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UBC Theses and Dissertations

Microyielding and flow in niobium alloy crystals Wilson, F. Graham 1969

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MICROYIELDING AND FLOW IN NIOBIUM ALLOY CRYSTALS by F.GRAHAM WILSON B.A., University of Cambridge, 1966 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, 1969 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 advanced degree 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 ag ree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a 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 h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no 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 . Depar tment 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 Co lumbia Vancouver 8, Canada i ABSTRACT Oriented single crystals of niobium and dilute alloys with molybdenum and tantalum were deformed in tension between 77°K and 500°K, and the macroflow and s l i p parameters established. At high temperatures the main effect of alloying was to increase the flow stress, considerably more with molybdenum than with tantalum. The observed yield drop and subsequent plastic flow were explained in terms of a s t a b i l i t y theory relating changes in yield and work hardening parameters with temperature and addition of solute. A technique was developed for measuring small plastic strains in the microflow region, and for recording the dynamic transition to macroflow. From studies on pure niobium between 77°K and 295°K, the nature of dislocation motion at small strains was established; microflow was explained in terms of a transition from edge dislocation motion to screw dislocation motion at the macroflow stress. I n t e r s t i t i a l effects were found to be particularly significant during microflow, and are probably important in determining the low temperature flow stress in: even the highest purity bcc metals. A further low temperature contri-bution comes from a directional component of the internal stress f i e l d which depends on the distribution of dislocations rather than on their density. Microflow curves were obtained for niobium alloy crystals,land the interaction of dislocations with substitutional solute atomsi i i established. In contradiction to previous suggestions, substitutional solute was found to restrict the mobility rather than the multiplication of dislocations. The elastic contribution of solute atoms to the internal stress field was confirmed, although a quantitative theory for bcc alloys does not yet exist. Peierls stress considerations alone were found to be incapable of explaining either the temperature sensitivity of flow or the low temperature solution softening; the short range interaction of interstitials with the dislocation core was considered to be more significant. i i i TABLE OF CONTENTS PAGE 1. INTRODUCTION . 1 2. EXPERIMENTAL PROCEDURE 3 2.1 Specimen preparation 3 2.1.1 Materials 3 2.1.2 Melting procedure 3 2.1.3 Impurities 4 2.1.4 Oriented single crystals 5 2.1.5 Specimen shaping 5 2.1.6 Crystal perfection. . 6 2.2 Analysis of alloys 7 2.2.1 Substitutional elements 7 2.2.2 In t e r s t i t i a l elements 9 2.3 Mechanical testing 11 2.3.1 Specimen deformation 11 2.3.2 Strain measurement 11 2.3.3 Computations 12 3. GENERAL DEFORMATION CHARACTERISTICS OF PURE NIOBIUM 14 3.1 Effect of purity 14 3.2 Effect of orientation 14 3.3 Effect of temperature 16 3.4 Effect of strain-rate 21 4. MACR0DEF0RMATI0N OF NIOBIUM ALLOY CRYSTALS 23 4.1 Tensile behaviour at 295°K 23 4.1.1 Yield points 23 4.1.1.1 Results 23 iv PAGE 4.1.1.2 Discussion .24 4.1.1.2.1 Dynamical theory 24 4.1.1.2.2 Stability theory 26 4.1.2 Flow curves ' .. . . . . . . . . .31 4.1.2.1 Results 31 4.1.2.2 Discussion 36 4.1.2.2.1 Uniformity of deformation 36 4.1.2.2.2 Flow parameters 40 4.2 Tensile behaviour at other than 295°K 42 4.2.1 Slip 44 4.2.1.1 Results 44 4.2.1.2 Discussion 46 4.2.1.2.1 Work hardening and uniformity of deformation 46 4.2.1.2.2 Solute hardening and softening. . . . 49 4.2.2 Twinning 52 4.2.3 Cleavage 55 4.3 Slip line observations at 295°K 56 4.3.1 Results . . . . . . . . 56 4.3.2 Discussion 59 5. MICRODEFORMATION OF PURE NIOBIUM CRYSTALS 62 5.1 Introduction 62 5.1.1 Microstrain observations 62 5.1.2 Previous work 65 5.1.3 Experimental procedure 67 5.1.4 Reproducibility of microflow 68 5.1.5 Discussion of microflow 70 V PAGE 5.2 Results 73 5.2.1 Deformation at 295°K 73 5.2.2 Deformation at 160°K . 76 5.2.3 Strain-rate during microflow. . 80 5.3 Discussion 82 5.3.1 Deformation at 295°K 82 5.3.2 Deformation at 160°K . 85 5.3.3 Temperature sensitivity 86 6. MICR0DEF0RMATI0N OF NIOBIUM ALLOY CRYSTALS 88 6.1 Results. 88 6.1.1 Deformation at 295°K 88 6.1.2 Deformation below 295°K 88 6.2 Discussion 5^ 6.2.1 Deformation at 295°K 95 6.2.2 Deformation below 295°K 100 7. THEORY OF PLASTIC FLOW IN NIOBIUM AND NIOBIUM ALLOY CRYSTALS . . 102 7.1 Mechanisms of deformation in pure niobium 102 7.1.1 Introduction 102 7.1.2 Discussion 103 7.2 Mechanisms for deformation of niobium alloys 107 7.2.1 Substitutional effects 107 7.2.1.1 Introduction 1°7 ; 7.2.1.2 Long range interactions 107 7.2.1.3 Short range interaction 112 7.2.2 In t e r s t i t i a l effects 113 8. SUMMARY AND CONCLUSIONS 1 1 9 v i PAGE APPENDICES 122 A.l Properties of relevant bcc metals 122 A.2 Purity of vacuum-melted niobium 122 A.3 Crystallography of sli p . 125 A. 3.1 Definitions of s l i p parameters 125' A.3.2 Orientation dependence 128 A.3.3 Determination of s l i p systems , 130 A.4 Details of microstrain testing , 134 A. 4.1 Introduction 134 A.4.2 Equipment • 135 A.4. 3 Design of extensometer 135 A.4.4 Testing , . 138 A.5 Elastic constants in cubic single crystals , 143 A.5.1 Youngs modulus 148 A.5.2 Shear modulus , • 149 REFERENCES . . .152 v i i LIST OF TABLES PAGE Table I. Analyses of Nb alloys 8 Table II. Crystallography of s l i p in Nb alloys at 295°K 58 Table III. Comparison of microflow data for FeC alloys at 300°K. . . 98 Table IV. Possible interactions between solute atoms and dislocations in fee and bcc metals 108 Table V. Non-zero terms for use in transformation equation in the case of cubic crystals 147 v i i i LIST OF FIGURES PAGE Fig 1. Analysis of Nb" a l l o y s : a) NbTa b) NbMo .10 Fig 2. Flow curves at 295°K for pure Nb c r y s t a l s i n d i f f e r e n t orientations 15 Fi g 3. Orientation dependence of s l i p systems 17 Fig 4. Flow curves for Nb c r y s t a l s at d i f f e r e n t temperatures. . . 18 Fig 5. V a r i a t i o n of y i e l d stress with temperature f o r Nb c r y s t a l s of d i f f e r e n t p u r i t i e s 19 Fi g 6. Strain-rate s e n s i t i v i t y of Nb c r y s t a l s prestrained into stage I . 22 Fi g 7. V a r i a t i o n of y i e l d point drop with add i t i o n of solute i n Nb a l l o y s , . 25 Fig 8. Predicted form of i n i t i a l y i e l d drop as a function of y i e l d load and true work hardening rate 29 Fig 9. Flow curves for Nb and Nb 4.8 Ta c r y s t a l s at 295°K. . , . 32 Fig 10. Flow parameters for NbTa a l l o y s deformed at 295°K . . . . 33 Fig 11. Conventional stress - conventional s t r a i n curves for NbMo a l l o y s at 295°K . 34 Fi g 12. Y i e l d stress of NbMo a l l o y s deformed at 295°K . . . . . 35 Fig 13. Idealized flow curve expressed as x(y) and T(0) 37 Fig 14. E f f e c t of increase i n y i e l d stress on s t a b i l i t y of t e n s i l e deformation 39 Fig 15. Comparison of y i e l d stress data f o r NbTa a l l o y s with: a) l i n e a r hardening b) parabolic hardening . . . . 41 Fig 16. Comparison of y i e l d stress data f o r NbMo a l l o y s . . . . ,43 Fi g 17. Y i e l d stress as a function of temperature for Nb and Nb a l l o y s ,45 Fig 18. I n i t i a l part of load-elongation p l o t obtained during deformation of Nb 0.5 Ta at 77°K .47 i x PAGE F i g 19. Composition dependence of y i e l d stress for TaRe a l l o y s 51 F i g 20. Resolved twin stresses i n Nb a l l o y s at 77°K 53 F i g 21. Intersecting twins i n Nb 4.8 Ta a l l o y . . 54 F i g 22. (001) Cleavage plane i n Nb 6.6 Mo a l l o y 54 F i g 23. S l i p l i n e observations i n Nb a l l o y s deformed into stage I. . 57 F i g 24. Results of s l i p l i n e analyses on Nb a l l o y s expressed as * K x ) - 60 F i g 25. Model for formation of a stable hysteresis loop 64 F i g 26. Types of hysteresis loops observed i n : a) tension b) compression . . . . 66 F i g 27. Technique for obtaining microflow curves 69 F i g 28. Representation of microflow data 69 F i g 29. Schematic d e s c r i p t i o n of microflow behaviour observed during stage I deformation at 295°K 71 F i g 30. Model f or movement of a d i s l o c a t i o n half-loop i n the microflow region 71 F i g 31. Microflow curves at 295°K f o r Nb c r y s t a l s i n d i f f e r e n t conditions 74 F i g 32. Microflow curves at 295°K for a Nb c r y s t a l with d i f f e r e n t prestrains 75 F i g 33. Microflow curves at 160°K for Nb c r y s t a l s i n d i f f e r e n t conditions 77 F i g 34. Microyield stresses at 160°K and 295°K for Nb c r y s t a l s i n d i f f e r e n t conditions 78 F i g 35. Flow stress at various s t r a i n s as a function of temperature . 79 F i g 36. Instantaneous s t r a i n and s t r a i n - r a t e during microflow at 295°K and 77°K 81 F i g 37. Comparison of observed microflow data at 295°K with macroflow predictions 84 F i g 38. Microflow curves f o r NbTa a l l o y s deformed at 295°K . . . .89 X PAGE Fig 39. Microflow stress at different strains for NbTa alloys deformed at 295°K. 90 Fig 40. Microflow curves for NbMo alloys deformed at 295°K . . . . 91 Fig 41. Microflow stress at different strains for NbMo alloys deformed at 295°K 92 Fig 42. Microflow curves for a Nb 4.8 Ta alloy at low temperatures . 93 Fig 43. Microyield values for Nb and Nb 4.8 Ta crystals as a function of temperature 94 Fig 44. Microflow data for polycrystalline Fe alloys deformed at 295°K 96 Fig 45. Temperature variation, i n various prestrained Fe alloys, of: a) microyield stress b) macroflow stress 99 Fig 46. Yield stress of NbW alloys at low temperatures 114 Fig 47. Section through ^[111] screw dislocation: a) approaching possible i n t e r s t i t i a l sites b) showing a l l the possible {011} and {112} s l i p planes and two energetically favourable dissociations 117 Fig 48. Sieverts Law plots of the solubility of a) oxygen b) nitrogen in Nb at high temperatures . . . . 124 Fig 49. Calculated equilibrium so l u b i l i t i e s of oxygen and nitrogen in Nb at high temperatures 125 Fig 50. (001) Stereographic projection showing parameters for s l i p line analysis 131 Fig 51. Plot of <j>($) showing information required for determination of one sl i p system 133 Fig 52. Schematic circuit diagram for microstrain testing . . . . 136 Fig 53. Distance meter reading as a function of plate separation . . 139 Fig 54. Displacement sensitivity of extensometer as a function of distance meter reading 140 Fig 55. Schematic diagram of section through gas-cooling cryostat. . 141 Fig 56. Test assembly for microstraining at low temperatures . . . 142 Fig 57. Idealized experimental X-Y recorder chart from microstrain test 144 ACKNOWLEDGEMENT It is a pleasure to acknowledge the helpful discussions with my research director, Professor E. Teghtsoonian, and with Dr. R.D. Warda. I am also grateful for the secretarial assistance of my wife, who no longer thinks that "microstraining" involves tiny sieves. 1 1. INTRODUCTION It is only within the last decade that fundamental studies of deformation processes in bcc crystals have been undertaken. Attention had.previously been restricted to fee and hep metals because they can easily be obtained as high purity single crystals. The development of electron beam melting in the late f i f t i e s enabled niobium, tantalum, molybdenum, and tungsten to be similarly prepared; consequently the r deformation characteristics of these metals have received considerable attention. It has been shown that their behaviour is often similar to that of the other metal structures, but there are important differences. In particular, bcc crystals show a very marked strengthening at low temperatures, a complex s l i p behaviour, marked orientation effects, and discontinuous yielding in a l l but the purest materials. Since these effects are a result of the motion of dislocations, i t has been necessary to consider the detailed properties of dislocations i n bcc crystals. Much of the theoretical interest and accompanying controversy has been concerned with determining the rate-controlling mechanisms of plastic deformation. A majority opinion considers the intrinsic nature of dislocations in the bcc la t t i c e to be the most important feature. However, even the purest crystals contain significant amounts of i n t e r s t i t i a l impurities, and i t is possible that impurity effects may be more important than intrinsic l a t t i c e effects. The application of thermal activation analysis and dislocation dynamics has proved very useful in relating many of the observed effects, but i t has not been possible to distinguish between int r i n s i c and impurity-controlled behaviour. Several new experimental techniques have 2 been used, particularly the etch-pit and microstrain techniques; the etch-pit technique has usually employed single crystals, while most microstrain tests to date have been performed on polycrystals. Whereas the effect of substitutional elements has been well established for fee crystals, most of the data on pure bcc alloys has. been published only since the beginning of this work. The microstrain technique offers particular advantages in the study of alloy effects and this work probably represents i t s f i r s t application to the deformation of bcc substitutional alloy crystals. , From the family of bcc refractory metals, niobium was selected, as a solvent having a melting point and vapour pressure most suitable for the preparation of pure single crystals. Molybdenum and tantalum were selected as sufficiently different solutes which both show complete solid solubility in niobium and have similar vapour pressures (see A . l ) . < The i n i t i a l part of the program was concerned with the conventional tensile testing of pure niobium single crystals over a range of variables such as orientation, temperature and strain-rate. These experiments ; served to establish and evaluate procedures, and provided a basis for selecting specific conditions for the subsequent deformation of alloy s crystals. The deformation properties of pure niobium and niobium alloy crystals were investigated in detail using macro and microstrain testing techniques. Since the microstrain technique is new, considerable effort was spent in interpreting the results of the technique and in assessing i t s importance. It was hoped that by applying the technique to a wide range of alloys, more insight could be gained into the interpretation of results from observations on pure bcc crystals. 3 2. EXPERIMENTAL PROCEDURE 2.1 Specimen preparation 2.1.1 Materials Polycrystalline rod, 1/8 inch in diameter was obtained from two sources. The i n i t i a l experiments were performed on pure Nb (minimum 99.82%) from Kawecki Chemical Co, New York. The subsequent experiments were performed on high purity material obtained from the Materials Research Corporation, New York. The six Nb alloys contained nominally 0.5, 2 and 8 atomic percent each of Mo and Ta as solute. A pure Nb sample (containing i n i t i a l l y about 40 ppm i n t e r s t i t i a l impurities) had undergone the same melting and fabrication procedure. For each alloy series, this material acted as a control in the evaluation of i n t e r s t i t i a l and substitutional solute effects. The i n t e r s t i t i a l content of the polycrystalline alloy rods after fabrication was. not known. 2.1.2 Melting procedure Single crystals, 1/8 inch in diameter and up to 8% inches long, were grown in an electron beam zone melter based on the original design (Calverley et al (1957)). Using an emission current of about 100 mA at 2000 V potential, single crystals were obtained by passing the molten zone at 25 cm hour \ This high speed was used to suppress any redistribution of solute in the liquid zone. Purification by electron beam zone melting occurs mainly by vacuum d i s t i l l a t i o n (Votava (1965)), so the distribution of i n t e r s t i t i a l impurities along 4 the rods can be regarded as sufficiently homogeneous. There is also a possibility of preferential loss of one of the alloy constituents, but from the relative and absolute values of the vapour pressures of the components at the melting temperature of the alloy (A.l), this loss i s expected to be slight. However this was not the case with the attempted preparation of some NbV alloy crystals. The V loss could have been reduced by operating at high pressures, but there is an upper pressure limit set by the operation of the electron emission technique and by the possibility of gaseous contamination. 2.1.3 Impurities The dynamic vacuum employed during zone melting was in a l l cases from 2 - 8 x 10 ~* torr, this value being a compromise between the loss of substitutional solute and the gain of i n t e r s t i t i a l impurities. However, the high pressure might have removed some carbon as carbon monoxide (Taylor and Christian (1965)) and hydrogen is relatively easily removed by vacuum d i s t i l l a t i o n . Pemsler (1961) has given data for the thermodynamics of the, interaction of ^ and 0^ with Nb at high temperatures in the form of Sieverts Law plots. The concentration of N and 0 in equilibrium with, _5 liquid Nb at 2470°C under an applied pressure of 5 x 10 torr has been calculated to be 27 and 12 wt ppm of N and 0 respectively (see A.2). Although the equilibrium concentration in the hot solid region w i l l be higher than this, the i n i t i a l material w i l l contain far more than 40 ppm, so contamination during electron beam melting should not be a problem,. However i t is unlikely that very much purification could be expected under the above melting conditions. 5 To reduce contamination from the tungsten cathode, i t was heated to a high temperature for some time before melting the specimen and then operated at the lowest possible emission temperature during melting. 2.1.4 Oriented single crystals A technique for producing oriented seed crystals was devised and used for each alloy to eliminate any possible composition gradients caused by melting together different alloys. The method involved melting part of a short length of rod to produce a single crystal, mounting the specimen for Laue back-reflection photography, and then bending the polycrystalline end to give the desired orientation relative to the X-ray beam. A permanent seed crystal was then grown from the oriented crystal and, once prepared, could be used repeatedly. The as-grown rods were confirmed to be single crystals of the desired orientation by taking Laue back-reflection photographs at points along their lengths. They were carefully cut with a jeweller's saw ., into pieces about 4 cm long, after having discarded the immediate beginning and end of the melted region. 2.1.5 Specimen shaping A spark lathing technique using deionized water as dielectric was developed to produce a reduced gauge section on the crystal lengths. A tapered copper electrode was passed alongside the rotating crystal and 4 the spark energy gradually reduced during the operation from 4 x 10 to 3 5 x 10 ergs. A final hand polish was given with 2/0 and 4/0 emery papers. About 0.25 mm (10 thou) was then removed from the gauge section 6 by chemical polishing in a solution of 1 part HF / 3 parts HNO^ . Gauge diameters were in general uniform to within 0.01 mm and typical specimen dimensions were as follows: gauge diameter 2.00 mm gauge length 20 mm shoulder diameter 3 mm shoulder radius 2 mm 2.1.6 Crystal perfection The extent of spark erosion damage on single crystals w i l l depend on the material, the spark energy, the cutting conditions, and the crystal orientation. Damage up to depths of 300]j has been reported in materials softer than Nb (Ahktar (1968)). To evaluate the effect in Nb, successive layers 0.05 mm deep were chemically polished from the surface until 0.3 mm had been removed. At each stage a Laue back-reflection photograph was taken and the specimen was examined optically. After the very f i r s t polish, the Laue spots were always exactly the same as on the as-grown crystal. This indicates either that the damage was very slight or that absence of asterism in Laue photographs i s no evidence of lack of strain in Nb crystals. The optical observations revealed damage after 0.05 mm had been removed from the surface, but a constant pattern of dislocation etch pits was observed i f 0.1 mm or more was removed. The completed specimens were X-rayed. Some of the large-angle reflections were observed to be composed of several small discrete spots, indicating the presence of a sub-structure of less than Jg° misorientation. This was also indicated by the presence of longitudinal striations observed 7 in the microscope after etching to reveal dislocations. This sub-structure was also seen on the as-grown crystals. A similar sub-structure has been recognized in W and other bcc metals (Koo (1963)). 