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Measurement and statistical interpretation of slip line length and microstrain in copper single crystals Garner, Andrew 1974

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MEASUREMENT AND STATISTICAL INTERPRETATION  OF SLIP LINE  LENGTH AND MICROSTRAIN IN COPPER SINGLE  CRYSTALS  by ANDREW GARNER B.Sc,  University  A THESIS SUBMITTED  o f L i v e r p o o l , 1968  IN PARTIAL  THE REQUIREMENTS  FULFILMENT OF  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n t h e Depa r t m e n t of METALLURGY  We a c c e p t  this  t h e s i s as c o n f o r m i n g  to the  required standard  THE UNIVERSITY  OF BRITISH COLUMBIA  December, 197^  In presenting this thesis  in partial fulfilment of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this  thesis  for scholarly purposes may be granted by the Head of my Department or by his representatives.  It  is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of B r i t i s h Columbia Vancouver 8, Canada  ABSTRACT  In o r d e r t o t e s t the a p p a r e n t l y c o n f l i c t i n g p r e d i c t i o n s of some c u r r e n t t h e o r i e s of s t r a i n hardening,  slip  line  l e n g t h measurements were made on a s e r i e s of o r i e n t e d copper s i n g l e c r y s t a l s , i d e n t i c a l l y p r e s t r a i n e d a t 673°K, p o l i s h e d and i n c r e m e n t a l l y s t r a i n e d a t temperatures  between 573°K and  s l i p l i n e s formed during low temperature  increments  were found  to be longer than those formed d u r i n g s t r a i n increments h i g h e r temperature  4.2°K;  at  (Garner and Alden, 1974).  The r e s u l t i s shown to be i n c o n f l i c t w i t h any of  s t r a i n hardening  i n which s l i p l i n e s are b l o c k e d by  theory  specific  o b s t a c l e c o n f i g u r a t i o n s , such as L o m e r - C o t t r e l l b a r r i e r s , ribbons of converted p i l e - u p s or d i s l o c a t i o n c e l l w a l l s .  In  c o n t r a s t , the r e s u l t i s shown to be c o n s i s t e n t w i t h t h e o r i e s of s t r a i n hardening  i n which s l i p l i n e s are b l o c k e d by  i n t e r a c t i o n between expanding  g l i d e loops and f o r e s t  statistical disloca-  t i o n s , on the c o n d i t i o n t h a t , w i t h i n the framework of such a theory, the g l i d e loops are able to expand a t h e r m a l l y over a newly a v a i l a b l e f r e e area of s l i p p l a n e , a f t e r a t h e r m a l l y activated process. are d i s c u s s e d .  Two  possible thermally activated  A u n i f i e d view of s l i p  lines properties i s  presented which i s shown to p r o v i d e a s e l f - c o n s i s t e n t t i o n o f the temperature  processes  v a r i a t i o n of s l i p  explana-  l i n e length, s l i p  band f o r m a t i o n , the e x i s t e n c e of m u l t i p o l e c a r p e t s and  the  v a r i a t i o n of flow s t r e s s with temperature. The investigated  s t a t i s t i c a l aspects of t h i s i n t e r p r e t a t i o n were f u r t h e r by o b t a i n i n g  a s e r i e s of o r i e n t e d  77°K m i c r o s t r a i n  curves from  copper s i n g l e c r y s t a l s , p r e s t r a i n e d  temperatures between 1000°K and  at  77°K, to produce d i s l o c a t i o n  m i c r o s t r u c t u r e s w i t h d i f f e r i n g degrees of r e g u l a r i t y , y e t approximately the  same o v e r a l l d e n s i t y .  The  forest  dislocation  m i c r o s t r u c t u r e s of an i d e n t i c a l l y prepared s e r i e s of were examined u s i n g a d i s l o c a t i o n e t c h on plane.  used to measure l o c a l d i s l o c a t i o n d e n s i t i e s .  new  parameter i s i n t r o d u c e d , namely the  quantifies  slip  d e v i s e d , which  was  standard d e v i a t i o n ,  crystals  the primary  A s t a t i s t i c a l sampling technique was  with  In  addition,  r a t i o of the  sampled  to mean l o c a l d i s l o c a t i o n d e n s i t y ,  which  the degree of r e g u l a r i t y of a d i s l o c a t i o n m i c r o -  structure.  A l l m i c r o s t r u c t u r e s were found to have a s m a l l e r  degree of r e g u l a r i t y than a random d i s t r i b u t i o n . For at any  crystals prestrained  g i v e n f r a c t i o n of the  microstrain  was  77°K y i e l d s t r e s s , the  found to i n c r e a s e as the  became l e s s r e g u l a r . exhibited  a t temperatures above 293°K,  Crystals  microstructures  prestrained  the Haasen-Kelly e f f e c t , which was  r e s t r i c t e d source o p e r a t i o n . operate, the  a t and  below 293°K  attributed  to  However, once sources begin  amount of m i c r o s t r a i n  of r e g u l a r i t y was  amount of  indeed d e t e c t e d .  anticipated  from the  to  degree  TABLE OF CONTENTS  Page ABSTRACT .  i i  LIST OF TABLES  v i i  LIST OF FIGURES  viii  ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . .  xiii  PART 1  SLIP LINE STUDIES  .1  1.1  Introduction . . .  1  1.2  E x p e r i m e n t a l Techniques  5  1.3  1.2.1  Specimen P r e p a r a t i o n .  . . . . . . . .  1.2.2  Mechanical T e s t i n g . . .  10  1.2.3  S l i p L i n e Measurement  14  Results  5  19  1.3.1  S l i p L i n e Lengths . . . . . . . . . .  19  1.3.2  V a r i a t i o n i n Flow S t r e s s w i t h Temperature  31  1.3.3  S l i p L i n e Density  31  1.3.4  S l i p Step Height. . . . . . . . . . .  33  1.3.5  Sources o f E r r o r  33  iv  Part  Page 1.4  Discussion 1.4.1 . 1.4.2 1.4.3  . . . . . . . . . .  . . .  36  S l i p L i n e B l o c k i n g by S p e c i f i c Dislocation Arrays. . . . . .  36  Flow S t r e s s and the G l i d e / Forest Interaction. . . . . .  52  S t a t i s t i c a l Blocking o f S l i p Lines  55  A U n i f i e d View o f S l i p L i n e Formation, M i c r o s t r u c t u r a l Observations and Flow S t r e s s .  71  Transition Effects.  76  MICROSTRAIN AND ETCH PIT STUDIES .  78  1.4.4  1.4.5  2.1  Introduction  2.2  Experimental Technique . .  81  2.2.1  Specimen P r e p a r a t i o n  81  2.2.2  Crystal Orientation  81  2.2.3  Mechanical T e s t i n g  83  2.2.4  Metallography  86  .2.3.  . .  l  78  . .  Results.  97  2.3.1  M i c r o s t r a i n Curves  97  2.3.2  Metallography . .  109  2.3.3  Average D i s l o c a t i o n Density and Flow S t r e s s .  121  2.3.4  Local Dislocation Densities .  121  2.3.5  The E f f e c t o f Subgrain Boundar es on the D i s t r i b u t i o n s . . . . .  136  2.3.6 • R e p r o d u c i b i l i t y  v  137  Part  Page 2.4  Discussion  137  2.4.1  Introduction  137  2.4.2  C r y s t a l s with above 293°K  2.4.3 2.4.4  Prestrains . . . . . . 138  Q u a n t i t a t i v e Estimates from Theories of M i c r o s t r a i n C r y s t a l s P r e s t r a i n e d a t 293°K and Below  . 141 150  SUMMARY AND CONCLUSIONS  153  REFERENCES  157  APPENDICES 1  C a l c u l a t i o n o f Work Hardening Rate i f S l i p L i n e s are Blocked by C e l l Walls.  163  D e f i n i t i o n s of Terms Used t o Describe Deformation . . . . . . .  166  3  Details of Microstrain Testing  170  4  O p t i c a l Microscopy with Normarski I n t e r f e r e n c e Contrast* . .  175  A P r o b a b i l i t y Model f o r Random Distributions .  17 8  2  5  vi  LIST OF TABLES  Table 1  '  .  Page  P a i r s of S l i p Systems which '  could  Produce L o m e r - C o t t r e l l Locks . . . . .  . . . . .  .40  2  D i s l o c a t i o n E t c h Compositions  92  3  Area S i z e s used to Sample Dislocation Densities.  95  vii  LIST OF FIGURES  Figure 1  2  .  Page  G r a p h i t e mold assembly used to grow s i n g l e crystals . . . . . . . . . . . . . . . . . . . . . ,. O r i e n t a t i o n of t e n s i l e a x i s f o r c r y s t a l s used i n s l i p l i n e work . . . . . . . . . .  3  673°K p r e s t r a i n  4  13  Acetone r e f l u x apparatus used i n p r e p a r a t i o n of r e p l i c a s . . . . .  . . . .  20  5  Histogram of s l i p l i n e lengths a f t e r 0.5% i n c r e m e n t a l s t r a i n a t 573°K. . . . . . . .  . . . .  22  6  Histogram of s l i p l i n e l e n g t h s a f t e r 0.5% i n c r e m e n t a l s t r a i n a t 293°K. . . . . . . . . . . .  23  7  Histogram of s l i p l i n e lengths a f t e r 0.5% i n c r e m e n t a l s t r a i n a t 190°K. . . . . . .  24  8  Histogram of s l i p l i n e lengths a f t e r i n c r e m e n t a l s t r a i n a t 77°K  0.5% . . . . . .  25  Histogram of s l i p l i n e lengths a f t e r 0.5% i n c r e m e n t a l s t r a i n at 4.2°K. . . . . . . . . . . .  26  Average s l i p l i n e l e n g t h versus temperature of i n c r e m e n t a l s t r a i n i n g . . . . . . . . . . . . .  27  O p t i c a l micrographs showing change i n s l i p l i n e l e n g t h w i t h temperature . . . . .  28  10  11  ...  viii  . . ...  . . . . . 9  . . ."• . . .  9  curve.  7  Figure  Page  12  13  E l e c t r o n micrographs o f s l i p l i n e s formed a f t e r s t r a i n increments a t (a) 4.2°K and (b) 293°K r e s p e c t i v e l y . . . . . . . . . . .  . . .  Histogram o f s l i p l i n e d e n s i t y , l i n e s formed a t 293°K . ... . . ... . . . . . . . .  . . .  14 .  Shadowed r e p l i c a w i t h carbon c a l i b r a t i o n spheres . . . . . . . . . . . . . . . . . . . .  15  T r a n s i t i o n and steady s t a t e s t r a i n observed  ;  :  3  0  32  34  d u r i n g i n c r e m e n t a l t e s t a t 4.2°K. .  37  16  S l i p l i n e b l o c k i n g according t o Seeger (1957) . .  41  17  S l i p l i n e b l o c k i n g a c c o r d i n g to H i r s c h and M i t c h e l l (1967) ... . . . . . . . . . . . . . . Average d i s l o c a t i o n c e l l diameters i n copper versus normalized flow s t r e s s . . . . . . . . . .  45  18  19  20  21  22  23  24  . A v e r a g e s l i p l i n e l e n g t h and average c e l l diameter i n copper . . . . . . . . . . . . .  48  .  50  Diagrams from Hocks' o r i g i n a l s t a t i s t i c a l . a n a l y s i s (1966) . . . . . . . . . . . . . . . . .  58  Schematic of primary s l i p plane showing . s t a t i s t i c a l b l o c k i n g of s l i p l i n e s d u r i n g steady s t a t e flow . . ... . . . . . . . . . . . . .  61  Data from a n a l y s i s of M o r r i s and Klahn (1974)... . . . . . . . . . . . . . . . . . . .  68  C r o s s - s e c t i o n p e r p e n d i c u l a r t o primary g l i d e plane showing s l i p l i n e b l o c k e d i n schematic c e l l s t r u c t u r e . . . . . . . . . . .  74  .  O r i e n t a t i o n of t e n s i l e a x i s f o r c r y s t a l s used i n m i c r o s t r a i n / e t c h p i t work . . . . . . . .  ix  82  Figure 25  Page P r e s t r a i n curves f o r c r y s t a l s used i n m i c r o s t r a i n / e t c h p i t work  85  26  77°K m i c r o s t r a i n curve f o r 68B prestrain).  (1000°K  98  27  77°K m i c r o s t r a i n curve f o r 65B prestrain)  (850°K  77°K m i c r o s t r a i n curve f o r 69B prestrain).  (750°K  77°K m i c r o s t r a i n curve f o r 78B prestrain)....  (293°K  77°K m i c r o s t r a i n curve f o r 70B prestrain)  (77°K  28  29  30  31  32  99  100  101  O r i g i n a l load-elongation 68B (1000°K p r e s t r a i n )  curve f o r  O r i g i n a l load-elongation  curve f o r  70B  (77°K p r e s t r a i n )  105  Normalized m i c r o s t r a i n curves  34  Etched d i s l o c a t i o n m i c r o s t r u c t u r e s t a t i c anneal Etched d i s l o c a t i o n m i c r o s t r u c t u r e c y c l i c anneal . . . .  36  37  38  102  104  33  35  .  Etched d i s l o c a t i o n m i c r o s t r u c t u r e c y c l i c anneal  .  108  after 110 after  ......  m  after . . . . .  112  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 6 8A (1000°K p r e s t r a i n ) xl560. . . . . . . . . . . . .  115  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 63A (850°K p r e s t r a i n ) xl560 . . . .  116  x  Figure 39  40  41  42  43  44  45  46  47  48  49  Page T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 69A (700°K p r e s t r a i n ) xl560 .  .  117  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 78A (293°K p r e s t r a i n ) xl560  118  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 70A (77°K p r e s t r a i n ) xl560  119  Etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 78A (293°K p r e s t r a i n ) . . . . . . . .  120  D i s l o c a t i o n d e n s i t y versus r e s o l v e d shear s t r e s s . .  .  122  Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area d e n s i t i e s from 68A (1000°K p r e s t r a i n ) . . . . .  124  Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area d e n s i t i e s from 63A (850°K p r e s t r a i n ) .  125  Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area d e n s i t i e s from 69A (700°K p r e s t r a i n ) . . . . . . . . . . . .  126  Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area d e n s i t i e s from 78A (293°K p r e s t r a i n )  127  Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area d e n s i t i e s from 70A (77°K p r e s t r a i n )  128  Schematic o f r e g u l a r , random and more c e l l u l a r d i s t r i b u t i o n s , t h e i r corresponding histograms and degrees of r e g u l a r i t y (s/x ) . . .  129  xi  Figure 50  51  52  53  54  55  56  Page C a l c u l a t e d histograms f o r random d i s t r i b u t i o n of p o i n t s  .  131  Degree of r e g u l a r i t y , s/x, f o r a range of sample s i z e s f o r upper t h r e e p r e s t r a i n temperatures. . . . . . . . . . . . . . . . . . .  133  Degree of r e g u l a r i t y , s/x, sample s i z e s f o r lower two temperatures  134  f o r a range of prestrain . . ... . . . .  Dot p a t t e r n taken from micrograph by S t r u t t and Gupta (1967) . . . . . . . .  . . . . .  135  Change i n normalized f r e e area, a/ao, w i t h r e l a t i v e a p p l i e d s t r e s s from Kocks (1967) and corresponding s t r a i n . . . . . . . . .  144  C a l c u l a t e d m i c r o s t r a i n curves u s i n g Alden's f r e e area f u n c t i o n . . . . . . . . . . . . . . . .  146  C a l c u l a t e d m i c r o s t r a i n curves u s i n g Ashby's function. . . . . . . . . . . . . . ... ...  57  Replotted  normalized m i c r o s t r a i n curves . . . . .  58  Schematic s t r e s s s t r a i n curves.  59  M i c r o s t r a i n t e s t i n g apparatus . . . .  60  O p t i c a l micrographs of the same area showing advantage of Normarski Interference Contrast  xii  . . .  .148 149 167  . . . . . .  172  176  ACKNOWLEDGEMENTS  The ment g i v e n other  author i s g r a t e f u l f o r the advice  by D r . T.H. A l d e n .  f a c u l t y members,  also greatly  Thanks a r e a l s o e x t e n d e d t o  and t o f e l l o w g r a d u a t e s t u d e n t s , f o r  many h e l p f u l d i s c u s s i o n . is  The a s s i s t a n c e o f t e c h n i c a l  Council  staff  appreciated.  Financial assistance Fellowship  and encourage-  i n t h e f o r m o f an  ( f o r two y e a r s ) , a n d f r o m t h e N a t i o n a l  (N.R.C. G r a n t No. A-4991) i s g r a t e f u l l y  xiii  Alcan Research  acknowledged.  P A R T  1  SLIP LINE STUDIES  1.1  Introduction A u n i v e r s a l l y observed f e a t u r e o f deformation i n  metals  i s the p r o d u c t i o n of s l i p l i n e s .  In pure f a c e c e n t r e d  c u b i c metal s i n g l e c r y s t a l s , the study of s l i p p a r t of the l a r g e body of experimental work attempt  l i n e s has been  performed  i n an  t o formulate t h e o r i e s of s t r a i n hardening, and t o  d i s t i n g u i s h between the t h e o r i e s of flow s t r e s s which each of  the s t r a i n hardening t h e o r i e s i n c o r p o r a t e .  Since  slip  l i n e s are a s u r f a c e m a n i f e s t a t i o n of m i c r o s t r u c t u r a l events, and are u s u a l l y observed to have l i m i t e d l e n g t h , an  essential  f e a t u r e of any theory of s t r a i n hardening i s a mechanism by which s l i p l i n e s are b l o c k e d .  The p r e s e n t experiments  were  designed t o d i s t i n g u i s h between s t r a i n hardening t h e o r i e s by i n v e s t i g a t i n g the temperature  dependence of s l i p  line  lengths. Some of the e a r l i e s t work i n the f i e l d of metal deformation i n c l u d e d the study of s l i p l i n e s , s i n c e they were  1  2  one of the few d i r e c t l y observable of a p l a s t i c a l l y s t r a i n e d metal.  microstructural features The  observation of s l i p  began w i t h the work of Ewing and Rosenhain the i n i t i a l o b s e r v a t i o n s Elam, 1923), the f i r s t by Yamaguchi (192 8).  (1899), and  from s i n g l e c r y s t a l s  comprehensive study was A major impetus f o r s l i p  after  (Taylor  and  carried line  lines  out  research  came with the f i r s t use of t r a n s m i s s i o n e l e c t r o n microscopy to  examine r e p l i c a s from metal s u r f a c e s  (Heidenreich  and  Shockley, 1947), which s t i m u l a t e d a g r e a t d e a l o f subsequent work (Read, 1952; et al.,  1955;  Kuhlman-Wilsdorf and W i l s d o r f , 1953;  Noggle and Koehler,  1957;  Blewitt  Seeger et a l .  1957;  3  F o u r i e and W i l s d o r f , 1959). Since s l i p  l i n e s were found to have f i n i t e  a mechanism f o r b l o c k i n g t h e i r growth was key  f a c t o r i n any  1957).  I t was  thought to be  e x p l a n a t i o n of s t r a i n hardening  shown t h a t s l i p  length,  (Seeger,  l i n e s formed a t h i g h e r  (Mader, 1957;  experimental  o b s e r v a t i o n together w i t h a p o s t u l a t e d b l o c k i n g  hardening  3  1957)  and  strains  were s h o r t e r  mechanism was  Seeger et al.  a  i n c o r p o r a t e d i n t o an e a r l y theory of  this  strain  (Seeger, 1957). Another important  > slip  l i n e p r o p e r t y , namely t h a t  s l i p l i n e s are formed i n a s h o r t time i n t e r v a l , was by cinematography  (Maddin and Chen, 1954), and the more r e c e n t  a c o u s t i c emission work of F i s h e r and L a l l y t h i s observation.  revealed  (L967)  confirms  S t u d i e s u s i n g both these techniques  indicate  3  t h a t s l i p l i n e s are formed by d i s l o c a t i o n s moving w i t h t i v e l y high v e l o c i t i e s example).  (up to 10  This observation  3  cm  sec  - 1  rela-  i n copper f o r  i s s i g n i f i c a n t since i t implies  t h a t the g l i d e process by which s l i p l i n e s are formed, i s essentially  athermal.  D i r e c t and d e t a i l e d o b s e r v a t i o n s  of 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 became p o s s i b l e f o r the f i r s t time when t r a n s mission  e l e c t r o n microscopy was  i n i t i a l l y by Bollman  (1956) and by H i r s c h et al.  In the subsequent search blocked  a p p l i e d to t h i n metal f o i l s , (1956).  f o r c o n f i g u r a t i o n s or o b s t a c l e s  which  s l i p i n pure metals, the study of s l i p l i n e s per  se  declined. The may  d e c l i n e i n p o p u l a r i t y of s l i p l i n e  research  be r e l a t e d a l s o to the work of Kramer (1961) and  t h a t of F o u r i e  (1968).  Both showed t h a t the s u r f a c e  later region  of a deformed c r y s t a l has p h y s i c a l p r o p e r t i e s which d i f f e r from the i n t e r i o r .  That the r e s u l t s o f Kramer  apparently  showed the s u r f a c e t o be harder w h i l e those of F o u r i e seemed to i n d i c a t e the reverse, only added to the growing doubts over the r e l i a b i l i t y  of s u r f a c e o b s e r v a t i o n s .  p r e v i o u s l y mentioned p r o p e r t i e s of s l i p length d e c r e a s i n g  with s t r a i n ;  And lines  w h i l e the (finite  l i n e s formed i n s h o r t  i n t e r v a l s ) were not s e r i o u s l y c h a l l e n g e d  length; time  by these experiments,  few q u a n t i t a t i v e s l i p l i n e s s t u d i e s were subsequently undertaken.  three  Recently on the formation  however, the e f f e c t of the s u r f a c e l a y e r  of s l i p l i n e s i n copper was  by Himstedt and Neuhauser  (1972).  demonstrated  By c o r r e l a t i n g the r e s u l t s  of t h e i r d i f f e r e n t i a l p o l i s h / i n c r e m e n t a l s t r a i n experiments with e l e c t r o n microscopy s t u d i e s , i t may  be concluded  that  s l i p l i n e s produced a f t e r the removal of a s u r f a c e l a y e r (which i s formed d u r i n g p r e s t r a i n ) are c h a r a c t e r i s t i c of interior slip  events.  In a d d i t i o n to measurements of s l i p l i n e observations  of v a r i o u s other c h a r a c t e r i s t i c s of s l i p  have been r e p o r t e d i n the l i t e r a t u r e .  of s l i p l i n e s and Hargreaves  Clarebrough  (1959), and the study by Mader (1957) i s o f However, i n r e l a t i o n to t h e o r i e s of  length.has-  emerged as the most s i g n i f i c a n t  Hence the present  the o b s e r v a t i o n of t h i s  study  e i t h e r by s p e c i f i c o b s t a c l e s  and M i t c h e l l , 1967)  or by s t a t i s t i c a l  property  i s d i r e c t e d towards  theories s l i p (Seeger, 1957;  lines  are  Hirsch  i n t e r a c t i o n between  f o r e s t and g l i d e d i s l o c a t i o n s (Alden, 1972). d i f f e r e n c e , these two  strain  property.  In c u r r e n t s t r a i n hardening blocked  the f i n e s t r u c t u r e  (most of t h i s work i s reviewed by  p a r t i c u l a r note).  of s l i p l i n e s .  lines  These c h a r a c t e r i s t i c s  i n c l u d e s l i p l i n e h e i g h t and width, and  hardening,  length,  Because of  types of t h e o r i e s make d i f f e r i n g  this  pre-  d i c t i o n s about the v a r i a t i o n i n s l i p l i n e l e n g t h w i t h  tempera-  t u r e , at constant  slip  structure.  The  former propose t h a t  l i n e l e n g t h i s a f u n c t i o n of m i c r o s t r u c t u r e alone, and i s  5  independent  o f the deformation temperature.  In c o n t r a s t ,  the l a t t e r p o s t u l a t e s t h a t the area swept out by  expanding  d i s l o c a t i o n loops v a r i e s i n v e r s e l y w i t h temperature. s l i p l i n e l e n g t h should i n c r e a s e w i t h d e c r e a s i n g In  S e c t i o n 1.4  Hence,  temperature.  these c o n f l i c t i n g p r e d i c t i o n s , the d i f f e r e n t  t h e o r i e s of flow s t r e s s on which they are based, and o t h e r p o s s i b l e t h e o r e t i c a l developments are d i s c u s s e d more The i n t e n t o f the f o l l o w i n g experiments  was t o t e s t  the c o n t r a r y predictions o f these t h e o r i e s by measuring l i n e l e n g t h s i n copper s i n g l e c r y s t a l s . technique o f Seeger  fully.  slip  Using the c l a s s i c a l  (1957) c r y s t a l s were p r e s t r a i n e d , p o l i s h e d  and i n c r e m e n t a l l y s t r a i n e d t o r e v e a l c h a r a c t e r i s t i c s l i p lengths.  In t h i s case, however, the structure  and the temperature the p r i o r study  1.2  constant,  r a t h e r than the r e v e r s e , as i n  (Seeger, 1957).  Experimental 1.2.1  variable,  is  line  Techniques  Specimen P r e p a r a t i o n 1.2.1.1  Material  P o l y c r y s t a l l i n e copper r o d 3/8 i n c h i n diameter was o b t a i n e d from American Smelting and R e f i n i n g Company.  I t had  a nominal composition o f 99.999% copper w i t h the f o l l o w i n g quoted i m p u r i t y c o n c e n t r a t i o n s i n ppm:  6  Fe  Sb  Pb  Sn  Ni.  <0.7  <1  <1  <1  <1  3i  Bi  Ag  As  Cr  Si  Te  Se  <0.1  <0.3  <2  <0.5  <0.1  <2  <1 . <1  i n c h lengths were cleaned by immersion  S  i n concentrated  n i t r i c a c i d , f o l l o w e d by a thorough wash, f i r s t  i n water  and  then i n pure e t h a n o l .  1.2.1.2  M e l t i n g Procedure  S i n g l e c r y s t a l s w i t h 3/16  i n c h square c r o s s  section  by 7 inches long were grown u s i n g a h o r i z o n t a l f u r n a c e which t r a v e l l e d ait 5 inches h o u r torr.  - 1  under a dynamic vacuum o f  10~  5  The s i n g l e zone furnace i s r e s i s t a n c e wound w i t h  Kanthal A l over a r e c r y s t a l l i z e d alumina tube.  I t was  operated w i t h a maximum temperature of 1170°C and had a temperature g r a d i e n t a t the s o l i d l i q u i d i n t e r f a c e of about 84°C  inch  - 1  .  In  order to o b t a i n square s e c t i o n e d , seeded, pure  copper s i n g l e c r y s t a l s a s p e c i a l mold was  designed and con-  s t r u c t e d from 99.9995% s p e c t r o s c o p i c g r a p h i t e s u p p l i e d by the  U l t r a Carbon Corp., F i g u r e 1.  i s used i n the f o l l o w i n g manner: i s p l a c e d above the 3/8 rod, the  This graphite initially  i n the g r a p h i t e mold.  assembly  a double wedge  i n c h diameter p o l y c r y s t a l l i n e The whole assembly  copper  i s placed i n  q u a r t z tube which i s then evacuated and the f u r n a c e i s  a c c u r a t e l y p o s i t i o n e d over the mold. mold  mold  Copper  flows i n t o the  as the temperature i n the r e g i o n o f the rod exceeds i t s  4  D  l  ryul  i\ \ \\\ \ \ \ u( (\ \ \\ (\ \ a fflimm  Plan of Mold  uwuwwwmmw  :  —  —  'A .  .  .  .  .  .  Before Melting  -.y- - ...  Y Y Y Y T W Y Y YYT YTV\ \ \ \ T U \ \ U UTVY W \ \ W \ \ W \ \ W \ W \ \ \ \ \ W \ W  Sections:  1  8  inn  IJMC  A<,ERGROW,H  fe  1 2 3 4 5 6 7 8  seed polycrystal stock singie aystal graphite mold upper wedge lower end stop quartz tube  Before Melting  F i g u r e 1.  G r a p h i t e mold assembly used t o grow s i n g l e c r y s t a l s .  8  m e l t i n g p o i n t , and the double wedge f a l l s this liquid.  i n t o p l a c e above  The upper wedge i s then manually pushed  forward  by means o f a s t a i n l e s s s t e e l rod, so t h a t on c o n t a c t w i t h the top  o f the q u a r t z tube, a n e t downward f o r c e i s e x e r t e d  through the lower wedge onto the l i q u i d copper. the  l i q u i d i s squeezed  i n t o the mold c o r n e r s .  In t h i s way Excess  copper  i s pressed out a t the end remote from the seed, and a c t s as a r e s e r v o i r to f i l l solidification.  the mold as the copper c o n t r a c t s on  The s t a i n l e s s s t e e l push r o d s l i d e s i n a  Wilson s e a l , and thus can be manipulated from o u t s i d e the vacuum system. With a c c u r a t e p o s i t i o n i n g o f the furnace a s m a l l but s u f f i c i e n t amount o f seed m e l t s , and the seed i s wetted.  interface  A t t h i s time the motor which moves the f u r n a c e  i s switched on manually, and the c r y s t a l i s grown.  1.2.1.3  Crystal Orientation  A l l crystals  used i n the s l i p l i n e s t u d i e s had a  t e n s i l e a x i s o r i e n t e d as shown i n F i g u r e 2.  The o r i e n t a t i o n  was chosen so t h a t s i n g l e g l i d e would be the dominant mode of  deformation f o r a l l s t r a i n s used i n these experiments  ( B a s i n s k i and B a s i n s k i , 1970). i s a l s o shown on t h i s  The p o s t - p r e s t r a i n o r i e n t a t i o n  diagram.  The l a t e r a l f a c e s o f these c r y s t a l s were s e l e c t e d such t h a t one p a i r was p a r a l l e l t o the primary Burgers v e c t o r  9  Figure  2.  O r i e n t a t i o n of t e n s i l e a x i s f o r c r y s t a l s used i n s l i p l i n e work. 0 = i n i t i a l orientation, X = orientation after prestrain.  10  w i t h i n 3°.  