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The effect of a thoria dispersion on the yielding and flow of iron Place, Thomas Alan 1969

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THE EFFECT OF A THORIA DISPERSION ON THE YIELDING AND FLOW OF IRON BY THOMAS ALAN PLACE B.Sc. University of Nottingham, 1961 M.Eng. McMaster University, 1966 A.I.M. 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 stancla^d THE UNIVERSITY OF BRITISH COLUMBIA August 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 of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood t h a t copying or p u b l i c a t i o n of 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 not 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 . Department of Metallurgy The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date September 15, 1969 i i ABSTRACT A technique for the preparation of iron containing a fin e dispersion of thoria p a r t i c l e s has been developed. The method involves co-precipitation, hydrogen reduction and hot r o l l i n g i n a reducing atmosphere at 1373°K. The mechanical properties of iron and iron-thoria s t r i p have been investigated over the temperature range 77-373°K. The dispersion increased the strength and work hardening rate of iron, and reduced the d u c t i l i t y . The y i e l d strength of the a l l o y has been explained i n terms of dislocation multiplication rates and the stress dependence of dislocation velocity. The work hardening rate i s regarded as a balance between the y i e l d stress behaviour at low temperatures and recovery processes at the higher temperatures. The mechanics of neck and Luders band formation have been examined, and the observed fracture angles explained i n terms of this analysis. The d u c t i l e - t o - b r i t t l e t r a n s i t i o n temperature was found to be improved in the iron-thoria, as a consequence of both a finer grain size and the direct influence of the p a r t i c l e s . i i i ACKNOWLEDGEMENTS The author i s indebted to Dr. J.A. Lund for his advice and assistance throughout the duration of this work. Thanks are also due t o n y fellow graduate students and the members of faculty for innumerable helpful discussions. The assistance of the technical s t a f f , throughout the experimental program, has been greatly appreciated. Financial assistance i n the form of a National Research Council Scholarship is gratefully acknowledged. i v CONTENTS Page Section 1 INTRODUCTION 1 Section 2 EXPERIMENTAL PROCEDURE 2 2.1. Materials 2 2.2. Co-Precipitation l 2 2.3. Oxide Reduction 1 2 2.4. Consolidation 13 2.4.1. Compact preparation 13 2.4.2. Hot r o l l i n g 1 6 2.5. Preparation of Tensile Samples 21 2.6. Tensile Tests 2 1 2.7. Metallography • 2 2 2.7.1. Optical metallography • 2 2 2.7.2. Electron microscopy replicas 2 2 2.7.3. Transmission microscopy 24 Section 3 RESULTS AND OBSERVATIONS 2 6 3.1. Characteristics of the Materials • 26 3.1.1. Chemical analyses • 26 3.1.2. Quantitative metallography. ^6 3.1.2.1. Optical metallography •••• • 2 ^ V Page 3.1.2.2. Replica microscopy 31 3.1.2.3. Transmission electron microscopy 34 3.2. Tensile Properties 34 3.2.1. Yield stress and ultimate t e n s i l e stress. 38 3.2.2. D u c t i l i t y 38 3.2.3. Fracture 44 3.2.4. Load-elongation curves 52 3.3. Work Hardening 56 3.3.1. Analysis of load-elongation curves 5 6 3.3.2. Work hardening behaviour 66 3.4. Transmission Electron Microscopy 66 3.4.1. Structure 66 3.4.2. Dislocation density 80 Section 4 DISCUSSION ... 85 4.1. Yielding Process i n Iron 86 4.1.1. Upper y i e l d point and y i e l d drop behaviour 89 4.1.2. The lower y i e l d point and Luders band ... 97 4.2. Yielding Processes i n Iron-Thoria Alloys 102 4.2.1. Theories of the y i e l d stress of dispersion hardened materials 103 v i Page 4.2.2. Assessment of the Ansell and Orowan theories ^ ® 4.2.3. Yield behaviour of iron thoria 1 1 4 4.2.4. Temperature dependence of the y i e l d stress 1 2 4 4.3. Work Hardening ^ 2^ 4.3.1. Work hardening of iron 4.3.2. Work hardening of iron-thoria 4.3.3. Luders band f a i l u r e s 4.4. Fracture 4.4.2. Low temperature f a i l u r e s 4.4.3. The duct i l e to b r i t t l e t r a n s i t i o n Section 5 SUMMARY AND CONCLUSIONS Appendix I INTERPARTICLE SPACING PARAMETERS ............ Appendix II COMPUTATION OF WORK HARDENING RATES... Appendix III SYMBOLS BIBLIOGRAPHY . 125 128 134 135 4.4.1. Ductile f a i l u r e • • 1 3 5 140 140 142 145 148 151 154 v i i LIST OF FIGURES Number Page 1. Flowsheet of fabrication methods 3 2. Weight loss-time curve for hydrogen reduction of ¥e^Q^ at 800°C. Hydrogen flow rate approximately 200 s.c.c.m. 14 3. Double acting die used for powder compaction 15 4. Hot r o l l i n g assembly 17 5. Internal arrangement of hot r o l l i n g assembly 18 6. Optical micrograph of iron 1.7% thoria. X1950 29 7. Optical micrograph of iron 1.7% thoria. X150 30 8. Replica, of iron 1.7% thoria 32 9. Replica of iron 1.7% thoria 33 10. Variation of y i e l d stress with temperature. Strain rate 4.2 x l O " 4 sec.-l 39 11. Variation of ultimate t e n s i l e stress with temperature. Strain rate 4.2 x 10~ 4 s e c . _ l 40 12. Qualitative plot of y i e l d behaviour 41 13. Variation of uniform elongation with temperature. Strain rate 4.2 x 10-4 sec. "1....... 42 14. Variation of t o t a l elongation with temperature. Strain rate 4.2 x 10~ 4 s e c . - 1 43 15. Diagram of a combined 90°-57° fracture. Two Luders bands run from the arrest point of a transverse crack, one of the bands forming a neck , . . 46 16. Thoria stringers (black lines) i n a broken te n s i l e sample. Rolling direction l e f t to ri g h t . X1600 47 17. Same as Fig. (16), but taken i n dark f i e l d to show thoria p a r t i c l e d i s t r i b u t i o n . X1600 49 18. Crack associated with large thoria stringer. X1000 ... 50 v i i i Number Page 19. Fracture edge of an iron t e n s i l e specimen. Pulled at 77°K, photographed i n polarised l i g h t . Twins aligned with fracture edge. X200 51 20. True stress-true p l a s t i c s t r a i n curves for iron. Strain rate 4.2 x 10" 4 sec." 1 53 21. True stress-true p l a s t i c s t r a i n curves for iron 0.9% thoria. Strain rate 4.2 x 1 0 - 4 s e c . - 1 54 22. True stress-true p l a s t i c s t r a i n curves for iron 1.7% thoria. Strain rate 4.2 x 10" 4 sec." 1 55 23. Variation of work hardening rate with temperature, at 1% true p l a s t i c s t r a i n 67 24. Variation of work hardening rate with temperature, at 2% true p l a s t i c s t r a i n 68 25. Variation of work hardening rate with temperature, at 4% true p l a s t i c s t r a i n 69 26. Variation of work hardening rate with temperature, at 8% true p l a s t i c s t r a i n . 70 27. Transmission electron micrograph of Fe 1.7% ThC^. As r o l l e d 71 28. Transmission electron micrograph of Fe 0.9% ThC^. As r o l l e d . 73 29. Transmission electron micrograph of Fe 1.7% ThO . Strained 4% 74 30. Transmission electron micrograph of iron. As r o l l e d .. 75 31. Transmission electron micrograph of iron. As r o l l e d .. 76 32. Transmission electron micrograph of Fe 0.9%Th02.As rolled 77 33. Transmission electron micrograph of iron. Strained 4%. 78 34. Transmission electron micrograph of iron. Strained 4%. 79 35. Variation of y i e l d drops with temperature 88 36. Theoretical y i e l d behaviour of iron, 94 ix Number 37. Sources of mobile dislocations at the y i e l d point: a) Pile-up at a sub-boundary b) Annihilation of the sub-boundary 38. Grain boundary ledge acting as a dis l o c a t i o n donor 40. A dislocation bowing out between 3 p a r t i c l e s . After Ashby (55) 46. Mohrs' c i r c l e construction to determine angle at which neck occurs 47. Computer program used for determining work hardening rates and coefficients Page 100 39. The Orowan by-passing process 105 107 41. Alternate methods of by-passing p a r t i c l e s . After Kocks (57) 113 42. 550 random points, connected by lines i f they cannot be by-passed at a stress level of cr/xg = 0.74. Heavy lines outline the "free" regions. After Kocks (57) 116 43. As Fig. (42), stress level a/xg = 0.90 117 44. As Fig. (43), stress level a/x B = 1.04 118 45. Directions of stress and necks i n sheet material 137 139 150 LIST OF TABLES Number Page 20 1. 2. 23 3. 27 4. Analyses of similar dispersion hardened alloys 28 5. 34 6. 35 7. ' Tensile properties of iron 0.9% thoria 36 8. Tensile properties of iron 1.7% thoria 37 9. Work hardening rates at 1% true p l a s t i c s t r a i n 59 10. Work hardening rates at 2% true p l a s t i c s t r a i n 60 11. Work hardening rates at 4% true p l a s t i c s t r a i n 62 12. Work hardening rates at 8% true p l a s t i c s t r a i n ..... 64 13. 82 14. Average dislocation densities and flow stresses .... 84 Orowan plot data 108 15. . 16. Data for Fisher, Hart and Pry (4) theory 131 1 1. INTRODUCTION There has been a recent upsurge of interest i n deformation processes i n B.C.C. metals (see, for example, Can. J . Phys. 1967 45 (2)). The causes of the y i e l d behaviour have given r i s e to considerable dispute and there are as yet no entirely satisfactory theories for either y i e l d or work hardening processes. The y i e l d strength of dispersion hardened materials i s usually explained in terms of either the Orowan (1) or the Ansell (2) theories. The y i e l d stress i n the former case i s controlled by a dislocation bowing process, and by p a r t i c l e fracture i n the l a t t e r case. Two theories of the work hardening of single crystals of d i s -persion hardened materials have been proposed (4,57) t and no theory has been put forward to explain polycrystal behaviour. The purpose of this work was to (i) prepare iron and iron-thoria alloys, ( i i ) examine the mechanical properties of iron-thoria and compare them with those of the iron, which was manufactured i n an identical fashion. The results enabled some insight to be gained into the deformation processes of these alloys. Many of the symbols in the text are defined, at the appropriate places, when they f i r s t appear. A l l of the symbols used as l i s t e d alphabetically and defined i n Appendix 3. 2 2. EXPERIMENTAL PROCEDURE Dispersion hardened iron i s not available commercially, and therefore had to be made i n the laboratory. A number of techniques for producing a dispersion and fabricating the resulting material were t r i e d , using a number of oxides as dipersoids. Dispersions were produced by both co-precipitation from solution, and mechanical mixing of powders. The end products were oxide mixtures which were s e l e c t i v e l y reduced i n hydrogen at 800°C. Fabrication techniques which were t r i e d included cold pressing, sintering, extrusion i n evacuated cans, hot r o l l i n g i n steel sheaths and hot r o l l i n g i n hydrogen-argon atmospheres at 1100°C. Most of the procedures tested are included i n Fig. 1, which is a flowsheet of the fabrication methods. Various systems or methods were eliminated on the basis of poor dispersions or manufacturing d i f f i c u l t i e s . The most successful combination of techniques was to co-precipi-tate iron and thorium hydroxides, reduced to Fe-TM^ powder, cold press and then hot r o l l to s t r i p i n hydrogen-argon mixtures at 1100°C. The production of iron and iron-thoria w i l l be examined i n d e t a i l . 2.1. Materials The starting materials for preparation of the F e - T l ^ alloys were: 1. Baker and Anderson reagent 1739 Fe (NO,^ ) ^ F^O of 99.0% minimum purity. 2. Baker and Anderson reagent 2386: Th(NO^)^.411^0 of minimum 96.2% purity. 3 Mechanical Mixing Sizing of Powders, Andreasen Pipette Mixing of Powders Agglomeration Problems Manufacture, Small Quantities Fabrication Pressing of 803 Cy l i n d r i c a l Slugs Welded Mild Steel Cans _FeCl 3 Th(N0 3) 4.4H 20 Coprecipitation FeCN0 3) 3.9H 20 Th(N0 3) 4.4H 20 Figure 1. Flowsheet of fabrication methods 4 Bulk Fe 20 3 1 . Too Large Stop Wet Blending Dry, H2 Reduction Bake Time/Temp./Wt. Change Tests Ball M i l l Pyrophoricity Problems Hot Rolling 1100°C in A i r Surface Grinding to Remove Can Sample Cutting Tensile Tests N H 4 C I Problems F e2°3_ Th0 2 ' f N H 4 0 H Fe(OH) 3 Th(OH) 4 Decant Dry Bake Fe 20 3 Th0 2 5 Fe A1 20 3 Optical Metallography Electron Microscopy Mechanical Mixing Fe 203 + A1 20 3 Manufacturing Problems Weaker than Fe-Th0 2 Coprecipitation ______ Fe C l 3 + Th(N0 3) 4 Manufacturing Problems Poor Dispersion Coprecipitation Fe(N0 3) 3 + Th(N0 3) 4 Dispersion Good Ball H 2 Reduction: Fe M i l l Time/Temp./Wt. Th0 2 Change Tests Pyrophoricity Problems Stop Stop Bulk B u l k F e Manufacture Fabrication Fe-l%Th0 2 Fe-3%Th03 Large Reduction Fe-6%Th02 Furnace Manufacture Die Design Cold Hot Rolling of Rolling a n d Pressing Compacts in Manufacture of >70% H 2/Ar, 1100°C Dense Compacts Can Design Cold Cold Pressing Extrusion a n d Pressing of Slugs into Machining of 80% Cans, Final Dense Slugs Density >70% Controlled Atmosphere Oxidation Problem Furnace Mk.l: 1200°C too long to Roll Furnace and Receiving too long to Cool Chamber Overheating Welding Evacuate to Weld Evacuation of Can ~ Su Hg Cold Tube closed, L i d s Heat to 1000°C Machine Nose of for 15 Mins. C a n 9 Controlled Atmosphere Furance MKII 0.625 mm Strips of Double Furnaces, 1200°C Fe, 0.9% and 1.7% Th0 2 Additional Gas Inlets Cracking Problems Additional Insulation with 3 % Cooling Chamber Extension, Water Cooled Flow-Metered Gas Improved Gate Design Extrude at 20:1 Specimen Optical , in 950 Ton Press, Cutting Metallography 1200°C Optical Metallography-Specimen Cutting Results Electron -Microscopy Tensile Tests 77°K - 373°K Heavy Oxidation Stop Grain Size - Uniform ^ 14u No Oxidation Dispersion Satisfactory Smallest Particles ^ 0.05u, Average ^ O.ly. 12 3. Ammonium hydroxide. 4. D i s t i l l e d water. 2.2. Co-precipitation Weighed quantities of the nitrates i n calculated proportions were dissolved i n d i s t i l l e d water. The solution was s t i r r e d for an hour. Ammonium hydroxide was added u n t i l the solution was just alkaline. The bath was strongly agitated during the process. The result of the addition was to produce iron and thorium hydroxides: Fe(NC> U + 3NH.0H — > Fe(OH)_ + 3NH.NCL 3 3 4 o 4 J Th(N0 3) 4 + 4NH40H — > Th(0H) 4 + 4NH4N03 The precipitate was allowed to sett l e and the supernatant l i q u i d was siphoned off. The precipitate was then heated to evaporate most of the free water, then baked to drive off combined water and the ammonium nitra t e : 2Fe[0H) + 6NH4N03 — > F e ^ + 3H20 + 6NH4N03T . Th (0H) 4 + 4NH4N03 >• ThC>2 + 2^0 + 4NH4N03t In this way batches were prepared, which corresponded to iron and 2 iron-thoria alloys. Each batch of dry oxide mixture was milled with ceramic b a l l s for 16 hours to produce a fine powder. 2.3. Oxide reduction A large tube furnace was constructed for hydrogen reduction of the oxide powders. The powder was put on trays i n batches of about 250 grams, and reduced in hydrogen at 800°C for 16 hours: 13 R 0 0 ° C Th0 2 + Fe 2 0 3 + 3H 2 > 2Fe + 3H20 + Th0 2 Completion of the reduction was estimated by weight loss. The actual reduction was effected i n 2 hours as shown i n Fig. 2, but the resulting iron powder was often pyrophoric. The 16 hour treatment caused the powder par t i c l e s to sinter,which reduced the problem. Pyrophoricity was eliminated by taking the tray straight from the cold furnace and immediately quenching in a shallow bath of l i q u i d nitrogen. 2.4. Consolidation The powders were stored i n sealed jars containing s i l i c a gel. When powder was required for further work i t was heated and held at 800°C for 2 hours in a hydrogen atmosphere to ensure complete reduction of any s u p e r f i c i a l oxide. 2.4.1. Compact preparation A double acting die for the cold pressing of powder compacts was designed and made (Fig. 3). Batches of powder weighing 50 gms were put into the die and pressed at 28 kg mm . The resultant compact was about 70% dense and measured 38 x 32 x 10 mm. A 1 mm diameter hole was d r i l l e d close to each end of the compact. A two meter length of 0.45 mm Nichrome wire was looped through each hole. The compact was then ready for threading through a r o l l i n g m i l l furnace assembly. 