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The deformation characteristics of zinc and cadmium Risebrough, Neil Reesor 1965-09-01

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THE DEFORMATION CHARACTERISTICS OF ZINC AND CADMIUM by NEIL REESOR RISEBROUGH B.A.Sc.,.University of Toronto,. i 960 M.A.Sc, U n i v e r s i t y of Toronto, 196I A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of METALLURGY We accept t h i s t h e s i s as conforming, to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, I965 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r  m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t , c o p y i n g o r p u b l i  c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n , . D e p a r t m e n t o f M e t a l l u r g y The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8 . C a n a d a D a t e F e b r u a r y 1 5 t h , 1966 The U n i v e r s i t y o f B r i t i s h C o l u m b i a FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of NEIL REESOR RISEBROUGH B.A.SCc, U n i v e r s i t y o f T o r o n t o , 1960 M.A.Sc.p U n i v e r s i t y o f T o r o n t o , 1961 TUESDAY, FEBRUARY 15, 1966 AT 2:30 P.M. IN ROOM 210 METALLURGY BUILDING COMMITTEE IN CHARGE Chairman: J . Ross MacKay W. M. A r m s t r o n g J . A. Lund C, A, B r o c k l e y E. T e g h t s o o n i a n L. G. H a r r i s o n D. Tromans D. L. W i l l i a m s E x t e r n a l E x a m i n e r : J , E, Dorn P r o f e s s o r o f M a t e r i a l s S c i e n c e Lawrence R a d i a t i o n L a b o r a t o r y B e r k e l e y , C a l i f o r n i a . THE DEFORMATION CHARACTERISTICS OF ZINC AND CADMIUM ABSTRACT T h i s work was u n d e r t a k e n t o s t u d y the n a t u r e o f the d e f o r m a t i o n mechanisms i n p o l y c r y s t a l l i n e z i n c and cadmium over a t e m p e r a t u r e range from 77°K t o 300°K. I t has been o b s e r v e d t h a t the o n l y non b a s a l s l i p s y s t e m w h i c h i s o b s e r v e d under normal l i g h t _ m i c r o s c o p y i s t h a t of second o r d e r p y r a m i d a l £ll22j <1123>. A t temperatures above T = T = ,4 } the amount H T M o f non b a s a l s l i p i s g r e a t e r i n z i n c t h a n i n cadmium. The amount o f t w i n n i n g , s u b s t r u c t u r e f o r m a t i o n and g r a i n boundary m i g r a t i o n i s comparable i n b o t h systems. N e g a t i v e work h a r d e n i n g beyond the U.T-. S. a t t e m p e r a t u r e s above T = .4 i s a s s o c i a t e d w i t h r e - i i . . H c r y s t a l l i z a t i o n . I n b o t h systems a t t e m p e r a t u r e s below T = ,26 a H r e g i o n o f t e m p e r a t u r e and s t r a i n r a t e i ndependent l i n e a r work h a r d e n i n g o c c u r s . The e x t e n t o f l i n e a r h a r d e n i n g i n c r e a s e s w i t h d e c r e a s i n g t e m p e r a t u r e below T = .26. Above T = .26 s p o l y c r y s t a l l i n e h a r d e n i n g i n . H H b o t h systems i s p a r a b o l i c from y i e l d on and the r a t e o f h a r d e n i n g a t a g i v e n v a l u e o f s t r a i n d e c r e a s e 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 . Cadmium s i n g l e c r y s t a l s showed a s i m i l a r t r e n t i n t h a t below .26 b o t h 0^ . and 0^ .^  remained c o n s t a n t . However, above .26 t h e r e was a s t e a d y d e c r e a s e i n t h e shear h a r d e n i n g r a t e s . I t was o b s e r v e d t h a t the C o t t r e l l - S t o k e s law i s obeyed o n l y i n the l i n e a r h a r d e n i n g r e g i o n s o f p o l y c r y s t a l s and i n Stage I I h a r d e n i n g o f s i n g l e c r y s t a l s below .26. When dynamic r e c o v e r y o c c u r s A C T i n c r e a s e s w i t h i n c r e a s i n g s t r a i n . °~ I t has been o b s e r v e d t h a t below .26 the l i n e a r h a r d e n i n g r a t e i n cadmium d e c r e a s e d w i t h i n c r e a s i n g g r a i n s i z e ( c o n s t a n t specimen d i m e n s i o n s ) so t h a t e = 0 O + kd The v a l u e o f 8 0 was shown t o c o r r e s p o n d t o the t e n s i l e h a r d e n i n g r a t e d u r i n g Stage I I s i n g l e c r y s t a l d e f o r m a t i o n . The t e n s i l e h a r d e n i n g r a t e was used because of t h e e x t e n s i v e t w i n n i n g found t o be a s s o c i a t e d w i t h S t a g e I I h a r d e n i n g . The g r a i n s i z e dependence o f 8 has been i n t e r p r e t e d i n terms o f a g r a i n s i z e dependence of t h e e x t e n t o f {1122} <1123> , s l i p . I t was found t h a t d u r i n g l i n e a r h a r d e n i n g i n b o t h z i n c and cadmium the d i f f e r e n c e i n f l o w s t r e s s a t two d i f f e r e n t t e m p e r a t u r e s i s a r e v e r s i b l e d i f f e r e n c e i m p l y i n g t h a t t h e d i s l o c a t i o n c o n f i g u r a t i o n s p r o d u c e d w i t h i n c r e a s i n g s t r a i n do not v a r y i n n a t u r e o r e x t e n t w i t h t e m p e r a t u r e . Under such c o n d i t i o n s i t i s p o s s i b l e t o f o r m u l a t e a m e c h a n i c a l e q u a t i o n o f s t a t e . E x t e n s i v e r a t e t h e o r y measurements have been made i n b o t h systems i n o r d e r t o att e m p t an e v a l u a t i o n o f the, r a t e c o n t r o l l i n g mechanisms b o t h d u r i n g l i n e a r h a r d e n i n g and d u r i n g dynamic r e c o v e r y . The former has t e n t a t i v e l y been a s s o c i a t e d w i t h i n t e r s e c t i o n . Dynamic r e c o v e r y on the o t h e r hand has been l i n k e d t o t h e l o o p a n n e a l i n g o b s e r v a t i o n s o f P r i c e . GRADUATE STUDIES F i e l d o f Study: M e t a l l u r g y M e t a l l u r g i c a l Thermodynamics C. S. Samis M e t a l l u r g i c a l K i n e t i c s E. P e t e r s H y d r o m e t a l l u r g y E. P e t e r s S t r u c t u r e o f M e t a l s I I I E. T e g h t s o o n i a n T o p i c s i n P h y s i c a l M e t a l l u r g y J. A. Lund R e l a t e d S t u d i e s ; S t a t i s t i c a l M e c h a n i c s R. F. S n i d e r E l e m e n t a r y Quantum Me c h a n i c s F. W„ D a l b y Computer Programming C. F r o e s e Chairman: Dr. E . Teghtsoonian ^ ABSTRACT This work was undertaken to study the nature of the deformation mechanisms i n p o l y c r y s t a l l i n e zinc and cadmium over a temperature range from 77°K to 300°K. I t has been observed that the only non basal s l i p system which i s observed under normal l i g h t microscopy i s that of second order pyramidal [ l l 2 2 ] <(ll23>. At temperature above T = T = .k, the amount of non basal H <V s l i p i s greater i n zinc than i n cadmium. The amount of twinning, substructure formation and grain boundary migration i s comparable i n both systems. Neg ative work hardening beyond the U.T.S. at temperatures above T = .k i s H associated with r e c r y s t a l l i z a t i o n . In both systems at temperatures below T = .26 a region of H temperature and s t r a i n rate independent l i n e a r work hardening occurs. The extent of l i n e a r hardening increases with decreasing temperature below .. = T = .26 . Above T n = .26 , p o l y c r y s t a l l i n e hardening i n both systems i s H n parabolic from y i e l d on and the rate of hardening at a given value of s t r a i n decreases with • increasing temperature. Cadmium single c r y s t a l s showed a s i m i l a r trend i n that below .26 both 0^ and • remained constant. However above .26 there was a steady decrease i n the shear hardening rates. I t was observed that the Cot t r e l l - S t o k e s law i s obeyed only i n the l i n e a r hardening regions of p o l y c r y s t a l s and i n Stage II hardening of single c r y s t a l s below .26 ..When dynamic recovery occurs A°~ increases cr with increasing s t r a i n . I t has been observed that below .26 the l i n e a r hardening rate i i i n cadmium decreased with increasing grain s i z e ( constant specimen dimen sions) so that 1 -2 0 = Qo + kd The value of 0 owas shown to correspond to the t e n s i l e hardening rate during Stage II single c r y s t a l deformation. The t e n s i l e hardening rate was used because of the extensive twinning found to be associated with Stage II hardening. The grain s i z e dependence of 0 has been interpreted, ih.iterms of a grain s i z e dependence of the extent of £ll2*2]^1123^ s l i p . I t was found that during l i n e a r hardening i n both zinc and cadmium the difference i n flow stress at two d i f f e r e n t temperatures i s a re v e r s i b l e difference implying that the d i s l o c a t i o n configurations produced with increasing s t r a i n do not vary i n nature or extent with temperature. Under such conditions i t i s .possible to formulate ai.mechanical equation of state. . Extensive rate theory measurements have been made i n both systems i n order to attempt an evaluation of the rate c o n t r o l l i n g mechanisms both during l i n e a r hardening and.during dynamic recovery.. The former has t e n t a t i v e l y been associated with intersection.. Dynamic recovery on the other hand has been l i n k e d to the loop annealing observations of P r i c e . ACKNOWLEDGEMENT The author i s g r a t e f u l f o r the advice and encouragement given by his research director,.Dr..E.•Teghtsoonian, and f o r h e l p f u l discussions with fellow graduate students. F i n a n c i a l assistance was received i n the form of an International N i c k e l Company of Canada Limited Fellowship. : i v TABLE OF CONTENTS ., Page 1. • DEFORMATION CHARACTERISTICS' OF ZINC AND CADMIUM 1 1 . 1 . INTRODUCTION 1 1 . 2 . EXPERIMENTAL PROCEDURE , 3 1 . 2 . 1 . Materials and Specimen-Preparation 3 1 . 2 . 2 . Testing Procedure 6 1 . 3 . .STRESS-STRAIN RELATIONSHIPS • 7 1 . 3 - 1 . Nature of the S t r e s s - S t r a i n Curves 7 1 . 3 . 2 . D u c t i l i t y . 14 a) Cadmium • 14- b> Zinc :• 16 1.3.. 3 . Grain Boundary E f f e c t s 16 1.3-4. . R e c r y s t a l l i z a t i o n 25 1 . 3 - 5 - .Maximum Stress V a r i a t i o n with Temperature and S t r a i n Rate. 27 1.4. DEFORMATION MODES IN ZINC AND CADMIUM 30 1 . 4 . 1 . S l i p -50 a) Zinc -. . 3 1 b) Cadmium • 37 1.4.2. Twinning 4l 1.4.3. The Formation of Low Angle Boundaries during' Deformation. 46 1 . 5 . YIELD STRESS AND WORK HARDENING 50 I .5..I. The Temperature Dependence of Y i e l d 50 I . . 5 . 2 . Temperature S e n s i t i v i t y of the Flow. Stress 53 a) Cadmium 55 b) Zinc ' 57 TABLE OF CONTENTS (continued) Page 1 - 5 -. 3 - S t r a i n R a t e ' S e n s i t i v i t y of the Flow Stress 6 l I . 5 A . The Deformation of Cadmium Single C r y s t a l s 66 1 . 5 . 5 . Temperature Dependence of Work Hardening 7 1 1 . 5 . 6 . The Grain Size Dependence of Hardening at -196°C 71 2. MECHANISMS - OF HARDENING IN ZINC AND CADMIUM 78 2 . 1 . INTRODUCTION 78 2 . 2 . TEMPERATURE CHANGE TESTS 85 2 . 2 . 1 . Procedure 85 2 . 2 . 2 . Cottre11-Stokes Tests 85 2 . 2 . 3 . The Mechanical Equation of State 88 a) Cadmium 90 b) . Zinc 90 2 . 2 . 4 . Equivalent States above T = .26 • 9^ H 2 . 5 . STRAIN RATE CHANGE TESTS 98 2 . 5 . 1 . Procedure 99 2 . 3 . 2 . C o t t r e l l - S t o k e s Behaviour i n Cadmium at -196°C 99 a) .Single C r y s t a l s 99 b) . P o l y c r y s t a l s 101 2 . 3 . 3 . The E f f e c t of Temperature on Co t t r e l l - S t o k e s Behaviour . .103 2 . 4 . HARDENING AT -196°C IN CADMIUM 107 2 . 4 . 1 . -Activation Volume 107 2 . 4 . 2 . • A c t i v a t i o n Energy ' H I 2 . 5 . HARDENING ABOVE -196°C IN ZINC AND CADMIUM 112 2 . 5.I. . Y i e l d Behaviour i n Cadmium 112 V I TABLE OF CONTENTS (continued) Page 2 . 5 . 2 . The V a r i a t i o n of AH with£Strain i n 25u Cadmium '. 118 2 . 5 . 3 . Y i e l d behaviour i n Zinc 119 3 . DISCUSSION 123 3 . 1 . LOOP FORMATION AND ANNEALING 123 3 . 2 . DYNAMIC RECOVERY 128 3 . 2 . 2 . Cross s l i p 129 3 . 2 . 3 . D i f f u s i o n C o n t r o l l e d Processes 131 3 . 3 . THE MECHANICAL EQUATION OF STATE 135 3 . 4 . THE COTTRELL-STOKES LAW 136 3 - 4 . 1 . • Obeyance 137 3 . 4 . 2 . Dynamic Recovery 138 3 . 5 . . RATE CONTROLLING PROCESSES 'BELOW T = .26 138 3 . 5 . 1 . .Peierls Stress 138 3 . 5 . 2 . Cross S l i p .' 139 3 . 5 . 3 . .The Non Conservative-Motion of Jogs 140 3 . 5 . 4 . I n t e r s e c t i o n l 4 l 4. - SUMMARY AND CONCLUSIONS 143 5. - SUGGESTIONS FOR FUTURE WORK 146 6. APPENDICES 147 6 . 1 . RATE THEORY 147 6 . 2 . UNLOADING YIELD POINTS IN CADMIUM . . 154 6 . 3 . THE DETERMINATION OF A C T FROM- STRAIN RATE CHANGE TESTS 160 v i i TABLE OF CONTENTS (continued) Page BIBLIOGRAPHY 164 v i i i LIST OF FIGURES No. Page 1. P o l y c r y s t a l l i n e specimen 8 2. .Polycrystal t e s t i n g apparatus 8 J . Single c r y s t a l t e s t i n g apparatus 9 4 . S t r e s s - s t r a i n curves f o r 25u cadmium . 1 0 - 5 . S t r e s s - s t r a i n curves f o r 400u cadmium.. 11 6. S t r e s s - s t r a i n curves f o r 2 0 u zinc 12 7. S t r e s s - s t r a i n curves f or .400u zinc . .• " 13 8. The e f f e c t of temperature on the d u c t i l i t y of 25u cadmium .... 15 9 . The e f f e c t of temperature on the d u c t i l i t y of 400u cadmium ... 15 10. The e f f e c t of grain s i z e on the d u c t i l i t y of zinc 17. 11. The e f f e c t of temperature on the d u c t i l i t y of 20u zinc ....... 18 12. Grain boundaries i n 25u cadmium before and a f t e r 7$ deformation at 20°C 20 13. Grain boundary motion i n 400u zinc and cadmium . 21 1.4. Grain boundary shear i n 400u zinc and cadmium at +20°C 22 15. R e c r y s t a l l i z a t i o n i n 25u cadmium deformed 20$ at +20°C 26 16. The temperature dependence of the maximum stress i n cadmium .. 28 17. The temperature dependence of the maximum stress i n zinc .. ... 28 18. The e f f e c t of s t r a i n rate on the maximum stress i n 25u cadmium . ~ 29 19. The e f f e c t of s t r a i n rate . on the maximum stress;:.in': 20u.-,zinc , . . 29 2 0 . Pyramidal g l i d e systems i n zinc and cadmium ( a f t e r Price) ..... 32 2 1 . { l l 2 2 l <112~3> s l i p i n 400u zinc at various s t r a i n s 35 22. { l l 2 2 ] < H 2 3 > s l i p i n 400u zinc at -196°C 39' 2 3 . [ l l 2 2 ] <1123> traces on .400u cadmium deformed 7$ at -196°C.. 40 24. Microstructure of 400u cadmium deformed 7/0 at +20°C 42 LIST OF FIGURES (continued) ...No. Page 2 5 . Microstructure of 400u cadmium deformed 7$ at -196"C Kj> 26. Lack of twinning i n the presence of boundary migration 45"" " 27. Twin basal s l i p and twin nucleation 45" 28. Low angle boundaries i n cadmium deformed 7$ a * -J0 oC 48 29. The formation of non cr y s t a l l o g r a p h i c boundaries, i n cadmium.due to underlying small grains 48 3 0 . C r y s t a l l o g r a p h i c boundary formation i n 400u zinc and cadmium . 49 .31. Flow stress-temperature r e l a t i o n s h i p s f o r 0.020 inch grain diameter cadmium as found by S t o l o f f 51 3 2 . Temperature dependence of the y i e l d stress i n p o l y c r y s t a l l i n e zinc and cadmium 52 33- The temperature'dependence of the s t r a i n hardening parameter i n p o l y c r y s t a l l i n e copper as found by R u s s e l l 54 3 4 . Flow stress-temperature r e l a t i o n s h i p s f o r 25)i cadmium 55""" 35- Flow stress-temperature r e l a t i o n s h i p s f o r 400u cadmium 56 .36. The v a r i a t i o n with temperature of the work hardening •parameter i n 25u cadmium 58 37- The v a r i a t i o n with temperature of the work hardening parameter i n 400u cadmium 59 3 8 . Flow stress-temperature r e l a t i o n s h i p s f o r 20u zinc 60 39• The v a r i a t i o n with temperature of the work hardening parameter i n 20u zinc 60 4 0 . Linear hardening of p o l y c r y s t a l l i n e zinc and cadmium below T = .26 62 11 41. The e f f e c t of s t r a i n rate on the flow stress of 20u zinc at +20°C 63 42. The v a r i a t i o n of the s t r a i n rate parameter Ao~ with .>':• temperature and s t r a i n i n 25>i cadmium 64 4 3 . The v a r i a t i o n of the s t r a i n rate parameter Ao~ with temperature and s t r a i n i n 20u zinc 65 •LIST OF FIGURES (continued) x . No. Page 44. • Single c r y s t a l stress s t r a i n curve f o r cadmium at +20°C 68 4 5 . S t r e s s - s t r a i n curves f o r cadmium single c r y s t a l s at various temperatures 69 4 6 . The temperature dependence of Stage-I hardening i n cadmium ... 74 47. The temperature dependence of Stage II hardening i n cadmium .. 74 4 8 . Temperature dependence of the rate of work hardening during Stage I deformation 75 4 9 . S t r e s s - s t r a i n curves f o r cadmium of various grain sizes at -196°C 76 5 0 . The e f f e c t of grain s i z e on the rate of l i n e a r hardening at -196°C 77 51 . •The temperature dependence of y i e l d i n terms of the stress components • 79 52 . The components of t h e . t o t a l flow stress i n f . c . c . c r y s t a l s ... o u 53. Components, of the differ e n c e i n flow stress when two i d e n t i c a l specimens are deformed at d i f f e r e n t temperatures ... 81 54. The relationship, between the change i n flow stress per °K and the resultant stress at -196°C obtained during the temperature c y c l i n g of cadmium 87 55 . The e f f e c t of elevated. temperature pr e s t a i n i n g on the '• stress s t r a i n curve of p o l y c r y s t a l l i n e copper at 77°K 89 56. Temperature c y c l i n g of 25u cadmium between - l40°C and -196°C . 91 57. The e f f e c t of p r e s t r a i n i n g at -95°C on the subsequent ... . !>..,.-.a deformation behaviour at -120°C i n 20u zinc 92 58 . The e f f e c t of p r e s t r a i n i n g at -70°C on the subsequent deformation behaviour at -120°C i n 20u zinc 93 59. Equivalent states at d i f f e r e n t s t r a i n s 95 6 0 . Reversible temperature change tes t s at equivalent states i n p o l y c r y s t a l l i n e copper 96 61 . The c o r r e l a t i o n of s t r a i n s at d i f f e r e n t temperatures i n 20u zinc 97 x i LIST OF. FIGURES (continued) No. Page 62. The v a r i a t i o n pf the Co t t r e l l - S t o k e s parameter during the deformation of cadmium single c r y s t a l s at -196°C 100 63. The grain size dependence of A°~ at -196°C 102 c r 6 4 . The v a r i a t i o n with stress of A°~ obtained from 25u cr cadmium at d i f f e r e n t temperatures 104 65. The stress dependence of Ao~ (400u cadmium) 105 cr 66. The f a i l u r e of the Co t t r e l l - S t o k e s law at -50°G 105 67. • The v a r i a t i o n with stress of the Co t t r e l l - S t o k e s parameter obtained from s t r a i n rate change t e s t s on 20u zinc 106 68. The stress dependence of the a c t i v a t i o n volume (cadmium) ..... 108 69. • Force-distance curves 115 70. The v a r i a t i o n of /\G with temperature for - 2 5 u cadmium 116 71 . The v a r i a t i o n of AH with s t r a i n and temperature i n 25u cadmium • 120 72. The v a r i a t i o n of AG with, temperature f or 20u zinc 122 73.. The formation of a prismatic d i s l o c a t i o n loop by an edge d i s l o c a t i o n which i s held up at an obstacle .123 74. The e f f e c t of jogs of various heights on screw d i s l o c a t i o n motion 124 75. Stages i n the formation of an elongated loop on the basal plane by the cross g l i d e of a £ l l 2 2 | 0-123,) screw d i s l o c a t i o n . 126 76. D i s l o c a t i o n s i n hexagonal close packed c r y s t a l s 127 7 7 ' The conservative climb of a basal d i s l o c a t i o n loop 132 78. T y p i c a l force-distance curve f o r a thermally a c t i v a t e d deformation process 153 79. Unloading y i e l d point i n p o l y c r y s t a l l i n e cadmium 154 8 0 . Unloading y i e l d point terminology 156 81 . .The v a r i a t i o n of <J\Jc^ with s t r a i n at -196°C 158 x i i LIST OF FIGURES (continued) Page No. 8 2 . The decrease i n flow stress due to s t a t i c recovery during interrupted t e s t i n g of 25u cadmium at -95°C 159 83. The nature of the flow stress obtained during s t r a i n rate change t e s t s i n p o l y c r y s t a l s , l 6 l x i i i LIST OF TABLES No. P a § e 1. Physical properties of hexagonal metals k 2. Non basal s l i p systems observed i n zinc 33 3. Non basal s l i p systems observed i n cadmium $k .k. S l i p systems i n hexagonal metals 38 5 . Upper temperature l i m i t s f o r l i n e a r hardening i n p o l y c r y s t a l l i n e zinc and cadmium 6 l 6. Stress required f o r twin formation i n cadmium single c r y s t a l s 70 7. Incubation s t r a i n required i n p o l y c r y s t a l l i n e copper p r i o r to the appearance of an i r r e v e r s i b l e component of the differ e n c e i n flow stress 89 8 . Grain size dependence of C o t t r e l l - S t o k e s behaviour 101 9 . Grain si z e dependence of the a c t i v a t i o n volume at y i e l d i n cadmium at -196°C 109 10. Energy values at y i e l d f o r cadmium deformed at -196°C m 11. Energy values at y i e l d f o r 2^yi cadmium 113 12. .Rate parameters at y i e l d i n 20u zinc 121 13. Loop formation i n zinc and cadmium 127 PART I -1- 1. DEFORMATION CHARACTERISTICS OF ZINC AND CADMIUM 1 .1 INTRODUCTION In the past, the major emphasis i n the f i e l d of p l a s t i c deformation has been placed on face centered cubic metals. Much of t h i s work has been r e s t r i c t e d to the use of single c r y s t a l s i n an attempt to determine the d i s l o c a t i o n mechanisms which con t r o l the work hardening processes. The study of single c r y s t a l behaviour has the advantage of avoiding the d i f f i c u l t problem . of dealing w i t h the constraining conditions imposed by grain boundaries during the deformation of p o l y c r y s t a l l i n e aggregates. This s i m p l i c i t y , which enables one to obtain a better.knowledge of shear stress conditions on p a r t i c u l a r s l i p planes has disadvantages due to the poor r e p r o d u c i b i l i t y of flow stress values of d i f f e r e n t single c r y s t a l s because of the e f f e c t s of s l i g h t changes i n such f a c t o r s as substructure, impurity concentration, o r i e n t a t i o n and i n i t i a l d i s l o c a t i o n density. The c o r r e l a t i o n of single c r y s t a l and p o l y c r y s t a l l i n e data f o r face centred metals i s a t o p i c of some current i n t e r e s t 1 " 5 and should i n the future provide a groundwork f o r the understanding of macroscopic deformation c h a r a c t e r i s t i c s . I n the p a s t few years the study of body centred cubic-metals,,' n o t a b l y , i r o n ; , has been i n t e n s i f i e d . C o n r a d 6 - 1 0 , Gregory 1 1, B a s i n s k i 1 2 and others have attempted to determine the rate c o n t r o l l i n g mechanism over a wide range o f temperaturei Much of t h i s work involved the use of p o l y c r y s t a l l i n e specimens. - 2 - The f i e l d of hexagonal metal deformation has "been somewhat ignored due i n most part to the anisotropic nature of the hexagonal system and the resultant d i f f i c u l t i e s encountered i n mechanical working and engineering use. With p o s s i b l y the exception of zinc and cadmium, the more prominent hexagonal metals in c l u d i n g T i , Be, Mg, Co, and Zr are associated with d i f f i c u l t m e t a l l u r g i c a l procedures during e x t r a c t i o n and r e f i n i n g which tend to make them, expensive and hence undesirable as engineering materials except i n ap p l i c a t i o n s where d e f i n i t e advantages occur. Of a l l . the hexagonal metals, titanium and-magnesium have been studied i n the greatest d e t a i l because of t h e i r favourable strength to weight r a t i o . . Because of gaseous embrittlement and d i f f i c u l t i e s encountered during mechanical forming, titanium has not become as widely used as once predicted. Magnesium i s d i f f i c u l t to produce i n wrought form due to embrittlement during cold working. Therefore the problem of determining the deformation mechanisms which c o n t r o l flow i n magnesium has been a subject of considerable i n t e r e s t . On the other hand very l i t t l e work has been done on zinc and 2 3 cadmium since the c l a s s i c works of Scamid and Boas- i n the e a r l y 1930's. Recently considerable i n t e r e s t has been shown i n the development of new wrought zinc a l l o y s to compete commercially with some aluminum and copper a l l o y s . Poor creep properties due to i t s high e f f e c t i v e temperature at" room temperature and reduced d u c t i l i t y because of cleavage fracture at reduced temperatures have so f a r l i m i t e d i t s use. I t i s necessary therefore to have a more d e t a i l e d knowledge of the deformation mechanisms to provide a ground work f o r future alloy- develop ment. Most inv e s t i g a t i o n s to date have been concerned with e i t h e r the hardening mechanisms during basal g l i d e 2 4 - 2 5 or the nature of eleavage f r a c t u r e . 