2.2 Analysis of alloys 2.2.1 Substitutional elements X-ray fluorescence analysis was performed on each of the alloys before and after melting. A measure of solute content was obtained from a "fluorescence parameter", R, defined as the ratio of the sees/count produced on a solute peak to the sees/count produced on a Nb peak. The ratios used were for Mo and ^f^" f° r Ta. r NbK3 NbKa It was better to time counts rather than count for a fixed time, since the solute counts were then taken over a longer period and the dilute element received s t a t i s t i c a l l y more weight. In a l l cases the variation of R with position along an as-grown rod was less than the standard deviation of R (maximum 5%). Therefore, > the composition along the rods can be regarded as uniform within the sensitivity of the fluorescence technique. Absolute analysis i s not possible unless standards are available. An analysis of the as-received material was given by the suppliers. After crystal growth, a further analysis was performed by Ledoux and Co, New Jersey. The results are given in Table I, together with the letter used throughout for alloy identification and the parameter R obtained from X-ray fluorescence. Because of the inconsistencies ; in the Mo results, a repeat analysis on this series was performed by , Coast Eldridge, Vancouver, and these results are also included in the table. Table I. Analyses of Nb alloys. Alloy identification B C D J K L Nominal at% alloy 0.5 Ta 2.0 Ta 8.0 Ta 0.5 Mo 2.0 Mo 8.0 Mo As-supplied: MRC 0.31 1.53 5.04 0.72 1.84 6.15 Coast Eldridge - - - 0.79 4.84 5.93 As-grown: Ledoux 0.55 2.21 4.41 1.31 6.38 7.62 Coast Eldridge - - - 0.69 4.74 5.99 Fluorescence parameter,R: As-supplied 7.1 14.8 33.6 8.9 27.7 37.7 As-grown 7.3 14.8 33.9 9.7 31.5 40.6 Best value, as-grown 0.5 Ta 1.7 Ta 4.8 Ta 0.9 Mo 4.9 Mo 6.6 Mo Specimen identification: Each alloy series was given an identifying letter as above; pure Nb was A. A digit following the letter identifies a particular crystal; a second digit identifies the position of the specimen within the crystal. Thus, K41 i s the f i r s t tensile specimen taken from the fourth crystal of Nb 4.9 Mo. oo The values given for the NbMo alloys present a rather confusing picture. In particular i t appears that the as-supplied alloy K must have contained far more than the 1.84 at% Mo quoted by MRC, yet the as-grown value of 6.38 at% by Ledoux seems improbably high. In an attempt to clarify the situation, the parameter R was plotted against the various quoted values and the most reasonable calibration curve drawn (Fig 1). The fluorescence results were then used to arrive at the best analysis values shown in Table I. 2.2.2 I n t e r s t i t i a l elements An analysis of C, H, N and 0 was performed on a selection of alloys by Ledoux. The results are shown below with the i n t e r s t i t i a l contents given in wt ppm. I n i t i a l melting Crystal C H N 0 pressure (x A4 10 5.8 8 102 7.6 D7 13 3.4 3 89 5.3 K3 10 3.7 9 28 2.3 L5 16 3.1 11 42 3.0 10 torr) Also included is the pressure in the electron beam melter when melting was commenced. There is a possible correlation between this pressure and the i n t e r s t i t i a l oxygen concentration. The analysis figures for N and 0 show no resemblance to the values calculated from equilibrium thermodynamics (see A.2). The total i n t e r s t i t i a l content is frequently quoted when comparing materials; in this case a typical value would be about 80 ppm total i n t e r s t i t i a l s . ,-> 30 20 10 1 i 1 1 y y y _ y -y y y y v y y y y *• y y y * A y OA y i i i • Composition, at% Ta O M R C • Ledoux B Coast Eldridge 2 3 4 5 Composition, at% Mo Fig 1. Analysis of Nb alloys: a) NbTa b) NbMo 2.3 Mechanical testing 2.3.1 Specimen deformation In a tensile test i t is necessary to ensure that loading is uniaxial and that no bending moments are introduced into the specimen. During single crystal deformation there is continuous reorientation of the latt i c e in the gauge length so that to completely prevent bending, the axis of a universal joint must l i e at the ends of the gauge length. For the small strains mainly used in this study, the problem is less serious. However to reduce bending and ensure the colinearity of tensile and specimen axes, universal joints were placed outside the specimen shoulders. The specimens themselves were held by the shoulders in split-jaw grips and loaded into the machine using a special j i g to prevent accidental damage. Straining was performed on a Floor Model Instron. 2.3.2 Strain measurement Two regions of strain sensitivity were investigated: - 4 a) Macrostrain region (e>5 x 10 ) In this region i t is usual to measure strain indirectly from the motion of the Instron crosshead, which effectively gives a load-time plot on the Instron chart. This is satisfactory for materials showing discontinuous yielding or for low strain sensitivities of the order -3 of 10 (ie a 0.1% proof stress). In order to investigate the use of the crosshead displacement - 4 as a measure of strain at sensitivities of 10 , an Instron extensometer was attached directly to the specimen and the true displacement recorded on the X-axis of an Instron X-Y recorder. Using a f u l l y elastic specimen 12 in the inverted-cage tensile j i g , a non-linear load-time plot was obtained - 4 at strain sensitivities of 10 with loads of 40 lbs. This was attributed to an inherent non-linearity in the Instron crosshead movement when under load. It was also observed htat, in spite of a l l precautions, a significant amount of plastic deformation occurred at the specimen .shoulders during straining. For these reasons, whenever necessary, strain was recorded on an extensometer attached directly to the specimen. In the conventional testing using load-time curves, temperature control was obtained by immersing the specimen in various hot and cold liquid baths. b) Microstrain region (e = 10 ^  - 5 x 10 ^ ) A high sensitivity extensometer was designed and built,consisting essentially of two parallel plates secured to the tensile specimen., The separation was monitored using a Wayne Kerr DM100B distance meter and the output amplified by a high sensitivity X-Y recorder. The signal from the Instron load c e l l was fed directly to the Y-axis of the recorder. Details of the design and construction of the extensometer are given in A.4. The recorder gave a continuous plot of load and of extension of the specimen gauge. The construction of the extensometer did not enable i t to be . used above room temperature. For use at low temperatures, a nitrogen gas-cooling cryostat was designed and built (see A.4). 2.3.3 Computations A special j i g was constructed for use with an optical microscope and X-ray set which allowed s l i p (or twin) systems to be determined on any cylindrical specimen. The method involved measuring the angle from the sl i p trace to the tensile axis at any point around the crystal, and 13 employed a two-surface trace analysis. A graphical extension of the technique (see A.3) enabled two or even three systems operating together to be identified. A program was written for an IBM 360 computer which enabled coordinates taken directly from the Instron chart to be converted to resolved shear stress versus shear strain on the observed s l i p system. These results were obtained graphically as a flow curve. This procedure assumed that only one sl i p system was operative, and that deformation was uniform, which was not always the case. 14 3. GENERAL DEFORMATION CHARACTERISTICS OF PURE NIOBIUM The flow properties of pure Nb single crystals have been established by Mitchell et al (1963) over a range of purities, orientations, temperatures and strain-rates. The geometry of the operative slip plane has been studied in detail by Foxall et al (1967). In order to study the effect of solute on pure Nb i t was necessary to restrict many of the above variables. A few experiments were performed on pure Nb to confirm the general deformation, characteristics which are summarized below. 3.1 Effect of purity The yield stress and flow stress of Nb decrease with increasing purity and the shape of the flow curve is altered (Mitchell et al (1963)). Bowen et al (1967) have prepared high purity crystals by ultra-high vacuum annealing. Although the impurity content could not be determined, the crystals aged significantly within a few weeks in air at room temperature. The crystals used here did not age detectably over a period of four months. 3.2 Effect of orientation Much discussion has centred on the dependence of the yield stress and the temperature sensitivity of yield, on the orientation of the tensile axis (see Bowen et al (1967)). The effect on the flow curve is very marked for crystals oriented near the corners of the stereographic triangle. Fig 2 shows flow curves for three crystals of extreme orientations. The stress has been resolved onto the (Oil) plane in each case. Within the stereographic triangle the effect of orientation is less marked, although different slip systems operate in different regions of the triangle. Foxall et al (1967) CN bO w co cu u *J CD as <D CO 8 \-4 k 2 k 0.2 0.4 0.6 Shear strain, y 0.8 1.0 1.2 1.4 Fig 2. Flow curves at 295°K for pure Nb crystals in different orientations. 16 found that s l i p could always be described as taking place on a plane of the type {110} or {112}. If sli p takes place on a plane having the highest resolved shear stress, and the c r i t i c a l resolved shear stress i s the same for both types of plane, then the regions i n which the different s l i p systems should operate (see A.3) would be as shown in Fig 3. Also shown in the figure is the direction of motion of the tensile axis during straining of three representative crystals. This motion illustrates that the predicted Burgers vector is operative, although the observed s l i p planes are not precisely of the type {110} or {112}. For this reason, the operative s l i p planes in this work are described more generally in terms of the s l i p . , plane parameter iji, defined as the angle between the observed s l i p plane and the (Oil) reference plane (Fig 3). An orientation in the middle of the stereographic triangle was selected for a l l the alloy crystals. This case was the simplest and allowed a considerable amount of s l i p on a single system, with a [ i l l ] Burgers vector. 3.3 Effect of temperature The marked increase in strength at low temperatures i s the most obvious feature of bcc deformation. Fig 4 shows flow curves at different temperatures; i t is clear that both the form of the curve and the stress level are changed. This makes i t d i f f i c u l t to compare yield parameters; the yield stress, T , is obtained by extrapolating the load-elongation curve in the linear region following yield back to zero plastic strain. The marked variation of T q with temperature is shown in Fig 5 and, for comparison, the data obtained by Duesbery (1967) and Ravi and Gibala (1969) is included. The lower values can be attributed to the higher purity, of material used by these workers. , 17 Region Observed slip system, [b] \> A [111] -28 B [ i l l ] -8 C [111] 28 Fig 3. Orientation dependence of slip systems. Fig 4. Flow curves for Nb crystals at different temperatures. 19 Fig 5. Variation of yield stress with temperature for Nb crystals of different purities. 20 At room temperature and up to about 450°K, a curve resembling the fee three-stage curve i s observed for orientations within the stereo-graphic triangle. The flow curve for the specimen in Fig 2 i s typical. Following the convention established for fee crystals, the curve is divided into distinct regions: •s ^ — ~ i i s t a g e i s t a g e , s t a g e i s t a g e i 0 , i i 1 ! II i 1 , 1 Stage 0 (y<0.1) consists of an upper yield point and yield point extension. Stage I (0.1<Y<0.3) and stage II (y>0.3) are linear and f a i r l y distinct. Plastic instability occurred soon after the end of stage II (ie stage III was very small). The form of the observed flow curve i s very dependent on temperature; in particular the following effects are observed in crystals of similar orientation as the temperature is reduced: a) yield and flow stresses are increased b) elongation to failure i s reduced c) upper yield becomes more pronounced ,• d) deformation becomes unstable directly after yield. 3.4 Effect of strain-rate Mitchell et a l (1963) deformed as-grown Nb crystals at room temperature using strain-rates between 1 x 10 and 4.5 x 10 ^ sec \ and found that the effect was similar to a decrease in temperature. Their results have been confirmed here over a smaller range of strain-rates; an increase in the nominal strain-rate of an order of magnitude at room temperature increases the yield and flow parameters by about 20%. \ Because of the importance of prestraining the crystals before microstrain testing, the strain-rate sensitivity of prestrained crystals was investigated. Crystals were prestrained into stage I and deformed at 295°K and 140°K at various strain-rates. The values of the flow stresses are shown in Fig 6, which also includes the yield stress data of Mitchell et al for as-grown crystals. At small plastic strains, or i f localized yielding occurs, the, actual plastic strain-rate might differ considerably from the nominal value (see 5.2.3). The crosshead speed selected for a l l the alloy tests was 0.005 in min \ because i t was suitable for both micro and macrostrain -4 -1 testing. The corresponding nominal strain-rate was 1.0 x 10 sec . 22 Fig 6 . Strain-rate sensitivity of Nb crystals prestrained into stage I. 23 4. MACRODEFORMATION OF NIOBIUM ALLOY CRYSTALS 4.1 Tensile behaviour at 295°K Macrostrain t e n s i l e tests were performed on specimens of pure MRC Nb and d i l u t e a l l o y s of Nb with Mo and Ta. A l l the c r y s t a l s were oriented i n the middle of the standard stereographic t r i a n g l e . Specimens were taken from d i f f e r e n t parts of the same c r y s t a l and from d i f f e r e n t c r y s t a l s of a p a r t i c u l a r a l l o y . The i n i t i a l y i e l d was observed with an Instron extensometer and the whole flow curve established from the Instron chart recorder. In many cases, the operative s l i p systems were determined. 4.1.1 Y i e l d points 4.1.1.1 Results A l l the c r y s t a l s showed an upper and lower y i e l d point at 295°K. In a l l normal cases, the y i e l d point was well-rounded with about y = 0.5% pr e - y i e l d microstrain. There were differences i n the magnitude of the y i e l d drop, Ax: a small v a r i a t i o n among specimens of a given a l l o y but a large v a r i a t i o n among d i f f e r e n t a l l o y s . For any given a l l o y , the value of Ax was very dependent on the l o c a l conditions of y i e l d : Ax was largest i n specimens having the greatest v a r i a t i o n i n diameter along the gauge section. This behaviour d i f f e r s from that expected from considerations of the mobile d i s l o c a t i o n density (Hutchison (1963)). I t was also found that Ax showed a c o r r e l a t i o n with the magnitude of the upper y i e l d stress T , but not with the lower y i e l d stress T£y-This means that, for a p a r t i c u l a r a l l o y , specimens which had a rather high 24 upper y i e l d s t r e s s s u b s e q u e n t l y e x h i b i t e d a r a t h e r l a r g e y i e l d d r o p . There was a l a r g e d i f f e r e n c e i n the a b s o l u t e y i e l d drop AT o f AT d i f f e r e n t a l l o y s a l t h o u g h the r e l a t i v e y i e l d drop remained f a i r l y uy c o n s t a n t (^5%). The mean v a l u e o f AT f o r a l l the a l l o y s t e s t e d , t o g e t h e r w i t h t h e s t a n d a r d d e v i a t i o n , i s shown i n F i g 7. I n b o t h a l l o y systems t h e r e i s an i n i t i a l i n c r e a s e i n AT which l e v e l s o f f a t h i g h e r c o n c e n t r a t i o n s . D i r e c t e v i d e n c e f o r non-uniform y i e l d i n g was o b t a i n e d i n two cases when the p o i n t o f minimum specimen d i a m e t e r l a y o u t s i d e the e x t e n s o -meter gauge l e n g t h . In t h e s e cases the l o a d passed through a maximum b e f o r e any p l a s t i c s t r a i n was r e c o r d e d . P l a s t i c s t r a i n was observed when.the oncoming L u d e r s f r o n t moved i n t o the extensometer gauge l e n g t h . Because of the inhomogeneous n a t u r e o f y i e l d , the r e p r e s e n t a t i o n o f the f l o w c u r v e i n s t a g e 0 i s n o t v e r y m e a n i n g f u l . However the l e n g t h o f s t a g e 0 was always about y = 10%, which s u g g e s t s t h a t a t the b e g i n n i n g o f s t a g e I a l l the c r y s t a l s were i n the same p h y s i c a l s t a t e whatever the d e t a i l s o f y i e l d . 4.1.1.2 D i s c u s s i o n 4.1.1.2.1 Dynamical t h e o r y A d i s l o c a t i o n dynamics approach has been s u c c e s s f u l l y used i n r e c e n t y e a r s to e x p l a i n many o f the f e a t u r e s o f y i e l d i n g i n b c c m e t a l s (Hahn (1962), C o t t r e l l (1963), S t e i n (1968)). The t h e o r y e x p l a i n s the y i e l d drop i n terms o f the i n i t i a l d e n s i t y o f m o b i l e d i s l o c a t i o n s and t h e i r subsequent v e l o c i t y and m u l t i p l i c a t i o n c h a r a c t e r i s t i c s . However the t h e o r y c o n t a i n s so many unknown parameters t h a t i t i s p o s s i b l e to f i t a c a l c u l a t e d s t r e s s - s t r a i n c u r v e to v i r t u a l l y any o b s e r v e d y i e l d i n g b e h a v i o u r . . The d y n a m i c a l approach was taken by Szkopiak (1966) to e x p l a i n 25 F i g 7. V a r i a t i o n of y i e l d point drop with a d d i t i o n of solute i n Nb a l l o y s . the linear increase of yield point drop with added solute observed in polycrystalline NbO alloys containing up to 800 ppm oxygen. According to this theory, an increase in AT could be largely due either to a decrease in the density of mobile dislocations or to an increase in the dislocation velocity exponent with increasing oxygen concentration. Since the dislocation velocity exponent obtained from strain-rate sensitivity experiments was observed to decrease as the oxygen content increased, the observed yield point behaviour was explained by Szkopiak in terms of a reduction in the i n i t i a l mobile dislocation density. In the case of substitutional solid solutions there is no reason to expect any marked reduction in the mobile dislocation density with i alloying. It is possible that the dislocation velocity exponent might be decreased by addition of substitutional solute, but there is no way of testing this supposition. It therefore appears that, while the dislocation dynamics approach might be capable of describing the yield point behaviour in pure Nb, i t cannot usefully be applied to substitutional alloys. Its applicability to this work is further limited because i t assumes that yielding is uniform and that no pre-yield microstrain occurs. f Since yielding in pure Nb was observed to be non-uniform, the following theory, based simply on the s t a b i l i t y of the i n i t i a l plastic; region, has been developed to predict the dependence of the yield drop on the magnitude of the yield stress and true work hardening rate. It is a continuum approach and i t s applicability derives from the general nature of discontinuous yielding in both crystalline and non-crystalline materials. 4.1.1.2.2 Stability theory It i s to be expected that the work hardening characteristic of ; 27 a material w i l l influence the nature of the y i e l d point. However, since deformation i n the region of an upper y i e l d point i s f a r from uniform, the true work hardening rate cannot be obtained from the observed load versus elongation curve. In f a c t , i n the e l a s t i c region, the apparent work hardening rate i s i n f i n i t e and a f t e r the upper y i e l d point the material i s apparently work softening. A more useful d e f i n i t i o n has therefore been adopted f o r , the "true work hardening ra t e " , Q. This i s supposed to be an i n t r i n s i c material parameter, definable i n p r i n c i p l e , i f not i n p r a c t i c e , as a function of past h i s t o r y , s t r a i n , and s t r a i n - r a t e . The true work hardening rate can be defined only for that region of the material which deforms homogeneously. In the case of the discontinuous y i e l d i n g observed during t e n s i l e deformation of Nb c r y s t a l s , i t r e f e r s to the p l a s t i c nucleus forming an i n c i p i e n t : Luders band. to be constant. Under a load L, the nucleus extends p l a s t i c a l l y by an amount dx i n time dt: Consider a nucleus of length x and of volume V which i s assumed L+dL< dx x Then the true work hardening rate i s defined by (1) cf da de x where A i s the instantaneous cross-sectional area given by A V (2) x 28 If the crosshead v e l o c i t y i s X and e l a s t i c e f f e c t s are neglected, dx = X.dt (3) provided the p l a s t i c nucleus i s the only region that deforms. Substituting (2) and (3) i n equation (1) gives « = — 4r (L.x) v.x d t or -a = f [ L + ] (4) X From t h i s d e f i n i t i o n , i t can be seen that i n the e l a s t i c region since x = 0, Q i s zero; a f t e r y i e l d Q can s t i l l be p o s i t i v e even though the load i s decreasing. It w i l l now be assumed that y i e l d i n g occurs suddenly at a load L q and that a f t e r y i e l d , Q i s constant. Equation (4) can be used to predict the subsequent flow behaviour (ie ^  ) f or given values of L q and fi. The c r i t i c a l condition for the p l a s t i c nucleus to be stable i s given by dt Therefore x.L fi = — — = a (cf Consideres c r i t e r i o n ) V o where a i s the true t e n s i l e stress at y i e l d , o J If Q > a , deformation w i l l be stable. In this case, there w i l l o be no y i e l d drop and x must be i d e n t i f i e d with the specimen gauge length. For n < a , deformation w i l l be unstable and F i g 8 shows the o predicted i n i t i a l forms of the load versus time curves, for three d i f f e r e n t true work hardening rates. Values for the constants are t y p i c a l of t h i s work. The actual magnitude of the t o t a l y i e l d drop w i l l depend s i g n i f i c a n t l y 40 30 _ 60 13 o 20 10 !-D. - 3 kg.mm - 2 _L £1 = 6 kg.mm - 2 X2 = 9 kg.mm - 2 Curves p l o t t e d from e q u a t i o n (4) u s i n g f o l l o w i n g c o n s t a n t s : 25 0 25 Time, t sees 25 x = 1 mm V = 3 mm" X = 0.02 i n . m i n -1 F i g 8. P r e d i c t e d form o f i n i t i a l y i e l d drop as a f u n c t i o n o f y i e l d l o a d and t r u e work h a r d e n i n g r a t e . 