h r e e a l p h a ( c o p p e r - z i n c ) b r a s s e s has been s t u d i e d i n t h e homologous t e m p e r a t u r e range o f 0 .4 t o 0 .9 and a t -4 -2 -1 s t r a i n r a t e s o f 10 and 10 s D i s c o n t i n u o u s y i e l d i n g was o b s e r v e d a t t h e l o w e r s t r a i n r a t e , but o n l y a t t e m p e r a t u r e s up t o 0.45 T m . In t h e f u l l r ange o f 0 .4 t o 0.6 T , t h e r e was a h i g h r a t e o f work h a r d e n i n g and v i r t u a l l y ho dependence o f f l o w s t r e s s on s t r a i n r a t e o r t e m p e r a t u r e . B e h a v i o u r i n t h i s r e g i o n has been i n t e r p r e t e d m a i n l y u s i n g dynamic s t r a i n a g i n g a r g u m e n t s , a l t h o u g h an a l t e r n a t i v e e x p l a n a t i o n based on t h e p o s s i b l e e x i s t e n c e o f l o n g range o r d e r i n g i n b r a s s e s a t t h e s e t e m p e r a t u r e s has been o f f e r e d . By c o n t r a s t , i n t h e 0 .7 < J ^ < 0 . 9 r a n g e , t h e r e was a s t r o n g s t r a i n r a t e and t e m p e r a t u r e dependence o f f l o w s t r e s s , and no work h a r d e n i n g o c c u r r e d . A n a l y s i s o f m i c r o s t r u c t u r e and o f d e f o r m a t i o n k i n e t i c s were c o n s i s t e n t w i t h t h e o c c u r r e n c e o f dynamic r e c r y s t a l l i z a t i o n a t t h e h i g h e r t e m p e r a t u r e s , wh i ch i s i n agreement w i t h t h e o b s e r v a t i o n s o f o t h e r wo r ke r s w i t h a l p h a b r a s s e s . A minimum i n t h e t e n s i l e f r a c t u r e d u c t i l i t y was o b s e r v e d a t i n t e r m e d i a t e t e m p e r a t u r e s i n t h e range 0.6 <T H < 0 . 7 5 . M e t a l l o g r a p h y r e v e a l e d t h a t t h e low d u c t i l i t y f r a c t u r e i n v o l v e d t he deve l opment and g rowth o f m i c r o c r a c k s ( v o i d s ) a t g r a i n b o u n d a r i e s t r a n s v e r s e t o t h e t e n s i l e a x i s . Mode ls a r e p roposed f o r t h e mechanism o f v o i d n u c l e a t i o n and g r o w t h , and t o a c c oun t f o r t h e s t r o n g t e m p e r a t u r e dependence o f t h e phenomenon. I t i s p roposed t h a t n u c l e a t i o n o f t h e m i c r o c r a c k s i s due t o m i g r a t i o n and c o a l e s c e n c e o f v a c a n c i e s a t g r a i n b o u n d a r i e s under c e r t a i n c o n d i t i o n s o f t e m p e r a t u r e and s t r e s s , p o s s i b l y a s s i s t e d by g r a i n boundary s l i d i n g . V a l i d a t i o n o f t h e t h e o r y was p r o v i d e d by e x p e r i m e n t s i n wh i ch spec imens were s u b j e c t e d t o a " s e n s i t i z i n g " t he rmo-m e c h a n i c a l t r e a t m e n t . " i i i T a b l e o f C on t en t s Page TITLE PAGE i ABSTRACT i i TABLE OF CONTENTS i i i ACKNOWLEDGEMENT I . INTRODUCTION 1 1. The D u c t i l i t y Minimum a t E l e v a t e d Tempe ra tu r e s 1 2 . D e f o r m a t i o n Mechanisms a t E l e v a t e d Tempe ra tu r e s 6 3. Dynamic S t r a i n Ag i ng 14 4 . O r d e r - D i s o r d e r T r a n s f o r m a t i o n i n B r a s s e s 15 I I . OBJECTIVES OF THE PRESENT WORK 16 I I I . EXPERIMENTAL PROCEDURE 17 1. M a t e r i a l s 17 2. Spec imen P r e p a r a t i o n 18 3. T e n s i l e T e s t i n g 20 IV RESULTS 3 1 1. T e n s i l e P r o p e r t i e s 31 2. E f f e c t o f Tempera tu re and S t r a i n Ra te on F low S t r e s s 32 3. E f f e c t o f Tempera tu re and S t r a i n Rate on D u c t i l i t y 44 4 . F i t o f t h e Data t o E q u a t i o n o f S t a t e 47 5. M e t a l l o g r a p h i c O b s e r v a t i o n s 51 i v fa^e V. DISCUSSION 68 1. Low Tempera tu re D e f o r m a t i o n Mode 68 2. H igh Tempera tu re D e f o r m a t i o n Mode 76 3. The D u c t i l i t y Minimum 77 4 . H igh Tempera tu re D u c t i l i t y 86 V I . CONCLUSIONS 90 LIST OF REFERENCES 1 92 APPENDIX I 97 APPENDIX I I 1 0 5 ACKNOWLEDGEMENT The a u t h o r w i s he s t o t hank D r . J . A . Lund f o r h i s engag i ng d i s c u s s i o n s and g u i d a n c e i n t h i s wo rk . F u r t h e r acknowledgement i s made t o P r o f . R. B u t t e r s f o r u s e f u l s u g g e s t i o n s i n equ ipment p r e p a r a t i o n , t o Mr . Ed K l a s s e n and Mr . H o r s t Tump f o r h e l p w i t h b u i l d i n g t h e equ ipment and t o Mr . N e i l Wa l k e r and M r s . Donna Schmid f o r h e l p w i t h e d i t o r i a l wo rk . F i n a n c i a l a s s i s t a n c e f rom t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g Re sea r ch C o u n c i l o f Canada i s g r a t e f u l l y a c know l e dged . 1 I . INTRODUCTION Two c h a r a c t e r i s t i c s t h a t d e t e r m i n e t h e f o r m i n g p r o p e r t i e s o f a me ta l a r e i t s r e s i s t a n c e t o p l a s t i c f l o w ( s t r e n g t h ) and i t s d u c t i l i t y under t h e c o n d i t i o n s o f d e f o r m a t i o n . . The f i r s t d e t e r m i n e s t h e s i z e o f t h e equ ipment needed f o r t h e f o r m i n g o p e r a t i o n , whereas t h e second d e t e r m i n e s t he maximum a l l o w a b l e d e f o r m a t i o n w i t h o u t r i s k o f f r a c t u r e . R a i s i n g t h e t e m p e r a t u r e o f w o r k i n g o f t e n has t h e b e n e f i c i a l e f f e c t o f l o w e r i n g s t r e n g t h and i n c r e a s i n g d u c t i l i t y . By no means i s t h i s u n i v e r s a l l y t r u e , however . Even f o r s i m p l e s o l i d s o l u t i o n a l l o y s t h e dependence o f d e f o r m a t i o n b e h a v i o u r on t e m p e r a t u r e i s complex and i n some r e s p e c t s p o o r l y u n d e r s t o o d . 1. The D u c t i l i t y Minimum a t E l e v a t e d Tempe ra tu r e s S e v e r a l n o r m a l l y p l a s t i c m e t a l s and a l l o y s e x h i b i t a d u c t i l i t y minimum i n t h e i n t e r m e d i a t e t e m p e r a t u r e range 0.4 - 0.6 Tm, where Tm i s t he m e l t i n g t e m p e r a t u r e i n deg rees K e l v i n . Hot w o r k i n g i s g e n e r a l l y c onduc t ed a t a t e m p e r a t u r e above t h i s range and t h e d u c t i l i t y " t r o u g h " does not t h en p r e s e n t a p r o b l e m . Moreove r t h e d u c t i l i t y t r o u g h i s deepe r f o r v e r y low s t r a i n r a t e s i n a l m o s t e v e r y r e p o r t e d c a s e , whereas most t r a d i t i o n a l ' ho t -wo rk o p e r a t i o n s i n v o l v e r e l a t i v e l y h i g h s t r a i n r a t e s . However i n some ca se s o f t h e hot r o l l i n g o f a l p h a b r a s s e s e m b r i t t l e m e n t o f t h i s t y pe has been i d e n t i f i e d , and t a k e s t h e form o f edge c r a c k i n g [ 1 ] . A l s o a fo rm o f l ow s t r a i n - r a t e hot d e f o r m a t i o n can a r i s e i n c o n t i n u o u s c a s t s t r a n d s and i n g o t s as a r e s u l t o f t h e t h e rma l c o n t r a c t i o n s t r e s s e s wh i ch a r e g e n e r a t e d as t h e c a s t i n g c o o l s , o r as a r e s u l t o f s t r e s s e s i n t r o d u c e d i n t h e hot b end i ng and ho t s t r a i g h t e n i n g o f t h e c a s t s t r a n d . One p o s s i b l e m a n i f e s t a t i o n o f t h e phenomenon a r i s e s i n c a s t 2 s t e e l i n g o t s and i s d e s c r i b e d as " pane l c r a c k i n g " [ 2 ] . A l t h o u g h t he d u c t i l i t y minimum was a p p a r e n t l y f i r s t r e p o r t e d by G.D. Bengouh i n 1912 t h e o r i g i n s o f t h e phenomenon a r e s t i l l no t e n t i r e l y u n d e r s t o o d . S e v e r a l phenomeno l og i c a l c o n c e p t s o f t h e b e h a v i o u r have been s ugge s t e d but t h e y a r e not m u t u a l l y c o n -s i s t e n t . In f a c t , as t h e d i s c u s s i o n wh i ch f o l l o w s w i l l i n d i c a t e , t h e r e i s c o n s i d e r a b l e c o n f u s i o n and c o n t r o v e r s y i n t h e p u b l i s h e d l i t e r a t u r e . D u c t i l i t y i s a f u n c t i o n not o n l y o f t e m p e r a t u r e and c o m p o s i t i o n but a l s o o f t h e s t a t e o f s t r e s s , s t r a i n and s t r a i n r a t e . As a r e s u l t , s i m p l e e n g i n e e r -i n g e l o n g a t i o n - t o - f r a c t u r e v a l u e s o b t a i n e d f rom a t e n s i l e t e s t c anno t be used t o p r e d i c t q u a n t i t a t i v e l y t h e maximum d e f o r m a t i o n a meta l can s u s t a i n , f o r i n s t a n c e , i n a s i n g l e r o l l i n g p a s s . The most r e l i a b l e method f o r s t u d y i n g w o r k a b i l i t y i s t o p r o c e s s a m a t e r i a l under p l a n t c o n d i t i o n s where a l l t h e v a r i a b l e s i n f l u e n c i n g d u c t i l i t y a r e s i m u l -t a n e o u s l y i n c l u d e d . However , such an app roach i s e x p e n s i v e and t h e r e s u l t s a r e d i f f i c u l t t o g e n e r a l i z e so t h a t t h e y can be a p p l i e d t o o t h e r p l a n t c o n d i t i o n s . T h i s has l e d t o t h e deve l opment o f s e v e r a l l a b o r a t o r y t e s t s . Fo r p r a c t i c a l r e a s o n s , t h r e e t e s t s t e n s i o n , c o m p r e s s i o n and t o r s i o n , a r e most w i d e l y used t o measure d u c t i l i t y . However , each o f t h e s e methods has i t s a dvan t age s and 1 i m i t a t i o n s . In t h e t e n s i o n t e s t , a common measure o f d u c t i l i t y i s t h e amount o f e l o n g a t i o n wh i ch p r e cede s f r a c t u r e . S i n c e t h i s i s composed o f a u n i f o r m e l o n g a t i o n and a h i g h l y l o c a l i z e d e l o n g a t i o n a s s o c i a t e d w i t h n e c k i n g , i t i s d i f f i c u l t t o a p p l y when compa r i ng d i f f e r e n t m a t e r i a l s . Ano t he r measure i s t h e p e r c e n t a g e r e d u c t i o n i n spec imen c r o s s s e c t i o n a t f r a c t u r e . However t h e h y d r o s t a t i c t e n s i l e s t r e s s d e v e l o p e d d u r i n g t e n s i l e n e c k i n g i n c r e a s e s t h e p r o b a b i l i t y o f f r a c t u r e . T h i s e f f e c t t e nd s t o p r e v e n t l a r g e d u c t i l i t y v a l u e s f rom b e i n g o b t a i n e d . P e r c e n t a g e r e d u c t i o n i n a r e a d a t a do not t h e r e f o r e 3 discr iminate well between very duct i le metals. Compression tes t ing suffers from s i m i l a r disadvantages. The complications of necking are avoided and thus larger to ta l s t ra ins are poss ib le . However the f r i c t i o n between platens and specimen leads to "bar re l ing" of the specimen, with consequent inhomogeneous deformation and some ambiguity in precise determination of s t ra in to f rac ture . In the tors ion t e s t , large s t ra ins can be achieved without p l a s t i c i n s t a b i l i t y , and are read i l y measured by the number of rotat ions to f a i l u r e . Results obtained at various temperatures can be used to define optimum hot working condi t ions . C lear ly however, s t r e s s , s t ra in and s t r a i n rate vary with rad ia l posit ion and care must be taken in the ca lcu la t ion of true stress and s t r a i n ra tes . Bearing in mind these l i m i t a t i o n s in character iz ing the d u c t i l i t y of metals , some published d u c t i l i t y vs temperature data can be examined. Figure 1 reveals the d u c t i l i t y minimum which has been observed for a range of non-ferrous metals and a l loys at intermediate temperatures. While the d u c t i l i t y minimum is more c l e a r l y recognizable in metals of low stacking fau l t energy, there are reports [3] of the ef fect in high stacking fau l t energy materials l i k e aluminum and Armco i r o n . S imi lar d u c t i l i t y vs temperature data for some steels which exhib i t a "trough" are presented in Figure 2 . It i s not recognized that the phenomenon in steels i s the same as that in non-ferrous a l l o y s . In fact most of the published work concerning the d u c t i l i t y trough in low a l l o y steels re lates the e f fect to grain boundary prec ip i tates of hard par t i c les [ 2 ] . By contrast , for low stacking fau l t f . c . c . s ingle phase materials l i k e alpha-brasses or copper-aluminum a l loys the d u c t i l i t y minimum is thought to be an i n t r i n s i c property of the pure s o l i d solut ion [ 3 ] , [ 7 ] , [ 8 ] , [ 9 ] . 4 F i g . 1 D u c t i l i t y o f v a r i o u s metals as a f u n c t i o n of temperature , a , c , d - a f t e r [ 3 ] , b - a f t e r [ 4 ] . 5 100 [ F i g . 2 Hot d u c t i l i t y o f s t e e l s (a) - Hot d u c t i l i t y o f two a u s t e n i t i c s t a i n l e s s s t e e l s [5] [b) - Hot d u c t i l i t y o f c a r bon s t e e l s [6] I00| 50 S l ra in Rat? 5.5 5 x !0~Ss - •— G r a m D ia . 23 Mm — » — G r a i n D ia . 400/Jm 300 1 00 500 600 700 800 900 1000 Temperature! K) ( a ) F i g . 3 ( a ) (b) (_ |~Cu-6*<.AI ( P r e s e n t W o r k ) L°4.76xl6*/S »476xl(?/s ,\ °4Ax1CT/s •4Axl03/s / \ L«2.2x1(52/s 300 400 500 600 700 BOO 900 1000 TOO 1200 Temoerature fK ) (b) The e f f e c t o f g r a i n s i z e on t h e d u c t i l i t y o f 70/30 b r a s s [10] The e f f e c t o f s t r a i n r a t e on t h e d u c t i l i t y o f Cu-6%A1 and Cu-30%Zn p o l y c r y s t a l s [11] 6 In t h e c a s e o f t h e work done w i t h a u s t e n i t i c s t a i n l e s s s t e e l s , ( F i g u r e 2a) t h e r e does no t appea r t o have been a s t u d y made o f t h e o r i g i n o f t h e d u c t i l i t y min imum. I t s h o u l d be no ted however t h a t s t e e l D i n F i g u r e 2a had a h i g h e r c o n t e n t o f n i t r o g e n (0.16% N compared w i t h 0.05% N i n s t e e l C ) . T h i s m igh t s u g g e s t a c o n t r i b u t i o n o f n i t r i d e p r e c i p i t a t e s t o d u c t i l i t y . The e x t e n t and dep th o f t h e d u c t i l i t y t r o u g h a l s o depend on g r a i n s i z e and s t r a i n r a t e . T h i s i s i l l u s t r a t e d i n F i g u r e 3 f o r a b r a s s and'.an a luminum b r o n z e . The d u c t i l i t y minimum i s i n t e n s i f i e d by i n c r e a s e d g r a i n s i z e and by l o w e r s t r a i n r a t e . The m e t a l l o g r a p h y o f m a t e r i a l s a t t h e d u c t i l i t y minimum has been s t u d i e d by s e v e r a l wo r ke r s [ 3 ] , [ 7 ] , [ 9 ] , [ 1 2 ] , [13] w i t h g e n e r a l agreement i n t h e o b s e r v a t i o n s . The d u c t i l i t y minimum c h a r a c t e r i s t i c o f low s t a c k i n g f a u l t , e ne r gy m a t e r i a l s i s a lway s a s s o c i a t e d w i t h i n t e r g r a n u l a r f r a c t u r e . Be l ow and above t h e t e m p e r a t u r e o f t h e min imum, f r a c t u r e i s t r a n s g r a n u l a r . The ma in e x p e r i m e n t a l o b s e r v a t i o n s can be summar ized as f o l l o w s : 1. Reduced d u c t i l i t y and g r a i n - b o u n d a r y c r a c k i n g a r e a s s o c i a t e d w i t h t h e p r i o r f o r m a t i o n o f s m a l l c a v i t i e s i n t h o s e g r a i n b o u n d a r i e s wh i ch a r e t r a n s v e r s e t o t h e a p p l i e d s t r e s s . The c a v i t i e s a r e not n e c e s s a r i l y seen a t t r i p l e j u n c t i o n s . 2. C a v i t i e s a r e no t found i n t w i n b o u n d a r i e s . At t h e i n t e r s e c t i o n o f t w i n s w i t h a g r a i n b ounda r y , c a v i t i e s i n t h e boundary o f t e n appear o n l y on one s i d e o f t h e t w i n n i n g p l a n e , i . e . , i t appea r s t h e r e i s a dependence on o r i e n t a t i o n d i f f e r e n c e a c r o s s t h e g r a i n b ounda r y . 2. D e f o r m a t i o n Mechanisms a t E l e v a t e d Tempe ra tu r e s I t i s r e a s o n a b l e t o deduce t h a t t h e c r a c k i n g mechan ism w h i c h o p e r a t e s a t t h e d u c t i l i t y minimum i s c onne c t e d w i t h t h e p r o c e s s o f d e f o r m a t i o n . 7 For t h i s r e a s on i t i s d e s i r a b l e t o examine p r o b a b l e d e f o r m a t i o n mechanisms i n t h e v i c i n i t y ^ o f t h e d u c t i l i t y min imum. The d i s l o c a t i o n t h e o r y assumes t h a t c r e a t i o n o f new g l i s s i l e d i s l o c a t i o n s i s r e l a t i v e l y e a s y , bu t t h a t t h e i r f u r t h e r movement can be s e v e r e l y o b s t r u c t e d . Ma j o r p r o c e s s e s i n t h e w o r k i n g o f m e t a l s a t low homologous t e m p e r a t u r e s (T^, t h e t e m p e r a t u r e as a f r a c t i o n o f Tm) a r e seen as a ) c r e a t i o n and movement o f d i s l o c a t i o n s , b) b l o c k i n g o f m o b i l e d i s l o c a t i o n a t o b s t a c l e s g i v i n g r i s e t o work h a r d e n i n g . I m p l i c i t l y , work h a r d e n i n g i n v o l v e s a l s o a l i m i t a t i o n on d u c t i l i t y s i n c e i n s e v e r e l y worked m a t e r i a l t he d e n s i t y o f d i s l o c a t i o n o b s t a c l e s i s so h i g h t h a t f u r t h e r d i s l o c a -t i o n movement becomes v e r y d i f f i c u l t . In o r d e r t o r e l a x s t r e s s e s a t d i s l o c a t i o n p i l e - u p s , t h e me ta l s i m p l y c r a c k s . At h i g h e r t e m p e r a t u r e s o t h e r p r o c e s s e s , known c o l l e c t i v e l y as "dynamic r e s t o r a t i o n " can o p e r a t e and r e l i e v e t h e e f f e c t o f work h a r d e n i n g . They a r e r e s p o n s i b l e f o r l o w e r i n g t h e d i s l o c a t i o n o b s t a c l e d e n s i t y , t hu s e n a b l i n g f u r t h e r d e f o r m a t i o n t o o c c u r . One i m p o r t a n t r e s t o r a t i o n p r o c e s s i s known as dynamic r e c o v e r y . A q u a l i t a t i v e model o f t h i s p r o c e s s has been s ugge s t e d by McQueen [ 1 4 ] . The b l o c k i n g o f d i s l o c a t i o n s a t t he o b s t a c l e s l e a d s t o a deve l opment o f i r r e g u l a r a r e a s o f h i g h d i s l o c a t i o n d e n s i t y . In o r d e r t o r educe t he a c cumu l a t ed ene rgy o f t h e s t r e s s f i e l d d e v e l o p e d i n t h e s e a r e a s t h e d i s l o c a t i o n s c r o s s g l i d e o r c l i m b c r e a t i n g d i s l o c a t i o n w a l l s i n wh i ch a s u b s t a n t i a l amount o f e l a s t i c ene rgy i s r e l e a s e d due t o t h e o v e r l a p p i n g o f t e n s i l e and c o m p r e s s i v e s t r e s s f i e l d s f rom n e i g h b o u r i n g d i s l o c a t i o n s . T h i s l e a d s t o t h e deve l opment o f d i s l o c a t i o n s u b g r a i n s w i t h i n g r a i n s o f t h e m e t a l . The s u b g r a i n b o u n d a r i e s : a r e a r e a s o f t a n g l e d d i s l o c a t i o n s , v e r y o f t e n o f d i f f e r e n t B u r g e r s v e c t o r s . When a d d i t i o n a l d e f o r m a t i o n i s i m p a r t e d , new d i s l o c a t i o n s a r e g e n e r a t e d by t he l o n g e r l i n k s bowing ou t f rom e x i s t i n g s u b - b o u n d a r i e s . R e p e t i t i o n o f t h i s t e nd s t o b reak down and d e s t a b i l i z e t h e a r r a y . In any one s u b g r a i n , many 8 s i m i l a r d i s l o c a t i o n s a r e mov ing a c r o s s i t on one o r more p a r a l l e l s l i p p l a n e s d e f i n e d no t o n l y by t h e a p p l i e d s t r e s s but a l s o by t h e s t r e s s f i e l d o f t h e s u r r o u n d i n g s u b - b o u n d a r i e s . Some d i s l o c a t i o n s i n p a s s i n g t h r o u g h a s u b g r a i n w a l l become j o gged w i t h consequen t r e d u c t i o n i n t h e i r m o b i l i t y . O t he r d i s -l o c a t i o n s i n t e r a c t and ann'ihi.1 a t e . w i t h d i s l o c a t i o n s i n t h e w a l l r e d u c i n g i t s s t a b i l i t y . In s t i l l o t h e r c a s e s t h e y combine w i t h d i s l o c a t i o n s i n a s u b -boundary i n c r e a s i n g i t s s t a b i l i t y . However , i t i s l i k e l y t h a t d i s l o c a t i o n s o f o p p o s i t e s i g n a r e mov ing i n t o , and combine w i t h , t h e same boundary f rom t h e o p p o s i t e s i d e and on d i f f e r e n t s l i p p l a n e s . The two o p p o s i t e s e t s o f d i s l o c a t i o n s i n t h e boundary a t t r a c t each o t h e r and c l i m b o r g l i d e w i t h i n t o a n n i h i l a t e . Thus t h e c o n t i n u o u s p r o c e s s o f b u i l d i n g up and d e s t r o y i n g sub -b o u n d a r i e s t a k e s p l a c e , wh i c h may be r e s p o n s i b l e f o r a s t e a d y s u b g r a i n s i z e and ~ a c o n s t a n t f l o w s t r e s s a t h i g h e r t e m p e r a t u r e s and l a t e r s t a g e s o f d e f o r m a t i o n . The c r i t i c a l f a c t o r s d e t e r m i n i n g t h e e x t e n t o f dynamic r e c o v e r y a r e t e m p e r a t u r e , s t r a i n , s t r a i n r a t e , and an i n h e r e n t p r o p e r t y o f t h e m e t a l , s t a c k i n g f a u l t ene rgy [ 1 5 ] . The l a s t o f t h e s e i s r e l a t e d t o t h e p r o p e n s i t y f o r c r o s s g l i d e and c l i m b o f d i s l o c a t i o n s and i t i s t h e r e f o r e t h e main f a c t o r i n d e t e r m i n i n g wh i ch m e t a l s a r e l i k e l y t o d y n a m i c a l l y r e c o v e r . The e x t e r n a l p a r ame te r s o f d e f o r m a t i o n o n l y l i m i t t h e deg ree o f dynamic r e c o v e r y f rom p r a c t i c a l l y z e r o a t l ow t e m p e r a t u r e t o f u l l r e s t o r a t i o n o f l ow d i s l o c a t i o n d e n s i t y a t h i g h e r t e m p e r a t u r e s and l a r g e r s t r a i n s . For m a t e r i a l s o f low s t a c k i n g f a u l t e ne r gy where c r o s s g l i d e and c l i m b o f d i s l o c a t i o n s a r e d i f f i c u l t , o n l y a l i m i t e d deg ree o f dynamic r e c o v e r y t a k e s p l a c e even a t h i g h t e m p e r a t u r e s . Such a m a t e r i a l t e nd s t o work harden much f a s t e r t h an one o f h i g h s t a c k i n g f a u l t e n e r g y . For example a t about 0.7 Tm a luminum e x h i b i t s l a r g e s u b g r a i n s w i t h s h a r p b o u n d a r i e s , whereas n i c k e l and c oppe r w i t h o n e - h a l f t o o n e - q u a r t e r t h e s t a c k i n g f a u l t e ne r gy o f a l um inum, e x h i b i t " f i n e r s u b g r a i n s w i t h much more ragged b o u n d a r i e s . 