2 6 2 7 Very l i t t l e work has been done with p o l y c r y s t a l l i n e zinc to determine such factors as the s t r a i n rate and temperature s e n s i t i v i t y of the flow s t r e s s , the mode and degree of non basal s l i p , the nature of dynamic recovery,, the relevance of g r ain boundary e f f e c t s etc. Although s i m i l a r i t i e s e x i s t between zinc and cadmium as shown i n Table I, the apparent deformation c h a r a c t e r i s t i c s d i f f e r i n that cadmium does not f a i l by cleavage fracture and maintains s u b s t a n t i a l d u c t i l i t y down to 4 .2°K„ However a quasi d u c t i l e - - b r i t t l e t r a n s i t i o n involving a change i n fracture mode from d u c t i l e shear to intergranular fracture has been reported by S t o l o f f 2 8 and Magnussen. 2 5 . Therefore by comparing zinc and cadmium under standard conditions of p u r i t y , grain s i z e , temperature, s t r a i n rate and specimen geometry i t may be possible to obtain a more d e t a i l e d knowledge of the flow and fracture mechanisms. The f i r s t part of t h i s t h e s i s i s concerned with the more macroscopic flow parameters whereas part two describes attempts to determine the rate c o n t r o l l i n g mechanisms by the use of rate theory. 1.2 EXPERIMENTAL PROCEDURE 1.2 .1 Materials and Specimen Preparation The zinc and cadmium used i n the course of t h i s work was of 99.999% p u r i t y and was supplied i n the form of one-half inch rods by the Consolidated Mining and Smelting Company, T r a i l , B.C. These rods were remelted i n a i r and cast into graphite moulds to give ingots with dimensions 5" x 2^" x . These were then r o l l e d into TABLE I P h y s i c a l Properties of Hexagonal Metals Element Zn Cd Mg Be Co T i Zr Melting Temperature °C 420 320 65O 1277 1495 1668 1852 Density gms/c.c. 7.14 8.65 1 .74 I . 8 5 8.9 4.51 6.49 c / a r a t i o I . 8 5 6 1 .886 1 .624 I . 5 8 6 1.623 1.588 1 .590 Stacking Fault Energy* medium medium high high low high high Periodic Group No. 11B 11B 11A 11A VIII IVB.-- IVB Stacking Fault Energies Low - under 25 ergs/cm 2 Medium - 2 5 - 1 0 0 ergs/em 2 High - > 100 ergs/cm 2 sheet. The i n i t i a l r o l l i n g passes f o r zinc were c a r r i e d out at 150°C i n order to avoid cracking. The grain refinement which occurred during t h i s hot r o l l i n g operation was such as to allow further reductions to be ca r r i e d out at room temperature. R e c r y s t a l l i z a t i o n during r o l l l i n g r e a d i l y occurred at room temperature because of the small grain s i z e . Cadmium was r o l l e d at room temperature throughout the-reduction process. .In both cases reduction was c a r r i e d out i n .010 inch steps from .150 inches to the f i n a l sheet size of .031 inches. This treatment produced a.very uniform and f i n e grained r e c r y s t a l l i z e d sheet with a grain size of 20u f o r zinc and 25u f o r cadmium. The grain si z e could be varied s i g n i f i c a n t l y by changing the amount of reduction i n the f i n a l r o l l pass. For instance a f i n a l grain size of 50;i instead of 25u could be obtained i n cadmium by using a f i n a l reduction of 51/0 instead of 25%. For t h i s reason a l l specimens were cut from the same sheet i n order to eliminate small differences i n grain size and preferred o r i e n t a t i o n . Tensile specimens were punched to give a reduced gauge length of .8 inches with cross s e c t i o n a l dimensions of .200" x .031" ( F i g . l ) . Since t h i s procedure, caused s l i g h t deformation around the specimen edges i t was necessary to chemically p o l i s h the surface p r i o r to t e s t i n g . Approximately .0015 inches were removed to give a f i n a l specimen thickness of .0280 inches. The p o l i s h i n g s o l u t i o n used f o r both zinc and cadmium was as follows - 320 gms C r 0 3 20 gms Na 2 S04 1000 mis H 2 0 , . 30 This represents a s l i g h t modification of Gilmans s o l u t i o n 1. - Besides - 6 - . p o l i s h i n g the surface, t h i s treatment caused grain boundaries to become s l i g h t l y grooved thereby f a c i l i t a t i n g metallographic examination a f t e r t e s t i n g . Large grained specimens were obtained by annealing punched specimens i n ai r . under the following conditions: Cadmium - 2 hrs at 230°C Zinc - 2 hrs at.l80°C The specimens were then furnace cooled from temperature over a period of 1 hour. The resultant grain size i n each case was 400u- - 25u. This produced specimens with only 1 to 2 grains across the specimen thickness. Because of t h i s , the s c a t t e r i n flow stress values was somewhat greater than f o r f i n e grained specimens. The purpose of producing such specimens was to f a c i l i t a t e metallographic observations and to provide an intermediate t e s t specimen between the normal f i n e grained material and single c r y s t a l s . The single c r y s t a l s used were grown by a modified Bridgman technique i n evacuated $mm diameter pyrex glass tubes. Extruded .100 inch lengths of cadmium were placed i n the tubes which were subsequently lowered at the rate of 1" per hour through a. 12" v e r t i c a l tube furnace. C r y s t a l s up to 18 inches i n length could be grown providing numerous samples of the same o r i e n t a t i o n . 1.2.2 Testing Procedure Specimens were deformed on a Floor Model Instron using s t r a i n rates that varied from 4.0 x 10 3 s e c 1 to 4.0 x 10 5 s e c *. Testing media included l i q u i d nitrogen^cooled petroleum ether (-140 to +20°C), hot water (+20 to + 100°C) and s i l i c o n e oil(above 100°C). The bath temperature - 7 - i n each case could be c o n t r o l l e d to i 2°C. S p l i t jaw grips which produced a very r i g i d t e s t i n g apparatus were used f o r the p o l y c r y s t a l l i n e specimens. (F i g . 2 ) . •Single c r y s t a l s were mounted i n solder i n aluminum grips and were deformed i n a t e s t i n g r i g which allowed complete freedom of r o t a t i o n ( F i g . 3 ) . The distance between grips was between 30 and 35 g i v i n g a length to diameter r a t i o of 10/1. . In order to f a c i l i t a t e comparison with previous work a l l data on p o l y c r y s t a l s i s expressed i n terms pounds per square inch (p.s.i.) whereas single c r y s t a l r e s u l t s are given i n terms of c.g.s. u n i t s . 1..3 STRESS ^ -STRAIN RELATIONSHIPS 1.3-1 Nature of the Stress S t r a i n Curves True s t r e s s - t r u e • s t r a i n curves f o r both grain sizes i n zinc and cadmium at a few selected temperatures are shown i n Figures k,, 5 , 6 and 7> The cadmium curves are q u a l i t a t i v e l y the same as those observed by S t o l o f f and Gensamer 2 8 f o r .020 inch grain size material. - S p e c i f i c a l l y , work hardening i n the e a r l y regions of s t r a i n i s parabolic at high e f f e c t i v e temperatures and tends to become more•linear with decreasing temperature. ( E f f e c t i v e temperature i s given by Tg .= T_ where T^ i s T M the melting p o i n t ) . At temperatures above approximately T^ = .kO there i s a large amount of s t r a i n beyond the point of maximum stress which was not observed to be associated with necking. I t was also observed that t h i s e f f e c t i s gr e a t l y reduced i n the UOOu material. - 8 - F i g . 2 P o l y c r y s t a l t e s t i n g apparatus. - 9 - F i g . 3 Single c r y s t a l t e s t i n g apparatus. F i g . 5 . S t r e s s - s t r a i n curves f o r k00u cadmium. - ST Fig.7 .Stress - s t r a i n curves f o r kOOn zinc 1.3-2 • D u c t i l i t y - ^ D u c t i l i t y w i l l be given only i n terms, of percent elongation or true s t r a i n as i t was not p r a c t i c a l to obtain reduction i n area values because of the specimen dimensions. .As an a i d to i n t e r p r e t a t i o n , d u c t i l i t y i s given not only as the true s t r a i n to fracture but also :in terms of the true s t r a i n to maximum stress conditions. This i s important when considering deformation at high values of T^ because of the large amounts of deformation associated with negative work hardening. 1 .3 .2 a) Cadmium The e f f e c t of temperature on the d u c t i l i t y of 25p and 400u cadmium i s shown i n Figures 8 and 9- I t i s observed that there i s an increase i n the s t r a i n to frac t u r e as the temperature decreases to -120°C. This increase is more pronounced with an increase i n grain s i z e . At -120°C both grain sizes have approximately the same s t r a i n to frac t u r e of about kO$>. Below -120°C there i s a steady decrease i n d u c t i l i t y independent of grain s i z e , i n agreement with the r e s u l t s of S t o l o f f and Gensamer. This decrease corresponds to a change i n the fracture mode from d u c t i l e shear to in t e r g r a n u l a r . f r a c t u r e . From Figures 8 and 9±t i s also observed, that above -120°C 7 ,1.) the true s t r a i n to maximum stress i s greater f o r kOOp. than f o r 25u cadmium 2) the percentage of the t o t a l d u c t i l i t y which i s associated with negative hardening a f t e r maximum stress conditions have been r e a l i z e d i s greater f o r the 25u cadmium. I I I I I I I I I I L I 1 1 -160 -120 -80 -40 0 +40 +80 Temperature °C F i g . 9 The e f f e c t of temperature on the d u c t i l i t y of 400j j cadmium. - 16 - •1.3.2 b). Zinc The r e s u l t s f o r zinc are q u a l i t a t i v e l y the same as those f o r cadmium except f o r the occurrence of a d i s t i n c t d u c t i l e to b r i t t l e t r a n s i t i o n due to cleavage f r a c t u r e . From Figure 10 i t i s observed that t h i s t r a n s i t i o n i s s h i f t e d about 50°C as the grain s i z e i s increased from 25u to 400u. Above the t r a n s i t i o n temperature i t i s observed that both the t o t a l s t r a i n to fracture and the true s t r a i n to maximum stress decrease with increasing temperature i n a manner s i m i l a r to cadmium. Likewise the percentage of the t o t a l d u c t i l i t y at fr a c t u r e which i s associated with negative work hardening i s greater f o r 20u zinc than f o r 400u'zinc. Also the true s t r a i n to maximum stress i s greater at the same temperature, f o r hOOu zinc than for 20u z i n c . This i s i n agreement with observations (1) and (2) of the previous section on cadmium. The e f f e c t of s t r a i n rate on the d u c t i l e to b r i t t l e t r a n s i t i o n i s shown i n Figure 11. . Changing the s t r a i n rate by a f a c t o r of 10 s h i f t e d the t r a n s i t i o n f o r 20u zinc by about 25°C. 1 .3 .3 Grain Boundary E f f e c t s The observations of the previous section with regard to grain size e f f e c t s indicate that some form of recovery and r e c r y s t a l l i z a t i o n are operative during deformation at elevated temperatures. These processes . are expected because of the high p u r i t y and high e f f e c t i v e temperatures. However S t o l o f f found no evidence of r e c r y s t a l l i z a t i o n during t e s t i n g at 20°C . - 17 - F i g . 1 0 The e f f e c t of grain s i z e on the d u c t i l i t y of z i n c . - 1 8 - . Temperature • °C F i g . 11 The e f f e c t of temperature on the d u c t i l i t y of 20u, z i n c . - .19 - Figure 12 shows the r e l a t i v e grain boundary structures i n 25u cadmium before and a f t e r 7$ deformation at room temperature. The boundaries have become very jagged i n appearance i n d i c a t i n g considerable grain boundary migration during deformation. Figure 13 shows other examples o f boundary motion i n kOOp zinc and cadmium. With decreasing temperature the migrating boundaries show a more continuous type of migration along the boundary (Fig.13b) opposed to the corrugated type observed at room, temperature and above (Fig. 1 3 d ) . At a given temperature these boundary e f f e c t s were much more prominent i n fine grained material. Figure 13(b)-shows a "stepped" type of boundary migration s i m i l a r to that observed by Chang and' G r a n t 3 1 during creep of p o l y c r y s t a l l i n e aluminum a t elevated temperatures. They interpreted such observations i n terms of alternate processes of shear and migration. In order to obtain a q u a t i t a t i v e assessment of the importance .of shear as a deformation process, polished kOOp specimens were f i n e l y marked by means of a soft brush and then deformed. . As shown i n Figure lk, shear was observed i n both systems at room temperature. I t was not ne c e s s a r i l y associated with migration. Only a few boundaries showed, v i s i b l e o f f s e t s and such o f f s e t s could only be seen at rather .high magnifications. / The e f f e c t of boundary migration and shear processes on the stress s t r a i n curve i s not f u l l y understood. Shear i s b a s i c a l l y a work hardening process and involves the deformation and subsequent hardening of areas adjacent to the boundaries. On the other hand migration occurs - 20 - X 2k0 (a) undeformed X 240 (b) deformed and immediately repolished Fig.12 Grain boundaries i n 25u cadmium before and a f t e r 7% deformation at +20°C. f * 21 - (a) corrugated grain boundary i n cadmium deformed 7$ at +20°C and repolished. X 2k0 X 2k0 Fig. 1 $ Grain boundary motion i n kOOu zinc and cadmium. - 22 - x 700 (a) cadmium at 10$ s t r a i n . X 700 (b) zinc at 7$ s t r a i n . F i g . l 4 Grain boundary shear i n kOOy zinc and cadmium at +20°C. - 23- - because of a net difference i n free energy across the boundary caused by differences i n the d i s l o c a t i o n configurations associated with the work hardened states on each side of the boundary.. I f when a boundary moves i t leaves behind a s t r a i n free region into which deformation can then proceed, the migration i s e s s e n t i a l l y a recovery process a c t i n g as a prelude to r e c r y s t a l l i z a t i o n . . Therefore any co-operative process of shear and migration represents a hardening and recovery c y c l e . Very small amounts of migration and shear were observed i n cadmium at -95°C (^ V. = . 3 0 ) , where migration was observed more as a s l i g h t grain boundary corrugation and did not involve gross boundary movement. More d e t a i l e d studies w i l l have to be made to accurately determine the temperature dependence of these 22 processes..Dorn observed s u b s t a n t i a l shear i n p o l y c r y s t a l l i n e magnesium at +20°C (T = .32). The p o s s i b i l i t y e x i s t s therefore that the d u c t i l i t y H t r a n s i t i o n i n cadmium at -120°C i s associated with the cessation of grain boundary recovery.. I f d u c t i l e shear as opposed to intergranular fracture occurs because of boundary recovery then i t i s necessary that the a c t i v a t i o n energy associated with migration be a v a i l a b l e down, to T R = .26 . The a c t i v a t i o n energy c o n t r o l l i n g boundary m o b i l i t y i s not 95 known. However Winegard has shown that the a c t i v a t i o n energy associated with grain growth i n u l t r a pure metals can be r e l a t e d to the l i q u i d s e l f d i f f u s i o n energy. This i s usually i n the range from .1 e.v. to .k e.v. Although the d r i v i n g force i s somewhat d i f f e r e n t i n each case i t i s not expected that the act u a l rate c o n t r o l l i n g mechanism during boundary migration w i l l be d i f f e r e n t than that associated with grain growth. There:- fore i t i s probable that the energy required f o r boundary migration w i l l be s u f f i c i e n t l y low that some boundary i n s t a b i l i t y can occur i n cadmium .- 2k - o at temperatures down to -120 C. Boundary migration i n p o l y c r y s t a l l i n e zinc poses a d i f f i c u l t problem f o r the development of successful zinc a l l o y s with good creep r e s i s t a n c e . I t i s desirable to obtain as f i n e a grain si z e as possible i n order to improve strength and mechanical working c h a r a c t e r i s t i c s . With such a structure however i t i s then necessary to s t a b i l i z e the boundaries by a suitable a l l o y i n g technique. • o Since climb i s known to occur at temperatures above -30 C, i t i s probable that the d u c t i l e - b r i t t l e t r a n s i t i o n i s r e l a t e d to the ease with which climb can occur as a dynamic recovery mechanism to allow f o r the circumvention of points of stress concentration. This i n t e r p r e t a t i o n however does not explain the known grain si z e dependence of the t r a n s i t i o n temperature. This dependence on grain size may be explained i n terms of easier crack propagation i n large grainedrmaterial. I t may also be explained i n terms of the r e l a t i v e amount of material being recovered by the action of boundary migration . Cleavage cracks are usually nucleated at grain boundaries, twin i n t e r s e c t i o n s etc. where stress concentration occurs. I f these points of stress concentration can be removed by boundary migration then cleavage f r a c t u r e should not occur. I f i t i s assumed that the def ormation processes do not change s i g n i f i c a n t l y with grain s i z e , then the increased d u c t i l i t y of zinc with decreasing grain si z e at low temperatures may be r e l a t e d to the increased importance of grain boundary migration as a recovery mechanism. At any given value of s t r a i n and boundary migration rate, the amount of recovered material will:.increase as the grain si z e decreases. - 25 - 1.3,k R e c r y s t a l l i z a t i o n Grain boundary migration always preceded the introduction of new r e c r y s t a l l i z e d grains which had t h e i r o r i g i n on the.grain boundaries. A r e c r y s t a l l i z e d grain which subsequently deformed i s shown growing from the boundary i n Fibure 15(b). The growth of the new grain appears to have been stepped s i m i l a r to the migration of e x i s t i n g boundaries. F i g . 15(a) i l l u s t r a t e s the movement of twin boundaries i n t o the parent grain by a d i f f u s i o n process. This process i s d i s t i n c t from stress dependent twin growth which i s a shear transformation. The o r i g i n a l twin which formed i s s t i l l v i s i b l e due to surface d i s t o r t i o n . R e c r y s t a l l i z e d grains were never observed before maximum stress conditions, but were associated with the negative work hardening slope obtained at elevated temperatures above T^ = .kO'. At a l l temperatures studied, r e c r y s t a l l i z a t i o n never went to completion during t e s t i n g except i n the f i n a l necked area. At room temperature i n 20u zinc ?about 50$ of the specimen volume remained u n r e c r y s t a l l i z e d a f t e r 60% deformation. R e c r y s t a l l i z e d grains were r a r e l y observed i n 400u specimens at any temperature. This i s r e f l e c t e d by the smaller amount of d u c t i l i t y from maximum stress to f a i l u r e (Figure 9)• The lower d u c t i l i t y to maximum stress f o r 20n Zn and 25u Cd at high temperatures as opposed to that f o r 400u material i s merely a r e f l e c t i o n of more pronounced boundary migration and the e a r l i e r introduction of r e c r y s t a l l i z a t i o n i n the f i n e grained material. 1.3-5 - 27 - Maximum Stress-Variation'with Temperature and S t r a i n Rate S t o l o f f using 500u cadmium found a plateau i n ultimate t e n s i l e strength below about - l 6 0°C. -The r e s u l t s of the present work are shown i n Figure 16 and indicate that the occurrence- of t h i s plateau i s grain s i z e dependent. .Continuously increasing strength values can be obtained to lower temperatures with a f i n e grain s i z e . S i m i l a r r e l a t i o n s h i p s are observed f o r zinc at high temperatures (Fig. 1 7 ) . However with the onset of cleavage fr a c t u r e that accompanies decreasing temperature, there i s a gradual decrease i n maximum.stress which l e v e l s • o f f at a more or l e s s constant value when d u c t i l i t y becomes less than 1$, i n d i c a t i n g a temperature independent fra c t u r e process. The e f f e c t of s t r a i n rate on fra c t u r e stress i s shown i n Figures 18 and 19. I t i s seen from Figure 19 that i n zinc when frac t u r e occurs by cleavage as opposed to d u c t i l e shear, the macroscopic fracture -stress i s s t r a i n rate independent over the range of s t r a i n rates used. With cadmium ( F i g . 18) the s t r a i n rate dependence of maximum stress varies only s l i g h t l y above 77°K. .At 77°K where fracture i s completely i n t e r c r y s t a l l i n e , there i s v i r t u a l l y no s t r a i n rate dependence of maximum str e s s . - 28 - O J I 1 I I I I I I I I I I I L -160 -120 -80 -hO 0 +k0 +80 Temperature °C F i g . l 6 The temperature dependence of the maximum stress i n cadmium. o X •H ft ra ra <u u t 20 -p S.10 I J L -k "I 6 = 4.0 x 10 sec. • 20M'Zn. O k00n Zn. J L J L J L J L -16 o -120 -8o -ho o +ko +8o •Temperature °C F i g . 1.7 The temperature dependence of the maximum stress i n z i n c . -29 - •o 40 H ft CQ CQ 0) rl -P CQ V U -P a •H X 30 20 10 4.0 x 10" -1 ;_4 s e c « _ j _ 4.0 x 10_^ sec. ^ 4.0.x 10 sec.~ -200 -160 -120 -80 -4o +40 Temperature °C F i g . 