30 on the e l a s t i c properties of the t e n s i l e t e s t i n g apparatus, and on the change i n fi with s t r a i n and s t r a i n - r a t e . However the trend i s quite c l e a r from F i g 8: i f there i s no s i g n i f i c a n t change i n true work hardening rate, then the expected y i e l d drop i s greater i n those c r y s t a l s having the higher y i e l d s t r e s s . I t w i l l be seen i n the next section that a l l o y i n g of Nb produces a large increase i n y i e l d stress without much e f f e c t on the work hardening parameters. Therefore, for the s u b s t i t u t i o n a l a l l o y s examined here, the s t a b i l i t y theory has explained the increase i n y i e l d drop with increasing solute content shown i n F i g 7. In the case of dispersion hardened a l l o y s i t i s expected that the work hardening rate w i l l be increased s i g n i f i c a n t l y without much e f f e c t on the y i e l d s t r e s s . In t h i s case the theory predicts a decrease i n y i e l d drop with increasing a l l o y content as can again be seen from F i g 8. This behaviour has recently been observed i n Fe.ThO^ a l l o y s (Place (1969)). A f t e r y i e l d , the values of L and ^ i n equation (4) are indeter-minate, but ^— w i l l increase and may reach zero while L i s s t i l l p o s i t i v e . In t h i s case, the Luders band w i l l become stable and spread along the specimen. This was the usual case with pure Nb and NbTa a l l o y s . However, i n cases where ^ 7 at y i e l d i s large and negative, i t i s possible that L w i l l f a l l to zero while i s s t i l l negative. This s i t u a t i o n corresponds to a Luders f a i l u r e , and was observed i n the high Mo a l l o y s as mentioned below. The success of the simple s t a b i l i t y theory indicates that the importance of d i s l o c a t i o n m u l t i p l i c a t i o n may have been overstressed i n the past. It i s suggested that the d i s l o c a t i o n dynamics approach and the. s t a b i l i t y approach represent two d i f f e r e n t aspects of discontinuous y i e l d i n g . Cases i n which discontinuous y i e l d i n g i s uniform (eg L i F ) must be expressed 31 in terms of dislocation dynamics. However in cases where yielding i s non-uniform , the stability approach clearly makes an important contribution. 4.1.2 Flow curves 4.1.2.1 Results Fig 9 shows the computed flow curve for a Nb 4.8 Ta crystal. It is very similar to the pure Nb flow curve; no differences i n form were observed with any of the Ta alloys. Deformation appeared completely uniform except for the i n i t i a l discontinuous yield. The yield stress, T , was obtained by extrapolating the flow curve in stage I back to zero plastic strain. The important parameters of the Ta alloy flow curves are shown i n Fig 10. The effect of Ta on T q is most noticeable at small concentrations but is never very marked. There is possibly an accompanying slight increase in QJJ, but no definite effect of Ta on the work hardening parameters i s evident. The order of magnitude of 6^ and is indicated. In the case of the NbMo alloys the hardening effect was much, greater than in the NbTa alloys; deformation was observed to become increasingly non-uniform as the Mo content increased. For this reason, the tensile results for the NbMo alloys are shown in Fig 11 as conventional stress versus conventional strain curves. The variation of T with Mo content is shown o in Fig 12. It can be seen that the hardening is linear, with a slope of -2 -1 2.1 kg mm at% . The hardening effect of Mo is therefore about 35 times greater than that of Ta, for which the corresponding slope i s only -2 -1 0.06 kg mm at% in the linear region after the i n i t i a l r ise. i 1 1 1 1 r Fig 9. Flow curves for Nb and Nb 4.8 Ta crystals at 295°K. OJ 33 i 1 1 1 1 r I I I I I L 0 1 2 3 4 5 6 C o m p o s i t i o n , a t % Ta F i g 10. Flow parameters f o r NbTa a l l o y s deformed a t 295°K. I 60 CO to u o • r l 4-1 a cu % o u 30 20 10 Nb 6 6 Mo Nb 4-9Mo _L Nb 0-9 Mo Nb 20 40 Conventional strain, % 60 80 Fj-g 11- Conventional stress - conventional strain curves for NbMo alloys at 295°K. u> 35 T 1 1 r 0 1 2 3 4 5 6 Composition, at% Mo Fig 12. Yield stress of NbMo alloys deformed at 295°K. 36 4.1.2.2 Discussion 4.1.2.2.1 Uniformity of deformation I t was shown above that a condition f o r deformation at y i e l d to be stable i s _ ^  fi > a o A f t e r the i n i t i a l perturbation at y i e l d , the same c r i t e r i o n should apply, and i f deformation does subsequently remain uniform then fi i s given by ~ , which can be obtained from the observed flow curve a(e). At any stress a, the s t a b i l i t y condition therefore becomes da —- = a de Tn terms of x(y), since T a = — s e = s.y the condition i s given by -.— = 0 = S . T dy where 0 i s the slope of the flow curve and the Schmid factor s i s approximately constant (^0.5) during flow. Consider F i g 13 which i s an i d e a l i z e d flow curve x(y) for a Nb sin g l e c r y s t a l , obtained under conditions of uniform flow. This curve can also be expressed as a plo t of T ( 9 ) , shown as case A i n the f i g u r e . I t can be seen that deformation remains stable (ie uniform) u n t i l T(6) crosses the l i n e x = — at a stress , at which point deformation becomes non-uniform. , The form of x(8) can be considered to represent the basic response of the material to deformation. This response w i l l only be obtained; i n a t e n s i l e test i f deformation remains st a b l e , ^ 9 > „ | T = T ( / ) 38 For the Nb and NbTa c r y s t a l s t h i s condition was s a t i s f i e d i n stage II but not always i n stage I (Fig 10). There i s therefore a p o s s i b i l i t y that deformation i n stage I might have been unstable. The e f f e c t of an increase i n a l l o y content or a decrease i n temperature i s to increase the l e v e l of the flow s t r e s s . Unless there i s a corresponding increase i n the work hardening parameters, deformation may become unstable at a much e a r l i e r stage as shown by the diagrams i n F i g 14. In case B, necking s t a r t s to occur during stage I but the l o c a l i z e d onset of stage II s t a b i l i z e s the neck. Therefore a deformation front w i l l pass down the gauge length during "stage I". The observed work hardening slope i n t h i s region i s meaningless. In case C, necking w i l l occur immediately a f t e r y i e l d . Deformation w i l l never become s t a b i l i z e d , so a Luders f a i l u r e w i l l r e s u l t . Again the observed work hardening slope i s meaningless. For the NbMo a l l o y s , deformation became more non-uniform as the solute content increased, and the behaviour predicted i n cases B andC , was i n f a c t observed. It also follows from the s t a b i l i t y argument that the t o t a l s t r a i n to f a i l u r e w i l l depend on the true work hardening rate rather than on the i n t r i n s i c d u c t i l i t y of the material; t h i s i s confirmed by observations of the reduction of area at f a i l u r e , which i n a l l cases was 100%. Similar considerations can be expected to apply during low temperature deformation. The arguments which have been expressed above are applicable i n p r i n c i p l e to both sin g l e c r y s t a l and p o l y c r y s t a l deformation. However the s i t u a t i o n represented by case B can occur only i n the deformation of sin g l e c r y s t a l s , because t h i s requires a work hardening rate which increases with s t r a i n . ; Fig 14. E f f e c t o f i n c r e a s e i n y i e l d s t r e s s on s t a b i l i t y o f t e n s i l e d e f o r m a t i o n . 40 4.1.2.2.2 Flow parameters Peters and Hendrickson (1966) have obtained c r i t i c a l shear stress data for c r y s t a l s having a composition range extending from pure Nb to pure Ta; t h e i r r e s u l t s are shown i n F i g 15a. They obtained a value of -2 -2 2.5 kg mm for T q i n pure Nb compared with T q = 3.4 kg mm obtained i n t h i s work. The lower value i s a consequence of the [001] o r i e n t a t i o n which they used (Hendrickson (1969)). Their c r y s t a l s were i n general s l i g h t l y l e s s pure than those used i n t h i s study. To f a c i l i t a t e comparison -2 of the hardening e f f e c t s , 0.9 kg mm has been subtracted from the values i n F i g 15a to give agreement with the T q values for pure Nb i n F i g 10. Peters and Hendrickson claimed that t h e i r r e s u l t s indicated a l i n e a r v a r i a t i o n i n T q from pure Nb to pure Ta. This gives a hardening -2 -1 -2 rate of 0.023 kg mm at% , which predicts a hardening of 0,11 kg mm. i n a 5 at% Ta a l l o y . Since the corresponding increase obtained i n t h i s work -2 was 0.9 kg mm , i t i s evident that most of the hardening occurs at .low solute concentrations. The data of Arsenault (1969) ind i c a t e s that (addition of Nb to pure Ta at room temperature also produces hardening, so there e x i s t s a p o s i t i v e deviation from the l i n e a r r e l a t i o n s h i p suggested by Peters and Hendrickson. A s o l u t i o n hardening study of Nb with Ta has recently been performed by Kostorz (1968). Crystals containing 1.8 at% Ta and 11 a t % Ta were deformed i n compression between 295°K and 573°K. Kostorz used a -2 d i f f e r e n t y i e l d parameter and obtained a value of T q = 2.2 kg mm for pure Nb at 295°K. The y i e l d parameters obtained by him at 295°K are shown as a function of c 2 i n F i g 15b and compared with the t e n s i l e data -2 obtained i n t h i s study, a f t e r subtraction of 1.2 kg mm to give agreement with pure Nb. The hardening e f f e c t s are very s i m i l a r and i n d i c a t e parabolic 4 1 4 _ / e • 60 A! CO CO cu u •U CO T J i H fl) •rl 2 _ • Present work • Peters and Hendrickson (1966) Nb 20 40 60 80 Ta C o m p o s i t i o n , a t % Ta CM i 60 Ai CO CO CU u CO T J rH Q> •rl - o 2 h 0 Present work O Kostorz (1968) (Composition, a t % Ta) F i g 15. Comparison o f y i e l d s t r e s s d a t a f o r NbTa a l l o y s w i t h : a) l i n e a r h a r d e n i n g , P e t e r s and H e n d r i c k s o n b) p a r a b o l i c h a r d e n i n g , K o s t o r z 42 hardening, although i t should be r e c a l l e d that the o v e r a l l e f f e c t i s small. Milne and Smallman (1968) have studied the t e n s i l e deformation c h a r a c t e r i s t i c s of NbMo a l l o y s over the complete composition range. Although the c r y s t a l s werenot seeded for a s p e c i f i c o r i e n t a t i o n , several specimens were oriented i n the middle of the stereographic t r i a n g l e . The values ;,of X Q for these specimens are shown i n F i g 16, together with.the r e s u l t s pf t h i s study. The values of x^ for pure Nb are i n agreement. The trend observed at low Mo concentrations i s seen to continue up to a maximum at the , equiatomic composition. Very few r e s u l t s have been published on the e f f e c t of solute on the hardening parameters of bcc metals: most tests have been performed i n compression; i n tension, a l l o y i n g usually induces non-uniform deformation, preventing a determination of 0^ and 6 . In the NbTa a l l o y s studied here, deformation did remain uniform so that the hardening r e s u l t s , though, l i m i t e d , are of i n t e r e s t . There appears to be no e f f e c t of Ta i n the easy g l i d e region; indeed, no s o l u t i o n hardening theory would predict an influence of solute on the hardening rate i n stage I. Since solute might influence the operation of secondary systems, there could be an e f f e c t on stage I I : the r e s u l t s i n d i c a t e a s l i g h t increase i n 0^ upon a l l o y i n g . This i s i n agreement with the very l i m i t e d data obtained by Arsenault and Lawley (1967) for addition of Nb to Ta at t h i s temperature. 4 .2"Tensile behaviour at other than 295°K The Nb a l l o y s were deformed i n tension at temperatures between 77°K and 500°K. Several d i f f e r e n t modes of p l a s t i c deformation were observed: the NbTa al l o y s deformed by s l i p at a l l temperatures, and i n addition i 1 1 r Nb 20 40 60 80 Mo Composition, at% Mo Fig 16. Comparison of yield stress data for NbMo alloys with data of Milne and Smallman. L o 44 twinning was common at 77°K; i n the NbMo a l l o y s twinning or cleavage f a i l u r e , without any evidence of s l i p , always occurred at low temperatures. 4.2.1 S l i p 4.2.1.1 Results Fi g 17 shows the values of T q obtained at d i f f e r e n t temperatures for those specimens which showed a detectable amount of p l a s t i c flow. In the case of the NbTa a l l o y s at 77°K, T q was taken to be the stress at 0.1% shear s t r a i n . The curves for Nb 0.5 Ta and Nb 1.7 Ta have been omitted for c l a r i t y . Above room temperature, deformation was uniform and Nb 6.6 Mo was the only a l l o y to show an upper y i e l d point. At low temperatures, , deformation was observed to become incr e a s i n g l y non-uniform: a f t e r a large i n i t i a l y i e l d point, flow often took place under a continuously decreasing load. I t can be seen from F i g 17 that as the temperature i s reduced, the NbTa a l l o y s show a t r a n s i t i o n from solute hardening to solute softening, although the e f f e c t i s never very marked. There i s no evidence for a s i m i l a r behaviour i n the NbMo a l l o y s . It i s customary to divide the T q (T) curve into two regions,: a) a thermal region, where the flow stress i s a s e n s i t i v e function of temperature, b) an athermal region, where the flow stress i s r e l a t i v e l y i n s e n s i t i v e to temperature, and var i e s i n the same manner as the shear modulus, u . Fig 17. Yield stress as a function of temperature for Nb and Nb alloys. 46 The components of the flow stress i n the thermal region are c a l l e d the thermal stress x* and athermal stress x . The t r a n s i t i o n y temperature T , at which x* f a l l s to zero, depends on the mathematical d e f i n i t i o n used to determine i t . Using the condition 1 . dx > _ 1 , dy . x dT T y dT ;T o o the following values for T q were obtained: Nb 420°K Nb 5 Ta 460 Nb 0.7 Mo 500 Nb 6 Mo 550 From F i g 17 i t can now be seen that the main e f f e c t of a l l o y i n g i s on the o v e r a l l stress l e v e l determined by T , rather than on the temperature s e n s i t i v i t y of the flow stress determined by x*. At 77°K, two of the s i x NbTa specimens did not twin before y i e l d so that a complete flow curve was obtained. Flow was wavy r i g h t up to the maximum load, as i l l u s t r a t e d i n F i g 18. The fractured specimen showed a l o c a l reduction i n area at three places along the gauge. One of these was the fr a c t u r e s i t e , which showed intense s l i p and about 90% reduction in, area. It i s notable that l o c a l i z e d f a i l u r e did not occur a f t e r the f i r s t upper y i e l d point, and that the specimen was subsequently capable of e x h i b i t i n g a high work hardening rate. 4.2.1.2 Discussion >• 4.2.1.2.1 Work hardening and uniformity of deformation The arguments presented to explain the change of y i e l d point with 48 a l l o y i n g ( s t a b i l i t y theory) can also p r e d i c t the increase i n y i e l d drop at low temperatures and i t s disappearance at high temperatures, i f i t i s assumed that the work hardening rate does not change s i g n i f i c a n t l y with temperature. Since two-stage hardening i s observed only at room temperature and above, the hardening rates below room temperature must be compared with stage II hardening rates above 295°K. However, because of the onset of non-uniform deformation, the determination of work hardening rates becomes doubtful at low temperatures. This d i f f i c u l t y i s r e f l e c t e d i n the r e s u l t s of M i t c h e l l et a l (1963), who obtained scattered data consistent with either of the following p o s s i b i l i t i e s : a) the values of 6 ^ may increase s l i g h t l y at low temperatures. They rejected t h i s p o s s i b i l i t y on the grounds that the apparent values of 0 j j were too high because of the observed occurrence of non-uniform deformation. However i t i s also possible that non-uniform deformation may lead to apparent values of 0.^ which are too low. b) a maximum i n the value of 0 ^ may occur at about 250°K... They favoured t h i s p o s s i b i l i t y ; s i m i l a r behaviour has been reported by Mordike (1962) and by Keh and Weissman (1963). However the maximum always appears to coincide with the point of rapid increase i n y i e l d stress (and onset of r e s t r i c t e d d u c t i l i t y ) , with the accompanying p o s s i b i l i t y of, non-uniform deformation. In compression t e s t i n g , deformation may i n i t i a l l y be stable even at low temperatures, and i t has then been observed that the, work hardening rate (in Mo) does increase with decreasing temperature CPr.ekel and ..Conrad (1968)). r 49 In the case o f t h e two NbTa a l l o y s which were d u c t i l e a t 77°K, the work h a r d e n i n g r a t e must have been s u f f i c i e n t l y h i g h to s t a b i l i z e a neck once i t had formed. I t i s s u g g e s t e d t h a t t h e t h r e e l a r g e y i e l d d rops observed i n F i g 18 c o r r e s p o n d to the o n s e t of l o c a l i z e d d e f o r m a t i o n i n the t h r e e r e g i o n s observed on the specimen a f t e r f a i l u r e . A t 77°K, b o t h i n t e r s t i t i a l and s u b s t i t u t i o n a l atoms a r e e s s e n t i a l l y immobile (Thomas and Leak (1954)); i t t h e r e f o r e appears t h a t no mechanism such as dynamic s t r a i n a g i n g c o u l d account f o r the wavy f l o w . The e f f e c t c o u l d be a r e s u l t o f s u c c e s s i v e l o c a l i z e d s l i p and h a r d e n i n g , s u g g e s t i n g t h a t d e f o r m a t i o n o c c u r r e d v e r y c l o s e to the s t a b i l i t y c o n d i t i o n . S i n c e t h e y i e l d s t r e s s was h i g h , s t a b i l i t y would r e q u i r e an i n c r e a s e i n work h a r d e n i n g r a t e above the room temperature v a l u e . 4.2.1.2.2 S o l u t e h a r d e n i n g and s o f t e n i n g The o n s e t of s o l u t i o n s o f t e n i n g a t low temperatures and a t s m a l l s o l u t e c o n c e n t r a t i o n s has been f r e q u e n t l y o b s e r v e d i n s i n g l e c r y s t a l b c c a l l o y systems (eg M i t c h e l l and R a f f o (1967)). The reduced temperature s e n s i t i v i t y o f the d i l u t e a l l o y produces a minimum i n the y i e l d s t r e s s v e r s u s c o m p o s i t i o n c u r v e . The e f f e c t has been o b s e r v e d up to 175°K i n the c a s e of TaRe a l l o y s ( R a f f o and M i t c h e l l (1968)), and up to 220°K i n WRe a l l o y s ( R a f f o (1969)). T h i s l a t t e r o b s e r v a t i o n was u n u s u a l i n t h a t a t 77°K, co n t i n u o u s s o f t e n i n g o c c u r r e d up to a c o m p o s i t i o n o f 25 a t % Re; the minimum y i e l d s t r e s s i n o t h e r systems i s u s u a l l y a t 1 - 4 a t % s o l u t e . S o l u t i o n s o f t e n i n g has a l s o been r e p o r t e d i n b c c i n t e r s t i t i a l s o l u t i o n s a t low temperatures ( C h r i s t e t a l (1969)). The e f f e c t has been s t u d i e d r e c e n t l y i n Nb s i n g l e c r y s t a l s by R a v i and G i b a l a (1969), who added c o n t r o l l e d amounts o f oxygen. 50 As F i g 17 shows, the e f f e c t was observed i n the NbTa a l l o y s , i n which s l i g h t continuous softening occurred up to 4.8 at % Ta at 77°K; there must be a minimum at some higher Ta concentration because pure Ta i s stronger than pure Nb. Unfortunately, because of the onset of b r i t t l e f r a c t u r e at low temperatures, i t was not possible to inves t i g a t e a so l u t i o n softening e f f e c t i n the NbMo a l l o y s . However Statham (1968) has observed softening of Nb c r y s t a l s by addition of 2 at % Mo when tested i n compression at 77°K. I t i s therefore probable that a softening would have been observed i n NbMo at 77°K i f p l a s t i c flow had occurred; a reduction i n twinning stress was i n fact observed (see F i g 20). Various arguments have been presented to explain the so l u t i o n softening e f f e c t . One group of authors (Raffo and M i t c h e l l (1968), , Arsenault (1969)) considers that solute atoms produce a l o c a l i z e d reduction i n the P e i e r l s s t r e s s . The minimum i n the stress versus composition curve T q ( C ) , i s then a consequence of a continuously decreasing thermal component T*(C), and a continuously increasing athermal component T^(C). At low temperatures, x*(c) i n i t i a l l y predominates; at high temperatures i t i s * n e g l i g i b l e and so l u t i o n hardening occurs. However the published data i s not completely consistent with t h i s i n t e r p r e t a t i o n . F i g 19 shows the composition dependence of y i e l d stress for TaRe a l l o y s observed by M i t c h e l l and Raffo (1967) at 77°K and 623°K. According to the above i n t e r p r e t a t i o n , the two curves are represented by the equations: I x ( c ) ? 7 = T * ( C ) 7 7 + T ^ ( C ) 7 7 (1) and t ( c ) 6 2 3 = T * ( c ) 6 2 3 + T y ( c ) 6 2 3 •  ( 2 ) since T = 0 for T > T' . • o Since T i s not very dependent on temperature, T u ( G ) 7 7 * T U ( C ) 6 2 3 51 Fig 19. Composition dependence of yield stress for TaRe alloys. (after Mitchell and Raffo (1967)) therefore from equations (1) and (2) x * ( c ) 7 7 - T ( C ) 7 7 - T ( C ) & 2 3 (3) This equation (3) i s represented by the dotted curve i n F i g 19. I t can be seen that there i s no possible way i n which x* can continuously decrease with a d d i t i o n a l solute, as the theory assumes. The microstrain technique can u s e f u l l y be applied to t h i s problem and further discussion w i l l be delayed u n t i l the m i crostrain results, have been presented. , 4.2.2 Twinning At 77°K, sporadic twinning was observed i n pure Nb and i n the. NbTa a l l o y s . The accompanying load drop and audible c l i c k were correlated with the formation of a v i s i b l e twin r i g h t across the specimen. Frequently the operation of one twin system was followed by a second. F i g 21 i s a micrograph showing two i n t e r s e c t i n g sets of twins. The twin systems were i d e n t i f i e d using the technique outlined i n A.3.3 and were found to be (112) [111] and (112) [111]. For the o r i e n t a t i o n used, there are s i x possible twin systems which give an extension on twinning; the observed systems were the two with the larges t Schmid factors (0.378 and 0.359 r e s p e c t i v e l y ) . In the remaining cases, the stress at which twinning started was resolved on the most favourable twin system. Although twinning started at d i f f e r e n t points on the flow curve, the values of resolved twinning stress were quite consistent t h i s indicates that a c r i t i c a l stress e x i s t s i n order for twinning to occur. The r e s u l t s are shown i n F i g 20. I t can be seen that addition of solute reduces the twinning stress at 77°K, j u s t as i t reduces the y i e l d stress at that temperature. Fig 20. Resolved twin stresses in Nb alloys at 77 °K. F i g 22. (001) Cleavage p l a n e i n Nb 6.6 Mo a l l o y (x230). 