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Yielding and flow in vanadium 1963

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YIELDING AND FLOW IN VANADIUM BY LEONARD ANGUS SIMPSON Sc. , The University of B r i t i s h Columbia, 1961 A THESIS SUBMITTED.IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of METALLURGY We accept this thesis as conforming to the .standard required from candidates for the degree of MASTER OF SCIENCE. Members of the Department of Metallurgy THE UNIVERSITY OF BRITISH.COLUMBIA A p r i l , I963 In presenting this thesis in partial fulfilment of .the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be.granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Metallurgy The University of British Columbia, Vancouver 8, Canada. Date April llth, 1963 ABSTRACT An investigation of the characteristics of yielding and flow in polycrystall ine and single crystal vanadium has been carried out. The effect of grain size, temperature and strain rate on these properties was studied. It was found that there is no s imi lari ty between the mechanisms of yie lding and flow, in vanadium which is in disagreement with work on iron. The results of tensile tests suggest that the mechanism controll ing thermally activated flow is probably either the Peierls-Nabarro force or the non^-conservative motion of vacancy jogs. Some inadequacies of these mechanisms suggest that there may not be a single mechanism controll ing thermally activated flow. Yie ld points were observed in the stress-strain curve upon re- loading a tensile specimen under certain conditions and these are explained in terms of Snoek ordering. i i ACKNOWLEDGEMENT The author wishes to thank Dr. J . A. Lund and Dr. E . Teghtsoonian for their, guidance and assistance in the interpretation of this research. Thanks are also due to Mr. E . J . Richter for technical assistance and to fellow graduate students for many helpful discussions. Financial aid for this project was received in the form of The Aluminium Laboratories Limited. Fellowship and National Research Council Grant No. A -1^63. This assistance is gratefully acknowledged. i i i TABLE OF CONTENTS Page I. INTRODUCTION AND PREVIOUS WORK 1 A. Introduction 1 B. Previous Work 2 II. EXPERIMENTAL 11 A. Material 11 B. Zone Refining . 12 C. Specimen Preparation . 13 1. Machining 13 2 . Electropolishing . . . 13 3* Grain Size Determination and Control 13 D. Tensile Testing 1^ E. Temperature Control 17 III. EXPERIMENTAL RESULTS 19 A. Polycrystalline Material 19 1. Grain Growth 19 2 . Hall-Petch Equation 20 3. Strain Rate Change Tests 26 h. Temperature Change Tests 27 5 . "Pseudo" Yield Points . . . . . 31 6 . Rate Equation 3^ B. Single Crystals • ^2 1. Zone Refining ^2 iv TABLE OF CONTENTS Continued Page 2 . S l ip Systems kk 3 . Strain Rate Change Tests k6 k. Temperature Change Tests 51 5. Rate Equation . . 51 6. Yie ld Points 5k C. Summary of Results 5$ IV. DISCUSSION . . 58 A. Hall-Petch Equation 58 B. Thermally Activated Flow 59 1. Intersection With a Forest 60 2 . Impurity Atoms 62 3 . Cross Slipping of Screw Dislocations 6k k. Peierls-Nabarro Force 6k 5. Non-Conservative Motion of Jogs in Screw Dislocations 67 6 . Comparison Between Polycrystals and Single Crystals 69 7 . Mechanism 70 C. Yie ld Points 73 V. CONCLUSIONS 77 VI. RECOMMENDATIONS FOR FURTHER WORK 78 VII. APPENDICES 79 VIII. BIBLIOGRAPHY 87 V LIST OF FIGURES No. Page o ... 1. . Effect of the concentration of C + N in solution on Q ~ in Fe ; after Heslop and Petch ° . . 3 2 . I l lustrat ion of grain-toj-grain y ie lding 5 3» Force-distance curve for thermally activated flow 7 k. Method of grain size measurement, specimen 9B} 210X . . . . 15 5. Drawing of grips used in tensile testing 16 6 . Assembly for testing at 100°C 17 7 . Inhomogeneous grain sizes after drawing, rod #2, 110X . . . 20 8. Typical load-elongation curve for vanadium polycrystal . . 21 9- Dependence of y i e l d stress and flow stress on grain size at 25°C 22 10. Dependence of lower y i e ld stress on grain size 2k 11 . Typical load-elongation curve for a strain rate change test 25 12. Method of extrapolation to obtain change in flow stress . . 26 13- /S ' f f o r a n increase in £ v s . ^ for polycrystals . . . 28 l ^ * A *Y ^ o r a n increase in £ vs. ^ and £. for polycrystals 29 v_. a a 15 • / v ' T ^ o r a n increase in strain rate vs g. for polycrystals 30 *" a 16. tC\ for a: decrease in temperature vs £ and. ̂  for polycrystals, 32 17- Load-elongation curves showing y ie ld points 33 18. Method of extrapolation through y i e ld points 3^ 19- Activation volume vs. flow stress 36 20 . Activation volume vs. temperature .' 37 21. Energy, H, vs. £ 38 • 22 . Thermally activated component of H°, Zs. Q, vs. £. 39 v i LIST OF FIGURES Continued No. Page 23• Pre-exponential factor A v s . J for polycrystals ko 2ka. Strain rate sensit ivi ty of ^ vs. temperature kl a 2kb. Z^Q vs. temperature for single crystals kl 25a. Laue photograph of suff ic iently polished specimen . . . . i+3 25b. Laue photograph of insuff ic ient ly polished specimen . . . ^3 26. Orientation of tensile axis of single crystals kk 27• Gas holes in single crystal specimens 1+5 28 . Typical s l ip traces on single crysta l specimen 6A, 210X . . kj 29* *" o r a n increase in £ vs. % for single crystals . k^ 30. / ^ Q ^ ^ f o r an increase in £ vs. £. for single crystals . . kQ 31 . Twin markings on surface of single crystal 50 32 . Laue photograph showing s l i t s in spots due to twins . . . . 50 33« / \ / T / a for a decrease in temperature vs ^ a for. single.. crystals '. 52 , A /7T o for a decrease in temperature vs. 