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Plasticity of [Beta]'AuZn single crystals Schulson, Erland Maxwell 1967

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The University of British Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of -ERLAND MAXWELL SCHULSON B.A.Sc. (Hons.), The University of British Columbia THURSDAY, JANUARY 4, 1968 AT 3:30 P.M. IN ROOM 201, METALLURGY BUILDING COMMITTEE IN CHARGE Chairman: B.N.Moyls CA. Brockley D.L. Williams L.C. Brown N.R. Risebrough E. Teghtsoonian D. Tromans External Examiner: Dr. R.E. Smallman Department of Physical Metallurgy and Science of Materials The University of Birmingham Birmingham, England Research Supervisor: Dr. E. Teghtsoonian THE PLASTICITY OF Q> 'AuZn SINGLE CRYSTALS ABSTRACT Single crystals of the CsCl type intermetallic compound 'AuZn were prepared and tested in tension ; over a wide range of temperatures, strain rates and orientations for three compositions, Au-rich (51.0 at .% Au), stoichiometric and Zn-rich (51.0 at.% Zn). Slip surfaces are generally non-crystallographic planes in the zone of the sl i p direction [001], and are temperature, strain rate and orientation sensitive. A model based on thermally activated s e s s i l e - g l i s s i l e transformations of screw dislocations has been proposed to explain non-crystallographic s l i p . Multi-stage work-hardening is observed over the temperature range 0.2*^ T/T ^0.35. In stage I the work-hardening rate is low \^ /*. /1000 to/*A./5000) but rises sharply during stage II ( Q - Q ^ _/A/500) . Stage III is characterized by a rapidly decreasing hardening rate coincident with the onset of profuse large-scale cross-slip. Surface sl i p line studies revealed that the end of easy glide is coincident with the onset of localized s l i p on non-crystallographic planes in the [lOd] zone. Thin f o i l electron microscopy was carried out on c r i t i c a l l y chosen crystallographic sections from annealed and deformed crystals. At the beginning of stage I clusters of edge dislocation dipoles were revealed, forming walls perpendicular to the glide plane. The dislocation density of the walls increases during easy glide. During testing at intermediate temperatures ('v.3 to .4 T ) serrated yielding was detected in non-stoichiometric crystals and was attributed to dislocation-solute atom interactions. Under special testing conditions (77°K or near<001> orientations) s l i p occurs in <111> directions. The associated work-hardening rates are very high and ductility is low. Thermal activation studies were made to deter-mine the dislocation mechanism responsible for the temperature sensitivity of yield in stoichiometric crystals below <"v 220 K. Activation volume measure-ments are consistent with both the Peierls-Nabarro and cross-slip mechanisms below ^ 150 K. GRADUATE STUDIES Field of Study: Physical Metallurgy Metallurgical Thermodynamics Structure of Metals Advanced Physical Metallurgy Diffusion Electron Microscopy Related Studies: Quantum Theory of Solids Theory of Plasticity C. S. Samis E. Teghtsoonian E. Teghtsoonian L.C. Brown L.C. Brown D. Tromans R. Haering H. Ramsey THE ; PLASTICITY OF ^' AuZn SINGLE CRYSTALS by . ERLAND .-MAXWELL'. SCHULSON B.A.Sc, ..University of British Columbia, 1 9 6 4 A 'THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE.REQUIREMENTS' FOR'THE DEGREE OF DOCTOR OF PHILOSOPHY in- the Department of METALLURGY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ' December, I967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i 1 a b 1 e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by hils r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M e t a l l u r g y  The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date A p r i l 1, 1968 i Supervisor: Dr. E. Teghtsoonian ABSTRACT Single crystals of the CsCl type intermetallic compound Q*AuZn were prepared and tested in tension over a wide range of temperatures, strain rates and orientations for three compositions,,Au-rich ( 5 1 . 0 at.$> Au), stoichiometric and Zn-rich ( 5 1 > 0 at.$ Zn). Slip surfaces.are generally non-crystallographic planes in the zone of the s l i p direction [ 0 0 1 ] , .and are temperature, strain rate and orientation sensitive. A model based on thermally activated ses s i l e - g l i s s i l e transformations of screw dislocations has been proposed to explain non-crystallographic s l i p . Multi-stage work-hardening is observed over the temperature range 0 . 2 ^T/T 'Z. O . . 3 5 . In stage I the work-hardening rate is low'( A/lOOO to / f / 5 0 0 0 ) but rises sharply during stage TI (On ~ /'/5OO). Stage III is characterized by a rapidly decreasing hardening rate coincident with the onset of profuse large-scale cross-slip. Surface s l i p line studies revealed that the end of easy glide is coincident with the onset of localized s l i p on non-crystallographic planes in the [ 1 0 0 ] zone. Thin f o i l electron microscopy was carried out on c r i t i c a l l y chosen crystallographic sections from annealed and deformed crystals. At the beginning of stage I clusters of edge dislocation dipoles were revealed, forming walls perpendicular to the glide plane. The dislocation density of the walls increases during easy glide. During testing at intermediate temperatures ( ~ . 3 to .h T ) m serrated yielding was detected in non-stoichiometric crystals and was attributed to dislocation-solute atom interactions. i i •Under special testing conditions (77°K or near K. OOl"/* orientations) s l i p occurs in <111> directions. The associated work-hardening rates are very high and d u c t i l i t y is low. Thermal activation studies were made to determine the dislocation mechanism responsible for the temperature sensitivity of yield in stoichiometric crystals below <^220°K. Activation volume measurements are consistent with both the Peierls-Nabarro and cross-slip.mechanisms below ">-150oK. ACKNOWLEDGEMENT The author gratefully acknowledges the advice and encouragement given by his director, Dr. E. Teghtsoonian. Helpful discussions with various members of the faculty and fellow graduate students are also acknowledged. Financial assistance was received from the International Nickel Company of Canada Limited in the form of a post-graduate fellowship. iv TABLE OF CONTENTS Page THE PLASTICITY OF @' AuZn SINGLE CRYSTALS 1. INTRODUCTION AND OBJECTIVES . 1 2 . DEFORMATION CHARACTERISTICS-OF (3»' AuZn;SINGLE CRYSTALS... 7 2 . 1 . EXPERIMENTAL PROCEDURE 7 2 . 1 . 1 . Alloy Preparation.and Crystal Growth .7 2 . 1 . 2 . Tensile Specimen Preparation • 8 2 . . 1 . . 3 . Testing Procedure 9 2 . 2 . GENERAL DESCRIPTION OF THE SHEAR STRESS-SHEAR STRAIN CURVES 11 2 . 2 . . 1 . Experiments ..... .11 2 . 2 . 2 . Definition of Work-Hardening Parameters 11 2 . 2 . 3 . Temperature and Strain Rate Dependence .12 2 . 2 . 4 . Effect of Deviations from Stoichiometry .18 2 . 2 . 5 . Orientation Dependence 2 1 2 . 3 . SERRATED FLOW 2k 2 . 3 . 1 . Occurrence 2k 2 . 3 . 2 . Origin . . . 2 8 2 . 3 . 3 - Dislocation-Solute Atom Interactions 2 9 2 . 3 . 4 . Segregating Species 3 2 2.k. DEFORMATION MODES . 3 6 2 . 4 . 1 . Introduction . 3 6 2 . 4 . 2 . Procedure . 4 3 2 . 4 . 3 . Definitions 43 2 . 4 . 4 . Slip Direction .44 2 . 4 . 5 . Primary-Slip Plane ...... 47 2 . 4 . 5 . 1 . Temperature Dependence .47 TABLE OF CONTENTS (continued) Page 2 . 4 . 5 . 2 . Strain Rate Dependence 56 2 . 4 . 5 - 3• Orientation Dependence 57 2 . 4 . 5 . 4 . Composition Dependence 6 4 2 . 4 . 6 . . Discussion 64 2 . 4 . 6 . 1 . < 0 0 1 > Zonal Slip 6 5 2 . 4 . 6 . 2 . { hkl] < 1 1 1 > Slip 72 2 . 5 . WORK-HARDENING'BEHAVIOUR 7 8 2 . 5 . 1 . .Flow Parameters . 7 8 2 . 5 . 1 . 1 . Yield Stress . 7 8 2 . 5 . 1 . 2 . The Work-Hardening Rate in-Stage I , Q± 8 6 2 . 5 . 1 . 3 . ..The End of Stage I 8 8 2 . 5 . 1.4. The Work-Hardening Rate in Stage II, On 92 2 . 5 . 1 . 5 . Stage III .97 2 . 5 . 1 . 6 . Maximum Shear Stress and Ductility .. . .98 2 . 5 . 2 . Slip Line Variation During Deformation . 1 0 0 2 . 5 . 2 . 1 . Procedure 1 0 1 2 . 5 . 2 . 2 . Observations . 101 2 . 5 . 2 . 3 . Deformation Bands .110 2 . 5 . 2 . 3 . 1 . Characteristics 1 1 0 2 . 5 . 2 . 3 . 2 . Crystallographic Nature... .111 2 . 5 . 2 . 3 . 3 . Mechanism of Formation.... 112 2 . 5 . 2 . 4 . Microcracks 113 2 . 5 . 3 . Transmission Electron Microscopy of.Thin Films. 115 2 . 5 . . 3 . 1 . Introduction 115 2 . 5 . 3 . 2 . Procedure 116 2 . 5 . 3 . 3 . Observations 117 v i TABLE OF CONTENTS (continued) .' Page 2 . 5 . 3 . 3 . 1 . As-Grown Structure . 1 1 7 2 . 5 . 3 . 3 . 2 . Variation in Dislocation Structure During Defor-mation, at 2 9 3 ° K . 1 1 9 . 2 . 5 . 3 . . 3 . 2 . I . (hko) Section 1 1 9 2 . 5 . . 3 - 3 . 2 . 2 . ( 1 1 0 ) Section 1 2 4 2 . 5 . 3 . 3 . 2 . 3 . Perpendicular (hko) Section 1 2 6 2 . 5 . 3 . 4 . -Discussion 1 2 6 2 . 5 . 4 . Discussion . .... . . 1 3 1 2 . 5 . 4 . 1 . Yield Stress Variation with Orientation 1 3 1 2 . 5 . 4 . 2 . Work-Hardening 1 3 3 2 . 6 . THERMALLY .ACTIVATED YIELD l 4 0 2 . 6 . 1 . Introduction 140 2 . 6 . 2 . Activation Volume l 4 4 2 . 6 . 3 . Activation Energy I5O 2 . 6 . 4 . Discussion 1 5 3 2 . 6 . 4 . 1 . Impurity Obstacles 1 5 3 2 . 6 . 4 . 2 . Peierls-Nabarro (PN) Mechanism 1 5 3 2 . 6 . 4 . 3 . Cross-Slip . 1 5 8 3 . SUMMARY AND CONCLUSIONS . 1 6 2 4 . SUGGESTIONS FOR FUTURE WORK . I 6 5 5 . APPENDICES 1 . Crystal Homogeneity 1 6 6 2 . Evaluation of Machining Damage . 1 7 0 3 . Equations for Resolved Shear Stress and Resolved Shear-Strain 1 7 3 v i i TABLE OF CONTENTS (continued) Page k. Taylor Rotation. Axes 175 5. Shear Modulus as a Function of Slip System .178 6. BIBLIOGRAPHY l8h v i i i • LIST OF FIGURES Page 1. Common types of superlattices i n which c r y s t a l structure does not change upon the formation of long-range order .1 2. Equilibrium diagram f o r the Au-Zn system. 6 3 . Diagram of a te n s i l e specimen 10 h. Showing specimen gripping arrangement 10 5 . Schematic resolved shear stress-shear str a i n curve .12 6. Resolved shear stress-shear s t r a i n curves of Au-rich (3* • AuZn crystals as a function of temperature between 77°K and 488°K ( f t = 2.5 x 1 0" 3/sec). .14 7. Resolved shear stress-shear s t r a i n curves of'stoichiometric AuZn crystals as a function of temperature between 77°K and 443°K ( Tf = 2.5 x 10 3/sec) 15 8. Resolved shear stress-shear s t r a i n curves of Zn-rich AuZn crystals as a function of temperature between 7 7 ° K and 473°K ( ft = 2.5 x 1 0 _ 3/sec) 16 9. .Resolved shear stress-shear s t r a i n curves of Au-rich AuZn crystals as a function of s t r a i n rate from 2.5 x 10" 4/sec to 2 .5 x 10" 2/sec (T = 293°K).... 17 10. Resolved shear stress-shear s t r a i n curves of J^' AuZn crystals as a function composition. 1 0 . 1 . at 293°K and'373°K. 19 1 0 . 2 . at 77°K and ~220°K 20 11. Resolved shear stress-shear s t r a i n curves of Au-rich (5,' AuZn crystals as a function of orientation. (T - 295°K; = 2 .5 x 10" 3/sec)...- 22 12. Schematic representation of a serrated flow curve 25 13. Photograph of segments of a serrated load-elongation curve during stage I and stage I I deformation 27 14. Showing the v a r i a t i o n i n c r i t i c a l s t r a i n with c r i t i c a l s t r a i n rate 15. Showing the effect of temperature on c r i t i c a l s t r a i n .34 16. A (001) stereographic projection showing the parameters-characterizing the specimen orientation and the s l i p plane r e l a t i v e to (110) .45 17. "Stereographic representation of specimen axis reorientation during p l a s t i c deformation as a function of temperature..... .46 ix LIST OF FIGURES (continued) Page 18. Crystal orientations used in s l i p plane-temperature study.... .48 1 9 . Photomicrographs of typical s l i p traces on.orthogonal faces A and B where A /^(2"01) -and B ~ ( 0 1 0 ) . 4 9 2 0 . -Replicas of surface sl i p traces on orthogonal faces A and B 48 2 1 . -A sketch of s l i p trace development on orthogonal surfaces •when one surface is parallel to the Burgers vector of the mobile dislocations . 5 1 2 2 . •Back-reflection Laue X-ray pattern from surface shown in Figure I9.A.I . . 53 2 3. Showing the variation in the s l i p plane parameter with temperature 5 5 24. Showing specimen orientations used in sl i p plane analysis.... .57 2 5 . Showing the variation of sl i p plane parameter with orientation 5 9 2 6 . Showing duplex s l i p in crystal oriented - along 1 0 1 - 111 boundary 6 0 2 7 . -A ( 0 0 1 ) stereographic projection showing the most highly stressed system of the form {lio} C 0 0 l ) . as a function of orientation 6 2 2 8 . Multiple s l i p observed near [ 0 0 l ] orientations 63 2 9 . -Schematic ill u s t r a t i o n of continual cross-slip on . orthogonal ^110} planes 6 8 3 0 . Schematic representation, of dissociation reactions 6 8 3 1 . -A sketch of the sessile to g l i s s i l e transformation sequence.. .70 32. 'Schematic ill u s t r a t i o n of the continual cross-slip cycle defining the s l i p plane parameter. 7 0 33- Showing the variation of yield stress with temperature for Au-rich,stoichiometric and Zn-rich ^>AuZn single crystals... 83 3 4 . Showing the resolved yield stress dependence on orientation.. 85 3 5 - "Showing effect of specimen geometry on inhibiting (hko) [ 0 0 1 ] s l i p 86 3 6 . Showing the variation in stage I hardening rate with temperature 87 X LIST OF FIGURES (continued) Page 37• Showing the variation in stage T work-hardening rate with orientation 88 38. Showing the effect of temperature and composition.on the extent of easy glide 89 39- Showing the effects of temperature and composition on the stress at the end of easy glide .90 kO. Showing the variation on the extent of easy glide with orientation .92 kl. Showing the variation of stage. II work-hardening rate with temperature .93 k2. Showing the orientation dependence of stage II work-hardening rate .. .95 43. Showing the effect of temperature on the stress at the end of stage II .98 kk. Showing the variation of maximum shear stress with temperature 99 45. Showing the variation of total d u c t i l i t y with temperature 99 46, 47, Variation in sl i p line structure with strain at 77°K, .102 48, 49, l40°K, 293°K, 398°K and 473°K ... -50. 106 51. Stereographic projection of deformation band poles versus crystal orientation and test temperature I l l 5 2 . Schematic representation of dislocations in deformation bands 1 1 2 Electron micrographs of dislocation structure: 5 3 • in as -grown crystals . 1 1 8 5 4 . at the beginning of stage I, (hko) section 1 2 0 , 1 2 1 5 5 - at the end of stage I, (hko) section 1 2 2 , 1 2 3 5 6 . at the beginning of stage I, ( 1 1 0 ) section..... .. 1 2 5 5 7 - at the beginning of stage I, section perpendicular to (hko).. 1 2 7 5 8 . . a t the beginning of stage I, section approximately perpendicular to ( 1 1 0 ) . 1 2 8 5 9 - Showing the experimental compared with the predicted values of c r i t i c a l resolved yield stress ratio versus s l i p plane parameter 1 3 2 6 0 . Illustrating the athermal and thermal components of the yield stress 1 4 3 x i LIST OF FIGURES (continued) Page 61. Schematic representation of changes i n the flow curve accompanying s t r a i n rate change t e s t s IU5 62. A c t i v a t i o n volume against shear s t r a i n at temperatures between 77°K and 213°K lk6 63. -Showing the v a r i a t i o n i n e f f e c t i v e stress with temperature.. lk& 6k. Showing the v a r i a t i o n of a c t i v a t i o n volume with e f f e c t i v e stress f o r ^ 1 AuZn and bcc metals iky 65. Showing the va r i a t i o n , of a c t i v a t i o n enthalpy with e f f e c t i v e s t r e s s . . . I52 66. Schematic i l l u s t r a t i o n of the Peierls-Nabarro mechanism 154 67. Schematic i l l u s t r a t i o n of the sinusoidal and quasi-parabolic P e i e r l s " h i l l " p r o f i l e s 156 68. Showing the v a r i a t i o n i n a c t i v a t i o n enthalpy with temperature 157 69. Showing the v a r i a t i o n i n a c t i v a t i o n volume with temperature.. .159 70. Showing the f u n c t i o n a l dependence of a c t i v a t i o n volume on e f f e c t i v e stress l 6 l A l . Showing composition gradients i n as-grown ^>'AuZn single c r y s t a l s 168 A2.1, Showing the v a r i a t i o n i n degree of asterism with reduction A2.2, of diameter of machined t e n s i l e specimens 171 .-A2.-3. A2.4. Showing t e n s i l e strength of machined c r y s t a l versus amount removed from the specimen diameter 172 A3.1. . 174 Ak.l. Sketch of asterism from Figure 22 175 A4 .2.' Stereographic-projection of a l l £L10) and (21lj poles with respect to the indexed d i f f r a c t i o n from Figure 22 177 A5.I.-Showing the x^ reference frame r e l a t i v e to the x^ frame 179 A5.2. 182 x i i LIST OF TABLES No. Page 1. Common Super-lattices 3 2. Showing the variation in c r i t i c a l strain with temperature and strain rate for non-stoichiometric (3' AuZn crystals... 26 3 . Slip systems in Cs'Cl type compounds 37 k. Comparison of line energies and mobilities of low energy dislocations in CuZn, NiAl, CsBr and AuZn kO 5. -Results of sl i p line analysis of temperature effect on sli p plane parameter 52 6. Results of sl i p trace analyses of strain rate effect on sli p plane parameter .56 7. Results of sl i p plane analyses of orientation.effect on sli p plane parameter 58 8. Correlation of sl i p direction with heats of formation and electronegativity differences in CsCl type compounds........ 75 9,.10, Work-hardening parameters as a function of temperature, 79 11,12. composition and orientation. -82 13. Ratio of the stress at the end of easy glide to the yield stress.as a function of temperature, composition and orientation 91 lh. Variation in stage II hardening rate with strain rate .9^  15. Comments on sl i p line variation during deformation 107, 108, 109 16. -Activation parameters / i H and v 0 * at temperatures between-77°K and 175°K for stoichiometric crystals... .151 • A l . l . Chemical analysis of as-grown crystals 167 A2..1. -Effect of annealing temperature and time on the strength of two tensile specimens relative to the unannealed condition 170 A5.I. Non-zero suffixes l 8 l A 5 . 2 . Shear moduli! for-various s l i p systems in cubic structures 183 1 1. •INTRODUCTION AND OBJECTIVES Intermetallic compounds can be defined as intermediate phases in binary or higher order metal - metal systems. Such compounds may possess long range order at a l l temperatures.in the solid state or undergo an order-disorder transformation at a c r i t i c a l temperature above which the compound adopts a nearly random structure. In the latter case, short range order persists in a decreasing degree up to the melting point. When highly ordered, intermetallic compounds have lower symmetry than the corresponding disordered alloy, leading to extra reflections, termed superlattice reflections, in their diffraction patterns. The four most common superlattices in which the crystal structure does not change upon the formation of long range order are shown in Figure 1 and their common names, corresponding disordered structure and examples of each are given in Table 1. The most frequently occurring super-< * > I Figure 1. Common types of superlattices in which crystal structure does not change upon the formation of long-range order: (a)B2, fb) L l 2 fc).D0i9 (d),D03. (after Stoloff and Davies 1! 3) lat t i c e type is the B2 or CsCl type structure which takes the form of two interpenetrating simple cubic lattices. The symmetry of the B2 structure is 2 lowered from a bcc in which a l l sites are equivalent to a simple cubic in which the ( 0 , 0, 0) and ( 1 / 2 , 1 / 2 , . 1 / 2 ) sites are different. The consequence of reduced symmetry is that two types of regionsrnay occur within a crystal: where the A atoms and B atoms are in their respective <=<. (0,0,0) and ( 1 / 2 , l / 2 , 1 / 2 ) sites and where the A and B atoms interchange sites such that A occupy sites and B occupy cX. sites. The boundary, between two such regions w i l l contain bonds between like atoms and w i l l be a surface of higher energy. The regions, are known as antiphase domains (APD) and the boundary as an antiphase boundary (APB). Because a stable domain structure . 1 1 4 * requires at least four sublattices, Bragg suggested that APB s in the B 2 structure(having two sublattices) which form during an ordering process should disappear as a result of domain growth. -Recent transmission electron microscopy studies on the B 2 compound NiAl 5 (ordered up to i t s melting point) led to the conclusion that' A P D ' S do not exist in annealed or deformed * > samples, substantiating Bragg s suggestion. In similar studies APB s have been detected in the B 2 compound Q-brass quenched from above the c r i t i c a l •116*117 ordering temperature. However, experiments were not carried out to * determine whether or not the APB s would disappear on prolonged annealing. The second most common superlattice type is the L l 2 structure, related to fee in much the same way as the B 2 superlattice is to bcc. The face centered sites are occupied by A atoms and the corner sites by B atoms. Symmetry is again lowered to simple cubic and antiphase domains can exist since of the four i n i t i a l l y equivalent sites ( 0 , 0 , 0 ) , ( 0 , 1 / 2 , l / 2 ) , ( l / 2 , 0, l / 2 ) and ( l / 2 , l / 2 , 0 ) , any one may be occupied by a B atom and the other three by A atoms. APB^s have been detected i n - L l 2 compounds 118* us* 1 2 0 i 2 i Cu3Au and Ni3Mn . Less commonly observed superlattices are the DO3 and D O 1 9 types. The most complex is the D0 3.type which is built up of eight bcc unit cells and may be considered as being composed of four inter-3 TABLE 1  Common Superlattices Structure Type C ommon • Name Disordered Structure Examples L 2 0 or B2 CsCl bcc CuZn AgZn AuZn AuCd AgCd NiTi AgMg NiAl CuAl FeAl FeCo L l 2 Cu3Au fee Cu3Au Au3Cu Ni3Mn Ir 3Cr Ni 3Fe Ni 3 A l •Pt3Fe D03 •Fe 3Al bcc Fe 3Al Fe 3Si Fe 3Be Cu 3Al D0 1 9 . Mg3Cd hep MgaCd Cd3Mg T i 3 A l Ni 3Sn penetrating fee lattices. For the DO19 structure, the ordered unit c e l l may be compared to four unit cells of the disordered hep structure. In some instances ordering also effects a change in crystal structure. The most common superlattice of this type is typified by the CuAu structure in which alternate layers of Cu and Au atoms form on (001) planes of the fee disordered solution, distorting i t into a f c t structure. The fourfold axis of symmetry is normal to the alternating planes of Au and Cu atoms and the axial ratio is usually between 0.9 and 1.0. Other examples of the fct ordered structure (classified as L l 0 type superlattice) are CoPt, FePt and FePd. Deformation studies on intermetallics have centered almost wholly on the behaviour of polycrystalline material under various conditions k of temperature, strain rate, grain size, defect structure created by depar-ture from stoichiometry and degree of long range order. In a comprehensive 1 2 2 treatment of the subject prior to 1 9 5 9 * Westbrook reviewed a large number of papers devoted to fabrication, testing procedures and the properties of very high melting point compounds as well as a few concerned with surface s l i p line structure and antiphase boundary observations. General interest in associated dislocation structure and behaviour was not apparent at that time. In recent years with the improvement of fabrication and testing techniques and the advent of sophisticated transmission electron microscopy contrast theory, much progress has been made from both the experimental and theoretical approaches in understanding the plastic behaviour and fracture of intermetallics. Also many papers have appeared in the literature relating strength and degree of long-range order. 113 In their recently published review, 'Stoloff and Davies point out that the mechanical behaviour of alloys that form superlattices at a c r i t i c a l temperature below the melting point can be understood mainly in terms of changes in dislocation configuration with degree of order. The ordered materials deform by the movement at relatively low stresses of superlattice dislocations which consist generally of closely spaced pairs of unit dislocations. Since the dislocations are constrained to move as a group to preserve the ordered arrangement of the l a t t i c e , cross s l i p i s 1 2 3 * 1 2 4 1 2 5 hindered thereby leading to high work-hardening rates (Cu3Au FeCo ) and b r i t t l e fracture (FeCo ). Decreasing the degree of long range order brings about an increase in the separation of superlattice partials which can explain the peak in yield stress manifested near the c r i t i c a l 7 6 _ n • 1 2 5 * 1 2 7 7 5 * 1 2 8 ordering temperature by many superlattices (Cu3Au, ieuo, CuZn 1 2 7 * 1 2 9 * 1 3 0 and Fe 3Al ). Variations in long range order, however, cannot explain similar intermediate-temperatures strengthening effects in alloys 5 1 0 1 131>132>>133 1 3 4 ordered up to the melting point (AgMg, Ni 3 A l and NiAl ). The resolution of this phenomenon necessitates detailed single crystal studies. In CsCl type superlattices (including intermetallic compounds as well as ionic materials such as Csl and CsBr) a distinction may be made between compounds on the basis of s l i p direction. When bonding is of ionic character, electrostatic forces prevent like ions from becoming nearest neighbors and K001^slip occurs, but when bonding is of metallic character and the two kinds of "ions" that alternate along close packed rows are practically indifferent to their neighbors, <111> sli p occurs. The poly-crystalline mechanical consequence of s l i p direction is reflected in the extreme brittleness of the "ionic" compounds compared to the relative d u c t i l i t y of the "metallic" compounds. Further discussion of s l i p modes in CsCl type superlattices with direct reference to the literature w i l l appear in the appropriate sections of the thesis. Perhaps the major obstacle in the path of more widespread industrial use of intermetallies is their extreme brittleness at low tern-13 R 13P5 1 3 6 peratures. Origin of brittleness in AgMg ,,NiAl ' and NiGa has been attributed to the segregation of i n t e r s t i t i a l impurities to grain boundaries. Grain boundary contamination, however, cannot be the only reason for b r i t t l e -ness. In NiAl for instance, single crystal studies show that sl i p occurs on £L10} <"00lj> systems. Since there are only three independent i l 2 9 110] <"001>modes , general polycrystalline d u c t i l i t y is not possible, and hence, even in the absence of grain boundary contamination, NiAl would be b r i t t l e at low temperatures. This example illustrates the value of single crystal studies in uncovering the nature of the deformation behaviour of intermetallics. The object of the work presented in the thesis was to study the plastic deformation of an intermetallic compound in single crystal form. The B2 compound 'AuZn was chosen for three reasons: (l) The moderate and congruent melting temperature of 725°C would ease the preparation of homogeneous single crystals; . (2) The solubility range from ..5 to 5I.O at. % Au, Figure 2, would permit an evaluation of the effects of deviations from stoichiometry on the plastic behaviour; (3) The crystal structure is relatively simple and remains 1 O T ) 1 3 T highly ordered to the melting point, thereby simplifying subsequent analysis of the results. Because the work represents one of the f i r s t deformation studies of inter-metallic crystals, the thesis is' not limited to the investigation of a single phenomenon. Instead the effects were noted of a wide range of variables that include temperature, strain rate, orientation and deviations from stoichiometry. Justification for the project stems from an academic . interest in the fundamental deformation behaviour of ordered alloys. Figure 2 . Equilibrium diagram for the Au-Zn system. 