55 The only Mo a l l o y tested twinned without y i e l d i n g , at a much lower stress than did pure Nb. Twinning has frequently been reported i n Nb c r y s t a l s deformed at 77°K (eg Bowen et a l (1967)). The incidence of twinning appears to increase as i n t e r s t i t i a l impurities are removed. Although twinning was observed i n both the NbMo and NbTa a l l o y s , i t may be suppressed at higher solute concentrations since Milne and Smallman (1968) did not observe twinning i n Nb 50 Mo. Twinning at 77°K could be prevented by p r e s t r a i n i n g the c r y s t a l s at room temperature. 4.2.3 Cleavage At 190°K, the more concentrated Mo a l l o y s f a i l e d by cleavage at the specimen shoulders, without any i n d i c a t i o n of p l a s t i c flow. Even prestra i n i n g the c r y s t a l s at room temperature did not prevent b r i t t l e f r a c t u r e . Since the stress concentration at the grips i n i t i a t e d the f a i l u r e , cleavage stresses were not determined. The cleavage plane was found by Laue b a c k - r e f l e c t i o n photography to be (001) i n a l l cases. This i s the usual cleavage plane i n bcc metals. A photomicrograph of the f r a c t u r e surface i s shown i n F i g 22. The surface i s not smooth but i s covered with steps or " r i v e r l i n e s " r a d i a t i n g from the fracture i n i t i a t i o n s i t e . These observations are i d e n t i c a l to those ( reported by Raffo and M i t c h e l l (1968) on cleavage i n TaRe s i n g l e c r y s t a l s . 4.3 S l i p l i n e observations at 295°K 4.3.1 Results S l i p l i n e observations were made on the c y l i n d r i c a l t e n s i l e specimens, and the operative s l i p systems were determined by measuring the traces (see A.3.3). Even though the specimens were oriented f o r s i n g l e s l i p , s l i g h t a c t i v i t y was often observed on a second system immediately a f t e r y i e l d However the amount of t h i s s l i p did not appear to increase, and i n stage I there was always one p r i n c i p a l system, [111]^. The value of \j> was determined for a l l the a l l o y s . : The s l i p traces were observed to be wavy on one face and s t r a i g h t on a face at 90° to the f i r s t . The wavy l i n e s appear on the face from which edge d i s l o c a t i o n s emerge, and trace the motion of screw d i s l o c a t i o n s . S i m i l a r l y the s t r a i g h t l i n e s depict the motion of edge d i s l o c a t i o n s , and these l i n e s disappear at a p o s i t i o n exactly perpendicular to the Burgers vector. F i g 23 shows t y p i c a l micrographs, taken from the posit i o n s shown, approximately 90° apart. There were no s i g n i f i c a n t differences i n the form of s l i p traces observed i n pure Nb and i n the a l l o y s . . With pure Nb and the NbTa a l l o y s , the onset of stage II agreed t h e o r e t i c a l l y with that expected from the movement of the t e n s i l e axis during deformation. However the p o s i t i o n of the t e n s i l e axis subsequently overshot the [001] -r [101] symmetry boundary by 7 - 10°. S l i p on the conjugate system was observed i n stage I I . The r e s u l t s of the s l i p determinations i n the a l l o y s are shown i n Table I I . The o r i g i n a l o r i e n t a t i o n i s indicated by the angle x» which i s the angle between the maximum resolved shear stress plane and the (011) reference plane. Since s l i p was observed to be always i n the [111] d i r e c t i o n F i g 23. S l i p l i n e observations i n Nb a l l o y s deformed into stage I, from (A) and (B). 58 Table II. Crystallography of slip in Nb alloys at 295°K. Specimen Orientation Slip parameter Stage II X° V overshoot (°) A3 3 6 6 9 A34 6 4 7 B42 10 14 7 B43 10 20 9 C41 3 0 10 C44 3 13 9 D42 10 14 8 J22 -4 3 -J24 -4 -4 -K22 8 14 -K23 8 6 • -L31 7 7 -L33 7 2 59 the t e n s i l e a x i s moves d u r i n g d e f o r m a t i o n a l o n g a g r e a t c i r c l e towards [111] and x i s t h e r e f o r e independent o f s t r a i n . F i g 24 i n c l u d e s t h e o b s e r v e d s l i p parameter IJJ compared w i t h the v a l u e s e x p e c t e d i f s l i p were c r y s t a l l o g r a p h i c . I t can be seen t h a t s l i p d i d not u s u a l l y o c c u r on ( O i l ) . I n a d d i t i o n , t h e r e c o u l d be some c l u s t e r i n g about (143) and ( 1 3 2 ) . However, i f an e r r o r o f ±2° i s ad m i t t e d i n ip, then o n l y 54% of the ca s e s can be d e s c r i b e d as c r y s t a l l o g r a p h i c compared w i t h 48%; i f xl> was randomly d i s t r i b u t e d between -5° and +20°. T h e r e f o r e s l i p cannot be d e s c r i b e d as b e i n g c r y s t a l l o g r a p h i c . There i s no e v i d e n c e f o r any dependence o f \p on a l l o y c o n t e n t < 4.3.2 D i s c u s s i o n There appear to be no r e p o r t s o f s l i p l i n e s t u d i e s on NbTa or NbMo a l l o y s . T h i s i s because most workers have been concerned w i t h , measurements of x . I t i s u s u a l to r e s o l v e the y i e l d s t r e s s onto the ( O i l ) p l a n e , which i s a r e a s o n a b l e a p p r o x i m a t i o n f o r c r y s t a l s o r i e n t e d i n the mi d d l e of the s t e r e o g r a p h i c t r i a n g l e . M i l n e and Smallman (1968) found s l i p l i n e a n a l y s i s on NbMo a l l o y s to be i m p o s s i b l e because o f t h e c o n f u s i o n of s u r f a c e markings on t h e i r samples. They determined s l i p systems by a s t e r i s m a n a l y s i s . I n no eases d i d they o b s e r v e s l i p on ( O i l ) ; the u s u a l s l i p p l a n e was r e p o r t e d t o be (132) o r ( 1 4 3 ) . T h i s i s i n agreement w i t h the p r e s e n t r e s u l t s a l t h o u g h ; t h e p o s s i b i l i t y o f c r y s t a l l o g r a p h i c s l i p on t h e s e p l a n e s i s d i s c o u n t e d . . S l i p on any p l a n e c o u l d be produced by composite s l i p on d i f f e r e n t {011} p l a n e s (Maddin and Chen ( 1 9 5 3 ) ) , but t h e r e i s some e v i d e n c e t h a t {112} c o u l d a l s o be a d i s c r e t e s l i p p l a n e (see A . 3 . 2 ) . I t i s e v i d e n t from,the s l i p l i n e r e s u l t s and from F i g 23(A) t h a t c r o s s - s l i p was p r e v a l e n t i n the d e f o r m a t i o n o f the Nb a l l o y s . T h i s d i f f e r s from t h e o b s e r v a t i o n s o f j M i t c h e l l 60 3 0 - r .(121) 25 20-15 I 10 + 5 + (132) (143) o - 0 — • + - 5 / / )/ y - 5 io o V • Nb Nb 0.5 Ta Nb 1.7 Ta Nb 4.8 Ta Nb 0.9 Mo Nb 4.9 Mo Nb 6.6 Mo (Oil) F i g 2 4 . R e s u l t s o f s l i p l i n e a n a l y s e s on Nb a l l o y s e x p r e s s e d as * K x ) • and R a f f o (1967) on TaRe a l l o y s . They r e p o r t e d s l i p on ( O i l ) i n a l l the a l l o y s , whereas pure Ta s l i p p e d on t h e maximum r e s o l v e d shear s t r e s s p l a n e . I t i s u s u a l to e x p r e s s an ob s e r v e d o r i e n t a t i o n dependence o f s l i p i n terms o f i/i(x) c u r v e s . A l t h o u g h the c r y s t a l s t e s t e d h e r e had s i m i l a r o r i e n t a t i o n s , t h e r e was a sp r e a d o f 14° i n the measured x v a l u e s . The p o s i t i o n s o f >Kx) a r e shown i n F i g 24 and t h e r e i s seen t o be a d e f i n i t e o r i e n t a t i o n dependence, a l t h o u g h i t does n o t take a s i m p l e form. F o r i n t e r m e d i a t e o r i e n t a t i o n s (5° < X < 10°) the s l i p p l a n e i s c l o s e to the maximum r e s o l v e d shear s t r e s s p l a n e (\j> = x)« A t low x t h e r e a r e d e v i a t i o n s towards ( O i l ) and a t h i g h x> d e v i a t i o n s away from ( O i l ) . 62 5. MICRODEFORMATION OF PURE NIOBIUM CRYSTALS 5.1 Introduction 5.1.1 M i c r o s t r a i n observations By load c y c l i n g specimens to successively higher str e s s e s , d i f f e r e n t points i n the y i e l d i n g process can be i d e n t i f i e d : the e l a s t i c l i m i t , x , i s the stress at which r e v e r s i b l e d i s l o c a t i o n motion f i r s t e occurs: the a n e l a s t i c l i m i t ; T , i s the stress at which i r r e v e r s i b l e a d i s l o c a t i o n motion f i r s t occurs; the macroscopic flow s t r e s s , x^°, i s the r e l a t i v e l y constant stress at which gross p l a s t i c flow occurs. The e l a s t i c and a n e l a s t i c l i m i t s depend on the s e n s i t i v i t y of the s t r a i n measurement; a s t r a i n s e n s i t i v i t y of 10 ^ has usually been taken as standard. In order to e s t a b l i s h a hysteresis loop and obtain r e v e r s i b l e , d i s l o c a t i o n motion i t i s necessary to create a " d i r e c t i o n a l " i n t e r n a l stress f i e l d (with stress component x/*) which w i l l allow d i s l o c a t i o n s to return to t h e i r o r i g i n a l positions when the applied stress i s removed. This may be accomplished by p r e s t r a i n i n g the annealed material; f u r t h e r , a supply of mobile d i s l o c a t i o n s i s thereby obtained i n c r y s t a l s where the d i s l o c a t i o n s may have been locked. Before p r e s t r a i n i n g , an i n t e r n a l stress f i e l d e x i s t s i n an-; ; annealed c r y s t a l . This w i l l be termed an " a d i r e c t i o n a l " stress f i e l d (with stress component x^°) i n that i t opposes the motion of d i s l o c a t i o n s h i r r e s p e c t i v e of t h e i r d i r e c t i o n of motion (cf a "drag s t r e s s " ) . The r e l a t i v e magnitudes of and x^° are not s p e c i f i e d . The following i d e a l i z e d model i l l u s t r a t e s the manner i n which a stable hysteresis loop i s established i n an annealed c r y s t a l where x ^ i s initially zero. Suppose dislocations move when the effective stress o on them (T ) reaches a critical value x ( > T . ) . At any applied stress x s - x x,the effective stress is given by d T T - T. X 1 Consider load cycling to successively higher stresses as shown in the diagrams of Fig 25. In A, at a stress x = Ax , x.^ = 0 . Therefore x = Ax , which 1 x is less than x and so no dislocation motion occurs, s In B, dislocations first move when x = x = x and they continue X s to do so as long as the load increases. When straining is stopped at x = x + Ax , x.^  = Ax . If there is no relaxation on unloading then s i x.^ = Ax at x = 0 and therefore l x = -Ax x This stress is s t i l l less than x and the dislocations do not move. s No reversible dislocation motion has yet occurred. In C, at x = x + Ax , x.^  = Ax . Therefore x = x and dislocations ' s i s move again. When straining is stopped at x = 2xg + Ax , = x g + Ax. On unloading to x = Ax , x = Ax - (x + Ax) = -x . & x s s Therefore dislocations now move backwards until at x = 0 x = -x and x.^  = x . x s l s A stable hysteresis loop, as in D, has now been formed and reversible dislocation motion will always occur when x = x = x - (-x ) x s s By definition this applied stress is x and x = 2x ^2x.9 e s l S t r e s s S t r a i n F i g 25. Model f o r f o r m a t i o n of a s t a b l e h y s t e r e s i s l o o p . 65 The hysteresis loop w i l l remain stable u n t i l i r r e v e r s i b l e d i s l o c a t i o n motion occurs at a stress x , at which the loop does not close. 5.1.2 Previous work The microstrain technique was f i r s t applied to the study of y i e l d i n g and flow i n bcc metals by Brown and h i s co-workers (Brown and Ek v a l l (1962)). The procedure they adopted i s indicated schematically i n Fig 26a. The e l a s t i c l i m i t , T , i s the stress at which a loop was f i r s t formed and the a n e l a s t i c l i m i t , T , i s the stress at which permanent set f i r s t occurred. Brown and E k v a l l and l a t e r , Kossowsky and Brown (1966), investigated the dependence of x and x on p u r i t y , temperature, and; e a p r e s t r a i n . They i d e n t i f i e d X £ with the stress to move e x i s t i n g kinks on d i s l o c a t i o n s ; x was i d e n t i f i e d with a P e i e r l s stress to move screw a d i s l o c a t i o n s . ;, The uniqueness of x g depends very much on the form of the hyst e r e s i s loops; i n tension, specimen alignment i s c r i t i c a l and i t i s unlikely.-.that a n e l a s t i c deformation i s completely uniform. In compression, Meakin (1967) observed hysteresis loops that were p a r a l l e l i p i p e d s rather than being of l e n t i c u l a r shape (see F i g 26b); the values of T £ were t y p i c a l l y l e s s ;than one-tenth of the values reported f o r tension. , Meakin suggested that the x g observed i n compression represents the true e l a s t i c l i m i t . In f a c t , up to th i s stress the modulus agreed with the "dynamic modulus" calculated from the dynamically determined e l a s t i c constants (see A,5.1). Above x the extensive l i n e a r a n e l a s t i c s t r a i n e contribution gave a smaller "relaxed modulus" which was t y p i c a l l y 30% - 50% of the dynamic value. Meakin further suggested that i n the case of t e n s i l e deformation the true e l a s t i c l i m i t was not observed and that the apparent value reported by Brown and E k v a l l coincided with a deviation from the 66 Strain Fig 26. Types of hysteresis loops observed in: a) tension b) compression (Meakin (1967)) 67 relaxed modulus rather than from the dynamic modulus. I t i s quite possible that i n tension the true e l a s t i c l i m i t i s not observed, but nevertheless the observed modulus cannot be i d e n t i f i e d with the relaxed modulus observed i n compression. In a l l the cases examined i n t h i s work, the observed modulus i n tension agreed with the calculated dynamic modulus to wi t h i n 10%. In t h i s study no attempt was made to i d e n t i f y an e l a s t i c l i m i t . The nature of p l a s t i c flow from the a n e l a s t i c l i m i t , which can be i d e n t i f i e d unequivocally, up to macroflow was considered to be of greater i n t e r e s t . Previous methods for determining such a microflow curve have had disadvantages. For example, S t o l o f f et al(1965) and Davies and Ku (1966) constructed curves by load c y c l i n g r i g h t up to macroflow, but t h i s method records some s t r a i n which occurs at stresses below the maximum reached i n any cycle., On the other hand, Carnahan et a l (1967) used continuous s t r a i n i n g but,, i n the determination of p l a s t i c shear s t r a i n , could not allow for non-linear a n e l a s t i c deformation occurring before microyield. Such s t r a i n s are.: of the same order as the p l a s t i c s t r a i n s i n the region of microyield. The technique adopted i n t h i s work overcomes both of these d i f f i c u l t i e s . 5.1.3 Experimental procedure The t r a n s i t i o n from micro to macroyielding was observed during t e n s i l e deformation of c r y s t a l s which had a capacitance extensometer attached d i r e c t l y to the specimen gauge length (see A.4 for experimental d e t a i l s ) . The onset of y i e l d i n g was determined by load c y c l i n g up to the stress at which permanent set f i r s t occurred (anelastic l i m i t or microyield s t r e s s , T ). This point also defined the e l a s t i c slope (or more c o r r e c t l y , 3. the " a n e l a s t i c slope") of the microflow curve which was subsequently, obtained by continuous s t r a i n i n g up to the macroflow s t r e s s , T °. This i s defined as 68 the applied stress when gross p l a s t i c deformation occurs at a rate determined by the crosshead speed. Thus i s asymptotically approached a f t e r y i e l d i n g , but the s t r a i n to reach i s not s p e c i f i e d . At room temperature x^° i s the same as the p r e s t r a i n s t r e s s , x^, as obtained from the macroscopic flow curve. The method used to determine the microflow curve x(y) i s indicated schematically i n F i g 27. A small p o s i t i v e load was taken as the base stress l e v e l to maintain alignment. A r e s u l t i n g microflow curve, p l o t t e d as x(log y)> i s shown i n F i g 28. For any t e s t , the v e r t i c a l l i n e forming part of the "L" motif represents the s t r a i n s e n s i t i v i t y l i m i t for the t e s t ; the h o r i z o n t a l l i n e represents the highest stress reached before a permanent set was observed. The c i r c l e d point represents the value obtained by load c y c l i n g , and the remaining points have been calculated from the continuous loading curve. This method of expressing microflow curves avoids the tendency to extrapolate the microyield stress to zero s t r a i n , which i s tempting when curves are plotted on a l i n e a r s t r a i n scale (see 6.2.1). ; . 5.1.4 Re p r o d u c i b i l i t y of microflow Although some microflow experiments were performed d i r e c t l y on as-grown c r y s t a l s , i t was usual to p r e s t r a i n the c r y s t a l s into stage I before t e s t i n g . The microflow curve was found to be reproducible i n t h i s region provided the c r y s t a l was prestrained at the flow stress x^ before each determination. I f the specimen was given a small p r e s t r a i n to a stress ,x^  below the flow s t r e s s , then the subsequent apparent microyield stres x • was 3. increased. The behaviour i s indicated schematically i n F i g 29. This means that, i n any microyield determination, the apparent microyield stress passes through a maximum as the p r e s t r a i n stress increases from X q to x^. Such behaviour, though not investigated q u a n t i t a t i v e l y , i s important as a Stress Elongation Fi g 27. Technique for obtaining microflow curves, --5 -4 log (shear s t r a i n , y) -3 F i g 28. Representation of microflow data. manifestation of a hardening process i n the microstrain region which ceases at the flow s t r e s s . I t i s also suggested that the maximum value of x 1 3. would represent a stress at which d i s l o c a t i o n m u l t i p l i c a t i o n becomes s i g n i f i c a n t . 5.1.5 Discussion of microflow It has been seen that x ' i s very s e n s i t i v e to the p r e s t r a i n stress T^, although the s t r a i n involved i n pr e s t r a i n i n g to x^ i s very small (y - 10 The d i s l o c a t i o n motion involved i s so small r e l a t i v e to that required to produce a s i m i l a r hardening i n the macroflow region, that the e f f e c t cannot be due to any gross change i n d i s l o c a t i o n density. Furthermore, even when the microyield stress i s r a i s e d , the subsequent asymptotic flow stress i s unaffected, which again suggests that the e f f e c t i s due to a s i g n i f i c a n t change i n a small f r a c t i o n of the d i s l o c a t i o n s : rather than to a change i n the o v e r a l l d i s l o c a t i o n structure. One of the d i f f i c u l t i e s associated with the interpretation, of microyield values i s that i t i s not known how many d i s l o c a t i o n s are involved i n microyielding. Thus the p a r t i c u l a r s t r a i n at which x i s measured 3 could be produced by a small number of d i s l o c a t i o n s moving a large distance, or by a large number of d i s l o c a t i o n s moving a small distance. Since only a small f r a c t i o n of the t o t a l d i s l o c a t i o n s move, a change i n th i s proportion could account for the observed change i n microyield stress between x. and x ' 3 3 One p o s s i b i l i t y i s that p r e s t r a i n i n g to x^ a f t e r unloading from x^ increases the number of d i s l o c a t i o n s which can move at a given s t r e s s , but i t i s not p h y s i c a l l y acceptable to suppose that they then require a higher stress to produce the same s t r a i n . On the other hand p r e s t r a i n i n g could decrease the number of di s l o c a t i o n s which can move at the p a r t i c u l a r s t r e s s . I t would then be required that the stress to move them increase F i g 29. Schematic d e s c r i p t i o n of microflow behaviour observed during stage I deformation at 295 °K. F i g 30. Model f o r movement of a d i s l o c a t i o n h a l f - l o o p i n the m i c r o f l o w r e g i o n . 72 quite r a p i d l y with the distance moved. There appear to be two possible explanations for th i s stress increase: a) The mean value of the f l u c t u a t i n g i n t e r n a l stress f i e l d could increase as d i s l o c a t i o n s move from t h e i r equilibrium p o s i t i o n s . On s t r a i n i n g above T some d i s l o c a t i o n s would be stopped or pass out of the c r y s t a l , c l thereby reducing the mobile d i s l o c a t i o n density. On unloading and reloading, the remaining d i s l o c a t i o n s would have to move further through the i n t e r n a l stress f i e l d , producing a larger microyield stress x '. On r e s t r a i n i n g 9. at the flow s t r e s s , d i s l o c a t i o n m u l t i p l i c a t i o n would again produce a d i s t r i b u t i o n of moveable d i s l o c a t i o n s i n equilibrium with the i n t e r n a l stress f i e l d . ; b) I f i t i s r e a l i z e d that a d i s l o c a t i o n l i n e cannot end in s i d e a c r y s t a l , and that consequently most d i s l o c a t i o n s must be i n the form of loops or networks, then the p o s s i b i l i t y e x i s t s that a loop may not expand uniformly at a given s t r e s s . Thus i f only a p a r t i c u l a r part of a d i s l o c a t i o n loop can move at x , the proportion of t h i s part w i l l decrease as the loop expands. Therefore a f t e r unloading and reloading i t w i l l be necessary to apply a higher stres x ' to move a d i f f e r e n t portion of the d i s l o c a t i o n , loop. Further s t r a i n i n g at the flow stress w i l l move the whole loop, which w i l l subsequently take up an "equilibrium" configuration again. ; The second suggestion has fewer independent requirements and can be supported i n d i r e c t l y by evidence obtained from transmission electron microscopy. Solomon and McMahon (1968) have examined d i s l o c a t i o n r e -arrangements at 77°K i n the microstrain region of Fe p o l y c r y s t a l s prestrained 73 at room temperature. They observed that there i s a tendency for the d i s l o c a t i o n s to a l i g n i n the screw o r i e n t a t i o n . Thus at 77°K the edge components of the d i s l o c a t i o n tangles move p r e f e r e n t i a l l y and consequently become exhausted. On continued s t r a i n i n g at higher stresses, the screw d i s l o c a t i o n s move. These observations have also b een reported i n Mo sin g l e c r y s t a l s by Lawley and Gaigher (1964) and i n Fe s i n g l e c r y s t a l s by Keh (1968). On the basis of these experiments, Solomon and McMahon : ; accounted for the observed decrease i n m i c r o p l a s t i c response on successive loading cycles at 77°K; i t i s therefore proposed that a s i m i l a r exhaustion hardening of edge d i s l o c a t i o n s accounts for the preliminary microflow observations reported here at 295°K. The suggested model i s shown i n Fig 30 which i l l u s t r a t e s the r e l a t i v e motion of edge and screw components i n a d i s l o c a t i o n loop during s t r a i n i n g i n the microflow region. . 