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A study of slip line lengths in aluminum single crystals during transient deformation De Larios, John Martin 1973

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A STUDY OF SLIP LINE LENGTHS IN ALUMINUM SINGLE CRYSTALS DURING TRANSIENT DEFORMATION by JOHN MARTIN DE LARIOS B.Sc., University of California, Berkeley A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of METALLURGY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1973 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the reguirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission f o r ext e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of M e t a l l u r g y The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date April 30, 1973 i A B S T R A C T Slip line length measurements have been carried out on oriented single crystals of high purity aluminum to qualitatively check the predictions of a new theory of plastic flow in strain hardened metals. This theory states that the slip line length L will be a function of the stress, a, the yield strength, T y , and the magnitude of the variation in the obstacle spacing, xv. To achieve this purpose, the specimens were prestrained at room temperature to the same stress level insuring that they had the same structure. They were then electropolished and given a small increment of strain at a temperature between 298°K and 4°K or a strain rate from e = 1x10 Vmin to e = 2x10 "''/min. The slip line lengths for these increments were found to increase with increasing strain rate and to de-crease with increasing temperature, in agreement with the theory. By applying the increment of strain in small, divisions, the slip line length was found to increase during the region of transient deformation following a quench. This increase in L was due to an increase in stress. Lowering the prestrain temperature showed that the slip line length depended on the structure parameter T . A low temperature pre-strain resulted in a smaller T v and therefore shorter slip lines. From the 4°K increment, the parameters characterizing the prestrained structure and the structure after the increment could be determined. Using equations of the theory, these values were related to the experimental slip line length data with good quantitative agreement. i i TABLE OF CONTENTS Page INTRODUCTION 1 1.1 PREVIOUS SLIP LINE STUDIES 1 1.2 THE THEORIES OF HIRSCH AND SEEGER 4 1.3 SCOPE OF PRESENT WORK 7 ,1.4 RECOVERY THEORY 8 1.5 REARRANGEMENT-RECOVERY THEORY T = 0°K 10 1.5.1 Equations of P l a s t i c Flow 10 1.5.2 S l i p Lines and the Relative Area Function . . 14 1.5.3. A n a l y t i c a l Development 17 1.6 REARRANGEMENT-RECOVERY THEORY T > 0°K 20 1.6.1 Equations of P l a s t i c Flow 20 1.6.2 The Relative Area Function 21 EXPERIMENT 24 2.1 SINGLE CRYSTAL PREPARATION 24 2.1.1 M a t e r i a l 24 2.1.2 Orientation 24 2.1.3 Growth 25 2.1.4 Annealing and P r e s t r a i n 26 2.1.5 E l e c t r o p o l i s h i n g 28 2.2 TESTS CONDUCTED 28 2.2.1 Temperature and S t r a i n Rate Changes 28 2.2.2 S l i p Line Length i n the Transient 28 2.2.3 P r e s t r a i n at 77°K 29 2.3 METHODS 29 2.3.1 Equipment 29 2.3.2 S l i p Line Measurement 29 2.3.3 Presentation of Data 30 i i i Page RESULTS . 31 3.1 TEMPERATURE CHANGES 31 3.2 STRAIN RATE CHANGES 38 3.3 SLIP LINE LENGTH IN THE TRANSIENT 43 3.4 PRESTRAIN AT 77°K 47 3.5 GENERAL RESULTS 47 3.5.1 S l i p Lines A f t e r P r e s t r a i n 47 3.5.2 Grown-In D i s l o c a t i o n Substructure 51 DISCUSSION 52 4.1 TEMPERATURE AND STRAIN RATE CHANGE TESTS 52 4.1.1 Increase i n A r and a During the Transient . . . 52 4.1.2 Ca l c u l a t i o n of Ar from the Stres s - S t r a i n Curve . 56 4.1.3 Ca l c u l a t i o n of A r from S l i p Line Lengths . . 61 4.1.4 S l i p Line Length versus Stress. . . . 66 4.2 SLIP LINE LENGTH IN THE TRANSIENT "" 68 4.3 PRESTRAIN AT 77°K 71 4.4 MAXIMUM SLIP LINE LENGTH 74 4.5 THE SLOPE OF THE STRESS-STRAIN CURVE 75 SUMMARY . . . . . . . . . . . . . . 7 9 CONCLUSIONS 8 2 SUGGESTIONS FOR FUTURE WORK 8 4 APPENDICES 8 5 BIBLIOGRAPHY . 9 4 i v LIST OF FIGURES Figure Page 1. Schematic diagram of the number N(T V;L) of "elements" with l o c a l y i e l d strength T y L . The y i e l d strength i s T y 12 2. The r e l a t i v e free area function Aj. increases with stress and equals unity when a - T . The r a t i o x v / X y determines the rate at which A r r i s e s with a . . . . . 15 3. Schematic of the s t r e s s - s t r a i n curve calculated from equation (15). As T V / T v increases there i s more transient deformation 19 4. Orientation of s i n g l e c r y s t a l s before (•) and a f t e r (+) p r e s t r a i n . ; 24 5. Schematic diagram of the r e l a t i v e o r i e n t a t i o n of the primary s l i p plane and Burgers vector to the t e n s i l e axis 25 6. Cross-section of graphite mold 27 7. Average s l i p l i n e length versus temperature . . . . 32 8a. Histogram of s l i p l i n e length d i s t r i b u t i o n , 298°K . . 33 8b. Histogram of s l i p l i n e length d i s t r i b u t i o n , 190°K . . 34 8c. Histogram of s l i p l i n e length d i s t r i b u t i o n , 4 PK . . . 35 9. Steady state slope of incremental s t r e s s - s t r a i n curves for temperature tests 37 10. Extrapolated stress for temperature decreases. . • . . 39 11. Temperature dependent r e v e r s i b l e flow stress r a t i o . . 40 12 Average s l i p l i n e length versus s t r a i n rate . . . . 41 13a. S l i p l i n e s f o r a 77°K increment with e = 1.2 x 10 "'"/min 42 o 13b. S l i p . l i n e s for a 77°K increment with e = 1.2 x 10 /min 42 14a. Histogram,of s l i p l i n e length d i s t r i b u t i o n , 77°K, 1.2 x 10 /min 44 V Figure Page 14b. Histogram of s l i p l i n e length d i s t r i b u t i o n , 77°K, 1.1 x lO-Vmin 45 15. Increase i n s l i p l i n e length during transient deformation, 77°K. When the steady state begins at e - 0.4% the s l i p l i n e length decreases 46 16a. Histogram of s l i p l i n e length d i s t r i b u t i o n f o r the f i r s t s t r a i n d i v i s i o n , 48 16b. Histogram of s l i p l i n e length d i s t r i b u t i o n for the l a s t s t r a i n d i v i s i o n i n the tr a n s i e n t . 49 17. Increase i n s l i p l i n e length during transient deformation at room temperature . 50 18. Measurement of " r e v e r s i b l e " change i n stress a f t e r a i temperature decrease from to . , 54 19. S t r e s s - s t r a i n curve for 4°K increment . 58 20. Increase i n A r during 4°K increment. The value of A,, for the room temperature p r e s t r a i n i s found at e = 0 . . . 59 21. Deer ease i n T v during 4°K increment calculated from equation (10) 60 22a. S l i p area diameters calculated from s l i p l i n e lengths, room temperature increment. For c l a r i t y , d i s t r i b u t i o n s are shown as being smooth 62 22b. S l i p area diameters calculated from s l i p l i n e lengths, 4°K 63 23 S l i p l i n e length versus stress f or temperature and s t r a i n rate change 67 24. S l i p l i n e length versus stress during transient deformation, 77°K 69 25. S t r e s s - s t r a i n curve f o r incremental and " s t r a i n d i v i s i o n " deformation, 77°K. 70 26. Schematic of T v r e l a t i v e to T at temperatures T, > T . 73 v i Figure Page 27. Change i n the slope of the s t r e s s - s t r a i n curve versus temperature as predicted by the concept of a tempera-ture i n s e n s i t i v e stage I I . Below T c the slope i s equal to 6JJ 77 28. C a l c u l a t i o n of the p r o b a b i l i t y that a random plane w i l l i n t e r s e c t a c i r c l e of diameter D t giving a chord length of L ± AL . 91 v i i A C K N O W L E D G E M E N T The author i s g r a t e f u l f o r the advice and encouragement given by Dr. T.H. Alden. Thanks are also extended to other f a c u l t y members and fellow graduate students f o r many h e l p f u l discussions. F i n a n c i a l a s s i s t -ance from the National Research Council (NRC Grant A-4991) i s g r a t e f u l l y acknowledged. 1 I N T R O D U C T I O N 1.1. PREVIOUS SLIP LINE STUDIES The formation of s l i p l i n e s , which i s a fundamental r e s u l t of the p l a s t i c deformation of many c r y s t a l l i n e s o l i d s , has been the subject of many investigations [Fourie and Wilsdorf 1959, Mader 1963]. Though a great deal of work has been done, there i s no general agreement i n the i n t e r p r e t a t i o n of the r e s u l t s . This i s due, i n part, to the complex nature of s l i p l i n e s and the existence of c o n f l i c t i n g theories of s t r a i n hardening. Studies on s l i p l i n e s can be divided into two general categories. Those studies which deal with the observations of s l i p l i n e s a f t e r t h e i r formation are termed s t a t i c . If the experiment i s concerned with the growth of s l i p l i n e s , then i t i s c a l l e d dynamic. One of the f i r s t s t a t i c studies on aluminum involved measuring the s l i p l i n e density i n the d i r e c t i o n normal to the s l i p plane and r e l a t i n g i t to the s t r a i n [Yamaguchi 1928]. It was found that the number of l i n e s per unit length was not proportional to the s t r a i n , which implied that the amount of s l i p i n each l i n e did not remain constant as the s t r a i n increased. The reason for t h i s non-constant r e l a t i o n s h i p became apparent when the electron microscope showed that a f i n e structure could be resolved within an 2 i n d i v i d u a l s l i p l i n e [Hiendrich and Shockley 1947]. The f i n e l i n e s contained i n a s l i p l i n e had a spacing of approximately 200A and a step height between 1600A and 2200A [Brown 1951]. There was no evidence for multiple s l i p , which therefore suggests that f i n e s l i p took place i n one avalanche. At higher s t r a i n s the number of f i n e l i n e s contained i n a s l i p l i n e increased but the amount of s l i p i n each f i n e l i n e remained constant. The e f f e c t of temperature was shown i n the tendency f o r the f i n e l i n e s to c l u s t e r together at higher temperatures and to form between the s l i p l i n e s at lower temperatures. More recent studies show that the step heights of the f i n e l i n e s vary over a wide range. The step heights of aluminum deformed at room temperature were found to be between a few Angstroms and as much as 1200A, the average being 300 to 400A [Noggle and Koehler 1957]. Seeger and h i s collaborators [1963] show that the step heights i n Cu, Au, and Ni are not greater than 120 atomic spacings. Probably the most studied aspect of s l i p l i n e s i s the depen-dence of t h e i r length on s t r a i n [Mader 1963, Hirsch 1964]. It i s experimentally observed that the s l i p l i n e length, L, i s r e l a t e d to the s t r a i n , e, by the following equation: where A i s a constant. L can r e f e r to ei t h e r the s l i p l i n e length measured i n an o p t i c a l microscope or the length of the f i n e s l i p l i n e s that are resolved by electron microscopy [Seeger 1956]. In Seeger's theory of s t r a i n hardening [Seeger, Mader, and Kronmuller 1963], the constant A i s important i n determining the value of the slope of the s t r e s s - s t r a i n curve 3 during stage II deformation. In order to better understand the way i n which s l i p l i n e s form, i t i s necessary to do dynamic t e s t s . The usual method i s cinematography. Experiments on high p u r i t y aluminum sing l e c r y s t a l s showed that s l i p f i r s t appeared as a short l i n e which then grew i n both d i r e c t i o n s [Maddin and Chen 1954]. As the l i n e s lengthened they became wider i n d i c a t i n g that the amount of s l i p on each l i n e increased with s t r a i n . The s l i p l i n e s did not grow to t h e i r f i n a l length i n one continuous process, but remained s t a t i o n -ary f o r a period of time and then grew with a speed of nearly 10^ microns/-sec. As the s l i p l i n e lengths were on the order of 500u, the spurts of growth took place i n l e s s than 1/10 of a second. Another dynamic method involved detecting small s l i p events by an acoustic technique [Fisher and L a l l e y 1967]. I t was found that specimens with a wide range of c r y s t a l structures and s t r a i n hardening behavior a l l emitted bursts of acoustic energy as they deformed p l a s t i c a l l y . The frequency of the pulses was proportional to the s t r a i n rate although i t was not possible to exactly determine the amount and the duration of s l i p responsible for a pulse. The speed at which the pulses developed suggested a d i s l o c a t i o n v e l o c i t y near lo'' microns/sec. If the study of s l i p l i n e s i s to be us e f u l i n understanding the process of s t r a i n hardening i n the i n t e r i o r of a c r y s t a l , then defor-mation on the surface must be representative of that i n the bulk material. Since s l i p l i n e s e x i s t on the surface, they can be compared with deformation i n the i n t e r i o r only i f the paths of moving d i s l o c a t i o n s can be made v i s i b l e inside the c r y s t a l . It i s possible to observe these paths using neutron i r r a d i a t e d c r y s t a l s . It was found that s l i p i n the i n t e r i o r of a c r y s t a l 4 removed the defects caused by the neutrons [Seeger 1963]. The lack of defects i n the s l i p paths was v i s i b l e using electron microscopy. A s t a t i s t i c a l comparison of the s l i p l i n e s and the paths showed that the s l i p l i n e s were equivalent to deformation i n the i n t e r i o r . Other studies i n d i c a t e that deformation on the surface and i n the bulk i s not the same. It was shown that the y i e l d strength of a deformed c r y s t a l i s not constant but that i t decreases near the surface. Electron microscopy revealed that the d i s l o c a t i o n density i s l e s s and the c e l l s i z e l a rger at the surface [Fourie 1968]. Chemically p o l i s h i n g 1 mm of the surface and giving a small- increment of s t r a i n produced a large decrease i n s l i p l i n e length [Himstedt and Neuhauser 1972]. These r e s u l t s throw doubt on Seeger's use of the s l i p l i n e parameter A. Although the surface may a f f e c t the absolute length of the s l i p l i n e s , the study of the v a r i a t i o n of s l i p l i n e length i s meaningful i f the r a t i o of the surface s l i p l i n e length to the i n t e r i o r s l i p length i s a constant. Observations reported i n the present study are s t a t i c i n nature, but deal with the e f f e c t of stress rather than s t r a i n changes on the s l i p l i n e length. The stress change i s achieved " r e v e r s i b l y " [ C o t t r e l l and Stokes 1955] by decreases i n temperature or increases i n s t r a i n rate. As w i l l be shown, e x i s t i n g theory [Seeger et a l . 