7 2. DEFORMATION CHARACTERISTICS OF @' AuZn'SINGLE CRYSTALS 2.1 EXPERIMENTAL PROCEDURE 2.1.1 Alloy Preparation and Crystal Growth The gold and zinc used in this investigation was of 99-999$ purity and was supplied by.the Cominco Ltd., T r a i l , B.C. in the form of gold splatter and one-half inch diameter zinc rod. Alloys weighing approximately 75 grams were prepared by encapsulating gold and zinc of accurately weighed amounts.under reduced pressure in 11 mm. diameter carefully cleaned fH 2S0 4-Cr 203 hot solution) quartz tubes then fusing at 800°C. The melts were repeatedly agitated to ensure thorough mixing then quenched in cold water to minimize segregation on solidification. To remove the de-zinced surface zone, castings were electrochemically polished in 5$ KCN solution (12 volts, approximately -l^ amp/cm2,.40°C). The piped end was cropped off each casting. The b i l l e t s were reduced to 0.108 inch wire f i r s t by hot swaging at 300°C to 0.16J inch rod then cold drawing to the f i n a l size. The wire was straightened while held in the die by heating with a propane torch after the f i n a l draw. Segments to be grown into single crystals approximately 16 inches in length, were thoroughly cleaned (degreased, abraded, and electrochemically polished in 5$ KCN solution), charged into pre-cleaned, close f i t t i n g 0.113 inch quartz quills then sealed under reduced pressure. Single crystals were grown in the standard Bridgman manner by superheating the melt 50°C to 775°0 then lowering the charge at 7cm/hr. through a temperature gradient of 25°C/cm. The quartz sheaths were subsequently removed in HF. Crystal orientations were determined at three points along their lengths from back-refleetion-Laue X-ray diffraction 8 patterns. I t was found that t h i s technique yielded c r y s t a l s bearing random a x i a l orientations, and that a common ori e n t a t i o n could be preserved by adopting a standard seeding technique. During preliminary.investigations of the c r y s t a l growth conditions, i t was found that c r y s t a l s having smooth surfaces would be obtained only i f the wire were c l o s e l y charged into the s i l i c a q u i l l s . Loose f i t t i n g , charges yielded c r y s t a l s with severely cavitated surfaces. •Crystals were analyzed for composition v a r i a t i o n s r e s u l t i n g from solidification-,^.:.for i n t e r s t i t i a l content, .and f o r trace element impurities. The a n a l y t i c a l procedure and r e s u l t s are reported i n Appendix 1. For the purposes of obtaining f a i r l y uniform compositions, only the f i r s t two-thirds of as-grown c r y s t a l s was accepted. Because the actual composition was quite close to the composition intended on a l l o y i n g , a l l compositions stated i n the thesis w i l l r e f e r to the i n i t i a l a l l o y composition. 2.1.2 Tensile Specimen Preparation "Dumb-bell" shaped t e n s i l e specimens, Figure 3., were prepared by c a r e f u l l y machining single c r y s t a l s i n a jewellers lathe. The gauge diameter was reduced from 0.113 inch to 0.090 inch through a series of cute les s than 0.0005 inch deep. Specimens were subsequently hand-polished i n the lathe with w e l l - l u b r i c a t e d 0 and .3/0 emery papers. Machining damage was evaluated and the r e s u l t s are reported i n Appendix 2. I t was found that damage was minimized upon removing 0.005 inch from the abraded surface. N o n - e l l i p i t i c a l cross-sections and taper-free gauge sections were effected by r a p i d l y r o t a t i n g and repeatedly turning the specimen end f o r end during p o l i s h i n g . Gauge diameters were accurate 9 + to -0.0005 inch. Machining damage was completely eliminated by subsequently annealing specimens in evacuated pyrex capsules at 300°C for one hour. 2.1.3 Testing Procedure Experiments were performed by straining specimens in a Floor Model.Instron tensile machine at strain rates varying from 2.5 x 10~4/sec to 2.5 xlO~ 2/sec and temperatures ranging from 77°K to 488°K. Load-elongation curves were autographically recorded during straining. Liquid testing environments accurate to ^2° included nitrogen (77°K), oxygen (90°K), petroleum ether cooled with nitrogen (133 to 293°K) and heated silicone o i l (293 to 488°K). Specimen dimensions were carefully measured using a Gaertner travelling microscope with, a 10X eyepiece. Diameters, accurate to -O.OOO5 inch were averaged from six reading along the gauge section and across two perpendicular diameters. Specimens were successfully gripped in a self-aligning pin-chuck and threaded collet system., Figure k.. 11 2 . 2 GENERAL DESCRIPTION OF THE SHEAR STRESS-SHEAR STRAIN CURVES 2 . 2 . 1 Experiments Specimens of three compositions (Au-rich = 5 1 . 0 at.$> Au, Stoichiometric = 5O.O at.$ Au and Zn-rich = 4 9 . O a t A u ) oriented near the centre of the standard stereographic triangle were prepared and tested in tension at temperatures from 77°K to 488CK. Cross-head speed was con-stant at O.O5 inch per minute corresponding to a strain-rate of 2 . 5 x 1 0 3/sec. Experiments were also performed to determine the effects of strain rate and orientation on the room temperature deformation behaviour of 5I.O at.$> Au-•crystals. Excluding the strain rate study, all.tests were done in duplicate and found to be reproducible in most cases to within five percent. An IBM computer was programmed to calculate resolved shear-stresses and shear-strains from equations presented in Appendix 3 . For these calculations i t was necessary to know the sli p system operative under the experimental conditions investigated. A thorough analysis of deforma-tion modes is presented in section 2 . 4 of the thesis where i t is shown that for specimens oriented within the [ 0 0 l ] - [ l 0 l ] - [ i l l ] stereographic triangle,slip occurs on planes belonging to the [ 0 0 1 ] zone, except for orientations very near the [ 0 0 1 ] corner where [ i l l ] zonal planes become operative. The results of the sli p mode study have been incorporated in the computations of resolved shear stress and shear strain. 2 . 2 . 2 Definition, of Work Hardening Parameters A schematic resolved shear-stress shear-strain curve typical of crystals oriented near the centre of the primary stereographic triangle and deformed at room temperature is shown in Figure 5 - The curve w i l l be divided 1 2 according to a scheme given by Mitchell et a l . Stage 0, a region of decreasing hardening rate is followed by two stages of linear hardening, I and II, then by a second region, of decreasing hardening rate, stage III. stage III 03 W <U U -P CO u CO Shear Strain Figure 5- Schematic resolved shear stress-shear strain curve, *X0 is defined as the yield stress at the f i r s t detectable departure from linearity;"Ci is the i n i t i a l flow stress obtained by extrapolating stage I to zero strain. Stresses T„ , '"fc^ / and correspond to shear strains V), , TV and V),| j f m . i s defined as the.maximum flow stress. Total d u c t i l i t y - i s given as Y f . The work-hardening rate in stage I is defined as Ox = "^u - "Ci and during Stage II as 0 n = ^.n -TTii T i . , - r„' 2 . 2 . 3 Temperature and Strain Rate Dependence The temperature dependence of the shear stress-shear strain 1 3 curves for the Au-rich, stoichiometric and Zn-rich crystals is illustrated in Figures 6 , , 7< a n & 8 respectively. In the range of intermediate tempera-tures from approximately 2 0 0 to 3 5 0 ° K i t can be seen that the flow curves are very similar to those classically observed for face-centered cubic metals 2' 3'' and those more recently reported for body-centered cubic m e t a l s . 1 , 5 , 6 , 7 , 8 , 9 , 1 < A low work-hardening, easy glide stage I is followed by a higher work-hardening stage II region. Total duc t i l i t y is very large f - 3 0 0 $ > shear strain). As the temperature is either increased or decreased, the length of stage I is decreased u n t i l a parabolic type of flow curve is obtained; .ductility is also reduced in both cases. Similar temperature effects have been observed in Nb,1 Ta 9 and Fe1-1 single crystals. The extent of stage II and III decrease and increase respectively with increasing temperature above approximately 2 ° 3 ° K . At temperatures below approximately 1 5 0 ° K , rapid hardening i a terminated in b r i t t l e fracture, while at temperatures greater than approximately < 3 7 5 ° K , i n i t i a l hardening is followed by general work-^softening u n t i l chisel-edge type ductile fracture occurs. Over the tempera-ture range in which stage III flow occurs, the maximum flow stress decreases with increasing temperature. In the range of intermediate temperatures stage 0 tends to decrease with increasing temperature. This is particularly noticeable for the Au-rich crystals, Figure 6. It is also apparent that the slow transition zone between stages I and TI tends to decrease with increasing temperature. The effect of strain rate on the room temperature p l a s t i c i t y of Au-rich crystals.is shown in Figure 9 . It is apparent that over the range of strain rates investigated 2 . 5 x 1 0 4 to 2 . . 5 x 1 0 ~ 2 / s e c , .the general flow behaviour is rather strain rate insensitive. There i s , however, an effect on stage II hardening which w i l l be discussed accordingly.in section 2 . 5 . 1 . -f=-Figure 6 Resolved shear stress-shear strain curves of' Au-rich $'AuZn crystals as a function of temperature between 7 7 ° K and 488°K. (Y = 2 . 5 x 1 0 " 3 / s e c ) Figure 7. Resolved shear stress-shear strain curves of stoichiometric AuZn crystals as a function of temperature between 77°K and khyYL. ( If = 2.5 x 10" 3/sec) 5 0 1 0 0 1 5 0 2 0 0 2 5 O 3 0 0 3 5 0 4 0 0 Resolved Shear Strain T (#) Figure 8 . .Resolved shear stress-shear strain curves of Zn-rich (2>'.AuZn crystals as a function of temperature between 7 7 ° K and kj3°K. ("fl* = 2 . 5 x 1 0 " 3 / s e c ) 25 50 100 150 200 250 JOO 350 400 Resolved Shear S t r a i n V ($>) Figure 9- Resolved shear"stress-shear s t r a i n curves of A u - r i c h C^ 'AuZn c r y s t a l s as a f u n c t i o n of s t r a i n r a t e from 2.5 x 10" 4/sec t o 2.5 x 10" 2/sec. (T = 293°K) 2 . 2 . 4 Effect of Deviations from'Stoichiometry 1 8 To illustrate the effect of deviations from stoichiometry on the shear stress-shear strain curves. at 77°K, 223°K, 293°K and 373°K,curves from Figures 6 , . 7 and 8 are superimposed on Figure 10,where the shortened notation' +Au, St. and +Zn refer to the Au-rich, stoichiometric and zinc-rich compositions respectively. It can be seen that at a l l temperatures, .yield ~ and flow stresses are higher for the non-stoichiometric material with approximately equal hardening on both sides of stoichiometry, similar to the behaviour of polycrystalline AuZn, 1 2 AgMg,13 NiAl 1 4', 8and Cu 3Au. 1 5 It appears that the difference in flow stress between stoichiometric and non-stoichiometric material is more pronounced during easy glide. In the intermediate temperature range deviations from stoichiometry increased the length of .stage I, with the most pronounced lengthening occurring for the Au-rich crystals. Similar stage I lengthening has been observed with is 1 7 increasing Zn content in «X-brass-and-in Ni-Co alloys. It-is.-observed, too, that stage III begins, at lower temperatures for the stoichiometric crys-t a l s . . It is also evident during straining at intermediate tempera-tures that flow in Zn-rich crystals occurs in a wavy manner prior to the high hardening rate, stage II region (for.instance, at 293°K and 348°K, Figure 8).Wave "frequency" and "height" were observed to increase and decrease respectively with increasing temperature then disappear just above ~-3k8°K. By carefully observing the specimen during straining at room temperature, i t could be seen.that each "wave" was coincident with localized one dimensional thinning within the gauge section. Once the gauge section was uniformly thinned, stage II hardening began. Stoichiometric and Au-rich crystals deformed uniformly throughout,.the gauge at a l l temperatures. 25 Figure -1.0.1. Resolved shear stress-shear strain curves, of composition at 293°K and 373°K. ^'AuZn crystals as a function of Figure 10.2. Resolved shear stress-shear strain curves of AuZn crystals as a function of composition at 77°K and — 220°K. 21 It appears that work-hardening rates during stage I and II are not greatly affected by deviations from stoichiometry, but at 77°K the i n i t i a l hardening rate appears to be higher for the non-stoichiometric crystals. Deviations from stoichiometry seem to have l i t t l e effect on total d u c t i l i t y , except at 77°K where Zn-rich material is considerably less • 1 2 ductile, similar to polycrystalline AuZn behaviour. It is possible that Zn-rich material undergoes a low temperature phase transformation similar 18 to that reported in polycrystalline AuZn which could account for the relative brittleness. Because the transformation temperature decreases with increasing Au, stoichiometric and Au-rich crystals may retain the parent structure at 77°K. Upon metallographic examination of deformed crystals, no unusual markings were observed on the surface of the b r i t t l e Zn-rich crystals. The possibility of transformation induced brittleness was not further investigated during the course of this work. .2.2.5 Orientation Dependence The orientation dependence of the stress-strain curves is shown in Figure 11. Within the stereographic triangle i t is apparent that as the position of the specimen axis is varied from the [001] - [ i l l ] boundary towards the centre, the flow curve changes from two stages of linear hardening to three—stage hardening at the expense of both stages I and II. .The work hardening rate appears unchanged but On decreases slightly. With increasing distance from the [101] - [ i l l ] boundary, both stages I and II and the transition region are shortened and total d u c t i l i t y is decreased. This behaviour is somewhat analogous to the orientation dependence of easy 3 1 glide in fee single crystals and bcc niobium where the extent of easy S3 2 3 glide decreases as the orientation approaches the symmetry boundaries [001]-[111] and [001]-[101] respectively. Whereas specimens oriented well within the stereographic triangle give rise to multi-stage work-hardening curves,.those oriented along the [101]-[ill] boundary near the •[101] corner display parabolic type hardening, and near [ i l l ] , semi two-stage hardening. Curves from these orientations are very much like the most commonly observed flow curves for 1 9 _ 2 4 bcc single crystals and that observed for the bcc ordered alloy NiAl deformed in compression along the [101] d i r e c t i o n . 2 5 An exception to the general multi-stage behaviour for orienta-tions within the triangle has been observed very near the [001] corner (orientation 10 Figure 11) . Plastic flow.is. characterized by an extremely high hardening rate and fracture occurs after very l i t t l e deformation, somewhat similar to the low temperature behaviour of AuZn single crystals. 2k 2 ..3 • SERRATED FLOW 2 . 3 . 1 Occurrence An interesting feature disclosed by tensile tests, not shown in Figures 6, 7> 8, 9 and 11 is the serrated nature of the plastic flow in non-stoichiometric crystals at intermediate temperatures. .The same 1 2 phenomenon has been reported by Causey during room temperature deforma-tion of polycrystalline (jJ'AuZn. Au-rich crystals displayed serrated load-elongation curves from 260°K to 403°K and Zn-rich crystals from 293°K to 408°K. (These temperatures must be taken as approximate limits; i.e. Au-rich crystals, for instance, when tested at 260°K display serrated flow curves whereas at 217°K show smooth curves; at high temperature serrations are observed at ir03°K but not at if33°K.) A typically serrated resolved shear-stress shear-strain curve is shown schematically in-Figure 12. Serrations begin after a few percent pre-strain in the stage 0 region and become f u l l y developed at the onset of stage I. The c r i t i c a l strain, V* in Figure 12, necessary for the-first u c jerk was observed to decrease with increasing temperature and decreasing strain rate, Table 2 . During stage I, the amplitude and frequency of the load-drops fluctuate slightly, but on the average remain approximately constant. • It was also observed that both the frequency and amplitude increase slightly with increasing temperature. Perhaps the most striking observation is that the amplitude is decreased abruptly during the transi-tion region. During stage II the amplitude is greatly reduced and the frequency tends to zero near the onset of stage III. .No serrations are observed during stage III deformation. Well-defined serrated yielding was observed during stage I deformation for 1Au-rich crystals at a l l orientations within the stereo-Tc Shear S t r a i n .Figure 1 2 . Schematic representation of a serrated flow-curve t y p i c a l for non-stoichiometric c rys ta ls tested around 300°K. i s the c r i t i c a l s t r a i n before f i r s t jerk? 2 6 TABLE 2 Showing the Variation in C r i t i c a l Strain with Temperature and Strain Rate for Non-stoichimetric ft1 AuZn Crystals  Temp. • Strain Rate C r i t i c a l Strain Composition •Test Wo. '. °K If/sec 5 1 . 0 at.i Au 84 2 6 0 2 . 5 x 1 0 " 3 0 . 1 7 - 0 . 2 3 i t 8 5 1 1 0 . 2 0 - 0 . 2 6 1 1 6 0 2 9 3 0 . 0 7 - 0 . 1 3 i t 6 1 11 0 . 0 7 - 0 . 1 3 1 1 7 5 3 7 3 " 0 . 0 2 - 0 . 0 6 1 1 7 6 1 1 " 0 . 0 2 - 0 . 0 5 t i 8 2 4 0 3 " 0 . 0 1 8 1 1 8 3 1 1 " 0 . 0 1 6 i t - 9 . 0 at.i Au 8 9 2 9 3 2 . 5 x 1 0 " 3 0 . 0 9 - 0 . 1 1 1 1 9 0 1 1 0 . 0 7 - 0 . 1 1 1 1 1 0 3 348 " 0 . 0 4 - 0 . 0 6 1 1 1 0 4 0 . 0 4 - 0 . 0 6 1 1 9 1 3 7 3 0 . 0 3 - 0 . 0 4 1 1 9 2 t i 0 . 0 3 - 0 . 0 4 1 1 1 0 5 4 0 8 0 . 0 1 6 5 1 . 0 a t . i Au. 1 1 2 2 9 3 2 . 5 x 1 0 ~ 4 0 . 0 5 - 0 . 0 7 I T 1 1 0 1 1 2 . 5 x 1 0 " 2 0 . 2 0 - 0 . 2 8 27 \ graphic triangle except near the [OOl] apex where fracture occurred after very l i t t l e deformation (section 2.2). For orientations along the [lOl]-[ l l l ] boundary, however, minor serrations were observed after very l i t t l e prestrain (2% shear strain) but were not detected past the high hardening region, disappearing at shear strains greater than approximately ^>Of>. This behaviour is identical with the stage II damping out and stage III absence of serrated flow reported above. A strain rate cycling experiment was performed at room tempera-ture on an Au-rich ( 5 1 - 0 a t $ Au) crystal by cycling the cross-head speed of the Instron between 0 . 0 0 2 to 0 . 2 0 inch per minute, corresponding to a strain rate variation from 1 x 1 0 4/sec to 1 x 1 0 ~ 2 / s e c . Part of the strain-rate-change flow curve was photographed and is shown in Figure 1 3 . It can be seen that both the amplitude and frequency of the serrations decrease with increasing strain rate. Figure 13. Photograph of segments of a serrated load-elongation curve during stage I and stage II deformation. 28 Metallographic examination of an1 Au-rich specimen strained well into stage I revealed a very even distribution of strain markings throughout the gauge which coincided with the operative s l i p plane. Since the strain markings would not reappear when electro-chemically polished in 5$ KCN and etched in the same solution (for one minute at 1.5 V potential), they were assumed to be sli p traces. From these observations i t appears that three conditions must be met for the occurrence of well-defined serrated flow in ,@*AuZn single crystals: (1) .The composition must deviate from stoichiometry; (2) The resolved shear stress-shear strain curve must display a well-developed easy glide region; (3) .The test temperature must l i e within a c r i t i c a l range centered approximately about 325°K which is equivalent to 0.33 T m where T m is the absolute melting point. 2.3.2 Origin Serrated flow has been observed during plastic deformation of many intermetallic compounds of various superlattice types. During compression testing from. 77°K to 350°K'Fe 3Be single crystals (DO3 super-lattice) undergo continual mechanical, twinning, giving rise to serrated 7 2 flow curves. Load drops during tensile experiments have been detected in the B2 compounds N i T i 6 and Cu-saturated (60 at $ Cu) (^-brass crystals 7 and were attributed to strain-induced transformations. -During tensile deformation at intermediate temperatures, serrated yielding observed in 1 3 non-stoichiometric AgMg (B2) and N i A l 1 4 (B2) and in crystals of (3-brass. (B2) and Cu3Au (Ll 2) was attributed to dislocation-solute 2 9 7 7 atom interactions, commonly termed the Portevin-LeChetelier effect. It is believed that the present phenomenon can also be explained in terms of solute interactions with moving dislocations. Since serrated yielding was restricted to non-stoichiometric compositions, i t appears that the 12 solutes interacting are either excess Au or Zn atoms. Causey has shown that a substitutional type defect structure exists on both sides of stoichiometry suggesting that the segregating species diffuses through a vacancy-type mechanism. The interaction mechanism w i l l be discussed in the next section. 2 . 3 . 3 .Dislocation-Solute Atom Interaction It is generally believed that plastic deformation may increase the density of mobile dislocations ^ according to the empirical relation-ship: ^ = KiY* (1) where Y is the plastic shear strain and Kx and m are constants. During deformation at a constant strain rate 0 , the product of the average dislocation velocity.v and mobile dislocation density remains constant, according to: - f ^ b v ( 2 ) where b is the Burgers vector of the mobile dislocations. ..Therefore v is a decreasing function of "8*. Under suitable conditions of temperature and strain rate, the dislocation velocity approaches a c r i t i c a l value v (at a c r i t i c a l strain ), sufficiently low to enable solute atoms to c c segregate towards the mobile dislocations; the strain fields, of the segregating species and the dislocation interact, atmospheres are formed and the dislocation is either slowed down or pinned. The c r i t i c a l 7 9 conditions for pinning were f i r s t expressed by C o t t r e l l in the relationship 3 0 where D is the self-diffusion coefficient of the segregating species and I is the atmosphere radius. In order to maintain the applied strain rate, the v e l o c i t y of the unrestricted dislocations must increase, necessitating a localized increase in stress. -At a sufficiently high stress level, the pinned dislocations either break-away from their atmospheres and/or fresh dislocations are generated. At this point, the f i r s t load drop occurs. •The process of solute locking followed by dislocation break-away and/or generation is repeated during subsequent deformation, giving rise to the observed serrated flow curve. From this description of serrated yielding, the experimentally observed effects, of temperature, strain rate and strain can be explained. Because of the marked effect of temperature on diffusion coefficients, v > 4_D at low temperatures for a l l strains beyond yield. Because of the I low D values,atmospheres are not formed around mobile dislocations, motion is not impeded and consequently serrated yielding does not occur. At high temperatures 4p_ >• v at a l l strains and atmospheres are formed, I but because of the larger D values,move along with dislocations and do not 'impede their motion. In the intermediate temperature range where serrated flow occurs, v^ increases proportional to the increase in D with tem-perature necessitating decreasing values of prestrain before f i r s t jerk. To understand the effect of strain rate on the c r i t i c a l strain, equation ( 3 ) is rewritten by incorporating equation ( 2 ) and 82 equating I to 4b, to give: y c = D 3 (4) Because vacancy-type defects are created during i n i t i a l straining the 80 value of D is believed to increase according to: a -Q/kT D = a 0 Z Ce (5) 3 1 where a is the lattice parameter, O the Debye frequency, Z the coordination number, C the concentration of vacancy-type defects created during strain-ing, Q the activation energy for motion of the segregating species, k the Boltzmann constant, and T the absolute temperature. The parameter C is 8 3 given by the empirical relationship: n C = K2t (6) where K 2 and n are constants. By substituting ( 6 ) into ( 5 ) a n d C l ) into 7 8 (4), i t is seen that at constant temperature: (n + m) Yc = K 3 ^ C ^ ( 7) where K 3 is a constant. It is immediately apparent from (7) that the c r i t i c a l strain increases with strain rate, in line with the experimental observations. In discussing strain rate effects on amplitude and frequency of serrations, one must be careful to distinguish between true strain rate effects and apparent strain rate effects. Apparent effects are noted directly from the autographically recorded load-elongation curves, and are recorded in the thesis. It is quite possible that at high strain rates, because of poor pen response on the testing machine, a stress drop which is coincident with a burst of dislocations w i l l not be detected. Consequently, an apparent large decrease in amplitude with strain rate does not necessarily mean that the number of dislocations released or generated per burst has been greatly reduced. Because of the uncertainty in relating •the apparent with the. true effect, strain rate dependence on serrated flow w i l l not be discussed further. During stage I deformation, the serration amplitude remains relatively constant, suggesting that an equilibrium exists between the mobile dislocation density and the vacancy-type defect concentration. 32 During stage II, however, because of increased dislocation-dislocation interactions (section 2 . 5 . 2 ) , the defect production rate i s increased upsetting the equilibrium. Diffusion rates may then be large enough to allow atmospheres to move with dislocations which would account for the gradual decay of serrated yielding. Some basis for this belief i s derived from the following rough calculations. Assuming that the mobile dislocation 9 / 2 density at the end of stage I is ~ 1 0 /cm , the average dislocation velocity v' necessary to maintain the applied strain rate of ~ 1 0 3/sec was calculated from equation (2) to be v ~ 1 0 cm/sec where b was taken as 3 x 10 cm. Furthermore, assuming that the defect concentration C as a function of - 4 . I l l strain may be given roughly as C ~-10 <y , then the diffusion coefficient D* at the end of easy glide calculated from equation (5) is given as D ' ~ 10 cm /sec, where a was taken as 3 x 10 cm, as 1, Z as 8, 0 as I 0 1 3/sec and Q as "~0.25 ev. (see next section). On substituting D ' 82 _ 3 into equation (3) and letting I = 4b , i t is seen that 4 0 ^ 1 0 cm/sec .1 and therefore somewhat greater than v , consistent with the suggestion that atmospheres diffuse along with moving dislocations at the end of stage I. It must be emphasized that these calculations are only crude approximations 9 / 2 since they are based on the questionable assumption that ^ mobile m ~4 . and that C ~ 1 0 j at large strains. Since specimens oriented along the [ l 0 l ] - [ l l l ] boundary deform by multiple s l i p (section 2 . 4 . 5 . 3 ) the explana-tion based on an increased vacancy production rate can probably account for the greatly reduced amplitude of serrations for these orientations. 2 . 3 . 4 Segregating Species The problem remains to identify the segregating species. • An analysis w i l l be carried out to determine the activation energy of motion, Q, with the view that knowledge of such a parameter is valuable in this regard. Although successful in estimating Q, the analysis does not permit an unambiguous identification, of the species. Since the variation i n " c 33 T with "o was not determined for Zn-rich crystals, the energy Q w i l l be evaluated for Au-rich crystals only. By equating D in equations (4).and (5) and substituting equa-tion ( 1 ) and ( 6 ) for and G respectively, i t can be shown that at constant strain rate: m + n Q/kT = K4.e (8) c where K 4 is constant. The sum fm + n) may be obtained by plotting l o g 1 0 c against log 1 0 QC (equation 7) and reciprocating the slope. Because of d i f f i c u l t i e s in obtaining accurate values of from load-elongation curves, i t was decided to plot the range of strain, Table 2 , in which ~Y fell,.Figure ih, where the bracketed points locate the c r i t i c a l strain range. At strains less than those depicted by the lower brackets, serra-tions were not detected whereas at strains greater than those specified by the upper brackets, serrated flow had definitely begun. To determine the value (m + n) from the points in Figure Ik, two extreme lines were drawn and the corresponding slopes measured then reciprocated to give an average value of (m + n) with a deviation term. It was found that (m+n)=3.6 - 0.9,compar-able -With the values 2 . 