18 The e f f e c t of s t r a i n rate on the maximum stress i n 25w cadmium. ft CQ CQ <u u -p CQ cu pi u -p 30 20 1.0 O - €'= 4.0 x 10_| sec."1 4.0 x 10 4.0 x 10" sec. _j_ sec. D u c t i l e shear f r a c t u r e Cleavage fracture -160 -120 -80 -40 0 -+4o +80 Temperature • °C. Fig.19 The e f f e c t of s t r a i n rate on the maximum stress i n 20p z i n c . - 30 - 1.4 DEFORMATION MODES IN.ZINC AND CADMIUM In I928 Von M i s e s 3 2 showed that i t was necessary f o r f i v e independent s l i p systems to be operative before a p o l y c r y s t a l l i n e material can undergo a general homogeneous s t r a i n by s l i p . In many systems however other mechanisms such as twinning 7bending and. kinking may occur thereby reducing the number of d i s t i n c t s l i p systems required. 1 .4 .1 S l i p The predominant s l i p system i n zinc and cadmium i s the basal system (0001) ^l^oy . I t has been a popular b e l i e f that i f and when non basal s l i p does occur i t w i l l do so much more r e a d i l y i n systems which have °/a r a t i o s equal to or l e s s than i d e a l . This argument follows from 33 r e l a t i v e close packing considerations. Seeger has suggested however that i n a d d i t i o n to the c / a r a t i o , the stacking f a u l t energy w i l l have a strong influence, on deformation modes i n as much as i t controls the cross s l i p process. S t o l o f f and D a v i e s 3 4 using hexagonal close packed Zn-Ag a l l o y s of s l i g h t l y varying a x i a l r a t i o s showed that the c / a r a t i o i s not the only c r i t e r i o n . f o r non basal s l i p . The possible pyramidal s l i p systems are indicated i n Figure 2 0 . Tables 2 and 3 show the r e s u l t s of previous investigations i n t o the non basal s l i p c h a r a c t e r i s t i c s of zinc and cadmium. . A l l studies were c a r r i e d out using single c r y s t a l s . Oilman 4 3 observed prismatic s l i p ^lOlO^ O-120/"- i n cadmium, but only at higher temperatures than those used i n the present i n v e s t i g a t i o n . The only non basal system that has been r e l i a b l y reported as being operative i n zinc i s the second order pyramidal £ l l 2 2 j < 1^12~3^ . - 31 - However both pyramidal systems [ l O l l j < 1120/" and £ l l22J <C 1123/> have been observed i n cadmium by Price using d i s l o c a t i o n free p l a t e l e t s . 4 0 4 1 This i s i n t e r e s t i n g i n that the f i r s t order pyramidal i s a possible cross . s l i p plane. .Alden using zinc and cadmium, single c r y s t a l s found that zinc has a. much higher net hardening rate than cadmium during a l t e r n a t i n g tension-compression t e s t s . This he interpreted to be due to the greater ease of cross s l i p i n cadmium. . However the 1st- order pyramidal system has never been observed during the deformation of bulk c r y s t a l s . On the other hand Price f a i l e d to observe any cross s l i p from basal to pyramidal planes. His pyramidal d i s l o c a t i o n s were a l l nucleated on the pyramidal planes. . Metallographic studies f o r - t h i s work indicated that non basal s l i p i s a common occurrence. A l l observations were-made using 400u material because of the d i f f i c u l t y i n r e s o l v i n g traces-on f i n e grained surfaces. Q u a l i t a t i v e l y i t appeared that the occurrence of non basal s l i p increased i n frequency with decreasing grain s i z e . .However any q u a n t i t a t i v e . r e s u l t s w i l l have to wait an extensive e l e c t r o n microscopy r e p l i c a study. 1.4.1 a) Zinc Non basal traces were much more prevalent at +20°C i n zinc than i n cadmium at an equivalent temperature. Figure 21 shows s l i p l i n e traces i n zinc over a series of s t r a i n s at +20°C. . Approximately•50$ of the grains i n zinc showed non basal traces at fr a c t u r e . These traces were u s u a l l y wavy and discontinuous i n nature at +20°C and d i d not completely traverse a grain. They would s t a r t near grain boundaries and gradually progress across the grain (Grain A, Fig.. 21). Many grains showed two d i f f e r e n t traces and a few three (Grain B). - 32 - (a) First-order pyram idal glide occurs when_dis- locations with a $[1210] Burgers vector move on (lOTl) planes, (b) Second- order pyramidal glide oc curs when dislocations_with a Burgers vector J[ll23] = c+a move on (1122) planes. (b) F i g . 20 Pyramidal g l i d e systems i n zinc and cadmium ( a f t e r Price ' ). TABLE 2 Non Basal S l i p Systems Observed i n Zinc Author Temperature Loading Non Basal System Remarks Rosenbaum 3 5 +20°C Bending {ll22} <112J> etch p i t t i n g on basal planes 3 6 B e l l and Cahn +20°C Tension 11 to basal planes [ l l 2 2 j <^1125/' s l i g h t departure from {ll22} - s l i p plane may be i r r a t i o n a l P r i c e 3 7 ' 3 8 ' 3 9 +20°C-^-150°C Tension {ll22]<1123> d i s l o c a t i o n f r ee p l a t e l e t s 4 2 Predvoditelev +20°C Compression 11 to c axis (1122} <1123> etch p i t studies Oilman +20°C-^ +150°C Compression 11 to basal planes {ll22}<1123> [1010]<1120> Prismatic only at elevated temperatures S t o f e l , Wood 4 4 and Clark 25°C and ^78°C Tension and Compression 11 to c axis {1122] <ii23 y rate of work hardening = 7.5 x. 10 5 psi.. decreases with decreasing temperature TABLE 3 Non Basal S l i p Systems Observed i n Cadmium Author' Temperature Loading Non Basal System Remarks S t o l o f f and Gensamer 4 7 +20°C -*- -196°C bending and compression £ll22j <1123> Single c r y s t a l s used P r i c e 4 0 ~ 4 1' +20°C -*- -150°C tension (1122] <1123> [ l O l l ] <11207 D i s l o c a t i o n free p l a t e l e t s [ l O l l j predominates at elevated temperatures G i l m a n 4 8 +150°C -»-2750C tension {lOlo} <1120> Only at elevated temperatures Wernick and Thomas 49 +25°C —- -150°C compression £1122} <1123> etch p i t studies (b) 4 .7 $ s t r a i n (c) 7 • ! io s t r a i n F i g . 21 £l_122] ^1123^ s l i p i n 400u zinc at various s t r a i n s . Temperature = +20°C Magnification X kQO - 37 - By a trace analysis technique s i m i l a r to that described by Reed-Hill and Baldwin, 6 5 a l l non basal traces were i d e n t i f i e d as o r i g i n a t i n g from £ll22^ <C 1123 s l i p . The technique can only be used i f at l e a s t 3 twin traces and the basal s l i p traces can be observed. Then by an appropriate measurement of angles i t i s possible to a r r i v e at a possible s o l u t i o n f o r the non basal traces. Because- of the s l i g h t ambiguity associated with t h i s procedure, i t was checked against X-Ray determination of the o r i e n t a t i o n f o r two selected grains showing non basal traces. In both cases, the two techniques indicated 2nd order pyramidal s l i p . Price observed that screw d i s l o c a t i o n s with a Burgers vector 3 {^ 11233 cross g l i d e d frequently between planes containing the ^ 1123^ ] d i r e c t i o n . This could explain the wavy nature of the traces at room temperature. With decreasing temperature the non basal traces became st r a i g h t e r and tended to concentrate i n t o bands, an observation made previously by S t o l o f f 4 7 i n cadmium and Gilman 4 3 i n z i n c . These bands may e a s i l y be mistaken f o r f i n e twins. However as seen i n Figure 22 they r e a d i l y i n t e r s e c t one another and were, removed by p o l i s h i n g . 1.4.1. b) Cadmium At 30°C and above non basal s l i p was l e s s extensive i n cadmium than i n z i n c . -With decreasing temperature i t became more prevalent as shown i n F i g . 23. These low. temperature markings were very s i m i l a r to those observed i n zinc at equivalent temperatures. As i n zinc-trace - 38 - analysis showed them to originate from { 1 1 2 2 } ^ 1 1 2 3 s l i p . In no case was f i r s t order pyramidal observed. The observation of extensive {1122} 0-123^ s l i p i n zinc and 86 cadmium i s not unexpected.•As reviewed by Dorn , t h i s system i s the only one which can promote extensive deformation p a r a l l e l to the c-axis. The contribution that twinning can make i s not s i g n i f i c a n t i n terms of the t o t a l s t r a i n . No combination of £000l]<1120>, £l010}<1120>, and{l01l}<1120> can provide the f i v e independent systems required f o r homogeneous deformation. However the operation of £L122^<1123> i s s u f f i c i e n t by i t s e l f i n providing the number of required systems. Table 4 shows the number of independent systems f o r each of the prominent s l i p systems . TABLE 4 S l i p Systems i n Hexagonal Metals No. S l i p System Burgers Vector Number of Independent Systems 1 (000l]O.120> a 2 2 [1010] CL120? a 2 3 £l011^  <1120> a 4 4 [1122] <1123> c + a 5 5 1 + 2 + 3 a 4 ( a f t e r Dorn ). twins 39 - cleavage crack X 240 (a) Non basal traces near fra c t u r e surface. twins X 240 (b) As above a f t e r p o l i s h i n g . F i g . 22 £ll22] ^ 1123> s l i p i n 400ju zinc at -196°C F i g . 23 ^1122J<1123> traces on UOOJJ cadmium deformed 7$ at -196°C. -.41 - 1.4.2 Twinning In both zinc and cadmium twinning occurs on the £l012^ planes i n the <1011^ d i r e c t i o n s . However s l i g h t differences occur i n the two systems i n the frequency of twinning. In the absence of general grain boundary migration the amount of twinning d i d not vary appreciably with temperature. As the temperature decreased the twins became f i n e r i n d i c a t i n g r e s t r i c t e d twin growth with decreasing temperature. A comparison of the deformation markings a f t e r 7$ s t r a i n at +20°C and -196°C in-400 u cadmium i s shown i n Figures 24 and 25. The amount of twinning under equivalent conditions of temperature and s t r a i n was s l i g h t l y l e s s i n zinc than i n cadmium. However Price observed that i n d i s l o c a t i o n free p l a t e l e t s , twins formed more r e a d i l y and grew to l a r g e r sizes i n zinc than they did i n cadmium. This he explained i n terms of the greater shear associated with twinning i n cadmium ( .171) as opposed to z i n c ( .139) a n c i "the p o s s i b i l i t y therefore of a more d i f f i c u l t process of twin nucleation i n cadmium. However one could e a s i l y argue that twins should nucleate and grow more r e a d i l y i n cadmium because they represent a more e f f e c t i v e deformation mechanism. The lower frequency of twinning i n zinc may a l s o be due to the fact that second order pyramidal s l i p occurs more r e a d i l y i n p o l y c r y s t a l l i n e zinc than i n cadmium and the amount of twinning needed to meet Von Mises' requirements i s therefore reduced. F i g . 2k Microstructure of 400JJ cadmium deformed 7$ at +20°C. - 43 - X 100 F i g . 25 Micro-structure of 400/4 cadmium deformed 7$ at -196°C -.kh - The amount of twinning i n both systems was always.governed by the r e l a t i v e temperature and s t r a i n . . In the region of room temperature with increasing boundary migration accompanying i n c r e a s i n g ' s t r a i n , twin formation became much l e s s frequent. Once d i s t i n c t r e c r y s t a l l i z a t i o n started twin formation was not observed. Therefore cadmium which had fractured at 20$ s t r a i n at -196°C always showed more twinning than cadmium deformed to 20$ s t r a i n at +20°C since at the l a t t e r temperature migration i s occurring. Migration a f f e c t s twinning due to i t s recovery e f f e c t on areas of stress concentration required f o r twin nucleation. Therefore true comparisons of r e l a t i v e twinning could only be made e i t h e r at low values of s t r a i n before extensive migration had started, or at low temperatures. On t h i s basis of comparison and keeping i n mind the greater tendency f o r migration and r e c r y s t a l l i z a t i o n with decreasing grain s i z e i t was observed that the amount of twinning did not vary s i g n i f i c a n t l y with'grain i .size .• . At temperatures above +20°C7where extensive migration occurs- very l i t t l e twinning occurred as shown i n F i g . 26. S t o l o f f observed l e s s basal s l i p i n twinned regions with a decrease i n temperature. This was not observed as indic a t e d i n Figure 27 which shows extensive twin basal s l i p at -196°C. The f i n e r nature of basal traces at low temperatures makes them more d i f f i c u l t to resolve. When twinning on one p a r t i c u l a r twin plane was stopped by a twin on another plane as shown i n Figure 27, extensive s l i p would occur i n basal planes of the l a t t e r and eventually cause twin nucleation on the opposite side of the twin. .This process of twin nucleation was a common occurrence e s p e c i a l l y i n cadmium. - +5 - X 260 F i g . 26 Lack of twinning i n the presence of boundary migration. ( 25ji cadmium deformed 10$ at +60°C.) X 120 Fig.27 Twin basal s l i p and twin nucleation. ( 400jj cadmium deformed 7$ at +20°C ) - .46 - 1.4.3 The Formation of Low Angle Boundaries during Deformation Low angle boundaries have been observed to form during the deformation of both single c r y s t a l s and p o l y c r y s t a l s i n many materials. Such boundaries have been r e f e r r e d to by a .variety of nomenclatures which has confused t h e i r nature of formation and. t h e i r importance as a deformation process. The formation of "kinks" i n single c r y s t a l s of cadmium was f i r s t observed and discussed by Orowan 5 4. Hess and B a r r e t t 5 2 , G i l man 5 5 5 6 and Washburn and P a r k e r 5 - ' ' 5 3 studied the nature of kinking i n zinc single c r y s t a l s . .Gilman distinguished between compression kinks and tension kinks, p o s t u l a t i n g that the l a t t e r form only due to c r y s t a l inhomogeneities. Compression kinks were further subdivided into "ortho" (formed, under low stress conditions) and"para" (formed at high stresses i n c r y s t a l s which have been extensively deformed). Compression ortho kinks were observed by Gilman i n which the kink planes were always perpendicular to surface basal, traces. They were observed to form i n zinc at temperatures down to -196°C i n d i c a t i n g that the process of t h e i r formation i s more l i k e l y one of stress induced d i s l o c a t i o n rearrangement on the basal planes-than one of d i s l o c a t i o n climb which i s thermally a c t i v a t e d . Boundary formation i n p o l y c r y s t a l s i s more complicated because of the nature of the stresses. I t has therefore been re f e r r e d to by a v a r i e t y of terms such as mosaic w a l l s , c e l l formation and non c r y s t a l l o  graphic boundary formation. G i f k i n s 5 0 reported the formation of " c e l l s " during the deformation of p o l y c r y s t a l l i n e zinc above 200°C. D o r n 2 1 2 2 studying magnesium observed temperature independent "non c r y s t a l l o g r a p h i c " -A7 - boundary formation which i n many cases crossed grain boundaries. He postulated that these boundaries formed because of the bending of the l a t t i c e associated with the non homogeneous deformation of underlying grains. The observations of the present work indicate that the formation of boundaries.is s i m i l a r i n degree and type i n both zinc and cadmium. "Non c r y s t a l l o g r a p h i c " boundaries s i m i l a r . t o those observed by Dorn are shown i n Figure 28 and 29. From Figure 28 i t i s a l s o observed that these boundaries can cross grain boundaries. On close examination of Figure 28 i t i s observed that some boundaries seem to be " c r y s t a l l o g r a p h i c " i n that they are perpendicular to the basal s l i p traces. S i m i l a r boundaries are shown i n Figure 30. These boundaries are s i m i l a r to the ortho kink planes of Gilman. They were distinguished by three d i s t i n c t features: 1) The misorientation of the b a s a l traces was always greater across the c r y s t a l l o g r a p h i c "kinks" than across non c r y s t a l l o g r a p h i c boundaries. 2) C r y s t a l l o g r a p h i c boundaries formed i n those grains which had the most prominent basal traces and few.if any twins. 3) The boundaries were sometimes observed to be composed of two or more smaller boundaries.(Figure 30) s i m i l a r to those of Washburn and Parker. I t would appear that the " c r y s t a l l o g r a p h i c kinks" from only under f a i r l y simple stress conditions such as ..bending or compression of the basal planes. The boundaries become non c r y s t a l l o g r a p h i c however where x4oo c r y s t a l l o g r a p h i c non " F i g . 28 Low angle boundaries i n cadmium deformed 7% a"t -30°C X 1.20 F i g . 29 The formation of non c r y s t a l l o g r a p h i c boundaries i n cadmium due to underlying small grains. ( deformed 15$ at +20°C.) - k9 - X240 (a) cadmium deformed 15$ at +20°C. X240 (b) zinc deformed 7$ at +20°C. Fig.30 Crystallographic boundary formation i n 400p Zn and Cd. -50 - boundary r e s t r a i n t s on a given grain become more complicated. The extensive non c r y s t a l l o g r a p h i c boundary formation of Figure 29 i s due to the r e s t r a i n t s imposed by the underlying.fine grain structure. 1.5 YIELD STRESS AND WORK HARDENING 1.5-1 The Temperature Dependence of Y i e l d The flow stress-temperature r e l a t i o n s h i p s found by S t o l o f f are shown i n Figure 31- He found that the y i e l d stress was independent of temperature below about -80°C and that the work hardening rate was constant over an increasing s t r a i n region as the temperature decreased. . The. r e s u l t s at k.2°K are somewhat i n doubt due to the d i f f e r e n t specimen geometry. The d e f i n i t i o n of y i e l d i n p o l y c r y s t a l l i n e zinc and cadmium i s d i f f i c u l t because of the gradual nature of the. y i e l d process". Therefore the y i e l d stress f o r the purpose of t h i s work was defined by an o f f s e t technique using .Yfo s t r a i n as the y i e l d s t r a i n . . In order to compare the temperature dependence of y i e l d i n zinc and cadmium, the y i e l d stresses were normalized i n each case by d i v i d i n g by the shear modulus G;. -Shear'modulus, values were "57 obtained from the tables of Koster . The y i e l d s t r e s s - e f f e c t i v e temperature r e l a t i o n s h i p s are shown i n Figure 32. I t i s seen that below the c r i t i c a l temperature where y i e l d i s completely thermally acti v a t e d the y i e l d stress appears to increase l i n e a r l y with decreasing temperature. The temperature dependence of y i e l d f o r both grain sizes i s s l i g h t l y greater f o r zinc than f o r cadmium. The normalized y i e l d stress i s also somewhat higher f o r zin c . .In 400u material the - 51 - 28 r 24 - -250 - 200 -150 -100 - 50 0 50 TEMPERATURE,°C Fig.51 Flow stress-temperature r e l a t i o n s h i p s f o r 0.020 inch grain diameter cadmium as found by S t o l o f f . O zinc Fig.32 .Temperature dependence of the y i e l d stress i n p o l y c r y s t a l l i n e zinc and cadmium - 53 - c r i t i c a l temperature (Tc) i s approximately the same f o r both zinc and cadmium(Tjj = .520). Tests could not be done on f i n e grained material above about Tg = .5 due to grain growth at these temperatures. I.5.2 Temperature S e n s i t i v i t y of the Flow Stress B u l l e n 5 8 6 0 using p o l y c r y s t a l l i n e copper.found a l i n e a r r e l a t i o n s h i p between the flow stress (at constant s t r a i n ) , and temperature over a range of temperature from h.2°K to q-50°K. He postulated that the hardening mechanism i n copper remained the same over the temperature range studied but that the lower flow stress values obtained with increasing temperatures were due to e i t h e r a. d i f f e r e n t rate of obstacle formation with s t r a i n , or to a temperature dependent dynamic recovery process which tended to remove obstacles once formed. On the other hand R u s s e l l 4 , again using p o l y c r y s t a l l i n e copper found that the work hardening rate was linear,- and temperature i n s e n s i t i v e below a c e r t a i n c r i t i c a l temperature. The amount of s t r a i n involved with t h i s constant l i n e a r hardening was a l s o a function of temperature, and increased with decreasing temperature. His r e s u l t s are shown i n Figure 33- S t o l o f f (Figure 31) a l s o indicated a l i n e a r and temperature i n s e n s i t i v e work hardening rate f o r cadmium at low temperatures where the flow stress i t s e l f was not a function of temperature. I.5.2 a) Cadmium Flow stress-temperature r e l a t i o n s f o r 25^. and hOOyx cadmium obtained during t h i s study are shown i n Figures 3^ a n o - 35-- 5+ - Fig-33 The temperature dependence of the s t r a i n hardening parameter of p o l y c r y s t a l l i n e copper as found by Russell" . F i g . $4 .Flow stress-temperature r e l a t i o n s h i p s f o r 25p cadmium. - 56 - . "jo s t r a i n -200 -160 -120 • -80 -40 0 +k0 +80 Temperature °C F i g . 35 .Flow stress-temperature r e l a t i o n s h i p s f o r 400p cadmium. - 57 - I t i s seen that as•opposed to S t o l o f f ' s r e s u l t s , the flow stress continues to increase with decreasing temperature down to -1^6°C. U n t i l t e s t s can be performed below -196°C i t i s not known whether t h i s trend continues. In order to obtain a more d i r e c t comparison of the stress s t r a i n r e l a t i o n s h i p s , the y i e l d stress was subtracted from the flow-' stress f o r each point g i v i n g p l o t s of the work hardening parameter ( °"flow ~ ^.1$ s t r a i n ) vs.temperature." These are shown i n Figures 36 and 37- . I t I s seen that i n both cases, a .region of temperature independent work hardening develops below -120°C s i m i l a r to that observed by R u s s e l l . .With increasing temperature the amount of s t r a i n involved decreases. The temperature independent work hardening region covers, the same temperature range f o r both grain s i z e s . However la r g e r amounts of s t r a i n i n hOOyx material show temperature independent work hardening than that found, f o r 25u ( i . e . l4fo s t r a i n at -196°C f o r 400u and 7$ f o r 25u). 1.5.2 b). Zinc Because of the l i m i t e d d u c t i l i t y of ,400u z i n c , no s i m i l a r evaluation could be made. However the r e s u l t s obtained f o r 20u zinc are shown i n Figures 38 and 39- Temperature independent work hardening i s observed below approximately -95°C• Table k shows a comparison between zinc and cadmium of the maximum temperature f o r t h i s region at the a r b i t r a r y value of 1$ s t r a i n . . I t i s seen that temperature independent work hardening occurs below a common equivalent temperature of Tg = .26 i n both systems. - 58 - F i g . 56 The v a r i a t i o n with temperature of the work hardening parameter i n 25/j cadmium. - 59 - F i g . 3 7 The v a r i a t i o n with temperature of the work hardening parameter i n k-00[l cadmium. - 60 - 20 H x 16 ID ft w to <D SH -P w H 12 • A A^ \ A c K " \ ' v XA . " \ ^ \ -A symbol $ s t r a i n o \ A \Q A V 10 ° \ » \ ^ \ ^ A • 5 ^ - - o A ,3 ^ ^ o _ A . . 2 _k x • 1 C = 4.0 x 10' sec. O >.(.5) I I I I I I I I I I I I L_ -160 .. . . -120 . -80 -4o ... o . +4o : :, +8o Temperature °C F i g . 38 Flow stress-temperature r e l a t i o n s h i p s f o r 20j_l z i n c . 