In t h i s way  p a i r of faces was  the s l i p step h e i g h t on the  maximized.  A seed was  obtained by  growing a randomly o r i e n t e d s i n g l e c r y s t a l . d e s i r e d o r i e n t a t i o n was  other first  From t h i s  reach by a p p r o p r i a t e r o t a t i o n  s u c c e s s i v e growth c y c l e s .  the and  The Laue back r e f l e c t i o n X-ray  technique, u s i n g a copper tube, enabled  the o r i e n t a t i o n to  be determined w i t h i n 2°. Specimens were cut to l e n g t h i n a s t r a i n f r e e  way  by u s i n g an e l e c t r o l y t i c d e v i c e which allowed a p l a n a r stream of  e l e c t r o l y t e to impinge on the  specimens 3/16 cut  (anodic) c r y s t a l .  i n c h square s e c t i o n by 3 i inches long were  from each as-grown c r y s t a l .  1.2.2  Mechanical 1.2.2.1  Testing  Prestrain  The v a r i a t i o n i n s l i p l i n e l e n g t h w i t h was  Two  temperature  f i r s t d e t e c t e d i n a s e r i e s of p r e l i m i n a r y experiments.  In order to study t h i s v a r i a t i o n , a s e r i e s of samples with same m i c r o s t r u c t u r e of  the  (which would be s t a b l e over a l a r g e range  temperatures) was  required.  With t h i s i n mind, the i d e n -  t i c a l l y o r i e n t e d c r y s t a l s were s t r a i n e d to a r e s o l v e d shear s t r e s s of 1.85 673  ±  Kg mm  -2  w h i l e immersed i n a s a l t bath at  3°K. The  c r y s t a l s were deformed i n t e n s i o n i n a f l o o r  model I n s t r o n machine at a crosshead  speed of 0.01  inches m i n " . 1  11  Two Dr. J.D. two  part s t a i n l e s s s t e e l g r i p s , kindly supplied  L i v i n g s t o n , h e l d the specimen.  parts  formed a 3/16  which g r i p p e d the  i n c h square cross  c r y s t a l on  two  h a l f with r e s p e c t  cavity  gripping,  A further  to the other d u r i n g  g r i p d e s i g n , together w i t h the use  j i g , ensured t h a t the the  the  modification  the a d d i t i o n of l o c a t i n g studs to prevent minor r o t a t i o n  of one The  section  s i d e s ; to a i d  f i l e i n s e r t s were added on these s i d e s . was  When assembled  by  Instron p u l l  c r y s t a l was  tightening.  of a s p e c i a l mounting  initially  c o l i n e a r with  rod.  Most s i n g l e c r y s t a l t e n s i l e t e s t i n g produces a continuous r e o r i e n t a t i o n of the c r y s t a l l a t t i c e , which  rotates  the t e n s i l e a x i s away from the o r i g i n a l c r y s t a l l o g r a p h i c tion. and  orienta-  I d e a l l y ; i n order to avoid misalignment between t e n s i l e  c r y s t a l axes, the a x i s of a u n i v e r s a l j o i n t must of the gauge l e n g t h .  be  placed  at each end  i t was  considered s u f f i c i e n t to i n c l u d e u n i v e r s a l j o i n t s near  the ends of the g r i p s , s i n c e the  In t h i s case however,  s t r a i n s used were r e l a t i v e l y  small. . Care was way.  C l e a r l y a c r y s t a l unloaded i n the s a l t bath c o u l d  a microstructure On  taken to unload a l l specimens i n the same  d i f f e r e n t from one  the other hand, almost any  unloaded a f t e r  have  cooling.  s i n g l e procedure would p r o v i d e  acceptable  ( i . e . i s o s t r u c t u r a l ) specimens.  method was  chosen:  once the r e q u i r e d  The  following  s t r e s s l e v e l was  reached  12  the c r o s s head d i r e c t i o n was r e v e r s e d and d u r i n g u n l o a d i n g the s a l t bath was removed. In o r d e r t o a t t a i n the chosen p r e s t r e s s , a r e s o l v e d shear s t r a i n of about 20% was r e q u i r e d :  t h i s p r e s t r a i n curve  i s shown i n F i g u r e 3.  1.2.2.2  Strain  Specimens  Increments  were p o l i s h e d  (see S e c t i o n 1.2.3) t o  remove the accumulated s l i p l i n e s .  They were then i n c r e m e n t a l l y  s t r a i n e d 0.5%, a t a s t r a i n r a t e o f approximately 2.5 x 10"  3  m i n " , and a t s e v e r a l d i f f e r e n t temperatures i n the environments 1  listed  below. 57 3°K 293°K I90°K  Vacuum  Furnace  Air Petroleum  Ether  77°K  Liquid  Nitrogen  42°K  Liquid  Helium  Temperatures below the p r e s t r a i n temperature were chosen t o minimize annealing ing e f f e c t s  (Berghout, 1956) and work s o f t e n -  ( C o t t r e l l and Stokes, 1955).  The s t a b i l i t y of the  m i c r o s t r u c t u r e was demonstrated by a s t a t i c anneal f o r 4 hours a t 573°K which produced no decrease i n the y i e l d Since d e t a i l e d comparisons o f the c r y s t a l  stress.  surfaces  were t o be made, non-corroding environments were e s s e n t i a l .  13  14  In f a c t no o x i d a t i o n of the e l e c t r o p o l i s h e d c r y s t a l s u r f a c e could be observed,  u s i n g an o p t i c a l microscope a t x600 m a g n i f i  c a t i o n , a f t e r p r e s t r a i n i n any o f the environments. The  i n t e n t i n these experiments was  t o compare s l i p  l i n e l e n g t h at constant s t r u c t u r e , e s t a b l i s h e d by the above p r e s t r a i n treatment.  To c o n f i r m t h a t the s t r a i n  increment  i t s e l f produced no s i g n i f i c a n t s t r u c t u r a l change, s e v e r a l c r y s t a l s were g i v e n two an e l e c t r o p o l i s h ) , one i n e i t h e r order.  The  0.5%  s t r a i n increments  a t low temperature, slip  (separated by  and one  at high,  l i n e lengths o b t a i n e d i n t h i s  f a s h i o n were i d e n t i c a l t o those o b t a i n e d i n c r y s t a l s g i v e n a s i n g l e s t r a i n increment  (except i n one  case which was  assumed  to be s p u r i o u s ) . Some s t r a i n increments  were performed a t d i f f e r e n t  s t r a i n r a t e s a t a s i n g l e temperature,  namely 293°K.  was  t o 3 inches m i n .  v a r i e d from 3 x 10"  5  the former case a.7-inch  inches m i n  - 1  long c r y s t a l was  The r a t e In  - 1  used.  In the  latter  case an o s c i l l o s c o p e recorded the l o a d p u l s e which l a s t e d approximately  0.15  seconds, and  a u t o m a t i c a l l y unloaded  1.2.3  a f t e r which the I n s t r o n  the specimen.  S l i p L i n e Measurement 1.2.3.1 In any  Chemical  Polish  study of s l i p l i n e l e n g t h , i t i s important  t h a t s u r f a c e s be f l a t and t h a t they r e v e a l the s l i p  processes  15  which are c h a r a c t e r i s t i c of the bulk m a t e r i a l . requirements i n mind, i n i t i a l p o l i s h i n g was  With  done chemically-  using a s a t u r a t e d s o l u t i o n of c u p r i c c h l o r i d e i n hydrochloric acid. f i r s t used by J.W.  The  technique  used was  M i t c h e l l et al. . (1967).  cotton p o l i s h i n g c l o t h  (Buehler Metcloth)  over a f l a t p o l y e t h y l e n e  these  concentrated  modified  from  one  A f i n e no-nap stretched t i g h t l y  slab, c a r r i e s the chemical  polish.  The slab i s r u l e d w i t h f i n e p a r a l l e l grooves t o a i d l a t e r a l d i s p e r s i o n of the s o l u t i o n .  S l i p l i n e s formed d u r i n g  pre-  s t r a i n are removed by drawing the c r y s t a l by hand g e n t l y backwards and angles  forwards over the c l o t h , i n a d i r e c t i o n a t r i g h t  to the grooves.  When the s u r f a c e markings appeared  to have been removed, methanol  i s g r a d u a l l y added t o the  c l o t h so as to reduce the c o n c e n t r a t i o n of the p o l i s h i n g s o l u t i o n w h i l e p o l i s h i n g continued. products bright  corrosion  were u n i f o r m l y removed from the s u r f a c e , l e a v i n g a finish. All  to  In t h i s way  prevent  f o u r faces were p o l i s h e d , arid w i t h care taken  r o c k i n g of the c r y s t a l d u r i n g the above procedure,  very l i t t l e edge rounding  occurred.  1.2.3.2 E l e c t r o p o l i s h .As  a . f i n a l step i n the s u r f a c e p r e p a r a t i o n ,  c r y s t a l s were g i v e n a l i g h t e l e c t r o p o l i s h . c o n s i s t e d of two  the  The p o l i s h i n g c e l l  c o n c e n t r i c s t a i n l e s s s t e e l cathodes between  16  which the c r y s t a l was  suspended.  A s o l u t i o n of two  methanol and one p a r t concentrated the e l e c t r o l y t e , with vigorous -30°C was  maintained  nitric  a c i d was  stirring.  The  parts used as  temperature of  by u s i n g an outer bath of c o o l e d a l c o h o l .  At t h i s temperature, the F l a d e p o t e n t i a l of the c e l l i s about 7 volts. The  c e l l i s designed  d i s t a n t from two  cathodes, to ensure t h a t o p p o s i t e  c r y s t a l are simultaneously at  so t h a t the c r y s t a l i s e q u i -  polished.  faces of  Four r o t a t i o n s of  the  90°  30 second i n t e r v a l s were u s u a l l y found s u f f i c i e n t to produce  an e x c e l l e n t p o l i s h :  the  procedure was  repeated  E l e c t r o p o l i s h times of g r e a t e r than about two  minutes  per f a c e produced long range s u r f a c e u n d u l a t i o n s only be removed by going back to the chemical r e p e a t i n g the whole procedure.  which c o u l d  polish  E l e c t r o p o l i s h i n g was  by a thorough r i n s e , f i r s t i n methanol and The. w h o l e . p o l i s h i n g  i f necessary.  and followed  then i n water.  o p e r a t i o n removed not l e s s than  0.010  inches from each f a c e . A r e c e n t study of s l i p and Neuhauser, 1972)  has  l i n e s i n copper  (Himstedt  shown t h a t a s u r f a c e l a y e r forms  during p r e s t r a i n which g i v e s r i s e to longer s l i p l i n e s than those  formed under i d e n t i c a l c o n d i t i o n s , except w i t h  l a y e r removed.  T h e i r r e s u l t s confirmed  that t h i s e f f e c t  e s s e n t i a l l y e l i m i n a t e d by the removal of 0.010 a f t e r p r e s t r a i n , and incremental  inches of  f u r t h e r s u r f a c e l a y e r formation  s t r a i n i n g was  considered  this  negligible.  was surface  during  17  1.2.3.3  O p t i c a l Microscopy  A Z e i s s I n t e r f e r e n c e Microscope, o p e r a t i n g i n the n o n - i n t e r f e r e n c e mode a t about x600 m a g n i f i c a t i o n , was to measure s l i p l i n e l e n g t h .  O p t i c a l microscopy was  so t h a t r e l i a b l e s t a t i s t i c a l data could be  used  chosen  conveniently  c o l l e c t e d , and the Z e i s s gave r e p r o d u c i b l e r e s o l u t i o n , and contrast. The measurements were taken by scanning  a fiducial  mark and a s c a l e l o c a t e d i n the microscope e y e p i e c e , to the s l i p t r a c e s .  Whenever the mark met  l i n e , i t s l e n g t h was  measured  piece  parallel  the end of a  slip  (by comparing i t to the eye-  scale). Using  t h i s technique  b o l d , f a i n t , s h o r t and being measured.  The  o p e r a t o r b i a s was  minimized:  long l i n e s have equal p r o b a b i l i t y  of  scans began at randomly chosen p o i n t s  w e l l away from the c r y s t a l edge, and the focus was a d j u s t e d to allow f o r any  s l i g h t departure  continually  from o p t i c a l  flatness. S e v e r a l o p t i c a l micrographs were taken a f t e r s t r a i n increments.  Of p a r t i c u l a r i n t e r e s t are those  different  taken  a f t e r a s t r a i n increment at 4.2°K, and then the same area f o l l o w i n g a p o l i s h and  a s t r a i n increment a t 293°K.  In order to achieve  a depth of f i e l d  i n the m i c r o -  graphs s i m i l a r to t h a t obtained by r o c k i n g the focus  while  t a k i n g the measurements d i r e c t l y , a lower m a g n i f i c a t i o n  was  18  used.  The use o f a newly a v a i l a b l e e x t r a f i n e g r a i n 35 mm  f i l m , H + W C o n t r o l , allowed subsequent  enlargement  o f the  n e g a t i v e , w h i l e m i n i m i s i n g photographic emulsion g r a i n  size i  effects. S l i p l i n e s appeared  t o be d i s t r i b u t e d u n i f o r m l y  over the s u r f a c e o f the c r y s t a l , without the c l u s t e r i n g sometimes observed i n o t h e r metals q u a n t i f y t h i s o b s e r v a t i o n some s l i p were taken.  ( d e L a r i o s , 1972).  To  l i n e d e n s i t y measurements  Scanning p e r p e n d i c u l a r t o t h e s l i p l i n e s , t h e  number o f l i n e s f a l l i n g w i t h i n 10 d i v i s i o n s  o f the eyepiece  scale (68y) was counted f o r s e v e r a l f i e l d s s e l e c t e d a t random. Using the microscope  i n i t s i n t e r f e r e n c e mode,  maximum s l i p step h e i g h t was measured.  1.2.3.4  E l e c t r o n Microscopy  I t i s i m p l i c i t l y assumed i n t h i s study t h a t t h e lengths o f the v i s i b l e " s l i p l i n e s "  (more a c c u r a t e l y  called  s l i p bands) v a r y i n the same manner as t h e l e n g t h s o f the unresolved elementary s l i p l i n e s , which make up the f i n e of the s l i p ' b a n d s . able assumption  Mader (1957) showed t h a t t h i s was a reason-  i n s t r a i n hardened  copper c r y s t a l s .  to q u a l i t a t i v e l y c o n f i r m t h i s o b s e r v a t i o n , two stage (acetate, chromium and 293°K s t r a i n  structure  shadowed  increments.  carbon) were taken a f t e r  In o r d e r replicas 4.2°K  19  These r e p l i c a s , having been taken from a f l a t p o l i s h e d c r y s t a l s u r f a c e , tended to break up p a r a l l e l t o the s l i p l i n e s when the a c e t a t e backing was c o n v e n t i o n a l r e p l i c a techniques.  d i s s o l v e d away u s i n g  In p a r t i c u l a r , carbon f i l m s  c o n t a i n i n g long s h a r p l y d e f i n e d steps were d i f f i c u l t  to  produce at the t h i c k n e s s e s r e q u i r e d f o r good q u a l i t y shadowed replicas.  An acetone r e f l u x u n i t was  overcome t h i s problem  (Figure 4).  b u i l t and used t o  I n s i d e the apparatus the  r e p l i c a , together with copper support g r i d i s p o s i t i o n e d on a c o l d f i n g e r , and surrounded by acetone vapor. backing  l e a v i n g the carbon r e p l i c a c l e a n , i n t a c t and Carbon spheres with diameters  o  i n the s i z e range  A to 5000 A were p l a c e d on the s u r f a c e of some r e p l i c a s By comparing t h e i r shadows w i t h the width  shadows from s l i p s t e p s , or f o r a more p r e c i s e e s t i m a t e ,  by measuring the i n c r e a s e i n l e n g t h of a shadow sphere,  from a  as i t crosses t h a t of a s l i p l i n e , the s l i p  h e i g h t may 1.3  supported.  o  b e f o r e shadowing. of  acetate  of the r e p l i c a i s g r a d u a l l y d i s s o l v e d by the condensing  acetone,  1000  The  be  step  obtained.  Results 1.3.1  S l i p L i n e Lengths 1.3.1.1 The  Temperature Changes  lengths of 100  s t r a i n increment,  two  s l i p l i n e s were measured a f t e r each  or more d i f f e r e n t specimens being used  20  Condenser-J  Water Water Specimen Cold Support Heating Mantle  Figure  4.  Acetone r e f l u x replicas.  apparatus used  i n preparation of  21  f o r each t e s t temperature.  Histograms o f these l i n e l e n g t h s  f o r each t e s t temperature are shown (Figures 5-9), n o r m a l i z e d to 250 r e a d i n g s f o r v i s u a l  comparison.  I t can be seen t h a t temperature has a marked e f f e c t on the s l i p l i n e p a t t e r n s a t c o n s t a n t s t r u c t u r e . mean l e n g t h and d i s t r i b u t i o n are changed.  Both the  F o r example, most  of the l i n e s formed a t 573°K are below 50u i n l e n g t h , whereas a t 4.2°K the m a j o r i t y are w e l l above 50y; f u r t h e r , w h i l e very few of the l i n e s formed a t 573°K are above 70u i n l e n g t h , 20% o f those formed a t 4.2°K are l o n g e r than 150y.  The  histograms f o r the i n t e r m e d i a t e temperatures, 293°K,  190°K  and 77°K show a continuous i n c r e a s e i n the number o f long y e t they a l s o c o n t a i n a s i g n i f i c a n t p r o p o r t i o n o f s h o r t A p l o t of the mean s l i p l i n e  length versus  lines,  lines.  tempera-  t u r e i s shown i n F i g u r e 10, t o g e t h e r w i t h t h e i r 95% c o n f i d e n c e i n t e r v a l s and the upper and lower q u a r t i l e s d e r i v e d from the histograms. ficiently  The c o n f i d e n c e i n t e r v a l s were c o n s i d e r e d s u f -  s m a l l t h a t 100 r e a d i n g s were adequate t o e s t a b l i s h  the mean l i n e l e n g t h , and t h a t the s t a t i s t i c a l e r r o r below the l e v e l of the major e x p e r i m e n t a l e r r o r s .  was  I t can be  seen t h a t s l i p l i n e s formed a t low temperatures are l o n g e r than those formed at a h i g h temperature, i n the range s t u d i e d ; f o r example, a t 4.2°K the average l i n e i s about t h r e e times longer than a t room temperature. The o p t i c a l micrographs these o b s e r v a t i o n s , showing  (Figure 11) d i r e c t l y  confirm  s l i p l i n e s formed on the same a r e a  22  50r  40[  573 K 6  UJ  -i30  r  u. o  |.20f  10 JT 0  50  100 SLIP LINE LENGTH (MICRONS)  I50~  F i g u r e 5, Histogram o f s l i p l i n e lengths a f t e r 0.5% i n c r e m e n t a l s t r a i n a t 573°K.  23  50  SLIP LINE LENGTH (MICRONS)  Figure Histogram of s l i p  line  lengths  6.  after  0.5% i n c r e m e n t a l ' s t r a i n  a t 293°K.  24  50 SLIP LINE LENGTH  100 (MICRONS)  150  F i g u r e 7. Histogram o f s l i p l i n e lengths a f t e r 0.5% i n c r e m e n t a l s t r a i n a t 190°K.  25  200 SLIP LINE LENGTH (MICRONS)  F i g u r e 8, Histogram of s l i p  line  lengths  after  0.5% i n c r e m e n t a l  strain  a t 77' K.  SLIP LINE LENGTH (MICRONS)  F i g u r e 9. istogram o f s l i p . l i n e lengths a f t e r 0.5% i n c r e m e n t a l s t r a i n a t 4.  1  1  L J  1  200 400 TEMPERATURE °K  I  I  I  600  Average s l i p l i n e l e n g t h versus temperature o f incremental s t r a i n i n g . Dashed l i n e s s p a n t h e m i d d l e two q u a r t i l e s f r o m a c c u m u l a t e d m e a s u r e ments. S o l i d e r r o r b a r s r e p r e s e n t 95% c o n f i d e n c e limits for individual points.  28  F i g u r e 11.  O p t i c a l micrographs showing change i n s l i p l i n e l e n g t h w i t h temperature. The same area o f c r y s t a l w i t h hardness marker f o r r e f e r e n c e , (a) a f t e r s t r a i n increment a t 4.2°K, (b) a f t e r s t r a i n increment a t 293°K.  29  of a c r y s t a l s u r f a c e t o be l o n g e r a t 4.2°K than a t 293°K. Another n o t i c a b l e d i f f e r e n c e i s the d i f f u s e nature o f the room temperature  l i n e s when compared t o the sharper 4.2°K  lines. A t h i g h e r m a g n i f i c a t i o n and r e s o l u t i o n , the e l e c t r o n micrographs and l o n g :  (Figure 12) r e v e a l the 4.2°K l i n e s as very f i n e few l i n e s end w i t h i n t h e frame.  In c o n t r a s t many  of the 293°K l i n e s end, and they are a p p a r e n t l y c l u s t e r e d i n narrow bands, w i t h spacings l e s s than the r e s o l u t i o n  limit  of the technique.; However, the spacing o f the more prominent l i n e s i n the r e p l i c a s corresponds t o t h a t o f t h e o p t i c a l l y observed l i n e s being i n the range 2-10y.  1.3.1.2  S t r a i n Rate Changes  Over the range o f s t r a i n r a t e s s t u d i e d , no s i g n i f i cant d i f f e r e n c e s i n the average s l i p  l i n e l e n g t h was measured.  I t should be noted however t h a t changes i n s t r a i n r a t e  produce  s t r e s s changes which a r e s m a l l compared t o those observed i n v a r i a b l e temperature  experiments  Adams and C o t t r e l l , 1955).  (Thornton et al.  3  1961;  F o r example, although s t r e s s d a t a  were not o b t a i n e d from these s t r a i n r a t e change  experiments,  a s t r e s s change o f about 7% i s p r e d i c t e d under these c o n d i t i o n s , u s i n g data from Thornton et al. (1961).  In c o n t r a s t major  changes i n average s l i p l i n e l e n g t h a r e observed i n t h e v a r i a b l e temperature  experiment o n l y when accompanied by  stress, changes o f g r e a t e r than 20%.  30  (a)  (b)  F i g u r e 12.  E l e c t r o n micrographs of s l i p l i n e s formed a f t e r s t r a i n increments a t (a) 4.2°K and (b) 293°K respectively. Both are chromium shadowed (two stage) carbon r e p l i c a s .  31  1.3.2  V a r i a t i o n i n Flow S t r e s s w i t h Temperature The i s o s t r u c t u r a l  ("reversible") increase of flow  s t r e s s w i t h d e c r e a s i n g temperature, i n a v a r i e t y of metals C o t t r e l l , 1955; experiments.  which has been e s t a b l i s h e d  ( C o t t r e l l and Stokes, 1955;  M i t c h e l l , 1964)  was  Adams and  observed i n the p r e s e n t  However, a c c u r a t e data were not o b t a i n e d ,  p r i m a r i l y because the c r o s s - s e c t i o n a l a r e a o f the samples was  not known p r e c i s e l y a f t e r v a r y i n g amounts of p o l i s h i n g .  Some examples of approximate mm""  2  a t 673°K, 2.04  and 2.36  Kg mm  1.3.3  -2  Kg mm  -2  flow s t r e s s v a l u e s are 1.85 a t 293°K, 2.26  Kg mm"  2  Kg  a t 77°K  at 4.2°K.  S l i p Line Density While s l i p l i n e d e n s i t y has np d i r e c t b e a r i n g on  the t h e o r e t i c a l i m p l i c a t i o n s of the main experiments, important t o show t h a t deformation was  i t is  r e l a t i v e l y homogeneous.  Luders band p r o p a g a t i o n , and s l i p l i n e c l u s t e r i n g would make the m i c r o s t r u c t u r a l i n t e r p r e t a t i o n more d i f f i c u l t . S l i p l i n e d e n s i t y readings were made on two c r y s t a l s , s t r a i n e d a t 293°K.  A t o t a l of 165  separate  independent  measurements were taken, and a histogram of these d e n s i t i e s i s shown i n F i g u r e 13. The b e l l shaped histogram i l l u s t r a t e s a p p r o p r i a t e l y the homogeneous nature of deformation.  Very few areas on the  c r y s t a l s u r f a c e were devoid of s l i p l i n e s :  few were densely  32  100 Slip 10  13.  300 ( lines • mm ')  5 Slip  Figure  200 Line Density  Line  400  500  2 5 Spacing  Histogram of s l i p a t 293°K.  line  (/xm)  density,  lines  formed  covered.  The m a j o r i t y o f  the  7 5-300 l i n e s m m " .  range  The a v e r a g e o f  1  corresponds the  l i n e d e n s i t i e s measured a r e  to  spacings  a mean l i n e  spacing of  Slip  an i n c r e m e n t a t  Interference  l i n e s mm  comparable  electron  -1  to  micrographs.  Step Height  O p t i c a l measurement after  5.9y,  o f p r o m i n e n t l i n e s on t h e  1.3.4  170  in  293°K,  Microscope.  of  slip  step h e i g h t s ,  were made u s i n g t h e  T h e s e measurements  formed  Zeiss  showed maximum o  slip  step heights  t o be o f  the  resolution limit  the  chromium f r e e  to  the  slip  of  the  the  of  instrument  shaddows o f  lines reveals  order  500 (Z  A , which i s  A/10).  carbon spheres  near  Comparing  i n Figure  maximum s t e p h e i g h t s  i n the  14 range  o 250  to  400  A, providing  qualitative  agreement  w i t h the  previous  technique. 1.3.5  Sources  of  Error  In a study of may be d i v i d e d ferences  this  type,  i n t o two g r o u p s .  from c r y s t a l  to  the  experimental  problems  T h o s e i n t r o d u c e d by  crystal,  or  from increment  difto  i n c r e m e n t were c o n s i d e r e d n e g l i g i b l e , s i n c e good c o n t r o l  was  exercised  the  over o r i e n t a t i o n ,  amount o f p r e s t r a i n  crystal  and s t r a i n  increment,  The s e c o n d g r o u p c o v e r s inherent  i n any one p r e s t r a i n ,  purity,  temperature,  and s u r f a c e  quality.  t h o s e , e x p e r i m e n t a l p r o b l e m s ,,  polish,  strain  increment,  34  F i g u r e 14.  Shadowed r e p l i c a w i t h c a r b o n c a l i b r a t i o n spheres (diameters i n the range 1000A t o 5000 A ) . 0  35  s l i p l i n e measurement sequence. experimental  l i m i t a t i o n was  Here the most  critical  the o p t i c a l f l a t n e s s of the s u r f a c e .  When measuring l i n e l e n g t h , a l i n e was  taken to begin or  at any d e v i a t i o n from l i n e a r i t y , or at any A p r o p o r t i o n of these d e v i a t i o n s were due  end  l a c k of continuity.. to s u r f a c e  irregu-  l a r i t i e s as. d i s t i n c t from m i c r o s t r u c t u r a l phenomena.  Hence,  l a c k of o p t i c a l f l a t n e s s would b i a s each s e t of r e s u l t s towards s h o r t e r l i n e l e n g t h s . . But t h i s b i a s i s g r e a t e r f o r longer mean l i n e l e n g t h .  A long l i n e would have a g r e a t e r  chance of c r o s s i n g a s u r f a c e i r r e g u l a r i t y than a s h o r t so the' long l i n e might be counted i n c o r r e c t l y as two lines.  line,  or more  Thus i t seems l i k e l y t h a t i f specimens had p e r f e c t  o p t i c a l f l a t n e s s , the observed e f f e c t t h a t s l i p l i n e s formed at lower temperatures are l o n g e r , would be more s t r o n g l y revealed. Another l i m i t a t i o n a r i s e s from the compromise made i n choosing  0.5%  as the s t r a i n increment.  On the one  hand  the s t r a i n increment must be l a r g e enough so t h a t most of the s t r a i n occurs  (and t h e r e f o r e most of the s l i p l i n e s  formed) at the steady  are  s t a t e flow s t r e s s ; then o n l y a s m a l l  f r a c t i o n of the l i n e s are formed d u r i n g the t r a n s i t i o n s t r a i n a t a lower s t r e s s .  On  elastic-plastic  the other hand a  l a r g e s t r a i n increment w i l l i n v a l i d a t e the assumption of constant s t r u c t u r e . In the m a j o r i t y of cases, as estimated  from the  l o a d e l o n g a t i o n curves, the t r a n s i t i o n s t r a i n occupied  less  36  than 0.15%  strain  (Figure 15).  However, t h i s e f f e c t  p r o v i d e a background of s h o r t e r l i n e s  (Alden, 1972)  may on a l l  t e s t s , thus l i m i t i n g and not enhancing the observed  effect.  W h i l s t f u r t h e r d i s c u s s i o n of t h i s matter i s presented the preceding  s e c t i o n , i t i s assumed t h a t i n the  in  present  experiments, most s l i p l i n e s are formed a t the steady  flow  1.4  state  stress.  Discussion 1.4.1  S l i p L i n e B l o c k i n g by S p e c i f i c D i s l o c a t i o n A r r a y s A major o b j e c t i v e of any  theory of work  hardening  i s to s u c c e s s f u l l y l i n k m i c r o s t r u c t u r a l phenomena t o mechanical properties.  Such a. theory  should e x p l a i n the nature of  l i n e s s i n c e these are s u r f a c e m a n i f e s t a t i o n s events.  Of the e x i s t i n g work hardening  to e x p l a i n s l i p l i n e l e n g t h , two those of Seeger  slip  of m i c r o s t r u c t u r a l  t h e o r i e s which attempt  prominent examples are  (1957) and of H i r s c h and M i t c h e l l (1967).  i.4.1.1  S l i p L i n e B l o c k i n g by L o m e r - C o t t r e l l Locks  Seeger's theory  (1957) r e l a t e s s l i p l i n e l e n g t h L  to s t r a i n e, from the experimental  observation that  slip  l i n e l e n g t h decreases with i n c r e a s i n g s t r a i n , namely,  L =  e  -  e*  (1.1)  ELONGATION  F i g u r e 15.  