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

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MEASUREMENT AND STATISTICAL INTERPRETATION OF SLIP LINE LENGTH AND MICROSTRAIN IN COPPER SINGLE CRYSTALS by ANDREW GARNER B . S c , U n i v e r s i t y of L i v e r p o o l , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Depa r tment of METALLURGY We accep t t h i s t h e s i s as c on f o rm ing to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 197^ In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada ABSTRACT In order to t e s t the apparently c o n f l i c t i n g predic-tions of some current theories of s t r a i n hardening, s l i p l i n e length measurements were made on a series of oriented copper single c r y s t a l s , i d e n t i c a l l y prestrained at 673°K, polished and incrementally strained at temperatures between 573°K and 4.2°K; s l i p l i n e s formed during low temperature increments were found to be longer than those formed during s t r a i n increments at higher temperature (Garner and Alden, 1974). The r e s u l t i s shown to be i n c o n f l i c t with any theory of s t r a i n hardening i n which s l i p l i n e s are blocked by s p e c i f i c obstacle configurations, such as Lomer-Cottrell b a r r i e r s , ribbons of converted pile-ups or d i s l o c a t i o n c e l l walls. In contrast, the r e s u l t i s shown to be consistent with theories of s t r a i n hardening i n which s l i p l i n e s are blocked by s t a t i s t i c a l i n t e r a c t i o n between expanding g l i d e loops and f o r e s t d i s l o c a -tions, on the condition that, within the framework of such a theory, the g l i d e loops are able to expand athermally over a newly available free area of s l i p plane, aft e r a thermally activated process. Two possible thermally activated processes are discussed. A u n i f i e d view of s l i p l i n e s properties i s presented which i s shown to provide a s e l f - c o n s i s t e n t explana-ti o n of the temperature v a r i a t i o n of s l i p l i n e length, s l i p band formation, the existence of multipole carpets and the v a r i a t i o n of flow stress with temperature. The s t a t i s t i c a l aspects of t h i s i n t e r p r e t a t i o n were investigated further by obtaining 77°K microstrain curves from a series of oriented copper single c r y s t a l s , prestrained at temperatures between 1000°K and 77°K, to produce d i s l o c a t i o n microstructures with d i f f e r i n g degrees of r e g u l a r i t y , yet with approximately the same o v e r a l l density. The f o r e s t d i s l o c a t i o n microstructures of an i d e n t i c a l l y prepared series of c r y s t a l s were examined using a d i s l o c a t i o n etch on the primary s l i p plane. A s t a t i s t i c a l sampling technique was devised, which was used to measure l o c a l d i s l o c a t i o n d e n s i t i e s . In addition, new parameter i s introduced, namely the r a t i o of the sampled standard deviation, to mean l o c a l d i s l o c a t i o n density, which quantifies the degree of r e g u l a r i t y of a d i s l o c a t i o n micro-structure. A l l microstructures were found to have a smaller degree of r e g u l a r i t y than a random d i s t r i b u t i o n . For c r y s t a l s prestrained at temperatures above 293°K, at any given f r a c t i o n of the 77°K y i e l d s t r e s s , the amount of microstrain was found to increase as the microstructures became less regular. Crystals prestrained at and below 293°K exhibited the Haasen-Kelly e f f e c t , which was attributed to r e s t r i c t e d source operation. However, once sources begin to operate, the amount of microstrain anticipated from the degree of r e g u l a r i t y was indeed detected. TABLE OF CONTENTS Page ABSTRACT . i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . x i i i PART 1 SLIP LINE STUDIES .1 1.1 Introduction . . . 1 1.2 Experimental Techniques 5 1.2.1 Specimen Preparation. . . . . . . . . 5 1.2.2 Mechanical Testing. . . 10 1.2.3 S l i p Line Measurement 14 1.3 Results 19 1.3.1 S l i p Line Lengths . . . . . . . . . . 19 1.3.2 Va r i a t i o n i n Flow Stress with Temperature 31 1.3.3 S l i p Line Density 31 1.3.4 S l i p Step Height. . . . . . . . . . . 33 1.3.5 Sources of Error 33 iv Part 1.4 Discussion . . . . . . . . . . . . . 1.4.1 S l i p Line Blocking by S p e c i f i c Dislocation Arrays. . . . . . . 1.4.2 Flow Stress and the Glide/ Forest Interaction. . . . . . 1.4.3 S t a t i s t i c a l Blocking of S l i p Lines 1.4.4 A Unified View of S l i p Line Formation, Microstructural Observations and Flow Stress. 1.4.5 Tra n s i t i o n E f f e c t s . MICROSTRAIN AND ETCH PIT STUDIES . 2.1 Introduction . . 2.2 Experimental Technique . . 2.2.1 Specimen Preparation 2.2.2 Crystal Orientation 2.2.3 Mechanical Testing 2.2.4 Metallography . . .2.3. Results. 2.3.1 Microstrain Curves 2.3.2 Metallography . . 2.3.3 Average Dislocation Density and Flow Stress . 2.3.4 Local Dislocation Densities . 2.3.5 The E f f e c t of Subgrain Boundar on the Di s t r i b u t i o n s . . . . . 2.3.6 • Reproducibility es Page 36 36 52 l 55 71 76 78 78 81 81 81 83 86 97 97 109 121 121 136 137 v Part Page 2.4 Discussion 137 2.4.1 Introduction 137 2.4.2 Crystals with Prestrains above 293°K . . . . . . 138 2.4.3 Quantitative Estimates from Theories of Microstrain . 141 2.4.4 Crystals Prestrained at 293°K and Below 150 SUMMARY AND CONCLUSIONS 153 REFERENCES 157 APPENDICES 1 Calculation of Work Hardening Rate i f S l i p Lines are Blocked by C e l l Walls. 163 2 Definitions of Terms Used to Describe Deformation . . . . . . . 166 3 Details of Microstrain Testing 170 4 Optical Microscopy with Normarski Interference Contrast* . . 175 5 A Probabil i t y Model for Random Distributions . 17 8 v i LIST OF TABLES Table ' . Page 1 Pairs of S l i p Systems which could ' Produce Lomer-Cottrell Locks . . . . . . . . . . . 4 0 2 Dislocation Etch Compositions 92 3 Area Sizes used to Sample Dislocation Densities. 95 v i i LIST OF FIGURES Figure . Page 1 Graphite mold assembly used to grow single c r y s t a l s . . . . . . . . . . . . . . . . . . . . . 7 2 ,. Orientation of t e n s i l e axis for cr y s t a l s used i n s l i p l i n e work . . . . . . . . . . . . . . . 9 3 673°K prestrain curve. . . . . . . . . . . ."• . . . 13 4 Acetone reflux apparatus used i n preparation of r e p l i c a s . . . . . . . . . 20 5 Histogram of s l i p l i n e lengths aft e r 0.5% incremental s t r a i n at 573°K. . . . . . . . . . . . 22 6 Histogram of s l i p l i n e lengths aft e r 0.5% incremental s t r a i n at 293°K. . . . . . . . . . . . 23 7 Histogram of s l i p l i n e lengths a f t e r 0.5% incremental s t r a i n at 190°K. . . . . . . 24 8 Histogram of s l i p l i n e lengths aft e r 0.5% incremental s t r a i n at 77°K . . . . . . 25 9 Histogram of s l i p l i n e lengths aft e r 0.5% incremental s t r a i n at 4.2°K. . . . . . . . . . . . 26 10 Average s l i p l i n e length versus temperature of incremental s t r a i n i n g . . . . . . . . . . . . . 27 11 Optical micrographs showing change i n s l i p l i n e length with temperature . . . . . 28 v i i i Figure Page 12 Electron micrographs of s l i p l i n e s formed after s t r a i n increments at (a) 4.2°K and (b) 293°K respectively. . . . . . . . . . . . . . 3 0 13 Histogram of s l i p l i n e density, l i n e s formed at 293°K . ... . . ... . . . . . . . . . . . 32 14 Shadowed r e p l i c a with carbon c a l i b r a t i o n ; :. spheres . . .... . . . . . . . . . . . . . . 34 15 Transition and steady state s t r a i n observed during incremental t e s t at 4.2°K. . 37 16 S l i p l i n e blocking according to Seeger (1957) . . 41 17 S l i p l i n e blocking according to Hirsch and M i t c h e l l (1967) ... . . . . . . . . . . . . . . 45 18 Average d i s l o c a t i o n c e l l diameters i n copper versus normalized flow stress . . . . . . . . . . 48 19 . A v e r a g e s l i p l i n e length and average c e l l diameter i n copper . . . . . . . . . . . . . 50 20 . Diagrams from Hocks' o r i g i n a l s t a t i s t i c a l . analysis (1966) . . . . . . . . . . . . . . . . . 58 21 Schematic of primary s l i p plane showing . s t a t i s t i c a l blocking of s l i p l i n e s during steady state flow . . ... . . . . . . . . . . . . . 61 22 Data from analysis of Morris and Klahn ( 1 9 7 4 ) . . . . . . . . . . . . . . . . . . . . . . 68 23 Cross-section perpendicular to primary gl i d e plane showing s l i p l i n e blocked i n schematic c e l l structure. . . . . . . . . . . . 74 24 Orientation of tensile axis f o r cr y s t a l s used i n microstrain/etch p i t work . . . . . . . . 82 ix Figure Page 25 Prestrain curves for c r y s t a l s used i n microstrain/etch p i t work 85 26 77°K microstrain curve for 68B (1000°K pr e s t r a i n ) . 98 27 77°K microstrain curve for 65B (850°K prestrain) 99 28 77°K microstrain curve for 69B (750°K pre s t r a i n ) . 100 29 77°K microstrain curve for 78B (293°K p r e s t r a i n ) . . . . 101 30 77°K microstrain curve for 70B (77°K prestrain) . 102 31 O r i g i n a l load-elongation curve for 68B (1000°K prestrain) 104 32 O r i g i n a l load-elongation curve for 70B (77°K prestrain) 105 33 Normalized microstrain curves . 108 34 Etched d i s l o c a t i o n microstructure after s t a t i c anneal 110 35 Etched d i s l o c a t i o n microstructure a f t e r c y c l i c anneal . . . . . . . . . . m 36 Etched d i s l o c a t i o n microstructure af t e r c y c l i c anneal . . . . . 112 37 Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 6 8A (1000°K prestrain) xl560. . . . . . . . . . . . . 115 38 Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 63A (850°K prestrain) xl560 . . . . 116 x Figure Page 39 Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 69A (700°K prestrain) xl560 . . 117 40 Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 78A (293°K prestrain) xl560 118 41 Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 70A (77°K prestrain) xl560 119 42 Etched d i s l o c a t i o n microstructure and corresponding dot pattern from 78A (293°K p r e s t r a i n ) . . . . . . . . 120 43 Dislocation density versus resolved shear stress.. . 122 44 Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area densities from 68A (1000°K prestrain) . . . . . 124 45 Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area densities from 63A (850°K p r e s t r a i n ) . 125 46 Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area densities from 69A (700°K p r e s t r a i n ) . . . . . . . . . . . . 126 47 Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area densities from 78A (293°K prestrain) 127 48 Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l area densities from 70A (77°K prestrain) 128 49 Schematic of regular, random and more c e l l u l a r d i s t r i b u t i o n s , t h e i r corresponding histograms and degrees of r e g u l a r i t y (s/x ) . . . 129 x i Figure Page 50 Calculated histograms for random d i s t r i b u t i o n of points . 131 51 Degree of r e g u l a r i t y , s/x, for a range of sample sizes for upper three prestrain temperatures. . . . . . . . . . . . . . . . . . . 133 52 Degree of r e g u l a r i t y , s/x, for a range of sample sizes for lower two prestrain temperatures . . . . . . . . . 134 53 Dot pattern taken from micrograph by Strutt and Gupta (1967) . . . . . . . . . . . . . 135 54 Change i n normalized free area, a/ao, with r e l a t i v e applied stress from Kocks (1967) and corresponding s t r a i n . . . . . . . . . 144 55 Calculated microstrain curves using Alden's free area function. . . . . . . . . . . . . . . . 146 56 Calculated microstrain curves using Ashby's function. . . . . . . . . . . . . . . . . . . . . 1 4 8 57 Replotted normalized microstrain curves . . . . . 149 58 Schematic stress s t r a i n curves. . . . 167 59 Microstrain testing apparatus . . . . . . . . . . 172 60 Optical micrographs of the same area showing advantage of Normarski Interference Contrast 176 x i i ACKNOWLEDGEMENTS The author i s g r a t e f u l f o r the advic e and encourage-ment g i v e n by Dr. T.H. Alden. Thanks are a l s o extended t o othe r f a c u l t y members, and t o f e l l o w graduate s t u d e n t s , f o r many h e l p f u l d i s c u s s i o n . The a s s i s t a n c e of t e c h n i c a l s t a f f i s a l s o g r e a t l y a p p r e c i a t e d . F i n a n c i a l a s s i s t a n c e i n the form o f an A l c a n F e l l o w s h i p ( f o r two y e a r s ) , and from the N a t i o n a l Research C o u n c i l (N.R.C. Grant No. A-4991) i s g r a t e f u l l y acknowledged. x i i i P A R T 1 SLIP LINE STUDIES 1.1 Introduction A universally observed feature of deformation i n metals i s the production of s l i p l i n e s . In pure face centred cubic metal single c r y s t a l s , the study of s l i p l i n e s has been part of the large body of experimental work performed i n an attempt to formulate theories of s t r a i n hardening, and to distinguish between the theories of flow stress which each of the s t r a i n hardening theories incorporate. Since s l i p lines are a surface manifestation of microstructural events, and are usually observed to have limited length, an e s s e n t i a l feature of any theory of s t r a i n hardening i s a mechanism by which s l i p l i n e s are blocked. The present experiments were designed to distinguish between s t r a i n hardening theories by investigating the temperature dependence of s l i p l i n e lengths. Some of the e a r l i e s t work i n the f i e l d of metal deformation included the study of s l i p l i n e s , since they were 1 2 one of the few d i r e c t l y observable microstructural features of a p l a s t i c a l l y strained metal. The observation of s l i p l i n e s began with the work of Ewing and Rosenhain (1899), and a f t e r the i n i t i a l observations from single c r y s t a l s (Taylor and Elam, 1923), the f i r s t comprehensive study was carried out by Yamaguchi (192 8). A major impetus f o r s l i p l i n e research came with the f i r s t use of transmission electron microscopy to examine re p l i c a s from metal surfaces (Heidenreich and Shockley, 1947), which stimulated a great deal of subsequent work (Read, 1952; Kuhlman-Wilsdorf and Wilsdorf, 1953; Blewitt et al., 1955; Noggle and Koehler, 1957; Seeger et a l . 3 1957; Fourie and Wilsdorf, 1959). Since s l i p l i n e s were found to have f i n i t e length, a mechanism for blocking t h e i r growth was thought to be a key factor i n any explanation of s t r a i n hardening (Seeger, 1957). I t was shown that s l i p l i n e s formed at higher st r a i n s were shorter (Mader, 1957; Seeger et al.3 1957) and t h i s experimental observation together with a postulated blocking mechanism was incorporated into an early theory of s t r a i n hardening (Seeger, 1957). > Another important s l i p l i n e property, namely that s l i p l i n e s are formed i n a short time i n t e r v a l , was revealed by cinematography (Maddin and Chen, 1954), and the more recent acoustic emission work of Fisher and L a l l y (L967) confirms t h i s observation. Studies using both these techniques indicate 3 that s l i p l i n e s are formed by dislocations moving with r e l a -t i v e l y high v e l o c i t i e s (up to 10 3 cm s e c - 1 i n copper for example). This observation i s s i g n i f i c a n t since i t implies that the g l i d e process by which s l i p l i n e s are formed, i s e s s e n t i a l l y athermal. Direct and detailed observations of d i s l o c a t i o n configurations became possible for the f i r s t time when trans-mission electron microscopy was applied to thin metal f o i l s , i n i t i a l l y by Bollman (1956) and by Hirsch et al. (1956). In the subsequent search for configurations or obstacles which blocked s l i p i n pure metals, the study of s l i p l i n e s per se declined. The decline i n popularity of s l i p l i n e research may be related also to the work of Kramer (1961) and l a t e r that of Fourie (1968). Both showed that the surface region of a deformed c r y s t a l has physical properties which d i f f e r from the i n t e r i o r . That the r e s u l t s of Kramer apparently showed the surface to be harder while those of Fourie seemed to indicate the reverse, only added to the growing doubts over the r e l i a b i l i t y of surface observations. And while the three previously mentioned properties of s l i p l i n e s ( f i n i t e length; length decreasing with s t r a i n ; l ines formed i n short time intervals) were not seriously challenged by these experiments, few quantitative s l i p l i n e s studies were subsequently undertaken. Recently however, the e f f e c t of the surface layer on the formation of s l i p l i n e s i n copper was demonstrated by Himstedt and Neuhauser (1972). By c o r r e l a t i n g the r e s u l t s of t h e i r d i f f e r e n t i a l polish/incremental s t r a i n experiments with electron microscopy studies, i t may be concluded that s l i p l i n e s produced aft e r the removal of a surface layer (which i s formed during prestrain) are c h a r a c t e r i s t i c of i n t e r i o r s l i p events. In addition to measurements of s l i p l i n e length, observations of various other c h a r a c t e r i s t i c s of s l i p l i n e s have been reported i n the l i t e r a t u r e . These c h a r a c t e r i s t i c s include s l i p l i n e height and width, and the f i n e structure of s l i p l i n e s (most of t h i s work i s reviewed by Clarebrough and Hargreaves (1959), and the study by Mader (1957) i s of p a r t i c u l a r note). However, i n r e l a t i o n to theories of s t r a i n hardening, length.has- emerged as the most s i g n i f i c a n t property of s l i p l i n e s . Hence the present study i s directed towards the observation of t h i s property. In current s t r a i n hardening theories s l i p l i n e s are blocked either by s p e c i f i c obstacles (Seeger, 1957; Hirsch and M i t c h e l l , 1967) or by s t a t i s t i c a l i n t e r a c t i o n between forest and g l i d e dislocations (Alden, 1972). Because of t h i s difference, these two types of theories make d i f f e r i n g pre-dictions about the v a r i a t i o n i n s l i p l i n e length with tempera-ture, at constant structure. The former propose that s l i p l i n e length i s a function of microstructure alone, and i s 5 independent of the deformation temperature. In contrast, the l a t t e r postulates that the area swept out by expanding d i s l o c a t i o n loops varies inversely with temperature. Hence, s l i p l i n e length should increase with decreasing temperature. In Section 1.4 these c o n f l i c t i n g predictions, the d i f f e r e n t theories of flow stress on which they are based, and other possible t h e o r e t i c a l developments are discussed more f u l l y . The intent of the following experiments was to t e s t the contrary predictions of these theories by measuring s l i p l i n e lengths i n copper single c r y s t a l s . Using the c l a s s i c a l technique of Seeger (1957) cry s t a l s were prestrained, polished and incrementally strained to reveal c h a r a c t e r i s t i c s l i p l i n e lengths. In this case, however, the structure is constant, and the temperature variable, rather than the reverse, as i n the p r i o r study (Seeger, 1957). 1.2 Experimental Techniques 1.2.1 Specimen Preparation 1.2.1.1 Material P o l y c r y s t a l l i n e copper rod 3/8 inch i n diameter was obtained from American Smelting and Refining Company. I t had a nominal composition of 99.999% copper with the following quoted impurity concentrations i n ppm: 6 Fe Sb Pb Sn Ni. B i Ag As Cr S i Te Se S <0.7 <1 <1 <1 <1 <0.1 <0.3 <2 <0.5 <0.1 <2 <1 . <1 3i inch lengths were cleaned by immersion i n concentrated n i t r i c acid, followed by a thorough wash, f i r s t i n water and then i n pure ethanol. 1.2.1.2 Melting Procedure Single c r y s t a l s with 3/16 inch square cross section by 7 inches long were grown using a horizontal furnace which t r a v e l l e d ait 5 inches h o u r - 1 under a dynamic vacuum of 10~ 5 t o r r . The single zone furnace i s resistance wound with Kanthal A l over a r e c r y s t a l l i z e d alumina tube. I t was operated with a maximum temperature of 1170°C and had a temperature gradient at the s o l i d l i q u i d interface of about 84°C i n c h - 1 . In order to obtain square sectioned, seeded, pure copper single c r y s t a l s a special mold was designed and con-structed from 99.9995% spectroscopic graphite supplied by the U l t r a Carbon Corp., Figure 1. This graphite mold assembly i s used i n the following manner: i n i t i a l l y a double wedge is placed above the 3/8 inch diameter p o l y c r y s t a l l i n e copper rod, i n the graphite mold. The whole assembly i s placed i n the quartz tube which i s then evacuated and the furnace i s accurately positioned over the mold. Copper flows into the mold as the temperature i n the region of the rod exceeds i t s D 4 Plan of Mold l 'A uwuwwwmmw Sections: ryul : — — . . . . . . -.y- - ... i\ \ \\\ \ \ \ u( (\ \ \\ (\ \ a fflimm YYYYTWY Y YYT YTV\ \ \ \ T U \ \ U UTVY W \ \ W \ \ W \ \ W \ W \ \ \ \ \ W \ W 1 Before Melting 8 fe i n n IJMC A<,ER GROW,H Before Melting 1 seed 2 polycrystal stock 3 singie aystal 4 graphite mold 5 upper wedge 6 lower 7 end stop 8 quartz tube F i g u r e 1. Graphite mold assembly used to grow s i n g l e c r y s t a l s . 8 melting point, and the double wedge f a l l s into place above thi s l i q u i d . The upper wedge i s then manually pushed forward by means of a sta i n l e s s s t e e l rod, so that on contact with the top of the quartz tube, a net downward force i s exerted through the lower wedge onto the l i q u i d copper. In t h i s way the l i q u i d i s squeezed into the mold corners. Excess copper i s pressed out at the end remote from the seed, and acts as a reservoir to f i l l the mold as the copper contracts on s o l i d i f i c a t i o n . The stainl e s s s t e e l push rod s l i d e s i n a Wilson seal, and thus can be manipulated from outside the vacuum system. With accurate positioning of the furnace a small but s u f f i c i e n t amount of seed melts, and the seed interface i s wetted. At thi s time the motor which moves the furnace i s switched on manually, and the c r y s t a l i s grown. 1.2.1.3 Cr y s t a l Orientation A l l crystals used i n the s l i p l i n e studies had a te n s i l e axis oriented as shown i n Figure 2. The orientation was chosen so that single g l i d e would be the dominant mode of deformation for a l l strains used i n these experiments (Basinski and Basinski, 1970). The post-prestrain orientation i s also shown on this diagram. The l a t e r a l faces of these cr y s t a l s were selected such that one pair was p a r a l l e l to the primary Burgers vector 9 F i g u r e 2. O r i e n t a t i o n of t e n s i l e a x i s f o r c r y s t a l s used i n s l i p l i n e work. 0 = i n i t i a l o r i e n t a t i o n , X = o r i e n t a t i o n a f t e r p r e s t r a i n . 10 within 3°. In t h i s way the s l i p step height on the other pair of faces was maximized. A seed was obtained by f i r s t growing a randomly oriented single c r y s t a l . From th i s the desired orientation was reach by appropriate rotation and successive growth cycles. The Laue back r e f l e c t i o n X-ray technique, using a copper tube, enabled the o r i e n t a t i o n to be determined within 2°. Specimens were cut to length i n a s t r a i n free way by using an e l e c t r o l y t i c device which allowed a planar stream of e l e c t r o l y t e to impinge on the (anodic) c r y s t a l . Two specimens 3/16 inch square section by 3i inches long were cut from each as-grown c r y s t a l . 