16 ^ 12 1 I i 8 ft b 1 8 H 4 h —A—A—A $> s t r a i n -m 7 • 5 - A 3 A 2 • 1 -O .5 -160 -120 -80 -40 0 .Temperature °C +40 +80 Fig.3 9 The v a r i a t i o n with temperature of the work hardening parameter i n 20jj z i n c . - 61 - The comparison of the work hardening behaviour of zinc and cadmium.in the temperature i n s e n s i t i v e hardening' region i s shown i n Figure 40. Taking i n t o account the shear•modulus of each system, i t i s seen that the work hardening of f i n e grained zinc and cadmium i s i d e n t i c a l . Except f or a parabolic region below 1% s t r a i n , the hardening i s also l i n e a r . TABLE 5 Upper temperature l i m i t s for, l i n e a r hardening i n p o l y c r y s t a l l i n e zinc and cadmium Zinc Cadmium . Maximum Temperature for' l i n e a r . hardening -95°C -120°C Maximum Equivalent Temperature T .26 .. .26 1.5-3 Strain Rate S e n s i t i v i t y of the Flow Stress The e f f e c t of s t r a i n rate on the flow stress was . investigated for 20u Zn and 25u Cd over the temperature.range from - I 9 6 to +20°C.' Tests were not done with 400u material because of the poorer r e p r o d u c i b i l i t y of flow stress values. Five s t r a i n rates between 4.0 x 10 3 sec 1 and 4.0 x 10 5 sec 1 were used. Linear r e l a t i o n s h i p s were obtained f o r a l l test conditions between flow stress at constant s t r a i n , and the na t u r a l logarithm of the s t r a i n rate as i l l u s t r a t e d f o r 20u zinc i n Figure 4 l . A C T was chosen as a s t r a i n rate parameter and values obtained as a Aln£* function of s t r a i n are shown i n Figures 42 and 4j for cadmium and zinc r e s p e c t i v e l y . - 62 - 1 2 3 + 5 6 i ' °jo s t r a i n F i g . hO Linear hardening of p o l y c r y s t a l l i n e zinc and cadmium below TTJ=.26'1 - 6k - k 8 - .12 16 •% s t r a i n Fig-42 .The v a r i a t i o n of the s t r a i n rate parameter AO" with A l n £ " temperature and s t r a i n i n 25p cadmium. i6oo - 65 - • o . . . . -120°C • ... . -105°C A . . . . -95°C • • ' • . -6o°c • . . . . -30°c • • . . .  +20°C 12 $ s t r a i n 16 20 Fig.^ 3 The v a r i a t i o n of the s t r a i n rate parameter ACT A l n g ' with temperature and s t r a i n i n 20(j z i n c . - 66 - From Figure 42 f o r 25u cadmium i t i s seen that at -]_96°C there i s an almost constant value of s t r a i n rate s e n s i t i v i t y up to about f$. s t r a i n above which t h e . s e n s i t i v i t y increases. This implies a s t r a i n rate i n s e n s i t i v e rate of work hardening i n t h i s region. This region of s t r a i n at -196°C i s approximately the same that showed temperature i n s e n s i t i v i t y of the hardening rate. Between -95°C and -196°C there i s a continual increase of s e n s i t i v i t y with increasing temperature and s t r a i n . At about -60°C and above however i t i s seen that, at high values of s t r a i n the increase i n s e n s i t i v i t y with s t r a i n i s much l e s s . I t i s i n t h i s region that grain boundary migration i s known to occur to a s i g n i f i c a n t degree. The r e s u l t s f o r zinc are q u a l i t a t i v e l y the same ( F i g . 45). Below -95°C "the s t r a i n rate s e n s i t i v i t y does not vary with s t r a i n . This again i s the same temperature region which gives a temperature independent work hardening rate. 1.5.4 The Deformation of Cadmium Single C r y s t a l s R u s s e l l has proposed that the deviations which occur from l i n e a r hardening at high values of s t r a i n or with increased temperature i n copper are due to a dynamic recovery mechanism in v o l v i n g cross s l i p . He has associated the i n i t i a l l i n e a r hardening region of p o l y c r y s t a l s with second stage single c r y s t a l hardening. Therefore i n order to have a b e t t e r understanding of the deformation behaviour of. cadmium i t was decided to undertake some study of single c r y s t a l behaviour. - 67 - • Resolved shear stress-shear s t r a i n curves f o r cadmium single c r y s t a l s of i d e n t i c a l o r i e n t a t i o n s are shown i n Fig s , hk and h^. The o r i e n t a t i o n used was as shown i n F i g . hh with the angle between t e n s i l e axis and s l i p plane and t e n s i l e axis and s l i p d i r e c t i o n being 36 0 and 38° r e s p e c t i v e l y . At +20°C a three stage hardening curve was obtained (Fig.U5) i n which the i n i t i a l stage i s subdivided i n t o two regions s i m i l a r to that observed by See g e r ^ i n zinc at room temperature. He also postulated that the t r a n s i t i o n from Stage I to Stage I I hardening i s due to the establishment of a c r i t i c a l , density of immobile d i s l o c a t i o n loops due to vacancy condensation. At -50°C and below the nature of the curves i s somewhat d i f f e r e n t ( F i g . kk\. Stage I was l i n e a r at a l l temperatures down to -196°C and d i d not show the "S" "type of hardening observed i n magnesium^. Of about 20 c r y s t a l s tested a l l showed extensive twinning during Stage I I . A l l possible care was taken i n order to avoid c r y s t a l damage p r i o r to t e s t i n g and no observable twins were present. Therefore at low temperatures i t must 89 be concluded that twinning i s a general feature of Stage I I hardening. L a l l y has reached a s i m i l a r .conclusion with respect to magnesium. An i n t e r e s t i n g observation i s that the resolved shear stress on the b a s a l plane associated with i n i t i a l twin formation was v i r t u a l l y independent of temperature as shown i n Table 6. The shear s t r a i n increased with i n c r e a s i n g temperature due to the lower rate of work hardening. Although the maximum rate of hardening during Stage II i s a function of the nature- and extent of twinning, the t r a n s i t i o n from Stage I to Stage I I cannot be associated with twinning. This follows from the experimental observation that a considerable proportion of the c r y s t a l remains untwinned i n the t r a n s i t i o n region. The flow stress i s derived - 70 - therefore only from the nature of the d i s l o c a t i o n configuration i n untwinned regions. Due to the above reasoning .i t i s more accurate to say that i t i s the stress associated with the end of Stage I ffb.iffih.lis,. independeht'npf, - temperature .Twinning therefore i s an " a f t e r the fact" consideration 1; 10U Somewhat s i m i l a r observations were made by Lucke .eit. al.i;v during the deformation of zinc single c r y s t a l s . They found that the stress associated with the end of Stage I hardening i s independent of s t r a i n rate at +20°C. The rela t e d shear s t r a i n increased with decreased s t r a i n f a t e s . TABLE 6 Basal shear stress required f o r i n i t i a l twin formation. Temperature Shear stress on basal plane gm/mm2 Shear s t r a i n +20°C 260 550 -50°C 280 2^0 -78°C 280 . 190 -120°C 285 1^5 -196°C 300 130 Wo attempt was made to calculate the macroscopic stress on the twinning system.. I t i s known 3 9 that such c a l c u l a t i o n s produce values that are at l e a s t an order of magnitude lower than the stress thought to be required f o r twin n u c l e a t i o n . l i t i s impossible to estimate with any degree of accuracy the stress at points of stress concentration which i s required -.71 - f o r twin nucleation i n bulk c r y s t a l s . .However i f one assumes a .constant r e l a t i o n s h i p between macroscopic shear stress and the value of stress at points of stress concentration i t i s then possible to p r e d i c t a temperature independent twinning s t r e s s . 1.5.5 Temperature Dependence of Work Hardening The temperature dependence of the rate of work' hardening i n Stage I (&xj and Stage II ^Qgjj i s shown i n Figures 4 6 and 4 7 - I t i s seen that below -120°C the work hardening rates are constant. Above t h i s temperature f o r both hardening regions the rate of hardening decreases. Fahrenhorst and Schmid 6 2, and Seeger and T r a u b l e 2 5 have found s i m i l a r r e l a t i o n s h i p s f o r Stage I.hardening of zi n c . The data of Seeger a f t e r normalizing f o r shear modulus changes i s compared with t h a t obtained during t h i s study of cadmium i n Figure 4 8 . I t i s seen that the drop i n the hardening rates occurs at the same equivalent temperature of Tg =. .26 i n bo£h systems. This i s the same temperature below which temperature and s t r a i n rate i n s e n s i t i v e hardening began i n p o l y c r y s t a l l i n e zinc and cadmium. I f t h i s decrease i n hardening' rate i s associated with some dynamic recovery mechanism i t would appear that such a mechanism, i s s i m i l a r i n both p o l y c r y s t a l s and single c r y s t a l s . • From Figure 4 8 i t i s also noted that t h i s work hardening t r a n s i t i o n region has an upper temperature l i m i t of about Tg = - . 4 0 i n both systems. 1.5.6 The Grain Size Dependence.of Hardening at -196°C 2 Clarebrcugh and Hargreaves have attempted an analysis of the s i m i l a r i t i e s of Stage II hardening of .f.c.c. single c r y s t a l s and the i n i t i a l deformation c h a r a c t e r i s t i c s of p o l y c r y s t a l s . This was based to a large - 72 - 3 extent on the r e s u l t s of Feltham and Meakih who observed a l i n e a r hardening region during the e a r l y regions of s t r a i n associated with the deformation of p o l y c r y s t a l l i n e copper. They also observed that the magnitude of the p o l y c r y s t a l l i n e l i n e a r hardening rate was comparable to that of Stage II hardening. There was therefore no appreciable e f f e c t of grain s i z e on the k work hardening rate. S i m i l a r conclusions, were reached by R u s s e l l who also correlated the stress, at the end of p o l y c r y s t a l l i n e hardening to that at the end of Stage I I single c r y s t a l hardening. Parabolic, hardening as opposed to l i n e a r hardening, occurs i n f . c . c . p o l y c r y s t a l s therefore, due to the action of cross s l i p i n a s i m i l a r manner to single c r y s t a l s . In order to t e s t t h i s concept, cadmium of 5 grain sizes was tested at -196°C. Since the specimen dimensions were held constant i t was thought that any consistent trend towards single c r y s t a l data.might be observed. Due to the extensive twinning during 2nd stage single c r y s t a l deformation at -196°C the meaning of the resolved shear stress- on the basal planes i s clouded. Therefore for comparison with p o l y c r y s t a l l i n e material i t was decided to merely report the single c r y s t a l Stage II hardening i n terms of the t e n s i l e hardening rate. This also avoids the d i f f i c u l t y associated with assigning some average shear stress value f o r p o l y c r y s t a l s . The p o l y c r y s t a l l i n e stress s t r a i n curves obtained are shown i n Figure ^9- I t i s seen that with increasing grain size a two stage. hardening curve gradually appears. Since i t was desired to compare the 2nd•stage hardening, the maximum hardening.rate was used i n a l l cases as shown. With decreasing grain size the region of l i n e a r hardening decreased to smaller - 73 - values of s t r a i n . The comparison of hardening rates i s made by a d~ 2 p l o t as shown _ J L i n F i g . 50- I t i s seen.that the rate of hardening varies l i n e a r l y with d 2 and extrapolates to the sin g l e c r y s t a l value at d 2= 0. There i s an increase by a f a c t o r of about J i n the hardening rate from the single c r y s t a l value to 25u material. - Since a l l c r y s t a l s are twinned to about the same degree, t h i s increase r e f l e c t s the nature of the hardening change with increasing grain boundary area per unit volume and decreasing proportion of grains with a free surface. Since no pyramidal s l i p was observed i n single crystals;, and an increasing amount occurs i n p o l y c r y s t a l s as the grain s i z e decreases, the increased hardening (rate can be explained i n terms of a gradual change i n the nature and extent of the deformation mechanisms. However i n copper single c r y s t a l and p o l y c r y s t a l hardening rates can be compared d i r e c t l y because the deformation mechanisms do not change with grain s i z e . Temperature f C g. ^ 7 The temperature dependence of stage I I hardening i n cadmium. - 75 - F i g . 48 Temperature dependence of the r a t e of work hardening during Stage I deformation F i g . 5° The e f f e c t of gr a i n size on the rate of l i n e a r hardening of cadmium at -196°C. - 78 - PART II 2. MECHANISMS OF HARDENING IM ZINC AND CADMIUM 2.1 INTRODUCTION Many techniques have been used i n recent years _in an-effort to / evaluate the hardening mechanisms that c o n t r o l deformation. Some of the more prominent include the use of transmission electron microscopy,.the study of Cot t r e l l - S t o k e s Law obeyance, and. the a p p l i c a t i o n of rate theory to determine the rate c o n t r o l l i n g mechanisms. A l l have t h e i r l i m i t a t i o n s depending upon experimental procedures and t h e o r e t i c a l assumptions. •Although transmission microscopy techniques have proven to be valuable i n observing d i s l o c a t i o n motion and behaviour, considerable d i f f i c u l t y i s encountered i n preparing specimens which t r u l y represent bulk samples Several authors have made d e t a i l e d studies, of the C o t t r e l l - Stokes • Law 6' 1 1 1 3 ; ' 6 8 7 2 , However there i s considerable controversy as- to the exact meaning of Cottrell-Stokes.obeyance. The a p p l i c a t i o n of rate theory to deformation processes .has been 1 1 9 73~* 7 8 plagued by a multitude of formulations a l l of which .require c e r t a i n s i m p l i f y i n g assumptions to a r r i v e at mathematical expressions which may be. e a s i l y used to i n t e r p r e t experimental data. 7 9 Seeger o r i g i n a l l y postulated.that the applied stress could be considered as the sum of two components such that where "X ra = rG + r* . . . . . . . . . . . . . . . . . ( D i s associated with short range obstacles which can be - 7 9 - overcome with the a i d of thermal energy ( forest d i s l o c a t i o n s ) . . Therefore i t maybe re f e r r e d to as the thermal component of the applied s t r e s s . and i s the athermal component of stress which a r i s e s due -to long range e l a s t i c i n t e r a c t i o n s such as those between p a r a l l e l g l i d e d i s l o c a t i o n s at distances large compared with "b". Such obstacles cannot be overcome with the a i d of thermal energy. .The temperature dependence of y i e l d as postulated by Seeger when the mechanism of y i e l d does not change with temperature, i s shown i n F i g . 51 The i n t e r n a l stress Tn v a r i e s with temperature only through a change i n the shear modulus. The increase i n T a below the c r i t i c a l temperature r e f l e c t s the decrease i n the amount of thermal energy a v a i l a b l e and subsequently the increased e f f e c t i v e stress necessary f o r activation.- Y i e l d stress 7* 1 k c Temperature F i g . 51 The temperature dependence of y i e l d i n terms of the stress components ( a f t e r Seeger). Much of the work i n deformation i n recent years has been concerned with obtaining a better knowledge of the nature and o r i g i n of the stress components i n various systems. - .80 - 1 7 Basinski postulated that because of., Cottrell-Stokes obeyance i n c e r t a i n f . c . c . metals,, the two components ^ and J*~ a r i s e from a single 6 6 source ( the i n t e r a c t i o n of glide and f o r e s t d i s l o c a t i o n s ). Seeger has since modified h i s o r i g i n a l d e f i n i t i o n of 7G to include a short range e l a s t i c i n t e r a c t i o n term . However he maintains that a major po r t i o n of the flow stress of s i n g l e c r y s t a l s i s s t i l l derived from longe range int e r a c t i o n s between p a r a l l e l d i s l o c a t i o n s (Fig.52 ). Shear s t r a i n J F i g . 52 The components of the t o t a l flow stress in. f . c . c . c r y s t a l s . 3g l,= c o n t r i b u t i o n of the long range i n t e r n a l stress Tc/^= e l a s t i c i n t e r a c t i o n between g l i d e and f o r e s t d i s l o c a t i o n s T * = thermal component of the stress ( e f f e c t i v e stress); In the only consistent study to date, Mitra and Dorn have with the a i d of rate theory separated the two components of the athermal stress i n 5) BO aluminum and copper single c r y s t a l s . .It would appear, that short range e l a s t i c stresses as proposed by Basinski account .for,a ;'greater proportion of the - 81 - t o t a l flow stress than has been indicated by Seeger. In 195^  C o t t r e l l and Stokes using aluminum sin g l e c r y s t a l s found that the r e v e r s i b l e change i n flow stress ( A T ) during a. temperature change t e s t was d i r e c t l y proportional t o the t o t a l flow stress and that the value of AO"* w a s not only independent of s t r a i n but also of the p r i o r thermal and 7 mechanical h i s t o r y . C o t t r e l l - S t o k e s obeyance occurs therefore when with increasing s t r a i n A T = QTJ ~ OTI. = 1 - 3T> = a constant (2) * %, Tfr, C o t t r e l l a l s o recognized that when two i d e n t i c a l specimens are deformed to the same s t r a i n at d i f f e r e n t temperatures, the t o t a l flow stress d i f f e r e n c e i s made up of a r e v e r s i b l e component due to the d i f f e r e n t amount of thermal energy a v a i l a b l e and an i r r e v e r s i b l e component due to the d i f f e r e n t d i s l o c a t i o n configurations produced at the d i f f e r e n t temperatures ( F i g . 53 ). S t r a i n Components of the dif f e r e n c e i n flow stress when two i d e n t i c a l specimens are deformed at d i f f e r e n t temperatures. - 82 - 6 9 C o t t r e l l - S t o k e s obeyance has been interpreted. to mean that the type of d i s l o c a t i o n configuration must remain constant during deformation with only the scale changing. Obeyance has a l s o been shown to require that a constant p r o p o r t i o n a l i t y e x i s t s between T/Q and *J during deformation. T8 Basinski showed that a C o t t r e l l - S t o k e s t e s t could be carried., out at a constant temperature by p e r i o d i c a l l y varying the s t r a i n rate and that such .tests might be more accurate i n that they eliminate some of the d i f f i c u l t i e s associated with temperature change t e s t s ( the necessity of stopping the t e s t to change temperature and the d i f f i c u l t y i n d e f i n i n g flow stress due to y i e l d point e f f e c t s • ) . .The o r i g i n of 0" the thermal component may be due to a number of processes a l l of which can be thermally a c t i v a t e d . Therefore rate theory has been used i n an attempt to e s t a b l i s h the rate c o n t r o l l i n g process f o r various systems.- Since 7 * i s the stress associated with thermal a c t i v a t i o n , studies of the s t r a i n rate and temperature dependence of can be c a r r i e d out i n order to determine such rate parameters as a c t i v a t i o n energy, a c t i v a t i o n volume and a c t i v a t i o n distance. By comparing experimental values with those t h e o r e t i c a l l y predicted, i t i s sometimespossible to postulate the rate c o n t r o l l i n g mechanism. Mechanisms that can be thermally a c t i v a t e d and there fore rate c o n t r o l l i n g include the following: •1) cross s l i p 2) f o r e s t i n t e r s e c t i o n 3) the non conservative motion of jogs i n screw, d i s l o c a t i o n s k) climb 5) overcoming of the. E e i e r l s stress - 83 I f a si n g l e a c t i v a t i o n process i s rate c o n t r o l l i n g over a c e r t a i n temperature range then the s t r a i n rate associated with deformation may be expressed as • w - A G/kT 6 = uc,e (3) - A G / k T j[ = NAb V e (4) where N = number of s i t e s per u n i t volume where a c t i v a t i o n occurs A = area swept out per successful a c t i v a t i o n event b = Burgers vector V = frequency with which b a r r i e r i s attempted G = Gibbs free energy of a c t i v a t i o n The development of equation 5 to give u s e f u l mathematical expressions i s outlined i n Appendix I..The r e l a t i o n s h i p s to be used i n the present work include the following: . A c t i v a t i o n volume v = b d l (5) = kT I £ In j/L \ (6) *-2kT / A In 6 /g 0 \ (7) l ACT* ! T A c t i v a t i o n enthalpy A H = - k ^ / (j In t/j0 \ I c)"J* \ (8) a7* h I 6T /ty. o - 8k - Thermal component of the a c t i v a t i o n energy- AG = AH - T A s (10) = A H + - L * - 4 .Tv . . (11) . Apparent a c t i v a t i o n energy = AG = AG + v (12) where d = a c t i v a t i o n distance 1 = d i s l o c a t i o n length involved i n thermal a c t i v a t i o n • J( = shear s t r a i n rate <5 = t e n s i l e s t r a i n rate T = shear stress CT = .tensile .. stress u = ..shear modulus AS = entropy change during thermal a c t i v a t i o n I t has been assumed that the shear stress i n : p o l y c r y s t a l s can be approximated by taking one-half of the t e n s i l e s t r e s s . .Such an approximation w i l l not a f f e c t the calculated values of AH, AG, • or AG since the O conversion must be made i n both numerator and denominator. I t w i l l however a f f e c t the values of the a c t i v a t i o n volume i n that ^f* appears in. the denominator of expression .6 . This w i l l be discussed i n more d e t a i l l a t e r . - 8 5 - 2 . 2 , .TEMPERATURE CHANGS TESTS Temperature change t e s t s were undertaken not only to check the v a l i d i t y of the Cot t r e l l - S t o k e s law but also , t o obtain a better, knowledge of the e f f e c t of p r e s t r a i n at elevated, temperatures on the subsequent def ormation behaviour at low.temperatures. 2 . 2 . 1 . Procedure In cadmium -196°C was used as a base temperature while.the upper c y c l i n g temperature varied from - l40°C to -30°C .-Strain increments between 1.5$ and .2.O/0 were used at each temperature. A f t e r deformation at the upper temperature, the specimens were cooled to -196°C as r a p i d l y as possible i n order to minimize recovery e f f e c t s . This cooling could u s u a l l y be accom p l i s h e d within 30 seconds. During the temperature change operation, the load was maintained at about 20% of the flow s t r e s s . Since s t a t i c recovery i s n e g l i g i b l e at -196°C, the specimens were held f o r 15 minutes p r i o r to resumption of t e s t i n g i n order to e q u i l i b r a t e the t e s t i n g device. A.O" values could not be obtained during an increase i n temperature due to recovery during the time necessary.for temperature e q u i l i b r a t i o n . Because of the l i m i t e d d u c t i l i t y of zinc , the Co t t r e l l - S t o k e s law f o r temperature change t e s t s could not be investigated. Therefore t e s t s were l i m i t e d to p r e s t r a i n i n g 20u zinc to a given value of s t r a i n at some elevated temperature between +20°C and -95°C and subsequently deforming the specimen to frac t u r e at -120°C. 2 . 