T r a n s i t i o n and steady s t a t e s t r a i n observed incremental, t e s t a t 4.2°K.  during  38  where A and e* a r e e m p i r i c a l slip  l i n e s a r e b l o c k e d by a r r a y s  Lomer-Cottrell ously  reaction  intersecting  continu-  region. fault  energy,  such  l o c k s may be g e n e r a t e d by a com-  between two e x t e n d e d d i s l o c a t i o n s o n two  g l i d e planes.  with Burgers vectors sociated  that  o f d i s l o c a t i o n s p i l e d up a t  a m e t a l w i t h a low s t a c k i n g  copper, L o m e r - C o t t r e l l  bination  He p r o p o s e s  l o c k s , w h i c h , he s u b m i t s a r e f o r m e d  i n the stage I I hardening In  as  constants.  Consider  f o r example,  dislocations  j a[101] a n d i a [ l ! o ] w h i c h h a v e d i s -  t o f o r m S h o c k l e y p a r t i a l d i s l o c a t i o n s as  |  a [101]  -> | a [112]  |  a[110]  follows,  +  | a[2Tl]  +  | a[211]  and  and  which l i e on the primary  planes r e s p e c t i v e l y . two  d i s l o c a t i o n from each  (111)  slip  dissociated  follows,  \ a[112] b  form a s t a i r  (111) a n d c o n j u g a t e  At the l i n e of i n t e r s e c t i o n of these  p l a n e s , one p a r t i a l  p a i r may combine as  to  | a[121]  +  J a[12l]  r o d type p a r t i a l  Cottrell  dislocation.  location  i s coplanar with neither  b  +  Ja[0ll]  o  d i s l o c a t i o n c a l l e d a Lomer-  Since the Burgers vector  of t h i s d i s -  the primary nor conjugate  s l i p plane, f t i s unable to g l i d e  and thus i s termed s e s s i l e .  The L o m e r - C o t t r e l l s e s s i l e d i s l o c a t i o n , t o g e t h e r w i t h two r e m a i n i n g  the  p a r t i a l s are c o l l e c t i v e l y r e f e r r e d t o as a  L o m e r - C o t t r e l l l o c k , which i s a wedge shaped s t a c k i n g r i b b o n , l y i n g , i n t h i s case, i n the L o m e r - C o t t r e l l d i s l o c a t i o n forming  fault  [Oil]-direction with  the  the edge of the wedge.  C l a r e b r o u g h and Hargreaves (1959) have i d e n t i f i e d the c o n d i t i o n s f o r p o s s i b l e p r o d u c t i o n o f  Lomer-Cottrell  d i s l o c a t i o n s w h i c h l i e i n the p r i m a r y s l i p p l a n e , namely t h a t the B u r g e r s v e c t o r s o f t h e o r i g i n a l p a i r o f d i s l o c a t i o n s make an a n g l e of 120°  w i t h each o t h e r .  P a i r s of s l i p  systems  s a t i s f y i n g t h i s c o n d i t i o n are l i s t e d i n T a b l e 1,  together  w i t h the d i r e c t i o n a l o n g w h i c h t h e c o r r e s p o n d i n g  sessile  dislocations l i e .  From t h i s t a b l e we  can see t h a t Lomer-  C o t t r e l l d i s l o c a t i o n s (and hence L o m e r - C o t t r e l l l o c k s )  may  be formed i n t h r e e c l o s e packed d i r e c t i o n s . I n Seeger"s t h e o r y , g l i d e d i s l o c a t i o n s , g e n e r a t e d from a Frank-Read s o u r c e , p i l e up i n the s l i p p l a n e a t LomerC o t t r e l l l o c k s i n t h e s e t h r e e c l o s e packed d i r e c t i o n s , F i g u r e 16.  C l a r e b r o u g h and Hargreaves (1959) l a t e r  pointed  out t h a t L o m e r - C o t t r e l l l o c k s would more l i k e l y be formed i n two of those d i r e c t i o n s (those i n w h i c h one of the two systems was  the p r i m a r y s l i p s y s t e m ) .  However, s i n c e  parent two  are s u f f i c i e n t t o b l o c k any p r i m a r y s o u r c e ( F r i e d e l , 1955), i n essence Seeger's p r o p o s i t i o n remains v a l i d .  40  Table Pairs of S l i p  Pairs  1  Systems w h i c h c o u l d P r o d u c e L o m e r - C o t t r e l l  of S l i p  Systems  Direction of Lomer-Cottrell Dislocation  (111) [10.1] , (111) [IlO]  [011]  (111) [ I l O ] ,  [011]  (111) [ l o i ]  Locks  (111) [101], (111) [ O i l ]  [110]  (111) [ O i l ] ,  (111) [ l o i ]  [110]  (111) [ O i l ] ,  ( i l l ) [110]  [101]  (111) [110], (111) [011]  [101]  41  primary  F i g u r e 16,  slip  plane  S l i p l i n e b l o c k i n g a c c o r d i n g t o Seeger (1957). Expanding d i s l o c a t i o n loops are blocked by • ' L o m e r - C o t t r e l l l o c k s formed i n three close-packed directions.  42  Since these l o c k s form c o n t i n u o u s l y d u r i n g stage I I hardening,  t h e r e i s a s s o c i a t e d w i t h any s t r a i n an a r r a y of  p i l e - u p groups which, by means of t h e i r long range e l a s t i c s t r e s s f i e l d s , r e s t r i c t the expansion  of loops from newly  a c t i v e sources to d i s t a n c e s somewhat l e s s than the pileup spacing.  In other words, the s l i p  s t r a i n increment  depends only on the s t r a i n i t s e l f  existing  l i n e l e n g t h i n any  and i s  (more  fundamentally  on the s t r u c t u r e ) , Eq. 1.1,  temperatures-  independent.  In the present experiments the s t r u c t u r e i s  constant, and t h i s theory p r e d i c t s an i n v a r i a n t s l i p length.  line  T h i s p r e d i c t i o n i s not i n agreement w i t h the p r e s e n t  results. * While the s t r u c t u r e i s constant, the flow a , i n c r e a s e s w i t h d e c r e a s i n g temperature.  stress,  In Seeger's theory  i  a = a  s  + a  (1.2) g  and the i n c r e a s e i n o i s e n t i r e l y due a  s  , the f o r e s t i n t e r s e c t i o n s t r e s s .  to the i n c r e a s e i n a„ i s t h a t p a r t o f the . g c  a p p l i e d s t r e s s a v a i l a b l e to d r i v e d i s l o c a t i o n loops i n t o  the  p i l e up a r r a y , and i s temperature independent; t h i s must f o l l o w from the long range nature of the i n t e r a c t i o n s which determine o„.  These c o n s i d e r a t i o n s r e i n f o r c e the c o n c l u s i o n  Appendix 2 o u t l i n e s the d e f i n i t i o n s used to d e s c r i b e deformation phenomena.  of  terminology  43  t h a t t h i s theory p r e d i c t s a c o n s t a n t s l i p the  l i n e length i n  p r e s e n t experiments. I t may be argued t h a t e x i s t i n g p i l e - u p s can r e -  arrange o r a n n i h i l a t e d u r i n g a h i g h e r temperature  strain  increment, w h i l s t a t the lower temperature, t h e i r d e n s i t y and d i s t r i b u t i o n remains unchanged.  However t h i s  possibility  i s not a d m i s s i b l e i n the theory; p i l e - u p s are supposed t o be * stable  i n stage I I .  Moreover  i f such a n n i h i l a t i o n should  occur, l o n g e r s l i p l i n e s would form a t h i g h e r temperatures, c o n t r a r y t o the r e s u l t s o f the p r e s e n t experiments. should be emphasized  t h a t s t r u c t u r e change d u r i n g  It  strain  increments a t v a r i o u s temperatures was shown t o be s m a l l e x p e r i m e n t a l l y by i n c r e m e n t a l s t r a i n i n g o f i n d i v i d u a l c r y s t a l s a t two d i f f e r e n t temperatures.  In each case  slip  l i n e l e n g t h s were the same as i n c r y s t a l s g i v e n a s i n g l e s t r a i n increment, and d i d not depend on the o r d e r i n which the  temperatures were s e l e c t e d . 1.4.1.2  S l i p L i n e B l o c k i n g by Ribbons o f Converted Pile-ups  H i r s c h and M i t c h e l l  (1967) o b j e c t t o Seeger's theory  on the grounds t h a t the s t r e s s f i e l d from the proposed p i l e - u p  A c c o r d i n g to S e e g e r et al. ( 1 9 5 7 ) L o m e r - C o t t r e l l b l o c k i n g becomes l e s s e f f e c t i v e d u r i n g s t a g e III deformation, where d i s l o c a t i o n s may b y - p a s s t h e s e b a r r i e r s by c r o s s - s l i p . However, in the p r e s e n t e x p e r i m e n t s where c r y s t a l s were p r e s t r a i n e d in s t a g e II, i t is u n l i k e l y t h a t a c r o s s - s l i p by-pass mechanism w i l l b e g i n t o o p e r a t e d u r i n g t h e s t r a i n i n c r e m e n t s a l l o f which a r e at t e m p e r a t u r e s below the p r e s t r a i n t e m p e r a t u r e .  44  arrays w i l l b l o c k only one  s l i p p l a n e , namely the plane  on  which the p i l e - u p i s formed; under the i n f l u e n c e of the a p p l i e d s t r e s s , glide  on neighbouring, p a r a l l e l planes  will  s t i l l be p o s s i b l e over long d i s t a n c e s , and i n c e r t a i n w i l l even by enhanced by the presence  of the p i l e - u p .  present c a l c u l a t i o n s o f long range i n t e r n a l s t r e s s to  areas They  fields  support t h i s argument. In order to overcome t h i s problem, they propose a  theory i n which the s l i p l i n e s are again blocked by o b s t a c l e s surrounding o b s t a c l e s are now  linear  the d i s l o c a t i o n source' however these  "converted p i l e - u p s , " i . e . p i l e - u p s which  are s t a b i l i z e d by secondary  slip.  The  converted p i l e - u p s  are i n the form of long ribbons p a r a l l e l to the primary  slip  plane and composed o f h i g h d e n s i t i e s of d i s l o c a t i o n s o f s e v e r a l Burger's ing  vectors.  The ribbons are e f f e c t i v e i n b l o c k -  a number of adjacent s l i p p l a n e s , F i g u r e 17, and  thus  are d e s c r i b e d as having a r a d i u s of i n t e r a c t i o n p e r p e n d i c u l a r to  the s l i p plane.  These o b s t a c l e s m u l t i p l y w i t h s t r a i n  as  g l i d i n g d i s l o c a t i o n loops are blocked by e x i s t i n g o b s t a c l e s and the r e s u l t i n g p i l e - u p s are then s t a b i l i z e d by secondary  slip.  As a consequence of t h i s m u l t i p l i c a t i o n of  o b s t a c l e s the s i z e of the areas of primary for  local  g l i d e i s reduced,  the s t r a i n i n c r e a s e s .  s l i p plane  available  and the s l i p l i n e l e n g t h decreases  as  45  S l i p l i n e b l o c k i n g a c c o r d i n g t o H i r s c h and M i t c h e l l (1967). Schematic c r o s s s e c t i o n p e r p e n d i c u l a r to s l i p plane showing r i b b o n - l i k e o b s t a c l e s (shaded and appearing as e l l i p s e s i n t h i s c r o s s - s e c t i o n ) b l o c k i n g d i s l o c a t i o n loops expanding from source S. Obstacles have t h i c k n e s s R p e r p e n d i c u l a r t o s l i p plane.  46  T h e i r i n t e r p r e t a t i o n of the temperature dependence of the flow s t r e s s i m p l i c i t l y  f o l l o w s t h a t of Seeger.  r e v e r s i b l e s t r e s s i n c r e a s e on d e c r e a s i n g  The  temperature i s a  consequence of the i n c r e a s e d s t r e n g t h of f o r e s t d i s l o c a t i o n s ; none of t h i s e x t r a s t r e s s i s a v a i l a b l e to overcome the obstacles.  Therefore  areas of primary plane  at constant  linear  s t r u c t u r e the s i z e of  the  a v a i l a b l e f o r g l i d e i s constant,  and  no s i g n i f i c a n t temperature dependence of s l i p l i n e l e n g t h i s predicted.  Attempts to r e c o n c i l e the theory  and  experiment  by means of assumed s t r u c t u r e change d u r i n g s t r a i n  increments  meet the o b j e c t i o n s p r e v i o u s l y s t a t e d . Some evidence  f o r the e x i s t e n c e of long  ribbon-  l i k e o b s t a c l e s can be found i n the e l e c t r o n microscope s t u d i e s of copper  (Steeds, 1966).  These o b s e r v a t i o n s  suggest t h a t  elongated m u l t i p o l e c l u s t e r s are formed i n ribbons by end of stage I.  However, the m i c r o s t r u c t u r e  the  characteristic  of stage I I c o n s i s t s of " c a r p e t s " of d i s l o c a t i o n s a p p r o x i mately p a r a l l e l to the primary g l i d e p l a n e , t o g e t h e r  with  s h o r t e r , more d i f f u s e " w a l l s " o f d i s l o c a t i o n s p e r p e n d i c u l a r to the g l i d e plane B a s i n s k i , 1964;  (Howie, 1960;  Esseman, 1965;  H i r s c h and Steeds,  Steeds,  1966;  Seeger, 1968).  A f t e r d i s l o c a t i o n e t c h p i t t i n g the t r a c e s of the may  "carpets"  a l s o be seen on a c r y s t a l face not coplanar w i t h  primary s l i p plane 1964).  1963;  the  (see f o r example B a s i n s k i and B a s i n s k i ,  With i n c r e a s i n g flow s t r e s s these  c a r p e t - l i k e patches  47  of d i s l o c a t i o n s become more numerous and more c l e a r l y  defined.  A t the same time i t i s e v i d e n t t h a t t h e d i s l o c a t i o n d e n s i t y w i t h i n these patches i n c r e a s e s , s i n c e the l a t t i c e r o t a t i o n across the dense regions becomes g r e a t e r .  By the beginning  of stage I I I , r e l a t i v e l y w e l l d e f i n e d c e l l s have formed which may be elongated observations  p a r a l l e l t o the s l i p p l a n e .  are q u a l i t a t i v e i n nature,  e l e c t r o n microscopy f o i l s prepared  Although  these  and many are from  without  the use o f r a d i a t i o n  p i n n i n g , they appear t o show t h a t the o b s t a c l e s t r u c t u r e proposed by H i r s c h and M i t c h e l l (1967) i s a p p r o p r i a t e at  the beginning  1.4.1.3  o f the stage I I hardening  only  region.  S l i p L i n e B l o c k i n g by D i s l o c a t i o n Cell  Walls  Since c e l l s t r u c t u r e s are a prominent f e a t u r e o f t h e , m i c r o s t r u c t u r e o f s t r a i n hardened metals,  i t should be  i n q u i r e d whether the c e l l w a l l s can a c t as o b s t a c l e s which block s l i p  lines.  study, Staker  I n a r e c e n t t r a n s m i s s i o n e l e c t r o n microscope  and H o l t  (1972) have measured d i s l o c a t i o n  cell  s i z e s i n copper deformed a t temperatures between 298°K and 973°K.  Using data from the l i t e r a t u r e together w i t h  own r e s u l t s they concluded  their  the the d i s l o c a t i o n c e l l s i z e was  i n v e r s e l y p r o p o r t i o n a l t o the ( C o t t r e l l - S t o k e s c o r r e c t e d ) shear s t r e s s .  These data are p l o t t e d i n F i g u r e 18.  p a r i n g the flow s t r e s s used i n the present  By com-  experiments, we  48  a  Feltner & Laird (1967)  A  Staker & Holt (1972)  o  Pratt (1966) (1967)  1.0 Average  18.  Cell  5  10  Diameter (microns)  Average d i s l o c a t i o n c e l l diameters  i n copper  versus  normalized (and C o t t r e l l - Stokes c o r r e c t e d ) flow s t r e s s , as c o r r e l a t e d by Staker and H o l t , 1972. Flow s t r e s s and deduced c e l l s i z e f o r the p r e s e n t study a r e i n d i c a t e d by arrows.  49  would expect  the p r e s t r a i n e d m i c r o s t r u c t u r e to have an average  d i s l o c a t i o n c e l l s i z e of about 5y.  However the average  length of s l i p l i n e s formed i n t h i s m i c r o s t r u c t u r e was the range 40 to 120y, than the a n t i c i p a t e d  one  in  order of magnitude or more g r e a t e r  dislocation cell size.  Thus i t appears  t h a t the s l i p l i n e s are not blocked by c e l l w a l l s  (see Appendix  1) . D i s c r e p a n c i e s between the apparent s l i p l i n e l e n g t h , and d i s l o c a t i o n most r e c e n t l y electron  c e l l s i z e have been noted  by Ambrosi et al.  (1974).  i n previous s t u d i e s , Using  microscopy they measured d i s l o c a t i o n  transmission  c e l l sizes i n  copper deformed a t room temperature up to a flow s t r e s s 7 Kg mm".  When these r e s u l t s are compared w i t h  2  o b s e r v a t i o n s of the lengths o f s l i p conditions  (Vorbrug  et al.,  previous  l i n e s formed under i d e n t i c a l  1971), F i g u r e 19, i t can be  t h a t the average c e l l diameters  seen  are s m a l l e r than the average  s l i p l i n e lengths f o r the complete range of flow Moreover, the r e l a t i v e values of d i s l o c a t i o n and s l i p l i n e l e n g t h at ~2Kg mm  of  -2  cell  stress. diameter  show good agreement w i t h  the deduced c e l l s i z e and measured s l i p l i n e lengths i n the present  study. While these r e s u l t s appear to show t h a t the m a j o r i t y  of c e l l w e l l s per  se p r o v i d e i n e f f e c t i v e b l o c k s f o r s l i p  B a s i n s k i and B a s i n s k i (1964) have argued t h a t s l i p  lines  blocked only by the "carpet" s e c t i o n s of the c e l l w a l l s .  lines, are  50  Shear F i g u r e 19.  Stress ( K g . m m  2  )  Average s l i p l i n e l e n g t h and average c e l l diameter i n copper f o r a range of shear (flow) s t r e s s as c o r r e l a t e d by Ambrosi et al. (1974).  51  Since the carpets are approximately s l i p p l a n e , primary  p a r a l l e l t o the  d i s l o c a t i o n s may  primary  g l i d e over d i s t a n c e s  l a r g e compared t o the c a r p e t s p a c i n g .  However, as w i t h  the  p r e v i o u s t h e o r i e s which- p o s t u l a t e b l o c k i n g by f i x e d o b s t a c l e a r r a y s , t h i s model i n c o r r e c t l y p r e d i c t s a constant s l i p  line  l e n g t h i n an i s o s t r u c t u r a l experiment. I t i s of i n t e r e s t to note t h a t the estimated c e l l diameter of  average  i n the p r e s e n t work, 5y, l i e s w i t h i n t h e range  spacing of the more prominent s l i p  l i n e s observed  the o p t i c a l and the e l e c t r o n micrographs  (2-10y).  in.both  It will  be r e c a l l e d t h a t d i s l o c a t i o n " c a r p e t s " p a r a l l e l t o the  primary  g l i d e plane and made up of primary d i s l o c a t i o n s i n t a n g l e d itiultipoles,  are a prominent f e a t u r e o f stage I I m i c r o s t r u c t u r e s .  Since such carpets i n any non-coplanar  cross-section w i l l  appear as c e l l w a l l s , i t i s reasonable to assume t h a t the carpet spacing i s of the order of the c e l l diameter. i t i s tempting  Then  to s p e c u l a t e t h a t s l i p l i n e s are formed by  d i s l o c a t i o n s , which, when g l i d i n g p a r a l l e l to these c a r p e t s , are able to penetrate the more d i f f u s e p o r t i o n s o f c e l l which are p e r p e n d i c u l a r to the s l i p p l a n e . :  "wall"  In t h i s model,  n e i t h e r the c a r p e t s , nor the connecting p o r t i o n s of  cell  " w a l l " pe3? se, b l o c k g l i d e , although f o r e s t - t y p e d i s l o c a t i o n s , , of which the l a t t e r r e g i o n s are composed, may  interact  statis-  t i c a l l y t o b l o c k g l i d e over d i s t a n c e s somewhat l a r g e r than c e l l w a l l spacing.  In t h i s way  the  s l i p l i n e s would be formed w i t h  lengths l a r g e compared w i t h the apparent  dislocation c e l l  size.  52  1.4.2  Flow S t r e s s and the G l i d e / F o r e s t Most of, the d i s c u s s i o n  Interaction.  so f a r has c e n t r e d on the  p o s s i b i l i t y t h a t s l i p l i n e s are b l o c k e d by s p e c i f i c d i s l o c a tion  arrangements  which are l i n e a r i n form on the s l i p p l a n e .  We have seen t h a t t h i s approach p r e d i c t s  constant s l i p  l e n g t h i n the p r e s e n t i s o s t r u c t u r a l experiments.  line  The f l o w  s t r e s s theory on which t h i s approach i s based d i v i d e s  the  s t r e s s opposing d i s l o c a t i o n motion i n t o a d d i t i v e  components  which are due t o e i t h e r long range back s t r e s s e s  from these  s p e c i f i c d i s l o c a t i o n arrays,  forest  interaction  o r t o a s h o r t range  stress.  As an a l t e r n a t i v e premise, we may flow s t r e s s  assume t h a t the  i s s o l e l y determined by i n t e r a c t i o n s  and f o r e s t d i s l o c a t i o n s .  between g l i d e  The f o r e s t d i s l o c a t i o n s ^ w h i c h  may  be p a r t o f a r e l a t i v e l y immobile d i s l o c a t i o n network o r c e l l u l a r structure, are then the o b s t a c l e s to g l i d e and one need hot r e f e r t o s p e c i f i c groupings o f o b s t a c l e  dislocations.  T h i s premise i s supported by r e s u l t s from two independent experimental f i e l d s , namely by l a t e n t hardening experiments, and e t c h p i t s t u d i e s . The l a t e n t hardening experiments Basinski the  ( f o r example  and Jackson, 1965; Kocks and Brown, 1966)  importance of the c o n t r i b u t i o n  a c t i o n by t e s t i n g the o r i e n t a t i o n s t r u c t u r a l flow s t r e s s .  illustrate  of the g l i d e / f o r e s t i n t e r dependence of the i s o -  In these experiments c r y s t a l s  53  prestrained tion  in single  such t h a t  another s i n g l e  When t h e s e two  is  the  expected  same f o r  stresses  directional w o u l d be two  the  since  the  the  However we  a factor  of  are the  assumed t h a t  the  The  low  temperature  for  single  density plane.)  the  importance of  glide  the  the any  pile-ups  to  the  distribution  provide  i s f o u n d t o be  a  inherently such  component  S i n c e no  do  flow  for  forest  dislocation  i s c o n v e n i e n t l y m e a s u r e d by  not  exist. do  However, s u c h a r e l a t i o n s h i p  not  stress. and  Basinski, with  1964 this  interaction.  to  the  density.  etching  of  interaction.  copper c r y s t a l s  proportional  of  difference  with a view  evidence consistent  glide  is  result  ~ c o s 6 0 ° = i when e i t h e r  (of B a s i n s k i  flow s t r e s s  average f o r e s t  this  these back s t r e s s e s  etch p i t studies  view of  flow stress  and  glide/forest  cannot conclude t h a t  for  o t h e r hand, i f b a c k  consistent  example) p r o v i d e a d d i t i o n a l  of  the  because of  make a s i g n i f i c a n t c o n t r i b u t i o n  root  On  operate.  interaction  c o p l a n a r s y s t e m s were u s e d .  R a t h e r i t must be  for  I f the  such p i l e - u p s  d e t e r m i n e d by  The  same.  orienta-  flow stress  piled-up dislocations  i s observed, these r e s u l t s flow s t r e s s  the  forest density  flow s t r e s s ,  r e d u c e d by  i n a new  system would  glide/forest  character of  other  the  coplanar systems.  from a r r a y s o f  component o f  the  f o u n d t o be  d e t e r m i n e d by  w o u l d be  glide  s y s t e m s were c o p l a n a r ,  e a c h s y s t e m was solely  g l i d e , were r e s t r a i n e d  the  oriented square  (This  primary  i s less well  slip  defined  54  f o r d e n s i t i e s which i n c l u d e  glide dislocations  (densities  determined by e t c h i n g non-primary s l i p p l a n e s ) . r e s u l t s are  also consistent  Hence these  w i t h the view t h a t the  f o r e s t i n t e r a c t i o n i s flow s t r e s s  glide/  determinant.  A g a i n s t t h i s background of experimental evidence, t h e o r e t i c a l support f o r the  importance of f o r e s t  as o b s t a c l e s to g l i d e i s p r o v i d e d by i n t e r s e c t i o n mechanisms. a f o r e s t , one  of two  analyses of  dislocations dislocation  When a g l i d e d i s l o c a t i o n encounters  events may  occur, depending on  combination of d i s l o c a t i o n Burgers v e c t o r s : d i s l o c a t i o n cuts through the  f o r e s t and  or t h e r e i s a d i s l o c a t i o n r e a c t i o n  the  particular  e i t h e r the  jogs are  (a j u n c t i o n  glide  produced,  reaction) i n  which a s e s s i l e segment of d i s l o c a t i o n i s produced, and g l i d e d i s l o c a t i o n must bow  round or break t h i s pinned seg-  ment i f i t i s t o proceed f u r t h e r .  D e t a i l s of p o s s i b l e  l o c a t i o n i n t e r s e c t i o n mechanisms have been summarized H i r t h and  Lothe  calculations although on  (1970) who  several  by  important r e a c t i o n ,  that,  i t i s not any  one  possible of  i n t e r s e c t i o n mechanisms c o u l d account f o r macroscopic  deformation phenomena.  Hence, w i t h p a r t i c u l a r r e f e r e n c e  g l i d e / f o r e s t i n t e r s e c t i o n , any f o r the  They conclude  of t h e i r c a l c u l a t i o n s  to u n i q u e l y i d e n t i f y the  dis-  a l s o present some attempted  of i n t e r a c t i o n e n e r g i e s . the b a s i s  the  such mechanism c o u l d account  r e l a t i o n s h i p between f o r e s t d i s l o c a t i o n d e n s i t y  flow s t r e s s .  to  and  55  Whilst a single f o r e s t d i s l o c a t i o n represents a plausible  o b s t a c l e f o r a segment of g l i d e d i s l o c a t i o n , a  g l i d e d i s l o c a t i o n loop w i l l i n g e n e r a l encounter a l a r g e number of f o r e s t d i s l o c a t i o n s the i n t r o d u c t i o n  simultaneously.  of s t a t i s t i c s  T h i s leads t o  to analyse the g l i d e of a d i s -  l o c a t i o n through a f i e l d of o b s t a c l e s .  Moreover s i n c e  strong  g l i d e / f o r e s t i n t e r a c t i o n mechanisms occur over d i s t a n c e s compared t o the average f o r e s t s p a c i n g , the f o r e s t t i o n s may be t r e a t e d  as a f i e l d of p o i n t - l i k e  line  disloca-  obstacles.  Then i f a g l i d e d i s l o c a t i o n loop can be blocked f i e l d , a s t a t i s t i c a l analysis  small  by such a  should l e a d to a theory o f s l i p  length.  1.4.3  Statistical 1.4.3.1  B l o c k i n g of S l i p  Lines  Introduction  The s t a t i s t i c a l t h e o r i e s 1966; Alden, 1972) propose that  of s t r a i n hardening  (Kocks,  s l i p l i n e s are b l o c k e d by  i n t e r a c t i o n between g l i d e d i s l o c a t i o n s and non-regular d i s t r i butions of f o r e s t d i s l o c a t i o n s ; viously  discussed theories,  no p a r t i c u l a r g l i d e - b l o c k i n g  i n contrast  w i t h the p r e -  these s t a t i s t i c a l t h e o r i e s d i s l o c a t i o n arrays.  such a framework, i t i s s t i l l p o s s i b l e  specify  Within  f o r s l i p l i n e s t o be  "blocked," f o r example a t unpenetrable areas of p a r t i c u l a r l y high f o r e s t d i s l o c a t i o n d e n s i t y the  (Kocks, 1966).  However,  important f e a t u r e of a s t a t i s t i c a l theory i s t h a t  56  the  size  of these  Consequently,  impenetrable  areas is stress  dependent.  i n a g i v e n m i c r o s t r u c t u r e , the s l i p l i n e  length  w i l l depend upon the a p p l i e d s t r e s s a t which the s l i p was formed.  line  Furthermore, w i t h p a r t i c u l a r r e f e r e n c e t o the  p r e s e n t experiments,  f o r two c r y s t a l s having the same micro-  s t r u c t u r e , but which have undergone steady s t a t e at d i f f e r e n t r e s p e c t i v e temperatures  (and hence a t two d i f f e r e n t  l e v e l s o f a p p l i e d s t r e s s ) , the r e s u l t i n g s l i p l i n e should d i f f e r  deformation  lengths  (Alden, 1972).  In the proceeding d i s c u s s i o n i t w i l l be shown t h a t the temperature dependence o f s l i p l i n e with s t a t i s t i c a l  length i s consistent  t h e o r i e s o f s t r a i n hardening  i n which  g l i d e loops are a b l e t o expand over a newly a v a i l a b l e  "free"  area o f s l i p p l a n e , a f t e r a t h e r m a l l y a c t i v a t e d p r o c e s s . A t present, only one such theory e x i s t s However there i s another,  (Alden, 1972).  