5.2 Results 5.2.1 Deformation at 295°K Fig 31 includes microflow curves for a Nb c r y s t a l (A62) i n two d i f f e r e n t conditions: as-grown and prestrained 10% to the beginning of stage I. I t can be seen that although s t r a i n i n g has increased the macroflow stress as expected, i t has decreased the microyield value. F i g 32 shows microflow curves for the same c r y s t a l at s t r a i n s corresponding roughly to the beginning and end of uniform stage I deformation. Further s t r a i n i n g has increased both the microyield and macroflow values. F i g 31 also shows microflow data f o r two as-grown c r y s t a l s to i l l u s t r a t e the e f f e c t of p u r i t y . The lowest curve has been plotted from the data of Bowen et a l (1967) who used u l t r a - h i g h vacuum p u r i f i e d material (specimen 55/6, tested i n compression). For the le s s pure as-grown c r y s t a l , A62, As-grown, present work • Bowen et a l (1967) A P r e s t r a i n e d , present work • L A -A -r=io°/< r=o -I 55/6 -5 -3 ...log (shear s t r a i n , y) 31. M i c r o f l o w c u r v e s a t 295 °K f o r Nb c r y s t a l s i n d i f f e r e n t c o n d i t i o n s . 76 both the micro and macroflow values are at l e a s t three times greater than for specimen 55/6. Specimen A62 subsequently showed a y i e l d drop; specimen 55/6 did not. 5.2.2 Deformation at 160°K Fig 33 shows microflow curves for c r y s t a l s i n d i f f e r e n t conditions tested at 160°K: as-grown (A63), prestrained at 295°K (A64), and high pu r i t y as-grown (55/4) which has again been plotted from the data of Bowen et a l . I t can be seen that at 160°K the e f f e c t of p u r i t y i s f a r l e s s pronounced than at 295°K (Fig 31) and that the impure prestrained c r y s t a l has deformed at a lower stress than the high p u r i t y as-grown specimen. Microyield values at the given s t r a i n s e n s i t i v i t i e s are shown i n F i g 34 as a function of temperature. The temperature s e n s i t i v i t y of the dT microyield stress ( -j^— ) i n the prestrained c r y s t a l s i s considerably le s s than i t i s i n the as-grown c r y s t a l s which i n turn i s less than the temperature dT s e n s i t i v i t y of the macroyield stress ( -pp- ) (see F i g 35). Since the stress T^° i s , by d e f i n i t i o n , asymptotically approached a f t e r y i e l d i n g , i t can be seen that the p l a s t i c s t r a i n to reach T^° increases as the temperature decreases. This i s a consequence of the greater s l o p e ' O f the microflow curve at low temperatures. For example, at 295°K (Fig 31) the asymptotic flow stress T^° i s reached a f t e r about y = 0.1%, but at 160°K (Fig 33) the stress does not reach T^° (which i s close to the unprestrained flow stress at 160°K) u n t i l about y = 5%. These e f f e c t s are i l l u s t r a t e d i n F i g 35 which shows the flow stress at various s t r a i n s as a function of temperature. I t can be seen that at very low temperatures the asymptotic flow stress a f t e r p r e s t r a i n i n g w i l l never reach the unprestrained flow s t r e s s . , ; -5 -4 -3 l o g (shear s t r a i n , y) F i g 33. M i c r o f l o w c u r v e s a t 160°K f o r Nb c r y s t a l s i n d i f f e r e n t c o n d i t i o n s . 100 150 200 250 300 350 Temperature, °K Mic r o y i e l d stresses at 160°K and 295°K f o r Nb c r y s t a l s in. d i f f e r e n t conditions. 35. Flow s t r e s s a t v a r i o u s s t r a i n s as a f u n c t i o n o f temperature. 80 5.2.3 Strain-rate during microflow For a given s t r a i n the slope pf the microflow curve v a r i e s with temperature, and as a consequence the true p l a s t i c s t r a i n - r a t e w i l l also vary with temperature. This i s true because the constant crosshead rate X can always be expressed as the sum of the p l a s t i c s t r a i n - r a t e of the specimen e ^ and the e l a s t i c s t r a i n - r a t e of the specimen plus machine E 0 : By Hookes Law, 1 • ex. M where M i s an e l a s t i c modulus. Changing from a(e) to T(Y), and s u b s t i t u t i n g (2) i n (1) gives x = ^ _ dr + s dY M.s dt dt B u t A A A d_r _ dj_ dy_ dt ~ dy*dt therefore Equation (3) shows that the true p l a s t i c s t r a i n - r a t e i s a function dx of the slope of the microflow curve ( )• In f a c t , throughout s t r a i n i n g i n a t e n s i l e t e s t , the p l a s t i c s t r a i n - r a t e w i l l increase continuously from dx zero i n the e l a s t i c region ( »-<*>) to a constant value determined by dY dx the crosshead speed at the asymptotic flow stress ( = 0 ). In t h i s work, i t was possible to determine the true p l a s t i c s t r a i n -rate at any s t r a i n by combining the load-elongation data from the X-Y recorder with the load-time data from the Instron chart. I t was found that at room temperature, the s t r a i n - r a t e was a s i m i l a r function of s t r a i n for a l l specimens. An example i s shown i n F i g 36. At 77°K the s t r a i n - r a t e 81 10 -2 u cu Xi CO 4-1 CO n) i H 10 -3 10 -4 10 -5 10 -7 10 -6 10 -5 10 -4 10 -3 Plastic shear strain-rate, y sec -1 Fig 36. Instantaneous strain and strain-rate during microflow at 295°K and 77°K (Nb 4.8 Ta alloy). 82 does not increase as r a p i d l y with s t r a i n as at 295°K, although the s t r a i n -rate at microyield i s about the same at both temperatures. The maximum range of s t r a i n - r a t e over the whole microflow curve i s about two orders of magnitude. 5.3 Discussion 5.3.1 Deformation at 295°K The difference between the microflow curves f o r as-grown and prestrained c r y s t a l s i s of considerable i n t e r e s t (Fig 31). The increase i n -3 stress l e v e l at large s t r a i n s (y - 10 ) i s r e a d i l y explained by the expected increase i n t o t a l d i s l o c a t i o n density with p r e s t r a i n . However the stress to produce a permanent p l a s t i c s t r a i n of y = 10 has been almost halved by the p r i o r deformation. As suggested i n 5.1.5, t h i s behaviour could be a consequence of a large increase i n the number of d i s l o c a t i o n s which can move at low stresses. The behaviour i s therefore s i m i l a r to the e f f e c t s observed i n the preliminary microflow investigations of s t r a i n i n g i n stage I (see 5.1.4). I t was found that the apparent microyield stress was very s e n s i t i v e to the i n i t i a l d i s t r i b u t i o n of mobile d i s l o c a t i o n s . , It i s generally supposed that d i s l o c a t i o n s move whenever the e f f e c t i v e stress on them (T*) reaches a c r i t i c a l value given by the .difference between the applied stress T and the i n t e r n a l stress T . However, since the microflow stress i s very dependent on the d i s t r i b u t i o n of mobile, d i s l o c -ations, i t i s c l e a r l y impossible to assign a constant value to the i n t e r n a l stress during microflow. Only at large s t r a i n s i s i t possible to assign a value to the i n t e r n a l s t r e s s ; i t has been confirmed that x i s then u independent of the i n i t i a l d i s t r i b u t i o n of mobile d i s l o c a t i o n s . . 83 The concept of x^ . could however be retained by considering both the "directional" and "adirectional" aspects of the internal stress f i e l d (on a scale involving larger dislocation motion than that previously considered in 5.1.1), defined by and x^°, respectively. Thus, the value of x ^ would be very dependent on the instantaneous distribution of moving dislocations: it would increase during microflow and decrease on unloading through relaxation. The adirectional component x^° would be independent of strain in the microflow region and would be influenced by, for example, the overall dislocation structure and the presence of solute atoms. It further follows that i t is not possible to consider the effective stress x* i n the microflow region of prestrained crystals. The dependence of microyield on the local mobile dislocation density is not usually considered when interpreting microflow observations. For example, Prekel and Conrad (1968) and Stein (1968) have assumed that the same velocity versus stress relation for dislocations can account for the whole flow curve from microyield to macroflow. In other words, the whole microflow curve is simply a consequence of the increasing strain-rate during microflow (see 5.2.3) together with the strain-rate sensitivity (or dislocation velocity characteristics) of the material. In order to examine this hypothesis the strain-rate sensitivity of macroflow for prestrained crystals has been investigated in 3.4. (It was, in fact, found to be jthe same as that of as-grown crystals, which shows immediately that the as-grown<and prestrained microflow curves cannot both be accounted for in terms of a strain-rate effect.) Since the strain-rate during microflow has been established as a function of strain (Fig 36), i t is possible to compare the stress versus strain-rate relation expected from macroflow experiments, with that observed from the microflow curves. This comparison is shown in Fig 37: the dashed 3 r " r=o 2 \-r=10% Microflow data: O as-grown • prestrained 10% Macroflow data 10 - 6 10 -5 10 -4 Strain-rate, y sec -1 37. Comparison of observed microflow data at "295°K with macroflow predictions. 85 curve represents the expected s t r a i n - r a t e s e n s i t i v i t y of the flow s t r e s s (obtained from F i g 6) for c r y s t a l A62 which has an as-grown macroflow stress of 3.2 kg mm (Fig 31) at a s t r a i n - r a t e of log y = -4.55 (Fig 36); the s o l i d curves show the observed stress versus s t r a i n - r a t e r e l a t i o n s (obtained from Figs 31 and 36) for the same c r y s t a l i n the as-grown and prestrained conditions. I t can be seen that the s t r a i n - r a t e s e n s i t i v i t y hypothesis (dashed curve) has predicted the observed microflow curve for the as-grown c r y s t a l , but there i s a marked discrepancy i n the case of the prestrained c r y s t a l . It therefore appears that for a prestrained c r y s t a l , the d e s c r i p t i o n of m i c r o p l a s t i c flow purely i n terms of a s t r a i n - r a t e e f f e c t i s i n c o r r e c t . The discrepancy becomes more noticeable i f comparisons are made at lower temperatures. The agreement between microflow and macroflow data i n the case of as-grown c r y s t a l s probably indicates that the same d i s l o c a t i o n motion i s involved i n both cases. This would be expected i f most d i s l o c a t i o n s i n the as-grown c r y s t a l were i n i t i a l l y locked and i f microflow then ;, represented mainly the m u l t i p l i c a t i o n and motion of screws. 5.3.2 Deformation at 160°K As F i g 33 shows, the d i f f e r e n c e between the microflow c h a r a c t e r i s t i c s of as-grown (A63) and prestrained (A64) c r y s t a l s becomes more marked at low temperatures. Since the microyield stress i n the prestrained c r y s t a l i s about one-third of the value i n the as-grown c r y s t a l , p r e s t r a i n i n g has c l e a r l y increased the number of d i s l o c a t i o n s capable of moving at low stresses. However i t i s quite s t r i k i n g to observe that at 160°K the prestrained impure c r y s t a l (A64) has a lower microyield stress than even the high p u r i t y specimen (55/4). This was not the case at 295°K. I t therefore appears that at 160°K, prestr a i n i n g i s a more e f f e c t i v e way of reducing, the 86 e f f e c t of i n t e r s t i t i a l s on t h o se d i s l o c a t i o n s t h a t move, t h a n i s i n c r e a s i n g the p u r i t y . T h i s would seem to be s t r o n g s u p p o r t f o r the r e c e n t s u g g e s t i o n o f K e l l y (1969) t h a t even i n the h i g h e s t p u r i t y b c c m a t e r i a l s , t h e r e a r e more than enough i n t e r s t i t i a l s to e f f e c t i v e l y l o c k most of the d i s l o c a t i o n s , a t l e a s t a t low t e m p e r a t u r e s . The r o l e o f i n t e r s t i t i a l s i n the d e f o r m a t i o n mechanisms o f bcc m e t a l s may i n d e e d be v e r y i m p o r t a n t . Those workers who have s t u d i e d o n l y as-grown c r y s t a l s and who c l a i m t h a t a P e i e r l s mechanism i s o p e r a t i v e have taken s u p p o r t from the agreement betweeen d a t a i n th e m i c r o f l o w and macroflow r e g i o n s (Bowen e t a l (1967), P r e k e l and Conrad (1968)) w h i l e a c c e p t i n g o t h e r e v i d e n c e f o r a P e i e r l s mechanism i n the macroflow r e g i o n . However the a l t e r n a t i v e c l a i m can now be made t h a t , s i n c e i m p u r i t y e f f e c t s a r e so dominant i n the m i c r o s t r a i n r e g i o n , they may be the dominant e f f e c t i n macroflow as w e l l . 5.2.3 Temperature s e n s i t i v i t y I t can be seen from F i g 34 t h a t the temperature s e n s i t i v i t y o f the m i c r o y i e l d s t r e s s i s the same i n as-grown c r y s t a l s o f d i f f e r e n t p u r i t y , but i s much l e s s i n impure p r e s t r a i n e d c r y s t a l s . The d e c r e a s e d temperature s e n s i t i v i t y i n p r e s t r a i n e d c r y s t a l s was f i r s t r e p o r t e d by Brown and E k v a l l (1962). They c o n c l u d e d t h a t the s t r e s s to move d i s l o c a t i o n s ( P e i e r l s s t r e s s ) i s not v e r y temperature s e n s i t i v e , and t h a t t h e i n c r e a s i n g temperature dependence a t h i g h e r s t r a i n s i s a consequence o f the temperature dependence o f d i s l o c a t i o n m u l t i p l i c a t i o n i n the m i c r o f l o w r e g i o n . However another i n t e r p r e t a t i o n i s t h a t the f r a c t i o n o f j d i s l o c a t i o n l i n e which moves may v a r y w i t h temperature. As d i s c u s s e d i n 5.1.5, 87 there is evidence (Solomon and McMahon (1968)) that at low temperatures the edge components have a higher mobility at small stresses. As the edge components become exhausted i t is necessary for the screw components to move as well. For reasons s t i l l to be discussed, the screws require considerably higher stresses to move. This behaviour would give rise to the variation with strain of the temperature sensitivity of microflow as shown in Fig 35. , It therefore appears that i f microyield in a fairly pure prestrained material represents the relatively uncomplicated motion of edge dislocations, then microstrain experiments will be very useful in determining the , effects of impurities and alloying elements on dislocation motion. 88 6. MICRODEFORMATION OF NIOBIUM ALLOY CRYSTALS 6.1 R e s u l t s 6.1.1 D e f o r m a t i o n a t 295°K M i c r o f l o w c u r v e s were o b t a i n e d f o r the Nb a l l o y s a f t e r p r e s t r a i n i n g F = 15% i n t o s t a g e I . T y p i c a l curves f o r NbTa c r y s t a l s a r e shown i n F i g 38. The e f f e c t o f the Ta a d d i t i o n has been to i n c r e a s e b o t h the m i c r o y i e l d and macroflow s t r e s s e s by a comparable amount. T h i s means t h a t the e f f e c t o f s o l u t e on the f l o w s t r e s s i s n o t v e r y dependent on the s t r a i n s e n s i t i v i t y , as i l l u s t r a t e d i n F i g 39. I t s h o u l d be noted t h a t t h e r e i s no e v i d e n c e f o r a d e c r e a s i n g s o l u t e dependence o f f l o w a t h i g h s t r a i n s e n s i t i v i t i e s . T y p i c a l m i c r o f l o w c u r v e s f o r NbMo a l l o y s a r e shown i n F i g 40. A l l o y i n g has produced a v e r y l a r g e i n c r e a s e i n m i c r o y i e l d s t r e s s and, as w i t h the NbTa a l l o y s , the c o n c e n t r a t i o n dependence o f the f l o w s t r e s s i s independent o f the s t r a i n s e n s i t i v i t y ( F i g 41). 6.1.2 D e f o r m a t i o n below 295°K Because o f the low temperature b r i t t l e n e s s o f t h e NbMo a l l o y s , experiments were performed o n l y on the NbTa a l l o y s . T y p i c a l m i c r o f l o w c u r v e s f o r a Nb 4.8 Ta a l l o y deformed a t v a r i o u s temperatures a r e shown i n F i g 42. The form o f t h e c u r v e s i s q u a l i t a t i v e l y the same as t h a t o f pure Nb. In o r d e r t o a s c e r t a i n any e f f e c t o f Ta, the m i c r o y i e l d v a l u e s a r e compared w i t h pure Nb i n F i g 43. I t can be seen t h a t the temperature s e n s i t i v i t y o f m i c r o y i e l d a t v e r y low temperatures i s s m a l l e r f o r the a l l o y , which has a lower m i c r o y i e l d v a l u e a t 77°K. T h i s b e h a v i o u r i s s i m i l a r t o t h a t observed i n t h e macroflow r e g i o n ( F i g 17). -5 -4 log (shear strain, y) Fig 38. Microflow curves for NbTa alloys deformed at 295°K. F i g 39. Microflow st r e s s at d i f f e r e n t s t r a i n s for NbTa a l l o y s deformed at 295°K. 20 15 \-10 5 L • L_ 6 6 Mo v 4-9 Mo A o — o 0 9 Mo © Nb -5 -4 - 3 log (shear s t r a i n , y) F i g 40. Microflow curves f o r NbMo alloy s deformed at 295°K. T F i g 41. Microflow stress at d i f f e r e n t s t r a i n s f o r NbMo a l l o y s deformed at 295°K. -5 -4 -3 log (shear strain, y) 42. Microflow curves for a Nb 4.8 Ta alloy at low temperatures. 94 F i g 43. M i c r o y i e l d v a l u e s ( l o g y = -4.8) f o r Nb and Nb 4.8 Ta c r y s t a l s as a f u n c t i o n o f temperature. 95 6.2 D i s c u s s i o n 6.2.1 D e f o r m a t i o n a t 295°K M i c r o s t r a i n experiments on b c c a l l o y s have p r e v i o u s l y been performed by workers a t the F o r d Motor Company ( S t o l o f f e t a l (1965), D a v i e s and Ku (1966)) on s u b s t i t u t i o n a l a l l o y s , and by Solomon and McMahon (1968) on i n t e r s t i t i a l a l l o y s . In a l l cases the t e s t s were performed on Fe-base p o l y c r y s t a l s . The r e s u l t s o b t a i n e d by D a v i e s and Ku were i n g e n e r a l d i f f e r e n t b o t h from those o b t a i n e d i n the p r e s e n t work on s u b s t i t u t i o n a l a l l o y s and from those of Solomon and McMahon on i n t e r s t i t i a l a l l o y s . The t e c h n i q u e used by D a v i e s and Ku (and by S t o l o f f e t a l ) t o o b t a i n m i c r o f l o w c u r v e s has a l r e a d y been c r i t i c i z e d . However the s t r e s s a t which they f i r s t r e c o r d e d p l a s t i c s t r a i n s h o u l d be a v a l i d o b s e r v a t i o n ( a l t h o u g h even t h i s w i l l be q u e s t i o n e d l a t e r ) . D a v i e s and Ku e x p r e s s e d t h e i r m i c r o f l o w c u r v e s on a l i n e a r s t r a i n s c a l e and f r e q u e n t l y made , u n j u s t i f i a b l e e x t r a p o l a t i o n s to zero s t r a i n . I n some c a s e s , a l t e r n a t i v e c o n c l u s i o n s , become t e n a b l e when t h e i r d a t a i s p l o t t e d more s u i t a b l y . . However i n the case of F e N i a l l o y s t h e r e s u l t s a r e s t i l l v e r y d i f f e r e n t from those o b t a i n e d h e r e f o r NbTa a l l o y s ( F i g 38) and NbMo a l l o y s ( F i g 40); F i g 44a i l l u s t r a t e s a s t r o n g c o n c e n t r a t i o n dependence o f f l o w a t h i g h s t r a i n s which i s c o n s i d e r a b l y reduced a t m i c r o y i e l d . A s i m i l a r though l e s s pronounced b e h a v i o u r was o b s e r v e d w i t h FeC a l l o y s . From t h e i r treatment o f the d a t a , D a v i e s and Ku c l a i m e d t h a t t h e s t r e s s t o move d i s l o c a t i o n s i n Fe i s independent of s o l u t e ; s o l u t e .,. r e s t r i c t s the a b i l i t y o f screws to c r o s s s l i p and m u l t i p l y so t h a t macroflow ds v e r y c o n c e n t r a t i o n dependent. ( T h i s p r o p o s i t i o n would r e q u i r e t h a t the d i r e c t i o n a l ZiSH^' " Microf lowdat'a fox poly crystalline Fe alloys deformed at 295°K. i n t e r n a l stress component be very concentration dependent.) They further proposed that the importance of d i s l o c a t i o n m u l t i p l i c a t i o n i s a f a i r l y general phenomenon i n determining the macroflow c h a r a c t e r i s t i c s of bcc metals. Thus Davies and G i l b e r t (1967) found no o r i e n t a t i o n dependence of flow at microyield i n Mo s i n g l e c r y s t a l s and a t t r i b u t e d the e f f e c t at macroyield to d i s l o c a t i o n m u l t i p l i c a t i o n during microflow. The importance of d i s l o c a t i o n m u l t i p l i c a t i o n as a r a t e - c o n t r o l l i n g mechanism i n bcc deformation i s an i n t e r e s t i n g suggestion, but i t i s not supported by the r e s u l t s of the present work or by those of Solomon and McMahon. These workers obtained values of a and a f o r various i e a , prestrained p o l y c r y s t a l l i n e FeC a l l o y s , and found a marked dependence pf a on C content. Their r e s u l t s are shown i n Table III and are compared with those of Davies and Ku. Figs 45a and 45b have been taken from the work of Solomon and McMahon and show the microyield stress a and macroflow stress a at 77°K and 300°K for d i f f e r e n t FeC a l l o y s . The r e s u l t s i n d i c a t e that when the amount of C i n the l a t t i c e increases, the microyield stress increases and approaches the flow s t r e s s . (The FeTi a l l o y contains,, r e l a t i v e l y l i t t l e C i n the l a t t i c e since i t i s associated with the T i . ) On the other hand Davies and Ku did not observe a s i g n i f i c a n t increase-, i n microyield stress with added C. : J In t r y i n g to resolve the above discrepancies, i t i s considered s i g n i f i c a n t that Davies and Ku did not use prestrained specimens. They did not observe an i n i t i a l y i e l d point and therefore concluded that there was no d i s l o c a t i o n locking. However, they used specimens i n the.form of s t r i p only 0.006 inches t h i c k , and i t i s common f o r y i e l d points to be obscured i n the presence of stress concentrations (Hutchison (1963)). As was indicated i n 5.1.1, i t i s p o s s i b l e , i n specimens which have not been prestrained, to obtain some p l a s t i c s t r a i n on the f i r s t loading at a stress 98 Table I I I . Comparison of microflow data for FeC a l l o y s at 300°K obtained by Davies and Ku (1966) and Solomon and McMahon (1968). Davies and Ku (unprestrained) Solomon and McMahon (prestrained) a a a a a e Fe "pure" (Q) 4 FeTi (FC) 7 4 Fe 0.008% C (Q) 3 Fe 0.006% (C + N) (FC) 18 5 Fe 0.014% C (Q) 5 (QA) 18 5 Fe 0.02% C (Q) 8 Fe 0.035% (C + N) (FC) (QA) 27 50 7 5 Fe 3.2% S i 32 Fe 3.1% S i (FC) 40 27 Units: 10 p s i , compositions i n wt% Q: quenched and held at -80°C QA: quenched and aged 1 hour at 60°C FC: furnace cooled Fig-45. -Temperature-variation, i n various.prestrained Fe a l l o y s , of: a) microyield stress b) macroflow stress (after Solomon and McMahon (1968)) (Alloys i d e n t i f i e d i n Table III) corresponding to a , before the establishment of a stable hysteresis loop. It i s therefore quite possible that the "microyield" values of Davies and Ku correspond to a rather than to a . In f a c t , Solomon and McMahon e a found that was r e l a t i v e l y independent of C content, and there i s remarkable agreement between their o* values and the a values of Davies e a and Ku, as shown i n Table I I I . This explanation of the anomalous microflow curves obtained by Davies and Ku i s further supported by previous r e s u l t s reported . by the same workers (Stoloff et a l (1965)). In a study of microyielding i n FeV and FeCo a l l o y s , they used c y l i n d r i c a l specimens and prestrained them before t e s t i n g . Their microflow curves f o r FeV a l l o y s have been replotted i n F i g 44b ( i n the manner adopted for t h i s work), from which i t can be seen that the microyield stress i s very dependent on solute . concentration, up to additions of 10%. This i s now i n agreement with the r e s u l t s of the present work, i n which solute has been observed to r a i s e the o v e r a l l l e v e l of the microflow curve. I t follows that the e f f e c t of solute i s to increase the a d i r e c t i o n a l i n t e r n a l stress component x^°, with l i t t l e e f f e c t on T ^. This i s contrary to the proposition of Davies and Ku, and t h e i r suggestion that the main e f f e c t of solute i s to r e s t r i c t c r o s s - s l i p of screw d i s l o c a t i o n s must be refuted. The present conclusion i s i n agreement with the r e s u l t s of s l i p l i n e observations on the Nb a l l o y s . There was no v i s i b l e change i n s l i p l i n e structure and no tendency for s l i p to be r e s t r i c t e d to {Oil}. 