2 . 2 . C o t t r e l l - S t o k e s Tests For the various temperature change t e s t s on 25u and kOOp cadmium, the Ac"* values obtained were corrected to take i n t o account the change i n - 86 - ACT due to the temperature dependence of the shear modulus kShear modulus 57 values were obtained from the work of Koster The A,cr values were then normalized to Ao~ i n order, to give A T values of the r e v e r s i b l e change i n flow stress per °K..When.these are exam ined .in terms of the flow stress at the standard temperature of -196°C, Cot t r e l l - S t o k e s p l o t s as shown i n F i g . 5^ 4 a r e obtained. It i s observed from F i g . 54 that the C o t t r e l l - S t o k e s law i s not s t r i c t l y obeyed f o r temperature change t e s t s . . Ao~* values decrease c r 7 7 A^T s l i g h t l y during.the e a r l y stages of deformation and increase again at higher- values o f Oyy. This i s true f o r both grain s i z e s . The dotted l i n e s indicate i d e a l Cottrell-Stokes, obeyance. From F i g . 54 i t i s also noted that the experimental value of A_o~* at a given value of flow stress at -196°C i s independent- of the A T upper c y c l i n g temperature. This i s true f o r both grain sizes although the values of A 0"* a r e considerably lower f o r ,40Qu material. CT77 A T The r e s u l t s of the temperature change -tests are therefore very s i m i l a r to those reported by Bullen et a l f o r p o l y c r y s t a l l i n e copper deform ed between 173°K and 373°K and subsequently, deformed at Y ^ K ^ 5 8 ' 5 9 ' 6 0 ^ They noted a deviation from.ideal Cottrell.-Stokes behaviour at high values of stress which gave increased values of Ao~* . They also found values o> 7 A T of A C T which at a given value of stress at 77°K were independent of the A T upper c y c l i n g temperature between 173 °K and 37-5 °K. They therefore concluded that the same sequence o f o b s t a c l e " formation occurs during deformation independent of the temperature but that the rate of obstacle production with increasing s t r a i n may be temperature dependent due to, the removal of obstacles by some process of dynamic recovery. They made no attempt to i d e n t i f y the Symbol Temp, cycle °C O (-140..-196) • (-120..-196) D (-95-..-196) A (-60. ..-196) A (-30...-196) -k -1 £ = k.O x 10 sec. o / • 2514 cadmium 400^1 cadmium Broken l i n e s indicate C o t t r e l l - S t o k e s obeyance. 1.0 2.0 3.0 k .True stress at -196°C p . s . i . x 10 The r e l a t i o n s h i p between the change i n flow stress per °K and the resultant stress at -196°C 1 obtained during the temperature c y c l i n g of. cadmium. 2i 0 0 ^ 1 ' " " - 8 8 - "obstacles" but postulated thaf'recovery was associated with the a n n i h i l a t i o n and.rearrangement of d i s l o c a t i o n s by the ac t i o n of point defects. A * o~ , only GT A T a rather narrow range of grain sizes was used. I t also appeared that there was no e f f e c t of preferred o r i e n t a t i o n . 2,2.3. The Mechanical Equation of State I f the sequence of events occurring during deformation i s the same at any two temperatures, then one would expect that the i r r e v e r s i b l e compon ent of the flow stress d i f f e r e n c e as shown i n F i g . 53- would be zero. The d i f f  erence i n flow stress at a given s t r a i n at two d i f f e r e n t temperatures i s therefore due only to a differ e n c e i n o~ .. Under such conditions i t i s ex pected that a mechanical equation- of state might be v a l i d . The flow stress can then be expressed as a unique function of the instantaneous value o f the s t r a i n , s t r a i n rate and temperature and i s independent of the p r i o r s t r a i n 8 1 h i s t o r y . Therefore cr = a(6,6 T) Bullen i n f a c t d i d observe that during temperature c y c l i n g below approximately 300°K there was always an 'Incubation s t r a i n " during which the i r r e v e r s i b l e component of the flow stress difference as shown i n F i g . 53 was zero. His r e s u l t s f o r various temperature cycles are shown i n Fig.55- The incubation s t r a i n s taken from h i s r e s u l t s are shown i n Table 7 . I t i s observed that the magnitude of the incubation s t r a i n increased as the upper " cy c l i n g temperature decreased. In t h i s region of s t r a i n therefore, the d i s  l o c a t i o n configuration at a given value of s t r a i n i s independent of temper ature which leads to temperature i n s e n s i t i v e work hardening and the obeyance - 89 - 20 30 PRE-STRAIN 1% F i g . 55 The e f f e c t of elevated temperature p r e s t r a i n i n g on the s t r e s s - s t r a i n curve of p o l y c r y s t a l l i n e copper at 7 7°K ( a f t e r B u l l e n 5 8 ) . TABLE 7 Incubation s t r a i n required- i n p o l y c r y s t a l l i n e - copper p r i o r to the appearance of an i r r e v e r s i b l e component of the difference i n flow stress Lower c y c l i n g Upper c y c l i n g Incubation temperature temperature s t r a i n 77°K 77°K 77°K 173°K 233 °K 293 °K 9$ 5% I t - 90 - of"the mechanical equation of state. The i n s e n s i t i v i t y of the hardening rate to temperature and the increase in,..incubation s t r a i n witbJdecre.asingi,temper- ature are i n q u a l i t a t i v e agreement with jthe' results, of R u s s e l l although the temperature range of i n s e n s i t i v i t y as found by R u s s e l l extended to somewhat higher temperatures than those found by Bullen. 2 . 2 . J . a ) Cadmium F i g . 56 i l l u s t r a t e s the flow stress c y c l i n g obtained when 25u cadmium i s deformed a l t e r n a t e l y at - l40°C and -196°C. For comparison the n normal s t r e s s - s t r a i n curves at the two temperatures are also indicated. I t i s seen that i n the e a r l y regions of s t r a i n where dynamic recovery does not occur at e i t h e r of the temperatures involved,, the t o t a l flow stress d i f f - erence i s due e n t i r e l y to the d i f f e r e n c e i n o~ . The s l i g h t y i e l d points obtained on reloading at -196°C are due to an unloading e f f e c t as described i n Appendix II and can be ignored. As deformation proceeded i n t o the region of dynamic recovery, the r e v e r s i b l e flow stress difference could not account f o r t h e ' t o t a l d i f f  erence i n the stress at a given value of s t r a i n . As the upper c y c l i n g temp erature was increased above -120°C, i r r e v e r s i b l e components of the flow stress d i f f e r e n c e were evident immediately a f t e r y i e l d i n g . 2.2.3 . "b) Zinc The r e s u l t s of the p r e s t r a i n t e s t s on 20u zinc are shown i n F i g s . 57 and 58. F i g . % i l l u s t r a t e s the e f f e c t of p r e s t r a i n i n g at -95°C to f i v e d i f f e r e n t s t r a i n s on the subsequent deformation behaviour at i:120°C. I t i s observed that i n a l l cases the flow stress at -120°C a f t e r p r e s t r a i n i n g at -95°C , was exactly that found, on s t r a i n i n g e x c l u s i v e l y at -120°C to that p a r t i c u l a r value of s t r a i n . - 91 - F i g . 56 Temperature c y c l i n g of 25)J cadmium between -lkO°C and -196°C. - 9 3 - I 1_ '. I I L _ .5 1.0 1.5 2.0 2.5 $ s t r a i n F i g . 58 The e f f e c t of p r e s t r a i n i n g at -70°C on.the subsequent deformation behaviour at -120°C of 20Llzinc. 9k - The r e s u l t s are therefore s i m i l a r to cadmium in.that the flow stress at a p a r t i c u l a r value of s t r a i n at temperatures below.T^. = .26 a r i s e s due to a common d i s l o c a t i o n configuration and the change i n stress with temperature merely r e f l e c t s a change i n the thermal component of stress cr*~ . However i n t h i s temperature range f o r z i n c , fracture occurs before an i r r e v e r s i b l e e f f e c t i s obtained with increasing s t r a i n as i s observed with cadmium. .Prestraining above = .26 as shown i n F i g . §>S, produced an i r r e v e r s i b l e component of the flow stress d i f f e r e n c e at a l l values of s t r a i n . I t would therefore appear that i n the regions of s t r a i n below T^ = .26 where l i n e a r hardening occurs i n both zinc and cadmium,,that"equi valent states" are obtained at equal strains.. In t h i s region therefore i t i s probable that a mechanical equation of state could be formulated. A mechanical equation of state f o r metals, i s r a r e l y v a l i d except during Stage I and Stage II hardening of f . c . c . single c r y s t a l s at low temperatures. Once parabolic hardening associated with dynamic recovery begins, the mech a n i c a l equation of state becomes i n v a l i d . • 5 Mitra and ;Dorn have stated that equivalent states are obtained i n p o l y c r y s t a l s only when O g and "1" (the average d i s l o c a t i o n length being thermally activated) are constant. Since the r e s u l t s below :T = .26 indicate that the flow stress difference i s just due to a diff e r e n c e i n cr* , then O~Q Hp must be constant independent of temperature at a given value of s t r a i n . Tests were not comprehensive enough to e s t a b l i s h the constancy of "1' 2.2.4. Equivalent States above TH=.26 I t may be as suggested by Bullen that i n a given system the - 95 - sequence of events occuringo. during deformation does not change with temper ature but the rate at which the sequence proceeds, might be temperature dep endent .. Under such a d e f i n i t i o n , the r e s u l t s . o f the previous section would be interpreted i n terms of a constant rate of obstacle, production below T = .26 i n the temperature i n s e n s i t i v e hardening region. However above T^ . = .2.6 where dynamic recovery occurs at a l l values of s t r a i n , i t i s necess ary to equate states at d i f f e r e n t values of s t r a i n at any two temperatures- i n order t o . s a t i s f y Bullen s postulate. This.condition i s i l l u s t r a t e d i n F i g . 5.9... Stress S t r a i n F i g . 59 Equivalent states at d i f f e r e n t strains. The i r r e v e r s i b l e difference i n flow stress then develops because of a d i f f e r e n t rate of obstacle production at d i f f e r e n t temperatures.•The state of the c r y s t a l at "E" deformed at T 2 i s the same as.the state obtained at "A" when deformed at Tj_. ,CE represents the difference i n the thermal com-; ponent of stress and CD can be r e l a t e d d i r e c t l y to AB. The equivalent s t r a i n - 96 - values at Tx and T 2 are then given by <g and (5m r e s p e c t i v e l y . Using t h i s above method of a n a l y s i s , the amount of p r e s t r a i n at various elevated temperatures was r e l a t e d to an equivalent value of s t r a i n at -120°C i n 20u z i n c . The r e s u l t s are shown i n F i g . 61.. With increas ing temperature of prestrain,an,:.increasing amount of s t r a i n i s required to give a c e r t a i n equivalent s t r a i n at -120°C. .When the sections CD and AB were superimposed, the hardening rates i n a l l cases were the same. T i e t z and Dorn using aluminum found that the hardening rate CD was s l i g h t l y greater, than AB 8 3 On the other hand Sylwestrowicz using p o l y c r y s t a l l i n e copper and aluminum was able to q u a l i t a t i v e l y show, that the Acr value (AA) obtained- on decreas ing the t e s t temperature from 300°K to 77°K was i d e n t i c a l to the her value (BB) obtained on increasing the temperature at a stress corresponding to A ( F i g . 60). This suggests equivalence of states at A and B. ol 1 I I ! 0 5 10 15 20 2 5 % STRA IN —Stress -strain curves of copper specimens, strained at different temperatures. Curve No. 1-at 300°K; Curve No. 3 — at 76° K; Curve RN — at 300° K after pre-strainingat 76* K; Curve NR —at 76° K after presenting at 300° K. F i g . 60 Reversible temperature change t e s t s at equivalent states i n p o l y c r y s t a l l i n e copper. O O \ ( a f t e r Sylwestrowicz ). F i g . 6l The c o r r e l a t i o n of strains, at d i f f e r e n t temperatures i n 2Cu zi n c . Although such an analysis might appear unwarranted at t h i s point because of the lack of d e t a i l e d experimental data, i t i s thought that there i s some merit i n the idea of equivalent states at various s t r a i n s at d i f f e r e n t temperatures i n zinc and cadmium above T .= .26.-This i s not to H be construed as applying to a l l metal systems where dynamic recovery can a f f e c t the nature of the hardening.-Much would depend on the exact nature of the recovery process. I f i t involved a process which caused a d i s t i n c t i v e change i n the nature and character of o^ then i t i s doubtful i f such an analysis can be made. However i f recovery i s lin k e d to the observations of 39 Price concerning the rearrangement and gradual disappearance of debris then the above ideas may have some merit. Price observed that basal d i s l o c a t i o n s and debris i n t e r a c t strongly. Because of the rearrangement and removal of debris i n a temperature region of recovery, l a r g e r values of s t r a i n are therefore required to a r r i v e at an equivalent obstacle density- to that produced a t a given value of s t r a i n when recovery does not occur. Since the rate of rearrangement of debris as observed by Price increased with increasing temperature, i t i s expected that the "equivalent s t r a i n " as observed i n F i g . 6 l w i l l , increase w i t h increasing t e m p e r a t u r e . 2.3. -STRAIN PATE CHANGE TESTS The a c c u r a t e determination of the r e v e r s i b l e change i n f l o w s t r e s s accompanying a change i n s t r a i n rate i s of prime importance since the e x p e r i m e n t a l v a l u e o f /\cr as: used not only i n determining the, v a l i d i t y of t h e C o t t r e l l - S t o k e s law but also to determine the experimental values of a c t i v a t i o n volume and a c t i v a t i o n energy.-A discussion of the d i f f i c u l t i e s e n c o u n t e r e d i n c o r r e c t l y determining Acr* i s given i n Appendix I I I . - 99 -2.3.1. Procedure 7 y For the purpose of t h i s work, Acr*"was obtained during an increase i n s t r a i n rate. For p o l y c r y s t a l l i n e zinc and cadmium the s t r a i n -5 -h -1 rate was cycled between 4.0 x 10 and 4.0 x 10" sec corresponding to a crosshead speed change from .002 to .02 inches per minute. Cadmium single c r y s t a l s were cycled using crosshead speeds of .02 and .2 inches per minute corresponding to i n i t i a l shear s t r a i n rates of approximately 3-8 x 10 -1 n • -3 - 1 sec. and 3-8 x 10 sec. 2.3.2. C o t t r e l l - S t o k e s Behaviour i n Cadmium at -196°C. Single c r y s t a l s (#o= 36°, Ao= 38°) and p o l y c r y s t a l s of three d i f f e r e n t g r a i n sizes ( 25u., 400u 1250u ) were tested f o r Co t t r e l l - S t o k e s behaviour at -196°C. 2.3.2. a) Single C r y s t a l s Three single c r y s t a l s of i d e n t i c a l o r i e n t a t i o n were tested and a l l showed s i m i l a r behaviour to that shown i n F i g . 62-. ..It i s seen that during basal g l i d e i n Stage I, there i s a gradual decrease i n the value of AX with increasing s t r a i n . However i n Stage II the- C o t t r e l l - S t o k e s law y 6 i i s obeyed confirming the r e s u l t s of Davis . With the beginning of Stage I I I i t would appear that there i s a s l i g h t but consistent increase i n A T • Although i t may be argued that t h i s apparent increase could just be a r e s u l t of the experimental e r r o r i n determining AT > i t w i l l be shown i n the ensuing r e s u l t s f o r p o l y c r y s t a l s that dynamic recovery following l i n e a r hardening i s associated with c o n t i n u a l l y increasing values of AQ~ o~ 016 Ql4 \ r- O O \ 012 010 008 006 b o oQ?-° Stage I Stage II Temperature •= -196°C Cadmium single c r y s t a l Ko= 36° Ao= 38° _L 400 800 1200 1600 Resolved shear stress T gm/ mm. 2000 Fig.62 The v a r i a t i o n of the Cottrell-Stokes parameter during the deformation of a cadmium single c r y s t a l at -196°C. H o o - 101 - 2 . 3 . 2 . b) Po l y c r y s t a l s The three stage behaviour f o r single c r y s t a l s i s a l s o observed during the deformation of p o l y c r y s t a l s ( F i g . 63). In a l l cases there i s a region of s t r a i n i n the f i r s t few percent of deformation during which A o~ cr decreases. The extent of t h i s i n i t i a l region increases with increasing grain size and at 1250n corresponds rather w e l l to the change i n work hardening slopes observed i n Fig..4 9 .This i n i t i a l region i n p o l y c r y s t a l s may therefore be r e l a t e d i n some manner to the basal g l i d e region of single c r y s t a l deform ation. From F i g . 63 i t ' i s seen that the C o t t r e l l - S t o k e s law i s obeyed in intermediate s t r a i n regions s i m i l a r - t o that observed f o r Stage II of single c r y s t a l s . This region of obeyance ends with a gradual increase i n the value of A Q~ • The stress and s t r a i n values at which t h i s occurs f o r each cr grain s i z e are summarized i n Table 8 . They correspond very w e l l to the beginning of parabolic hardening observed i n F i g . . 4 9 . I t would therefore appear.that the i n i t i a t i o n of dynamic recovery at -196°C i n single c r y s t a l s and p o l y c r y s t a l s i s associated with increasing values of the Cot t r e l l - S t o k e s r a t i o . . TABLE 8 Grain si z e dependence of C o t t r e l l - s t o k e s behaviour (cadmium at -I96 Grain Size S t r a i n at beginning of C .S.. obeyance S t r a i n at beginning of recovery Stress at beginning of recovery Constant C o t t r e l l - S t o k e s r a t i o 25u 2.5$ 7$ 19,000 p. s . i . .0195 400u 4 .5$ 13$ 14,500 .0180 125 Qu 10$ 20$ 11,000 •0155 Single c r y s t a l 130$ shear 100$ t e n s i l e l 8 0 $ shear 5 , 5 0 0 p . s . i . 1,100 gm/mm. (shear) .0115 .023 .021 ACT .019 .017 ,015 .013 .011 - O n O - f f O O O O . . . 25y Cd A . . • kOOu Cd Q . .. 1250p Cd Numbers at arrows i n d i c a t e % s t r a i n £, = 4.0 x 10 sec._^ Cz = 4.0 x 10" sec. Stage II sing l e c r y s t a l (from f i g . 62 L 1.0 2.0 jj_ True stress p . s . i . x 10 F i g . 63 The grain s i z e dependence of ACT at -196°C.(cadmium) 3-0 4.0 O ro 2.3.3. The E f f e c t of Temperature on C o t t r e l l - S t o k e s Behaviour _ " The r e s u l t s of s t r a i n rate changes on p o l y c r y s t a l s above -196°C are shown i n F i g s . 64 and 65 f o r 25u and 400u cadmium r e s p e c t i v e l y . Regard l e s s of the temperature or grain s i z e there i s a decrease i n ACT i n the cr i n i t i a l regions of deformation previously r e l a t e d to Stage I deformation of single c r y s t a l s . F i g . 66 shows that A C T decreases during'Stage I deform- cr a t i o n of a si n g l e c r y s t a l deformed at -50°C where recovery i s known to a f f e c t hardening..Therefore whether dynamic recovery occurs or not, the C o t t r e l l - Stokes r a t i o decreases during Stage I deformation. As the temperature increases above -196°C,.there i s a decrease' i n the. amount of s t r a i n showing Cottrell-Stokes, obeyance u n t i l above -120°C ( T „ = .26), t h i s region disappears completely. Above T = .26 f o r cadmium H regardless of the grain s i z e , the values of ACT increase Immediately cr a f t e r the i n i t i a l region associated with Stage I. Therefore the s t r a i n rate and temperature i n s e n s i t i v e hardening regions.below Tg = ,26: described i n Part I would seem to be associated with C o t t r e l l - S t o k e s law obeyance. Several s t r a i n rate change t e s t s were performed•on 20u zinc at temperatures between -70°C and -120°C. The r e l a t i o n s h i p s shorn i n F i g • 67 indicate that the r e s u l t s are q u a l i t a t i v e l y the same as those f o r cadmium. However at -95°C a nd. -120°C (below Tg = .26) fracture occurs before any general increase i n A cr i n d i c a t i n g the absence of dynamic recovery. cr At -70°C there was a s l i g h t increase i n Acr s i m i l a r to that observed cr during dynamic recovery above Tv, = .26 i n cadmium. .05 .04 .03 A C T .02 .01 -95°c £,= 4.0 x 10" 5 sec. 1 £,= 4.0 x 10 sec. Numbers at arrows i n d i c a t e % s t r a i n -120°C 5>P x 5o 1.0 2-0 ^ True stress p . s . i . x 10 3-0 -196°C 4.0 Fig.64 The v a r i a t i o n with stress of Ao~ obtained from 25JJ cadmium at d i f f e r e n t temperatures. 105 Acr cr - .06 .04 .02 400JJ cadmium -5 -1 6.= 4.0 x 10 sec. -4 -1 - \Q -95 °C rCTL -i4o°c -196°C 4.0 8.0 12.0 16.0 20.0 •True stress p . s . i . x 10 F i g . 65 The stress dependence of A C T (400JJ cadmium) .04 .05 .02 A T 7 .01 Single c r y s t a l cadmium Oo= 36° Ao= 38° Temperature = -50°C End of stage I _L 800 1000 2 200 400 600 Resolved shear stress gm/mm. F i g . 66 The f a i l u r e of the Co t t r e l l - S t o k e s law at -50°C, - .106 - .06 .05 .04 A C T CT .05 .02 ,01 g a — • — • — a a - • -70° o o — ° o — o- -95°C -120°C -5 £,= 4.0 x 10 sec, £,= 4.0 x 10 sec, 1.0 1.5 k True stress p . s . i . x 10 2.0 F i g . 67 The v a r i a t i o n with stress of the. Co t t r e l l - S t o k e s parameter A C T obtained from s t r a i n rate change tes t s O " on 20p zinc . 2.k. .HARDENING AT -196°C IN CADMIUM " 1 0 7 " Since l i n e a r hardening i n p o l y c r y s t a l s at -196°C i s not a f f e c t e d by dynamic recovery, the experimental determination of a c t i v a t i o n volume and a c t i v a t i o n energy i s s i m p l i f i e d and more accurate than at higher temperatures. 2.4.1. A c t i v a t i o n Volume In order to calculate the a c t i v a t i o n volume i t i s assumed that the shear stress i n p o l y c r y s t a l s i s equal to one-half of the t e n s i l e stress y = o~ . The correct f a c t o r f o r the conversion i s co n t r o l l e d by the degree 2 8 4 of preferred o r i e n t a t i o n and therefore can vary with grain s i z e . However i t 8 5 w i l l have a value somewhere between 1/2 and l/k . Since the a c t i v a t i o n volume i s experimentally determined from the expression v = kT /_Aln5/eo) \ AT * IT the values of "v" obtained represent the lowest possible values i f the con version f a c t o r of 1/2 i s used. The manner i n which the a c t i v a t i o n volume va r i e s with the applied stress f o r p o l y c r y s t a l s i s shown i n F i g . 66J . TO a f i r s t approximation, v i s a function of the str e s s , decreasing i n an almost exponential manner with increasing s t r e s s . At any constant value of stress, v increases with increas ing grain s i z e , an observation expected due to the decreasing values of A Q" cr with.an increase i n grain s i z e ( F i g . 