as y e t unproposed v a r i a t i o n o f  t h i s theory which i s a l s o c o n s i s t e n t w i t h the p r e s e n t and t h i s w i l l a l s o be d i s c u s s e d . shown t h a t , although Kocks  1  results,  In a d d i t i o n i t w i l l be  statistical  a n a l y s i s (1966)  forms the b a s i s f o r s u c c e s s f u l i n t e r p r e t a t i o n o f the p r e s e n t r e s u l t s , Kocks  1  (1966) theory i t s e l f p r e d i c t s no i s o s t r u c t u r a l  change i n s l i p l i n e  1.4.3.2  length with  temperature.  Kocks' S t a t i s t i c a l  Statistical  Model  a n a l y s i s o f the expansion  d i s l o c a t i o n loop through  of a g l i d e  a f i e l d of randomly p l a c e d p o i n t  57  obstacles  was  and  (1966).  Makin  f i r s t performed by Kocks  (1966) and  by Foreman  In.these a n a l y s e s , the p o i n t o b s t a c l e s  act  i n p a i r s to b l o c k segments of expanding g l i d e d i s l o c a t i o n s , and  the  segments can  s u f f i c i e n t l y high.  continue to move only i f the While t h e r e are  i n the d e t a i l s of these two e s s e n t i a l l y the  same.  a n a l y s e s , the  to g l i d i n g d i s l o c a t i o n s . glide  stress,  differences  results  are  Both show t h a t there e x i s t s a  s t r e s s above which such an o b s t a c l e  athermal  some minor  stress i s  critical  f i e l d becomes t r a n s p a r e n t  (This s t r e s s may  be c a l l e d  the  s i n c e a t t h i s s t r e s s g l i d e can  occur  without the need f o r t h e r m a l l y a c t i v a t e d processes.) at lower s t r e s s the  s i z e of the p e n e t r a b l e area, from hereon  r e f e r r e d to as the free applied  stress.  Moreover,  area,  i s shown to i n c r e a s e  Since Kocks' a n a l y s i s  with  forms p a r t of  the  his  theory of s t r a i n hardening, h i s r e s u l t s w i l l be used to i l l u s t r a t e t h i s s t r e s s dependence. The F i g u r e 20  t h r e e diagrams from Kocks' a n a l y s i s , shown i n  (a), (b),  ( c ) , i l l u s t r a t e the  statistically  f r e e areas at three d i f f e r e n t l e v e l s of a p p l i e d (which i s expressed as a f r a c t i o n of the stress, y ) • T  Whereas the  athermal g l i d e  are j o i n e d by  whereas at CT/T  =  0.74  obstacles. the  those  l i n e s , the  areas are shown surrounded by t h i c k e r l i n e s and mostly unconnected p o i n t  s t r e s s , cr  i n p e n e t r a b l e r e g i o n s are  i n which most or a l l p o i n t s  determined  I t can be  free  contain seen  that  f r e e areas are separate and  are  58  Figure  20.  Diagrams from Kocks' o r i g i n a l (1966) .  statistical  analysis  59  completely plane  surrounded  by an impenetrable  region of g l i d e  (Figure 20 ( a ) ) , a t a / x = 1.04 (Figure 20 (c)) , t h e y  f r e e area i s l a r g e , c o v e r i n g a l l o f the s l i p plane the e x c e p t i o n o f a few remaining of  with  s m a l l impenetrable  e x c e p t i o n a l l y high obstacle density.  areas  F i g u r e 20 (b) shows  the f r e e area a t an i n t e r m e d i a t e s t r e s s , a/x =0.90. y f r a c t i o n o f o b s t a c l e f i e l d which i s f r e e area was  The  determined  by Kocks a t f i v e s t r e s s l e v e l s and i s shown i n F i g u r e 20  (d).  Thus the amount o f f r e e area can be seen t o change from an i n d e t e r m i n a t e l y . s m a l l v a l u e a t low s t r e s s , t o a l a r g e v a l u e at  a s t r e s s near the athermal g l i d e  1.4.3.3  stress.  S l i p L i n e B l o c k i n g i n Kocks' Theory o f S t r a i n Hardening  Using the diagrams reproduced  i n F i g u r e 20 t o r e p r e  sent a p o r t i o n o f g l i d e plane threaded by f o r e s t  dislocation,  Kocks takes t h i s a n a l y s i s as a b a s i s f o r h i s theory o f flow s t r e s s and s t r a i n hardening  (1966), which, s i n c e i t i n c l u d e s  no t h e r m a l l y a c t i v a t e d processes On t h i s theory, the mechanical deformation of  i s taken to apply a t 0°K.  phenomenon o f steady s t a t e  (see Appendix 2) i s a s s o c i a t e d w i t h the g l i d e  d i s l o c a t i o n s over i n d e f i n i t e l y l a r g e areas o f s l i p  which occurs when the.athermal  plane,  glide stress i s attained.  T h i s theory e x p l a i n s the l i m i t e d l e n g t h o f s l i p l i n e s by t h e e x i s t e n c e o f areas which have p a r t i c u l a r l y  high  60  local dislocation densities; has  athermally  a f t e r the g l i d e  dislocation  swept almost the whole s l i p plane  (at the  athermal g l i d e s t r e s s ) , s m a l l " i s l a n d s " of u n s l i p p e d  area  remain surrounded by p o r t i o n s of g l i d e d i s l o c a t i o n s . a slip  Thus  l i n e formed on a p a r t of the c r y s t a l s u r f a c e which  happened to i n t e r s e c t an u n s l i p p e d area would be broken as shown i n F i g u r e 21  (a).  In a l a t e r paper, Kocks to the treatment of thermal steady  (1967) adapts t h i s a n a l y s i s  of o b s t a c l e s which may  activation.  s t a t e flow s t r e s s  Nonetheless,  he s t i l l views the  as "the s t r e s s  d i s l o c a t i o n can proceed i n d e f i n i t e l y . " n e i t h e r of the two blocked  be overcome w i t h the a i d  a t which a  (glide)  As a r e s u l t , i n  v a r i a n t s of t h i s theory  are s l i p  lines  i n the u s u a l sense; they have l i m i t e d l e n g t h because  of the e x i s t e n c e of s m a l l areas of h i g h d i s l o c a t i o n d e n s i t y surrounded by  'negative p i l e - u p s , which happen t o be  inter-  1  sected by the c r y s t a l s u r f a c e  (Figure 21a).  Hence s l i p  line  * l e n g t h i s a s t r u c t u r e dependent phenomenon o n l y ; d i s c u s s e d f o r the t h e o r i e s of Seeger Mitchell  (1967), a theory of s l i p  as  was  (1957) and o f H i r s c h  and  l i n e l e n g t h i n which l i n e s  are blocked by s p e c i f i c d i s l o c a t i o n a r r a y s cannot account for  changes i n s l i p  l i n e l e n g t h i n i s o s t r u c t u r a l experiments.  5*  A  and  Makin  parallel development w o u l d be subject  (1966)  of to  the the  analysis of Foreman same objections.  61  F i g u r e 21.  Schematic o f Primary s l i p p l a n e showing s t a t i s t i c a l b l o c k i n g o f s l i p l i n e s d u r i n g steady s t a t e flow, (a) as p o s t u l a t e d by Kocks (1966), (b) an a d d i t i o n a l b l o c k i n g mechanism f i r s t p o s t u l a t e d by A l d e n (1972), which p r e d i c t s temperature dependent s l i p l i n e length.  62  1.4.3.4  S t a t i s t i c a l I n t e r p r e t a t i o n o f the V a r i a t i o n of S l i p L i n e Length w i t h Temperature  (a)  Slip  Line  Strain  Blocking  in Alden's  Theory  of  Hardening  The p r e v i o u s l y d i s c u s s e d a n a l y s i s of Kocks (1966) has been developed  i n t o a theory of temperature dependent  flow s t r e s s and s t r a i n hardening Kocks' o r i g i n a l be athermal.  by Alden  (1972).  As  with  (1966) theory, the o b s t a c l e s are taken t o  However, i n Alden's  a d i s l o c a t i o n may  theory, the area over which  g l i d e i s i n d i r e c t l y thermally  controlled.  , This temperature dependence o f the f r e e area r e p r e sents a s i g n i f i c a n t d i f f e r e n c e between the t h e o r i e s of Kocks and Alden.  In the l a t t e r theory, steady s t a t e flow occurs  at a temperature dependent s t r e s s , a ( T ) , g e n e r a l l y the athermal g l i d e s t r e s s , T . occur f o r example a t O{1\)/T^ areas may  T ( < T ) and X  = 0.74,  may  where the s i z e of f r e e ( a ) , and  (b), (c) c o u l d i l l u s t r a t e the f r e e area d u r i n g  steady s t a t e deformation 2  Thus steady s t a t e flow  be s i m i l a r to those shown i n F i g u r e 20  F i g u r e s 20  below  0°K  The  a t an. i n t e r m e d i a t e temperature  respectively.  temperature dependence of s l i p l i n e l e n g t h i s  a d i r e c t consequence of t h i s steady s t a t e flow a t v a r i a b l e s t r e s s e s below the athermal g l i d e s t r e s s .  F i g u r e s 21 ( a ) ,  (b) s c h e m a t i c a l l y i l l u s t r a t e s l i p l i n e p r o d u c t i o n d u r i n g steady  s t a t e flow, r e s p e c t i v e l y a t and below the  athermal  63  glide stress. 0°K).  (and  thus  r e s p e c t i v e l y a t and above  I t can be seen t h a t , whereas a t 0°K (a/x  = 1.04)  slip  l i n e s are "blocked" only when the s m a l l , h i g h d e n s i t y " i s l a n d s " of  u n s l i p p e d g l i d e plane happen t o c o i n c i d e w i t h the s u r f a c e  (Kocks' b l o c k i n g c o n d i t i o n ) , a t a h i g h e r temperature sented here by o*/x in.a  = 0.74) s l i p  l i n e s may a l s o be b l o c k e d  somewhat d i f f e r e n t sense, being completely  by the unpenetrable produced  region.  surrounded  In consequence, the s l i p  a t h i g h e r temperatures  w i t h the p r e s e n t  (repre-  lines  are s h o r t e r , i n agreement  results.  In Alden's theory the Kocks' a n a l y s i s  (1966) i s  taken t o be d i r e c t l y a p p l i c a b l e t o a m i c r o s t r u c t u r e a t 0°K, where the m i c r o s t r u c t u r e i s s t a b l e , and no thermal t i o n o f f o r e s t d i s l o c a t i o n s can occur.  penetra-  Furthermore, f o l l o w i n g  Saada (1963), who has shown t h a t the a t t r a c t i v e t r e e s o f t h e f o r e s t i n t e r a c t w i t h g l i d e d i s l o c a t i o n s over a l o n g  range,  i t i s assumed t h a t the e f f e c t i v e s t r e n g t h o f the d i s l o c a t i o n f o r e s t does not vary w i t h temperature for  e l a s t i c modulus  (after a correction  change).  The athermal g l i d e s t r e s s , x , i s c a l l e d the y i e l d strength  (a p r o p e r t y o f the s o l i d ) , and d e f i n e d by  x y or  i n a s t r a i n hardened c r y s t a l  (1.3)  64  T  = a" p)* u Gb (vp;  (1.4)  In Eq. (1.3) 1 i s the average n e a r e s t neighbour of  spacing  the o b s t a c l e s ; i n Eq. (1.4) p i s the average d e n s i t y o f  forest dislocations; a  1  and a" are e m p i r i c a l constants  with  values c l o s e t o u n i t y . In the p r e s e n t experiments,  the s t r u c t u r e i s  n e a r l y constant, but the s t r e s s v a r i e s w i t h  temperature.  I t i s h i g h e r a t low temperature and as a r e s u l t most o b s t a c l e p a i r s w i l l be p e n e t r a b l e ; the f r e e area w i l l be l a r g e and the s l i p l i n e s long.  When we c o n s i d e r deformation  same m i c r o s t r u c t u r e a t h i g h temperatures  i n the  because t h e flow  s t r e s s i s lower, areas through which d i s l o c a t i o n s may  glide  w i l l be s m a l l e r and hence, s l i p l i n e s w i l l be s h o r t e r . A n a l y t i c a l l y , the f r e e area a a t a g i v e n temperat u r e and s t r e s s i s contained i n the r e l a t i v e area f u n c t i o n A__  A  r  = A  r  t— a  AI  (1-5)  A c c o r d i n g to A l d e n ' s t h e o r y , the change in f l o w s t r e s s w i t h t e m p e r a t u r e is due to the change in rates of r e a r r a n g e m e n t and r e c o v e r y of f o r e s t d i s l o c a t i o n s . Clearly s t r u c t u r e changes per se cannot be r e s p o n s i b l e f o r a r e v e r s i b l e i s o s t r u c t u r a 1 change in f l o w s t r e s s . The p r e s e n t e x p e r i m e n t s were d e s i g n e d to m i n i m i z e s t r u c t u r e c h a n g e s , by u s i n g s m a l l s t r a i n i n c r e m e n t s and h i g h p r e s t r a i n t e m p e r a t u r e s . However the changes in rates o f r e a r r a n g e m e n t and r e c o v e r y w i l l be l a r g e s i n c e the s t r a i n i n c r e m e n t s a r e c a r r i e d out o v e r a l a r g e temperature range.  65  where A i s a l a r g e area o f s l i p plane c o n t a i n i n g a 'true average  1  density of obstacle d i s l o c a t i o n s .  s t r e s s a and s t r u c t u r e suggested f o r A  The f u n c t i o n o f  is r  T  A^ = exp -  i s d e f i n e d by Eq. (1.4) and x  -T  v  — 0~'  (1.6)  1  V  decreases w i t h  increasing  r e g u l a r i t y of the s t r u c t u r e . In the 0°K case the s t r e s s a equals  x  during  steady s t a t e , and when steady s t a t e i s a t t a i n e d , the f r e e area equals A.  Thus the s l i p l i n e s are p r e d i c t e d t o have  t h e i r maximum l e n g t h . thermally  A t higher  temperature, t h e r a t e o f  a c t i v a t e d s t r u c t u r e change, i n c l u d i n g l o s s  (recovery)  and rearrangement of o b s t a c l e d i s l o c a t i o n s , i n c r e a s e s . Regions o f lower o b s t a c l e d e n s i t y w i l l tially.  then deform p r e f e r e n -  In other words, a i s l e s s than A and the s l i p  line  l e n g t h i s reduced. This concept of a changing f r e e area leads to a m i c r o s t r u c t u r a l e x p l a n a t i o n  of the ( i s o s t r u c t u r a l )  v a r i a t i o n i n flow s t r e s s w i t h temperature. and  From Eqs. (1.5)  (1.6) i t i s seen t h a t when a < A, a < x . ' y  steady s t a t e flow occurs  also  So w h i l e  a t a s t r e s s below the athermal  g l i d e s t r e s s , a t the same time d i s l o c a t i o n loops expand over areas o f s l i p plane c o n t a i n i n g an o b s t a c l e d e n s i t y l e s s than the average d e n s i t y .  In t h i s manner the mechanical  66  e f f e c t , namely the  temperature dependent flow s t r e s s ,  the m i c r o s t r u c t u r a l are  e f f e c t , v a r i a t i o n i n s l i p l i n e length  coupled.  (b)  S l i p Line  Blocking  of Alden's As change as the one  may  rather  using  a  Variation  Theory  an a l t e r n a t i v e to t h e r m a l l y a c t i v a t e d  structure  o r i g i n of temperature dependent p l a s t i c i t y ,  consider thermally activated than t a k i n g  the  thermally activated  glide.  In o t h e r words,  " g l i d e " of f o r e s t d i s l o c a t i o n s  p r o c e s s , we  the t h e r m a l l y a c t i v a t e d it  and  should be noted t h a t ,  may  alternatively  as  consider  g l i d e of g l i d e d i s l o c a t i o n s . i n order to s u c c e s s f u l l y  the  However,  account  f o r the p r e s e n t r e s u l t s , t h i s a l t e r n a t i v e p r o p o s a l remains within  the  framework of Alden's t h e o r y .  Hence, t h i s  new  thermal a c t i v a t i o n process i n i t s e l f produces l i t t l e Rather i t releases  g l i d e d i s l o c a t i o n s i n t o newly  f r e e areas, which are An has  recently  This a n a l y s i s activated of Kocks activated  then t r a v e r s e d  athermally.  i n t e r e s t i n g s t a t i s t i c a l basis been p u b l i s h e d by M o r r i s and  strain.  available ' .  f o r such a p r o p o s a l Klahn  (1973).  adds a s t a t i s t i c a l c r i t e r i o n f o r a t h e r m a l l y  g l i d e / f o r e s t i n t e r a c t i o n , to the (1966), and  Foreman and  step i s taken to be  d i s l o c a t i o n l i n e through one  Makin  o r i g i n a l analyses  (1966).  The  thermally  the movement of a segment of of the o b s t a c l e s at which  the  67  g l i d e d i s l o c a t i o n has been t e m p o r a r i l y s t a b i l i z e d . f o r c e d i s t a n c e curve f o r the d i s l o c a t i o n / o b s t a c l e  The interaction  i s taken as a s t e p f u n c t i o n , w i t h thermal a c t i v a t i o n being treated  as a s t o c h a s t i c ,  thermally activated  Moreover, the  step i s assumed t o b e . r a t e c o n t r o l l i n g ,  i n t h a t the time r e q u i r e d positions  random p r o c e s s .  f o r d i s l o c a t i o n g l i d e between  stable  i s n e g l i g i b l e compared t o t h e time r e q u i r e d f o r  thermal a c t i v a t i o n .  The d i s l o c a t i o n l i n e may then proceed  from one s t a b l e p o s i t i o n t o another along a s t a t i s t i c a l l y chosen path, under the combined i n f l u e n c e  o f the a p p l i e d  s t r e s s , and thermal a c t i v a t i o n . In a computer s i m u l a t i o n and  Klahn  (1974) s u c c e s s f u l l y  of t h i s p r o c e s s , M o r r i s  i l l u s t r a t e the p o s s i b i l i t y o f  steady s t a t e , thermal a c t i v a t i o n c o n t r o l l e d g l i d e a t s t r e s s e s w e l l below the athermal g l i d e s t r e s s , by o b t a i n i n g  average  g l i d e v e l o c i t y / t e m p e r a t u r e data f o r a g i v e n o b s t a c l e s t r e n g t h , at d i f f e r e n t a p p l i e d  stress  levels  (Figure  22). W h i l s t i t  i s i s d i f f i c u l t to l i n k t h i s k i n d o f d a t a t o s l i p  line  i t should be p o s s i b l e  analysis  t o use t h e M o r r i s and Klahn  lengths,  to compute the area swept by a g l i d e d i s l o c a t i o n when i t moves between s t a b l e p o s i t i o n s ,  as a f u n c t i o n  of applied  s t r e s s , f o r a given obstacle strength, at various  temperatures.  T h i s f u n c t i o n w i l l then be d i r e c t l y analogous t o t h a t posed by Alden  (Eq. 1.6), s i n c e  pro-  i n both cases the g l i d e  processes which generate s l i p l i n e s are i n i t i a t e d by thermal  _  60  •  <x>  aj  E. S. 5 0 o CL  0.45  Ty/  increasing  S  40  velocity  > o </>  83  10 10  30  c o  E o  (/)  c • <u  ~ 1.8 x 10 o  E  20  >  0  G CD >  V c  increasing 0  0  •00  200 (Dimensionless  CL  a = 260 O"  Figure  22.  = 0.90Ty  300  a  = 68  O-=0.69xy  a =  Temperature  temperature 400  500  Parameter)  35  O"=0.45Ty 00  D a t a f r o m a n a l y s i s o f M o r r i s and K l a h n (1974) s h o w i n g velocity/temperature r e l a t i o n s h i p s f o r f o u r s e p a r a t e random o b s t a c l e a r r a y s a t t h r e e d i f f e r e n t applied stress levels. T h e i r s c h e m a t i c below shows p r e d i c t e d a p p e a r a n c e o f slip lines. An e s t i m a t i o n o f a c t u a l v e l o c i t i e s u s i n g c o n d i t i o n s f r o m p r e s e n t e x p e r i m e n t s i s added (left).  69  activation; specified  the only difference  thermal  activation  1.4.3.5  to  step.  T e m p e r a t u r e D e p e n d e n t S l i p Band W i d t h s and  An  Heights  extension  the simultaneous  of the Morris  deformation  l e a d s t o an i n t e r e s t i n g lines  Included  22 a r e d a t a  i n Figure  four separate  velocities  produced  By i m p o s i n g  a fixed  uniform  distribution  of glide  (1974) show t h a t a s t r a i n these  four planes  hence a s i n g l e ture  and t h u s  distributed.  Strain  a combination  o f these  rate,  by g l i d e  Morris  a t an i n t e r m e d i a t e two r e s u l t s ,  containing  distribution  of glide  increasingly  homogeneous d i s t r i b u t i o n  dislocations,  only; tempera-  glide  i s equally  temperature  producing  surface.  Klahn  At a higher  the s t r a i n  microstructure.  and  (0.45 T ) a l l f o u r  at the c r y s t a l  i n a given  differ  on one p l a n e  be f o r m e d .  step heights  ture,  temperatures,  and a s s u m i n g a  increment i n a c r y s t a l  line will  are equal,  A t low  dislocation,  and r e l a t i v e l y l o w e r s t r e s s  velocities  movement  (0.9 r ) i s r e q u i r e d , t h e  strain  i s achieved  slip  temperatures.  f o r the four arrays  widely.  planes of the  at different  for dislocation  stress  determined  analysis  description  obstacle arrays.  where a r e l a t i v e l y h i g h glide  and K l a h n  of p a r a l l e l s l i p  qualitative  appearance o f s l i p  through  between t h e m o d e l s i s t h e  gives  a range o f  Thus a s s u m i n g a  uniform  t h i s model p r e d i c t s of s t r a i n with  an  tempera-  70  Micrographs of s l i p l i n e s slip  (more p r e c i s e l y termed  "bands"), formed i n the same m i c r o s t r u c t u r e a t d i f f e r e n t  temperatures are shown i n F i g u r e 11.  I t can be seen t h a t  s l i p bands formed a t 4.2°K appear to be s h a r p l y d e f i n e d when compared w i t h the r e l a t i v e l y d i f f u s e bands formed a t 293°K. I f the M o r r i s and Klahn model i s a p p l i e d to a number of neighbouring s l i p planes which are a c t i v e d u r i n g the 293°K increment, and i n which a r e l a t i v e l y uniform of mobile d i s l o c a t i o n s may  distribution  be a n t i c i p a t e d a t both tempera-  t u r e s , the r e l a t i v e appearance of s l i p bands i s c o r r e c t l y predicted. However, as can be seen i n both f i g u r e s , the s l i p band d e n s i t y to  (the number o f bands per u n i t l e n g t h p e r p e n d i c u l a r  the band d i r e c t i o n ) i s apparently lower a f t e r the 293°K  increment than t h a t of bands formed a t 4.2°K.  Thus w h i l e  s t r a i n i n neighbourhood of a 293°K s l i p band has been more homogeneous than i t would have been a t 4.2°K, s t r a i n i n g e n e r a l has been l e s s so.  T h i s l a t t e r e f f e c t may  provide  a c l u e to the predominant d i s l o c a t i o n m u l t i p l i c a t i o n which w i l l operate to supply s u f f i c i e n t g l i d e to m a i n t a i n the imposed s t r a i n r a t e .  mechanism  dislocations  A t low temperatures,  where steady s t a t e flow occurs a t s t r e s s e s approaching the athermal g l i d e s t r e s s , Frank-Read sources are assumed t o be o p e r a t i v e , and these may  supply g l i d e d i s l o c a t i o n s on a  l a r g e r p r o p o r t i o n of s l i p p l a n e s .  In c o n t r a s t a t h i g h e r  71  temperatures,  steady s t a t e flow occurs a t lower  stresses,  so there w i l l be fewer a v a i l a b l e Frank-Read s o u r c e s . t h i s case a m u l t i p l e proposed by Koehler  In  c r o s s - g l i d e mechanism o f the type (1952) may  supply a d d i t i o n a l sources on  adjacent s l i p p l a n e s , s i n c e c r o s s - s l i p i s a dependent phenomenon.  first  temperature  As a d i r e c t consequence of such a  mechanism, t h e r e w i l l be more s l i p l i n e c l u s t e r i n g bands) at h i g h e r temperatures.  (into  Whereas t h i s e f f e c t cannot  be a n t i c i p a t e d from a t h e o r e t i c a l model which assumes a uniform d i s t r i b u t i o n of g l i d e d i s l o c a t i o n s , these two m u l t i p l i c a t i o n mechanisms l e a d to an a p p r o p r i a t e d e s c r i p t i o n of the nature of s l i p a t d i f f e r e n t  1.4.4  temperatures.  A U n i f i e d View of S l i p L i n e Formation, Microstructural  Observations and Flow S t r e s s  In the preceding d i s c u s s i o n  several  deductions  have been made from the r e s u l t s of t h i s s l i p l i n e study  and  other work, concerning the nature of p l a s t i c flow i n copper. Since the d i s c u s s i o n was  n e c e s s a r i l y extended,  t i o n s might appear t o be r e l a t i v e l y indpendent.  these deducHowever,  i n t h i s s e c t i o n i t w i l l be shown t h a t , when taken t o g e t h e r , they form a new  and s e l f - c o n s i s t e n t p i c t u r e of p l a s t i c  deformation i n copper. S l i p l i n e l e n g t h s i n copper are g e n e r a l l y much g r e a t e r than the average diameter of d i s l o c a t i o n c e l l s .  72  The reason f o r t h i s e f f e c t i s r e v e a l e d on c l o s e r of  the c e l l c o n s t r u c t i o n .  examination  Only the s i d e s of c e l l w a l l s  l y i n g p a r a l l e l to primary s l i p plane are w e l l - d e f i n e d 'strong' o b s t a c l e s , being formed  from the dense c a r p e t s of primary  d i s l o c a t i o n m u l t i p o l e s which are c h a r a c t e r i s t i c of stage I I m i c r o s t r u c t u r e s i n copper 1968).  (Kuhlman-Wilsdorf,  1968;  Seeger,  Since they are r e a d i l y i d e n t i f i e d i n m e t a l l o g r a p h i c  s e c t i o n s , i t i s the spacing of these c a r p e t s which i s u s u a l l y taken as a measure of the c e l l diameter. completed by more d i f f u s e a r r a y s of f o r e s t  The  cells  are  dislocations  l y i n g roughly p e r p e n d i c u l a r to the m u l t i p o l e c a r p e t s . g l i d e d i s l o c a t i o n loops w i l l  expand on the primary  plane i n the r e l a t i v e l y s o f t r e g i o n s between move through a r r a y s of f o r e s t d i s l o c a t i o n s high speed by c e l l  ( F i s h e r and L a l l y , 1967),  loops expand and s l i p  slip  these c a r p e t s , at r e l a t i v e l y  and be b l o c k e d not  " w a l l s " per se, but by s t a t i s t i c a l  w i t h the d i s l o c a t i o n f o r e s t .  New  interactions  In t h i s manner, d i s l o c a t i o n  l i n e s are formed  compared with the apparent c e l l  over d i s t a n c e s l a r g e  size.  S l i p l i n e s are u s u a l l y c l u s t e r e d i n t o bands, and i t was  observed i n t h i s study t h a t the range of spacing o f  the more prominent  bands as seen i n both o p t i c a l and  electron  micrographs, namely 2-10y, c o i n c i d e s w i t h the independently estimated spacing of the dense c a r p e t s of d i s l o c a t i o n s  found  i n the m i c r o s t r u c t u r e . The occurrence and s p a c i n g of these  73  bands can be e x p l a i n e d w i t h i n the framework of the p r e c e d i n g analysis.  In a d d i t i o n  a s i g n i f i c a n t f e a t u r e o f the proceeding  explanation i s that g l i d e loops, i n t e r a c t i n g with f o r e s t d i s l o c a t i o n s  statistically  expand through r e g i o n s o f the  c r y s t a l u n c l u t t e r e d by d i s l o c a t i o n d e b r i s , and hence the p r e v i o u s e x p l a n a t i o n o f the v a r i a t i o n o f s l i p l i n e l e n g t h w i t h temperature  i s supported.  The bands are formed and widened as a r e s u l t o f the accumulation  of d i s l o c a t i o n debris during p l a s t i c flow.  I n i t i a l l y g l i d e d i s l o c a t i o n loops expand o n l y on one s l i p plane as shown i n F i g u r e 23 ( e i t h e r because t h i s plane i s intersected  by f o r e s t d i s l o c a t i o n s more a p p r o p r i a t e l y  placed f o r s t a t i s t i c a l  interaction  (Morris and Klahn,  1974),  or because a f a v o u r a b l y p o s i t i o n e d source i s a v a i l a b l e on this plane).  After  a number o f loops have expanded i n  t h i s p l a n e , accumulated  d i s l o c a t i o n d e b r i s prevents the  f u r t h e r o p e r a t i o n o f t h i s g l i d e source, and the mechanism of m u l t i p l e c r o s s g l i d e 1959)  (Koehler, 1952; Johnson and Gilman,  p r o v i d e s a new source on the neighbouring s l i p  The process i s repeated on s u c c e s s i v e neighbouring and thence the s l i p band i s formed. l i n e length i s controlled  planes  Thus w h i l e the s l i p  by the s t a t i s t i c a l  a d i s l o c a t i o n loop expanding  plane.  i n t e r a c t i o n of  a t h e r m a l l y through an area  of s l i p plane u n c l u t t e r e d by d i s l o c a t i o n d e b r i s , t h i s d e b r i s causes d i s l o c a t i o n s  t o c l u s t e r i n t o bands of f i n i t e  width.  s Figure  23.  B  Cross-section perpendicular to primary glide plane showing slip  line blocked xn schematic cell structure: source S near surface of multipole carpets fSresl' secSons of cell wall, f , and.is ^ / ^ ^statistically - P a n d s , threading'tS^gh^??