1.2.2 Mechanical Testing 1.2.2.1 Prestrain The v a r i a t i o n i n s l i p l i n e length with temperature was f i r s t detected i n a series of preliminary experiments. In order to study t h i s variation, a series of samples with the same microstructure (which would be stable over a large range of temperatures) was required. With t h i s i n mind, the iden-t i c a l l y oriented c r y s t a l s were strained to a resolved shear stress of 1.85 Kg mm-2 while immersed i n a s a l t bath at 673 ± 3°K. The c r y s t a l s were deformed i n tension i n a f l o o r model Instron machine at a crosshead speed of 0.01 inches min" 1. 11 Two part s t a i n l e s s s t e e l g r i p s , kindly supplied by Dr. J.D. Livingston, held the specimen. When assembled the two parts formed a 3/16 inch square cross section cavity which gripped the c r y s t a l on two sides; to aid gripping, f i l e inserts were added on these sides. A further modification was the addition of locating studs to prevent minor ro t a t i o n of one h a l f with respect to the other during tightening. The grip design, together with the use of a s p e c i a l mounting j i g , ensured that the c r y s t a l was i n i t i a l l y colinear with the Instron p u l l rod. Most single c r y s t a l t e n s i l e t e s t i n g produces a con-tinuous reorientation of the c r y s t a l l a t t i c e , which rotates the tensile axis away from the o r i g i n a l c r y s t a l l o g r a p h i c o r i e n t a -t i o n . Ideally; i n order to avoid misalignment between t e n s i l e and c r y s t a l axes, the axis of a universal j o i n t must be placed at each end of the gauge length. In t h i s case however, i t was considered s u f f i c i e n t to include universal j o i n t s near the ends of the grips, since the strains used were r e l a t i v e l y small. . Care was taken to unload a l l specimens i n the same way. Clearly a c r y s t a l unloaded i n the s a l t bath could have a microstructure d i f f e r e n t from one unloaded aft e r cooling. On the other hand, almost any single procedure would provide acceptable ( i . e . isostructural) specimens. The following method was chosen: once the required stress l e v e l was reached 12 the cross head d i r e c t i o n was reversed and during unloading the s a l t bath was removed. In order to atta i n the chosen prestress, a resolved shear s t r a i n of about 20% was required: t h i s p r e s t r a i n curve i s shown i n Figure 3. 1.2.2.2 St r a i n Increments Specimens were polished (see Section 1.2.3) to remove the accumulated s l i p l i n e s . They were then incrementally strained 0.5%, at a s t r a i n rate of approximately 2.5 x 10" 3 min" 1, and at several d i f f e r e n t temperatures i n the environments l i s t e d below. 5 7 3 ° K Vacuum F u r n a c e 293°K A i r I90°K P e t r o l e u m E t h e r 77°K L i q u i d N i t r o g e n 42°K L i q u i d H e l i u m Temperatures below the pre s t r a i n temperature were chosen to minimize annealing (Berghout, 1956) and work soften-ing effects ( C o t t r e l l and Stokes, 1955). The s t a b i l i t y of the microstructure was demonstrated by a s t a t i c anneal for 4 hours at 573°K which produced no decrease i n the y i e l d s t ress. Since detailed comparisons of the c r y s t a l surfaces were to be made, non-corroding environments were e s s e n t i a l . 13 14 In f a c t no oxidation of the electropolished c r y s t a l surface could be observed, using an o p t i c a l microscope at x600 magnifi cation, after p r e s t r a i n i n any of the environments. The intent i n these experiments was to compare s l i p l i n e length at constant structure, established by the above prestrain treatment. To confirm that the s t r a i n increment i t s e l f produced no s i g n i f i c a n t s t r u c t u r a l change, several crystals were given two 0.5% s t r a i n increments (separated by an e l e c t r o p o l i s h ) , one at low temperature, and one at high, i n either order. The s l i p l i n e lengths obtained i n t h i s fashion were i d e n t i c a l to those obtained i n c r y s t a l s given a single s t r a i n increment (except i n one case which was assumed to be spurious). Some s t r a i n increments were performed at d i f f e r e n t s t r a i n rates at a single temperature, namely 293°K. The rate was varied from 3 x 10" 5 inches min - 1 to 3 inches m i n - 1 . In the former case a.7-inch long c r y s t a l was used. In the l a t t e r case an oscilloscope recorded the load pulse which lasted approximately 0.15 seconds, and after which the Instron automatically unloaded the specimen. 1.2.3 S l i p Line Measurement 1.2.3.1 Chemical P o l i s h In any study of s l i p l i n e length, i t i s important that surfaces be f l a t and that they reveal the s l i p processes 15 which are c h a r a c t e r i s t i c of the bulk material. With these requirements i n mind, i n i t i a l polishing was done chemically-using a saturated solution of cupric chloride i n concentrated hydrochloric acid. The technique used was modified from one f i r s t used by J.W. M i t c h e l l et al. . (1967). A f i n e no-nap cotton polishing cloth (Buehler Metcloth) stretched t i g h t l y over a f l a t polyethylene slab, carries the chemical p o l i s h . The slab i s ruled with f i n e p a r a l l e l grooves to aid l a t e r a l dispersion of the solution. S l i p l i n e s formed during pre-s t r a i n are removed by drawing the c r y s t a l by hand gently backwards and forwards over the c l o t h , i n a d i r e c t i o n at r i g h t angles to the grooves. When the surface markings appeared to have been removed, methanol i s gradually added to the cloth so as to reduce the concentration of the polishing solution while polishing continued. In t h i s way corrosion products were uniformly removed from the surface, leaving a bright f i n i s h . A l l four faces were polished, arid with care taken to prevent rocking of the c r y s t a l during the above procedure, very l i t t l e edge rounding occurred. 1.2.3.2 E l e c t r o p o l i s h . A s a . f i n a l step i n the surface preparation, the crystals were given a l i g h t e l e c t r o p o l i s h . The polishing c e l l consisted of two concentric s t a i n l e s s s t e e l cathodes between 16 which the c r y s t a l was suspended. A solution of two parts methanol and one part concentrated n i t r i c acid was used as the e l e c t r o l y t e , with vigorous s t i r r i n g . The temperature of -30°C was maintained by using an outer bath of cooled alcohol. At this temperature, the Flade p o t e n t i a l of the c e l l i s about 7 v o l t s . The c e l l i s designed so that the c r y s t a l i s equi-distant from two cathodes, to ensure that opposite faces of the c r y s t a l are simultaneously polished. Four rotations of 90° at 30 second inter v a l s were usually found s u f f i c i e n t to produce an excellent p o l i s h : the procedure was repeated i f necessary. E l e c t r o p o l i s h times of greater than about two minutes per face produced long range surface undulations which could only be removed by going back to the chemical p o l i s h and repeating the whole procedure. Electropolishing was followed by a thorough r i n s e , f i r s t i n methanol and then i n water. The. whole.polishing operation removed not less than 0.010 inches from each face. A recent study of s l i p l i n e s i n copper (Himstedt and Neuhauser, 1972) has shown that a surface layer forms during prestrain which gives r i s e to longer s l i p l i n e s than those formed under i d e n t i c a l conditions, except with t h i s layer removed. Their results confirmed that t h i s e f f e c t was e s s e n t i a l l y eliminated by the removal of 0.010 inches of surface after prestrain, and further surface layer formation during incremental s t r a i n i n g was considered n e g l i g i b l e . 17 1.2.3.3 Optical Microscopy A Zeiss Interference Microscope, operating i n the non-interference mode at about x600 magnification, was used to measure s l i p l i n e length. O p t ical microscopy was chosen so that r e l i a b l e s t a t i s t i c a l data could be conveniently co l l e c t e d , and the Zeiss gave reproducible resolution, and contrast. The measurements were taken by scanning a f i d u c i a l mark and a scale located i n the microscope eyepiece, p a r a l l e l to the s l i p traces. Whenever the mark met the end of a s l i p l i n e , i t s length was measured (by comparing i t to the eye-piece s c a l e ) . Using t h i s technique operator bias was minimized: bold, f a i n t , short and long l i n e s have equal p r o b a b i l i t y of being measured. The scans began at randomly chosen points well away from the c r y s t a l edge, and the focus was continually adjusted to allow for any s l i g h t departure from o p t i c a l f l a t n e s s . Several o p t i c a l micrographs were taken after d i f f e r e n t s t r a i n increments. Of p a r t i c u l a r i n t e r e s t are those taken after a s t r a i n increment at 4.2°K, and then the same area following a p o l i s h and a s t r a i n increment at 293°K. In order to achieve a depth of f i e l d i n the micro-graphs similar to that obtained by rocking the focus while taking the measurements d i r e c t l y , a lower magnification was 18 used. The use of a newly available extra f i n e grain 35 mm film, H + W Control, allowed subsequent enlargement of the negative, while minimising photographic emulsion grain s i z e e f f e c t s . i S l i p l i n e s appeared to be d i s t r i b u t e d uniformly over the surface of the c r y s t a l , without the clu s t e r i n g sometimes observed i n other metals (deLarios, 1972). To quantify t h i s observation some s l i p l i n e density measurements were taken. Scanning perpendicular to the s l i p l i n e s , the number of li n e s f a l l i n g within 10 d i v i s i o n s of the eyepiece scale (68y) was counted for several f i e l d s selected at random. Using the microscope i n i t s interference mode, maximum s l i p step height was measured. 1.2.3.4 Electron Microscopy I t i s i m p l i c i t l y assumed i n t h i s study that the lengths of the v i s i b l e " s l i p l i n e s " (more accurately c a l l e d s l i p bands) vary i n the same manner as the lengths of the unresolved elementary s l i p l i n e s , which make up the fin e structure of the slip'bands. Mader (1957) showed that t h i s was a reason-able assumption i n s t r a i n hardened copper c r y s t a l s . In order to q u a l i t a t i v e l y confirm t h i s observation, two stage r e p l i c a s (acetate, chromium shadowed carbon) were taken after 4.2°K and 293°K s t r a i n increments. 19 These r e p l i c a s , having been taken from a f l a t polished c r y s t a l surface, tended to break up p a r a l l e l to the s l i p l i n e s when the acetate backing was dissolved away using conventional r e p l i c a techniques. In p a r t i c u l a r , carbon films containing long sharply defined steps were d i f f i c u l t to produce at the thicknesses required for good q u a l i t y shadowed r e p l i c a s . An acetone r e f l u x unit was b u i l t and used to overcome t h i s problem (Figure 4). Inside the apparatus the r e p l i c a , together with copper support g r i d i s positioned on a cold finger, and surrounded by acetone vapor. The acetate backing of the r e p l i c a i s gradually dissolved by the condensing acetone, leaving the carbon r e p l i c a clean, i n t a c t and supported. Carbon spheres with diameters i n the s i z e range o o 1000 A to 5000 A were placed on the surface of some r e p l i c a s before shadowing. By comparing t h e i r shadows with the width of shadows from s l i p steps, or for a more precise estimate, by measuring the increase i n length of a shadow from a sphere, as i t crosses that of a s l i p l i n e , the s l i p step height may be obtained. 1.3 Results 1.3.1 S l i p Line Lengths 1.3.1.1 Temperature Changes The lengths of 100 s l i p l i n e s were measured a f t e r each s t r a i n increment, two or more d i f f e r e n t specimens being used 20 Condenser-J Water Specimen Water Cold Support Heating Mantle F i g u r e 4. Acetone r e f l u x apparatus used i n p r e p a r a t i o n of r e p l i c a s . 21 for each test temperature. Histograms of these l i n e lengths for each test temperature are shown (Figures 5-9), normalized to 250 readings for v i s u a l comparison. It can be seen that temperature has a marked e f f e c t on the s l i p l i n e patterns at constant structure. Both the mean length and d i s t r i b u t i o n are changed. For example, most of the l i n e s formed at 573°K are below 50u i n length, whereas at 4.2°K the majority are well above 50y; further, while very few of the l i n e s formed at 573°K are above 70u i n length, 20% of those formed at 4.2°K are longer than 150y. The histograms for the intermediate temperatures, 293°K, 190°K and 77°K show a continuous increase i n the number of long l i n e s , yet they also contain a s i g n i f i c a n t proportion of short l i n e s . A p l o t of the mean s l i p l i n e length versus tempera-ture i s shown i n Figure 10, together with t h e i r 95% confidence int e r v a l s and the upper and lower qu a r t i l e s derived from the histograms. The confidence inter v a l s were considered suf-f i c i e n t l y small that 100 readings were adequate to est a b l i s h the mean l i n e length, and that the s t a t i s t i c a l error was below the l e v e l of the major experimental errors. I t can be seen that s l i p l i n e s formed at low temperatures are longer than those formed at a high temperature, i n the range studied; for example, at 4.2°K the average l i n e i s about three times longer than at room temperature. The o p t i c a l micrographs (Figure 11) d i r e c t l y confirm these observations, showing s l i p l i n e s formed on the same area 22 50r 40[ UJ - i30 r u. o |.20f 10 573 6 K JT I50~ 0 50 100 SLIP LINE LENGTH (MICRONS) Figure 5, Histogram of s l i p l i n e lengths aft e r 0.5% incremental s t r a i n at 573°K. 23 50 SLIP LINE LENGTH (MICRONS) F i g u r e 6. Histogram of s l i p l i n e lengths a f t e r 0.5% incremental' s t r a i n a t 293°K. 24 50 SLIP LINE LENGTH 100 (MICRONS) 150 Figure 7. Histogram of s l i p l i n e lengths aft e r 0.5% incremental s t r a i n at 190°K. 25 SLIP LINE LENGTH (MICRONS) 200 F i g u r e 8, Histogram of s l i p l i n e lengths a f t e r 0.5% in c r e m e n t a l s t r a i n at 77' K. SLIP LINE LENGTH (MICRONS) Figure 9. istogram of s l i p . l i n e lengths aft e r 0.5% incremental s t r a i n at 4. 1 1 L J 1 I I I 200 400 600 TEMPERATURE °K Average s l i p l i n e l e n g t h versus temperature of inc r e m e n t a l s t r a i n i n g . Dashed l i n e s span the middle two q u a r t i l e s from accumulated measure-ments. S o l i d e r r o r bars r e p r e s e n t 95% c o n f i d e n c e l i m i t s f o r i n d i v i d u a l p o i n t s . 28 Figure 11. Optical micrographs showing change i n s l i p l i n e length with temperature. The same area of c r y s t a l with hardness marker for reference, (a) after s t r a i n increment at 4.2°K, (b) aft e r s t r a i n increment at 293°K. 29 of a c r y s t a l surface to be longer at 4.2°K than at 293°K. Another noticable difference i s the d i f f u s e nature of the room temperature l i n e s when compared to the sharper 4.2°K l i n e s . At higher magnification and resolution, the electron micrographs (Figure 12) reveal the 4.2°K l i n e s as very f i n e and long: few l i n e s end within the frame. In contrast many of the 293°K l i n e s end, and they are apparently clustered i n narrow bands, with spacings less than the resolution l i m i t of the technique.; However, the spacing of the more prominent li n e s i n the re p l i c a s corresponds to that of the o p t i c a l l y observed l i n e s being i n the range 2-10y. 1.3.1.2 Strain Rate Changes Over the range of s t r a i n rates studied, no s i g n i f i -cant differences i n the average s l i p l i n e length was measured. I t should be noted however that changes i n s t r a i n rate produce stress changes which are small compared to those observed i n variable temperature experiments (Thornton et al.3 1961; Adams and C o t t r e l l , 1955). For example, although stress data were not obtained from these s t r a i n rate change experiments, a stress change of about 7% i s predicted under these conditions, using data from Thornton et al. (1961). In contrast major changes i n average s l i p l i n e length are observed i n the variable temperature experiment only when accompanied by stress, changes of greater than 20%. 30 (a) (b) Figure 12. Electron micrographs of s l i p l i n e s formed a f t e r s t r a i n increments at (a) 4.2°K and (b) 293°K respectively. Both are chromium shadowed (two stage) carbon r e p l i c a s . 31 1.3.2 Variation i n Flow Stress with Temperature The i s o s t r u c t u r a l ("reversible") increase of flow stress with decreasing temperature, which has been established i n a variety of metals ( C o t t r e l l and Stokes, 1955; Adams and C o t t r e l l , 1955; M i t c h e l l , 1964) was observed i n the present experiments. However, accurate data were not obtained, primarily because the cross-sectional area of the samples was not known precisely a f t e r varying amounts of p o l i s h i n g . Some examples of approximate flow stress values are 1.85 Kg mm""2 at 673°K, 2.04 Kg mm-2 at 293°K, 2.26 Kg mm"2 at 77°K and 2.36 Kg mm-2 at 4.2°K. 1.3.3 S l i p Line Density While s l i p l i n e density has np d i r e c t bearing on the t h e o r e t i c a l implications of the main experiments, i t i s important to show that deformation was r e l a t i v e l y homogeneous. Luders band propagation, and s l i p l i n e c l u s t e r i n g would make the microstructural interpretation more d i f f i c u l t . S l i p l i n e density readings were made on two separate c r y s t a l s , strained at 293°K. A t o t a l of 165 independent measurements were taken, and a histogram of these densities i s shown i n Figure 13. The b e l l shaped histogram i l l u s t r a t e s appropriately the homogeneous nature of deformation. Very few areas on the c r y s t a l surface were devoid of s l i p l i n e s : few were densely 32 100 2 00 3 0 0 4 0 0 Slip Line Density ( lines • mm ') 5 0 0 10 5 Slip Line Spacing (/xm) 2 5 F i g u r e 13. Histogram of s l i p l i n e d e n s i t y , l i n e s formed a t 293°K. c o v e r e d . The m a j o r i t y o f l i n e d e n s i t i e s measured a r e i n the range 7 5-300 l i n e s m m " 1 . The average o f 170 l i n e s m m - 1 c o r r e s p o n d s to a mean l i n e s p a c i n g o f 5 . 9 y , comparable t o the s p a c i n g s o f prominent l i n e s on the e l e c t r o n m i c r o g r a p h s . 1 . 3 .4 S l i p Step H e i g h t O p t i c a l measurement o f s l i p s t e p h e i g h t s , formed a f t e r an i n c r e m e n t a t 2 9 3 ° K , were made u s i n g t h e Z e i s s I n t e r f e r e n c e M i c r o s c o p e . These measurements showed maximum o s l i p s t e p h e i g h t s to be o f the o r d e r o f 500 A , w h i c h i s near the r e s o l u t i o n l i m i t o f the i n s t r u m e n t (Z A / 1 0 ) . Comparing the chromium f r e e shaddows o f carbon spheres i n F i g u r e 14 t o t h e s l i p l i n e s r e v e a l s maximum s t e p h e i g h t s i n t h e range o 250 t o 400 A , p r o v i d i n g q u a l i t a t i v e agreement w i t h the p r e v i o u s t e c h n i q u e . 1 . 3 . 5 Sources o f E r r o r In a s tudy o f t h i s t y p e , the e x p e r i m e n t a l problems may be d i v i d e d i n t o two g r o u p s . Those i n t r o d u c e d by d i f -f e r e n c e s from c r y s t a l to c r y s t a l , o r f rom i n c r e m e n t t o i n c r e m e n t were c o n s i d e r e d n e g l i g i b l e , s i n c e good c o n t r o l was e x e r c i s e d o v e r o r i e n t a t i o n , c r y s t a l p u r i t y , t e m p e r a t u r e , t h e amount o f p r e s t r a i n and s t r a i n i n c r e m e n t , and s u r f a c e q u a l i t y . The second group c o v e r s t h o s e , e x p e r i m e n t a l problems ,, i n h e r e n t i n any one p r e s t r a i n , p o l i s h , s t r a i n i n c r e m e n t , 34 Figure 14. Shadowed r e p l i c a with c a r b o n 0 c a l i b r a t i o n spheres (diameters i n the range 1000A to 5000 A). 35 s l i p l i n e measurement sequence. Here the most c r i t i c a l experimental l i m i t a t i o n was the o p t i c a l flatness of the surface. When measuring l i n e length, a l i n e was taken to begin or end at any deviation from l i n e a r i t y , or at any lack of continuity.. A proportion of these deviations were due to surface i r r e g u -l a r i t i e s as. d i s t i n c t from microstructural phenomena. Hence, lack of o p t i c a l flatness would bias each set of results towards shorter l i n e lengths. . But t h i s bias i s greater f o r longer mean l i n e length. A long l i n e would have a greater chance of crossing a surface i r r e g u l a r i t y than a short l i n e , so the' long l i n e might be counted i n c o r r e c t l y as two or more l i n e s . Thus i t seems l i k e l y that i f specimens had perfect o p t i c a l f l a t n e s s , the observed e f f e c t that s l i p l i n e s formed at lower temperatures are longer, would be more strongly revealed. Another l i m i t a t i o n arises from the compromise made in choosing 0.5% as the s t r a i n increment. On the one hand the s t r a i n increment must be large enough so that most of the s t r a i n occurs (and therefore most of the s l i p l i n e s are formed) at the steady state flow stress; then only a small f r a c t i o n of the lines are formed during 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 s t r a i n at a lower stress. On the other hand a large s t r a i n increment w i l l i n v a l i d a t e the assumption of constant structure. In the majority of cases, as estimated from the load elongation curves, the t r a n s i t i o n s t r a i n occupied less 36 than 0.15% s t r a i n (Figure 15). However, t h i s e f f e c t may provide a background of shorter l i n e s (Alden, 1972) on a l l te s t s , thus l i m i t i n g and not enhancing the observed e f f e c t . Whilst further discussion of this matter i s presented i n the preceding section, i t i s assumed that i n the present experiments, most s l i p l i n e s are formed at the steady state flow stress. 