2 - 0 . 1 and 1 . 9 - 0 . 2 determined for Cu-Sn 7 1 and 8 4 Cu-Zn alloys respectively. It i s now possible t o determine Q. from the slope of the graph In "ft c against the reciprocal of the absolute temperature, Figure 1 5 . Again, the c r i t i c a l strain range i s plotted and the bracketed points have the same significance as described above. To determine the slope from the points i n Figure 1 5 , the two-extreme line technique was again used,then to obtain the maximum and minimum, and hence most probable activation energy, the highest and lowest slopes were multiplied by the highest and lowest (m+ n) values. The most probable activation energy Q obtained in Figure iK. Showing the variation.in C r i t i c a l Strain Y with C r i t i c a l Strain Rate c 3^  (T = 293°K; 5 1 . 0 at 4 Au) ° o H IS -p CO 03 o •H •p • H in O -0 .5 -- 1 . 0 -1 .5 -3.5 • -3.0 -2T5 -2.0 -r.5 Critical Strain Rate (logm^) Figure 1 5 . Showing the effect of temperature on c r i t i c a l strain ( ft = 2 . 5 x 1 0 " 3 / s e c , 5 1 . 0 at.#Au) this way was found to be: Q = 0 . 2 6 - 0 . 0 8 ev. The activation energy for motion of vacancies in material of the same 85 , composition was given by Mukherjee et a l as 0 . ^ 7 - 0 . 0 5 ev., .determined from isothermal annealing studies on quenched-in defects in J mm. diameter single crystals. It is apparent that the energy for motion of vacancies does not f a l l within the range determined for the present activation energy. Due to the paucity of data in the literature l i s t i n g the activation energies for the motion of defects in intermetallic compounds in general, and AuZn in particular, i t is impossible at this juncture to associate Q = 0 . 2 6 - 0 . 0 8 ev. with the motion of any defect species. The identity of the segregating species, therefore, remains unknown. 3 6 2.4 DEFORMATION MODES 2.4.1 Introduction The operative sl i p systems in CsCl type' superlattices :are strongly dependent on the degree of ionic bonding between the component atoms. In considering the question how large must the ordering energy be to change the sli p direction from the usual bcc "metallic" < 1 1 1 > type to the "ionic" < 0 0 1 > - t y p e , , Rachinger and C o t t r e l l 2 6 (RC) begin with Nabarro's 2 7 .postulation that <11I> slip should not occur unless the total a < 1 1 1 ^ dislocation- dissociates into two superlattice partial dislocations according to: a < 1 1 1 7 - = a < 1 1 1 > + a < l l l > ( 9 ) •2 2 Nabarro s postulation is based.on the fact that total a < 1 1 1 > dislocations are able to dissociate into three perfect a < 0 0 1 > type dislocations with no change in elastic energy, and consequently, an applied stress, that acts strongly.on one of the three a < 0 0 1 / > components w i l l move this component independent of the others. RC point out that even i f dissociation ( 9 ) occurs, <001y s l i p 'may' s t i l l be favourable i f the stacking fault energy linking the two superlattice partials is high enough so that their e q u i l i -brium, spacing is only ~a, one lattice spacing. The c r i t i c a l factor, then, in determining whether or not < 1 1 1 > slip occurs is the c r i t i c a l stacking fault energy COQ above which, < 0 0 1 ^ is the favoured mode and below which the equilibrium separation of the partials is greater than a so that ^ 1 1 1 > is favoured. As a quantitative criterion, RC assume that 0) exerts a force on the dislocation equal to the theoretical shear strength of the lat t i c e , and hence may be expressed as: •v uJc =°<.f/\> ( 1 0 ) where is the theoretical shear strength ( CX ~ l / 3 0 ) and b is the Burgers vector. Giving typical values of ' 3 x 1 0 u dynes/cm and 2.5 x 1 0 cm 37 to /f and b respectively, uJ was calculated as ~ 2 5 O ergs/cm2. Then considering the atomic density on ( 1 1 0 ) planes Nf27 a2 a n d t h a t e a c n a t o m forms bonds with two nearest neighbours in parallel layers, RC calculate that on the nearest neighbour bond approximation: VAB " i (VAA + VBB) = S!_ ^ - - 0 6 ev. ( 1 1 ) 2 c 2 Y _ 2 ~ A which i s the c r i t i c a l ordering energy per atomic,bond, equivalent to a charge of only - 0 . 1 electron on each"ion'.' They conclude that since ionic character as small as this i s quite possible, even in highly "metallic" alloys, < 1 0 0 > should be the common mode of sli p in CsCl type structures. In an attempt to evaluate their hypothesis, RC studied sl i p modes in several CsCl compounds and compared the sli p direction with that expected on the basis of bond type. In compounds undergoing an order-disorder reaction, kT c was taken as a measure of the ordering energy per bond, where T c is the ordering temperature in degrees Kelvin. Their results are given in Table 3 . For the ionic compounds noted, < C l 0 0 ^ i s the sli p direction, as predicted. For CuZn, <"111> s l i p was observed, again in agree--TABLE 3 Slip Systems in CsCl Type Compounds (Rachinger and C o t t r e l l 2 6 ) Compound Plane Direction Bond Character  T c (°K) Ordering Energy (ev.) = kTc/4 CuZn AgMg TIBr, T i l , T1C1 • TIBr, L i T l , MgTl AuZn AuCd 1 1 0 1 1 1 7 3 8 0 . 0 1 5 3 2 1 1 1 1 1 0 9 3 (Tm.p.) 0 . 0 2 3 1 1 0 0 0 1 ionic compounds 1 1 0 0 0 1 9 9 8 (Tm.p.) 0 . 0 2 2 1 1 0 0 0 1 9 0 0 (Tm.p.) 0 . 0 2 0 5 8 raent with their hypothesis, since the low ordering energy of 0 . 0 1 5 ev. is less than the c r i t i c a l energy 0 . 0 6 ev. But, sli p directions < " 0 0 1 / > in the compounds.AuZn. and: AuCd appear to invalidate the hypothesis since the ordering energies are less than 0 . 0 6 ev. However, as RC point out, the fact that AuCd and AuZn are ordered up to their melting points suggests that the use of:T mp as T c in computing the ordering energy probably results in a figure less than the true value. 2 9 •Ball and Smallman (BS) have recently reported s l i p systems in the B2 compound NiAl to be of the form {llo}<001>. Since NiAl, like AuCd, AgMg and AuZn, i s ordered up to i t s melting point, BS calculated that the ordering energy per atomic bond^from the relationship kTmpjis 0.0k ev, somewhat less than the c r i t i c a l 0.06 ev. for <100? s l i p according to 'RC. •.BS conclude that the RC criterion f a i l s again to satisfactorily predict s l i p modes in CsCl type compounds. The problem of sli p systems in CsCl compounds is then reconsidered in terms of the dislocation elastic energy E 30 which is given by the relationship.derived by Foreman: E = Kbf_ In R ( 1 2 ) kir r o where R and r Q are the outer and inner cut-off radii respectively, b the Burgers•vector and K the energy factor which is a function of the elastic constants c 1 ; L, c 1 2 , c 4 4 and the direction cosines , ^  , of the dis-location line with respect to the cube axis. Energy factors for screw dislocations lying along ^ l l ^ directions have been calculated by Head and for screws along < " 1 0 0 > and (HOy as well as edges lying along ^ lOO^ and ^ H O , by Foreman. 3 0 Since the relative mobility-S (i.e. the ratio of the stress required to move a dislocation to that necessary to make the atomic planes move rigidly over one another) of the lower energy dislocations in possible glide planes w i l l determine their rate of multiplication and 3 9 hence the predominant sli p system, the S values of the lower energy 3 3 dislocations must also be compared. Eshelby has proposed an equation to determine S: - 2 1 T ? S. = k-rr ±_e (13) b where is the width of the dislocation and can be calculated from the 3 3 expression: J?j= I K d (Ik) b 2/5, b where K is the energy factor, ^  the shear modulus in the sli p direction on the glide plane and d the spacing between the glide planes. To compare the experimentally observed s l i p modes in some CsCl compounds with those predicted by the elastic energy-dislocation mobility criterion, BS calculated E and S for screw and edge dislocations lying along <• 1117, <1017 and <^100>directions in CuZn, N iA l and CsBr. For comparison the author calculated the corresponding terms for s i m i l a r d i s l o c a t i o n arrangements i n the i s o - s t r u c t u r a l compound AuZn. The lowest energy conf igurat ions, the r e l a t i v e mob i l i t i es and the experimentally .determined s l i p modes are given i n Table k. Although a ^100>>appears to be the most favourable t o t a l d i s -location in CuZn, BS show that a<lll>' dissociates into two a O - l l > 2 superlattice partials having total energy lower than that for the aO-00> dislocation. The predicted system [lio ) ( l l l X s therefore consistent with experimental results. In CsBr, energetically both ^ l i o ] ( 001>and [lOO) ('OOl^are possible systems, but because of the favourable mobility term, [ l i o ] ('OOl^is predicted, again in agreement with experiment. Clearly, ( 0 0 1 > is the favourable s l i p direction in NiAl and the energetics suggest that sl i p should occur on {lOOJ planes. However, because the mobility term 4 o TABLE.k Comparison of Line Energies E and Mobilities 'S of Low.Energy-Dislocations in CuZn, NiAl, CsBr (after Ball and'Smallman29) and-AuZn Slip Burgers Dislocation E*x 10 " 4 System Ref. C ompound Plane Vector Character (ergs/cm) S 1 1 0 111 26 CuZn 0 1 0 1 0 0 e 3.24 . 7 2 3 1 1 0 111 s k.k • 7 1 1 0 1 1 0 s k.k . 0 8 6 1 1 0 1 0 0 e 4 . 8 6 . 6 2 6 1 1 0 0 0 1 2 9 NiAl 1 0 0 1 0 0 e 7 - 6 9 • 7 1 110 1 0 0 e 8 . 8 3 - . 5 1 3 1 1 0 1.00 s 9 . 2 9 .46 1 0 0 1 0 0 s 9 . 2 9 . 6 7 110 0 0 1 2 6 AuZn 1 0 0 1 0 0 e 2 . 3 . 4 3 1 1 1 0 1 0 0 e 2.64 . 2 7 6 1 1 0 110 s 2 . 7 8 . 4 2 6 1 1 0 1 0 0 s 3 - 0 . 4 8 2 1 1 0 0 0 1 7 0 CsBr 1 1 0 1 0 0 s . 8 2 5 .46 1 0 0 1 0 0 s . 8 2 5 • 6 7 E - line energy where 1 In R is taken as unity. % r Q e = edge dislocation s = screw dislocation i s more favourable on {ll6} planes, the predicted system is {lio} ^ 0 0 1 ^ consistent with the experimental observations. In• AuZn as well, < 0 0 1 ^ is clearly the preferred sl i p direction. Since mobility appears to dominate in low energy configurations, {lip} <001> is the predicted s l i p system, in 2 6 agreement with the results, of'Rachinger and Cottrell and consistent with the observations to be presented in sections 2 . 4 . 3 and 2 . 4 . 4 of the thesis. The latest discussion of sli p systems in CsCl type compounds suggests that the atom size ratio R^/Rg may be an important factor governing kl the choice of s l i p system. Lautenschlager et a l grouped B2 compounds into three classes: .A, mainly ionic with some degree of covalent bonding; • B, metallically bonded and reinforced by a strong covalent component; C, wholly metallic with a slight covalent tendency. •In class 'A, .ionic characteristics are believed to dominate the s l i p direc-tion and < 0 0 1 / ' is the preferred mode, while in class C, an orientation criterion dominates and on the basis of Schmid factor calculations for the systems {lio} <112>, $ 2 1 1 } < 1 1 1 > , { ' 3 2 l } < 1 1 1 > , (lio) £LLO>, {lio} < 0 0 1 ? a n d [lOO} ^ 0 0 1 } , the most favoured s l i p direction is along < 1 1 1 ? . (It must be pointed out that the orientation criterion necessarily assumes that the c r i t i c a l resolved shear stress is the same on a l l possible sl i p systems.) In class B, though, Lautenschlager et a l state that 'R^ /^ B S o v e r n s the choice of s l i p mode. The criterion for classifying a particular compound was not stated. Hard sphere CsCl models were constructed 3 4 and the effect of atom size ratio from 1 . 0 0 0 to 0 . 7 3 2 (i.e. range in which CsCl structure is stable 3 5) on the choice of s l i p system was evaluated in the following manner. Choosing either < 1 0 0 > , < 1 1 0 > or < 1 1 1 > as the sli p direction, possible s l i p planes were either accepted or rejected i f the maximum displacement normal to the shear plane, d x were less than or greater than 0 . 3 a , where a is the lattice parameter. The only systems f u l f i l l i n g this criterion were (lOO} < 0 0 1 ) , (HO) (001}, { 1 1 0 } £L10>, (lio) < 1 1 1 > , { 2 1 l } ^lll>and £2l] < 1 1 1 > . For a specific R^/Rg ratio, the most favourable system from these six poss i b i l i t i e s is associated with the smallest value. On this basis i t was shown that < 1 1 1 > is the favourable sl i p direction as R^ tends to 1 . 0 0 0 R~B while < 0 0 1 > is preferred when the ratio tends to 0 . 7 3 2 . For values k2 between 1.000 and 0.732, however, the d x criterion is less discriminative in .its choice for s l i p direction. Also, i t appears that the d j _ criterion is not able to distinguish unambiguously between the possible sl i p planes in the <11L> or ^OOl? zone. •A second important deformation mode that is often encountered in bcc metals and alloys is that produced by mechanical twinning. Upon 4 1 ordering, however, Laves pointed out that a crystal may lose i t s twinning 4 2 a b i l i t y . Marcinkowski and Fisher considered the number of nearest neighbor A-B bonds broken and A-A and B-B bonds formed during mechanical twinning in a CsCl type compound, assuming the twin mechanism to be that invariably observed in bcc lattices, namely £211} ^ U l X Unlike the case for s l i p , a suitable pair of dislocations that might create a twin without leading to any subsequent disorder does not exist, because i f the .a <111> 6 twin dislocation dissociates or combines with another, the necessary atom movements for twin formation would not be provided. Consequently the stress necessary to move the twinning dislocation, and thereby disorder the l a t t i c e , is believed to be very high. Marcinkowski and Fisher conclude, therefore that <111> s l i p is generally preferable to twinning in the B2 l a t t i c e . In alloys that exhibit <001> s l i p , , twinning on the £211~J ^ l l l ^ system is probably even less l i k e l y than in systems undergoing <111> s l i p . • As noted, considerable information is available on the deformation modes in the various CsCl type intermetallic compounds. However, no detailed account has appeared in the literature concerning the effects of temperature, strain rate and crystal orientation (on deformation modes). •Such a study has been carried out during the course of the present investi-gation with the view that such information would aid in understanding the general deformation characteristics of ^'AuZn and possibly elucidate the underlying dislocation mechanisms responsible for the,observed behaviour. 43 2.4.2 Procedure For the purposes of trace analysis, highly polished f l a t surfaces were prepared by spark-machining square cross-sections on the J>mm. diameter crystals over a 2 cm. gauge length then electrochemically removing approximately 0.004 inch from the eroded surface in fresh yjo KCN solution. A l l specimens were subsequently encapsulated under reduced pressure and annealed one hour at 300°C to remove any slight residual strains that may have been incurred from the spark-machining operation. A l l specimens were examined under the optical microscope prior to straining. In the experiments designed-to study temperature effects on deformation modes, the same pair of crystallographic faces were exposed on a l l specimens. This procedure i s , of course, not possible when studying orientation effects. In determining the orientation dependence of the sli p plane, specimens were strained in tension u n t i l well-defined strain markings could be seen under the optical microscope. Shear strains from 5 "to 10 percent were found to be adequate. For metallographic examination, speci-mens were supported in.a specially designed " j i g " then examined under green-fi l t e r e d oblique lighting using a Reichert metallograph. Strain markings on adjacent surfaces were carefully paired, noted and then photographed. The orientation of the surfaces examined was determined from back-reflection Laue X-ray patterns taken after straining. To minimize experimental errors extreme care was taken to align the crystal face parallel to the X-ray film plane. -The operative sl i p plane was determined by a two-surface trace 43 analysis according to a procedure.given by Barrett . -Reproducibility was found to be within three degrees. 2.4.3 Definitions In studying sl i p systems i t was useful to characterize the kk orientation of the specimens by the angles p and %• in the manner f i r s t 4 4 s proposed by Taylor, shown in Figure 16,.and recently employed in similar studies on Ta, 4 8 Nb 4 5 and Fe-Si a l l o y s . 4 6 ' 4 7 Accordingly, ^ i s the angle between the sli p direction b (shown later to be [ 0 0 1 ] ) and the tensile axis Cfand }C is the angle between a reference plane in the [ 0 0 1 ] zone, taken as ( 1 1 0 ) , and the maximum resolved shear stress plane M in the zone,which is normal to the plane containing b and C~„ In Figure 1 6 , '\f/'is defined as the angle between the macroscopic sl i p plane and the reference plane ( 1 1 0 ) and w i l l subsequently be termed the sli p plane parameter. 2.k.k .Slip Direction The Burgers vector of the mobile dislocations was determined from changes in crystal orientation caused by plastic deformation.since . i t can be shown49 that the direction in the glide plane to which the longi-tudinal axis moves is the s l i p direction. The changes in orientation for a s l i g h t l y A u - r i c h c r y s t a l (~50.3 at #Au) during deformation at 77°K, lkO°K, 293°K, 398°K and if73°K are shown in Figure 17. .The corresponding load-elongation curves are also shown so that a rough idea may be gained of the amount of strain induced prior to each re-orientation. It is evident that the specific direction of specimen axis re-orientation is temperature dependent. At 77°K, two experiments were performed from which s l i g h t l y d i f f e r e n t r e s u l t s were obtained. I t i s observed that the specimen a x i s i n one case r o t a t e d towards [100] throughout deformation (Figure 17.1.a) but i n the second case r o t a t e d toward [100] i n i t i a l l y then towards [001] during the l a t e r stages of deformation (Figure 17.I.b). I n both cases the specimen a x i s followed a great circular route to e i t h e r [100] or [00l], During straining at temperatures lU0°K, 298°K and 398°K (Figure 17.2.a; Figure 1 7 . 3 .a; .Figure 17 .4.a) the specimen axis rotated directly towards [ 0 0 1 ] 0 0 1 Figure 1 6 . A ( 0 0 1 ) stereographic projection showing the parameters characterizing the specimen orientation and the s l i p plane relative to ( 1 1 0 ) . 7? = angle between s l i p direction [ 0 0 1 ] and tensile axis 0"~. angle between the ( 1 1 0 ) reference plane and most highly stressed plane (of pole M) in the [ 0 0 1 ] zone. = sli p plane parameter; angle between the ' ( 1 1 0 ) ' " reference plane and the observed s l i p plane of pole S. 17.5a Extension (inch) .Figure 17. Stereographic representation of specimen axis reorientation during plastic deformation as a function of temperature. (0 denotes i n i t i a l orientation) 47 throughout deformation, while at 473°K (Figure 17 .5.a) towards [OOl] i n i t i a l l y then towards [lOO]. From these results, i t could be inferred that the re-orientation along two directions during straining is indicative of a change in the operative s l i p system. Slip systems have been studied during deformation and the results are reported in section 2 . 5 . 2 where i t is shown that the most prominent system at 473°K changes during deformation. At this juncture, i t may be concluded that the most prominent slip direction in (^ 'AuZn is the < 0 0 1 > type, in agreement with the observations of Rachinger and C o t t r e l l 2 and the predictions based on e l a s t i c i t y theory given in section 2 . 4 . 1 . . 2 . 4 . 5 Primary Slip Planes 2 . 4 . 5 . 1 Temperature Dependence The primary s l i p plane dependence on temperature from 77°K to 473°K was investigated for two orientations within the stereographic triangle, Figure 1 8 , using slightly. Au-rich ( ~ 5 ° - 5 at $ Au) crystals. Typical s l i p traces from orientation 1 observed at low magnification are shown in-Figure 1 9 , and at high magnification, Figure 2 0 . The high magnifi-cation structures were obtained from transmission electron microscopy studies of replicated surfaces using chromium-shadowed carbon replicas taken from cellulose-acetate impressions of the crystal surfaces. It is seen that s l i p traces on faces A are generally short and wavy while those on faces B are long and relatively straight. This differenc is particularly evident in Figure 2 0 . Since the operative s l i p direction above 77°K is [ 0 0 1 ] , i t was possible to calculate that the inclination of the sl i p vector to faces A and B(which were oriented from back-reflection photographs) is 2 8 ° and 1 0 ° respectively. .The short wavy traces,.therefore, h8 001 2 0 . B 1 m i c r o n F i g u r e 20. R e p l i c a s o f s u r f a c e s l i p t r a c e s on o r t h o g o n a l f a c e s A and B. 50 were created by edge dislocations tracing the paths of screws and the long straight traces were formed by screw dislocations tracing the motion of edges. Slip trace development is sketched in Figure 21.. These observations 1; 45 are very similar to s l i p traces observed in Wb single crystals and appear analogous to those of slip-bands in Fe-3.2$> Si crystals5°revealed by etching. The presence of the short, wavy traces suggests that the screw dislocations travel over relatively short distances before they either cross-slip onto other planes or stop, while the long, straight traces imply that the edges travel over quite long distances. . At 77°K i t is apparent that three systems are operative, although system 1,dominates; :.Figiire ;19;1, while at higher temperatures, only one system operates. Schematic pairing is illustrated under each photo-micrograph in Figure 19. Because of the profuse waviness of face A traces i t was decided to characterize the respective markings by a narrow wedge rather than a line; .in this way they could be r e a l i s t i c a l l y paired with the straight B traces and then analyzed. Markings labelled 2 in Figure 19.2 will'be discussed in section 2.5.2 under the t i t l e of deformation bands. At temperatures above 77°K> i t was found that the sl i p sur-faces are non-crystallographic planes (i.e. high index) in the [001] zone and that the macroscopic sl i p plane varied with temperature. The results of these.analyses, including the range, are given in Table 5 under column Ajy . At 77°K (Figure 19-1) the dominant sli p plane was. found to be a non-crystallographic plane in the [TOO] zone lying within 3 or k degrees of (Oil) and therefore could not be expressed in terms of AJT" as i t is presently defined; trace - 2 , is a non-crystallographic plane in the [001] zone which can be expressed in terms of the s l i p plane parameter \|/" and is given in Table 5- Since the sl i p direction must be parallel to the zone 51 5 2 axis, .the Burgers vectors of dislocations giving rise to primary deformation are parallel to [ 1 0 0 ] at 7 7 ° K a n d parallel.to [ 0 0 1 ] at higher temperatures. These results are in agreement with the primary sl i p direction determined from axial rotations during plastic deformation, section 2.k.k. TABLE 5 Results of Slip Trace Analyses of Temperature Effect on Slip Plane ParameterAf/" ( - j * = 2 . 5 x lQ-3/sec.) Orientation (Figure 1 8 ) NO. y. ?• •' Test No. Test Temp. °K (deg.) 1 2 0 2k 1 5 3 7 7 5 - 6 1 5 5 iko 1 3 - 1 7 146 2 9 3 2k - 29 1 3 5 2 9 3 2 2 - 2 7 1 5 7 3 9 8 2 9 - 3 7 1 5 6 q-73 ko - 5 0 2 6 k2 1 3 0 7 7 1 - 2 1 3 2 2 1 0 3 - 5 1 2 7 . 4 5 3 1 1 - 1 5 1 3 3 4 7 5 1 3 - 1 7 Prior to discussing the variation in ~y with temperature, one further feature of the 7 7 ° K deformation traces must be noted. The pole of trace - 3 in Figure 1 9 . 1 l i e s within two degrees of (Oil), the most highly stressed plane of the { l i o } ^ l l l ^ s y s t e m for the given orientation. An examination, of the back-reflection 'Laue photograph obtained from the surface in Figure I9.A.I revealed two distinct branches of asterism, -Figure 2 2 , 53 Figure 22. Back-reflection Laue X-ray pattern from surface shown in Figure I9.A.I. Note two branches of asterism. 54 indicating that deformation on two systems has occurred. The Taylor rotational axis for each branch is determined in Appendix k where i t is shown that the axes are consistent with sl i p on the two systems, ( 0 1 1 ) [ l 0 0 ] which is very close to the primary system! and (Oil) [ i l l ] , which probably defines system 3. It appears, therefore, that at 77°K, s l i p wanders slightly in the [ i l l ] direction from st r i c t [ 1 0 0 ] s l i p . An examination of Figure 19.A.I reveals that the straight trace-3 lines are in fact branches of the wavy trace - 1 lines. The variation in the macroscopic sli p plane with temperature can now be considered further. It is observed that i f the sl i p plane parameter ~y is plotted against the absolute temperature T, a f a i r l y good straight line connects the points and goes through the origin, Figure 2 3 . •It can be seen, too, that i f an error of - 3 degrees was incurred in indexing the s l i p planes, suitable adjustment would give an even better straight lin e . In any case several features are worth noting: ( 1 ) The sl i p surface is generally a non-crystallographic plane in the [ 0 0 1 ] zone; ( 2 ) The s l i p surface is not the most highly stressed plane in the [ 0 0 1 ] zone, except at approximately 2 2 0 ° K , where "VJ/ = X f ° r both orientations 1 and 2 . Clearly, more information must be gained before i t is known with any certainty whether or not the temperature at which "ty" ="X is unique and independent of orientation; ( 3 ) The temperature sensitivity of the sl i p plane parameter is 'an increasing function of . In section 2 . 4 . 5 - 3 i t w i l l be shown that the sl i p plane is not a function of <j° , thereby validating comparison ( 3 ) . To the author's knowledge, the above results represent the 55 0 1 0 0 2 0 0 3 0 0 4 o o 5 0 0 Temperature T°K Figure 2 3. Showing the variation of sli p plane parameter with temperature T. ( T = 2 . 5 x 1 0 " 3 /sec ) 56 f i r s t documented study of temperature effects in a material exhibiting non-crystallographic s l i p . Although i t is known that bcc metals deform on planes in the (111"} zone and tend to s l i p on {llOJ planes at low 5 1 temperatures, the "\y(T) curves have never been reported. A possible reason for this oversight may be that because of the wavy nature of non-crystallographic traces, authors believed that slip traces close to £licty £ 3 2 l } and £ 2 1 l ] were actually these planes, since the angle between the planes is quite small ( 1 9 ° and 1 1 ° respectively). 2 A . 5 „ 2 Strain Rate Dependence Specimens from orientation 1 were prepared from the same as-grown crystal used in the temperature study and strained a few percent at room temperature at two additional strain rates, g = 1 x 10 /sec and 0 = 2.5 x 1 0 x/sec, corresponding to cross-head speeds of 0 . 0 0 0 2 and 5 - 0 inch per minute respectively. The sli p traces were analyzed and the results are list e d in.Table 6 which includes for comparison sake the room temperature results from the previous tests where f = 2 . 5 x 10 /sec. •It is readily apparent that the sli p plane parameter is strain rate sensi-tive. As increases, decreases, similar to the effect of decreasing temperature. TABLE 6 Results of Slip.Trace Analyses of Strain Rate Effect on Slip Plane Parameter "u/ (T = 293°K) Orientation (Figure 18) No. X f Test No. "^(sec X) 1 2 0 24 1 8 0 1 x 1 0 " 5 53 - 3 6 1 4 6 2 . 5 x 1 0 3 24 - 2 9 155 2 . 5 x 1 0 " 3 22 - 27 143 2 . 5 x 1 0 " 1 2 0 - 23 57 Slip plane dependence on strain rate has been observed in , 52 Fe- -3% Si single crystals tested at room temperature under three-point bending. At strain rates above 10/sec. only {llO) slip was observed whereas at lower strain rates, s l i p occurred on the most highly stressed plane in the <111> zone. In-AuZn i t is apparent that the most highly stressed plane in the <001> zone is the s l i p plane only.at intermediate -i , strain rates of approximately 2.5 x 10 /sec. 2.4.5.3 Orientation Dependence The s l i p plane dependence on orientation has been studied at - 3 , room temperature and at a strain rate of 2.5 x 10 /sec for the specimen orientations shown in'Figure 24. 