9 I f t h e t e m p e r a t u r e i s h i g h enough , such i l 1 - r e c o v e r e d s t r u c t u r e s have a g r e a t e r t e n d e n c y t o undergo a n o t h e r r e s t o r a t i o n p r o c e s s , dynamic r e c r y s -t a l l i z a t i o n . Dynamic r e c r y s t a l 1 i z a t i o n i s u s u a l l y r e c o g n i z e d by changes i n t h e s t r e s s - s t r a i n r e s p o n s e o f t h e m a t e r i a l . D u r i n g c r e e p d e f o r m a t i o n dynamic r e c r y s t a l 1 i z a t i o n i s m a n i f e s t e d by one o r more t r a n s i e n t p e r i o d s o f a c c e l e r a t e d c r e e p . In c o n s t a n t s t r a i n r a t e t e s t s ana l ogou s f l u c t u a t i o n s i n f l o w s t r e s s a r e o b s e r v e d a t r e l a t i v e l y l ow s t r a i n r a t e s [ 1 6 ] . A q u a l i t a t i v e model o f t h e r o l e o f dynamic r e c r y s t a l 1 i z a t i o n has been p roposed by S e l l a r s [ 1 7 ] . T y p i c a l s t r e s s - s t r a i n c u r v e s f o r n i c k e l a t 0.7 Tm a r e shown i n F i g u r e 4 f o r two s t r a i n r a t e s . The s t r e s s i n F i g u r e 4 a) r i s e s i n i t i a l l y as a r e s u l t o f a b a l a n c e between work h a r d e n i n g and r e c o v e r y p r o -c e s s e s , but a t a c r i t i c a l s t r a i n , yc, dynamic r e c r y s t a l l i z a t i o n i s n u c l e a t e d and t h e c u r v e pa s s e s t h r o u g h a maximum and t h en o s c i l l a t e s s e v e r a l t i m e s b e f o r e s e t t l i n g t o a s t e a d y s t a t e v a l u e . The dashed c u r v e i n d i c a t e s t h e . . . • s t r e s s - s t r a i n b e h a v i o u r wh i ch wou ld be e x p e c t e d i f r e c o v e r y were t h e o n l y o p e r a t i v e dynamic s o f t e n i n g p r o c e s s . The i n s e t i n F i g u r e 4 shows t he r e c r y s -t a l l i z a t i o n c u r v e s f rom wh i ch t h e o b s e r v e d s t r e s s - s t r a i n b e h a v i o u r can be computed . When t he s h e a r s t r a i n r e a che s y , new s t r a i n f r e e g r a i n s a r e n u c l e a t e d and w i t h i n c r e a s i n g s t r a i n ( t i m e ) r e c r y s t a l ! i z a t i o n p r o ceeds a l o n g c u r v e 1 l e a d i n g t o a d e c r e a s e i n f l o w s t r e s s . However , t h e . c o n c u r r e n t d e f o rm-a t i o n c ause s work h a r d e n i n g o f t h e g r a i n s so t h a t as t he r a t e o f r e c r y s t a l l i z a -t i o n f a l l s , t h e f l o w s t r e s s passes t h r o u g h a minimum and s t a r t s t o r i s e a g a i n . When Y c i s a g a i n r e a ched i n t h e g r a i n s t h a t r e c r y s t a l 1 i z e d e a r l i e s t i n t h e f i r s t c y c l e , t h e s e g r a i n s r e c r y s t a l l i z e once more a l o n g c u r v e 2, l e a d i n g t o a se cond maximum i n t h e f l o w s t r e s s . T h i s p r o c e s s i s r e p e a t e d each t i m e Y C i s r e a c h e d i n t h e r e c r y s t a l 1 i z e d g r a i n s , l e a d i n g t o f u r t h e r o s c i l l a t i o n s i n f l o w s t r e s s u n t i l t h e p r o c e s s becomes s u f f i c i e n t l y o u t - o f - p h a s e i n d i f f e r e n t 10 F i g . 4 S h e a r s t r e s s - s h e a r s t r a i n c u r v e f o r 99.9% n i c k e l de fo rmed i n O 1 t o r s i o n a t a s u r f a c e s h e a r s t r a i n r a t e : a) 3 .5 x 10 s b) 1.15 x 1 0 _ 1 s - 1 . The i n s e t shows t he c y c l e s o f dynamic r e c r y s t a l l i z a t i o n d u r i n g d e f o r m a t i o n . A c c o r d i n g t o S e l l a r s [17] 11 l o c a l r e g i o n s o f t h e m a t e r i a l t o make r e c r y s t a l 1 i z a t i o n e f f e c t i v e l y " c o n t i n u o u s " . T h i s r e s u l t s i n an o v e r a l l s t e a d y - s t a t e f l o w s t r e s s a t h i g h e r s t r a i n s . At h i g h e r s t r a i n : r a t e s ( F i g u r e 4 b) o n l y one maximum i n f l o w s t r e s s i s o b s e r v e d . As shown by t h e i n s e t t h e r e i s s t i l l r e p e a t e d r e c r y s t a l l i z a t i o n . The d i f f e r e n c e i n t h i s case i s t h a t t h e s t r a i n i n t e r v a l o v e r wh i ch t h e f i r s t c y c l e o f r e c r y s t a l l i z a t i o n o c c u r s , y > i s n o w c o n s i d e r a b l y l a r g e r t h an y X c so t h a t s e v e r a l c y c l e s o f r e c r y s t a l 1 i z a t i o n o v e r l a p , g i v i n g e f f e c t i v e l y c o n t i n u o u s r e c r y s t a l 1 i z a t i o n and a smoother f a l l i n f l o w s t r e s s t o s t e a d y s t a t e . A c c o r d i n g t o t h i s m o d e l , t h e e x t e n t t o wh i ch dynamic r e c o v e r y t a k e s p l a c e depends on t h e c o r r e l a t i o n between y^ ( F i g u r e 4 a . ) and y . When y c i s much s m a l l e r t h an y , dynamic r e c r y s t a l l i z a t i o n dom ina t e s t h e f l o w s t r e s s , whereas when y c > y^ t he o n l y s o f t e n i n g mechanism i s r e c o v e r y . When dynamic r e c r y s t a l 1 i z a t i o n i s t h e ma in s o f t e n i n g mechan ism o p t i c a l m i c r o s c o p y r e v e a l s t h e p r o g r e s s i v e deve lopment o f t h e r e c r y s t a l l i z e d s t r u c t i i r e w i t h i n c r e a s i n g s t r a i n beyond y , u n t i l i t has e n t i r e l y r e p l a c e d t he o r i g i n a l g r a i n s t r u c t u r e . W i t h ' f u r t h e r s t r a i n t h e r e i s no a p p a r e n t change i n g r a i n s t r u c t u r e , and t h e g r a i n s a lway s appea r n e a r l y e q u i a x e d . The s t e a d y s t a t e g r a i n s i z e i s i n d ependen t o f t h e meta l h i s t o r y and depends o n l y on t he compo-s i t i o n , t e m p e r a t u r e and s t r a i n r a t e . E l e c t r o n m i c r o s c o p i c o b s e r v a t i o n o f d y n a m i c a l l y r e c r y s t a l 1 i z e d g r a i n s r e v e a l t h a t t h e y c o n t a i n a t a n g l e d d i s l o c a 1 ' . -t i o n s u b s t r u c t u r e ana l ogous. t o t h a t o b s e r v e d a t a s t r a i n l e s s t h an y c [ 1 8 ] . The ma jo r me ta l 1 o g r a p h i c d i f f e r e n c e between dynamic r e c o v e r y and r e c r y s t a l l i -z a t i o n i s i n t h e appea rance o f t h e g r a i n s . In t h e c a s e o f dynamic r e c o v e r y , g r a i n s a r e e l o n g a t e d i n t h e d e f o r m a t i o n d i r e c t i o n (no g r a i n boundary m i g r a t i o n i n v o l v e d ) whereas i n t he ca se o f dynamic r e c r y s t a l l i z a t i o n one o b s e r v e s new e q u i a x e d g r a i n s ( e x t e n s i v e g r a i n boundary m i g r a t i o n i s i n v o l v e d ) . 12 The model o f dynamic r e c y r s t a l l i z a t i o n d e s c r i b e d above , a l t h o u g h w i d e l y a c c e p t e d , has r e c e n t l y been s u b j e c t e d t o s e v e r e c r i t i c i s m . S a k a i e t a l . [19] have made a c c u r a t e measurement o f y and y' > and t h e i r o b s e r v a t i o n s a r e not C X c o n s i s t e n t w i t h t he p r e d i c t i o n o f t h e model o f " c y c l i c " dynamic r e c r y s t a l l i z -a t i o n : - In t h e i r s ubsequen t work [ 2 0 ] , r a t h e r t han abandon ing t h e c o n c e p t , t h e y s i m p l y m o d i f i e d t h e p h y s i c a l mean ings o f y and y . C X ' Based upon c e r t a i n d i s a g r e e m e n t s between t h e a c t u a l appea rance o f a d y n a m i c a l l y r e c r y s t a l 1 i z e d s t r u c t u r e and t h a t p r e d i c t e d by c y c l i c r e c r y s t a l -1 i z a t i o n [ 2 1 ] , [ 2 2 ] , t h e p r e s en ce o f n onun i f o rm d e f o r m a t i o n i n . a s i m p l e hot c o m p r e s s i o n t e s t [ 2 1 ] , and t he absence o f r andomly d i s t r i b u t e d new g r a i n s i n ho t -wo r ked c oppe r s i n g l e c r y s t a l s [ 2 3 ] , B l a z and Ko r be l [24] have d eve l o ped a new app roach t o the e x p l a n a t i o n o f dynamic r e c r y s t a l l i z a t i o n . They s ugge s t t h a t t h e o p e r a t i n g mechanism o f d e f o r m a t i o n c o n s i s t s o f t h e group movement o f " o r d e r e d " d i s l o c a t i o n w a l l s wh i ch can combine i n t o new h i g h a n g l e g r a i n b o u n d a r i e s , and t h a t t h e m i g r a t i o n o f p r e - e x i s t i n g h i g h a n g l e b o u n d a r i e s p l a y s an i m p o r t a n t r o l e i n k e e p i n g t h e g r a i n s e q u i a x e d . One must o b s e r v e t h a t t h i s c o n c e p t r e q u i r e s f u r t h e r e x p e r i m e n t a l v e r i f i c a t i o n . As no ted e a r l i e r , t h e s i g n i f i c a n t f e a t u r e o f f l o w c u r v e s under hot w o r k i n g c o n d i t i o n s i s t h e s t e a d y - s t a t e r e g i o n , where f l o w s t r e s s i s i n d ependen t o f s t r a i n . V a r i o u s a t t e m p t s have been made t o f o r m u l a t e a r e l a t i o n between s t r e n g t h ( a ) , t e m p e r a t u r e ( T ) , and s t r a i n r a t e (e) f o r t h i s r e g i o n . The most s u c c e s s f u l one appea r s t o be t h a t o f S e l l a r s and T e g a r t [ 2 5 ] : k = A ( s i n h a a ) n ' e x p ( - Q / R T ) (1) where A, a , n' a r e t e m p e r a t u r e - i n d e p e n d e n t , c o n s t a n t s and Q i s an a c t i v a t i o n e n e r g y . At low s t r e s s e s [ a o <0.8) Eq . 1 r educes t o a power r e l a t i o n k = A ' a n ' e x p ( - Q / R T : ) (2) 13 At h i g h s t r e s s e s ( a a - M . 2 ) i t r e du ce s t o an e x p o n e n t i a l r e l a t i o n : e = A" exp(gc ) exp ( -Q/RT) (3 ) The c o n s t a n t s a , 3, and n 1 a r e r e l a t e d by B = a n ' so t h a t a and n ' can be s i m p l y d e t e r m i n e d f rom e x p e r i m e n t a l d a t a f o r h i g h and low s t r e s s e s . Hot w o r k i n g d a t a f o r s e v e r a l m e t a l s have been s a t i s f a c t o r i l y c o r r e l a t e d by E q . ' l [ 2 1 ] , [ 2 4 ] , [25] , [ 2 6 ] . Rear rangement o f Eq . 1. t o t h e form p e r m i t s d a t a f o r d i f f e r e n t t e m p e r a t u r e s and s t r a i n r a t e s t o be p l o t t e d , o n a s i n g l e s t r a i g h t l i n e . Z, t h e " Z e n e r - H o l l o m a n p a r a m e t e r " , i s a t e m p e r a t u r e -compensated s t r a i n r a t e . T h i s t y p e o f p l o t p r o v i d e s a method f o r i n t e r p o l a t i n g d a t a t o o b t a i n v a l u e s o f s t r e n g t h a t any t e m p e r a t u r e o r s t r a i n r a t e w i t h i n t h e r anges s t u d i e d . The a c t i v a t i o n e n e r g y Q has an i m p o r t a n t p h y s i c a l m e a n i n g . I t can be a s s o c i a t e d w i t h t h e r a t e - c o n t r o l l i n g dynamic s o f t e n i n g p r o c e s s , s i n c e t h e s l i p p r o c e s s i n v o l v e d i n d e f o r m a t i o n has a much s m a l l e r a c t i v a t i o n e n e r g y t han t h a t d e t e r m i n e d i n hot w o r k i n g [ 2 7 ] . A c a r e f u l e x a m i n a t i o n o f d a t a summar i zed i n r e f e r e n c e [28] l e a d s t o t h e c o n c l u s i o n t h a t i n t h e c a s e o f h i g h s t a c k i n g f a u l t e n e r g y m a t e r i a l s t h e a c t i v a t i o n e n e r g y o f hot w o r k i n g i s c l o s e t o t h a t o f ' s e l f - d i f f u s i o n , wh i ch i s t o be e x p e c t e d f rom t h e r e c o v e r y model p r e s e n t e d above . In t h e ca se o f low s t a c k i n g f a u l t e ne r gy m a t e r i a l s t h e a c t i v a t i o n ene r gy o f hot d e f o r m a t i o n i s o b s e r v e d t o be h i g h e r t han t h a t o f s e l f - d i f f u s i o n . D e t e r -m i n a t i o n o f Q p r o v i d e s a method o f d i s t i n g u i s h i n g t h e two dynamic r e s t o r a t i o n mechan isms u s i n g o n l y d a t a f rom m e c h a n i c a l t e s t s . The p h y s i c a l mean ing o f Q i n t h e ca se o f dynamic r e c r y s t a l 1 i z a t i o n i s s t i l l no t e n t i r e l y u n d e r s t o o d , due t o c o n t r o v e r s y o v e r t h e dynamic r e c r y s t a l l i z a t i o n mechanism as d i s c u s s e d ' b r i e f l y above . 1 1 == e exp(Q/RT) = A ( s i n h « 0 ) n (4) 1 4 3. Dynamic S t r a i n Ag i ng In t h e t e rm "dynamic s t r a i n a g i n g " (DSA) a r e i n c l u d e d a number o f e f f e c t s wh i ch a r e o b s e r v e d when c e r t a i n m e t a l s undergo t e n s i l e d e f o r m a t i o n a t e l e v a t e d t e m p e r a t u r e s . A c c o r d i n g t o R e e d - H i l l [29] t h e i m p o r t a n t e f f e c t s a r e : a) Lack o f dependence o f t h e y i e l d s t r e s s o f t h e me t a l on t e m p e r a t u r e w i t h i n t h e DSA t e m p e r a t u r e i n t e r v a l . b) Lack o f dependence o f t h e f l o w s t r e s s on s t r a i n r a t e . c ) Nonun i f o rm d e f o r m a t i o n , m a n i f e s t e d i n t h e P o r t e v i n - L e C h a t e l i e r e f f e c t . d) Abnormal d e c r e a s e i n d u c t i l i t y i n t h e c e n t e r o f t h e t e m p e r a t u r e range where DSA i s o b s e r v e d . S i n c e some o f t h e s e a n o m a l i e s a r e w e l l known i n a l p h a b r a s s [30] i t i s u s e f u l t o r e v i e w t he t h e o r y o f t h e o r i g i n o f dynamic s t r a i n a g i n g . The o r i g i n a l model f o r t h e phenomenon was t h a t advanced by C o t t r e l l [ 3 1 ] , who assumed t h a t d i s l o c a t i o n s move w i t h a s t e a d y v e l o c i t y and t h a t t h e y d r ag a s o l u t e a tmosphere a t t h e same s p e e d . I n t e r a c t i o n s between s o l u t e and d i s l o -c a t i o n s a r e h e l d t o be r e s p o n s i b l e f o r a l l t h e e f f e c t s d e s c r i b e d above . The c on cep t t h a t d i s l o c a t i o n and s o l u t e v e l o c i t i e s a r e equa l was t h e s u b j e c t o f l a t e r c r i t i c i s m , p a r t i c u l a r l y i n r e l a t i o n t o s u b s t i t u t i o n a l a l l o y s [ 3 2 ] , [ 3 3 ] , [ 3 4 ] , [35] . R e f e r e n c e s [ 3 2 ] , [ 3 3 ] , [34]; ; dea l o n l y w i t h t h e P o r t e v i n - L e C h a t e l i e r e f f e c t and n e g l e c t o t h e r a n o m a l i e s c h a r a c t e r i s t i c o f DSA. However , a new ';' t h e o r e t i c a l model was ;p r oposed by Van den Beuke l [ 3 5 ] . "The t h e o r y i s based on t h e a s s u m p t i o n t h a t t h e m o t i o n o f a d i s l o c a t i o n im. i t s s l i p p l ane i s i n g ene r a l a d i s c o n t i n u o u s p r o c e s s . The d i s l o c a t i o n meets - ob s t a c l e s wh i ch have t o be s u rmoun t ed , p o s s i b l y w i t h t h e h e l p o f t h e rma l a c t i v a t i o n . A d i s -l o c a t i o n segment has t o w a i t f o r a c e r t a i n a ve r age t i m e t u n t i l t h e o b s t a c l e i s o ve r come , f o l l o w i n g wh i ch i t jumps a t a h i g h v e l o c i t y t o t h e nex t o b s t a c l e . 15 T h i s t r e a t m e n t i s based on t he i d e a t h a t t he mov ing d i s l o c a t i o n s " s e e " a s o l u t e c o n c e n t r a t i o n wh i ch i s dependent on t h e w a i t i n g t i m e and t he s o l u t e d i f f u s i o n c o e f f i c i e n t . The c on cep t was d eve l o ped i n t o m a t h e m a t i c a l f o r m u l a s d e s c r i b i n g t h e dependence o f s t r e s s on s t r a i n r a t e , s t r a i n , and t e m p e r a t u r e i n t h e p r e s e n c e o f dynamic s t r a i n a g i n g . However i t s h o u l d be p o i n t e d ou t t h a t e x p e r i m e n t a l v e r i f i c a t i o n o f t h e s e f o r m u l a s i s ' r a t h e r d i f f i c u l t s i n c e t h e y c o n t a i n p a r ame t e r s not d i r e c t l y mea su r ab l e e x p e r i m e n t a l l y . The t h e o r y does not r e l a t e t h e p r e s en ce o f DSA t o t h e e x i s t e n c e o f a d u c t i l i t y min imum. In f a c t , no such r e l a t i o n s h i p has been p roposed by any p r e v i o u s i n v e s t i g a t o r s . 4 . O r d e r - D i s o r d e r T r a n s f o r m a t i o n i n B r a s s e s There a r e s e v e r a l p u b l i s h e d works wh i ch s u g g e s t , but f a i l t o p rove c o n c l u s i v e l y , t h a t a l p h a b r a s s e s may e x p e r i e n c e an o r d e r - d i s o r d e r t r a n s f o r m -a t i o n . For e xamp l e , t h e r e a r e r e p o r t s o f b r a s s e s e x h i b i t i n g an abnormal dependence on t e m p e r a t u r e o f such p h y s i c a l p r o p e r t i e s as l a t t i c e pa rame te r [ 3 6 ] , t h e rma l c a p a c i t y [37] and e l e c t r i c a l r e s i s t i v i t y [ 3 8 ] . The r e s p o n s e o f X - r a y d i f f r a c t i o n p a t t e r n s t o c e r t a i n h e a t - t r e a t m e n t s was i n t e r p r e t e d by P r e snyakov e t a l . [39] as e v i d e n c e f o r t he p r e s e n c e o f an o r d e r e d s t r u c t u r e i n s i n g l e phase b r a s s . L a t e r a c o p p e r - z i n c phase d i ag ram was p u b l i s h e d [40] wh i ch s u g g e s t e d t h e p r e s e n c e o f an o r d e r e d s t r u c t u r e i n a l p h a b r a s s e s a t t e m p e r a -t u r e s as h i g h as:500° C . P r e snyakov a l s o a t t emp t ed q u a l i t a t i v e l y t o r e l a t e t h e o r d e r i n g o b s e r v a t i o n w i t h t he d u c t i l i t y min imum. S u r p r i s i n g l y , no a t t emp t has a p p a r e n t l y been made t o r e l a t e o r d e r i n g and DSA o b s e r v a t i o n s i n a l p h a b r a s s e s . 1 6 I I OBJECTIVES OF THE PRESENT WORK The o r i g i n a l and p r i m a r y o b j e c t i v e o f t h i s work was t o i n v e s t i g a t e t h e " d u c t i l i t y minimum" i n a l p h a b r a s s e s . The phenomenon has been examined t o v a r i o u s deg rees by p r e v i o u s i n v e s t i g a t o r s , but t h e r e a r e w ide d i f f e r e n c e s i n i n t e r p r e t a t i o n o f t h e r e s u l t s . M o r e o v e r , i t i s not c l e a r t h a t a s i m i l a r b e h a v i o u r seen i n s t e e l s and o t h e r a l l o y s has t h e same o r i g i n s as t h e phenom-enon i n b r a s s . E x t e n s i v e p r e v i o u s e f f o r t has been devo t ed t o e x p 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 o f t h e hot w o r k i n g o f s t e e l s and o t h e r n o n - f e r r o u s a l l o y s . From t h i s work has come an a p p r e c i a t i o n o f dynamic r e s t o r a t i o n p r o c e s s e s and t h e i r r o l e i n h i g h t e m p e r a t u r e d e f o r m a t i o n . However , r a t h e r l i t t l e by c ompa r i s on has been done by way o f b a s i c s t u d i e s o f t h e ho t d e f o r m a t i o n o f c o p p e r a l l o y s ' •"" i n c l u d i n g a l p h a b r a s s e s . S i n c e t h e d u c t i l i t y minimum i n b r a s s e s o c c u r s a t t e m p e r a t u r e s where dynamic r e s t o r a t i o n may be i m p o r t a n t l y i n v o l v e d , t h e scope o f t h e e x p e r i m e n t a l work was ex t ended t o i n c l u d e t he hot w o r k i n g r a n g e . Much p r e v i o u s work has been r e p o r t e d on dynamic s t r a i n a g i n g and t h e P o r t e v i n - L e C h a t e l i e r e f f e c t i n a l p h a b r a s s e s . The phenomena a r e g e n e r a l l y a p p a r e n t a t t e m p e r a t u r e s l owe r t h an t h o s e o f t h e d u c t i l i t y min imum, a l t h o u g h t h e r e can be an o v e r l a p o f t h e e f f e c t s f o r c e r t a i n s t r a i n r a t e s and b r a s s com-p o s i t i o n s . T e n s i l e d e f o r m a t i o n s t u d i e s i n e v i t a b l y i n c l u d e d t h e DSA r a n g e . The s cope o f t h e p r e s e n t work was t h e r e f o r e l o g i c a l l y e x t ended t o i n c l u d e a c o n s i d e r a t i o n o f t h e p o s s i b l e l i n k between s t r a i n a g i n g and t h e d u c t i l i t y min imum. The o v e r a l l o b j e c t i v e o f t h e work h e r e i n r e p o r t e d t hu s was t o s t u d y s e v e r a l a s p e c t s o f t h e e l e v a t e d t e m p e r a t u r e d e f o r m a t i o n b e h a v i o u r o f a l p h a b r a s s e s , w i t h p a r t i c u l a r emphas i s on t h e d u c t i l i t y min imum. 17 I I I EXPERIMENTAL PROCEDURES 1. M a t e r i a l s Th ree a l p h a b r a s s c o m p o s i t i o n s were used i n t he p r e s e n t wo rk . The a l l o y s were p r e pa r ed f rom coppe r and z i n c , bo th 99.99% p u r e . M e l t i n g , a l l o y -i n g and c a s t i n g were c a r r i e d ou t i n an i n d u s t r i a l l a b o r a t o r y . B i l l e t s o f 120 mm d i a m e t e r were hot e x t r u d e d t o p roduce r e c t a n g u l a r ba r s o f 8 x 20 mm. The ba r s were r educed t o s t r i p s o f 1.8 mm . t h i c k n e s s by c o l d r o l l i n g . The f i n a l z i n c c o n t e n t s o f t h e t h r e e a l l o y s , f rom c h e m i c a l a n a l y s i s * on f o u r s t r i p spec imens r andom ly s e l e c t e d f rom each a l l o y w e r e : C o m p o s i t i o n 1 - 33.1% by w e i g h t 2 - 24.1% by w e i g h t 3 - 19.3% by we i gh t S p e c t r o g r a p h i c a n a l y s e s * * r e v e a l e d t he p r e s en ce o f t r a c e amounts o f a l um inum, c a l c i u m , and t i t a n i u m , p l u s l e a d and s i l v e r a t l e s s t h an 0.01%. Thus v e r y l i t t l e c o n t a m i n a t i o n had o c c u r e d d u r i n g m e l t i n g o r subsequen t p r o c e s s i n g o f t h e a l l o y s , and t he t o t a l i m p u r i t y c o n t e n t i s b e l i e v e d t o be l e s s t h a n 0 .03 wt.