^ e rorest sections blocked at B! T f ' T e l f t t  1  0  75  T h i s account of s l i p band formation i n t u r n e x p l a i n s the f o r m a t i o n and t h i c k e n i n g o f dense c a r p e t s o f d i s l o c a t i o n s . The d e b r i s c r e a t e d a f t e r the expansion of a number o f d i s l o c a t i o n loops on neighbouring s l i p planes w i l l  contain a  l a r g e p r o p o r t i o n o f primary d i s l o c a t i o n s arranged i n the form o f m u l t i p o l e s .  As the s l i p band i s p r o g r e s s i v e l y formed  the m u l t i p o l e s accumulate i n h i g h d e n s i t y p l a n a r a r r a y s , p a r a l l e l to the primary  s l i p plane.  With f u r t h e r  deformation,  pinned d i s l o c a t i o n s on the s u r f a c e o f these a r r a y s w i l l p r o v i d e a s i g n i f i c a n t p r o p o r t i o n o f the new s o u r c e s .  As a  r e s u l t , more m u l t i p o l e s w i l l be swept i n t o a p r o g r e s s i v e l y denser  c a r p e t of d i s l o c a t i o n s .  Thus the c a r p e t s are i n f a c t  the somewhat d i s t o r t e d remnants o f p r e v i o u s l y formed  slip  lines. •A s e l f - c o n s i s t e n t view has been presented which, couples the p r o p e r t i e s of s l i p  l i n e s with m i c r o s t r u c t u r a l  observations. As was shown i n the p r e v i o u s d i s c u s s i o n , a view of  p l a s t i c flow i n which expanding  interact statistically  g l i d e d i s l o c a t i o n loops  with f o r e s t d i s l o c a t i o n s , can a l s o  account f o r the v a r i a t i o n o f flow s t r e s s w i t h Hence t h i s micromechanical  temperature.  p i c t u r e of the process o f p l a s t i c  deformation s u c c e s s f u l l y l i n k s m i c r o s t r u c t u r a l o b s e r v a t i o n s w i t h both the p r o p e r t i e s o f s l i p dependence of flow s t r e s s .  l i n e s , and the temperature  76  1.4.5  Transition  Effects  D i s c u s s i o n so f a r has been e x c l u s i v e l y  concerned  w i t h an e x p l a n a t i o n of the p r o p e r t i e s of s l i p l i n e s formed during steady s t a t e flow, and w h i l e i t i s apparent t h a t a statistical  approach  i s f o l l o w e d by  i n which a t h e r m a l l y a c t i v a t e d process  athermal g l i d e , can s u c c e s s f u l l y d e s c r i b e  the p r o p e r t i e s of s l i p l i n e s formed d u r i n g steady flow, some mention should be made of t r a n s i t i o n (microstrain).  state  effects  I f a t any i n s t a n t i n the deformation a  sudden change i s made i n e i t h e r s t r a i n r a t e or  temperature  (which i s i d e a l l y what has been done i n these  experiments),  no sudden change i n s t r e s s i s p r e d i c t e d by such a t h e o r y . S p e c i f i c a l l y , on a decrease of temperature,  t h e r e w i l l be a  t r a n s i t i o n s t r a i n d u r i n g which the s t r e s s and f r e e area r i s e to t h e i r new  steady s t a t e v a l u e  (Figure 15).  (This  t r a n s i t i o n s t r a i n on i n c r e a s e of s t r a i n r a t e has been observed f o r deformation of Pb a t v a r i o u s (Alden, 1973;  C l a r k and Alden, 1973;  temperatures  Alden, 1974).)  Hence  l i n e s produced  d u r i n g the t r a n s i t i o n w i l l be s h o r t e r than  those produced  d u r i n g steady s t a t e flow, .and these a d d i t i o n a l  s h o r t l i n e s w i l l appear along w i t h l i n e s whose l e n g t h s r e p r e s e n t the steady s t a t e f r e e area. Figure  9,  As can be seen i n  a number of s h o r t l i n e s are indeed observed a t  4.2°K,in a d d i t i o n to the high p r o p o r t i o n of long l i n e s , which are presumed t o be produced d u r i n g steady s t a t e when the f r e e area i s l a r g e .  77  An i n c r e a s e i n s l i p l i n e l e n g t h d u r i n g the t r a n s i t i o n s t r a i n has a l s o been observed  ( d e L a r i o s , 1973).  Pre-  s t r a i n e d and p o l i s h e d aluminum s i n g l e c r y s t a l s were s t r a i n e d i n a sequence  o f s m a l l increments through the t r a n s i t i o n  region.  The average l i n e l e n g t h was  found t o i n c r e a s e ,  reaching  a maximum immediately a f t e r the t r a n s i t i o n ; w i t h  f u r t h e r s t r a i n the average l e n g t h decreased as expected (Seeger, 1957).  P A R T  2  MICROSTRAIN AND ETCH PIT STUDIES  2.1  Introduction The  r e s u l t s of the s l i p l i n e s t u d i e s  t h i s t h e s i s are c o n s i s t e n t w i t h s t a t i s t i c a l  i n Part I of  theories  p l a s t i c flow i n which, a t f i n i t e temperatures, a  of  thermally  a c t i v a t e d process enables a g l i d e d i s l o c a t i o n loop to athermally expand i n t o a newly a v a i l a b l e area o f s l i p p l a n e (the " f r e e a r e a " ) .  Moreover, the r e l a t i v e l e n g t h s o f  l i n e s formed i n a g i v e n m i c r o s t r u c t u r e considering  the  slip  can be e x p l a i n e d  s t r e s s dependence of t h i s f r e e a r e a .  by  It is  a l s o a n t i c i p a t e d t h a t the s i z e of t h i s f r e e area at a f i x e d s t r e s s w i l l depend on the d e n s i t y dislocation  and  d i s t r i b u t i o n of  the  microstructure.  I t has  already  been shown (Kocks,  1966)  that  the  c r i t i c a l s t r e s s a t which the f r e e area becomes l a r g e i s a f u n c t i o n of the o v e r a l l d i s l o c a t i o n d e n s i t y , f i e l d of o b s t a c l e s .  f o r a random  However, d i s l o c a t i o n s i n e x p e r i m e n t a l l y  observed m i c r o s t r u c t u r e s  u s u a l l y appear to be  being commonly d e s c r i b e d  as t a n g l e d ,  78  f a r from random,  c l u s t e r e d or  cellular,  79  and i t i s reasonable t o expect t h a t the f r e e area w i l l be a f u n c t i o n o f the degree o f such c e l l u l a r i t y .  Clearly, at  constant d i s l o c a t i o n d e n s i t y the more c e l l u l a r the m i c r o s t r u c t u r e , the g r e a t e r w i l l be the area f r a c t i o n w i t h low d i s l o c a t i o n density.  Given t h a t the c e l l w a l l s do not form i n t o  b a r r i e r s that block s l i p  contiguous  (which i s shown i n the p r e v i o u s  d i s c u s s i o n t o be u n l i k e l y i n the stage I I hardening  region  of copper c r y s t a l s ) , the f r e e area should i n c r e a s e more r a p i d l y with s t r e s s i n a more c e l l u l a r m i c r o s t r u c t u r e (Kocks, Alden,  1966;  1972). The  s i z e of the f r e e area w i l l i n f l u e n c e the m i c r o -  p l a s t i c response  o f a c r y s t a l t o an i n c r e a s e i n a p p l i e d s t r e s s .  When t h i s s t r e s s r i s e s from zero t o t h e y i e l d s t r e s s , t h e f r e e area i s expected  t o i n c r e a s e i n s i z e continuously, r e a c h -  i n g a maximum a t steady s t a t e flow deformation  terminology  (for d e f i n i t i o n s of  see Appendix 2 ) . Assuming t h a t d i s -  l o c a t i o n sources are s c a t t e r e d and r e l a t i v e l y abundant, the amount o f m i c r o s t r a i n ( p l a s t i c deformation  i n the p r e y i e l d  region) w i l l be a d i r e c t f u n c t i o n o f the s i z e o f the f r e e area a t any g i v e n l e v e l o f a p p l i e d s t r e s s . Taken together, these arguments mean t h a t c e l l u l a r m i c r o s t r u c t u r e s should e x h i b i t more pronounced m i c r o s t r a i n . In order t o t e s t t h i s p r e d i c t i o n , h i g h r e s o l u t i o n  stress-  s t r a i n curves were o b t a i n e d from specimens which d i f f e r e d mainly  i n the degree of c e l l u l a r i t y o f t h e i r d i s l o c a t i o n  80  microstructure.  Almost a l l previous  microstrain studies i n  copper have been performed on annealed c r y s t a l s ( R o s e n f i e l d Averbach, 1960; and use  Hordon, 1962;  Tinder  a m u l t i p l e load c y c l e  and  and Washburn, 1964),  ( B a n e r j i et cel., 1970), or a  h y s t e r e s i s technique, o f t e n i n an attempt to d i s c o v e r a lower limit  f o r the c r i t i c a l r e s o l v e d shear s t r e s s .  In t h i s case  however, i n order t o i n v e s t i g a t e p l a s t i c flow i n c r y s t a l s i n a work hardened s t a t e , the samples are p r e s t r a i n e d .  Moreover  i n the present work, the m i c r o s t r a i n curve i s obtained a s i n g l e load c y c l e , to avoid any  t r a n s i e n t or  from  irreversible  e f f e c t s from repeated load c y c l i n g . In order t h a t the degree of c e l l u l a r i t y of these specimens could be q u a n t i f i e d , the m i c r o s t r u c t u r e s  of  i d e n t i c a l s e r i e s of c r y s t a l s were examined u s i n g the l o c a t i o n etch p i t technique.  an dis-  Whereas d i s l o c a t i o n arrangements  i n copper have been e x t e n s i v e l y s t u d i e d i n a q u a l i t a t i v e manner, only s i n c e the beginning o f t h i s work have sampling techniques been used to measure l o c a l d i s l o c a t i o n d e n s i t y variations  (Donner et al.  1974).  3  d i s l o c a t i o n arrangements may mechanical p r o p e r t i e s , and sents  A d e t a i l e d knowledge of  be v a l u a b l e  the present  f o r the a n a l y s i s o f  work probably  repre-  the f i r s t use of a s t a t i s t i c a l technique to q u a n t i f y  the degree of c e l l u l a r i t y  of a d i s l o c a t i o n m i c r o s t r u c t u r e .  81  2.2  Experimental 2.2.1  1  Technique  Specimen P r e p a r a t i o n Seeded copper s i n g l e c r y s t a l s were grown u s i n g  technique  previously described.  the  Since the e t c h p i t method  of measuring d i s l o c a t i o n d e n s i t i e s i s l i m i t e d t o r e l a t i v e l y low d e n s i t i e s , i t i s important  to reduce d e n s i t i e s of grown-in  d i s l o c a t i o n s and d i s l o c a t i o n s i n t r o d u c e d by h a n d l i n g . t h i s case specimens were cut t o l e n g t h u s i n g an a c i d (South Bay  Technology, Model No.  750)  i n order to  In saw  prevent  any damage to the as-grown c r y s t a l s .  Previous work ( L i v i n g s t o n ,  1962)  (annealing a t a tempera-  has  shown t h a t c y c l i c annealing  ture which c o n t i n u o u s l y  c y c l e s between a maximum and minimum)  i s more e f f e c t i v e than a s t a t i c anneal dislocation density.  i n reducing p o s t  Consequently a l l c r y s t a l s were annealed  f o r 72 hours i n a vacuum of 10" c y c l e d h o u r l y between 795°C and  5  t o r r , with 1045°C.  temperatures  As w i l l be  l a t e r , t h i s technique  was  l o c a t i o n content,  i n c r e a s i n g the grown-in  and  anneal  s u c c e s s f u l i n reducing the  discussed dis-  subgrain  size .  of annealed c r y s t a l s .  2.2.2  Crystal Orientation C r y s t a l s used i n the m i c r o s t r a i n and e t c h p i t study  had t e n s i l e axes o r i e n t e d as shown i n F i g u r e 24. c o n v e n i e n t l y grown o r i e n t a t i o n which maintained  T h i s was  a  single glide  82  F i g u r e 24.  O r i e n t a t i o n of t e n s i l e a x i s f o r c r y s t a l s used i n m i c r o s t r a i n / e t c h p i t work. 0 = i n i t i a l o r i e n t a t i o n , X = o r i e n t a t i o n a f t e r p r e s t r a i n at 1000°K.  83  as the dominant d e f o r m a t i o n mode t h r o u g h o u t the used i n t h i s experiments has  prestrains  S i n g l e g l i d e was chosen s i n c e i t  been shown ( B a s i n s k i and B a s i n s k i , 1964) t h a t  dislocation densities  forest  i n s i n g l e s l i p c r y s t a l s , d e t e r m i n e d on  a cross section p a r a l l e l t o the primary g l i d e plane,  correlate  w e l l w i t h low temperature f l o w s t r e s s i n copper.  2.2.3  Mechanical Testing 2.2.3.1  Prestrain  In o r d e r t o e x p l o r e t h e i n f l u e n c e l o c a l d i s l o c a t i o n density  on the m i c r o s t r a i n  of variation i n behaviour of  copper c r y s t a l s , a s e r i e s o f specimens was r e q u i r e d p a r a b l e d i s l o c a t i o n d e n s i t i e s , b u t d i f f e r e n t local densities.  I t was a n t i c i p a t e d  w i t h comdislocation  that substructures with  d i f f e r e n t d i s l o c a t i o n arrangements c o u l d be o b t a i n e d by changing t h e temperature o f p r e s t r a i n .  W i t h t h i s i n mind,  p r e s t r a i n s were c a r r i e d out a t t e m p e r a t u r e s , and i n t h e environments shown below: I000°K 850°K 700°K .293° K 77°K  Vacuum  Furnace "  11  "  " Air  Nitrogen  Gas  Cryostat  84  By u s i n g p u b l i s h e d  data  ( M i t c h e l l / 1964)  on t h e  r e v e r s i b l e change i n flow s t r e s s w i t h temperature, the attempt was  made t o produce specimens w i t h t h e same 77°K flow  (and o v e r a l l d i s l o c a t i o n d e n s i t y ) .  stress  Some attempt was a l s o  made to a n t i c i p a t e d i f f e r e n c e s i n c r o s s - s e c t i o n area r e s u l t i n g from the d i f f e r i n g amounts o f p r e s t r a i n (greater a t higher temperatures).  Hence a t any g i v e n temperature and p r e s t r a i n ,  the load r e q u i r e d t o g i v e s i m i l a r d i s l o c a t i o n d e n s i t i e s was determined. '  The p r e s t r a i n s were performed on f l o o r model I n s t r o n  machines, using the lowest a v a i l a b l e c r o s s head speed.  Above  273°K t h i s was 2 x l O * i n c h m i n " , and a t 293°K and below, -1  1  i t was 2 x 10" i n c h m i n . 3  - 1  S t r a i n s were measured from  cross,..head movement f o r the former temperatures:  f o r the  l a t t e r an I n s t r o n Extensometer was used, together  with a  Honeywell x-y r e c o r d e r . :'./v-..--/-/Plots  o f r e s o l v e d shear s t r e s s versus shear s t r a i n  wereCcomputed w i t h ' t h e " a i d Packard/ 9100 A c a l c u l a t o r .  2.2.3.2 One  o f a program w r i t t e n f o r a HewlettThese curves are shown i n F i g u r e 25.  Microstrain Testing  o f the o b j e c t i v e s o f t h i s study was t o o b t a i n  s t r e s s s t r a i n curves o f the v a r i o u s p r e s t r a i n e d samples a t low temperature, and a t a s e n s i t i v i t y s u f f i c i e n t t o r e v e a l q u a l i t a t i v e d i f f e r e n c e s i n the shapes o f those  curves.  Percent  Figure  25.  P r e s t r a i n curves  Resolved Shear Strain  f o r c r y s t a l s used  i n microstrain/etch  p i t work.  86  A s e r i e s o f p r e l i m i n a r y experiments showed t h a t s t r a i n d i f f e r e n c e s o f up t o 2 x 10 * were to be expected -1  i n the m i c r o s t r a i n r e g i o n . to,use  A f t e r some t r i a l s ,  i t was d e c i d e d  a 1/2 i n c h , 10% I n s t r o n extensometer, which was  modified  f o r use with s i n g l e c r y s t a l s .  T h i s d e v i c e was  capable o f measuring s t r a i n s o f l e s s than 10" environment.at 77°K.  5  i n a gaseous  Other c o n s i d e r a t i o n s i n the c h o i c e o f  t h i s extensometer, together w i t h d e t a i l s o f the d e s i g n , t e s t i n g and c a l i b r a t i o n procedure are d i s c u s s e d i n Appendix 3; an o u t l i n e o f p r e c a u t i o n s  taken t o guard a g a i n s t  spurious  s t r a i n measurements i s a l s o g i v e n . The  ,  s i g n a l from the extensometer was a m p l i f i e d  u s i n g an Instron Load C e l l a m p l i f i e r a t maximum  sensitivity,  and the output was monitored on a low-response-time x-y r e c o r d e r . : By t h i s means the s t r a i n was measured t o a s e n s i t i v i t y o f 7.4 x 10~  6  ( e q u i v a l e n t t o one s m a l l d i v i s i o n o f  the e l o n g a t i o n s c a l e ) .  The a m p l i f i e d s i g n a l from the l o a d  c e l l was used t o d r i v e the y a x i s o f the r e c o r d e r .  Using  a C - c e l l loads o f up t o 40 l b s were measured w i t h a f u l l s c a l e d e f l e c t i o n o f 5 l b s and 7 stages o f zero (achieved by o p e r a t i n g the coarse balance  . • • 2  2  4  suppression  control).  Metallography 2.2.4.1  Introduction  M i c r o s t r u c t u r a l s t u d i e s were undertaken i n conjunct i o n with the mechanical t e s t i n g .  The p r i n c i p a l o b j e c t i v e o f  87  these s t u d i e s was microstructure.  t o s p e c i f y the degree  of regularity  of the  For every c r y s t a l prepared f o r m i c r o s t r a i n  t e s t i n g , another was  i d e n t i c a l l y p r e s t r a i n e d and used f o r  m i c r o s t r u c t u r a l examination.  In most c a s e s , these two  specimens were c u t from the same p a r e n t c r y s t a l . The two p r i n c i p a l techniques used t o observe d i s l o c a t i o n m i c r o s t r u c t u r e s i n copper are t r a n s m i s s i o n  electron  microscopy, and d i s l o c a t i o n e t c h p i t t i n g . The former technique has two important d e f i c i e n c i e s which make i t u n s u i t a b l e f o r t h i s work.  Firstly, i tis  d i f f i c u l t to i n s u r e t h a t no d i s l o c a t i o n movement o r l o s s occurs d u r i n g the p r e p a r a t i o n of t h i n f o i l s f o r t r a n s m i s s i o n e l e c t r o n microscopy. 1967)  Other workers  (Esseman,  1965;  Ramsteiner,  sought to a v o i d t h i s problem u s i n g low temperature  neutron i r r a d i a t i o n to p i n d i s l o c a t i o n b e f o r e f o i l  preparation.  Even i f such a technique i s attempted, a second o b j e c t i o n remains:  s i n c e low m a g n i f i c a t i o n i n s p e c t i o n of the m i c r o -  s t r u c t u r e i s not p o s s i b l e , i t i s d i f f i c u l t to d e c i d e i f the areas chosen f o r l o c a l d i s l o c a t i o n d e n s i t y measurements are r e p r e s e n t a t i v e o f the s t r u c t u r e as a whole.  Without  this  p o s s i b i l i t y , a l a r g e number o f f o i l s would be needed t o o b t a i n a s t a t i s t i c a l l y v a l i d sample  (Washburn and Murty, 1967).  should be noted, however, t h a t t h i s problem i s l e s s acute when measuring overall  dislocation densities, particularly  a f t e r p r e s t r a i n to h i g h e r s t r e s s  levels.  It  88  The e t c h p i t technique overcomes both these d i f ficulties.  Low m a g n i f i c a t i o n i n s p e c t i o n i s p o s s i b l e , p r o -  v i d i n g d i s l o c a t i o n d e n s i t i e s are low enough t o a l l o w use of o p t i c a l microscopy; and o b s e r v a t i o n s are taken from the s u r f a c e of a bulk specimen l o s s i s minimised. '. Any  so t h a t d i s l o c a t i o n movement and  l o c a l d i s l o c a t i o n movement which  may  occur on unloading cannot be avoided.  However, s i n c e both  the  are unloaded b e f o r e  m i c r o s t r a i n and e t c h p i t specimens  t h e i r subsequent  t e s t s , comparison o f r e s u l t s remains  The o p t i c a l r e s o l u t i o n of about 0.5y  valid.  s e t s an e f f e c -  t i v e upper l i m i t t o measurement of e t c h p i t d e n s i t i e s by o p t i c a l microscopy a t 4 x 10  cm .  8  Furthermore,  -2  since  observed d i s l o c a t i o n s u b s t r u c t u r e s are not completely homogeneous, and o f t e n i n c l u d e c e l l w a l l s and s u b g r a i n boundaries a d e n s i t y an o r d e r of magnitude l e s s than t h i s i s d e s i r a b l e . D e n s i t i e s g r e a t e r than 4 x 10  8  d i s l o c a t i o n s cm  -2  can be  measured u s i n g shadowed r e p l i c a s , but t h i s method s u f f e r s from the sampling problems In  previously  the l i g h t of these c o n s i d e r a t i o n s , a d i s l o c a -  t i o n d e n s i t y of about 3 x 10 c r y s t a l s by p r e s t r a i n i n g . s t r e s s was  outlined.  425g mm  -2  7  cm"  2  was  i n t r o d u c e d i n t o the  The nominal low temperature f l o w  ( r e s o l v e d shear s t r e s s ) .  89  2.2.4.2  P r e p a r a t i o n of C r y s t a l s f o r E t c h i n g  Orientation c r y s t a l was tion.  - The o r i e n t a t i o n of a p r e s t r a i n e d  confirmed u s i n g Laue back r e f l e c t i o n X-ray  I t was  subsequently c u t p a r a l l e l to the primary  plane u s i n g an a c i d saw. c r y s t a l was  diffrac-  F o l l o w i n g t h i s o p e r a t i o n , the  a t t a c h e d to a 45° s t e e l mount w i t h  conducting adhensive  silverprint  and mounted on a goniometer  back r e f l e c t i o n apparatus.  i n the Laue  A sequence of adjustments  d i f f r a c t o g r a p h s then served to a l i g n the primary s l i p p a r a l l e l to the back p l a t e of the goniometer, The goniometer  slip  and plane  w i t h i n 1°.  d e s i g n i s such t h a t i t i s t r a n s -  f e r a b l e t o a c r y s t a l f a c i n g instrument, and a smooth, a c c u r a t e l y o r i e n t e d s u r f a c e may  be o b t a i n e d .  Subsequent stages, to which the goniometer  could  be a t t a c h e d , to permit f i n a l e l e c t r o p o l i s h , i n v e r t e d normal stage o p t i c a l microscopy, were designed and By u s i n g the goniometer  i n t h i s way,  built.  any o r i e n t e d p l a n e c o u l d  be observed with h i g h m a g n i f i c a t i o n o p t i c a l  Polishing  and  microscopy.  - The use of o p t i c a l microscopy  together  w i t h a d i s l o c a t i o n e t c h r e q u i r e s t h a t the s u r f a c e t o be etched be o p t i c a l l y f l a t y e t f r e e from spurious damage. criteria  are f u l f i l l e d u s i n g a c r y s t a l f a c i n g  (Model 451 from South Bay Technology)  These  instrument  together with a  final  90  electropolish.  In t h i s apparatus a 10 i n c h diameter s t a i n l e s s  s t e e l r o t a t i n g cathode i s covered cloth  (Beuchler Metcloth)  with a f i n e no-nap c o t t o n  which c a r r i e s a s o l u t i o n of  p a r t s methanol t o one p a r t concentrated The  crystal  nitric  two  a c i d a t 35°C.  (the anode) i s mounted on the goniometer, which  r o t a t e s i n the o p p o s i t e sense, and h e l d a g a i n s t the wheel with minimum p r e s s u r e . specimen and  The  r o t a t i o n r a t e s of the wheel  the c u r r e n t d e n s i t y are optimized  p o l i s h e d s u r f a c e w i t h l i t t l e edge  and  to g i v e a  flat  rounding.  I d e a l s u r f a c e s produced i n t h i s way  have good  f l a t n e s s and p o l i s h but e x h i b i t l i n e a r t r a c e s whose depth was  of the order of the t h i c k n e s s of the t w i l l e d threads  the c o t t o n c l o t h . for  of  These were removed by e l e c t r o p o l i s h i n g  about one minute i n a separate, r a p i d l y s t i r r e d s o l u t i o n  of the same e l e c t r o l y t e .  A ten minute  rinse i n stirred  e t h y l a l c o h o l completed the s u r f a c e p r e p a r a t i o n .  2.2.4.3  Etching  A d i s l o c a t i o n e t c h f o r copper was by L o v e l l and Wernick w i t h g r e a t e r success p i t s to be i n 1-1  (1959) and  discovered  l a t e r developed and  by L i v i n g s t o n (1960).  used  He showed e t c h  correspondence with edge d i s l o c a t i o n s ,  i n t r o d u c e d by bending. to  first  Furthermore, he presented  evidence  show t h a t screw d i s l o c a t i o n s were r e v e a l e d by the  etchant.  91  Since used t h i s 1961;  this  etchant,  1967;  Table  list  of  i n v e s t i g a t o r s have  slight modifications  Basinski  Vallaikal,  A representative  and  1969;  B a s i n s k i , 1964;  Van  etching  D r u n e n and  close  some p r e l i m i n a r y  t o t h a t u s e d by  etching  t i m e was  2.2.4.4  t i o n of  and  Saimoto,  experimentation,  about f i v e  found  t o be  seconds at  -5°C.  S p e c i f i c a t i o n of D i s l o c a t i o n  of. Dot-Patterns  the m i c r o s t r u c t u r e s ,  the  - Before  1971).  known.  T h i s was  done t o m i n i m i z e o p e r a t o r  of r e p r e s e n t a t i v e  areas  Microstructures examined w i t h etch.  revealed  by  the  a bench m i c r o s c o p e t o a s s e s s  Ideal metallography requires  t i o n was  For  higher  relaxed  by  most s u c c e s s f u l .  Microstructures any  be  observaso  immediately  bias  i n the  e t c h were the  selec-  first  q u a l i t y of  distinguishable as  an  for a shorter  pits  equilaterial  dislocation densities this  etching  composition  for analysis.  w h i c h i n t e r s e c t a smooth p o l i s h e d s u r f a c e triangle.  a  c r y s t a l s were r e l a b l e d  t h a t t h e i r p r e s t r a i n temperatures would not  the  Gupta  s o l u t i o n s i s shown i n  L i v i n g s t o n was  Preparation  tion  (Young,  2. After  The  often with  H o r d o n , 1962;  Strutt,  e a r l y work, o t h e r  last  t i m e , so  condi-  that  the  p i t s would remain d i s t i n g u i s h a b l e . The and  microstructures  a representative  were t h e n c a r e f u l l y  s e r i e s . o f p h o t o m i c r o g r p a h s was  surveyed taken,  92  Table 2 D i s l o c a t i o n E t c h Compositions  ^** *<«*^ £onstitutent  Parts  ,,  >  B Source Livingston  , ,  ,  (1960)  2  r  ^^*"*'*'''"''»«^ ,  H0  HC1  Glacial Acetic  1  250  45  30  1  250  60  50  1  90  25  15  VanDrunen and Saimoto (197 0) ( i )  1  90  25  15  (ii)  1  220  25  15  1  175  50  35  Hordon  (1962)  Gupta and S t r u t t  Present Work  (1967)  Other  130 methanol  u s i n g a Z e i s s U l t r a p h o t I I microscope w i t h h i g h p r e s s u r e mercury i l l u m i n a t i o n , and a Normarski I n t e r f e r e n c e Planochromat * objective  (x40, 0.85 N.A.)  (see Appendix 4 ) .  The micrographs i n  the form o f 4" x 5" n e g a t i v e s were p h o t o g r a p h i c a l l y e n l a r g e d t o 8" x 10".  These p r i n t s had a m a g n i f i c a t i o n of x l 5 6 0 .  such micrographs were o b t a i n e d f o r each specimen.  About 20  From these  the 10 b e s t were s e l e c t e d , j u d g i n g only on the b a s i s o f o p t i c a l quality.  From the c e n t r e s o f each o f these p r i n t s , u s i n g  this  same c r i t e r i o n , an area 4" x 5" was s e l e c t e d from d i s l o c a t i o n d e n s i t y measurements.  Actual.measurements  dot t r a c i n g s of the micrographs  were made on b l a c k  (from here on r e f e r r e d t o as  "patterns") which were e a s i l y counted u s i n g a Quantimet.  On  these, p a t t e r n s , each dot marks the p o s i t i o n o f one d i s l o c a t i o n at i t s p o i n t o f i n t e r s e c t i o n w i t h the primary s l i p plane^  Measurement  of Looal  Dislocation  Densities  - The  p a t t e r n s taken from the micrographs were used t o determine dislocation densities.  