00.1 101 .Figure 24.' Showing specimen orientations used in slip plane analysis. For a l l orientations except No. -1, the primary slip surface was found to be a non-crystallographic plane in the*[00l] zone. The results of the two-surface analyses are given i n Table 7. On comparing the s l i p plane parameters for two orientations of constant ~)C but differing $ values (for instance Wo. 6 and 9> Figure 24) i t appears that ~\jf is independent of ^ , implying that the stress normal to the macroscopic s l i p plane has negligible effect on the s l i p surface. The results in Table 7 were therefore plotted as ~\y versus 'X- , Figure 25. The dashed line represents an ideal case of non-crystallographic s l i p where the macroscopic s l i p surface i s the plane of highest resolved shear stress in the [001] zone. It'appears.that for orientations near both the [ 0 0 l ] - [ l l l ] and [OOl]-[lOl] boundaries, ideal behaviour is approached whereas for other orientations (approximate limits: 5°^^<40°) macroscopic sli p occurs on less highly stressed planes. Slip plane dependence on orientation in both tension and com-pression has been studied in-Fe-3$ and Fe-6.5$ Si single crystals'at room temperature, 4 6' 5 3 in Fe-3$ S i at 77°K v and in Nb single crystals at 295°K. With reference to s l i p planes in the ^111> zone inclined at an angle T^'to a reference { l l O J plane, the Fe-Si alloys displayed ljf ( ]C ) curves similar to those observed in AuZn but Wb showed preference for either (21lj or {lio} s l i p . TABLE 7 Results of Slip Plane Analyses of Orientation 'Effect on Slip Plane Parameter (T=293°K; - ^ 2 . 5 x 10"3/sec.) Test Orientation (Ref. Fig. .24) Y (cleg.) | (deg.) Y (deg 147 2 2 •18 2 - <k - 5- 6 42 7* 144 3 . 15 23 22 - 28 135 • 4 20 24 24 - 29 146 4 20 24 22 - 27 152 8 28 34 30 - 36 123 6 32 44 3^ - 39 138 9 32 22 32 - 37 148 7 38 39 38 - 40 150 10 4 i 19 4 l X-Extrapolated from Figure 23. Figure 2^. Showing the variation of s l i p plane parameter ~\y with orientation "X. (T.= 293°K; Ys.2.5 x 10"3/sec) Two s l i p systems, shown i n Figure 2 6 , were operative i n orientation 6 which i s close to the [ l O l ] - [ l l l ] boundary. On analysis i t was found.that trace - 2 belonged to the [001] zone and consequently the corresponding s l i p plane parameter, "U/ = J>k to 3 9 degrees, was l i s t e d in Table 7 ; trace - 1 belonged to the [100] zone, f a l l i n g ^ 3 9 degrees from (Oil) and 6 degrees from ( 0 0 1 ) . The reason both [001] and [100] zonal s i Figure 2 6 . Showing duplex s l i p i n c r y s t a l oriented along 101-111 boundary, (orientation 6, Figure 2k) 61 occurs along the boundary'[101]-[ill] and single [001] s l i p occurs along [ 0 0 l ] - [ l l l ] becomes apparent when one considers the variation in Schmid factor on systems of the form £L10} ^00l]>. Although this system is a special case of non-crystallographic <001> s l i p , i t can be considered to illustrate the point at hand i f i t is assumed that the fundamental planes on which dislocations move are of this form. Unlike the case of {ill} O l O ^ s l i p in fee metals a specific system of the form ( l i o ) <001> is not limited to orientations within one of the 2k primary stereographic triangles; .instead, each system operates within two adjacent triangles equivalent to a stereographic quadrangle, Figure 2 7 . It can be seen that along the [ 0 0 l ] - [ l l l ] t i e - l i n e , (110).[001] is the most highly stressed system giving rise to the observed traces of that form. • Along. [ 1 0 1]-[ill] both (110)[001] and (011)[100] are equally stressed which is consistent with the observation that both [00l] and [100] zonal s l i p is active in these orientations. The remaining primary sli p systems to be reported are those active in the near'[001] orientations. Multiple s l i p shown i n Figure 2 8 has been observed -in orientation l a n d is of a completely different nature from that observed in other orientations. Strain markings are coarse and not as evenly spread throughout the gauge section as were those in other orientations. The markings shown in Figure 2 8 are typical of a localized area near the middle of the gauge. From two-surface analysis, i t was found that the most prominent set, trace -1, was within 2 degrees of (112). Because of uncertainties in pairing, the extra traces were indexed from single surface analyses; trace -2 was found 'to be consistent with either {21l} or [ l i o } and trace -3 , with {321} . To determine the sli p direction associated with the most prominent markings, a cylindrical specimen was 62 010 100 1 1 0 / DI / Dl . F l \ F l N ^ l l O / \ / E2 / ^ 1 101(E) . I l l / E2 * \ A3 B3 / I E 2 A3 Nv E2 \ w I O i l J / 3 0 1 ^ 1 1011 (F) ) \ . C2 B3 / \ ^ A3 C2 \ no _ i i : \ / B 5 A3 \ 111 no / \ \ 101 \ (CO \sd j D l \ D1 F l / F l i i o \ (B) \ / 110 / (A) •100 [1] 010 [2] .Plane s Directions A (110) •1 B ( i io) 2 C (101) 3 D ( O i l ) E (101) F ( O i l ) [100] [010] [001] Figure 27. A (001) stereographic projection showing the most highly stressed system of the form {llO} as a function of orientation. <001> Figure 28. Multiple sl i p observed in near [001] orientations. (No. 1, Figure 2k) 64 prepared and strained u n t i l (112) traces were easily detected. The position round the specimen axis at which the s l i p traces disappeared was noted then indexed from a back-reflection Laue X-ray pattern since in this position the Burgers vector lying in (112) is parallel to the specimen surface. The cross product of the slip-plane normal [112] and the specimen-surface normal [110] gave the s l i p direction, i.e. [112] x [llO] .= [ I l l ] It i s therefore concluded that the most prominent sl i p system in orienta-tion 1 is ( 1 1 2 ) [ i l l ] . It should be noted that these traces are rather wavy suggesting that- ( l l 2 ) may not be an elementary plane on which dis-locations move but a macroscopic s l i p plane made up of composite s l i p on other, more fundamental, , [ l l l ] zonal planes. 2.4.5.4 Composition Dependence Although a complete study of composition effects on the sl i p plane parameter "Vj/ was not undertaken, a few experiments were performed on Au-rich (51.0 at $ Au) and Zn-rich (49.0 at $> Au) crystals. Essentially no composition dependence was detected, and therefore, further experiments were abandoned. Composition effects have been reported in'Fe-Si alloys where increased Si content from 5 .0 to 6.5 wt. .increased the tendency for £llo) O-H^ crystallographic s l i p . 5 3 . However, because of an ordering , 5 4 reaction reported in :Fe-Si at approximately 5-5% S i , the nature of the composition effect is not known. 2.4.6. Discussion It is now possible to compare the primary deformation modes in ^AuZn with those operative in systems of different and similar structures. .The non-crystallographic nature of the <001^ zonal s l i p is 65 distinctly•different from the octahedral {ill} <110> s l i p systems, observed in fee metals and ordered alloys Cu3Au and Ni 3A l , different from cube (l00)^110^ s l i p observed in Al 5 7and N i 3 A l 5 6 , and different from the generally observed { 3 2 l } < l l l > , 2 6 ' 3 8 ' 3 9 { 2 1 l ) < l l l > 4 0 and (lio) <lll}4° crystallographic s l i p in the bcc ordered alloy AgMg. The observed non-crystallographic s l i p closely resembles the <lll"> zonal sl i p in bcc metals CuZn, T a 9 ' 4 8 , C r 2 3 5 9 W,5 Fe, Fe-Si alloys,'partially ordered FeCo 3 6 and ordered Fe3Al., and is very much the same as the {210} , {310) and {100} dislocation traces observed 2 9 in thin films of NiAl during transmission electron microscopy studies. The systems {21l} < l l l > and {lid] <111> observed in AuZn under the special testing conditions mentioned are typical of the more commonly reported s l i p modes in bcc metals and ordered alloys. Both <^ 001>and <111> zonal s l i p w i l l now be considered in more detail. .2.4.6.1 <001> Zonal Slip . The immediate task arising from the observation of non-crystallographic s l i p . i s to distinguish between the macroscopic sl i p planes, which are the planes observed under the optical microscope, and the funda-mental s l i p planes, which should be the planes on which .individual dis-locations move. In bcc metals three views are prevalent; f i r s t , elementary 6 0 slip.planes are generally non-crystallographic; second, they are only {llO} planes and different macroscopic s l i p planes are the result of composite s l i p on \110J planes; . and third, the elementary planes are [lid} as well as {21l} planes and different macroscopic sl i p planes are the 6 2 * 5 1 result of composite s l i p on both types of crystallographic planes. 5 1 Adopting the third approach Kroupa and Vitek have quite successfully calculated the~Vj/Q3 curve for an Fe-3$. Si alloy based on a thermally activated cross-slip model of dissociated screw dislocations. Considering 66 screw dislocation partials lying in sessile configurations on \llO~] and {21l] planes, they show that the thermal energy necessary to transform the sessile configurations into g l i s s i l e ones, bowed out onto {211} and {llOJ glide planes decreases as the force on the partials, due to an applied shear stress, increases. In this way, as the position of the tensile axis varies, so does the force on individual partials and as a result, some recombinations are more l i k e l y than others. Consequently, the macroscopic sl i p plane, which .is believed to be governed by the motion of recombined screw dislocations, is orientation sensitive. 2 9 Ball and SmalLman have suggested that the \210}, {310} and {IOC1} sl i p traces-observed in NiAl are the result of continual cross-slip of screw dislocations on orthogonal (llO) planes, shown schematically in Figure 29. Combining this general description of non-crystallographic <Q01^ s l i p with the dissociated screw dislocation concept of Kroupa and Vitek, an attempt w i l l be made to obtain a more detailed account of ^ 001^ zonal s l i p in ^J'AuZn. That -Q-IO)- planes are believed to be the elementary s l i p planes was deduced from the observations that\J/ degenerates to zero at very low temperatures (and•presumably high strain rates) and small "X values; .that thermally activated cross-slip of screw dislocations governs the macroscopic s l i p plane i s concluded from the pronounced influence of tem-perature and deformation rate on the sli p plane parameter The possibility of s p l i t t i n g of <.001> dislocations in bcc metals 63 has recent ly been discussed by. V i tek . Using isotop ic e l a s t i c i t y theory, he found that d issoc iat ions of the d i s loca t ion <001>according to the react ion : a <001> =a/8 <011> + a / 8 <017> (15) are stable for an a rb i t ra ry or ientat ion of the d i s loca t ion l i n e in a l l bcc metals studied, ranging f rom'Li and (3 brass with anisotropy rat ios Adhere 67 A i s defined as the r a t i o of the shear modulus on the { 1 0 0 ) < 0 0 1 > system to that i n {llO~)<ilO> and i s given as 2 C 4 4 / ( C i 1 - C i 2 ) ) of 9.1+ and 8.8 res-p e c t i v e l y to Nb with a r a t i o of 0 . 5 . Furthermore, Vitek found that d i s s o c i a t i o n s of the type: a <001> = a/ < 1 1 0 > + a / ^ l l 2 > + a / ^ 1 1 2 ? + a/ Q < 1 1 0 > ( 1 6 ) are stable only f o r an edge d i s l o c a t i o n ; f o r an a r b i t r a r y o r i e n t a t i o n of the d i s l o c a t i o n l i n e they are stable only i n L i , ^-brass, Na and K; i . e . , those metals which possess an extremely high anisotropy f a c t o r . Since the anisotropy f a c t o r f o r ^'AuZn i s 3 - 3 (calculated from e l a s t i c s t i f f n e s s 64 constants given by Schwartz and Muldawer ), reaction( 1 5)applies but rea c t i o n ( 1 6 ) f o r the case of screw d i s l o c a t i o n s i s doubtful. - Reactions ( 1 5 ) and ( l 6)are sketched i n Figure J>0. Considering r e a c t i o n ( 1 5 ) i n more d e t a i l i t i s seen that the extended d i s l o c a t i o n defines a f a u l t on a (lOO)plane. Since the fundamental s l i p plane i s believed to be a £ 1 1 0 } plane, dissociation ( 1 5)renders d i s -l o c a t i o n s s e s s i l e with respect to ( l l O ) motion. ' In order to f i n d the average motion of screw d i s l o c a t i o n s which governs the s l i p geometry i t i s then necessary to study the transformation of the s e s s i l e d i s l o c a t i o n into a d i s l o c a t i o n g l i s s i l e on [llO] planes as a thermally activated event i n a stress f i e l d . The important parameters that must be determined are the a c t i v a t i o n energy f o r the s e s s i l e - g l i s s i l e transformation and the distance t r a v e l l e d by the d i s l o c a t i o n during an a c t i v a t i o n event. • If the ses s i l e - g l i s s i l e transformation is considered as the reverse of reaction(15)plus the bowing out of the recombined length on a (llOJ glide plane, then the a c t i v a t i o n energy is the sum of the c o n s t r i c -t i o n energy U c, the recombination energy U r and the increase, i n the l i n e energy due to bowing of the d i s l o c a t i o n A U L ^ inus the work done by the 6 8 Figure 29. Schematic representation of continual cross-slip on orthogonal (llO"} planes. (After Ball and Smallman29) a [OOI3 I a < 0 1 1 > { 1 0 0 } .a < 0 1 7 > a £ 0 0 l } . k < l l d a < 1 1 0 > 8 \ > a<n ; 8 > A < 1 1 0 > {ii°} Reaction ( 1 5 ] Reaction ( 1 6 ) a < 0 0 1 > = a < 0 1 1 > + a < 0 1 7 > 8 8 a < 0 0 1 > = a< 1 1 0 > + a< 1 1 2 > + a< 1 1 2 > + a < 1 1 0 > 8 k k 8 •Figure 3 0 . Schematic representation of dissociation reactions (P5)and ( 1 6 ) 69 local stress "C= (X. a -T^ ) acting on the {lio} plane in the direction of the Burgers vector, where t" a is the resolved shear stress and 7^ is the back stress acting on the glide plane. Entropy assistance is neglected. The transformation sequence is illustrated in Figure 31. Because the g l i s s i l e dislocation is believed to be a <0(XL> unstable dislocation, i t is assumed that the distance of glide on the {_110} plane w i l l be equal to the bowing-out distance. Consequently i t is assumed that the transformations of the sessile into the g l i s s i l e configuration and vice versa are continually repeated for the occurrence of plastic flow. In the [001] zone, the most probable elementary glide plane, either (110) or ( 1 1 0 ) , w i l l be determined by the activation energy necessary for cross s l i p from the (100) and (010) planes onto the (110) and (110) planes which in turn depends on the effective stress acting on the glide 6 5 planes. The probability p for activation is given by: p = e ( 1 7 ) where AG is the Gibbs free energy of activation and k and T have their usual meaning. AG is that energy which must be supplied by a thermal fluctuation of the dislocation before the transformation is completed. The average number of activation events N on the (110) plane before' an event occurs on the (110) plane is then given by the probability ratio: . _ - [ A G 1 1 0 - A Gi Io 3 A m N = Piio •= e K i . (18) Pilo where subscripts 110 and 110 refer to the two planes of interest. .The average motion of the screw dislocations can then be considered to be composed of N units of slip on the (110) plane followed by one unit of s l i p on the (110) plane. The unit of s l i p on each plane is 70 .[oil] a [ 0 0 l ] *a[017T ,8 (100) Extended Constricted Recombined Bowed-out Figure ^>1. A sketch of the sessile to g l i s s i l e transformation sequence. '1 J [001] .Figure J>2. Schematic illustration of the continual cross-slip cycle defining the sl i p plane parameter \J7 . The circled regions represent the transformation sites (i.e. sessile to g l i s s i l e and vice versa). 71 given by the bowing-out distance d, which is related to the local stress on the two planes. This "cycle" is shown diagramatically in Figure 32, where i t is apparent that the s l i p plane parameter^ can be given by the relationship: tan V = dxlo ( 1 9 ) N.d 1 1 0 In order to account for the dependence of on temperature, strain rate and orientation, the parameters N and d must be expressed in.terms of T, "o and}£. (Note: X specifies the orientation dependence since £ was observed to have negligible effects.) Because the effects of temperature and strain rate are believed to be similar, "\|/ need be expressed only as a function of temperature and orientation. It can be shown that the free energy of activation for the s e s s i l e - g l i s s i l e transformation c r i t i c a l for the nucleation of cross s l i p 6 5 is given by the expression: AG = U** 1 _ 2/(fg- r s f rgV ( 2 0 ) ^»b / where b is the Burgers vector erf the screw dislocation [ 0 0 1 ] , fB is the o line tension of the perfect dislocation and f"s is the line tension of the extended dislocation. •It should be noted that since an entropy term does not appear in equation ( 2 0 ) , AG is really an activation enthalpy. The 6 5 corresponding value of d is given by the expression: a = r g - re ( 2 1 ) On substituting ( 1 8 ) , ( 2 0 ) and ( 2 1 ) into ( 1 9 ) i t is found that: tanV :;T*,o(X) e T L ^ i i o T?io J TTio( %) 1 3 where' A is a constant equal to (2 [ r - T ] [ ) 2 and g s g U is the constriction energy (22) 72 M where T110OO =Ta Cos ^. - T~ 1 1 D f 23) y F''-' M and T-iio(X) =?; S i n /* " £ x£o (24) M in which (Ja is defined as the resolved shear stress acting on the most highly stressed plane on the [001] zone. Since the back stresses i„ 1 1 0 and x£ 0 are not known, i t is d i f f i c u l t to test the validity.of equation (22) quantitatively. Qualitatively, however, i t is in agreement with the experimental observations. -As temperature decreases, tan "^f and hence Y tends to zero for a constant value of^jL; the exact (^") behaviour cannot be stated since the variation of the local stresses with temperature is not known. As "X increases and approaches- 45° the applied stresses and hence the local effective stresses *^"no and T ^ i l o become equal so that tan "\|^  approaches unity and ~\y approaches 45°. Since'Tj" i£o tends to zero as r~ approaches zero, equation (22) becomes undefined in this limit and hence cannot be compared with the observation that ~\J/ tends to zero. The essential feature in this description is the assumption of dissociated screw dislocations. Since stacking fault energies in bcc metals are considered generally to be higher than those in fee metals, i t is l i k e l y that fault energies in ordered bcc alloys w i l l be even higher. Consequently the extent of partial dislocation separation may be very small, in which case an anisotropic dislocation core would develop.instead of a true faulted plane. Whether or not an anisotropic core offers significant resistance to dislocation motion is not known; .the above description assumes i t does. -The attractive feature, however, is that the description appears to account for the observations. 2.4.6.2 -Chkl) < l l l > S l i p 6 6 Using a method given by Groves and Kelly for determining the 2 9 number of independent sl i p systems in crystals, Ball and Smallman have 75 recently shown that the continual cross-slip process on orthogonal {llO} planes of the <001> zone does not increase the number of independent s l i p systems in NiAl, leaving only three independent ^llO^OOl^ systems to operate. .The same result applies to-AuZn. -Whereas the total ductility to fracture in polycrystalline NiAl decreases abruptly to less than three percent at temperatures below 0.4-5' Tm, total ductility in polycrystalline AuZn remains in excess of ten percent at temperatures as low as 77°K 12 (0.077-Tm). On single surface analyses of s l i p traces in AuZn grains, general "thko} traces were evident within the grains, and £ 2 1 ] } and { 5 2 l } traces were observed near the boundaries. Since general pl a s t i c i t y of a polycrystalline aggregate necessitates the operation of at least five Q Q independent deformation modes, the d u c t i l i t y of polycrystalline AuZn was . attributed to s l i p on the extra { 2 1 l } and { 5 2 l ) planes. The single crystal observations are direct proof for the operation of { 2 1 l } < 1 1 1 > and {llO) <111> s l i p systems i n AuZn; the { 5 2 l } trace is also believed to be associated with <C 1U> s l i p . Examination of NiAl single crystals compressed along < 0 0 l " ^ axis revealed no extra sl i p 2 5 modes; instead, evidence, of kinking was observed similar to that in the iso -structural compounds C s l 6 9 and C s Br. 6 9' 7 0 It appears, therefore, that for crystal orientations in which the Schmid factors on the ^lloJ^OOl^ systems are near zero (i.e. .the near <001> orientations) CsCl type compounds either s l i p on < 1 1 1 > zonal planes- or undergo kinking. If <111> s l i p occurs the corresponding polycrystalline aggregates are ductile, but i f kinking occurs, the aggregates are b r i t t l e . It remains to be shown whether or not <111^ slip can be explained in terms of the three models reviewed earlier, viz. those based on ordering energy per atomic bond, dislocation line energy and atom size 74 2 5 ratio. As pointed out by Rachinger and Cottrell (RC) the fact that <Clll> s l i p occurs in a CsCl type superlattice implies that a <111^ superlattice 2 partials are present connected by a ribbon of stacking fault wider than about one lattice spacing. Obviously, AuZn is a near borderline case between "metallic" and "ionic" bonding nature since the crystal slips in both the "metallic" <111^ and "ionic" (001> directions under special testing conditions (77°K; near <001>orientations). AuZn, then, is an almost ideal compound i n which to evaluate the RC criterion which states that compounds with ordering energies greater or less than 0.06 ev. per atomic bond s l i p along <001> or <'111> respectively and conversely, those with energies of ^0.06 ev. s l i p in both directions. However since accurate ordering energies (not estimates using kT mp) for-AuZn are not known at present, the true ~4~~ evaluation of the RC criterion remains to be performed. Therefore, i t is impossible to state whether.or not <111> s l i p in the AuZn superlattice can be explained in terms of the c r i t i c a l ordering energy criterion. It is interesting to compare heats of formation, electro-negativity differences and s l i p directions for several CsCl type compounds, Table §. If i t is assumed that the heats of formation and the electro-negativity difference between component species is an indication of the strength of the A-B bond, then i t can be seen that bond strength has a direct effect on the sl i p direction. As a very rough estimate, i t appears that compounds having heats of formation greater than "-6000 cal/mole and electronegativity differences (on the A-R scale) in excess of~ 0 . 2 4 , s l i p along <0017 l a t t i c e directions whereas those with lower heats of formation and associated electronegativity differences sl i p along <111>. The transi-t i o n range in which both <001>and <111> s l i p may occur appears to l i e somewhere between formation energies, of 4500 to 6000 cal/mole and electro-negativity differences of 0.21 to 0.24. Anomalous in this respect is AuCd 75 TABLE 8 Correlation of Slip Direction with Heats of Formation and Electronegativity . Differences in CsCl Type Compounds Compound Electronegativity Difference A X Heat of Formation (cal/mole) ref. •Slip Direction uvw ref. NiAl .28 14000 105 001 29 L i T l M 6400 106 001 26 AuZn ..24 6150 107 001 , 111 26, present work MgTl .21 5000 to 65OO 106 001 26 AuCd .Ok 4660 106 001 and 26 possibly 111 AgMg .19 4380 108 111 26, .39, ko CuZn .09 2900 106 111 26 ~Allred-Rochow Electronegativities for component species used to determine A x . with A X ^ '0.04. Since the Allred-Rochow electronegativity scale is only one of many (used here because i t gives the best A X -slip direction correlation), .it was found that on other scales (Pauling, 1 0 9 Mullikeri 1 1 0) A X for AuCd f e l l close to AuZn. Qualitative though i t i s , the above comparison does lend some support to the basic idea of Rachinger and Cottrell that bond strengths play a very important role in determining sl i p directions in CsCl type compounds. 2 9 If the line energy model of Ball and Smallman is to account for both <001 > and <111/' s l i p directions, then i t must be shown that the total energy of the superdislocation (i.e. self-energies of both a X l l l > 2 partials plus the interaction energy between the partials plus the energy of the stacking fault linking the partials) as well as mobility are very 76 near the values f o r the ( O O l ) d i s l o c a t i o n s (Table 4). Accurate computa-t i o n s n e c e s s a r i l y require a knowledge of s t a c k i n g f a u l t energy and width, which are not known f o r AuZn. .Again, q u a n t i t a t i v e assessments between theory and experiment cannot be performed. As noted e a r l i e r , the r e l a t i v e atom s i z e model of 3 4 . . :Lautenschlager et a l p r e d i c t s O O l / s l i p d i r e c t i o n s when R^ approaches .HI 0.732, but (111/ when the r a t i o approaches u n i t y . C l e a r l y there i s an intermediate r a t i o where the model p r e d i c t s both <00l) and ( i l l ) s l i p . •Lautenschlager et a l c a l c u l a t e d R^ f o r a s e r i e s of C s C l type compounds, i n c l u d i n g AuZn, i n the f o l l o w i n g manner. Given the atomic and i o n i c r a d i i of each component,lattice parameters f o r the ordered bcc s t r u c t u r e s may be c a l c u l a t e d assuming both s p e c i e s , i n one case, are present as ions and i n a second case, as atoms. The measured l a t t i c e parameter i s then compared w i t h the c a l c u l a t e d ' " i o n i c " and "atomic" parameters. .Assuming a l i n e a r r e l a t i o n s h i p between R^ and l a t t i c e parameter, the e f f e c t i v e r a t i o i s determined since R^ and R^ are known. C a l c u l a t e d i n t h i s Rg i o n i c Rg atomic f a s h i o n .Rzn/pA /"s" From t h e i r measured v a r i a t i o n of d A (defined i n s e c t i o n 2.4.1) against R^y^ , i t appears t h a t both ( i l l ) and (001) s l i p d i r e c t i o n s are favourable when the r a t i o i s i n the v i c i n i t y of 0.9. The occurrence of both (OOl)and ( i l l ) s l i p i n AuZn i s the r e f o r e c o n s i s t e n t w i t h the p r e d i c t i o n s based on the atom s i z e r a t i o . One f u r t h e r c r y s t a l l o g r a p h i c feature of ( i l l ) s l i p must be noted. At both 77°K and i n near ( O O l ) o r i e n t a t i o n s at room temperatures ( c o n d i t i o n s under which <111> s l i p was detected) the operative s l i p plane was close t o the maximum resolved shear s t r e s s plane i n the ( i l l ) zone. The Schmid f a c t o r was~0.49 f o r both the ( O i l ) [ i l l ] system operative at 77°K and 77 the (112) [ i l l ] system operative at room temperature. Combined with the fact that the (112) traces were somewhat wavy (Figure 28) these observa-tions tentatively suggest that s l i p occurs on non-crystallographic planes in the <111> zone. 7 8 2 . 5 WORK-HARDENING BEHAVIOUR 2.5.1 Flow Parameters The work-hardening parameters as defined i n section 2.2.2 are compiled in Tables 9 > -10 and .11 for Au-rich, stoichiometric and Zn-rich crystals respectively (Figures 6 , 7 * and 8 ) and Table 12 for Au-rich crystals tested at room temperature in various orientations (Figure 11). The following discussions centre on.the variation of the hardening parameters with temperature, composition and orientation. 2.5.1.1 Yield Stress The temperature dependence of*f 0 is illustrated in Figure 3 3 • Points denoted as crosses were obtained from multiple tests on one crystal by subtracting the total work-hardening due to straining at prior tempera-tures from the yield stress.at the test temperature. There is good agreement between yield stresses determined from the work-hardening curves and those determined from multiple tests on one specimen. T" 0 for stoichiometric:AuZn decreases slightly above 220°K but approximately triples between 220°K and 7 7 ° K . For the non-stoichiometric crystals, the general shape of theT*0-T curve is different. Two regions of weak temperature-sensitivity exist, above approximately 3 0 0 ° K and below about 140°K. •From.300°K to lkO°K f0 increases about three times for the Zn-rich crystals but less than two times for the Au-rich alloy. It is also seen that yield stress for Zn-rich crystals displays a second region of temperature sensitivity above approximately k00°K. Similar temperature dependence of yield -stress has been found in polycrys-12 talline AuZn. .If the data shown in Figure 3 5 are used to compare the TABLE 9 Work-Hardening Parameters for 51.0 at. $Au$'AuZn Single Crystals ( T = 2.5 x 10" 3/sec) Tercp to T i T11 ^111" T i T n TTii i T i n Tm Q i QiT Test ° K psi ^ ~ fo ! - psi 71 72 77 6 7 0 0 6 4 0 0 -— -•21100 1 8 6 0 0 — -5 0 42 - .. 