'TJ a for single crystals 53 35- Strain rate sensit iv i ty of the flow stress vs. temperature sensit iv i ty to f ind ln(A/jbj . (equation (15)) 55 36. Transmission electron micrograph of dislocations i n s i l i con iron , after Low and Turkalow 65 37• Formation of a dislocation kink in a Peierls force f i e l d . 66 v i i LIST OF TABLES No. Page I. Summary of Previous Work 10 II. Analysis of Material 11 III. Possible Slip Systems k6 IV. Summary of Results 57 I. INTRODUCTION AMD PREVIOUS WORK A. Introduction One of the most remarkable properties inherent in a body centered cubic metal is the marked temperature dependence of i t s strength. This has evoked considerable interest and many investigators have tr i ed to analyze and explain this property. Most of the investigations have been centered about the temper- ature dependence of the y i e ld strength and have not been very productive in explaining the detailed mechanisms of deformation. Most of this work has been concerned with iron while the other body centered cubic metals have received re lat ive ly l i t t l e attention. The adoption of rate theory to explain thermally activated flow has led to an intensif ied attack on the problem of determining the defor- mation mechanisms in these metals. This method, f i r s t used to study f . c . c . metals, has been adapted to b . c . c . metals during the past four years. This has resulted in a number of publications on the flow mechanisms in iron, tantalum and columbium. The problem of establishing a single mechanism as the one con- t r o l l i n g thermally activated flow is rather d i f f i c u l t and current authors are in disagreement on many points. However, i t must be borne in mind that this study is s t i l l in the embryo stage and a vast quantity of experi- ments ia required before a sound theory can be devised. It i s , therefore, the object of this thesis t ° a<3-<i to this pool of knowledge a description of the temperature dependent deformation - 2 - Properties of vanadium. It is intended to make a comparison of these prop- erties between polycrystals and single crystals and also to investigate the role of grain size in the yielding and flow of vanadium. B. Previous work This investigation w i l l be concerned with the two experimentally derived equations: o o* - l / 2 , , (TLY= c± + cr + k L Y d (1) di = Ci + a* + * f i d " 1 / 2 ( 2 ) where 0~LY ^ s ^ e l ° w e r y i e l d stress, that i s , the lowest point on the stress-strain curve after the beginning of plast ic deformation. 0*̂ 2. is the flow stress, that . i s , the stress required to main- ta in plast ic deformation at/ any given strain. 0"i> 0~L° a r e athermal f r i c t i o n stresses opposing dislocation motion. a r e temperature dependent f r i c t i o n stresses opposing dis - location motion. . - l / 2 k f 1> L̂Y a r e s l ° P e s °^ plots of O f 1 a n d Qjjf against d - 1 / 2 d r ' / i s the mean grain diameter. o o* In ear l ier work QT .and Q~ have been studied as a single o parameter, f ^ , and equation ( l ) has been used in the form QLY = CTo + k L V d - 1 / / 2 (3) This is known as the Hall-Petch equation and was obtained experi- 1 ' 2 mentally by Petch who followed up some ear l ier work by Hal l . Petch, who worked with polycrystall ine iron supplemented his or ig inal work with - 3 - two more papers in which he discussed the term (T~ . By varying the o C + N content in solution he obtained the plots shown in Figure 1. Figure 1. : Effect of the concentration of C + N in solution on (y° in Fe, after Heslop and Petch. Because the slopes were the same at a l l temperatures, Petch con- cluded that the effect of i n t e r s t i t i a l atoms on the f r i c t i o n stress is i n - dependent of temperature. Therefore the in t er s t i t i a l s would contribute to Q"T in equations ( l ) and (2) . - k - Other factors contributing to C T \ are long range elast ic stress f ie lds which are created .by the dislocation networks. CT* . therefore, w i l l depend on temperature only through the e last ic modulus. A recent publication by Conrad and Schoeck^ gives evidence that the mechanisms for yielding and flow might be the same. A series of tests was performed on polycrystall ine iron from which the lower y i e ld stress and - l /2 the flow stress at 5$ s train were plotted against d~ ' . This resulted i n similar values for k in the two plots. It was also observed that k was independent of temperature over a range from 100°K to 300°K. Data from Petch^ also indicates a temperature independence of k^y from 19^ K to 300°K. Conrad and Schoeck interpreted the s imilari ty between k and kp, , and the J_iX identical temperature dependences of C T ^ , O^LY an<^" ®~ 1' a s indicating that the same dislocation mechanism is involved in yie lding as in flow. The temperature independence of k^y has evbked considerable inter- 7 est. Cot tre l l has explained the significance of k^y in the following way. When a dislocation source is unpinned, i t releases an avalanche of dis loca- tions into the grain and these dislocations pi le up at a grain boundary. Their stress concentration acts on sources in the next grain and the process is repeated. According to C o t t r e l l , the sources are pinned by i n t e r s t i t i a l atpms. The stress concentration due to a pi le up on a source a distance 1 ahead of the p i le up i s given by (o-LY- (rt) (hf « - 5 - YIELDED GRAIN HMYIEI DFH ORAIN GRAIN BDY. Figure 2. I l lustrat ion of grain-to-grain yie lding Yielding w i l l occur when the stress concentration at the next source is the unlocking stress, that is 1/2 (la) ( &x \ ^) + ^ = ^ (5) and since 2 l / d <̂  1, •1/2 :.