14 F i g u r e 2. Weight l o s s - t i m e curve f o r hydrogen r e d u c t i o n o f F e ^ at 800°C. Hydrogen flo w r a t e a p p r o x i m a t e l y 1000 s.c.c.m. 15 R A M e 3. Double a c t i n g d i e used f o r powder compaction 16 2.4.2. Hot Rolling The hot r o l l i n g assembly which was b u i l t for this project i s shown i n Fig. 4. The equipment consists of 2 furnaces capable of attai n -ing 1100°C, situated one on either side of the r o l l i n g m i l l . The Inconel furnace tubes are of rectangular section, and one has a prolong which, i s surrounded by a water cooling jacket. A prototype assembly, consisting of 1 furnace on one side and a receiving tube containing an inert atmosphere on the other, had proved unsatisfactory. The holding tube imparted severe c h i l l i n g , and took a long time to cool down after completion of a r o l l i n g sequence. The inert atmosphere equipment was inadequate to prevent oxidation of the s t r i p . The second design, as shown i n Fig. 4, i s much more e f f i c i e n t . The second furnace eliminates the c h i l l i n g effect and the water cooling jacket allows the finished s t r i p to be removed within 5 minutes of the commencement of operation of the water jacket. There are 3 atmosphere i n l e t s on each furnace, two at the r o l l s and one at the entry gates. A diagram of the internal arrangement of the assembly, with a sample i n the charging and cooling position, i s shown in Fig. 5. The furnaces were raised to a temperature of 1100°C and the tubes flushed with argon through a flow meter system. One of the compact wires was then threaded through the whole assembly, by attaching one end to a notched steel tube and pushing the tube through the 4 gates and the open r o l l s . The V-notched gates at the ends were then shut down, leaving a small hole to allow free movement of the wire. The sample was l e f t i n the water jacket prolong whilst the assembly was flushed of any a i r which entered during threading. The hydrogen was then turned F i g u r e 4. Hot r o l l i n g assembly Figure 5. Internal arrangement of hot r o l l i n g assembly oo 19 on and adjusted so that the volume r a t i o of hydrogen and argon entering the 2 i n l e t s was about 1:2. The t o t a l flow rate must not be less than 3000 s.acm. of argon to the r o l l s and 2240s.cam.of hydrogen/argon to each furnace i f oxidation of the s t r i p i s to be prevented. The compact was then pulled through into the furnace hot zone, which contained 3 thermocouples, and allowed to soak for 15 minutes. The r o l l i n g sequence was then started by p u l l i n g the compact through the preset r o l l s , and into the hot zone of the second furnace. As the hot compact s l i d through the hinged gate for the f i r s t time, the hydrogen a i r mixture beyond the gate exploded, and then burned contin-uously. The r o l l s were kept turning from this point on in order to avoid lo c a l heating. The compact was soaked for one minute and then pulled back through the r o l l s . The pass sequence was continued u n t i l a 0.635 mm s t r i p was obtained. The s u s c e p t i b i l i t y of the compacts to fracture during r o l l i n g increased rapidly with the ThO^ content of the a l l o y . The highest ThO^ content a l l o y , containing about 3% by volume of ThC^ proved to be impossible to fabricate by this method. The pass sequences were arrived at by t r i a l and error, and represent the heaviest deforma-tions^and hence the quickest method, of r o l l i n g each alloy to s t r i p without serious edge cracking. The sequences are shown in Table 1. After the f i n a l pass the s t r i p was pulled into the prolong. The prolong was cooled to room temperature in 5 minutes by means of the water jackets. The hydrogen flow was turned o f f at the start of this period, and the argon was l e f t flowing,. 20 Table 1 ROLLING SEQUENCES Starting Weight 50 gms Starting Thickness 3.8 mm Material Fe Fe 3%Th0 Reduction Range Draft per Pass Thickness, mm 2 Compact shattered mm 3.80 - 1.52 0.51 1.52 - 0.63 0.25 Fe 0.9%Th09 3.80 - 1.52 0.51 z i <;? - n.63 0.25 Fe 1.7%Th02 3.80 - 1.52 0.25 1.52 - 0.63 0.12 1.00 - 0.63 0.08 3.80 - 3.00 0.08 21 The resulting s t r i p was bright and smooth and about 45 cm long. 2.5. Preparation of Tensile Samples Samples of 2.02 cm gauge length and 0.53 cm width were cut from the r o l l e d s t r i p with a special die mounted on a mechanical press. They were then deburred with 3 '0' emery paper and electropolished i n a chromic-acetic solution, of composition 133 ml Acetic acid 25 gms Chromic acid 7 ml D i s t i l l e d water 2.6. Tensile Tests Testing was carried out i n an Instron machine, using a cross-head speed of 0.508 mm min *, which for these samples gives a strai n -1 rate of 0.025 min The specimens were mounted in f l a t wedge grips, which i n turn were mounted in a cage underneath the crosshead which allowed t o t a l immersion of the sample i n a temperature bath. Alignment of the samples i n the grips was carried out by marking the centre l i n e of the grips and also that of the sample. The two centre lines were then matched when mounting the sample, thus ensuring that the sample was central. The grip was then placed on graph paper with one edge placed p a r a l l e l to one set of li n e s . The grip was then tightened onto the sample, with the gauge length being judged p a r a l l e l by eye to the same set of graph lines as the grip edge. 22 The grip was then screwed into the cage and the sample lowered into the second grip. The grip was tightened onto the sample with the centre lines once again aligned. Overloading of the sample during tightening of the second grip was avoided by running the machine on a 2-10 kg load cycle. Tensile tests were carried out over the range of temperatures 77-373°K, using the temperature bath f l u i d s shown i n Table 2. Temperature measurement was carried out using a copper-constan-tan thermocouple with the hot junction attached to the gauge length of the sample, and the cold junction i n ice-water. A Cambridge potentiometer was used to measure the E.M.F. The cage was immersed i n the temperature bath whilst the sample was under low load cycling. In the low-temperature tests, the whole assembly was cooled 5-10°K below the required temperature and then allowed to warm up slowly to the required temperature. The gauge lengths of the samples tested i n b o i l i n g water were f i r s t coated i n grease to minimise corrosion. 2.7. Metallography 2.7.1. Optical metallography A solution of 5% n i t r i c acid i n amyl alcohol was found to give a satisfactory grain structure etch, and was used for optical metallography i n the present work. 2.7.2. Electron microscopy replicas Quantitative metallography was carried out using electron 23 Table 2 CONSTANT TEMPERATURE BATHS IE Fluid 373 Water 298 A i r 253 Ethyl alcohol, l i q u i d nitrogen cooled 233 213 Petroleum ether, " " 173 133 77 Liquid nitrogen 24 micrographs of re p l i c a s . Ordinary opt i c a l metallography samples were used in the etched state. The etching allowed such features as ThO^ p a r t i c l e s , grain boundaries and twins to be reproduced. The replicas were made by impressing acetone soaked cel l u l o s e acetate onto the sample and allowing the acetate to harden. The f i l m was then removed, coated with carbon from a carbon arc and shadowed with evaporated chromium. The acetate film was then dissolved away i n acetone. The r e p l i c a was placed on a water surface to fl a t t e n i t out, picked up on copper grids and examined in a Hitachi HU IIA electron microscope. 2.7.3. Transmission microscopy Sheet material was chemically thinned from 0.06 cm to about 0.02 cm thick i n a solution consisting of 30 ml N i t r i c acid 10 ml Hydrochloric acid 10 ml Phosphoric acid 50 ml Acetic acid 3 mm diameter discs were then spark machined from the sheet. The edges of the discs were stopped o ff with lacquer. The discs were electropolished for 10 minutes i n chromic-acetic acid solution. The lacquer was then dissolved away in acetone and the discs polished generally u n t i l a central hole appeared. The i n i t i a l edge lacquering ensured that the hole would be within the viewing area permitted by the specimen holder of the electron microscope. 25 A number of problems were encountered during attempts to produce good f o i l s . The best polish resulted from the use of conditions s l i g h t l y above the voltage plateau on the voltage-current curve. 20 volts and 0.02 amps were commonly used. The thoria p a r t i c l e s were not attacked by the solution, and tended to adhere to the f i l m . It i s possible to remove them by rapidly switching the current o f f and on at the end of the polishing period. The f o i l s corrode rapidly and cannot be easily kept. The method used to counter this was to wash the fresh f o i l i n water, and then i n absolute methyl alcohol. The f o i l i s then given 2 acetone washes to remove any lacquer and then immersed i n absolute methyl alcohol. The f o i l i s then transferred immediately from the alcohol to a f i l t e r paper for drying and thence straight into the specimen holder. Handling of iron f o i l s i s f a c i l i t a t e d by the use of magnetic tweezers. Despite the fact that f o i l s made by this method f i t the specimen holder exactly with no further mechanical damage such as cutting, they must be mounted between two grids. The available area for viewing i s cut down, but the dist o r t i o n of a free f o i l i n the heat of the electron beam, which often results i n badly out-of-focus micro-graphs, i s minimised by the use of the double grid. 26 3. RESULTS AND OBSERVATIONS 3.1. Characteristics of the Materials 3.1.1. Chemical Analyses A l l the alloys were subjected to chemical analysis for i n t e r s t i t i a l elements, and thoria, and the iron-1.7% ThO^ a l l o y was analysed spectrographically. The results are shown i n Table 3. Comparison with other dispersion hardened materials i s d i f f i c u l t since f u l l analyses are rarely quoted, but some figures from the l i t e r a t u r e are given i n Table 4. 3.1.2. Quantitative metallography 3.1.2.1. Optical metallography The optical microstructure of iron was featureless apart from grain boundaries. The iron-thoria microstructure revealed the coarser thoria p a r t i c l e s and grain boundaries. Some strings of thoria p a r t i c l e s were revealed by metallography on longitudinal sections, and the influence of these i s discussed i n 3.2.3. A high magnification photomicrograph of an iron 1.7% Th0 2 a l l o y i s shown i n Fig. 6. Fig. 7 is a low magnification photograph showing the grain size and the strings of thoria p a r t i c l e s . The grain sizes were measured by use of the Heyn intercept method (A.S.T.M. Standard E112), using 5 random lines per micrograph. The average grain size of the iron was 66p and that of the iron-thoria alloys 14y. 27 Table 3 ANALYSES OF MATERIALS Substance N C H Al Cr Cu Si T i ThO„ Unit p.p.m. wt.! Fe 108 32 Material Fe 0.9%ThO„ 111 26 20 volume fraction 0.009 Fe 1.7%Th02 113 21 12 0.03 0.04 0.002 0.003 0.001 0.017 Other elements not detected 28 Table 4 ANALYSES OF SIMILAR DISPERSION HARDENED MATERIALS Maker Material C N Th0 2 p.p.m. p.p.m. v o l . fraction Du Pont Ni-Th0 2 40 0.020 Du Pont Ni-Cr-Th0 2 247 60 0.020 Zwilsky § Grant (76) Fe-Th0 2 300 0.1 Hahn § Rosenfield (3) Fe-Th0 2 200 0.025 Ferrovac E 50-100 5-10 29 F i g u r e 6. O p t i c a l micrograph of i r o n 1.7% t h o r i a . X1950 30 — > R o l l i n g D i r e c t i o n F i g u r e 7. O p t i c a l micrograph of i r o n 1.7% t h o r i a . X150 31 3.1.2.2. Replica Microscopy Carbon-chromium replicas were taken from the surface of etched optical metallography samples. The etching procedure caused the iron to be attacked and l e f t the embedded thoria p a r t i c l e s protruding from the surface. Typical micrographs of an Fe - 1.7% ThO sample are shown in Figs. 8 and 9. The replicas were used to determine A for the iron thoria a l l o y s . A i s the planar mean free path between p a r t i c l e s , and i s given by (5) where f = volume fraction of dispersed phase N = number of pa r t i c l e s per unit length of random l i n e i-i A, therefore, i s the mean separation of particles along a random l i n e . N was determined by drawing 10 random lines across each of 5 enlarged micrographs from 5 samples, and counting the par t i c l e s along the t o t a l l i n e length. f was calculated from the chemical analysis, using a density for ThC>2 of 10.03 gms c c " 1 . The A values were then used to calculate true p a r t i c l e diameter values, 2r y, which are given (6) by 3f A m v 2(1 - ±J Figure 8. Replica of iron 1.7% thoria. -F i g u r e 9. R e p l i c a o f i r o n 1.7% t h o r i a . 34 The results obtained for X and 2r are shown in Table 5. The data of v Table 5 were used to calculate the mean planar centre to centre p a r t i c l e spacing, R , using the equation of Kocks (7) (see Appendix 1), R = l-18.2r /T^ (3) 3.1.2.3. Transmission electron microscopy True p a r t i c l e diameters were measured d i r e c t l y from micrographs of thin f o i l s . Care was taken to avoid measuring those pa r t i c l e s which protruded from the surface of the f o i l , since these were subject to heavy carbon shadowing. 200 pa r t i c l e s were measured using a number of micrographs, and the average obtained for each alloy i s shown i n Table 5. Table 5 DISPERSION PARAMETERS Material Fe - 1.7%ThO, Fe - 0.9%ThO„ microns 2.52 5.93 2r .calculated v microns 0.07 0.08 2r .measured v microns 0.07 0.12 3.2. Tensile Properties A l l the load-elongation curves were analysed to give upper and lower y i e l d stress or 0.2% proof stress, U.T.S., and uniform and to t a l elongations. The results for Fe, Fe - 0.9%Th02 and Fe - 1.7%ThO are shown in Tables 6, 7, and 8 respectively. Table 6 TENSILE PROPERTIES OF IRON Upper Lower .2% Proof Ultimate Uniform Total Y i e l d Yield Stress Tensile Elongation Point Point Stress kg mm -2 23.1 22.5 22.2 21.9 24.5 23.6 24.4 23.6 34.2 31.8 32.9 32.6 43.5 40.6 38.5 35.4 15.2 14.8 57.6 19.0 90.0 15.9 24. 4 12.5 16.4 13.2 19. .6 5.7 11.1 17.6 24. ,2 9.0 24.1 18.0 24, .9 21.5 33.9 20.8 25 .1 11.7 21.9 28 .1 19.7 28.5 27. 9 16. 0 29. 3 33. 4 12. 5 25. 3 32. 6 9. 0 21. 1 43. 5 6. 5 15. 6 38. ,6 6. .5 12. ,6 57. ,6 0. .2 14. ,0 63 .3 0. ,0 1, .8 19 .6 1. .4 1. .9 26 .4 2 .0 2 .9 43 .3 0 .0 1 .2 90 .0 0 .3 0 .3 88 .5 0 .0 0 .0 36 Table 7 TENSILE PROPERTIES OF IRON 0.9% THORIA Test Number Temperature .2% Proof Ultimate Uniform Total Stress Tensile Elongation 34 35 22 41 81 46 Stress °K kg mm"2 % 373 27.3 36.5 5.5 9.7 373 38.6 41.6 5.8 10.5 298 29.1 41.7 7.2 11.9 23 298 29.6 40.8 10.0 17.1 40 253 32.4 43.0 12.5 19.0 253 31.6 42.3 10.7 16.0 233 36.2 49.4 11.0 16.0 213 43.7 49.8 11.5 26.7 47 213 45.4 50.6 8.5 16.6 50 173 59.7 56.7 5.5 11.4 52 173 68.0 63.2 3.6 8.5 57 133 70.3 74.8 3.5 16.2 58 133 69.2 76.0 3.5 7.4 26 77 98.2 98.2 0.2 3.0 29 77 97.3 98.2 0.2 3.4 37 Table 8 TENSILE PROPERTIES OF IRON 1.7% THORIA Test Number Temperature .2% Proof Ultimate Uniform Total Stress Tensile Elongation Stress 'K kg mm 2 37 373 28 298 48 213 56 133 31 77 36 373 30.7 42.0 6.2 9.3 38.6 48.0 4.4 7.0 27 298 34.4 47.8 7.5 10.7 33.9 47.8 8.0 12.0 42 253 36.4 49.5 9.0 12.0 4 3 253 37.7 51.3 6.2 10.3 8 0 233 38.1 53.3 10.0 13.6 48.2 57.2 7.0 11.0 49 213 45.0 55.0 6.0 8.9 53 173 64.7 69.0 6.4 11.3 54 173 67.7 69.8 5.7 10.'2 55 133 70.3 73.2 3.1 5.4 69.2 70.3 5.5 8.7 30 77 102.5 102.6 0.4, 3.7 101.2 101.2 0.7 5.7 38 -2 3.2.1. Yi e l d stress and ultimate t e n s i l e stress The variations in y i e l d stress and U.T.S. with temperature are plotted in Figs. 10 and 11 respectively. The temperature dependence of the y i e l d stress i s about the same i n the iron and iron-thoria alloys. The difference in y i e l d stress values between the iron and the iron-1.7% ThO^ remains at about 12 kg mm over the test temperature range. This observation contrasts with that of Hahn and Rosenfield (3), who found that an iron-2.5% ThO^ a l l o y showed a s l i g h t l y greater temperature dependence than their iron. The temperature dependence of the U.T.S. values i s stronger in the iron than in the Fe-ThO^ alloys. The U.T.S. difference between iron and iron-1.7% ThC>2 i s 23 kg mm"2 at 298°K and 12.7 kg mm~2 at 77°K. The curves for iron are drawn i n a smooth line to 77°K, though twinning fa i l u r e s are observed in iron at this temperature. Twinning was not observed in iron-thoria. The nature of the y i e l d point, changes with temperature and thoria content, i n a manner indicated q u a l i t a t i v e l y i n Fig. 12. A rounded y i e l d point i s observed in a l l the materials at 298°K and above. A peaked y i e l d point, characteristic of i n t e r s t i t i a l impurity locking of dislocations, is observed in a l l materials at 77°K. The transition between the two types depends on the thoria content, the sharp y i e l d point being most strongly suppressed in the Fe-1.7% Th0 2 alloy. 