2 0 0 0 0 17500 79 8 0 l k l 6 0 0 0 6 5 0 0 --- -I78OO 21700 - - - - - 5 4 72 - -10800 9300 87 8 8 •181 5 6 5 0 5 l 4 o 10150 10150 14100 14300 -- -I63OO 17700 44 50 106 116 - _ _ 152 178 3 7 5 0 3 5 5 0 _ 77 86 217 3 3 4 0 5 0 8 0 8 8 0 0 9 8 0 0 1 2 8 0 0 1 4 2 0 0 1 3 9 0 0 15300 -2 0 7 0 0 2 3 2 0 0 56 44 160 152 202 190 - - 270 2 8 0 2500 2 9 3 0 10000 8 8 0 0 84 85 2 6 0 4 1 5 0 4 0 2 0 8 5 0 0 8 5 0 0 11900 11900 1 3 0 0 0 1 3 0 0 0 -2 2 8 0 0 21700 46 46 148 148 198 198 - -3 0 8 292 2 3 0 0 2 3 0 0 8 7 3 0 914 0 60 61 293 4 2 2 0 3 7 2 0 7 6 0 0 7 3 0 0 1 0 5 0 0 1 0 3 0 0 11600 11700 -1 8 0 0 0 17500 42 4 0 152 158 211 232 - _ 302 312 1 9 0 0 1900 7 0 0 0 7180 75 76 373 3 2 3 0 3 4 i o 7 1 0 0 7100 8 3 0 0 8 3 0 0 8 6 0 0 8 5 0 0 11100 1 1 0 0 0 11300 11100 29 3 0 116 126 146 150 194 198 209 209 2 2 0 228 1 0 0 0 1 0 0 0 5160 5260 82 83 403 3 4 3 0 3 8 0 0 6300 6 6 0 0 7 3 5 0 7 7 0 0 7 5 5 0 7 8 0 0 9 3 0 0 8 9 0 0 :_9550 9 2 5 0 14 2 0 43 45 52 49 76 64 85 83 144 97 2 3 7 0 2 5 0 0 7 2 0 0 7 0 0 0 81 433 3 1 3 0 .- - - 7 8 0 8 8 0 0 0 - - - - - 47 - -73 74 488 3 1 0 0 3 4 0 0 • - - - - 5 7 0 0 5 9 0 0 - - - - - 6 0 5 0 - -TABLE 1 0 Work-Hardening Parameters for Stoichiometric (5 ' AuZn'Single Crystals ( IT = 2 . 5 x 1 0 " 3 / s e c ) Temp. Ti r i n r m r 1 r 1 1 . 1 1 Y i n m 6 1 On Test °K psi psi 7 8 7 7 7 7 5 9 0 0 5 8 0 0 _ -_ _ 1 6 0 0 0 1 9 7 0 0 _ - - - - 1 + 6 7 0 - 2 2 1 + 0 0 - 2 3 2 0 0 9 1 0 1 5 3 1 5 3 . 2 8 8 0 2 5 3 0 5 0 0 0 i+ooo 9 0 0 0 8 0 0 0 - -II+5OO 1 2 0 0 0 3 0 2 0 7 2 6 0 --- 1 5 1 + 1 1 0 1 + 7 0 0 56OO -1 8 3 1 8 3 2 3 0 0 1 + 8 0 0 7 1 5 0 87OO - 1 2 7 0 0 3 0 8 6 l l + 7 - - 1 9 5 2 5 0 0 -3 1+ 2 2 3 2 2 3 i 9 6 0 1 1 + 1 0 3 6 0 0 2 5 0 0 1 + 1 + 0 0 1 + 6 0 0 5 3 0 0 5 2 0 0 1 8 7 0 0 1 7 0 0 0 2 1 1 + 0 0 2 0 9 0 0 1 6 2 0 6 0 9 0 9 0 1 2 0 2 5 8 2 6 0 -3 1 6 . 3 1 + 0 2 0 0 0 2 2 0 0 7 7 0 0 8 5 0 0 5 6 2 9 3 2 9 3 1 8 8 0 1 5 4 0 3 3 0 0 2 1 + 0 0 1 + 5 0 0 1 + 5 0 0 5 1 0 0 5 0 0 0 9 1 + 0 0 8 7 0 0 1 3 1 0 0 1 2 5 0 0 1 1 + 1 0 5 0 7 0 6 5 8 7 11+1+ 1 5 8 - 2 9 2 3 5 2 2 9 0 0 3 0 0 0 5 1 7 0 5 1 2 0 2 0 2 1 3 7 3 3 7 3 1 8 9 0 1 7 8 0 - --5 8 0 0 5 7 0 0 7 5 0 0 8 0 0 0 1 2 1 0 --9 1 + 8 8 2 5 0 2 2 0 2 9 0 2 1 + 3 -2 3 0 0 2 1 + 0 0 1 8 1 9 4 4 3 1+1+3 1 8 3 0 1 8 2 0 -- --1 + 9 0 0 1 + 6 0 0 ---- 8 0 8 0 1 2 2 1 6 0 --C o o TABLE 11 Work-Hardening Parameters f o r 5 1 - 0 a t Z n @ ' AuZn Single Crystals ( T = 2 „ 5 x 1 0" 3/sec) Test Temp. °K -t o f i x ^111 m ^m e i psi .On psi 9 5 77 8 2 0 0 2 4 5 0 0 1 8 9 6 77 8 8 5 0 - - - - 2 4 6 0 0 - - - - - 1 8 - -1 0 1 i4o 8 5 1 0 _ _ _ 2 8 3 0 0 _ _ 8 0 I67OO 1 0 2 1 4 0 7 6 7 0 - - - - 2 3 8 0 0 - - - - - 5 0 - I65OO 9 9 192' 5 1 9 0 1 0 6 0 0 1 4 7 0 0 _ _ I97OO 5 2 1 1 0 _ _ _ 1 9 0 3 7 0 0 _ 1 0 0 1 9 2 4 1 3 0 10400 1 4 3 0 0 - - I65OO 5 2 9 8 - - - 1 4 0 3 8 9 0 -97 2 4 3 4420 1 0 0 0 0 1 0 6 0 0 1 0 8 0 0 1 6 2 0 0 1 7 7 0 0 40 94 _ 1 8 0 247 2 5 6 2 0 0 0 7 3 0 0 9 8 2 ^ 3 4 3 7 0 1 2 0 0 0 1 2 5 0 0 - 1 7 8 0 0 1 9 0 0 0 5 0 - - l 4 o 2 0 0 2 2 2 - 1 0 0 0 0 8 9 2 9 3 3 2 2 0 78OO 85OO _ 1 3 1 0 0 1 4 3 0 0 2 6 7 0 _ 2 1 8 2 6 0 2 7 4 2 9 0 3 8 7 0 9 0 2 9 3 4 0 0 0 8 2 0 0 85OO - 1 4 7 0 0 1 5 1 0 0 2 0 7 0 - 2 2 4 2 4 0 2 4 5 4 2 0 4 0 5 0 103 348 2 7 8 0 6 6 0 0 7 6 0 0 7 6 0 0 9 4 5 0 1 0 2 0 0 2 8 79 8 0 104 1 3 8 1 5 6 1 2 0 0 7 7 8 0 104 348 3 0 4 0 65OO 7 4 0 0 7 4 0 0 9 4 5 0 985O 4 2 7 8 8 0 106 122 1 3 2 1 1 0 0 7 7 0 0 9 1 3 7 3 2 5 1 0 5 2 0 0 5 9 0 0 6 0 5 0 8 2 0 0 9 2 0 0 14 4 0 52 64 9 0 1 5 0 1 6 2 0 1 7 0 0 0 9 2 3 7 3 3 3 2 0 65OO 7 1 0 0 7 4 0 0 9 7 0 0 1 0 0 0 0 12 3 8 55 72 9 0 1 4 6 1 6 4 0 1 3 6 0 0 1 0 5 4 0 8 2 8 4 0 _ _ _ 6 2 0 0 _ _ _ 2 2 1 7 5 _ _ 1 0 6 4 0 8 3 2 5 0 - - - - 5 7 0 0 - - - - 2 0 1 5 2 - -93 4 7 3 2 5 1 0 _ _ _ 3 4 5 0 _ _ _ _ 12 2 2 5 _ _ 9k 4 7 3 2 2 4 0 - - - 3 5 0 0 - - - - 12 2 1 8 -TABLE 1 2 Work-Hardening Parameters as a Function of Orientation for >^ AuZn Single Crystals (T = 2 9 3 ° K ; ~fT= 2 . 5 x 1 0 " 3 / s e c ; , 5 1 . 0 at.#Au) Orientation T o fx T11 T11 ^111 T m Y i Y11 "fill T f i n Xn 7f> 0i Q11 Test (Fig. i i) "X f psi £ psi 1 1 5 1 1 6 l 6 k6 3 7 0 0 4120 8 3 0 0 85OO 1 1 7 0 0 1 2 2 0 0 i 4 o o o i 4 o o o — 2 3 8 0 0 2 4 0 0 0 5 6 5 6 1 2 0 1 4 0 2 1 0 2 1 0 — - 3 4 0 - 340 2 7 2 0 2 7 4 0 7 9 2 0 7 9 2 0 1 2 1 1 2 2 2 1 3 k9 4 i 6 o 3 9 1 0 - -- 1 8 5 0 0 1 6 4 0 0 20400 2 1 4 0 0 -- -I5O 1 0 0 2 2 0 1 2 6 - 3 0 0 - 2 5 0 --• 1 1 9 1 2 0 3 3 0 k6 3 6 5 0 3 7 0 0 _ --1 2 0 0 0 -II3OO 1 7 1 0 0 1 9 8 0 0 --8 0 3 5 2 4 - 1 2 4 - 1 5 2 - -• 1 1 7 1 1 8 4 3 6 kk 4 0 0 0 3 7 4 0 -- - 1 2 8 0 0 II7OO 1 9 4 0 0 1 6 2 0 0 - - 8 6 2 4 3 5 - 1 6 0 - 1 8 5 - -6 0 6 1 5 1 4 2 6 4 2 2 0 3 7 2 0 7 6 0 0 7 3 0 0 1 0 5 0 0 1 0 3 0 0 1 1 6 0 0 1 1 7 0 0 - • 1 8 0 0 0 1 7 5 0 0 4 2 40 1 5 2 1 5 8 2 1 1 2 3 2 - - 3 0 2 - - 3 1 2 1 9 0 0 1 9 0 0 7 0 0 0 7 1 8 0 2 5 6 1 8 3 1 3 9 0 0 7 8 O O 1 0 6 0 0 1 1 2 0 0 - 1 7 0 0 0 5 4 1 4 8 1 8 0 - - 2 8 0 1 8 0 0 5 8 0 0 1 3 7 7 3 2 2 3 3 0 0 0 7 4 0 0 9 4 0 0 9 8 0 0 1 3 6 0 0 1 4 4 0 0 3 4 1 1 0 1 3 2 2 1 4 2 6 0 2 8 5 1 8 5 0 5 0 0 0 6 7 6 8 8 1 2 12 3 9 8 0 3 9 8 0 7 5 0 0 7 4 0 0 1 0 1 0 0 1 0 0 0 0 1 0 2 0 0 1 0 1 0 0 - 1 5 3 0 0 1 5 2 0 0 3 0 3 0 6 2 6 2 6 6 6 6 - - 1 5 8 - 1 5 8 3 9 0 0 3 9 0 0 5 8 5 0 5 8 5 0 1 3 9 9 1 6 10 3 7 0 0 6 4 0 0 8 0 0 0 8 3 0 0 1 1 8 0 0 1 2 5 0 0 1 4 5 0 6 0 1 2 6 1 5 4 1 6 4 3 6 0 0 5 6 0 0 6 9 7 0 1 0 1 5 5 3 3 5 0 0 3 6 5 0 0 - - - - 4 5 0 0 0 4 9 5 0 0 - - - - - 1 0 - 1 8 -3 8 5 0 0 CO ro • o A 4 - 9 . 0 at. % Au 5 0 . 0 at. io Au 5 1 . 0 at. i Au I I I I 1 1 1 1 J i 1 0 5 0 1 0 0 1 5 0 2 0 0 . 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 Temperature T°K Figure 3 3 . Showing the variation of yield stress with temperature for Au-rich,Stoichiometric and Zn-rich @xAuZn single crystals. ( t = 2 . 5 x lO^/sec; ? ~ 2 6 ° , X ~~ 2 0 ° ) 8k strengthening effect of the excess Au and Zn, i t is seen that at temperatures above approximately 2 0 0 ° K both species impart similar hardening o f ~ 1 7 0 0 psi/at. whereas below 2 0 0 ° K Zn appears to be the more potent strengthening agent. For example, at 1 5 0 ° K , .At"/at. deviation is ^ 5 1 0 0 psi for Zn-rich crystals and jkOO psi for Au-rich while at 7 7 ° K , is ^ 2 7 0 0 for Zn-rich and 1 2 only 5 0 0 psi for Au-rich. In polycrystalline material, Causey found that hardening above 1 3 3 ° K was approximately equal on both sides of stoichiometry with A 0 ~ ~ 9 0 0 0 psi/at for'Zn-rich alloys and /V.75OO psi/at. i for Au-rich A C alloys; at 7 7 ° K a pronounced minimum exists at ^ 5 0 - 5 at. $ Au so that the hardening per at. "jo deviation from stoichiometry is considerably less for Au-rich than Zn-rich alloys. It appears, therefore, that the effects of deviations from stoichiometry on yield stress are the same in single and polycrystalline AuZn. Since the absolute melting point of AuZn is 9 9 8 ° K the homologous temperature is directly proportional to temperature in degrees K. Over the same range of homologous temperatures 0 . 0 8 to 0 . 5 T m single crystals of • I O O * 1 0 1 2 5 * 3 7 stoichiometric AgMg and NiAl were observed to display yield stress variations similar to AuZn increasing approximately three times in the tem-perature interval . 2 5 Tffl to . 0 8 T . Polycrystalline AgMg 1 3' 3 8, N i A l 2 B ' 6 7 6 7 and NiTi exhibited comparable yield stress temperature dependence. In 9 4 general the behaviour is similar to bcc metals (for review see Conrad ) although the dependence is not quite as strong. The temperature dependence of yield w i l l be further considered in.the light of thermally activated defor-mation mechanisms, Section 2 . 6 of the thesis. The orientation dependence o f T 0 resolved on the macroscopic sli p plane in the [ 0 0 1 ] direction is shown in Figure . Although some scatter in the results is evident, i t appears that'CTn is approximately 8 5 001 3 - 6 5 3 - 6 7 Figure 5 4 . Showing the resolved y i e l d stress dependence on o r i e n t a t i o n . ( 5 1 . 0 at. % Au; T = 2 9 3 ° K ) independent of o r i e n t a t i o n and may be given as ^ 3 8 5 0 p s i . The s i g n i f i c a n c e of these observations and t h e i r r e l a t i o n to the v a r i a t i o n i n s l i p plane parameter"V*/ with o r i e n t a t i o n i s . discussed i n section 2 . 5 . 4 . 1 . The value o f f a i n o r i e n t a t i o n 1 0 (Figure 1 1 ) was resolved on the operative (112)-[ill] system and found to be ~ 3 5 0 0 0 p s i which i s about nine times higher than T o f o r {hko} ( O O l ) s l i p . It i s believed that the (hko)[OOl] deformation mode that might be expected to operate was suppressed i n o r i e n t a t i o n 1 0 because of the geometry of the t e s t specimens. In Figure 3 5 ' i t i s shown that f o r the given specimen diameter/gauge length r a t i o of 0 . 1 g r i p constraints do not allow [OOl] s l i p f o r orientations i n which *f i s l e s s than approximately 6 ° . In o r i e n t a t i o n 1 0 where ^ i s 5 ° , [ 0 0 1 ] s l i p i s thus r e s t r i c t e d . I t can be seen, too, from'Schmid fa c t o r and resolved shear-stress c a l c u l a t i o n s that s l i p on the second most highly stressed- {hkoj^OOl) system, namely ( o k l ) [ l 0 0 ] , was not as favourable as s l i p on the observed system. For the given o r i e n t a t i o n 1 0 the Schmid factor on a representative ( o k l ) [ l 0 0 ] system taken as ( 0 1 1 ) [ 1 0 0 ] i s O.O5 while the f a c t o r on ( 1 1 2 ) - [ i l l ] 86 Figure 35 - Showing effect of specimen geometry on inhibiting (hko)[00l] s l i p . is 0 .48 , a difference of 9-5 times. This is greater than the difference of about.9 times in c r i t i c a l resolved shear stress for s l i p on the two systems. Although (112)-[lll] is the most prominent system i t was observed in Figure 28 that some -TllOj s l i p may have occurred as well. .The occurrence of this extra mode can most l i k e l y be accounted for by the fact that stress on the (011)[l00] system probably becomes c r i t i c a l in the i n i t i a l hardening stages past yield. 2.5.1.2 The Work-Hardening Rate in Stage I, Qt shown in Figure 36 for the three compositions tested where the results are plotted as Q±/M , where is the shear modulus characteristic of the operat of s l i p system in Appendix 5- -It is apparent that Qx is a minimum at inter-mediate temperatures of fv210°K, 295°K and 370°K for the stoichiometric, •Zn-rich and Au-rich crystals respectively. The variation of Qx with tem-perature appears to be linear for the non-stoichiometric crystals. In fee metals, stage T hardening rates have been observed to either increase The work-hardening rates during stage I deformation 01} are modulii have been calculated as a function 1 6 * 8 3 13'F> j 13 B 1 4 0 slightly, remain constant or decrease slightly in a monotonic 87 150 200 250 300 350 400 450 Temperature T°K Figure 36. Showing the variation in stage I work-hardening rate Qj with temperature. fashion with increasing temperature, while in bcc crystals Nb and Ta , Q1 was found to increase with increasing temperature, from near zero at ^ .08 T . The slope of the Ox>T curves in AuZn crystals is similar to Nb and Ta and considerably stronger than fee metals. The occurrence of a minimum hardening rate at intermediate temperature appears to be unique. The very low values of 0 X about room temperature for Zn-rich crystals is believed responsible for the wavy nature of the corresponding flow curves (Figure 8 ) . What probably happens is that early in the test, some part of the gauge section deforms at a slightly faster rate than the rest. The cross-sectional area in this region becomes slightly smaller than in the rest of the specimen giving rise to a corresponding increase in the shear stress. Since the rate at which s l i p planes harden with increasing strain is low, then continued sl i p in this reduced section w i l l occur more easily than in the remainder of the specimen. In this manner localized 88 thinning occurs along the gauge- analogous to a necking mechanism. Once the primary sl i p planes harden to such an extent that further sl i p requires an applied tensile stress greater than is necessary to promote new s l i p packets, deformation in the localized area- ceases. A new "neck, is formed and the procedure repeats i t s e l f u n t i l the gauge section is uniformly thinned. The dependence of 0X on i n i t i a l orientation is shown in Figure 37- While insufficient experiments were performed to permit unambiguous comments on orientation effects, i t is noted that the hardening rate near the 001 Figure 57- Showing the variation in stage I work-hardening rate with orientation. (51.0 at. i Au; T = 293°K) [001] corner is approximately two times greater than that near the middle of the stereographic triangle. 2.5.1.3 The End of Stage I The strain at the end of easy glide ~$ 1 X is shown in Figure 38. The extent of stage I is considerably greater for non-stoichiometric compositions by the approximate ratios of :1 for Au-rich, Zn-rich and stoichiometric crystals respectively. Similar effects have been observed on 8 9 2001-1 5 0 % 1 0 0 •H H M W CH O -P c <u -p X 5 0 0/-_L 1 5 0 2 0 0 2 5 0 3 0 0 Temperature T°K • 4 9 . 0 a t . % Au O 5 0 . 0 at.$ Au A 5 1 . 0 a t . i Au 3 5 0 4 0 0 450 Figure 3 8 . Showing the e f f e c t of temperature and composition on the extent of easy g l i d e . adding solute elements to fee m e t a l s . 1 6 ' 1 7 ' 1 4 1 An e x p l a n a t i o n 1 4 1 i s based on the assumption that easy glide ends when the stress concentration around cl u s t e r s of d i s l o c a t i o n s on the primary s l i p plane are s u f f i c i e n t l y large to i n i t i a t e s l i p on c r y s t a l l o g r a p h i c a l l y s i m i l a r secondary systems. Since the y i e l d stress i s raised on both sides of stoichiometry, the c l u s t e r s need greater stress f i e l d s to move secondary d i s l o c a t i o n s and since the work hardening rates are about the same, then easy glide must be more extensive to e f f e c t the l a r g e r c l u s t e r s . Thin f o i l transmission electron microscopy studies (section 2 . 5 - 3 ) v e r i f y the assumption that d i s l o c a t i o n c l u s t e r s are present on primary s l i p planes and suggest that c l u s t e r s give r i s e to the macroscopically detected deformation bands (section 2 . 5 . 2 ) i n the v i c i n i t y of which l o c a l i z e d secondary s l i p i s detected near the end of stage I. 9 0 Except at temperatures below ~ 2 0 0 ° K , the extent of easy glide increases with decreasing temperature, which is similar to the effect of deviations from stoichiometry and can be explained in a similar manner. Since the yield stress increases with decreasing temperature (Figure 3 3 ) larger clusters and hence more extensive easy glide is necessary to init i a t e secondary s l i p . The effects of temperature and composition on the stress at the end of easy glide are shown in Figure 3 9 - For the stoichiometric • Temperature T°K Figure 3 9 - Showing the effects of temperature and composition on the stress at the end of easy glide. crystals i t is seen that lTi\ increases ~ 2 . 5 times between 2 2 0 ° K and 1 5 0 ° K but remains constant at temperatures above 2 2 0 ° K , similar to the yield stress temperature variation. Non-stoichiometric alloys display a monotonic variation 91 in T i i with temperature, decreasing by ~2 times over the range l80°K to 400°K, . An interesting result is obtained when one calculates the stress ratios T"n/^ for the three compositions at various temperatures and orientations, Table 13. It is found that the average ratio is 2.4 - 0.2, 2.9 - 0.2 and 2.7 - 0.4 for the Zn-rich, stoichiometric and Au-rich alloys respectively: within the limits statedT"-^/ appears to be independent of CO temperature,composition and orientation • suggesting that the work hardening mechanism is unchanged over this range of temperature, composition and orientation. TABLE 13 '^11/7s- as--a Function of Temperature, ^° Composition and Orientation Table 15.1 T°K 192 243 293 348 375 2.8 2.4 2.4 2.7 2.5 Avg Zn-Rich 3-5 - 2.1 2.4 2.1 2.4 ? .2 Table 15.2 T°K 155 183 223 293 5-1 5.1 2.2 2.4 Avg, . t i i / r 0 Stoichiometric 3-1 - 5.2 2.9 2.9 t .2 Table 13.3 T°K 181 217 260 293 373 403 2.5 3.8 2.9 2.5 2.6 2.2 Avg Au-Rich 2.8 2.8 2.0 2.8 2.4 2.0 2.7 + .4 Table 13-4 Orientation (Figure 11) 1 •5 6 7 8 9 W t b 3-1 2.5 2.7 3-1 2.5 2.2 Avg • W r 0 Au-Rich •3.0 2.8 - - 2.5 - 2.7 t .4 The effect of orientation on the extent of easy glide is shown in Figure 40. It is apparent that "jTn increases as ^ increases, 92 c o n s i s t e n t w i t h the observa t ions t ha t 0 X decreases w i t h i n c r e a s i n g •TV-wh i l e C X 1 remains approx imate ly unchanged. 001 101 F igure 4 0 . Showing the v a r i a t i o n i n the extent o f easy g l i d e w i t h o r i e n t a t i o n . (51 .0 a t . •% Au ; T = 293°K) 2.5.1.4 The Work-Hardening Rate i n Stage I I , 0 X 1 - The temperature and compos i t ion dependence of stage I I work-hardening ra te i s shown i n F igure kl where the r e s u l t s are g iven i n terms of QX1. Sample f low curves from which 0 n va lues were r e l i a b l y taken are a l s o shown. S ince p a r a - l i n e a r type hardening was observed at temperatures below ~150°K i t was ques t ionab le whether or not the s lope 0^ of the l i n e a r segment of the f low curve could be r e l a t e d t o stage I I harden ing . Th is d i f f i c u l t y was p a r t i a l l y r eso l ved by s tudy ing the v a r i a t i o n i n s l i p l i n e s t r u c t u r e d u r i n g s t r a i n i n g . The r e s u l t s o f t h i s study are repor ted i n a l a t e r s e c t i o n of the t h e s i s where i t w i l l be shown tha t m u l t i p l e s l i p occurs c o n t i n u a l l y a t 77°K, but on ly i n minor amounts du r ing stage I I at h igher temperatures . P a r a - l i n e a r hardening was then b e l i e v e d t o be r e l a t e d , i n pa r t a t l e a s t , t o stage I I hardening ra tes s ince both types of deformat ion are man i f e s t a t i ons of m u l t i p l e s l i p p rocesses . The corresponding l i n e a r work-9 3 I I _ J _ ; 1 J 1 1 1 0 0 1 5 0 2 0 0 . 2 5 0 3 0 0 35O 1 + 0 0 Temperature T°K Figure 41.1. Showing the variation of stage II work-hardening QX1 rate with temperature. .Strain Figure 4l.2. Schematic flow curves. .Type A analyzed for linear hardening rates and Types C and D for stage II hardening rates; • Type B not analyzed". 9k hardening rates at 77°K are plotted, therefore, in Figure kl where they are connected with the Oix values, by a dashed line. An explanation of the higher work-hardening rates during para-linear flow is given in section 2.5.4. Over the temperature range - -225°K to 350°K in which 0 X i could be reliably evaluated, i t is apparent that hardening rates during stage II decrease by 2.5 times from M to /•• with increasing temperature. 500 1200 This behaviour is different from the very minor decrease with temperature in fee metals and alloys, but resembles the quite pronounced decrease above 142*143 0.27 T in Cd. It is also•interesting to note that 0 n is strain m rate sensitive, decreasing with decreasing strain rate. The flow curves in Figure 9 were analyzed for this feature and the results are given in Table Ik, TABLE 14 Variation in Stage II Hardening Rate with Strain Rate (T = 295°K, g.= 26°, X= 20°)  Test Y f s e c . " 1 ) ©ii (psi) Oil//' x<103 112 2.5 x 10" 4 5900 1.36 60, 61 2.5 x 10" 3 7100 1.63 110 2.5 x 10" 2 9100 2.09 It appears, therefore, that thermally activated recovery processes can take place concurrent with plastic deformation. Below 350°K, the effect of composition on S l x is very small, 16*17*141 similar to fee systems. ' At higher temperatures, however, non-stoichiometric crystals exhibit an unusual sudden rise in hardening rate, 1*9 which is somewhat similar to behaviour of bcc metals near .1 T . • m These results suggest that•the work-hardening mechanism is probably the same 1 95 for a l l compositions below 550°K, but differs for stoichiometric and non-stoichiometric . crystals above this temperature. The dependence of 0 n on crystal orientation is shown in Figure k-2. It can be seen that On decreases as the specimen orientation approaches the [001]-[101] boundary (i.e. as "X-increases) which is similar to the effect of increasing temperature and decreasing strain rate. 001 Figure k-2. Showing the orientation dependence of stage II work-hardening rate (51.0 at. $ Au;.T = 293°K) •An intimate relationship between temperature, strain rate and orientation was noted earlier in the discussion on deformation modes. • It was shown that the sl i p plane parameter "\|/. increases with increasing temperature, increasing"X and decreasing strain rate which was interpreted in terms of an increasing tendency for cross-slip of screw dislocations. It is once more apparent that these variables are related through their combined effect on stage II hardening rates. With increasing temperature, increasing ")£ and decreasing strain rate, 0 n decreases. It is not known i f the observed variation in Q1± is simply a manifestation of the changing s l i p plane and hence a property of the macroscopic sli p plane per se, or i f i t 96 suggests that cross-slip plays a dual role in the deformation of AuZn, active as a dynamic recovery mechanism as well as governing the choice of sl i p plane. Work-hardening rates in specimens oriented near the [OOl] corner (Figure 1 1 , orientation . 1 0 ) must be discussed separately. As already noted, in these orientation {hko} < 0 0 L > systems no longer serve as the primary s l i p modes; instead s l i p occurs on a { 2 1 l } - (ill)system. Secondary systems, probably { 3 2 l } < 1 1 1 > and (llO) (OOl), operate as well (Figure 2 8 ) but observations at fracture show that their contribution to the total ductility is negligible with respect to the { 2 1 l } < l l l ) primary s l i p . It i s to be concluded therefore that the very high work-hardening rate and limited d u c t i l i t y are direct results of < 1 1 1 > s l i p . The linear hardening slopesOg wereanalyzed giving values of ^ 3 8 , 5 0 0 psi. In terms of shear modulus, this is equal to Si wherewas calculated from the relationship [ 2 1 1 } ( i l l ) , - 6 0 .. • = 1 ( c n - c i 2 + c 4 4 ) , Appendix 5- The elastic constants c 1 1 ; c X 2 and c 4 4 were taken from the data of Muldawer and Schwartze . It is immediately apparent that hardening .rates resulting from ( 1 1 1 ) s l i p are over 1 0 times greater than those associated with general £hko} ( 0 0 1 ) s l i p modes. The reason for this unusually high value must in some way be asso-ciated with antiphase boundaries created by the motion of l a ( 1 1 1 ) type 2 superlattice partial dislocations. Possible superdislocation hardening mechanisms are discussed in section 2..5..U. It is significant that the linear hardening rate 0 ^ / l 5 12 observed in polycrystalline AuZn is considerably nearer the value // 6 0 in single crystals undergoing O-ll "> sl i p than the values . ^  to /<Y 5 0 0 1 2 0 0 in crystals deforming along ^ 0 0 1 > direction. (It should be noted that quoted in reference 1 2 was obtained using the shear modulus for { 2 1 1 } ( 1 1 1 > s l i p rather than for (hko} (.001)1) While sl i p on the [hko) < T0 0 1 ) systems is 9 7 believed to account for most of the pl a s t i c i t y observed in polycrystalline material, { 2 1 1 } and { 3 2 l ] traces were observed in the v i c i n i t y of grain boundaries, suggesting that < 1 1 1 > s l i p does occur.under rather high stress conditions. In view of the very high hardening rate associated with < 1 1 1 > s l i p in single crystals i t appears as. though.the rapid hardening in poly-crystalline material below <v-300°K is controlled by dislocation motion on {hkl| < 1 1 1 > systems. Hence, grain boundary hardening, rather than hardening within the grain is believed to control the flow stress. This is consistent ' 12 with the Hall-Petch behaviour of AuZn , where the Petch slope k^ .. relating the flow stress to the reciprocal of the square root of the grain size is observed to increase with increasing strain. . 2 . 5 . 1 . 5 Stage III Stage III is characterized by rapidly decreasing values of work-hardening rate with increasing strain. In Figure i t is shown that the stress T i n a"t the end of stage "II decreases with increasing temperature, suggesting that a thermally activated recovery mechanism is responsible for the breakdown of linear hardening. Since only a few orientations investigated gave.rise to three-stage work-hardening curves, i t is d i f f i c u l t to comment with certainty on the quantitative effects of specimen orientation o n T j i i i -There is some evidence to suggest thatT"m decreases as X increases, but u n t i l further experiments can be carried out, this must remain as a tentative observation only. •H [fl CO I 98 20 r 15 10 H H • o A 49.0 at. $ Au 50.0 at. <?o Au 51.0 at. io Au 200 250 3OO 350 Temperature T°K 1+00 1+50 Figure 1+3. Showing the, effect of temperature on the stress at the end of stage II. 2.5-1.6 .Maximum Shear Stress and Ductility The maximum shear-stress (~ and total shear-strain to fracture m 2(j are shown in Figures 1+1+ and 1+5 as functions of temperature and composition. tends to decrease with increasing temperature except over the intermediate temperature range 200°K to 300°K where a peak is detected. Polycrystalline • 12 AuZn "' shows exactly the same variation in ultimate tensile strength versus temperature. In both polycrystalline and single crystal material, the maximum flow stress decreases, by about 5 times between 77°K and 500°K. Although?"^ increases with deviations from stoichiometry in both materials, the effect is not nearly as pronounced as i t is on the yield stress. The shear strain to fracture VV may be taken as a measure of 99 •H CO P. o I CO CO cu u - P CO u o3 JB CO X I 30 25 20 15 10 1 4 9 . 0 at. i Au 5 0 . 0 at. # Au 5 I . O at. i Au 1 0 0 200 300 Temperature"T°K 400 500 Figure 44, Showing the variation of maximum shea'r stress with temperature. 4 9 . 0 at. % Au 50.0 at. % Au 0 at. <f> Au 200 300 Temperature T°K 500 Figure 4 5 . Showing the variation of total d u c t i l i t y with temperature. 100 duc t i l i t y in single crystals. From Figure 45 i t can be seen that the temperature dependence of du c t i l i t y may be divided into three distinct regions: (A) from 77°K to 250°K, in which Y increases by about 6 to 10 times depending on composition; (B) from 250°K to 400°K where Y decreases by^ 2 times for Zn-rich crystals and about 5 times for stoichiometric and Au-rich alloys; (C) above 400°K where du c t i l i t y appears to increase in both Zn-rich and Au-rich crystals. Over the temperature ranges A and B, st o i -chiometric crystals are most ductile and Zn-rich crystals are the least ductile,.but in region C, Zn-rich alloys display greatest ductility. The temperature range of maximum duc t i l i t y . i s coincident with the peak flow stress. • 12 Polycrystalline du c t i l i t y exhibits identical behaviour over the same temperature range suggesting that similar mechanisms may be responsible for fracture. Until some crystallographic aspects of single crystal fracture are presented in section 2.5-2, further'discussion is terminated. 2.5-2 • Slip Line Variation'During Deformation In addition to the direct evaluation of work-hardening rates, supplementary studies are often performed to arrive at possible hardening models. Slip.line studies on the surfaces of deformed crystals, transmission electron microscopy in thin films, X-ray diffraction, e l e c t r i c a l r e s i s t i v i t y and magnetic properties of. ferromagnetic materials are useful experiments from this viewpoint. Indeed detailed studies of the f i r s t two types have led to the development of two very prominent work-hardening theories of 1 4 4 > 1 4 5 stage II deformation, the long range hardening theory of Seeger and 1 4 6 the pile-up theory of Hirsch. Consequently information from both surface sli p line variation during deformation and the corresponding dislocation structure in thin films is believed to be of utmost importance in under-stand ing "the gsnersil dsfopina-'tion behaviour of metal crystals. Both, surface 101 s l i p line studies and transmission electron microscopy experiments were carried out during the present investigations and the results are reported in their respective sections 2.5-2 and 2.5-3• While these studies were not intended to give a detailed account of the work-hardening mechanisms, they were designed to add information which would aid in understanding the plastic behaviour of ^ AuZn single crystals. 