>i/2 (6) According to equation (6) and the Cottre l l -Bi lby theory, the unlocking stress, should decrease rapidly with increasing temperature. This would require that k^y also decreases with temperature which i s not generally observed.. The quantity 0~ i s known as the thermal component of the flow stress. It arises from short range forces on moving dislocations which can be overcome by thermal activation. - 6 - The use of rate theory as a means of studying thermally activated flow became firmly established after the publication of Basinski's classic g paper ,in 1959- His analysis was. developed for f. c. c. metals but i t is also adaptable to b. c. c. metals. Various refinements and modifications have since been made to •9 • the rate analysis by such workers as Conrad and Wiedersich, Mordike and Haasen^ and D. P. Gregory"*"̂ ". Let us assume that a dislocation i s held up by an obstacle, the nature of which i s s t i l l not defined. This w i l l create an energy barrier which we can depict by the force-distance curve in Figure 3- To surmount this barrier a dislocation must acquire an energy H over a length L . If the distance travelled by the dislocation during activation is d, the activation volume i s defined as v* = b L d (7) where b is theStoEgers'vector. If the dislocation cannot overcome the barrier with the aid of thermal energy alone i t . i s assumed that H can be reduced with the aid of an applied stress, rCQ, by an amount v* TT to a value. AQ, given by A Q = H - v * T (8) a - Assuming the deformation process i s thermally activated one obtains the following Arrhenius type equation for the strain rate, £ : A e ^ (9) - 7 - where A = N>»>\ (lO) where N i s the number of activation sites per unit volume, is the frequency with which the dislocations "try" the barr ier , ^ is the average strain contributed by a dislocation when i t overcomes the barr ier . £ _ i : ^ DISTANCE Figure 3'.' Force-distance curve for thermally activated flow - 8 - Gregory remarks that the internal stress f i e l d due to a randomly oriented dislocation network should average to zero over long distances but should give, r ise to a loca l stress within the dimensions of the network i t s e l f . Therefore, the energy, H, should be made up of two parts given by H = H* + v* f (11) where H* is defined in figure (3) and v* ^ is the energy required to overcome the internal stress f i e l d over a .distance . Substituting equation ( l l ) into equation (8) one obtains A Q = H* - v* - X±) ='H* - v* <\* (12) Here * is the effective stress operating against the energy barr ier . It would be desirable to calculate H* directly, for this would y i e ld a good approximate value for the activation energy H°. However, there is no direct method of doing this because i t requires an evaluation of ^ . and therefore, one calculates H using a a n < l estimates > t . by methods of extrapolation. If the temperature dependence of the shear modulus is neglected and i f A is not a significant function of stress or temperature then the 11*9 following equations can be derived : " - - ( ^ H T ( 1 3 ) - 9 - M = k f l >̂ T IE (15) T These equations along with equation (9) provide the means of analysis. Several workers during the last two or three years have attempted to establish a mechanism for thermally activated flow by comparing measured values of v*, H°, Z^Q and A with theoretically calculated values. 12 Basinski and Christian observed that the change in flow stress for a change in strain rate decreased with increasing deformation for iron. They concluded that the Peierls-Nabarro force is the dominant factor after ruling out an i n t e r s t i t i a l mechanism and a forest intersection mechanism. However, they did- not consider any other possible mechanisms. 1^ ,15,16,19 • '. Conrad favours a Peierls mechanism also. He selects this one after rul ing out a l l others on the basis of disagreement between theory and experiment on values of H° and v*. Mordike and Haasen"^ favour an impurity mechanism for iron in which f inely divided precipitates are responsible for the thermal component of the flow stress. They explain an increase in the strain rate sensit ivi ty of the flow stress with strain in terms of a divis ion of precipitates during straining due to intersection with dislocations. This results in a greater density of obstacles and hence a smaller activation volume. - 1 0 - In another publication, Mordike, referring to his work on tantalum, was unable to determine any definite mechanism although he suggests a mech- anism involving the conservative motion of jogs. However, this mechanism, 20 which was f i r s t suggested by Hirsch ' for f . c c . metals, involves the con- s tr ic t ion of extended jogs which are not generally observed in b . c . c . metals. Gregory et al"^ favour a mechanism involving the non-conservative motion of jogs for columbium on the grounds that their measured activation energy was too high to be compatible with any other mechanism. A summary of the previous work is given in Table I . TABLE I Author Material H° (ev) L (cm) Mechanism Ref. Honrad Fe .5 - . 