1963, Hirsch and M i t c h e l l 1967] predicts no change i n l i n e length f o r such a r e v e r s i b l e stress change, while a new theory [Alden 1972a] predicts that the l i n e length w i l l increase. 1.2. THE THEORIES OF HIRSCH AND SEEGER The e x i s t i n g theories that deal the most extensively with s l i p l i n e lengths are those of Seeger [Seeger et a l . 1963] and Hirsch [Hirsch 5 and M i t c h e l l 1967]. Though these theories a t t r i b u t e the stopping of d i s l o c a t i o n s to d i f f e r e n t kinds of obstacles, they are quite s i m i l a r i n t h e i r explanation of the experimentally observed decrease i n s l i p l i n e length with s t r a i n . In Seeger's model the y i e l d strength i s determined by two types of d i s l o c a t i o n i n t e r a c t i o n s . Only one of these i n t e r a c t i o n s i s capable of blocking s l i p l i n e s . This i n t e r a c t i o n i s the s t r e s s between nearly p a r a l l e l d i s l o c a t i o n s l y i n g on d i f f e r e n t s l i p planes and, because t h i s stress acts over a long range, i t i s independent of temperature. From the o r i e n t a t i o n dependence of the y i e l d strength, Seeger concludes that the obstacles i n i t i a l l y blocking the primary d i s l o c a t i o n s are Lomer-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 s . He assumes the density of these obstacles increases with s t r a i n so the length of the pile-ups behind them, and therefore the s l i p l i n e length, w i l l decrease correspondingly. The other component of the y i e l d strength i s due to i n t e r - . sections between primary and forest d i s l o c a t i o n s . If the d i s l o c a t i o n s a t t r a c t each other, then they i n t e r a c t over a long range and the force between them i s large enough that thermal f l u c t u a t i o n s w i l l have no e f f e c t [Saada 1963]. Dislocations that r e p e l each other do not i n t e r a c t as strongly so the energy required to form the jog, which r e s u l t s from the i n t e r s e c t i o n , can be supplied thermally. Although t h i s type of i n t e r a c t i o n r e s u l t s i n an increase i n y i e l d strength with decreasing temperature, there i s no influence on the s l i p l i n e length because Seeger claims that the forest d i s l o c a t i o n s are incapable of blocking primary d i s l o c a t i o n s around t h e i r e n t i r e perimeter. The increase i n y i e l d strength following a quench does not change the s l i p l i n e length because the Lomer-Cottrell obstacles are impenetrable. 6 Hirsch's c a l c u l a t i o n s [1964] show another mechanism of hardening i s necessary because the back stresses from Seeger's pile-ups cannot account f o r more than 50% of the flow stress [Hirsch and Hazzledine 1966] while Seeger a t t r i b u t e s much more to i t . According to Hirsch [Hirsch and M i t c h e l l 1967], at the end of stage I primary d i s l o c a t i o n s p i l e up behind long continuous obstacles, which are p o s s i b l y d i s l o c a t i o n dipoles. The r e s u l t i n g i n t e r n a l stresses cause s l i p to occur on secondary systems near the head of the pile-up. There w i l l then be a region of high d i s l o c a t i o n density that can impede s l i p on other primary planes. Hirsch assumes that i n stage II the structure maintains the same form but i s only reduced i n scale so that h i s theory predicts a l i n e a r decrease i n s l i p l i n e length with s t r a i n . He does not d e t a i l the r o l e of the forest d i s l o c a t i o n s i n c o n t r o l l i n g the thermal component of the flow s t r e s s , but i t i s most l i k e l y that he accepts Seeger's explanation. Hirsch does state that the f o r e s t d i s l o c a t i o n s are i n s u f f i c i e n t f or block-ing the primary d i s l o c a t i o n s ( i . e . s l i p l i n e s ) because according to Hirsch the s o l i d s described by e x i s t i n g f o r e s t theories [Basinski 1959] have no hard or soft spots. Hirsch's obstacles are not impenetrable, Their strength depends on the distance from the obstacles' center. They contain a large proportion of secondary d i s l o c a t i o n s so t h e i r strength w i l l depend on temperature i n the same way as the f o r e s t d i s l o c a t i o n s . Because of t h i s , the increase i n y i e l d strength with decreasing temperature w i l l not enable primary d i s l o c a t i o n s to overcome obstacles that they could not penetrate at a higher temperature. Therefore, t h i s theory predicts that the s l i p l i n e length w i l l not depend on temperature or s t r a i n rate. 7 1.3. SCOPE OF PRESENT WORK Since s l i p l i n e s are associated with p l a s t i c deformation, t h e i r study i s important and they form a basis f o r several theories of s t r a i n hardening [Seeger et a l . 1963, Hirsch and M i t c h e l l 1967]. A recent theory [Alden 1972a] predicts that the s l i p l i n e length i s a function not only of the stress a and the obstacle density, but also the obstacle arrangement. The obstacles are f o r e s t d i s l o c a t i o n s , as proposed by Basinski [1959] and Saada [1963] except that t h e i r arrangement as w e l l as t h e i r density i s e x p l i c i t l y defined. These f o r e s t d i s l o c a t i o n s are part of the r e l a t i v e l y immobile network, s t a b i l i z e d i n a t t r a c t i v e junctions [ F r i e d e l 1964]. Because the obstacles are not uniformly spaced, i t i s necessary to define two structure parameters, X y dependent on the l o c a l density and x v on the arrangement of obstacles. The i n c l u s i o n of the parameter x v and i t s v a r i a t i o n with s t r a i n , time, and temperature, means that Alden's theory predicts c e r t a i n s l i p l i n e phenomena that are not consistent with the theories of Hirsch and Seeger. The new theory i s s i m i l a r i n concept to the recovery theory [Orowan 1947, F r i e d e l 1964]; i n both theories the obstacles are "thermally-unstable" and can a n n i h i l a t e one another with time and temperature, producing a decrease i n obstacle density. However, Alden's theory also states that the obstacles change t h e i r d i s t r i b u t i o n due to the same thermal i n s t a b i l i t i e s . This process i s c a l l e d "rearrangement". Because the obstacles can undergo both recovery and rearrangement, i t may be c a l l e d the recovery-rearrangement theory. Hereafter i t w i l l be referred to as the RR theory. This work was c a r r i e d out to evaluate c e r t a i n predictions of 8 the RR theory, i n p a r t i c u l a r a change i n s l i p l i n e length during an increment of deformation following changes i n temperature and s t r a i n rate. The e f f e c t of the i n i t i a l structure on the s l i p l i n e length was also established. 1.4. RECOVERY THEORY Because the recovery theory forms the basis f o r the RR theory, i t w i l l be discussed f i r s t . The recovery theory provides a simple d e s c r i p t i o n of the temperature-dependent p l a s t i c flow i n s o l i d s which s t r a i n harden. While i t i s able to pr e d i c t secondary creep [McLean 1968] and most of the c h a r a c t e r i s t i c s of the s t r e s s - s t r a i n curve [Alden 1972a], i t cannot account f o r r e v e r s i b l e ( i s o s t r u c t u r a l ) changes i n flow stress [ C o t t r e l l and Stokes 1955] or the rapid and non-linear changes i n s t r a i n rate follow-ing a change i n temperature or stress [Garofalo 1965]. Though i t does have these f a u l t s , the recovery theory i s often used because of i t s s i m p l i c i t y . The recovery theory applies i f the s o l i d contains a set of thermally-impenetrable obstacles, uniformly d i s t r i b u t e d . The distance between these obstacles, I, i s re l a t e d to the y i e l d strength T by the equation Gb T y " ° X ( 2 ) where a i s a constant, G i s the shear modulus, and b i s the Burgers vector. Since the obstacles may be l o s t with time and temperature, the y i e l d strength i s lowered by recovery, the rate of which i s defined by 9 ,3x (it 1), ' "ry • (3> A solid matching the description of this theory will not yield plastically i f i t is subjected to a stress less than the i n i t i a l yield strength, x°. As a becomes equal to T ° , the solid yields abruptly. However, i t is not possible for the stress to become greater than the yield strength so as the stress is increased the solid muqt strain harden. The yield strength increases with strain as follows where 6^. is the coefficient of strain hardening. The total change in x^ is written as a sum of the partial differential equations (3) and (4). dx Z = fl - r v , . (5a) de - °y " y/e At 0°K there is no thermal activation to aid recovery so Xy = 0 and dx Z - 6 „ . (5b) de y When the solid is flowing plastically there is always an equality between X y and a so equations (5a) and (5b) can also be written as de y y/-10 and at 0°K da . = 6 . (6b) de y It i s because of t h i s equality that the recovery theory predicts a r e v e r s i b l e flow stress r a t i o of unity [Alden 1972a]• On quenching to 0 9K a f t e r a p r e s t r a i n at some higher temperature, x , and therefore a, w i l l be unaltered a f t e r c o r r e c t i o n f o r the change i n shear modulus. Because of the regular arrangement of obstacles, the recovery theory i s unable to make meaningful s l i p l i n e p r e d i c t i o n s . Independent of s t r e s s , s t r a i n , temperature, or s t r a i n rate, the d i s l o c a t i o n loops expanding on the primary s l i p plane are able to sweep out the e n t i r e plane. The stress that enables a d i s l o c a t i o n to overcome one p a i r of obstacles i s s u f f i c i e n t f o r i t to pass through a l l obstacles. The s l i p l i n e length of a s i n g l e c r y s t a l would then depend only on the specimen's dimensions, assuming there are no subgrain boundaries. In contrast, s l i p l i n e s i n r e a l c r y s t a l s are short and are known to vary with s t r a i n [Mader 1963]. It i s cl e a r then that the recovery theory must be modified i n some way i f i t i s to agree with experimental observations of s l i p l i n e lengths. 