1.4 Discussion 1.4.1 S l i p Line Blocking by S p e c i f i c Dislocation Arrays A major objective of any theory of work hardening i s to successfully l i n k microstructural phenomena to mechanical properties. Such a. theory should explain the nature of s l i p lines since these are surface manifestations of microstructural events. Of the existing work hardening theories which attempt to explain s l i p l i n e length, two prominent examples are those of Seeger (1957) and of Hirsch and M i t c h e l l (1967). i.4.1.1 S l i p Line Blocking by Lomer-Cottrell Locks Seeger's theory (1957) relates s l i p l i n e length L to s t r a i n e, from the experimental observation that s l i p l i n e length decreases with increasing s t r a i n , namely, L = e - e* (1.1) ELONGATION Figure 15. Transition and steady state s t r a i n observed during incremental, test at 4.2°K. 38 where A and e* are e m p i r i c a l c o n s t a n t s . He proposes t h a t s l i p l i n e s are bloc k e d by a r r a y s o f d i s l o c a t i o n s p i l e d up a t L o m e r - C o t t r e l l l o c k s , which, he submits are formed c o n t i n u -o u s l y i n the stage I I hardening r e g i o n . In a metal w i t h a low s t a c k i n g f a u l t energy, such as copper, L o m e r - C o t t r e l l l o c k s may be generated by a com-b i n a t i o n r e a c t i o n between two extended d i s l o c a t i o n s on two i n t e r s e c t i n g g l i d e p l a n e s . Consider f o r example, d i s l o c a t i o n s w i t h Burgers v e c t o r s j a[101] and i a[l!o] which have d i s -s o c i a t e d t o form Shockley p a r t i a l d i s l o c a t i o n s as f o l l o w s , | a [101] -> | a [112] + | a[2Tl] and | a[110] | a[121] + | a[211] and which l i e on the primary (111) and conjugate (111) s l i p planes r e s p e c t i v e l y . At the l i n e o f i n t e r s e c t i o n o f these two p l a n e s , one p a r t i a l d i s l o c a t i o n from each d i s s o c i a t e d p a i r may combine as f o l l o w s , \ a[112] + J a[12l] + J a [ 0 l l ] b b o to form a s t a i r rod type p a r t i a l d i s l o c a t i o n c a l l e d a Lomer-C o t t r e l l d i s l o c a t i o n . S i n c e the Burgers v e c t o r of t h i s d i s -l o c a t i o n i s coplanar w i t h n e i t h e r the primary nor conjugate s l i p plane, f t i s unable to g l i d e and thus i s termed s e s s i l e . The L o m e r - C o t t r e l l s e s s i l e d i s l o c a t i o n , together w i t h the two remaining p a r t i a l s are c o l l e c t i v e l y r e f e r r e d to as a L o m e r - C o t t r e l l l o c k , which i s a wedge shaped s t a c k i n g f a u l t r i b b o n , l y i n g , i n t h i s case, i n the [ O i l ] - d i r e c t i o n w i t h the L o m e r - C o t t r e l l d i s l o c a t i o n forming the edge of the wedge. Clarebrough and Hargreaves (1959) have i d e n t i f i e d the c o n d i t i o n s f o r p o s s i b l e production of L o m e r - C o t t r e l l d i s l o c a t i o n s which l i e i n the primary s l i p plane, namely t h a t the Burgers vectors of the o r i g i n a l p a i r of d i s l o c a t i o n s make an angle of 120° w i t h each other. P a i r s of s l i p systems s a t i s f y i n g t h i s c o n d i t i o n are l i s t e d i n Table 1, together w i t h the d i r e c t i o n along which the corresponding s e s s i l e d i s l o c a t i o n s l i e . From t h i s t a b l e we can see t h a t Lomer-C o t t r e l l d i s l o c a t i o n s (and hence L o m e r - C o t t r e l l locks) may be formed i n three c l o s e packed d i r e c t i o n s . In Seeger"s theory, g l i d e d i s l o c a t i o n s , generated from a Frank-Read source, p i l e up i n the s l i p plane at Lomer-C o t t r e l l l ocks i n these three c l o s e packed d i r e c t i o n s , Figure 16. Clarebrough and Hargreaves (1959) l a t e r pointed out that L o m e r - C o t t r e l l locks would more l i k e l y be formed i n two of those d i r e c t i o n s (those i n which one of the two parent systems was the primary s l i p system). However, s i n c e two are s u f f i c i e n t to block any primary source ( F r i e d e l , 1955), i n essence Seeger's p r o p o s i t i o n remains v a l i d . 40 Table 1 P a i r s of S l i p Systems which c o u l d Produce L o m e r - C o t t r e l l Locks P a i r s of S l i p Systems D i r e c t i o n o f L o m e r - C o t t r e l l D i s l o c a t i o n (111) [10.1] , (111) [IlO] [011] (111) [ I l O ] , (111) [loi] [011] (111) [101], (111) [ O i l ] [110] (111) [ O i l ] , (111) [loi] [110] (111) [ O i l ] , ( i l l ) [110] [101] (111) [110], (111) [011] [101] 41 primary slip plane Figure 16, S l i p l i n e blocking according to Seeger (1957). Expanding d i s l o c a t i o n loops are blocked by • ' Lomer-Cottrell locks formed i n three close-packed d i r e c t i o n s . 42 Since these locks form continuously during stage II hardening, there i s associated with any s t r a i n an array of pile-up groups which, by means of t h e i r long range e l a s t i c stress f i e l d s , r e s t r i c t the expansion of loops from newly active sources to distances somewhat less than the e x i s t i n g pileup spacing. In other words, the s l i p l i n e length i n any s t r a i n increment depends only on the s t r a i n i t s e l f (more fundamentally on the structure), Eq. 1.1, and i s temperatures-independent. In the present experiments the structure i s constant, and t h i s theory predicts an invariant s l i p l i n e length. This prediction i s not i n agreement with the present r e s u l t s . * While the structure i s constant, the flow stress, a, increases with decreasing temperature. In Seeger's theory i a = a + a (1.2) s g and the increase i n o i s e n t i r e l y due to the increase i n a , the forest i n t e r s e c t i o n stress. a„ i s that part of the . s g c applied stress available to drive d i s l o c a t i o n loops into the p i l e up array, and i s temperature independent; t h i s must follow from the long range nature of the interactions which determine o„. These considerations reinforce the conclusion Appendix 2 outlines the definitions of terminology used to describe deformation phenomena. 43 that t h i s theory predicts a constant s l i p l i n e length i n the present experiments. It may be argued that e x i s t i n g pile-ups can re-arrange or annihilate during a higher temperature s t r a i n increment, whilst at the lower temperature, t h e i r density and d i s t r i b u t i o n remains unchanged. However th i s p o s s i b i l i t y i s not admissible i n the theory; pile-ups are supposed to be * stable i n stage I I . Moreover i f such a n n i h i l a t i o n should occur, longer s l i p l i n e s would form at higher temperatures, contrary to the results of the present experiments. I t should be emphasized that structure change during s t r a i n increments at various temperatures was shown to be small experimentally by incremental s t r a i n i n g of i n d i v i d u a l crystals at two d i f f e r e n t temperatures. In each case s l i p l i n e lengths were the same as i n cr y s t a l s given a single s t r a i n increment, and did not depend on the order i n which the temperatures were selected. 1.4.1.2 S l i p Line Blocking by Ribbons of Converted  Pile-ups Hirsch and M i t c h e l l (1967) object to Seeger's theory on the grounds that the stress f i e l d from the proposed pile-up Accord ing to Seeger et al. (1957) L o m e r - C o t t r e l l b l o c k i n g becomes l e s s e f f e c t i v e dur ing stage III d e f o r m a t i o n , where d i s l o c a t i o n s may by-pass these b a r r i e r s by c r o s s - s l i p . However, in the present experiments where c r y s t a l s were p r e -s t r a i n e d in stage II, i t is u n l i k e l y that a c r o s s - s l i p by-pass mechanism w i l l begin to operate dur ing the s t r a i n increments a l l of which are at temperatures below the p r e s t r a i n tempera ture . 44 arrays w i l l block only one s l i p plane, namely the plane on which the pile-up i s formed; under the influence of the applied stress, glide on neighbouring, p a r a l l e l planes w i l l s t i l l be possible over long distances, and i n certain areas w i l l even by enhanced by the presence of the pile-up. They present calculations of long range i n t e r n a l stress f i e l d s to support this argument. In order to overcome th i s problem, they propose a theory i n which the s l i p l i n e s are again blocked by l i n e a r obstacles surrounding the d i s l o c a t i o n source' however these obstacles are now "converted pile-ups," i . e . pile-ups which are s t a b i l i z e d by secondary s l i p . The converted pile-ups are i n the form of long ribbons p a r a l l e l to the primary s l i p plane and composed of high densities of d i s l o c a t i o n s of several Burger's vectors. The ribbons are e f f e c t i v e i n block-ing a number of adjacent s l i p planes, Figure 17, and thus are described as having a radius of i n t e r a c t i o n perpendicular to the s l i p plane. These obstacles multiply with s t r a i n as g l i d i n g d i s l o c a t i o n loops are blocked by ex i s t i n g obstacles and the r e s u l t i n g pile-ups are then s t a b i l i z e d by l o c a l secondary s l i p . As a consequence of this m u l t i p l i c a t i o n of obstacles the size of the areas of primary s l i p plane available for g l i d e i s reduced, and the s l i p l i n e length decreases as the s t r a i n increases. 45 S l i p l i n e blocking according to Hirsch and M i t c h e l l (1967). Schematic cross section perpendicular to s l i p plane showing ribbon-like obstacles (shaded and appearing as e l l i p s e s i n t h i s cross-section) blocking d i s l o c a t i o n loops expanding from source S. Obstacles have thickness R perpendicular to s l i p plane. 46 Their interpretation of the temperature dependence of the flow stress i m p l i c i t l y follows that of Seeger. The reversible stress increase on decreasing temperature i s a consequence of the increased strength of forest d i s l o c a t i o n s ; none of t h i s extra stress i s available to overcome the l i n e a r obstacles. Therefore at constant structure the s i z e of the areas of primary plane available for g l i d e i s constant, and no s i g n i f i c a n t temperature dependence of s l i p l i n e length i s predicted. Attempts to reconcile the theory and experiment by means of assumed structure change during s t r a i n increments meet the objections previously stated. Some evidence for the existence of long ribbon-l i k e obstacles can be found i n the electron microscope studies of copper (Steeds, 1966). These observations suggest that elongated multipole clusters are formed i n ribbons by the end of stage I. However, the microstructure c h a r a c t e r i s t i c of stage II consists of "carpets" of dislocations approxi-mately p a r a l l e l to the primary g l i d e plane, together with shorter, more d i f f u s e "walls" of dislocations perpendicular to the g l i d e plane (Howie, 1960; Hirsch and Steeds, 1963; Basinski, 1964; Esseman, 1965; Steeds, 1966; Seeger, 1968). After d i s l o c a t i o n etch p i t t i n g the traces of the "carpets" may also be seen on a c r y s t a l face not coplanar with the primary s l i p plane (see for example Basinski and Basinski, 1964). With increasing flow stress these carpet-like patches 47 of dislocations become more numerous and more c l e a r l y defined. At the same time i t i s evident that the d i s l o c a t i o n density within these patches increases, since the l a t t i c e r o t a t i o n across the dense regions becomes greater. By the beginning of stage I I I , r e l a t i v e l y well defined c e l l s have formed which may be elongated p a r a l l e l to the s l i p plane. Although these observations are q u a l i t a t i v e i n nature, and many are from electron microscopy f o i l s prepared without the use of ra d i a t i o n pinning, they appear to show that the obstacle structure proposed by Hirsch and M i t c h e l l (1967) i s appropriate only at the beginning of the stage II hardening region. 1.4.1.3 S l i p Line Blocking by Di s l o c a t i o n  C e l l Walls Since c e l l structures are a prominent feature of the,microstructure of s t r a i n hardened metals, i t should be inquired whether the c e l l walls can act as obstacles which block s l i p l i n e s . In a recent transmission electron microscope study, Staker and Holt (1972) have measured d i s l o c a t i o n c e l l sizes i n copper deformed at temperatures between 298°K and 973°K. Using data from the l i t e r a t u r e together with t h e i r own res u l t s they concluded the the d i s l o c a t i o n c e l l size was inversely proportional to the (Cottrell-Stokes corrected) shear stress. These data are plotted i n Figure 18. By com-paring the flow stress used i n the present experiments, we 48 a Feltner & Laird (1967) A Staker & Holt (1972) o Pratt (1966) (1967) 1.0 5 10 Average Cell Diameter (microns) 18. Average d i s l o c a t i o n c e l l diameters i n copper versus normalized (and C o t t r e l l - Stokes corrected) flow stress, as correlated by Staker and Holt, 1972. Flow stress and deduced c e l l size for the present study are indicated by arrows. 49 would expect the prestrained microstructure to have an average d i s l o c a t i o n c e l l size of about 5y. However the average length of s l i p l i n e s formed i n t h i s microstructure was i n the range 40 to 120y, one order of magnitude or more greater than the anticipated d i s l o c a t i o n c e l l s i z e . Thus i t appears that the s l i p l i n e s are not blocked by c e l l walls (see Appendix 1) . Discrepancies between the apparent s l i p l i n e length, and d i s l o c a t i o n c e l l size have been noted i n previous studies, most recently by Ambrosi et al. (1974). Using transmission electron microscopy they measured d i s l o c a t i o n c e l l sizes i n copper deformed at room temperature up to a flow stress of 7 Kg mm"2. When these r e s u l t s are compared with previous observations of the lengths of s l i p l i n e s formed under i d e n t i c a l conditions (Vorbrug et al., 1971), Figure 19, i t can be seen that the average c e l l diameters are smaller than the average s l i p l i n e lengths for the complete range of flow stress. Moreover, the r e l a t i v e values of d i s l o c a t i o n c e l l diameter and s l i p l i n e length at ~2Kg mm-2 show good agreement with the deduced c e l l size and measured s l i p l i n e lengths i n the present study. While these results appear to show that the majority of c e l l wells per se provide i n e f f e c t i v e blocks for s l i p l i n e s , Basinski and Basinski (1964) have argued that s l i p l i n e s are blocked only by the "carpet" sections of the c e l l walls. 50 Shear Stress (Kg.mm 2 ) Figure 19. Average s l i p l i n e length and average c e l l diameter i n copper for a range of shear (flow) stress as correlated by Ambrosi et al. (1974). 51 Since the carpets are approximately p a r a l l e l to the primary s l i p plane, primary dislocations may g l i d e over distances large compared to the carpet spacing. However, as with the previous theories which- postulate blocking by fi x e d obstacle arrays, this model i n c o r r e c t l y predicts a constant s l i p l i n e length i n an i s o s t r u c t u r a l experiment. I t i s of i n t e r e s t to note that the estimated average c e l l diameter i n the present work, 5y, l i e s within the range of spacing of the more prominent s l i p l i n e s observed in.both the o p t i c a l and the electron micrographs (2-10y). I t w i l l be r e c a l l e d that d i s l o c a t i o n "carpets" p a r a l l e l to the primary glide plane and made up of primary dislocations i n tangled itiultipoles, are a prominent feature of stage II microstructures. Since such carpets i n any non-coplanar cross-section w i l l appear as c e l l walls, i t i s reasonable to assume that the carpet spacing i s of the order of the c e l l diameter. Then i t i s tempting to speculate that s l i p l i n e s are formed by di s l o c a t i o n s , which, when g l i d i n g p a r a l l e l to these carpets, are able to penetrate the more d i f f u s e portions of c e l l "wall" which : are perpendicular to the s l i p plane. In t h i s model, neither the carpets, nor the connecting portions of c e l l "wall" pe3? se, block g l i d e , although forest-type dislocations,, of which the l a t t e r regions are composed, may i n t e r a c t s t a t i s -t i c a l l y to block g l i d e over distances somewhat larger than the c e l l wall spacing. In t h i s way s l i p l i n e s would be formed with lengths large compared with the apparent d i s l o c a t i o n c e l l s i z e . 52 1.4.2 Flow Stress and the Glide/Forest Interaction. Most of, the discussion so far has centred on the p o s s i b i l i t y that s l i p l i n e s are blocked by s p e c i f i c d i s l o c a -t i o n arrangements which are l i n e a r i n form on the s l i p plane. We have seen that t h i s approach predicts constant s l i p l i n e length i n the present i s o s t r u c t u r a l experiments. The flow stress theory on which t h i s approach i s based divides the stress opposing d i s l o c a t i o n motion into additive components which are due to either long range back stresses from these s p e c i f i c d i s l o c a t i o n arrays, or to a short range forest interaction stress. As an alternative premise, we may assume that the flow stress i s solely determined by interactions between g l i d e and forest d i s l o c a t i o n s . The forest dislocations^which may be part of a r e l a t i v e l y immobile d i s l o c a t i o n network or c e l l u l a r structure, are then the obstacles to g l i d e and one need hot refer to s p e c i f i c groupings of obstacle d i s l o c a t i o n s . This premise i s supported by results from two independent experimental f i e l d s , namely by latent hardening experiments, and etch p i t studies. The latent hardening experiments (for example Basinski and Jackson, 1965; Kocks and Brown, 1966) i l l u s t r a t e the importance of the contribution of the g l i d e / f o r e s t i n t e r -action by te s t i n g the orientation dependence of the i s o -s t r u c t u r a l flow stress. In these experiments c r y s t a l s 53 p r e s t r a i n e d i n s i n g l e g l i d e , were r e s t r a i n e d i n a new o r i e n t a -t i o n such t h a t another s i n g l e g l i d e system would operate. When these two systems were co p l a n a r , the flow s t r e s s f o r each system was found to be the same. I f the flow s t r e s s i s s o l e l y determined by the g l i d e / f o r e s t i n t e r a c t i o n t h i s r e s u l t would be expected s i n c e the f o r e s t d e n s i t y and d i s t r i b u t i o n i s the same f o r coplanar systems. On the other hand, i f back s t r e s s e s from a r r a y s of p i l e d - u p d i s l o c a t i o n s p r o v i d e a component of the flow s t r e s s , because of the i n h e r e n t l y d i r e c t i o n a l c h a r a c t e r of such p i l e - u p s any such component would be reduced by a f a c t o r of ~cos60° = i when e i t h e r of the two other coplanar systems were used. Since no d i f f e r e n c e i s observed, these r e s u l t s are c o n s i s t e n t w i t h a view o f flow s t r e s s determined by the g l i d e / f o r e s t i n t e r a c t i o n . However we cannot conclude t h a t p i l e - u p s do not e x i s t . Rather i t must be assumed t h a t these back s t r e s s e s do not make a s i g n i f i c a n t c o n t r i b u t i o n to the flow s t r e s s . The e t c h p i t s t u d i e s (of B a s i n s k i and B a s i n s k i , 1964 f o r example) p r o v i d e a d d i t i o n a l evidence c o n s i s t e n t w i t h t h i s view of the importance of the g l i d e f o r e s t i n t e r a c t i o n . The low temperature flow s t r e s s f o r copper c r y s t a l s o r i e n t e d f o r s i n g l e g l i d e i s found to be p r o p o r t i o n a l to the square r o o t of the average f o r e s t d i s l o c a t i o n d e n s i t y . (This d e n s i t y i s c o n v e n i e n t l y measured by e t c h i n g the primary s l i p plane.) However, such a r e l a t i o n s h i p i s l e s s w e l l d e f i n e d 54 for densities which include g l i d e dislocations (densities determined by etching non-primary s l i p planes). Hence these results are also consistent with the view that the g l i d e / forest i n t e r a c t i o n i s flow stress determinant. Against t h i s background of experimental evidence, t h e o r e t i c a l support for the importance of forest d i s l o c a t i o n s as obstacles to g l i d e i s provided by analyses of d i s l o c a t i o n i n t e r s e c t i o n mechanisms. When a gl i d e d i s l o c a t i o n encounters a forest, one of two events may occur, depending on the p a r t i c u l a r combination of d i s l o c a t i o n Burgers vectors: either the g l i d e d i s l o c a t i o n cuts through the forest and jogs are produced, or there i s a d i s l o c a t i o n reaction (a junction reaction) i n which a s e s s i l e segment of d i s l o c a t i o n i s produced, and the glide d i s l o c a t i o n must bow round or break this pinned seg-ment i f i t i s to proceed further. Details of possible d i s -location i n t e r s e c t i o n mechanisms have been summarized by Hirth and Lothe (1970) who also present some attempted calculations of i n t e r a c t i o n energies. They conclude that, although on the basis of t h e i r calculations i t i s not possible to uniquely i d e n t i f y the important reaction, any one of several i n t e r s e c t i o n mechanisms could account for macroscopic deformation phenomena. Hence, with p a r t i c u l a r reference to g l i d e / f o r e s t i n t e r s e c t i o n , any such mechanism could account for the relationship between forest d i s l o c a t i o n density and flow stress. 55 Whilst a single forest d i s l o c a t i o n represents a plausible obstacle for a segment of g l i d e d i s l o c a t i o n , a glide d i s l o c a t i o n loop w i l l i n general encounter a large number of forest dislocations simultaneously. This leads to the introduction of s t a t i s t i c s to analyse the g l i d e of a d i s -location through a f i e l d of obstacles. Moreover since strong g l i d e / f o r e s t i n t e r a c t i o n mechanisms occur over distances small compared to the average forest spacing, the forest d i s l o c a -tions may be treated as a f i e l d of p o i n t - l i k e obstacles. Then i f a glide d i s l o c a t i o n loop can be blocked by such a f i e l d , a s t a t i s t i c a l analysis should lead to a theory of s l i p l i n e length. 1.4.3 S t a t i s t i c a l Blocking of S l i p Lines 1.4.3.1 Introduction The s t a t i s t i c a l theories of s t r a i n hardening (Kocks, 1966; Alden, 1972) propose that s l i p l i n e s are blocked by inter a c t i o n between gli d e dislocations and non-regular d i s t r i -butions of forest dislocations; i n contrast with the pre-viously discussed theories, these s t a t i s t i c a l theories specify no p a r t i c u l a r glide-blocking d i s l o c a t i o n arrays. Within such a framework, i t i s s t i l l possible for s l i p l i n e s to be "blocked," for example at unpenetrable areas of p a r t i c u l a r l y high forest d i s l o c a t i o n density (Kocks, 1966). However, the important feature of a s t a t i s t i c a l theory i s that 56 the size of these impenetrable areas is stress dependent. Consequently, i n a given microstructure, the s l i p l i n e length w i l l depend upon the applied stress at which the s l i p l i n e was formed. Furthermore, with p a r t i c u l a r reference to the present experiments, for two c r y s t a l s having the same micro-structure, but which have undergone steady state deformation at d i f f e r e n t respective temperatures (and hence at two d i f f e r e n t levels of applied s t r e s s ) , the r e s u l t i n g s l i p l i n e lengths should d i f f e r (Alden, 1972). In the proceeding discussion i t w i l l be shown that the temperature dependence of s l i p l i n e length i s consistent with s t a t i s t i c a l theories of s t r a i n hardening i n which glide loops are able to expand over a newly available "free" area of s l i p plane, after a thermally activated process. At present, only one such theory exists (Alden, 1972). However there i s another, as yet unproposed v a r i a t i o n of th i s theory which i s also consistent with the present r e s u l t s , and t h i s w i l l also be discussed. In addition i t w i l l be shown that, although Kocks 1 s t a t i s t i c a l analysis (1966) forms the basis for successful i n t e r p r e t a t i o n of the present r e s u l t s , Kocks 1 (1966) theory i t s e l f predicts no i s o s t r u c t u r a l change i n s l i p l i n e length with temperature. 