6 30 b . .. Peierls Ik Basinski + Christian Fe Did not Calculate Did not Calculate Peierls 12 Conrad + Frederick Fe Did not Calculate : .12 b' Peierls 19 Vfordike + Baasen Fe • 5 17 b .Impurity Mechanism 10 Mordike ' Ta 55". b Conservative jog mech. (not definite) 13 Gregory et a l Cb 2 50 b Non conservative jog. 11 - 11 - . Ii:. EXPERIMENTAL A. Material The vanadium for this project was supplied by Union Carbide Canada Limited, of Toronto, Ontario. It was in the form of cylindrical polycrystalline rods 0.250 inches in diameter. The results of an analysis for i n t e r s t i t i a l gases and carbon are given in Table II. TABLE II Analysis of Material Impurity "As Received" (p.p.m) Zone Refined (p.p.m.) Oxygen 680 160 Carbon 31^ 136 Nitrogen 259 318 Eydrogen 8.9 7 .2 - 12 - B. Zone Refining Single crystals were grown in the electron beam floating zone refiner described by Snowball"^. "As-received" rods were used in nine inch lengths. A portion of oneoehd,roughly 0.75 inches in length,was machined down to a diameter of 0.125 inches to f i t the lower grip on the zone refiner. The other end was faced off squarely in a lathe. A seed crystal, about one inch in length, was prepared in a similar fashion and f i t t e d to the upper grip of the zone refiner such that there was a gap of approximately l / l 6 " between the seed and the rod. The filament was lined up so that i t was just above the gap and the power was turned on. It was found necessary to use a current of 130 ma. and a poten- t i a l difference of 1.6 kv. to maintain a stable molten zone. Minor ad- justments were necessary during the pass due to fluctuations i n the power supply, gas bursts from the rod and other minor effects. Two passes were given to each rod. The f i r s t pass was done at 25 cm. per hour and was primarily intended to outgas the specimen. The second pass was intended to purify the rod and was carried out at 10 cm per hour. The vacuum maintained was generally better than 10"^ mm of Hg. An analysis for i n t e r s t i t i a l s in zone refined material i s given in Table II. - 13 - C. Specimen Preparation 1. Machining Polycrystall ine and single crystal specimens were both prepared for testing by the same procedure. The polycrystall ine specimens were manufactured from"as-received"rod. A 1 1 /V ' length was cut from a rod and machined down to a diam- eter of from .110" to .130" with a gauge length of about one inch. The i n i t i a l cuts were .003" deep. At a diameter of about . 150" the depth was reduced after every four cuts to . 0 0 2 " , .001" and .0005". The specimen was then hand polished in the lathe with 0 and 000 emery paper. 2 . Electropolishing To remove any surface deformation, a l l specimens were electro- polished in a solution of the following composition: 320 cc methyl alcohol, 80 cc cone, sulfuric acid* The polishing was carried out at room temperature with a poten- t i a l difference of about 10 volts and a current of 1.5 amperes. In a l l cases a minimum of .005" was removed from the surface of the specimen. This required a polishing time of from fifteen to twenty minutes. 3- Grain Size Determination and Control Grain size was controlled by heat treatment in the vacuum an- 18 nealing furnace described by Fraser . The vacuum was generally kept well below 10~5 mm of Hg. A table of temperatures, times and resultant grain sizes is given in Appendix I . - ih - Samples were cut from the annealed specimens, mounted and pol- ished for metallographic examination. The polished specimens were etched in a solution of 15 cc. cone, la c t i c acid, 15 cc. cone, n i t r i c acid, 2-3 cc. cone, hydrofluoric acid,: and photomicrographs were taken. The mean grain diameter was determined by the intercept method. Six lines were drawn across the photograph as in Figure k. The number of grain boundaries cutting each line was counted and, knowing the magnification, the mean grain diameter was calculated. D. Tensile Testing Specimens were tested on.an Instron tensile testing machine. The mounting device consisted of the set of universal grip holders described 17 by Snowball and a set of grips designed specifically for the type,of specimen used. A diagram of the grips i s shown in Figure 5. Three types of tensile tests were performed: (1) Continuous tests on polycrystals. (2) Strain rate change tests on single crystals and polycrystals. (3) Temperature change tests on single crystals and polycrystals. In the continuous tests the specimens were strained continuously to fracture at a constant rate of crosshead travel of .02 inches per :• minute. Figure k. Method of grain size measurement, specimen 9B, 210X. - 16 - Figure 5. Drawing of grips used in tensile testing. In the strain rate change tests the specimens were strained past i n i t i a l yielding u n t i l the specimen began to work harden. Then the cross- head was stopped and the load relaxed, usually to about eighty percent of the flow stress. The crosshead speed was increased by a factor of ten or one hundred and the straining was resumed for another 0.5$ strain. Then the crosshead was stopped, the load was relaxed and the crosshead was restarted at the i n i t i a l strain rate. This cycling process was continued u n t i l the specimen failed. These experiments were carried out over a temperature range of -72°C to 100°C. - 17 - The strain rates employed here were mainly those corresponding to a crosshead speed of .01 inches per minute for the basic rate with increases of 10 or 100 times. The temperature change tests were similar except that 1 the temp- erature was cycled at a constant strain rate. This involved straining past the y i e ld point up to a given strain at one temperature, stopping the cross- head, relaxing the load, changing the temperature bath and reloading at the same strain rate after allowing time for thermal equilibrium to be reached. This cycling process was,: continued u n t i l the specimen fa i l ed . E . Temperature Control Three principal temperature baths were used. A boi l ing water bath was used to maintain a temperature of 100 C. A photograph of this assembly is shown in Figure 6 . Figure 6 . Assembly for testing at 100 C. - .18 An ice water bath was used to obtain a temperature of 0°C and a mixture of acetone and sol id CCvj provided a temperature bath of -72°C. A salt bath of calcium chloride and ice water was used for two tests on single