63 -). .The grain size (stress) dependence of the a c t i v a t i o n volume at y i e l d i s shown i n Table 9 . I f a simple assumption i s now made that the a c t i v a t i o n distance "d" can be approximated by the Burgers vector b, then 2 v = l b TABLE 9 Grain s i z e dependence of the a c t i v a t i o n volume at y i e l d i n cadmium at -196°C. Grain Size A c t i v a t i o n volume at y i e l d (cm. 3) A c t i v a t i o n volume at y i e l d 3 (* ) Forest density at y i e l d 2 lines/cm. Average activated length "1" at y i e l d (cm.) A c t i v a t i o n volume at s t a r t of dyn. recovery (cm. 3) 25u .30 x 10" 2 0 110 10 9.0 x 10 3-3 x 10" 6 -20 .20 x 10 kOOp. -20 1.20 x 10 450 9 5.5 x 10 -5 1.3 x 10 -20 .26 x 10 125 Ou -20 . 2.70 x 10 IgOO 9 1.1 x 10 3.0 x 10 5 _20 .38 x 10 Single c r y s t a l *o= 36°, \>= 38° -20 • 30.0 x 10 11,000 7 1.0 x 10 3-3 x 10 * -20 1.75 x 10 o V O - 110 - From t h i s an estimate of some smeared average of d i s l o c a t i o n length being activ a t e d per event at y i e l d can be obtained. These values along with single c r y s t a l data are shown i n Table 9 . I t i s seen that there i s a consistent increase i n 1 from 3.J x 10 cm. to 3-3 x . 1 0 cm. as the grain size increases from 25u to single c r y s t a l dimensions. I f i t i s further assumed that the rate c o n t r o l l i n g mechanism i s one of f o r e s t i n t e r s e c t i o n , then an estimate of the f o r e s t density at y i e l d may be obtained since p = 1 i s a good approximation of the forest ~ ^ 1 0 2 density. From Table 9 i t i s seen that p varies from 9-0 x 10 lines/cm . 7 2 f o r 25u cadmium to approximately 10 lines/cm. f o r single c r y s t a l s . These values are r e a l i s t i c f o r the i n i t i a l f o r e s t density. Mitra and Dorn found that i n t e r s e c t i o n i s the rate c o n t r o l l i n g process during the deformation of aluminum and copper single c r y s t a l s at rJrJ°K. For c r y s t a l s i n i t i a l l y oriented f o r easy g l i d e , they c a l c u l a t e d a f o r e s t density at y i e l d of the 9 2 5 order of 10 lines/cm. For aluminum p o l y c r y s t a l s they found an i n i t i a l 1 0 2 density of about 10 lines/cm. The a c t i v a t i o n volume at the end of Stage I deformation i n ~ 2 0 3 cadmium was found to be 5 . 0 x 10 cm. On the basis of the previous.assump- 8 tions t h i s would indicate a f o r e s t density of approximately 3 x 10 l i n e s 2 per cm. This represents an increase i n the f o r e s t density during Stage I. of s l i g h t l y more than an order of magnitude. By assuming an a c t i v a t i o n distance of d = b the density values calculated f o r cadmium may be s l i g h t l y low i n that "d"" may be somewhat la r g e r . Price found that the stacking f a u l t energy f o r cadmium i s probably 2 between 15 and 30 ergs/cm. • This i s considerably lower than previously b e l i e v  ed. Therefore before i n t e r s e c t i o n can occur there must be a recombination of the basal p a r t i a l s . This tends to give a more gradual slope to the f o r c e -- I l l - distance curve than would be expected i f the stacking f a u l t energy was high. This i n turn e f f e c t i v e l y increases the possible a c t i v a t i o n distance.•How ever at 77°K, a s i g n i f i c a n t proportion of the energy required f o r i n t e r s e c t  ion i s supplied by the e f f e c t i v e stress since the thermal component of the a c t i v a t i o n energy /\G should be quite small. This therefore l i m i t s the value of the a c t i v a t i o n distance during thermal a c t i v a t i o n and the assump t i o n that d = b i s not u n r e a l i s t i c . I t has been assumed up to t h i s point that i n t e r s e c t i o n i s the rate c o n t r o l l i n g process governing y i e l d and l i n e a r hardening at -196°C. This w i l l be discussed more f u l l y i n subsequent sections with regard to possible a l t e r n a t i v e mechanisms. 2.4.2. • A c t i v a t i o n Energy With the known values of the a c t i v a t i o n volume i t i s now poss i b l e to c a l c u l a t e a value of the apparent activation'energy AG D at y i e l d f o r 25u and 400u cadmium. The Ao~* values required f o r the c a l c u l a t i o n AT were obtained from the y i e l d stress-temperature r e l a t i o n s h i p s of Fig.,32. They agreed very w e l l with the extrapolated values at y i e l d of A o~ AT obtained from C o t t r e l l - S t o k e s temperature change t e s t s . The values obtained f o r the various rate parameters are shown i n Table 10 and were calculated using equations 9,\1 .and 12 . TABLE 10 Energy values at y i e l d f or cadmium deformed at -196°C. Grain s i z e AH e.v. AG e.v. vT* e.v. A G o e.v. 25u .10 .08 .28 • 36 400u .08 .06 • 33 • 39 • - 112 - The values shown i n Table 10 can be considered-accurate to at best ± 10$. Within experimental e r r o r the apparent a c t i v a t i o n energy AGo appears to be independent of grain size..This suggests that the f o r c e - distance curve does not change s i g n i f i c a n t l y with grain s i z e and therefore that the mechanism, of y i e l d i s independent of grain size..The change i n a c t i v a t i o n volume with grain s i z e therefore merely r e f l e c t s the v a r i a t i o n of the f o r e s t spacing "1" as previously assumed. The s i g n i f i c a n c e of AG o w i l l be discussed later..However i t i s noted, that the thermal component A G i s much smaller than, the v j term. This i s to be expected since with decreasing temperature an increasing proportion of the energy A G w i l l o be associated with the work done by the e f f e c t i v e s t r e s s . 2.5. HARDENING ABOVE -196°C IN ZINC AND. CADMIUM 2.5.I. Y i e l d Behaviour i n Cadmium The values of v, A H , A G , A G D and v T at y i e l d i n 25u cadmium at temperatures above -196°C are shown i n Table 11. Values, are quoted to two s i g n i f i c a n t figures which i s u n j u s t i f i e d because of experimental l i m i t a t i o n s . This procedure was followed only, to provide a method of comparison between values calculated i n the same manner using consistent techniques of analysing experimental data. It i s seen, that below -120°C there i s a steady increase i n AGo and vX with increasing temperature. This increase i s consistent with an energy b a r r i e r as shown i n F i g . 69a i n which there i s a f a i r l y gradual slope of the force distance relationship.• With t h i s type of b a r r i e r i t i s expected that both A G 0 and v T w i l l increase with increasing temperature as ind i c a t e d TABLE 11 Energy values at y i e l d i n 25u cadmium. Temperature (°c) AH (e.v.) A G (e.v.) v T (e.v.) A G (e.v.) A c t i v a t i o n volume g (cm.) -196 .10 .08 .28 • 36 -20 .30 x 10 -lUO .ko • 35 .h2 • 77 .60 x 10 _ 2° -120 M .Ul M .82 -20 .70 x 10 -95 M •57 .26 • 63 .55 x 10~ 2 0 -60 •50 M .20 .63 -20 .55 x 10 -50 •57 .50 .12 .62 -20 .50 x 10 by. the two random temperatures T x and T 2. Under such conditions i t i s therefore impossible to p r e d i c t a .. rate c o n t r o l l i n g mechanism s t r i c t l y from the values of A G . . I t was f o r t h i s o reason that AG was l a b e l l e d "apparent"..It does not include the work done o by.the e f f e c t i v e stress before thermal a c t i v a t i o n . In order to calculate AG0, the t o t a l a c t i v a t i o n energy, i t i s necessary to know the c r i t i c a l temperature T c where T = 0. Under such conditions AG = AGd = AGo* A * A l t e r n a t i v e l y AGO may be calculated simply, by knowing the s t r a i n rate de- 86 pendence of the c r i t i c a l temperature .Therefore a l l that can be said about A G 0 below -120°C i s that i t i s something i n excess of .8e.v.. (the value of AGo at -120°C) . Above -120°C A G 0 tends to remain constant at approximately .6 e.v..This temperature independence might suggest a rate c o n t r o l l i n g pro cess above -120°C which i s associated with an energy b a r r i e r as shown i n F i g . 69-b f o r which neither A G q nor d changes appreciably with temperature. Under such conditions A G * may be approximated by AG o and.is therefore equal to .6 e.v. + .1 e.v. This proposed change i n the rate c o n t r o l l i n g mechanism at -120°C i n 25u cadmium i s i l l u s t r a t e d quite c l e a r l y by the dependence of A G at y i e l d on temperature as shown i n F i g . 7/Q. Before such a proposed i n t e r p r e t a t i o n can become acceptable two major inconsistencies with theory must be explained. F i r s t of a l l the rate theory r e s u l t s i n d i c a t e a change i n the rate c o n t r o l l i n g mechanism of y i e l d at a .temperature of -120°C. However the Force AG = bcdb \ abdea / c AG0= abcdea / AG = b'cd'b' } / 1 \ vT* = ab'dea ) / \ 1 \ AGo = ab'cde'a' / 1 1 1 sd a e Distance Force AG = bcdb vT*= abdea AG0= abcdea Distance F i g . 69 Force-Distance curves - 116 - ho 80 120 160 .200 2ko 280 Temperature °K g. 7° The v a r i a t i o n of AG with temperature f o r 25/j cadmium. - 117 - y i e l d stress has been found to v a r y • l i n e a r l y with temperature (Fig..32). This l i n e a r i t y suggests a common mechanism c o n t r o l l i n g y i e l d . I t may be however that the change at -120°C causes only a s l i g h t change i n the y i e l d stress-temperature slope which f a l l s w ithin the experimental scat t e r of the l i n e a r r e l a t i o n s h i p shown i n Fig..32. Further t e s t s below -196°C should pro vide a more complete understanding of the temperature dependence of y i e l d . S t o l o f f a c t u a l l y did observe a change i n slope at approximately -120°C but h i s experimental scat t e r was too great to accurately e s t a b l i s h the change i n Ao~* . AT The second inconsistency concerns the nature of the temperature dependence of AG below -120°C..From the r e l a t i o n s h i p AG = -kT. In _t_ to i f $ 0 i s not a.function of temperature,.then AG should vary i n a l i n e a r manner with temperature.I.However from F i g . 72 i t i s seen that a l i n e a r r e l  a tionship does not e x i s t below -120°C. E i t h e r deformation i s not c o n t r o l l e d by a single mechanism which would place i n doubt any c a l c u l a t i o n based on rate theory or the nature of the force -distance curve changes with temper ature. This l a t t e r p o s s i b i l i t y could occur without a change i n the basic rate process i f f o r instance ths stacking f a u l t energy changes with temper- i s ature . At t h i s point there i s considerable doubt as to the magnitude of the stacking f a u l t energy i n cadmium l e t alone any possible temperature dependence. 68 Thornton and. Hirsch have proposed that the a c t i v a t i o n d i s  tance w i l l vary with temperature due to a change i n the stacking f a u l t energy. This would a l t e r the shape of the force-distance curve without a f f e c t i n g the basic rate mechanism. - 118 - 1 4 1 5 Conrad ' using an i n t e r s e c t i o n model f o r magnesium single c r y s t a l s has shown that the nature of the force-distance curve may vary s l i g h t l y with temperature, (stress) due to the influence of stress on the amount g l i d i n g d i s l o c a t i o n s how out i n the s l i p plane thereby changing-the e f f e c t i v e f o r e s t spacing "1". • Another p o s s i b i l i t y i s that the pre-^exponential term J(i= NAby varies with temperature. However i n the development of rate theory express ions, Xo m u s " t he assumed temperature independent i n order to a r r i v e at use f u l mathematical expressions. I f i n fa c t )(0 does vary ?then the derived form u l a t i o n s must be modified. At t h i s time riot enough experimental data are a v a i l a b l e . i . . because of the l i m i t e d t e s t temperatures a v a i l a b l e below -120°C i n order- to d i s t i n g u i s h between the above possible causes. A l i q u i d helium crydstat i s now nearing completion which w i l l allow f o r a more extensive t e s t i n g program. At t h i s point no attempt has been made to discuss the sign i f i c a n c e of the experimental value of AG D = .6 + .1 e.v. at. temperatures above T = .26. This discussion w i l l follow i n section 3.2. which includes H a. comprehensive survey of the possible mechanisms of dynamic recovery. 2.5.2. The V a r i a t i o n of AH with S t r a i n i n 2^u Cadmium. The manner i n which AH varies with s t r a i n at d i f f e r e n t temperatures i s shown i n F i g . 71- Because of the unknown v a r i a t i o n of T with s t r a i n the values of AG and AGo cannot be obtained. However the s t r a i n dependence of AG w i l l be s i m i l a r to that of AH i n that the entropy f a c t o r i s not expected to lower AH by more than about 20%. - 119 - From F i g . 71 i t i s observed that the l i n e a r hardening regions below -120°C are associated with s t r a i n independent values of AH. This i s as expected when the Co t t r e l l - S t o k e s law i s obeyed. When dynamic recovery occurs AH decreases s u b s t a n t i a l l y . 2.5.3. Y i e l d Behavior i n Zinc. The experimental values of- the various components of the a c t i v a t i o n energy at y i e l d i n 20u zinc are given i n Table 12..It i s observed that the r e s u l t s f o r zinc are s i m i l a r to those f o r cadmium. Both indicate a temperature independent value of AG o of about .6 e.v. i n the recovery range above T = .26. The values of AG f o r zinc are s l i g h t l y lower than H those f o r cadmium but do show a s i m i l a r temperature r e l a t i o n s h i p as shown i n F i g . 72. The break i n the AG - temperature r e l a t i o n s h i p f o r zinc at T = .26 i s not as obvious because of the l i m i t e d data below, t h i s H temperature.. The a c t i v a t i o n volume values f o r zinc are comparable to those of cadmium with the zinc values, beingaabout..25% smaller. - 120 Linear hardening Dynamic recovery k 8 12 16 20 .2k "jo s t r a i n 71 The v a r i a t i o n of AH with strain.and temperature i n 25/j cadmium. • TABLE 12 Rate parameters at y i e l d i n 20u zinc. Temperature °C TH AH e.v. A G e.v. * v T e.v. AG 0 e.v. V 3 cm. -120 .26 .21 • 17 .40 •57 -20 .36 x 10 -105 .28 .27 • 23 .42 .65 -20 .41 x 10 -95 •30 .29 .24 .40 .64 -20 .43 x 10 -70 • 34 • 33 .27 .34 .61 -20 .40 x 10 -30 .41 .42 • 36 .24 .60 -20 .40 x 10 - 122 - CD < .26 • 5 • 3 .2 .1 .0 25JJ cadmium / J_ 20p zinc 40 80 120 160 . 200 2k0 Temperature °K F i g . 72 .The v a r i a t i o n of AG with temperature for' 20p. z i n c . 3- DISCUSSION -123 - Before any discussion of the present r e s u l t s i s attempted, i t i s desirable to review the e l e c t r o n transmission observations of Pric e on d i s l o c a t i o n structure and motion i n p l a t e l e t s of zinc and cadmium. Since the flow stress i s generally thought to a r i s e from some combination of long and short range1 e l a s t i c i n t e r a c t i o n s between d i s l o c a t i o n s and fromaa short range thermal component of s t r e s s , P r i c e ' s observations on t,he formation i and behaviour of d i s l o c a t i o n loops and t h e i r subsequent i n t e r a c t i o n with d i s l o c a t i o n s are thought to be s i g n i f i c a n t with regard to the mechanisms of hardening i n zinc and cadmium. 3.1 Loop Formation and Annealing D i s l o c a t i o n loops can form on s p e c i f i c atomic planes by a v a r i e t y of processes. ( i ) P r i c e observed that prismatic loops can form behind moving b a s a l edge d i s l o c a t i o n s by a process i l l u s t a t e d i n F i g . 73 F i g . 73 The formation of a prismatic d i s l o c a t i o n loop by an edge d i s l o c a t i o n which i s held up at an obstacle. - 124 - This loop formation i s thought to be associated with the cross g l i d e of the screw components (c) and the subsequent a n n i h i l a t i o n of these components by glide on a p a r a l l e l g lide plane. This type-of•loop w i l l be s e s s i l e because of i t s edge components on non b a s a l planes. The kinks on the o r i g i n a l d i s l o c a t i o n (d) are expected to r e t a r d . i t s f u r t h e r motion. Loops thought to be formed by t h i s type of mechanism have been observed by L a l l y 89 during the Stage I deformation of magnesium at 20°C. ( i i ) P r i c e also observed loop formation behind moving b a s a l screwvrdisloca- t i o n s . A jog on a screw may a f f e c t the-motion of the screw depending on the size of the jog as shown i n F i g . 7U (c) F i g . 7^ The e f f e c t of jogs of various heights on screw d i s l o c a t i o n motion. Small jogs (lb-2b) can move non conservatively along with the d i s l o c a t i o n leaving a row of point defects behind ( a ) . This process may be thermally a c t i v a t e d and therefore rate, c o n t r o l l i n g . Very large jogs do not move and can act as pinning points f o r single ended Frank-Read sources.(b). Intermediate jogs (3b-300b) however can lead to the formation of an edge -125 - d i s l o c a t i o n dipole (c) and a f t e r pinching off:,, to the formation of an elongated loop on a non b a s a l plane. Such dipoles and loops have been observed i n a v a r i e t y of metal systems. ( i i i ) S e s s i l e d i s l o c a t i o n loops which contain a stacking f a u l t and have a Burgers vector \ c + p , may be produced by the collapse of vacancy discs as postulated by S e e g e r 2 5 . Berghezan 6 3 has observed such loops o on b a s a l planes i n zinc f o i l s which were heavily deformed at +20 C. S i m i l a r loops were observed to form by Priccf 3due to ion damage i n the e l e c t r o n microscope. The temperature range studied by P r i c e extended down o to -100 C. Whether such loops w i l l contribute to work hardening w i l l depend on the rate of loop production compared to the s t r a i n rate imposed on the system. The rate of loop production i n turn w i l l depend on the super saturation of vacancies i n a given area and the thermal energy- av a i l a b l e f o r vacancy migration. ( i v ) A fourth type of loop formation was observed i n considerable d e t a i l by P r i c e . Elongated s e s s i l e loops with B'urgers vectors c + a were observed to form on the b a s a l planes by the multiple cross g l i d e of £ll22^ <^1123^ d i s l o c a t i o n s . The stages i n the formation of these loops are i l l u s t r a t e d i n F i g . 75 • I t was observed that the loops acted as strong b a r r i e r s to the motion of b a s a l d i s l o c a t i o n s on the same glide plane and also produced a strong e l a s t i c i n t e r a c t i o n with other b a s a l d i s l o c a t i o n s on p a r r a l e l glide planes as long as the distance between the plane of the d i s l o c a t i o n and the loop plane was not greater than the loop width. A summary of the various types of loops i s given i n Table 13 The d i f f e r e n t types of d i s l o c a t i o n s possible i n the hexagonal system are i l l u s t r a t e d f o r c l a r i t y i n F i g . 76 -126 - (b) (c) F i g . 75 Stages i n the formation of an elongated loop on the b a s a l plane by the cross g l i d e of a £ll22? <1123^ screw d i s l o c a t i o n The various types of loop formation have been described i n d e t a i l because i t i s thought that t h i s debris may be a p r i n c i p a l source of obstacles leading to hardening due to the lack i n the hexagonal system of strong Cottrell-Lomer b a r r i e r s . Comprehensive studies concerning the annealing behaviour of loops during deformation have only been c a r r i e d out f o r those formed as a r e s u l t of £ll22^ \1123"^ d i s l o c a t i o n motion. Pric e has come to the following conclusions regarding the behavior of these loops i n zinc and cadmium. ( i ) At temperatures below T = .27, the loops are completely stable and can act as strong b a r r i e r s to the motion of b a s a l d i s l o c a t i o n s . - 1 2 7 -TABLE 13 Loop formation i n zinc and cadmium. Method of Formation Plane of Loop Burgers Vector .,. Behind moving ba s a l Non b a s a l a v 1 ^ edges as i n Fig.73 /..\, Cross g l i d e of b a s a l Non b a s a l v 1 1 / screws , . Condensation of Basal -ic + p ( 1 1 1 ) vacancies .. . Cross glide of Basal c + a (IvJ £1122} <1123> screws can decompose to c or -§c + p - 128 - (ii)'' At temperatures between T = .27 and T = AO the elongated loops H H break up i n t o rows of c i r c u l a r loops by a process of pipe d i f f u s i o n . The d r i v i n g force f o r s p l i t t i n g i s supplied by the p o t e n t i a l decrease i n l i n e energy.. The area within the elongated loops i s conserved, during the s p l i t t i n g operation and the rate at which s p l i t t i n g occurs increases with increasing temperature. Pric e envisaged that the a c t i v a t i o n energy f o r such a process w i l l be equal to U + U where U. i s associated with the required jog j P J formation and U i s the pipe d i f f u s i o n energy..Implicit i n the concept of a? loop s p l i t t i n g as opposed to loop shrinkage i s that U •+ must be less than the s e l f d i f f u s i o n energy U ( i i i ) At temperatures above T = A,0. the c i r c u l a r loops gradually d i s - H appeared by a process of climb. The measured a c t i v a t i o n energy f o r the shrinkage of loops was found to be equal to the s e l f d i f f u s i o n energy. The observations of Price are of p a r t i c u l a r s i g n i f i c a n c e with regard to the present work i n that the temperature region above which loop s p l i t t i n g occurs (.27), i s s i m i l a r to that above which temperature and s t r a i n rate independent l i n e a r work hardening disappear i n p o l y c r y s t a l l i n e zinc and cadmium and above which there i s a decrease i n the hardening rate associated with both Stage I and Stage II deformation of single c r y s t a l s . 3.2. DYNAMIC RECOVERY In a very broad sense dynamic recovery may be re l a t e d to e i t h e r of the following processes: i ) cross s l i p i i ) d i f f u s i o n c o n t r o l l e d processes ; - 129 - 3.2.1. Cross S l i p Cross s l i p i s known to be a dynamic recovery mechanism i n f . c c . metals.. In hexagonal metals, cross s l i p must be looked at from a s l i g h t l y d i f f e r e n t point of view..It may be a.required mechanism to permit basal d i s l o c a t i o n s to move r e a d i l y onto non basal planes and thereby allow fo r the operation of a d d i t i o n a l s l i p systems. On the other hand i t may operate as an adjunct to the operating systems i n order to allow d i s l o c a t i o n s to move around b a r r i e r s and therefore r e l i e v e points of stress concentration. Only under, the l a t t e r condition would cross s l i p be c l a s s i f i e d as a dynamic recovery process. Cross s l i p has not been observed under normal l i g h t microscopy i n e i t h e r zinc cadmium or magnesium. However L a l l y using r e p l i c a s has observed i t during the stage I deformation of magnesium at +20°C. Therefore r e p l i c a studies on p o l y c r y s t a l l i n e zinc and cadmium are c u r r e n t l y i n progress to e s t a b l i s h whether c r o s s - s l i p occurs to a s i g n i f i c a n t degree. I t must occur on a l i m i t e d scale i n order t o account f o r the loop formation behind moving basal d i s l o c a t i o n s described i n the previous s e c t i o n . However such i s o l a t e d instances are not expected to m a t e r i a l l y a f f e c t the flow stress.