2.5-2.1 Procedure Specimen preparation for sl i p line analyses has been given in section 2.4. Crystals of orientation 1 in Figure 18 previously used to determine primary sl i p traces versus temperature have been used to study •the variation of sl i p lines during tensile deformation at temperatures between 77°K and 473°K. Specimen surfaces were examined optically under oblique f i l t e r e d lighting with the Reichert metallograph. 2.5.2.2 Observations In presenting photomicrographs showing the development of sl i p l i n e s , i t was decided for the purposes of comparison to include the series already presented in section 2.4 since they are characteristic of the s l i p line appearance during i n i t i a l flow. Slip.traces on orthogonal faces A and B (identified in section 2.4) are shown as a function of strain at 77°K, l40°K, 293°K, ,398°K and'473°K in Figures 46 to 50 respectively. Schematic flow curves noting the strains at which observations were made are also shown. Because of the non-crystallographic nature of s l i p traces, systems are indexed with respect to approximate -{hkoji planes in the (001) zone. For easy comparison, comments on the observations are summarized in Table 15. The results are discussed in the context of a general discussion on work-hardening, section 2 .5-4.2. 1 0 2 46.A.l 46.A.2 46.A.3 F i g u r e 4 6 . V a r i a t i o n i n s l i p l i n e s t r u c t u r e w i t h s t r a i n a t 77°K. X 1 0 0 20 120 F i g u r e 4 7• V a r i a t i o n i n s l i p l i n e s t r u c t u r e w i t h s t r a i n a t l40°K. X 100 F i g u r e 48. V a r i a t i o n i n s l i p l i n e s t r u c t u r e w i t h s t r a i n a t 2 9 3 ° K ; X 100 ( e x c e p t where n o t e d ) . 1 0 5 h ^5 Ico r a) Figure 4 9 . Variation in slip line structure with strain at 3 9 8 ° K . X 1 0 0 (except where noted) 1 0 6 5 0 . A.l 5 0 . B . l 5 0 . A . 2 Figure 5 0 . Variation in sl i p line structure with strain at 4 7 3 ° K . X 1 0 0 TABLE 15 Comments on Slip Line Variation During Deformation Primary Slip System Secondary Slip System System Description Description Contribution to Total Strain -)f Temp. Ref. (approximate °K Fig. with respect Variation Yes First Variation £ final (degree) to plane) In i t i a l with Strain No Detected Systems Initial with Strain Measured Calculated Amount 77 1*6 (Oil)I100J face A-coarse, coarsen on Yes at (Oil)I111J minor amount coarsen slightly wavy; both faces yield (110)[001] slightly face B-fine, A and B 1  TI coarsen and 51 ~xaf, straight increase in (from (from number [100]) [100]) ll+O 47 (210)1001J on both A and development of Yes during {321} short, straight remain fine, B, banded fine traces last and detected near straight, slip lines between few {211) shadowed areas widely "bands" percent parallel to spaced lh 15 Negligible strain microcracks 293 1+8 (310)L001J wavy on A face A traces Yes at end (021)U00j short, curved coarsen, new and straight coarsen,become of appearance, traces on B; rela- wavy on a large- stage I occurring in develop around tively fine scale with wave localized smalle r traces lying regions in deformation 11 10 Negligible close to (100); vicinity of hands traces B adopt deformation slight wavy hands appearance 598 1*9 f510)I001J increased both traces A No - - - - - - -waviness of and B coarsen face A traces; while traces A straight become profusely traces on face wavy B and finely spaced it-73 50 (100)[001] increased tt Yes at ~50jt (010)[100] long, remain waviness of strain slightly app rox imat ely face A traces; (i.e. at wavy, constant in face B traces peak flow coarse number, but remain straight stress) coarsen - -and finely spaced See note, p.XC9 TABLE 15 (Continued) Other Crystallographic. Features .Fracture Appearance Temp. °K Type First Detected •Plane Variation with Strain 77 micro- at kO& (001) apparent increase fracture surface cracks within in length pa r a l l e l to 5 degrees microcracks iko micro- during last few- (001) fracture surface cracks percent strain within pa r a l l e l to .5 degrees - microcracks deformation Section bands yield . 2.5.2.3. -LJ-U.. 293 s l i p line onset of (100) fracture surface clusters stage III within pa r a l l e l to visible on 5 degrees coarsen fissures, which face B appear at begininj deformation Section of stage III in bands yield 2.5.2.3 v i c i n i t y of sli p clusters 398 s l i p line hQPJo (100) coarsen; fracture plane clusters (near maximum within at fracture gauge par a l l e l to visible on flow stress) 5 degrees section "saturated" cluster face B with clusters deformation Section bands yield 2.5.2.3 Note re Table 15: Contribution of secondary systems to total strain was estimated by calculating specimen reorientations assuming various amounts of secondary s l i p , then comparing the calculated with the measured values. .Reorientation calculations were performed using the relationship: 4 9 Sin jrt .= lo Sin <£0 1. 1 where the suffixes o and i refer to the original and instantaneous values of gauge length 1 and angle £ between the tensile axis and the sl i p direction. 110 • 2 . 5 . 2 . . 3 Deformation Bands 2 . 5 . 2 . 3 . 1 Characteristics A f a i r l y common feature of the surface appearance of the deformed crystals is the occurrence of a series of markings traversing the primary s l i p traces, termed deformation bands. The bands can be des-cribed as striations which can be seen under very low magnification; at high magnification, they seem to disappear. Deformation bands were detected in a l l orientations at room temperature, except those near the [001] corner where (112)[ill] s l i p predominates, and at a l l temperatures between l40°K and 398°K for a constant orientation near the middle of the stereo-graphic triangle. Bands form ir. the very early stages of deformation. Their variation in appearance during straining is best illustrated in the room temperature observations,-Figure 48. -Although the distance between any two is not quite uniform, the average spacing is approximately 0.2 to 0 . 3 mm. and the average width about 0 . 0 3 to 0.04 mm. As strain increases, the bands increase in intensity and become wavy (Figure 48.B.2),while the average separation appears unchanged. It can be seen that the primary sli p traces crossing deformation bands change their direction, slightly at f i r s t (Figure 48.B.l),but quite markedly at higher strains (Figure 48;B.':2). The appearance of the sli p lines suggests a change of elevation at the band site and the change in sli p line direction indicates that the material within the band lags behind the matrix during lattice reorientations. These markings closely resemble similar structures observed in aluminum single c r y s t a l s . 1 4 7 I l l 2.5.2 .3 .2 Crystallographic Nature During the early stages of deformation, the boundaries of the bands may be defined as plane surfaces. Two surface trace analyses were subsequently performed and the results are shown stereographically in Figure 51-1 0 1 Figure 5 1 . Stereographic projection of deformation band poles versus crystal orientation and test temperature. It i s apparent that the poles cluster round the [OOl] sli p direction to within, an average of~8°, suggesting that the sli p direction plays an important role in deformation band formation. No systematic effects of orientation or temperature were detected. 112 2.5.2.3.3 Mechanism of Formation Deformation bands are a f a i r l y common feature of the deformati of cubic crystals. Two explanations have been advanced to account for 147*148*149 their formation. From a macroscopic aspect i t is believed that bands are caused by bending which occurs as a result of constraints at the specimen loading grips, or by inhomogeneous lattice rotations where one section of the crystal slips more than i t s neighbours. The band planes produced move in the direction of sl i p u n t i l two of opposite sign meet and become stuck. .A planar obstacle is consequently formed against which later dislocations pile up and create the deformation band. This explanation accounts for the observed scale, but does not explain the fundamental dislocation processes by which bend planes are formed. 1 5 0 . Mott has proposed a more detailed theory to account for band formation. His idea of the dislocation arrangement within a band i s shown in Figure 52. -Walls of positive edge dislocations are pushed by the applied shear stress and arrive from the right while walls of edge disloca-Figure 52. Schematic representation of dislocations in deformation bands, (after Mott 1 5 0) 115 tions of opposite sign and roughly equal strength arrive from the l e f t , thereby forming a deformation band. The positive wall causes the lattice to t i l t downward while the negative wall effects.an upward curvature. The mutual attraction of positive and negative dislocations makes these con-1 5 1 figurations f a i r l y stable. •Mott's edge dislocation wall-model necessarily implies that the deformation bands are perpendicular to the sl i p direction. In • I 4 y y 1 4 8 aluminum deformation bands l i e along ( 1 1 0 ) planes perpendicular to the s l i p direction in support of Mott's theory. In-AuZn i t was shewn that although the bend planes are not quite perpendicular to the [ 0 0 1 ] s l i p direction, they tend to adopt orientations centered about this pole. The view is taken therefore, .that a mechanism similar to that proposed by Mott can account for the band formation in j^'AuZn. . 2 . 5 - 2 . 4 Microcracks The nucleation and propagation of microcracks along {lOO} planes is a commonly observed fracture mode in bcc metals deformed at low 1 5 2 * 1 5 3 * 1 5 4 temperatures. In considering the i n i t i a t i o n of a crack in bcc structures, C o t t r e l l 1 5 5 has suggested that the formation of dislocations with a[OOl] Burgers vectors .is an important step. According to the reaction: , l a [ l l l ] - + la [ 1 1 1 ] = a [ 0 0 l ] ( 2 5 ) 2 2 an edge dislocation of Burgers vector a [ 0 0 l ] can form from a combination of two dislocations with Burgers vectors l a [ i l l ] and l a [ i l l ] gliding in the 2 2 ( 1 0 1 ) and ( 1 0 1 ) planes respectively. The product dislocation l i e s along [ 0 1 0 ] and is a pure edge with glide plane ( 1 0 0 ) . Since a [ 0 0 l ] dislocations are normally not mobile in bcc lattices, Cottrell'suggests that they act as barriers against which other dislocations pile up and eventually nucleate a Ilk crack. Cottrell's mechanism is not likely to be responsible for crack nucleation in AuZn since slip trace analyses show that [OOl] dislocations are highly mobile. The crystallographic similarity between the high temperature deformation bands and the low temperature microcracks (i.e. both are approximately coincident with (001) planes) suggests that similar edge dislocation wall mechanisms may be responsible for their formation. Whereas stress concentrations resulting from wall formation may be relieved through the operation of secondary {hko} (00]) slip systems at 293°K (Figure k&.B.2-), they appear to be relieved through crack formation at low temperatures, presumably because the increased critical shear stress for {hko] (001) • slip renders secondary slip unfavourable. The increasing susceptibility for crack formation at low temperatures, therefore, probably accounts for the decreasing ductility, region A in Figure k^>. 115 2.5-3 Transmission Electron•Microscopy of Thin Films 2 ..5.3 .1 Introduction Considerable dispute has arisen in the literature concerning the degree to which dislocation arrangements observed in.thin films are 1 4 4 * 1 7 2 * 1 7 3 1 4 4 representative of configurations in the bulk material. Seeger has cri t i c i z e d the use of thin film microscopy observations to support work-hardening theories, on the grounds that long-range stress fields present in the cold-wbrked state and extending over distances large with respect to the usual f o i l thickness are to a large extent relaxed during the process of preparing a f o i l from the bulk, since the surface of the f o i l s must be stress-free. The relaxation in stresses must then effect a change in the dislocation arrangement. The degree of rearrangement is believed to depend•on the stacking fault energy of a material being greater 1 7 3 .the higher the energy. Hirsch, on the other hand, takes the view that since dislocation distributions obtained from etch-pit studies are in s u f f i -ciently good agreement with the results of thin film microscopy, the latter may. be regarded as being reasonably representative of the bulk, at least as regards the overall dislocation distribution. In recent years more elaborate specimen preparation techniques have been established in attempts to preserve the dislocation arrangements that are truly characteristic of crystals undergoing plastic deformation. 174 Dislocation arrays have been pinned by both precipitation techniques 1 7 5 * 1 7 6 and neutron irradiation exposures. In the case of Cu crystals irradiation pinning did not significantly affect the dislocation arrange-ment since similar structures were observed in f o i l s prepared in the more 1 6 4 conventional manner. Until more experiments of this nature are carried 116 out, i t must "be assumed that the unpinned structure is f a i r l y characteristic of the pinned bulk arrays. What must also be evaluated is the effect of irradiation and subsequent point defect creation on dislocation structure. The purpose of the present study is to examine the dislocation distribution in annealed and deformed' AuZn single crystals tested at room temperature. Irradiation experiments were not carried out and i t is therefore not possible to estimate the degree to which the f o i l structure is characteristic of the bulk. Recently, martensitic transformation products have been observed near the edges of Zn-saturated (53.9 at. $ Zn) 177 1 2 AuZn f o i l s similar to the twin-like marking observed by Causey . In the present study, however, similar markings were not detected for any f o i l orientation. Any rearrangements that occurred during thinning, therefore, could not be attributed to stress fields of a transformation product. 2.5-3-2 Procedure Because of d i f f i c u l t i e s experienced in preparing thin f o i l s from.3 mm. diameter crystals, large 5 m m - diameter Au-rich (51-0 at.$> Au) crystals oriented near the middle of the stereographic triangle were employed to study dislocation distribution. The room temperature work-hardening behaviour of the larger crystals agreed well with the small crystal behaviour. Specimens were mounted .in epoxy resin and strained in the usual manner. To obtain information about the three-dimensional nature of dislocation distributions in deformed crystals, i t is necessary to examine f o i l s of several orientations. Specimens of three orientations were obtained from deformed crystals by spark machining discs /"ulmm. in thickness: (1) parallel to the (hko) glide plane and parallel to the [001] 117 •Burgers vector, termed (hko) section; (2) parallel to the (110) plane and parallel to the Burgers vector, called (110) section; and (3) perpendicular to the glide plane.and parallel to the Burgers vector, termed perpendicular (hko).section. Discs were cut using surface sl i p traces as a guide; the orientation of sections (2) and (3) was checked with the back-reflection Laue X-ray technique. Thinning was achieved by repeatedly jet-machining and electro-chemically p o l i s h i n g 1 7 8 the discs in a mixture of \yjo hydrochloric acid, 50fo ethyl alcohol and 5$ glycerine at -20°C and 12 volts.. The electrolyte was contained in a pyrex beaker cooled in a bath of methyl alcohol and solid C0 2. -To avoid straining the f o i l during thinning the usual technique of lacquering the outer rim of the specimen was abandoned. After thinning, specimens were thoroughly washed in d i s t i l l e d water then rinsed in ethyl alcohol'and kept in a dessicator u n t i l examined. Observations were made on a Hitachi Hu 11A electron micro-scope operated at 100 KV. Contrast was varied by t i l t i n g the specimens through~10° during examination. A selected area diffraction pattern was obtained from each area photographed to permit subsequent analyses of dislocation arrangements relative to prominent crystal directions. 2.5.3.3 Observations 2.5-3.3.1- As-Grown Structure Dislocation structures observed in f o i l s prepared from as-grown crystals are shown in Figure 53- Dislocation density in as-grown 1 1 8 ZS'ooox 3&000X 5 3 . 1 5 3 - 2 Figure 5 5 • Electron micrographs of dislocation structure i n as-grown crystals. 7 / 3 crystals i s estimated as ~10 cm/cm since on the whole only a few disloca-tions were v i s i b l e i n any photomicrograph corresponding to a f o i l area of - 7 P *»10 cm . Figure 5 5 - 1 represents the most commonly observed as-grown structure, showing rather long, straight dislocations, while Figure 5 5 - 2 represents a less commonly observed zig-zag dislocation structure si m i l a r n 1 7 9 * 1 8 0 to equilibrium configurations found i n p -brass. The unusually high density seen i n Figure 55.2 i s not to be associated with zig-zag dislocations since si m i l a r shapes were observed i n other low density areas. In an e l a s t i c a l l y isotropic c r y s t a l with no applied stress, the concept of dislocation l i n e tension implies that the equilibrium position of a dislocation running between pinning points (for instance, f o i l surfaces) i s a straight l i n e , similar to the structures i n Figure 5 3 . 1 . On the other hand, zig-zagged dislocations are considered to be a direct 1 7 9 result of c r y s t a l anisotropy. A straight dislocation which i s i n a high energy direction may be unstable with i t s t o t a l energy decreasing i f 1 1 9 ' 3 1 i t changes to a zig-zag shape. Calculated from Zener s relationship 64 A = 2 C 4 4 and using that data of Schwartz and Muldawer, the degree of ( C l i - C 1 2 ) elastic anisotropy A in (5AuZn is 3 - 3 > which is probably high enough to render certain dislocations unstable, and hence account for their zig-zag shape. 2 . 5 . 3 - 3 . 2 Variation in Dislocation Structure During Deformation at 2 9 5 ° K 2 . 5 . 3 . 3 . 2 . 1 fhko) Section The effect of room temperature deformation on dislocation structure in f o i l s cut approximately parallel to the glide plane and containing the [ 0 0 1 ] Burgers vector is shown in Figures 5 ^ a n d 5 5 - The structures shown are typical of those observed in several f o i l s prepared from crystals strained 3 5 $ "to the beginning, of easy glide, Figure ^k, and 1 3 0 $ to the end of easy glide, Figure 5 5 - The most outstanding characteris-t i c of the dislocation structure is the long, generally straight arrays of dislocation bundles. With increasing deformation, the dislocation content of the bundles increases while the average distance between any two appears to decrease slightly from~ 2 to 1 . 5 microns. At the end of stage I, several areas along neighbouring bundles appear bridged by dislocations parallel to those within the bundle, giving rise to a rectangular shaped c e l l structure. •Structural developments were not followed into stage II because of the increasing d i f f i c u l t y experienced in preparing satisfactory f o i l s from the more heavily work-hardened crystals. Crystallographic reference directions are shown on most electron micrographs. It is readily apparent that the direction of the bundles is perpendicular to the operative sl i p direction [ 0 0 1 ] suggesting that the dislocations comprising the arrays possess predominantly edge 120 F i g u r e 54.1 F i g u r e 5 4 . E l e c t r o n m i c r o g r a p h s o f d i s l o c a t i o n s t r u c t u r e a t t h e b e g i n n i n g o f easy g l i d e ; (hko) s e c t i o n s p a r a l l e l t o m a c r o s c o p i c s l i p p l a n e . ( Y = 35$) 121 Figure 5*1.2 122 Figure 5 5 - 1 Figure 5 5 . Electron micrographs of dislocation structure at the end of easy glide; (hko) section parallel to macroscopic s l i p plane. ( V = 1 3 0 $ ) 1 2 3 124 character. The absence of screw dislocations is noticeable in a l l micro-graphs. Analysis of dark f i e l d patterns employing the g_'b = 0 i n v i s i b i l i t y 1 8 1 criterion ,where g is the diffraction vector responsible for the dis-location contrast and b i s the Burgers vector, showed that the Burgers vector of the dislocations in the bundles was consistent with [OOl]. However, detailed Burgers vector•analysis to determine possible non-[00l] dislocations in the clusters was not carried out since high resolution dark f i e l d photographs could not be obtained with the existing f a c i l i t i e s . 2 . 5 . 3 . 3 . 2 . 2 (110) Section Because of the non-crystallographic nature of the macroscopic sl i p plane, sections parallel to the fundamental (110) glide plane con-taining the s l i p direction [001] were also examined. Typical structures seen after 35$ deformation are shown in Figure 56. It can be seen that no major differences exist between the dislocation structures on the macro-scopic and fundamental glide planes, although the length of the clusters (bundles) on the (110) plane appears slightly greater than on the (hko) glide plane suggesting that the edge array may l i e along [llO] directions. The observation that the bundles penetrate the f o i l should not be taken as evidence against this suggestion, since even i n f i n i t e l y long bundles would pass through the f o i l which .deviated slightly h>k°) from (110). Because of this deviation the length of the bundle in.the parent crystal is uncertain. During examination in the microscope, f o i l s t i l t e d to bring many (110) reciprocal lattice spots into reflection showed considerably longer bundles than are shown in Figure 5 6 , often strung over distances of , suggesting that the bundles are at least this long. F i g u r e % . E l e c t r o n m e t a l l o g r a p h o f d i s l o c a t i o n s t r u c t u r e a t t h e b e g i n n i n g o f s t a g e I ; (11.0) s e c t i o n c o n t a i n i n g the Burge.rs v e c t o r [00l]. 1 2 6 2 . 5 . 3 . 3 . 2 . 3 Perpendicular (hko) Section Sections perpendicular to the macroscopic sli p planes and containing the [OOl] sl i p direction were examined to determine the extent to which these two-dimensional bundles formed walls perpendicular to the glide plane. Typical structures observed after 5 5 $ strain are shown in Figure 5 7 - In comparison with the bundled arrays on the glide planes, contrast effects from the perpendicular sections are due to dislocations passing almost ve r t i c a l l y through the f o i l s . The piercing character of the dislocations was verified by t i l t i n g the specimen stage and observing the decreasing projected length of dislocation onto the film plane, Figure 5 8 , as the f o i l approached a reflecting position effectively perpendicular to the ( 1 1 0 ) glide plane. From these sections dislocation density at the 9 / 3 start of easy glide is estimated as ~ 1 0 cm/cm . In Figure 5 7 i t can be seen that the [OOl] s l i p direction is perpendicular to the dislocations, again suggesting that predominantly edge dislocations are present in the f o i l . In Figure 5 7 - 1 dislocations are observed to be along rows approximately.parallel to both [ 2 1 0 ] and [OOl]. •The same trend is evident in Figure 5 7 . 2 , although the rows appear shorter. These observations suggest that the two-dimensional bundle arrays are in fact low walls ( ^ 2 ^ high) of edge dislocations. Since the projected dis-location length decreased near the ( 1 1 0 ) reflections i t is believed that the walls are perpendicular to the ( 1 1 0 ) fundamental glide plane. The [OOl] rows suggest that some of the bundles may be narrow carpets of dislocations lying in the glide plane. 2 . 5 . 3 . 4 Discussion Three points must be explained: ( 1 ) the nature of the bundled 1 2 7 5 7 - 2 F i g u r e 5 7 . E l e c t r o n m i c r o g r a p h s o f d i s l o c a t i o n s t r u c t u r e a t t h e b e g i n n i n g o f s t a g e I ; s e c t i o n p e r p e n d i c u l a r t o t h e m a c r o s c o p i c s l i p p l a n e and c o n t a i n s t h e [ 0 0 1 ] B u r g e r s v e c t o r . 57.1 57-2 F i g u r e 5 7 . E l e c t r o n m i c r o g r a p h s o f d i s l o c a t i o n s t r u c t u r e a t t h e b e g i n n i n g o f s t a g e Ij s e c t i o n p e r p e n d i c u l a r t o t h e m a c r o s c o p i c s l i p p l a n e and c o n t a i n s t h e [OOl] B u r g e r s v e c t o r . 128 129 array, (2) the mechanism of the wall formation, and (3) the absence of screw dislocations. These points w i l l be considered in.order. Edge dislocation bundles ("clusters", "strands" or "braids") are a common feature of the stage I dislocation structure on the primary 1 4 4 ; 1 6 4 ; 1 6 5 ) 1 8 2 7 , 8 > 1 6 0 s l i p plane in many metal crystals of fee , bcc and h e p 1 6 7 ' 1 8 3 structures. Characteristic,.too, is the absence of screw dislocations. The edge clusters are usually arrays of dislocation dipoles, i.e. close pairs of dislocations of opposite sign lying along roughly para-l l e l planes that may be frequently linked to form narrow closed loops, of dislocation line. The bundles observed in AuZn are also believed to be comprised almost entirely of dislocation dipoles which often link to form closed loops as is seen particularly well in Figure ^>k. The bundles are comprised of approximately equal numbers of edge dislocations of opposite sign, as no net contrast changes are observed' across the arrays, even at the higher stresses near the end of stage I. A model for dipole formation i n Mg crystals has been given by 1 6 7 Hirsch and Lally. Assuming that approaching edge dislocations of oppo-site signs on parallel s l i p planes trap one another, bands of dipoles form i f the distance between the sli p plane is less than a certain c r i t i c a l distance. This model is believed to account for dipole formation in AuZn as well. -Assuming that s l i p occurs in part through the operation of Frank-Read sources, then at higher stresses when more sources operate, the distance between active s l i p planes decreases. Once the applied stress reaches a c r i t i c a l value, thought to be near the stress at the onset of stage I, sufficient s l i p planes are active and edge trapping may begin, forming the f i r s t dipoles. With increasing deformation, clusters of dipoles. are expected to develop in the vi c i n i t y of the originals, giving 130 rise to an increased density of dislocations within the clusters, and subsequent wall growth. The similarity- between the crystallographic nature of the previously reported macroscopic deformation bands and the microscopically observed edge dislocation walls suggests that the mechanism of formation of both structures may be similar. If edge dislocation trapping can account for both structures,,as suggested, the difference in scale between the bands and clusters may be explained i f i t is assumed that periodically extra-heavily populated clusters form which can give rise to an overall lattice t i l t , large enough to be detected optically. Whether or not screw dislocation annihilation by cross-slip 1 6 7 as suggested by-Hirsch and Lally can account for the apparent absence of screws in AuZn is not known since sections inclined to the Burgers vector were not studied. Because of the ease with which cross-slip occurs in AuZn and since the f o i l s examined were a l l parallel to the Burgers vector of the primary dislocations, i t is highly probable that screw dis-locations escape by cross-slip during the thinning process as suggested by 1 4 4 Seeger. The foregoing discussion presupposes that the dipoles are formed in the bulk of the crystal during deformation. Evidence for this 1 7 = ; has been observed in Cu crystals- where i t was shown that dislocation dipoles. are present in f o i l s prepared from specimens subjected to neutron irradiation pinning while s t i l l under stress. Clearly, similar irradiation under load of AuZn crystals is necessary before i t can be ascertained whether or not the dipole structure is truly representative of the bulk arrangements, or whether i t forms, in larger part, due to a relaxation of stresses within the f o i l s during thinning. 