crystals but i t was inconvenient to use and was not em- ployed in further tests. The temperatures of the water baths were measured with a mercury in-glass thermometer. The temperatures of the low temperature baths were measured with a copper^constantan thermocouple.. - 19 - III. EXPERIMENTAL RESULTS A. Polycrystalline Material 1. Grain Growth Using the annealing procedures outlined in the introduction, a set of specimens was obtained with mean grain diameters ranging.from 15 microns to 90 microns. I n i t i a l l y , attempts were made to obtain polycrystals of higher purity by zone refining the "as-received" material. This process yielded a single crystal and attempts were made to make polycrystals of controlled grain size by wire-drawing and annealing processes. Two zone refined bars were machined to a constant diameter and drawn down to 50$ and 30$ of the original cross sectional areas.. After annealing,the grain size was small at the periphery and large near the center of the rod, owing to the inhomogeneity of.deformation by drawing. A typical structure i s shown in Figure ( 7 ) . Because of the non-uniformity, this procedure was abandoned and specimens were made from."as received" material. A Laue back-reflection photograph did not reveal any preferred orientation. - 20 - H H H B H H H H B H B I Figure 7« Inhomogeneous grain sizes after drawing, rod #2, 110X. 2. Hall-Petch Equation A typica l load-elongation curve for a continuous test on a polycrystal is shown in Figure (8). This diagram also i l lustrates the method of obtaining the parameters Q£ and. fffv • XI ... ̂  J_)JL •1/2 Figure (9a) shows the behavior of G~ T V and (y~ with respect J-iJL 1 1 to d By comparing the two plots i t i s d i f f i c u l t to see the re lat ion- ship between Of»i a n d ^ L Y ^ e c a u s e °^ ^ke scatter. It is more enlighten- ing to study Ofi - OIy = «To • (To ) + ( k f i • k L Y ) d •1/2 (16) Figure 8. Typical load-elongation curve for vanadium polycrystal.  - 23 -• It i s evident that i f k^y - k ^ then O f i " 0~Ly will'."be a constant, i n - dependent of grain size. From Figure (9b) i t is evident that k ^ y ^ kf]_- ( V , - 0*TV- decreases with decreasing grain size. This method of plotting the differences i s the most accurate one because the maximum error in these - 1 / 2 results l i e s in the measurement of d . A comparison of the difference between the yie l d and flow stresses for a given grain size i s independent of the measurement of the grain size. It would have been desirable to investigate.this property further but this would have required testing a large number of specimens to fracture at a constant strain rate. Because of the high cost of vanadium, i t was decided to perform only tests which gave a maximum amount of information from each specimen. A l l subsequent tests pn polycrystals were of the strain rate change or temperature change type. This made i t impossible to obtain con- sistent flow stress values at 5$ strain because the deformation history varied from specimen to specimen. The lower y i e l d stress was measured during a l l of these tests, however, and is plotted in Figure (10) . There i s a trend for the slopes of the plots in Figure (lO) to increase with.temperature. One might question this trend on the grounds of the uncertainty of grain size measurement. However, the specimens of various grain sizes were made in batches and a specimen from each batch was tested at each temperature. The grain size was measured and a single value was allotted to each batch. Therefore, i f there was an error i n the Figure 10. Dependence of lower yi e l d stress on grain size.  - 26 - grain size measurement i t would be the same for a l l the specimens of the given batch, and the relat ive slopes of the curves would not be affected. 3- Strain Rate Change Tests A typical stress-strain curve involving strain rate changes is shown in Figure ( l l ) . The stress leve l at the i n i t i a t i o n of plast ic flow was found by extrapolating the two straight portions of the curve in Figure (12) and calculating the strain:.rate sensit ivity A CT" from the equation ACT = cr (B) - cr (A). (17) S T R A I N £ Figure 12. Method of extrapolation to obtain change in flow stress - 27 - the change in the resolved shear stress, was calculated assuming a.Schmid factor of 0 . 5 . was then plotted as a function a of ^ and £ in Figures ( 1 3 ) , (l*0 and ( 1 5 ) ' The general trend in these plots is for *d ^ to decrease s l ight ly with deformation in disagree- a ment with the Cottrell-Stokes Law. It is evident that there is no measureable dependence of ^ ^ a on.grain size over the range studied. This is in agreement with the work of Conrad and Schoeck on polycrystall ine iron. Temperature, in the range studied, has no measureable effect on the slopes of the plots in Figures ( l3)> (l*0 and ( 1 5 ) ' However, one test at room temperature yielded a zero slope which suggests that the scatter in these types of measurements is sufficient to mask a small temperature dependence of the slope. k.. Temperature Change Tests A series of temperature change tests was performed on the poly- crystal l ine material between 100°C and 8°C. The reason for using 8°C was one of convenience. It was found that upon removal of the bo i l ing water bath and switching.to an ice water bath.the assembly and bath came to temporary thermal equilibrium at about 8°C. The var iab i l i ty of this pro- o cess was rarely more than + 1 C and i t was easier to adjust the bath temp- erature to 8°C than to cool the system to 0°C. The specimen was allowed at least ten minutes to attain the. bath temperature at 8°C and at least five minutes at 100°C. The shorter time was considered sufficient for the high temperature because the heating .45- 40- . 