1.5. REARRANGEMENT-RECOVERY THEORY T = 0°K 1.5.1. Equations of P l a s t i c Flow. The RR theory i s s i m i l a r to the recovery theory i n the nature of i t s obstacles except that t h e i r d i s t r i b u t i o n i s no longer regular. Consequently, the structure cannot be characterized by one parameter because there i s no unique value for I. cannot be defined by equation (2) since the v a r i a t i o n i n obstacle spacing gives the s o l i d a range of hard and soft "elements" each with t h e i r own l o c a l y i e l d strength, T v T j » which i s equal to Gb ,_v T y L = 0 1 17 ( 7 ) L i where I i s the l o c a l obstacle spacing. However, i t i s s t i l l p ossible Li to define a y i e l d strength by T, • «• & (8) a' i s only weakly dependent on the obstacle d i s t r i b u t i o n and H i s the average obstacle spacing [Kocks 1966]. Like the recovery theory, x^ i n the RR theory indicates the maximum stress the sample can support without an increase i n the density of obstacle d i s l o c a t i o n s , i . e . without s t r a i n hardening. However, unlike the recovery theory, l i m i t e d p l a s t i c flow can occur for a < x by l o c a l s l i p i n elements of small x _. y yL In describing the obstacle d i s t r i b u t i o n , the RR theory includes a second s t r u c t u r a l parameter, T , which gives an i n d i c a t i o n of the magni-tude of the v a r i a t i o n i n x . Figure 1 shows schematically the number of y L elements N(x^) that have a l o c a l y i e l d strength Ty^' The width pf t h i s curve determines the value of x v. As the curve becomes wider, x v increases. The necessity of including a second parameter i s shown by the r e s u l t s of electron microscopy studies of d i s l o c a t i o n arrangements i n both p o l y c r y s t a l s [Segall and Partridge 1959, Swann 1963] and s i n g l e c r y s t a l s [Howie 1962] of aluminum. Disl o c a t i o n s are observed to have a wide range 12 13 of configurations. At low s t r a i n s they are often i r r e g u l a r l y d i s t r i b u t e d while at high stresses and temperatures they are found tangled i n c e l l walls surrounding e s s e n t i a l l y d i s l o c a t i o n free volumes. The s o l i d described by the RR theory w i l l not y i e l d abruptly at 0°K as i n the recovery theory but w i l l begin to deform l o c a l l y when a equals the smallest value of the l o c a l y i e l d strength. S t r a i n w i l l occur only over that f r a c t i o n of the s l i p plane where a > Ty-^' This f r a c t i o n i s c a l l e d the r e l a t i v e free area, A . An increment of s t r e s s added to a ' r specimen that i s deforming when A^ < 1 w i l l r e s u l t i n l e s s s t r a i n than that from a s i m i l a r increment given to an equivalent specimen deforming oyer a l l of i t s s l i p plane. Therefore, the slope of the s t r e s s - s t r a i n curve w i l l depend on what f r a c t i o n of the s l i p plane i s deforming. Equation (6b) of the recovery theory must be modified to include t h i s e f f e c t . This can be done by m u l t i p l y i n g the r i g h t hand side of the equation by A^. At 0°K j 9 • 42- = (9) de A w r ( As a increases more of the obstacles w i l l become "transparent" and the expanding d i s l o c a t i o n loops w i l l sweep out a l a r g e r f r a c t i o n of the s l i p plane. w i l l increase and the slope of the s t r e s s - s t r a i n curve decrease. The RR theory predicts that i n t h i s region the r e l a t i o n s h i p between stress and s t r a i n i s non-linear even though 8^ i s constant; t h i s i s c a l l e d the ( e l a s t i c - p l a s t i c ) t r a n s i t i o n region. A l l transients cannot be explained by u n c e r t a i n t i e s such as unwanted bending and misalignment of the experimental apparatus. Any change i n the conditions of t e s t i n g such as unloading, annealing, and 14 reloading [Tietz et a l . 1962, Cherian et a l . 1949], or a s t r a i n rate change [Weinberg 1968] produces a region of rapid change i n the slope of the s t r e s s - s t r a i n curve with s t r a i n . Thus the theory must account for t h i s e f f e c t . A specimen undergoing t h i s kind of transient deformation continues to y i e l d i n i t s s o f t e r spots and harden them u n t i l eventually the d i s l o c a t i o n loops are able to move across the e n t i r e s l i p plane. At t h i s point the r e l a t i v e area i s equal to unity. The s l i p plane w i l l have a uniform hardness and a = T . A cannot change with a d d i t i o n a l s t r a i n so y r the s t r e s s - s t r a i n curve becomes l i n e a r . This condition, when the moving s l i p d i s l o c a t i o n s are sweeping out the maximum area that can become accessible to them, i s characterized by dA^/de = 0, and i s c a l l e d the "steady state". 1.5.2. S l i p Lines and the Relative Area Function. S l i p l i n e s are the r e s u l t of d i s l o c a t i o n loops expanding on the s l i p plane and emerg-ing from the c r y s t a l at the surface. Since the r e l a t i v e area function i s a factor i n determining the area that the loops sweep out, the s l i p l i n e length w i l l be dependent on A^. Therefore, whenever there i s a region of transient deformation and A^ changes, the s l i p l i n e length i s expected to vary. The r e l a t i v e free area w i l l depend on both of the structure parameters and the stress. Alden suggests the following function for A^: / T - a v A r = exp - f J . (10) v A r p l o t t e d against a at d i f f e r e n t i n i t i a l values of T v / T i s shown i n Figure 2. This i l l u s t r a t e s that equation (10) meets the 15 Fig.2. The r e l a t i v e free area function A r increases with stress and equals unity when a = xy. The r a t i o xv/xy determines the rate at which A,, increases with a. 16 requirements of the RR theory. A^ . increases with increasing a and becomes equal to unity when a = T y . If T V = 0, there should be no deformation u n t i l a = X y . Therefore A = 0 i f 0 < x r y A = 1 i f a = x r y This i s the s i t u a t i o n i n the recovery theory, which i s a s p e c i a l case of the RR theory. From a m i c r o s t r u c t u r a l point of view, A^ i s defined as the f r a c t i o n of the s l i p plane that i s accessible to expanding d i s l o c a t i o n loops. The r e l a t i v e free area w i l l then be equal to a/A, where a i s the area undergoing deformation within area A. A probably does not represent the e n t i r e s l i p plane: i t i s the minimum area that can enclose a "true" 5 2 average obstacle density, p. If the obstacle density i s 10 obstacles/cm., 2 A w i l l be much smaller than i f p = 10 obstacles/cm.. The f a c t that dA /dc = 0 i s the c r i t e r i o n for the existence of r the steady state does not require that the s l i p l i n e length remain constant a f t e r the t r a n s i t i o n region. This would be i n c o n f l i c t with the accepted inverse r e l a t i o n s h i p between s l i p l i n e length and s t r a i n [Mader 1963]. Appendix A shows that the average s l i p l i n e length L can be r e l a t e d to the r e l a t i v e area function by the following equation: a 3 i r n L" 2 /UN A r = 8" "A" where n i s the number of a c t i v e s l i p areas i n s i d e A. D i f f e r e n t i a t i n g equation (11) with respect to s t r a i n , s e t t i n g i t equal to zero, and solving 17 * d L f o r gives ^ = L/2A M _ L / 2 n ^ (12) de de de As deformation proceeds the f o r e s t d i s l o c a t i o n density w i l l dA become la r g e r . This r e s u l t s i n a decrease i n A, that i s a negative — , because as the density of obstacles increases, i t takes a smaller area to encompass the same "average" density of obstacles. I t i s expected also dn that -7— i s greater than zero since new sources w i l l become act i v e as de 1 s t r a i n proceeds. Therefore, the r i g h t hand side of equation (12) i s negative and tjie s l i p l i n e length w i l l decrease with s t r a i n . 1.5.3. A n a l y t i c a l Development. The shape of the s t r e s s -s t r a i n curve w i l l depend on the properties of the i n i t i a l structure and how they change with s t r a i n . A specimen subjected to a constant s t r a i n rate test w i l l begin to deform p l a s t i c a l l y at a stress lower than x y. This i s the region of transient deformation where the slope of the s t r e s s -s t r a i n curve decreases due to a changing r e l a t i v e area function. As a approaches x , A^ increases according to equation (10), and decreases ^ dx £ and asymptotically approaches . When a = and A^ = 1, the steady state condition i s obtained and further deformation does not a l t e r the value of A^. At 0°K i t i s possible to c a l c u l a t e the shape of the s t r e s s -s t r a i n curve by s u b s t i t u t i n g equation (10) into equation (9) and then rearranging i t . d e = i - e x p - f - ^ Ida (13) y v Since the y i e l d strength i s not strongly dependent on the obstacle d i s t r i b u t i o n , equation (5b) i s v a l i d for the RR theory as w e l l as f o r the 18 recovery theory. This equation can be integrated and solved f o r T . This gives T = x ° + 9 e (14) where x^° i s the i n i t i a l y i e l d strength. Equation (13) i s then integrated a f t e r equation (14) i s substituted f or x^ and the v a r i a b l e s a and e are separated. x v and 9 y are assumed constant f o r s i m p l i c i t y . ea / x v = ( e e y e / T v _ i ) e T y ° / T v + ± ( 1 5 ) From t h i s equation, the stress can be calculated f or any value of s t r a i n and the s t r e s s - s t r a i n curve can be constructed. This i s i l l u s t r a t e d i n Figure 3 for d i f f e r e n t values of x /x . ° v y Although x v i s not constant i n the tr a n s i e n t , i t s v a r i a t i o n w i l l not q u a l i t a t i v e l y change the above discussion. The hardening of the softer areas causes x^ to decrease which, from equations (10) and (9), tends to made — increase. The only e f f e c t of t h i s increase i n de de w i l l be a shortening of the transient region. The change i n x v with s t r a i n i s defined as (16) 6^ must be a function of s t r a i n because i n the steady state when dA^/de =0, x^ i s a constant. This i s shown by d i f f e r e n t i a t i n g equation (10) with respect to s t r a i n . 19 F i g . 3. Schematic of the s t r e s s - s t r a i n curve calculated from equation (15). As T v / r y ° increases there i s more transient deformation. 20 dA i de 3A da r l da de 3A + f ^ 9x dx _ J de 3A x dx + / r \ v = A da de x de v dx dx d f f i - In A V de de r de + 8x / de v x - a x dx -| v N X V de (17) dA d 0 dx Since -r-^- - 0 and — = -r-^- i n the steady state at 0°K de de de dx v de = 0 (18) In the steady state at 0°K the s o l i d has a uniform hardness so the minimum value of the l o c a l y i e l d strength w i l l be nearly equal to the y i e l d strength i t s e l f . This means that the obstacle d i s t r i b u t i o n ; curve w i l l be extremely narrow, and x^ i s approximately zero. 1.6. REARRANGEMENT-RECOVERY THEORY T > 0°K 1.6.1. Equations of P l a s t i c Flow. At temperatures greater than 0°K both x^ and x^ vary thermally. Equation (3) s t i l l defines the recovery rate, and the increase i n x v with time and temperature, c a l l e d the rearrangement rate r v i s defined as / 3 t \ dt J V e (19) x^ increases with time, as the obstacle structure becomes less regular. That i s , soft spots are created i n the s o l i d so, at constant a, r e -arrangement can produce flow. This means that f o r a given rate of flow, a need not increase as f a s t . When the equivalent of equation (6a) i s 21 written for deformation above 0°K, the rearrangement rate (as a subtractive term) as w e l l as the r e l a t i v e area function, must be included. , 6 r r £ l . (20) de A e e r Rewriting equation (20) gives the basic equation of p l a s t i c flow for the RR theory. / x - a \ de = 1/6 (da + r dt + r^dt) exp - ( J (21) y y v \ x^ Since the y i e l d strength i s not strongly dependent on the d i s t r i b u t i o n of d i s l o c a t i o n s [Kocks 1966], i t follows that rearrangement has l i t t l e influence on the y i e l d strength; equation (5a) s t i l l holds dr r - r - * = 6 - (5a) de y e The t o t a l d i f f e r e n t i a l equation f o r x^ i s written by combining equations (16) and (19) dx r - r ^ = - 9 + (22) de v e 1.6.2. The Relative Area Function. Above 0°K, a transient w i l l e x i s t f o r the same reasons that one did at 0°K. As the stress r i s e s the s o f t e r areas on the s l i p plane deform and harden, and more of the s l i p plane i s " f r e e " to moving d i s l o c a t i o n s . The r e l a t i v e area function increases and 2^. w i n f a l l according to equation (20). At 0°K, ^ £ f i n aiiy dx y became constant and equal to -^p- when a equaled x . This was c a l l e d the 22 dA r steady state and was defined by =0. I t i s anticipated that a steady state should e x i s t at temperatures above 0°K and may also be dA x defined by — = 0. J de If T v i s a constant, t h i s d e f i n i t i o n of steady state defor-mation r e s u l t s i n the equality 77 = HT* ( 2 3 ) de de and i t i s possible to solve for A^ from equations (5a) and (20), 8 A = Q — rr (24) r 9 + r /e > / y v Since r ^ i s thermally-activated, r /e i s greater than zero, and the steady state value of A^ w i l l be l e s s than unity. This i s the most important d i s t i n c t i o n between deformation at T = 0°K and T > 0°K. Although equation (24) i s a t t r a c t i v e l y simple and may be u s e f u l as an approximation, from mic r o s t r u c t u r a l evidence i t i s u n l i k e l y that i t applies exactly at temperatures greater than 0°K. As shown dx v e a r l i e r , -:— = 0 when equation (23) i s v a l i d . A constant x i s not de v consistent with observations showing that structures become increasingly c e l l u l a r with s t r a i n [Segall and Partridge 1959]. The formation of a dx more c e l l - l i k e structure implies that — > 0 since c e l l u l a r structures de seem to have larger x v values than non-^cellular structures [Alden 1973]. The consequences of an increasing x^ are apparent by con-s i d e r i n g the e f f e c t on equation (17). Rearranging t h i s equation and dA r s e t t i n g -j-jT- - 0 gives 23 , dx /dx \ £ - ^ + >» \ (zf) • «« dx Because i s always l e s s than or equal to one and > 0, i n the steady state above 0°K This i n e q u a l i t y i s i n agreement with the Cottrell-Stokes law. I f the change i n stress Aa found by quenching to 0°K divided by the stress at 0°K ( i . e . x ) i s to be independent of s t r a i n , then Aa must increase with s t r a i n . Since Aa = x^ - a, t h i s would be possible only i f x^ i s increasing f a s t e r with s t r a i n than a. The i n e q u a l i t y of equation (26) i n d i c a t e s that equation (24) i s i n c o r r e c t . I t must be written e A < r — t 1 rr (27) r 9 + r /e y v ; Although i t i s not possible to solve for the exact value of the r e l a t i v e free area function, the RR theory does state that w i l l be les s than unity during steady state deformation above 0°K. When the , s t r a i n rate i s decreased or the temperature increased, equation (27) implies that the steady state value of A r decreases. 