1.4.3.2 Kocks' S t a t i s t i c a l Model S t a t i s t i c a l analysis of the expansion of a gli d e d i s l o c a t i o n loop through a f i e l d of randomly placed point 57 obstacles was f i r s t performed by Kocks (1966) and by Foreman and Makin (1966). In.these analyses, the point obstacles act i n pairs to block segments of expanding g l i d e d i s l o c a t i o n s , and the segments can continue to move only i f the stress i s s u f f i c i e n t l y high. While there are some minor differences i n the d e t a i l s of these two analyses, the r e s u l t s are e s s e n t i a l l y the same. Both show that there exists a c r i t i c a l stress above which such an obstacle f i e l d becomes transparent to g l i d i n g d i s l o c a t i o n s . (This stress may be c a l l e d the athermal g l i d e stress, since at t h i s stress g l i d e can occur without the need for thermally activated processes.) Moreover, at lower stress the size of the penetrable area, from hereon referred to as the free area, i s shown to increase with the applied stress. Since Kocks' analysis forms part of his theory of s t r a i n hardening, his results w i l l be used to i l l u s t r a t e this stress dependence. The three diagrams from Kocks' analysis, shown i n Figure 20 (a), (b), (c), i l l u s t r a t e the s t a t i s t i c a l l y determined free areas at three d i f f e r e n t levels of applied s t r e s s , cr (which i s expressed as a f r a c t i o n of the athermal g l i d e stress, T y ) • Whereas the inpenetrable regions are those i n which most or a l l points are joined by l i n e s , the free areas are shown surrounded by thicker l i n e s and contain mostly unconnected point obstacles. I t can be seen that whereas at CT/T = 0.74 the free areas are separate and are 58 F i g u r e 20. Diagrams from Kocks' o r i g i n a l s t a t i s t i c a l a n a l y s i s (1966) . 59 completely surrounded by an impenetrable region of g l i d e plane (Figure 20 (a)), at a/x y = 1.04 (Figure 20 (c)) , the free area i s large, covering a l l of the s l i p plane with the exception of a few remaining small impenetrable areas of exceptionally high obstacle density. Figure 20 (b) shows the free area at an intermediate stress, a/x =0.90. The y f r a c t i o n of obstacle f i e l d which i s free area was determined by Kocks at f i v e stress levels and i s shown i n Figure 20 (d). Thus the amount of free area can be seen to change from an indeterminately.small value at low stress, to a large value at a stress near the athermal g l i d e stress. 1.4.3.3 S l i p Line Blocking i n Kocks' Theory of  Strain Hardening Using the diagrams reproduced i n Figure 20 to repre sent a portion of glid e plane threaded by forest d i s l o c a t i o n , Kocks takes this analysis as a basis for his theory of flow stress and s t r a i n hardening (1966), which, since i t includes no thermally activated processes i s taken to apply at 0°K. On thi s theory, the mechanical phenomenon of steady state deformation (see Appendix 2) i s associated with the gl i d e of dislocations over i n d e f i n i t e l y large areas of s l i p plane, which occurs when the.athermal g l i d e stress i s attained. This theory explains the limited length of s l i p l i n e s by the existence of areas which have p a r t i c u l a r l y high 60 l o c a l d i s l o c a t i o n densities; a f t e r the g l i d e d i s l o c a t i o n has athermally swept almost the whole s l i p plane (at the athermal glide s t r e s s ) , small "islands" of unslipped area remain surrounded by portions of g l i d e d i s l o c a t i o n s . Thus a s l i p l i n e formed on a part of the c r y s t a l surface which happened to in t e r s e c t an unslipped area would be broken as shown i n Figure 21 (a). In a l a t e r paper, Kocks (1967) adapts t h i s analysis to the treatment of obstacles which may be overcome with the aid of thermal a c t i v a t i o n . Nonetheless, he s t i l l views the steady state flow stress as "the stress at which a (glide) d i s l o c a t i o n can proceed i n d e f i n i t e l y . " As a r e s u l t , i n neither of the two variants of t h i s theory are s l i p l i n e s blocked i n the usual sense; they have limited length because of the existence of small areas of high d i s l o c a t i o n density surrounded by 'negative p i l e - u p s 1 , which happen to be i n t e r -sected by the c r y s t a l surface (Figure 21a). Hence s l i p l i n e * length i s a structure dependent phenomenon only; as was discussed for the theories of Seeger (1957) and of Hirsch and M i t c h e l l (1967), a theory of s l i p l i n e length i n which l i n e s are blocked by s p e c i f i c d i s l o c a t i o n arrays cannot account for changes i n s l i p l i n e length i n i s o s t r u c t u r a l experiments. 5* A p a r a l l e l d e v e l o p m e n t o f t h e a n a l y s i s o f F o r e m a n a n d M a k i n (1966) w o u l d be s u b j e c t t o t h e s ame o b j e c t i o n s . 61 Figure 21. Schematic of Primary s l i p plane showing s t a t i s t i c a l blocking of s l i p l i n e s during steady state flow, (a) as postulated by Kocks (1966), (b) an addi t i o n a l blocking mechanism f i r s t postulated by Alden (1972), which predicts temperature dependent s l i p l i n e length. 62 1.4.3.4 S t a t i s t i c a l Interpretation of the Var i a t i o n of S l i p Line Length with Temperature (a) Slip Line Blocking in Alden's Theory of  Strain Hardening The previously discussed analysis of Kocks (1966) has been developed i n t o a theory of temperature dependent flow stress and s t r a i n hardening by Alden (1972). As with Kocks' o r i g i n a l (1966) theory, the obstacles are taken to be athermal. However, i n Alden's theory, the area over which a d i s l o c a t i o n may g l i d e i s i n d i r e c t l y thermally controlled. , This temperature dependence of the free area repre-sents a s i g n i f i c a n t difference between the theories of Kocks and Alden. In the l a t t e r theory, steady state flow occurs at a temperature dependent stress, a(T), generally below the athermal g l i d e stress, T . Thus steady state flow may occur for example at O{1\)/T^ = 0.74, where the size of free areas may be s i m i l a r to those shown i n Figure 20 (a), and Figures 20 (b), (c) could i l l u s t r a t e the free area during steady state deformation at an. intermediate temperature T 2(<T X) and 0°K respectively. The temperature dependence of s l i p l i n e length i s a d i r e c t consequence of t h i s steady state flow at variable stresses below the athermal g l i d e stress. Figures 21 (a), (b) schematically i l l u s t r a t e s l i p l i n e production during steady state flow, respectively at and below the athermal 63 glide stress. (and thus respectively at and above 0°K). I t can be seen that, whereas at 0°K (a/x = 1.04) s l i p l i n e s are "blocked" only when the small, high density "islands" of unslipped g l i d e plane happen to coincide with the surface (Kocks' blocking condition), at a higher temperature (repre-sented here by o*/x = 0.74) s l i p l i n e s may also be blocked in.a somewhat d i f f e r e n t sense, being completely surrounded by the unpenetrable region. In consequence, the s l i p l i n e s produced at higher temperatures are shorter, i n agreement with the present r e s u l t s . In Alden's theory the Kocks' analysis (1966) i s taken to be d i r e c t l y applicable to a microstructure at 0°K, where the microstructure i s stable, and no thermal penetra-ti o n of forest dislocations can occur. Furthermore, following Saada (1963), who has shown that the a t t r a c t i v e trees of the forest i n t e r a c t with g l i d e dislocations over a long range, i t i s assumed that the e f f e c t i v e strength of the d i s l o c a t i o n forest does not vary with temperature (after a correction for e l a s t i c modulus change). The athermal gl i d e stress, x , i s c a l l e d the y i e l d strength (a property of the s o l i d ) , and defined by x y (1.3) or i n a s t r a i n hardened c r y s t a l 64 T = a" Gb (p)* - u vp; (1.4) In Eq. (1.3) 1 i s the average nearest neighbour spacing of the obstacles; i n Eq. (1.4) p i s the average density of forest dislocations; a 1 and a" are empirical constants with values close to unity. In the present experiments, the structure i s nearly constant, but the stress varies with temperature. I t i s higher at low temperature and as a r e s u l t most obstacle pairs w i l l be penetrable; the free area w i l l be large and the s l i p l i n e s long. When we consider deformation i n the same microstructure at high temperatures because the flow stress i s lower, areas through which di s l o c a t i o n s may g l i d e w i l l be smaller and hence, s l i p l i n e s w i l l be shorter. A n a l y t i c a l l y , the free area a at a given tempera-ture and stress i s contained i n the r e l a t i v e area function A__ A = A r r t— a AI (1-5) Accord ing to A l d e n ' s t heo ry , the change in f low s t r e s s with temperature is due to the change in rates of r e -arrangement and recovery of f o r e s t d i s l o c a t i o n s . C l e a r l y s t r u c t u r e changes per se cannot be r e s p o n s i b l e f o r a r e v e r s i b l e i s o s t r u c t u r a 1 change in f low s t r e s s . The present exper iments were des igned to min imize s t r u c t u r e changes, by us ing smal l s t r a i n increments and high p r e s t r a i n temperatures . However the changes in rates of rearrangement and recovery w i l l be l a r ge s i nce the s t r a i n increments are c a r r i e d out over a l a rge temperature range. 65 where A i s a large area of s l i p plane containing a 'true average 1 density of obstacle d i s l o c a t i o n s . The function of stress a and structure suggested for A i s r A^ = exp - T — 0~' - 1 (1.6) T V i s defined by Eq. (1.4) and x v decreases with increasing reg u l a r i t y of the structure. In the 0°K case the stress a equals x during steady state, and when steady state i s attained, the free area equals A. Thus the s l i p l i n e s are predicted to have t h e i r maximum length. At higher temperature, the rate of thermally activated structure change, including loss (recovery) and rearrangement of obstacle d i s l o c a t i o n s , increases. Regions of lower obstacle density w i l l then deform preferen-t i a l l y . In other words, a i s less than A and the s l i p l i n e length i s reduced. This concept of a changing free area leads also to a microstructural explanation of the (isostructural) v a r i a t i o n i n flow stress with temperature. From Eqs. (1.5) and (1.6) i t i s seen that when a < A, a < x . So while ' y steady state flow occurs at a stress below the athermal g l i d e stress, at the same time d i s l o c a t i o n loops expand over areas of s l i p plane containing an obstacle density less than the average density. In t h i s manner the mechanical 66 e f f e c t , namely the temperature dependent flow st r e s s , and the microstructural e f f e c t , v a r i a t i o n i n s l i p l i n e length are coupled. (b) S l i p Line Blocking using a Variation  of Alden's Theory As an alternative to thermally activated structure change as the o r i g i n of temperature dependent p l a s t i c i t y , one may consider thermally activated g l i d e . In other words, rather than taking the "glide" of forest dislocations as the thermally activated process, we may a l t e r n a t i v e l y consider the thermally activated g l i d e of g l i d e d i s l o c a t i o n s . However, i t should be noted that, i n order to successfully account for the present r e s u l t s , t h i s alternative proposal remains within the framework of Alden's theory. Hence, t h i s new thermal ac t i v a t i o n process i n i t s e l f produces l i t t l e s t r a i n . Rather i t releases g l i d e dislocations into newly available free areas, which are then traversed athermally. ' . An i n t e r e s t i n g s t a t i s t i c a l basis for such a proposal has recently been published by Morris and Klahn (1973). This analysis adds a s t a t i s t i c a l c r i t e r i o n for a thermally activated g l i d e / f o r e s t i n t e r a c t i o n , to the o r i g i n a l analyses of Kocks (1966), and Foreman and Makin (1966). The thermally activated step i s taken to be the movement of a segment of d i s l o c a t i o n l i n e through one of the obstacles at which the 67 glid e d i s l o c a t i o n has been temporarily s t a b i l i z e d . The force distance curve for the dislocation/obstacle i n t e r a c t i o n i s taken as a step function, with thermal a c t i v a t i o n being treated as a stochastic, random process. Moreover, the thermally activated step i s assumed to be.rate c o n t r o l l i n g , i n that the time required for d i s l o c a t i o n g l i d e between stable positions i s n e g l i g i b l e compared to the time required f o r thermal ac t i v a t i o n . The d i s l o c a t i o n l i n e may then proceed from one stable p o s i t i o n to another along a s t a t i s t i c a l l y chosen path, under the combined influence of the applied stress, and thermal a c t i v a t i o n . In a computer simulation of t h i s process, Morris and Klahn (1974) successfully i l l u s t r a t e the p o s s i b i l i t y of steady state, thermal ac t i v a t i o n controlled g l i d e at stresses well below the athermal g l i d e s t r e s s , by obtaining average glide velocity/temperature data for a given obstacle strength, at d i f f e r e n t applied stress l e v e l s (Figure 22). Whilst i t i s i s d i f f i c u l t to l i n k t h i s kind of data to s l i p l i n e lengths, i t should be possible to use the Morris and Klahn analysis to compute the area swept by a gli d e d i s l o c a t i o n when i t moves between stable positions, as a function of applied stress, for a given obstacle strength, at various temperatures. This function w i l l then be d i r e c t l y analogous to that pro-posed by Alden (Eq. 1.6), since i n both cases the g l i d e processes which generate s l i p l i n e s are i n i t i a t e d by thermal o </> E o 8 3 ~ 1.8 x 10 o G CD > _ 6 0 • <x> aj E . S. 5 0 o CL S 4 0 > 10 10 c o (/) c • <u E > V c 3 0 2 0 0 0 0.45 T y / increasing velocity 0 increasing temperature • 0 0 2 0 0 3 0 0 4 0 0 CL (Dimensionless Temperature Parameter) 5 0 0 F i g u r e 22. a = 260 a = 68 a = 35 O" = 0 .90Ty O-=0.69xy O"=0.45Ty Data from a n a l y s i s o f M o r r i s and Klahn (1974) showing v e l o c i t y / t e m p e r a t u r e r e l a t i o n s h i p s f o r f o u r separate random o b s t a c l e a r r a y s a t three d i f f e r e n t a p p l i e d s t r e s s l e v e l s . T h e i r schematic below shows p r e d i c t e d appearance of s l i p l i n e s . An e s t i m a t i o n of a c t u a l v e l o c i t i e s u s i n g c o n d i t i o n s from p r e s e n t experiments i s added ( l e f t ) . 00 69 a c t i v a t i o n ; the only d i f f e r e n c e between the models i s the s p e c i f i e d thermal a c t i v a t i o n s t e p . 1.4.3.5 Temperature Dependent S l i p Band Widths  and Heights An e x t e n s i o n of the M o r r i s and Klahn a n a l y s i s to the simultaneous deformation of p a r a l l e l s l i p planes leads t o an i n t e r e s t i n g q u a l i t a t i v e d e s c r i p t i o n o f the appearance of s l i p l i n e s produced at d i f f e r e n t temperatures. Included i n F i g u r e 22 are data f o r d i s l o c a t i o n movement through f o u r separate o b s t a c l e a r r a y s . At low temperatures, where a r e l a t i v e l y h i g h s t r e s s (0.9 r ) i s r e q u i r e d , the g l i d e v e l o c i t i e s determined f o r the fo u r a r r a y s d i f f e r w i d e l y . By imposing a f i x e d s t r a i n r a t e , and assuming a uniform d i s t r i b u t i o n o f g l i d e d i s l o c a t i o n , M o r r i s and Klahn (1974) show t h a t a s t r a i n increment i n a c r y s t a l c o n t a i n i n g these f o u r planes i s achieved by g l i d e on one plane only; hence a s i n g l e s l i p l i n e w i l l be formed. At a h i g h e r tempera-tu r e and r e l a t i v e l y lower s t r e s s (0.45 T ) a l l f o u r g l i d e v e l o c i t i e s are equal, and thus the s t r a i n i s e q u a l l y d i s t r i b u t e d . S t r a i n a t an i n t e r m e d i a t e temperature g i v e s a combination o f these two r e s u l t s , producing a range of step h e i g h t s a t the c r y s t a l s u r f a c e . Thus assuming a uniform d i s t r i b u t i o n of g l i d e d i s l o c a t i o n s , t h i s model p r e d i c t s an i n c r e a s i n g l y homogeneous d i s t r i b u t i o n o f s t r a i n w i t h tempera-t u r e , i n a g i v e n m i c r o s t r u c t u r e . 70 Micrographs of s l i p l i n e s (more pre c i s e l y termed s l i p "bands"), formed i n the same microstructure at d i f f e r e n t temperatures are shown i n Figure 11. I t can be seen that s l i p bands formed at 4.2°K appear to be sharply defined when compared with the r e l a t i v e l y d i f f u s e bands formed at 293°K. If the Morris and Klahn model i s applied to a number of neighbouring s l i p planes which are active during the 293°K increment, and i n which a r e l a t i v e l y uniform d i s t r i b u t i o n of mobile dislocations may be anticipated at both tempera-tures, the r e l a t i v e appearance of s l i p bands i s c o r r e c t l y predicted. However, as can be seen i n both figu r e s , the s l i p band density (the number of bands per u n i t length perpendicular to the band direction) i s apparently lower a f t e r the 293°K increment than that of bands formed at 4.2°K. Thus while s t r a i n i n neighbourhood of a 293°K s l i p band has been more homogeneous than i t would have been at 4.2°K, s t r a i n i n general has been less so. This l a t t e r e f f e c t may provide a clue to the predominant d i s l o c a t i o n m u l t i p l i c a t i o n mechanism which w i l l operate to supply s u f f i c i e n t g l i d e d i s l o c a t i o n s to maintain the imposed s t r a i n rate. At low temperatures, where steady state flow occurs at stresses approaching the athermal gl i d e stress, Frank-Read sources are assumed to be operative, and these may supply g l i d e dislocations on a larger proportion of s l i p planes. In contrast at higher 71 temperatures, steady state flow occurs at lower stresses, so there w i l l be fewer available Frank-Read sources. In this case a multiple cross-glide mechanism of the type f i r s t proposed by Koehler (1952) may supply additional sources on adjacent s l i p planes, since c r o s s - s l i p i s a temperature dependent phenomenon. As a d i r e c t consequence of such a mechanism, there w i l l be more s l i p l i n e c l u s t e r i n g (into bands) at higher temperatures. Whereas this e f f e c t cannot be anticipated from a t h e o r e t i c a l model which assumes a uniform d i s t r i b u t i o n of g l i d e d i s l o c a t i o n s , these two m u l t i p l i -cation mechanisms lead to an appropriate description of the nature of s l i p at d i f f e r e n t temperatures. 1.4.4 A Unified View of S l i p Line Formation, Microstructural Observations and Flow Stress In the preceding discussion several deductions have been made from the results of this s l i p l i n e study and other work, concerning the nature of p l a s t i c flow i n copper. Since the discussion was necessarily extended, these deduc-tions might appear to be r e l a t i v e l y indpendent. However, in t h i s section i t w i l l be shown that, when taken together, they form a new and self-consistent picture of p l a s t i c deformation i n copper. S l i p l i n e lengths i n copper are generally much greater than the average diameter of d i s l o c a t i o n c e l l s . 72 The reason for this e f f e c t i s revealed on closer examination of the c e l l construction. Only the sides of c e l l walls lying p a r a l l e l to primary s l i p plane are well-defined 'strong' obstacles, being formed from the dense carpets of primary d i s l o c a t i o n multipoles which are c h a r a c t e r i s t i c of stage II microstructures i n copper (Kuhlman-Wilsdorf, 1968; Seeger, 1968). Since they are r e a d i l y i d e n t i f i e d i n metallographic sections, i t i s the spacing of these carpets which i s usually taken as a measure of the c e l l diameter. The c e l l s are completed by more di f f u s e arrays of forest dislocations lying roughly perpendicular to the multipole carpets. New gli d e d i s l o c a t i o n loops w i l l expand on the primary s l i p plane i n the r e l a t i v e l y s o f t regions between these carpets, move through arrays of forest dislocations at r e l a t i v e l y high speed (Fisher and L a l l y , 1967), and be blocked not by c e l l "walls" per se, but by s t a t i s t i c a l interactions with the d i s l o c a t i o n forest. In this manner, d i s l o c a t i o n loops expand and s l i p l i n e s are formed over distances large compared with the apparent c e l l s i z e . S l i p l i n e s are usually clustered into bands, and i t was observed i n t h i s study that the range of spacing of the more prominent bands as seen i n both o p t i c a l and electron micrographs, namely 2-10y, coincides with the independently estimated spacing of the dense carpets of dislocations found in the microstructure. The occurrence and spacing of these 73 bands can be explained within the framework of the preceding analysis. In addition a s i g n i f i c a n t feature of the proceeding explanation i s that g l i d e loops, i n t e r a c t i n g s t a t i s t i c a l l y with forest dislocations expand through regions of the c r y s t a l uncluttered by d i s l o c a t i o n debris, and hence the previous explanation of the v a r i a t i o n of s l i p l i n e length with temperature i s supported. The bands are formed and widened as a r e s u l t of the accumulation of d i s l o c a t i o n debris during p l a s t i c flow. I n i t i a l l y g l i d e d i s l o c a t i o n loops expand only on one s l i p plane as shown i n Figure 23 (either because t h i s plane i s intersected by forest dislocations more appropriately placed for s t a t i s t i c a l i n t e r a c t i o n (Morris and Klahn, 1974), or because a favourably positioned source i s available on this plane). After a number of loops have expanded i n th i s plane, accumulated d i s l o c a t i o n debris prevents the further operation of th i s g l i d e source, and the mechanism of multiple cross g l i d e (Koehler, 1952; Johnson and Gilman, 1959) provides a new source on the neighbouring s l i p plane. The process i s repeated on successive neighbouring planes and thence the s l i p band i s formed. Thus while the s l i p l i n e length i s controlled by the s t a t i s t i c a l i n t e r a c t i o n of a d i s l o c a t i o n loop expanding athermally through an area of s l i p plane uncluttered by d i s l o c a t i o n debris, t h i s debris causes dislocations to cluster into bands of f i n i t e width. s B F i g u r e 23. Cross-section perpendicular to primary glide plane showing slip line blocked xn schematic cell structure: source S near surface of multipole carpets fSresl'secSons T f ' T e l f t t f , 1 0 ^ / ^ ^ - P a n d s , threading'tS^ gh^ ??^  e rorest sections of cell wall, and.is statistically blocked at B! 75 This account of s l i p band formation i n turn explains the formation and thickening of dense carpets of d i s l o c a t i o n s . The debris created after the expansion of a number of d i s -location loops on neighbouring s l i p planes w i l l contain a large proportion of primary di s l o c a t i o n s arranged i n the form of multipoles. As the s l i p band i s progressively formed the multipoles accumulate i n high density planar arrays, p a r a l l e l to the primary s l i p plane. With further deformation, pinned dislocations on the surface of these arrays w i l l provide a s i g n i f i c a n t proportion of the new sources. As a re s u l t , more multipoles w i l l be swept into a progressively denser carpet of di s l o c a t i o n s . Thus the carpets are i n f a c t the somewhat distorted remnants of previously formed s l i p l i n e s . •A self-consistent view has been presented which, couples the properties of s l i p l i n e s with microstructural observations. As was shown i n the previous discussion, a view of p l a s t i c flow i n which expanding g l i d e d i s l o c a t i o n loops intera c t s t a t i s t i c a l l y with forest d i s l o c a t i o n s , can also account for the v a r i a t i o n of flow stress with temperature. Hence this micromechanical picture of the process of p l a s t i c deformation successfully l i n k s microstructural observations with both the properties of s l i p l i n e s , and the temperature dependence of flow stress. 