35 - 5 .251- * ^.15 ^ 3,20 4 .35- .30- .25- .20- J 5 _ O • 3.40 _L 3.60 3.80 4.00 4.20 (PS./. * /0*) 4.40 7 &^=/00 P—o—o T= -72°C O IIC • /2A O I2f • I3B • 13 £ T= 0°C O I IA • / I E O 12 C • 13 A • I3D 2.40 2.60 2 .80 3-60 3.00 3.2 0 3.40 r* CPS./. x/o*) Figure 13« / ^ ^ a for an increase in £ vs. a for polycrystals. ro 00 .25 o O; .20 .15 ± T = o°c o If A • (If o /2C /3A = /0 ,02 .04 06 .08 STRAIN .10 .12 J 4 ./6 4 .05 0 • o o o o J- 1.60 ISO Figure lh. 2.00 2.20 2.40 260 (PS.I*I0V 280 3.00 A ^F- for an increase in £ vs. ^ and £. for poly cry stals. ro vo T = -72°C O IIC • 12A o 12 E • I3B 10 o 7 = /oo°c O I/O • 12 b o 12 F • I3C • I5F 10 0 .02 Figure 15. .04 .06 .10 ,/2 .08 STftAIN S for an increase in strain rate vs £. for polycrystals. ./4 ./6 - 31 - process was slow. This made i t unlikely, that the specimen was ever more than a few degrees below the bath temperature. Plots of ZL\ against / f „• and £ are shown i n Figure (l6). a a The trend here is for A t „ to decrease with increasing stress and strain. It i s note able that the magnitude of A t i s roughly twice that for a 61 o strain rate change of a.factor of ten at 0 C. Therefore, there would be a greater likelihood of detecting a grain size sensitivity of A t l n these tests. However, such a sensitivity i s not observed. 5. "Pseudo" Yield Points Under certain conditions y i e l d points were observed upon reload- ing the specimen after a strain rate or temperature change. A typical stress-strain curve showing this property is included in Figure (l7)- These yie l d points were observed;.under the following conditions: i (1) i n strain rate change tests at 100°C, (2) In temperature change tests involving 100°C. In the temperature change tests the yi e l d drop was observed on both the low temperature and the high temperature.portions of the stress- strain curve, but the y i e l d drops were slightly larger on the former portion. The magnitude of the yi e l d drops was roughly the same for both the upper and lower strain rates i n the rate change tests. The size of these y i e l d drops was in a l l cases independent of strain. Because of the temperature dependence of the y i e l d drops i t is unlikely that "tĥ y could be due to any characteristic of the testing apparatus. ° • .3* o o V30L CO .50- .02 .04 I : _ L _ .06 08 STRAIN E 10 .12. 14 T = 8°C~~I00 C O IIB • M F O I2D 40[ .301 _L 1.60 1.80 2.00 2.80 2.20 2 4 0 2.60 (P.S.I. x | 0 4 ) Figure 16 . A t ^ o r a decrease in temperature vs. 6- and for polycrystals 3.00 IZ A - 3^ - The occurrence of these y i e l d points created the problem of how to measure the flow stress upon reloading. For reasons discussed later the method of extrapolation illustrated in Figure (l8) was used. Figure l8. Method of extrapolation through yi e l d points. 6. Rate Equation The activation volume was calculated for each specimen and plotted as a function of stress. To do this, the strain rate sensitivity was found from the points in Figures (13) and (lk-) and used in equation (13)- - 35 - Results of these calculations are plotted in Figure (19). There appears to he a slight increase i n v* with stress at 0°C and -72°C. This increase is small, however, and therefore an average value of v* was taken and plotted against temperature. This i s shown in Figure (20 ) . The energy, H, was calculated using data from the temperature change tests and equation (l k ) . The value of v* used in this equation was that for the mean temperature of the test, 327°K. Since no strain rate change test was doner at 327°K, v* was found by interpolation on Figure ( 20 ) . H is plotted as a function of strain i n Figure ( 21 ) . It i s seen that H increases somewhat with strain. Using equation (15), the thermally activated component of the activation energy, A Q, was calculated for a temperature of 327°K and at 10$ strain. The value obtained was .60 e.v. A Q was also calculated from equation (8) and plotted as a function of strain i n Figure (22). The value f o r - t . .in this equation was taken as the average of the applied stresses at the two testing temperatures. From Figure (22<), A Q at 10$ strain i s 0.72 e.v. This i s somewhat different from the value obtained in equation (15) and serves as a rough measure of the accuracy of this type of analysis. The factor. A i n the rate equation was determined.from equation (9) and is plotted in Figure (23) as a function of strain. o A rough estimate of the activation energy, H , may be made with the aid of Figure (2*4-8). When the strain rate sensitivity i s equal to zero / 4 0 * 120 loa- 80 60f- 40 20 100°c- I /oo° c o • SINGLE C f t V V T ^ o •7Z°C -72 C JL /.o 2.0 3.0 (PS.I. v IO 4 ) 40 OA Figure 19• Activation volume vs. flow stress Figure 20. Activation volume vs. temperature. 2.5 2.0 POLYCR Y.ST A LS 32 7° K SINGLE CRYSTALS > 15 .0 287°K 237°K 356°K 32 7°K 0.5 L o 1 0 .02 £)4 .06 .08 STRAW JO ,12 .14 Figure 21. Energy, H, vs. £  Figure 23» Pre-exponential factor A vs. £. for polycrystals. 