6.2.2 Deformation below 295°K The r e s u l t s of low temperature microstrain experiments on NbTa c r y s t a l s show that the softening observed i n macroflow i s also observed i n the microflow region. Although the error i n any y i e l d determination 101 i s greater at microyield than at macroyield, i t i s probable that the softening i s greater at high s t r a i n s e n s i t i v i t i e s . The softening of bcc metals has been frequently reported even at room temperature. For example, F i g 44b showed the e f f e c t of small additions of V to Fe p o l y c r y s t a l s . S t o l o f f et a l (1965) performed s i m i l a r microstrain experiments at 77°K and found an even greater softening e f f e c t . Transmission electron microscopy revealed an increasing volume f r a c t i o n of p a r t i c l e s i n the higher a l l o y specimens. Although these could not be analyzed i t was concluded that they were compounds of V with C and N. An in t e r a c t i o n between the solvent metal or solute metal with i n t e r s t i t i a l s has been suggested for many other bcc systems; f o r example, VTi alloys. (Fraser and Lund (1962)), and many a l l o y s of Cr (Allen and Jaffee (1963)). Since softening i s observed during microstrain as we l l as; macrostrain, i t i s suggested that the motion of both edge and screw di s l o c a t i o n s i s affected. This suggestion eliminates any in t e r p r e t a t i o n s based s o l e l y on the properties of screw d i s l o c a t i o n s . 102 7. THEORY OF PLASTIC FLOW IN NIOBIUM AND NIOBIUM ALLOY CRYSTALS 7.1 Mechanisms of deformation i n pure niobium 7.1.1 Introduction The exact nature of the mechanisms c o n t r o l l i n g the low temperature flow stress of bcc metals has s t i l l to be determined. As recent discussions have shown (Nabarro (1968)), there i s at present no theory capable of explaining a l l the observations. Two extreme views are that the flow stress i s c o n t r o l l e d s o l e l y either by the i n t r i n s i c nature of d i s l o c a t i o n s i n the bcc l a t t i c e (Peierls mechanism) (Conrad (1963)), or by the , i n t e r a c t i o n of r e s i d u a l impurities with d i s l o c a t i o n s (impurity mechanism) (Fleischer (1967)). In most cases i t i s not possible to d i s t i n g u i s h between the predictions of these theories by conventional mechanical t e s t i n g . This has led to the use of many a d d i t i o n a l techniques such as transmission electron microscopy, d i s l o c a t i o n displacement, i n t e r n a l f r i c t i o n , m icrostrain, and stress r e l a x a t i o n studies. However the ; l i m i t a t i o n s of such supplementary techniques are not s u f f i c i e n t l y w e l l understood for conclusive i n t e r p r e t a t i o n of the r e s u l t s to be made-.; i n favour of either theory. The d i f f i c u l t i e s have been complicated by the recent discovery of unique c h a r a c t e r i s t i c s of bcc metals such as the asymmetry of s l i p and the f a i l u r e of the Schmid law of resolved shear s t r e s s . Furthermore, the experiments have been performed on a wide v a r i e t y of both pure and impure bcc s i n g l e c r y s t a l s and p o l y c r y s t a l s . As a r e s u l t , the proponents have been forced to modify t h e i r p a r t i c u l a r theories to such an extent that they cease to have any p r e d i c t i v e c a p a b i l i t i e s ; i t becomes hard to define what the theories r e a l l y are. This complexity i s r e f l e c t e d in.the 103 t e r m i n o l o g y which has d e v e l o p e d : m o d i f i e d - , pseudo-, or m o d i f i e d pseudo-P e i e r l s s t r e s s . The l i m i t a t i o n s o f t e r m i n o l o g y a r e p r o b a b l y s i g n i f i c a n t i n the c u r r e n t c o n t r o v e r s y . F o r example, i f i t were e s t a b l i s h e d t h a t screw d i s l o c a t i o n m o tion i n v o l v e d an i m p u r i t y - c o n t r o l l e d recombination, o f p a r t i a l s and subsequent t r a n s f e r to the next p o t e n t i a l energy v a l l e y , would the p r o c e s s be d e s c r i b e d as a P e i e r l s mechanism or an i m p u r i t y mechanism? T h i s comment i l l u s t r a t e s the p o s s i b l e i n t e r d e p e n d e n c e o f s p e c i f i c mehanisms; p r e s e n t t h e o r i e s a r e i n c a p a b l e o f a c c o u n t i n g f o r t h i s p o s s i b i l i t y . The f o l l o w i n g d i s c u s s s i o n w i l l c o n s i d e r a l l t h o s e f a c t o r s w h i c h a r e s i g n i f i c a n t i n c o n t r o l l i n g d e f o r m a t i o n , r a t h e r than assuming t h a t one mechanism i s the complete s o l u t i o n . 7.1.2 D i s c u s s i o n The m i c r o s t r a i n e v i d e n c e p r e s e n t e d f o r pure Nb has i n d i c a t e d t h a t , even i n the h i g h e s t p u r i t y m a t e r i a l s t e s t e d to d a t e , the e f f e c t o f i m p u r i t i e s i s s i g n i f i c a n t . I t i s t h e r e f o r e f a l l a c i o u s to assume t h a t i m p u r i t y e f f e c t s can be n e g l e c t e d a p r i o r i , even i f the m a t e r i a l i s e x t r e m e l y pure. A l s o the " e f f e c t i v e " i m p u r i t y c o n t e n t can d i f f e r from the a n a l y z e d i m p u r i t y c o n t e n t : i n t e r s t i t i a l s may be a s s o c i a t e d w i t h s o l u t e atoms (eg F e T i a l l o y s ) , o r w i t h s o l v e n t atoms (depending on the magnitude o f the s o l u b i l i t y l i m i t ) , o r w i t h each o t h e r ( i n the form o f c l u s t e r s r a t h e r t h a n as s i n g l e atoms). The common p r a c t i c e o f q u o t i n g i m p u r i t y c o n t e n t s i n weight f r a c t i o n s (wt ppm) and s o l u t e c o n t e n t s i n atom f r a c t i o n s appears d e l i b e r a t e l y d e c e p t i v e . Thus an i m p u r i t y l e v e l o f o n l y 5 ppm hydrogen i n n i o b i u m a c t u a l l y c o n s t i t u t e s a Nb 0.05 a t % H a l l o y . I n t h e work on FeC a l l o y s , t h e r e was found t o be a v e r y marked e f f e c t o f i n t e r s t i t i a l s on the temperature dependence o f T^, which-has 104 been i d e n t i f i e d with a stress at which edge d i s l o c a t i o n s move. On the other hand, the e f f e c t of i n t e r s t i t i a l s on the temperature dependence of the flow stress x^ was r e l a t i v e l y small. I t i s possible that the d.TF r e l a t i v e i n s e n s i t i v i t y of to impurities could r e s u l t from the s a t u r a t i o n of a screw d i s l o c a t i o n - i n t e r s t i t i a l i n t e r a c t i o n at very low concentrations. This would require that the i n t e r a c t i o n between d i s l o c a t i o n s and i n t e r s t i t i a l s be much stronger for screws than for edges. I t would then be necessary to achieve a considerable increase i n o v e r a l l p u r i t y for the flow stress x^ to show a s i m i l a r dependence on p u r i t y as the microyield stress T . In f a c t , as was shown i n F i g 5, there i s strong a • « dx ^  evidence for a decrease i n jrjr™ with increasing p u r i t y at low temperatures. Although the x^(T) curve can be divided into thermal and athermal regions, the assignment of thermal and athermal stress components at temperatures below T q requires caution. I t has been seen that a, unique value cannot be ascribed to the i n t e r n a l stress f i e l d x^ during microflow, because the flow stress i s influenced by the p a r t i c u l a r d i s t r i b u t i o n of moving d i s l o c a t i o n s . The constant e f f e c t of x^ can be f e l t only whens d i s l o c a t i o n s move long distances through the c r y s t a l at the asymptotic flow stress x^° i n a prestrained c r y s t a l . At 295°K t h i s stsess i s reached a f t e r -3 a s t r a i n of about y = 10 . However at low temperatures the strain-, to< reach x^° may be much la r g e r , so the influence of the p a r t i c u l a r j _3 d i s l o c a t i o n d i s t r i b u t i o n may be f e l t even at macrostrains (Y > 10 ). To the extent that t h i s i s true, the apparent value of x^ w i l l be temperature dependent. In f a c t the assumption that x v a r i e s with temperature only as the shear modulus implies that work hardening (given by ) i s independent of temperature; t h i s i s c e r t a i n l y not the case (see F i g 42) Thus a l t h o u g h the d i s l o c a t i o n d e n s i t y (which i n f l u e n c e s x^ ) has been found to be independent of d e f o r m a t i o n temperature (Keh and Weissman (1963) ), t h e r e i s a change i n d i s l o c a t i o n c o n f i g u r a t i o n (Lawley and G a i g h e r (1964) ) which c o u l d i n f l u e n c e x ^. A l s o the r e l a x a t i o n of d i s l o c a t i o n groups d u r i n g d e f o r m a t i o n a t low temperatures c o u l d be s l o w e r t h a n a t h i g h t e m p e r a t u r e s . Another d i f f i c u l t y w i t h the i n t e r p r e t a t i o n of x^ even a t temperatures near T q , i s t h a t x^ can be a p p l i e d to d i s l o c a t i o n i n t e r a c t i o n s over b o t h s h o r t and l o n g d i s t a n c e s . C o n s i d e r the e f f e c t o f two o b s t a c l e s to d i s l o c a t i o n m o t i o n which c o u l d i n f l u e n c e x : the o v e r a l l d i s l o c a t i o n s t r u c t u r e and s i n g l e s o l u t e atoms. The i n f l u e n c e o f a d i s l o c a t i o n group can be f e l t over l o n g d i s t a n c e s , and i s u s u a l l y d e s c r i b e d as the i n t e r n a l 1^  s t r e s s f i e l d ( p r o p o r t i o n a l to p 2 ) . T h i s i n t e r a c t i o n s h o u l d be s i m i l a r ; f o r b o t h edges and screws. However another c o n t r i b u t i o n to the i n t e r n a l s t r e s s f i e l d a r i s e s from the p r e s e n c e of s o l u t e atoms. The e f f e c t o f a s i n g l e s o l u t e atom on a d i s l o c a t i o n l i n e i s f e l t o n l y a t c l o s e d i s t a n c e s of approach and the i n t e r a c t i o n c o u l d w e l l be d i f f e r e n t f o r edges and screws (eg F l e i s c h e r (1963) model). Both t h e s e i n t e r a c t i o n s a r e e f f e c t i v e a t , T temperatures above T q ; they can be overcome o n l y by an a p p l i e d s t r e s s . They a r e t h e r e f o r e d e s c r i b e d as " l o n g r a n g e " i n t e r a c t i o n s , a l t h o u g h . t h e d i s l o c a t i o n may f e e l the s o l u t e atom o n l y a t v e r y c l o s e d i s t a n c e s . u ( I t i s o f i n t e r e s t to note t h a t t h i s d i f f i c u l t y w i t h the s p a c i a l e x t e n t o f an i n t e r a c t i o n does not a r i s e i n the s t a t i s t i c a l t h e o r y of the f l o w s t r e s s developed by Kocks (1966, 1967). H i s t h e o r y d e s c r i b e s , the f l o w s t r e s s i n terms of an e f f e c t i v e o b s t a c l e s p a c i n g through which d i s l o c a t i o n s must pass and the hardness o f the o b s t a c l e s themselves,,. When, as i n the above c a s e , d i f f e r e n t k i n d s o f o b s t a c l e s can c o n t r i b u t e to the f l o w s t r e s s , Kocks (1968) found t h a t t h e i r c o n t r i b u t i o n s a r e a d d i t i v e whenever the spacings of the respective obstacles d i f f e r s i g n i f i c a n t l y , i r r e s p e c t i v e of the distance over which the o b s t a c l e - d i s l o c a t i o n i n t e r a c t i o n occurs. There appear to be the following possible contributions to the temperature dependence of the flow stress of bcc metals: ,< a) short range i n t e r s t i t i a l - d i s l o c a t i o n i n t e r a c t i o n , which w i l l be further discussed l a t e r , b) temperature dependent "long range" s t r e s s , which has been discussed above , c) inherent l a t t i c e f r i c t i o n . This l a s t p o s s i b i l i t y i s often expressed i n terms of the r e s t r i c t i o n of screw d i s l o c a t i o n motion at low temperatures. Although d i s s o c i a t i o n of d i s l o c a t i o n s i n bcc c r y s t a l s must be extremely s l i g h t i f i t occurs at a l l , the concept of p a r t i a l s e x i s t i n g on planes other than the g l i d e plane has been quite successful i n explaining both the anisotropy of c r i t i c a l shear stress with respect to the sense of shear, and the o r i e n t a t i o n dependence of the resolved shear stress (Kroupa and Vitek (1967)). Screw d i s l o c a t i o n motion has been thought to occur by thermally activated t r a n s i t i o n s from s e s s i l e to g l i s s i l e configurations of the , disso c i a t e d screws (Bowen et a l (1967)). This theory i s i n agreement with the observed r e s t r i c t i o n of s l i p to {011} planes at low temperatures, since i t i s possible that the reduced a b i l i t y f o r c r o s s - s l i p produces, a large increase i n flow s t r e s s . I t i s agreed that the motion of screw d i s l o c a t i o n s i s r a t e - c o n t r o l l i n g at low temperatures, but i t i s no;t ne c e s s a r i l y true that the high flow stress i s a consequence of the ; r e s t r i c t e d a b i l i t y of screws to cross s l i p . For instance the addition of 6.6 at% Mo to Nb produced a s i x times increase i n flow stress ajt room temperature without any apparent change i n the wavy s l i p traces l e f t by screw d i s l o c a t i o n s . 107 7.2 Mechanisms for deformation of niobium a l l o y s 7.2.1 S u b s t i t u t i o n a l e f f e c t s 7.2.1.1 Introduction The change of flow stress on a l l o y i n g , a pure metal i s due to an i n t e r a c t i o n between the solute atoms and the d i s l o c a t i o n s . The possible i n t e r a c t i o n s have been established for fee metals and to a l e s s e r extent for. hep metals (Haasen (1965)). There i s r e l a t i v e l y l i t t l e data a v a i l a b l e on bcc s u b s t i t u t i o n a l a l l o y s , but some attempts have been made to determine the important i n t e r a c t i o n s (Harris (1966), Kostorz (1969)). Since there appear to be no a d d i t i o n a l e f f e c t s involved i n bcc a l l o y s , the mechanisms established for fee systems have been tabulated and t h e i r r e l a t i v e importance indicated for both fee and bcc s u b s t i t u t i o n a l a l l o y s (Table IV). ,, j The table shows that i n bcc a l l o y s the important i n t e r a c t i o n s are due to the s i z e and modulus differences of the solute atom and > possibly also to an i n t e r a c t i o n with the d i s l o c a t i o n core. The s i z e and modulus differences are "long range" e f f e c t s , i n that the i n t e r a c t i o n can be overcome only by an applied s t r e s s , at l e a s t at moderate temperatures. Therefore i n the region of T q (450 - 550°K for Nb a l l o y s ) only the long range int e r a c t i o n s can produce hardening. The core i n t e r a c t i o n i s a "short range" e f f e c t and, being capable of thermal a c t i v a t i o n , w i l l be apparent only at low temperatures. 7.2.1.2 Long range int e r a c t i o n s The e f f e c t of the d i f f e r e n t s i z e of the solute atom i s measured by the s i z e m i s f i t , 6 , defined as the r e l a t i v e change of l a t t i c e parameter 108 T a b l e IV. P o s s i b l e i n t e r a c t i o n s between s o l u t e atoms and d i s l o c a t i o n s i n f e e and b c c m e t a l s . Range o f Importance o f i n t e r a c t i o n S o l u t e e f f e c t ^ i _ i n t e r a c t i o n f e e b c c S i z e d i f f e r e n c e l o n g Modulus d i f f e r e n c e l o n g Core i n t e r a c t i o n s h o r t C h emical i n t e r a c t i o n s h o r t S h o r t range o r d e r s h o r t Grown-in d i s l o c a t i o n d e n s i t y ( i n d i r e c t e f f e c t ) moderate l a r g e moderate moderate s m a l l l a r g e s m a l l may be moderate n e g l i g i b l e v e r y s m a l l v e r y s m a l l v e r y s m a l l 109 on a l l o y i n g : S i m i l a r l y , the modulus defect, n, i s defined as the r e l a t i v e change i n shear modulus on a l l o y i n g : The necessity for both these parameters was f i r s t recognized by Fl e i s c h e r (1961,1963). He used a model for the long range e l a s t i c i n t e r a c t i o n between d i s l o c a t i o n s and solute atoms to derive an expression for the c r i t i c a l resolved shear stress of an a l l o y , given by T q = Z.u.e / 2 . c / 2 (3) where c i s the atom f r a c t i o n of solute u i s shear modulus 1 Z i s a numerical constant ( - f60 ^ e i s a m i s f i t parameter. Equation (3) has been found to be a good d e s c r i p t i o n of the parabolic hardeing behaviour observed i n copper-base and sil v e r - b a s e a l l o y s (Haasen (1968)). To obtain the best c o r r e l a t i o n , the m i s f i t parameter combines the e f f e c t of s i z e and modulus d i f f e r e n c e s , and i s given by e = | n' - a. 6 | j, where n' = — 1 + W 2 and a = +3 i s appropriate f o r an i n t e r a c t i o n between solute atoms and screw d i s l o c a t i o n s . It has therefore been concluded that i n Cu and Ag a l l o y s the hardening i s produced by an e l a s t i c i n t e r a c t i o n between solute atoms 110 and screw d i s l o c a t i o n s . Furthermore, the modulus e f f e c t produces about 75% of the t o t a l hardening. (To a f i r s t approximation, a s i z e e f f e c t alone does not predict any i n t e r a c t i o n with screws; the o r i g i n a l Mott and Nabarro (1948) theory considered only s i z e e f f e c t s . ) The a p p l i c a b i l i t y of equation (3) has since been tested f o r bcc a l l o y s . Harris (1966) found that i n p o l y c r y s t a l l i n e Nb s o l i d s o l u t i o n s , i n contrast to fee a l l o y s , the s i z e m i s f i t rather than the modulus , defect was the dominant f a c t o r . Raffo and M i t c h e l l (1968) reviewed several s i n g l e c r y s t a l studies and concluded that hardening was due only to the i n t e r a c t i o n of m i s f i t t i n g solute atoms with edge d i s l o c a t i o n s (cf Mott and Nabarro). However t h i s c l e a r l y cannot account for the observed hardening i n NbTa a l l o y s , f o r which 6 i s zero. A more recent attempt by Kostorz (1969) to c o r r e l a t e experimental data with equation (3) has indicated the importance of both 6 and n,in determining the s o l u t i o n hardening of bcc s o l i d s o l u t i o n s . However so many ad hoc modifications were required for p a r t i c u l a r a l l o y systems, that serious doubts must be expressed as to the a p p l i c a b i l i t y of the F l e i s c h e r a n a l y s i s . Although, as Kostorz suggested, i t i s true that much more data on bcc s o l i d solutions i s required, i t also appears that c o r r e l a t i o n of data on the l i n e s he has attempted w i l l become even more complicated than at present. The following are s p e c i f i c c r i t i c i s m s of the methods •, adopted to date: a) I t i s clear that hardening does not always follow a parabolic law (NbMo, present work; TaRe, M i t c h e l l and Raffo (1967)); therefore i n such cases c o r r e l a t i o n with equation (3) should not be attempted. I l l b) In the absence of experimental data, the modulus defect i s usually calculated assuming a l i n e a r v a r i a t i o n of the modulus between the pure components of an a l l o y showing complete s o l i d s o l u b i l i t y . This could be a source of error, but more important i s the lack of consistency i n the expressions used f o r the modulus. The following table shows the d i f f e r e n t expressions which have been used by various workers; there i s a large discrepancy i n the value of n c a l c u l a t e d , as an example, for NbTa a l l o y s . ; n Ha r r i s : Youngs modulus = = -0.07 s l l M i t c h e l l and Shear modulus on 1 Raffo: (Oil) i n [111] = 3 ( C l 1 " ° 1 2 + C ^ + ° ' 3 6 c l l ~ c12 j, <finMr = ) • = +0.64 C _ Q Kostorz: Voigts average _ ( r 1 1 ] \ \ shear modulus It i s suggested that Voigts average shear modulus (see A,5.2) i s the best expression to use; the expression used by M i t c h e l l and Raffo concerns the operative s l i p system, which does not enter into the c a l c u l a t i o n of the d i s l o c a t i o n - s o l u t e i n t e r a c t i o n i n the F l e i s c h e r model. c) The F l e i s c h e r model allows v a r i a t i o n s i n the m i s f i t parameter e depending on whether edge or screw d i s l o c a t i o n i n t e r a c t i o n s are considered to be dominant. However i f i t becomes necessary to consider d i f f e r e n t combinations of edge and screw i n t e r a c t i o n s f or d i f f e r e n t a l l o y systems (Kostorz (1969)), there are then so many independent v a r i a b l e s that the model ceases to have any p r e d i c t i v e value. I t i s important to r e c a l l that the microstrain experiments pn NbTa and NbMo a l l o y s revealed the same concentration dependence of microyield as macroflow. Assuming that microyield represents the motion of edges 112 and macroflow represents the motion of screws, t h i s i s experimental evidence for a s i m i l a r i n t e r a c t i o n of solute with both edges and screws. At present there seems to be no s a t i s f a c t o r y theory for s u b s t i t u t i o n a l s o l u t i o n hardening i n bcc a l l o y s . I t i s possible that the s t a t i s t i c a l theory of a l l o y hardening, which has been applied to dispersion and p r e c i p i t a t i o n hardened a l l o y s (Kocks (1969)), could be extended to the case of s o l u t i o n hardening. However the a p p l i c a b i l i t y of the theory becomes incr e a s i n g l y l i m i t e d as the p e n e t r a b i l i t y of the obstacles to d i s l o c a t i o n motion increases; s i n g l e solute atoms represent the most unfavourable case. Kocks has recognized that the s t a t i s t i c a l theory i s not applicable to deformation by the spreading of a Luders band, which often occurs i n a l l o y s . His theory suggests that t h i s mode of deformation may be a consequence of the greater mobility of screw d i s l o c a t i o n s so that the edges are r a t e - c o n t r o l l i n g . This i s not supported by the r e s u l t s of the present work: the formation of a Luders band has been shown to be a geometrical phenomenon determined by the r e l a t i v e ;. values of y i e l d stress and work hardening rate; screw d i s l o c a t i o n s are r a t e - c o n t r o l l i n g i n both pure Nb and the a l l o y s . 