76 1.4.5 Transition Effects Discussion so far has been exclusively concerned with an explanation of the properties of s l i p l i n e s formed during steady state flow, and while i t i s apparent that a s t a t i s t i c a l approach i n which a thermally activated process i s followed by athermal g l i d e , can successfully describe the properties of s l i p l i n e s formed during steady state flow, some mention should be made of t r a n s i t i o n e f f e c t s (microstrain). If at any instant i n the deformation a sudden change i s made i n either s t r a i n rate or temperature (which i s i d e a l l y what has been done i n these experiments), no sudden change i n stress i s predicted by such a theory. S p e c i f i c a l l y , on a decrease of temperature, there w i l l be a t r a n s i t i o n s t r a i n during which the stress and free area r i s e to t h e i r new steady state value (Figure 15). (This t r a n s i t i o n s t r a i n on increase of s t r a i n rate has been observed for deformation of Pb at various temperatures (Alden, 1973; Clark and Alden, 1973; Alden, 1974).) Hence lines produced during the t r a n s i t i o n w i l l be shorter than those produced during steady state flow, .and these addi t i o n a l short lines w i l l appear along with l i n e s whose lengths represent the steady state free area. As can be seen i n Figure 9, a number of short l i n e s are indeed observed at 4.2°K,in addition to the high proportion of long lines, which are presumed to be produced during steady state when the free area i s large. 77 An increase i n s l i p l i n e length during the t r a n s i -t i o n s t r a i n has also been observed (deLarios, 1973). Pre-strained and polished aluminum single c r y s t a l s were strained i n a sequence of small increments through the t r a n s i t i o n region. The average l i n e length was found to increase, reaching a maximum immediately af t e r the t r a n s i t i o n ; with further s t r a i n the average length decreased as expected (Seeger, 1957). P A R T 2 MICROSTRAIN AND ETCH PIT STUDIES 2.1 Introduction The r e s u l t s of the s l i p l i n e studies i n Part I of t h i s thesis are consistent with s t a t i s t i c a l theories of p l a s t i c flow i n which, at f i n i t e temperatures, a thermally activated process enables a g l i d e d i s l o c a t i o n loop to athermally expand into a newly available area of s l i p plane (the "free area"). Moreover, the r e l a t i v e lengths of s l i p lines formed i n a given microstructure can be explained by considering the stress dependence of t h i s free area. I t i s also anticipated that the s i z e of t h i s free area at a f i x e d stress w i l l depend on the density and d i s t r i b u t i o n of the d i s l o c a t i o n microstructure. I t has already been shown (Kocks, 1966) that the c r i t i c a l stress at which the free area becomes large i s a function of the o v e r a l l d i s l o c a t i o n density, for a random f i e l d of obstacles. However, dislocations i n experimentally observed microstructures usually appear to be f a r from random, being commonly described as tangled, clustered or cellular, 78 79 and i t i s reasonable to expect that the free area w i l l be a function of the degree of such c e l l u l a r i t y . C l e a r l y , at constant d i s l o c a t i o n density the more c e l l u l a r the microstructure, the greater w i l l be the area f r a c t i o n with low d i s l o c a t i o n density. Given that the c e l l walls do not form into contiguous barriers that block s l i p (which i s shown i n the previous discussion to be unlikely i n the stage II hardening region of copper crystals), the free area should increase more rapidly with stress i n a more c e l l u l a r microstructure (Kocks, 1966; Alden, 1972). The size of the free area w i l l influence the micro-p l a s t i c response of a c r y s t a l to an increase i n applied stress. When t h i s stress r i s e s from zero to the y i e l d stress, the free area i s expected to increase i n size continuously, reach-ing a maximum at steady state flow (for d e f i n i t i o n s of deformation terminology see Appendix 2). Assuming that d i s -location sources are scattered and r e l a t i v e l y abundant, the amount of microstrain ( p l a s t i c deformation i n the p r e y i e l d region) w i l l be a d i r e c t function of the size of the free area at any given l e v e l of applied stress. Taken together, these arguments mean that c e l l u l a r microstructures should exhibit more pronounced microstrain. In order to tes t t h i s prediction, high resolution s t r e s s -s t r a i n curves were obtained from specimens which d i f f e r e d mainly i n the degree of c e l l u l a r i t y of t h e i r d i s l o c a t i o n 80 microstructure. Almost a l l previous microstrain studies i n copper have been performed on annealed c r y s t a l s (Rosenfield and Averbach, 1960; Hordon, 1962; Tinder and Washburn, 1964), and use a multiple load cycle (Banerji et cel., 1970), or a hysteresis technique, often i n an attempt to discover a lower l i m i t for the c r i t i c a l resolved shear stress. In t h i s case however, i n order to investigate p l a s t i c flow i n c r y s t a l s i n a work hardened state, the samples are prestrained. Moreover i n the present work, the microstrain curve i s obtained from a single load cycle, to avoid any transient or i r r e v e r s i b l e effects from repeated load c y c l i n g . In order that the degree of c e l l u l a r i t y of these specimens could be quantified, the microstructures of an i d e n t i c a l series of c r y s t a l s were examined using the d i s -location etch p i t technique. Whereas d i s l o c a t i o n arrangements i n copper have been extensively studied i n a q u a l i t a t i v e manner, only since the beginning of t h i s work have sampling techniques been used to measure l o c a l d i s l o c a t i o n density variations (Donner et al.3 1974). A detailed knowledge of d i s l o c a t i o n arrangements may be valuable for the analysis of mechanical properties, and the present work probably repre-sents the f i r s t use of a s t a t i s t i c a l technique to quantify the degree of c e l l u l a r i t y of a d i s l o c a t i o n microstructure. 81 2.2 Experimental Technique 1 2.2.1 Specimen Preparation Seeded copper single c r y s t a l s were grown using the technique previously described. Since the etch p i t method of measuring d i s l o c a t i o n densities i s l i m i t e d to r e l a t i v e l y low d ensities, i t i s important to reduce densities of grown-in dislocations and dislocations introduced by handling. In th i s case specimens were cut to length using an acid saw (South Bay Technology, Model No. 750) i n order to prevent any damage to the as-grown c r y s t a l s . Previous work (Livingston, 1962) has shown that c y c l i c annealing (annealing at a tempera-ture which continuously cycles between a maximum and minimum) is more e f f e c t i v e than a s t a t i c anneal i n reducing post anneal d i s l o c a t i o n density. Consequently a l l c r y s t a l s were annealed for 72 hours i n a vacuum of 10" 5 t o r r , with temperatures cycled hourly between 795°C and 1045°C. As w i l l be discussed l a t e r , this technique was successful i n reducing the d i s -location content, and increasing the grown-in subgrain s i z e . of annealed c r y s t a l s . 2.2.2 Crystal Orientation Crystals used i n the microstrain and etch p i t study had t e n s i l e axes oriented as shown i n Figure 24. This was a conveniently grown orientation which maintained single g l i d e 82 Figure 24. Orientation of t e n s i l e axis f o r c r y s t a l s used i n microstrain/etch p i t work. 0 = i n i t i a l o r i e n t a t i o n , X = orientation a f t e r p r e s t r a i n at 1000°K. 83 as the dominant deformation mode throughout the p r e s t r a i n s used i n t h i s experiments S i n g l e g l i d e was chosen s i n c e i t has been shown ( B a s i n s k i and B a s i n s k i , 1964) t h a t f o r e s t d i s l o c a t i o n d e n s i t i e s i n s i n g l e s l i p c r y s t a l s , determined on a cross s e c t i o n p a r a l l e l to the primary g l i d e plane, c o r r e l a t e w e l l w i t h low temperature flow s t r e s s i n copper. 2.2.3 Mechanical T e s t i n g 2.2.3.1 P r e s t r a i n In order to explore the i n f l u e n c e of v a r i a t i o n i n l o c a l d i s l o c a t i o n d e n s i t y on the m i c r o s t r a i n behaviour of copper c r y s t a l s , a s e r i e s of specimens was re q u i r e d w i t h com-parable d i s l o c a t i o n d e n s i t i e s , but d i f f e r e n t local d i s l o c a t i o n d e n s i t i e s . I t was a n t i c i p a t e d t h a t substructures w i t h d i f f e r e n t d i s l o c a t i o n arrangements could be obtained by changing the temperature of p r e s t r a i n . With t h i s i n mind, p r e s t r a i n s were c a r r i e d out a t temperatures, and i n the environments shown below: I000°K Vacuum F u r n a c e 8 5 0 ° K 11 " 700°K " " .293° K A i r 77°K N i t r o g e n Gas C r y o s t a t 84 By using published data ( M i t c h e l l / 1964) on the reversible change i n flow stress with temperature, the attempt was made to produce specimens with the same 77°K flow stress (and o v e r a l l d i s l o c a t i o n density). Some attempt was also made to anticipate differences i n cross-section area r e s u l t i n g from the d i f f e r i n g amounts of pr e s t r a i n (greater at higher temperatures). Hence at any given temperature and pr e s t r a i n , the load required to give s i m i l a r d i s l o c a t i o n densities was determined. ' The prestrains were performed on f l o o r model Instron machines, using the lowest available cross head speed. Above 273°K th i s was 2 x lO - 1* inch min" 1, and at 293°K and below, i t was 2 x 10" 3 inch m i n - 1 . Strains were measured from cross,..head movement for the former temperatures: f o r the l a t t e r an Instron Extensometer was used, together with a Honeywell x-y recorder. :'./v-..--/-/Plots of resolved shear stress versus shear s t r a i n wereCcomputed with'the"aid of a program written for a Hewlett-Packard/ 9100 A calculator. These curves are shown i n Figure 25. 2.2.3.2 Microstrain Testing One of the objectives of thi s study was to obtain stress s t r a i n curves of the various prestrained samples at low temperature, and at a s e n s i t i v i t y s u f f i c i e n t to reveal q u a l i t a t i v e differences i n the shapes of those curves. Percent Resolved Shear Strain F i g u r e 25. P r e s t r a i n curves f o r c r y s t a l s used i n m i c r o s t r a i n / e t c h p i t work. 86 A series of preliminary experiments showed that s t r a i n differences of up to 2 x 10-1* were to be expected i n the microstrain region. After some t r i a l s , i t was decided to,use a 1/2 inch, 10% Instron extensometer, which was modified for use with single c r y s t a l s . This device was capable of measuring strains of less than 10" 5 i n a gaseous environment.at 77°K. Other considerations i n the choice of this extensometer, together with d e t a i l s of the design, testing and c a l i b r a t i o n procedure are discussed i n Appendix 3; an outline of precautions taken to guard against spurious s t r a i n measurements i s also given. , The signal from the extensometer was amplified using an Instron Load C e l l amplifier at maximum s e n s i t i v i t y , and the output was monitored on a low-response-time x-y recorder.: By thi s means the s t r a i n was measured to a sensi-t i v i t y of 7.4 x 10~ 6 (equivalent to one small d i v i s i o n of the elongation s c a l e ) . The amplified s i g n a l from the load c e l l was used to drive the y axis of the recorder. Using a C - c e l l loads of up to 40 lbs were measured with a f u l l scale d e f l e c t i o n of 5 lbs and 7 stages of zero suppression (achieved by operating the coarse balance c o n t r o l ) . . 2 • 2 • 4 Metallography 2.2.4.1 Introduction Microstructural studies were undertaken i n conjunc-tion with the mechanical t e s t i n g . The p r i n c i p a l objective of 87 these studies was to specify the degree of regularity of the microstructure. For every c r y s t a l prepared for microstrain te s t i n g , another was i d e n t i c a l l y prestrained and used for microstructural examination. In most cases, these two specimens were cut from the same parent c r y s t a l . The two p r i n c i p a l techniques used to observe d i s -location microstructures i n copper are transmission electron microscopy, and d i s l o c a t i o n etch p i t t i n g . The former technique has two important d e f i c i e n c i e s which make i t unsuitable for t h i s work. F i r s t l y , i t i s d i f f i c u l t to insure that no d i s l o c a t i o n movement or loss occurs during the preparation of thin f o i l s for transmission electron microscopy. Other workers (Esseman, 1965; Ramsteiner, 1967) sought to avoid this problem using low temperature neutron i r r a d i a t i o n to pin d i s l o c a t i o n before f o i l preparation. Even i f such a technique i s attempted, a second objection remains: since low magnification inspection of the micro-structure i s not possible, i t i s d i f f i c u l t to decide i f the areas chosen for l o c a l d i s l o c a t i o n density measurements are representative of the structure as a whole. Without th i s p o s s i b i l i t y , a large number of f o i l s would be needed to obtain a s t a t i s t i c a l l y v a l i d sample (Washburn and Murty, 1967). I t should be noted, however, that t h i s problem i s less acute when measuring overall d i s l o c a t i o n d e n s i t i e s , p a r t i c u l a r l y after prestrain to higher stress l e v e l s . 88 The etch p i t technique overcomes both these d i f -f i c u l t i e s . Low magnification inspection i s possible, pro-viding d i s l o c a t i o n densities are low enough to allow use of o p t i c a l microscopy; and observations are taken from the surface of a bulk specimen so that d i s l o c a t i o n movement and loss i s minimised. '. Any l o c a l d i s l o c a t i o n movement which may occur on unloading cannot be avoided. However, since both the microstrain and etch p i t specimens are unloaded before t h e i r subsequent te s t s , comparison of re s u l t s remains v a l i d . The o p t i c a l resolution of about 0.5y sets an e f f e c -t i v e upper l i m i t to measurement of etch p i t densities by o p t i c a l microscopy at 4 x 10 8 cm - 2. Furthermore, since observed d i s l o c a t i o n substructures are not completely homo-geneous, and often include c e l l walls and subgrain boundaries a density an order of magnitude less than t h i s i s desirable. Densities greater than 4 x 10 8 dislocations cm - 2 can be measured using shadowed r e p l i c a s , but t h i s method suffers from the sampling problems previously outlined. In the l i g h t of these considerations, a d i s l o c a -tion density of about 3 x 10 7 cm"2 was introduced into the crystals by prestraining. The nominal low temperature flow stress was 425g mm-2 (resolved shear s t r e s s ) . 89 2.2.4.2 Preparation of Crystals for Etching Orientation - The orien t a t i o n of a prestrained c r y s t a l was confirmed using Laue back r e f l e c t i o n X-ray d i f f r a c -t i o n . I t was subsequently cut p a r a l l e l to the primary s l i p plane using an acid saw. Following t h i s operation, the c r y s t a l was attached to a 45° s t e e l mount with s i l v e r p r i n t conducting adhensive and mounted on a goniometer i n the Laue back r e f l e c t i o n apparatus. A sequence of adjustments and diffractographs then served to a l i g n the primary s l i p plane p a r a l l e l to the back plate of the goniometer, within 1°. The goniometer design i s such that i t i s trans-ferable to a c r y s t a l facing instrument, and a smooth, accurately oriented surface may be obtained. Subsequent stages, to which the goniometer could be attached, to permit f i n a l e l e c t r o p o l i s h , inverted and normal stage o p t i c a l microscopy, were designed and b u i l t . By using the goniometer i n t h i s way, any oriented plane could be observed with high magnification o p t i c a l microscopy. Polishing - The use of o p t i c a l microscopy together with a d i s l o c a t i o n etch requires that the surface to be etched be o p t i c a l l y f l a t yet free from spurious damage. These c r i t e r i a are f u l f i l l e d using a c r y s t a l facing instrument (Model 451 from South Bay Technology) together with a f i n a l 90 e l e c t r o p o l i s h . In t h i s apparatus a 10 inch diameter s t a i n l e s s s t e e l rotating cathode i s covered with a f i n e no-nap cotton cloth (Beuchler Metcloth) which carr i e s a solution of two parts methanol to one part concentrated n i t r i c a cid at 35°C. The c r y s t a l (the anode) i s mounted on the goniometer, which rotates i n the opposite sense, and held against the wheel with minimum pressure. The rotation rates of the wheel and specimen and the current density are optimized to give a f l a t polished surface with l i t t l e edge rounding. Ideal surfaces produced i n t h i s way have good flatness and p o l i s h but exhibit l i n e a r traces whose depth was of the order of the thickness of the t w i l l e d threads of the cotton c l o t h . These were removed by e l e c t r o p o l i s h i n g for about one minute i n a separate, r a p i d l y s t i r r e d s o l u t i o n of the same e l e c t r o l y t e . A ten minute rinse i n s t i r r e d ethyl alcohol completed the surface preparation. 2.2.4.3 Etching A d i s l o c a t i o n etch for copper was f i r s t discovered by L o v e l l and Wernick (1959) and l a t e r developed and used with greater success by Livingston (1960). He showed etch p i t s to be i n 1-1 correspondence with edge d i s l o c a t i o n s , introduced by bending. Furthermore, he presented evidence to show that screw dislocations were revealed by the etchant. 91 Since t h i s e a r l y work, other i n v e s t i g a t o r s have used t h i s etchant, o f t e n w i t h s l i g h t m o d i f i c a t i o n s (Young, 1961; Hordon, 1962; B a s i n s k i and B a s i n s k i , 1964; Gupta and S t r u t t , 1967; V a l l a i k a l , 1969; Van Drunen and Saimoto, 1971). A r e p r e s e n t a t i v e l i s t of e t c h i n g s o l u t i o n s i s shown i n Table 2. A f t e r some p r e l i m i n a r y e x p e r i m e n t a t i o n , a composition c l o s e to t h a t used by L i v i n g s t o n was found to be most s u c c e s s f u l . The e t c h i n g time was about f i v e seconds at -5°C. 2.2.4.4 S p e c i f i c a t i o n of D i s l o c a t i o n M i c r o s t r u c t u r e s Preparation of. Dot-Patterns - Before any observa-t i o n of the m i c r o s t r u c t u r e s , the c r y s t a l s were r e l a b l e d so t h a t t h e i r p r e s t r a i n temperatures would not be immediately known. T h i s was done t o minimize o p e r a t o r b i a s i n the s e l e c -t i o n of r e p r e s e n t a t i v e areas f o r a n a l y s i s . M i c r o s t r u c t u r e s r e v e a l e d by the e t c h were f i r s t examined wi t h a bench microscope to assess the q u a l i t y of the e t c h . I d e a l metallography r e q u i r e s d i s t i n g u i s h a b l e p i t s which i n t e r s e c t a smooth p o l i s h e d s u r f a c e as an e q u i l a t e r i a l t r i a n g l e . For h i g h e r d i s l o c a t i o n d e n s i t i e s t h i s l a s t c o n d i -t i o n was r e l a x e d by e t c h i n g f o r a s h o r t e r time, so t h a t the p i t s would remain d i s t i n g u i s h a b l e . The m i c r o s t r u c t u r e s were then c a r e f u l l y surveyed and a r e p r e s e n t a t i v e s e r i e s . o f photomicrogrpahs was taken, 92 Table 2 Dislocation Etch Compositions ^**,,*<«*^>£onstitutent Parts Source ^^ *"*',*',','",''»«^  B r H20 HC1 G l a c i a l A cetic Other Livingston (1960) 1 250 45 30 Hordon (1962) 1 250 60 50 Gupta and St r u t t (1967) 1 90 25 15 130 methanol VanDrunen and Saimoto (197 0) (i) 1 90 25 15 ( i i ) 1 220 25 15 Present Work 1 175 50 35 using a Zeiss Ultraphot II microscope with high pressure mercury il l u m i n a t i o n , and a Normarski Interference Planochromat * objective (x40, 0.85 N.A.) (see Appendix 4). The micrographs i n the form of 4" x 5" negatives were photographically enlarged to 8" x 10". These prin t s had a magnification of xl560. About 20 such micrographs were obtained f o r each specimen. From these the 10 best were selected, judging only on the basis of o p t i c a l q u a l i t y . From the centres of each of these p r i n t s , using t h i s same c r i t e r i o n , an area 4" x 5" was selected from d i s l o c a t i o n density measurements. Actual.measurements were made on black dot tracings of the micrographs (from here on referred to as "patterns") which were e a s i l y counted using a Quantimet. On these, patterns, each dot marks the po s i t i o n of one d i s l o c a t i o n at i t s point of inters e c t i o n with the primary s l i p plane^ Measurement of Looal Dislocation Densities - The patterns taken from the micrographs were used to determine l o c a l d i s l o c a t i o n densities. Using a Quantimet, the o v e r a l l d i s l o c a -t i o n densities were f i r s t computed by counting a l l points on each set of ten patterns, and div i d i n g by the appropriate specimen area. Then a sampling technique was employed whereby a sample square was scanned stepwise across each pattern, and the number of points covered by the square was counted for each step. ' L i m i ted areas of e x c e p t i o n a l l y high d i s l o c a t i o n dens i t y were sometimes observed on i n s p e c t i o n of the etched pr imary s l i p p l ane . However, they were regarded as a t y p i c a l , and hence were not sampled. It is thought that such areas are ev idence of d i s l o c a t i o n c a r p e t s , wh ich, in these l i g h t l y s t r a i n e d c r y s t a l s , would be in the e a r l y stages of f o r m a t i o n . Moreover, as d i s cu s sed in S e c t i o n \.k.k i t is not expected that such areas wi11 be t r a v e r s e d by g l i d e d i s -l o c a t i o n s ; , hence t h e i r omiss ion is j u s t i f i e d on t h e o r e t i c a l grounds 94 I t should be noted t h a t s t a t i s t i c a l d i s t r i b u t i o n s determined i n t h i s way are s u b j e c t to scaling effects: f o r any p a r t i c u l a r m i c r o s t r u c t u r e the s t a t i s t i c a l l y determined d i s t r i b u t i o n (of l o c a l d i s l o c a t i o n d e n s i t y ) w i l l vary w i t h the sample square s i z e . In oth e r words the d i s t r i b u t i o n s vary w i t h the scale of sampling and hence.change wi t h the sample mean. In the i n i t i a l t r i a l s , a square o f fixed s i z e was used t o scan all the se t s of p a t t e r n s . In a case where each of the m i c r o s t r u c t u r e s have the same o v e r a l l d i s l o c a t i o n d e n s i t y , the r e s u l t i n g d i s t r i b u t i o n s would each have the same mean. In the presen t study, however, the o v e r a l l d i s l o c a t i o n d e n s i t i e s are d i f f e r e n t f o r each of the m i c r o s t r u c t u r e s , and use o f a f i x e d sample area leads t o d i f f e r i n g sample means; hence these d i s t r i b u t i o n s cannot be compared. In order to avoid t h i s s c a l i n g e r r o r , sample s i z e s were chosen to g i v e the same mean f o r each m i c r o s t r u c t u r e , i n the f o l l o w i n g manner. A f t e r f i r s t measuring the overall d i s l o c a t i o n d e n s i t y on each of the p a t t e r n sets, the s i z e o f the sample area r e q u i r e d to g i v e the d e s i r e d sample mean was determined by simple d i v i s i o n ; the p a r t i c u l a r area s i z e s f o r each p a t t e r n s e t , together w i t h the corresponding sample means are g i v e n i n Table 3. Given t h a t the r e l a t i v e sample area s i z e s can be determined (from the o v e r a l l d e n s i t y ) the c h o i c e Of a b s o l u t e . Table 3 Area Sizes Used to Sample Dislocation Microstructures Sample Number Overall D i s l o c a t i o n Density (cm - 2) S A M P L E A R E A S I Z E S 68A (1000°K) 2.83 x 10 7 Area i n ppt 2 Area i n y -— ( A v e r a g e N o . o f p o i n t s p e r A r e a ) 78 x 78 5.882 10.0 108 x 108 8.142 18.5 117 x 133 9.402 24.2 158 x 158 11. 