5.0 100 200 300 400 T (°K) Figure 2ha.. Strain rate sensitivity of /S.'X? vs. temperature. - k2 - the deformation process w i l l be completely thermally activated and A Q = H° Thus i f Figure (2k) is extrapolated to the temperature, T , at which T ~2Tir\ir~ = °> "t^sn H can be calculated from equation (9). H° = kT c m(A/^) (18) Taking a value for ln(A/g) of 25 from Figure (23) and a value for T Q of klO°K from Figure (2^),H° is found to be approximately 0.9 e.v. Since H = 2 .3 e.v. at 10$ strain, this suggests that about l.k e.v goes into overcoming the internal stress t ^ . B. Single Crystals 1. ' Zone Refining Laue back reflection photographs were taken of each single crystal specimen after machining and electropolishing to determine the orientation and to ensure that any surface deformation caused by machining was removed. Figures (25a) and (25b) show Laue spots for a sufficiently polished and an Insufficiently polished specimen. It was necessary to remove about .005" to 'obtain sharply defined spots. The orientation of each specimen was plotted and a l l specimens were found to be of the same orientation to within 2 ° . The position of the tensile axis with.respect to the standard st.ereographic triangle i s shown in Figure ( 2 6 ) . Figure 25b. Laue photograph of an insuff ic iently polished specimen. - kk - The single crystal rods were a l l radiographed and found to contain a number of small gas holes. These were also observed while the specimens were being polished during which the holes would appear as small surface pits. A photograph of a pitted specimen i s shown i n Figure (27) - 2 . Slip Systems Metallographic examination of the.deformed single crystals revealed that at least two s l i p systems were operative during deformation. Near the necked region evidence of three or more systems was frequently observed. Typical s l i p traces are shown.in Figure ( 28 ) . - 45 - Figure 27. Gas holes in single crystal specimens, Figure 28. Typical s l ip traces on single crysta l , specimen 6A, 210X. The s l ip direction was found to be <Clll> . Due to the complex- i ty of separating the s l ip markings of the systems involved, an attempt to determine the s l ip planes was abandoned. It is already known that the s l ip planes in a b. c. c. metal are {lio} , {ll2} and [123} • Using the known orientation and a stereographic projection, the angles between the tensile axis and a l l of the possible s l ip planes were measured. From these angles Schmid factors were calculated and the largest of these are tabulated in Table III . - 46 - TABLE III Possible Slip Systems Slip Direction Slip Plane Schmid Factor •{ml (101) .468 [In] (101) .482 [111] (112) .468 [In] ( l l 2 ) .456 [ i n ] (213) .483 [In] (213) • 500 The Schmid factors are roughly the same for each type of s l i p plane. Because of this, an average factor of 0.48 was used to calculate the resolved shear stress in the single crystals. 3. Strain Rate Change Tests These tests were carried out using the same strain rates as for the polycrystals. The results are plotted in Figures (29) and ( 30 ) . Z \ / T j a . i s essentially independent of strain and stress at 0°C o 0 and 100 C. At -72 C A 'r appears to decrease with deformation. The u a increase of A °£ with a and £ shown by specimen 5A.is too small to be regarded as definite. .14 . / 3 l /.6 0 .10 2 .05 « 6 0 <j ./5L ./o o O /.80 2 0 0 -o *- o .80 /.oo 66, -/9°C 5A, 2 5 ° C T = O O. -72°C S B 7£ 2.20 T= IOO°C • 6 A O SD I.2Q 7A . 0°C -o- o •<D- .051 .90 J. 10 -50 Figure 29 . 1-30 or* (PS./. x 10*) /\^T ,a for an increase in £ vs. t a for single crystals. ,25L .20L x v. cc .10 I9°C 0°C =fl3= _0_ o • • • -G G- O SB O 7E <D 7A • 6 B © 5A • 6A e 5D 0 25°C o- -05L /oo°c 0 .02 .04 .06 .08 S T R A I N E .12 .14 Figure 30 • A^Co for an increase i n £ vs. £.for single crystals. 3. J6 4=- cx> It i s evident that the results for single crystals are not as reproduceable as those for polycrystals. The.reason for this may he the gas holes in the single crystal specimens. o Due to the limited du c t i l i t y of vanadium below 200 K most of the tests were performed above this temperature. However, one crystal was tested at 78°K to determine the strain rate sensitivity of the flow stress at, this temperature. It was possible to cycle the strain rate only twice before the. .specimen, f a i l e d so determination of the; dependence of • /\ * t on and f was not possible... . The specimen showed evidence of twinning at this temperature. Sudden reductions of stress occurred in the elastic region accompanied by a sharp cracking sound. Metallographic examination revealed twin like markings on the surface of the specimen. (Figure (3l))- Also a Laue back reflection photograph of the supposed twinned region was taken using a beam in the form of a long narrow s l i t (Figure 32 ) . Each spot on the film i s s l i t in an identical manner. Small spots corresponding to the gaps were not definitely observed but they could have been masked by the background radiation. The two s l i t s seen in.the photograph correspond to two bands of narrow twins with a spacing of about 2 mm. between them. 17 It should be noted that Snowball i n his work on vanadium, did not observe twinning down to 78°K« However, his specimens v/eosEof a differ- ent orientation, which could explain this disagreement. - 50 - - 51 - h. Temperature Change Tests o The temperature change tests were done mainly between 100 C and some lower temperature although one test was performed between 0°C and -7? C. The results of these tests are plotted in Figures (33) and (34). In a l l cases, except for specimen 6D, / \ ^ decreases slightly . . . a with deformation. Since only four points could be obtained for 6D, i t can- not be definitely concluded that A ^ increases with r C Q .for this specimen. 5. Rate Equation • The activation volume for each specimen which was subjected to a strain rate change test was calculated and plotted as a function of stress and temperature in Figures ('19) and (20). As for the polycrystals, an average value for v* was used in the plot of v* against temperature. The activation volume i s generally larger for the single crystals than for the polycrystals at any given temperature. This difference de- creases as the temperature increases to 100°C where v* i s the same for both types of specimen. The energy, H, was calculated from equation (l4) and is plotted as a function of strain i n Figure (2l). A Q at 10$ strain was calculated from equation (15) for the various mean testing temperatures and i s plotted as a function of temper- ature in Figure (24b) assuming = 0 at 0°K. A Q was also calculated from equation (8) and is plotted as a function of strain i n Figure (22). A Q again shows a small tendency to decrease with strain although.this decrease i s not as marked as for the polycrystals. 