% i n a l l a l l o y s . No i m p u r i t y phases were e v e r d e t e c t e d i n t h e c o u r s e o f h i g h - r e s o l u t i o n m i c r o s c o p y on t h e s p e c i m e n s . Pe r f o rmed by C a n t e s t L t d . , V a n c o u v e r , B .C . Pe r f o rmed by Gene ra l T e s t i n g L a b o r a t o r i e s , V an c ouve r , B .C . 18 2. Spec imen P r e p a r a t i o n F l a t t e n s i o n t e s t spec imens o f t h e d e s i g n shown i n F i g u r e 5 were mach ined f rom t h e a s - r o l l e d 1.8 mm s t r i p , w i t h t h e l o n g ( t e n s i l e ) axes o f t h e spec imens a lways a l i g n e d i n t h e d i r e c t i o n o f r o l l i n g . No t e s t s were made f o r p r e f e r r e d o r i e n t a t i o n , b u f l i t t l e wou ld be e x p e c t e d i n t h e s t r i p i n v i ew o f t h e use o f hot e x t r u s i o n t o p e r f o rm most o f t h e r e q u i r e d d e f o r m a t i o n o f t h e b i l l e t , and t h e r e l a t i v e l y s m a l l amount o f t o t a l shape change i m p a r t e d by r o l l i n g . In any c a s e , t h e e x i s t e n c e o f c r y s t a l l o g r a p h i c t e x t u r e wou ld no t l i k e l y a f f e c t t h e c o n c l u s i o n s o f t h i s work s i n c e a l l spec imens were o f common o r i e n t a t i o n . I t was i m p o r t a n t t o en su r e t h a t a l l t e n s i l e spec imens were i n e x a c t l y t h e same i n i t i a l temper c o n d i t i o n . By t r i a l and e r r o r , i t was e s t a b l i s h e d what a n n e a l i n g c o n d i t i o n s wou ld be r e q u i r e d t o p roduce a u n i f o r m , f u l l y r e c r y s t a l 1 i z e d g r a i n s i z e o f a p p r o x i m a t e l y 30 ym d i a m e t e r i n each o f t h e t h r e e a l l o y s t r i p s . These c o n d i t i o n s , and t h e g r a i n s i z e s o b t a i n e d , a r e l i s t e d i n T a b l e I . T a b l e I A n n e a l i n g c o n d i t i o n s and f i n a l g r a i n s i z e o f t h e a l l o y s i n v e s t i g a t e d . C o m p o s i t i o n number Tempe ra tu re o f a n n e a l i n g [°K] " T i m e i n f u r n a c e [min] F i n a l g r a i n s i z e [ym] 1 1123 30 29 ± 3 2 1123 37 30 + 3 3 1123 40 29 ± 3 19 32 — 20—| t o OJ 1 e Thickness = 1-8 F i g . 5 T e n s i l e spec imen used i n t h e p r e s e n t work r—Steel Box Heavy Brass Sections Specimens Preheated Alumina F i g . 6 S c h e m a t i c d r a w i n g o f t h e p a c k - a n n e a l i n g a r r angemen t 20 D e t e r m i n a t i o n s o f g r a i n s i z e were based on m e t a l l o g r a p h i c measurements u s i n g t h e l i n e a r i n t e r c e p t method w i t h a t l e a s t ? 3 0 0 i n t e r c e p t s pe r measurement . Twin b o u n d a r i e s were i n c l u d e d i n t h e i n t e r c e p t s . To r educe z i n c l o s s e s d u r i n g a n n e a l i n g , and t o e n su r e u n i f o r m i t y o f s t r u c t u r e t h r o u g h o u t each spec imen and f rom spec imen t o spec imen w i t h i n a b a t c h , a n n e a l i n g was done by t h e pack t e c h n i q u e shown i n F i g u r e 6. A bund l e o f t e n mach ined t e n s i l e spec imens was packed between h e a v i e r b r a s s s e c t i o n s , and i n v e s t e d i n p r e h e a t e d a l u m i n a powder i n a box . The a s s emb l y was t hen b r ough t t o t h e r e q u i r e d a n n e a l i n g t e m p e r a t u r e i n a box f u r n a c e . T o t a l h e a t i n g t i m e s a r e i n d i c a t e d i n T ab l e I . The c o n t e n t s o f t h e pack were quenched i n w a t e r t o t e r m i n a t e a n n e a l i n g . O p t i c a l m e t a l l o g r a p h y r e v e a l e d t h a t g r a i n s i z e w i t h i n t h e gage l e n g t h o f each a nnea l e d s p e c i m e n , and f o r a l l spec imens i n a b a t c h , was u n i f o r m w i t h i n t h e l i m i t s i n d i c a t e d i n T ab l e I . L i n e s cah a n a l y s e s f o r z i n c were made on p o l i s h e d spec imen s e c t i o n s u s i n g an e l e c t r o n beam m i c r o p r o b e a n a l y s e r . These r e v e a l e d t h a t l o s s e s o f z i n c d u r i n g a n n e a l i n g were c o n f i n e d t o a l a y e r o f a few m i c r o m e t e r s t h i c k n e s s a t spec imen s u r f a c e s . There was no i n d i c a t i o n i n subsequen t work t h a t t h e s e t h i n d e z i n c i f i e d s k i n s a f f e c t e d t h e b u l k m e c h a n i -c a l p r o p e r t i e s wh i ch were mea su r ed . 3. T e n s i l e T e s t i n g The e l e v a t e d - t e m p e r a t u r e t e n s i l e t e s t s were c a r r i e d ou t on an I n s t r o n t e s t i n g m a c h i n e . In one s e r i e s o f e x p e r i m e n t s , a t h r e e - z o n e h i g h - t e m p e r a t u r e f u r n a c e s u p p l i e d by I n s t r o n C o r p o r a t i o n was u s e d . For a l l o t h e r e x p e r i m e n t s a s p e c i a l f u r n a c e was d e s i g n e d and b u i l t a c c o r d i n g t o F i g u r e 7. In o r d e r t o a s s e s s t h e t r u e m i c r o s t r u c t u r e o f a spec imen a t t h e end o f a t e s t , i t was i m p o r t a n t t o be a b l e t o quench t he spec imen v e r y r a p i d l y . W i t h t h e I n s t r o n f u r n a c e t h e quench i ng t i m e was 25-30 seconds whereas w i t h t h e s p e c i a l l y 21 designed furnace the quenching time did not exceed 3 seconds. The rap id quench was achieved by quick ly removing the furnace and rep lac ing i t with a bucket of water. A l l metal lographic observations reported in t h i s thes i s are from specimens tested in the rapid-quench furnace of F igure 7. The var iab le parameters in the tes ts were t e m p e r a t u r e i n i t i a l s t r a i n ra te , and the s t r a i n at which deformation was terminated. A broad out l ine of the experiments is given in Table I I . Table II Out l ine of t e n s i l e tes t parameters Sample code Composition [weightXZn] S t ra in rateLV1] Temperature [°K] S t ra in e Remarks 111 33.1 l . l x l O " 2 473 Fracture 112 33.1 l . l x l O " 2 650 Fracture Ser ies #1 113 33.1 l . l x l O " 2 750 Fracture 114 33.1 l . l x l O " 2 850 Fracture Three samples pei 115 33.1 l . l x l O " 2 950 Fracture code number, 216 116 33.1 l . l x l O " 2 1050 Fracture specimens in a l l 121 33.1 l . lx lO" 1 * 473 Fracture 122 33.1 l . lx lO" 1 * 650 Fracture 123 33.1 l . l x lO- 1 * 750 Fracture 124 33.1 l . l x lO- 1 * 850 Fracture 125 33.1 l . lx lO" 1 * 950 Fracture 126 33.1 l . l x lO- 1 * 1050 Fracture 311 24.1 l . l x l O " 2 473 Fracture 312 24.1 l . l x l O " 2 650 Fracture 22 Table II continued Sample code Composition [weightXZn] S t r a i n r a t e C s " 1 ] Temperature [°K] S t r a i n e Remarks 313 24.1 l . l x l O " 2 750 Fracture 314 24.1 l . l x l O " 2 850 Fract u r e 315 24.1 l . l x l O - 2 950 Fracture 316 24.1 l . l x l O " 2 1050 Fract u r e 321 24.1 l . l x l O " 1 * 473 Fracture 322 24.1 l . l x l O - 4 * 650 Fr a c t u r e 323 24.1 l . l x l O " 1 1 750 Fracture 324 24.1 1.1x10" h 850 Fract u r e 325 24.1 l . l x l O - 1 * 950 Fracture 326 24.1 l . l x l O " 1 * 1050 Fract u r e 411 19.3 l . l x l O " 2 473 Fracture 412 19.3 l . l x l O " 2 650 Fract u r e 413 19.3 l . l x l O " 2 750 Fracture 414 19.3 l . l x l O " 2 850 Fract u r e 415 19.3 l . l x l O " 2 950 Fracture 416 19.3 l . l x l O " 2 1050 Fract u r e 421 19.3 l . l x l O - 4 473 Fracture 422 19.3 l . l x l O " 1 * 650 Frac t u r e 423 19.3 l . l x l O - 1 * 750 Fracture 424 19.3 l . l x l O " 1 * 850 Frac t u r e 425 19.3 l . l x l O " 1 * 950 Frac t u r e - 426 19.3 l . l x l O " 1 * 1050 Fract u r e 23 Table II continued Sample code Composition [weight%Zn] S t r a i n r a t e [ s " * ] Temperature [°K] • S t r a i n e Remark 511 33.1 l . l x l O " 2 473 0.08 Ser i e s #2 512 33.1 l . l x l O " 2 573 0.08 One sample per sam-513 33.1 l . l x l O - 2 673 0.08 ple code. Tests 514 33.1 l . l x l O " 2 773 0.08 performed with 515 33.1 l . l x l O " 2 873 0.08 rapid-quench 521 33.1 l . l x l O " 1 * 473 0.08 apparatus of F i g . 7 . 523 33.1 l . l x l O " 1 * 573 0.08 Quenching time<3s. 524 33.1 l . l x l O - 1 * 673 0.08 This s e r i e s was per-525 33.1 l . l x l O " 1 * 873 0.08 formed to i n v e s t i g a t e 611 19.3 l . l x l O " 2 473 0.08 the s t r u c t u r e of 612 19.3 l . l x l O " 2 573 0.08 a-brasses at constant 613 19.3 l . l x l O " 2 673 0.08 s t r a i n but varying 614 19.3 l . l x l O " 2 773 0.08 temperature. Two d i f -615 19.3 l . l x l O " 2 873 0.08 fe r e n t s t r a i n rates 621 19.3 l . l x l O " 1 * 473 0.08 were used. The eng-622 19.3 l . l x l O " 1 * 573 0.08 i n e e r i n g s t r a i n was 623 19.3 l . l x l O - 1 * 673 0.08 roughly equal to th a t 624 19.3 l . l x l O " 1 * 773 0.08 at f r a c t u r e i n the d u c t i l i t y minimum region. 24 Table II continued Sample code Composition [weightXZn] Strain ra te [s - 1 ] Temperature [°K] Strain e Remark 11 24.1 l . l x l O - 4 673 0.003 Series #3 12 24.1 l . l x l O " 4 673 0.022 Samples deformed at 13 24.1 l . l x l O - 4 673 0.028 the temperature ofthe 14 24.1 l . l x l O " 4 673 0.037 duct i l i ty minimum. 15 24.1 l . l x l O " 4 673 0.047 Amount of strain was 16 24.1 l . l x l O " 4 673 0.053 varied. Series was 17 24.1 l . l x l O " 4 673 0.063 performed with the apparatus of F ig .7. Quenching time <3s. The f la t surfaces of the samples were pre-polished to 6ym in order to investigate the surface' effects of deformation. In one experiment designed to check the influence of the oxidizing environment, a sample of alloy 2 was deformed using a heated sil icone o i l bath instead of the furnace in Fig.7. The test temperature was 633°K and the strain rate 1.1 x 1 0 " 4 s - 1 . 25 To load cell F i g . 7 S p e c i a l l y - D e s i g n e d Fu rnace f o r R a p i d Quench ing o f T e s t Spec imens 1. I n s t r o n c r o s s h e a d , 2. I n s t r o n f r a m e , 3 . Removable s t a g e , 4 . Fu rnace s h e l l , 5 . I n s u l a t i o n , 6 . H e a t i n g e l e m e n t s , 7 . H o l l o w s t a i n l e s s s t e e l l o w e r p u l l r o d , p e r f o r a t e d t o a l l o w f r e e e n t r y o f w a t e r d u r i n g q u e n c h i n g , 8 . S t a i n l e s s s t e e l p u l l r o d , 9 . Upper g r i p , 1 0 . Lower g r i p , 1 1 . C o u n t e r a c t i n g n u t , 1 2 . S a m p l e , 1 3 . The rmocoup l e , 1 4 . C o l d j u n c t i o n , 1 5 . C h a r t r e c o r d e r f o r c o n t i n u o u s t e m p e r a t u r e r e c o r d i n g . 26 Load-crosshead displacement curves were autographically recorded on the Instron testing machine. From these plots were derived, using the formulas below, the values of a number of tensile deformation properties. (i) Yield stress (0.2 Percent "Offset" Flow Stress) ''o ? "0.2 - -TT <5> where: PQ 2 = Load corresponding to 0.2 percent plastic strain (e = 0.002) A Q = In i t ia l cross section area of the sample ( i i ) Ultimate tensile stress P UTS = (6) A o where: P = Maximum load reached on the load-crosshead displacement max y curve. ( i i i ) Instantaneous true flow stress ° = T - ' J - ( 7 ) o o where: P = Instantaneous deforming load 1 = Instantaneous gage length 1 = In i t ia l gage length (iv) Instantaneous true strain e = ln i- (8) o 27 (v) Engineering elongation to fracture [%) 1* - 1-e f = T 1 0 • 100[%] (9) o where: l f = Final length of the broken sample between original gage marks. (vi) Uniform reduction in area (%) A - A A ™ = — i — - • ioon%] d o ) ru o where: A^ . = cross section area of the broken sample measured well outside the necked area. (v i i ) Reduction in area at fracture {%) A n - A A r f i = A * 1 0 0 C % ] where: A = cross section area at fracture surface; measured with the r aid of a travell ing microscope. ( v i i i ) Instantaneous strain-hardening rate v.-<&>. . ( 1 2 ) i . e . The slope of the true stress-true strain curve'measured graphic-al ly at a true strain e . 28 (ix) Strain-hardening exponent n=3-4^ (13) d loge derived from the Ludwik description of the flow curve; i .e . a = Ke11, and obtained from the slope of a plot of logo vs loge. (x) Strain rate sensit ivity coefficient m = £7 (14) log e2 Derived assuming that flow stress and strain rate are related according to the power law c ( e J ) = Ke"1 where a ( e J ) -s & flw s t r e s s at a fixed strain and temperature. Thus the coefficient m was calculated using points obtained from flow stress-strain curves from tests which employed two different strain rates. The use of Eq.14 was just i f ied by the work of Gryziecki [41], who successfully applied the power law to alpha brasses at temperatures ranging from 77°K to 723°K and for a strain rate range of three orders of magnitude. The coefficient m can also be determined by making abrupt changes in the crosshead speed of the tensile machine during a test and subsequently using Equation 14 for calculations. This method differs from the previous one since i t contains information about the response of the metal to strain rate change, whereas the f i r s t method is influenced by different deformation history. It should be pointed out that the values of m vary with strain. However, for the investigated strain range of 0.002 to 0.08 the difference between m values was negligible. Therefore for the sake of c la r i t y , values of m for the y ie ld stress were used. 29 Calculation examples together with a ful l l i s t of the results are presented in Appendix I. F ina l ly , the activation energy of deformation for the three investigated compositions was calculated according to the method described in chapter I. (Formulas 1,2,3). A detailed sample calculation is presented in Appendix II together with a fu l l table of results. 4. Metallography Procedures used to prepare specimens for metallographic observation are outlined ful ly in Table III. Table III Details of metallographic sample preparation Series.mechanical # polish electro-polishing electro-etching chemica' etching Details of polishing procedure Mechanical polishing: 1 yes yes no yes (i) Emery papers of different grades ( i i ) 6 urn diamond paste 2 yes yes yes no on a cloth Electropolishing: electrolyte - 50% H 3P0 l t 3 yes yes no yes cathode _ copper 3' yes no no no temperature _ ambient current density «0 .1A/cm 2 Table III continued 30 De ta i l s of po l i sh ing procedure E l ec t roe tch ing : e l e c t r o l y t e -3g N a 2 S 2 0 3 . 5 H 2 0 , l g N H 4 C 2 H 3 0 2 5 7ml NH^ OH 1300 ml water cathode - copper temperature - ambient current density «10mA/cm2 time: composition 1 - 2 0 sec. composition 3 - 4 0 sec . Chemical e tch ing: s o l u t i o n : 30ml HCl . lOg FeCl 3 120ml Ch30H temperature _ ambient time: composition 1 _ 20 sec. composition 2 - 3 0 sec. composition 3 - 3 5 sec. Etching by immersion 31 IV RESULTS 1. T e n s i l e P r o p e r t i e s Append i x I c o n t a i n s t a b u l a t e d d a t a f rom t e n s i l e t e s t s . Of t h e v a r i o u s p r o p e r t i e s c a l c u l a t e d o r d e r i v e d f rom t h e s e d a t a , some were found t o be more u s e f u l and m e a n i n g f u l t h an o t h e r s . To d e s c r i b e t h e w o r k - h a r d e n i n g b e h a v i o u r o f m e t a l s , many i n v e s t i g a t o r s used t he s t r a i n - h a r d e n i n g e x p o n e n t , n , d e f i n e d i n E q u a t i o n 13 above . In u s i n g t h i s p r o p e r t y , t h e a s s u m p t i o n i s made t h a t t h e e m p i r i c a l Ludwik r e l a - . t i o n , o = K £ n , a p p l i e s o v e r an a p p r e c i a b l e range o f s t r a i n . In t h e p r e s e n t work t he a s s u m p t i o n was found t o be s i m p l y i n v a l i d . For t h i s and o t h e r r e a s o n s , t h e exponen t n i s o f h i g h l y q u e s t i o n a b l e v a l u e as a q u a n t i t a t i v e i n d i c a t o r o f s t r a i n - h a r d e n i n g b e h a v i o u r . Even as a c o m p a r a t i v e p r o p e r t y , i t c a n ' l e a d t o m i s l e a d i n g c o n c l u s i o n s . A c c o r d i n g l y , t h e pa r ame te r v £ ( i n s t a n t a n e o u s s l o p e o f t h e s t r e s s - s t r a i n c u r v e a t a s p e c i f i e d t r u e s t r a i n ) was adop t ed i n t h i s work as t h e p r e f e r r e d i n d i c a t o r o f s t r a i n - h a r d e n i n g b e h a v i o u r . The s e v e r a l q u a n t i t i e s c o n v e n t i o n a l l y used t o d e s c r i b e t e n s i l e " d u c t i l i t y " f a l l i n t o two g r o u p s : t h o s e wh i ch i n c l u d e t he s t r a i n wh i ch i s c o n c e n t r a t e d i n t h e necked p o r t i o n o f t h e spec imen ( e l o n g a t i o n t o f r a c t u r e , r e d u c t i o n i n a r e a a t f r a c t u r e ) , and t h o s e wh i ch do not ( u n i f o r m e l o n g a t i o n , u n i f o r m r e d u c t i o n i n a r e a ) . Most i n v e s t i g a t o r s r e p o r t v a l u e s wh i ch a r e i n one o r o t h e r g r o u p , and c ompa r i s on s o f d a t a can be m i s l e a d i n g . In t h e p r e s -en t wo rk , bo th t y p e s o f measurements a r e r e p o r t e d , a l t h o u g h a c a se i s l a t e r made t h a t one i s more m e a n i n g f u l t h an t he o t h e r as a d e s c r i p t o r o f d u c t i l i t y . 32 2. E f f e c t o f Tempera tu re and S t r a i n Rate on F low S t r e s s T e n s i l e t e s t d a t a a r e p l o t t e d as t r u e s t r e s s vs t r u e s t r a i n i n F i g u r e 8 . S e r r a t e d y i e l d i n g (P -LeC e f f e c t ) was ob s e r v ed a t t h e l o w e s t t e m p e r a t u r e s , but o n l y a t t h e l o w e r o f t h e two s t r a i n r a t e s u s e d . The e f f e c t p e r s i s t e d t o h i g h e r t e m p e r a t u r e s f o r t h e 19 In a l l o y t h a n f o r t h e 24 o r 33 Zn a l l o y s . No a t t e m p t has been made t o t r a n s f e r t h e s e r r a t i o n s on l o a d - e l o n g a t i o n p l o t s t o t h e o-e p l o t s . Two d i f f e r e n t k i n d s o f s e r r a t i o n s have been o b s e r v e d and a r e r e v e a l e d i n t h e l o a d - e l o n g a t i o n c u r v e s o f F i g u r e 9. Where s e r r a t e d y i e l d i n g was d e t e c t e d , t h e s t r e s s - s t r a i n c u r v e s i n F i g u r e 8 a r e marked " P - L e C " . Under no t e s t c o n d i t i o n s were o s c i l l a t i o n s o b s e r v e d i n t h e s t r e s s -s t r a i n b e h a v i o u r . M o r e o v e r , i n o n l y a few t e s t s was t h e r e a maximum, o r peak , found i n t h e s t r e s s - s t r a i n c u r v e . These two f e a t u r e s have been p r e v i o u s l y i d e n t i f i e d w i t h t h e o c c u r r e n c e o f dynamic r e c r y s t a l l i z a t i o n , . as d i s c u s s e d i n t h e I n t r o d u c t i o n . S i n c e t h e l i q u i d u s t e m p e r a t u r e f o r a l p h a b r a s s e s d e c r e a s e s w i t h i n c r e a s i n g z i n c c o n t e n t i t i s i n some r e s p e c t s more m e a n i n g f u l t o use homologous t e m p e r a t u r e i n r e p o r t i n g t h e e f f e c t o f c o m p o s i t i o n on m e c h a n i c a l p r o p e r t i e s a t h i g h e r t e m p e r a t u r e s . Y i e l d s t r e s s ( a n ? ) i s p l o t t e d a g a i n s t t e s t t e m p e r a t u r e i n F i g u r e 10 . True Strain F i g u r e 8 T rue s t r e s s - t r u e s t r a i n c u r v e s as o b t a i n e d f rom t e n s i l e t e s t s . C o n d i t i o n s o f t e s t i n g s p e c i f i e d on t he g r a p h s . (a) 160 True Strain (d) 39 F i g . 