3.2.2. D u c t i l i t y The variation of uniform and t o t a l elongations with temperature are shown in Figs. 13 and 14 respectively. 39 100 200 °K 3 0 0 4 0 0 Figure 10. Variation of y i e l d stress with temperature. -4 -1 Strain rate 4.2 x 10 sec. 40 41 Figure 1 2 . Qualitative plot of y i e l d behaviour. 42 43 44 Both plots show that the iron is generally more ductile than the iron-thoria except at temperatures of 133°K and below. The position of the uniform elongation curve of Fe below 173°K is not certain. Seven tests were carried out on iron at 133°K and 77°K, and in 5 of these the d u c t i l i t y was essentially zero. Two tests showed 1-2% d u c t i l i t y , and both were on samples showing low U.T.S. twinning f a i l u r e s at 77°K. A peak i n elongation values occurs around 250°K, and coincides with the peak work hardening rate which w i l l be discussed later. 3.2.3. Fracture The specimens which exhibited the highest d u c t i l i t y f a i l e d by a 3-stage process. Two stages of necking were following by f i n a l rupture. The three stages were: a) General necking, or tapering, over at least half the gauge length. b) A second state of necking i n a localised band near the centre of the large neck. The band ran across the plane of the specimen at an average angle of 60° to the tensile axis in iron, and 57° in iron-thoria. c) Shear rupture occurred in the necked band. The rupture took place ina plane inclined to the plane of the sheet by roughly 45°. Reduced d u c t i l i t y resulted on increasing the thoria content or using the lower test temperatures. Under these circumstances stages a) and c) were very much reduced, and the amount of necking in stage b) was less pronounced. 45 Further reduction i n temperature caused the disappearance of stage a), and the appearance of a fracture which was p a r t i a l l y at 90° to the tensile axis and p a r t i a l l y at about 60°. The 90° area appeared at the apex of two stage b) necks, as indicated i n Fig. 15. At 77°K, the fractures of a l l materials were at 90° to the ten s i l e axis and showed no evidence of necking at a l l . The 90° fractures were always bright, and the 57° fractures were always d u l l grey i n appearance to the unaided eye. The fracture of iron followed the sequence described above for decreasing temperatures, with the 90° fractures f i r s t appearing at 133°K. On the basis of fracture angle and appearance, the ductile-t o - b r i t t l e t r a n s i t i o n temperature for iron is estimated to l i e between 173 and 133°K, and that of the iron-thoria between 133 and 77°K. The elongation data (see Figures 13 and 14) reinforce the observation that the transition temperature is lower in the iron-thoria alloys, although this effect may not be exclusively due to the presence of thoria, as w i l l be discussed later. Observation under the binocular microscope revealed occasional surface cracks along the gauge length of most of the broken iron thoria specimens. Optical metallography revealed that these were associated with stringers of thoria p a r t i c l e s . The agglomerates of particles were not always strung out in the direction of r o l l i n g . Fig. 16 shows the gauge length of a broken tensile sample i n which the thoria stringers appear as black lines. The thoria appears black since the matrix i n ome these areas appears to be preferentially etched, and also because s of the lines are r e a l l y cracks. Point X i s an agglomerate running transverse to the r o l l i n g direction, which has a crack running through 46 figure 15. Diagram of a combined 90°-57° fracture. Two Luders bands run from the arrest point of a transverse crack, one of the bands forming a neck. X Rolling Direction Figure 16. Thoria stringers (black lines) in a broken te n s i l e sample. Rolling direction l e f t to right. X1600. 48 i t and down into the specimen. The top centre of the micrograph shows a long stringer which again turns transverse to the r o l l i n g d irection. The transverse stringers of particles were the preferential sites for cracking, but not a l l such strings were cracked even close to the neck. The black areas do not represent a l l the thoria present, as indicated by Fig. 17. The f i e l d of view i s the same as in Fig. 16, but was photographed using dark f i e l d illumination. The white spots indicate the thoria p a r t i c l e s , which protrude from the surface after etching the matrix. The spots indicate that a fine dispersion i s present as well as the stringers. Another crack associated with a string of p a r t i c l e s i s shown in Fig. 18. Such cracks were never observed in the iron samples. Fractures associated with twinning were observed i n some iron samples at 77°K, but not i n the iron-thoria. Twins were observed at a number of positions along the gauge length of the t e n s i l e samples, but most were in the fracture region and some were associated with the fracture. Fig. 19 shows a fracture edge of an iron sample, photographed using . polarised l i g h t since this technique gave a sharply defined edge. There are twins close to the fracture edge which run p a r a l l e l to i t , and the edge i t s e l f i s straight. There is one discontinuity i n the fracture edge across the grain, and this seems to be associated with a second set of twins running at an angle to the edge. Beyond this discontinuity the edge i s once again p a r a l l e l to the f i r s t set of twins. A number of such twinned grains could be seen along the fracture edge. Iron specimens which did not twin at 77°K gave strength , > R o l l i n g D i r e c t i o n F i g u r e 17. Same as F i g . (16), but t a k e n . i n dark f i e l d t o sh t h o r i a p a r t i c l e d i s t r i b u t i o n . X1600. 50 _ — . — ——> R o l l i n g D i r e c t i o n F i g u r e 18. Crack a s s o c i a t e d w i t h l a r g e t h o r i a s t r i n g e r . X1000. F r a c t u r e edge o f an i r o n t e n s i l e specimen. P u l l e d at 77°K, photographed i n p o l a r i s e d l i g h t . Twins a l i g n e d w i t h f r a c t u r e edge. X200. 52 values which lay on the curves of Figs. 10 and 11. The samples which twinned showed lower strengths. 3.2.4. Load-elongation curves The families of true stress-true p l a s t i c s t r a i n curves for iron, iron 0.9% thoria and iron 1.7% thoria are shown i n Figs. 20, 21 and 22 respectively. The sharp type of y i e l d point usually associated with inter-s t i t i a l locking did not appear in iron u n t i l the temperature had been reduced to 253°K. As the temperature was lowered further, the Luders extension increased and the subsequent work hardening rate dropped. At 213°K and 173°K i t was d i f f i c u l t to detect where Luders band propagation stopped and uniform work hardening commenced, and at 173°K the upper y i e l d load was higher than the subsequent ultimate ten s i l e load. At 133°K in iron, the load dropped after the upper y i e l d point and never rose again during the test, indicating a Luders band f a i l u r e . At 77°K the observations were complicated by the onset of twinning, which caused f a i l u r e both in the " e l a s t i c " range and just after y i e l d . The specimens which gave enough of a work hardening curve to permit a rate calculation showed sharp y i e l d points, l i t t l e i f any Luders extension, and then a very high work hardening rate. The iron 0.9% thoria did not begin to show y i e l d points and Luders ranges u n t i l a temperature of 213°K or below was reached. Yield points f i r s t appeared in iron-1,7% thoria at 133°K and below. Below these two temperatures the iron thoria alloys showed Luders band 53 90 7 77°K Fe 80 -70 -i e ~ 0 0.05 0.10 TRUE PLASTIC STRAIN Figure 20. True stress-true p l a s t i c strain curves for iron. -4 -1 Strain rate 4.2 x 10 sec. 90 P 7 7 °K 54 I E E 80 -70 6 0 CO CO ^ 5 0 to ID 4 0 30 20 10 Fe 0.9 % T h 0 2 133 °K 173 °K X " O 0.05 0.10 TRUE PLASTIC STRAIN Figure 21. True stress-true p l a s t i c strain curves for iron 0.9% -4 -1 thoria. Strain rate 4.2 x 10 sec. 55 0 0.05 0.10 T R U E P L A S T I C S T R A I N Figure 22. True stress-true p l a s t i c strain curves for iron 1.7% -4 -1 thoria. Strain rate 4.2 x 10 sec. 56 f a i l u r e s i . e . , the maximum load occurred at the upper y i e l d point. This behaviour remained the same down to 77°K. 3. 3 Work Hardening 3.3.1. Analysis of load-elongation curves Work hardening values were obtained from true stress-true p l a s t i c s t r a i n curves. The l a t t e r were obtained by using those points from the load-elongation curves which lay between the end of the Luders band, where present, and the U.T.S. The e l a s t i c slope was subtracted, true stress and true plastic s t r a i n values calculated, and these points then f i t t e d to the equation a = Ac n (4) P A least squares program was applied to the equation lna = InA + nine (5) P to give the A and n values for the particular specimen. Then the work hardening rate, 6, is given by $1 = Ane n _ 1 (6) de p 6 values were obtained for every specimen at true p l a s t i c strains of 1, 2, 4, and 8%. The presence of Luders bands meant that some of the 1% strain values were obtained by back extrapolation using equation (4) 57 Similarly, other values were obtained by extrapolation beyond the uniform elongation of the specimen. The extent of both types of extrapolation never exceeded about 1% strain. The extrapolation beyond the uniform elongation values was f e l t to be permissible in some cases, since there was appreciable scatter i n the elongation results. Samples showing the more limited d u c t i l i t y were subjected to extrapolation. The f i t of the data to equation (4) could not be extended to 0% z , since the correlation coefficient for the f i t t i n g procedure used on equation (5) decreased rapidly as 0% was approached. Wilson and Konnan (8) have worked with spheroidized steels and found that a complete f i t to the whole experimental curve i s best achieved with two f i t s to equation (4). One curve would be f i t t e d from 0% stra i n upwards using a high n value, and the second would f i t to the higher strain values using a small n value. Since the work hardening rates in this work were estimated at 1% and higher strains a l l the curves were f i t t e d from 0.5% upwards. This procedure gave correlation coefficients to equation (5) of better than 0.99 in a l l cases. A computer program was devised, such that a l l the manipulations described above were carried out using cartesian coordinates of points along the load elongation curve as starting data. The details of the program are set out in Appendix I. The values of the work hardening rate at strains of 1, 2, 4, and 8% are given i n Tables 9, 10, 11, and 12 respectively. Table 9 WORK HARDENING RATES AT 1% TRUE PLASTIC STRAIN -4 Starting s t r a i n 0.005. Strain rate 4.2 x 10 sec. Material Specimen Temperature Work Hardening Number °K Coefficient kg Fe 63 373 0.17 64 373 0.12 65 373 0.14 59 298 0.10 60 298 0.16 66 253 0.11 67 253 0.14 89 253 0.11 90 253 0.10 77 233 0.14 68 213 0.09 69 213 0.09 70 173 0.05 71 173 0.04 61 77 0.21 62 77 0.19 Fe 0.9% 3 4 3 7 3 0.13 Th0 2 i t 22 298 0". 13 11 23 298 0.14 Tab1e 9 (Continued) Material Specimen Temperature Work Hardening _ 2 Number °K Coefficient kg mm e Fe 0.9% 4 Q 2 5 3 0.13 468 Th0 o 41 253 Th0 2 81 233 46 213 47 213 28 298 42 253 43 253 79 233 80 233 48 213 49 213 53 173 54 173 56 133 0.13 467 0.13 530 0.12 488 0.09 387 Fe 1.7% 3 6 3 7 3 0.12 447 27 298 0.13 515 0.14 545 0.12 492 0.13 569 0.17 382 0.13 581 0.09 445 0.10 487 0.56 367 0.72 483 0.06 360 Table 10 WORK HARDENING RATES AT 2% TRUE PLASTIC STRAIN Starting s t r a i n 0.005. Strain rate 4.2 x 10 sec. Material Specimen Temperature Number °K Fe 63 373 " 64 373 " 65 373 " 59 298 60 298 " 66 253 " 67 253 " 89 253 " 90 253 " 77 233 " 68 213 " 69 213 " 70 173 " 71 173 Fe 0.9%ThO2 34 373 22 298 23 298 " 40 253 " 41 253 Table 10 (Continued) Material Specimen Number Fe 0.9%ThO2 81 46 47 Fe 1.7%Th02 36 " 27 " 28 " 42 " 43 79 80 48 I I 49 " 53 " 54 56 Temperature °K kg mm " e 233 290 213 264 213 207 373 244 298 281 298 300 253 267 253 312 233 214 233 319 213 236 213 261 173 191 173 254 133 188 62 Table 11 WORK HARDENING RATES AT 4% TRUE PLASTIC STRAIN -4 -1 Starting st r a i n 0.005. Strain rate 4.2 x 10 sec. p. Material Specimen Temperature 2 _ Number °K kg mm e : e 63 373 74 64 373 72 65 373 70 59 298 60 60 298 91 66 253 68 . 67 253 93 89 253 70 90 253 68 77 233 92 lt 68 213 82 69 213 77 70 173 54 71 173 40 Fe 0.9%ThO2 34 373 122 M 22 298 136 23 298 145 40 253 141 Table 11 (Continued) Material Specimen Temperature _ 2 Number °K kg mm Fe 0.9%ThO2 41 253 141 « 81 233 159 • • 46 213 144 n 47 213 HO Fe 1.7%Th02 36 373 133 M 27 298 154 i i 28 298 165 n 42 253 145 I I 43 253 171 I I 79 233 120 I I 80 233 175 n 48 213 125 ,i 49 213 140 H 53 173 99 54 173 133 56 133 98 64 Table 12 WORK HARDENING RATES AT 8% TRUE PLASTIC STRAIN -4 Starting st r a i n 0.005. Strain rate 4.2 x 10 sec. Material Specimen Temperature Number °K kg mm e Fe 63 373 42 » 64 373 39 " 59 298 32 " 60 298 51 66 253 37 " 67 253 51 » 77 233 51 t . 68 213 44 69 213 41 t i 70 173 28 71 173 21 • Fe 0.9%Th02 22 298 75 23 298 80 11 40 253 77 41 253 77 46 213 77 ., 47 213 85 Table 12 (Continued) Material Specimen Temperature ^ Number ^ 6 ,7%Th02 36 373 72 i t 27 298 84 u 28 298 90 I I 42 253 79 I I 79 233 68 I I 80 233 96 I I 48 213 66 I I 49 213 75 11 53 173 51 66 3.3.2. Work hardening behaviour The G values calculated as detailed in 3.3.1. are plotted i n Figs. 23, 24, 25, and 26,which show the families of curves for 1,2, 4, and 8% true p l a s t i c s t r a i n respectively. The highest work hardening rates at a l l strains i s shown by the iron 1.74% thoria, and the lowest i s shown by the iron. The peak in the work hardening rate at 250-300°K occurs in a l l three alloys and at a l l strains. The work hardening rates of a l l the alloys decline with increasing s t r a i n . The rate for the iron 1.74% thoria remains roughly double that of the iron at a l l strains. 3.4. Transmission Electron Microscopy 3.4.1. Structure Thin f o i l s were examined for a l l 3 materials. The objectives were to obtain true p a r t i c l e diameters, determine dislocation densities and distributions, and observe the nature of dislocation-particle interactions. True p a r t i c l e diameters were determined from micrographs such as Fig. 27. Such areas also give a general picture of the structure. The material in Fig. 27 i s as r o l l e d Fe-1.74%Th02. The scatter i n p a r t i c l e sizes can be readily appreciated, and there are a few particles with diameters of about 1 micron. A l l sizes of particles appear to be effective i n hindering dislocation motion. There are examples of single dislocations apparently being held up by a p a r t i c l e . 