Using a Quantimet, the o v e r a l l  local  disloca-  t i o n d e n s i t i e s were f i r s t computed by counting a l l p o i n t s on each s e t o f t e n p a t t e r n s , and d i v i d i n g by the a p p r o p r i a t e area.  specimen  Then a sampling technique was employed whereby a sample  square was scanned stepwise across each p a t t e r n , and the number of p o i n t s covered by the square was counted f o r each s t e p . ' L i m i ted areas of e x c e p t i o n a l l y high d i s l o c a t i o n density were sometimes o b s e r v e d on i n s p e c t i o n o f the e t c h e d p r i m a r y s l i p plane. However, they were r e g a r d e d as a t y p i c a l , and hence were not sampled. It is t h o u g h t t h a t such a r e a s a r e e v i d e n c e o f d i s l o c a t i o n c a r p e t s , w h i c h , in t h e s e l i g h t l y s t r a i n e d c r y s t a l s , would be in the e a r l y stages of f o r m a t i o n . M o r e o v e r , as d i s c u s s e d in S e c t i o n \.k.k i t is not e x p e c t e d t h a t such a r e a s wi11 be t r a v e r s e d by g l i d e d i s l o c a t i o n s ; , hence t h e i r o m i s s i o n is j u s t i f i e d on t h e o r e t i c a l grounds  94  It  particular  distribution the  scaling  (of l o c a l  the  distributions  effects:  microstructure the s t a t i s t i c a l l y  sample s q u a r e  vary with  that s t a t i s t i c a l  i n t h i s way a r e s u b j e c t to  determined any  s h o u l d be n o t e d  dislocation  size.  scale  for  determined  density) w i l l  vary  with  I n o t h e r words t h e d i s t r i b u t i o n s  o f sampling  and h e n c e . c h a n g e w i t h t h e  sample mean. In the i n i t i a l used t o scan of  mean.  all t h e s e t s o f p a t t e r n s .  the r e s u l t i n g  In the present  densities  distributions study,  are d i f f e r e n t  of a fixed  hence t h e s e  of  distributions  dislocation  was each  dislocation  w o u l d e a c h h a v e t h e same  leads t o d i f f e r i n g  dislocation  s a m p l e means;  c a n n o t be compared.  t h e same  t h e f o l l o w i n g manner.  size  I n a c a s e where  however, t h e o v e r a l l  to avoid t h i s  were c h o s e n t o g i v e  fixed  f o r e a c h o f t h e m i c r o s t r u c t u r e s , and  sample a r e a  In o r d e r  in  a square  t h e m i c r o s t r u c t u r e s have t h e same o v e r a l l  density,  use  trials,  scaling  error,  sample  sizes  mean f o r e a c h m i c r o s t r u c t u r e ,  After  first  overall  measuring the  d e n s i t y on e a c h o f t h e p a t t e r n s e t s , t h e s i z e o f  t h e sample a r e a r e q u i r e d t o g i v e t h e d e s i r e d sample mean was determined for  by s i m p l e  division;  the p a r t i c u l a r  each p a t t e r n s e t , t o g e t h e r w i t h  means a r e g i v e n Given determined  (from  i n Table  area  sizes  the corresponding  sample  3.  t h a t t h e r e l a t i v e sample a r e a s i z e s the o v e r a l l  density)  c a n be  t h e c h o i c e Of a b s o l u t e  .  Table 3  Area S i z e s Used t o Sample D i s l o c a t i o n M i c r o s t r u c t u r e s  Sample  Number  68A (1000°K)  Overall Dislocation D e n s i t y (cm )  S A M P L E  -2  2.83 x 1 0  7  Area i n p p t Area i n y -  2  2  —(Average No. of points per Area)  63A (850°K)  2.43 x 1 0  7  Area i n p p t Area i n y  2  4.54 x 1 0  7  Area i n p p t Area i n y  2  9.63 x 1 0  6  Area i n p p t Area i n y  2  2  7OA  (77°K)  2.21 x 1 0  7  Area i n p p t Area i n y X  2  2  117 x 133 9.40 2  24.2  S I Z E S  158 x 158 11. 9 2  40.0  185 x 185 13. 9 2  55.6  78 x 90 6.32 9.82  108 x 108 8.14 16.5  134 x 135 10.14 24.3  170 x 170 12.8 40.0  200 x 200 15.1 54.8  62 x 62 4.67 10.2  80 x 80 6.03 16.8  98 x 99 7.42 23.4  125 x 125 9.42 41.0  146 x 146 11. 0 55.2  108.x 108 . 8.14 6.30 .  134 x 135 10.1 10.0  214 x 214 16.3 26.0  216 x 216 16.3 26.0  271 x 271 20.4 41.0  87 x 90 6.67 10.6  115 x 115 8.67 17.5  137 x 138 10.4 23.5  174 x 174 13.1 36.0  204 x 204 15.4 54.6  2  X  108 x 108 8.14 18.5  2  X  78A (293°K)  10.0  2  X  69A (700°K)  78 x 78 5.88  A R E A  2  2  2  2  2  2  2  2  2  2  2  2  2  228 x 228 17.2 2  83.3  2  2  2  2  317 x 317 23. 9 58.7 2  96  s i z e remains open.  C l e a r l y the sample area must be  smaller  than the t o t a l area of the p a t t e r n s e t , and an obvious l i m i t i s the area covered by one p o i n t . these two  lower  Yet n e i t h e r of  extremes w i l l r e v e a l d i f f e r e n c e s between d i s s i m i l a r  d i s t r i b u t i o n s ; so some i n t e r m e d i a t e s i z e must be From a m e t a l l u r g i c a l viewpoint, strong t h e o r e t i c a l reasons  f o r choosing  chosen.  there are  not  any p a r t i c u l a r  inter-  mediate s i z e d sample area.  However, the s c a l e of the  l o c a t i o n m i c r o s t r u c t u r e may  provide a p h y s i c a l c r i t e r i o n :  sample areas greater  than any  dis-  substructure c e l l s i z e w i l l  be i n s e n s i t i v e to the e x i s t e n c e of such c e l l s .  Sample  areas s m a l l e r than t h i s c e l l s i z e should r e v e a l d e n s i t y d i f f e r e n c e s between the c e l l w a l l and the c e l l However, m i c r o s t r u c t u r e s which e x h i b i t e d marked  interior. cellularity  showed a range of c e l l s i z e s and p r o v i d e d no obvious  upper  bound f o r the sample area. A f t e r some p r e l i m i n a r y t r i a l s with d i f f e r e n t  sample  areas, i t became apparent t h a t a s e r i e s of area s i z e s would best i l l u s t r a t e d i f f e r e n c e s i n l o c a l d i s l o c a t i o n d e n s i t y . Each s e t of p a t t e r n s was s i z e was 25 p o i n t s  f i r s t scanned w i t h a square whose  determined such t h a t on an average i t would c o n t a i n (the a p p r o p r i a t e s i z e s being c a l c u l a t e d from the  i n i t i a l o v e r a l l d e n s i t y measurements). then repeated  T h i s process  was  f o u r times, with four other sample means.  Histograms were c o n s t r u c t e d showing the f r a c t i o n of samples c o n t a i n i n g a range of numbers o f p o i n t s , and w i t h the a i d of  97  a program w r i t t e n f o r the Hewlett-Packard and standard d e v i a t i o n of t h i s grouped  9100A, the mean  d a t a were c a l c u l a t e d .  In u s i n g the Quantimet f o r t h i s sampling  procedure,  some o v e r l a p p i n g o f unsampled areas between s u c c e s s i v e square p o s i t i o n s was d i f f i c u l t t o a v o i d (because the frame p o s i t i o n c o n t r o l i s not c o n t i n u o u s l y v a r i a b l e ) .  However,  the major p o r t i o n o f the p a t t e r n s was covered once w i t h sample s i z e .  Since o v e r l a p p i n g and l a c k o f sampling was  randomly determined, tributions  each  i t was not c o n s i d e r e d t o b i a s the d i s -  significantly. The  e f f e c t of s u b g r a i n boundaries  (dislocation  boundaries p r e s e n t a f t e r annealing) on the d i s t r i b u t i o n s was a l s o assessed by e l i m i n a t i n g d i s l o c a t i o n s which formed subg r a i n boundaries,from  the p a t t e r n s e t s , and r e p e a t i n g the  analysis.  2.3  Results 2.3.1  Microstrain  Curves  * Microstrain  curves were o b t a i n e d a t 77°K f o r f i v e  c r y s t a l s , each o f which had a d i f f e r e n t p r e s t r a i n  temperature.  The curves shown i n F i g u r e s 26-30 are d e r i v e d from l o a d t i o n curves by a p p r o p r i a t e s t r e s s and s t r a i n  elonga-  resolutions,  Appendix 2 g i v e s the d e f i n i t i o n s of t e r m i n o l o g y used t o d e s c r i b e d e f o r m a t i o n p r o c e s s e s and m e c h a n i c a l phenomena.  98  66  100  TOT  F i g u r e 30.  77°K m i c r o s t r a i n curve f o r 70B  (77°K p r e s t r a i n ) .  x  = 408  (g mm" ). 2  103  and  s u b t r a c t i o n o f the  original  31 and  load elongation  of t h e i r  d e v i a t i o n f r o m an  Such a c u r v e plastic. rising  The  are  steady  has  two  distinct  in  the  Figures  two  represents  the  given  steady  strain.  (ideally)  and to  elastic region  a  of  constant  t h e work h a r d e n i n g s t r e s s of  point of  rate):  the  i n t e r s e c t i o n of  l i n e s d e f i n e s the y i e l d s t r e s s . e x p e r i m e n t s , where s t r a i n  sensitivity,  d e t e r m i n e d by  specimens are  an  i n which e x p e r i m e n t a l l y idealized  i s the  for  2).  response  t o the  s t a t e flow  e l a s t i c b e h a v i o u r below t h e y i e l d t h e n be  initial  latter  The  curve  namely e l a s t i c  slope equal  f l o w w h i c h has  i n terms  (see A p p e n d i x  the  ( o f t e n r e f e r r e d t o as  In these at a high  analysed  stress strain  regions,  s t r e s s e s , the  state p l a s t i c  line  be  idealized  a constant  At higher  lower s l o p e  curve,  at these  the  i s measured  found to d e v i a t e  stress:  this  determined  curves  micros train  stresses w i l l  These e x p e r i m e n t a l l y  be  region,  deviate  called  and  from  s t r e s s must  extrapolation procedure.  a t s t r e s s e s below the y i e l d  r e f e r r e d t o as  strain  examples o f  reproduced  may  former, r e p r e s e n t i n g  s p e c i m e n a t any  be  Two  ( p r e s t r a i n e d ) specimen  s t r e s s , has  modulus.  these  curves  m i c r o s t r a i n curves  a work h a r d e n e d  this  strain.  32. The  but  elastic  The from  region the  stress, will non-elastic  microstrain.  determined curves,  Figures  26-30, show t h a t t h e s h a p e o f t h e m i c r o s t r a i n r e g i o n v a r i e s with  the  temperature of p r e s t r a i n .  Microstructures  formed  104  SI  x5.  x2  i i i  i  t t ft  2 Figure  3  31.  4  5  t  t  7  6  O r i g i n a l load ( v e r t i c a l ) - e l o n g a t i o n (horizontal) c u r v e f o r 68B ( 1 0 0 0 ° K p r e s t r a i n ) . Numbers 1-7 show l o a d z e r o - s u p p r e s s i o n s t e p s . Scale units a r e shown b o t t o m c e n t r e : v e r t i c a l = 0.5 l b s , h o r i z o n t a l = 3.7 x 1 0 " inches. 5  105  I  I xlO./ 7  /  x5-  x2  t t 1 2  t 3  F i g u r e 32.  t 4  t 5  f 6  7  O r i g i n a l load ( v e r t i c a l ) - e l o n g a t i o n (horizontal) curve f o r 70B (77°K p r e s t r a i n ) . Numbers 1-7 show l o a d zero-suppression s t e p s . S c a l e u n i t s are shown bottom c e n t r e : v e r t i c a l = 0.5 l b s , h o r i z o n t a l = 3.7 x 10~ i n c h e s . 5  106  at h i g h e r temperatures  d e v i a t e from e l a s t i c behaviour a t  s t r e s s e s ,well below the y i e l d s t r e s s , i . e . they a r e associated with gradual y i e l d i n g .  By c o n t r a s t , lower  temperature  m i c r o s t r u c t u r e s y i e l d more a b r u p t l y , showing l e s s m i c r o s t r a i n . For the upper t h r e e p r e s t r a i n temperatures, the amount o f m i c r o s t r a i n decreases c o n t i n u o u s l y w i t h p r e s t r a i n temperature. (1000°K), 10~ 10  T h i s s t r a i n i s approximately 1.5 x 1 0 3  f o r 65B (850°K), and 0.5 x 10"  y i e l d stress  (the s t r e s s a t which 10  3  - 3  f o r 68B  f o r 69B (700°K).  strain i s  detected) f o r these t h r e e c r y s t a l s i s o f the order o f h a l f the macroflow  s t r e s s , i n c o n t r a s t w i t h 78B (293°K) and 70B  (77°K) whose 10~ stress.  5  y i e l d s t r e s s e s a r e ~0.85 o f the m a c r o y i e l d  However the l a t t e r two c r y s t a l s s t i l l  show a p p r e c i -  able m i c r o s t r a i n , which i s f o r example approximately 10" f o r 78B  3  (293°K). A d i s t i n c t i v e f e a t u r e o f the m i c r o s t r a i n curves  from these l a t t e r two c r y s t a l s i s the r e g i o n o f low work hardening r a t e immediately  following macroyield.  This  phenomenon i s known as the Haasen-Kelly e f f e c t and i s a r e l o a d i n g t r a n s i e n t observed i n s e v e r a l pure F.C.C. metals  It is of i n t e r e s t to note that only about 1000 d i s l o c a t i o n loops, need t r a v e r s e the c r y s t a l to produce 10" s t r a i n in a 2 cm gauge length: i t i s a n t i c i p a t e d that many more than 1000 d i s l o c a t i o n loops w i l l expand over areas c o n s i d e r a b l y smaller than the whole g l i d e plane.  The  107  (Haasen and K e l l y , 1957,  Cupp and Chalmers,  C o t t r e l l and Stokes, 1955;  Noggle,  1955)  1954;  Dehl,  1955,  a t low homologous  temperatures. A t s t r a i n s above 5 x 10" r a t e i s observed f o r a l l c r y s t a l s .  3  a c o n s t a n t work hardening T h i s l i n e , when extrapo-  l a t e d back t o zero s t r a i n i d e n t i f i e s the steady s t a t e flow s t r e s s a t any s t r a i n .  By p l o t t i n g the f r a c t i o n a l d i f f e r e n c e  between t h i s s t r e s s and the measured s t r e s s f o r any  strain,  the m i c r o s t r a i n r e g i o n may  I t can  be seen t h a t 70B by ~2.5%, and 78B  be i s o l a t e d , F i g u r e 33.  (77°K) overshoots i t s expected flow s t r e s s (293°K) exceeds  i t by a m a r g i n a l  0.5%.  I t i s a l s o apparent i n F i g u r e 33 t h a t none of the c r y s t a l s with h i g h e r p r e s t r a i n temperature  ever exceed t h e i r  expected  steady s t a t e flow s t r e s s ; i n o t h e r words no Haasen-Kelly e f f e c t i s observed. This f i g u r e  (33) a l s o i l l u s t r a t e s more d i r e c t l y  the g e n e r a l c o n c l u s i o n s drawn from F i g u r e s 26-30. upper t h r e e p r e s t r a i n temperatures  For the  the amount of 77°K m i c r o -  s t r a i n i n c r e a s e s w i t h p r e s t r a i n temperature.  And  lower two p r e s t r a i n temperatures, w h i l e the 10~  5  f o r the flow s t r e s s  i s h i g h e r than t h a t f o r the o t h e r t h r e e c r y s t a l s , some microstrain i s s t i l l  observed.  A l l the above o b s e r v a t i o n s were q u a l i t a t i v e l y confirmed w i t h f u r t h e r experiments, a t l e a s t two being used f o r each p r e s t r a i n  temperature.  specimens  109  2.3.2  Metallography 2.3.2.1  M i c r o s t r u c t u r e s from Annealed  Microstructures shown i n F i g u r e 34 static  anneal  35;  1045°C.  u n s t r a i n e d specimens  the former  a t 1045°C, the  between 7 50°C and sect  and  from  Specimens  latter  was  obtained  after  cyclic  are after  anneal  They show d i s l o c a t i o n s w h i c h  a c l o s e p a c k e d p l a n e , most f o r m i n g  A  cyclic-  anneal m i c r o s t r u c t u r e at higher m a g n i f i c a t i o n f u r t h e r this  boundaries  anneal  by  point  ( F i g u r e 36),  are a l s o  counting  intersections with  After  static  e i g h t micrographs anneal  similar  the average  total  cyclic  dislocation  2  those  each type  cm  circle  t o F i g u r e s 34  anneal  t o F i g u r e 36,  x 10 /cm , i n c l u d i n g 6  similar  a 30  s u b g r a i n s were f o u n d  o f 74.6u, a f t e r  micrographs  1.2  evident. s i z e s were m e a s u r e d a f t e r  on  illu-  some p o l y g o n i z e d  Subgrain  1964)  diameter  and  inter-  clearly delineated  t h o u g h somewhat d i s c o n t i n u o u s s u b g r a i n b o u n d a r i e s .  strates  a  after  151u.  35. average  Taking  a cyclic  d e n s i t y was  (Hilliard  and  t o have an  seven  anneal,  measured t o  dislocations  of  be  i n subgrain  boundaries.  2.3.2.2  M i c r o s t r u c t u r e s and strained  Five with  five  Pre-  .  Crystals  sets of micrographs  different  P a t t e r n s from  prestrain  were p r o d u c e d  temperatures,  each  from set  crystals  110  F i g u r e 34.  Etched d i s l o c a t i o n m i c r o s t r u c t u r e a f t e r anneal (72 hours a t 1045°C) x305.  static  Ill  F i g u r e 35.  Etched d i s l o c a t i o n m i c r o s t r u c t u r e a f t e r c y c l i c anneal (72 hours c y c l e d hourly between 750°C and 1045°C) x305.  112  F i g u r e 36.  Etched d i s l o c a t i o n m i c r o s t r u c t u r e a f t e r c y c l i c anneal (72 hours c y c l e d h o u r l y between 750°C and 1045°C) x595.  113  c o n t a i n i n g 10 p r i n t s . were recorded  From these p r i n t s e t c h p i t p o s i t i o n s  by hand, i n the manner p r e v i o u s l y o u t l i n e d .  This gave f i v e s e t s o f 10 p a t t e r n s c o n t a i n i n g 500 t o 2000 p o i n t s per p a t t e r n ; i n t o t a l the p o s i t i o n s o f about 70,000 dislocationswere  recorded.  Examples o f micrographs and t h e i r  corresponding  p a t t e r n s f o r each p r e s t r a i n temperature are shown i n F i g u r e s 37-41 Of the f i v e m i c r o s t r u c t u r e s , the only one which d i s played c l e a r c e l l u l a r i t y temperature  (Figure 37).  i s 68A.with a 1000°K p r e s t r a i n Here the m a j o r i t y o f d i s l o c a t i o n s  l i e within p a r t i a l l y or f u l l y the m i c r o s t r u c t u r e s  formed c e l l w a l l s .  In c o n t r a s t ,  from 63A (850°K), F i g u r e 38, and 69A  (700°K), F i g u r e 39, show no d i s t i n c t l y formed c e l l w a l l s , although Using  some degree of d i s l o c a t i o n c l u s t e r i n g i s e v i d e n t .  a p u r e l y v i s u a l assessment, these two m i c r o s t r u c t u r e s  are d i f f i c u l t  t o d i s t i n g u i s h from each o t h e r .  78A (293°K), F i g u r e 40, shows a m i c r o s t r u c t u r e with a lower d i s l o c a t i o n d e n s i t y and w i t h l a r g e r e t c h p i t s . In t h i s specimen there appears t o be aggregation t i o n s without  full  c e l l formation.  of d i s l o c a -  The 77°K m i c r o s t r u c t u r e  (70A), F i g u r e 41, again shows some c l u s t e r i n g , but with areas  free of d i s l o c a t i o n s .  fewer  T h i s m i c r o s t r u c t u r e i n many  ways resembles those of 63A (850°K), F i g u r e 38, and 69A (700°K), F i g u r e 39.  However i n t h i s case and i n 78A (293°K)  there appears t o be a g r e a t e r predominance of s h o r t -dislocation arrays.  linear  114  Figure  37.  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 68A (1000°K) p r e s t r a i n ) xl560.  F i g u r e 38.  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e corresponding dot p a t t e r n from 63A (850°K p r e s t r a i n ) xl560.  and  Figure  39.  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e c o r r e s p o n d i n g d o t p a t t e r n f r o m 69A (700°K prestrain) x 1560.  and  117  Figure  40.  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 78A (293°K p r e s t r a i n ) xl560.  F i g u r e 41.  T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e corresponding dot p a t t e r n from 7OA (77°K p r e s t r a i n ) xl560.  and  119  I t was  p r e v i o u s l y noted t h a t although the o v e r a l l  d i s l o c a t i o n d e n s i t i e s f o r each s e t of micrographs are s i m i l a r (within a f a c t o r of f i v e ) , they are not equal.  This i n e q u a l i t y  i n t e r f e r s w i t h accurate  v i s u a l comparison, s i n c e the  higher  density microstructures  were etched f o r s h o r t e r t i m e s .  These  s h o r t e r etch times were used because the e t c h p i t s must be s m a l l e r to remain d i s t i n g u i s h a b l e . The micrographs i n F i g u r e s  37-41  were s e l e c t e d  as t y p i c a l on the b a s i s of both v i s u a l assessment and t i c a l analysis.  statis-  In other words, not o n l y do they d i s p l a y  c h a r a c t e r i s t i c m i c r o s t r u c t u r a l f e a t u r e s , but  they a l s o have  a d i s t r i b u t i o n of l o c a l d i s l o c a t i o n d e n s i t i e s c l o s e to d i s t r i b u t i o n s of t h e i r r e s p e c t i v e p a t t e r n s e t . f e a t u r e , which does not appear i n F i g u r e occuasionally occurred,  The  i s the grown-in subgrain  boundary.  slightly  An  extreme example of  occurrence of these boundaries i s shown i n F i g u r e (293°K).  As  i n d e n s i t y measurements.  On  the  42,  can be seen from the corresponding  below, d i s l o c a t i o n s i n the subgrain  from  pattern  boundaries were i n c l u d e d  the whole, the p r o p o r t i o n  d i s l o c a t i o n s i n the boundaries was  small and  made l i t t l e d i f f e r e n c e to the shape of the distribution.  greater  boundaries, g i v i n g a d i s t i n g u i s h a b l e  c o l o u r i n g to t h e i r c o n t r a s t .  78A  one  Interference  depth of a t t a c k by the e t c h was  at grown-in subgrain  However  37 to 41 but which  These were r e a d i l y i d e n t i f i e d under Normarski contrast.  the  of  t h e i r absence  sampled  120  F i g u r e 42.  Etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 78A (293°K) p r e s t r a i n ) showing extreme example o f i n c i d e n c e of subgrain boundaries.  121  2.3.3  Average D i s l o c a t i o n Density and Flow S t r e s s I t w i l l be r e c a l l e d t h a t flow s t r e s s v a r i e s as the  square r o o t o f the f o r e s t d i s l o c a t i o n d e n s i t y f o r a number of metals  ( B a s i n s k i and B a s i n s k i , 1964).  r e l a t i o n s h i p a l s o holds f o r copper  Although  this  f o r a wide range o f flow  s t r e s s v a l u e s , the experimental d a t a a t low d i s l o c a t i o n d e n s i t i e s i s somewhat s c a t t e r e d .  This i s i l l u s t r a t e d i n  F i g u r e 43, which shows flow s t r e s s and d i s l o c a t i o n * from v a r i o u s sources, experiments.  While  only approximately  together w i t h those from the p r e s e n t  the data from the p r e s e n t study shows the expected  c o r r e l a t i o n between flow  s t r e s s and d i s l o c a t i o n d e n s i t y , the experiments, low,  densities  and as n e a r l y as p o s s i b l e equal  using a  s e t o f flow s t r e s s e s ,  were not designed w i t h t h i s o b j e c t i v e i n mind.  Nonetheless  the r e s u l t s can be seen to f a l l w i t h i n the range o f those obtained by other workers.  2.3.4  Local Dislocation Densities L o c a l d i s l o c a t i o n d e n s i t i e s were measured u s i n g the  p r e v i o u s l y e x p l a i n e d area sampling  technique, and sample area  s i z e s were used which contained a sample average, x, o f  Because o f the r e l a t i v e s c a r c i t y o f f o r e s t d i s l o c a t i o n d e n s i t y measurements from c r y s t a l s o r i e n t e d f o r s i n g l e s i i p , d e n s i t y measurements o f " f o r e s t " d i s l o c a t i o n s in d o u b l e and m u l t i p l e s l i p c r y s t a l s a r e a l s o i n c l u d e d i n t h i s d i ag ram.  122  Bosinski  [OIIJ (III)  A  Basinski 8  O  VanDrunen a Saimoto [OOQ(III)  o  Livingston  e°off  (110)  (III)  •  "  [112]  •  "  QMQ(IIO)  •  Bailey Present  Polycrystal data  O  o  10'  CP  10  'E  o  •  A*T°O  10  A  10  AA  AA  10' Resolved  Figure  43.  Shear  Stress  (g.mm" ) 2  D i s l o c a t i o n d e n s i t y versus r e s o l v e d (Cottrell-Stokes corrected)  I  0  shear s t r e s s  123  between 6.5  and  55 p o i n t s per area.  F i v e area s i z e s were  s e l e c t e d f o r each of the f i v e s e t s o f p a t t e r n s each p r e s t r a i n temperature).  (i.e. for  Histograms g i v i n g the  fraction  of squares w i t h a g i v e n number of p o i n t s are shown i n F i g u r e s 44-48, f o r sample area s i z e s with x - 25. t y p i c a l of those obtained  The  data  are  over a range of sample area  I t i s expected t h a t the breadth  of these  histograms  w i l l i n c r e a s e as the s t r u c t u r e changes from a f a i r l y d i s t r i b u t i o n of p i t s  sizes.  uniform  (a " r e g u l a r " array) t o a d i s t r i b u t i o n  i n which the p i t s are c l u s t e r e d , l e a v i n g s u b s t a n t i a l areas n e a r l y f r e e of p i t s  (a "non-regular"  structure).  This  trend i s i l l u s t r a t e d i n F i g u r e 49 which shows t y p i c a l from r e g u l a r , random and more c e l l u l a r  patterns  microstructures,  together w i t h c h a r a c t e r i s t i c histograms which would be obtained  u s i n g the present  From t h i s f i g u r e (49)  l o c a l area sampling procedure.  i t can be seen t h a t m i c r o s t r u c t u r e s  which are r e s p e c t i v e l y r e g u l a r , random, and more c e l l u l a r , have l o c a l area d e n s i t y d i s t r i b u t i o n s which p r o g r e s s i v e l y i n c r e a s e i n breadth.  Thus the degree  of r e g u l a r i t y  a l t e r n a t i v e l y , c e l l u l a r i t y ) i s q u a n t i f i e d by the  (or  statistical  sampling procedure. On  t h i s b a s i s i t i s expected t h a t some of  the  obvious f e a t u r e s of the micrographs w i l l emerge from the statistical  analysis.  A histogram  for a highly  cellular  m i c r o s t r u c t u r e should i l l u s t r a t e the l a r g e p r o p o r t i o n o f  20  40  60  x (Number of Points per Area)  Figure  44.  H i s t o g r a m showing sampled f r e q u e n c y d i s t r i b u t i o n o f l o c a l areas f r o m 68A (1000°K p r e s t r a i n ) . Sample a r e a s i z e (9.40) y . 2  2  densities  140  120  in  100  o cu 80  W a>  60  40  20  0  20  i 40  60  80 to  x (Number of Points per Area)  F i g u r e 45.  Histogram showing sampled frequency d i s t r i b u t i o n o f l o c a l areas from 63A (850°K prestrain). Sample area s i z e ( 1 0 . 1 4 ) y . 2  2  densities  120  100  g 801 I  60  £ 3  40  20  0  10  1 1  20  30 .  40  1  50  1  60  70  x (Number of Points per Area)  Figure  46.  CT.  H i s t o g r a m showing sampled f r e q u e n c y d i s t r i b u t i o n o f l o c a l f r o m 69A (700°K p r e s t r a i n ) . Sample a r e a s i z e (7.42) y . 2  2  areas  densities  lOOr-  j  D  |  80|—  S  60  e  • 3 .  40  20  0  1  10  20  30  40  50  1  60  70  x (Number of Points per Area) Figure  47.  H i s t o g r a m showing sampled f r e q u e n c y d i s t r i b u t i o n o f l o c a l a r e a s f r o m 78A (293°K p r e s t r a i n ) . Sample a r e a s i z e ( 1 6 . 3 ) y . 2  2  densities  120  100  </> 8 0 o cu  60 cu  -Q  E - Z3  40  20  0  1 10  20  30  40  50  60  70  80  x ( N u m b e r of Points per A r e a ) F i g u r e 48.  00  Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l areas from 70A (77°K p r e s t r a i n ) . Sample area s i z e ( 1 0 . 4 ) y . 2  2  densities  129  N(x)  s/x =  0  Regular  S/X  = X  1/2  N(x)  Random  N(x) s/x>x  l/2  More Cellular  F i g u r e 49.  Schematic o f r e g u l a r , random and more c e l l u l a r d i s t r i b u t i o n s , t h e i r corresponding histograms and degrees o f r e g u l a r i t y (s/x).  130  sampled areas densities.  c o n t a i n i n g e i t h e r high or low  dislocation  Thus the c l e a r c e l l u l a r i t y seen i n 68A  (1000°K),  F i g u r e 37, i s a l s o apparent i n F i g u r e 44 where 20% of sample areas than 40.  c o n t a i n 10 or l e s s p o i n t s , and  12%  the  c o n t a i n more  Comparing the three h i g h e r temperature m i c r o s t r u c t u r e s ,  the histograms become more c l o s e l y d i s t r i b u t e d about the mean as the p r e s t r a i n temperature decreases. sents a t r e n d away from c e l l u l a r i t y and an •increasing  degree  of regularity  This e f f e c t  repre-  can be d e s c r i b e d  as  i n the m i c r o s t r u c t u r e .  I f the histograms f o r the two  lower p r e s t r a i n  temperatures are compared, the same g e n e r a l t r e n d i s observed. The histogram  f o r 70°C (77°K), F i g u r e 48,  i s more c l o s e l y  d i s t r i b u t e d about the mean than t h a t of 78A 47.  (293°K), F i g u r e  However on the b a s i s of t h i s s t a t i s t i c a l a n a l y s i s , the  l a t t e r d i s t r i b u t i o n appears s i m i l a r to t h a t of the h i g h l y c e l l u l a r 68A  (1000°K), F i g u r e 44,  arid the former 77°K d i s t r i -  b u t i o n i s almost i n d i s t i n g u i s h a b l e from those o b t a i n e d microstructures  formed at 850°K and  700°K.  A parametric measure of the degree may  of regularity  be i n t r o d u c e d , namely the r a t i o of the standard  to mean , s/x,  f o r any p a r t i c u l a r d i s t r i b u t i o n .  