9 S e r r a t i o n s on l o a d - e l o n g a t i o n c u r v e s . ( a ) - Type 1 c h a r a c t e r i s t i c o f l o w e r t e m p e r a t u r e (b) - Type 2 c h a r a c t e r i s t i c o f h i g h e r t e m p e r a t u r e 40 0 1 " 1 1 i i I 0-3 0-4 0-5 0-6 0-7 0-8 0-9 Homologous Temperature T H (a ) F i g . 10 Y i e l d s t r e s s o f a l p h a b r a s s e s vs homologous t e m p e r a t u r e T H ( a ) s t r a i n r a t e - 1.1 x 1 0 " 2 s _ 1 (b ) s t r a i n r a t e - 1.1 x 1 0 " 4 s 41 100 80 o Q_ 60 00 6 4 0 20 h 0 0-3 strain rate-O 3 3 % Zn • 24% Zn A 19% Zn X I0 - 4 sH CrA 0-4 0-5 0-6 0 7 0-8 Homologous Temperature T, 0-9 H (b) 42 The d a t a r e v e a l : (a ) W i t h i n e x p e r i m e n t a l e r r o r , t h e y i e l d s t r e s s e s o f t h e t h r e e b r a s s e s show t he same t e m p e r a t u r e dependence a t t h e h i g h e r t e m p e r a t u r e s . (b) In t h e l o w e r t e m p e r a t u r e r a n g e , y i e l d s t r e s s i s r e l a t i v e l y i ndependen t o f t e m p e r a t u r e . The e x t e n t o f t h e r e g i o n i s g r e a t e r a t t h e h i g h e r s t r a i n r a t e . ( c ) In t h e l o w e r t e m p e r a t u r e range y i e l d s t r e s s i n c r e a s e s w i t h i n c r e a s i n g z i n c c o n t e n t . (d) At l o w e r t e m p e r a t u r e s , t h e y i e l d s t r e s s i s e s s e n t i a l l y i n dependen t o f s t r a i n r r a t e i n t h e r a n g e . At t h e upper t e m p e r a t u r e s , y i e l d s t r e s s i s much h i g h e r a t t h e h i g h e r o f t h e two s t r a i n r a t e s u s e d . S i m i l a r o b s e r v a t i o n s were made when t h e t r u e f l o w s t r e s s a t a h i g h e r s t r a i n ( e . g . e = 0 . 0 5 ) i n s t e a d o f OQ ^ w a s p l o t t e d a g a i n s t t e m p e r a t u r e . At l o w e r t e s t t e m p e r a t u r e s , t h e f l o w s t r e s s r o s e r a p i d l y i n r e s p o n s e t o s t r a i n h a r d e n i n g . At t h e h i g h e r t e m p e r a t u r e s , dynamic r e s t o r a t i o n p r o c e s s e s competed w i t h s t r a i n h a r d e n i n g t o r educe m a r k e d l y t h e s l o p e o f t h e s t r e s s - s t r a i n c u r v e a t a l l p l a s t i c s t r a i n s . The e f f e c t o f dynamic s o f t e n i n g i s seen more q u a n t i t a t i v e l y by o b s e r v i n g how t he i n s t a n t a n e o u s s t r a i n h a r d e n i n g r a t e a t a g i v e n s t r a i n (vg) v a r i e s w i t h t e s t t e m p e r a t u r e . T h i s i s shown i n F i g u r e 1 1 . f o r a t r u e s t r a i n o f 0 . 0 5 . The f o l l o w i n g f e a t u r e s a r e s e e n : (a ) At t h e l o w e s t t e s t t e m p e r a t u r e s , V £ i s h i g h and i s e s s e n t i a l l y i n dependen t o f s t r a i n r a t e and a l l o y c o m p o s i t i o n i n t h e range s t u d i e d . (b) A t t h e h i g h e s t t e s t t e m p e r a t u r e s , an e s s e n t i a l l y s t e a d y - s t a t e ( s t r a i n - i n d e p e n d e n t ) f l o w s t r e s s . c o n d i t i o n i s r e a ched f o r a l l a l l o y s and bo th s t r a i n r a t e s ; i . e . v £ app roaches z e r o . 43 10 8 6 4 CL C. 0 g. 8 " 6 4 2 0 0-3 high strain rate O 33% Zn • 24% Zn A 19 %Zn -o low strain rate O 33% Zn • 24% Zn A 19% Zn 0-4 0-5 Homologous 0-6 0-7 Temperature 0-8 0-9 T. H F i g . 1 1 S t r a i n h a r d e n i n g r a t e vs homologous t e m p e r a t u r e T| 44 0*31 0-2 E 01 0 O 33% Zn • 24% Zn A 1 9 % Zn -A- A -01 0-3 i 0-4 0-5 0-6 0-7 Homologous Temperature 0-8 T, H 0-9 F i g . 12 S t r a i n r a t e s e n s i t i v i t y c o e f f i c i e n t vs homologous t e m p e r a t u r e T 45-(c) At the lower strain rate the effects of dynamic restoration are evident at temperatures above 0.55 T , and v £ decreases sharply over a narrow temperature range. At the higher strain rate dynamic softening processes are delayed up to a temperature of about 0.65 T f f l and the decrease in v g is more gradual. (d) There is no apparent influence of composition on the homologous temperature at which dynamic softening processes commence, and the rate of softening is closely similar for al l three compositions. As expected, the strain rate sensit iv ity of the flow stress is greatest at the higher test temperatures, where dynamic restoration compensates for strain hardening. In Figure 12 the strain rate sensit iv i ty parameter, m, is plotted against homologous temperature. Again the behaviour shows l i t t l e dependence on alloy composition. The temperature above which the strain rate sensit iv i ty is high is about 0.65 T . 3. Effect of Temperature and Strain Rate on Ducti l i ty Figure 13 reveals the observed variation with temperature of uniform reduction in area A r ( j (one of several properties used to describe duct i l i t y ) . At the lower strain rate (Figure 13 a) there is a sharp decline in A r u between 470 and 650°K. Above 650°K, however, the value does not change s igni f icant ly . The behaviour is the same for al l compositions, although the 33% Zn alloy has a higher A r u at al l temperatures than the other two al loys. At the higher strain rate (Figure 13 b), the absolute values of A r u for a given alloy are higher at al l temperatures above 470°K. For 19% and 24% zinc brasses there is a gradual decrease in A r u with increasing temperature. For the 33% zinc alloy no clear trend is evident. 46 40 30 20 10 0 40 30 20 10 • 1— r (a) low strain rate O 33% Zn • 24%Zn A 19% Zn 0 L L (b) high strain rate O 33%Zn • 24%Zn A 19% Zn JD. O *A-470 570 670 7 70 870 9 70 1070 Temperature (°K) F i g . 13 Un i f o rm r e d u c t i o n i n a r e a vs t e m p e r a t u r e T 47 (a) low strain rate J 1 1 1 I (b) high strain rate j\ — " 1 1 1 1 I I 470 570 670 770 870 970 1070 Temperature (°K) 14 R e d u c t i o n i n a r e a a t f r a c t u r e A . vs t e m p e r a t u r e 48 For none o f t h e b r a s s e s , and f o r n e i t h e r s t r a i n r a t e , i s a " d u c t i l i t y minimum" i n d i c a t e d by t h e p r e s e n t d a t a , when u n i f o r m r e d u c t i o n i n a r e a i s used as t h e measure o f d u c t i l i t y . In F i g u r e 14 a d i f f e r e n t d u c t i l i t y p r o p e r t y , r e d u c t i o n i n a r e a a t t h e f r a c t u r e A ^ , i s p l o t t e d a g a i n s t t e m p e r a t u r e . The re i s a "minimum" i n e v e r y p l o t somewhere i n t h e t e m p e r a t u r e range 700 - 850°K. Fo r some c o m b i n a t i o n s o f c o m p o s i t i o n and s t r a i n - r a t e , t h e minimum i s i l l - d e f i n e d because A ^ does not r i s e c o n t i n u o u s l y - beyond t h i s min imum. In a l l but one c a s e , t h e minimum wou ld be more a p p a r e n t i f t h e d a t a f o r t h e h i g h e s t t e m p e r a t u r e were o m i t t e d . T h i s b e h a v i o u r i s t y p i c a l o f t h a t r e p o r t e d f o r a l p h a b r a s s e s . S i m i l a r p l o t s were p u b l i s h e d by Izumi and Yamagata [8] and a r e r e p o r t e d i n F i g u r e 1 (b ) o f t h i s t h e s i s . The f o l l o w i n g a d d i t i o n a l t r e n d s a r e e x h i b i t e d by F i g u r e 14 . ( a ) The " t r o u g h " i n t h e L\ ^  vs t e m p e r a t u r e p l o t s i s s h a r p e r a t t h e h i g h s t r a i n r a t e . (b) A ^ i s g e n e r a l l y h i g h e r a t h i g h s t r a i n r a t e s . ( c ) A c o m p o s i t i o n - d e p e n d e n c e o f A .p i s not a p p a r e n t , i n c o n t r a s t t o t h e c a s e o f A r u ( F i g u r e 1 3 ) . V a l u e s o f e l o n g a t i o n t o f r a c t u r e (e^) f o l l o w e d c l o s e l y t h e t r e n d s o f r e d u c t i o n i n a r e a a t f r a c t u r e ( A ^ ) . I t was t h e r e f e l t t o be u n n e c e s s a r y t o p r e s e n t p l o t s o f t h e e^ . d a t a . 4 . F i t s o f Data t o an E q u a t i o n o f S t a t e The v a l i d i t y o f E q u a t i o n (4 ) i s t e s t e d i n F i g u r e 15 a , b , c . The s t r a i g h t s o l i d l i n e r e p r e s e n t s t h e r e g i o n o v e r wh i ch E q u a t i o n (4 ) can be c o n s i d e r e d v a l i d . T h i s r e g i o n e x t e n d s t o s l i g h t l y h i g h e r v a l u e s o f Z f o r t h e 33% z i n c a l l o y t h an f o r t h e o t h e r two c o m p o s i t i o n s . Va l u e s o f Q, t h e a p p a r e n t 60 49 -re-56 52 48 CuZn 33% € =A(sinh(a-o-))nexp-j^ A = l-757XI0l4s"' n = 4-4 a =00121 MPo Q = 2 6 0 ^ r - i O I I 44 a g 40 36 0 € = 11X10 #€=11X10 - 4 - 2 o I o o I 32 28 24 / O 20 -12 -8 -4 0 4 8 12 nlnsinh ( a c r ) F i g . 15 (a ) C o r r e l a t i o n o f h i g h t e m p e r a t u r e d a t a t o E q u a t i o n o f S t a t e ( E q . l ) 50 68 TX 64 60 56 52 p 48 or o §" 44 c M 40 c 36 32 CuZn 24% € = A (sinh(acr))n exp—^ A = 2042XI0 l 4s~' n=5-2 a =00184 MPa"1 Q=289 ^ O € = IIXI0 -4 #€ =IIXI0-2 o / / O / / y 28 O 24 20 -12 -8 -4 0 4 ntn s in h (a- cr) 8 12 F i g . 15(b) 51 16 -12 '6 CuZn 19% : A(sinh(a-cr))nexp-^-RT A = 1-776 Xlo'V' n = 4-8 a = 00144 MPa"1 235 o 40 O €=I|><I0"4 • € =IIXI0"2 36 32 -28 / O o o - 8 4 0 4 nln s i n h ( a c r ) 8 12 F i g . 1 5 ( c ) 52 a c t i v a t i o n ene r gy o f d e f o r m a t i o n , were d e t e r m i n e d f o r each a l l o y a c c o r d i n g t o Append i x I I , and a r e i n d i c a t e d on F i g u r e 1 5 . The re was no a p p a r e n t ' t r e n d i n t h e v a r i a t i o n o f Q w i t h c o m p o s i t i o n , and i t i s c o n s i d e r e d l i k e l y t h a t d i f f e r e n c e s i n t h e v a l u e s a r e w i t h i n t h e l i m i t s o f e x p e r i m e n t a l e r r o r . 5. M e t a l l o g r a p h i c O b s e r v a t i o n s C h a r a c t e r i s t i c m i c r o s t r u c t u r e s appea r i n F i g u r e 16 , 17 , 18 and 19 . The m e t a l l o g r a p h i c o b s e r v a t i o n s may be summar ized as f o l l o w s : 1. Reduced d u c t i l i t y i s a s s o c i a t e d w i t h t h e f o r m a t i o n o f s m a l l c a v i t i e s , wh i ch a r e g e n e r a l l y found o n l y i n t h e g r a i n b o u n d a r i e s t r a n s v e r s e t o t h e a p p l i e d t e n s i l e s t r e s s , and no t n e c e s s a r i l y a t t r i p l e j u n c t i o n s ( F i g u r e s 1 6 , 1 7 ) . 2. The c a v i t i e s a r e d i s t r i b u t e d t h r o u g h o u t t h e gage l e n g t h o f t h e s p e c i m e n , i . e . even i n r e g i o n s remote f rom t h e f r a c t u r e . 3. At h i g h e r t e m p e r a t u r e s t h e s l i p t r a c e s v i s i b l e i n t h e s t r u c t u r e s a r e r e p l a c e d by s u b b o u n d a r i e s ( F i g u r e s 16 a , b ) . 4 . Due t o t h e s m a l l amount o f d e f o r m a t i o n i n t h e samp les i t was not p o s s i b l e t o s ay w i t h c o n f i d e n c e whe the r o r not t h e g r a i n s r ema ined e q u i a x e d a f t e r s t r a i n i n g a t h i g h e r t e m p e r a t u r e s . However , t h e r e was some d i r e c t e v i d e n c e f o r t h e o c c u r r e n c e o f g r a i n boundary m i g r a t i o n . Assuming t h a t c r a c k n u c l e a t i o n a lway s t a k e s p l a c e a t g r a i n b o u n d a r i e s , i f t h e r e were no boundary m i g r a t i o n , c r a c k s s h o u l d be found o n l y a t g r a i n b o u n d a r i e s . In f a c t i s o l a t e d rounded c a v i t i e s were o b s e r v e d i n spec imens de fo rmed a t t h e l owe r s t r a i n r a t e ; e . g . a t 770 and 870°K f o r t h e 33% z i n c a l l o y ( F i g u r e 1 6 a ) . In t h e ca se o f h i g h e r s t r a i n r a t e t e s t s , o n l y t h e 33% z i n c a l l o y e x h i b i t e d a few s m a l l c a v i t i e s w i t h i n t h e g r a i n s a f t e r d e f o r m a t i o n a t 870°K/(Figure 1 6 b ) . 53 870°K F i g . 1 6 M i c r o s t r u c t u r e s o f a l p h a b r a s s e s e l o n g a t e d 8% ( s e r i e s 2) a t d i f f e r e n t t e m p e r a t u r e s 4 •> (a ) 33% z i n c a l l o y s t r a i n r a t e - 1.1 x 1 0 " s " , M a g n i f i c a t i o n 500X 54 55 870°K F i g . 16 ( c ) 19% z i n c a l l o y , s t r a i n r a t e - 1.1 x 10 " s " , M a g n i f i c a t i o n 500X 56 870°K F i g . 16 (d) 19% z i n c a l l o y , s t r a i n r a t e - 1.1 x 1 0 " 2 s " 1 , M a g n i f i c a t i o n 500X 57 F i g . 17 M i c r o - s t r u c t u r e s o f 24% z i n c b r a s s de fo rmed t o d i f f e r e n t d e g r e e s a t 670°K. S t r a i n r a t e 1.1 x 1 0 " 4 s _ 1 . MAG. 200X 58 F i g . 17 c o n t i n u e d 59 A t v e r y e a r l y s t a g e s o f d e f o r m a t i o n , i n t h e low d u c t i l i t y t e m p e r a t u r e r e g i o n , t h e r e i s d i r e c t e v i d e n c e o f g r a i n boundary s l i d i n g , p a r t i c u l a r l y a t t h o s e g r a i n b o u n d a r i e s wh i ch a r e t r a n s v e r s e t o t h e t e n s i l e s t r e s s . T h i s can be c o n c l u d e d f rom t he e x a m i n a t i o n o f p r e - p o l i s h e d sample s u r f a c e s i n t h e SEM ( F i g u r e 1 8 ) , where one can a l s o see g r a i n e d g e s . F i g u r e 17 r e v e a l s s t r u c t u r e s o f S e r i e s 3 spec imens (24% z i n c , sma l l s t r a i n s , 670°K). There i s no e v i d e n c e o f c r a c k i n g up t o 3.7% e l o n g a t i o n , a l t h o u g h g r a i n boundary s l i d i n g was v i s i b l e ( F i g u r e 18) a t s t r a i n s as l ow as 0.2%. T h i s may be more a p p a r e n t t h an r e a l , due t o t h e method o f sample p r e p a r a t i o n . Ve ry s m a l l c a v i t i e s a r e l i k e l y t o be masked o r even e l i m i n a t e d by e l e c t r o p o l i s h i n g . Above a c e r t a i n c r i t i c a l s i z e a c a v i t y s t a r t s t o r e t a i n a s h a r p edge and may a c t u a l l y i n c r e a s e s l i g h t l y i n s i z e d u r i n g e l e c t r o p o l i s h i n g . Above a s t r a i n o f 3.7% t h e r e was o b s e r v e d a g r a d u a l i n c r e a s e , w i t h i n c r e a s i n g s t r a i n , i n t h e s i z e o f t h e i n t e r -g r a n u l a r c r a c k s . F i g u r e 19 compares s t r u c t u r e s o f spec imens quenched a f t e r 30 s e c . and 3 s e c . The o b v i o u s d i f f e r e n c e can be a t t r i b u t e d t o t h e o c c u r r e n c e o f s t a t i c r e c r y s t a l 1 i z a t i o n i n t h e c a s e o f t h e more d e l a y e d quench . A d d i t i o n a l i n f o r m a t i o n was o b t a i n e d f rom SEM e x a m i n a t i o n o f f r a c t o g r a p h s ( F i g u r e 2 0 ) . For low t e m p e r a t u r e d e f o r m a t i o n ( h i g h d u c t i l i t y ) a t bo th h i g h and low s t r a i n r a t e s , t h e f r a c t u r e s can be i d e n t i f i e d as t r a n s -g r a n u l a r , c o n s i s t i n g m o s t l y o f s h e a r d i m p l e s . In spec imens f r a c t u r e d a t i n t e r m e d i a t e t e m p e r a t u r e s , t h e shapes o f g r a i n s were somet imes v i s i b l e , s u g g e s t i n g t h a t f r a c t u r e was a t l e a s t p a r t l y i n t e r g r a n u l a r . However t h e r e were no smooth f r a c t u r e f a c e s t y p i c a l o f i n t e r g r a n u l a r c l e a v a g e ; r a t h e r , t h e f r a c t u r e s u r f a c e s were c o v e r ed w i t h d i m p l e s and s h a r p i r r e g u l a r i t i e s . 60 U n f o r t u n a t e l y , f o r h i g h t e m p e r a t u r e spec imens t he f r a c t u r e f e a t u r e s c o u l d not be p o s i t i v e l y i d e n t i f i e d due l a r g e l y t o t h e o x i d a t i o n wh i ch t ook p l a c e d u r i n g f r a c t u r e and i m m e d i a t e l y a f t e r . T h i s i s seen by c ompa r i ng t h e ' f r a c t u r e s u r f a c e o f t h e sample de fo rmed i n o i l ( F i g u r e 21) w i t h t h e o t h e r s i n F i g u r e 20 . 61 F i g . 18 24% z i n c b r a s s de fo rmed t o d i f f e r e n t d eg r ee s a t 670°K As po l i s h e d - a n d - d e f o r m e d l o n g i t u d i n a l s u r f a c e Obse rved w i t h SEM ( s e r i e s 3 ' ) M a g n i f i c a t i o n 950X 62 ( a ) (b) F i g . 19 M i c r o s t r u c t u r e s o f 33% z i n c b r a s s de f o rmed a t 870°K a t a -2 -1 s t r a i n r a t e 1.1 x 10 s . M a g n i f i c a t i o n 800X ( a ) quenched w i t h i n 3 s e c , (b) quenched w i t h i n 30 s e c . 63 950°K 1050°K F i g . 20 F r a c t u r e s u r f a c e s o f spec imens de fo rmed i n t e n s i o n ( a ) 33% z i n c , s t r a i n r a t e 1.1 x 1 0 - 4 s - 1 . M a g n i f i c a t i o n 1000X 64 950°K 1050°K 2 1 F i g . 20(b) 33% z i n c , s t r a i n r a t e 1.1 x 10 s , M a g n i f i c a t i o n 1000X 65 950°K 1050°K F i g . 2 0 ( c ) 19% z i n c , s t r a i n r a t e 1.1 x 10 " s~ , M a g n i f i c a t i o n 400x 66 950°K 1050°K F i g . 2 0 ( d ) 19% z i n c , s t r a i n r a t e 1.1 x 10 s , M a g n i f i c a t i o n 400X 67 68 V DISCUSSION Two s t r a i n - r a t e dependent r e g i o n s o f e l e v a t e d t e m p e r a t u r e d e f o r m a t i o n b e h a v i o u r a r e r e p r e s e n t e d by t h e p r e s e n t r e s u l t s : 1. Lower T^ r e g i o n [0 .37 <J^ < 0 . 6 f o r t h e h i g h e r s t r a i n r a t e and 0..37<Tp| <0 .5 f o r t h e l o w e r s t r a i n r a t e ] . T h i s r e g i o n i s c h a r a c t e r i z e d by: - H i gh work h a r d e n i n g r a t e s ; i . e . h i g h v a l u e s ° f v e = j^f" - Y i e l d s t r e s s r e l a t i v e l y i n dependen t o f t e m p e r a t u r e w i t h i n t h e r e g i o n . - D i s c o n t i n u o u s y i e l d i n g (P - LeC e f f e c t ) a t t h e l o w e s t t e m p e r a t u r e s i n t h e r e g i o n . - Y i e l d and f l o w s t r e s s i n d ependen t o f s t r a i n r a t e (m=0) - D e c r e a s i n g d u c t i l i t y w i t h i n c r e a s i n g t e m p e r a t u r e . 2. H i g h e r T R r e g i o n [0 .75 <T^ < 0 .89 f o r h i g h s t r a i n r a t e and 0 .65 <T^ < 0 . 8 9 f o r l o w e r s t r a i n r a t e ] . T h i s r e g i o n i s c h a r a c t e r i z e d by : - Absence o f ne t s t r a i n - h a r d e n i n g - H igh s e n s i t i v i t y o f f l o w s t r e s s t o s t r a i n r a t e (m - 0 .2 t o 0 . 2 5 ) . - Data f i t an e q u a t i o n o f s t a t e ( E q . 4 ) . 