7.2.1.3 Short range i n t e r a c t i o n At low temperatures as the e f f e c t of thermal a c t i v a t i o n i s , reduced, a possible i n t e r a c t i o n between solute atoms and the d i s l o c a t i o n core becomes evident. Thus s o l u t i o n softening has been observed In NbTa all o y s at 77°K and has been reported i n many other bcc systems. It has been commonly supposed that addition of solute leads to a reduction i n the P e i e r l s stress ( M i t c h e l l and Raffo (1967)). However i t was shown i n 4.2.1.2.2 that t h i s p o s s i b i l i t y would s t i l l require a minimum i n the P e i e r l s stress versus concentration curve; arguments to 113 e x p l a i n t h e e f f e c t of s o l u t e can o n l y p r e d i c t a c o n t i n u o u s d e c r e a s e i n P e i e r l s s t r e s s ( A r s e n a u l t (1967)). R a v i and G i b a l a (1969) have p r e s e n t e d d a t a o b t a i n e d by Koss (1969) f o r the c r i t i c a l r e s o l v e d shear s t r e s s o f NbW s i n g l e c r y s t a l s a t low temperatures ( F i g 46). I t i s p o s s i b l e to e x t r a p o l a t e the a l l o y c u r v e s to zero s o l u t e , g i v i n g a v a l u e f o r pure Nb which i s v e r y c l o s e to the p u r e s t m a t e r i a l s t e s t e d to d a t e . S i m i l a r e x t r a p o l a t i o n s were p o s s i b l e -w i t h the NbO s i n g l e c r y s t a l a l l o y s s t u d i e d by R a v i and G i b a l a . I n . p a r t i c u l a r , they observed t h a t t h e e f f e c t was n o t r e v e r s i b l e : a d d i t i o n of s o l u t e produced s o f t e n i n g but subsequent p u r i f i c a t i o n d i d n o t produpe h a r d e n i n g . They c o n c l u d e d t h a t i n b o t h cases the s o f t e n i n g was due, to a s s o c i a t i o n of i n t e r s t i t i a l atoms (wi t h e i t h e r s u b s t i t u t i o n a l s o l u t e atoms or w i t h o t h e r i n t e r s t i t i a l atoms), l e a d i n g to a r e d u c t i o n i n the number of b a r r i e r s o v e r which the d i s l o c a t i o n must p a s s . A t higher-s o l u t e c o n c e n t r a t i o n s the a s s o c i a t i o n r e a c t i o n becomes s a t u r a t e d and h a r d e n i n g o c c u r s . , T h i s c o n c l u s i o n i s i n agreement w i t h the r o l e of i n t e r s t i t i a l s a l r e a d y o u t l i n e d i n t h i s d i s c u s s s i o n . P o s s i b l e d i s l o c a t i o n - i n t e r s t i t i a l i n t e r a c t i o n s w i l l be f u r t h e r d i s c u s s e d below. ,,; 7.2.2 I n t e r s t i t i a l e f f e c t s The importance o f i n t e r s t i t i a l s i n Nb has become i n c r e a s i n g l y e v i d e n t d u r i n g the p r e c e e d i n g d i s c u s s i o n s o f p l a s t i c d e f o r m a t i o n . The r o l e o f i n t e r s t i t i a l s has, however, a l s o been I n f e r r e d from o t h e r e x p e r i m e n t s . In i n t e r n a l f r i c t i o n s t u d i e s , a Snoek damping peak i s observed when the f r e q u e n c y o f the s t r e s s e q u a l s the f r e q u e n c y a t which the i n t e r s t i t i a l s jump from one s i t e to a n o t h e r . Of p a r t i c u l a r i n t e r e s t i s the r e s u l t of d e t a i l e d a n a l y s i s of damping c u r v e s ( G i b a l a and Wert (196 6)), 114 1 1 — r o I I I I _ i 0 2 3 4 5 I Composition, at% W ! | Fig 46. Yield stress of NbW alloys at low temperatures, (after Koss (1969)) 115 namely that i n t e r s t i t i a l atoms i n bcc metals are present both as s i n g l e unassociated atoms and as c l u s t e r s containing from two to four atoms. Similar r e s u l t s have been obtained from d i r e c t studies i n the f i e l d - i o n microscope (Nakamura and Muller (1965)). Such associations could influence the e f f e c t i v e i n t e r s t i t i a l content, as already suggested, As i n the case of s u b s t i t u t i o n a l atoms, there could be an, i n t e r a c t i o n between d i s l o c a t i o n s and i n t e r s t i t i a l s due to both the s i z e and modulus di f f e r e n c e s . However the modulus e f f e c t i s very small; the s i z e e f f e c t can be evaluated on an i s o t r o p i c continuum approximation, i f i t i s assumed that the i n t e r s t i t i a l produces i s o t r o p i c d i l a t a t i o n only ( C o t t r e l l and Bilby (1949)). However i n actual anisotropic c r y s t a l s , the stable i n t e r s t i t i a l configuration can have tetragonal symmetry and.the i n t e r s t i t i a l would then react with shear stresses as w e l l as normal stresses. An approximate e l a s t i c i t y c a l c u l a t i o n was performed by Cochardt et a l (1955) and was l a t e r applied by Schoeck and Seeger (1959) ands Fl e i s c h e r (1962). The large tetragonal d i s t o r t i o n i n bcc metals i s ; produced by i n t e r s t i t i a l s i n the (%,0,0) octahedral p o s i t i o n s ; a smaller d i s t o r t i o n i s possible i n the (h,ht0) tetrahedral p o s i t i o n s . I n t e r s t i t i a l s i n the octahedral s i t e s occupy equivalent p o s i t i o n s , 120° apart, i n the neighbourhood of a screw d i s l o c a t i o n . The maximum i n t e r a c t i o n energy has been calculated to be about the same f o r both edge d i s l o c a t i o n s and screws. In addition to i n t e r s t i t i a l i n t e r a c t i o n s with.the e l a s t i c s t r a i n f i e l d s of d i s l o c a t i o n s , i n t e r a c t i o n s are also possible with the d i s l o c a t i o n core, where e l a s t i c c a l c u l a t i o n s break down. Atomic c a l c u l a t i o n s f o r possible core states are very d i f f i c u l t to perform; attempts have been made by assuming a p a r t i c u l a r representation for - the i n t e r a c t i o n between atoms, such as a Morse function (Doyama and C o t t e r i l l (1968). 116 At low temperatures, i n t e r s t i t i a l s are potent hardeners of the hep metals Zr, T i , and Hf i n which the e f f e c t i s mainly on the thermal component of the flow stress (Tyson (1967)). Tyson found that a p p l i c a t i o n of the e l a s t i c c a l c u l a t i o n s of Cochardt et a l resulted i n an i n t e r a c t i o n which was too small to account for the observed e f f e c t s . The hardening was concluded to be due to a short range interference by solute atoms,with the atomic core of g l i d i n g d i s l o c a t i o n s , although no qu a n t i t a t i v e | d e s c r i p t i o n was pos s i b l e . The s i t u a t i o n i s more complicated i n bcc metals because i n t e r s t i t i a l atoms have an athermal as well as a thermal e f f e c t . Furthermore the i n t e r a c t i o n of i n t e r s t i t i a l s must be stronger with screws than with edges to explain the microstrain evidence. In the absence of c a l c u l a t i o n s involving the d i s l o c a t i o n core, there e x i s t s no s a t i s f a c t o r y theory at present. However, further attempts at e l a s t i c approximations haver » revealed some i n t e r e s t i n g agreements with experimental observations: ,• a) An attempt to r e l a t e the i n t e r s t i t i a l - d i s l o c a t i o n i n t e r a c t i o n to the c r o s s - s l i p of screw d i s l o c a t i o n s has been made by Formby (1966). Fi g 47a shows a section through a [111] screw dislocation-;on ( O i l ) . The possible c r o s s - s l i p planes are taken to be (110) and (101). As the d i s l o c a t i o n moves from A to B the stress at the d i s l o c a t i o n due to« , impurity atoms i n the three equivalent (^,0,0) positions w i l l change. > The actual forces w i l l depend on the p a r t i c u l a r occupation of s i t e s but i t can be seen that there are s i t u a t i o n s when the forces w i l l be applied to the d i s l o c a t i o n i n a c r o s s - s l i p d i r e c t i o n . Formby obtained experimental support for the mechanism of impurity-induced c r o s s - s l i p from observations on c r o s s - s l i p within s l i p bands and from changes dn d u c t i l i t y . F i g 47 . S e c t i o n through - ^ [ l l l ] screw d i s l o c a t i o n : a) a p p r o a c h i n g p o s s i b l e i n t e r s t i t i a l s i t e s b) showing a l l the p o s s i b l e {011} and {112} s l i p p l a n e s and two e n e r g e t i c a l l y f a v o u r a b l e d i s s o c i a t i o n s i—* 118 b) F i g 47b shows a section through a [111] screw d i s l o c a t i o n , together with a l l the possible {011} and {112} s l i p planes on which d i s s o c i a t i o n of the d i s l o c a t i o n would be po s s i b l e . Certain e n e r g e t i c a l l y favourable d i s l o c a t i o n reactions involve s e s s i l e and g l i s s i l e configurations; an example of each i s shown. Using anisotropic e l a s t i c i t y theory, Vitek and Kroupa (1966) calculated the p r o b a b i l i t i e s f o r t r a n s i t i o n s betweeen {112} and {110} planes and found good c o r r e l a t i o n with s l i p l i n e and asymmetric c r i t i c a l shear stress data. However i t must be recognized that the magnitude of the d i s s o c i a t i o n i s too small to admit the existence of d i s c r e t e p a r t i a l d i s l o c a t i o n s . i I t i s conceivable that a combination of the above two approaches could explain the impurity i n t e r a c t i o n as w e l l as the manner of screw d i s l o c a t i o n motion. However i n each model, the l i m i t of a p p l i c a b i l i t y i s set by the extent to which e l a s t i c i t y theory can be applied, since both are concerned with l o c a l i z e d e f f e c t s i n the region of the d i s l o c a t i o n core. I t i s rather discouraging to r e a l i z e that t h i s d e f i c i e n c y would become even more pronounced i n any combined approach. 119 8. SUMMARY AND CONCLUSIONS a) Oriented s i n g l e c r y s t a l s of Nb and d i l u t e a l l o y s of Nb with Mo and Ta were deformed i n tension at various temperatures over a wide range of s t r a i n s e n s i t i v i t i e s , from microyield to macroscopic f a i l u r e . b) At 295°K, a small i n i t i a l y i e l d drop was followed by a m u l t i -stage flow curve. Addition of Ta to Nb produced a s l i g h t parabolic hardening e f f e c t i n sp i t e of no atomic s i z e d i f f e r e n c e ; a very large l i n e a r hardening was produced by Mo additions. The primary s l i p system and the nature of s l i p were found to be unaffected by a l l o y i n g ; the, active s l i p plane was non-crystallographic and was rela t e d to the ori e n t a t i o n of the c r y s t a l . , c) At low temperatures or with large Mo additions, non-uniform deformation became evident. This behaviour, and the magnitude of the i n i t i a l y i e l d drop, was explained i n terms of a s t a b i l i t y c r i t e r i o n r e l a t i n g the y i e l d stress and true work hardening rate. d) The NbTa a l l o y s deformed by s l i p at a l l temperatures, although sporadic twinning was i n evidence at 77°K. At t h i s temperature Nb was softened by the addition of Ta. At low temperatures twinning or cleavage f a i l u r e without evidence of s l i p always occurred i n NbMo a l l o y s . e) A high s e n s i t i v i t y m i c r o strain technique was developed to give the stress versus s t r a i n r e l a t i o n from microyield (e - 10 ^ ) to _3 macroflow (e > 10 ). Microyield was i d e n t i f i e d with the motion of edge d i s l o c a t i o n segments; at higher stresses the screws begin to move and d i s l o c a t i o n m u l t i p l i c a t i o n occurs. At macroflow, both edges and screws 120 move l o n g d i s t a n c e s through the c r y s t a l . f ) The m i c r o y i e l d s t r e s s was found to be v e r y s e n s i t i v e to the i n i t i a l d i s t r i b u t i o n of m o b i l e d i s l o c a t i o n s . P r e s t r a i n i n g was n e c e s s a r y to e l i m i n a t e the d i r e c t e f f e c t of i m p u r i t i e s , w hich a r e b e l i e v e d to be o f s i g n i f i c a n c e even i n the h i g h e s t p u r i t y bcc m e t a l s t e s t e d to d a t e . g) The temperature s e n s i t i v i t y o f m i c r o y i e l d was deduced to be a consequence of an i n t e r a c t i o n between i n t e r s t i t i a l i m p u r i t i e s and t h e , c o r e of. edge d i s l o c a t i o n s . I t i s s u g g e s t e d t h a t t h e i n t e r a c t i o n w i t h screw d i s l o c a t i o n s i s s t r o n g e r and becomes s a t u r a t e d a t v e r y low i n t e r s t i t i a l c o n c e n t r a t i o n s . The P e i e r l s mechanism a l o n e was found to be i n c a p a b l e of e x p l a i n i n g the low temperature d e f o r m a t i o n of Nb. F urthermore the low temperature s o f t e n i n g i n NbTa a l l o y s was a l s o o b s erved a t m i c r o y i e l d , and an i n t e r s t i t i a l a s s o c i a t i o n e f f e c t , r a t h e r than a r e d u c t i o n i n P e i e r l s s t r e s s upon a l l o y i n g , was c o n s i d e r e d to be r e s p o n s i b l e . . . h) The m i c r o s t r a i n r e s u l t s i n d i c a t e d t h a t the a l l o y e f f e c t s were f a i r l y independent of s t r a i n s e n s i t i v i t y ; s u b s t i t u t i o n a l s o l u t e had a. s i m i l a r e f f e c t on the m o b i l i t y o f edges and screws. P r e v i o u s s u g g e s t i o n s t h a t the c o n c e n t r a t i o n dependence of f l o w i s a consequence o f the c o n c e n t r a t i o n dependence of d i s l o c a t i o n m u l t i p l i c a t i o n have been r e f u t e d . The F l e i s c h e r a n a l y s i s o f s o l i d s o l u t i o n h a r d e n i n g was found t o be ; u n s u c c e s s f u l when a p p l i e d to b c c s u b s t i t u t i o n a l a l l o y systems. A s t a t i s t i c a l t h e o r y o f a l l o y h a r d e n i n g may be more p r o m i s i n g . i ) Two components o f the i n t e r n a l s t r e s s f i e l d were r e c o g n i z e d : one component always opposes d i s l o c a t i o n motion and i s d e t e r m i n e d by the d i s l o c a t i o n d e n s i t y and s o l u t e c o n c e n t r a t i o n ; the e f f e c t o f t h e o t h e r 121 component depends on the d i r e c t i o n o f d i s l o c a t i o n motion and i s dete r m i n e d by t h e i n s t a n t a n e o u s d i s t r i b u t i o n o f m o b i l e d i s l o c a t i o n s . C o n s e q u e n t l y the i n t e r n a l s t r e s s does n o t have a unique v a l u e below the macroflow s t r e s s and can i t s e l f be temperature dependent. j ) The s i g n i f i c a n c e o f screw d i s l o c a t i o n m o t i o n i n the d e f o r m a t i o n o f b c c me t a l s and a l l o y s was r e c o g n i z e d . However the , d i s t i n c t i o n between a " P e i e r l s e f f e c t " and an " i m p u r i t y e f f e c t " was c o n s i d e r e d semantic r a t h e r than b a s i c i n view o f the predominance o f i n t e r s t i t i a l - d i s l o c a t i o n c o r e i n t e r a c t i o n s a t a p o i n t where e l a s t i c c a l c u l a t i o n s b r e a k down. The p o s s i b l e r e s u l t s o f an ex a c t c a l c u l a t i o n were suggested by means o f d i f f e r e n t e l a s t i c a p p r o x i m a t i o n s . ; 122 APPENDICES A . l P r o p e r t i e s o f r e l e v a n t b c c m e t a l s Niobium Molybdenum Tantalum Atomic number 41 42 73 Atomic weight 92.9 95.9 180.9 o M e t a l l i c r a d i u s (A) 1.43 1.37 1.43 V a l e n c e e l e c t r o n s 5 6 5 M e l t i n g p o i n t (°C) 2468 2610 2996 Vapour p r e s s u r e a t 2468°C ( t o r r ) 1 x 10~3 3 x 10~2 8 x 10' E l a s t i c c ompliances a t 300°C: „-12 2, -1, (x 10 cm dyn ) Su 0.687 0.291 0.699 ( C a r r o l l (1965), s 12 -0.248 -0.818 -0.379 F e a t h e r s t o n e and Neighbours (1963)) 3.41 0.823 1.22 A.2 P u r i t y o f vacuum-melted n i o b i u m The m e c h a n i c a l p r o p e r t i e s o f the bcc m e t a l s a r e v e r y s e n s i t i v e to the p r e s e n c e o f s m a l l amounts of i m p u r i t i e s , p a r t i c u l a r l y the gaseous i n t e r s t i t i a l elements, oxygen and n i t r o g e n . S i n c e a n a l y s i s o f t h e s e elements becomes i n c r e a s i n g l y u n r e l i a b l e as t h e i r c o n c e n t r a t i o n i s d e c r e a s e d , i t i s u s e f u l to e s t i m a t e the amounts p r e s e n t under g i v e n m e l t i n g c o n d i t i o n s , by u s i n g c a l c u l a t i o n s based on e x i s t i n g thermodynamic d a t a . S i e v e r t s Law, which i s a m o d i f i c a t i o n o f Henrys Law, s t a t e s t h a t a t c o n s t a n t temperature the s o l u b i l i t y o f a d i a t o m i c gas i n a m e t a l i s p r o p o r t i o n a l to the square r o o t o f the p r e s s u r e . Pemsler (1961) has published data f o r the i n t e r a c t i o n of Nb with 0 and N at temperatures near the melting point. This data i s shown i n F i g 48 i n the form of Sieverts Law p l o t s . From the slopes of the l i n e s at 2470°C, the s o l u b i l i t i e s of 0 and N at a p a r t i a l pressure p t o r r are given by: t 0 ] a t % - ^ 5 p Q J (1) [ N ] a t % (06 *N 2 <2> I f P t o r r i s the operating pressure i n the melting chamber (ie dynamic vacuum) and the gases are present i n the proportions they are i n a i r , then p n = 0.2 P (3) u2 P N 2 = 0.8 P (4) Also, since [0]at% = ~^[0]VVm (5) [N]at% = Y ^ f N l p p m (6) s u b s t i t u t i n g equations (3) and (5) i n (1), and (4) and (6) i n (2), gives the amount of 0 and N ( i n wt ppm) i n equilibrium with l i q u i d Nb at 2470°C: [0]ppm = 1730 P^ (7) [N]ppm = 3800 P** (8) Equations (7) and (8) are plotted as f u l l l i n e s i n F i g 49, from which i t can be seen, for example,that at an operating pressure of 5 x 10 ^ t o r r , the equilibrium concentrations of 0 and N are 12 and 27 ppm r e s p e c t i v e l y . 124 0 1 2 3 4 5 6 7 8 9 Oxygon Concentration, Atomic Porcont 48. Sieverts Law plots of solubility of a) oxygen b) nitrogen in Nb at high temperatures (from Pemsler (1961)). 