131 2.5-4 Discussion 2.5.4.1 Yield Stress Variation with Orientation It was shown that the yield stress resolved on the macroscopic sl i p system {hko} (001) is apparently independent of orientation (section 2.5-1-1); while the sli p plane is orientation sensitive (section 2.4.5-3)-However, the sl i p plane is not always the most highly stressed plane in the (001) zone suggesting that the yield stress on non-crystallographic planes should be a continuous function of their position in the zone. To resolve this apparent paradox, an attempt w i l l be made to predict the relative variation of yield stress with orientation and then to note to what extent the apparent yield stress "invariance" with orientation is consistent with the predictions. This treatment is based on the Taylor 4 4 analysis used recently to analyze non-crystallographic sl i p in Fe-3% 4R , rp 4 8 ana ia single crystals. The resolved shear stress necessary for the onset of plastic flow on the potential macroscopic sli p plane is given by the expression (Appendix .3) : T ( " y 0 Y = <r s i n £ 0 cos cos ("X-V) (26) where 0~ is the tensile stress at yield ("l|/", "y. and ^  are defined (in section 2.4.3). Assuming that tj"("l/)* is independent of % , as suggested by Figure 34, then on differentiating (26) with respect t o * y and rearrang-ing, i t is found that: . d = tan CX-T|/) (27) T m n Y which gives the variation of the shear resistance with the angle . Integrating (27) between the limits 0 and gives the espression: In n o * t a n ( y - y ) < a y 132 ( 2 8 ) where is the resistance to shear on the ( 1 1 0 ) plane. From the experimentally determined (p£ )j relationship, Figure 2 5 , equation ( 2 8 ) was solved for T ( V ) * hy employing- Simpson's method for graphical integration. H O The resulting curve is shown in Figure 59 and is compared with the experimentally determined values of yield stress versus orientation from Figure 3 4 . The experimental values are given relative to the yield stress for ( 1 1 0 ) s l i p , taken as 3 8 5 O psi. 1 1 0 O 0 0 O Experimental Data o n V ( X ) data 1 05 1 00 i io 0 9 5 9 0 - O 0 ° O — 8 0 8 5 8 0 1 1 1 1 1 0 I I I ! 5 1 0 •15 2 0 2 5 3 0 3 5 4 0 4 5 Figure 5 9 • Showing the experimental compared with.the predicted values of c r i t i c a l resolved yield stress ratio versus ' W . Since the predicted orientation dependence of the relative yield stress varies by less than 5 $ over the orientation range 0 4 "Vi/^.45 0 i t is d i f f i c u l t to ascertain whether or not the somewhat scattered experi-.133 mental results display a similar trend. The apparent "invariance" in yield stress with orientation is therefore not in disagreement with the predictions. Clearly, before definite statements can be made concerning the yield stress dependence on orientation, experiments giving highly reproducible results must be performed. .If i t is assumed that the fundamental glide system is (110)[001] as suggested earlier in the discussion of non-crystallographic slip (section 2.4 .6 . 1 ) , then the present results imply that as X increases the c r i t i c a l resolved shear stress for (110)[001] sli p decreases. For constant X and varying ^ , the c r i t i c a l resolved shear stress on (110)[001] is constant. This may be explained by considering again the concept of dissociated screw dislocations contained on the (100) planes (section 2 .4 .6 .1) . As "X-increases, so does the ratio of the shear stress resolved on (100)[001] to that on (110)[001] implying an increasing tendency for dislocation recombina-tion relative to that for bowing-out in the sessile to g l i s s i l e transforma-tion process (Figure 31)- Consequently, i f recombination is the mechanism controlling the yield stress, then the c r i t i c a l resolved shear stress for (110)[001] s l i p should decrease with increasing X > a s observed. Since t ( 100)[001]/T( 110)[001] is independent of £ , then the tendency for recombination relative to the bowing-out process .is unchanged and the yield stress resolved on (110)[001] should be constant, again in agreement with experiment. 2.5.4.2. Work Hardening It was shown that the form of the work hardening curves of ^J'AuZn single crystals changes markedly over the temperature range 77°K to 475°K from para-linear at low temperatures (below ~-.15 T ) to multi-134 stage at intermediate temperatures ( ^ . 2 0 to .35 T m) a n (^ then to parabolic followed by work-softening at higher temperatures (above'*-.4 T ) . In this respect AuZn is entirely different from fee and hep crystals which are known to display multi-stage work-hardening curves at temperatures as low l as 4°K. Recently deformation studies on crystals of bcc metals Nb and Ta have shown that the form of the work-hardening curves in these materials i s also highly temperature sensitive, varying in much the same 9 fashion as AuZn. The temperature range of three-stage hardening in Ta corresponds to T m ^ 0.10 to 0 .18 while in Nb1 to ^0.10 to 0 . 2 5 , only slightly lower than the multi-stage hardening range in AuZn. The apparent likeness in the deformation behaviour of the ordered bcc compound and ordinary bcc metals suggests that similar hardening mechanisms may be controlling the flow stress in both instances. The occurrence of cross-slip immediately upon the onset of plastic flow and the formation of dislocation dipole clusters on the primary glide planes during easy glide are features common to both AuZn and bcc metals, signifying a fundamental similarity in dislocation behaviour during deformation. Since dipole clusters have been observed in fee and hep crystals as well then i t is believed that the fundamental AuZn-bcc metal relationship is linked mainly through the common continual cross-slip of screw dislocations. The motion of screw dislocations is known to be thermally activated. Consequently, the exact path along which screws move w i l l depend on temperature and the extent to which temperature i n -fluences this motion could very well be the c r i t i c a l factor in subsequent work-hardening behaviour. Since observations of three-stage hardening in bcc metals are relatively new, authors are concerned primarily with noting the rather 135 specific conditions of temperature, strain rate, orientation and impurity level under which multiple stage flow occurs. Effort is also being directed at studying the general work-hardening characteristics and attempts are being made to correlate dislocation structure observed in thin-foils with stage I and stage II hardening rates J > B : i l e c " l e l However, detailed models to account for stage I and stage T I hardening such as those presented for the classically studied fee and hep metals have not been proposed. Based predominantly on electron metallography observations in thin f o i l s , the general concensus of opinion seems to be that work-hardening in bcc metals can be explained in a similar way to fee metals since analogous dislocation clusters and subsequent c e l l formation are common structural features of the work-hardened state. However because 0 U ) for instance, i, 2 * 7 * 8 9 , 1 0 is approximately half that for fee metals (Nb ~- /»/600 , .Ta ~/V/600 1 6 2 Fe /* /k-00 to /f /900 compared with 0 1 X ~ft /200 t o / * /300) and because i t is quite strongly temperature dependent, .it is probable that some important differences exist in hardening mechanisms for the two structures. Certainly, any model for stage TI hardening in bcc metals must diff e r from fee models by including temperature sensitive parameters. Flow curves from crystals of the highly ordered bcc compounds 4 0 2 5 * 3 7 AgMg and NiAl show essentially para-linear type hardening, although 3 7 crystals of NiAl compressed at 25°C in <lll>and <112> orientations tend to deform in a two-stage manner. Work-hardening was not discussed. Some discussion, however, has arisen on the work-hardening mechanism in poly-3 8 1 2 5 crystalline bcc ordered AgMg and FeCo which undergo (111) s l i p . Based on the creation of anti-phase boundaries resulting from the motion of superlattice partial dislocations 1 a <111^ these mechanisms predict 2 an unusually high hardening rate, since as well, as interacting with each other, superlattice dislocations, on intersecting, disorder the lattice 136 125*127 which alone is enough to increase the flow stress substantially. This mechanism probably accounts for the exceptionally high work-hardening rate observed in.AuZn crystals oriented near the [OOl] corner which deform on the { 2 1 l ) <111> system ( 0 "~ /f/60 compared with On ^-/f /5OO in orien-tations near the middle of the triangle.) It may also account for the higher linear hardening rate in specimens tested at 77°K (0^ ^/f/200) which are believed to undergo a minor amount of < 1 1 1 ) s l i p . The disordering mechanism, however, is not applicable to AuZn tested under conditions giving rise to a ^ O O l ^ s l i p since this vector does not disorder the l a t t i c e . The mechanisms of work-hardening in ordered bcc crystals undergoing < 0 0 1 ) slip.are in a l l likelihood, as complex, i f not more so because of the two atom types, as in bcc metals and close-packed structures. To avoid speculation, a model w i l l not be proposed. It w i l l simply be shown where the present observations f i t the framework of existing hardening theories. As shown stage I deformation is characterized by hardening rates in the order of ///5OOO to fl /lOOO which are not greatly different from those in easy glide of fee crystals 6 ( ^ 10 A . ) . Slip occurs on a single non-crystallographic system giving rise to 'dislocation dipoles which form walls perpendicular to the primary ( 1 1 0 ) glide plane. With increasing deformation the dislocation density within the walls increases and the average wall spacing tends to decrease slightly. These 144*164*165 observations are very similar to those reported in fee metals 7*8*160 and bcc Nb suggesting that similar mechanisms may be responsible for the observed hardening. Two principal theories have been proposed to explain work-1.66 hardening during easy glide in close packed crystals,that of Seeger et a l 167 and of Hirsch and'Lally. The essential difference between the theories 137 is that of dislocation distribution. Based on slip.line studies, Seeger et a l consider that hardening arises chiefly from the long range inter-actions of the stress fields of individual dislocations randomly arranged in the l a t t i c e , while Hirsch and -Lally rely on electron metallography observations and take the opposite view that stress fields associated with dislocation clusters present the dominant barrier to dislocation motion. If the clustered arrays observed in (jj'AuZn can be taken as truly represen-tative of the work-hardened structure, then i t would appear that the latter description is consistent with the observations. .The nature of stage TI hardening is perhaps one of the most disputed topics in deformation theory today. Several mechanisms have been proposed in which the flow stress is controlled either by long range 1 4 4 1 1 6 8 stresses from dislocations piled up at insurmountable obstacles, by 1 6 H interactions with forest dislocations, by sessile jogs on gliding 1 7 0 1 7 1 . , . ., , . dislocations, by bowing-out of dislocation loops or by dislocations piling-up at long continuous barriers whose effectiveness varies with 1 7 2 distance from them. • The view is taken here that any of these models could probably account for the flow stress in AuZn, since essential to them a l l are elastic interactions between primary and secondary disloca-tions which are manifested in.AuZn through sl i p on secondary systems detected at the end of stage I. .The detailed differences between disloca-tion interactions probably accounts for the quite low values of Q n in AuZn compared with fee metals'( /i /5OO to // /l200 compared with /y /200 to /f /300). Essential to a deeper understanding of hardening mechanisms in. AuZn is a knowledge of the Burgers vectors of dislocations comprising the clusters. "Since unequivocal Burgers vector analyses were not performed during these investigations possible dislocation interactions w i l l not be speculated upon. 138 Stage III hardening also remains to be explained. The decrease in work-hardening rate at the onset of stage III in fee metals is explained by the occurrence of thermally.activated cross-slip of screw dislocations. Although thermal activation does play a role as evidenced by T ^ i i decreasing with increasing temperature, i t is not known whether the cross-slip explanation applies to AuZn in which cross-slip occurs continually throughout deformation. It is possible, as suggested by 1 Mitchell et a l for Nb, that either large-scale cross-slip or the breakdown of dislocation barriers causes the onset of stage III. Slip line obser-vations did, in fact, show that large-scale cross-slip occurs quite early .in the deformation of AuZn crystals (i.e. near the end of stage I), and becomes profuse during stage III, suggesting that this may be the dynamic recovery mechanism. It was shown that specimens oriented along the [ l O l ] - [ l l l ] boundary of the stereographic triangle work harden in a parabolic or semi-two-stage manner, similar to the stage II-stage III region of flow exhibited by crystals oriented well within the triangle. Subsequent metallographic examination, of deformed [ l O l ] - [ l l l ] crystals revealed that duplex s l i p occurs throughout deformation, thereby accounting for the absence of easy glide. Since the two systems operating are of the same general form as the primary and secondary stage TI systems, viz. (hko ) [ 0 0 l ] and (okl)[lOO], similar dislocation interaction mechanisms are probably responsible for the high hardening rates in both the early stages of parabolic flow and stage II deformation. Whether the hardening results from long-range interactions of a [ 0 0 l ] and a[lOO] dislocations or from obstacles created by reactions of the type a [ 0 0 l ] + a[lOO] a[lOl] against which dislocations pile up, is not known. It might be expected, though, that since the Burgers 139 vectors of the mobile dislocations are mutually perpendicular, the long-range interaction may be weak and hence not an effective hardening mechani 140 2.6 THERMALLY ACTIVATED YIELD 2.6.1 Introduction Plastic deformation is now generally recognized as a dynamic, i.e. time dependent, process that may be thermally activated. Since plastic flow occurs through the movement of dislocations, i t is believed that during the course of passage through the la t t i c e , dislocations periodically contact obstacles that, unless.overcome, cause the dislocation to stop. Obstacles are essentially of two types, short range, extending over distances in the order of ten atomic diameters and long range that possess stress fields of the order ten atomic diameters or greater. Thermal energy is able to assist the applied stress in pushing the disloca-tion past short range obstacles but, because of their extent, cannot aid in getting dislocations past the stronger long range obstacles. Hence the names thermal and athermal.barriers to flow. -At sufficiently high tem-peratures, a l l thermal barriers.become transparent and dislocations pass through the lattice unimpeded once the applied stress has overcome the athermal barriers. At lower temperatures, however, thermal barriers are present and must be overcome by the assistance of stress and thermal energy i f an applied strain rate is to be maintained. Usually several thermal, barriers are encountered, the strongest of which determines the rate at which dislocations can move under the given conditions of temperature, stress and strain rate. Assuming that thermally activated processes occur sequentially, i.e. one after another, and assuming that the same event is rate controlling throughout the l a t t i c e , then i t is generally accepted that the macroscopic shear strain rate t may be expressed by the relationship: 8 7 lkl AG f = Vo e k T (29) where 0 is a parameter which depends on the number and arrangement of the dislocations and their vibrational frequency, A G is the change in Gibbs free energy of the system during an activated event and k and T have their usual significance. The process having the highest "activation energy" AG therefore controls the strain rate. If, on the other hand, thermally activated processes occur independently and the rate controlling step is not the same at a l l points in the lattice then the shear strain . . . 6 5 rate is given as: AG-• f ^ f . ^ Y . e k T (30) 0 1 : th where i refers to the i — kind of mechanism. In this case the application of activation theory for the purposes of identifying the rate controlling mechanism is not possible since one can no longer extrapolate macroscopic thermodynamic measurements to a single activated event occurring in a specific region of the l a t t i c e . Schoeck has recently obtained an expression for AG which has been modified slightly by.Risebrough and stands presently as; AG = AH + T ^/i T*v* d T yv (31) 1 - T i where^-j is the shear modulus on the s l i p plane in the s l i p direction, and AH is the activation enthalpy , given by: AH = -kT^jln t / f 0 \ / \ , and (32) U x r " ) T P * - ) f / T o v * i s the activation volume defined as: v* = _ / ^  AG I - - w / d m t/1To\ (53) lk2 and £"ls the stress component e f f e c t i v e i n a s s i s t i n g the thermal energy i n 8 8 pushing the d i s l o c a t i o n past the l a r g e s t thermal obstacle and i s defined as the difference between the applied stress t a and the long range stress " T ^acting i n the v i c i n i t y of the b a r r i e r . The concept of a c t i v a t i o n volume a r i s e s when one considers the work W done on the system during an a c t i v a t i o n event, by the stress pushing a d i s l o c a t i o n segment of length 1 a distance d : W = C b" 1 d (jlj.) where b i s . the Burgers vector of the d i s l o c a t i o n . A c t i v a t i o n volume, then, i s simply the term b i d . •To account f o r the temperature dependence of y i e l d S e e g e r 8 8 postulated that the applied stress C a consists of two (previously defined) components "£"'*and such that: r a - T*.+ TA ( 3 5 ) .For an i d e a l system i n which the rate c o n t r o l l i n g thermally activated mechanism does not change below a c r i t i c a l temperature T c (above which, thermal obstacles are transparent), .the components of T/ m a y be i l l u s t r a t e d schematically, Figure 60. Since the rate c o n t r o l l i n g mechanism i s no longer thermally activated above T c, y i e l d i s only s l i g h t l y dependent on temperature, decreasing through the small temperature changes i n the shear modulus. .Below T c, however, the decreasing amount of thermal energy a v a i l a b l e f o r a s s i s t i n g the overcoming of short range b a r r i e r s necessitates an increase i n the e f f e c t i v e stress term L and hence a r i s e i n y i e l d stress, varies through changes i n the shear modulus. .To evaluate the a c t i v a t i o n parameters A G , A H and v i t i s necessary to evaluate the p a r t i a l d i f f e r e n t i a l s and / c>ln t / f'o \ at a constant structure,• i ,e. density, arrangement and ' c9 tr * I T 143 i ff CO CO 0) u -p CO cu •H Figure' 60. Temperature Illustrating the athermal and thermal components of the yield stress ' f . (after Seeger 8 8) a the number of mobile dislocations remaining essentially constant. Assuming the yield stress to be governed by the same thermally activated mechanism over the range of temperatures of interest, then one can consider that the structure is relatively constant at y i e l d . 8 6 Hence,on subtracting TT/y from C (Figure 60) a plot of C against temperature is realized and the slope /A't"* | may be taken as / j't"* ] . Likewise plots of ^ versus \ A1 T / T 1 c)T /T 0 at constant temperature can.be adjusted (by subtracting C/f which is not strain rate sensitive, assuming a constant mechanism) to give X against o and hence allow / e)ln / " j ^ n |_ to be evaluated. Instead of \ d x r * /T studying strain rate effects on yield stress i t is common practice to perform differential strain rate change tests on a single specimen during flow at a fixed temperature then to extrapolate, to zero strain to evaluate the second partial. lkk .The object of the experiments reported here was to measure the activation parameters in AuZn single crystals then to suggest a possible rate controlling mechanism responsible for the temperature dependence of yield (section 2.5.T.I). .The discussion w i l l be limited to stoichiometric crystals oriented near the middle of the stereographic triangle. 2.6.2 Activation Volume and Effective Stress Flow stress differences accompanying instantaneous changes of strain rate may be taken as variations in effective stress At*with i f i t is- assumed that the structure remains constant during the instant of change. Differential tests corresponding to strain rate changes from TO 4 to 10~2 per sec were carried out by varying the cross-head speed on the Instron from 0.002 to 0.20 inch per.minute, and vice versa,.using a push-button speed selector. Experiments were performed at temperatures ranging from 77°K to 213°K, i.e. over the temperature interval corresponding to temperature sensitive yield (Figure -33)- ..Typical flow curve variations accompanying a strain rate change are shown schematically in Figure 6 l . Tn principle, both stress increments and decrements may be taken as A C However, because of less uncertainty in measurement, stress increments were taken in this work, by subtracting the flow stress at the lower strain rate from the flow stress at the higher rate. 2 was taken at 9 0 the f i r s t deviation from linearity, following Basinski and Christian. 145 1 A T A l l T. Low Strain i d High Strain #. T4 145°K< T< 215°K T A T i Figure 6 l . Schematic representation of changes in the flow curve accompanying strain rate change tests. From the relationship presented in 2.6.1, activation volumes were calculated from: (36) y * = k T / m V * a / i \ { A C and are plotted in Figure 62 (in units of b 3) as a function of shear strain; b is the Burgers vector of the mobile [001] dislocations and was , - 8 1 2 taken as the lat t i c e parameter:3.14 x 10 cm. At 77°K the activation volume at yield, v is ~-30b 3 and is constant within 5b 3 throughout deformation. As the temperature increases, v n also increases and at the same time, becomes somewhat more strain (stress) sensitive, decreasing with increasing strain. At 215 K for instance, v D decreases from 525b at Figure 62. Activation volume against shear strain at temperatures between 77°K and 213°K. lA7 yield to 130b3 at fracture. In Figure 62 i t can be seen that activation volumes determined just past yield at a l l temperatures are slightly lower than those obtained by extrapolating later values to zero strain. In a l l likelihood, the i n i t i a l l y lower volumes are associated with uncertainties in determining A T * since the departure from linearity with a strain rate increase just past yield is not nearly as marked as i t is at higher strains (Figure 6 l ). For the purposes.of comparison and for calculating activation energies in section 2.6.3 activation volumes v D extrapolated to zero strain w i l l be employed. Activation volumes are commonly plotted against the effective To separate ( from the applied stress, the athermally attributed linear portion of the yield stress-temperature curve for stoichiometric crystals (Figure 33 ) was extrapolated to 0°K and at each temperature of interest below the c r i t i c a l temperature T c '—220°K the corresponding, athermal stress component'was subtracted from the yield. stress. The subsequent --T results are plotted in Figure 63 and extra-- 7 - * polated to 0 K for future reference. • Activation volumes, then, versus <• are shown in Figure 6k . Also shown are the variations for polycrystalline AuZn and the bcc metals (V, ,Nb, . Ta, Cr,Mo;W,, Fe) over the same range of t 9 1 taken from.the compiled data of Conrad and Hayes. It is apparent that single and polycrystalline AuZn display exactly the same v D - C behaviour that f a l l s within.the range for the bcc transition metals, suggesting that similar mechanisms may be rate controlling. 148 Figure 6$. Showing the variation in effective stress with temperature. v 1^ 9 Figure 6k . Showing the variation of activation volume with effective stress for Q 1 AuZn and bcc metals. 150 2.6.3 Activation Energy The expression given for the activation free energy A G in equation(31)when rewritten to include the terms for A H and v , gives the relationship: A G = -kT2 (^inT/To) I"/ a t ' L . . - 4 * T * l (57) \ <s>r» H I a T /V^o /» -I 1 - T ^ It is assumed that the temperature variation of the elastic stiffness constants in AuZn is small implying that changes in shear modulus with temperature are also small. This assumption is justified on the basis that 7^, , which is related to the shear modulus variation with temperature, increases only approximately 10 percent from 300°K to 0°K. Hence, the term d/r T**in equation(37)will be small and may be neglected with respect to the larger term giving the effective stress variation with temperature. A G may then be approximated as A H ; i.e.: A G ~ - V*T / a r * \ ^ , . = A H ( 5 8 ) I a T / / y o .As a measure of activation energy, therefore, activation enthalpies were calculated rather than activation free energies and are reported in Table. 16 The term / <5£** \ -fr/^A at different temperatures was evaluated from the slope of the T" - T curve, Figure 63. .To determine the height of the rate controlling thermal 9 1 - 9 4 barrier, A H ^ i t is necessary to measure A H when C is zero. Figure 65 shows A H versus C and, when extrapolated to zero stress, gives 9 1 - 9 4 3 A H 0 ~ 0.9' ev. For bcc metals, A H 0 is approximately 0.1^ b . On 3 this basis,,the controlling energy barrier in AuZn crystals is •^-0.15/^b (where b = a <00T> = 3.14 x 10 8 cm:and f, = C 4 4 = 3.0 x 1 0 1 1 dynes/cm2) 151 and, therefore, of the same order of magnitude as the pure bcc metals, again suggesting that similar mechanisms may be rate controlling. TABLE 16 Activation Parameters AH and v G at Temperatures between 77°K and 175°K for Stoichiometric Crystals T (°K) vo/b (from -Fig.69) r*(psi) (from Fig.65) AH (ev.) (from •Equation(38) 77 30 4100 • 23 90 33 3200 .29 100 50 2600 -35 125 76 1600 .46 132 90 1300 • 49 143 125 1000 .66 150 137 800 • 73 172 170 400 .72 175 200 350 • 79 In polycrystalline AuZn, i t was found that A H o ' v ' 0 . 4 3 ev., about half of the single crystal value. Since activation volumes for both systems are in good agreement, .part of the discrepancy may l i e in the certainty of measuring / A'C'* \ . In both instances the partial was 1 AT / T measured from the slope of the corresponding yield stress-temperature curve. In polycrystals only five temperatures were used to establish the shape of the yield stress curve over the range of interest below fw250°K, whereas in single crystals yield stresses were measured more frequently giving a total of 13 points over the same temperature range, interjecting a greater degree of certainty in the measured / AT* \ terms. 152 .Figure 65. Showing the variation of activation enthalpy with effective stress. 153 2.6.4 Discussion Since the present activation parameters are in accord with the results for bcc metals suggesting that similar mechanisms may be responsible for yield of AuZn, the commonly discussed bcc rate controlling mechanisms w i l l be reviewed. 2.6.4.1. Impurity Obstacles In view of the rather high i n t e r s t i t i a l content (Oxygen plus Nitrogen ^-.300 ppm Appendix 1) , i t is possible that dislocation-interstitial . interactions could be controlling the yield stress. Assuming that the dislocation is bent to make nearest neighbour contacts with the impurity atom, then the distance 1 between obstacles is related to the impurity ST 2 concentration C through the Burgers vector , b ~ 'C . For an impurity 1* content of 300 ppm the shear stress necessary to push the dislocation between the obstacles, given by V =/* b, is approximately/*/50, which is I** equivalent to <~-90,000 psi. On extrapolating the tT - T curve to 0°K (Figure 63) i t was found that C . 