1. Low Tempera tu re D e f o r m a t i o n Mode In t h e low homologous t e m p e r a t u r e r e g i o n t h e r e was e v i d e n c e o f t h e phenomenon o f dynamic s t r a i n a g i n g . The e v i d e n c e t ook t h e forms o f : (a ) . independence o f t h e y i e l d s t r e s s on t e m p e r a t u r e ; F i g u r e 10 a , b , (b) i ndependence o f t h e y i e l d s t r e s s on s t r a i n r a t e ; F i g u r e 1 2 a , ( c ) d e c r e a s e i n d u c t i l i t y ; F i g u r e 1 4 a , (d) p r e s en ce o f t h e P o r t e v i n -L e C h a t e l i e r e f f e c t ; F i g u r e 8 b , d , f (marked by l e t t e r s P - L e C ) . However t h e 69 P-LeC e f f e c t was o b s e r v e d o n l y a t t h e l o w e s t t e m p e r a t u r e s and o n l y a t t h e l o w e r s t r a i n r a t e , wh i ch i s somewhat c o n t r a d i c t o r y t o t h e c on cep t d i s c u s s e d by r e f e r e n c e [ 2 9 ] . T h i s a p p a r e n t s e p a r a t i o n o f t h e P-LeC e f f e c t f rom o t h e r DSA e v i d e n c e , a t l e a s t i n t h e c a s e o f a l p h a b r a s s e s , i s a l s o f ound i n t h e r e s u l t s p r e s e n t e d by r e f e r e n c e s [ 8 ] ; [ 1 1 ] , [ 4 2 ] . The o b s e r v a t i o n was not men t i o ned i n t h e c r i t i c i s m o f C o t t r e l l ' s model o f P-LeC e f f e c t , as a p p l i e d t o i n t e r s t i t i a l a l l o y s by K o r b e l [33] , [ 3 4 ] . However i t wou ld s u p p o r t t h e Ko r be l m o d e l , wh i ch r e l a t e d t he P-LeC e f f e c t t o t h e dynamic p r o p e r t i e s o f d i s l o c a t i o n p i l e - u p s . The t h e o r e t i c a l model o f Van Den Beuke l [35] d e s c r i b e d i n t h e I n t r o d u c -t i o n seems t o p r e d i c t q u a l i t a t i v e l y t h e o b s e r v e d b e h a v i o u r a t l o w e r t e m p e r a -t u r e . I t w i l l he re be d i s c u s s e d i n more d e t a i l . When a d i s l o c a t i o n overcomes an o b s t a c l e by t h e combined a c t i o n o f e f f e c t i v e s t r e s s and t he rma l a c t i v a t i o n t h e s t r a i n r a t e i s d e s c r i b e d by t he f o r m u l a . e = L e x p ( - ! L ) (15) where e 0 - a c o n s t a n t , H - an a c t i v a t i o n e n t h a l p y , k - B o l t z m a n ' s c o n s t a n t . P h y s i c a l l y t h e a c t i v a t i o n e n t h a l p y r e p r e s e n t s t h e ene rgy wh i ch s h o u l d be p r o v i d e d , by means o f t h e rma l v i b r a t i o n s o f atoms and by e f f e c t i v e s t r e s s , t o a m o b i l e d i s l o c a t i o n w a i t i n g a t an o b s t a c l e . Thus i t depends on bo th t h e e f f e c t i v e s t r e s s a and t he s o l u t e c o n c e n t r a t i o n a round t he d i s l o c a t i o n . e The l a t t e r dependence i s caused by c h e m i c a l a t t r a c t i o n o f s o l u t e atoms t o t h e s t r e s s f i e l d a round a d i s l o c a t i o n . Du r i n g d e f o r m a t i o n o f an a l l o y t h e s o l u t e c o n c e n t r a t i o n u s u a l l y can be c o n s i d e r e d c o n s t a n t and equa l t o c . I f , howeve r , s o l u t e atoms d i f f u s e t o d i s l o c a t i o n s wh i ch a r e w a i t i n g i n f r o n t o f o b s t a c l e s t o be s u rmoun t ed , t h e l o c a l c o n c e n t r a t i o n o f s o l u t e a t t h e w a i t i n g d i s l o c a t i o n i s not c o n s t a n t but a f u n c t i o n o f t h e w a i t i n g t i m e t u i 70 and t h e s o l u t e c o e f f i c i e n t D: c = c ( D t w ) (16) By u s i n g Orowan ' s e x p r e s s i o n f o r s t r a i n r a t e and t h e A r r h e n i u s r e l a t i o n s h i p wh i ch d e s c r i t e x p r e s s e d by i d e s c r i b e s t h e dependence o f D on t e m p e r a t u r e t h e q u a n t i t y Dt may be m+B Q D t w - — e x p ( - ^ ) (17) whe re : m and 8 a r e c o e f f i c i e n t s i n e m p i r i c a l e q u a t i o n s r e l a t i n g m o b i l e d i s l o c a t i o n d e n s i t y and v a c an c y c o n c e n t r a t i o n t o s t r a i n , and Q m i s t h e a c t i v a t i o n ene rgy f o r s e l f d i f f u s i o n . U s i ng a t h e o r e t i c a l c o n s i d e r a t i o n o f F r i e d e l [ 4 3 ] , i t i s p o s s i b l e t o d e s c r i b e t h e dependence o f c on Dt as i n F i g u r e 2 2 . From E q u a t i o n s 16 and 17 i t f o l l o w s t h a t t h e s o l u t e c o n c e n t r a t i o n a t t h e w a i t i n g d i s l o c a t i o n i s a f u n c t i o n o f e, £, and T. S u b s t i t u t i o n o f t h i s r e s u l t i n E q u a t i o n 15 p e r m i t s t h e a c t i v a t i o n e n t h a l p y t o be e x p r e s s e d , a s a f u n c t i o n o f d e f o r m a t i o n p a r a m e t e r s , and t h e r e b y p e r m i t s t h e dependence o f s t r e s s on e, e, and T t o be p r e d i c t e d . I t was shown by Pen i ng [44] t h a t inhomogeneous d e f o r m a t i o n , as o b s e r v e d i n t h e P o r t e v i n - L e C h a t e l i e r e f f e c t , can be e x p l a i n e d e l e g a n t l y by a s sum ing t h a t t h e s t r a i n r a t e h a r d e n i n g do/de i s n e g a t i v e i n a f i n i t e i n t e r v a l o f s t r a i n r a t e s . Inhomogeneous d e f o r m a t i o n s h o u l d s t a r t when t h e spec imen e n t e r s t h e r e g i o n o f n e g a t i v e s t r a i n r a t e h a r d e n i n g , i . e . when ^ = 0 (18 ) de Keep ing t h i s i n mind one can f i n d by d i f f e r e n t i a t i n g E q u a t i o n 15 and s u b s t i -t u t i n g E q u a t i o n 16 a f u n c t i o n d e s c r i b i n g t h e dependence o f s t r e s s on s t r a i n r a t e . 3 £ B J S I _ X a H D t dc ( l g ) where : V i s t h e a c t i v a t i o n v o l ume. 71 22 S c h e m a t i c dependence o f t h e s o l u t e c o n c e n t r a t i o n a t t h e d i s l o c a t i o n s , c , and i t s d e r i v a t i v e c 1 , on t h e f u n c t i o n Dt w 1 kT Q / A B \ • SH/SC Q D t w -23 Dt c ' ( D t ) as a f u n c t i o n o f Dt . In t h e r e g i o n o f Dt v a l u e s W W w w between A and B da /de i s n e g a t i v e . 72 Thus t h e c o n d i t i o n f o r t h e P-LeC e f f e c t w i l l be: D t w sm • W 5 c <20> w G r a p h i c a l l y t h i s i s p r e s e n t e d i n F i g u r e 2 3 . Between A and B a range o f Dt v a l u e s e x i s t s where do/dk i s n e g a t i v e . S i n c e Dt i s a f u n c t i o n o f w a w t e m p e r a t u r e ( E q u a t i o n 17) t h i s d e t e r m i n e s t h e range o f t e m p e r a t u r e f o r wh i ch t h e P-LeC e f f e c t s h o u l d o c c u r . A s i m i l a r m a t h e m a t i c a l p r o c edu r e can be employed f o r c a l c u l a t i o n o f t he t e m p e r a t u r e dependence o f f l o w s t r e s s . From E q u a t i o n 15 i t f o l l o w s : o H(a ) = kT In % (21) D i f f e r e n t i a t i o n a n d v s u b s t i t u t i o n i n E q u a t i o n s 16 and 17 l e a d t o t h e e x p r e s s i o n rJ* - H + 3" | H D t dc ( 2 2 ) o T T ^jZ 8c w d (D t ) The r e g i o n o f r e v e r s e dependence o f a on t e m p e r a t u r e i s c a l c u l a t e d i n e x a c t l y t h e same way as i n t he ca se o f s t r a i n r a t e dependence , l e a d i n g t o a r e s u l t : D tw dTory= T W9c" ( 2 3 ) w m I t may be seen t h a t t h e o n l y d i f f e r e n c e between t h i s c r i t e r i o n and t h e p r e v i o u s one f o r t h e P-LeC e f f e c t ( E q u a t i o n 20) i s t h e r a t i o . In t h i s c on cep t m o f ^ d e f o r m a t i o n t h e s i n g l e u n i t d i s l o c a t i o n movement i n v o l v e s o n l y t h e o ve r c om ing o f l a t t i c e f r i c t i o n under t h e combined a c t i o n o f t he e f f e c t i v e s t r e s s a and a t h e rma l a c t i v a t i o n e n e r g y . Thus t he v a l u e o f H can be c o n n e c t e d w i t h t h e o ve r c om ing by d i s l o c a t i o n s o f t h e P e i e r l s - N a b a r r o . b a r r i e r s , a t l e a s t i n t h e e a r l y s t a g e s o f d e f o r m a t i o n . A c c o r d i n g t o c a l c u l a t i o n s done on c oppe r by Seeger e t a l . [45] H i s equa l t o a p p r o x i m a t e l y 9 .6 k j /mo l whereas t h e v a l u e o f Q m a s . r e p o r t e d by T e g a r t [28] i s about 210 k j / m o l . 73 One can e x p e c t t h e v a l u e s o f H and Q m f o r a l p h a b r a s s t o d i f f e r . f r o m t h o s e H o f c o p p e r ; however t h e r a t i o jr- s h o u l d r ema in r e l a t i v e l y unchanged i . e . much l e s s t h an u n i t y . For t h i s r e a s on a t c o n s t a n t t e m p e r a t u r e t h e v a l u e o f dc d ( D t w ) wh i c h i s equa l t o Q— ^ i s l o w e r t h a n i n t h e c a s e o f s t r a i n r a t e m dependence . Thus , l o o k i n g a t F i g u r e 22 t h e c o r r e s p o n s i n g range o f tempera^ t u r e ( p o i n t s A and B) as d e t e r m i n e d by Dt i s b r o ade r f o r t h e r e v e r s e dependence o f f l o w s t r e s s on t e m p e r a t u r e , wh i c h i s i n a c c o r d a n c e w i t h t h e p r e s e n t e x p e r i m e n t s , a t l e a s t on t h e ' h i g h e r t e m p e r a t u r e s i d e . Ano t he r app roach t o a c c o u n t i n g f o r t h e r e l a t i v e i ndependence o f t h e f l o w s t r e s s o f a l p h a b r a s s on t e m p e r a t u r e may be i n v o k e d i f i t i s a c c e p t e d t h a t some l o n g - r a n g e o r d e r i n g i s p r e s e n t . As summar ized by N i c h o l s o n & K e l l y [46] t h e t e m p e r a t u r e dependence o f y i e l d i n g i n many o f t h e f . c . c . s u p e r l a t t i c e a l l o y s so f a r examined appea r s t o be n o r m a l ; t h e r e i s a s m a l l d e c r e a s e i n f l o w s t r e s s w i t h i n c r e a s i n g t e m p e r a t u r e . However , t h e r e a r e some r e p o r t e d e x c ep t t i o n s l i k e N i^Al and Cu-^Au i n wh i ch t h i s t e nden cy i s r e v e r s e d . Mo reove r t h e e f f e c t o f d i f f u s i o n - r e l a t e d mechanisms has been r u l e d ou t by t h e absence o f o t h e r s t r a i n a g i n g e f f e c t s [ 4 7 ] . I f o r d e r i n g o c c u r s i n a l p h a b r a s s e s as b r i e f l y d i s c u s s e d i n t h e I n t r o -d u c t i o n t he s t o i c h i o m e t r y o f t h e o r d e r e d domains wou ld be r e p r e s e n t e d by Cu^Zn a c c o r d i n g t o r e f e r e n c e [ 4 0 ] . The a p p l i c a t i o n t o b r a s s e s o f an argument due t o J o h n s t o n e t :al . [48] i s i l l u s t r a t e d w i t h t h e a i d o f F i g u r e 24 . A b a s i c c e l l o f t h e Cu^.Z.ii l a t t i c e i s d e p i c t e d t o g e t h e r w i t h an e x t e n s i o n t o a d j a c e n t c e l l s o f {111} p l a n e . As shown {111} c o n s i s t s o f c l o s e - p a c k e d rows i n wh i ch coppe r and z imc atoms a l t e r n a t e . Le t us assume t h a t t h e b i n d i n g ene rgy o r e l e c t r o n d e n s i t y between coppe r and z i n c atoms i n t h e d i r e c t i o n o f row B-B d i f f e r s . : f r o m t h a t between coppe r atoms and z i n c atoms i n t h e d i r e c t i o n o f row A-A , and t h a t t h e c o p p e r - c o p p e r b i n d i n g i s s i m i l a r t o t h a t wh i ch a p p l i e s 74 O Cu © Zn F i g . 24 A b a s i c c e l l o f t h e Cu^Zn l a t t i c e w i t h a { 1 1 1 } p l a n e e x t e n d e d t o a d j a c e n t c e l l s . 75 i n pure c o p p e r . As t h e t e m p e r a t u r e i s i n c r e a s e d each o f t h e s e rows may be e x p e c t e d t o e x h i b i t d i f f e r e n t e x p a n s i o n c h a r a c t e r i s t i c s , w i t h t h e o v e r a l l m a c r o s c o p i c e x p a n s i o n b e i n g an a v e r a g e o f t h e t w o . S i n c e t h e o b s e r v e d t h e rma l e x p a n s i o n c o e f f i c i e n t f o r a CuZn50 a l l o y i s about 22% h i g h e r t h an t h a t o f pure c o p p e r , i t f o l l o w s t h a t e x p a n s i o n a l o n g row A-A i s g r e a t e r t h an a l o n g row B-B. T h e r e f o r e , i n o r d e r t o p r e s e r v e t he c u b i c c r y s t a l s t r u c t u r e , a sys tem o f e q u i l i b r a t e d l a t t i c e f o r c e s must be i n t r o d u c e d . As l o n g as t h e Cu^Zn a l l o y r ema i n s o r d e r e d , t h e magn i t ude o f t h e s e l a t t i c e f o r c e s w i l l i n c r e a s e w i t h t e m p e r a t u r e . These l a t t i c e f o r c e s can be accommodated a l o n g t h e m i s f i t r e g i o n o f a d i s l o c a t i o n . C o n s e q u e n t l y , d i s l o c a t i o n s i n t h e Cu'^Zn l a t t i c e w i l l be s u b j e c t e d t o a l a t t i c e r e s i s t a n c e t h a t w i l l i n c r e a s e w i t h t e m p e r a t u r e u n t i l t h e r m a l l y a c t i v a t e d p r o c e s s e s such as k i n k g e n e r a t i o n and r e c o v e r y beg i n t o d o m i n a t e , o r u n t i l o r d e r i n g i s l o s t , g i v i n g r i s e t o a peak i n t h e f l o w s t r e s s - t e m p e r a t u r e c u r v e . The e f f e c t s h o u l d be more p ronounced i n t h e c a s e o f a s i n g l e c r y s t a l , where t h e p o s s i b l e r e l a x i n g r o l e o f g r a i n b o u n d a r i e s i s e l i m i n a t e d . I n d e e d , a v e r y d i s t i n c t peak i n f l o w s t r e s s vs t e m p e r a t u r e has been r e p o r t e d f o r a l p h a b r a s s s i n g l e c r y s t a l s [49]whereas t h e p r e s e n t d a t a f o r p o l y c r y s t a l s show no e q u i v a l e n t maximum. The above t h e o r i e s seem a t f i r s t g l a n c e t o be q u i t e d i s t i n c t and d i f f e r e n t . However , i f we s u b s t i t u t e t h e c on cep t o f d i s l o c a t i o n s " w a i t i n g " a t o b s t a c l e s f o r t h a t o f d i s l o c a t i o n s " a r r e s t e d " i n an o r d e r e d domain and w a i t i n g t o be f r e e d by p o s s i b l e d i s o r d e r i n g o f t h e domain ( o n l y s h o r t r ange d i f f u s i o n i s i n v o l v e d ) t h e two mode l s c o u l d have much i n common. No d i r e c t s u p p o r t f o r t h e e x i s t e n c e o f o r d e r e d domains i n a l p h a b r a s s e s was sough t a c t i v e l y i n t h e p r e s e n t s t u d i e s . E v i d e n c e f o r o r d e r i n g , and the c o n d i t i o n s n e c e s s a r y t o i n t r o d u c e i t i n b r a s s e s , a r e a t b e s t s k e t c h y i n t h e p u b l i s h e d l i t e r a t u r e . I t i s t h e r e f o r e pe rhaps p rematu re t o d e v e l o p t he o r d e r i n g model f u r t h e r . 76 2. H igh Tempera tu re D e f o r m a t i o n Mode S i n c e a l p h a b r a s s e s have low s t a c k i n g f a u l t e ne r gy dynamic r e c r y s t a l l i -z a t i o n s h o u l d be t h e dominant r e s t o r a t i o n p r o c e s s i n h i g h t e m p e r a t u r e d e f o r m -a t i o n [ 1 7 ] . C o n f i r m a t i o n o f t h i s was not p r o v i d e d i n t h e p r e s e n t work by t h e appea r ance o f t h e s t r e s s - s t r a i n c u r v e s s i n c e t h e r e were no o s c i l l a t i o n s i n f l o w s t r e s s , and o n l y i n a few c a s e s was a maximum f l o w s t r e s s o b s e r v e d . The p r e s en ce o f s t r a i g h t s l i p t r a c e s i n spec imens de fo rmed a t t e m p e r a t u r e s be low 770°K ( F i g u r e 16) p r o v i d e d p r o o f t h a t t h e b r a s s e s i n v e s t i g a t e d do have low s t a c k i n g f a u l t e ne r gy and d i d no t undergo f u l l dynamic r e c o v e r y . Compar i son o f m i c r o s t r u c t u r e s w i t h s t r e s s - s t r a i n c u r v e s r e v e a l t h a t when s t r a i g h t s l i p t r a c e s v a n i s h t h e m a t e r i a l c ea se s t o work h a r d e n ; t hu s a dynamic r e s t o r a t i o n p r o c e s s has become a c t i v e . The sudden change i n work h a r d e n i n g r a t e wh i ch was o b s e r v e d ( F i g u r e 12) wou ld a l s o s ugge s t t h a t dynamic r e c r y s t a l 1 i z a t i o n i s t h e a c t i v e s o f t e n i n g p r o c e s s . U n f o r t u n a t e l y , d i r e c t s u p p o r t f o r r e c r y s t a l 1 i z a t i o n was not o b t a i n e d f rom o b s e r v a t i o n s o f t h e m i c r o s t r u c t u r e o f spec imens de fo rmed a t t h e h i g h e r t e m p e r a t u r e s . Changes i n . g r a i n s i z e were t oo s m a l l t o be a t t r i b u t e d unamb ig -u o u s l y t o dynamic r e c r y s t a l 1 i z a t i o n . However t h e p r e s en ce o f v o i d s w i t h i n g r a i n s ( F i g u r e 16) i n d i c a t e d t h a t g r a i n bounda ry m i g r a t i o n had a c t u a l l y t a k e n p l a c e , s i n c e n u c l e a t i o n o f v o i d s d u r i n g h i g h t e m p e r a t u r e d e f o r m a t i o n was o b s e r v e d o n l y a t g r a i n b o u n d a r i e s . F u r t h e r s t r o n g s u p p o r t f o r t h e dominance o f dynamic r e c r y s t a l l i z a t i o n was, m o r e o v e r , p r o v i d e d by a c t i v a t i o n ene r gy v a l u e s . In t h e ca se o f t h e 33% z i n c a l l o y t h e a c t i v a t i o n ene r gy o f d e f o r m a t i o n was 260 k J / m o l . T h i s i s i n e x c e l l e n t agreement w i t h t h e v a l u e o f 255-264 kJ/mo l r e p o r t e d f o r a 34% z i n c a l l o y by B l a z e t a l . [ 2 4 ] , who i d e n t i f i e d t h e o p e r a t i n g s o f t e n i n g mechanism as dynamic r e c r y s t a l 1 i z a t i o n on t h e b a s i s o f s e v e r a l t y p e s o f e v i d e n c e . The s t r e s s - s t r a i n c u r v e s d e s c r i b e d by B l a z e t a l . a g a i n f a i l e d t o 77 show o s c i l l a t i o n s o r peak s t r e s s e s ? . . The a c t i v a t i o n e n e r g y o f 235 kJ/mo l wh i ch was found i n t h e p r e s e n t work f o r Cu'Znl9 b r a s s i s s u b s t a n t i a l l y h i g h e r t h a n t h a t o f s e l f d i f f u s i o n i n b r a s s . The l a t t e r s t a t e m e n t i s based on a c o m p a r i s o n w i t h pure c o p p e r , s i n c e no v a l u e s f o r t h e s e l f d i f f u s i o n a c t i v a -t i o n ene r gy o f a l p h a b r a s s e s c o u l d be found i n t h e l i t e r a t u r e . For pure c oppe r t h e a c t i v a t i o n ene r gy i s r e p o r t e d t o be equa l t o about 300 kJ/mo l f o r hot w o r k i n g [ 2 4 ] , [ 2 8 ] , and 210 kJ /mo l f o r s e l f d i f f u s i o n [ 2 8 ] . S i n c e s e l f d i f f u s i o n i n b r a s s e s s h o u l d be s u b s t a n t i a l l y e a s i e r due t o t h e h i g h e r m o b i l i t y o f z i n c atoms [50] one can e x p e c t t h a t a c t i v a t i o n ene r gy f o r s e l f d i f f u s i o n o f b r a s s e s s h o u l d be l o w e r t h a n t h a t f o r c o p p e r . A l s o one m i gh t e x p e c t t h a t t h e r a t i o o f t h e a c t i v a t i o n e n e r g y o f dynamic r e c r y s t a l l i z a t i o n t o t h a t o f s e l f d i f f u s i o n s h o u l d be a p p r o x i m a t e l y t h e same f o r coppe r as f o r b r a s s e s . T h i s r a t i o f o r c oppe r i s 1 .4 . Thus assuming t h a t 260 kJ /mo l i s t h e c o r r e c t v a l u e f o r t h e a c t i v a t i o n ene rgy o f dynamic r e c r y s t a l 1 i z a t i o n i n t h e 33% z i n c a l l o y , one m i gh t e x p e c t t h e v a l u e f o r s e l f d i f f u s i o n t o be about 185 k J / m o l . T h i s s u p p o r t s t h e s u g g e s t i o n t h a t f o r a l l t h e a l l o y s t h e ma in dynamic s o f t e n i n g mechanism was r e c r y s t a l l i z a t i o n , s i n c e t h e a c t i v a t i o n ene rgy f o r dynamic r e c o v e r y wou ld be e x p e c t e d t o be equa l t o t h a t o f s e l f d i f f u s i o n . 3 . The D u c t i l i t y Minimum In pure t e n s i l e d e f o r m a t i o n , f r a c t u r e i s p receded by p l a s t i c i n s t a b i l i t y ( n e c k i n g ) . I t can be a rgued t h a t t h e s t r a i n t o n e c k i n g , as measured by A r u i n t h e p r e s e n t wo rk , i s t h e most s i g n i f i c a n t measure o f t h e " d u c t i l i t y " , s i n c e f o r p r a c t i c a l pu rposes f a i l u r e has o c c u r r e d once t h e m a t e r i a l n e c k s . However , t h e amount o f s t r a i n t h a t a m a t e r i a l can a b s o r b b e f o r e i t f r a c t u r e s i s a more u s e f u l c r i t e r i o n o f d u c t i l i t y when m e c h a n i c a l w o r k i n g p r o c e s s e s l i k e r o l l i n g and f o r g i n g a r e c o n c e r n e d . E l o n g a t i o n t o f r a c t u r e ( e f ) o f 78 r e d u c t i o n i n a r e a a t f r a c t u r e ( A ^ ) o f t e n s i o n spec imens a r e t h e r e f o r e commonly used t o d e s c r i b e d u c t i l i t y . U n f o r t u n a t e l y , t h e s t a t e o f s t r e s s changes d u r i n g t h e g rowth o f necks and a l s o t h e v a l u e o f t h e s e l a t t e r p r o p e r t i e s i s i n f l u e n c e d by m i c r o s t r u c t u r a l f a c t o r s . Owing t o t h e c o m p l e x i t y o f g e o m e t r i c a l c o n s i d e r a t i o n s , no a t t e m p t has been made t o i n c l u d e them i n t h e a n a l y s i s wh i ch f o l l o w s . The f r a c t u r e d u c t i l i t y min ima i n t h e p r e s e n t work o c c u r r e d i n t h e c a s e o f h i g h s t r a i n r a t e at . t e m p e r a t u r e s between t h e two r eg imes d e s c r i b e d i n t h e p r e c e d i n g d i s c u s s i o n ; i . e . a t t h e t o p end o f t h e l o w - t e m p e r a t u r e r e g i o n and a t t h e l owe r end o f t h e h i g h t e m p e r a t u r e r e g i o n . In t h e case o f t h e l o w e r s t r a i n r a t e t h e d u c t i l i t y minimum f o r t h e 33% z i n c a l l o y i s w e l l w i t h i n t h e h i g h t e m p e r a t u r e r e g i o n . T h i s l e a v e s open a l a r g e r ange o f p o s s i b i l i t i e s f o r f a c t o r s wh i c h may i n f l u e n c e t he d u c t i l i t y minimum s i n c e dynamic s t r a i n a g i n g , o r o r d e r i n g , o r dynamic r e s t o r a t i o n , o r c o m b i n a t i o n s o f t h e s e phenomena may be i n v o l v e d . Most n o t a b l e o f t h e m e t a l l o g r a p h i c o b s e r v a t i o n s i n t h i s t e m p e r a t u r e -s t r a i n r a t e r e g i o n was t h e o c c u r r e n c e o f g r a i n boundary m i c r o c r a c k s . These were found t o have formed a f t e r s m a l l p l a s t i c s t r a i n s ( F i g u r e 17) and t h e g rowth o f t h e c r a c k s was c l e a r l y an i m p o r t a n t p a r t o f t h e f r a c t u r e p r o c e s s . There a r e numerous r e p o r t s o f s i m i l a r m i c r o c r a c k g rowth i n m e t a l s and a l l o y s u n d e r g o i n g e l e v a t e d t e m p e r a t u r e d e f o r m a t i o n , i n c l u d i n g c r e e p . A g e n e r a l i z e d d e s c r i p t i o n o f m e c h a n i c a l f a i l u r e a t h i g h e r t e m p e r a t u r e s was s ugge s t e d by T e g a r t [ 2 8 ] . A c c o r d i n g t o t h i s model t h e s t r e s s r e q u i r e d t o de fo rm a me t a l p l a s t i c a l l y a t t h e imposed s t r a i n r a t e i s a l s o s u f f i c i e n t t o fo rm c r a c k s a t t h e o r i g i n a l g r a i n b o u n d a r i e s , bo th a t t r i p l e j u n c t i o n s and a t i r r e g u l a r i t i e s d e v e