- Although i t i s true that such cross s l i p does allow f o r the circumvention of obstacles the net r e s u l t of the o v e r a l l process i s the production of s e s s i l e loops which w i l l act as strong b a r r i e r s to further d i s l o c a t i o n motion. 89 The observations of L a l l y and Hirsch indicate that during the Stage I deformation of magnesium, the d i s l o c a t i o n structure consists of a high density of elongated edge dipoles..Very few screws were observed..It was postulated that the edge dipoles were formed by the trapping of edge components from d i f f e r e n t sources on nearby g l i d e planes. Screws of opposite -•130 - sign.on the other hand can a n n i h i l a t e leading to a low screw density. Under such conditions, i t could be postulated, that dynamic recovery i n magnesium i s associated with the temperature at which cross s l i p can occur leading.to a lower o v e r a l l d i s l o c a t i o n density and a d i f f e r e n t d i s l o c a t i o n configuration. S i m i l a r observations t o those of L a l l y cannot be made on zinc or cadmium because of the higher e f f e c t i v e temperature at +20°C which w i l l give r i s e to a considerable d i s l o c a t i o n rearrangement during the time necessary f o r f o i l preparation. The observations on magnesium however are of importance i n that i t i s tempting to invoke cross s l i p as the mechanism of dynamic recovery. However on a more macroscopic scale the i n t e r p r e t a t i o n becomes more complex. Conract 6'"^as observed a s i m i l a r temperature dependence, of 9 /G i n magnesium to that observed i n zinc and cadmium. S p e c i f i c a l l y Q/Gc remains constant below approximately T^ = .23 and decreases above t h i s temperature'.... . Therefore the e f f e c t i v e temperature above which the hardening rate decreases i n a l l three metal systems i s approximately the same. However the ease of cross s l i p i s r e l a t e d to the stacking f a u l t energy i n that the p a r t i a l s must recombine before the process can occur. Although there i s considerable controversy with regard to the magnitude of the stacking f a u l t energies in. the three systems, i t ; i s generally thought that the stacking f a u l t energy of magnesium i s appreciably higher than that of e i t h e r zinc or cadmium. Because o f the a s s o c i a t i o n between the stacking f a u l t energy and cross s l i p i t would not be expected that a l l three metals w i l l undergo dynamic recovery at the same e f f e c t i v e temperature..Therefore i t i s not possible to l i n k dynamic recovery to cross s l i p . - 131 - 3.2.2. .Diffusion C o n trolled Processes D i s l o c a t i o n climb when governed by the s e l f d i f f u s i o n energy U was not observed by Price at temperatures below T ..= .4 ..Therefore D H i t cannot be considered as a dynamic recovery mechanism i n the region of T H = .26 . •Kroupa and P r i c e 8 8 have observed, that above TTT = .26 i n zinc n c i r c u l a r c + a or c loops produced behind £L122|0-123^ screws can move under the stress associated with the i n t e r a c t i o n of the loops with approach ing basal d i s l o c a t i o n s ( F i g . 78 ). This motion was termed.conservative climb since i t d i d not involve s e l f d i f f u s i o n but rather the generation of vacancies on one side of the loop and t h e i r subsequent motion along the loop to the other side.. The a c t i v a t i o n energy f o r conservative climb w i l l therefore be the pipe d i f f u s i o n energy U . • Such loop i n s t a b i l i t y i s a mechanism of dynamic recovery since, as observed by P r i c e , the normally s e s s i l e loops act as strong b a r r i e r s to basal d i s l o c a t i o n motion. Loop motion by conservative climb should therefore r e s u l t i n a considerable r e l i e f of the back stress associated with pile-ups behind the loops. In the present work, extensive [1122}(112.3} s l i p was observed i n p o l y c r y s t a l l i n e zinc and cadmium. Therefore a s i g n i f i c a n t concentration of basal loops should be formed.. The value of AG Q at y i e l d f o r poycrystals at temperatures above T = .26 was determined to be almost i d e n t i c a l f o r H 96 both zinc and cadmium at .6 ± .1 e.v. F r i e d e l has stated that the expected values of the pipe d i f f u s i o n energy.in zinc and cadmium are .62 e.v..and .57 e.v. re s p e c t i v e l y . From an energetic point of view i t i s therefore - 132 - F i g . 77 ( c ) Sequence of transmission electron micrographs showing the ' conservative climb ' motion of a dislocation loop with a [0001] Burgers vector due to its interaction with a moving edge dislocation with a [^1120] Burgers vector. The plane of the micrograph is parallel to the basal plane of the zinc platelet. ( a f t e r Price ). possible to postulate that dynamic recovery i s associated with the conserv ative climb of basal loops. Since the energy required f o r elongated loop breakup ( U + U. ) i s greater than that of conservative climb (U ), i t p J j P may be argued that the rate c o n t r o l l i n g process i s a c t u a l l y that of loop breakup.However because of the higher d i s l o c a t i o n density i n r e a l c r y s t a l s as opposed to Price's specimens, i t i s probable that the edge dipole w i l l i j tend to pinch o f f e a r l i e r i n the sequence of loop formation.. Therefore the process of loop s p l i t t i n g i s not as important a process. In order f o r such a conservative climb i n t e r p r e t a t i o n to be v a l i d f o r single c r y s t a l s , i t i s neccessary to postulate that some £ll22} ^1123^> s l i p can occur during both Stage I and Stage II single c r y s t a l deformation. 1 3 Basinski has shown that the C o t t r e l l - S t o k e s law i s only s t r i c t l y obeyed i n magnesium at temperatures below 46°K.•However the dev ia t i o n s that occur at higher temperatures are not p a r t i c u l a r l y severe..In any case there i s a considerable increase i n y during both stages of def ormation which was interpreted by Basinski i n terms of an increasing forest concentration. E s s e n t i a l l y s i m i l a r r e s u l t s were obtained during t h i s work on cadmium single c r y s t a l s . Although the C o t t r e l l - S t o k e s law was not obeyed during Stage I, there was a considerable decrease i n the activation, volume at a l l temperatures studied. At -196°C "v" decreased from 30 x 10 2 ° cm.3 -20 3 to 5 x 10 cm. Based on an i n t e r s e c t i o n mechanism t h i s corresponds to approximately a 30 f o l d increase i n the forest density during Stage I. It would appear therefore that there i s some non basal a c t i v i t y during Stage I and t h i s could lead to a s i g n i f i c a n t concentration of basal loops. - 134 - Dynamic recovery to t h i s point has been associated only with nd the conservative climb of basal loops produced behind 2 order pyramidal screws. However the pipe d i f f u s i o n process may a l s o have a s i g n i f i c a n t e f f e c t on the nature and mobility of the other types of loop debris men tioned i n the previous section. Since t h i s debris i s produced only as a r e s u l t of basal d i s l o c a t i o n motion, i f s i m i l a r processes i n v o l v i n g conserv ati v e climb can occur, then . i t i s not necessary to postulate a change i n the forest density during single c r y s t a l deformation i n order to explain dynamic recovery. Before t h i s i n t e r p r e t a t i o n can proceed i t w i l l be necessary to know more about the annealing c h a r a c t e r i s t i c s of debris. As an a l t e r n a t i v e to a pipe d i f f u s i o n mechanism, there i s a d i s t i n c t p o s s i b i l i t y that recovery may be associated with the annealing c h a r a c t e r i s t i c s of excess vacancies produced during deformation. R e s i s t i v i t y 90 studies made by Sharp, M i t c h e l l and C h r i s t i a n on cadmium , single c r y s t a l s and p o l y c r y s t a l s deformed 12$, indicated.the existence of an annealing peak i n the v i c i n i t y of T = .25 . They determined the a c t i v a t i o n energy H to be .25 * .2 e.v. and associated the peak with si n g l e vacancy migration. . 9 1 9 2 P e i f f e r and Stevenson i n a s i m i l a r study observed two annealing peaks, one at T = .23 and another at T = .28. The a c t i v a t i o n H H energies were.24 and .30.e.v. re s p e c t i v e l y . They believed that one of the peaks was associated with single vacancy migration although they were not sure which one. The expected value f o r the energy associated with vacancy 1 9 7 motion i n cadmium i s .41 e.v. This i s somewhat lower than the .6 * .1 e.v. a c t i v a t i o n energy found to control the dynamic recovery.of zinc and cadmium. However due to the approximations involved in. the rate theory expressions when applied t o deformation, i t i s not outside the realm of p r o b a b i l i t y that vacancy migration and dynamic recovery are somehow l i n k e d . The exact nature of such a r e l a t i o n s h i p i s rather vague. I f the net r e s u l t of migration i s the production of basal d i s l o c a t i o n loops by vacancy condensation, then i t would be expected that such loops w i l l con t r i b u t e to hardening and w i l l not lead to a recovery e f f e c t . On the other hand by annealing out at edge d i s l o c a t i o n s excess vacancies can cause climb allowing for. the circumvention of obstacles. Climb therefore can occur at lower temperatures than T = .k and under such conditions i s c o n t r o l l e d H only by the vacancy migration energy.- Although Price did not observe climb below T = .k , i t i s probable that the excess vacancy concentration i n the H p l a t e l e t s used f o r study was quite low due to the low d i s l o c a t i o n density and the a v a i l a b i l i t y of the specimen surface. . In summary.it must be concluded that the exact cause of the dynamic recovery occuring i n the v i c i n i t y of T^ = .26 cannot be d e f i n i t e l y established at t h i s time. However because of the b e t t e r c o r r e l a t i o n of energies, i t i s thought that the most l i k e l y process i s one in v o l v i n g pipe d i f f u s i o n leading to the conservative climb of normally s e s s i l e basal d i s l o c a t i o n loops. .3.3. -THE MECHANICAL EQUATION OF STATE It was shown i n section 2.2.3. that a mechanical equation of state could be formulated f o r :polycrystals i n the regions of l i n e a r hardening below T = .26 .-Only under these conditions i s i t possible to H obtain equivalent states at an equal value of s t r a i n when deformation occurs at d i f f e r e n t temperatures. The concept of dynamic recovery being associated with the conservative climb of loops i s consistent with the above observations. Basal loops should remain stable i n the l i n e a r hardening regions.- The -136 - r ' o v e r a l l d i s l o c a t i o n configuration at a given value of s t r a i n w i l l therefore be independent of temperature. However once conservative climb can o c c u r , . i t i s not expected that equivalent s t a t e s . w i l l be found, at equal s t r a i n s because of the temp erature dependence of the rate of loop annealing as observed by P r i c e . 3.4. THE COTTRELL-STOKES LAW I t was observed that at a l l temperatures. the C o t t r e l l - S t o k e s law was not s t r i c t l y obeyed during the Stage I deformation of cadmium single c r y s t a l s or during the early, s t r a i n regions of p o l y c r y s t a l s . In both cases the Acr r a t i o decreased, with increasing s t r a i n . In e f f e c t t h i s cr means that the rate of increase of cr was somewhat les s than the rate of increase of cr . In p o l y c r y s t a l s t h i s may be r e l a t e d . to., grain boundary G e f f e c t s . • T h i s e a r l y region of s t r a i n i s associated with the gradual buildv. up of a stable d i s l o c a t i o n configuration at the grain boundaries ( p i l e ups). This c o n t r i b u t i o n to stress w i l l be athermal, contributing only to cr G It i s therefore expected that the increase i n cr w i l l be somewhat i n excess G of cr which i s associated.with intragranular processes. Although A T decreases during Stage I hardening.there i s X a s i g n i f i c a n t decrease i n the a c t i v a t i o n volume. This implies.that there _ *- i s an increase i n the value of T with increasing s t r a i n although not as great as. the increase i n *X . This observation i s not consistent with the G 2 4 theory of Seeger regarding Stage I hexagonal metal deformation. He assumes that %/G can be calculated, by only considering•the e l a s t i c i n t e r a c t i o n s between i n d i v i d u a l p a r a l l e l d i s l o c a t i o n s and that the contribution to ©j/G from J i s n e g l i g i b l e . For such' a theory to be correct, the a c t i v a t i o n volume must remain constant during Stage I. Because of the marked change - 137 - i n "v" i n cadmium, i t must be concluded that Qj/G represents an important contribution to the observed rate of hardening and.cannot be neglected. 3.4.1. Obeyance The C o t t r e l l - S t o k e s law i s obeyed only-during. the l i n e a r hardening regions of p o l y c r y s t a l s and during the Stage II hardening of single c r y s t a l s below T = .26 ..In order to postulate the o r i g i n and H re l a t i o n s h i p between C*and cr i t i s necessary to know how. twinning may f \ . G a f f e c t the two stress components.-As observed i n single c r y s t a l s the formation of a twin does not affect- t h e flow stress required f o r furth e r deformation. Basal s l i p . w i t h i n a twinned region therefore must represent an important contribution t o deformation only immediately a f t e r the twin formation before the macroscopic observed stress returns to i t s previous value. Therefore although twinning may. a f f e c t the work, hardening r a t e , i t does not changexthe instantaneous r e l a t i v e values of the. two stress com ponents. This ^status quo" condition during.deformation may al s o be applied to the nature of the d i s l o c a t i o n configuration i n the neighbourhood of grain boundaries.- Once a stable configuration i s obtained i t w i l l be assumed that hardening becomes intragranular and the component of <3~ associated with boundaries need not be considered•with regard,to the Cot t r e l l - S t o k e s law. The athermal component of stress is. therefore associated with some combination of the following components: i ) the i n t e r a c t i o n of forest and g l i d e d i s l o c a t i o n s i i ) the i n t e r a c t i o n of loops and gl i d e d i s l o c a t i o n s - 138 - I t w i l l be assumecL that cr a r i s e s due to an i n t e r s e c t i o n mechanism. The C o t t r e l l - S t o k e s obeyance can therefore be interpreted i n terms, of an increasing forest density with increasing s.train . • o~ remains pro p o r t i o n a l to both athermal stress components because the basal loop density w i l l be a function of the f o r e s t density. 3.4.2. - Dynamic Recovery It has been shown that i n both si n g l e c r y s t a l s and poly c r y s t a l s , dynamic recovery i s associated with increasing values of Ao~ o~ This means, that the rate of increase of cr i s somewhat l e s s than that of G 0~* . . I f recovery i s related, to loop i n s t a b i l i t y and i f a s i g n i f i c a n t proportion of cr i s derived, from the nature of l o o p - d i s l o c a t i o n i n t e r - G -actions, then i t i s expected that c T ^ . w i l l increase at a lower rate because of the r e l i e f of back stress which occurs because of obstacle^ motion. 3.5. -RATE CONTROLLING '.PROCESSES BELOW. TTT- = .26 _ ^ I t was proposed,that forest i n t e r s e c t i o n i s the rate cont r o l l i n g process governing y i e l d i n single c r y s t a l s and p o l y c r y s t a l s below T = .26 . This was done without any. consideration of possible alternate H- mechanisms.. These w i l l now be discussed. 3.5.I. .Peierls Stress The P e i e r l s stress f o r the motion of basal d i s l o c a t i o n s i s very low because of the close-packed nature of the basal plane.. I t cannot therefore be considered as a rate c o n t r o l l i n g process. However during p o l y c r y s t a l l i n e deformation when non basal s l i p must occur, the P e i e r l s - 139 - stress associated with movement on the corrugated pyramidal planes may be s i g n i f i c a n t and rate c o n t r o l l i n g . The a c t i v a t i o n -volume at y i e l d i n 25ji cadmium at -196°C based _ -20 3 on a shear stress conversion of y = cr , was .30 x 10 cm. . In terms of . 2 3 a basal d i s l o c a t i o n where a = 2.97 A* , t h i s gave a value of 110b . •However 3 i n terms of a c+a pyramidal d i s l o c a t i o n t h i s reduces to about 17b a much more acceptable value for.the P e i e r l s mechanism. However i t was a l s o observed.(Table 9) that the a c t i v a t i o n volume increased with increasing grain s i z e , a trend not expected f o r the P e i e r l s mechanism. From F i g . 68 i t was observed, that f o r a given grain size "v" decreased s u b s t a n t i a l l y • w i t h increasing s t r a i n . - I f the P e i e r l s mechanism i s rate c o n t r o l l i n g .the a c t i v a t i o n volume should not vary with s t r a i n . -Since the P e i e r l s mechanism i s not compatible with a l l of the experimental observations,.it can be rejected as a possible c o n t r o l l i n g mechanism. 3.5.2. . Cross. S l i p C r o s s s s l i p has been considered i n section 3-3» as a possible mechanism of dynamic recovery.. I t may a l s o be argued that cross s l i p could c o n t r o l y i e l d at temperatures below T . = .26 i f i t i s required.in order f o r H basal d i s l o c a t i o n s to move onto non basal planes and thereby c o n t r o l the extent of non basal s l i p . However-this argument cannot be v a l i d a t e d i n that the only non basal traces that are observed i n zinc and cadmium.arise from [ll22J0-123^ s l i p . This i s not a cross s l i p system. Also as pointed out i n section 1.4.1., when [ll22J<(ll23>sl .ip.occurs, the number of independent systems that can operate i s s u f f i c i e n t ..to promote extensive p o l y c r y s t a l l i n e - iko - deformation. Therefore cross s l i p i s not a necessary process i n order f o r deformation to proceed. 3 . 5 - 3 - The Non Conservative-Motion of Jogs 9 8 Frank f i r s t postulated that a jogged screw d i s l o c a t i o n can move only i f the jog leaves behind i t e i t h e r a row of vacancies or i n t e r - s t i t i a l s depending on its . sign and d i r e c t i o n of motion.- The conservative motion of vacancy jogs i s thought to be associated with a r e l a t i v e l y high a c t i v a t i o n energy, and therefore need not be considered. . I t i s u s u a l l y not possible to d i s t i n g u i s h between a jog or an i n t e r s e c t i o n mechanism merely from the values of rate theory parameters. Both processes are expected; to have s i m i l a r values of a c t i v a t i o n volume 2 3 4 3 i n the range from -10 b to 10 b . The concept of a jog mechanism being rate c o n t r o l l i n g i s not - 99 strongly supported by experimental observations..It was advanced by Mott mainly to explain the nature of the flow stress v a r i a t i o n s i n copper single c r y s t a l s . . In copper i t has been observed that the flow stress i s almost temperature independent between T = .2 and T = .5 .-At higher temperatures there i s a s i g n i f i c a n t drop i n s t r e s s . Mott therefore proposed that at temp eratures below TJJ = -5 > "the s e l f d i f f u s i o n process required f o r ..the mech anism to procede cannot occur at an appropriate rate i n terms of the applied s t r a i n rate. The vacancy nucleation at the jog i s therefore completely athermal leading to a temperature independent flow s t r e s s . . I t was al s o assumed by Mott that energy i s not a v a i l a b l e f o r vacancy migration away from the region of the jog. Under such conditions any thermal f l u c t u a t i o n w i l l tend to move the jog forward a sing l e atomic spacing, but i f the vacancy -.141- produced i s not mobile, the jog can be pu l l e d back by the i n t e r a c t i o n with the vacancy produced. Therefore the flow stress should not be temperature dependent below T =. .5 . Above t h i s temperature however where the energy . H f o r s e l f d i f f u s i o n i s a v a i l a b l e , the vacancies produced should be mobile and a temperature dependent flow stress should r e s u l t . There are some questionable features of t h i s theory. F i r s t of a l l i t i s not c l e a r why the nucleation of a vacancy at a jog should be a completely athermal process. The thermal energy a v a i l a b l e i n t h i s temperature range should be s u f f i c i e n t to provide a p o r t i o n of the energy needed f o r vacancy nucleation. Secondly, even i f vacancy nucleation i s an athermal process, the temperature at which a temperature dependent flow stress occurs should be associated only with the energy f o r vacancy migration. This should occur at temperatures w e l l below T ..= .5 H .In zinc and cadmium.it i s believed; that single vacancy motion can occur at-appreciable rates above T = .25 .Therefore below t h i s temperature according to the jog mechanism, the flow stress of zinc and cadmium w i l l be governed by the athermal process of vacancy nucleation. This should lead to a temperature independent flow s t r e s s . This was not observed .There i s a considerable increase i n the flow stress i n both systems below T = .26. I t i s therefore u n l i k e l y that the non conservative H motion of jogs i s the rate c o n t r o l l i n g process at low temperatures i n zinc and cadmium. 3.5.4. I n t e r s e c t i o n The postulate of forest i n t e r s e c t i o n was not made merely - 1 4 2 - because other mechanisms could not explain a l l of the experimental observat ions. There are no major inconsistencies with the forest mechanism. The estimate of f o r e s t spacings of Table 9 made from .activation volume data are reasonable f o r the systems involved. The v a r i a t i o n of a c t i v a t i o n volumes 2 3 4 ..,3 at -196°C from 1.1 x 10 b for. 25u cadmium to 1.1 .x 10 bb f o r si n g l e c r y s t a l s :.is. w ithin the expected range f o r i n t e r s e c t i o n . - The v a r i a t i o n has been shown to a r i s e from the expected difference i n the forest spacing as the grain s i z e changes and i s not due to a change i n the force distance curve.. The non l i n e a r v a r i a t i o n of AG with temperature can be interpreted from the fo r e s t mechanism to. be due e i t h e r to a change i n the stacking f a u l t energy with temperature or to a change i n the e f f e c t i v e value of the f o r e s t spacing due to the manner i n which a d i s l o c a t i o n bows out under the influence of a s t r e s s . 