125 Since i t i s c e r t a i n that the as-received Nb contained more than t h i s amount of 0 and N, the calculated figures must represent the highest possible p u r i t y . The figures are a minimum because, as can be seen from F i g 48, the a f f i n i t y of Nb f o r 0 and N increases at temperatures below the melting point. Pemsler's data has also been used to c a l c u l a t e the dashed curves i n F i g 49, which show the amounts of 0 and N i n equilibrium with s o l i d Nb at 2170°C. I t i s evident that contamination of the melted Nb w i l l occur a f t e r s o l i d i f i c a t i o n . I t i s possible to determine the magnitude of the dynamic vacuum required for p u r i f i c a t i o n by high temperature annealing. For example, a pressure of l e s s than 5 x 10 ^ t o r r would be required to p u r i f y an a l l o y containing 12 ppm 0 by vacuum annealing at 2170°C. ,-A. 3 Crystallography of s l i p A.3.1 D e f i n i t i o n s of s l i p parameters Consider a d i s l o c a t i o n l i n e with Burgers vector b_, which can move conservatively on a general plane of normal vector p_, i n a c r y s t a l subjected to a t e n s i l e s t r e s s , a. r~~ 1 — r 1 r Nb(s) at Nb(l) at 2170 °C [0] 2470 °C i io~8 io"7 io"6 io - 5 • io"4 10 3 P r e s s u r e , P t o r r F i g 49. C a l c u l a t e d e q u i l i b r i u m s o l u b i l i t i e s o f oxygen and n i t r o g e n i n Nb a t h i g h temperatures. Glide motion i s represented by b.g = 0 Let a " p = 9 and a " b = £ , as shown, where 9 + £ > 90° In bcc c r y s t a l s , the Burgers vector i s always of the type <111> , so that f o r a p a r t i c u l a r orientation,£ i s f i x e d . The resolved shear stress i n the d i r e c t i o n b on the plane p_ i s given by T(9) = a.cos9.cos£ where cosG.cos? i s the Schmid factor f o r that s l i p system. I f no assumptions are made about s l i p planes being c r y s t a l l o g r a p h i c , then the plane on which T(9) i s a maximum i s of i n t e r e s t ; the plane p becomes the "maximum resolved shear stress plane", m . I t corresponds to a minimum value of 9 (9 + £ = 90°) which occurs when the vectors m a and b are coplanar. On the other hand, i f the observed s l i p traces are measured and the d i r e c t i o n of p_ i s determined, then g i s c a l l e d the "observed s l i p plane n , and 0 has the p a r t i c u l a r value, K. The shear stress resolved on the observed s l i p system i s then given by T = O.COSK.COSE ; K where COSK.COSC i s the observed Schmid f a c t o r , s. r These vectors can be represented as poles on a (001) stereographi pro j e c t i o n , i f the t e n s i l e axis a i s placed within the standard t r i a n g l e [001] - [101] - [111] : 128 12 / \pll \ % D / £ . -—" cr - - " 11 — • — 121 Because of the symmetry of the (111) pole, the (112) plane i s equivalent to the (121); so i t i s useful to take (Oil) as a reference plane. The positions of n and m can then be determined by the angles ^ and x that they make with the (Oil) pole, such that -30° < *,x * +30° A.3.1 Orientation dependence I t has been found experimentally (Sestak and Zarubova (1965)) that i n c r y s t a l s of d i f f e r e n t o r i e n t a t i o n s , the observed s l i p plane (measured by \p) does not depend on the angle but i s a function only of the p o s i t i o n of the maximum resolved shear stress plane (measured by x)-Thus the o r i e n t a t i o n dependence of the observed s l i p plane can be expressed by K x ) for -30 °< x < +30? If s l i p occurs on the maximum resolved shear stress plane, then <J> = X I f s l i p i s c r y s t a l l o g r a p h i c : on (Oil) then x\> = 0 (143) 4> = 14° (132) ip = 19° (121) i|> = 30° In the past, c r y s t a l l o g r a p h i c s l i p has been reported on a l l these planes (Bowen et a l (1967), Milne and Smallman (1968)). However the accuracy of the s l i p l i n e determinations i s never quoted, so that whether s l i p i s or i s not c r y s t a l l o g r a p h i c could depend on the s t a t i s t i c a l approach of the author. The Oxford school (eg F o x a l l et a l (1967)) take the observed asymmetry of iji with respect to x as evidence for d i s c r e t e c r y s t a l l o g r a p h i c s l i p on {112} planes. They divide the standard t r i a n g l e into regions of expected operation of s l i p systems having {011} and {112} s l i p planes. 001 Thus the enclosed areas represent regions where the Schmid ,factor f or a system with a [111] Burgers vector on the given s l i p plane i s a maximum; the dotted boundary l i n e s have equal Schmid factors on the adjacent systems. To allow for non-crystallographic s l i p , the s l i p systems are here described i n terms of the Burgers vector and the s l i p plane parameter 130 A.3.3 Determination of s l i p systems The s l i p plane can be determined by measuring the angle of i n c l i n a t i o n of the s l i p trace to the t e n s i l e axis (<|>) at various angles (8) around the c r y s t a l . n In the above diagram, the s l i p plane i s represented by i t s normal vector n ; C i s tangential to the s l i p plane at i t s point of maximum i n c l i n a t i o n to a ; A represents g = 0 and i s normal to C and n ; E, i s f i x e d for a given c r y s t a l o r i e n t a t i o n . I t i s desired to determine K from a knowledge of <j>(g). In F i g 50, the above angles are represented on a stereographic p r o j e c t i o n with a again i n the standard o r i e n t a t i o n and b = [111]., , Z(n) represents the zone of the s l i p plane n , and Z(b) and Z(a) represent the zones of b and a r e s p e c t i v e l y . Therefore, A l i e s at the i n t e r s e c t i o n of Z(a) and Z(n) B l i e s at g from A along Z(a) C l i e s at 90° from A on Z(n) „ D l i e s at <f> from a on Z(n) and at 90° to B Let D * C = X, then using Napiers Rule i n A DCa sin(90 - 0 = cos(90 - K).COSX therefore, cose}) = -sinic.cosX (1). 131 Fig 50. (001) Stereographic projection showing parameters for slip line analysis. 132 Using the Cosine Rule i n A DAB : cos 90 = cosg.cos(90 - X) + sing.sin(90 - X)cos(180 - K) therefore, COSK = cotg.tanX (2) Eliminating X from equations (1) and (2) gives: s i n ^ K .. , 2 ^ 2 . 2~ = 1 + cos K.tan 3 cos cb c . + sin< (r.. therefore, coscb = - 2 2 — \ - ^ ' (1 + cos K.tan 6 ) 2 When 6 = 0 , coscb = s i n K = cos (90 - K) therefore, cb = 90 - K which i s the minimum value of cb. When B = 90, coscb = 0 therefore, • <b - 90 When 3 = 180, cos<{> = -sinK = cos (90 + K) . therefore, c() = 90 + K which i s the maximum value of cf). Fig 51 shows a t h e o r e t i c a l p l o t of <j>(g) calculated from equation (3). Any two experimental observations separated by g 'v 90° would be s u f f i c i e n t to characterize the curve. In the case of s i n g l e s l i p the graphical approach i s unnecessary because an X-ray observation at known 3 can be used with a two-surface trace analysis to i d e n t i f y the s l i p plane. However i n the case of multiple s l i p , the cosine curves must be f i t t e d to the observed points. The cosine curves for each system w i l l have the same period, but may d i f f e r i n phase and amplitude. Once the purye i s established, the operative plane i s determined as for s i n g l e s l i p . The operative Burgers vector can be determined absolutely i f the point of disappearance of the s l i p l i n e s i s determined. Then D is, p a r a l l e l to b and so cf> = 5 , which i d e n t i f i e s b for a known o r i e n t a t i o n . > 133 F i g 51. P l o t of <|>(B) showing i n f o r m a t i o n r e q u i r e d f o r d e t e r m i n a t i o n o f one s l i p system. 134 A.4 D e t a i l s o f m i c r o s t r a i n t e s t i n g A.4.1 I n t r o d u c t i o n A m i c r o s t r a i n experiment r e q u i r e s an i n s t r u m e n t which i s -4 c a p a b l e o f measuring s t r a i n s l e s s than 10 . Some of the c h a r a c t e r i s t i c s of p o s s i b l e types o f extensometer a r e g i v e n i n the t a b l e below: C a p a c i t a n c e D i f f e r e n t i a l Bonded M e c h a n i c a l t r a n s f o r m e r r e s i s t a n c e o p t i c a l Maximum s t r a i n 10 s e n s i t i v i t y (1 in.gauge) Attachment to specimen d i r e c t O p e r a t i n g temperature room and below Range R e v e r s i b i l i t y L i n e a r i t y wide e x c e l l e n t good 10 i n d i r e c t room wide e x c e l l e n t v e r y good I O " 6 d i r e c t room and below s m a l l f a i r poor i o " 1 0 d i r e c t room v a r i a b l e p o o r poor I n a d d i t i o n to the i n h e r e n t s e n s i t i v i t y of the gauge, the o p e r a t i n g l i m i t i s determined by the m e c h a n i c a l , e l e c t r i c a l and th e r m a l s t a b i l i t y of the t e s t assembly. For t e s t i n g i n t e n s i o n , o v e r a range o f temperature, the c a p a c i t a n c e extensometer was s e l e c t e d as b e i n g the most s u i t a b l e . The d i f f e r e n t i a l t r a n s f o r m e r i s p r e f e r a b l e f o r compression t e s t i n g where d i r e c t attachment to the specimen i s not r e q u i r e d (Meakin (1967), Bowen e t a l (1967)). The c a p a c i t a n c e t e c h n i q u e was p i o n e e r e d by Brown and h i s co-workers a t the U n i v e r s i t y o f P e n n s y l v a n i a . The p l a t e s e p a r a t i o n was determined from a d i r e c t measurement o f the c a p a c i t a n c e C which depends on p l a t e 135 separation, SL, according to: This means that the output, x, varies with plate separation according to: dx d!L dC d!L « 1 The i n i t i a l gap was set to the desired s e n s i t i v i t y using a micro-meter attachment on the ground plate of the capacitance extensometer (Brown (1968)). A.4.2 Equipment The e l e c t r o n i c micrometer used i n t h i s work i s a commercial apparatus which gives a d i r e c t measurement of the separation between a dx ' probe and a grounded object. Thus x « SL and the s e n s i t i v i t y , , i s independent of p l a t e separation. Wayne Kerr Company Limited, New Maiden, Surrey, England and consists pf a s t a b i l i z e d power supply, 50 kHz o s c i l l a t o r , high gain a m p l i f i e r and. distance metering c i r c u i t . The output from the instrument was fed i n t o a Wayne Kerr low pass f i l t e r , F731A, through a simple reverse voltage c i r c u i t to the X-axis of a Honeywell model 520 X-Y recorder having a maximum s e n s i t i v i t y of 0.1 mV i n \ The output from the Instron load c i r c u i t was fed d i r e c t l y to the other axis of the recorder. The c i r c u i t i s shown schematically i n F i g 52. A.4.3 Design of extensometer The operation of the distance meter depends on comparing the test capacitance with an i n t e r n a l preset capacitor of value 0.35 pF. This value defines the maximum separation of the plates of the test The Wayne Kerr DM100B distance meter was manufactured by.; the 136 Load c e l l Ins tron recorder p 1 . 0 High Low Guard Ground Y-axis inputs 'Specimen xtensometer Load Honeywell model 520 X-Y recorder Extension X-axis inputs High Low Guard Ground Reverse voltage c i r c u i t Wayne Kerr DM 100 B Distance meter Low pass f i I t e r T T F i g 52. Schematic c i r c u i t diagram f o r m i c r o s t r a i n t e s t i n g . 137 capacitance, and i s d i r e c t l y proportional to the area of the a c t i v e plate. The displacement s e n s i t i v i t y i s i n v e r s e l y proportional to the area of the p l a t e . The a c t i v e p l a t e must be completely surrounded by a guard r i n g , and separated from i t by a t h i n layer of i n s u l a t i o n to ensure that the f i e l d i s normal (or nearly so) to the surface of the p l a t e . I t i s p ossible to relax t h i s condition and, by decreasing the area of the active plate, increase the s e n s i t i v i t y at the expense of l i n e a r i t y . The instrument must then be s p e c i f i c a l l y c a l i b r a t e d . This was the case i n t h i s design which gave adequate s e n s i t i v i t y without re q u i r i n g excessive s i g n a l a m p l i f i c a t i o n . Detailed i n s t r u c t i o n s f o r the design of s p e c i a l probes are given i n the DM100B Instr u c t i o n Manual, p.44. A section through the c y l i n d r i c a l extensometer i s shown diagrammatically below: Body of gauge Double-shielded coaxial cable Insulation Guard ring Ground plate Specimen M a t e r i a l : brass Insulation: mica and epoxy r e s i n Area of a c t i v e p l a t e : 0.039 i n ^ The output from the distance meter i s 0 - 1 mA at 1000 fi. The distance meter reading, d, i s i n the range 0 - 1 0 u n i t s . The extensometer 138 was c a l i b r a t e d using f e e l e r gauges and the r e s u l t s are shown i n F i g 53. This gives d = d(&) which i s the meter reading as a function of true plate separation i n thou. The output corresponding to a meter reading d = 100.d mV therefore, input to recorder = (100.d r- V) mV (where V i s the reverse voltage i n mV) * J J - i (100.d - V) therefore, recorder displacement x = ^ (where S i s recorder s e n s i t i v i t y i n mV i n ~ x ) therefore, recorder displacement s e n s i t i v i t y ^ = ins of chart / i n displacement. If E i s the displacement represented by one small d i v i s i o n of chart ( = 0.1 i n ) , then E = S ( ) inches displacement. This parameter has been calculated from the c a l i b r a t i o n curve i n F i g 53 and i s shown for d i f f e r e n t S values i n F i g 54. 1 A.4.4 Testing At room temperature, the microstrain test was performed using a conventional inverted t e n s i l e j i g , with e f f o r t s made to eliminate draughts and v i b r a t i o n s . Room temperature s t a b i l i t y was investigated p r i o r to each test using the time-base mode on the Y-axis of the recorder. For use at low temperatures, a gas-cooling cryostat was designed and b u i l t . The arrangement i s shown schematically i n Fi g 55. Fig, 56 i s a photograph of the testing assembly before lowering of the cryostat. The range of temperature operation was determined by the bath temperature. Fine adjustment was provided by c o n t r o l l i n g both the current through the grip heaters and the flow of cold gas, obtained by e l e c t r i c a l heating of 5 F i g 53. D i s t a n c e meter r e a d i n g as a f u n c t i o n of p l a t e s e p a r a t i o n . S = 0-2 S s 0 - 5 o •rt CO •rt > •rt T3 ca 6 CO a •rt 3. W •H > •H •U •H CO fl CU CO 4 J fl cu s cu o CO r H P -CO •rt Q 3.0 2.0 1.0 1 1 1 E = E(d,S) -I I ! I - 8.0 6.0 4.0 2.0 1.4 1.8 2.2 2.6 3.0 3.4 Distance meter reading, d 3.8 4.2 Fig "54.' Displacement sensitivity of extensometer" as a function of-distance meter reading. o A - u n i v e r s a l j o i n t s B - h e a t i n g c o i l s on g r i p s C - c o n t r a c t i o n j o i n t D - d r y gas i n l e t E - c a p a c i t a n c e extensometer F - b a s e p l a t e G - e x i t h o l e Thermocouples n o t shown F i g 55. Schematic diagram o f s e c t i o n through g a s - c o o l i n g c r y o s t a t . F i g 56. Test assembly f o r m i c r o s t r a i n i n g at low temperatures. ( cf F i g 55 ) 143 a c o i l immersed i n l i q u i d N^. The temperature was measured by two thermo-couples near the specimen shoulders, and required very accurate c o n t r o l . S t a b i l i t y was best gauged p r i o r to t e s t i n g , by monitoring the instantaneous stress and s t r a i n on the specimen during the operation of the Instron stress c y c l i n g procedure. During testing,, a continuous record of load versus time was. obtained from the Instron chart recorder. The load c e l l could be c a l i b r a t e d on e i t h e r the Instron chart of the X-Y recorder (Y-axis s e n s i t i v i t y = 1.0 mV inch X) and the r e l a t i o n between them obtained. The alignment was established by repeatedly applying small loads u n t i l an exactly r e v e r s i b l e l i n e a r trace was obtained. This could r e a l l y only be done s a t i s f a c t o r i l y on specimens which had been prestrained i n s i t u . The procedure adopted at room temperature i s indicated i n an i d e a l i z e d chart from the X-Y recorder i n F i g 57. A.5 E l a s t i c constants i n cubic s i n g l e c r y s t a l s In a s i n g l e c r y s t a l , many e l a s t i c constants may be required to define the anisotropic e l a s t i c behaviour. The value of Youngs modulus, E, and shear modulus, u, w i l l be a d i f f e r e n t combination of the constants for d i f f e r e n t c r y s t a l orientations and shear systems. The p a r t i c u l a r expressions can be derived by applying the tensor transformation law to the generalized Hookes Law equation: e = S a (1) mn mnpq pq where the repeated s u f f i x convention i s used and m n p q = 1,2,3. This equation expresses any component of the s t r a i n tensor e i n terms of the stress tensor a . The S are the " e l a s t i c compliances" and form pq mnpq a fourth order tensor. The tensors are ref e r r e d to a right-handed set of 144 A - c a l i b r a t i o n o f l o a d c e l l on X-Y r e c o r d e r ( L = f u l l s c a l e l o a d i n l b s ) B - s t a b i l i t y t e s t , w i t h Y - a x i s on time base C - al i g n m e n t check D - l o a d c y c l i n g u n t i l permanent s e t E - d e t e r m i n a t i o n of f l o w c u r v e F i g 57. I d e a l i z e d e x p e r i m e n t a l X-Y r e c o r d e r c h a r t from m i c r o s t r a i n t e s t . 145 orthogonal axes, chosen to correspond with symmetry axes i n the c r y s t a l . Because of the symmetry of e and o , equation (1) can be J J mn pq ^ expressed i n shorter matrix form by the equation: e. = S. . o\ (2) where i , j = 1... 6 Equations (1) and (2) are equivalent provided the following, scheme i s followed when changing from tensor to matrix notation (see Nye (1957)): The pai r s of s u f f i x e s mn become i , and pq become j according to mn 11 22 33 23,32 13,31 12,21 pq , i l 2 3 4 5 6 j , and i n addition factors of h and h ave introduced i n t o the S..: S S. . (for i , j <= 1, 2, 3) mnpq i j S = SfiS.. (either i or j = 4, 5, 6) mnpq i j S hS.. (both i and j = 4, 5, 6) mnpq xj The stress states of i n t e r e s t i n this study are u n i a x i a l tension and pure shear, but the r e s u l t i n g s t r a i n s can only be obtained d i r e c t l y from the published i f the stress state can be described d i r e c t l y by the o r i g i n a l co-ordinate frame, x^ ( i = 1, 2, 3). Thus Youngs modulus i n [100] i s defined by ; 1 3 E I and the shear modulus on (001) i n [010] i s given by - 9 E 1 4 6 The new co-ordinate frame, x. , i s chosen to coincide with 1 the direction of interest (for E) or the plane and direction of interest (for u). The transformed tensor equation ( 1 ) becomes S i j k l akl ( i , j , k s 1 , 2 , 3 ) The required S . . can be obtained from the S by applying M ijk& mnpq J r r J the tensor transformation law for fourth order tensors: S,.,rt ~™ 3 , • 3 , • cli • 3. rt • S ijk£ im in kp Jiq mnpq ( 5 ) where a.. is the transformation matrix which relates the new frame x^ to the old, x^. The components are defined as: a.. = cos(x. " x.) i j i 3 From equations ( 3 ) and ( 4 ) the required moduli are: 1 = S 11 = S 1111 (6) ( 7 ) and = S^it = S 3 232 + S 3 223 + S2332 + S2323 (8) In general, the symmetric matrix has 2 1 independent constants, but in the case of cubic crystals, this number i s reduced to three and '11 s12 s12 0 S12 S l l S12 0 S12 S12 S l l 0 0 \ 0 0 0 0 0 0 0 0 shk 0 0 0 0 0 0 0 0 0 0 0 0 s 4 i + From the non-zero values of S.. we can obtain the non-zero values of S to be used in the transformation equation ( 5 ) . The mnpq 147 T a b l e V. Non-zero terms f o r use i n t r a n s f o r m a t i o n e q u a t i o n (5) i n the case o f c u b i c c r y s t a l s . Non-zero i j C o r r e s p o n d i n g m n p q E q u i v a l e n t S ( c u b i c ) E q u i v a l e n t C „ ( c u b i c ) 1 1 1 2 1 3 1 1 1 1 2 3 1 2 3 S n s 1 2 S12 C l l C12 C12 2 1 2 2 2 3 2 2 2 s12 S l l S12 C12 C l l C 1 2 3 1 3 2 3 3 3 3 3 S 1 2 S l 2 S l l Cl2 C 1 2 C l l 2 2 3 3 3 3 2 2 2 3 2 3 3 2 3 2 Hsu* 3 3 1 1 1 1 3 3 3 1 3 1 1 3 1 3 Ci+i+ 2 2 1 1 1 2 1 2 2 1 2 1 Ci+4 p a r t i c u l a r value of the S i s then converted back to S.. for cubic mnpq I J c r y s t a l s . These operations have been performed i n Table V. A.5.1 Youngs modulus If the t e n s i l e axis l i e s i n a d i r e c t i o n z r e l a t i v e to the,old frame (which coincides with the tetrad axes i n a cubic c r y s t a l ) such that: z " X} = a z A x 2 = 3 z * x 3 = y then the new frame i s selected so that x.^  coincides with z . From equation (6), i t follows that the components of the i t n row of the transformation matrix are the components of a unit vector along x. r e l a t i v e to the x. frame. Thus a.. = x..x. and the relevant components i n this case become: a ^ = cosa a 1 2 = cosg a 1 3 = cosy From equations (5) and (7), — = S i i i i = a, .a, .a. .a, .S - 1 1 1 1 lm In l p lq mnpq The non-zero terms are obtained d i r e c t l y from the table by su b s t i t u t i n g the m, n, p, q and corresponding S ,^ for cubic c r y s t a l s . Thus, - = S l l ( a n + a 1 2 + a 1 3 ) E H" ( 2a.^ j^  "^12 2^12 "^13 2^13 "^11 ^ + J j S ^ ( 4 a n 2 . a 1 2 2 + 4 a 1 2 2 . a 1 3 2 + 4 a 1 3 2 . a n 2 ) I f n i s d e f i n e d as an o r i e n t a t i o n f a c t o r : 2 2. • 2. 2 , 2 2 n = cos a.cos 3 + cos p.cos y + cos y-cos a then the e x p r e s s i o n s i m p l i f i e s t o : - = S n . ( l - 2n) + (2S 1 2 .+ S ^ ) . T J A.5.2 Shear modulus The s h e a r modulus can be o b t a i n e d s i m i l a r l y from e q u a t i o n s (5) and (8) once the t r a n s f o r m a t i o n m a t r i x i s d e t e r m i n e d . I f the s h e a r i s on a p l a n e w i t h normal v e c t o r n i n d i r e c t i o n 3 then the frame i s s e l e c t e d so t h a t X3 c o i n c i d e s w i t h n and x 2 c o i n c i d e s w i t h 3 . A g a i n the components of the t r a n s f o r m a t i o n m a t r i x a r e g i v e n by a.. = x..x. , where x. and x. a r e u n i t v e c t o r s , i j 1 3 i j C o n s i d e r f o r example, a s h e a r on (112) i n [111] which i s a t w i n n i n g s h e a r i n b c c m e t a l s : K 3 = [112] x 2 = [111] t h e r e f o r e , a 3 1 - £ [112].[100] a 3 2 = £ [112].[010] a 3 3 = £ [112].[001] = J S i m i l a r l y , a n = - y j , a n = , &23 = I f t h e s e v a l u e s a r e s u b s t i t u t e d i n e q u a t i o n s (5) and (8) and a change made to matrix notation, the r e s u l t i n g expression for the shear modulus i s : - = -I- [Skk + 4 ( S n - S 1 2 ) ] Since the transformation matrix i s symmetrical with respect to n and 8 , V remains the same i f n and g are interchanged. Expressions for — for other shears i n cubic c r y s t a l s are: y (001) [110] s ^ (110) [110] 2 ( S n - S 1 2 ) (111) [ l l O ] i [ S 4 1 + + 4 ( S n - S 1 2 ) ] Similar equations to these can be obtained by expressing the stress components i n terms of the s t r a i n components a. = C.. e. (cf equation (2)) where the C . are the " e l a s t i c s t i f f n e s s constants". The r e l a t i o n s between the S.. and C.. for the cubic system are i j ±3 given by: s l l " s 12 - r. _ r , „ u l l ^12 S\i + 2S^ 2 = ^ 0 r , — c l l + 2 C 1 2 I t can be seen that for the shears (001) [110] and (110) [ l l O ] y i j but t h i s i s not the case for the shear (111) [110] which i s of 151 importance in bcc deformation. Since deformation experiments apply a particular stress and measure the resulting strains, the modulus should be expressed in terms of S „ rather than C\ , as is usually the case (Bowen et al (1967), Mitchell and Raffo (1967)). A shear modulus useful for solution hardening theory is Voigts'average: In.the case of an isotropic crystal (eg tungsten) or a polycrystal ( 2 8 ^ ( 5 1 ! - S i 2 ) 1 there are only two independent elastic constants E = 1 and 1 1 y = Si+4 2 ( S n - S 1 2 ) and Voigts expression reduces to u . REFERENCES Ahktar A. (1968). PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia A l l e n B.C. and Jaffee R.F. (1963). Trans ASM 56 387 Arsenault R.J. (1967). Acta Met 15 501 (1969). Acta Met 17 1291 A r s e n a u l t R.J. and Lawley A. (1967). P h i l Mag 15 549 Bowen D.K. , C h r i s t i a n J.W. and T a y l o r G. (1967). Can J . Phys 45_ 903 Brown N. (1968). M i c r o p l a s t i c i t y p.52, I n t e r s c i e n c e , New York Brown N. and E k v a l l R.A. (1962). A c t a Met 10 1101 C a l v e r l e y A., D a v i s M. and L e v e r R.F. (1957). J . S c i I n s t 34 142 Carnahan R.D., A r s e n a u l t R.J. and Stone G.A. (1967). Trans Met Soc AIME C a r r o l l K.J. (1965). J . 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