67 4 0 0 0 Fe 1.7 % T h 0 2 oFe 0 . 9 % T h 0 2 gure 24. Variation of work hardening rate with temperature, at 2% true p l a s t i c strain. gure 25. Variation of work hardening rate with temperature, at 4% true p l a s t i c strain 70 Figure 26. Variation of work hardening rate with temperature, at 8% true p l a s t i c strain. 71 Figure 27. Transmission electron micrograph of Fe 1.7%ThOr As r o l l e d . 72 Figs. 28 and 32 are high magnification micrographs of as-rolled Fe 0.9%ThC>2, and indicate the nature of the dis l o c a t i o n - p a r t i c l e interactions. A sub-boundary dislocation network can be seen i n the corner of Fig. 28. The great majority of dislocations are either associated with p a r t i c l e s or grain boundaries. The fabrication process i s carried out at 1100°C, and i t might be expected that on cooling the d i f f e r e n t i a l i n thermal contraction between the matrix and pa r t i c l e s would give r i s e to stresses. Such stresses can cause the punching out of dislocation loops on either side of the p a r t i c l e s , and the loops would have a strong influence on the mechanical behaviour of the material, perhaps i n the manner indicated by Ashby (9). There was no clear evidence of loops i n any of the many specimens examined, which indicates that the contraction stresses appear to be i n s u f f i c i e n t to produce deformation i n this system. The effect of 4% true p l a s t i c s t r a i n on the structure of iron 1.7% thoria i s shown i n Fig. 29. The density of dislocations has increased, but a c e l l structure has not developed. The structures of the iron 0.9% thoria alloy and the iron 1.7% thoria a l l o y are similar. The structure of the as hot-rolled iron i s shown in Figs. 30 and 31. There are few tangles, and most of the dislocations are either contained i n boundaries or scattered through the grains. The influence of 4% true p l a s t i c strain on iron i s shown by Figs. 33 and 34. The dislocation density i s higher, and the number of tangles has increased. The grain boundaries i n Fig. 33 have acquired 7 3 Figure 28. Transmission electron micrograph of Fe 0.9%ThO2. As r o l l e d . 74 Figure 29. Transmission electron micrograph of Fe 1.7%ThC>2. Strained 4%. 75 Figure 31. Transmission electron micrograph* of iron. As r o l l e d . 77 78 F i g u r e 33. T r a n s m i s s i o n e l e c t r o n micrograph o f i r o n . S t r a i n e d 4%. 79 80 the "furry" appearance described by Keh and Weissman (10), meaning that dislocations are apparently generated i n grain boundaries and emerge from them. The network of dislocations i n Fig. 34 may well be the basis for a c e l l structure, which at room temperature becomes well defined at 8% s t r a i n in iron. The 4% s t r a i n produced an appreciable number of dislocation-dislocation interactions, whereas the dislocations in the as-rolled iron were comparatively free from intersections. 3.4.2. Dislocation densities The technique used in determination of dislocation densities is detailed below. A transparent area of f o i l was found and a selected area d i f f r a c t i o n pattern was obtained. If there was one dominant spot i n the pattern the specimen was not moved. If there was no dominant spot, the specimen was t i l t e d u n t i l one spot became stronger than the rest. The d i f f r a c t i o n pattern was then photographed. The objective aperture was introduced and positioned over the bright spot. The bright f i e l d image was then restored and photographed. In this way an estimate of the number of i n v i s i b l e dislocations can be made. The d i f f r a c t i o n pattern was analysed and the strong spot found to be the {h k 1} plane, or the <h k 1> direction in reciprocal space. The Burgers vector of a l l dislocations in iron l i e s i n the <111> directions, and there are four such directions available. The disloca-tions with their Burgers vectors lying in the <uvw> direction w i l l be i n v i s i b l e i f hu + kv + lw = 0 (7) 81 This equation was solved for each direction and a correction applied to the dislocation densities according to the proportion of i n v i s i b l e dislocations. The dislocation densities were determined by using the random l i n e count method due to Ham (11), in which p = —— cm cm ^ (8) F p = dislocation density N = no. of dislocations unit random l i n e J-j t„ = thickness of f o i l F A minimum N value of 50 was used for each determination. The o value of t„ was taken to be 3500 A. Dingley and McLean (12) measured r o the thickness of 50 f o i l s of iron and found the average to be 3500 A. The dislocation density data is shown in Table 13. The average values of corrected dislocation density, and corresponding average flow stress values are shown in Table 14. 82 Table 13 DISLOCATION DENSITIES Material P l a s t i c s t r a i n Measured density Corrected density % 10 9 cm cm - 3 10 9 cm cm"3 Fe 0 2.3 2.3 n " 2.3 2.3 2.7 2.7 I I " 3.5 .3.5 I I " 1.7 2.1 , i " 1.6 1-6 .. 1.4 2.7 , i " 0.9 1-7 . t " 1.3 1-6 " 1.9 1-9 t i 4 1-7 3.3 „ '< 3 . 3 3.3 ,, " 3.8 3.8 2.7 2.7 3.2 3.2 2.3 2-9 2.1 4.3 Fe 1.74%Th02 0 2.2 2.7 2.2 2.8 4.1. 4.1 83 Table 13 (Continued) Material P l a s t i c s t r a i n Measured density Corrected density Q 3 9 - 3 % 10 cm cm 10 cm cm 1.74%Th02 0 3. 2 3. 2 II IT 7. 5 7. .5 n n 0. 7 1. .4 n I I 2. ,9 3. .6 n 11 2, ,0 4. .0 H H 3, ,1 3 .1 II 4 1 .7 3 .4 11 II 6 .0 6 .0 11 11 6 .8 6 .8 84 Table 14 AVERAGE DISLOCATION DENSITIES AND FLOW STRESSES Material Strain Average /p Gb /p Observed Flow Dislocation Stress, Density 10 9 cm cm - 3 x 10~ 4 kg mm-2 kg mm 2 Fe 0 2.2 4.7 9.4 22.2 F e 4 3.2 5.7 11.2 24.8 Fe 1.74%Th02 0 4.0 6.3 12.5 30.3 Fe 1.74%Th02 4 5 . 4 7.4 14.6 48.6 85 4. DISCUSSION Current theories relating to the y i e l d stress of p o l y c r y s t a l l i n e iron and dispersion hardened iron (1,2,13) are not entirely satisfactory, for reasons developed in 4.1. and 4.2. A more complete explanation of the y i e l d behaviour of iron is based on the stress dependence of dislocation v e l o c i t y and dislocation multiplication, f i r s t proposed by Johnston and Gilman (14), and developed for polycrystalline iron by Hahn (15). In Section 4.1. the Hahn theory of the upper y i e l d point and y i e l d drop i s presented. In connection with the present work the theory is modified, and considered i n r e l a t i o n to both room temperature and 77°K. The lower y i e l d stress and Luders band are then discussed in terms of dislocation multiplication theory, rather than unpinning (13), in order to be consistent with the theory of the upper y i e l d point. In Section 4.2. i t is proposed that the yielding behaviour of the iron-thoria i s controlled by the matrix, the influence of the p a r t i c l e s being largely i n d i r e c t . Thus the theory of 4.1. applies to iron-thoria as well as iron, although certain modifications are put forward. The propos-i t i o n represents a departure from current theories (1,2) in which the particles play a dominant role. The proposals of L i (66) are used to discuss the work hardening of iron in Section 4.3. The behaviour of iron-thoria i s considered i n terms of the Fisher, Hart and Pry (4) and Ashby (9) theories. The variation i n work hardening rate with temperature is related to changes in dislocation structure and the temperature dependence of the f r i c t i o n stress. The fracture behaviour i s discussed in Section 4.4. A calcula-tion is presented to explain the observed'fracture angle, and an improvement in d u c t i l e - t o - b r i t t l e transition temperature is attributed to the dispersion. 86 4.1. Yielding Processes in Iron Microstrain experiments on ferrous materials (16,17,18,19,20) indicate that considerable dislocation mobility can occur prior to macroyield at a s t r a i n of about 10 ^. The number of mobile disloca-tions i s d i f f i c u l t to assess in B.C.C. materials containing i n t e r s t i t i a l s . Dislocations can be pinned by the i n t e r s t i t i a l s , and assuming s u f f i c i e n t i n t e r s i t i t i a l atoms are always present the degree of locking depends on prior mechanical and thermal treatment. Fisher (21) found that weak locking was induced i n an iron containing about 10 ppm i n t e r s i t i a l s by quenching and ageing for 1 hr at 140°C. Strong locking was induced by a heat treatment of > 2 hrs at 140°C. The segregation of carbon and nitrogen atoms to dislocations in iron is rapid (22) at ordinary temperatures, and thus the cooling cycle of the present materials (5 minutes from 1100°C) must have produced strong locking. It appears l i k e l y that most of the dislocations were strongly locked in the as-r o l l e d materials, but there can be mobile dislocations produced in the pre-yield region.as w i l l be described later. It was previously considered (13) that a l l the dislocations remained locked unti1 the upper y i e l d point. At this point the stress was s u f f i c i e n t l y high to cause appreciable numbers of dislocations to be unlocked, and a sudden stress relaxation occurred. There are a number of d i f f i c u l t i e s with this theory. Iron should ord i n a r i l y be t o t a l l y e l a s t i c up to the upper y i e l d point, but p l a s t i c microstrain has been observed in Armco iron which had not been pre-strained (17). The theory also accounts for the y i e l d drop as being 87 the difference between the stress for unlocking dislocations, and that for their subsequent propagation. Johnston and Gilman (14), however, have observed large y i e l d drops in LiF crystals with no evidence of dislocation unlocking. There i s evidence that the y i e l d point and y i e l d drop are associated with production and multiplication of mobile dislocations (23,24,25). Mobile dislocations are created in the pre-yield region by the influence of stress concentrations. Small notches, foreign i n c l u -sions and particles can produce mobile dislocations (26,27). The magnitude of the y i e l d drop is strongly dependent on the mobile disloca-tion density L, and L can be markedly increased by the stress concentra-tions produced by the shoulders of the sample and misalignment in the machine (28,29). A high value of L has been shown to cause v i r t u a l elimination of the y i e l d drop (15,28,29,30). The present work involved the use of f l a t specimens with 90° corners and rounded shoulders. The effect of these, added to the i n f l u -ence of misalignments and smaller stress concentrations, suggest that y i e l d drop detection w i l l be insensitive. The observed yiel d drops (see Fig. 35) are somewhat smaller than those detected by Hahn and Rosenfield (3). They used Fe-2.5%Th02 formed into Hutchison wire specimens. An example of the low s e n s i t i v i t y i n the present work is that no y i e l d drop was detected i n iron at room temperature. The true -2 y i e l d drop for the Fe-ThO^ w i l l be about 2.8 kg mm greater than the observed ones, based on the Hahn and Rosenfield results. The specimen geometry and alignment procedure were identical for a l l tests. Thus whilst the stress concentration effects were 88 CM i E E o» Q . O CC Q Q _ J LLJ O O v Fe o Fe 0 . 9 % T h 0 2 D Fe I. 7 % T h 0 2 o 9 50 100 150 200 250 300 K Figure 35. Variation of y i e l d drops with temperature. 89 appreciable, they should be f a i r l y consistent. 4.1.1. Upper y i e l d point and y i e l d drop behaviour The theory to be presented was proposed by Johnston and Gilman (14), developed for iron and other B.C.C. materials by Hahn (15) and further considered by C o t t r e l l (22) . During a tensile test the t o t a l s t r a i n rate e i s given by the sum of the e l a s t i c and p l a s t i c s t r a i n rates where ' = 10) E M dt M = effective e l a s t i c modulus of the specimen and testing assembly combined a = applied stress t = time LbV ' £ = —rr- 11) p 2 J where -j = contribution to the strain rate of 1 dislocation of unit 1ength and unit velocity moving in a direction close to the maximum shear stress. L = tot a l length/unit volume of mobile dislocations of Burgers vector b and velocity V. To account for pre-yield p l a s t i c strain i t i s necessary to have L > 0, and in the presence of strong i n t e r s t i t i a l pinning explain how this can be so. L i s d i f f i c u l t to measure and there i s l i t t l e data 90 i n the l i t e r a t u r e , but i t has been estimated (14,15) that L ^ 0-1P 12) p = t o t a l dislocation density, and for 10" 3 < e < 10" 1 we have (10,14,31,32) P p = p + Ce^ H yo p 13) p = i n i t i a l d islocation density o C and A are constant, and t y p i c a l values for iron, mild steel and LiF 8 - 2 are C ^ 10 cm A ^ 1 The measured t o t a l p l a s t i c s t r a i n up to the upper y i e l d point i n the present materials was 0.05-0.3%, the majority being > 0.1%. The dislocation velocity V i s a sensitive function of stress, as indicated by the empirical r e l a t i o n o_.m !4) V = dislocation velocity a = stress applied to dislocation aD= the stress required to produced a dislocation velocity of 1 cm sec i n a particular structure at a particular temperature. The influence of work hardening on the behaviour described by equation 14) i s allowed for by assuming linear work hardening over the small str a i n range over which y i e l d occurs, so that 1 o = ne • P 91 The s t r a i n at which the lower y i e l d stress i s f i r s t achieved i s usually < 0.003 in the present work, so that the contribution due to 15) i s r e l a t i v e l y small. Gilman and Johnston (33) found that a given change in macro-scopic flow stress in LiF gave an approximately equal change to that required to maintain a given dislocation velocity. The assumption i s made that this i s also v a l i d for iron, whereupon a - ne or From (11) and (16) 'p T UD v Lb Then from (12), (13), and (17) a = a / 7 " 1 + ne p 16) 2e , , , P.^l/ m 1 7 1 2e , , a = ne + a [- j~\ 1 8 J P ° 0.1b(p„ + Ce*) At this point Hahn (15) and C o t t r e l l (22) redefine P q as being the i n i t i a l density of mobile dislocations. At the upper y i e l d point da dt 0 so that from equation (10) e£ = 0 and consequently e= e . 2z ,1/m 1 9 1 a = nep + a [—— r-J i y J U 0.1b(pQ + Ce p) Equation 19) defines the yielding behaviour. 