l o p e d i n t h e bounda r y . These c r a c k s can t hen grow under t h e combined a c t i o n o f v a c a n c y d i f f u s i o n a l o n g g r a i n b o u n d a r i e s and a p p l i e d t e n s i l e s t r e s s e s . For m a t e r i a l s i n wh i ch r e c r y s t a l 1 i z a t i o n i s 79 t h e s o f t e n i n g p r o c e s s , c o n d i t i o n s o f h i g h d u c t i l i t y c o r r e s p o n d t o t h e m i g r a t i o n o r r e c r y s t a l l i z a t i o n o f o r i g i n a l g r a i n b o u n d a r i e s , t hu s i s o l a t i n g t h e i n i t i a l c r a c k s and p r e v e n t i n g f u r t h e r immed ia te g r o w t h . F u r t h e r c r a c k g rowth o c c u r s by " c a p t u r i n g " a mov ing g r a i n boundary f o r a s u f f i c i e n t t ime f o r v a c a n c y d i f f u s i o n and t h e a p p l i e d s t r e s s t o l e n g t h e n t h e c r a c k s a l i t t l e b e f o r e t h e boundary a g a i n b r eak s away. Thus t h e c o n t r o l l i n g p r o c e s s i n c r a c k p r o p a g a t i o n i s seen t o be g r a i n boundary m i g r a t i o n . For m e t a l s where r e p o l y g o n i z a t i o n (dynamic r e c o v e r y ) i s t h e s o f t e n i n g p r o c e s s , a l t h o u g h t h e o r i g i n a l g r a i n b o u n d a r i e s t end t o l o s e ; t h e i r i d e n t i t i e s , t h e m i g r a t i n g s u b b o u n d a r i e s w o u l d . n o t be e x p e c t e d t o sweep t h r o u g h t h e m a t e r i a l i n t h e same way as g r a i n b o u n d a r i e s . F u r t h e r , t h e measured a ve r age low a n g l e s o f m i s o r i e n t a t i o n j o f t h e s u b b o u n d a r i e s wou ld not f a v o u r v a c an c y d i f f u s i o n a l o n g s u b b o u n d a r i e s o v e r v a c a n c y d i f f u s i o n t h r o u g h t he l a t t i c e , and hence g rowth o f t h e c r a c k by t h i s p r o c e s s wou ld be e x p e c t e d t o be s l o w . However , i n l o c a l i z e d zones a d j a c e n t t o t h e o r i g i n a l g r a i n b o u n d a r i e s , h i g h m i s o r i e n t a t i o n can be a c h i e v e d and f u r t h e r c r a c k i n g can o c c u r a t t h e s e new t r i p l e p o i n t s and boundary s e r r a t i o n s . These can l i n k up under t h e a p p l i e d s t r e s s g i v i n g a s e r r a t e d , c r a c k e d boundary r e g i o n . The c o n t r o l l i n g p r o c e s s i n c r a c k p r o p a -g a t i o n i s t hu s r e p o l y g o n i z a t i o n , wh i c h a c t s t o m a i n t a i n a l l s u b b o u n d a r i e s a t t h e same sma l l a v e r age m i s o r i e n t a t i o n . T h i s c on c ep t does not dea l i n d e t a i l w i t h c r a c k n u c l e a t i o n . F u r t h e r m o r e , t h e s u g g e s t e d method o f c r a c k p r o p a g a t i o n by v a c a n c y - d i f f u s i o n i s not commonly a c c e p t e d . I t i s t h e r e f o r e a p p r o p r i a t e t o p r e s e n t more d e t a i l e d t h e o r i e s c o n c e r n i n g t h e o r i g i n o f m e c h a n i c a l f a i l u r e a t e l e v a t e d t e m p e r a t u r e s . Three t h e o r i e s have been advanced t o e x p l a i n t h e o r i g i n o f g r a i n - b o u n d a r y m i c r o -c r a c k s : The f i r s t o f t h e s e t o be d i s c u s s e d i n v o l v e s a g r a i n - b o u n d a r y s l i d i n g mechan i sm. At c e r t a i n t e m p e r a t u r e s and s t r a i n r a t e s , d e f o r m a t i o n can 80 i n v o l v e s l i d i n g a t g r a i n b o u n d a r i e s . I f t h e c o n t i n u i t y o f t h e s o l i d i s t o be m a i n t a i n e d , changes i n t h e shape o f g r a i n s must o c c u r by p l a s t i c f l o w . I f t h e g r a i n i n t e r i o r s a r e r e l a t i v e l y h a r d , accommodat ion o f s l i d i n g w i l l t a k e t h e f o rm o f o p e n i n g s a t t r i p l e j u n c t i o n s [ 1 4 ] . However , i n t h e p r e s e n t work t r i p l e j u n c t i o n s were not t h e p r e f e r r e d l o c a t i o n o f t h e m i c r o c r a c k s t h a t were o b s e r v e d . O t he r s have a l s o r e p o r t e d t h a t g r a i n boundary c r a c k s a r e commonly remote f rom t r i p l e j u n c t i o n s . A m o d i f i e d model has been advanced by G i f k i n s [ 5 1 ] , and i s e x p l a i n e d i n F i g u r e 2 5 . To accommodate shape changes a s s o c i a t e d w i t h s l i d i n g between g r a i n s A and B i n r e s pon se t o a s t r e s s normal t o t h e s l i d i n g b ounda r y , s l i p o c c u r s i n g r a i n B as i n d i c a t e d . T h i s p roduces a s m a l l j o g i n t h e bounda r y , t h e s i z e o f wh i ch depends on the number o f d i s l o c a t i o n s wh i ch must pass i n t o t h e boundary t o i n i t i a t e " a c commoda t i ng " s l i p i n g r a i n A. D i s l o c a t i o n s a r e p i l e d up as i n d i c a t e d , and t h e r e w i l l be a s t r e s s f i e l d a round them a t t h e bounda r y . S i n c e t h e boundary i s s l i d i n g , a c a v i t y may open up as shown i n F i g u r e 2 5 ( c ) b e f o r e t h e boundary can become s t r a i g h t e n e d by boundary m i g r a t i o n . I t i s assumed i n t h e model t h a t t h e s t r e s s f i e l d a s s o c i a t e d w i t h d i s l o c a t i o n p i l e - u p can p roduce l o s s o f c o h e s i o n a c r o s s t h e j o gged p o r t i o n o f t h e boundary when t h e j o g i s s m a l l . A v o i d once n u c l e a t e d can grow due t o s t r e s s c o n c e n t r a t i o n a t t h e t i p o f t h e " c r a c k " as w e l l as by f u r t h e r g r a i n boundary s l i d i n g . T h i s model does not d i r e c t l y e x p l a i n t h e o b s e r v e d s t r a i n r a t e dependence o f t h e d u c t i l i t y min imum. A second model f o r t h e o r i g i n o f g r a i n boundary m i c r o v o i d s has emerged m o s t l y f rom c r e e p s t u d i e s . The s u g g e s t i o n was f i r s t made by Greenwood e t a l . [13] t h a t c a v i t y n u c l e i a r e fo rmed by t h e m i g r a t i o n o f v a c a n c i e s a l r e a d y e x i s t i n g i n t h e me ta l o r c r e a t e d d u r i n g d e f o r m a t i o n . Owing t o t h e rma l v i b r a t i o n s , and t h e movement o f d i s l o c a t i o n s under s t r e s s , t h e r e l a t i v e p o s i t i o n s o f t h e v a c a n c i e s a r e c o n s t a n t l y c h a n g i n g . I f a s u f f i c i e n t number 81 F i g . 25 V o i d n u c l e a t i o n by s l i d i n g g r a i n boundary [51] 82 a r r i v e s i m u l t a n e o u s l y a t a g i v e n l o c a t i o n a s t a b l e " h o l e " can be n u c l e a t e d . The g r a i n boundary t r a n s v e r s e t o t h e t e n s i l e s t r e s s a x i s a c t s as a t r a p i n wh i ch such v o i d s r e a d i l y f o r m . A more d e t a i l e d mechanism f o r v o i d c r e a t i o n due t o c o n c e n t r a t i o n o f v a c a n c i e s was p roposed by B a l u f f i and S e i g l e [ 5 0 ] . A c c o r d i n g t o t h e s e a u t h o r s , i n a l p h a b r a s s e s due t o t h e K i r k e n d a l l e f f e c t a s u p e r s a t u r a t i o n i n v a can cy c o n c e n t r a t i o n i s c r e a t e d wh i ch s u b s e q u e n t l y i s r e s p o n s i b l e f o r v o i d f o r m a t i o n . S e i g l e l a t e r advanced an h y p o t h e s i s o f h e t e r ogeneous n u c l e a t i o n , i n wh i ch o x i d e p a r t i c l e s c o u l d a c t as n u c l e i [ 5 2 ] . T h e o r e t i c a l l y t h i s c on c ep t was d e v e l o p e d by B a l u f f i and S e i g l e [ 5 3 ] . S u b s e q u e n t l y e x p e r i m e n t a l work was u n d e r t a k e n [ 5 4 ] , [55] wh i ch v e r i f i e d a p r e d i c t i o n t h a t a d e c i s i v e f a c t o r i n t h e g rowth o f c a v i t i e s i s t h e s t a t e o f s t r e s s . When h y d r o s t a t i c p r e s s u r e i s s upe r imposed on a t e n s i l e s t r e s s . t h e c a v i t y g r ow th i s r e t a r d e d f o r bo th magnesium and c oppe r unde r c r e e p c o n d i t i o n . R ' a t c l i f f e and Greenwood [55] a c c o r d i n g l y c o n c l u d e d t h a t c r a c k p r o p a g a t i o n i s due o n l y t o v a c an c y d i f f u s i o n . On t h e o t h e r hand t h e s t u d i e s o f T a p l i n [ 5 5 ] r e l a t i n g t o t h e shape o f i n t e r g r a n u l a r c a v i t i e s s u g g e s t s t h a t a p r o c e s s more l i k e m e c h a n i c a l t e a r i n g was i n v o l v e d i n t h e c r a c k p r o p a g a t i o n . A t h i r d t h e o r y o f g r a i n - b o u n d a r y m i c r o c r a c k i n i t i a t i o n and t h e d u c t i l i t y minimum d e v e l o p e d by Izumi e t a l . [ 30 ] a t t emp ted t o e x p l a i n some o b s e r v a t i o n s on a l p h a b r a s s e s . They n o t i c e d a s t e a d y i n c r e a s e i n g r a i n boundary s l i d i n g w i t h t e m p e r a t u r e as measured on b i c r y s t a l s . The amount o f s l i d i n g i n t h e i n t e r m e d i a t e t e m p e r a t u r e r e g i o n was o n l y about 1 ym but i t was r e a ched a t an e a r l y s t a g e o f d e f o r m a t i o n . Izumi c o n c l u d e d t h a t g r a i n boundary s l i d i n g i s not r e s p o n s i b l e f o r t h e d u c t i l i t y min imum, s i n c e a t h i g h e r t e m p e r a t u r e s ( h i g h d u c t i l i t y ) i t s c o n t r i b u t i o n t o t h e d e f o r m a t i o n p r o c e s s i s much l a r g e r . However t h e amount o f s l i d i n g t h e y o b s e r v e d c o u l d be s u f f i c i e n t t o n u c l e a t e a v o i d by t h e mechanism o f G i f k i n s [51] d i s c u s s e d above . On t h e o t h e r hand 83 m e t a l l o g r a p h i c s t u d i e s by Yamagata and Izumi [10] had r e v e a l e d changes i n t h e d i s t r i b u t i o n o f s l i p bands i n 70/30 b r a s s . The s l i p bands became c o a r s e r as t h e d e f o r m a t i o n t e m p e r a t u r e was i n c r e a s e d and t h e f r a c t u r e s u r f a c e s showed d i m p l e p a t t e r n s when o b s e r v e d by s c a n n i n g e l e c t r o n m i c r o s c o p y . On t h e b a s i s o f t h e s e o b s e r v a t i o n s a model was p roposed by Izumi a c c o r d -i n g t o wh i ch s l i p bands a r e t e r m i n a t e d at. g r a i n b o u n d a r i e s . At each s l i p band t h e d i s l o c a t i o n s p i l e up a t t h e g r a i n boundary and c r e a t e s t r e s s c o n c e n t r a t i o n s wh i ch a r e h i g h enough t o n u c l e a t e f i n e c r a c k s . The o b s e r v a t i o n o f c r a c k s i n ' g r a i n b o u n d a r i e s a t t h e ends o f s l i p t r a c e s c o u l d not be c o n f i r m e d i n t h e p r e s e n t work . Greenwood e t a l . [13] r e p o r t e d t h a t c o a r s e s l i p t r a c e s i n f a c t d i s a p p e a r i n t h e r e g i o n o f low d u c t i l i t y . A l s o t h e p r e s e n t o b s e r v a t i o n s t h a t l ow d u c t i l i t y can be o b s e r v e d i n s t r u c t u r e s u n d e r g o i n g some deg ree o f dynamic r e s t o r a t i o n a r e i n d i r e c t c o n f l i c t w i t h I z u m i 1 s m o d e l . To d i s c r i m i n a t e e x p e r i m e n t a l l y between t h e f i r s t two p roposed mechanisms o f m i c r o c r a c k n u c l e a t i o n i s d i f f i c u l t . Some g r a i n boundary s l i d i n g a t p r e -p o l i s h e d s u r f a c e s was d i r e c t l y o b s e r v ed i n t h e p r e s e n t work ( F i g u r e 1 8 ) . There i s a p o s s i b i l i t y , however , t h a t g r a i n b o u n d a r i e s a r e weakened by c o n c e n t r a t i o n s o f v a c a n c i e s ( v o i d s ) and t h a t s l i d i n g i s t h e r e f o r e an e f f e c t r a t h e r t h an a c a u s e . A s p e c i a l e x p e r i m e n t was d e v i s e d i n an a t t e m p t t o c l a r i f y t h i s i s s u e . A spec imen o f 24% z i n c b r a s s was hea ted f o r 30 m i n u t e s a t 670°K, t h e t e m p e r a t u r e o f t h e d u c t i l i t y minimum f o r t h i s a l l o y a t a s t r a i n r a t e o f -4 -1 1.1 x 10 s , t h en c o o l e d t o 470°K and t e s t e d t o f a i l u r e a t a s t r a i n r a t e -4 -1 o f 1.1 x 10 s . W h i l e a t t h e h i g h t e m p e r a t u r e , t h e spec imen was " s e n s i t i z e d " by s u b j e c t i n g i t t o a t e n s i l e s t r e s s equa l t o one q u a r t e r o f t h e y i e l d s t r e s s a t t h a t t e m p e r a t u r e , t hu s p r o v i d i n g encouragement f o r v a c an c y m i g r a t i o n . 84 The use o f t h e low s t r e s s was s u g g e s t e d by t h e t h e o r e t i c a l work o f B a l u f f i and S e i g l e [ 5 3 ] . The h o l d i n g t ime o f 30 m i nu t e s a t t h e h i g h e r t e m p e r a t u r e c o r r e s p o n d s c l o s e l y t o t h e t ime r e q u i r e d t o c omp l e t e a t e n s i o n t e s t t o -4 -1 f r a c t u r e a t t h e d u c t i l i t y min imum, and a t t h e s t r a i n r a t e o f 10 s The t e n s i l e d a t a i n T a b l e IV r e v e a l t h a t t h e f r a c t u r e d u c t i l i t y a t 470°K was s u b s t a n t i a l l y l owe r ed by t h e s e n s i t i z i n g t r e a t m e n t . T a b l e IV I n f l u e n c e o f s e n s i t i z i n g t r e a t m e n t on d u c t i l i t y o f 24% z i n c a l l o y de formed a t 470°K non s e n s i t i z e d sample ( a v e r age o f 5 t e s t s ) s e n s i t i z e d sample ( a ve r age o f 3 t e s t s ) e n g i n e e r i n g e l o n g a t i o n , e^[%] 50 40 u n i f o r m r e d u c t i o n i n a r e a A [%] ru 27 . 5 24 .5 r e d u c t i o n i n a r e a a t f r a c t u r e A . r f [%] I 55 43 . 5 M e t a l l o g r a p h y r e v e a l e d t h a t t h e r e were many i n t e r g r a n u l a r c r a c k s remote f rom t he necked f r a c t u r e r e g i o n i n t h e s e n s i t i z e d spec imen ( F i g u r e 26 a) but none i n an u n t r e a t e d s p e c i m e n . The o b s e r v a t i o n o f p r e p o l i s h e d s u r f a c e o f a spec imen s u b j e c t e d o n l y t o s e n s i t i z i n g t r e a t m e n t has no t r e v e a l e d any p r e s en ce o f g r a i n boundary s l i d i n g . I t may t h e r e f o r e r e a s o n a b l y be c o n c l u d e d t h a t v a c an c y movement i s t h e p r i m a r y o r i g i n o f m i c r o c r a c k n u c l e a t i o n . The i n t e n s i t y o f v o i d n u c l e a t i o n by v a c an c y c o n c e n t r a t i o n wou ld be e x p e c t e d t o depend s t r o n g l y on t e m p e r a t u r e and s t r a i n r a t e . At l ow t e m p e r a t u r e s t h e va can cy c o n c e n t r a t i o n and m o b i l i t y a r e no t s u f f i c i e n t t o c o n t r i b u t e t o g r a i n boundary v o i d f o r m a t i o n . The d i f f e r e n t p r o c e s s e s 85 (a) (b) F i g . 26 M i c r o s t r u c t u r e o f 24% z i n c b r a s s de fo rmed t o f r a c t u r e a t 470 K. S t r a i n r a t e 1.1 x 1 0 ~ 4 s _ 1 . M a g n i f i c a t i o n 180X. ( a ) Sample s e n s i t i z e d as d e s c r i b e d i n t e x t . (b ) C o n v e n t i o n a l l y - t r e a t e d sample (no s e n s i t i z a t i o n t r e a t m e n t ) . 86 by wh i ch m i c r o v o i d s may fo rm and c o a l e s c e t o l e a d to t r a n s g r a n u l a r f r a c t u r e a t l owe r t e m p e r a t u r e s have been d i s c u s s e d i n a t h e o r e t i c a l paper by Sm i th and Barn by [ 5 7 ] . 4 . H igh Tempera tu re D u c t i l i t y At t e m p e r a t u r e s above t h o s e o f t h e d u c t i l i t y minimum a s i m p l e i n c r e a s e o f d u c t i l i t y w i t h t e m p e r a t u r e was no t g e n e r a l l y o b s e r v e d ( F i g u r e 1 4 ) . In f a c t , w i t h t h e e x c e p t i o n o f o n l y one s e t o f t e s t c o n d i t i o n s , t h e f r a c t u r e d u c t i l i t y d e c r e a s e d a g a i n between 950 and 1050°K. An e x p l a n a t i o n f o r t h e i n i t i a l i n c r e a s e i n d u c t i l i t y o f t e m p e r a t u r e s above t h o s e o f t h e minimum was p r e v i o u s l y o f f e r e d . The g rowth o f g r a i n boundary m i c r o c r a c k s i s d e l a y e d when t h e y become i s o l a t e d f rom t he b ounda r i e s by g r a i n boundary m i g r a t i o n d u r i n g dynamic r e c r y s t a l l i z a t i o n . The p r o c e s s c o n t r o l l i n g f r a c t u r e d u c t i l i t y becomes p r o p a g a t i o n r a t h e r t h an n u c l e a t i o n o f m i c r o v o i d s . I t r ema i n s t o e x p l a i n why d u c t i l i t y does not c o n t i n u e t o r i s e as t h e t e m p e r a t u r e o f d e f o r m a t i o n f u r t h e r i n c r e a s e s . I t i s commonly assumed t h a t t h e o c c u r r e n c e o f dynamic r e c r y s t a l l i z a t i o n a lway s c ause s a d u c t i l i t y r e s t o r a t i o n . In f a c t i t i s n e c e s s a r y t o t a k e i n t o a c c oun t t h e r e l a t i v e r a t e s o f g r a i n boundary m i g r a t i o n and o f v o i d g r o w t h . One app roa ch t o t h e p rob lem has been e l a b o r a t e d by Evans e t a l . [58] who assumed t h a t t h e mechanism o f v o i d g rowth was g r a i n boundary s l i d i n g . Wh i l e t h e p r e s e n t i n v e s t i g a t i o n does not e x c l u d e t h a t mechan i sm, t h e r e i s e v i d e n c e t h a t c a v i t y g rowth was enhanced by v a c an c y c o n c e n t r a t i o n and by m e c h a n i c a l t e a r i n g due t o h i g h s t r e s s c o n c e n t r a t i o n s a t t h e t i p o f t h e c r a c k a t l a t e r s t a g e s o f c r a c k p r o p a g a t i o n . The l a t t e r c o n c l u s i o n i s s u p p o r t e d by t h e appea rance o f t h e f r a c t u r e s u r f a c e as seen by SEM, and a l s o by t h e e x p e r i -ments o f T a p ! i n [ 5 6 ] . C o n s i s t e n t w i t h t h e s e o b s e r v a t i o n s a m o d i f i c a t i o n 87 o f Evans and J o n e s ' model [58] has been d e v e l o p e d . I f g r a i n b o u n d a r i e s move away f rom d e v e l o p i n g c a v i t i e s , g rowth o f t h e c a v i t i e s i s s l owed down o r s t o p p e d . Growth i s a l s o no l o n g e r a s s i s t e d by g r a i n boundary s l i d i n g . A l a c k o f p r e f e r e n t i a l v a can cy d i f f u s i o n a l l o w s an i s o l a t e d c a v i t y t o become more s p h e r i c a l i n shape t hu s d e c r e a s i n g t he s t r e s s c o n c e n t r a t i o n i n i t s v i c i n i t y . I f c a v i t i e s become s u f f i c i e n t l y l a r g e t h e y w i l l i n s t e a d p i n t h e b o u n d a r i e s so t h a t any g r a i n boundary movement wh i ch i s t o a c c oun t f o r h i g h d u c t i l i t y must o c c u r b e f o r e a c r i t i c a l c a v i t y r a d i u s r c i s a t t a i n e d . The p rob lem o f a c c o u n t i n g f o r t h e d u c t i l i t y v a r i a t i o n o b s e r v e d i n F i g u r e 14 i s t hu s r educed t o e s t a b l i s h i n g whe the r g r a i n boundary m i g r a t i o n o v e r a c r i t i c a l d i s t a n c e r £ can o c c u r w i t h i n t h e t i m e r e q u i r e d t o c r e a t e c a v i t i e s o f r a d i u s r . Suppose t h a t , a t c o n s t a n t i n i t i a l g r a i n s i z e , t h e t i m e r e q u i r e d t o d e v e l o p such c a v i t i e s i s t . M a t h e m a t i c a l l y i t can be e x p r e s s e d by t h e f o r m u l a t = ^ (24) whe re : - c r i t i c a l c a v i t y r a d i u s , - r a t e o f c a v i t y g r o w t h . The q u a n t i t y i s a f u n c t i o n m a i n l y o f t e m p e r a t u r e ( t h e h i g h e r t h e t e m p e r a -t u r e , t h e f a s t e r i s c a v i t y g rowth) and c o m p o s i t i o n . The dependence o f on c o m p o s i t i o n m i gh t be e x p e c t e d t o be s i m i l a r t o t h a t o f t h e v a can cy d i f f u -s i o n c o e f f i c i e n t D on s o l u t e c o n c e n t r a t i o n . F o l l o w i n g Lucke and S t ' u re [59 ] t h e g r a i n - b o u n d a r y v e l o c i t y V i s g i v e n by V = M( c , T ) • P ( c,e , T ) (25) where M i s t h e g r a i n boundary m o b i l i t y and i s a f u n c t i o n o f t e m p e r a t u r e T and c o m p o s i t i o n c , and P i s t h e d r i v i n g f o r c e f o r movement. In d e f o r m i n g spec imens t h e ma j o r c o n t r i b u t i o n t o P w i l l be t h e s t o r e d ene r gy o f c o l d work wh i ch w i l l v a r y w i t h e, T , and c . 