24 E X P E R I M E N T 2.1. SINGLE CRYSTAL PREPARATION 2.1.1. Ma t e r i a l . The sin g l e c r y s t a l specimens were grown from the melt i n a h o r i z o n t a l graphite boat under an argon atmosphere. The aluminum was research grade (99.994% Al) supplied i n ingots which were r o l l e d and then machined to the desired s i z e . The as grown c r y s t a l s had eith e r a 1/4" x 1/4" or a 3/16" x 3/16" cross section and were cut into two three inch long specimens a f t e r growth. 2.1.2. Orientation. There were several reasons f or choosing the o r i e n t a t i o n shown i n Figure 4. In order to observe " s i n g l e s l i p " , the c r y s t a l s had to be oriented so that a f t e r a 20% p r e s t r a i n the t e n s i l e axis was s t i l l i n s i d e the standard t r i a n g l e . To ensure the greatest accuracy F i g . 4. Orientation of single c r y s t a l s before (•) and a f t e r (+) p r e s t r a i n . 25 i n the s l i p l i n e measurements the Burgers vector of the primary s l i p system had to l i e p a r a l l e l to one surface of the specimen (Figure 5). This gives a maximum step height on the other surface where measurements are being made. The f i n a l consideration was that the s l i p plane should i n t e r -sect the observed surface at such an angle that the distance between the s l i p l i n e s was large enough that they could be e a s i l y resolved. A seed of the desired o r i e n t a t i o n was obtained by f i r s t growing a random sin g l e c r y s t a l and then performing a serie s of r o t a t i o n s . A l l the seeds and specimens were checked by the Laue back r e f l e c t i o n technique to be c e r t a i n that t h e i r t e n s i l e axis was within 2° of the correct o r i e n t a t i o n . T.A. F i g . 5. Schematic diagram of the r e l a t i v e o r i e n t a t i o n of the primary s l i p plane and Burgers vector to the t e n s i l e a x i s . 2.1.3. Growing. The greatest d i f f i c u l t y i n growing the sing l e c r y s t a l s was obtaining complete contact between the molten material and the p a r t i a l l y molten seed. The surface tension and a thick oxide skin prevented 26 the l i q u i d s from mixing. Several methods of breaking through the oxide layer were t r i e d ; the most successful involved a s p e c i a l l y constructed mold (Figure 6). A gap was l e f t between the seed and p o l y c r y s t a l l i n e blank and the l i d of the mold was constructed with an i n s e r t that rested on the blank. Af t e r the blank melted, the l i d did not close t i g h t l y over the mold because i t s weight was p a r t i a l l y supported by the seed. When the mold and aluminum were brought up to temperature, the seed melted up to the end of the i n s e r t and the l i d dropped squeezing molten aluminum into the gap. The oxide skin was broken giving perfect contact. At t h i s point the motor that moved the furnace was manually switched on. The , furnace moved at 4 inches per hour. When the l i d dropped, the excess aluminum was pressed out the opposite end of the mold above the l e v e l of the blank to act as a re s e r v o i r to f i l l the mold as the aluminum s o l i d i f i e d and contracted. When the e n t i r e c r y s t a l froze, the sing l e c r y s t a l was then constrained between the res e r -v o i r at one end and the i n s e r t at the other. To prevent the sing l e c r y s t a l from being loaded the re s e r v o i r end of the l i d was cut o f f as shown i n Figure 6 so when the c r y s t a l contracted the wedge would r i s e . 2.1.4. Annealing and P r e s t r a i n . For the purposes of the experiment i t was necessary to produce a serie s of specimens with the same d i s l o c a t i o n microstructure. This was achieved by annealing the sing l e c r y s t a l s at 540°C for a minimum of 24 hours. In most cases, t h i s was followed by a room temperature p r e s t r a i n of approximately 20% resolved 2 shear s t r a i n to a stress l e v e l of 940 g/mm a f t e r which the samples were placed i n a l i q u i d nitrogen storage dewar. F i g . 6. Cross section of graphite mold. to 28 2.1.5. E l e c t r o p o l l s h l n g . To get a surface that was s u i t a b l e for s l i p l i n e studies, the c r y s t a l s were elect r o p o l i s h e d a f t e r p r e s t r a i n for t h i r t y to f o r t y - f i v e minutes. The f i v e parts methanol and one part p e r c h l o r i c a c i d s o l u t i o n was cooled to -20°C to keep p i t t i n g at a minimum. At 10 v o l t s the current was 0.4 amperes when the s o l u t i o n was slowly s t i r r e d with a magnetic s t i r r e r . Under these conditions the sample was kept at l e a s t below 0°C because moisture from the atmosphere froze on the end of the specimen which stuck above the p o l i s h i n g s o l u t i o n . The temperature of the specimen should be s u f f i c i e n t l y low that s t a t i c recovery during p o l i s h i n g can be neglected. In any case, a l l the samples were given nearly i d e n t i c a l treatments so i f t h e i r structures were changed, they should change the same amount. 2.2. TESTS CONDUCTED 2.2.1. Temperature and S t r a i n Rate Changes. Aft e r p o l i s h i n g , the specimens were subjected to a s t r a i n increment of approximately 1.0% i n order to produce a small number of s l i p l i n e s at one of f i v e temperatures, -2 with k = 1 x 10 /min , or at one or nine s t r a i n rates, with T = 77°K. This small amount of s t r a i n was s u f f i c i e n t l y large that the steady state c h a r a c t e r i s t i c of the t e s t i n g condition was reached. Evidence f o r t h i s was the observed l i n e a r i t y of the s t r e s s - s t r a i n curve which followed the transient. The s l i p l i n e s were then measured to see i f any dependence on temperature or s t r a i n rate could be detected. 2.2.2. S l i p Line Length i n the Transient. Two c r y s t a l s were given 77°K s t r a i n increments which were divided into small steps, c a l l e d s t r a i n d i v i s i o n s . No p o l i s h i n g was done between these d i v i s i o n s . Because 29 of t h e i r small s i z e , these d i v i s i o n s were n e c e s s a r i l y of a random s i z e . A f t e r each d i v i s i o n , the s l i p l i n e s were measured to check i f t h e i r length changed i n the transient region. A s i m i l a r test was done on one specimen at room temperature. 2.2.3. P r e s t r a i n at 77°K. A 77°K p r e s t r a i n was given to a specimen to check the dependence of the s l i p l i n e length on the i n i t i a l structure. The stress applied was equivalent to that found a f t e r a room temperature p r e s t r a i n and a quench to 77°K. Assuming that the C o t t r e l l -Stokes law applies, these specimens with d i f f e r e n t p r e s t r a i n s had the same a and x . Af t e r a s t r a i n increment at 77°K, measuring the s l i p l i n e s would show i f t h e i r length depended on x , which i s expected to be smaller f o r a 77°K p r e s t r a i n than a 298°K p r e s t r a i n . 2.3. METHODS 2.3.1. Equipment. Straining was done on a standard Instron machine. The specimens were held i n s p l i t s t a i n l e s s s t e e l grips f i t t e d with f i l e i n s e r t s to keep slippage at a minimum. Extension was measured from both the Instron recorder and gage marks on the specimens. The temperature tests were done using the following baths: l i q u i d helium at 4°K; l i q u i d nitrogen at 77°K; petroleum ether at 140°K and 190°K; and et h y l alcohol at 247°K. The petroleum ether and e t h y l alcohol baths were cooled using l i q u i d nitrogen; the temperatures were maintained within ±3°K. 2.3.2. S l i p Line Measurements. S l i p l i n e s were measured using an inverted-stage microscope with a graduated eyepiece. The s l i p l i n e s were moved across the f i e l d of view i n a d i r e c t i o n perpendicular to 30 t h e i r length u n t i l the end of one l i n e matched up with the zero point on the eyepiece; the length of t h i s l i n e was then recorded i n terms of the number of "spaces" i t equaled on the eyepiece. Each space was approxi-mately 13u wide, so the s l i p l i n e s were measured to an accuracy of ±6.5u. There w i l l be a sampling error (S.E.) i n the value of the measured average s l i p l i n e length, L, because only a small f r a c t i o n of the s l i p l i n e s were counted. This error w i l l be equal to the standard deviation f o r the d i s t r i b u t i o n of lengths divided by the square root of the number of l i n e s measured. There i s a p r o b a b i l i t y of 0.95 that the measured average length w i l l l i e within ±2 S.E. of the true average length [DeHoff 1968]. A s u f f i c i e n t number of l i n e s was counted that i n almost a l l cases the actu a l error i n measuring an i n d i v i d u a l s l i p l i n e was greater than the error due to the sampling e f f e c t s . 2.3.3. Presentation of Data. In a l l cases a and e are given i n resolved shear stress and resolved shear s t r a i n . The e l a s t i c component of the s t r a i n has been removed so the s t r e s s - s t r a i n curves are a c t u a l l y s t r e s s - p l a s t i c s t r a i n curves. When a modulus cor r e c t i o n i s necessary, the shear modulus f o r s l i p on a (111)[110] system has been used [Simmons and Wang 1971, Kamm and Alers 1964]. • - • 31 R E S U L T S 3.1. TEMPERATURE CHANGES Str a i n increments were done at f i v e d i f f e r e n t temperatures between 4°K and room temperature. As shown i n Figure 7, the average s l i p l i n e length increases markedly a f t e r the transient deformation following a decrease i n temperature. These r e s u l t s are i n q u a l i t a t i v e agreement with the predictions of the RR theory and cannot be accounted f o r by the theories of Hirsch and Seeger. Figure 7 indicates that the average s l i p l i n e length increases almost l i n e a r l y with temperature and that even near 0°K the s l i p l i n e s are not able to expand over the e n t i r e s l i p plane as suggested by the recovery theory. To give a better idea of the d i s t r i b u t i o n of s l i p l i n e lengths a f t e r the increment of s t r a i n , histograms representing the frequency that a length occurs are given for three temperatures i n Figures 8a, 8b, and 8c. Because the number of l i n e s counted was not the same i n a l l cases, the histograms have been normalized to show 100 l i n e s . By comparing the histograms, i t i s clear that long l i n e s appear i n greater numbers at lower temperatures. At 298°K there were only two l i n e s out of 100 longer than 200p, while at 190°K there were nine, and at 4°K, twenty-two. The length of these longer l i n e s may be more nearly c h a r a c t e r i s t i c of the steady state at 200 < 40 100 200 300 Temperature (°K) F i g . 7. Average s l i p l i n e length versus temperature. 20 _ L = 63y 16 12 _L 50 100 150 S l i p Line Length, L (y) F i g . 8a. Histogram of s l i p l i n e length d i s t r i b u t i o n , 298°K. 200 L O 20 -4-1 o u 1 16 L = 103y 13 0) u C 3 O u CD c 12 50 _1_ 100 S l i p Line Length, L (y) 150 200 F i g . 8b. Histogram of s l i p l i n e length d i s t r i b u t i o n , 190°K. u> 20 _ 16 L = 168u T3 0) 4-1 a I 12 cn cu C •H O p 8 1 50 100 S l i p Line Length, L (y) 150 200 F i g . 8c. Histogram of s l i p l i n e length d i s t r i b u t i o n , 4°K. 36 each temperature than the average length. The f i r s t l i n e s to form during the increment w i l l be those shorter l i n e s r e s u l t i n g from the steady state value of that was established during the p r e s t r a i n . The shorter l i n e s would lower the measured L so the temperature e f f e c t would be somewhat greater than that of Figure 7. It should be noted that these histograms cannot represent the true d i s t r i b u t i o n of the diameters of the areas undergoing deformation. Even i f the s l i p area had a constant diameter throughout the c r y s t a l , the s l i p l i n e s would have a d i s t r i b u t i o n of lengths since the p r o b a b i l i t y that a s l i p area would i n t e r s e c t the surface at i t s maximum dimension i s l e s s than one. Later i t w i l l be shown that the d i s t r i b u t i o n of s l i p area diameters can be calculated from the s l i p l i n e length histograms. The s t r e s s - s t r a i n curves as produced by the Instron recorder showed a short e l a s t i c - p l a s t i c t r a n s i t i o n followed by a region of nearly l i n e a r hardening ( i . e . "steady s t a t e " ) . The l i n e a r i t y i s probably due to the small magnitude of the s t r a i n increment so that r ^ and remain nearly constant. The steady state slope varied with temperature according to Figure 9.. Results of Noggle and Koehler [1957] are also shown on t h i s curve. I t i s not r e a l l y expected that there should be agreement between these sets of data because of the d i f f e r e n t experimental methods used. Noggle and Koehler deformed t h e i r specimens only at the temperature at which they measured ^ , while i n the present work, the majority of the deformation was done at room temperature with j u s t an increment of s t r a i n at the indicated temperatures. The dependence of ^ on the p r i o r h i s t o r y i s i l l u s t r a t e d by the deformation of one specimen at 77°K to the same stress l e v e l as that found a f t e r a room temperature p r e s t r a i n and a 77°K 37 38 increment. The r e s u l t i n g value of 4?1 ±s ^ n g 0 0 ( j agreement with the previous work (Figure 9), although the slope was not found to be constant. The change i n flow stress with temperature i s shown i n Figure 10. From these data i t i s possible to construct a graph of the r e v e r s i b l e flow stress r a t i o , Figure 11. According to the C o t t r e l l -Stokes law, t h i s quantity i s independent of s t r a i n . The r e s u l t s compare favorably with those of previous works [ C o t t r e l l and Stokes 1955, M i t c h e l l 1964]. 