92 The theory i s subject to error in certain respects. The mobile dislocation density i s f i r s t approximated at O.lp, which causes this factor to appear i n equation 19). However, the mobile dislocation 2 4 - 2 density at the upper y i e l d point i s later taken (15,22) to be 10 -10 cm 6 8 out of a t o t a l density of 10 -10 . In this case we should have L ^ 10"4p 20) A Multiplying Ce by 0.1 i s incorrect since these freshly produced dislocations are already mobile. The r e d e f i n i t i o n of p Q as an i n i t i a l mobile dislocation density gives r i s e to a similar error. The i n i t i a l mobile dislocation density is ^ 0 in strongly locked materials with s u f f i c i e n t i n t e r s i t i a l s , but i t w i l l be assumed 2 4 that p Q = 10 -10 at the upper y i e l d point as a consequence of pre-y i e l d flow, p a r t i c u l a r l y associated with stress concentrations. p Q w i l l be defined as the mobile dislocation density at the upper y i e l d point, these dislocations having a multiplication rate of Ce\ The factors 0.1 or 10 4 need not enter the calculations. Another error l i e s in the assumption that the strain rate i s constant during the yielding process. Bernstein (34) performed tensile tests on specimens cut from iron s t r i p , using an Instron machine and a range of strain rates and temperatures which encompassed those of the present work. The normal behaviour at a l l temperatures was that the specimen s t r a i n rate was about 0.1 of the applied strain rate at the upper y i e l d point. Between the upper and lower y i e l d points the specimen stra i n rate rose rapidly to equal the applied strain rate. It 93 is therefore assumed here that the p l a s t i c strain rate at the upper y i e l d point i s 0.1 of the applied s t r a i n rate 4.2 x 10 4 sec. 1 . The strain difference between the upper y i e l d stress, and the f i r s t achievement of the lower y i e l d stress is taken to be 0.001. At this and a l l greater strains the p l a s t i c s t r a i n rate i s assumed equal to the applied s t r a i n rate. Equation 19) now becomes a = ne + a n [ ^ ]l/m 21) P b(p + Cc A) o p^ The y i e l d curve can now be calculated by substituting the same values as used by Hahn for polycrystalline iron, with the exception of the strain rate correction described above and a work hardening rate taken from the present work; -2 n = 200 kg mm -2 = 20 kg mm b = 2.48 x 10" 7 mm C = 1.6 x 10 9 cm"2 . ' A = 0.8 m =35 e: (e = 0.0) = 4.2 x 10" 3 sec." 1 P e (e> 0.001) = 4.2 x 10" 4 sec." 1 P The value of p Q is fixed for a particular curve, and may be assigned 1 4 - 2 values of 10 to 10 cm . The theoretical y i e l d behaviour is shown in Fig. 36. The effect of removing the factor 0.1 from the equation is to raise both 95 the upper and lower y i e l d points above the values of the Hahn model. The correction for p l a s t i c s t r a i n rate leaves the upper y i e l d point unchanged, but reduces the lower y i e l d point to about the same value as the Hahn model. It i s observed i n the present work that reducing the tempera-ture of testing results i n i) larger y i e l d drops i i ) higher upper and lower y i e l d points Equation 21) should predict these changes, and to test this the li t e r a t u r e was searched to find values of a^, C, and m for iron at 77°K. The work hardening rate n was taken to be half the 298°K value, this being a reasonable approximation from the data of the present work. The value of m increases as the temperature is lowered (34,35, 36), and i n s i l i c o n iron changes from 35 at 298°K to 44 at 77°K (22, 31). Estimating C at 77°K i s d i f f i c u l t since the only data i n the li t e r a t u r e i s that of Stein (37) for iron single 11 -2 11 crystals, i n which C > 10 cm and a value of C = 1.6 x 10 i s assumed here. Substituting the above values, appropriate for 77°K, in equation 21) gives the y i e l d behaviour shown i n Fig. 36. When p o = 7 o -2 10 cm the upper y i e l d point i s 45.7 kg mm and the y i e l d drop i s 12 kg mm . Thus both the upper y i e l d point, and the y i e l d drop, change in the correct direction. In the absence of more exact data from the l i t e r a t u r e , these quantitative evaluations of the yield behaviour are far from precise. 96 The revised model, however, provides a more complete explanation of y i e l d behaviour than that based on unpinning (13). The predicted y i e l d drops are greater than those observed in the present work, but this i s a consequence of the stress concentration effects discussed e a r l i e r . The revised model has been shown to correctly predict the direction, as observed i n the present work, of the changes in y i e l d behaviour on changing the temperature. In summary, the yielding process i s considered to be a continuous one, with no s p e c i f i c event such as catastrophic unpinning being associated with the upper y i e l d point. The sequence of events i s as follows: i) At zero stress; dislocations strongly pinned. i i ) On straining up to the upper y i e l d point, mobile disloca-cations are created at stress concentrations. The stress opposing these dislocations i s a sensitive function of t h e i r velocity. The f i r s t dislocations w i l l move very rapidly i n an attempt to accommodate the applied s t r a i n rate. i i i ) The dislocations multiply according to equation 13), and at the upper y i e l d point there are s u f f i c i e n t of them contributing to the p l a s t i c strain to accommodate the whole of the applied strain rate. iv) Multiplication continues beyond the upper y i e l d point, supplying a r e l a t i v e l y large number of mobile dislocations. The average dislocation velocity thus f a l l s and the stress 97 therefore f a l l s i n accordance with equation 14), this giving r i s e to the yi e l d drop. 4-1.2. The lower y i e l d point and Luders band Thus far, nothing has been said of the uniformity of deforma-tion. In fact, there i s an embryonic Luders band at the upper y i e l d point. The band forms at 54°-60° to the ten s i l e axis (15,22) for reasons which are discussed i n the fracture section. The propagation of Luders bands i s often considered to be associated with dislocation pile-ups at grain boundaries (Fig. 37a) and unpinning processes (38,39,40). However, electron microscopy studies on the deformation behaviour of iron have never revealed the presence of pile-ups (10,41,42,43). Dislocations appear to be generated at grain boundaries (10, 41,42,43,44). Pile-ups were never observed in the present work, but many boundaries with a 'furry' appearance were seen, suggesting that dislocations could be emitted from them. Fisher (21) observed disloca-tion loops nucleated at grain boundaries at the Luders front, i n specimens which were strongly pinned. Such loops were never observed in iron with weakly pinned dislocations. The experimental evidence suggests that the Luders band i n the present materials must be explained without invoking dislocation p i l e -ups or unpinning processes, and i n a manner consistent with 4.1.1. J.C.M. L i (25) has developed a model of grain boundary dislocation generation which gives suitable values for the k of the Petch relation. Mobile dislocations are produced either by annihilation of sub-boundaries, 98 Figure 37. Sources of mobile dislocations at the y i e l d point: a) Pile-up at a sub-boundary b) Annihilation of the sub-boundary 99 or by generation of dislocations from grain boundary ledges i n high angle boundaries (Figs. 37b and 38). The theory gives satisfactory values for the y i e l d stress of iron when substituting i n the Petch r e l a t i o n . Consider, for example, the case of high angle grain boundaries with large ledges. J.P. Hirth and J. Lothe (45) found the L i model to be satisfactory for grain boundary ledges > 2 Burgers vectors i n width. Then D = grain size G = shear modulus a = constant, ^0.4 aQ = f r i c t i o n stress, incorporating a l l effects but those o the grain boundaries. Wb = constant ^0.02-0.2 for iron and steel Suitable values of the constants for iron can be taken from 22) Armstrong et a l (44) so that 4.8 kg mm -2 a o k G 0.27 x 10 -3 mm 1/2 Then W 3.28 x 10 6 100 e 38. Grain boundary ledge acting as a dislocation donor. 101 Substituting the above values i n (22) and assuming D = 30 y, -2 a = 18.8 kg mm ys to An iron sample with a measured grain size of 30y in the present work gave -2 a = 17 .8 kg mm ys 6 The L i model, considered b r i e f l y here, gives us an explanation of the y i e l d behaviour implied by the Petch relation by invoking dislocation generation processes, rather than unpinning processes and pile-ups. The propagation of the Luders band can now be considered i n terms of dislocation generation. The stress relaxation i n the yielded grains of the embryonic Luders band applies the stress to the adjacent, unyielded grains. The stress generates dislocations i n the grain boundaries, and these dislocations then run out into the unyielded grain. The process has been considered in terms of the motion of the whole Luders band front by E.W. Hart (46), and Hahn (15). The generated dislocations run a short distance ahead of the Luders front, thereby increasing P q and e p in this l o c a l i t y . The a b i l i t y to y i e l d i s thus conferred by the Luders band on the region immediately ahead of i t . The type of yi e l d point with a constant lower yield stress and steady state propagation of the Luders band front is possible only i f the p l a s t i c deformation occurring outside the band is small. If much p l a s t i c deformation occurs, then the classic y i e l d point i l l be less pronounced and perhaps eliminated. w 102 If the above description is correct, then the v e l o c i t y of the Luders front should be the same as that of the dislocations running ahead of i t . The plot of Luders band velocity against stress for a 0.06%C s t e e l , due to Winlock (47), i s almost identical to the dislocation velocity-stress plot for s i l i c o n - i r o n produced by Stein and Low (26), According to Hahn (15) eg = Ue 2 3 ) -4 -1 e = applied strain rate = 4.2 x 10 sec. U = velocity of the Luders band front = t o t a l Luders band strain g = gauge length of specimen = 2.04 cm Substituting the above values and assuming a Luders stra i n of 0.01 gives - ] U = 0.086 cm sec. From the results of Winlock (47) this is equivalent to a room tempera--2 ture y i e l d stress of about 19 kg mm , which i s a reasonable estimate for the iron in the present work. 4.2. Yielding Processes in Iron-Thoria Alloys The strength of the iron-thoria alloys was greater than that of the iron, as indicated by Figs. 10 and 11. There is a grain size effect which can be compensated for by using the results of Hall (38). The y i e l d stress increment for the 103 iron, to give corresponding grain size to the iron-thoria, i s about 4.6 kg mm . The iron curve i n Fig. 10 can be raised by t h i s amount. The gap between the iron and iron 0.9%ThC>2 alloys i s VI2 kg mm"2, which -2 means that 7.0 kg mm are d i r e c t l y or i n d i r e c t l y attributable to the _ 2 ThC>2 dispersion. Similarly, there is ^ 5.3 kg mm between the two iron-thoria curves, which i s a consequence of the different thoria levels. The difference i n strength w i l l be considered i n terms of the current theories of the y i e l d stress of dispersion hardened materials, and also on the basis of the dislocation multiplication theory of the previous section. The variation i n y i e l d behaviour, shown qu a l i t a t i v e l y i n Fig. 12, w i l l also be considered. The appearance of a peaked y i e l d point i s suppressed to progressively lower temperatures as the thoria level increases. There i s appreciable scatter i n the y i e l d drop data of Fig. 35 as predicted i n 4.1., but there i s a tendency for the y i e l d drops for a given material to increase as the temperature drops. The observed magnitudes are l i k e l y to be underestimates (Sec. 4.1.), and this view i s supported by the work of Hahn and Rosenfield (3) on iron-thoria wire formed into Hutchison samples. The reduced y i e l d drop i n the presence of p a r t i c l e s i s probably due to the relaxed stress being partly taken by the p a r t i c l e s . 4.2.1. Theories of the y i e l d stress of dispersion hardened materials Orowan (1) considered the case of uniformly distributed incoherent precipitates, where r y << Rg. r y i s the true p a r t i c l e radius and R i s the planar centre-centre p a r t i c l e spacing. The matrix stra i n s 104 associated with the p a r t i c l e s is assumed to be zero, so that a p a r t i c l e could only influence the progress of a dislocation i f i t actually occupied i t s s l i p plane. The entire process by which a dislocation by-passed par t i c l e s i n i t s s l i p plane i s shown in Fig. 39. The bowing process of Fig. 39 has been observed (48,49). The y i e l d stress i s that associated with the minimum radius of curvature (Fig. 39c), so that 2Gb aOrowan ^ R 2 43 Experimental v e r i f i c a t i o n of this relation for several systems has been provided by Gregory and Grant (50), Cremens and Grant (51), Ashby and Smith (48), Dew-Hughes and Robertson (52), and others. Kelly and Nicholson (53) employed the Nabarro expression (54) for the line tension of a dislocation to rewrite 14) in the form Vowan = § • ^  * l n 2 5 3 s s r = planar centre-centre p a r t i c l e radius s cf) = (1- sin 2B) a, 1.25 3 = the angle between the Burgers vector b and the dislocation l i n e . The equation gives the increment in flow stress due to the dispersion, and the f u l l expression for the yie l d stress of the material is 106 M Orowan s s = flow stress of the pure matrix Ashby (55) has pointed out that i) the c r i t i c a l bowed out shape i s a function of 3, and i s not generally a semicircle. The shape calculated by Ashby for 3 = 45° is shown in Fig. 40. i i ) In Fig. 40, there is an interaction between dislocation segments 1 and 2, such that there is a mutual lowering of line energy The effect of point i i ) i s to modify equation 26) so that 2r Gbc}> 2 -, , s. s s The value of a„ i s reduced by a factor of 0.5-0.7 as compared Orowan with 25). The theoretical slope of 27) i s independent of the dispersion parameters, and has a value G"^ = 3.94 x 10 4 kg mm 1 for iron. 2TT A plot of a « - 2 - ^ r — In & ) = X 28) R -2r 2b s s 107 108 should y i e l d the theoretical slope and give an intercept of a^. From Fig. 10., the average value of 2r^ = 0.075y, so that 2r = 2r / 4 = 0.061y (see Appendix 1) s v 3 Using the re l a t i o n of Kocks (56, Appendix 1) R = 1.18 2r / -rp s v 6i 29) Rs can be calculated from 29), and the values substituted in 28) to obtain X. X may then be compared with the average experimental y i e l d stress values obtained from Tables 7 and 8. The results are given in Table 15. Table 15 OROWAN PLOT DATA Observed Yield Stress kg mm -2 mm Fe 0.9%ThO, Fe 1.71%ThO, 29.4 34.1 0.6i 0.43 15.5 x 10" 24.1 x 10" The slope of 28), i s therefore 5.47 x 10~4 kg mm"1 and the intercept value i s 21.0 kg mm"2. The experimental value of au> when adjusted for grain size, is 22.4 kg mm"2. The f i t of the data to the theory is not 109 precise though i t i s comparable to that achieved in other tests of the Orowan r e l a t i o n (48,50,51,52). A more recent theory of the y i e l d stress was that of Ansel1 and Lenell (2). Dispersions were divided into coarse, in which 2r^ > , and fine, in which 2r < . Substituting the room temperature a v a y i e l d stress data for a, we have = 0.14M and 0.12y in the 0.9% Th0 2 and 1.7% Th0 2 alloys respectively. Since 2r^ ^ 0.075 y, the present dispersion c l a s s i f i e s as fin e . The process of y i e l d was described as follows; the by-passing process i n Fig. 39 proceeds in the pre-yield region. Loops are l e f t around the pa r t i c l e s and exert a back stress on the dislocation sources, which are eventually prevented from operating. The stress continues to r i s e u n t i l the y i e l d stress of the material i s reached. Macroyield occurs when the stress reaches the y i e l d stress or the U.T.S. of the p a r t i c l e s , defined by G.r. 30) a = p v 2X,C 1 p a = y i e l d stress of material G = shear modulus of particles P r^ = p a r t i c l e radius X^  = a p a r t i c l e spacing parameter based on a volumetric p a r t i c l e distribution model (see Appendix 1) C = a constant associated with the p a r t i c l e s , usually taken P as ^ 30. 110 The theory cannot be tested completely since the values of G p and C„ are d i f f i c u l t to assess, though the r e l a t i o n a a X ^ 1 31) can be tested. As i n the case of the Orowan theory, the ordinate intercept should equal the flow stress of the matrix. Using data from the present work i t is calculated that -2 o M = 27 kg mm wh ich i s a considerably greater than the experimental values. 