change with x p a r t i c u l a r l y f o r s m a l l s.  deviation  may  This v a r i a t i o n  f o r a random d i s t r i b u t i o n i s i l l u s t r a t e d i n F i g u r e 50, t h i s case s/x v a r i e s as  ,  I t should  be noted t h a t f o r any g i v e n d i s t r i b u t i o n t h i s r a t i o  _  for  i  (x)* (see Appendix 5 ) .  This  and i n  131  in a at  4>  E  20  30  40  x (Number of Points per Area)  Figure  50.  C a l c u l a t e d h i s t o g r a m s f o r random d i s t r i b u t i o n o f p o i n t s showing s c a l i n g e f f e c t : histograms with x = 10 and x = 25 h a v e d i f f e r e n t s t a n d a r d deviation/mean r a t i o .  132  r e l a t i o n s h i p , t o g e t h e r w i t h those measured by r e p e a t e d sampling f o r a wide range of sample and  area s i z e s , i s  shown i n F i g u r e s  51  52. I t can be seen i n F i g u r e 51  (1000°K,  850°K, 700°K),  t h a t the r e l a t i v e degree of r e g u l a r i t y measured a t x = 25 i s maintained throughout the range of sample between 10 and 55 52  (points per a r e a ) .  sizes, for x  Similarly  for Figure  (293°K, 77°K) f o r x between 10 and 59 the room temperature  m i c r o s t r u c t u r e remains more c e l l u l a r than t h a t formed a t 77°K.  T h i s range of sample  area s i z e s corresponds to a  specimen area range of between  (5y)  2  and  (25y)  2  (see T a b l e 3 ) .  A l s o shown i n F i g u r e 51 i s an a n a l y s i s o f a m i c r o graph o f Gupta and S t r u t t creep experiment at 82 3°K.  (1967) from a copper s i n g l e * A pattern  crystal  made from the m i c r o - .  graph i s shown i n F i g u r e 53, and appears t o be s i m i l a r to those o f 68A  (1000°K), F i g u r e 37.  And  as can be seen i n  F i g u r e 51, the s t a t i s t i c a l a n a l y s i s f o l l o w s c l o s e l y t h a t o f 6 8A.  These r e s u l t s show t h a t a l l the m i c r o s t r u c t u r e s are  f a r from random. u s i n g x = 6.3  For example, a ty t e s t of d a t a from 2  i n d i c a t e s t h a t t h e r e i s a l e s s than  p r o b a b i l i t y t h a t the m i c r o s t r u c t u r e i s randomly I t may  the  0.1%  distributed.  be r e c a l l e d t h a t the a n a l y s e s of d i s l o c a t i o n motion  through f i e l d s of p o i n t o b s t a c l e s  of  78A  (Kocks,  1966;  Foreman and  T h i s p a t t e r n is n e c e s s a r i l y a p p r o x i m a t e q u a l i t y of r e p r o d u c t i o n o f the m i c r o g r a p h .  because  x  (Average  Number  of  Points  per  Area) U)  Figure  51.  D e g r e e o f r e g u l a r i t y , s/x, f o r a r a n g e o f sample s i z e s f o r u p p e r t h r e e p r e s t r a i n temperatures. E r r o r b a r s r e p r e s e n t 95% c o n f i d e n c e l i m i t s c a l c u l a t e d from histograms.  T  293 °k. prestrain  •  293 k  A 77  0.8  -i  O  °k  without subgrain boundaries "  V  o.oh-  0.4  R ondom  0.2  Regular  0  0  20 Figure  52.  _ 30 40 x (Average Number of Points per Area.)  50  60  70  D e g r e e o f r e g u l a r i t y , s/x, f o r a r a n g e o f sample s i z e s f o r l o w e r two p r e s t r a i n temperatures. E r r o r b a r s r e p r e s e n t 95% c o n f i d e n c e l i m i t s c a l c u l a t e d from h i s t o g r a m s .  135  '1  2  'w.'.-.V...  •  *.  s•  .  ft-  *  v  . •  - • \ >. .  v.. •  v  '  •  • • •  •  TO--  •  • „  •*  '*  • •  •••  .  "ft • •  S  *\  •*  *A  •  \ •  •  J ? V . ••  •  .  Mb • 9  I  •  * t*  F i g u r e 53.  *  Dot p a t t e r n taken from micrograph by S t r u t t and Gupta (1967) from copper s i n g l e c r y s t a l a f t e r creep a t 823°K, r e s o l v e d shear s t r e s s ,250 g mm _ 2 , ( M a g n i f i c a t i o n x780).  136  Makin, 1966; M o r r i s and Klahn, 1973) use random o b s t a c l e t r i b u t i o n s , and i n one case regular d i s t r i b u t i o n .  dis-  (Foreman and Makin, 1966), a  Both random and r e g u l a r d i s t r i b u t i o n s  appear i n F i g u r e s 51 and 52, the l a t t e r d i s t r i b u t i o n f o l l o w ing  the a b s c i s s a f o r a l l x > 1.  I t can be seen t h a t none  of the m i c r o s t r u c t u r e s have o b s t a c l e d i s t r i b u t i o n s which f a l l between random and r e g u l a r :  a l l d i s t r i b u t i o n s show  d e v i a t i o n from the random away from t h e r e g u l a r  (see Appendix  5 f o r d i s c u s s i o n o f random and r e g u l a r a r r a y s ) . C e l l s i z e s were measured f o r 68A (1000°K) u s i n g 10 micrographs s i m i l a r t o F i g u r e 37. 20y,  The average c e l l s i z e was  with the m a j o r i t y w i t h i n the range 17y-25y.  be noted t h a t a sample area s i z e o f ( 2 0 y )  2  I t should  would g i v e a  d i s t r i b u t i o n with x ^ 115 which i s over twice t h e s i z e o f the l a r g e s t sample area used.  In other words, sample  were much s m a l l e r than the average c e l l  2.3.5  The E f f e c t o f Subgrain  areas  size.  Boundaries on t h e  Distributions By e l i m i n a t i n g from the p a t t e r n s d i s l o c a t i o n s i n subgrain boundaries and resampling, the d i r e c t e f f e c t o f these f e a t u r e s on the d e n s i t y d i s t r i b u t i o n s was assessed.  Using  p a t t e r n s from 78A (293°K), which was the specimen with  lowest  d i s l o c a t i o n d e n s i t y and the h i g h e s t p r o p o r t i o n o f d i s l o c a t i o n s i n subgrain boundaries samples w i t h x = 10 and x - 25 were  137  taken.  These r e s u l t s  distributions that  appear  i n Figure  become more random.  As  expected  However i t c a n be  t h e non-random n a t u r e o f t h e s e m i c r o s t r u c t u r e s  no means s o l e l y  2.3.6  due  specimen  t o specimen line  orientation,  s t u d y , good c o n t r o l was  g i v e n temperature and  also  small,  a n n e a l and p r e s t r a i n  prestrain ( w i t h one  c e d u r e was  followed  properties  c o u l d be r e l i a b l y  since,  exercised  crystal. and  the main 2.4  This  pro-  compared.  c o n f i r m e d by  temperature,  of  mechanical  additional  on a t l e a s t  two  either during  results  experiments:  e x c e p t i o n (293°K), m e t a l l o g r a p h i c o b s e r v a t i o n  each p r e s t r a i n  over  exception) the p a i r  so t h a t m i c r o s t r u c t u r e  t e s t s were p e r f o r m e d  as  conditions.  B o t h t h e m e c h a n i c a l and m i c r o s t r u c t u r a l were q u a l i t a t i v e l y  from  c u r v e s were i d e n t i c a l f o r  s p e c i m e n s were c u t f r o m t h e same o r i g i n a l  strain  i s by  i n t r o d u c e d by d i f f e r e n c e s  a r e c o n s i d e r e d t o be  crystal purity,  each specimen,  one  seen  Reproducibility  the s l i p  A t any  the  to the presence of s u b g r a i n boundaries.  Experimental errors  in  52.  and  with  micro-  specimens f o r  the p r e l i m i n a r y  or i n  experiments.  Discussion 2.4.1  Introduction The  results  o f t h e m i c r o s t r a i n work f a l l  g r o u p s , namely t h o s e f r o m c r y s t a l s  prestrained  into  two  above room  138  temperature, and  the remainder which are from c r y s t a l s p r e -  s t r a i n e d a t room temperature and below.  The  latter display  a r e l o a d i n g t r a n s i e n t known as the Haasen-Kelly e f f e c t , which i s c h a r a c t e r i z e d by a low or negative immediately f o l l o w i n g m a c r o y i e l d .  work hardening r a t e  For the'former group of  c r y s t a l s , however, as the a p p l i e d s t r e s s i n c r e a s e s , g r a d i e n t of the s t r e s s - s t r a i n curve f a l l s  the  continuously  through  the m i c r o s t r a i n r e g i o n , from the e l a s t i c modulus slope to  the  steady s t a t e work hardening r a t e . Whether or not t h i s d i s t i n c t i o n i s p u r e l y a r b i t r a r y i s a matter which w i l l be d e a l t w i t h i n the proceeding cussion.  However, f o r the p r e s e n t ,  made, and  the two  dis-  the d i s t i n c t i o n w i l l  sets of r e s u l t s w i l l be analysed  be  separately.  i  2.4.2  C r y s t a l s w i t h P r e s t r a i n s above 293°K Measurements have been d e s c r i b e d of the e a r l y  stages of low  temperature s t r e s s - s t r a i n curves f o r c r y s t a l s  w i t h d i f f e r i n g thermal-mechanical h i s t o r i e s . show t h a t the s p e c i f i c a t i o n of the low  The  results  temperature y i e l d  s t r e s s i s not s u f f i c i e n t to c h a r a c t e r i z e the s t r e s s s t r a i n curves of copper c r y s t a l s p r e s t r a i n e d at v a r i o u s temperatures. are d i v e r s e :  higher  In p a r t i c u l a r the e a r l y stages of these curves c r y s t a l s p r e s t r a i n e d at high temperature y i e l d  more g r a d u a l l y than those p r e s t r a i n e d a t a lower temperature, i . e . they e x h i b i t more microstrain.  This r e s u l t  i s evident  139  from F i g u r e s 26-28 and p a r t i c u l a r l y from F i g u r e 33 i n which the s t r e s s s t r a i n curves are shown as a group. The c o n c u r r e n t m i c r o s t r u c t u r a l s t u d i e s show a corresponding change i n the nature o f the m i c r o s t r u c t u r e w i t h p r e s t r a i n temperature: s i v e l y more regular  the m i c r o s t r u c t u r e s become p r o g r e s -  with decreasing p r e s t r a i n  temperature.  T h i s e f f e c t can be c l e a r l y r e c o g n i z e d i n F i g u r e s 45, 46 and 47, the histograms o f t h e l o c a l d i s l o c a t i o n d e n s i t y w i t h an x - 25 sample s i z e .  Moreover i n F i g u r e 51, t h i s  degree o f r e g u l a r i t y f o r these specimens  relative  (as seen by comparing  F i g u r e s 45, 46 and 47 f o r x - 25) i s c l e a r l y maintained  over  the whole range o f sample s i z e s . C o n s i d e r i n g t h i s m i c r o s t r u c t u r a l and mechanical evidence t o g e t h e r , i t i s suggested t h a t these r e s u l t s are c o r r e l a t e d , and t h a t c r y s t a l s w i t h a l e s s e r degree o f regularity  show more m i c r o s t r a i n .  T h i s means t h a t whereas the  l e v e l o f the s t r e s s - s t r a i n curve i s determined  by the o v e r a l l  d i s l o c a t i o n d e n s i t y , as p r e v i o u s l y shown by many workers ( L i v i n g s t o n , 1962; B a i l e y , 1963; B a s i n s k i and B a s i n s k i , 1964), the shape o f the e a r l y stages o f the curve may be c o n t r o l l e d by the d i s t r i b u t i o n of the o b s t a c l e d i s l o c a t i o n s . new r e s u l t i s among the most s i g n i f i c a n t o f t h i s This r e s u l t  This l a t t e r study.  i s c o n s i s t e n t w i t h the view o f p l a s t i c  flow developed i n d i s c u s s i o n o f the s l i p  l i n e study, whereby  s t r a i n occurs because o f the athermal expansion o f d i s l o c a t i o n loops through the e x i s t i n g m i c r o s t r u c t u r e .  At f i n i t e  140  temperature'this athermal g l i d e i s preceded by activation  (two  thermal  p o s s i b l e a c t i v a t i o n steps were d i s c u s s e d )  r e l e a s i n g a number of g l i d e d i s l o c a t i o n s which then sweep a newly a v a i l a b l e area of s l i p p l a n e , c a l l e d the free Whereas t h i s model was explain properties tures during  shown i n the  area.  s l i p line discussion  to  of s l i p l i n e s formed at d i f f e r e n t tempera-  steady s t a t e deformation, the m i c r o s t r a i n  curves  i l l u s t r a t e t r a n s i e n t deformation and were determined at a low homologous temperature activated effects  (0.057 T ) such t h a t any  (such as f o r e s t rearrangement or  are assumed to be n e g l i g i b l e . a c t i v a t i o n , d i s l o c a t i o n s may  Without the  recovery)  a i d o f thermal  g l i d e over a f r e e area whose  s i z e , i n a given microstructure, s t r e s s as d i s c u s s e d  thermally  i n Section  i s determined by 1.4.3.2.  I f we  the p r e s e n t t h a t the amount of m i c r o s t r a i n ,  the  assume f o r  produced i n a  c r y s t a l under a g i v e n a p p l i e d s t r e s s , i s p r o p o r t i o n a l t h i s f r e e area, then the may  be d e r i v e d  applied  to  s t r e s s dependence of t h i s f r e e area  from the m i c r o s t r a i n  This b a s i s f o r a 0°K  curve.  theory of m i c r o s t r a i n  the ideas f i r s t proposed by Kocks  uses  (1966, 1967), f o l l o w i n g  a n a l y s i s of the movement of g l i d e d i s l o c a t i o n s i n a f i e l d random p o i n t o b s t a c l e s , outlined i n Section  the  1.4.3.2.  of  r e s u l t s of which have been !  Whereas these r e s u l t s were p r e v i o u s l y with reference  his  discussed  to steady s t a t e deformation at f i n i t e  t u r e s , the same i n t e r p r e t a t i o n may  be a p p l i e d to  tempera-  transient  141  deformation a t low  temperature.  diagrams shown i n F i g u r e the f r e e area i n c r e a s e s o f the o b s t a c l e  field  Accordingly,  following  20,through the m i c r o s t r a i n from an i n d e t e r m i n a t e l y  a t low  the  region  small f r a c t i o n  s t r e s s , t o become " i n d e f i n i t e l y  l a r g e " a t a s t r e s s near the athermal g l i d e ' s t r e s s . While d i s l o c a t i o n motion has way  f o r g l i d e through random or r e g u l a r  point obstacles, microstructures randomly nor  been analysed i n t h i s arrangements of  f o r e s t d i s l o c a t i o n s i n the  as-prestrained  used i n t h i s study were shown to be  regularly distributed.  They have a l e s s e r degree  of r e g u l a r i t y than a random arrangement as can be Figure  51.  Although the Kocks a n a l y s i s has  on c e l l u l a r or c l u s t e r e d o b s t a c l e it  neither  not  arrangements  seen i n  been performed (Kocks, 1974),  i s assumed t h a t such an a n a l y s i s would p r e d i c t a  s t r e s s f o r athermal g l i d e through o b s t a c l e  critical  arrangements  w i t h the degree of r e g u l a r i t y of those found i n t h i s study. Moreover, i t i s expected"that the f r e e area w i l l increasing  f u n c t i o n of s t r e s s , although, as d i s c u s s e d  introduction  (Section  2.1),  g i v e n a p p l i e d s t r e s s and f r e e area should be  2.4.3  be a s i m i l a r l y the  i t i s a n t i c i p a t e d that, at a  o v e r a l l d i s l o c a t i o n density,  l a r g e r i n a more c e l l u l a r  Quantitative  in  the  microstructure.  Estimates from T h e o r i e s of  Microstrain  Although the o r i g i n a l o b j e c t i v e o f t h i s work  was  simply to demonstrate the q u a l i t a t i v e e f f e c t of departure from  142  r e g u l a r i t y on the shape of the low temperature  microstrain  curve, i t i s i n t e r e s t i n g to e x p l o r e the p o s s i b i l i t y o f a more quantitative d e s c r i p t i o n o f m i c r o s t r a i n .  F o r such a d e s c r i p -  t i o n i t i s necessary t o l i n k the f r e e area t o s t r a i n , and there are two p o s s i b l e ways t o do t h i s .  On the one hand t h e  s t r e s s dependence of the f r e e area may be determined a n a l y t i c a l l y f o r a given microstructure,  i n a manner s i m i l a r t o the  Kocks a n a l y s i s , and the f r e e area may then be taken as a measure o f s t r a i n through a r e l a t i o n s h i p o f the form y = ban (where n i s the number o f d i s l o c a t i o n loops which have swept f r e e area, a a t any g i v e n s t r e s s ) .  A l t e r n a t i v e l y the f r e e  area may be r e p r e s e n t e d by a s u i t a b l e f u n c t i o n o f s t r e s s and microstructure.  This free  area  function  to d i r e c t l y r e l a t e s t r e s s and s t r a i n .  may then be used I f this function i s  known, the second approach has the d i s t i n c t advantage t h a t the m i c r o s t r a i n  curve i s expressed i n terms, o f ( i n p r i n c i p l e )  measurable s t r u c t u r e  parameters.  A microstrain the  former approach, by t a k i n g  free area/stress be  curve w i l l f i r s t be d e r i v e d  area f u n c t i o n s , curves.  the a n a l y t i c a l l y determined  r e l a t i o n s h i p derived  compared w i t h m i c r o s t r a i n  using  by Kocks, and t h i s w i l l  curves from two suggested  free  together w i t h the e x p e r i m e n t a l l y measured  Using more c a r e f u l l y d e f i n e d  parameters o f a p p l i e d  s t r e s s and athermal g l i d e s t r e s s , Kocks  (1967) has determined  a s t r e s s versus normalized f r e e area r e l a t i o n s h i p , f o r a  "h6,n-wbrk/hardening -crystal l,  • t o F i g u r e 20(a)-(c)> the area.of  a t , 0°K, from diagrams s i m i l a r  and t h i s i s shown i n F i g u r e 54.  a is 0  s l i p p l a n e w h i c h c o n t a i n s on average one o b s t a c l e .  T h e . a d d i t i o n a l a b s c i s s a which.appears on t h i s diagram t r a n s form  i t i n t o a m i c r o s t r a i n c u r v e and have been c a l c u l a t e d  u s i n g a r e l a t i o n s h i p o f t h e form y = ban f o r n = 25.  This  assumes t h a t 25 d i s l o c a t i o n s have c r o s s e d a l l f r e e areas a t any g i v e n s t r e s s , i n a c r y s t a l whose s l i p p l a n e s c o n t a i n randomly d i s t r i b u t e d p o i n t o b s t a c l e s w i t h an o v e r a l l d e n s i t y of ~ 1 0 c m 7  -2  and h a v i n g  approximately  t h e same d i m e n s i o n s as  the samples used i n t h e p r e s e n t work. For comparison, m i c r o s t r a i n c u r v e s have been c a l c u l a t e d u s i n g t h e f r e e a r e a f u n c t i o n suggested by A l d e n  (1972)  which appears i n Eq. 1.6,, namely  (2.1)  T h i s f u n c t i o n behaves i n a manner a p p r o p r i a t e t o d e s c r i b e t h e change i n f r e e a r e a w i t h a p p l i e d s t r e s s :  i n the m i c r o s t r a i n  r e g i o n , a t an a p p l i e d . s t r e s s , a, below t h e 0°K  macroyield  s t r e s s , T^, ( i . e . a < 'x ) , A^ w i l l be s m a l l and p o s i t i v e : for  a near  as a ->• x^., A^ becomes u n i t y .  The degree o f  r e g u l a r i t y o f t h e m i c r o s t r u c t u r e i s e x p r e s s e d through t h e as y e t u n d e f i n e d  parameter x , which has dimensions o f s t r e s s ,  and. i s s m a l l compared t o T ^ .  L i k e the standard  d e v i a t i o n of  -  1.25  .00  ^  .075  0.50  0.25  0  0  XL  0  Figure  54.  50  25  (X.io"  75 a/oo  100  25  150  Change i n n o r m a l i z e d f r e e area, a / a , w i t h r e l a t i v e a p p l i e d s t r e s s from Kocks (1967) and corresponding s t r a i n c a l c u l a t e d assuming 25 d i s l o c a t i o n t r a v e r s e d c r y s t a l s i m i l a r t o those used i n present experiments. ;  0  145  the  sampled l o c a l d i s l o c a t i o n d e n s i t i e s ,  larger  f o r more c e l l u l a r m i c r o s t r u c t u r e s .  becomes  larger  with increased  e q u i v a l e n t l y with decreased consistent  w i t h the This  stress  strain  degree  degree  arguments  (Alden,  of  9  is,the  y  is  Through T , A cellularity  r  (or which  developed i n Section  f u n c t i o n forms p a r t equation  of  parameter  of r e g u l a r i t y ) ,  is  2.1.  an i n c r e m e n t a l ,  0°K  1972)  dy = | da A  where  this  (2.2)  r  work h a r d e n i n g r a t e ,  constant  at  constant  structure. Integration  o f E q . 2.2  T  V Y  —  =  ' This curves,  - a  J*- -  V  Y  an e q u a t i o n f o r  55 u s i n g T  s i m i l a r to  = 410  structure  2  -T  (2.3)  microstrain  w h i c h has  the  form of  by M . F . A s h b y  added advantage  plotted  and a work h a r d e n i n g  that determined experimentally  f u n c t i o n has b e e n s u g g e s t e d  gives  y exp * — T *V  a family of  ( i n gram" )  A s i m i l a r but a l t e r n a t e  vis,  constant  t h r e e o f w h i c h h a v e b e e n computed and a r e  in Figure rate  is  x  —Y T  exp  at  the  in this relative  (Alden,  study. area  1972),  of being zero v a l u e d at  a =  0,  0  0  8  F i g u r e 55.  C a l c u l a t e d m i c r o s t r a i n curves u s i n g Alden's f r e e a r e a f u n c t i o n , and x =. 410 (g mm" ) , x = Ty/100, Ty/20 and Xy/10, curves 1, 2 and 3 r e s p e c t i v e l y . 2  v  147  A r  = —  exp  (2.4) v  S u b s t i t u t i o n i n Eq.  Y  =  v  9 x y y  2.2  (a -  and i n t e g r a t i o n g i v e s  x ) v  exp  - a)  - (T  y T  +  These two  v  exp  —T  (2.5)  V  V  Three of the f a m i l y of m i c r o s t r a i n curves have a l s o been computed and  x  from t h i s  appear i n F i g u r e  equation  56.  s e t s of m i c r o s t r a i n curves may  now  be  compared with the one d e r i v e d d i r e c t l y from Kocks' a n a l y s i s . Although  the value of  a p p r o p r i a t e f o r a random o b s t a c l e  f i e l d i s not d e f i n e d , a comparison of F i g u r e 54 w i t h 55 and  Figures  56, would seem to i n d i c a t e t h a t , i f the f r e e area i s  t r a v e r s e d by a reasonable  number of d i s l o c a t i o n loops  f o r a random a r r a y of o b s t a c l e s , T  v  = T  (25) ,  7100. y  Since the d i s l o c a t i o n arrangements measured i n the p r e s e n t study had a l e s s e r degree of r e g u l a r i t y than a random array  (see F i g u r e 51), a somewhat l a r g e r v a l u e of T  seem to be a p p r o p r i a t e f o r these m i c r o s t r u c t u r e s . t h e o r e t i c a l m i c r o s t r a i n curves  v  would  When the  shown i n F i g u r e s 55 and  56  compared to the e x p e r i m e n t a l l y determined curves, f o r the specimens with higher p r e s t r a i n temperatures,  F i g u r e 57  t h i s deduction appears to be confirmed.  three  (re-  p l o t t e d from F i g u r e 33 with axes a p p r o p r i a t e f o r d i r e c t comparison),  are  In  F i g u r e 56.  C a l c u l a t e d m i c r o s t r a i n curves using Ashby's f u n c t i o n , and x = 410 (g mm" ), T = T /100, Ty/20 and x /10, curves 1, 2 and 3 r e s p e c t i v e l y . 2  y  y  v  y  T — —  075  b <  0  — r  r  50  025h  J  i_  4  5  Xp ( x l O )  i  t  8  -4  F i g u r e 57.  Normalized experimental m i c r o s t r a i n curves from specimens with p r e s t r a i n temperatures o f 1000°K, 850°K and 700°K, r e p l o t t e d from F i g u r e 33.  150 other words, the  f r e e area f u n c t i o n s  Eqs.  c o u l d make reasonably a c c u r a t e q u a n t i t a t i v e  2.1  and  2.4  in  p r e d i c t i o n s of the amount of m i c r o s t r a i n measured i n these crystals.  2.4.4  Crystals Prestrained  at 293°K and  Below  I t w i l l be r e c a l l e d t h a t w h i l e c r y s t a l s p r e s t r a i n e d a t 293°K and below gave l i t t l e m i c r o s t r a i n below a/x and  showed the Haasen-Kelly e f f e c t , the sampled  -  0.9,  microstructures  f o r these p r e s t r a i n s i n d i c a t e d a degree of r e g u l a r i t y comparable with c r y s t a l s prestrained other words, the m i c r o s t r a i n  at 700°K and  850°K.  In  curves from these c r y s t a l s  appear to be i n c o n s i s t e n t w i t h the c o r r e l a t i o n s  previously  discussed. I t has (Haasen and 1960)  been e s t a b l i s h e d by a number of workers  K e l l y , 1957;  Makin, 1958;  Birnbaum,  t h a t the Haasen-Kelly e f f e c t i s time-independent,  thus i t can not be e x p l a i n e d p i n n i n g mechanism.  by a s t r a i n ageing or  T h i s time-independence was  the p r e s e n t study, by was  B o i l i n g , 1959;  a s e r i e s of p r e l i m i n a r y  and  impurity  confirmed i n tests.  It  a l s o found i n agreement w i t h the p r e v i o u s work, t h a t  the e f f e c t occurs only on u n l o a d i n g , and the t e s t i s merely i n t e r r u p t e d by The  i s not observed i f  stopping  the  cross-head.  e f f e c t can be e x p l a i n e d i n terms of the i n t e r a c t i o n  of g l i d e and  f o r e s t d i s l o c a t i o n s on unloading  K e l l y , 1957;  Makin, 1958;  (Haasen  Birnbaum, 1960); d u r i n g  an  and elastic  r e l a x a t i o n i n the unloading c y c l e , some g l i d e d i s l o c a t i o n s  151  undergo e n e r g e t i c a l l y f a v o u r a b l e forest dislocations.  junction reactions  with  Because o f t h e s e j u n c t i o n r e a c t i o n s ,  upon r e l o a d i n g , t h e o p e r a t i o n  of sources w i t h i n the f r e e  areas i s r e s t r i c t e d . The s o u r c e s a r e r e s t r i c t e d because newly g e n e r a t e d g l i d e d i s l o c a t i o n s w i l l e x p e r i e n c e a back s t r e s s from t h o s e p r e v i o u s l y g e n e r a t e d , b u t w h i c h a r e now p i n n e d i n some i n t e r m e d i a t e  position.  G l i d e d i s l o c a t i o n s cannot move  t o t h e f r e e a r e a boundary u n t i l t h e s t r e s s i s s u f f i c i e n t l y high t o reverse  the j u n c t i o n r e a c t i o n s ; thus u n t i l t h e s t r e s s  reaches a c r i t i c a l l e v e l , s o u r c e o p e r a t i o n w i l l be r e s t r i c t e d , and  t h e r e w i l l be l i t t l e The  were o p e r a t i v e In Figure  microstrain.  m e c h a n i c a l response e x p e c t e d i f such a mechanism i s i n d e e d o b s e r v e d i n t h e present, e x p e r i m e n t s .  33 i t can be seen t h a t m i c r o s t r a i n i s measured i n  c r y s t a l s 78B (293°K) a n d 7OB (77°K) o n l y a t s t r e s s e s above .85 T '. Moreover once .85  i s exceeded (and s o u r c e s b e g i n  t o o p e r a t e ) t h e amount o f m i c r o s t r a i n produced i s comparable w i t h t h a t o b s e r v e d i n c r y s t a l s 65B (850°K) and 69B (700°K) respectively.  By comparing t h e degrees o f c e l l u l a r i t y o f  these c r y s t a l s , F i g u r e s 65B,  51, 5-2, i t can be seen t h a t 78B and  and 70B and 69B, have s t a t i s t i c a l l y s i m i l a r m i c r o s t r u c -  tures.  Thus once s u f f i c i e n t numbers o f g l i d e d i s l o c a t i o n s  are produced, t h e s e two p a i r s o f c r y s t a l s r e s p e c t i v e l y produce s i m i l a r amounts o f m i c r o s t r a i n . W h i l e t h e m i c r o s t r a i n c u r v e s o f 78B (293°K) and 70B  (77°K) have been shown t o be c o n s i s t e n t w i t h a r e s t r i c t e d  152  source h y p o t h e s i s ,  i n c o n t r a s t , the curves from 68B  65B (850°K) and 69B (700°K), which have 1 0  - 5  (1000°K),  microyield  s t r e s s e s o f ~0.5 x , have been shown t o be c o n s i s t e n t w i t h y a statistical  a n a l y s i s of m i c r o s t r a i n assuming a reasonably  l a r g e number o f d i s l o c a t i o n loops  t r a v e r s e each f r e e  I t i s f o r t h i s reason t h a t the s e p a r a t i o n r e s u l t s i s taken t o be v a l i d .  area.  o f the two s e t s o f  SUMMARY AND  (a)  Slip  line  CONCLUSIONS  l e n g t h measurements were made on  s e r i e s o f o r i e n t e d copper s i n g l e c r y s t a l s , 673°K, p o l i s h e d and between 573°K and ture s t r a i n  strain  (b)  by  theory  of  This  s t r a i n hardening  pile-ups  c e l l walls; the  these  present  (c) stress) with  glide  loops  those  temperature.  shown t o be  in conflict  i n which s l i p (linear  Mitchell,  lines i n the  1967)  theories p r e d i c t constant  Temperature  or  with  are  blocked  slip  slip  plane), of  dislocation line  line  ( o r more f u n d a m e n t a l l y l e n g t h was  s t r a i n hardening  statistical and  than  tempera-  b a r r i e r s (Seeger, 1957), r i b b o n s  ( H i r s c h and  dependent s l i p  by  longer  low  length  experiments.  