88 Low d u c t i l i t y w i l l t h e r e f o r e be e x p e c t e d i f c (26) s u b s t i t u t i n g E q u a t i o n 25 i n t o 26 , M P < ',C (27) g Thus when t he v a l u e o f M • P i s s m a l l and C g i s h i g h t he m a t e r i a l e x h i b i t s low d u c t i l i t y . On ly t r e n d s can be e x p l a i n e d by t h e t h e o r y a t t h i s s t a g e , :. >' s i n c e d a t a a r e not a v a i l a b l e t o t e s t t h e t h e o r e t i c a l e q u a t i o n s q u a n t i t a t i v e l y . At a c o n s t a n t t e m p e r a t u r e and s t r a i n r a t e , M i n c r e a s e s w i t h a l l o y c o n t e n t (as p roved by t he t i m e r e q u i r e d t o p roduce t he same g r a i n s i z e f o r d i f f e r e n t c o m p o s i t i o n s - T a b l e I , c h a p t e r I I ) , C g r ema in s a p p r o x i m a t e l y c o n s t a n t . P depends on t h e d e n s i t y and d i s t r i b u t i o n o f d i s l o c a t i o n s and w i l l d e c r e a s e w i t h t h e amount o f r e c o v e r y t h a t o c c u r s d u r i n g t h e t e s t . . The s t a c k i n g f a u l t ene rgy o f a Cu-Zn s i n g l e phase a l l o y d e c r e a s e s w i t h i n c r e a s i n g z i n c c o n t e n t t hu s r e s t r i c t i n g t h e deg ree o f recovery i n h i g h z i n c c o n t e n t a l l o y s . By t h i s argument t h e 33% z i n c a l l o y s h o u l d be t h e most d u c t i l e s i n c e bo th M and P v a l u e s a r e e x p e c t e d t o be l a r g e s t f o r t h e h i g h e s t z i n c c o n t e n t . T h i s i s c o n f i r m e d by u n i f o r m d e f o r m a t i o n - ( A ) d a t a i n F i g u r e 11 b. The r e d u c t i o n i n a r e a a t f r a c t u r e does not e x h i b i t t h i s c o m p o s i t i o n dependency; however , one s h o u l d t a k e i n t o a c c oun t t h a t t h e above t h e o r y d e a l s w i t h u n i f o r m d e f o r m -a t i o n . Once t h e . s t a b i l i t y o f d e f o r m a t i o n i s l o s t t h e v a l u e o f C„ i n c r e a s e s r a p i d l y due t o i n t e r n a l t e a r i n g , as does t h e v a l u e o f P due t o i n c r e a s e d s t r a i n r a t e , t hu s c hang i ng a l l t h e p r o p o r t i o n s i n E q u a t i o n 2 7 . ' At c o n s t a n t c o m p o s i t i o n and s t r a i n r a t e , M i n c r e a s e s e x p o n e n t i a l l y w i t h t e m p e r a t u r e whereas P d e c r e a s e s w i t h t e m p e r a t u r e due t o t h e i n c r e a s e d r e c o v e r y r a t e s . The v a l u e C a l s o i n c r e a s e s w i t h t e m p e r a t u r e . Thus f o r c e r t a i n c o n d i t i o n s ( h i g h M, not v e r y l ow P and r e l a t i v e l y low C n ) one may be a b l e t o g g 89 a n t i c i p a t e a maximum i n t h e d u c t i l i t y - t e m p e r a t u r e c u r v e . Such a maximum i s o b s e r v e d f o r 33% and 24% z i n c a l l o y s f o r bo th A , and A a t 950°K J r f r u ( F i g u r e 1 3 , " 1 4 ) . The subsequen t d e c r e a s e i n d u c t i l i t y w i t h i n c r e a s i n g t e m p e r a t u r e i s due t o a f u r t h e r d e c r e a s e i n P and an i n c r e a s e i n C^. In t h e c a s e o f t h e 19% z i n c a l l o y v a l u e s o f b o t h M and P a r e l ow t h u s no d u c t i l i t y maximum i s o b s e r v ed i n t h e i n v e s t i g a t e d range o f t e m p e r a t u r e and s t r a i n r a t e . At c o n s t a n t c o m p o s i t i o n and t e m p e r a t u r e , t h e v a l u e o f M r ema ins r e l a t i v e -l y c o n s t a n t f o r v a r i o u s s t r a i n r a t e s . The same s h o u l d be t r u e f o r C . 9 Changes i n P w i t h e a r e p r o p o r t i o n a l t o t h e deg ree o f dynamic r e c o v e r y . I f t h e d i s l o c a t i o n d e n s i t y i s r educed a t a s t e a d y r a t e by r e c o v e r y , t h e t o t a l d i s l o c a t i o n d e n s i t y s h o u l d be h i g h e r f o r h i g h s t r a i n r a t e a t a g i v e n s t r a i n ; t hus P s h o u l d a l s o be l a r g e r , c h a n g i n g t he p r o p o r t i o n i n E q u a t i o n 27 t owa rds h i g h e r d u c t i l i t y c o n d i t i o n s . In t h i s way t h e dependence o f d u c t i l i t y on s t r a i n r a t e o b s e r v e d e x p e r i m e n t a l l y can be a t l e a s t q u a l i t a t i v e l y e x p l a i n e d by t h e m o d e l . A d i f f e r e n t c on c ep t o f t h e s t r a i n r a t e dependence o f d u c t i l i t y has r e c e n t l y been p roposed by No r s t r om and J ohans son [ 5 ] . T h e i r model i s based on a s p e c u l a t i v e dependence o f h i g h d u c t i l i t y on dynamic work s o f t e n i n g mechan i sms . The t h e o r y does not seem t o be c o n f i r m e d by t h e p r e s e n t f i n d i n g s s i n c e as d i s c u s s e d above t he p r e s en ce o f dynamic r e c r y s t a l l i z a t i o n i s not a s u f f i c i e n t c o n d i t i o n f o r h i g h d u c t i l i t y . A l s o t h e i r i n t e r p r e t a t i o n o f t h e dependence o f t h e d u c t i l i t y o f s t a i n l e s s s t e e l on s t r a i n r a t e i s i n d i s a g r e e m e n t w i t h t h e f a c t t h a t s t a i n l e s s s t e e l under t h e i r e x p e r i m e n t a l c o n d i t i o n s does undergo dynamic r e c r y s t a l 1 i z a t i o n [ 2 7 ] , y e t a c c o r d i n g t o t h e i r t h e o r y i t s h o u l d n o t . 90 VI CONCLUSIONS At t e m p e r a t u r e s i n t h e range 0.4 <T^ <0 . 6 and a t s t r a i n r a t e s o f -4 -2 -1 10 and 10 s , a l p h a b r a s s e s e x h i b i t a h i g h r a t e o f work h a r d e n i n g and v e r y l i t t l e dependence o f f l o w s t r e s s on t e m p e r a t u r e o r s t r a i n r a t e . These o b s e r v a t i o n s can be i n t e r p r e t e d i n t e rms o f t h e modern t h e o r y o f dynamic s t r a i n a g i n g (DSA ) . An a l t e r n a t i v e e x p l a n a t i o n may be v a l i d , i f b r a s s e s e x h i b i t l o n g range o r d e r i n g o v e r t h i s r ange o f t e m p e r a t u r e s . D i s c o n t i n u o u s y i e l d i n g ( t h e P o r t e v i n - L e C h a t e l i e r e f f e c t ) was ob s e r v ed a t t e m p e r a t u r e s be low 0.45 T^, and t hen o n l y a t t h e l o w e r s t r a i n r a t e . S i n c e o t h e r e v i d e n c e o f DSA p r e v a i l e d t o 0 .6 T , i t i s c o n c l u d e d t h a t t h e P-LeC e f f e c t i s not a n e c e s s a r y m a n i f e s t a t i o n o f DSA i n b r a s s e s . At t e m p e r a t u r e s >0.7 T m t h e t e n s i l e d e f o r m a t i o n o f a l p h a b r a s s e s i s accompan ied by dynamic r e c r y s t a l l i z a t i o n p r o c e s s e s . The m e c h a n i c a l c o n d i t i o n s o f s t r e s s o s c i l l a t i o n w i t h s t r a i n , o r a s i n g l e f l o w s t r e s s d r o p , a r e not n e c e s s a r y m a n i f e s t a t i o n s o f dynamic r e c r y s t a l 1 i z a t i o n . Thus t h e model o f Lu ton and S e l l a r s [18] and i t s r e c e n t deve l opmen t by S a k a i e t a l . [ 1 9 ] , [49] i s q u e s t i o n a b l e . There i s a f r a c t u r e d u c t i l i t y minimum f o r a l p h a b r a s s e s a t i n t e r m e d i a t e t e m p e r a t u r e s as r e p o r t e d by o t h e r s . However , t h e minimum i s not s t r o n g l y d e f i n e d , p a r t i c u l a r l y a t l ow s t r a i n r a t e s . The t e m p e r a t u r e range wh i c h t h e minimum i s o b s e r v e d w i t h i n , o v e r l a p s bo th t h e range i n wh i ch DSA e f f e c t s a r e s e e n , and t h e range where dynamic r e c r y s t a l l i z a t i o n i s b e l i e v e d t o have o c c u r r e d . I t i s t h e r e f o r e c o n c l u d e d t h a t t h e phenom-enon o f a d u c t i l i t y t r o u g h i n b r a s s e s i s not d i r e c t l y c onne c t ed w i t h e i t h e r DSA o r dynamic r e c r y s t a l 1 i z a t i o n , as has been s ugge s t e d by p r e v i o u s i n v e s t i g a t o r s . 9 1 T e n s i l e f r a c t u r e o f a l p h a b r a s s e s i n t h e range o f t h e d u c t i l i t y minimum i s a r e s u l t o f t h e n u c l e a t i o n and g rowth o f m i c r o v o i d s a t g r a i n b o u n d a r i e s w h i c h a r e t r a n s v e r s e t o t h e t e n s i l e a x i s . N u c l e a t i o n o f t h e s e v o i d s i s t h e r e s u l t o f d i f f u s i o n and c o a l e s c e n c e o f v a c a n c i e s a t t h e b o u n d a r i e s wh i ch a r e s u b j e c t e d t o t e n s i l e s t r e s s under a p p r o p r i a t e c o n d i t i o n s o f t e m p e r a t u r e and s t r a i n . The t e m p e r a t u r e dependence o f f r a c t u r e d u c t i l i t y can be e x p l a i n e d q u a l i t a t i v e l y as t h e r e s u l t o f a b a l a n c e between t h e r a t e s o f g r a i n boundary m i g r a t i o n and c a v i t y g r o w t h . F u r t h e r e x p e r i m e n t s , d e s i g n e d t o t e s t q u a n t i t a t i v e l y t h e h i g h t e m p e r a t u r e d u c t i l i t y model p r e s e n t e d h e r e i n , m igh t p r o v i d e b e t t e r u n d e r s t a n d i n g o f . h i g h t e m p e r a t u r e d u c t i l i t y p r o b l e m s . Arguments p r e s e n t e d by p r e v i o u s i n v e s t i g a t o r s t o e x p l a i n t h e d u c t i l i t y minimum i n a l p h a b r a s s e s a r e not c o n s i s t e n t w i t h t h e p r e s e n t r e s u l t s o r t h e p r e s e n t i n t e r p r e t a t i o n . 92 LIST OF REFERENCES 1. R. C a r l s s o n , "Hot E m b r i t t l e m e n t o f Copper and B r a s s A l l o y s " , S c a n d i n a v i a n J o u r n a l o f M e t a l l u r g y 9 (1980) 2 5 . 2. B. Thomas, " I n v e s t i g a t i o n o f Pane l C r a c k i n g i n S t a t i c - C a s t S t e e l I n go t s U s i ng M a t h e m a t i c a l and P h y s i c a l M o d e l s " , Re sea r ch P r o p o s a l f o r Ph .D .  T h e s i s , U n i v e r s i t y o f B r i t i s h Co lumb ia Dep t . o f M e t a l l u r g i c a l E n g i n e e r i n g , (1982) U n p u b l i s h e d . 3 . F .N . R h i n e s , P . J . Wray, " I n v e s t i g a t i o n . o f t h e I n t e r m e d i a t e Tempera tu re D u c t i l i t y Minimum i n M e t a l s " , T r a n s a c t i o n s o f t h e ASM, 54 (1961) 117 . 4 . N. I g a t a , K. M i y a h a r a , K. T anaka , "The S t r a i n Ra te Dependence o f t h e D u c t i l i t y Minimum o f Copper and Cu-Zn A l l o y a t I n t e r m e d i a t e T e m p e r a t u r e s " , T r a n s a c t i o n s o f t h e J apanese I n s t i t u t e o f M e t a l s , 20 (1979) 344 . 5. L .A . N o r s t r o m , B. J o h a n s s o n , "Hot D u c t i l i t y - A M a t t e r o f S t r a i n R a t e " , S c a n d i n a v i a n J o u r n a l o f M e t a l l u r g y , 11 (1982) 139 . 6. G. B e r n a r d , I . B i r a t , B. C o n s e i l , J . C . Humber t , " S t u d y on t h e S e n s i t i v i t y t o C r a c k i n g o f C o n t i n u o u s l y Cas t S t e e l s U s i ng Hot D u c t i l i t y T e s t s " , Revue de M e t a l l u r g i e , 75 #7 (1978) 467 . 7. B . J . S u n t e r , N.M, Burman, "The Deve lopment o f Improved Hot W o r k a b i l i t y o f Some Copper A l l o y s " , The J o u r n a l o f t h e A u s t r a l i a n I n s t i t u t e o f M e t a l s , 17 #2 (.1972) 9 1 . ~~ 8 . 0. I z u m i , " I n t e r m e d i a t e Tempera tu re E m b r i t t l e n e s s and I n t e r g r a n u l a r C r a c k i n g o f Copper A l l o y s " , J apanese Me ta l S o c i e t y , 18 #1 (1979) 1 5 . 9. B. R u s s e l l , D. J a f f r e y , "H i gh Tempe ra tu re B r i t t l e F r a c t u r e i n Copper and C o p p e r - T i n S o l i d S o l u t i o n s " , The J o u r n a l o f t h e A u s t r a l i a n I n s t i t u t e o f  M e t a l s , 10 #4 (.1965) 349 . 10 . H. Yamagata , 0 . I z u m i , " I n t e r m e d i a t e Tempera tu re E m b r i t t l e m e n t o f A l p ha S o l i d S o l u t i o n Cu-Zn and Cu-A l A l l o y s " , The J o u r n a l o f t h e Japanese  I n s t i t u t e o f M e t a l s , 42 #12 (1978) 1167 . I T . . H. Yamagata , 0. I z u m i , "The E f f e c t o f C o m p o s i t i o n s on t h e D u c t i l i t y o f Copper a S o l i d S o l u t i o n s " , The J o u r n a l o f t h e J apanese I n s t i t u t e o f M e t a l s , 43 #3 (1979) 209 . 12 . H. Yamagata , 0. I z u m i , " E f f e c t s o f D e f o r m a t i o n Tempera tu re and A l l o i n g Con t en t on t h e D e f o r m a t i o n and F r a c t u r e B e h a v i o u r s o f a C u - Z n , a Cu-A l . A l l o y s " , The J o u r n a l o f t h e J apanese I n s t i t u t e o f M e t a l s , 42 #12 (1978) 1173 . 13 . J . N . Greenwood, D.R. M i l l e r , J .W. S u i t e r , " I n t e r g r a n u l a r C a v i t a t i o n i n S t r e s s e d M e t a l s " , A c t a M e t a l ! u r g i c a , 2 (1954) 250 . 93 14 . H . J . McQueen, "Dynamic Recove ry and i t s R e l a t i o n t o O the r R e s t o r a t i o n M e c h a n i s m s " , M e t a l u r g i a i Od l ewn i c two 6 (1979) 5. 15 . H . J . McQueen, "The P r o d u c t i o n and U t i l i t y o f Recove red D i s l o c a t i o n S u b s t r u c t u r e s " , M e t a l l u r g i c a l T r a n s a c t i o n s A, 8 (1977) 807 . 16 . C. R o s s a r d , P. B l a i n , " I n i t i a l R e s u l t s o f Re sea r ch on t he Hot D e f o r m a t i o n o f S t e e l s " , Revue de M e " t a l 1 u r g i e , 55 (1958) 5 7 3 . 17 . C M . S e l l a r s , " R e c r y s t a l 1 i z a t i o n o f M e t a l s Du r i n g Hot D e f o r m a t i o n " , P h i l o s o p h i c a l T r a n s a c t i o n s o f t h e Roya l S o c i e t y o f London A, 288 (1978) 18 . M . J . L u t o n , C M . S e l l a r s , "Dynamic R e c r y s t a l 1 i z a t i o n i n N i c k e l - I r o n A l l o y s D u r i n g H igh Tempera tu re D e f o r m a t i o n " , A c t a M e t a l l u r g i c a , 17 (1969) 1033 . 19 . T. S a k a i , M.G. Akben , J . J . J o n a s , "The Ro l e o f Dynamic R e c r y s t a l 1 i z a t i o n i n P r o d u c i n g G r a i n Re f i nemen t and G r a i n C o a r s e n i n g i n M i c r o a l l o y e d S t e e l s " , I n t e r n a t i o n a l C o n f e r e n c e on t h e Thermomechan i ca l P r o c e s s i n g o f  M i c r o a l l o y e d A u s t e n i t e . P i t t s b u r g h , PA, Augus t 1 9 8 1 . 2 0 . J . J . J o n a s , T . S a k a i , "The T r a n s i t i o n f rom M u l t i p l e t o S i n g l e Peak R e c r y s t a l 1 i z a t i o n Du r i ng H igh Tempera tu re D e f o r m a t i o n " , 24 §me C o l l o g u e  de M g t a l l u r g i e , INSTN ( S a c l a y ) June 1981 . 2 1 . L. B l a z , Ph .D . T h e s i s Academy o f M i n i n g and M e t a l l u r g y , Krakow, P o l a nd ( 1 9 7 8 ) . 2 2 . L. F r i t z m e i e r , M . J . L u t o n , H . J . McQueen, " D i s l o c a t i o n i n Dynamic Re cove r y and R e c r y s t a l 1 i z a t i o n o f Hot Worked A u s t e n i t i c S t a i n l e s s S t e e l " , S t r e n g t h  o f M e t a l s and A l l o y s , F i f t h I n t e r n a t i o n a l C o n f e r e n c e , Aa chen , W. Germany 1 (.1 979) 95": 2 3 . G. G a t t s t e i n , D. Z a b a r d j a d i , H. M e c k i n g , "Dynamic R e c r y s t a l 1 i z a t i o n i n T e n s i o n Deformed Copper S i n g l e C y r s t a l s " , Me ta l S c i e n c e 13 (1979) 222 . 24 . L. B l a z , A. K o r b e l , "H i gh Tempe ra tu re D e f o r m a t i o n o f Copper and 34% Zn B r a s s " , Arch iwum H u t n i c t w a 9 (1981) 3. 2 5 . C M . S e l l a r s , W . J . McG. T e g a r t , "Hot W o r k a b i l i t y " , I n t e r n a t i o n a l  M e t a l l u r g i c a l R e v i e w s , Rev i ew 158 , 17 (1972 ) 1 . 2 6 . H . J . McQueen, W.A. Wong, J . J . J o n a s , " D e f o r m a t i o n o f Aluminum a t H igh Tempe ra tu r e s and S t r a i n R a t e s " , Canad i an J o u r n a l o f P h y s i c s , 45 (1967) 1225 . 27 . J . J . J o n a s , C M . S e l l a r s , W . J . McG T e g a r t , " S t r e n g t h and S t r u c t u r e under Ho t -Wo r k i n g C o n d i t i o n s " , M e t a l l u r g i c a l R e v i e w s , Rev iew 130 , 14 (1969) 1. 94 28 . W . J . McG. T e g a r t , "The Ro l e o f D u c t i l i t y i n Hot W o r k i n g " , D u c t i l i t y , E d i t e d by ASM, M e t a l s P a r k , Ohio 1968 , p. 133 . 29 . R .E . R e e d - H i l l , P h y s i c a l M e t a l l u r g y P r i n c i p l e s , London 1973 p. 346 30 . 0 . I z u m i , H. Yamagata , "H i gh Tempera tu re D u c t i l i t y o f A l p ha B r a s s e s -D e c r e a s i n g and I n c r e a s i n g o f E l o n g a t i o n " , Sh i ndo G i j u t s u K e n k y u k a i s h i 17 #1 (1978) 165 . 3 1 . A . H . C o t t r e l l , " A . n o t e on t h e P o r t e v i n - L e C h a t e l i e r E f f e c t " , P h i l o s o p h i c a l  M a g a z i n e , 44 (1953) 8 2 9 . 32 . L . J . Cuddy, W.C. L e s l i e , "Some A s p e c t s o f S e r r a t e d Y i e l d i n g i n S u b s t i t u -t i o n a l S o l i d S o l u t i o n s o f I r o n " , A c t a Me ta l 1 u r g i c a , 20 (1 972) 1157 . 33 . A. K o r b e l , "The S t r u c t u r a l A s pe c t o f t h e P o r t e v i n - L e C h a t e l i e r E f f e c t i n A l p ha B r a s s e s " , S c r i p t a Me ta l 1 u r g i c a , 8 (1974) 609 . 34 . A. K o r b e l , "The D e n s i t y o f M o b i l e D i s l o c a t i o n s and t h e C r i t i c a l D i s l o c a -t i o n V e l o c i t y Du r i n g Non -Un i f o rm D e f o r m a t i o n o f A l pha B r a s s " , S c r i p t a  M e t a l l u r g i c a 9 (1975) 115 . 3 5 . A. Van Den Beuke l , " Theo ry o f t h e E f f e c t o f Dynamic S t r a i n Ag i ng on M e c h a n i c a l P r o p e r t i e s " , P h y s i c a S t a t u s S o l i d u s , 30 (1975) 197 . 3 6 . R. F ede r , A . S . Now i ck , D.B. R o s e n b l a t t , " S t udy o f Orde r i n Annea l ed and I r r a d i a t e d A l pha B r a s s by L a t t i c e Pa rame te r Mea su r emen t s " , J o u r n a l  o f A p p l i e d P h y s i c s 29 #6 (1958) 984 . 37 . L .M. C l a r e b r o u g h , M.H. L o r e t t o , " O r d e r - D i s o r d e r Phenomena i n a - B r a s s . I . Deve lopment o f O r d e r " , P r o c e e d i n g s o f t h e Roya l S o c i e t y A257 (1960) 326 . 38 . A . C . Damask, "Some R e s i s t i v i t y E f f e c t s o f S h o r t - R a n g e Orde r i n A l p ha B r a s s " , J o u r n a l o f A p p l i e d P h y s i c s , 27 #6 (1956) 610 . 3 9 . A . A . P r e s n y a k o v , L . J . Dau t ova , U.F . K l y u c n i k o v , "Homologous Ag i ng o f N o n - s a t u r a t e d S o l i d S o l u t i o n s " , F i z y k a M e t a l o v M e t a l o v e d e n i e 8 (1959) 394 . 4 0 . A . A . P r e s n y a k o v , V . V . C e r v y a k o v a , La tum ' ( M e t a l u r g i j a ) , Moskva 1969 . 4 1 . J . G r y z i e c k i " S t r u c t u r e and P r o p e r t i e s o f A l p ha B r a s s e s w i t h a D i f f e r e n t S t a c k i n g F a u l t Ene rgy Which Were Deformed i n t h e T e n s i l e T e s t " , M e t a l u r g i a i Od l ewn i c two B u l l e t i n 78 (1977) 1. 4 2 . L. A l m e i d a , S .M . M o n t e i r o , " S t u d y o f t h e P l a s t i c D e f o r m a t i o n o f A l pha B r a s s between -196°Cv400°C." 35 t h Annual Cong ress o f A s s o c . B r a s i l e i v a  de M e t a i s , ( P r o c . C o n f . ) , Sao P a u l o , B r a z i l 1980 , p. 593 . 95 4 3 . J . F r i e d e l , D i s l o c a t i o n s , Ox fo rd 1964 , p . 4 0 5 . 44 . P. P e n n i n g , "Ma t hema t i c s o f t h e P o r t e v i n - L e C h a t e l i e r E f f e c t " , A c t a  M e t a l ! u r g i c a , 20 (1972) 1169 . 4 5 . A. S e e g e r , H. Don t h , F. P f a l l , "The Mechanism o f Low Temperature.-M e c h a n i c a l R e l a x a t i o n i n Deformed C r y s t a l s " , D i s c u s s i o n s o f t h e Fa raday  S o c i e t y , 23 (1957) 19 . 46 . N .S . S t o l o f f , " I n t e r m e t a l l i c Compounds and Orde red P h a s e s " , S t r e n g t h -e n i n g Methods i n C r y s t a l s , E d i t e d by A. K e l l y , R.B.. N i c h o l s o n , London 1971 . 47 . R.G. D a v i e s , N .S . S t o l o f f , "On t h e Y i e l d S t r e s s o f Aged N i - A l A l l o y s . " , AIME M e t a l l u r g i c a l S o c i e t y T r a n s a c t i o n s , 233 (1965) 714 . 4 8 . T . L . J o h n s t o n , A . S . T e t e l m a n , A . J . M c e v i l y , H igh S t r e n g t h M a t e r i a l s , E d i t e d by V . F . Z a c k a y , New York 1965 . 49 . H. Yamaga ta , 0. I z u m i , " D e f o r m a t i o n and D u c t i l i t y o f A l p ha B r a s s S i n g l e C r y s t a l s a t E l e v a t e d T e m p e r a t u r e s " , The J o u r n a l o f t h e Japanese I n s t i t u t e  o f M e t a l s 42 #10 (1978) 1012 . 50 . R.W. B a l l u f f i , L . L . S k i g l e , " E f f e c t o f G r a i n B o u n d a r i e s upon Pore F o rma t i o n and D imens i ona l Changes Du r i n g D i f f u s i o n " , A c t a Me ta l 1 u r g i c a , 3 (1955) 170 . 5 1 . R.D. G i f k i n s , "A Mechanism f o r t h e F o rma t i o n o f I n t e r g r a n u l a r C r a c k s when Boundary S l i d i n g O c c u r s " , A c t a M e t a l l u r g i c a , 4 (1956) 98 . 52 . L . L . S e i g l e , R. R e s n i c k , "On Pore F o rma t i on d u r i n g D i f f u s i o n " , A c t a  M e t a l l u r g i c a , 3 (1955) 605 . 53 . R.W. B a l u f f i , L . L . S e i g l e , "Growth o f V o i d s i n M e t a l s Du r i ng D i f f u s i o n and C r e e p " , A c t a M e t a l l u r g i c a , 5 (1957) 4 4 9 . 54 . D. H u l l , D.E. Rimmer, "The Growth o f G r a i n - b o u n d a r y Vo i d s Under S t r e s s " P h i l o s o p h i c a l M a g a z i n e , 4 (1959) 673 . 