3.2. STRAIN RATE CHANGES The change i n s l i p l i n e length as a function of s t r a i n rate was measured from e = 1 x 10 /min to e = 3 x 10 /min at 77°K. The s l i p l i n e s formed at higher s t r a i n rates were longer as shown i n Figure 12. This i s i n q u a l i t a t i v e agreement with the RR theory because an increase i n s t r a i n rate has the same e f f e c t on the r e l a t i v e area function as does the decrease i n temperature, equation (24). The d i f f e r e n c e i n length over t h i s range of s t r a i n rates i s comparable to that observed i n the temperature change experiments. Figures 13a and 13b show the s l i p l i n e s r e s u l t i n g from a s t r a i n -1 -3 increment given at a s t r a i n rate of 1.2 x 10 /min and 1.2 x 10 /min r e s p e c t i v e l y . These increments were done on specimens cut from the same c r y s t a l . The f i g u r e i l l u s t r a t e s that although there i s a wide range of s l i p l i n e lengths, the higher s t r a i n rate specimen has quite a few longer l i n e s . The average s l i p l i n e length f o r the high s t r a i n rate increment i s 150y and that for the slow s t r a i n rate increment i s 95y. 39 40 0 100 200 Temperature (°K) F i g . 11. Temperature dependent r e v e r s i b l e flow stress r a t i o . S t r a i n Rate, t (/min) Fi g . 12. Average s l i p l i n e length versus s t r a i n rate. 42 Fig . 13a. S l i p l i n e s for a 77°K increment with t = 1.2 x 10 /min. (x200). F i g . 13b. S l i p l i n e s for a 77°K increment with e = 1/2 x 10~ 3/min.(x200). 43 Histograms equivalent to those of the temperature changes are reproduced i n Figures 14a and 14b. Analogous to the low temperature t e s t s , the high s t r a i n rate produced a l a r g e r number of longer s l i p l i n e s . In Figure 14a, where i = 1.2 x 10 ^/min , there are only two l i n e s longer than 200u , but when e = 1.1 x 10 "'"/min , over 16 l i n e s are longer than 200y (Figure 14b). The stress and <jid not vary as much i n these te s t s as i n the temperature change t e s t s . No r e l a t i o n s h i p between these quantities and s t r a i n rate i s presented because the experimental scat t e r was greater than t h e i r v a r i a t i o n . 3.3. SLIP LINE LENGTH IN THE TRANSIENT It was possible to measure the s l i p l i n e length v a r i a t i o n i n the transient i f the s t r a i n was not applied i n one continuous increment but was divided into smaller sections c a l l e d " s t r a i n d i v i s i o n s " . This was done for two specimens at 77°K and one at room temperature. The s l i p l i n e was found to increase at constant temperature i n t h i s early stage of the s t r e s s - s t r a i n curve. The r e s u l t s of the low temperature tests are shown i n Figure 15. L f i r s t increases r a p i d l y to a maximum at a s t r a i n of 0.40% which, according to the Instron recorder i s approximately the end of the transient, and then gradually decreases. Because the specimens were not polished between s t r a i n d i v i s i o n s , the average s l i p l i n e length at each s t r a i n d i v i s i o n represents the cumulative e f f e c t of the current d i v i s i o n and a l l previous d i v i s i o n s . The maximum value of L f o r these tests was 102y as compared to 120y f o r a specimen given the same amount of s t r a i n i n one increment. 20 16 T3 CD 4 J f o o cn 0) C •H P. •H L = 83y 12 u-i o u •i i s 50 100 150 S l i p Line Length (u) F i g . 14a. Histogram of s l i p l i n e length d i s t r i b u t i o n , 77°K, t = 1.2 x 10"^/min 200 20 16 L = 156u 12 _L_ 50 100 150 S l i p Line Length (y) F i g . 14b. Histogram of s l i p l i n e length d i s t r i b u t i o n , 77°K, i 200 1.1 x 10 "'"/min. 4 > U 1 47 This d i f f e r e n c e i s probably due to unwanted rearrangement and recovery that would occur on heating the specimens to room temperature for examination a f t e r each d i v i s i o n . As a r e s u l t , the steady state condition would not be achieved so r e a d i l y . Figures 16a and 16b show the histograms of the s l i p l i n e d i s t r i b u t i o n s f o r the f i r s t d i v i s i o n and the l a s t d i v i s i o n i n the transient r e s p e c t i v e l y . A comparison of these figures indicates that the d i s t r i b u t i o n for the l a s t d i v i s i o n i s s h i f t e d towards longer s l i p l i n e lengths. The s l i p l i n e length was found to increase also during the transient at room temperature (Figure 17). The maximum average s l i p l i n e length for t h i s test was 75y while L f o r a normal s t r a i n increment was 71y. This d i f f e r e n c e i s not s i g n i f i c a n t because the s l i p l i n e s could not be measured with s u f f i c i e n t accuracy. 3.4. PRESTRAIN AT 77°K The s l i p l i n e length was found to be dependent on the p r e s t r a i n temperature. The specimen prestrained at 77°K and given a 77°K increment had an average s l i p l i n e length of lOOy while a room temperature p r e s t r a i n and an increment at 77°K resulted i n an L of 140y. The y i e l d stress at the end of the 77°K p r e s t r a i n was equal to that found a f t e r the room temperature p r e s t r a i n and quench to 77°K. 3.5. GENERAL RESULTS 3.5.1. S l i p Lines a f t e r P r e s t r a i n . Since the t e n s i l e axis was w e l l within the standard t r i a n g l e at the end of the p r e s t r a i n (Figure 4), there should only have been s l i p on the primary s l i p system. However, the 20 T3 <D 4-1 G 3 O o CD <L> G o M CU •§ 3 16 12 J L L = 81y 50 150 100 S l i p Line Length, L (y) F i g . 16a. Histogram of s l i p l i n e length d i s t r i b u t i o n f o r the f i r s t s t r a i n d i v i s i o n . 200 00 20 _ 16 12 L = 103y JL I 150 J L 200 F i g . 16b. 50 100 S l i p Line Length, L (y) Histogram of s l i p l i n e length d i s t r i b u t i o n f o r the l a s t s t r a i n d i v i s i o n i n the transient. deformation was not homogeneous, but took place i n bands that l ay nearly p a r a l l e l to the s l i p plane. This i s i n agreement with e a r l i e r studies [Honeycombe 1951] which found that i n between the bands of sin g l e s l i p , refered to as deformation bands, there were regions where very l i t t l e deformation occurred. In these regions, very f a i n t s l i p l i n e s from the primary and secondary systems could be seen. Kink bands running perpen-d i c u l a r to the deformation bands were also observed. A l l microscopic evidence of the deformation bands was removed by the e l e c t r o p o l i s h . The s t r a i n given to the specimens from the incremental deformation was so small that the bands did not develop d i s t i n c t l y ; the s l i p l i n e s appeared to be much more homogeneously d i s t r i b uted than a f t e r the p r e s t r a i n . An attempt was made to measure only those s l i p l i n e s i n the regions of s i n g l e s l i p as they should be more represen-t a t i v e of the "normal" deformation of f . c . c . metals. 3.5.2. Grown-in D i s l o c a t i o n Substructure. I t was possible reveal the grown-in d i s l o c a t i o n substructure of the annealed specimens using an etch suggested by Barber [1962]. Etching resulted i n p i t s where d i s l o c a t i o n s intersected the surface. The majority of the d i s l o c a t i o n s lay i n randomly oriented subgrain boundaries. The etch was not able to show d i s l o c a t i o n s that formed during p l a s t i c deformation. Af t e r etching, a specimen was strained a small amount at room temperature. It was found that the subgrain boundaries were not capable of blocking s l i p ; the s l i p l i n e s only very r a r e l y terminated at a boundary. The width of the subgrains i n the d i r e c t i o n of the s l i p l i n e s averaged 400y. 52 D I S C U S S I O N 4.1. TEMPERATURE AND STRAIN RATE CHANGE TESTS 4.1.1. Increase In A and a During the Transient. Any theory attempting to describe the p l a s t i c deformation of pure f . c . c . metals must be able to account for a wide range of experimental phenomena. The recovery theory, the theories of Hirsch [Hirsch and M i t c h e l l 1967] and Seeger [Seeger et a l . 1963], and the RR theory [Alden 1972a] are examples of theories which can explain the most basic of experimental observations, namely the existence of a y i e l d strength, X y , and the increase i n T y with s t r a i n . A clos e r look w i l l be taken at these theories to see i f they predict the observed increase i n stress and s l i p l i n e length following a decrease of temperature or increase of s t r a i n rate. The recovery theory states that there i s always an equality between the stress and the y i e l d strength. Since x^, as defined i n equation (2), i s not a function of temperature (except through the shear modulus G), o i s not expected to increase with decreasing temperature at constant structure. This theory also f a i l s to account for any s l i p l i n e length observations because during p l a s t i c flow the e n t i r e s l i p plane i s free f or d i s l o c a t i o n motion. A temperature dependent y i e l d stress i s possible i n the 53 theories of Hirsch and Seeger because they both r e l a t e the "strength" of the f o r e s t d i s l o c a t i o n s to the temperature. At lower temperatures, there i s l e s s thermal energy a v a i l a b l e f o r jog formation; i f the s t r a i n rate i s to remain constant, the stress must r i s e instantaneously. However, the increase i n a with decreasing temperature w i l l have no e f f e c t on the s l i p l i n e length because the obstacles blocking s l i p l i n e s are " i n f i n i t e l y hard" i n Seeger's case and increase i n strength i n the same way as the f o r e s t d i s l o c a t i o n s i n Hirsch's case. These theories r e l a t e the s l i p l i n e length only to the density of the obstacles that are capable of blocking s l i p . This density w i l l increase with s t r a i n so the theories can account f o r the decreasing s l i p l i n e length with s t r a i n . Increasing the stress w i l l not decrease the obstacle density so these theories are i n disagreement with the r e s u l t s of the present work. This work shows that L increases along with the stress during the region of transient deformation following a quench. The RR theory r e l a t e s the y i e l d strength to the obstacle spacing i n an equation that i s s i m i l a r i n form to the recovery theory. The important d i f f e r e n c e between these theories i s that the stress a i s not equal to the y i e l d strength i n the RR theory at temperatures above 0°K. Although T y does not increase abruptly a f t e r the temperature i s lowered, i t i s possible for a to increase r a p i d l y . This i s shown i n Figure 18, where a specimen deforming i n the steady state at T^ i s quenched to and defor-mation i s continued. There i s not a "true" r e v e r s i b l e change i n s t r e s s , but since ^ i s large i n i t i a l l y and then quickly decreases to i t s new steady state value at T 2 , i t i s possible to define the " r e v e r s i b l e " change i n stress as (o^, - where cr-j^ i s measured j u s t p r i o r to the quench and 0 T i s the stress found by extrapolating the steady state s t r e s s - s t r a i n curve 55 at back to where the quench was applied (Figure 18). I t w i l l now be shown that, through the A r function, the RR theory predicts that the s l i p l i n e length increases due to the rapid increase i n stress a f t e r the temperature i s lowered. The RR theory states that the obstacles to p l a s t i c flow are not arranged uniformly so the s o l i d w i l l contain a range of hard arid s o f t "elements". When a s u f f i c i e n t l y large stress i s applied to such a s o l i d i t w i l l y i e l d p r e f e r e n t i a l l y i n the so f t e r areas ( i . e . where the l o c a l obstacle density i s le a s t ) and d i s l o c a t i o n loops w i l l begin to expand. The f r a c t i o n of the s l i p plane which i s accessible to these expanding loops w i l l depend on both the obstacle density and d i s t r i b u t i o n , as we l l as the stress. The RR theory r e l a t e s t h i s f r a c t i o n to the structure