9 2 40.0 185 x 185 13. 9 2 55.6 228 x 228 17.2 2 83.3 63A (850°K) 2.43 x 10 7 Area i n ppt 2 Area i n y X 78 x 90 6.322 9.82 108 x 108 8.142 16.5 134 x 135 10.142 24.3 170 x 170 12.8 2 40.0 200 x 200 15.1 2 54.8 69A (700°K) 4.54 x 10 7 Area i n ppt 2 Area i n y X 62 x 62 4.672 10.2 80 x 80 6.032 16.8 98 x 99 7.422 23.4 125 x 125 9.422 41.0 146 x 146 11. 0 2 55.2 78A (293°K) 9.63 x 10 6 Area i n ppt 2 Area i n y 2 X 108.x 108 . 8.142 6.30 . 134 x 135 10.1 2 10.0 214 x 214 16.3 2 26.0 216 x 216 16.3 2 26.0 271 x 271 20.4 2 41.0 317 x 317 23. 9 2 58.7 7OA (77°K) 2.21 x 10 7 Area i n ppt 2 Area i n y X 87 x 90 6.672 10.6 115 x 115 8.672 17.5 137 x 138 10.42 23.5 174 x 174 13.1 2 36.0 204 x 204 15.4 2 54.6 96 size remains open. Clearly the sample area must be smaller than the t o t a l area of the pattern set, and an obvious lower l i m i t i s the area covered by one point. Yet neither of these two extremes w i l l reveal differences between d i s s i m i l a r d i s t r i b u t i o n s ; so some intermediate size must be chosen. From a metallurgical viewpoint, there are not strong t h e o r e t i c a l reasons for choosing any p a r t i c u l a r i n t e r -mediate sized sample area. However, the scale of the d i s -location microstructure may provide a physical c r i t e r i o n : sample areas greater than any substructure c e l l s i z e w i l l be i n s e n s i t i v e to the existence of such c e l l s . Sample areas smaller than t h i s c e l l size should reveal density differences between the c e l l wall and the c e l l i n t e r i o r . However, microstructures which exhibited marked c e l l u l a r i t y showed a range of c e l l sizes and provided no obvious upper bound for the sample area. After some preliminary t r i a l s with d i f f e r e n t sample areas, i t became apparent that a series of area sizes would best i l l u s t r a t e differences i n l o c a l d i s l o c a t i o n density. Each set of patterns was f i r s t scanned with a square whose size was determined such that on an average i t would contain 25 points (the appropriate sizes being calculated from the i n i t i a l o v e r a l l density measurements). This process was then repeated four times, with four other sample means. Histograms were constructed showing the f r a c t i o n of samples containing a range of numbers of points, and with the aid of 97 a program written for the Hewlett-Packard 9100A, the mean and standard deviation of th i s grouped data were calculated. In using the Quantimet for th i s sampling procedure, some overlapping of unsampled areas between successive square positions was d i f f i c u l t to avoid (because the frame position control i s not continuously v a r i a b l e ) . However, the major portion of the patterns was covered once with each sample s i z e . Since overlapping and lack of sampling was randomly determined, i t was not considered to bias the d i s -tributions s i g n i f i c a n t l y . The e f f e c t of subgrain boundaries (dislocation boundaries present a f t e r annealing) on the d i s t r i b u t i o n s was also assessed by eliminating dislocations which formed sub-grain boundaries,from the pattern sets, and repeating the analysis. 2.3 Results 2.3.1 Microstrain Curves * Microstrain curves were obtained at 77°K for f i v e c r y s t a l s , each of which had a d i f f e r e n t prestrain temperature. The curves shown i n Figures 26-30 are derived from load elonga-tion curves by appropriate stress and s t r a i n resolutions, A p p e n d i x 2 g i v e s the d e f i n i t i o n s o f t e r m i n o l o g y used to d e s c r i b e d e f o r m a t i o n p r o c e s s e s and m e c h a n i c a l phenomena. 98 66 100 TOT Figure 30. 77°K microstrain curve for 70B (77°K pres t r a i n ) . x = 408 (g mm"2). 103 and s u b t r a c t i o n o f the e l a s t i c s t r a i n . Two examples of the o r i g i n a l l o a d e l o n g a t i o n curves are reproduced i n F i g u r e s 31 and 32. The m i c r o s t r a i n curves may be analysed i n terms of t h e i r d e v i a t i o n from an i d e a l i z e d s t r e s s s t r a i n curve f o r a work hardened ( p r e s t r a i n e d ) specimen (see Appendix 2). Such a curve has two d i s t i n c t r e g i o n s , namely e l a s t i c and p l a s t i c . The former, r e p r e s e n t i n g the i n i t i a l response to r i s i n g s t r e s s , has a c o n s t a n t s l o p e equal t o the e l a s t i c modulus. At h i g h e r s t r e s s e s , the l a t t e r i s the r e g i o n of steady s t a t e p l a s t i c flow which has ( i d e a l l y ) a c o n s t a n t but lower s l o p e ( o f t e n r e f e r r e d to as the work hardening r a t e ) : t h i s l i n e r e p r e s e n t s the steady s t a t e flow s t r e s s of the specimen at any g i v e n s t r a i n . The p o i n t of i n t e r s e c t i o n of these two l i n e s d e f i n e s the y i e l d s t r e s s . In these experiments, where s t r a i n i s measured at a h i g h s e n s i t i v i t y , specimens are found to d e v i a t e from e l a s t i c behaviour below the y i e l d s t r e s s : t h i s s t r e s s must then be determined by an e x t r a p o l a t i o n procedure. The r e g i o n i n which e x p e r i m e n t a l l y determined curves d e v i a t e from the i d e a l i z e d curve, a t s t r e s s e s below the y i e l d s t r e s s , w i l l be r e f e r r e d to as the micros train region, and n o n - e l a s t i c s t r a i n a t these s t r e s s e s w i l l be c a l l e d microstrain. These e x p e r i m e n t a l l y determined curves, F i g u r e s 26-30, show t h a t the shape of the m i c r o s t r a i n r e g i o n v a r i e s with the temperature of p r e s t r a i n . M i c r o s t r u c t u r e s formed 104 x5. SI x2 i i i i t 7 F i g u r e 31. O r i g i n a l l o a d ( v e r t i c a l ) - e l o n g a t i o n ( h o r i z o n t a l ) c u r v e f o r 68B ( 1 0 0 0 ° K p r e s t r a i n ) . Numbers 1-7 show l o a d z e r o - s u p p r e s s i o n s t e p s . S c a l e u n i t s a re shown bottom c e n t r e : v e r t i c a l = 0.5 l b s , h o r i z o n t a l = 3.7 x 1 0 " 5 i n c h e s . t t f 2 3 4 t 5 t 6 105 I I xlO./ 7 / x5-x2 t t t t t f 1 2 3 4 5 6 7 Figure 32. O r i g i n a l load (vertical)-elongation (horizontal) curve for 70B (77°K p r e s t r a i n ) . Numbers 1-7 show load zero-suppression steps. Scale units are shown bottom centre: v e r t i c a l = 0.5 l b s , horizontal = 3.7 x 10~ 5 inches. 106 at higher temperatures deviate from e l a s t i c behaviour at stresses ,well below the y i e l d stress, i . e . they are asso-ciated with gradual y i e l d i n g . By contrast, lower temperature microstructures y i e l d more abruptly, showing less microstrain. For the upper three p r e s t r a i n temperatures, the amount of microstrain decreases continuously with p r e s t r a i n temperature. This s t r a i n i s approximately 1.5 x 10 - 3 f o r 68B (1000°K), 10~ 3 for 65B (850°K), and 0.5 x 10" 3 for 69B (700°K). The 10 y i e l d stress (the stress at which 10 s t r a i n i s detected) for these three cr y s t a l s i s of the order of half the macroflow stress, i n contrast with 78B (293°K) and 70B (77°K) whose 10~ 5 y i e l d stresses are ~0.85 of the macroyield stress. However the l a t t e r two cr y s t a l s s t i l l show appreci-able microstrain, which i s for example approximately 10" 3 for 78B (293°K). A d i s t i n c t i v e feature of the microstrain curves from these l a t t e r two cry s t a l s i s the region of low work hardening rate immediately following macroyield. This phenomenon i s known as the Haasen-Kelly e f f e c t and i s a reloading transient observed i n several pure F.C.C. metals It is of interest to note that only about 1000 dislocation loops, need traverse the crystal to produce 10" strain in a 2 cm gauge length: i t is anticipated that many more than 1000 dislocation loops w i l l expand over areas considerably smaller than the whole glide plane. 107 (Haasen and K e l l y , 1957, Cupp and Chalmers, 1954; Dehl, 1955, C o t t r e l l and Stokes, 1955; Noggle, 1955) at low homologous temperatures. At str a i n s above 5 x 10" 3 a constant work hardening rate i s observed f o r a l l c r y s t a l s . This l i n e , when extrapo-lated back to zero s t r a i n i d e n t i f i e s the steady state flow stress at any s t r a i n . By p l o t t i n g the f r a c t i o n a l difference between th i s stress and the measured stress for any s t r a i n , the microstrain region may be i s o l a t e d , Figure 33. I t can be seen that 70B (77°K) overshoots i t s expected flow stress by ~2.5%, and 78B (293°K) exceeds i t by a marginal 0.5%. It i s also apparent i n Figure 33 that none of the cr y s t a l s with higher prestrain temperature ever exceed t h e i r expected steady state flow stress; i n other words no Haasen-Kelly e f f e c t i s observed. This figure (33) also i l l u s t r a t e s more d i r e c t l y the general conclusions drawn from Figures 26-30. For the upper three prestrain temperatures the amount of 77°K micro-s t r a i n increases with pr e s t r a i n temperature. And for the lower two pr e s t r a i n temperatures, while the 10~ 5 flow stress i s higher than that for the other three c r y s t a l s , some microstrain i s s t i l l observed. A l l the above observations were q u a l i t a t i v e l y confirmed with further experiments, at least two specimens being used for each prestrain temperature. 109 2.3.2 Metallography 2.3.2.1 M i c r o s t r u c t u r e s from Annealed Specimens M i c r o s t r u c t u r e s from u n s t r a i n e d specimens are shown i n F i g u r e 34 and 35; the former was o b t a i n e d a f t e r a s t a t i c anneal a t 1045°C, the l a t t e r a f t e r c y c l i c anneal between 7 50°C and 1045°C. They show d i s l o c a t i o n s which i n t e r -s e c t a c l o s e packed plane, most forming c l e a r l y d e l i n e a t e d though somewhat d i s c o n t i n u o u s s u b g r a i n boundaries. A c y c l i c -anneal m i c r o s t r u c t u r e a t h i g h e r m a g n i f i c a t i o n f u r t h e r i l l u -s t r a t e s t h i s p o i n t (Figure 36), and some p o l y g o n i z e d boundaries are a l s o e v i d e n t . Subgrain s i z e s were measured a f t e r each type of anneal by counting i n t e r s e c t i o n s with a 30 cm c i r c l e ( H i l l i a r d 1964) on e i g h t micrographs s i m i l a r to F i g u r e s 34 and 35. A f t e r s t a t i c anneal subgrains were found to have an average diameter of 74.6u, a f t e r c y c l i c anneal 151u. Taking seven micrographs s i m i l a r to F i g u r e 36, a f t e r a c y c l i c anneal, the average t o t a l d i s l o c a t i o n d e n s i t y was measured to be 1.2 x 10 6/cm 2, i n c l u d i n g those d i s l o c a t i o n s i n s u b g r a i n boundaries. 2.3.2.2 M i c r o s t r u c t u r e s and P a t t e r n s from Pre- . s t r a i n e d C r y s t a l s F i v e s e t s of micrographs were produced from c r y s t a l s with f i v e d i f f e r e n t p r e s t r a i n temperatures, each s e t 110 Figure 34. Etched d i s l o c a t i o n microstructure a f t e r s t a t i c anneal (72 hours at 1045°C) x305. I l l Figure 35. Etched d i s l o c a t i o n microstructure a f t e r c y c l i c anneal (72 hours cycled hourly between 750°C and 1045°C) x305. 112 Figure 36. Etched d i s l o c a t i o n microstructure a f t e r c y c l i c anneal (72 hours cycled hourly between 750°C and 1045°C) x595. 113 containing 10 p r i n t s . From these p r i n t s etch p i t positions were recorded by hand, i n the manner previously outlined. This gave f i v e sets of 10 patterns containing 500 to 2000 points per pattern; i n t o t a l the positions of about 70,000 dislocationswere recorded. Examples of micrographs and t h e i r corresponding patterns for each prestrain temperature are shown i n Figures 37-41 Of the f i v e microstructures, the only one which d i s -played clear c e l l u l a r i t y i s 68A.with a 1000°K prestrain temperature (Figure 37). Here the majority of di s l o c a t i o n s l i e within p a r t i a l l y or f u l l y formed c e l l walls. In contrast, the microstructures from 63A (850°K), Figure 38, and 69A (700°K), Figure 39, show no d i s t i n c t l y formed c e l l walls, although some degree of d i s l o c a t i o n c l u s t e r i n g i s evident. Using a purely v i s u a l assessment, these two microstructures are d i f f i c u l t to di s t i n g u i s h from each other. 78A (293°K), Figure 40, shows a microstructure with a lower d i s l o c a t i o n density and with larger etch p i t s . In this specimen there appears to be aggregation of d i s l o c a -tions without f u l l c e l l formation. The 77°K microstructure (70A), Figure 41, again shows some cl u s t e r i n g , but with fewer areas free of d i s l o c a t i o n s . This microstructure i n many ways resembles those of 63A (850°K), Figure 38, and 69A (700°K), Figure 39. However i n t h i s case and i n 78A (293°K) there appears to be a greater predominance of short l i n e a r -dislocation arrays. 114 Figure 37. Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 68A (1000°K) prestrain) xl560. Figure 38. Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 63A (850°K prestrain) xl560. F i g u r e 39. T y p i c a l etched d i s l o c a t i o n m i c r o s t r u c t u r e and corresponding dot p a t t e r n from 69A (700°K p r e s t r a i n ) x 1560. 117 Figure 40. Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 78A (293°K prestrain) xl560. Figure 41. Typical etched d i s l o c a t i o n microstructure and corresponding dot pattern from 7OA (77°K prestrain) xl560. 119 I t was previously noted that although the o v e r a l l d i s l o c a t i o n densities for each set of micrographs are s i m i l a r (within a factor of f i v e ) , they are not equal. This inequality i n t e r f e r s with accurate v i s u a l comparison, since the higher density microstructures were etched f o r shorter times. These shorter etch times were used because the etch p i t s must be smaller to remain distinguishable. The micrographs i n Figures 37-41 were selected as t y p i c a l on the basis of both v i s u a l assessment and s t a t i s -t i c a l analysis. In other words, not only do they display c h a r a c t e r i s t i c microstructural features, but they also have a d i s t r i b u t i o n of l o c a l d i s l o c a t i o n densities close to the d i s t r i b u t i o n s of t h e i r respective pattern set. However one feature, which does not appear i n Figure 37 to 41 but which occuasionally occurred, i s the grown-in subgrain boundary. These were readily i d e n t i f i e d under Normarski Interference contrast. The depth of attack by the etch was s l i g h t l y greater at grown-in subgrain boundaries, giving a distinguishable colouring to t h e i r contrast. An extreme example of the occurrence of these boundaries i s shown i n Figure 42, from 78A (293°K). As can be seen from the corresponding pattern below, dislocations i n the subgrain boundaries were included i n density measurements. On the whole, the proportion of dislocations i n the boundaries was small and t h e i r absence made l i t t l e difference to the shape of the sampled d i s t r i b u t i o n . 120 Figure 42. Etched d i s l o c a t i o n microstructure and corresponding dot pattern from 78A (293°K) prestrain) showing extreme example of incidence of subgrain boundaries. 121 2.3.3 Average Dislocation Density and Flow Stress It w i l l be r e c a l l e d that flow stress varies as the square root of the forest d i s l o c a t i o n density f o r a number of metals (Basinski and Basinski, 1964). Although t h i s r elationship also holds for copper for a wide range of flow stress values, the experimental data at low d i s l o c a t i o n densities i s somewhat scattered. This i s i l l u s t r a t e d i n Figure 43, which shows flow stress and d i s l o c a t i o n densities * from various sources, together with those from the present experiments. While the data from the present study shows only approximately the expected c o r r e l a t i o n between flow stress and d i s l o c a t i o n density, the experiments, using a low, and as nearly as possible equal set of flow stresses, were not designed with t h i s objective i n mind. Nonetheless the re s u l t s can be seen to f a l l within the range of those obtained by other workers. 2.3.4 Local Dislocation Densities Local d i s l o c a t i o n densities were measured using the previously explained area sampling technique, and sample area sizes were used which contained a sample average, x, of Because of the r e l a t i v e s c a r c i t y of f o r e s t d i s -l o c a t i o n dens i ty measurements from c r y s t a l s o r i e n t e d f o r s i n g l e s i i p , dens i t y measurements of " f o r e s t " d i s l o c a t i o n s in double and m u l t i p l e s l i p c r y s t a l s are a l s o i n c l uded in th i s d i ag ram. 122 10' 'E 10 10 10 10 ' A Basinski 8 Bosinski e°off [OIIJ (III) O VanDrunen a Saimoto [OOQ(III) o Livingston (110) • " [112] (III) • " QMQ(IIO) • Bailey Polycrystal Present data A AA A A O o CP o • A * T ° O Resolved Shear Stress (g.mm" 2) I 0 Figure 43. Dislocation density versus resolved shear stress (Cottrell-Stokes corrected) 123 between 6.5 and 55 points per area. Five area sizes were selected for each of the f i v e sets of patterns ( i . e . f o r each pr e s t r a i n temperature). Histograms giving the f r a c t i o n of squares with a given number of points are shown i n Figures 44-48, for sample area sizes with x - 25. The data are t y p i c a l of those obtained over a range of sample area s i z e s . I t i s expected that the breadth of these histograms w i l l increase as the structure changes from a f a i r l y uniform d i s t r i b u t i o n of p i t s (a "regular" array) to a d i s t r i b u t i o n in which the p i t s are clustered, leaving substantial areas nearly free of p i t s (a "non-regular" s t r u c t u r e ) . This trend i s i l l u s t r a t e d i n Figure 49 which shows t y p i c a l patterns from regular, random and more c e l l u l a r microstructures, together with c h a r a c t e r i s t i c histograms which would be obtained using the present l o c a l area sampling procedure. From t h i s figure (49) i t can be seen that microstructures which are respectively regular, random, and more c e l l u l a r , have l o c a l area density d i s t r i b u t i o n s which progressively increase i n breadth. Thus the degree of r e g u l a r i t y (or a l t e r n a t i v e l y , c e l l u l a r i t y ) i s quantified by the s t a t i s t i c a l sampling procedure. On t h i s basis i t i s expected that some of the obvious features of the micrographs w i l l emerge from the s t a t i s t i c a l analysis. A histogram for a highly c e l l u l a r microstructure should i l l u s t r a t e the large proportion of 2 0 4 0 6 0 x (Number of Points per Area) F i g u r e 44. Histogram showing sampled frequency d i s t r i b u t i o n o f l o c a l areas d e n s i t i e s from 68A (1000°K p r e s t r a i n ) . Sample area s i z e ( 9 . 4 0 ) 2 y 2 . 140 120 100 8 0 a> W 6 0 in o cu 4 0 2 0 0 i 2 0 4 0 6 0 x (Number of Points per Area) 8 0 to Figure 45. Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l areas densities from 63A (850°K p r e s t r a i n ) . Sample area size (10.14) 2y 2. 120 1 0 0 g 801 I 60 £ 3 4 0 2 0 0 1 1 1 1 10 2 0 3 0 . 4 0 5 0 6 0 x (Number of Points per Area) 7 0 CT. F i g u r e 46. Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l areas d e n s i t i e s from 69A (700°K p r e s t r a i n ) . Sample area s i z e ( 7 . 4 2 ) 2 y 2 . l O O r -D j | 8 0 | — S 6 0 e • 3 . 4 0 2 0 0 1 1 10 2 0 3 0 4 0 5 0 6 0 x (Number of Points per Area) 7 0 F i g u r e 47. Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l areas d e n s i t i e s from 78A (293°K p r e s t r a i n ) . Sample area s i z e ( 1 6 . 3 ) 2 y 2 . 120 1 0 0 </> 8 0 o cu 6 0 cu -Q E - Z3 4 0 2 0 0 1 10 2 0 3 0 4 0 5 0 6 0 7 0 x (Number of Points per A rea ) 8 0 00 Figure 48. Histogram showing sampled frequency d i s t r i b u t i o n of l o c a l areas densities from 70A (77°K p r e s t r a i n ) . Sample area size (10.4) 2y 2. 129 N(x) s/x = 0 Regular Random N(x) S / X = X 1/2 More Cellular N(x) s/x>x l / 2 Figure 49. Schematic of regular, random and more c e l l u l a r d i s t r i b u t i o n s , t h e i r corresponding histograms and degrees of reg u l a r i t y (s/x). 130 sampled areas containing either high or low d i s l o c a t i o n densities. Thus the clear c e l l u l a r i t y seen i n 68A (1000°K), Figure 37, i s also apparent i n Figure 44 where 20% of the sample areas contain 10 or less points, and 12% contain more than 40. Comparing the three higher temperature microstructures, the histograms become more closely d i s t r i b u t e d about the mean as the prestrain temperature decreases. This e f f e c t repre-sents a trend away from c e l l u l a r i t y and can be described as an •increasing degree of regularity i n the microstructure. If the histograms f o r the two lower p r e s t r a i n temperatures are compared, the same general trend i s observed. The histogram for 70°C (77°K), Figure 48, i s more closely d i s t r i b u t e d about the mean than that of 78A (293°K), Figure 47. However on the basis of this s t a t i s t i c a l analysis, the l a t t e r d i s t r i b u t i o n appears si m i l a r to that of the highly c e l l u l a r 68A (1000°K), Figure 44, arid the former 77°K d i s t r i -bution i s almost indistinguishable from those obtained for microstructures formed at 850°K and 700°K. A parametric measure of the degree of regularity , may be introduced, namely the r a t i o of the standard deviation to mean , s/x, f o r any p a r t i c u l a r d i s t r i b u t i o n . I t should be noted that for any given d i s t r i b u t i o n t h i s r a t i o may change with x p a r t i c u l a r l y for small s. This v a r i a t i o n for a random d i s t r i b u t i o n i s i l l u s t r a t e d i n Figure 50, and i n _ i this case s/x varies as (x)* (see Appendix 5). This 131 in a at 4> E 2 0 3 0 x (Number of Points per Area) 4 0 F i g u r e 50. C a l c u l a t e d histograms f o r random d i s t r i b u t i o n o f p o i n t s showing s c a l i n g e f f e c t : histograms w i t h x = 10 and x = 25 have d i f f e r e n t standard deviation/mean r a t i o . 