55 . R.T. R a t c l i f f e , G.W. Greenwood, "The Mechanism o f C a v i t a t i o n i n Magnesium d u r i n g C r e e p " , P h i l o s o p h i c a l M a g a z i n e , 12 (1965) 5 9 . 56 . D.M.R. T a p l i n , L . J . B a r k e r , "A S t udy o f t h e Mechanism o f I n t e r g r a n u l a r C reep C a v i t a t i o n by Shadowgraph i c E l e c t r o n M i c r o s c o p y " , A c t a M e t a l l u r g i c a 14 (1956) 1527 . 57 . E. S m i t h , J . T . B a r n b y , " C r a c k N u c l e a t i o n i n C r y s t a l l i n e S o l i d s " , Me ta l  S c i e n c e J o u r n a l , 1 (1 967) 56 . 96 R.W. E van s , F . L . J o n e s , "Hot D u c t i l i t y o f A u s t e n i t i c Fe -N i A l l o y s " , M e t a l s T e c h n o l o g y , 11 (1976) 494 . K. Lu'cke, H.P. St i /we, Re cove r y and R e c r y s t a l 1 i z a t i o n o f M e t a l s , ( E d i t e d by L. Himmel) New York 1963 , p . 171 : 97 APPENDIX I TENSILE TEST DATA Ca l cu la t ion Procedure Values of the f i n a l gage length and the f i n a l cross sect ion dimensions, both at the f rac ture and well outside the necking area were measured with an accuracy of 0.1 mm and 0.01 mm r e s p e c t i v e l y . Subsequently they were used to c a l c u l a t e e f (Eq. 9 ) , A r u (Eq. 10) and E r f (Eq. 11) The s t r a i n hardening slope v £ was ca l cu la ted fo r a constant true s t r a i n of 0.05 by drawing a tangent to the true s t ress - t rue s t r a i n curve at e = 0.05 using a manual method which employs the ass istance of a mi r ror . The s t r a i n rate s e n s i t i v i t y c o e f f i c i e n t m was ca l cu la ted as fo l lows: For a few values of true s t ra in the corresponding true stresses were obtained from the true s t r e s s - t r u e s t r a i n curves and were subsequently used for c a l c u l a t i o n s according to Equation 14. Typica l resu l t s are tabulated below: Table V Typica l Ca l cu la t ions of the C o e f f i c i e n t m True S t ra in Composition S t ra in rate Temperature 0.002 0.009 0.019 0.031 0.049 0.072 19% Zn l . lxlO-V 1 and l . ix io-V 1 470°K -0.01 0.01 0.01 0.01 0.01 0.01 33% Zn l . ix io-V 1 and l . lxlO-V 1 950°K 0.28 0.27 0.26 0.26 0.26 0.25 In drawing p lots such as F i g . 12 only the values of m for the o f f s e t y i e l d s t ra in (0.002) have been used for purposes of the t h e s i s . A f u l l table of the resu l t s i s presented below. Table VI Tensile Properties ei = l . l x l O - V 1 e 2 = l . l x l O - V 1 CuZn 33% alloy Temperature [°K] yield stress °0.2 [MPa] Eq.(5) U T S [MPa] Eq. 6 Engineering Elongation e f [ « Eq. 9 Uniform reduc-tion in area A ru [*] Eq. 10 Reduction in area at fracture A r f [ « Eq. 11 Strain harden-ing rate ve [MPa/*] Eq. ,12 strain harden-ing exponent n Eq. 13 strain rate sensitivity coefficient m Eq. 14 • el £2 ei 62 ei E2 ei 62 ei 62 ei £2 ei 62 470 90 91 279 284 60 61 29 34 68 61 8.6 9.3 0.39 0.40 0.00 570 69 99 9.4 8.8 0.24 0.18 -0.08 650 93 85 252 167 47 24 28 15 50 29 7.9 8.1 0.34 0.21 0.02 670 81 83 8.6 2.1 0.19 0.03 -0.01 750 86 45 145 54 45 29 28 16 36 26 6.2 1.3 0.20 0.07 0.14 770 77 44 4.0 0.4 0.14 0.07 -0.08 850 63 22 67 26 59 18 26 12 34 27 1.7 0.8 0.07 0.09 0.22 870 58 16 1.0 0.5 0.05 0.09 0.28 950 23 7 27 11 102 36 31 16 97 46 0.8 0.5 0.13 0.15 0.25 1050 11 4 14.5 5 45 22 19 9 48 25 0.4 0.3 0.10 0.10 0.25 Table VI continued ei = l . l x l O - V 1 h = l . l x l O - V 1 CuZn 24% alloy Temperature [°K] yield stress °0.2 [MPa] Eq.{5) U T S [MPa] Eq. 6 Engineering Elongation e f [ « Eq. 9 Uniform reduc-tion in area A ru [*] Eq. 10 Reduction in area at fracture A r f [X] Eq. 11 Strain harden-ing rate v e [MPa/*] Eq. 12 strain harden-ing exponent n Eq. 13 strain rate sensitivity coefficient m Eq. 14 ei 62 ei 62 • 61 62 • 61 62 61 62 61 62 61 • 62 470 80 87 270 267 56 52 30 30 65 58 8.5 7.7 0.45 0.41 -0.01 650 86 78 245 150 45 10 26 10 50 23 7.9 6.7 0.34 0.21 0.02 750 75 62 152 70 26 11 18 6 24 17 7.7 2.5 0.22 0.08 0.04 850 68 31 90 35 26 13 17 6 35 29 . 3.5 1.3 0.12 0.07 0.17 950 32 11 40 16 32 15 20 10 50 45 1.1 0.8 0.13 0.14 0.23 1050 17 5 22 8 43 9 15 7 14 28 0.5 0.1 0.10 0.13 0.25 Table VI continued ei = l . l x l O - V 1 « = l . l x l O - V 1 CuZn 19% alloy Temperature [°K] yield stress °0.2 [MPa] Eq.(5) U T S [MPa] Eq. 6 Engineering Elongation e f [X] Eq. 9 Uniform reduc-tion in area A ru [X] Eq. 10 Reduction 1n area at fracture \ f [X] Eq. 11 Strain harden-ing rate [MPa/S]• Eq. 12 strain harden-ing exponent n Eq. 13 strain rate sensitivity coefficient m Eq. 14 • 61 • 62 61 62 61 • 62 • 61 62 • 61 62 61 • 62 • 61 62 470 73 69 252 249 53 49 28 28 82 54 8.6 8.7 0. 47 0.45 -0.01 570 68 66 8.9 8.4 0. 25 0.23 0.00 650 81 74 235 164 45 14 26 12 53 35 8.0 8.3 0. 37 0.24 0.02 670 63 63 7.7 6.8 0. 24 0.22 0.00 750 67 59 159 75 35 9 22 7 39 22 7.5 1.4 0. 24 0.10 0.03 770 63 51 8.5 0.4 0. 23 0.05 0.04 850 66 29 99 35 30 12 19 7 28 23 4.1 0.4 0. 15 0.04 0.18 870 69 6.0 0. 19 950 34 14 45 17 31 11 18 8 40 13 1.0 0.6 0. 13 0.11 0.20 1050 18 7 25 10 37 11 17 6 54 26 0.7 0.3 0. 11 0.14 0.20 101 A computer program was used to ca l cu la te true s t r e s s - t r u e s t ra in curves and to determine the values of the s t r a i n hardening exponent, n. For the l a t t e r c a l c u l a t i o n a computer l i b r a r y sub-program was used to obtain the best f i t fo r the loga- loge s t r a i g h t l i n e . As a precaution in t h i s procedure the computer p lo t ted points using experimental data and a s t ra igh t l i n e using parameters of the best f i t in loga- loge coord inates . The computer-printed program wri t ten in FORTRAN and a typ i ca l p r in tout of r e s u l t s are presented on the fo l lowing pages. 1 REAL L(90).P(3,90).S(3),DEF(90).STR(3,90),AVST(90). 2 3 STRAIN(90).ER(3).SUM(3).AV(90).ST(90) 3 INTEGER C.R.T.A.IER 4 C C-CODE FOR COMPOSITION.R-CODE FOR RATE OF DEFORMATION.T-CODE FOR TEMPERATURE 5 CALL PLOT3(C) 6 DO 5 II-1.5 7 READ(5,100) C.R.T 8 tOO FORMATOII) 9 READ(5.200) O.N.PACE,NS 10 C D-INITIAL GAUGE LENGTH.N-NUMBER OF POINTS PER ONE CURVE,PACE-INCREAC 11 C E IN GAUGE LENGTH.NS-NUMBER OF SAMPLES 12 200 F0RMAT(F4.1.I2,F4.2,I1) 13 L(1)-D 14 DO 10 1-2,N 15 10 L(I)-L(I-1)+PACE 16 WRITE(6.400) C. R .T.D.N.NS 17 400 FORMAT(1H1.///44X,12HSAMPLE CODE-,311.///18X,16HINITIAL LENGTH-18 1.F4.1.4X,' NUMBER OF POINTS-',I2.4X,' NUMBER OF SAMPLES-',11) 19 WRITE(6,500) 20 500 FORMAT(1H0,9X,'CURRENT LENGTH',8X,'FORCE',1IX,'ELONGATION',6X, 21 1 'TRUE STRAIN',9X.'STRESS') 22 REA0(5,302)(S(d).d-1,NS) 23 302 FORMAT(3F6.3) 24 25 00 20 J-1.NS 26 READ(5.30O) (P(J.A).A-1,N) 27 C S(d)"INITIAL CROSS SECTION AREA OF SAMPLE J,P(J,A)-DEFORMING FORCESOF SAMPLEAS READ OUT FROM THE FORCE VS DISTANCE CUR 28 WRITE(6.301)S(d) 29 301 FORMAT(1H0.34X,'INITIAL CROSS SECTION AREA-',F5.2) 30 DO 30 K-1.N 31 STR(U.K)-(P<J,K)«L(K))/(S<d)*0)-9.81»0.4536 32 DEF(K)-(L(K)-D)/D-100. 33 STRAIN(K)-ALOG(L(K)/D) — 1 34 OEF(1)-0.2 £3 35 STRAIN)1)-AL0G(30.06/0) 1 X 3 36 30 VRITE(6,600)L(K),P(U.K),DEF(K).STRAIN(K).STR(d.K) 37 20 CONTINUE 38 C CALCULATION OF AVAREGE STRESS 39 WRITE(6.700) 40 700 FORMAT(1HI.28X.'AVAREGE TRUE STRESS VS STRAIN'/1H0.2IX 41 2 ,'ELONGATION',6X.'TRUE STRAIN',8X,'STRESS') 42 DO 40 K-1.N 43 IF(NS.GE.2) GO TO 21 44 AVST(K)-STR(1,K) 45 GO TO 22 46 21 SUM(1)>STR(1.K) 47 00 41 I-2.NS 48 SUM(I)-SUM(I-1)+STR(I,K) 48.2 41 CONTINUE 49 AVST(K)°SUM(NS)/NS 50 22 CONTINUE 51 C AVST-AVAREGE TRUE STRESS 52 40 WRITE(6.800) OEF(K).STRAIN(K),AVST(K) 53 300 FORMAT(40F6.1) 54 800 F0RMAT(24X,F4.1,•%•,11X.F5.3,10X.F5.1,'MPA') 55 600 FORMAT(13X.F5.2.'MM', 1 1X.F6.2,'Lb'.12X.F4.1.'%',11X.F5.3,»1X. 56 3 F5.1,'MPA') 57 C TRANSFORMATION OF CALCULATED CURVE TO FITING PROCEDURE 58 REAL*8 ERROR)1).Y1(90).X1(2.90) 59 M-2 60 DO 90 I-2.N 61 Y1(I-1)-AL0G10(AVST(D) 62 X1(2.I-1)-AL0G10(STRAIN(D) 63 90 X1(1,I-1)-1. 64 NF*N-1 65 LOGICAL COMP 66 COMP-.TRUE. 67 CALL DLSQHS(Y1,X1,NF.M,1.16.2.ER.COMP.IER,&160) 68 D1»10*«Y1(1) 69 CALL PL0T4(STRAIN.AVST,N.T) 70 WRITE(7,900)Y1(1),D1.Y1(2).ER(1) 7t 900 FORMAT(1H0.11X.F6.4,10X.F6.2,10X.F4.2,10X.F10.8) 72 00 50 1-2,N 73 AV(I-1)«AVST(I) 74 SO ST(I-1)«STRAIN(I) 75 WRITE(8.930)T.N 76 WRITE(8.940)(AV(J),J-1.NF) 77 WR1TE(8.950)(ST(J).J-1.NF) 78 WRITE(8,960)01,Y1(2) 79 930 F0RMAT(2I2) 80 940 FORMAT(13F6.2) 81 950 FORMATf13F5.3) 82 960 F0RMAT(F6.2.F4.2) 83 5 CONTINUE 84 WRITE(6,910) 85 910 FORMAT(1H1.11X,'LODIO(K)', 1 1X. 'K' . 14X.'M',9X, 86 2 'ERROR SUM OF SQUARES") 87 CALL PLOTND 88 STOP 89 160 WRITE(6,920) 90 920 FORMAT('OSOLUTION FAILED') 91 IF(IER.NE.O) GO T0160 92 93 STOP 94 END 95 SUBROUTINE PL0T3(C) 96 INTEGER C 97 CALL PLCTRL('METR'.1) 98 CALL AXPLOT('TRUE STRAIN;',0.,20..0..0.01) 99 CALL AXCTRL('DIGITS'.-1 ) 100 CALL AXPLOTC'TRUE STRESS (MPA);',90..20..0.,12.5) 101 CALL PLCTRL('METR'.O) 102 RETURN 103 END 104 SUBROUTINE PL0T4(STRAIN,AVST,N,T) 105 REAL STRAIN(90).AVST(90) 106 INTEGER T 107 REAL*4 X(90),Y(90) 108 CALL PLCTRL('METR' . 1 ) 109 L-T-1 1 10 DO 10 1-1.N 111 X(I)-STRAIN(I)»100. 112 Y(I)-AVST(I)/12.5 113 CALL SYMB0L(X(I) ,Y(I ),0. 1778 , L . 0. . - 1) 1 14 10 CONTINUE 1 15 CALL PLCTRL('METR',0) 1 16 RETURN 117 END End of F1 l i e 104 S A M P L E C O D E - 1 1 2 I N I T I A L L E N G T H = 3 0 . 0 NUMBER OF P O I N T S ' 1 3 NUMBER OF S A M P L E S = 3 C U R R E N T L E N G T H F O R C E E L O N G A T I O N T R U E S T R A I N S T R E S S I N I T I A L C R O S S S E C T I O N AREA = 1 1 . 5 9 1MPA 3 0 . OOMM 1 1 0 . OOKG 0 . 2% 0 . 0 0 2 9 3 . 3 0 . 50MM 127 . OOKG 1 . 7% 0 . 0 1 7 109 . 3MPA 31 . OOMM 1 4 0 . O O K G 3 . 3% 0 . 0 3 3 1 2 2 . 4MPA 31 . 5 0 M M 152 . OOKG 5 .0% 0 . 0 4 9 135 . 1MPA 3 2 . OOMM 164 . OOKG 6 . 7% 0 . 0 6 5 1 4 8 . 1MPA 3 2 . 5 0 M M 172 . O O K G 8 . 3% 0 . 0 8 0 157 . 7MPA 3 3 . OOMM 1 8 2 . OOKG 1 0 . 0 % 0 . 0 9 5 1 6 9 . 5MPA 3 3 . . 5 0 M M 1 9 0 . OOKG 1 1 . 7% 0 . 1 10 179 . 6MPA 3 4 . OOMM 2 0 1 . OOKG • 1 3 . 3 % 0 . 125 192 . 8MPA 3 4 . 5 0 M M 2 0 7 . , OOKG 1 5 . 0 % 0 . 1 4 0 201 . 5MPA 3 5 . OOMM 2 1 2 . OOKG 16 . 7% 0 . 154 2 0 9 . 3MPA 3 5 . , 5 0 M M 2 1 8 . OOKG 18 . 3% 0 . 168 2 1 8 . 3MPA 3 6 . . OOMM 2 2 6 . OOKG 2 0 . 0 % 0 . 182 2 2 9 . 5MPA I N I T I A L C R O S S S E C T I O N AREA = 1 1 . 4 9 3 0 .OOMM 1 0 8 . OOKG 0 . 2% 0 . 0 0 2 9 2 . 2MPA 3 0 . . 5 0 M M 1 2 2 . OOKG 1 . 7% 0 . 0 1 7 105 . , 9MPA 31 .OOMM 135 . .OOKG 3 . 3% 0 . 0 3 3 1 1 9 . , 1MPA 31 . . 5 0 M M 148 . .OOKG 5 . 0 % 0 . 0 4 9 132 . . 7MPA 3 2 . OOMM 161 . .OOKG 6 . 7% 0 . . 0 6 5 146 . . 6MPA 3 2 . . 5 0 M M 1 6 9 . .OOKG 8 . 3% 0 . . 0 8 0 156 . , 3MPA 3 3 .OOMM 181 . O O K G 1 0 . 0 % 0 . 0 9 5 1 7 0 . .OMPA 3 3 . . 5 0 M M 194 . OOKG 1 1 . 7% 0 . . 1 10 1 8 5 . ,0MPA 3 4 .OOMM 2 0 2 . OOKG 13 . 3% 0 . . 125 195 . 5MPA 3 4 . 5 0 M M 2 0 5 . O O K G 1 5 .0% 0 . 1 4 0 201 . 3MPA 3 5 .OOMM 2 1 0 . O O K G 1 6 . 7 % 0 . 154 2 0 9 . 2MPA 3 5 . 5 0 M M 2 1 7 . O O K G 18 . 3% 0 . 168 2 1 9 . 2MPA 3 6 . OOMM 2 2 5 . O O K G 2 0 . 0 % 0 . 182 2 3 0 . 5MPA I N I T I A L C R O S S S E C T I O N AREA = 1 1 . 4 0 3 0 . OOMM 1 10 . OOKG 0 . 2% 0 . 0 0 2 9 4 . 7MPA 3 0 . 5 0 M M 1 2 0 . O O K G 1 . 7% 0 . 0 1 7 105 .OMPA 31 .OOMM 134 . O O K G 3 . 3% 0 . 0 3 3 1 19 . 2MPA 31 . 50MM 146 . O O K G 5 .0% 0 . 0 4 9 131 . 9MPA 3 2 .OOMM 159 . O O K G 6 . 7% 0 . 0 6 5 145 . 9MPA 3 2 . 5 0 M M 168 . O O K G 8 . 3% 0 . 0 8 0 156 . 6MPA 3 3 .OOMM 178 . O O K G 1 0 . 0 % 0 . 0 9 5 168 . 5MPA 3 3 . 5 0 M M 187 . O O K G 1 1 . 7% 0 . 1 10 179 . 7MPA 3 4 . OOMM 196 . O O K G 13 . 3% 0 . 125 191 . 2MPA 3 4 . 50MM 2 0 6 . O O K G 15 .0% 0 . 1 4 0 2 0 3 . 9MPA 3 5 . OOMM 2 1 5 . OOKG 1 6 . 7 % 0 . 154 2 1 5 . 8MPA 3 5 . 5 0 M M 2 2 1 . O O K G 18 . 3% 0 . 168 2 2 5 . OMPA 3 6 .OOMM 2 3 0 . OOKG 2 0 . 0 % 0 . 182 2 3 7 . 5MPA A V A R E G E T R U E S T R E S S V S S T R A I N E L O N G A T I O N T R U E S T R A I N S T R E S S 0 . 2 % 0 . 0 0 2 9 3 . 3 M P A 1.7% O . 0 1 7 1 0 6 . 7 M P A 3 . 3 % 0 . 0 3 3 1 2 0 . 2 M P A 5 . 0 % 0 . 0 4 9 1 3 3 . 2 M P A . 6 . 7 % 0 . 0 6 5 1 4 6 . 9 M P A 8 . 3 % 0 . 0 8 0 1 5 6 . 9 M P A t O . 0 % 0 . 0 9 5 1 6 9 . 3 M P A 1 1 . 7 % 0 . 1 1 0 1 8 1 . 4 M P A 1 3 . 3 % O . 1 2 5 1 9 3 . 1 M P A 1 5 . 0 % 0 . 1 4 0 2 0 2 . 2 M P A 1 6 . 7 % 0 . 1 5 4 2 1 1 . 5 M P A 1 8 . 3 % 0 . 1 6 8 2 2 0 . 9 M P A 2 0 . 0 % 0 . 1 8 2 2 3 2 . 5 M P A 105 APPENDIX II CALCULATION OF ACTIVATION ENERGIES OF DEFORMATION Ca l cu la t ion example For low stresses range Equation 2 is employed c = A 1 a n exp(-which can be converted as fol lows e exp(-^-) = A^ 1 1 0 i lne + -py = l n A i + n lno 0 1 n lno = lne - In A^  + ^ • j l n f l s { 1 r \ } 1 n ' + TTR * T (28) To f i n d the value of n' the experimental data are p lo t ted in Figure 27 a , b , c , as lno vs y . The other two parameters e and e are kept constant . The exact values of lna and i for a true s t r a i n e = 0.07 are l i s t e d in Table V I I . 106 5-2 4-8 4-4 4 0 3-6 3-2 2-8 2-4 2 0 1-2 1 — i — § — — — § > 33% Zn • high strain rate O low strain rate 0-8 1 0 1-2 1-4 1-6 1-8 2 0 2-2 I / T X I O 3 (l/K°) F i g . 27 Dependence o f f l o w s t r e s s a t 7% s t r a i n o f a l p h a b r a s s e s t e m p e r a t u r e p l o t t e d i n c o o r d i n a t e s I n a v s y ( a ) .33% z i n c a l l o y on 107 5 0 4-6 4-2 3-8 34 * 3 0 2-6 2-2 1-8 T r 1 1 24%Zn • high strain rate O low strain rate J L 0-8 10 1-2 1-4 1-6 1-8 2 0 2 2 1/Tx lO 3 (l/K°) F i g . 27 (b) 24% z i n c a l l o y 108 5-2 b c 4-8 4-4 4 0 3-6 3-2 2-8 2-4 2 0 @ r r z r ^ § = B > -19% Zn • high strain rate O low strain rate 0-8 10 12 6 1-8 2 0 2 2 I / T X I O (l/K°) F i g . 27 ( c ) 19% z i n c a l l o y Tab l e V I I The v a l u e s o f l n a and - f o r 0 .07 s t r a i n T Temper-a t u r e T [°K] ,1 - l O 3 [1/deg] Z i n c c o n t e n t - 33% Z i n c c o n t e n t - 24% Z i n c c o n t e n t - 19% e = l . l x l O " 2 s _ 1 -4 -1 e = l . l x l O s £ = l . l x l 0 " 2 s " 1 £ = l . l x l 0 " 4 s _ 1 E = 1 . 1 X 1 0 " 2 S " 1 -4 - 1 £=1.1x10 s o[MPa] 1na a [MPa] l n a ; a[MPa] l n a a [MPa] l n a a [MPa] l n a a [MPa] l n a 470 2.11 150 151 5.01 5.02 149 . 130 5.00 4 .89 148 5.00 148 5.00 147 147 4 .95 4 .99 138 139 4 . 9 3 4 . 9 3 570 1.74 148 5.00 151 5.02 136 4 .91 137 4 . 92 650 1.53 150 5.01 151 5.01 148 5.00 139 4 .93 140 4 .94 135 4 .90 670 1.49 147 4 .99 96 4 .56 126 4 .84 118 4 .77 750 1.33 146 4 .98 55 4 .01 134 4 .90 76 4 . 33 127 4 .84 79 4 .37 770 1.29 125 4 .83 "39 3.66 126 4 .84 58 4 .06 850 1.17 70 4 .25 28 3.33 93 4 .53 39 3.66 102 4 .62 36 3 .58 '870 1.14 63 4 .14 20 3.00 127 4 .84 950 1.05 28 3.33 11 2.40 40 3.69 16 2 .77 45 3 .81 18 2 .89 1050 0 .95 16 2.77 5 1.61 21 3.04 8 2 .08 25 3.22 9 2 .20 110 The c o e f f i c i e n t n' can be ca l cu la ted assuming that A x does not vary with s t r a i n ra te . Th is i s qui te j u s t i f i e d since the slopes for high and low s t ra in rates in t h e i r s t ra igh t l i n e range are the same. Thus from the curve in Figure 27(a) slope = 2 , 3 4 , = 6985 deg. 0 .34-10" J For the higher s t r a i n rate , extrapo lat ion to zero y i e l d s : i e % i — - i = 3.8 - 6985 deg x 1.1 ' 1 0 " 3 = - 3.883 (1) n' For the lower s t ra in rate extrapo lat ion to zero y i e l d s : ln . 2.05 - 6985 deg x 1.1 •10"3- 1 Beg A l n 4.395 ( i i ) By combining ( i ) and ( i i ) 111 1 1 = n' -3.883 + 4.935 n' = 4 .4 . For the high stress range Equation 3 is employed: e = AgexpfBa) «exp(-which can be converted as fo l lows: e exp = A 2exp(3o) Ine + ^jl = lnA 2 + go " - ( 1 "AJ >/* + W ' T ( 2 9 ) To f i n d the value of 3 the experimental data are p lo t ted as a vs j in F igure 2 8 ( a ) , ( b ) , ( c ) . The other two parameters e and e are kept constant. The exact values of a and j for a true s t r a i n e = 0.07 are l i s t e d in table V I I . C o e f f i c i e n t 6 can be ca l cu la ted assuming that A 2 does not vary with s t r a i n r a t e . This i s qui te j u s t i f i e d s ince the slopes for high and low s t ra in rates in t h e i r s t r a i g h t l i n e range are the same. 112 0 8 10 12 1-4 1-6 1-8 2 0 2-2 I / T X I O 3 (l/K°) F i g . 28 Dependence o f f l o w s t r e s s o f a l p h a b r a s s e s on t e m p e r a t u r e p l o t t e d i n c o o r d i n a t e s a v s | (a ) 33% z i n c a l l o y 113 F i g . 28 (b) 24% z i n c a l l o y 114 F i g . 28 ( c ) 19% z i n c a l l o y 115 Thus from the curve in F igure 28(a) , 98MPa slope = * 0.24«10"J 1/deg slope = 408333 MPa•deg For the higher s t r a i n rate\ extrapo lat ion to zero y i e l d s : . e l A ? -3 1 3 131 MPa - 408333 MPa«deg«l .3-10 " . e l = - 399.8 MPa ( i i i ) 3 For the lower s t r a i n r a t e c extrapo lat ion to zero y i e l d s : , e l l n ^ -3 1 3 = 126 MPa - 408333 MPa-deg'1.5.10 " A 2 ( iv) - - 486.5 MPa 3 By combining ( i i i ) and ( iv) 116 lne, - l n e ? 1 = P -398.8 + 486.5 1 6 = 0.00531 MPa The same procedure was used in c a l c u l a t i n g c o e f f i c i e n t s p and n 1 for other composit ions. The resu l t s are summarized as fol lows B[l/MPa] n' a = [1/MPa] 33% zinc a l l o y 0.0531 4.4 0.0121 24% zinc a l l oy 0.0959 5.2 0.0184 19% zinc a l l oy 0.0825 4.3 0.0192 Having ca l cu la ted c o e f f i c i e n t s 8 and n 1 i t i s poss ib le to apply the general formula ( E q . l ) : e = A[sinh(A-o-)] n 'exp (- p^) where: a = , A - constant Th is can be converted as fo l lows: e exp(-^- ) = A[sinh(a»o-)] lne + -^jl = InA + n' ln[sinh(a»a)] ln[s1nh(a-a)] = (In j )/n + ^ • | (30) By p l o t t i n g the experimental data for constant s t ra in rate F igure 29 ( a ) , ( b ) , ( c ) in coordinates ln[sinh(a»a)] vs ^ one can . ca l cu la te the a c t i v a t i o n energy Q . 117 1-2 0-8 0-4 0 -0 -4 -0 -8 -1-2 -1-6 2 0 2-4 2-8 • T r i __.-•§—^ 8—-------i-/ / / 0-8 1 0 12 33%Zn • high strain rate O low strain rate I /TXIO (l/K°) 29 F i t o f e x p e r i m e n t a l d a t a t o e q u a t i o n o f s t a t e ( a ) 33% z i n c a l l o y 1-8 2 0 2-2 118 119 12Q T a b l e V I I I F i t o f d a t a t o t h e e q u a t i o n o f s t a t e 1 n [ s i n h ( n ' g ) ] I . 1 0 3 3 3 % Z n a = 0 . 0 1 2 1 2 4 % Z n a = 0 . 0 1 8 4 1 9 % Z n a = 0 . 0 1 9 2 I e = l . l x l 0 " 2 s _ 1 -4 - 1 E = l . l x l 0 S e = l . l x l O ' V 1 1 e= l . 1 x 1 0 " V 1 E - l . l x l O ' V 1 e - l . l x l O ' V 1 2 . 1 1 1 . 7 4 1 . 5 3 1 . 4 9 1 . 3 3 1 . 2 9 1 . 1 7 1 . 1 4 1 . 0 5 0 . 9 5 1 . 0 9 1 . 0 7 1 . 0 9 1 . 0 6 1 . 0 4 0 . 7 7 - 0 . 0 5 - 0 . 1 8 - 1 . 0 6 - 1 . 6 4 1 . 0 8 1 . 1 1 1 . 1 1 0 . 3 6 - 0 . 3 3 - 0 . 7 1 - 1 . 0 6 - 1 . 4 1 - 2 . 0 1 - 2 . 8 0 2 . 0 3 2 . 0 3 1 . 7 6 0 . 9 8 - 0 . 2 2 - 0 . 9 3 2 . 0 3 1 . 8 6 0 . 6 4 - 0 . 2 5 - 1 . 2 1 - 1 . 9 1 2 . 0 1 1 . 9 5 1 . 9 1 1 . 9 3 1 . 9 9 1 . 8 9 1 . 7 2 1 . 5 6 1 . 7 3 0 . 7 7 1 . 7 2 0 . 3 1 1 . 2 4 - 0 . 2 9 1 . 7 3 0 . 0 2 - 1 . 0 4 0 . 7 4 - 1 . 7 5 2 6 0 K J m o l Q = 2 8 9 2 3 5 The e x a c t v a l u e s o f l n [ s i n h ( a - a ) ] and j f o r a t r u e s t r a i n e = 0.07 a r e l i s t e d i n T a b l e V I I I . Example c a l c u l a t i o n o f t h e a c t i v a t i o n e n e r g y . From F i g u r e 29 ( a ) 1 1 . 2 * 0 . 2 s i ope 0 . 315 . 10 " s l o p e = 7111 Q = n' - R - s l o p e Q = 4.4«7111-8.314-10 KJ -3 KJ mol Q = 2 6 0 mol In o r d e r t o combine t h e c u r v e s f o r l ow and h i g h s t r a i n r a t e s t h e above r e s u l t s were r e c a l c u l a t e d t o draw a c u r v e o f l n ( e e x p ^y) vs n l n [ s i n h ( a - a ) ] . The e x a c t v a l u e s a r e l i s t e d i n T ab l e IX and t he r e s u l t i n g c u r v e s a r e p r e s e n t e d i n t h e main s e c t i o n o f t h e t h e s i s . 121 T a b l e IX F i t o f d a t a t o t h e e q u a t i o n o f s t a t e e m p l o y i n g p a r a m e t e r Z t—1 1 in CM 1 O I H X f—1 1—( II •OJ 3 3 % z i n c a l l o y 2 4 % z i n c a l l o y 1 9 % z i n c a l l o y l n f e e x p ^r) n I n [ s i n h ( a - a ) ] l n ( e e x p n I n [ s i n h ( o c 0 ) ] In(feexp A ) n I n [ s i n h ( a « a ) ] 6 2 . 0 2 4 . 8 0 4 . 8 8 6 9 . 4 5 1 0 . 6 5 5 . 6 8 . 6 5 0 . 3 5 4 . 7 1 4 5 . 1 8 . 2 4 3 . 6 0 4 . 8 0 4 9 . 0 1 0 . 6 3 9 . 0 8 . 5 4 2 . 1 6 4 . 6 6 3 7 . 7 7 . 4 3 7 . 1 9 4 . 5 8 4 1 . 8 9 . 1 3 3 . 2 7 . 4 3 6 . 1 0 3 . 3 9 3 2 . 2 7 . 4 3 2 . 2 8 - 0 . 2 2 3 6 . 4 5 . 1 2 8 . 7 5 . 3 3 1 . 4 3 - 0 . 7 9 2 7 . 0 7 . 4 2 5 . 2 7 - 7 . 2 2 2 8 . 6 - 4 . 8 2 2 . 4 - 3 . 2 cn «*• i o t-H X II • OJ 5 7 . 4 1 4 . 7 5 2 6 4 . 8 1 0 . 6 5 1 . 0 8 . 4 4 5 . 7 4 4 . 8 8 4 0 . 5 8 . 3 3 8 . 9 9 4 . 8 8 4 4 . 6 9 . 7 3 4 . 4 8 . 1 3 7 . 5 5 1 . 5 8 3 3 . 1 6 . 7 3 2 . 5 8 - 1 . 4 5 3 7 . 2 3 . 3 2 8 . 6 3 . 3 3 1 . 4 9 - 3 . 1 2 2 7 . 6 1 . 3 2 7 . 6 7 - 4 . 6 6 3 1 . 8 - 1 . 3 2 4 . 1 - 1 . 2 2 6 . 8 2 - 6 . 2 0 2 3 . 8 0 - 8 . 8 4 2 7 . 5 - 6 . 3 2 0 . 6 - 4 . 5 2 0 . 6 6 - 1 2 . 3 2 2 4 . 0 - 9 . 9 1 7 . 8 - 7 . 5 122 The o r i g i n o f t h e chosen c o o r d i n a t e s i s as f o l l o w s i = A [ s i n h ( a - 6 - ) ] n ' e x p ( - ^ ) e exp = A [ s i n h ( a * a ) ] In [e ' . e x p ^ j r ] = InA + n' 1 h [sinh(a«a)] . 

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