4.2.2. Assessment of the Ansel1 and Orowan theories A number of objections can be made to the Ansell theory, and these have been examined i n d e t a i l by Kelly and Nicholson (53). Only the major objections w i l l be l i s t e d here; i) the p l a s t i c strain associated with dislocation loop' formation (Fig. 39) i s said to be small, but that associated with loop collapse on yielding appreciable. However, a dislocation expanding between particles on loop formation sweeps out an area ^ R^ , whereas loop 2 collapse causes a sweep of ^ r . i i ) there is no evidence in the l i t e r a t u r e of the f a i l u r e of hard, incoherent p a r t i c l e s . I l l i i i ) the p o s s i b i l i t y of shear of the matrix rather than f a i l u r e of the par t i c l e s is ignored, even though the former seems more reasonable. iv) the v a l i d i t y of the volumetric model for calculating X^ is i n dispute (see Appendix 1). The objections given above, coupled with the poor prediction of matrix y i e l d strength indicated e a r l i e r , lead to the conclusion that the Ansell theory is unsatisfactory. The Orowan theory gives a better prediction of the matrix flow stress, though only because a correction for the difference i n grain sizes was made. The theory i t s e l f does not allow for the influence of particles on grain size and dislocation density, and i t would be d i f f i c u l t in many cases to produce identical structures i n a p a r t i c l e free matrix. The theory does not consider pre-yield s t r a i n , though this could be accounted for by the generation and movement of dislocations up to the p a r t i c l e barriers. R_ is a barrier spacing parameter which includes solely the part i c l e s and ignores other structural features, implying that the part i c l e s are a dominant barrier. A characteristic of dispersion hardened materials is the very high dislocation densities i n the as-manufactured state, and the great s t a b i l i t y of these structures even at elevated temperatures. In B.C.C. materials, i n t e r s t i t i a l locking can give strongly pinned dislocations which can act as strong barriers. The stress required to unpin a strongly locked dislocation i s of the 112 order of E/80 (22,40). The as-manufactured iron-thoria has a r e l a t i v e l y high dislocation density and strongly pinned dislocations. If the Orowan stress is to be meaningful, then the other structural features such as grain boundaries and strongly locked dislocations cannot hinder the i n t e r - p a r t i c l e bowing process of Fig. 39, or act as one or both of the pinning points. Since some hindrance undoubtedly occurs, then the matrix effect i s not additive i n the manner of equation 27. The i n f l u -ence of structural features other than the par t i c l e s i s d i f f i c u l t to define precisely, so that r e - d e f i n i t i o n of to include such features is not simple. It i s d i f f i c u l t to understand the existence of y i e l d drops i f the bowing process controls the y i e l d stress. After one dislocation has by-passed two p a r t i c l e s , i t leaves a loop around each which e f f e c t i v e l y reduces R . Subsequent dislocations require a higher stress to pass through, this being a powerful effect which forms the basis of the Fisher, Hart and Pry work hardening theory (4). At the upper y i e l d point dislocation accumulation is both heterogeneous and rapid. Thus the same par t i c l e s are by-passed many times, and i f the particles play a dominant role, a strong work hardening effect should be observed. In contrast to t h i s , Hahn and Rosenfield (3) observed large y i e l d drops, of up to 11 kg mm , in an iron-2.5% thoria alloy. A futher objection to the Orowan theory i s that there are alternate modes, requiring operating stresses different to c r Q r o w a n , of by-passing particles than that of Fig. 39, and these are i l l u s t r a t e d i n Fig. 41-113 Figure 41. Alternate methods of by-passing p a r t i c l e s . After Kocks ( 5 7 ) 114 It i s proposed in Section 4.2.3. that the matrix behaviour controls the flow stress, and that the influence of the p a r t i c l e s i s largely indirect. Consequently the quantitative aspects of the y i e l d behaviour of iron-thoria can be explained by equation 21), with certain adjustments to the parameters. The qualitative aspects describing the influence of the pa r t i c l e s are examined with the aid of a model proposed by Kocks (57). 4.2.3. Yield behaviour of iron-thoria Kocks (57,58) and Foreman and Makin (59,60) have treated flow stress and work hardening theory in a s t a t i s t i c a l manner. A l l obstacles are regarded as points, and dislocations may by-pass these points by bowing out. The method is applicable to penetrable and impenetrable obstacles and can be developed into an analysis of work hardening in the presence of 3-dimensional dislocation networks (58). For the present purposes the case of fixed impenetrable obstacles w i l l be considered (57), and these are the major barriers within the grains. The graphical evaluation (57) of dislocation-p a r t i c l e interactions w i l l be used, since this allows for a l l the by-passing modes of Fig. 41. The mathematical analysis also presented only includes the by-passing method of Fig. 41a. Kocks considered a s l i p plane containing 550 randomly distributed points, these representing impenetrable obstacles spaced 1 apart, where 115 1 = / L 32) N A = area of plane N = to t a l no. of points. Then an average by-passing stress, T , i s defined such that 5k 33) 1 The applied stress i s raised to some fixed value a, and the dislocation arrangement evaluated for the situations corresponding to — = 0.74, 0.90 and 1.04. The results are shown i n Figs. 42, 43 and 44 respectively, Lines are drawn joining those particles which cannot be by-passed at the appropriate — l e v e l . The heavily outlined open regions are those which can be swept by mobile dislocations. In Fig. 42, the dislocation source would feel a back stress from the f i r s t dislocation emitted, since this w i l l be stopped whilst s t i l l close to the source. In Fig. 42, the dislocations move appreciable distances, and subsequent generation can occur at almost the same stress. The subsequent disloca-tions are faced with a s l i g h t l y reduced 1 value due to loops l e f t around the p a r t i c l e s . The dotted regions are those which could be penetrated from the outside, but dislocations generated from within could not escape. Such a situation can arise because of dislocation bowing processes such as those 0 f pig. 4 1 b . At some value of a = a •_, the heavily outlined penetrable c r i t _ regions become contiguous. Dislocations can then sweep continuously over 116 Figure 42. 550 random points, connected by lines i f they cannot be by-passed at a stress level of a/x B = 0.74. Heavy lines outline the "free" regions. After Kocks (57). 117 118 Figure 44. As Fig. (43), stress level a/ig = 1-04. 119 the s l i p plane, despite the presence of impenetrable areas such as those i n Fig. 44, always provided that they can overcome the effect of the steadily decreasing 1 value. The l a t t e r effect i s a work hardening process, but i f this i s small, then Kocks has shown that ° " 9 V °crit < It i s considered that the nature of the d i s l o c a t i o n - p a r t i c l e interactions is q u a l i t a t i v e l y described above. There is nothing in this behaviour which can give r i s e to a y i e l d drop, but t h i s , and the quantitative aspects of yielding i n iron-thoria, are described by equation 21). One of the attractive features of 21) i s that we need not define a barrier spacing. Thus far, in the description of dislocation-particle interactions, i t has been assumed that the only obstacles are fixed and impenetrable pa r t i c l e s . Dislocations and grain boundaries can be penetrable or impenetrable obstacles, depending largely on stress l e v e l , and these must be considered in any y i e l d . stress theory. Any r e a l i s t i c barrier spacing must incorporate a l l appropriate structural features. This requirement is not met by existing theories of y i e l d , and need not be s a t i s f i e d by equation 21). The problem becomes much more d i f f i c u l t when discussing work hardening, when we have a continuously changing dislocation barrier spacing superimposed on a fixed p a r t i c l e spacing. Equation 21) describes the y i e l d of iron-thoria as well as that of iron, but the values of the various parameters w i l l be different. 120 The effect of these and other differences are discussed below. In iron, a dislocation could travel an appreciable distance and therefore not exert a back stress. In iron-thoria the dislocations can be stopped close to their source by the particles or other disloca-tions and provide a back stress. Thus a higher stress w i l l be needed to either i ) operate another source i i ) by-pass the obstacle Sources close to those regions impenetrable even at high stresses may be prevented from operating altogether. A dislocation, moving under a s u f f i c i e n t l y high stress to overcome a l l obstacles, w i l l nevertheless be hindered in i t s progress by the bowing processes occurring over i t s whole length. The a D parameter i n equation 21) w i l l be appreciably increased by the processes described above. Evidence for this concept i s found by col l e c t i n g the available dislocation velocity-stress data for iron and steel from the literature (15,26,47,61). It is found that higher stresses are required to achieve a given dislocation velocity as the carbon content increases. Thus to achieve a given dislocation velocity a higher applied stress w i l l be required for the iron-thoria.. Alternatively, using equation 11), we can say that to achieve the same p l a s t i c strain rate in both iron and iron-thoria, a higher stress i s required in the l a t t e r . Furthermore, w i l l increase with thoria content and disloca-tion density. A higher value of a Q substituted in 21) can account for the higher upper and lower y i e l d stresses in the iron-thoria alloys. As the specimen undergoes the y i e l d drop the influence of the 121 particles is similar to, but not as drastic as, reversing the events of Figs. 43 and 44. At the lower y i e l d stress there are s t i l l contiguous penetrable.regions although their t o t a l area i s reduced. The dislocations moving at this reduced stress encounter more impenetrable barriers than at the upper y i e l d stress. In iron a dislocation could theoretically run from one grain boundary to the next, thereby relaxing the associated shear stress against the next grain boundary. In iron-thoria appreciable lengths of dislocation w i l l be blocked by impenetrable regions, and the shear stress applied to the grain boundary dislocation sources of Figs. 37 and 38 w i l l be reduced. A higher applied stress w i l l be needed to operate these sources and thus the lower y i e l d stress w i l l be raised above that of iron under similar conditions. Unfortunately the y i e l d drop data for the 3 alloys i n Fig. 35 does not overlap very much, so the point cannot be conclusively proved. The y i e l d drops i n iron 1.7% thoria do appear to be quite small for such low temperatures, and Fig. 12 shows that the y i e l d point is suppressed to progressively lower temperatures as the ThC^ level increases. Felberbauer et a l . (62) have shown that the y i e l d point can be eliminated i n iron at 298°K simply by r a i s i n g the volume fraction of an A.1 0^ dispersion. There are other influences on the y i e l d drop, however, and these are discussed below. The influence of different grain sizes is found in the multiplication parameters, C and A, the values of which may be arrived at by means of plots of dislocation density and true p l a s t i c strain. The dislocation density increment p-P Q over a p l a s t i c s t r a i n i s due entirely to production of mobile dislocations according to 122 Once p-p Q has been determined, values of C and A are f i t t e d to the particular plot. The values are dependent on grain size, as shown by the figures below calculated from data on iron (10,15). GS, mm C A 0.015 2.4 x 10 9 0.9 0.020 2.0 x 10 9 0.7 0.100 1.6 x 10 9 0.8 Thus the smaller grain sizes have the higher dislocation multiplication rates, and this alone w i l l give a y i e l d point difference between iron and iron-thoria. The laborious procedure of determining a pv e p plot was not carried out in the present work, but values of p were determined after a strain of e =4% was applied to iron and iron 1.7% thoria. p Approximate C values can therefore be determined by using the data of Table 14 in equation 34) and assuming A = 1, which gives C = 2.6 x 10 1 0 (Fe) C = 3.6 x 10 1 0 (Fe-1.7%Th02) Thus the multiplication rate is higher in the iron-thoria alloys, which means that the y i e l d drop w i l l tend to be greater than that in iron, though the upper y i e l d point is not changed. 123 The multiplication process in iron thoria w i l l not be entirely-due to grain boundary sources as is the case for pure iron. There appears to be a decline in y i e l d drop as the thoria level increases, despite the fact that the grain sizes are the same. The observation i s reinforced by the results of Felberbauer et a l . (62). The p a r t i c l e s must as s i s t dislocation generation, and evidence for this can be found in the l i t e r a t u r e . Gilman (24) observed that during p l a s t i c deformation of LiF crystals, multiplication at particles occurred by the Koehler (63) c r o s s - s l i p process. Stein and Low (26) observed that s l i p i n s i l i c o n -iron crystals always began by the motion of dislocations near inclusions, and the f i r s t s l i p bands radiated from inclusions. Leslie (64) observed that dislocation loops were generated near carbide particles during the pastic deformation of s t e e l ; and this type of process was used by Ashby (9) in a work hardening theory, considered later. It has been observed that dislocation loops can be punched out around particles as a consequence of d i f f e r e n t i a l thermal contraction (65). If such dis-locations were produced around thoria particles during the manufacturing process, i t is l i k e l y that they would be strongly pinned by i n t e r s t i t i a l atoms. Stein and Low, however, consider that the stress concentrations associated with the particles may be s u f f i c i e n t to give unpinning on subsequent stress application. We need not specify the precise processes by which multiplica-tion occurs, since we need only find the change in dislocation density to obtain values of C and A. A l l the factors discussed above increase the multiplication rate and the mobile dislocation density at the upper y i e l d point. Exactly what their combined influence proves to be depends 124 on the precise values of the parameters of 21). 4.2.4. Temperature dependence The temperature dependence of the y i e l d stress i n iron-thoria is l i t t l e or no different to that of iron, as can be seen from Fig. 10. No deductions can be made concerning the v a l i d i t y of the Orowan theory since the bowing process is scarcely temperature dependent (53). The presence of a dispersion might induce more cr o s s - s l i p than occurs in iron, as a mechanism of avoiding p a r t i c l e s . Cross-slip can be thermally assisted, and i t would be expected i n this case that the temperature dependence of the y i e l d stress of iron-thoria would be greater than that of iron. At low temperatures the a b i l i t y to cross-s l i p around pa r t i c l e s would be very limited. Hahn and Rosenfield (3) claim that the temperature dependence of their Fe-2.5% ThO was s l i g h l y greater than that of their iron. The effect of temperature on the parameters of equation 21) has been described i n 4.1.1-125 4.3. Work Hardening Figs. 23, 24, 25, and 26 compare the work hardening rates of iron and iron-thoria at true p l a s t i c strains of 1, 2, 4, and 8%. The curves show a peak i n the 250-300°K region, with the peak occurring i n the same temperature range for a l l strains and a l l alloys. The work hardening behaviour of iron at room temperature w i l l be considered f i r s t , followed by the possible reasons for the observed changes i n 0 with temperature. The fact that the shape of the curves for both the iron and the iron-thoria i s similar i s taken as some indication that similar processes occur in both materials. The work hardening behaviour of the iron-thoria w i l l then be discussed, together with explanations of the differences between the iron and iron-thoria curves. 