4. SUMMARY AND.CONCLUSIONS - 1U3 , The observations and i n t e r p r e t a t i o n s of the deformation char a c t e r i s t i c s , of zinc and cadmium may be summarized.as follows: 1) Negative work hardening beyond .the point of maximum- stress i n poiycrys-: t a l l i n e zinc and cadmium at temperatures above T = ,4 i s associated •with r e c r y s t a l l i z a t i o n . However at temperatures up to at l e a s t T = .5 r e c r y s t a l l i z a t i o n does not go to completion during deformation. At l e a s t 50$ of the structure outside the necked area remains u n r e c r y s t a l l i z e d . 2) Grain boundary migration can occur i n the i n i t i a l hardening regions and i s p a r t i c u l a r l y important as a recovery mechanism above T = ,4 . H S l i g h t boundary corrugations were observed i n cadmium, at temperatures down to -95°C suggesting that the change i n fracture mode from d u c t i l e shear to intergranular fr a c t u r e which occurs at -120°C i s associated with the cessation of recovery by boundary migration. 3) J1122}0-123)" i s the only non basal s l i p system observed during poly c r y s t a l l i n e deformation..It i s more prevalent as the temperature decreases.-At +20°C i t i s more extensive i n zinc than i n cadmium.at an equivalent temperature. The non basal traces are wavy and discontinuous at elevated temperatures.-At low temperatures they are s t r a i g h t and.tend to concentrate i n t o bands. Q u a l i t a t i v e l y i t appears that the amount.of non basal s l i p increases as the grain s i z e decreases. 4) The formation of low angle boundaries during deformation i s s i m i l a r i n both systems.and does not vary i n nature or extent with temperature. -M - 5) I n cadmium s i n g l e c r y s t a l s t h e resolved b a s a l s h e a r s t r e s s a t w h i c h S t a g e I ends i s i n d e p e n d e n t o f t e m p e r a t u r e i n t h e range f r o m -50°C t o -196°C. 6) T w i n n i n g i s a g e n e r a l f e a t u r e o f S t a g e I I cadmium d e f o r m a t i o n a t temper a t u r e s below -50°C. 'a IV-.:., i j 7) A r e g i o n o f t e m p e r a t u r e and s t r a i n r a t e i n d e p e n d e n t l i n e a r work h a r d e n i n g d e v e l o p s below T = .26 i n b o t h p o l y c r y s t a l l i n e z i n c and cadmium. The H amount o f s t r a i n a s s o c i a t e d w i t h l i n e a r h a r d e n i n g i n c r e a s e s as t h e temp e r a t u r e d e c r e a s e s . The r a t e o f h a r d e n i n g i s s i m i l a r i n b o t h s y s t e m s . 8) Cadmium s i n g l e c r y s t a l s a l s o show c o n s t a n t S t a g e I and S t a g e I I h a r d  e n i n g r a t e s b e l o w T = .26 and c o n t i n u o u s l y d e c r e a s i n g h a r d e n i n g r a t e s H above t h i s t e m p e r a t u r e . T h i s i s s i m i l a r t o t h e b e h a v i o u r o f z i n c and magnesium. 9) The maximum l i n e a r h a r d e n i n g r a t e o f p o l y c r y s t a l l i n e cadmium a t -196°C - 1 _ l v a r i e s l i n e a r l y w i t h d . The e x t r a p o l a t e d v a l u e o f 0 t o d 2 = 0 c o r r e s p o n d s t o t h e t e n s i l e S t a g e I I h a r d e n i n g r a t e o f s i n g l e c r y s t a l s . T h i s change i n h a r d e n i n g r a t e can be e x p l a i n e d i n terms o f a change i n t h e f r e q u e n c y o f |jL122^<1123)slip. 10) The C o t t r e l l - S t o k e s law i s n o t s t r i c t l y obeyed f o r e i t h e r , t e m p e r a t u r e o r s t r a i n r a t e change t e s t s . Obeyance i s o n l y o b s e r v e d d u r i n g t h e l i n e a r h a r d e n i n g o f p o l y c r y s t a l s and d u r i n g S t a g e I I s i n g l e c r y s t a l h a r d e n i n g b e l o w T = .26. Dynamic r e c o v e r y i s a s s o c i a t e d w i t h i n c r e a s i n g v a l u e s H o f ACT , cr 11) I n p o l y c r y s t a l l i n e z i n c and cadmium e q u i v a l e n t s t a t e s a t e q u a l s t r a i n s - 14-5 - are only obtained during linear-hardening.- Therefore only i n these regions can a mechanical equation of state be formulated. 12) Y i e l d at temperatures below T = .26 can be interpreted i n terms of a . H fo r e s t i n t e r s e c t i o n mechanism.. The'total a c t i v a t i o n cannot be estimated but D . v c i s somewhat i n excess of . 8 e.v. i n cadmium . 13) I t i s probable that dynamic recovery-above TJJ = .26 i s associated with a d i f f u s i o n c o n t r o l l e d process. The most l i k e l y mechanism.involves.the conservative climb of normally s e s s i l e basal loops by a process of pipe - d i f f u s i o n . The experimental a c t i v a t i o n energy f o r y i e l d i n p o l y c r y s t a l l i n e zinc and cadmium i s .6 * .1 e.v. -ike - 5. • SUGGESTIONS FOR FUTURE WORK Several l i n e s of i n v e s t i g a t i o n are immediately recognized from the r e s u l t s of t h i s work. These include: 1) An extension of t e s t i n g t o temperatures below -196°C i n order to completely e s t a b l i s h the flow stress-temperature r e l a t i o n s h i p s . 2) . A thorough electron microscopy r e p l i c a study on s l i p traces i n order to e s t a b l i s h the s i g n i f i c a n c e of cross s l i p i n p o l y c r y s t a l l i n e zinc and cadmium. 3) An extensive r e s i s t i v i t y study of various deformed states i n order to e s t a b l i s h the relevance of sing l e vacancy motion to dynamic recovery. APPENDIX I - 1 ^ 7 - I ;.'.vi.:,Rate,, Theory As mentioned i n section 2 . 1 . when deformation i s goverened by a. si n g l e rate controlling..process, the shear s t r a i n rate may be expressed by )f = Jo e ( 1 ) 81 A more general expression as indicated.by Dorn , i s given by y SJ AQt/kT 5L= J 0 .e . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 ) t h where " i " r e f e r s to the i ' kind of mechanism. •Several d i f f e r e n t thermally activ a t e d processes can operate at the same time. I f they occur s e q u e n t i a l l y then the steady state s t r a i n rate observed during c r e e p / w i l l be associated with the slowest process, i . e . the process with the highest a c t i v a t i o n energy.' If-the s t r a i n rate i s f i x e d as i s usual i n a , t e n s i l e test,.then the magnitude of the stress w i l l r e f l e c t ' t h e p a r t i c u l a r c o n t r o l l i n g process i n that any a c t i v a t i o n process occurs under the combined influence of thermal energy and the e f f e c t i v e s t r e s s . An example of sequential processes would be the movement of a jogged screw d i s l o c a t i o n during pyramidal g l i d e i n a hexagonal metal. Energy must be supplied f o r the non-conservative motion of the jog, f o r f o r e s t i n t e r s e c t i o n , to overcome the P e i e r l s stress and f o r possible cross s l i p around obstacles.. I f th i s , sequence of processes, must occur before deformation can proceed, then one of the processes w i l l be rate controlling..Which one w i l l depend on the nature of the e f f e c t i v e stress on.the d i s l o c a t i o n at any stage of the process. The stress w i l l increase, to a value at which the thermal energy a v a i l a b l e at that temperature i s s u f f i c i e n t to continue deformation -148 - at the required s t r a i n r a te. Under such a sequential system i t i s quite possible that the rate c o n t r o l l i n g process can change during deformation from one of the aforementioned processes to another. - I f however, two or more independent processes contol the s t r a i n rate, then, the t o t a l s t r a i n rate i s given by ^ )( ^  and. the a p p l i c a t i o n of simple rate theory to deformation i s not p o s s i b l e . . I f i t assumed that a single process i s rate c o n t r o l l i n g then from (1) A G = -kT In-i/j . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 ) I f i t i s now assumed that at a constant d i s l o c a t i o n configuration thei. s t r a i n rate i s given bjp i = i ( T , ................(4) then by d i f f e r e n t i a t i n g . (4) withi.respect to temperature at constant e f f e c t i v e stress and recombining AG = - k T 2 / ^ m I c ) T * \ + T / G>AG\ Since i n general the a c t i v a t i o n entropy i s given by A s = - M A G \ .......(6) then the a c t i v a t i o n enthalpy i s given by AH = -kT 2 / 3 m i/i* \ I ................a) The a c t i v a t i o n volume defined as -•149 - v = b d l ..................... ...(8) where b = Burgers vector d = a c t i v a t i o n distance 1 = length of d i s l o c a t i o n . undergoing a c t i v a t i o n . i s given by the stress dependence of AG such that v =-/_p\AG_\ = kT / din jl \ .................. (|) . Therefore a c t i v a t i o n energy and a c t i v a t i o n volume can be determined from r e v e r s i b l e temperature and s t r a i n rate changes during deformation. Probably the l a r g e s t source of er r o r i n t h i s type of c a l c u l a t i o n can be traced to the lack of r e v e r s i b i l i t y i n some systems. I f , which i s a function of the active d i s l o c a t i o n density, changes during the s t r a i n rate change, then the measured value of A l and subsequent c a l c u l a t e d values of v, A G , and AH w i l l be i n error . In order to c a l c u l a t e AC-., i t i s necessary to develop an expression whereby the entropy term ... / c) AG \ can be r e a d i l y evaluated. from .experiment- so \ d T fa* a l data. Mitra and Dorn using a graphical technique have attempted such a c a l c u l a t i o n but i n the process appear to have interchanged free energy and 7 4 enthalpy. Schoeck ± n a consistent thermodynamic treatment has a r r i v e d at energy change (thermal % V d T n T dn P- d T AG = AH + ..(10) - 'I50 = This he claims makes possible a simple c a l c u l a t i o n of AG since i t only contains terms that can be e a s i l y determined from experimental data. However h i s formulation i s based on the questionable r e l a t i o n s h i p i = y ( T , T a ) . . . . . . . . . . . . . . . . . . . . . . ( i i ) This d i f f e r s from (4) i n the use of the applied stress ^ a instead of the e f f e c t i v e stress Q'"*" . The s t r a i n rate f o r a given system under c e r t a i n conditions of temperature and applied stress i s associated with a c e r t a i n . rate c o n t r o l l i n g mechanism. This mechanism operates under the combined influence of the thermal energy a v a i l a b l e and the e f f e c t i v e stress 7 • Ultimately therefore the s t r a i n rate and the e f f e c t i v e stress are dependent variables and the dependence of \ on the macroscopic flow stress f i s u n j u s t i f i e d . I f 1 i s substituted f o r T3. i n Schoeck s expression ( 1 0 ) , a re l a t i o n s h i p s i m i l a r to ( 1 0 ) but containing T instead of T i s obtained. a Because of the unknown nature of J during deformation, i t i s d i f f i c u l t t o make any reasonable estimate of A G . Much of the confusion i n the l i t e r a t u r e concerning rate expressions involves the statement of the basic rate equation (1).. I t has been common to substitute ZVH, the enthalpy change f o r /\G i n ( 1 ) . When t h i s i s done, • S/'k i t i s assumed that the entropy term e i s incorporated i n t o the pre- exponential term Vo . This approximation of the rate equation i s v a l i d only i f the entropy change does not represent a s i g n i f i c a n t contribution to the o v e r a l l free energy change and i f i t does not vary appreciably with temperature or s t r e s s . 73 As outlined by Conrad. , attempts have been made to calculate the energy of activation, including the work done by the e f f e c t i v e stress during thermal a c t i v a t i o n . In t h i s case t h i s " t o t a l " a c t i v a t i o n energy i s us u a l l y 151 expressed as AH0= AH + Fd : ....(12) where F= the force on the d i s l o c a t i o n segment Since Fd = lbd^f*= v j * , therefore AH 0 = AH + v i * . (15) where v j represents the work done by the applied stress during thermal a c t i v a t i o n . However the energies represented i n (15) should be free energies so that > AG D = AG + v!r* (14) A t y p i c a l force distance diagram i s shown i n F i g . 7 8 to i l l u s t r a t e the various energy terms. The term A;G0 must be l a b e l l e d only as the"apparent" a c t i v a t i o n energy since i t does not include the work done by the e f f e c t i v e stress before the a c t i v a t e d event. The true a c t i v a t i o n energy i s given by the t o t a l area under the force distance curve (AGo ). This can be a s c e r t - ained only i f the conditions under which T = 0 are known. When T = 0 , AG = AG 0 * AGo • However even i f the c r i t i c a l temperature where T = 0 can be accurately established, i t must be assumed that the force distance curve does not change with temperature i f the value of AG„ found at T c i s applied to other temperatures where the same rate c o n t r o l l i n g process i s thought to occur. Even i f the same process i s thought to occur over a range of temperatures, i f the stacking f a u l t energy changes with temperature then f o r some processes the shape of the force-distance diagram can change with temperature. The c a l c u l a t i o n of AG o Is therefore r e s t r i c t e d to conditions -152 - ft* of y i e l d when J can be estimated from the y i e l d stress-temperature r e l a t i o n  ship . 1 1 Gregory developed a s i m i l a r r e l a t i o n s h i p to expression (13) i n order to cal c u l a t e t h e " t o t a l " a c t i v a t i o n energy. However he used the applied stress T instead of y*~ i n the l a s t term. This i s u n j u s t i f i e d and can lead to serious errors e s p e c i a l l y when Ta >'* jT . - 153 - A G = ABCA VT* = ACxgX-jA A G Q = x-^Cx^x A G * = x 0ABCx^x B Distance F i g . 7 8 Typical.Force-Distance curve f o r a thermally a c t i v a t e d deformation process. - 15+ - APPENDIX 2  [ Unloading Y i e l d Points i n Cadmium. During the e a r l y stages of deformation when c y c l i n g 25p. and kOOp. cadmium between -140°C and--196°C, s l i g h t y i e l d points as shown i n Fig.79 were observed on reloading at T-196°C . s t r e s s s t r a i n F i g . 79 -Unloading y i e l d point i n p o l y c r y s t a l l i n e cadmium These made the determination of Aer d i f f i c u l t because of the ambiguity of the y i e l d stress at -196°C. Several authors have observed unloading y i e l d points i n f . c . c . 100 101 metals. Haasen and K e l l y and Makin observed y i e l d points i n sin g l e c r y s t a l s of aluminum, copper and n i c k e l produced by unloading and reloading. They postulated that Cottrell-Lomer s e s s i l e s are produced during unloading causing a higher y i e l d stress on reloading. 102 B o i l i n g using p o l y c r y s t a l l i n e Ag,"Al, Cu, Ni and Pb, found that - 155 - u n l o a d i n g y i e l d p o i n t phenomena i s a common o c c u r e n c e i n f . c . c . . m e t a l s . He f u r t h e r o b s e r v e d , t h a t t h e magnitude o f t h e s t r e s s i n c r e a s e was dependent on t h e amount o f u n l o a d i n g a n d -independent o f time.. Y i e l d p o i n t s were o n l y o b s e r v e d however when r e c o v e r y d u r i n g t h e u n l o a d i n g c y c l e Has n e g l i g i b l e . B irnbaum, t e s t i n g z i n c and magnesium s i n g l e c r y s t a l s between 7 7 C~ and 293°K f o u n d no y i e l d p o i n t s a f t e r r e l o a d i n g . F u r t h e r he o b s e r v e d that i n cop p e r s i n g l e c r y s t a l s , t h e magnitude o f t h e s t r e s s i n c r e a s e was o r i e n t a t i o n i n d e p e n d e n t , an o b s e r v a t i o n n o t c o n s i s t e n t w i t h t h e co n c e p t o f C o t t r e l l - "•- Lomer s e s s i l e p r o d u c t i o n . He t h e r e f o r e p o s t u l a t e d t h a t a change o c c u r s d u r i n g u n l o a d i n g i n t h e n a t u r e o f g l i d e - f o r e s t d i s l o c a t i o n i n t e r a c t i o n s . I n o r d e r t o o b t a i n a b e t t e r u n d e r s t a n d i n g o f t h e y i e l d phenomena i n cadmium, 25u and kOOp, specimens were examinedVat -196°C. They were d e f  ormed u s i n g 2$ s t r a i n i n c r e m e n t s f o l l o w e d b y u n l o a d i n g t o a g i v e n p e r c e n t a g e o f t h e f l o w s t r e s s . Specimens were t h e n h e l d f o r v a r i o u s p e r i o d s o f t i m e b e f o r e r e l o a d i n g . I t was f o u n d t h a t t h e h o l d i n g t i m e up t o f i v e m i n u t e s had no e f f e c t on t h e magnitude o f t h e f l o w s t r e s s on r e l o a d i n g . However t h e o c c u r r e n c e o f y i e l d p o i n t s was a f u n c t i o n o f t h e amount o f u n l o a d i n g as p r e v i o u s l y observed^ by B o i l i n g . No y i e l d phenomena was o b s e r v e d u n t i l a t l e a s t 40$ o f t h e l o a d was removed. - A t low v a l u e s o f s t r a i n o~ a= cr ( F i g . 80 ), However w i t h i n c r e a s i n g d e f o r m a t i o n , o*"a became somewhat g r e a t e r t h a n o~ c. B o i l i n g o b s e r v e d s i m i l a r b e h a v i o u r and e x p l a i n e d i t i n terms o f c r e e p d u r i n g t h e u n l o a d i n g c y c l e . F i g . 8 l i l l u s t r a t e s t h e v a r i a t i o n o f 0^/05 w i t h i n c r e a s i n g s t r a i n . I n a l l c a s e s specimens were u n l o a d e d t o 10$ o f t h e f l o w s t r e s s and h e l d f o r f i v e m i n u t e s b e f o r e r e l o a d i n g . The u n l o a d i n g o p e r a t i o n was c o n t i n u o u s and t o o k about t e n se c o n d s . -156 - Fig . Q p Unloading y i e l d point terminology - 157 - I t i s observed from F i g . 8 l that the y i e l d e f f e c t i s considerably greater in.400u cadmium than i n 25". The values of ^b/c% remain approx imately constant up to a value of s t r a i n which has been previously i d e n t i f i e d with the s t a r t of dynamic recovery at -196°C. Beyond t h i s value of s t r a i n , Ob/og decreased and the y i e l d e f f e c t gradually disappeared. Specimens were a l s o tested at -95°C and no y i e l d e f f e c t s were observed. In f a c t s l i g h t s t a t i c recovery occurred. The drop i n stress assoc} iated with unloading and: t h i r t y second holding at 10$ of the flow s t r e s s , i s shown as a function of the flow stress i n F i g . 82 .Therefore during C o t t r e l l - S t o k e s t e s t s , appropriate corrections were made to A o~ to take A T into account the s t a t i c recovery occurring during the operation of changing temperatures. -158 -- 159 - Fig.82 The decrease i n flow stress due to s t a t i c recovery during interrupted t e s t i n g of 25u cadmium at -95°C. ( 30 second holding at 10$ of the flow stress ). .-•i6o - APPENDIX j The Determination of A°~ from-Strain Rate Change Tests An i d e a l s t r a i n ; r a t e change should occur i n an instantaneous fashion without an intermediate drop i n load or a measurable time l a g during the change. These factors become of extreme importance i f one i s attempting to analyze materials at high e f f e c t i v e temperatures where recovery can occur. When using an Instron, mechanical d i f f i c u l t i e s are sometimes responsible f o r a considerable e r r o r i n the determination of ACT . Although i the Instron used during t h i s work was equipped with an automatic push button crosshead speed change, a considerable time l a g was observed during a de-r crease i n s t r a i n rate. When the crosshead speed was changed from .02"/min. to ,002"/min. there was a delay time of 1.6 seconds during which time the machine stopped. When changing from .2"/min. to .02"/min. the delay time was .8 seconds. However at these higher speeds there was also a s l i g h t r e v e r s a l of the screws during the change which caused a.drop i n load on the specimen. There was no measurable delay time associated with a change to an.increased crosshead speed. Therefore a l l j A 0 " values were obtained during an increase i n s t r a i n rate. In many materials i t i s necessary to obtain Ao~ from a decrease i n s t r a i n rate because of the appearance of d i s t i n c t y i e l d points on increasing the s t r a i n rate. For t h i s work i t was decided that any error a r i s i n g from any' s l i g h t y i e l d phenomenon would be much l e s s than that r e s u l t i n g from the time dellay during a decrease i n s t r a i n rate. - . : i 6 l - Fig.83 i l l u s t r a t e s the nature of the change i n flow stress during s t r a i n rate change t e s t s under various conditions. (a) below T = .26 i n l i n e a r hardening regions . Fig.83 v The nature of the flow stress obtained during s t r a i n rate change t e s t s i n p o l y c r y s t a l s - 162 - Obtaining values at low temperatures was r e l a t i v e l y easy because of the abrupt nature of y i e l d a f t e r a change i n s t r a i n rate ( F i g . 83 a) . With increasing s t r a i n however y i e l d became more gradual s i m i l a r to that obtained at a l l values of s t r a i n at temperatures above = .26 . Under these conditions /L\a~ was obtained by extrapolating the e l a s t i c and work hardening regions.(Fig.83 b) . As the temperature increased to the region of T = .hO a i t became extremely d i f f i c u l t to obtain r e l i a b l e values of Ao~ because of the almost completely parabolic nature of y i e l d a f t e r an increase i n s t r a i n rate (Fig.83 c ) . For t h i s reason rate theory c a l c u l a t i o n s were not ; attempted above T = .k-0 H Some work softening occurred on decreasing the s t r a i n rate at a l l temperatures above T .= .26 .=This gave f i s e . t o a more gradual H decrease i n the observed flow stress a f t e r the s t r a i n rate change had been made. This became more pronounced with increasing temperature ( F i g . 83 c ) . I f the load was removed and immediately reapplied ifenefe was no evidence of any y i e l d point which i s u s u a l l y associated with work softening. However y i e l d points were observed during s t r a i n rate changes ok on cadmium single c r y s t a l s s i m i l a r t o those reported by Langenecker i n aluminum and zin c . I t would therefore appear that i n p o l y c r y s t a l s at elevated temperatures considerable d i s l o c a t i o n rearrangement can occur i n the time necessary f o r a decrease i n s t r a i n rate. One i s faced with the d i f f i c u l t y of e s t a b l i s h i n g the flow stress along a-b (Fig.83 c) at which y i e l d i s occurring at the reduced s t r a i n rate. Since t h i s region u s u a l l y involves up to at l e a s t .5$ s t r a i n there i s considerable doubt i f any such determination r e a l l y r e f l e c t s a r e v e r s i b l e change. - 163 - Some ambiguity i s a l s o associated 1with an increase i n s t r a i n r a t e. 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