theories of  blocked  at  temperatures  formed d u r i n g  at high  configurations  Lomer-Cottrell  converted  lines  increments  r e s u l t was  specific obstacle  s u c h as  in  4.2°K; s l i p  s t r a i n e d at  i n c r e m e n t s were f o u n d t o be  formed d u r i n g  any  incrementally  prestrained  a  shown t o be  i n which s l i p  i n t e r a c t i o n between the  f o r e s t d i s l o c a t i o n s , on  153  isostructural  the  consistent  lines  are  expanding  condition that  154 w i t h i n the framework of such a theory, the g l i d e loops a r e a b l e t o expand a t h e r m a l l y over a newly a v a i l a b l e f r e e area of  s l i p plane, a f t e r a t h e r m a l l y a c t i v a t e d p r o c e s s .  (d)  Two p o s s i b l e t h e r m a l l y a c t i v a t e d  were d i s c u s s e d . g l i d e o f forest own  Firstly  i t i s proposed  dislocations  processes  that thermally activated  (over s m a l l d i s t a n c e s on t h e i r  'primary' and 'cross s l i p ' p l a n e s ) , enables these  dis-  l o c a t i o n s t o change p o s i t i o n and a n n i h i l a t e a t a f i n i t e r a t e (rearrangement  and recovery)  (Alden, 1972).  The second  p o s s i b l e process i s t h e r m a l l y a c t i v a t e d movement o f d i s l o c a t i o n through a r r a y s o f o b s t a c l e s .  glide  E i t h e r o r both o f  these processes c o u l d form p a r t of a theory which s u c c e s s fully  accounts  (e)  f o r the p r e s e n t  results.  A u n i f i e d view o f s l i p  l i n e p r o p e r t i e s , micro-  s t r u c t u r a l f e a t u r e s and flow s t r e s s i s presented i n which, by c o r r e l a t i n g the spacing o f prominent s l i p  l i n e s with  m i c r o s t r u c t u r a l o b s e r v a t i o n s from the l i t e r a t u r e , i t i s suggested of  t h a t g l i d e d i s l o c a t i o n loops expand between c a r p e t s  primary m u l t i p o l e s p a r a l l e l t o the primary s l i p  and i n t e r a c t s t a t i s t i c a l l y  w i t h the more d i f f u s e  p o r t i o n s o f the c e l l " w a l l s . "  slip  forest  T h i s model i s shown t o p r o v i d -  a s e l f - c o n s i s t e n t e x p l a n a t i o n of the temperature of  plane,  variation  l i n e l e n g t h , s l i p band f o r m a t i o n , the e x i s t e n c e o f  155  of m u l t i p o l e carpets and the v a r i a t i o n o f flow s t r e s s  with  temperature.  (f)  77°K m i c r o s t r a i n curves were obtained  from a  s e r i e s of o r i e n t e d copper c r y s t a l s , p r e s t r a i n e d at temperatures between 1000°K and  77°K to produce d i s l o c a t i o n  microstructures  w i t h d i f f e r i n g degrees of r e g u l a r i t y , y e t w i t h  approximately  the same o v e r a l l d e n s i t y .  (g)  The  f o r e s t d i s l o c a t i o n m i c r o s t r u c t u r e s of  i d e n t i c a l l y prepared a dislocation  s e r i e s of c r y s t a l s were examined u s i n g  etch on the primary s l i p p l a n e .  sampling technique  was  d e v i s e d , which was  local dislocation densities  A  statistical  used to measure  and hence to q u a n t i f y the degree  of r e g u l a r i t y of the f o r e s t d i s l o c a t i o n  (h)  an  microstructure.  A l l m i c r o s t r u c t u r e s were found t o have a  s m a l l e r degree of r e g u l a r i t y  ( i . e . a h i g h e r degree of  l a r i t y ) than a random d i s t r i b u t i o n .  Microstructures  cellufrom  c r y s t a l s p r e s t r a i n e d at temperatures above 293°K became l e s s regular  (more c e l l u l a r ) with i n c r e a s i n g  a p a r a l l e l t r e n d was or below 293°K.  prestrain  temperature;  observed f o r c r y s t a l s p r e s t r a i n e d at  156  (i)  For  crystals  room t e m p e r a t u r e , a t any stress,  the  given  (j) below e x h i b i t e d  became l e s s  regular  Crystals prestrained the  at temperatures  f r a c t i o n of  amount o f m i c r o s t r a i n was  microstructures  be  prestrained  Haasen-Kelly  the  77°K y i e l d  found to increase (more  junction reactions  a t room t e m p e r a t u r e  e f f e c t w h i c h was  which occurred  Once s o u r c e s b e g a n t o o p e r a t e , d u r i n g straining, degree of  the  amount o f m i c r o s t r a i n  regularity  of  as  on  reloading  was  and  deduced caused  to  by  unloading. for  a n t i c i p a t e d from  the m i c r o s t r u c t u r e  the  cellular).  a consequence of r e s t r i c t e d source o p e r a t i o n ,  dislocation  above  indeed  incremental the detected.  REFERENCES  Adams, M.A.  and C o t t r e l l , A.H.  1955.  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ARE  and  those of Staker  and  (1974) and Vorbrugg et al.  (1971)  i t might be argued t h a t the observed  slip  l i n e s are b u i l t up from many s h o r t e r l i n e s , formed when d i s l o c a t i o n s t r a v e r s e the 3u diameter c e l l s and blocked.  then are  In t h i s appendix i t w i l l be shown t h a t such a model  p r e d i c t s a work hardening r a t e higher F i s h e r and L a l l y  than g e n e r a l l y observed.  (1967) suggested t h a t long  l i n e s are b u i l t up from many s h o r t e r l i n e s i n order  slip  to  e x p l a i n the r e s u l t s of a c o u s t i c emission s t u d i e s made on copper s i n g l e c r y s t a l s d u r i n g burst-type  easy g l i d e .  Assuming t h a t  a c o u s t i c p u l s e s were generated by s m a l l r a p i d  increments of p l a s t i c s t r a i n , they c a l c u l a t e d the s t r a i n per p u l s e to be  ~10~  6  to 10" . 7  average  However e l e c t r o n  microscopy s t u d i e s i n d i c a t e d c e l l s i z e s s u f f i c i e n t to g i v e only 1 0  - 1 1  s t r a i n when t r a v e r s e d by 40 d i s l o c a t i o n s .  conclude t h a t each a c o u s t i c b u r s t i s due source o p e r a t i o n  to  They  co-operative  i n many c e l l s , whereby d i s l o c a t i o n s sweeping  163  164  out one c e l l are b l o c k e d by a c e l l w a l l and t r i g g e r a source i n a neighbouring c e l l .  In t h i s manner a s l i p l i n e  will  i be made up of the many s h o r t l i n e s produced increments  of p l a s t i c  by many sub-  strain.  T h i s view of p l a s t i c flow can be shown t o p r e d i c t too l a r g e a v a l u e f o r the work hardening  coefficient, since  the r a t e of d i s l o c a t i o n storage i s too h i g h .  In t h i s  i t w i l l be assumed t h a t the storage of both primary  context  and  secondary d i s l o c a t i o n s i n c r e a s e the flow s t r e s s through a r e l a t i o n s h i p of the form x = a Gb ( p )i where G i s the shear y 2  modulus and p the average  d i s l o c a t i o n d e n s i t y (primary or  secondary), which i n t h i s case has dimensions l e n g t h per u n i t volume.  of d i s l o c a t i o n  F o r example, c o n s i d e r a c r y s t a l  of l e n g t h L, and c r o s s s e c t i o n a l area A which c o n t a i n s n c i r c u l a r d i s l o c a t i o n l o o p s , and which has d i s l o c a t i o n with an average produced  r a d i u s r , then  the increment  when an i n c r e m e n t a l number of loops  cells  of s t r a i n / dn  ^Yr  traverse  d i s l o c a t i o n c e l l s i s g i v e n by bur dY = -jg2  , dn  -  Assuming t h a t a f t e r t h i s s t r a i n increment some secondary  ,,. (1)  dn  loops undergo  s l i p and become p a r t of the c e l l w a l l s , i n a  manner analogous  to the "converted p i l e - u p " concept of  H i r s c h and M i t c h e l l  (1967), t h e r e w i l l be an i n c r e m e n t a l  increase i n d i s l o c a t i o n density  dp  g i v e n by  165  J „ _ 27rr ,  d p - _ d n  The  ( 2  corresponding i n c r e m e n t a l i n c r e a s e i n flow s t r e s s ,  )  dx Y'  i s g i v e n by d i f f e r e n t i a t i n g equation  1.4  Thus -i^. aGb dx =  ni  (p>*  Y  Combining equations  dp  (1), (2) and  (  3  )  (3), the work hardening  c o e f f i c i e n t becomes  -JL^SG d  In the present study, p - 3 x 10 10~  h  ( 4 )  (p)*r  e  9  ( i n cm/cm ) and r = 2.5 3  ( i n cm), w i t h a - 1 which g i v e s dx^/de - G/15  higher  than the normally observed v a l u e o f ~G/200 by more than order of magnitude.  x  one  Hence i t i s deduced t h a t the d i s l o c a t i o n s  must move over d i s t a n c e s g r e a t e r than the average dimensions of d i s l o c a t i o n c e l l s i n the p r e s t r a i n e d m i c r o s t r u c t u r e .  APPENDIX 2  DEFINITIONS OF TERMS USED TO DECRIBE DEFORMATION  Some o f the terms used i n t h i s t h e s i s  to describe  deformation phenomena are d e r i v e d from t h e o r e t i c a l  concepts  of m i c r o s t r u c t u r a l events, w h i l e o t h e r s have t h e i r o r i g i n s i n experimental o b s e r v a t i o n . of t h i s appendix  By t r a c i n g  these o r i g i n s  i s t o c l a r i f y the use o f t h i s  the aim  terminology.  Terms D e r i v e d from Experimental Observations An i d e a l i z e d stress-strain  form o f the e x p e r i m e n t a l l y observed  curve from a p r e s t r a i n e d specimen i s shown as  curve GAB i n F i g u r e 58 ( a ) , w i t h the e l a s t i c  (OA) and p l a s t i c  (AB) r e g i o n s i n t e r s e c t i n g  (A): the s t r e s s  a t the yield  point  l e v e l corresponding t o A i s termed the yield  stress.  The  slope o f OA i s the e l a s t i c modulus and o f AB, the work hardening rate,which i s lower and which i n the p r e s e n t study became constant soon a f t e r the y i e l d p o i n t , as shown i n curve OCDB, Figure  58 (a).  T h i s second curve i l l u s t r a t e s i n more d e t a i l  the type o f s t r e s s - s t r a i n  relationship  166  usually  observed i n  Figure  58.  Schematic s t r e s s  strain  curves.  168  practice.  I t d i f f e r s from curve OAB  or microstrain  region  p l a s t i c behaviour.  curve,  being the  a  (10~  and a 10~  between the r e g i o n s of e l a s t i c  3  5  to 1 0 ~ ) , i t may  be  6  yield  stress  called a  10 .  to as the macroyield  the  by A,the l a t t e r i s sometimes r e f e r r e d  stress.  -  . ,  While the curve OCDB w i l l u s u a l l y envelope of curve OAB,  micro-  To d i s t i n g u i s h t h i s s t r e s s from  - 5  the y i e l d s t r e s s d e f i n e d  when the  l i e within  the  s t r e s s i s a l s o measured w i t h  h i g h s e n s i t i v i t y , ( i . e . at a s e n s i t i v i t y of l e s s than 1% the y i e l d s t r e s s ) , a f t e r low move o u t s i d e  of  temperature p r e s t r a i n s , i t  t h i s envelope as shown i n F i g u r e  58  (b).  curves of t h i s type, the y i e l d s t r e s s w i l l a l s o be a t the p o i n t  and  (C) can be i d e n t i f i e d ,  s t r e s s at which the measured s t r a i n exceeds  e l a s t i c s t r a i n by  transition  I f the whole curve i s measured w i t h a h i g h  strain.sensitivity  strain  (CD)  i n t h a t i t has  may In  defined  A.  Terms Derived from T h e o r e t i c a l Concepts of  Microstructural  Events In these experiments a r e g i o n hardening r a t e region, increase  (as DB)  of constant work  i s r e f e r r e d to as a steady  f o r i n such a r e g i o n  state  i t i s assumed t h a t the r a t e  of a p p l i e d s t r e s s w i t h s t r a i n i s equal to the  of i n c r e a s e  i n the  (0°K)  y i e l d stress with s t r a i n .  p a r t i c u l a r value of a p p l i e d  s t r e s s at which steady  of  rate  Any state  169  p l a s t i c deformation i s o c c u r i n g i s d e f i n e d as the steady  state  reloaded,  flow  stress.  the flow  stress  instantaneous  I f the specimen i s unloaded and may be d e f i n e d from the r e l o a d i n g  curve a t the i n t e r s e c t i o n o f the e x t r a p o l a t e d e l a s t i c and p l a s t i c r e g i o n s :  the flow s t r e s s i s so named  because i t i s the s t r e s s a t which l a r g e s c a l e d i s l o c a t i o n movement o c c u r s .  l i n e s f o r the  In g e n e r a l  irreversible  the i n i t i a l  flow  s t r e s s and y i e l d s t r e s s c o i n c i d e , as i n F i g u r e 58 ( a ) . in  the e x c e p t i o n a l case shown i n F i g u r e  s l i g h t l y , the flow  s t r e s s being  d e f i n e d a t E.  must be noted t h a t the terms athermal critical  stress,  58 ( b ) , they  glide  differ  Finally, i t  stress  f r e q u e n t l y used i n d i s c u s s i n g  and  statistical  t h e o r i e s o f p l a s t i c flow, are synonymous w i t h the yield at  0°k  3  T,; ( p o i n t A a t 0°K) .  However  stress  APPENDIX 3  DETAILS OF MICROSTRAIN TESTING  Choice of  Extensometer In choosing a s u i t a b l e extensometer  f o r the measure-  ment o f s t r a i n d u r i n g the i n i t i a t i o n of p l a s t i c flow, both t h e o r e t i c a l and e x p e r i m e n t a l f a c t o r s were taken i n t o sideration.  For t h e o r e t i c a l reasons i t was  con-  necessary t o  perform such measurements a t low temperature,  so t h a t  change through t h e r m a l l y a c t i v a t e d processes would be Furthermore,  structure negligible.  i n o r d e r to a v o i d the i r r e v e r s i b l e changes i n  s t r u c t u r e which might have o c c u r r e d i f the commonly employed l o a d - c y c l e , or h y s t e r e s i s m i c r o s t r a i n technique were used, the m i c r o s t r a i n curve was  determined d u r i n g a s i n g l e l o a d  The major experimental requirement, of 10" , 5  was  a strain  determined by p r e l i m i n a r y  An I n s t r o n 1/2", No. G-51-16) was  application.  sensitivity  tests.  10% S t r a i n Gage Extensometer  found to f u l f i l l these b a s i c  requirements  when used i n c o n j u n c t i o n w i t h an I n s t r o n Load C e l l o p e r a t i n g a t maximum s e n s i t i v i t y . t h i s extensometer  (Model  Amplifier  In order to a v o i d damaging  by thermal shock e f f e c t s d u r i n g c o o l i n g ,  170  and  171  at the same time, p r o v i d e a s t a b l e temperature under q u i e t , conditions, a  s t a i n l e s s s t e e l c r y o s t a t , c o n t a i n i n g a gaseous  environment a t 77°K, was  employed;  a schematic diagram o f  t h i s c r y o s t a t i s shown i n F i g u r e 5 9 ( a ) .  W h i l s t they are  s a t i s f a c t o r y i n most o t h e r r e s p e c t s , e l e c t r i c a l  resistance  s t r a i n gauge extensometers e x h i b i t poor r e v e r s i b i l i t y a t high s t r a i n s e n s i t i v i t i e s . c y c l e was  to be employed  However, s i n c e a s i n g l e  load  i n these experiments, t h i s drawback  was not c o n s i d e r e d important.  Using t h i s equipment  s t r a i n was measured to a s e n s i t i v i t y o f 7.4 x 10~ to one s m a l l d i v i s i o n of the e l o n g a t i o n  6  (unresolved) (equivalent  scale).  M o d i f i c a t i o n f o r Use w i t h S i n g l e C r y s t a l s The extensometer attachment p a r t s were r e d e s i g n e d so t h a t the extensometer was  connected t o the c r y s t a l by  four p i v o t screws, F i g u r e 59  (b).  The t i p s o f these screws  were c o n i c a l , having been c a r e f u l l y machined  to a s o l i d angle  of 60°, and were l o c a t e d i n s m a l l c o n i c a l d e p r e s s i o n s , which were made on o p p o s i t e faces of the c r y s t a l by screws w i t h 90° s o l i d angle, c o n i c a l t i p s .  The l a t t e r s e t of f o u r screws  were h e l d by a standard gauge b l o c k , used t o g i v e a r e p r o ducable i n i t i a l s e p a r a t i o n of the extensometer arms.  These  attachments and l o c a t i n g p o i n t s were designed so as to allow the c r y s t a l l a t t i c e t o r o t a t e w i t h o u t t r a n s m i t t i n g a torque to the harp of the extensometer.  Figure  59.  M i c r o s t r a i n t e s t i n g apparatus: (a) c r y o s t a t (b.) d e t a i l o f e x t e n s o m e t e r a t t a c h m e n t .  and p u l l  rod  assembly, H -J fx)  173  C a l i b r a t i o n , f o r Use a t 77°K F i r s t the extensometer was  s u i t a b l y c a l i b r a t e d at  room temperature by u s i n g a micrometer, and low amplification.  signal  Next, two s e t s o f l o c a t i n g p o i n t s were made  on a p o l y c r y s t a l l i n e copper sample, and these two gauge lengths were c a r e f u l l y measured a t r a v e l l i n g microscope.  (at room temperature) u s i n g  The two 77°K gauge l e n g t h s c o r r e -  sponding to those measured a t room temperature were then c a l c u l a t e d u s i n g the a p p r o p r i a t e c o e f f i c i e n t o f thermal expansion.  F i n a l l y , the corresponding e l e c t r o n i c s i g n a l s •  from the extensometer a t 77°K were r e c o r d e d , by a t t a c h i n g the extensometer to the specimen and c o o l i n g each o f the two gauge l e n g t h s .  (to 77°K) f o r  Having confirmed t h a t the  l i n e a r i t y of the extensometer was maintained a t 77°K  (by  extending an e l a s t i c specimen), the 77°K c a l i b r a t i o n o f the extensometer c o u l d be completed, and r e l i a b l y reproduced.  Use o f the Extensometer In the course of v a r i o u s p r e l i m i n a r y t r i a l s some procedures were i d e n t i f i e d which were c o n s i d e r e d to a v o i d spurious s t r a i n measurements.  essential  For example,  crystal  alignment problems were found t o be n e g l i g i b l e p r o v i d i n g the specimen was  not removed from the g r i p s a f t e r p r e s t r a i n  (and, o f course, p r o v i d i n g u n i v e r s a l j o i n t s were used).  This  174  was p a r t i c u l a r l y  relevant  a f t e r high temperature p r e s t r a i n  when the specimen was t r a n s f e r r e d t o the m i c r o s t r a i n from the high  temperature vacuum furnace.  specimen was c a r e f u l l y holding  transferred using  b l o c k , which was attached  specimen had c o o l e d  In t h i s case, the a s o l i d aluminum  t o the g r i p s a f t e r the  t o room temperature.  Care was taken i n a t t a c h i n g  the extensometer t o  the specimen; the four p i v o t screws were l i g h t l y (1/8  turn).  reflected elongation  cryostat  tightened  Any looseness i n t h i s attachment would be  i n the slope of the e l a s t i c p o r t i o n of the l o a d curve.  A c r i t e r i o n o f constant e l a s t i c  equal t o 4.9 ± 0.3 x 10  3  Kg mm"  2  was used t o v e r i f y the  r e l i a b i l i t y o f the s t r a i n measurement. used was as high  The s t r a i n  as p r a c t i c a l , w h i l s t s t i l l  f o r zero s u p p r e s s i o n and s c a l e changes. s t a b i l i t y was recognized  slope,  by the e x i s t e n c e  rate  allowing  Finally,  time  thermal  o f a s t a b l e balance  p o i n t f o r the extensometer and was a t t a i n e d about one hour after  the s t a r t of c o o l i n g .  APPENDIX 4  OPTICAL MICROSCOPY WITH NORMARSKI INTERFERENCE CONTRAST  The purpose o f t h i s appendix i s to i l l u s t r a t e the advantage of u s i n g Normarski I n t e r f e r e n c e C o n t r a s t , a t e c h nique which proved to be i n d i s p e n s i b l e i n t h i s work.  The  Normarski technique makes use of p o l a r i z i n g - i n t e r f e r e n c e c o n t r a s t , and i n consequence, m i c r o s t r u c t u r a l f e a t u r e s on the o b j e c t which are d i f f e r e n t i a t e d by s u r f a c e r e l i e f  (after  e t c h i n g , f o r example) are d i s t i n g u i s h a b l e i n the image through both c o l o u r and i n t e n s i t y .  An e x c e l l e n t e x p l a n a t i o n o f t h e  i n t e r f e r e n c e p r i n c i p l e t o g e t h e r w i t h d e t a i l s of the o p t i c s , may  be found i n G i f k i n s  1  book Optical  Microscopy  in  Metals  (1970). The dramatic i n c r e a s e i n r e s o l u t i o n , g a i n e d by use of Normarski I n t e r f e r e n c e can be seen i n F i g u r e 60 ( a ) , (b).  These two micrographs are taken from the same area o f  an etched s u r f a c e , and w i t h the same o p t i c a l except t h a t i n one case p o l a r i z e d l i g h t was  conditions, used t o g e t h e r  w i t h a Normarski i n t e r f e r e n c e attachment, whereas i n the o t h e r , i n the absence of an i n t e r f e r e n c e attachment, a green  175  176  advantage of Normarski I n t e r f e r e n c e C o n t r a s t x900: (a) b r i g h t f i e l d , (b) w i t h Normarski Interference Contrast.  177  f i l t e r was used w i t h u n p o l a r i z e d i l l u m i n a t i o n . Both, these micrographs were taken on a Z e i s s U l t r a p h o t I I microscope, using a.high p r e s s u r e mercury l i g h t source  together with a  planachromat o b j e c t i v e ( E p i p l a n x40, 0.85NA).  Assuming  that f o r each m i c r o g r a p h the a v e r a g e 'Wavelength  i s A = 5500  o  ( i n A),.the r e s o l u t i o n l i m i t  (the s m a l l e s t d i s t a n c e between  two p o i n t s on the o b j e c t , f o r which these p o i n t s can remain d i s t i n g u i s h a b l e i n the image), g i v e n by R = 0.61A/N.A., w i l l be the same i n each case, namely ~0.4u.  In the absence  o f i n t e r f e r e n c e c o n t r a s t , t h i s r e s o l u t i o n was i m p o s s i b l e t o achieve and the c o n s i d e r a b l e l o s s o f d e t a i l i s r e a d i l y apparent on comparison o f F i g u r e s  60 (a) and (b).  On the other  hand, an extremely s a t i s f a c t o r y r e s o l u t i o n and c o n t r a s t was obtained using Normarski I n t e r f e r e n c e with which i t was p o s s i b l e t o d i s t i n g u i s h e t c h p i t s having l e s s than 0.5u.  a separation of  APPENDIX 5  A PROBABILITY MODEL FOR RANDOM DISTRIBUTIONS  Introduction When s t u d y i n g t h e r e l a t i v e v a l u e s o f a p r o p e r t y o f a p h y s i c a l system, i t i s u s e f u l t o d e f i n e an a b s o l u t e v a l u e w i t h which t o compare those measured.  O f t e n t h e thermodynamic  e q u i l i b r i u m . s t a t e may be .used.as such a s t a n d a r d .  However,  i n t h e p r e s e n t s t u d y , where t h e p r o p e r t y o f i n t e r e s t i s d i s l o c a t i o n d i s t r i b u t i o n , t h e thermodynamic e q u i l i b r i u m s t a t e p r o v i d e s an empty s t a n d a r d , s i n c e d i s l o c a t i o n s are thermodynamically  unstable:  c o n t a i n s no d i s l o c a t i o n s .  themselves  a f u l l y annealed  crystal  So we must l o o k elsewhere t o  define a standard d i s l o c a t i o n c o n f i g u r a t i o n . In r e c e n t y e a r s t h e r e has been i n c r e a s i n g i n t e r e s t i n t h e s t a t i s t i c a l a n a l y s i s o f d i s l o c a t i o n motion fields ofpoint obstacles.  through  The model systems have o b s t a c l e s  which a r e p o s i t i o n e d randomly (Kocks, 1966;  Foreman and  M a k i n , 1966; M o r r i s and K l a h n , 1973); r e g u l a r  (triangular)  o b s t a c l e a r r a y s (Foreman and M a k i n , 1967) and i n t e r m e d i a t e  178  179  cases  (random a r r a y s w i t h s p e c i f i e d minimum o b s t a c l e spac-  ings)  (Foreman, unpublished,  1971)  r e p o r t e d i n Brown and  have a l s o been c o n s i d e r e d . P o i n t o b s t a c l e s may  r e p r e s e n t f o r example s o l u t e  atoms, s m a l l p r e c i p i t a t e s or f o r e s t d i s l o c a t i o n s . two to  Ham,  former examples, random and r e g u l a r a r r a y s  For  the  correspond  thermodynamic e q u i l i b r i u m s t a t e s i n r e a l systems.  In  the case of f o r e s t d i s l o c a t i o n s , where no such thermodynamic equilibrium configuration exists, i t i s d i f f i c u l t c a l l y determine even a p s e u d o - e q u i l i b r i u m  to t h e o r e t i -  configuration, since  the mechanisms which produce these a r r a y s are v a r i e d and complex.  However, i n the e a r l y stages of work  d i s l o c a t i o n a r r a y s may  hardening,  appear, to the c a s u a l o b s e r v e r , t o  be more or l e s s random, and f o r  convenience  t h i s may  be  used as a standard c o n f i g u r a t i o n .  The P r o b a b i l i t y Model I f a f i e l d of randomly p l a c e d p o i n t s i s sampled by c o v e r i n g i t w i t h a g r i d of square  sample a r e a s ,  the  p r o b a b i l i t y f u n c t i o n f o r the number of p o i n t s per area i s a s o l u t i o n of the C l a s s i c a l Occupancy Problem p.  1950,  54). I f a f i e l d of  into, of  (Feller,  n  r  randomly p l a c e d p o i n t s i s d i v i d e d  equal areas, t h e r e are  n  r  p o s s i b l e arrangements  p o i n t s over these areas, each w i t h a p r o b a b i l i t y of  n  r  .  180  The  p r o b a b i l i t y t h a t any g i v e n area has e x a c t l y  i s determined i n the f o l l o w i n g way. can be chosen from the t o t a l o f remaining  k  points  F o r t h i s area, k p o i n t s  r  r  in  (r-k) p o i n t s can be p l a c e d  \  r  ways.  The  i n the remaining n-1  r—k cells i n  (n-1)  ways.  So the p r o b a b i l i t y t h a t the c e l l  under c o n s i d e r a t i o n contains  P  This i s a b i n o m i a l  k  bution  points  i s then (5)  d i s t r i b u t i o n f u n c t i o n , which may be  k;  r  k  r-k r (n-1) kl r ' n  =  approximated by the Poisson  i n cases where  exactly  and  n  f u n c t i o n has a mean  distribution  e n  _r n  frl* — n  (6)  ki  are l a r g e .  The P o i s s o n  x , and standard  distri-  d e v i a t i o n s*  where r •>  x = — n  and  and  s =  r n  the f u n c t i o n has been t a b u l a t e d t o l a r g e values  and k (Molina,  (7),  of  — n  1943).  A l t h o u g h the mean and s t a n d a r d d e v i a t i o n o f a random v a r i a b l e a r e u s u a l l y d e s c r i b e d by the symbols \i and a r e s p e c t i v e l y , f o r c l a r i t y , the symbols x and s, conventionally used to d e s c r i b e sampled d a t a , a r e a l s o used h e r e .  181 r By choosing  d i f f e r e n t v a l u e s of  —  in this table,  sampling with d i f f e r e n t area s i z e s can be modelled.  For  example, an area s i z e can be chosen to g i v e a sample mean of — = 10.  The  histogram,  showing the f r a c t i o n of squares which  c o n t a i n a g i v e n number of p o i n t s , sampled w i t h t h i s  square  s i z e from a f i e l d of randomly p l a c e d p o i n t s , appears i n Figure  50. C l e a r l y the shape of such a histogram  will  depend  on the sample area s i z e chosen. To i l l u s t r a t e t h i s p o i n t the — = 25 histogram f o r random d i s t r i b u t i o n s i s a l s o shown i n n ^ F i g u r e 50. From equations (7) and (8), the v a r i a t i o n i n s =• with sample area s i z e i s g i v e n by X  t i o n s h i p appears i n F i g u r e 52 and  s X  =  n  and  this  rela-  51.  In c o n t r a s t i t should be noted t h a t samples from a vegulartion  array of p o i n t o b s t a c l e s w i l l have standard  s = 0 (for  — > 1  devia-  as i s the case i n t h i s work); thus  s s — = 0 a l s o . Hence, the p l o t of — shown f o r a r e g u l a r x array i n F i g u r e 51 and 52 w i l l f o l l o w the a b s c i s s a . x  

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