parameters and the stress through the r e l a t i v e area function A^_. Because s l i p l i n e s are a re s u l t of expanding loops i n t e r s e c t i n g the surface, at a given structure the s l i p l i n e length w i l l depend on what f r a c t i o n of the s l i p plane i s deforming. Therefore, any changes i n the r e l a t i v e area function should give a corresponding change i n the s l i p l i n e length. The RR theory predicts that varying the temperature or the s t r a i n rate w i l l a l t e r the s l i p l i n e length because of a change i n the steady state value of A^ .. When a specimen deforming i n the steady state at T^ i s quenched to T^ and deformation i s continued, the recovery and rearrangement rates w i l l be lowered from r ^ and r ^ to r ^ and r v 2 * P r i o r to quenching, the r e l a t i v e free area function was A ^ so equations (20) and (5a) are written 56 , r 8 - -r± (28b) Reducing the temperature does not produce an instantaneous change i n A^, since neither the structure nor the stress change abruptly; a f t e r the quench these equations w i l l be ^ - £ L _ _ 42 _ \ - ( 2 9 a ) de L A e e 1 1 r l ' d x ~ \ r y 2 9„ - -IT (29b) 2 As a r e s u l t of the temperature decrease both the slope of the s t r e s s - s t r a i n curve and are increased because the subtractive recovery and rearrange-ment terms are smaller. da d T The important r e s u l t of the changes i n — and -r-^- a f t e r a de de decrease i n temperature i s that A^ w i l l not remain constant. I t i s shown da d x y i n Appendix B that the quench increases — r e l a t i v e to -r-^- and, from de de equation (17), A^ w i l l increase. This v a r i a t i o n i n A^ w i l l cause the s l i p l i n e length to increase and also w i l l decrease 7^ (equation 20) so there w i l l be a region of transient deformation. The same argument can be applied to an increase i n the s t r a i n rate. 4.1.2. Ca l c u l a t i o n of A from the 4°K Stres s - S t r a i n Curve. r It can be seen from the above analysis and from the data of figures 7 and 12 that the RR theory predicts the experimental observations of an increase i n s l i p l i n e length with decreasing temperature or increasing s t r a i n rate. I t 57 i s d i f f i c u l t , however, to check the data q u a n t i t a t i v e l y because of the problems i n measuring the structure parameters. If i t i s assumed that a f t e r the quench the s t r a i n increments are so small that x and x do not y v change appreciably as compared to a, then i t i s possible to consider A r as being a function only of a. A quench to 4°K shows that t h i s may be v a l i d f o r x , but i t i s i n c o r r e c t for x . At 4°K A and x can be y v r v calculated since r y and r v are e s s e n t i a l l y equal to zero. The transient following the quench w i l l then be described by equation (9). By measuring the slope of the 4°K s t r e s s - s t r a i n curve (Figure 19), A^ can be c a l c u l a t e d . The increase i n A i s plotted i n Figure 20 f o r a specimen given such a quench. From t h i s curve, the steady state value of A r during room temperature deformation can be found by extrapolating back to zero s t r a i n ( i . e . zero s t r a i n at 4°K). This shows that the r e l a t i v e free area function i s approximately 0.2 at room temperature. Since ( A r ) ^ - 1» the r a t i o between the r e l a t i v e area function at 4°K and 298°K as calculated from the 4°K s t r e s s - s t r a i n curve i s ( Ar) v L>?98 / A / = 0.2 (30) [ r ) 4 The y i e l d strength at room temperature j u s t p r i o r to the quench can be found by extrapolating the 4°K steady state s t r e s s - s t r a i n curve back to zero s t r a i n . Comparing t h i s value of x^ to i t s value at the end of the transient shows that x increases by nearly 14%. This increase y may be s u f f i c i e n t l y large that x^ cannot be taken as a constant. Since a and A^ are known and x^ can be found from equation (14), i t i s possible to c a l c u l a t e x v from equation (10), as shown i n Figure 21. 59 cu H cu cu M fa > •H cu Pi 0.2 0.3 S t r a i n , E (10 - 2) 0.4 0.5 F i g . 20. Increase i n Aj. during 4°K increment. The value of Aj. for the room temperature p r e s t r a i n i s found at e = 0. 60 m o 0.2 0.3 Stra i n , e (10 - 2) 0.4 0.5 F i g . 21. Decrease i n T y during 4°K increment calculated from equation (10), 61 decreases with s t r a i n as predicted, but, since i t changes by at l e a s t one order of magnitude, i t cannot be considered as being a constant i n the t r a n s i t i o n region. In the case of a quench to 4°K, A r i s not a function of the stress alone. Quenching to any temperature appreciably above 0°K makes i t impossible to apply equation (9) because there are no experimental values for the recovery and rearrangement rates so A r cannot be calculated. 4.1.3. Ca l c u l a t i o n of A from S l i p Line Lengths. The r a t i o between the r e l a t i v e free area function at 4°K and 298°K can be calculated from the s l i p l i n e length d i s t r i b u t i o n s using equations developed i n Appendix A. A^ may be determined d i r e c t l y from the average s l i p l i n e lengths by equation (11) . _ JIT nL / i -i \ r 8 A~~ The quantitative use of equation (11) i s questionable, however, because i n writing t h i s equation, i t had to be assumed that the s l i p areas a l l had the same diameter D. I t would be more correct to use equation (31) (see Appendix A). ( 3 D If t h i s equation i s to be used the d i s t r i b u t i o n of s l i p area diameters must be determined from the s l i p l i n e lengths. I t i s possible to do t h i s by a method d e t a i l e d i n Appendix C. The s l i p l i n e length d i s t r i b u t i o n s and the r e s u l t i n g s l i p area diameters calculated f or the specimens deformed at room temperature and 4°K are shown i n Figures 22a and 22b. Because the 62 F i g . 22a. S l i p area diameters calculated from s l i p l i n e lengths, room temperature increment. For c l a r i t y , d i s t r i b u t i o n s are shown as being smooth. 63 64 value of n i n the summation i s not known, an a r b i t r a r y value n' must be chosen. This means that A can only be solved i n terms of the area A , r 1 n' which encloses the n 1 s l i p areas instead of the area A. However, since " D.2 i ' D 2 "-T- (32) . A T A i i n the p a r t i c u l a r value of n' w i l l not influence the value of A r. The summations are c a r r i e d out i n Table I with n' = 100. The values of A can r then be calculated from equation (31), (M = T T — ( 4 - 5 6 x 1 q 6 V2) 0 3 a ) V r / 4 ^ A100 ( A ) = TT~—' (1-01 x 10 6 u 2) (33b) Because the tests are incremental, n and A should be the same for both t e s t s ; therefore i s also the same and i t may be eliminated from equations (33a) and (33b) giving 298 ( Ar) = 0.22 (34) This i s i n good agreement with the A^ r a t i o determined from the 4°K s t r e s s - s t r a i n curve (equation 30). Since the two methods used to cal c u l a t e the r a t i o between A r at 4°K and 298°K are independent of one another, t h i s agreement indicates that the RR theory can q u a n t i t a t i v e l y r e l a t e the r e l a t i v e f ree area function to s l i p l i n e lengths for the temperature change t e s t s . 65 TABLE I n CALCULATION OF Z D? IN EQUATION (31) T (°K) D g (y) D 2 (10 4 y 2) N o N„D? (10 4 y 2 )  s  2 (10 4 s s 25 0.06 0 0.0 50 0.25 0 0.0 75 0.56 2 1.1 100 1.00 5 5.0 125 1.56 8 12.5 150 2.25 12 27.0 175 3.06 18 55.1 200 4.00 24 96.0 225 5.06 14 70.8 250 6.25 8 50.0 275 7.56 6 45.4 300 9.00 5 45.0 325 10.56 6 63.4 Z = 104 Z= 471.3 298 25 0.06 6 0.4 50 0.25 25 6.3 75 0.56 25 14.1 100 1.00 22 22.0 125 1.56 7 11.0 150 2.25 3 6.8 175 3.06 2 6.1 200 4.00 1.5 6.0 225 5.06 2 10.0 250 6.25 2.5 15.6 275 7.56 0 0.0 Z-96 Z-97-9 Normalization of N 2D g to 100 S l i p Areas. 2 _ 100 i 104 4°K- I D =IbT 471'3 <1()V> = 456-2 <i°v>. x 100 I i 298°K. J D 2 = i | | 9 7. 9 ( 1 0 4 u 2) = 1 0 1 . 4 (10 4y 2) 66 4.1.4. S l i p Line Length versus Stress. According to the RR theory, during transient deformation the s l i p l i n e length can increase even i f the obstacle density i s increasing. I t i s possible for t h i s increase i n L because the RR theory states that the s l i p l i n e length w i l l be a function of the stress as w e l l as the structure. From the d e f i n i t i o n of the r e l a t i v e area function (equation 10) i t i s c l e a r that the s l i p l i n e length w i l l increase with stress at constant structure. Although even i n these incremental t e s t s the structure parameters, e s p e c i a l l y T , are not constant, i t has been shown (Appendix B) that increases i n A^ following a quench or a s t r a i n rate increase are due only to an increase i n a. I t i s to be expected that two specimens given d i f f e r e n t types of increments which r e s u l t i n the same s l i p l i n e length changes should have nearly the same changes i n s t r e s s . Eventhough a true quantitative comparison between a and L i s not possible because the structure parameters are not constant, the experimental values of these v a r i a b l e s can be p l o t t e d against one another to see i f the q u a l i t a t i v e predictions of the RR theory are found. This i s done i n Figure 23 f o r the temperature and the s t r a i n rate change t e s t s . This curve shows that the temperature change experiments are i n good agreement with the RR theory; the s l i p l i n e length increases with increasing s t r e s s . However, the s t r a i n rate data does not follow the same q u a l i t a t i v e pattern. The large changes i n L f o r the s t r a i n rate tests are inconsistent with the small changes i n s t r e s s . The RR theory cannot explain why the s l i p l i n e lengths are more s e n s i t i v e to the stress f o r the s t r a i n rate t e s t s . 68 4.2. SLIP LINE LENGTH IN THE TRANSIENT As predicted by the RR theory, s l i p l i n e s have been found to increase i n length during the transient a f t e r a quench (fig u r e 15). By considering the form of the r e l a t i v e area function and the way and change i n the transient, i t has been concluded that the continuous increase i n L a r i s e s from an increasing a. This i s further i l l u s t r a t e d i n Figure 24 which shows the s l i p l i n e length increasing i n the transient as a function of a. At the end of the tran s i e n t , at approximately e = 0.4%, the s l i p l i n e length decreases because of the a f f e c t described by Hirsch and Seeger. A f t e r the transient, the stress i s not increasing f a s t enough to overcome the increase i n the obstacle density. In the context of the RR theory, the decrease i n L i s explained by a decreasing A i n equation (11). The theories of Hirsch and Seeger can account f o r the decreasing L, but they are not able to predict the i n i t i a l rapid increase because they r e l a t e the s l i p l i n e length only to the obstacle density and not to the st r e s s . Although the s t r e s s - s t r a i n curves for a 77°K increment and the composite curve for the serie s of s t r a i n d i v i s i o n s (Figure 25) shows that both specimens hardened i n a s i m i l a r manner, the maximum value of L f o r the s t r a i n d i v i s i o n s was found to be appreciably l e s s than that for a sin g l e increment at 77°K. This discrepancy i s expected because the specimens had to be warmed a f t e r each s t r a i n d i v i s i o n i n order to measure the s l i p l i n e lengths. While the specimens were at room temperature they experienced a c e r t a i n amount of recovery and rearrangement that would not have occured i f the 77°K s t r a i n were continuous. This resulted i n the formation of areas of decreased obstacle density. When the specimen was 69 Fig. 24. Slip line length versus stress during transient deformation 77°K. T 1 1 1 1 r O S t r a i n Increment A O S t r a i n D i v i s i o n 0.2 0.4 1.0 1.2 1.4 0.6 0.8 S t r a i n , e (10~ 2). F i g . 25. S t r e s s - s t r a i n curves for incremental and " s t r a i n d i v i s i o n " deformation, 77°K. o 71 reloaded at 77°K, p l a s t i c deformation took place at a stress l e v e l lower than the specimen supported at the end of the previous increment. This decrease i n stress w i l l cause the s l i p l i n e s i n the subsequent s t r a i n d i v i s i o n to be shorter. The f a c t that the unwanted recovery and r e -arrangement d i d not cause an o v e r a l l lowering of the s t r a i n d i v i s i o n s t r e s s -s t r a i n curve r e l a t i v e to that f or an increment of s t r a i n can be explained i n terms of "meta-recovery" [Cherian et a l . 1948, Alden 1972b]. During the i n i t i a l stages of annealing, there i s a greater tendency for the s t r e s s - s t r a i n curve a f t e r annealing to show a t r a n s i t i o n region than for i t to become lower i n the steady state region. 