4.3.1. Work hardening of iron Very l i t t l e has been published on the theory of work harden-ing of polycrystalline B.C.C. metals. Keh and Weissman (10) considered the available experimental data to be so inadequate as to render prema-ture the formulation of even a qualitative theory. Li (66) has put forward certain ideas which could form the basis of a theory, and Keh and Weissman consider his approach to offer the best explanation of their results on polycrystalline iron. Li considered the work hardening of a polycrystalline B.C.C. metal, starting with material in the f u l l y r e c r y s t a l l i s e d condition. Events are envisaged by Li as follows: 1. The grains w i l l have a high degree of perfection, and grown in Frank-Read sources are probably absent. Dislocations are generated 126 at grain boundary sources, perhaps by the mechanisms of Figs. 37 and 38. In the present material this process might be taking place i n the grain boundary at the s i d e of Fig. 33. 2. Straight dislocations move into the st r a i n free grains. The i n i t i a l state of the present material might so be described, as evidenced by Figs. 30 and 34. Wavy s l i p w i l l be due to cro s s - s l i p induced by interaction with other dislocations. Generation within the grain by the Koehler process (63) i s then possible. 3. A secondary s l i p system begins to operate and the dislocations tangle with those of the primary system. The start of such a process may be taking place i n Figs. 33 and 34, which i l l u s t r a t e iron which has undergone 4% true p l a s t i c s t r a i n . The network of Fig. 34 could well be the basis of a c e l l structure which would be well developed at about 10% s t r a i n . 4. Subsequent dislocations moving on the primary system push between the tangles. The equivalent events could be those described by Kocks (57) and i l l u s t r a t e d i n Figs. 43 and 44. 5. The average dislocation s l i p distance cannot account for the observed s t r a i n . To account for the additional s t r a i n , i t i s suggested that the dislocation tangles as a whole must move. These ideas are consistent with the multiplication model of 4.1. and also with the electron microscopy on iron i n Section 3.4. A similar curve, for 9 v K° for iron, to that of the present 127 work has been observed by Keh and Weissman (10), though the peak i n 6 occurred at room temperature. At small strains, the structure of iron at low temperatures consisted of long straight dislocations, but at room temperature the same number of dislocations form tangles. The long range stresses associated with the tangles were absent i n the more open low temperature structure. Such stresses were considered (10) to give r i s e to the increase in 6 from 120-300°K. The decline in 6 above 300°K was explained by the onset of recovery processes. The position of the peak is thus decided by the balance between long range stresses and recovery. The implication behind Keh and Weissman's reasoning i s that recovery processes commence at some temperature < 300°K, and are s u f f i c i e n t l y strong at 300°K to cause a decline i n 9. Applying the same arguments to the present data, which shows (Fig. 23) a peak at about 250°K implies that recovery processes start below 250°K, and have become dominant at this temperature. The position of the peak in Fig. 23 could also be influenced by the y i e l d stress behaviour, shown in Fig. 10. Below 250°K, the y i e l d stress increases rapidly with decrease i n temperature, whilst 6 i s decreasing, as can be seen by comparing Figs. 10, 20, and 23. Consider a small temperature drop, from some temperature T < 250°K, to T . The y i e l d stress at T w i l l be greater than that 1 2 ^ at T^. The dislocation arrangement w i l l be essentially the same in both cases, and the work hardening barriers w i l l be of about the same magnitude. The driving stress behind the dislocations at T , however, is greater than that at T^ and consequently the barriers are more 128 readily overcome. Thus 6 i s lower at T 2 than at T . The suggestion i s that the rapid r i s e i n y i e l d stress below 250°K more than compensates for any increase i n barrier strength which might occur. The influence of this process i s appreciably reduced at temperatures > 250°K, since the rate of change of the y i e l d stress i s much lower. 4.3.2. Work hardening of iron-thoria The Fisher, Hart and Pry theory (4) considered the work hardening effect of the loops l e f t around particles by dislocations passing across a s l i p plane. The theory has been applied to dispersion hardened polycrystals (53,67). The model of the by-passing of p a r t i c l e s is that of Fig. 39. It i s assumed that there i s no c r o s s - s l i p , and no interaction with dislocations in p a r a l l e l planes. Only 1 s l i p system operates and the dislocations leave loops around the p a r t i c l e s which 1. Make by-passing more d i f f i c u l t 2. Exert a back stress which raises the c r i t i c a l operating stress of the sources. The stress increment to be added to the corresponding flow stress of the pure matrix i s o^, and aH " 3 f r 35) s J fN = no. of dislocation loops' Equation 35) describes the work hardening behaviour for the f i r s t few percent s t r a i n , after which a c r i t i c a l stress increment a i t i s achieved, whereupon 129 % • 5 f 3 / 2 ' e x i t 5« When N reaches a value of several hundred, either the p a r t i c l e shears and causes loop collapse or the matrix shears, either event giving r i s e to a stress relaxation i n this v i c i n i t y . The sequence of Fig. 39 then repeats i t s e l f u n t i l ocT^ 1 S again achieved. The overall effect i s a steady state process of straining, and once a • i s generally achieved the value of q , , remains constant, c r i t & / H A number of points, should be made concerning the theory, these being: 1. Ignoring s l i p on other systems means that the theory can be v a l i d for polycrystals, i f at a l l , only at small strains. The theory w i l l be incorrect for a l l conditions where cross-sl i p occurs readily. 2. G should exhibit a small temperature dependence similar to that of the e l a s t i c moduli, and Figs. 25-28 show that this is not the case. 3. N cannot be measured, so that 36) cannot be tested, though the v a l i d i t y of c H a f 3 / 2 3 7 ) n can be examined. The- value of c*H at f = 0 should be equal to the appropriate flow stress of iron. 130 Ashby (9) considered that the theory should be v a l i d for dispersion hardened single crystals up to strains of about 2%. The results of calculations using the data of the present work i s shown in Table 16. It can be seen that the predicted flow stress of iron i s a considerable underestimate of the experimental values. Even using the small s t r a i n of 0.2%, and a temperature of 173°K to minimise the effect of c r o s s - s l i p , does not result in a reasonable answer for the flow stress. It i s concluded that the work hardening process envisaged by Fisher, Hart and Pry does not occur in the present materials, since 1. The quantitative predictions are poor. 2. S l i p on more than 1 system almost certainly occurs, even at small strains. 3. The observed temperature dependence of 0 i s not predicted by the theory. 4. Most dispersion-hardened materials have a high dislocation density in the as-manufactured condition, and the theory ignores a l l obstacles but the p a r t i c l e s . The work hardening rate w i l l be influenced by the i n i t i a l dislocation density and configuration. Ashby (9) proposed a work hardening theory for dispersion hardened single crystals, which gives reasonable prediction of the behaviour of these materials. The ideas presented in the model are of interest i n discussing the iron-thoria. -131 Table 16 DATA FOR FISHER, HART AND PRY (4) THEORY 3/2 Material Temperature Flow Stress at Strains f of — °K 0.2% 4% -2 -2 kg mm kg mm 2 2 Fe 298 17.8 -4 Fe-1.7% TMK 298 34.1 16.3 22.4 x 10 253 50.2 23.5 173 69.4 28.4 -4 Fe-0.9% Th0 o 298 29.4 11.6 8.68 x 10 253 41.8 15.2 173 63.8 22.8 Predicted a„ _2 kg mm 8.6 253 26.0 9-9 173 41.0 2 4 - 9 132 The theory proposes that s l i p commences on a primary system, and the dislocations by-pass and leave loops around the hard incoherent par t i c l e s i n the s l i p plane. The loops exert a shear stress on the p a r t i c l e s , which f i r s t d i s t o r t e l a s t i c a l l y and eventually relax the stress by rotating. The rotation causes dislocation loops to be punched out around the p a r t i c l e s , thus i n i t i a t i n g s l i p on secondary systems. A similar process has been observed by Leslie (64). The volume of p l a s t i c a l l y deformed material around the par t i c l e s increases as more loops are created, and the deformed zones of the various pa r t i c l e s ultimately overlap. Work hardening i s caused by the interaction of primary dislocations with the secondary dislocations. Deformation on the secondary system commences after a p l a s t i c shear stra i n of about 10% on the primary system. The increment in flow stress, o-o~M , above that of the pure matrix is given by r , bfa a-a M = aG / 38) s a = strai n on primary system Equation 38) i s derived on the basis of the number of loops opposing the motion of a straight dislocation, and allows for s l i p on several systems. The result is almost independent of the details of the model. If a dislocation forest of density p p is considered, rather than the loops, we obtain an equivalent and common (9,10,43) expression a-a^j = aGb /pp 39) 133 Equation 39) has been applied to polycrystals (10,12,43) and the precise mechanism by which secondary dislocations obstruct primary dislocations only changes the value of a s l i g h t l y (9). It i s considered that the Ashby theory presents a reasonable picture of the influence of par t i c l e s on the work hardening rate. 39) is equivalent to a = oQ + aGb /p 40) where p is now an average, and not a forest dislocation density. oq in this case i s a f r i c t i o n stress containing the effect of grain boundaries, impurities, l a t t i c e f r i c t i o n and the dispersion. Thus aGb/p represents purely the effect of dislocation-dislocation interactions. Alternatively o a Gb/p 41) a is a proportionality constant measuring the efficiency of dislocation strengthening in the material (12), and in the case of iron has a value of 1-4-1.6 with an average of 1.5. The value i s not altered by changing the temperature between 156°K and 673°K (12). This i n v a r i -ance is an indication that the long range stress theory of work hardening (10) discussed e a r l i e r i s not v a l i d , since a should i n that case show a similar temperature dependence to S. The data of Table 14 allows an estimate of a to be made for iron and iron-1.7% thoria. Using the iron data, the slope of 48) i s 1.6, and for the iron-thoria data is 6.4. The fact that the a value 134 for iron-thoria i s so much greater than that of iron shows that the influence of the p a r t i c l e s i s largely i n d i r e c t . This i s i n qualitative agreement with the Ashby (9) theory, which indicates that the sphere of influence of a p a r t i c l e , as a consequence of the loops punched out around i t , i s far greater than i t s own volume. The work hardening rate of the iron-thoria alloys i s always greater than that of the iron (Figs. 23-26). Whilst a l l the rates decrease sharply with increasing s t r a i n , the values for iron-thoria remain roughly double that of the iron, since the dislocation multi-p l i c a t i o n rates are higher (Tables 13 and 14). The multiplication rates are higher in the Fe-0.9% ThC>2 than i n the iron because of the particles and also because of the finer grain size, which contributes to a higher density of dislocation sources (Figs. 37 and 38). The rates are higher i n the Fe-1.7% Th0 2 than in the Fe-0.9% Th0 2 because of the greater number of p a r t i c l e s . Since the grain size of the two thoriated alloys i s the same, the difference in 6 values between the two i s a measure solely of the effect of a 0.8% volume fraction of particles* The work hardening processes i n iron-thoria are considered to be similar to those in iron, but with the added influence on dislocation multiplication rates of the finer grain size and the disper-sion . 4.3.3. Luders band failures Below about 250°K, the work hardening rate of a l l the alloys declines. A point i s reached where Luders band failures occur, which means that fracture occurs at the lower y i e l d point and ordinary 135 work hardening processes never commence. Since the ef f i c i e n c y of work hardening processes i n the alloys depends partly on thoria content, the onset of Luders band fai l u r e s is suppressed to progressively lower temperatures as the thoria level increases. A Luders band f a i l u r e may be defined, for convenience, as occurring when the load never rises above the lower y i e l d point apart from the upper y i e l d . Using this d e f i n i t i o n to establish the tempera-tures below which Luders failures occur, we obtain from load-elongation curves the following figures: Fe 173°K Fe 0.9% Th0 2 173°K Fe 1.7% Th0 2 77°K Luders band failures are discussed in more de t a i l i n Section 4.4. 4.4. Fracture 4.4.1. Ductile f a i l u r e The 3-stage necking process described in 3.2.3. was also observed in f l a t samples of iron by Crussard et a l . (68). In this case the necks formed at about 63° to the tensile axis as d i s t i n c t from the 60° of the present work. No explanation for this behaviour was offered. The lengths: width ra t i o of the present samples was only 4:1, and i t is considered that the general neck (stage (a) in 3.2.3.) was due to the constraint of the shoulders of the sample. A very much 136 longer sample would have exhibited uniform p l a s t i c strain over a l l but the ends of the gauge length. The type of localised necking described as stage (b) in 3.2.3. was always about 60° to the tensile axis in iron and 57° in the iron-thoria. It i s not possible to form a neck at 90° to the t e n s i l e axis (75). Consider the p o s s i b i l i t y of the formation of a transverse neck, in direction x , referring to the axes of Fig. 45a. The von Mises y i e l d c r i t e r i o n is a = J ( l / 2 [ ( a 2 - a 3 ) 2 + ( V a l ) 2 + ( ° r a 2 ) 2 ] ) 4 2 ) Under conditions of plane strain and uniaxial tension i n direction x_, a, '2 " ~.2 a. = 4 43) Substituting 43) in 42), we obtain / 3 0 = V 4 o r a < Oy a n i m P 0 S S 1 D i e condition. For necking to be permissible plane stra i n conditions must exist, so that we must define some other directions x* and x' , with stress components al and a'^ such that a' =- 2al 44) 137 Figure 45. Directions of stress and necks in sheet material: 138 The problems becomes quite simple with the aid of a Mohr's c i r c l e construction, as i n Fig. 46. The angle 6 between x^ and x 3 is obtained by measuring AB and BC on the construction, giving BC 2.125 tan 26 = M = 6 = 35° 16' Thus the angle between x^ and the direction of the neck, x!,, i s 90° - S = 54° 44' ^ 54.7° The neck therefore forms as depicted in Fig. 45b. The experimental results deviated from this theoretical angle by ^2° in the case of iron-thoria and ^5° in iron. The error i s due to the fact that the edges of the specimens were used as the measurement base-line for the fracture angles. The magnitude of the error i s dependent on the amount of stage (a) necking, which was less i n the iron-thoria than in the more ductile iron. If an accurate tensile axis could have been recorded, i t i s l i k e l y that the stage (b) neck would be found to form at 55° from this axis. The mechanics of Luders band formation is the same as that described above, so that both necks and Luders bands form at the same angle. Luders band failures occur when necking to fracture takes place at some position of the Luders band front. Such a process does not preclude the limited propagation of other Luders bands elsewhere in the sample. 