0 ^ 1 5 , 0 0 0 psi which is considerably less than the stress necessary to push dislocation through impurity obstacles spaced 50bin.theslip plane. Assuming that the dislocation makes three-1 9 8 3 dimensional nearest-neighbour contacts, then b ""C and the corresponding 1* stress necessary to push the dislocations through the obstacle f i e l d is /*/l5 and is greater than the theoretical shear strength of the l a t t i c e . In the light of an impurity model, i t would also be d i f f i c u l t to explain the increase in activation volume with increasing temperature. 2.6.4.2 • Peierls Nabarro (PN) Mechanism It has been the conclusion of a great many workers that the 91*94;95*96*99 rate controlling mechanism in bcc metals and ordered alloy 154 l O O ; 1 0 1 AgMg is the overcoming of the Peierls-Nabarro barrier to flow, i.e. the inherent lattice energy barrier that a dislocation, lying along close packed rows of atoms, overcomes as i t moves from one equilibrium "valley" to the next. In surmounting the Peierls " h i l l s " the configuration of atoms at the dislocation core is altered which therefore increases the dislocation energy. Hence the energy of a straight dislocation is a function of i t s displacement from the bottom of a "valley" and has the periodicity of the spacing between the parallel rows of atoms. The c r i t i c a l step in the PN mechanism is the thermally a c t i -vated nucleation of a pair of kinks that w i l l subsequently move apart bringing the dislocation into i t s next equilibrium position, Figure 66. Figure 66. Schematic il l u s t r a t i o n of the PN mechanism. 155 The dislocation line i n i t i a l l y l i e s along A 0BoC 0 in an equilibrium position in the Peierls "valley". Under the action of an effective stress (. the dislocation moves from i t s equilibrium position to a new position ABC part way up the " h i l l " . Thermal fluctuations occasionally supply enough energy to assist C in pushing a dislocation segment AB C over the ' h i l l " thereby creating a double kink, AB* and B'C, which on moving apart, brings the dislocation into a new position A B C ready for the next activated event. On the basis of a line energy model of a dislocation, Dorn 1 0 3 and Rajnak have recently derived an expression for the energy U to nucleate a kink. Assuming that the Peierls " h i l l " is sinusoidal: - l U = 2fn a I 2 (\ a b \ 2 ( 5 9 ) rr \ r r 0 / where p is the dislocation line energy per unit length in the low energy position A0BoCo, a is the spacing between parallel rows of closely spaced atoms in the s l i p plane, T" the Peierls stress (the stress necessary, in the absence of thermal energy, to push the dislocation over the barrier) 1 0 2 . and b the Burgers vector. For a quasi-parabolic " h i l l " , Guyot and Dorn have derived the following expression for U : 3 / 1 U k = TT 2 To a / 2 Tp a b \ 2 (kO ) 8 1 ^ j The sinusoidal and quasi-parabolic profiles are shown in Figure 67 . 156 Figure 67- Schematic il l u s t r a t i o n of the sinusoidal and quasi-parabolic Peierls " h i l l " profiles. To evaluate the kink energy i n : AuZn, values for \~Q and T P are necessary. Assuming that ["~0- — 5 . 0 x 10 ergs/cm (Table 4 ; (llOJCOOl) system) and that 7j" at 0°K may be approximated as 15,000 psi (obtained by 1? extrapolating L - T to 0°K) the kink energy is then given as ~- 0.24 ev. When C = 0 thermal fluctuations must supply enough energy to nucleate a double kink, i.e.- .2 ~~ 0.48 ev. If i t is assumed that the PN mechanism is rate controlling over the whole range of temperature sensitivity of yield, i.e. below 220°K, then AHo (i.e. at 220°K) should be equivalent to ~ 0.48 ev. Clearly the value 0.9 ev. obtained earlier is considerably larger than the energy estimated to nucleate a double kink. It appears therefore, that i f the PN mechanism acts as a rate controlling process, i t operates over;a lower temperature range. It is assumed that the PN mechanism is rate controlling below a c r i t i c a l temperature T Cp where T Cp may be estimate'd by noting the temperature at which the experimentally determined activation enthalpy is approximately 0.48 ev. From the data .in Table 16, A H i s plotted against T, Figure 68. Assuming that a straight line through the origin joins the points then T n T S —' 125°K. The p l a u s i b i l i t y of a -PN mechanism controlling yield at tempera-157 Figure 68. Showing the variation in activation enthalpy with temperature. 158 tures below '^"125°K becomes apparent when one examines the activation 1 0 2 volumes. Guyot and Dorn have, given an expression for the width w of the c r i t i c a l sized loop forming the embryonic kinks: I_ w • = jrf 2a F0 \ 2 (la ) S U T P ) where a l l parameters have the same meaning as defined earlier. The activa-tion volume for the PN process is then expressed as: (42) v = w a b * , 3 which upon substitution for [<•, T . a and b gives v 0 ~ 40b , similar to the activation volume in •AgMg 1 0 0' 1 0 1 below 250°K. On comparing the calculated with the experimentally determined values plotted against temperature in Figure 69, i t is apparent that below ^ 120°K, v* ~ 30-l+0b3 and is not strongly temperature sensitive, agreeing well with the dictates • of a PN process. Above 120 K, v 0 increases linearly with temperature suggesting that a different mechanism becomes rate controlling. It is noted that the c r i t i c a l temperature for a PN process estimated from the activation volume measurements is in good agreement with that deduced from the activation enthalpy calculations. That PN may be rate controlling below 'N-'125°K is further evidenced by the fact that activation volume remains approximately constant during deformation (Figure 62). Analysis of 12 polycrystalline data was consistent with the dictates of a PN mechanism controlling yield below a c r i t i c a l temperature of approximately 150°K, a c r i t i c a l temperature slightly higher than the present temperature of /^125°K but in quite good agreement in view of the uncertainties in determining T Cp. 2.6.4.3 Cross Slip In view of the extensive cross s l i p occurring continually during the deformation of AuZn, the possibility that i t may control the 159 ! . I I L_ I I 0 50 100 150 200 250 Temperature T°K Figure 69. Shewing the variation in activation volume with temperature. 160 deformation rate w i l l be considered. The analysis w i l l be a qualitative one, based on the experimentally determined activation volumes. Considering activation volume as the product of the activated length 1, the activated distance d* and the Burgers vector b and employing expressions given by 6 5 * * Dorn for the parameters d- and 1- for the cross s l i p process,, one arrives at an expression relating activation volume to effective stress: v* = -• A .(43) ( r * ) 2 where A is a constant incorporating the line tension of the bowing-out dislocation and i t s Burgers vector. Since T<\1 > then, qualitatively, ( r * ) 2 activation volume should increase with decreasing ^ (i.e. increasing temperature), as observed. On plotting In v Q against In C in Figure 70, two regions of behaviour are apparent, from 77°K to ~ 150°K in which v 0 V l and from ^150°K to 213°K over which v0*°<-l . It appears, therefore, that below 150°K the activation volume variation follows quite closely the dictates of the cross s l i p mechanism suggesting •that cross s l i p may be an alternative to the PN mechanism controlling yield at low temperatures. 161 Figure 70. Showing the functional dependence of activation volume on effective stress. 162 3. SUMMARY AND CONCLUSIONS 1. The primary sl i p surface is a non-crystallographic plane (hko) in the zone of the s l i p direction [OOl]. With increasing temperature, decreasing strain rate and increasing distance of the tensile axis from the [ 0 0 l ] - [ l l l ] boundary, the sl i p plane varies from (110) to (100). This phenomenon was interpreted in terms of continual cross-slip of screw dislocations on orthogonal planes (110) and (110). 2. Slip line observations indicate that edge dislocations can sl i p over large distances on {lio} planes while screw dislocations sl i p over much smaller distances. These observations are consistent with the continual cross-slip mechanism proposed to explain non-crystallographic s l i p . 3. •Multi-stage work-hardening and associated large d u c t i l i t y as high as 300$ shear strain has been observed for Au-rich (51-0 at.$ Au), stoichiometric and Zn-rich (51.0 at.fo Zn) crystals in the approximate temperature range 0.2 2 T / T 2 0.35 and for a l l orientations within m the standard stereographic triangle except near the [001] corner. k. Following a short transition region (stage 0 ) , stage I easy glide begins, characterized by relatively low hardening rates (/</l000 to Sf/'j000) . Transmission electron microscopy studies revealed walls of edge dislocations perpendicular to the primary glide plane which are believed to be the main cause of stage T hardening. 5. The principal effects of deviations from stoichiometry are to increase the flow stress and lengthen the extent of stage I. 6. The end of easy glide is coincident with sl i p on secondary systems of 163 the form (okl)[lOO] in localized regions of the crystal near deforma-tion bands. The contribution of the secondary system to the overall deformation is negligible. 7. Stage II deformation commences after a f a i r l y slow transition and the development of the secondary system probably accounts for the high work-hardening rate. 0X1 is nearly independent of composition, but is some-what sensitive to the other experimental variables,increasing with decreasing temperatureIncreasing strain rate and decreasing distance of the specimen axis from the [ 0 0 l ] - [ l l l ] boundary. © 1 1 A / 5 0 0 , about half that for fee metals. 8. The onset of stage III is coincident with sl i p line cluster formation a l l along the gauge section which is believed to result from large-scale cross-slip of screw dislocations. 9. At temperatures below ^-150°K, para-linear hardening is observed. The onset of minor amounts of ( O i l ) [ i l l ] s l i p at 77°K (as detected from Taylor analysis of asterism in a back-reflection Laue photograph) .is believed to be responsible, for the high linear hardening rate. 10. .Increasing susceptibility.to crack formation along (001) planes is believed to be responsible for decreasing ductility.below — 250°K. 11. Above ~350°K, parabolic type hardening occurs followed by work softening and total strain to fracture decreases. This behaviour was explained in terms of increasing propensity for large-scale cross-slip. 12. Ductility increases with temperature above ^  450°K, coincident with the operation, to about the same extent, of two sl i p systems of the general forms (hko)[00l] and (okl ) [ l 0 0 ] . 164 In orientations near the [001] corner of the stereographic triangle, ( i l 2 ) [ l l l ] s l i p occurs. Hardening rates are very high ( ///60) and d u c t i l i t y i s low (^20$ shear strain), presumably due to the disorder-ing of the lattice resulting from the motion of superdislocation partials a [ i l l ] . The yield stress is about ten times greater than 2 for (hko)[00l] s l i p . Serrated flow was detected at temperatures between 293°K and 400°K for non-stoichiometric crystals and is attributed to a dislocation-solute atom interaction. Deformation bands roughly parallel to (001) planes were detected in a l l orientations at temperatures above 140°K. It is suggested that the bands are a manifestation of the formation.of edge dislocation walls of opposite sign. It was found that the yield stress is approximately orientation insensitive and was shown to be consistent with predictions based on the observed variation of s l i p plane with orientation. These results were interpreted in terms of decreasing c r i t i c a l resolved shear stress for glide on (110)[001] with increasing distance from the [ 0 0 l ] - [ l l l ] boundary, arising from the increasing ease of recombination of dissociated screw dislocations lying on (100) planes. The temperature sensitivity of yield in stoichiometric crystals below *-220°K was subjected to thermal activation analysis. The dictates of either the Peierls-Nabarro mechanism or cross-slip of screw disloca-tions are consistent with the experimentally determined activation volumes and activation enthalpies below 150°K. 165 •k. -SUGGESTIONS FOR FUTURE WORK The following questions arise from this work and can be considered as the most natural extensions of i t : 1. -What are the effects, of a wider range of temperature and orientation on non-crystallographic sl i p and hence on the shape of the A// (~X.) •and "\J/(T) curves? At constant strain rate,, is the temperature at which the s l i p surface coincides, with the most highly stressed plane in the <001^ zone independent of ~)C ? 2. Assuming fundamental (110)[OOl] s l i p , how w i l l the c r i t i c a l resolved shear stress-orientation relationship T* ("ty/) vary with temperature and strain rate? 3. What are the dislocation reactions leading to stage II work-hardening as revealed by detailed Burgers vector analysis of dislocation clusters using high resolution dark f i e l d techniques. Also, does ( i l l ) s l i p contribute to general deformation for orientations other than those near (OOl) and at temperatures above 77°K? k. For near (001). orientations, is the s l i p plane in the zone of the sli p direction ( l l l ^ temperature sensitive? How does the c r i t i c a l resolved shear stress for ( i l l ) s l i p vary with temperature and strain rate? 5. Is serrated yielding a phenomenon inherent in.the deformation of non-stoichiometric (^ 'AuZn at intermediate temperatures and strain rates or is i t a manifestation of dislocation-impurity atom interactions? 166 APPENDIX I Crystal Homogeneity A.1.1 Chemical Analysis and Composition Gradients Specimens of approximately 300 mg. were accurately weighed then dissolved in .15 cc of warmed aqua regia. The solution was boiled to remove N02 then diluted to 200 cc with d i s t i l l e d water and warmed again. Hydrazine sulphate was carefully added to precipitate gold. The solution was l e f t over night then subsequently -filtered and the residue ashed at 600°C after drying at 150°C. .The gold was carefully weighed and the zinc was determined by difference. Crystals of three different nominal compositions, 4-9.0, 50.0 and 51.0 at $ Au were grown and analyzed chemically forAu at five points along their length. The results are given in Table A1..1 and are plotted in Figure A l . l . ..All analyses were in duplicate and are accurate to within ~0.04 at $. To a f i r s t approximation, the average composition over the f i r s t two-thirds of the crystal to solidify deviates only slightly from the nominal composition. In a l l crystals a region of high zinc content is noted over approximately the last third to solidify. In view of the large density difference between gold and zinc ( ^  Au = ^9«3 g/oc; ^.£n = T-l g/cc) and the high vapour pressure of zinc at melt temperatures (775°C), i t is not surprising that the last end to freeze should be zinc rich. Attempts to reduce the composition gradients by homogenizing the as-grown crystals in reduced pressure at a temperature sufficiently high for rapid di f f u s i v i t y of zinc vapour met with no success. TABLE A l . l Chemical Analysis of As-Grown Crystals Nominal Distance along Comp. the crystal Composition (at. % Au) • (inch) (Vft.i Au) 51.0 ' 0-5 75.90 75.89 ,3.6 . 75-88 75.85 9-8 75.85 75.84 • ik. 2 75-73 75.70 18.0 75.38 75.45 50.0 0.8 75-00 75.15 2.4. 75-16 75-12 7.0 75.13 74.99 11.6 75-06 74.97 14.2 74.42 74.33 49.. 0 0.3 74.63 74.61 3-5 74.45 74.62 9-8 74.38 74.32 14.4 74.32 74.29 17.5 73.90 73-88 I I 1 — — ' 5 10 15 20 Distance From First End to'Solidify (inch) Figure Al. Showing composition gradients in as-grown l^'AuZn single crystals. |_1 o> CD 169 A.1.2 Interstitials and Trace Elements Results are given in-Table A.1.2 for i n t e r s t i t i a l and trace element content in a typical i^'AuZn tensile specimen, analyzed by Ledoux Analysts, Teaneck, New Jersey-.. .TABLE A.1.2 Impurity Analysis in Typical ft'AuZn Tensile Specimen I n t e r s t i t i a l Content Element' (ppm) Carbon • 8 Oxygen -• 195 Nitrogen . 127 Hydrogen l . Trace Elements Not Detected 170 APPENDIX 2 Evaluation of Machining Damage Evidence that the cold-worked surface layers of the tensile specimens could be completely removed by polishing and annealing was obtained from back-reflection Laue diffraction patterns and room temperature tensile tests on stoichiometric specimens. The surface layers were removed in fresh 5$ KCN solution. The variation in asterism with amount polished from the surface, Ad, is shown in Figures A2.1,. A2.2 and A2 .5. Tensile strength versus Ad for two series of experiments is shown in Figure A2.4 and the effect of annealing time and temperature on the yield strength of two different crystals is reported in Table A..2.1. -For these latter tests, each specimen was strained just past yield then re-annealed and tested again. It is apparent that removing a surface layer 0.005 inch and annealing at 300°C for one hour renders the tensile specimens free from machining damage. TABLE A.2.1 Effect of annealing temperature and time on the strength of two tensile specimens relative to the unannealed condition Specimen Ad Annealing Annealing .Tensile Load (inch) Temp.(°C) Time (hr.) at Yield (lb.) E-l - 7 0.0100 - - 129 11 '300 1 h. 103 11 •300 3 h. 104 D-1-5 0.0094 - - 42 it 300 1 h. 35 11 300 3 1/2 35 11 425 1 34 Ad - 0 A2.1 Ad =0.0033 inch A2.2 Ad - O.OO98 inch A2.3 Figures A2.1, A2.2, A2.5. Showing the variation in degree of asterism with reduction of diameter of machined tensile specimens. H - ^ 1 172 Figure k2.k. Showing tensile strength of machined crystal versus amount removed from the specimen diameter. 173 APPENDIX 3 Equations: Resolved Shear Stress T~ (V) and Resolved Shear Strain The equations. used in computing resolved shear stress (^ (Tj/ ) and resolved shear strain Yflj/') on the operative sl i p plane defined by the parameter ~\lf (section 2.k) were obtained in the following manner. From 4 9 Schmid and Boas , the resolved shear stress and resolved shear strain acting on the operative sl i p system are given as: f = P_ Sin (90 - 0 O) Ao t [(1) - Sin £ 1 2 and s.2 2 ^ - i 1 Cos Po (2) - Sin )= c-Sin(9O-0o! where P is the instantaneous tensile load on the crystal., A 0 the i n i t i a l cross sectional area of the specimen, 1 G and 1 the i n i t i a l and instantaneous gauge lengths respectively, ^ 0 the i n i t i a l angle between the sl i p direction and the tensile axis and (9O-0o) "the i n i t i a l angle between the s l i p plane and the tensile axis. The angle 0 may be written in terms of the parameters Y > X A N <^ F ( s e c"tion 2.k) characterizing s l i p modes in (^AuZn. With the aid of Figure A3.1 and applying the cosine law from spherical trigonometry, i t is found that: Cos 0 O = Cos ( y - Y ) S i n f o (3) Therefore, in terms of the known parameters ^ Q , ~)C and l]/, ( and If from (1) and (2) may be rewritten in the form: ,2 -C- (l|r ) = P Sin £ o Cos {-% - \f) l c A 0 1 and - Sin" 2 CO - I 1-174 001 100 Figure A ^ . l . 175 APPENDIX k Taylor Rotation Axes '(•Re: Asterism at 77°K, Figure 22) The Taylor rotation axis l i e s in the sl i p plane and is perpendicular to the s l i p direction. It is that axis about which the lattice rotates during s l i p on a given system. The identification of rotation axes from asterism on back-reflection X-ray photographs of deformed crystals is a 1 5 6 * 1 5 7 * 1 5 8 commonly used technique for identifying s l i p systems. Asterism shown in Figure 22 was assumed to arise from.the operation of two s l i p systems operating simultaneously. It is known that asterism. may also arise ' 1 5 9 from deformation bands (for review, see Hirsch) , but this origin was ruled out since bands' were not detected at 77°K. A typical branch from Figure 22 is illustrated schematically in Figure A4.1. X Y Figure Ak.l. Sketch of asterism from Figure 22. V is the vertical axis in the film plane; X-X and Y-Y are associated with branches 1 and 2 respectively and are the projections of the rotation axes Ri and R2 onto the film plane. The primary sl i p system at 77°K was found to be a non-crystallographic plane in the [100] zone lying ^ 6 ° from (Oil); a secondary 176 trace was coincident with the most highly stressed plane of the -£L10} ( i l l ) system, namely (Oil). Representing the primary system as (011)[l00] and assuming that the secondary system is ( O i l ) [ i l l ] , the Taylor rotation axes are then given as [Oil] -and [211]respectively. To check the assumption a l l {llO} and {211} poles are plotted stereographically in Figure Ak.2 relative to the indexed diffraction spots from Figure 22. Also plotted are X-X and Y-Y, the film plane projections of the rotation axes Rx and R2 respectively, along which the poles of the axes must l i e . It is seen that [Oll]lies on the Y-Y projection implying that s l i p on the (011)[l00] system gave rise to asterism branch 2. The pole [211] is seen to l i e along X-X, and thereby consistent with the assumption that one of the branches of asterism in Figure 22 ( i ..e. -branch 1) is the result of (Oil) [ i l l ] s l i p . 177 Figure AU.2. Stereographic projection of a l l { l i o ) and {21l} poles plotted with respect to indexed diffraction spots from Figure 22. The base circle represents the film plane. Rotation axes "R± andR 2 l i e on the great circles X-X and Y-Y respectively. 0 178 APPENDIX 5 Shear Modulus as a Function of Slip System The work-hardening parameters <0, 0X and 0 1 X are often expressed in terms of the unitless quantities 0j__ and Q 1 X where /i is the shear modulus characteristic of a given sl i p system. Expressed in this way, the quantities can be readily compared with similar terms characterizing work-hardening in other systems. During the course of the present investi-gations, similar quantities were obtained only after a general method had •been established for evaluating ^ as a function of slip system ^hklj {uvwj. The method employed w i l l be outlined in it s general and hence most useful form using standard second order tensor notation and the repeated suffix convention. The components of the stress tensor T relative to a right-mn handed set of orthogonal axes x i , x 2, x 3 is given by the expression: ^ n n = c £ ; ^ , n = 1, 2 , .3) (i) mn mnpq pq where C represent the elastic stiffness constants of the material and mnpq €pq is the corresponding strain tensor. Equation (1) defines completely the stress state relative to the X^(i=l , 2 , 3 ) reference frame. In cal-culating shear modulii for a s l i p system defined by the unit vectors n and (given in the frame) where n is parallel to the normal of the {hkl} slip plane and is parallel to the sl i p direction <uvw> , i t is necessary to change the reference frame fromx^ to a new set of orthogonal axes Sc^-The new frame is oriented relative to the old frame in such a manner that x x is parallel to Q_ and x 2 i s parallel to n; .x 3is then given by the vector _g x n to form a right-handed system, Figure A5.1« Relative to the new system of axes the stress tensor 'Ci ^ is given by the second order tensor 179 Figure A^.l. Showing the X reference frame relative to the X frame. i i transformation law: r . 1. a. T im jn ran ( 2) where a. . is the transformation matrix which relates the new frame to the old, In order to expand equation (2) into a term consisting of elastic constants and strain, i t is necessary to express the strain tensor £ in terms of a PO. strain tensor in the new frame €. ^  by employing the reverse transformation method for second order tensors f = a, a. p v p q kp lq * k l (3) where a denotes the transformation matrix relating the old frame to the new. Combining equations (1), (2) and (3) , the stress tensor in the new system is given by the expression: <F. , = a. a a a C (L , n Ik) X J 1m j n kp l q mnpq C k i \ ' The component of which is of special interest in determiningyc- (n, (3 ) i j _ _ ~ corresponds to the stress acting on face x 2 in direction x x; i.e. f~ which 2 1 is given by: /?** = a a a a C £. 2 1 2m m kp l q mnpq k l (5] 180 Equation (5) w i l l be considered again. If i t is assumed that lattice rotations within a body can be neglected as infinitesimal, .then the stress tensor is symmetric regardless of the reference frame, i ,.e. f = T~ and . =?"..• Symmetry then .mn nm i j J 1 effects a reduction in the number:of elastic constants characteristic of a general material from 8 l to J>6. The stress state can then be written in terms of a shortened 'notation: T = C m n € n ' (n=l,2,...6) (6) Vm mn n The relationship between and is seen by comparing the tensor *7" m mn • mn with the vector in six-space ': T -mn T *• i i T K 1 2 r i 2 '-1 r6 f5 Tzi ^ 2 2 ^ 2 3 ^ 1 2 ^ 2 2 ' 2 3 % L 3 2 7-l 3 3 2 3 3 3 r 4 r 3 - r m Equation (5) w i l l now be evaluated for the special case of cubic symmetry. Use w i l l be made of the fact that a l l but 12 of the J6 elastic constants C are zero, and of the 12 non-zero terms, only 3 are mn independent, namely c x l , c 1 2 and c 4 4 . The elastic constant matrix is then given as: mn C i i C 1 2 C 1 3 CO. 0 0 \ C 2 1 !0 .' 0 0 C31 c 3 2 c 3 3 .00 0 0 0 0 0 c 4 4 . o 0 0 0 0 0. c 5 5 0 0 0 0 0 0 c 6 6 / C n C 1 2 C 1 2 0 0 0 C 1 2 C n C 1 2 0 0 0 C 1 2 G i 2 C X 1 0 0 0 0 0 0 C 4 4 0 '0 0 0 0 0 C 4 4 0 0 0 0 0 0 C 4 4 \ From the greatly reduced number of non-zero elastic constants i t can be s£en that C is non-zero only for the suffixes given in Table A5.1. mnpq 181 TABLE A5.I Non-Zero Suffixes C C Suffixes C n O 1 1 1 1 1 1 1 1 c 1 2 0n22 1 1 2 2 C1133 1 1 3 3 c 2 1 C 2 2 1 1 2 2 1 1 c 2 2 c 2 2 2 2 2 2 2 2 023 C 2 2 3 3 2 2 3 3 C31 C3311 y 3 1 1 0 3 2 C3322 3 3 2 2 0 3 3 C3333 3 3 3 3 C 4 4 C 2 3 2 3 2 3 2 3 =03232 3 2 3 2 =02332 2 3 3 2 =03223 3 2 2 3 c 5 5 C 1313 1 3 1 3 = C 3 1 3 1 3 1 3 1 =Cl331 1 3 3 1 =03113 3 1 1 3 ^ 6 6 0 l 2 l 2 1 2 1 2 =0 2 121 2 1 2 1 =01221 1 2 2 1 =02112 2 1 1 2 By substituting values of C and the appropriate suffixes from Table A5.I mnpq into equation (5) an expression for ^ 2 1 i s obtained in terms of the independent elastic constants c i x , c 1 2 and c 4 4 and the direction cosines between the X and x reference frames. Following this procedure i t was i i found that 2 1 is given by the relationship: ~~ P 2 2 2 ~1 Z* = C n ( a n a 2 i ) + ( a 2 2 a 1 2 ) + ( a 1 3 a 2 3 ) J ' 2 1 - L_ - J + 2C 1 2 £ a 1 1 a 1 2 a 2 1 a 2 2 •+ a 1 1 a 2 i a 1 3 a 2 3 + a 1 2 a 1 3 a 2 2 a 2 3 " J C 2 2 2 2 2 ( a 1 3 a 2 2 ) + ( a 1 2 a 2 3 ) .+ ( a 2 1 a 1 2 ) + ( a 1 3 a 2 1 ) + ( a l x a 2 3 ) 2 + ( a 1 1 a 2 2 ) ;+ 2 a 2 2 a 1 3 a 2 3 a 1 2 + 2 a 2 1 a 1 3 a 2 3 a 1 1 :+ 2 a 2 1 a 1 2 a 2 2 a i x ^ ] ^ 2 1 + other strain-terms (7) k / - 2 , . l ^ l ) v k l The shear m o d u l u s ( n , , | _ ) = T.gi i s computed directly from the coeffi-c) 6 2 1 cients of £ 2 1 in equation (7) Consider the shear modulus for the ^.lo) (001/*system. transformation matrix is given as (Figure A5:2) _ x 3 The Figure A5.2. (1, 0 0 0 1 1 \T2 0 -1 1 1 n f V2 (It should be noted that the components of the first row of a^j, namely aij are the components of a unit vector along x1 relative to the xk frame. Likewise a 2 . and a 3 . are the components of unit vectors along x 2 and x 3 J J respectively. In a similar fashion the components of the f i r s t , second and 183 third columns of a.jj, namely a i 2 a n c^ a i 3 ' a r e ^ e c o m P o n e n " t s o r" unit vectors along •x1, x 2 and x 3 respectively, relative to the x^ frame.) On substituting values of a. . from the above matrix into equation (7) i t is seen that: yy \llo) <001> = C 4 4 (8) For the sake of completion, shear modulii were calculated on other possible s l i p systems in cubic structures and are given in Table A5.2. TABLE A3.2 . Shear Modulii for Various Slip Systems in Cubic Structures Cubic Slip System A (n, <3> ) [hko] <001> . 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