132 relationship,together with those measured by repeated sampling for a wide range of sample area s i z e s , i s shown i n Figures 51 and 52. It can be seen i n Figure 51 (1000°K, 850°K, 700°K), that the r e l a t i v e degree of r e g u l a r i t y measured at x = 25 i s maintained throughout the range of sample s i z e s , for x between 10 and 55 (points per area). S i m i l a r l y for Figure 52 (293°K, 77°K) for x between 10 and 59 the room temperature microstructure remains more c e l l u l a r than that formed at 77°K. This range of sample area sizes corresponds to a specimen area range of between (5y) 2 and (25y) 2 (see Table 3). Also shown i n Figure 51 i s an analysis of a micro-graph of Gupta and Strutt (1967) from a copper single c r y s t a l * creep experiment at 82 3°K. A pattern made from the micro- . graph i s shown i n Figure 53, and appears to be s i m i l a r to those of 68A (1000°K), Figure 37. And as can be seen i n Figure 51, the s t a t i s t i c a l analysis follows cl o s e l y that of 6 8A. These re s u l t s show that a l l the microstructures are far from random. For example, a ty2 t e s t of data from 78A using x = 6.3 indicates that there i s a less than 0.1% pr o b a b i l i t y that the microstructure i s randomly d i s t r i b u t e d . I t may be r e c a l l e d that the analyses of d i s l o c a t i o n motion through f i e l d s of point obstacles (Kocks, 1966; Foreman and This pa t te rn is n e c e s s a r i l y approximate because of the q u a l i t y of r ep roduc t i on of the m ic rograph. x (Average Number of Points per A r e a ) U ) F i g u r e 51. Degree of r e g u l a r i t y , s/x, f o r a range of sample s i z e s f o r upper t h r e e p r e s t r a i n temperatures. E r r o r bars r e p r e s e n t 95% confidence l i m i t s c a l c u l a t e d from histograms. T 0 . 8 -i o . o h -O 293 °k. prestrain • 293 k without subgrain boundaries A 77 °k " V 0.4 0 . 2 R o n d o m 0 0 Regular 20 _ 30 40 50 x (Average Number of Points per Area.) 60 70 F i g u r e 52. Degree of r e g u l a r i t y , s/x, f o r a range of sample s i z e s f o r lower two p r e s t r a i n temperatures. E r r o r bars r e p r e s e n t 95% confidence l i m i t s c a l c u l a t e d from histograms. 135 ' 1 2 *. 'w.'.-.V... • s • . ft- * v . • v- • \ >. . v.. • ' • T O - - . "ft \ • . • • „ • • S • • • Mb • • • •* ••• *\ • *A • 9 • • '* •* • J?V . •• I * t * * Figure 53. Dot pattern taken from micrograph by St r u t t and Gupta (1967) from copper single c r y s t a l a f t e r creep at 823°K, resolved shear stress ,250 g mm , (Magnification x780). _ 2 136 Makin, 1966; Morris and Klahn, 1973) use random obstacle d i s -t r i b u t i o n s , and i n one case (Foreman and Makin, 1966), a regular d i s t r i b u t i o n . Both random and regular d i s t r i b u t i o n s appear i n Figures 51 and 52, the l a t t e r d i s t r i b u t i o n follow-ing the abscissa for a l l x > 1. I t can be seen that none of the microstructures have obstacle d i s t r i b u t i o n s which f a l l between random and regular: a l l d i s t r i b u t i o n s show deviation from the random away from the regular (see Appendix 5 for discussion of random and regular arrays). C e l l sizes were measured for 68A (1000°K) using 10 micrographs si m i l a r to Figure 37. The average c e l l size was 20y, with the majority within the range 17y-25y. I t should be noted that a sample area s i z e of (20y) 2 would give a d i s t r i b u t i o n with x ^ 115 which i s over twice the s i z e of the largest sample area used. In other words, sample areas were much smaller than the average c e l l s i z e . 2.3.5 The E f f e c t of Subgrain Boundaries on the  Distributions By eliminating from the patterns dislocations i n subgrain boundaries and resampling, the d i r e c t e f f e c t of these features on the density d i s t r i b u t i o n s was assessed. Using patterns from 78A (293°K), which was the specimen with lowest d i s l o c a t i o n density and the highest proportion of dislocations i n subgrain boundaries samples with x = 10 and x - 25 were 137 taken. These r e s u l t s appear i n F i g u r e 52. As expected the d i s t r i b u t i o n s become more random. However i t can be seen t h a t the non-random nature of these m i c r o s t r u c t u r e s i s by no means s o l e l y due to the presence of s u b g r a i n boundaries. 2.3.6 R e p r o d u c i b i l i t y Experimental e r r o r s i n t r o d u c e d by d i f f e r e n c e s from specimen to specimen are c o n s i d e r e d to be s m a l l , s i n c e , as i n the s l i p l i n e study, good c o n t r o l was e x e r c i s e d over o r i e n t a t i o n , c r y s t a l p u r i t y , anneal and p r e s t r a i n c o n d i t i o n s . At any g i v e n temperature p r e s t r a i n curves were i d e n t i c a l f o r each specimen, and a l s o (with one exception) the p a i r of specimens were cut from the same o r i g i n a l c r y s t a l . T h i s p r o-cedure was f o l l o w e d so t h a t m i c r o s t r u c t u r e and mechanical p r o p e r t i e s c o u l d be r e l i a b l y compared. Both the mechanical and m i c r o s t r u c t u r a l r e s u l t s were q u a l i t a t i v e l y confirmed by a d d i t i o n a l experiments: w i t h one e x c e p t i o n (293°K), m e t a l l o g r a p h i c o b s e r v a t i o n and micro-s t r a i n t e s t s were performed on a t l e a s t two specimens f o r each p r e s t r a i n temperature, e i t h e r d u r i n g the p r e l i m i n a r y or i n the main experiments. 2.4 D i s c u s s i o n 2.4.1 I n t r o d u c t i o n The r e s u l t s of the m i c r o s t r a i n work f a l l i n t o two groups, namely those from c r y s t a l s p r e s t r a i n e d above room 138 temperature, and the remainder which are from c r y s t a l s pre-strained at room temperature and below. The l a t t e r display a reloading transient known as the Haasen-Kelly e f f e c t , which i s characterized by a low or negative work hardening rate immediately following macroyield. For the'former group of c r y s t a l s , however, as the applied stress increases, the gradient of the s t r e s s - s t r a i n curve f a l l s continuously through the microstrain region, from the e l a s t i c modulus slope to the steady state work hardening rate. Whether or not t h i s d i s t i n c t i o n i s purely a r b i t r a r y i s a matter which w i l l be dealt with i n the proceeding d i s -cussion. However, for the present, the d i s t i n c t i o n w i l l be made, and the two sets of results w i l l be analysed separately. i 2.4.2 Crystals with Prestrains above 293°K Measurements have been described of the early stages of low temperature s t r e s s - s t r a i n curves for c r y s t a l s with d i f f e r i n g thermal-mechanical h i s t o r i e s . The r e s u l t s show that the s p e c i f i c a t i o n of the low temperature y i e l d stress i s not s u f f i c i e n t to characterize the stress s t r a i n curves of copper cr y s t a l s prestrained at various higher temperatures. In p a r t i c u l a r the early stages of these curves are diverse: c r y s t a l s prestrained at high temperature y i e l d more gradually than those prestrained at a lower temperature, i . e . they exhibit more microstrain. This r e s u l t i s evident 139 from Figures 26-28 and p a r t i c u l a r l y from Figure 33 i n which the stress s t r a i n curves are shown as a group. The concurrent microstructural studies show a corresponding change i n the nature of the microstructure with prestrain temperature: the microstructures become progres-s i v e l y more regular with decreasing prestrain temperature. This e f f e c t can be c l e a r l y recognized i n Figures 45, 46 and 47, the histograms of the l o c a l d i s l o c a t i o n density with an x - 25 sample s i z e . Moreover i n Figure 51, t h i s r e l a t i v e degree of re g u l a r i t y for these specimens (as seen by comparing Figures 45, 46 and 47 for x - 25) i s c l e a r l y maintained over the whole range of sample sizes. Considering this microstructural and mechanical evidence together, i t i s suggested that these re s u l t s are correlated, and that crystals with a lesser degree of regu-l a r i t y show more microstrain. This means that whereas the le v e l of the s t r e s s - s t r a i n curve i s determined by the o v e r a l l d i s l o c a t i o n density, as previously shown by many workers (Livingston, 1962; Bailey, 1963; Basinski and Basinski, 1964), the shape of the early stages of the curve may be controlled by the d i s t r i b u t i o n of the obstacle d i s l o c a t i o n s . This l a t t e r new r e s u l t i s among the most s i g n i f i c a n t of t h i s study. This r e s u l t i s consistent with the view of p l a s t i c flow developed i n discussion of the s l i p l i n e study, whereby s t r a i n occurs because of the athermal expansion of d i s l o c a t i o n loops through the existing microstructure. At f i n i t e 140 temperature'this athermal gl i d e i s preceded by thermal act i v a t i o n (two possible a c t i v a t i o n steps were discussed) releasing a number of g l i d e dislocations which then sweep a newly available area of s l i p plane, c a l l e d the free area. Whereas th i s model was shown i n the s l i p l i n e discussion to explain properties of s l i p l i n e s formed at d i f f e r e n t tempera-tures during steady state deformation, the microstrain curves i l l u s t r a t e transient deformation and were determined at a low homologous temperature (0.057 T ) such that any thermally activated effects (such as forest rearrangement or recovery) are assumed to be n e g l i g i b l e . Without the aid of thermal activation, dislocations may g l i d e over a free area whose siz e , i n a given microstructure, i s determined by the applied stress as discussed i n Section 1.4.3.2. If we assume for the present that the amount of microstrain, produced i n a c r y s t a l under a given applied stress, i s proportional to this free area, then the stress dependence of t h i s free area may be derived from the microstrain curve. This basis for a 0°K theory of microstrain uses the ideas f i r s t proposed by Kocks (1966, 1967), following his analysis of the movement of g l i d e dislocations i n a f i e l d of random point obstacles, the results of which have been outlined i n Section 1.4.3.2. ! Whereas these results were previously discussed with reference to steady state deformation at f i n i t e tempera-tures, the same interpretation may be applied to transient 141 deformation at low temperature. Accordingly, following the diagrams shown i n Figure 20,through the microstrain region the free area increases from an indeterminately small f r a c t i o n of the obstacle f i e l d at low stress, to become " i n d e f i n i t e l y large" at a stress near the athermal glide' stress. While d i s l o c a t i o n motion has been analysed i n t h i s way f o r glide through random or regular arrangements of point obstacles, forest dislocations i n the as-prestrained microstructures used i n t h i s study were shown to be neither randomly nor regularly d i s t r i b u t e d . They have a lesser degree of r e g u l a r i t y than a random arrangement as can be seen i n Figure 51. Although the Kocks analysis has not been performed on c e l l u l a r or clustered obstacle arrangements (Kocks, 1974), i t i s assumed that such an analysis would predict a c r i t i c a l stress for athermal gl i d e through obstacle arrangements with the degree of r e g u l a r i t y of those found i n t h i s study. Moreover, i t i s expected"that the free area w i l l be a s i m i l a r l y increasing function of stress, although, as discussed i n the introduction (Section 2.1), i t i s anticipated that, at a given applied stress and o v e r a l l d i s l o c a t i o n density, the free area should be larger i n a more c e l l u l a r microstructure. 2.4.3 Quantitative Estimates from Theories of Microstrain Although the o r i g i n a l objective of t h i s work was simply to demonstrate the q u a l i t a t i v e e f f e c t of departure from 142 regul a r i t y on the shape of the low temperature microstrain curve, i t i s in t e r e s t i n g to explore the p o s s i b i l i t y of a more quantitative description of microstrain. For such a descrip-tion i t i s necessary to l i n k the free area to s t r a i n , and there are two possible ways to do t h i s . On the one hand the stress dependence of the free area may be determined a n a l y t i -c a l l y f or a given microstructure, i n a manner s i m i l a r to the Kocks analysis, and the free area may then be taken as a measure of s t r a i n through a relat i o n s h i p of the form y = ban (where n i s the number of d i s l o c a t i o n loops which have swept free area, a at any given s t r e s s ) . A l t e r n a t i v e l y the free area may be represented by a suitable function of stress and microstructure. This free area function may then be used to d i r e c t l y r e l a t e stress and s t r a i n . If t h i s function i s known, the second approach has the d i s t i n c t advantage that the microstrain curve i s expressed i n terms, of (in p r i n c i p l e ) measurable structure parameters. A microstrain curve w i l l f i r s t be derived using the former approach, by taking the a n a l y t i c a l l y determined free area/stress r e l a t i o n s h i p derived by Kocks, and th i s w i l l be compared with microstrain curves from two suggested free area functions, together with the experimentally measured curves. Using more c a r e f u l l y defined parameters of applied stress and athermal gl i d e stress, Kocks (1967) has determined a stress versus normalized free area r e l a t i o n s h i p , for a "h6,n-wbrk/hardening l ,-crystal at, 0°K, from diagrams s i m i l a r • to F i g u r e 20(a)-(c)> and t h i s i s shown i n Figure 54. a 0 i s the area.of s l i p plane which contains on average one o b s t a c l e . T h e . a d d i t i o n a l a b s c i s s a which.appears on t h i s diagram t r a n s -form i t i n t o a m i c r o s t r a i n curve and have been c a l c u l a t e d using a r e l a t i o n s h i p of the form y = ban f o r n = 25. This assumes th a t 25 d i s l o c a t i o n s have crossed a l l f r e e areas at any given s t r e s s , i n a c r y s t a l whose s l i p planes c o n t a i n randomly d i s t r i b u t e d p o i n t obstacles w i t h an o v e r a l l d e n s i t y of ~10 7 cm - 2 and having approximately the same dimensions as the samples used i n the present work. For comparison, m i c r o s t r a i n curves have been c a l -c u l a t e d using the f r e e area f u n c t i o n suggested by Alden (1972) which appears i n Eq. 1.6,, namely (2.1) This f u n c t i o n behaves i n a manner appropriate to de s c r i b e the change i n f r e e area w i t h a p p l i e d s t r e s s : i n the m i c r o s t r a i n r e g i o n , at an a p p l i e d . s t r e s s , a, below the 0°K macroyield s t r e s s , T^, ( i . e . a < 'x ) , A^ w i l l be s m a l l and p o s i t i v e : f o r a near as a ->• x^ ., A^ becomes u n i t y . The degree of r e g u l a r i t y of the m i c r o s t r u c t u r e i s expressed through the as yet undefined parameter x , which has dimensions of s t r e s s , and. i s small compared to T ^ . L i k e the standard d e v i a t i o n of - 1.25 . 0 0 ^ . 0 7 5 0 . 5 0 0 . 25 0 0 XL ( X . io" 0 25 50 75 a/oo 100 25 150 Figure 54. Change in ;normalized free area, a/a 0, with r e l a t i v e applied stress from Kocks (1967) and corresponding s t r a i n calculated assuming 25 d i s l o c a t i o n traversed c r y s t a l s i m i l a r to those used i n present experiments. 145 the sampled l o c a l d i s l o c a t i o n d e n s i t i e s , t h i s parameter i s l a r g e r f o r more c e l l u l a r m i c r o s t r u c t u r e s . Through T , A r becomes l a r g e r w i t h i n c r e a s e d degree o f c e l l u l a r i t y (or e q u i v a l e n t l y w i t h d e c r e a s e d degree o f r e g u l a r i t y ) , w h i c h i s c o n s i s t e n t w i t h the arguments d e v e l o p e d i n S e c t i o n 2 . 1 . T h i s f u n c t i o n forms p a r t o f an i n c r e m e n t a l , 0 °K s t r e s s s t r a i n e q u a t i o n ( A l d e n , 1972) dy = | da A r (2.2) where 9 y i s , t h e work h a r d e n i n g r a t e , c o n s t a n t a t c o n s t a n t s t r u c t u r e . I n t e g r a t i o n o f E q . 2.2 a t c o n s t a n t s t r u c t u r e g i v e s T V Y = — ' Y exp x - a -T Y J y — *- - exp — * -T * T V V (2.3) T h i s i s an e q u a t i o n f o r a f a m i l y o f m i c r o s t r a i n c u r v e s , t h r e e o f w h i c h have been computed and a r e p l o t t e d i n F i g u r e 55 u s i n g T = 410 ( i n gram"2) and a work h a r d e n i n g r a t e s i m i l a r to t h a t d e t e r m i n e d e x p e r i m e n t a l l y i n t h i s s t u d y . A s i m i l a r b u t a l t e r n a t e form o f the r e l a t i v e a r e a f u n c t i o n has been sugges ted by M . F . Ashby ( A l d e n , 1972) , w h i c h has the added advantage o f b e i n g zero v a l u e d a t a = 0, v i s , 0 0 8 Figure 55. Calculated microstrain curves using Alden's free area function, and x =. 410 (g mm"2) , x v = Ty/100, Ty/20 and Xy/10, curves 1, 2 and 3 respectively. 147 Ar = — exp v (2.4) Substitution i n Eq. 2.2 and integration gives Y = v 9 x y y (a - x v ) exp - ( T - a) y T V + x v exp — T V (2.5) Three of the family of microstrain curves from t h i s equation have also been computed and appear i n Figure 56. These two sets of microstrain curves may now be compared with the one derived d i r e c t l y from Kocks' analysis. Although the value of appropriate for a random obstacle f i e l d i s not defined, a comparison of Figure 54 with Figures 55 and 56, would seem to indicate that, i f the free area i s traversed by a reasonable number of d i s l o c a t i o n loops (25) , for a random array of obstacles, T = T 7100. v y Since the d i s l o c a t i o n arrangements measured i n the present study had a lesser degree of r e g u l a r i t y than a random array (see Figure 51), a somewhat larger value of T v would seem to be appropriate for these microstructures. When the t h e o r e t i c a l microstrain curves shown i n Figures 55 and 56 are compared to the experimentally determined curves, for the three specimens with higher prestrain temperatures, Figure 57 (re-plotted from Figure 33 with axes appropriate for d i r e c t comparison), t h i s deduction appears to be confirmed. In Figure 56. Calculated microstrain curves using Ashby's function, and x y = 410 (g mm"2), Tv = T y/100, Ty/20 and x y/10, curves 1, 2 and 3 respectively. T — — — r b < 0 7 5 r 0 50 0 2 5 h J i_ 4 5 Xp (xlO - 4 ) i t 8 Figure 57. Normalized experimental microstrain curves from specimens with prestrain temperatures of 1000°K, 850°K and 700°K, replotted from Figure 33. 150 other words, the free area functions i n Eqs. 2.1 and 2.4 could make reasonably accurate quantitative predictions of the amount of microstrain measured i n these c r y s t a l s . 2.4.4 Crystals Prestrained at 293°K and Below It w i l l be r e c a l l e d that while c r y s t a l s prestrained at 293°K and below gave l i t t l e microstrain below a/x - 0.9, and showed the Haasen-Kelly e f f e c t , the sampled microstructures for these prestrains indicated a degree of r e g u l a r i t y com-parable with c r y s t a l s prestrained at 700°K and 850°K. In other words, the microstrain curves from these c r y s t a l s appear to be inconsistent with the correlations previously discussed. I t has been established by a number of workers (Haasen and K e l l y , 1957; Makin, 1958; B o i l i n g , 1959; Birnbaum, 1960) that the Haasen-Kelly e f f e c t i s time-independent, and thus i t can not be explained by a s t r a i n ageing or impurity pinning mechanism. This time-independence was confirmed i n the present study, by a series of preliminary t e s t s . I t was also found i n agreement with the previous work, that the e f f e c t occurs only on unloading, and i s not observed i f the test i s merely interrupted by stopping the cross-head. The e f f e c t can be explained in terms of the i n t e r a c t i o n of g l i d e and forest dislocations on unloading (Haasen and K e l l y , 1957; Makin, 1958; Birnbaum, 1960); during an e l a s t i c relaxation i n the unloading cycle, some g l i d e dislocations 151 undergo e n e r g e t i c a l l y favourable j u n c t i o n r e a c t i o n s w i t h f o r e s t d i s l o c a t i o n s . Because of these j u n c t i o n r e a c t i o n s , upon r e l o a d i n g , the operation of sources w i t h i n the f r e e areas i s r e s t r i c t e d . The sources are r e s t r i c t e d because newly generated g l i d e d i s l o c a t i o n s w i l l experience a back s t r e s s from those p r e v i o u s l y generated, but which are now pinned i n some intermediate p o s i t i o n . G l i d e d i s l o c a t i o n s cannot move to the f r e e area boundary u n t i l the s t r e s s i s s u f f i c i e n t l y high t o reverse the j u n c t i o n r e a c t i o n s ; thus u n t i l the s t r e s s reaches a c r i t i c a l l e v e l , source o p e r a t i o n w i l l be r e s t r i c t e d , and there w i l l be l i t t l e m i c r o s t r a i n . The mechanical response expected i f such a mechanism were ope r a t i v e i s indeed observed i n the present, experiments. In Figure 33 i t can be seen th a t m i c r o s t r a i n i s measured i n c r y s t a l s 78B (293°K) and 7OB (77°K) only at s t r e s s e s above .85 T '. Moreover once .85 i s exceeded (and sources begin to operate) the amount of m i c r o s t r a i n produced i s comparable w i t h t h a t observed i n c r y s t a l s 65B (850°K) and 69B (700°K) r e s p e c t i v e l y . By comparing the degrees of c e l l u l a r i t y of these c r y s t a l s , Figures 51, 5-2, i t can be seen th a t 78B and 65B, and 70B and 69B, have s t a t i s t i c a l l y s i m i l a r m i c r o s t r u c -t u r e s . Thus once s u f f i c i e n t numbers of g l i d e d i s l o c a t i o n s are produced, these two p a i r s of c r y s t a l s r e s p e c t i v e l y produce s i m i l a r amounts of m i c r o s t r a i n . While the m i c r o s t r a i n curves of 78B (293°K) and 70B (77°K) have been shown to be c o n s i s t e n t w i t h a r e s t r i c t e d 152 source hypothesis, i n contrast, the curves from 68B (1000°K), 65B (850°K) and 69B (700°K), which have 1 0 - 5 microyield stresses of ~0.5 x , have been shown to be consistent with y a s t a t i s t i c a l analysis of microstrain assuming a reasonably large number of d i s l o c a t i o n loops traverse each free area. I t i s for this reason that the separation of the two sets of results i s taken to be v a l i d . SUMMARY AND CONCLUSIONS (a) S l i p l i n e l e n g t h measurements were made on a s e r i e s o f o r i e n t e d copper s i n g l e c r y s t a l s , p r e s t r a i n e d a t 673°K, p o l i s h e d and i n c r e m e n t a l l y s t r a i n e d a t temperatures between 573°K and 4.2°K; s l i p l i n e s formed d u r i n g low tempera-t u r e s t r a i n increments were found to be l o n g e r than those formed d u r i n g s t r a i n increments a t h i g h temperature. (b) T h i s r e s u l t was shown to be i n c o n f l i c t w i t h any theory of s t r a i n hardening i n which s l i p l i n e s are b l o c k e d by s p e c i f i c o b s t a c l e c o n f i g u r a t i o n s ( l i n e a r i n the s l i p p l a n e ) , such as L o m e r - C o t t r e l l b a r r i e r s (Seeger, 1957), ribbons of converted p i l e - u p s ( H i r s c h and M i t c h e l l , 1967) o r d i s l o c a t i o n c e l l w a l l s ; these t h e o r i e s p r e d i c t constant s l i p l i n e l e n g t h i n the p r e s e n t experiments. (c) Temperature (or more fundamentally i s o s t r u c t u r a l s t r e s s ) dependent s l i p l i n e l e n g t h was shown to be c o n s i s t e n t with t h e o r i e s of s t r a i n hardening i n which s l i p l i n e s are blocked by s t a t i s t i c a l i n t e r a c t i o n between the expanding g l i d e loops and f o r e s t d i s l o c a t i o n s , on the c o n d i t i o n t h a t 