139 e 46. Mohrs' c i r c l e construction to determine angle at which gur neck occurs. 140 The f i n a l fracture occurs by shear i n the neck, in a plane at roughly 45° to that of the sheet from which the sample was cut. 4.4.2. Low temperature failures At the lowest temperatures a l l the fractures are at 90° to the te n s i l e axis and show no evidence of necking. The material i s b r i t t l e , and cracks i n i t i a t e d at stress concentrations propagate i n the transverse direction, which contains the plane of maximum normal stress. The crack i n i t i a t i o n sites can be thoria agglomerates, as in Figs. 16-18; twins as i n Fig. 19; or a surface stress concentration. In the temperature region 133-173°K, the type of fracture of Fig. 15 can be observed, this being a combination of the types discussed e a r l i e r . A crack can i n i t i a t e at or near one edge and start to propagate at 90° to the te n s i l e axis. The crack is then arrested by p l a s t i c deformation, which in this case occurs as twin Luders bands running from the crack t i p to the opposite edge of the sample. One of the bands becomes a neck along which f i n a l fracture occurs. 4.4.3. The ductile to b r i t t l e transition The transition temperature T^ can be estimated from the d u c t i l i t y curves of Figs. 13 and 14. The d u c t i l i t y of the iron reaches zero in both diagrams at about 133°K for iron and 77°K for iron-thoria. The i d e n t i f i c a t i o n of these temperatures as T £ values i s supported by the fracture appearance c r i t e r i o n . The 90° type of fracture predominates below 133°K in iron, but only at 77°K in iron-thoria. At f i r s t sight this appears to be an improvement in T £ due to the dispersion, but this i s only partly true. Cleavage fractures can 141 i n i t i a t e at the head of a s l i p band in iron (3,69,70). The p a r t i c l e s reduce the effective s l i p length, so that the stress concentrations at the head of the s l i p bands i n iron-thoria are less than those in iron. The value of T for iron-thoria should therefore be less than for iron, c However, grain boundaries also reduce the effective s l i p length, and the grain size of the iron-thoria alloys is less than that of the iron. The effect of grain size on T c for iron i s given by (3) dT „ C n = 90°K 45) dlog D J Adjusting T £ for iron to account for this grain size effect gives T = 133°-41° = 92°K c Since T for iron-thoria i s about 77°K, the improvement in T due purely c c to the direct effect of the dispersion i s about 15°K. It can be argued that the f u l l difference of 56°K should be quoted, since the f i n e r grain size is an inherent property of dispersion hardened materials. There i s no detectable improvement as a consequence of the different thoria levels. 142 5. SUMMARY AND CONCLUSIONS 1- Fe-ThO^ has been fabricated by a process involving i ) co-precipitation i i ) hydrogen reduction i i i ) hot r o l l i n g to s t r i p in a hydrogen-argon atmosphere at 1100°C. 2. The yielding processes in iron and iron-thoria can be explained in terms of dislocation multiplication and the stress dependence of dislocation velocity. 3. The dislocations i n the as-manufactured material are probably strongly locked, and mobile dislocations prior to y i e l d are created at stress concentrations. 4. The mobile dislocations multiply, and at the upper y i e l d stress there are s u f f i c i e n t of them to satisfy the applied s t r a i n rate. 5. Continuing multiplication provides more dislocations that .the minimum necessary to s a t i s f y the applied s t r a i n rate and the average dislocation velocity f a l l s . The applied stress level required to maintain the strain rate therefore f a l l s , giving a y i e l d drop. 6. Luders band propagation probably proceeds at the average dislocation velocity. Dislocations generated at grain boundaries run into unyielded grains, thus propagating the Luders band. 7. Neither the Ansell nor the Orowan theory provide a s a t i s f a c t -ory explanation of the results. 143 8. Yi e l d can be explained as being dominated by matrix behaviour, the effect of the particles being on rate of dislocation m u l t i p l i c a t i o n and f r i c t i o n stress. 9. The temperature dependence of the y i e l d stress of iron-thoria i s l i t t l e different to that of iron. 10. The work hardening rate between 250 and 373°K may decrease due to the onset of recovery processes such as cr o s s - s l i p . 11. The work hardening rate below 250°K may decrease as a conse-quence of the rapidly increasing y i e l d stress. The work hardening barriers are overcome by progressively increased stresses. 12. Work hardening processes in iron-thoria are similar to those in iron, but with the additional effects of i) increased dislocation multiplication rate due to the pa r t i c l e s . i i ) f i n e r grain size. 13. Necking and fracture at the higher temperatures employed, i s a 3-stage process: i ) general tapering from the shoulders of the sample. i i ) necking at 55° to the tensile axis i i i ) f i n a l shear i n the neck at about 45° to the plane of the specimen. 14, Fracture at low temperatures -is b r i t t l e , and occurs at 90° to the tensile axis. 144 15. A combined 55° and 90° fracture process can occur, starting with a b r i t t l e 90° crack, which i s arrested by the formation of two 55° Luders bands. One of the bands then forms a neck. 145 APPENDIX I INTERPARTICLE SPACING PARAMETERS The mean free path between the p a r t i c l e s , X, i s obtained by measurements on micrographs of replicas. Ashby (5) uses a random di s t r i b u t i o n of p a r t i c l e s . The mean free path is obtained by drawing a random straight line across a random section through the specimen, and counting the number of particles along i t . Then x - l ^ i ~ N L 46) and this is the value employed i n the present work. The correct value of R to be used i n various theories of s dispersion strengthening is under considerable dispute. For the purposes of theoretical models, the particles must be arranged in some way on the s l i p plane under consideration. The value of R s depends on the nature of the p a r t i c l e arrangement, and some of the equations which have been used are given below. Preston and Grant (71) assume a simple cubic arrangement on the plane, and % - ' s ^ 4 7 ) Kelly and Nicholson (53), Dew-Hughes and Robertson (52), and Ashby and Smith (48) assume an F.C.C. l a t t i c e on the plane so that 146 R = r / i 48) s s t Ashby (5) assumed a random di s t r i b u t i o n . Rg was defined as the radius of the smallest c i r c l e , lying i n the s l i p plane and surrounding a p a r t i c l e , i n which the probability of finding another p a r t i c l e was 1. Then R = ! l 49) s 4 1 Ansel1 and Hirschorn (72) assume a cubic arrangement of pa r t i c l e s throughout the volume, and then consider a random plane cut through the l a t t i c e . Then x i = r v 4 f ) 1 / 3 s°) -1/2 Most authors arrive at an f variation of R . Marcinkowski s and Wreidt (73) support the Kelly and Nicholson derivation of Rs, and point out that the random section in the Ansel1 and Hirschorn 3-dimensional model gives r i s e to a range of p a r t i c l e diameters, which are not averaged in the derivation. Ansel1 and Hirschorn suggest that a 2- dimensional model cannot give the best representation of a real 3- dimensional case. Westmacott, Fountain and Stirton (74) assumed a random d i s t r i -bution of p a r t i c l e s in the s l i p plane. -R was calculated as a mean nearest neighbour distance so that 147 R = r 51) s s 61 Kocks (7) started by defining a simple spacing i n the s l i p plane, R1 as the square root of the s l i p plane area, A, divided by the number of p a r t i c l e s i n this area, P, such that R1 = 2r / ^ 52) s P v 6f R1 i s compared with 12 values of R2, the l a t t e r being computed by mathematical techniques varying from simple to complex. 2 The most sophisticated methods gave values of Rg such that 2 R — = 1.18 R s 1 _ 2 Kocks concludes that for most purposes we can take Rg - Rg but a correction can be applied so that R = 1 • 18. 2r / 7 j 53) s v of The Kocks value of R s is the one used i n the present work. 148 APPENDIX II COMPUTATION OF WORK HARDENING RATES The values of 9 for a l l specimens was calculated with the aid of an I.B.M. 360 computer, using the program shown i n Fig. 47. The program follows the procedure outlined i n 3.3.1. The data cards are stacked i n the sequence of cards numbered 3, 4, and 5 in the program. The characters within these cards are defined below: SPECNO: specimen number A; cross-sectional area XI, Y l ; define the e l a s t i c slope. The load direction on an Instron chart i s the Y axis, and the extension direction i s the X axis. FSD; lbs. in 1 on the load scale, or ( f u l l - s c a l e deflection)/10 on a standard Instron chart. NP; number of points used for this specimen. STP; starting s t r a i n ; the strain above which curve-fitting i s applied. X,Y; cartesian coordinates of any number of data points taken from the Instron chart i n inches. In line 16 the factor 0.008 represents the strain/inch of Instron chart, and must be adjusted for the particular case. The data card sequence represented by cards 3, 4, and 5 repeats for every sample, and any number of samples can be processed. 149 The ordinate intercept of equation 5) i s printed out in logarithmic form. 150 F I * 1 R A \ IV 0 0 ML!" HA I s 0 6 - 0 6 -1 I :0] :3B PAGE G u O l 0 0 0 3 wet 0 0 0 5 c o o t C O C 7 O C C 8 coo C 0 1 0 o o n 0 0 1 2 0 0 1 3 __00 1 4 _ 0 0 1 5 0 0 16 0 0 1 7 00 I d 0 0 1 9 _ £ O i o _ 0 0 ? 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 1 0O26__ 0 0 2 7 0 0 2 3 0 0 2 9 0 0 3 0 0 0 3 1 J > 0 1 1 — 0 0 3 3 CO 14 0 0 35 O J \<-0 0 3 7 0 0 3 H O C 3 9 C 0 4 0 C 0 4 1 0 0 4 2 0 0 4 3 0 04 4 0 0 4 5 C 0 4 6 0 0 4 7 0 0 4 B 0 0 4 9 C 0 5 C h'AI'nF'j 1 h: i t i l t h r 10 I fcMT FROM IF fl S I L f T E S T OA T A n i i ' l SS I TV IS'-, ( S o u ) , S T N ( S o 0 > , l S M i 0 0 l , P ( S 0 G I . X ( 5 C O ) , Y < 5 C C I , U < 5 0 0 l , V l ( s i o i , . c i o o 1 , 11 loot , n ( i o o i ,r,( I O O I , E i l o o i i<m,vr SPFCM1 HI AIM '•, TO , f i n = 2 1 SPt C.,VO, A, X 1 , V 1 , r S O , *.P, S TP RE A I H S , 20 I ( V I I ) , 1 = I ,MPI Pt All I1) , 20) ( X II ) . 1 = 1 ,NR ) FORMAT( KF 10 . 2 > n . » A l ( I 1 0, f 1 0 . 5, 2F 10.2 ,F 10.0 , I 1 0 , F 10 . 3 I S 1 C P F = X 1 / Y 1 .. SU = 0 . SV = 0 . SUU=0. S V V = 0 . S U V = 0 . n (1B 0 J = 1 ,NP P | J 1 = V U 1 « F S H T = ( X 1 J ) - S L 0 P F * Y I J ) ) * C . 0 0 B IF(T.It.0.IT=0.OOCO1 STN(J)=T+1.0 rsN(JI=ALCG(STMJ I 1 _ I S S ( J ) = P ( J ) « S T H ( J ) / A _ _ N U = 1 DCB5NR=1,NP I F ( T S N ( N K I . L T.S T P > NC=NQ+ 1 CONTINUE 0 ( ) 9 0 M= N C , NP JJI «I=A LOCI0 ( T S N ( H ) I V I M ) = A L GG 1 0 ( I S S ( M l j SU = SU-H1(M) S V = S V * V ( K ) S U U= S U L H-U ( M)»U ( M ) S V V = S V V * V ( M ) * V ( C ) S IJ_V = SU V *-U ( M) * VI M ) ANP=Fl 0AT IN P+l - N C ) D t \ < C M = A N P * S U U - S U * S U AH =( SULK'S V - S U V * S U I / D E N O M B = ( A N P * S U V - S U * S V ) / O E N C M cr.=a«nEi«r,«/susT I O E N C M * ( A N P * S V V - 5 V « 5 V I t NP = K G H . * I F j X ( A N P ) _ O l i l 30K3 = 1 ,NP I F ( T S N ( K 3 ) . L E . 0 . ) T S N ( K 3 ) =-0.00001 Z ( K 3 ) = B * ( A E P G 1 0 ( T S N I K 3 ) ) ) + A 6 0(K31 = 10.«*Z <K3I W R I T E ( 6 , 7 0 ) S P E C N 0 , A , X 1 , Y 1 , F S 0 , N P , N C FIIRVAT ( / / J O X , B H S P F C . N0_ , 1 4/10 X j 7 H A R E A = ,F10.5,5X,5HX1 = , F 6 . 2 , 5 X , 15HY1 = , F 6 . 2 .SX , 1 IFhFSO/ 10.0 = ,F5.a7l0x, 19HNUMBER Of P O I N T S = ,12/ 1 1 0 X , 2 5 H L 0 G PLOT S T A R T S AT PCI NT , 1 2 / / ) » R [ i f c, looua.s.r.c F O R M A T ( H X , 2 1 H 0 R 0 I N A T C I N T E R C E P T = , E 16 . 7/1 OX,29HW0RK HARDENING CO 1 F F F 1 C I F N T = , F 1 0 . 3 / 1 0 X , 2 6 H C 0 R R E L A T I O N C O E F F I C I E N T = ,E16.7//> _G( 1 1=0.01 . _ _. AP1=10.**AB O r 2 5 0 * 2=1,4 E ( K 2)=APl»B«IGIK ? ) t'*IB-l.)) 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 45 46 47 48 49 50 51 52 53 54 55 IV f, r / : « , ' f l L F 0 6 - 0 6 - 6 9 H : 0 1 ; T I PAGE 0 0 0 2 24 J OM.'+l) -0 1 ^ 2 1 * 2 . * P 1 T i ( i , , I 10 ) ( 0 I K * ) , h ( » 1) , K J = 1 ,4 1 1].. F C P " A l I I.' X , 1 SHI Cl, IHOI S L U P F , F 6 . 2 , 3 H = , 0 1 0 . 0 1 h R 111 I (, , s.-.) '< 0 I r p V A T I / I 2X , I M V , i r x , 1-tY , 1 2X , A M I OAO, 'IX ,HHST^ AI 1 , r.x , I 1 HTKUT S T R E S S i i 4 x , 1 1 ' * 1 -' 0 F SIR A I \ , sx , I 2H1 (T, P|.i,T I S S / / 1 -. l II I ,6 0 i I X I " > , Y( K I , P ( K ) ,ST'I( M , I S S I K ) , I S M K ) ,0 ( K I , K, 1, NR | -: ' • - " M ( '-x , l 11 . ', sx , r o .2 ,s x , F l r . 1 , sx ,r I r . s , s V , F I o . u , s x , r 1,1.4 , 7x, F l o 1 . <. 1 lM I: 1 57 5B 59 60 61 Figure 47. Computer program used for determining work hardening rates and coefficients. 151 b C G P k L 1 nG N APPENDIX III SYMBOLS A,B,C, constants, often described further i n the text Burgers vector, usually of iron, for which b = 2.46 x 10 cm. constant depending on perfection of p a r t i c l e s , usually taken as ^ 30. D grain size E Young's modulus f volume fraction of particles g gauge length G shear modulus of matrix shear modulus of particles slope of the Petch equation to t a l length of mobile dislocation per unit volume, a spacing parameter between obstacles, of which there are N i n an area A. effective e l a s t i c modulus of the specimen and testing assembly combined, dislocation velocity exponent work hardening coefficient t o t a l number of grain boundaries along a random li n e , tot a l length of random line number of structural features, specified in text, per unit length of random l i n e . 152 mean centre-centre p a r t i c l e spacing. mean planar centre-centre p a r t i c l e spacing. mean planar p a r t i c l e radius i . e . , mean radius of those areas of par t i c l e s cut by a s l i p plane. mean p a r t i c l e radius. time. thickness of an electron microscopy thin f o i l , d u c t i l e - t o - b r i t t l e t ransition temperature, velocity of a Luders band front, dislocation velocity. density of ledges (dislocation sources) i n the grain boundary. Number 1 unit length or number/unit area, constant = 0.2-0.4 unless otherwise specified, angle between Burgers vector and dislocation l i n e , shear s t r a i n . angle between a neck or a Luders band and the tensile axis, true s t r a i n , e l a s t i c s t r a i n Luders band strain, true p l a s t i c s t r a i n work hardening rate i . e . , slope of true stress-true str a i n curve. planar mean free path. Ansell volumetric p a r t i c l e spacing parameter, dislocation density. 153 Po °F a., M aOrowan 0 o a c r i t °F i n i t i a l dislocation density, forest dislocation density. true tensile stress; flow stress; unless otherwise specified, structure and temperature dependent stress required to produce unit dislocation velocity. a y i e l d stress. ys matrix y i e l d stress. Orowan stress, to be added to a., to obtain the flow stress ' M of a dispersion hardened material. f r i c t i o n stress; further defined i n text for particular case, the stress increment to be added to the flow stress of the pure matrix, i n the Fisher, Hart and Pry (4) work hardening theory. maximum stress increment due to the par t i c l e s i n the Fisher, Hart, and Pry (4) work hardening theory flow stress. shear stress to by-pass obstacles of spacing 1 orientation factor = 0.5 unless otherwise specified. 154 BIBLIOGRAPHY 1. Orowan E., Symposium on Internal Stresses, Inst. Metals, London 1947, p. 451. 2. Ansell- G.S., Lenel F.V., Acta. Met., 1960, _8, 612. 3. Hahn G.T. and Rosenfield A.R., TAIME, 1967, 239, 668. 4. Fisher J.C., Hart E.W., Pry R.H. , Acta Met., May 1953, 1_, 336. 5. Ashby M.F., Z. Metallkunde, 1964, 55_, 5. 6. Edelson B.I. and Baldwin W.M., T.A.S.M., 1962, 55, 230. 7. Kocks U.F., Acta. Met., 1966, 14, 1629. 8. Wilson D.V. and Konnan Y.A. , Acta Met. 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