4.3. PRESTRAIN AT 77°K The specimen given the 77°K p r e s t r a i n was loaded to the same stress l e v e l as that found a f t e r a room temperature p r e s t r a i n and a quench to 77°K. Any diffe r e n c e i n s l i p l i n e length for these two specimens w i l l be due to the structure parameters x and x . y v I t i s possible to show that x i s also the same f o r both y treatments i f the Cottrell-Stokes law applies. This law states that the stress measured following a quench divided by the stress measured before the quench i s independent of the s t r a i n . Although no d i r e c t tests were ca r r i e d out to check t h i s law, i t can be assumed that t h i s " r e v e r s i b l e flow stress r a t i o " i s independent of s t r a i n because of the r e s u l t s presented i n Figure 11. This graph compares the r e v e r s i b l e flow stress changes of M i t c h e l l [1964] to those of the present work. These two sets of data were c o l l e c t e d from specimens that underwent d i f f e r e n t amounts of s t r a i n , but the dependence of t h e i r stress changes on temperature are very nearly 72 the same. This means that two specimens with the same yield stress at one temperature will have the same yield stress at 0°K regardless of their prior histories. According to the RR theory these specimens have identical Ty 's because the yield strength is numerically equal to the yield strength measured at 0°K. Two identical specimens strained to the same x at different y temperatures will not have the same value for x^ because at the higher temperature recovery and rearrangement are producing local decreases in obstacle density at a faster rate than at the lower temperature. The high temperature specimen will have more of a variation in local yield strength x a nd therefore a larger x^. This is shown schematically in Figure 26 for two specimens strained to the same x at temperatures T and T„, y^ J- ^ where T- > T„: their values of x will be x and x respectively. 1 2 ' v v v T Tl T2 If the relative area function is to be a function only of x , it is necessary to have the same values of (x - a) for both the high and low temperature samples. The 298°K specimen should not be prestrained to a x of x = x . I t must be prestrained such that on quenching to y y298 y77 77°K and giving an increment of strain it would have the same value of a, and therefore the same x , as the 77°K prestrain sample. Since it has been shown that x^ decreases after a decrease in temperature, x^ for the 298°K prestrain sample would be less after the increment at 77°K. It is unlikely, however, that its value or x will be as small as x after the increment. v77 Since after the 77°K increment (x - a) was the same for the , y 298°K and 77°K samples, any difference in L must be attributed to a > difference in x^. As mentioned above, the value of x^ should be greater for the 298°K prestrain. Since Ar is larger for larger values of x , the 73 74 RR theory predicts that the s l i p line lengths should be longer for the 298°K prestrain. This i s good agreement with the experimental results that the incremental test for the specimen prestrained at 77°K had an average sl i p line length of lOOy while L for the 298°K prestrain was 140u. 4.4. MAXIMUM SLIP LINE LENGTH One of the basic ideas of the RR theory i s that the distribution of obstacles i s not regular. As a result, the s l i p plane does not have a uniform strength and plastic deformation w i l l take place only in the softer areas where the obstacle density i s the least. The fraption of the s l i p plane yielding i s indicated by the value of the relative area function. At room temperature A r was calculated to be 0.2 (Figure 20). However, this does not mean that the same 20% of the sl i p plane continues to yield indefinately. The original soft regions w i l l strain harden and plastic deformation w i l l begin in "newer" soft areas. Although the actual location of deformation is constantly changing, at any given time, at room temperature, 20% of the s l i p plane w i l l be yielding p l a s t i c a l l y . According to the RR theory, the obstacle distribution i s not stable with respect to time and temperature; through rearrangement and recovery, local decreases in the obstacle density are produced. During deformation above 0°K, the formation of soft spots enables the solid to yield continuously at a stress less than T . Dislocation loops expanding as a result of the applied stress w i l l be able to sweep out only those areas where the local yield strength is less than a. Since recovery and rearrangement require thermal activation, r^ and r^ w i l l decrease with decreasing temperature and fewer soft areas w i l l be produced. At lower 75 temperatures the s l i p plane w i l l have a more uniform d i s t r i b u t i o n of obstacles because the s o f t e s t spots are s t r a i n hardening and are not being replaced at the same rate by thermally activated processes. Therefore, a larger f r a c t i o n of the s l i p plane w i l l have a x l e s s or equal to a and yL w i l l be larger at lower temperatures. At 0°K r and r are equal to zero and new soft areas are not v y created. A specimen which was deformed at T > 0°K and then quenched to 0°K w i l l p r e f e r e n t i a l l y harden i t s s o f t e r areas during the region of transient deformation. I d e a l l y , when the steady state i s reached the s l i p plane w i l l have a uniform strength (a = T ) and d i s l o c a t i o n loops w i l l be able to expand over the e n t i r e s l i p plane. This does not imply that the average s l i p l i n e lengths w i l l be on the order of the specimen s i z e . There w i l l be a maximum value of L determined by the number of areas n undergoing s l i p i n s i d e the area A. During steady state deformation at 0°K A =1; from equation (11) the maximum average s l i p l i n e length w i l l be (35) 4.5. THE SLOPE OF THE STRESS-STRAIN CURVE The s t r e s s - s t r a i n curve for f . c . c . metals i s often divided into three stages [Clarebrough and Hargreaves 1959]. There i s , however, no generally accepted t h e o r e t i c a l explanation of the shape of the curve and even the experimental data are contradictory. Stage I, which has not been discussed i n t h i s paper, i s a l i n e a r region characterized by a low s t r a i n hardening rate. Stage II i s the most studied and also the most 76 controversial. The majority of the previous works suggest that i t is linear with a high strain hardening rate, 6 ^, that is relatively temperature insensitive [Clarebrough and Hargreaves 1959, Noggle and Koehler 1957]. Other data shows that there is no linear stage II and that i s quite dependent on temperature [Hosford et a l . i960]. In stage III the work hardening rate i s not constant but decreases with increasing stress and the temperature and strain rate dependence are greater than i n stage II [Kocks et a l . 1968]. The extent of stage II depends on the temperature and the particular type of crystal tested. At room temperature copper has a definite linear region while with aluminum only stage I and III are present. Stage II in aluminum i s not well developed above 100°K [Honeycombe 1968]. The tests conducted on the series of specimens with identical structures provided an experimental method to check the above generalize ations. As expected, the prestrain at room temperature resulted iri a , typical stage III curve. However, the slope of the subsequent strain increments shown in Figure 9 are not in agreement with the concept of a temperature independent work hardening rate in stage II. If 0 ^ is not a function of temperature then the slope of the stress-strain curve should vary according to Figure 27 for the incremental tests. Below some c r i t i c a l temperature, T , which would depend on structure, 4^ should be equal to c de 8- while above T £ i t would decrease when stage III predominated. The observed temperature dependence is consistent with the RR theory and can be explained by the steady state value of equation (20). Similar tests on copper over the same temperature range show a strain hardening rate that is not a function of temperature [Garner 1973], This i s partially explained by the fact that the deformation of aluminum 77 b to 0) > u u c •H ca 4J to I CO CO <D M 0) P. o Temperature F i g . 27. Change i n the slope of the s t r e s s - s t r a i n curve versus temperature as predicted by the concept of a temperature i n s e n s i t i v e stage I I . Below T the slope i s equal to 78 at 77°K i s equivalent to that of copper near room temperature. The r e l a t i v e change i n ^— f o r aluminum between 0°K and 77°K should then be de approximately the same as found i n copper between 0°K and room temperature. Equation (20) also states that the slope of the steady state s t r e s s - s t r a i n curve should be strongly dependent on the s t r a i n rate. Experimentally, the v a r i a t i o n i n slope f o r the s t r a i n rate change increments was l e s s than would be expected from equation (20). 79 S U M M A R Y The s t r a i n hardening theories of Hirsch [Hirsch and M i t c h e l l 1967] and Seeger [Seeger et a l . 1963] r e l a t e the s l i p l i n e length, L, only to the density of the obstacles that can block s l i p . These theories state that the increase i n stress following a quench or an increase of s t r a i n rate w i l l not r e s u l t i n an increase i n L. A recent theory, the RR theory [Alden 1972a] predicts that such a stress increase w i l l cause the average s l i p l i n e length to become la r g e r . This i s explained i n terms of the r e l a t i v e free area function A which indicates the f r a c t i o n of the r s l i p plane that i s accessible to expanding d i s l o c a t i o n loops. This f r a c t i o n i s dependent on the stress and two structure parameters, x , which i s r e l a t e d to the o v e r a l l obstacle density, and x , which depends on the magnitude of the v a r i a t i o n i n obstacle spacing. This work was c a r r i e d out to check the RR theory's p r e d i c t i o n that L increases with increasing s t r e s s . A s e r i e s of aluminum sin g l e c r y s t a l s with i d e n t i c a l structures were given an increment of s t r a i n over a wide range of temperatures and s t r a i n rates. I t was found that L did change as predicted but that i t was more s e n s i t i v e to stress f o r the s t r a i n rate changes than f o r the tempera-ture changes. This l a t t e r a f f e c t could not be explained by the RR theory. The r a t i o of the r e l a t i v e free area at 4°K and 298°K was determined from two independent methods. The f i r s t involved measurements 80 of the slope of the stress-strain curve during the transient deformation after a quench to 4°K. The change in A r from the value i t had during the room temperature prestrain could be calculated by dividing the slope of the steady state stress-strain curve by the slope of the transient (equation (9)). The second method determined the ratio by measuring the sli p line lengths at 4°K and 298°K. From this data, the distributions of sl i p area diameters and then the relative area ratio could be calculated (equation (31)). The ratios calculated from these two methods agreed favorably. For several specimens the increments of strain were not given continuously; the strain was applied in smaller quantities called "strain divisions". In this way, the sl i p line length was measured as i t increased with the increasing stress through the transient region. This showed that L did not increase abruptly when the temperature was lowered. The strain divisions that were applied after the steady state had been reached showed that following the region of transient deformation the sli p line length gradually decreased. This decrease in L can be explained by the RR theory as well as by the theories of Hirsch and Seeger. The RR theory states that during the steady state A r is constant, but due to an increasing obstacle density, the area A in equation (11) decreases giving a decrease in L. The dependence of the sl i p line length on was investigated by prestraining a specimen at 77°K. This specimen was loaded to such a level that the value of (T - cr) in equation (10) was the same as that for the specimen given a room temperature prestrain and an increment at 77°K. According to the RR theory, the low temperature prestrain should result in a smaller and therefore shorter sl i p lines. The experimental results confirmed this prediction. 81 The steady-state slope of the incremental s t r e s s - s t r a i n curve was found to be strongly dependent on temperature i n disagreement with the idea of a temperature i n s e n s i t i v e stage I I work hardening rate. Equation (20) of the RR theory suggests that the slope should be a strong function of temperature and s t r a i n rate. The experiment could not detect any regular dependence of the slope on the s t r a i n rate. 82 C O N C L U S I O N S 1. The s l i p l i n e length, L, i s not a function only of the obstacle density. An increase i n stress following a decrease of tempera-ture or increase of s t r a i n rate w i l l cause the s l i p l i n e length to increase. This a f f e c t i s predicted by the RR theory through a change i n the r e l a t i v e area function, A^, but cannot be accounted f o r by the theories of Hirsch and Seeger or the recovery theory. 2. The r e l a t i v e area function seems to have a quan t i t a t i v e s i g n i f i c a n c e experimentally. The r a t i o of A r at 0°K and 298°K determined from the slope of the 4°K s t r e s s - s t r a i n curve was i n good agreement with the r a t i o calculated from the s l i p l i n e length data. 3. The s l i p l i n e length was larger f o r a given change i n stress for the s t r a i n rate t e s t s than f o r the temperature t e s t s . The RR theory could not account f o r t h i s d i f f e r e n c e i n s e n s i t i v i t y to st r e s s . 