153 154 within the framework of such a theory, the g l i d e loops are able to expand athermally over a newly available free area of s l i p plane, aft e r a thermally activated process. (d) Two possible thermally activated processes were discussed. F i r s t l y i t i s proposed that thermally activated g l i d e of forest dislocations (over small distances on t h e i r own 'primary' and 'cross s l i p ' planes), enables these d i s -locations to change p o s i t i o n and annihilate at a f i n i t e rate (rearrangement and recovery) (Alden, 1972). The second possible process i s thermally activated movement of glide d i s l o c a t i o n through arrays of obstacles. Either or both of these processes could form part of a theory which success-f u l l y accounts for the present r e s u l t s . (e) A u n i f i e d view of s l i p l i n e properties, micro-s t r u c t u r a l features and flow stress i s presented i n which, by cor r e l a t i n g the spacing of prominent s l i p l i n e s with microstructural observations from the l i t e r a t u r e , i t i s suggested that g l i d e d i s l o c a t i o n loops expand between carpets of primary multipoles p a r a l l e l to the primary s l i p plane, and inter a c t s t a t i s t i c a l l y with the more d i f f u s e forest portions of the c e l l "walls." This model i s shown to provid-a self-consistent explanation of the temperature v a r i a t i o n of s l i p l i n e length, s l i p band formation, the existence of 155 of multipole carpets and the v a r i a t i o n of flow stress with temperature. (f) 77°K microstrain curves were obtained from a series of oriented copper c r y s t a l s , prestrained at temperatures between 1000°K and 77°K to produce d i s l o c a t i o n microstructures with d i f f e r i n g degrees of r e g u l a r i t y , yet with approximately the same o v e r a l l density. (g) The forest d i s l o c a t i o n microstructures of an i d e n t i c a l l y prepared series of crystals were examined using a d i s l o c a t i o n etch on the primary s l i p plane. A s t a t i s t i c a l sampling technique was devised, which was used to measure l o c a l d i s l o c a t i o n densities and hence to quantify the degree of r e g u l a r i t y of the forest d i s l o c a t i o n microstructure. (h) A l l microstructures were found to have a smaller degree of r e g u l a r i t y ( i . e . a higher degree of c e l l u -l a r i t y ) than a random d i s t r i b u t i o n . Microstructures from cry s t a l s prestrained at temperatures above 293°K became less regular (more c e l l u l a r ) with increasing prestrain temperature; a p a r a l l e l trend was observed for crystals prestrained at or below 293°K. 156 ( i ) For c r y s t a l s p r e s t r a i n e d a t temperatures above room temperature, at any g i v e n f r a c t i o n o f the 77°K y i e l d s t r e s s , the amount of m i c r o s t r a i n was found to i n c r e a s e as the m i c r o s t r u c t u r e s became l e s s regular (more c e l l u l a r ) . (j) C r y s t a l s p r e s t r a i n e d a t room temperature and below e x h i b i t e d the Haasen-Kelly e f f e c t which was deduced t o be a consequence of r e s t r i c t e d source o p e r a t i o n , caused by d i s l o c a t i o n j u n c t i o n r e a c t i o n s which o c c u r r e d on u n l o a d i n g . Once sources began to operate, d u r i n g r e l o a d i n g f o r i n c r e m e n t a l s t r a i n i n g , the amount of m i c r o s t r a i n a n t i c i p a t e d from the degree of r e g u l a r i t y of the m i c r o s t r u c t u r e was indeed d e t e c t e d . REFERENCES Adams, M.A. and C o t t r e l l , A.H. 1955. P h i l . Mag., £6, 1187. Ambrosi, P., Gottler, E. and Schwink, Ch. 1974. Scripta M e t a l l . , 8, 1093. Alden, T.H. 1972. P h i l . Mag., 25, 785. • . 1973. Metall. Trans., 4, 1047. . 1974. Meta l l . Trans (in press). Alden, T.H. and Clark, M.A. 1972. John E. Dorn Memorial Symposium (in press). Bailey, J.E. 1963. P h i l . Mag., 8, 223. 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Tinder, R.F. and Washburn, J . 1964. Acta Met., 12, 129. Van Drunen, G. and Saimoto, S. 1971. Acta Met., 19, 213. V e l l a i k a l , G. 1969. Acta Met., r7, 1145. Vorbrugg, W. , Goetting, Chi H. and Schwink, Ch. 1971. Phys. Stat. Sol. (b), 46, 257. Washburn, J . and Murty, G. 1967. Can. J . Phys., 45, 523. Yamaguchi, K. 1928. S c i . Papers Inst. Phys. Chem. Research, 8, 289. Young, F.W. 1961. J . Appl. Phys., 32, 1815. APPENDIX 1 CALCULATION OF WORK HARDENING RATE IF SLIP LINES ARE BLOCKED BY CELL WALLS The present observations and those of Staker and Holt (1972), Ambrosi et al. (1974) and Vorbrugg et al. (1971) notwithstanding, i t might be argued that the observed s l i p l i n e s are b u i l t up from many shorter l i n e s , formed when dislocations traverse the 3u diameter c e l l s and then are blocked. In this appendix i t w i l l be shown that such a model predicts a work hardening rate higher than generally observed. Fisher and L a l l y (1967) suggested that long s l i p l i n e s are b u i l t up from many shorter l i n e s i n order to explain the r e s u l t s of acoustic emission studies made on copper single c r y s t a l s during easy g l i d e . Assuming that burst-type acoustic pulses were generated by small rapid increments of p l a s t i c s t r a i n , they calculated the average s t r a i n per pulse to be ~10~6 to 10" 7. However electron microscopy studies indicated c e l l sizes s u f f i c i e n t to give only 1 0 - 1 1 s t r a i n when traversed by 40 d i s l o c a t i o n s . They conclude that each acoustic burst i s due to co-operative source operation i n many c e l l s , whereby dislocations sweeping 163 164 out one c e l l are blocked by a c e l l wall and t r i g g e r a source in a neighbouring c e l l . In t h i s manner a s l i p l i n e w i l l i be made up of the many short l i n e s produced by many sub-increments of p l a s t i c s t r a i n . This view of p l a s t i c flow can be shown to p r e d i c t too large a value for the work hardening c o e f f i c i e n t , since the rate of d i s l o c a t i o n storage i s too high. In t h i s context i t w i l l be assumed that the storage of both primary and secondary dislocations increase the flow stress through a i relationship of the form x = a Gb (p) 2 where G i s the shear y modulus and p the average d i s l o c a t i o n density (primary or secondary), which i n t h i s case has dimensions of d i s l o c a t i o n length per unit volume. For example, consider a c r y s t a l of length L, and cross sectional area A which contains n c i r c u l a r d i s l o c a t i o n loops, and which has d i s l o c a t i o n c e l l s with an average radius r, then the increment of s t r a i n / ^Yr produced when an incremental number of loops dn traverse d i s l o c a t i o n c e l l s i s given by bur 2 , - ,,. dY = -jg- dn (1) Assuming that a f t e r this s t r a i n increment dn loops undergo some secondary s l i p and become part of the c e l l walls, i n a manner analogous to the "converted pile-up" concept of Hirsch and M i t c h e l l (1967), there w i l l be an incremental increase i n d i s l o c a t i o n density dp given by 165 J „ _ 27rr , d p - _ d n ( 2) The corresponding incremental increase i n flow st r e s s , dx Y' i s given by d i f f e r e n t i a t i n g equation 1.4 Thus -i^. aGb dp dx = n i Y ( p > * ( 3 ) Combining equations (1), (2) and (3), the work hardening c o e f f i c i e n t becomes -JL^SG ( 4 )  d e (p)*r In the present study, p - 3 x 10 9 (in cm/cm3) and r = 2.5 x 10~h (in cm), with a - 1 which gives dx^/de - G/15 higher than the normally observed value of ~G/200 by more than one order of magnitude. Hence i t i s deduced that the dislocations must move over distances greater than the average dimensions of d i s l o c a t i o n c e l l s i n the prestrained microstructure. APPENDIX 2 DEFINITIONS OF TERMS USED TO DECRIBE DEFORMATION Some of the terms used i n t h i s thesis to describe deformation phenomena are derived from t h e o r e t i c a l concepts of microstructural events, while others have t h e i r origins i n experimental observation. By tracing these ori g i n s the aim of t h i s appendix i s to c l a r i f y the use of thi s terminology. Terms Derived from Experimental Observations An i d e a l i z e d form of the experimentally observed s t r e s s - s t r a i n curve from a prestrained specimen i s shown as curve GAB i n Figure 58 (a), with the e l a s t i c (OA) and p l a s t i c (AB) regions intersecting at the yield point (A): the stress l e v e l corresponding to A i s termed the yield stress. The slope of OA i s the e l a s t i c modulus and of AB, the work harden-ing rate,which i s lower and which i n the present study became constant soon afte r the y i e l d point, as shown i n curve OCDB, Figure 58 (a). This second curve i l l u s t r a t e s i n more d e t a i l the type of s t r e s s - s t r a i n relationship usually observed i n 166 F i g u r e 58. Schematic s t r e s s s t r a i n curves. 168 practice. I t d i f f e r s from curve OAB i n that i t has a transition or microstrain region (CD) between the regions of e l a s t i c and p l a s t i c behaviour. If the whole curve i s measured with a high s t r a i n . s e n s i t i v i t y (10~ 3 to 10~ 6), i t may be c a l l e d a micro-strain curve, and a 10~5 yield stress (C) can be i d e n t i f i e d , being the stress at which the measured s t r a i n exceeds the e l a s t i c s t r a i n by 10 - 5. To d i s t i n g u i s h t h i s stress from the y i e l d stress defined by A,the l a t t e r i s sometimes referred to as the macroyield stress. - . , While the curve OCDB w i l l usually l i e within the envelope of curve OAB, when the stress i s also measured with high s e n s i t i v i t y , ( i . e . at a s e n s i t i v i t y of less than 1% of the y i e l d s t r e s s ) , a f t e r low temperature prestrains, i t may move outside t h i s envelope as shown i n Figure 58 (b). In curves of t h i s type, the y i e l d stress w i l l also be defined at the point A. Terms Derived from Theoretical Concepts of Microstructural  Events In these experiments a region of constant work hardening rate (as DB) i s referred to as a steady state region, for i n such a region i t i s assumed that the rate of increase of applied stress with s t r a i n i s equal to the rate of increase i n the (0°K) y i e l d stress with s t r a i n . Any p a r t i c u l a r value of applied stress at which steady state 169 p l a s t i c deformation i s occuring i s defined as the instantaneous steady state flow stress. If the specimen i s unloaded and reloaded, the flow stress may be defined from the reloading curve at the in t e r s e c t i o n of the extrapolated l i n e s for the e l a s t i c and p l a s t i c regions: the flow stress i s so named because i t i s the stress at which large scale i r r e v e r s i b l e d i s l o c a t i o n movement occurs. In general the i n i t i a l flow stress and y i e l d stress coincide, as i n Figure 58 (a). However i n the exceptional case shown i n Figure 58 (b), they d i f f e r s l i g h t l y , the flow stress being defined at E. F i n a l l y , i t must be noted that the terms athermal glide stress and c r i t i c a l stress, frequently used i n discussing s t a t i s t i c a l theories of p l a s t i c flow, are synonymous with the yield stress at 0°k3 T,; (point A at 0°K) . APPENDIX 3 DETAILS OF MICROSTRAIN TESTING Choice of Extensometer In choosing a suitable extensometer for the measure-ment of s t r a i n during the i n i t i a t i o n of p l a s t i c flow, both t h e o r e t i c a l and experimental factors were taken into con-sideration. For t h e o r e t i c a l reasons i t was necessary to perform such measurements at low temperature, so that structure change through thermally activated processes would be n e g l i g i b l e . Furthermore, i n order to avoid the i r r e v e r s i b l e changes i n structure which might have occurred i f the commonly employed load-cycle, or hysteresis microstrain technique were used, the micro-s t r a i n curve was determined during a single load a p p l i c a t i o n . The major experimental requirement, a s t r a i n s e n s i t i v i t y of 10" 5, was determined by preliminary t e s t s . An Instron 1/2", 10% Str a i n Gage Extensometer (Model No. G-51-16) was found to f u l f i l l these basic requirements when used i n conjunction with an Instron Load C e l l Amplifier operating at maximum s e n s i t i v i t y . In order to avoid damaging this extensometer by thermal shock effects during cooling, and 170 171 at the same time, provide a stable temperature under quiet, conditions, a s t a i n l e s s s t e e l cryostat, containing a gaseous environment at 77°K, was employed; a schematic diagram of this cryostat i s shown i n Figure 5 9 (a). Whilst they are sa t i s f a c t o r y i n most other respects, e l e c t r i c a l resistance s t r a i n gauge extensometers exhibit poor r e v e r s i b i l i t y at high s t r a i n s e n s i t i v i t i e s . However, since a single load cycle was to be employed i n these experiments, t h i s drawback was not considered important. Using t h i s equipment (unresolved) s t r a i n was measured to a s e n s i t i v i t y of 7.4 x 10~ 6 (equivalent to one small d i v i s i o n of the elongation s c a l e ) . Modification for Use with Single Crystals The extensometer attachment parts were redesigned so that the extensometer was connected to the c r y s t a l by four pivot screws, Figure 59 (b). The t i p s of these screws were coni c a l , having been c a r e f u l l y machined to a s o l i d angle of 60°, and were located i n small conical depressions, which were made on opposite faces of the c r y s t a l by screws with 90° s o l i d angle, conical t i p s . The l a t t e r set of four screws were held by a standard gauge block, used to give a repro-ducable i n i t i a l separation of the extensometer arms. These attachments and locating points were designed so as to allow the c r y s t a l l a t t i c e to rotate without transmitting a torque to the harp of the extensometer. F i g u r e 59. M i c r o s t r a i n t e s t i n g apparatus: (a) c r y o s t a t and p u l l rod assembly, (b.) d e t a i l o f extensometer attachment. H - J fx) 173 Calibration, for Use at 77°K F i r s t the extensometer was suitably c a l i b r a t e d at room temperature by using a micrometer, and low sig n a l amplification. Next, two sets of locating points were made on a p o l y c r y s t a l l i n e copper sample, and these two gauge lengths were c a r e f u l l y measured (at room temperature) using a t r a v e l l i n g microscope. The two 77°K gauge lengths corre-sponding to those measured at room temperature were then calculated using the appropriate c o e f f i c i e n t of thermal expansion. F i n a l l y , the corresponding e l e c t r o n i c signals • from the extensometer at 77°K were recorded, by attaching the extensometer to the specimen and cooling (to 77°K) for each of the two gauge lengths. Having confirmed that the l i n e a r i t y of the extensometer was maintained at 77°K (by extending an e l a s t i c specimen), the 77°K c a l i b r a t i o n of the extensometer could be completed, and r e l i a b l y reproduced. Use of the Extensometer In the course of various preliminary t r i a l s some procedures were i d e n t i f i e d which were considered e s s e n t i a l to avoid spurious s t r a i n measurements. For example, c r y s t a l alignment problems were found to be n e g l i g i b l e providing the specimen was not removed from the grips a f t e r p r e s t r a i n (and, of course, providing universal j o i n t s were used). This 174 was p a r t i c u l a r l y relevant af t e r high temperature p r e s t r a i n when the specimen was transferred to the microstrain cryostat from the high temperature vacuum furnace. In th i s case, the specimen was c a r e f u l l y transferred using a s o l i d aluminum holding block, which was attached to the grips a f t e r the specimen had cooled to room temperature. Care was taken i n attaching the extensometer to the specimen; the four pivot screws were l i g h t l y tightened (1/8 turn). Any looseness i n t h i s attachment would be re f l e c t e d i n the slope of the e l a s t i c portion of the load-elongation curve. A c r i t e r i o n of constant e l a s t i c slope, equal to 4.9 ± 0.3 x 10 3 Kg mm"2 was used to v e r i f y the r e l i a b i l i t y of the s t r a i n measurement. The s t r a i n rate used was as high as p r a c t i c a l , whilst s t i l l allowing time for zero suppression and scale changes. F i n a l l y , thermal s t a b i l i t y was recognized by the existence of a stable balance point for the extensometer and was attained about one hour after the s t a r t of cooling. APPENDIX 4 OPTICAL MICROSCOPY WITH NORMARSKI INTERFERENCE CONTRAST The purpose of th i s appendix i s to i l l u s t r a t e the advantage of using Normarski Interference Contrast, a tech-nique which proved to be indispensible i n this work. The Normarski technique makes use of polarizing-interference contrast, and i n consequence, microstructural features on the object which are d i f f e r e n t i a t e d by surface r e l i e f (after etching, for example) are distinguishable i n the image through both colour and i n t e n s i t y . An excellent explanation of the interference p r i n c i p l e together with d e t a i l s of the opt i c s , may be found i n G i f k i n s 1 book Optical Microscopy in Metals (1970). The dramatic increase i n resolution, gained by use of Normarski Interference can be seen i n Figure 60 (a), (b). These two micrographs are taken from the same area of an etched surface, and with the same o p t i c a l conditions, except that i n one case polarized l i g h t was used together with a Normarski interference attachment, whereas i n the other, i n the absence of an interference attachment, a green 175 176 advantage of Normarski Interference Contrast x900: (a) bright f i e l d , (b) with Normarski Interference Contrast. 177 f i l t e r was used with unpolarized i l l u m i n a t i o n . Both, these micrographs were taken on a Zeiss Ultraphot II microscope, using a.high pressure mercury l i g h t source together with a planachromat objective (Epiplan x40, 0.85NA). Assuming that for each micrograph the average 'Wavelength i s A = 5500 o (in A),.the resolution l i m i t (the smallest distance between two points on the object, for which these points can remain distinguishable i n the image), given by R = 0.61A/N.A., w i l l be the same i n each case, namely ~0.4u. In the absence of interference contrast, t h i s r e s olution was impossible to achieve and the considerable loss of d e t a i l i s r e a d i l y apparent on comparison of Figures 60 (a) and (b). On the other hand, an extremely s a t i s f a c t o r y resolution and contrast was obtained using Normarski Interference with which i t was possible to d i s t i n g u i s h etch p i t s having a separation of less than 0.5u. APPENDIX 5 A PROBABILITY MODEL FOR RANDOM DISTRIBUTIONS I n t r o d u c t i o n When studying the r e l a t i v e values of a property of a p h y s i c a l system, i t i s u s e f u l to de f i n e an absolute value w i t h which to compare those measured. Often the thermodynamic e q u i l i b r i u m . s t a t e may be .used.as such a standard. However, i n the present study, where the property of i n t e r e s t i s d i s l o c a t i o n d i s t r i b u t i o n , the thermodynamic e q u i l i b r i u m s t a t e provides an empty standard, s i n c e d i s l o c a t i o n s themselves are thermodynamically unstable: a f u l l y annealed c r y s t a l contains no d i s l o c a t i o n s . So we must look elsewhere to define a standard d i s l o c a t i o n c o n f i g u r a t i o n . In recent years there has been i n c r e a s i n g i n t e r e s t i n the s t a t i s t i c a l a n a l y s i s of d i s l o c a t i o n motion through f i e l d s of p o i n t o b s t a c l e s . The model systems have ob s t a c l e s which are p o s i t i o n e d randomly (Kocks, 1966; Foreman and Makin, 1966; Morris and Klahn, 1973); r e g u l a r ( t r i a n g u l a r ) o b s t a c l e arrays (Foreman and Makin, 1967) and intermediate 178 179 cases (random arrays with s p e c i f i e d minimum obstacle spac-ings) (Foreman, unpublished, reported i n Brown and Ham, 1971) have also been considered. Point obstacles may represent for example solute atoms, small p r e c i p i t a t e s or forest d i s l o c a t i o n s . For the two former examples, random and regular arrays correspond to thermodynamic equilibrium states i n r e a l systems. In the case of forest d i s l o c a t i o n s , where no such thermodynamic equilibrium configuration e x i s t s , i t i s d i f f i c u l t to t h e o r e t i -c a l l y determine even a pseudo-equilibrium configuration, since the mechanisms which produce these arrays are varied and complex. However, i n the early stages of work hardening, d i s l o c a t i o n arrays may appear, to the casual observer, to be more or less random, and for convenience t h i s may be used as a standard configuration. The P r o b a b i l i t y Model If a f i e l d of randomly placed points i s sampled by covering i t with a g r i d of square sample areas, the p r o b a b i l i t y function for the number of points per area i s a solution of the C l a s s i c a l Occupancy Problem ( F e l l e r , 1950, p. 54). If a f i e l d of r randomly placed points i s divided into, n equal areas, there are n r possible arrangements of points over these areas, each with a p r o b a b i l i t y of n r . 180 The p r o b a b i l i t y that any given area has exactly k points i s determined i n the following way. For t h i s area, k points ways. The r \ r can be chosen from the t o t a l of r i n remaining (r-k) points can be placed i n the remaining n-1 r—k c e l l s i n (n-1) ways. So the p r o b a b i l i t y that the c e l l under consideration contains exactly k points i s then P k = r (n-1) kl r ' n r-k (5) This i s a binomial d i s t r i b u t i o n function, which may be approximated by the Poisson d i s t r i b u t i o n k; n _ r n f r l * e — n ki (6) i n cases where r and n are large. The Poisson d i s t r i -bution function has a mean x , and standard deviation s* where x = — and n s = r •> r n (7), and the function has been tabulated to large values of — n and k (Molina, 1943). Although the mean and s tandard d e v i a t i o n of a random v a r i a b l e are u s u a l l y de s c r i bed by the symbols \i and a r e s p e c t i v e l y , f o r c l a r i t y , the symbols x and s, c o n v e n t i o n a l l y used to d e s c r i b e sampled da ta , are a l so used he re . 181 r By choosing d i f f e r e n t values of — i n t h i s table, sampling with d i f f e r e n t area sizes can be modelled. For example, an area size can be chosen to give a sample mean of — = 10. The histogram, showing the f r a c t i o n of squares which contain a given number of points, sampled with t h i s square size from a f i e l d of randomly placed points, appears i n Figure 50. Clearly the shape of such a histogram w i l l depend on the sample area size chosen. To i l l u s t r a t e t h i s point the — = 25 histogram for random d i s t r i b u t i o n s i s also shown i n n ^ Figure 50. From equations (7) and (8), the v a r i a t i o n i n s s =• with sample area size i s given by = X X n and t h i s r e l a -tionship appears i n Figure 52 and 51. In contrast i t should be noted that samples from a vegular- array of point obstacles w i l l have standard devia-tion s = 0 (for — > 1 as i s the case i n t h i s work); thus s s — = 0 also. Hence, the plo t of — shown for a regular x x array i n Figure 51 and 52 w i l l follow the abscissa. 

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