4. By c l o s e l y examining the transient region of the s t r e s s - s t r a i n curve following a decrease i n temperature, i t was found that the s l i p l i n e s r a p i d l y increased i n length during transient deformation i n agreement with the RR theory. After the transient L decreased gradually; t h i s a f f e c t could be explained by the theories of Hirsch and. Seeger and the RR theory. 83 5. The s l i p line length was found to be shorter i f the prestrain temperature was decreased. This was predicted by the RR theory in terms of the structure parameter T . 6. The temperature dependence of the strain hardening rate did not agree with the concept of stage II hardening. The RR theory could explain the strong dependence of the slope of the stress-strain curve on temperature through a change in the recovery and rearrangement rates. The slope did not show a strain rate dependence predicted by the RR theory. 84 SUGGESTIONS FOR FUTURE WORK The present work ind i c a t e s that the RR theory's q u a l i t a t i v e predictions of an increase i n s l i p l i n e length during transient defor-mation due to " r e v e r s i b l e " changes i n stress are i n agreement with the experimental r e s u l t s . However, the increases i n L were more s e n s i t i v e to stress f o r the s t r a i n rate t e s t s than f o r the temperature t e s t s . Because the methods of t h i s study were not i d e a l l y suited for measuring small r e v e r s i b l e changes In s t r e s s , i t may be u s e f u l to measure the stress changes independent of s l i p line'measurements. The r e l a t i o n between L and a could be determined more accurately than i t was i n the present work. Such studies could also be used to c o l l e c t more data on the s t r a i n rate dependence of the slope of the s t r e s s - s t r a i n curve. It would be possible to do further experiments t e s t i n g the quantitative r e l a t i o n s h i p between the r e l a t i v e free area function and s l i p l i n e lengths. In the present work was measured only at the p r e s t r a i n temperature, 298°K. By varying the p r e s t r a i n temperature and doing 4°K increments, A^ could be measured at any temperature. These values of A could then be compared to the A^ r a t i o calculated from s l i p l i n e length data to see i f the quantitative predictions of the RR theory are correct f o r temperature changes as suggested by the present work. From the methods used i n t h i s study, the v a r i a t i o n i n A^ with s t r a i n at a constant temperature could be measured. This would be done by giving d i f f e r e n t amounts of p r e s t r a i n before the 4°K increment. This could be combined with s l i p l i n e studies to determine the v a r i a t i o n i n •7- with s t r a i n . 85 APPENDIX A CALCULATION OF THE RELATIVE FREE AREA FROM THE SLIP AREA DIAMETER AND THE AVERAGE SLIP LINE LENGTH The s l i p area diameter D can be re l a t e d to the r e l a t i v e area function A f by considering the f r a c t i o n a l area a that i s accessible to expanding d i s l o c a t i o n s within area A. A r i s written as A = f . (36) r A If there are n act i v e s l i p areas i n s i d e area A, then a can be written a = a^ + + a^ ... + a n , (37a) or n a If the s l i p areas are assumed to be c i r c u l a r with a diameter th D., then the i area i s l a ± = J D 2 (38) Substituting t h i s into equation (37b) gives n a = J £ D 2 (39) i 86 The r e l a t i v e area function can then be written * D 2 Although t h i s equation w i l l be used to c a l c u l a t e the r e l a t i v e area from the s l i p area diameters, i t would be u s e f u l to q u a l i t a t i v e l y r e l a t e A^ to the average s l i p l i n e length. To do t h i s i s must be assumed that the s l i p areas are a l l the same s i z e with a diameter D. Equation (39) can then be written a = j nD 2 (40) Substituting t h i s into equation (36) gives -2 A = 7 n f . (41) r 4 A I t can be shown that the average s l i p l i n e length L r e s u l t i n g from n random i n t e r s e c t i o n s of s l i p areas of diameter D with the surface w i l l be [Fullman 1953] L = N3 Therefore, equation (41) i s written 4 D • (42) \--r4- w The assumption that a l l s l i p areas have the same s i z e w i l l n a t u r a l l y l i m i t the quantitative value of equation (11), but i t w i l l be useful i n q u a l i t a t i v e discussions. 87 APPENDIX B INCREASE IN THE RELATIVE AREA AFTER A QUENCH The purpose of t h i s appendix i s to show that during transient deformation a f t e r a quench 1) the r e l a t i v e area function increases due to an increase i n stress and 2) that a change i n x^ cannot keep A^_ constant. 1) The steady state equations describing p l a s t i c flow at T 1 are (28a) (28b) Quenching to a lower temperature T^ does not instantaneously change the structure or the s t r e s s , so immediately a f t e r the quench (29a) (29b) Subtracting equation (28b) from (28a) and (29b) from (29a) gives 88 ( £ - ^ ) 2 - y ( ^ - ) - e (42b) Subtracting equation (42a) from (42b) gives / da _ d T y \ _ , _ ^ , . \de de / 2 \de de J e ^ v 1 A d T V (43) Since rearrangement i s thermally activated r v ^ > r v ^ so / da \ , J d x s _ Z \ > /da y_\ de J2 \de de J (44) Therefore, quenching increases the value of -rr- r e l a t i v e to dT E -r-^- . The e f f e c t of t h i s on A can be seen from equation (17). de r dA A r r de T v dx . ,dx . (17) At T^ the deformation was steady state, so from equation (17) dA de 1^ x 1 v /da f^Xv\ [dl ~ de )1 " l n Av± \ dr)1 = 0 (45a) Immediately a f t e r the quench dA i de i s written 89 (45b) If the terms containing x^ are neglected, then the inequality of equation (44) indicates that ,dA » /dA i [df)2 > \if)1 (46) This shows that the rapid rise in stress during transient deformation after a quench is responsible for the increase in A . 2) The change in x^ i s now considered. Before the quench the change in x^ with strain was, from equation (22), ,dx > . v-m _ e + ,de ^ v 1 e (47a) /d T\ 6 v w i l l not change abruptly on quenching, so f^T" w i l l be ,dx > + —r= e (47b) Subtracting equation (47b) from (47a) gives 4x . . dx i de y, V de L e v, v ' (48) Since r > r v l V2 90 ,dx . /dx x ( s V l a r ) , • <«> dx The decrease i n w i l l tend to decrease (equation (17)) dx because -In A i s p o s i t i v e . It can be shown that t h i s decrease i n j — r r de cannot be s u f f i c i e n t l y large that A w i l l remain constant. This i s done dA . .dA . by f i r s t assuming that J = j and then checking i f t h i s i s consistent with a quench. ,dAx fdAr\ If j = } then immediately a f t e r the quench equation (17) shows ( H - £ V t a \ ( £ V ( £ - ^ 2 - > « \ . ( £ ) 2 • ( 5 0 > Rearranging t h i s gives do d T dcr d T " d T d T (de ~ d i ^ L " (de " d e ^ l = " l n A r , (de^), " (dT") '2 '1 "1 1 w " '2 (51) S u b s t i t u t i n g equations (43) and (48) into equation (51) * (r - r )= - l n A * (r - r ) . (52) e v 1 v 2' r 1 e v 1 v2 Since - l n A / 1 because A i s not n e c e s s a r i l y —, equation r r^ e • n (52) states that i f the r e l a t i v e area i s to remain constant then r v ^ = r V 2 « It i s not possible for the rearrangement rates to be the same i f there was a quench so A must change. Therefore, the change i n x^ following a quench cannot keep A^ at the same value i t had p r i o r to the quench. 91 APPENDIX C CALCULATION OF SLIP AREA DIAMETERS FROM SLIP LINE LENGTHS The c a l c u l a t i o n of the s l i p area diameters from the s l i p l i n e lengths i s based on the diameters having a stepwise instead of smooth d i s t r i b u t i o n . The s l i p areas are divided into groups each with a diameter that i s some multiple of the smallest group diameter. In t h i s case, the smallest diameter i s 25u and there are 13 groups. The number of s l i p areas i n each group i s determined f i r s t f o r the largest group and then f o r each progressively smaller group. The number of s l i p areas i n the group with diameter D g i s found by c a l c u l a t i n g the p r o b a b i l i t y P g fc that a s l i p area with diameter Dfc w i l l i n t e r s e c t the surface giving a s l i p l i n e of length L . The p r o b a b i l i t y p S S 5 w i l l be equal to the p r o b a b i l i t y that a random plane i n t e r s e c t i n g a c i r c l e F i g . 28. Cal c u l a t i o n of the p r o b a b i l i t y that a random plane w i l l i n t e r s e c t a c i r c l e of diameter D t giving a chord length of L s ± AL. 92 of diameter Dfc w i l l give a chord length of L g ± AL (Figure 28). This p r o b a b i l i t y w i l l be [Underwood 1968] P s , t = rT72 ' ( 5 2 ) I f there are Nfc s l i p areas with diameter Dfc, then the number N(s,t) of s l i p l i n e s with length L due to these N areas w i l l be s t N(s,t) = p c . (53) If there are T groups of s l i p diameters, then the t o t a l number N(s) of s l i p l i n e s with length L w i l l be a sum a l l the N(s,t) terms s where D > D . t s T N ( S ) = \ X P s , i N i • ( 5 4 ) i = s Since the longest s l i p l i n e s of length L T can come only from the s l i p area group with diameters D^ ,, there w i l l only be one term i n the summation: N(T) = p N T (55) The number N(T) of s l i p l i n e s with length L^ i s experimentally found and the p r o b a b i l i t y p i s calculated from equation (52), so the -I- > -l-number of s l i p areas with diameter D^ , w i l l be N - . (56) T. D T,T The number N^ , ^  of s l i p areas i n the next smaller group can then be determined from equation (54), 93 N(T-l) = p ^ ^ N T_ X + p T _ 1 > T N T . (57) The only unknown i n t h i s equation i s NT_^. In t h i s manner N g can be calculated f o r a l l the groups. 94 B I B L I O G R A P H Y Alden, T.H., 1972a, P h i l . Mag., 25, 785. Alden, T.H., 1972b, to be published, Met. Trans. Alden, T.H., 1973, to be published. Barber, D.J., 1962, P h i l . Mag., 7_, 1925. Basinski, Z.S., 1959, P h i l . Mag., 40, 393. Brown, A.F., 1951, J . Inst. Metals, 80, 115. Cherian, T.V., Pietrokowsky, P., and Dorn, J.E., 1949, Trans. A.I.M.E., 185, 1948. Clarebrough, L.M. , and Hargreaves, M.E., 1959, Proc. Met. Phys. , 8^, 1. C o t t r e l l , A.H. , and Stokes, R.J., 1955, Proc. R. S o c , A, 233, 17. DeHoff, R.T., 1968, Quantitative Microscopy, McGraw-Hill, New York, 12. Fisher, R.M., and L a l l e y , J.S., 1967, Can. J . Phys., 45, 1147. Fourie, J.T., 1968, P h i l . Mag., 17, 735. Fourie, J.T., and Wilsdorf, H.G.F., 1959, Acta Met., ]_, 339. F r i e d e l , J . , 1964, D i s l o c a t i o n s , Addison-Wesley, Reading, 211. Fullman, R.L., 1953, Trans. A.I.M.E., 197, 447. Garner, A.W., 1973, Ph.D. Thesis, Univ. of B r i t i s h Columbia. Garofalo, F., 1965, Fundamentals of Creep and Creep Rupture i n Metals, McMillan, New York. Heindrich, R.D., and Shockley, W., 1947, J . Appl. Phys., 18, 1029. Himstedt, N. , and Neuhauser, H. , 1972, Scrip t a Met., 6_, 1151. Hirsch, P.B., 1964, Dis. Faraday S o c , 38, 111. Hirsch, P.B., and Hazzledine, P.M., 1967, P h i l . Mag., 15_, 121. Hirsch, P.B., and M i t c h e l l , T.E., 1967, Can. J. Phys., 45_, 663. Honeycombe, R.K.W., 1951, J . Inst. Metals, 80, 49. Honeycombe, R.K.W., 1968, The P l a s t i c Deformation of Metals, Arnold, London. 95 Hosford, W.F. J r . , F l e i s c h e r , R.L., and Backofen, W.A., 1960, Acta Met., 8, 187. Howie, A., 1962, Direct Observations of Imperfections i n C r y s t a l s , Wiley New York, 283. Kocks, U.F., 1966, P h i l . Mag., 13, 541. Kocks, U.F., Chen, H.S., Rigney, D.A., and Schaefer, R.J., 1968, Work  Hardening, Gordon & Breach, New York, 151. Kamm, G.N., and Ale r s , G.A., 1964, J . App. Phys., 35, 327. Maddin, R., and Chen, N.K., 1954, Prog. Met. Phys., _5, 53. Mader, S., 1963, Electron Microscopy and Strength of Cr y s t a l s , Wiley, New York, 183. McLean, D., 1968, Trans. A.I.M.E., 242, 193. M i t c h e l l , T.E., 1964, Prog. App. Mat. Res., 6_, 119. Noggle, T.S., and Koehler, J.S., 1957, J . App. Phys., 2_8, 53. Orowan, E., 1946-47, J . W. Sc o t l . Iron Steel Inst., 54, 45. Saada, G., 1963, Electron Microscopy and Strength of C r y s t a l s , Wiley, New York, 651. Seeger, A., 1956, Dis l o c a t i o n s and Mechanical Properties of Cr y s t a l s , Wiley, New York, 243. Seeger, A., 1963, The Relation Between the Structure and Mechanical  Properties of Metals, H.M.S.O., London, 3. Seeger, A., Mader, S., and Kronmuller, H., 1963, Electron Microscopy  and Strength of Cr y s t a l s , Wiley, New York, 665. Segall, R.L., and Partridge, P.G. , 1959, P h i l . Mag., 4L, 912. Simmons, G., and Wang, H., 1971, Single C r y s t a l E l a s t i c Constants and  Calculated Aggregate Properties: A Handbook, The M.I.T. Press, Cambridge. Swann, P.R., 1963, Electron Microscopy and Strength of Crysta l s , Wiley New York, 131. T i e t z , T.E., Meyers, C.L., and Lytton, J.L., 1962, Trans. A.I.M.E., 224, 339. Underwood, E.E., 1968, Quantitative Microscopy, McGraw-Hill, New York, 149. Weinberg, F., 1968, Trans. A.I.M.E., 242, 2111. Yamaguchi, K., 1928, S c i . Papers Inst. Phys. Chem. Research (Tokyo), 8^, 289. 

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