60 fOO°C — -26°C -50 .401 .60 7 0 80 . 9 0 1.00 110 2.901 6C /00°C >-660C a; - -70[__ & 60 40k- TO B O .30 .301 -©- 5C \00°C~B°C -251 .60 1 -70 Figure 33- .80 .90 |.00 1.10 A f for a decrease i n temperature vs /tT a for single crystals. ro  - 54 - An average value for A at.10$ strain was calculated by plotting against =p | in Figure '(-35) • This yielded a value for ln^A/g) of 25 .9 which i s quite comparable to that for the polycrystalline material. . 6. Yield Points Yield points similar to those observed during the tests on poly- crystals also appeared in.the single crystal tests. They appeared under the same temperature conditions as for the polycrystals but.in this case they did not show up u n t i l after about 5$ strain. They increased in size somewhat with increasing strain but they were always smaller than those observed during the polycrystal tests. One test was performed by straining continuously to 20$ strain at the basic strain rate and then the rate was increased by a factor of ten; This was done to determine whether the size of the yi e l d drop depended on the amount of loading and unloading of the specimen/. A size- able y i e l d point was obtained which compared quite favourably with those at the same amount of strain in a normal test. This is shown.in Figure ( 1 7 ) . A summary of the results i s included in Table IV. - 55 - Figure 35» Strain rate sensitivity of the flow stress vs. temperature sensitivity to find ln(A/g) . (equation (15))• - 56 - C. Summary of Results 1. The lover y i e l d stress and the flow stress of vanadium both obey Hall- Petch equations. -1/2 2 . The slopes of the yie l d stress and flow stress versus d •' plots are not identical for vanadium. 3. The slope, k^y* increases with temperature. . k. Vanadium does not obey the Cottrell-Stokes Law. A, f i s constant or decreases slightly with ' t a » 5' Asrt'a is Independent of grain size over the range tested. 6 . The activation volume for single crystal specimens diverges from that for polycrystals at low temperatures. 7. The activation energy, H°, is roughly 1 e. v. for thermally activated flow in vanadium. 8. Yield points are observed upon reloading a specimen in tests involving a . temperature of 100°C. 9 . Vanadium deforms by twinning at -196°C when the orientation i s as in Figure ( 2 6 ) . - 57 - TABLE IV Summary of Results Specimen T H(e.v.),£ =0.1 Q(ew), £ = 0 . 1 H^e.v.^ 287 1-7 .76 Single Crystals 327 1-5 .81 310 1.5 •75 0.9 25.9 ' 356 • 1.1*. • 76 237 1.3 . .hi Poly- crystals 327 2 .3 • •72 0.9 25.2 - 58 - IV. DISCUSSION A. Hall-Fetch. Equation From the preceding results come two interesting observations: 1. k^y increases with temperature. 2. k^Y i s not equal to k^. Let us f i r s t nook at the temperature dependence of k^y On the basis of Cottrell's'theory, k^y should depend on the stress to unlock d i s - location sources pinned by impurity atoms. Hence, one expects that k LY should decrease with.increasing temperature. However, just the opposite i s observed. This suggests that the C o t t r e l l theory, i s an over-simplification. 5,6 These observations d i f f e r from those of other workers i n that 5 t h e i r k values were, independent of temperature. Conrad points out that a temperature independent k could be explained i n terms of an athermal unlocking process. If one attempts to use t h i s idea to explain a .k which increases with temperature one must have an 1 (the distance from the p i l e up to the nearest source i n the next grain) which increases with temperature. Such a si t u a t i o n •is d i f f i c u l t to understand. Conrad's suggestion that .the d i f f e r i n g heat treatments given to the specimens may be responsible for the behavior.of k ,. i s worthy of , LY discussion. . I f the sources are pinned dislocations,then those specimens annealed at a higher temperature (larger grains) might be expected to have fewer sources. This would result i n a larger value for 1 (equation (6)) - 59 - than i n fine grained specimens. However, for any given grain size 1 should be the same and while d i f f e r i n g heat treatments may affect the actual values f o r k^y* i t . i s hard to see how i t could affect the r e l a t i v e values, over a range of temperature. The difference between k-̂ y and k ^ i s not surprising. Conrad assumes .that because k ^ = k^y i n his measurements the same deformation mechanism i s involved i n y i e l d i n g as i n flow. His k was measured at a f l s t r a i n of .05 as i t was i n t h i s work. In a po l y c r y s t a l l i n e specimen considerable work hardening occurs near the grain boundaries. Hence, one might expect that the flow stress at give percent s t r a i n would depend on the grain size through the rate of work hardening i n the grains and t h i s i s not necessarily related to the lower y i e l d stress. An equivalence between y i e l d i n g and flow mechanisms would then suggest an equivalence between the act i v a t i o n and stopping of sources and there i s no- sound reason to make t h i s assumption. The results of t h i s work are i n disagreement with the work of Conrad and i t appears as i f his conclusions were premature. No alternative explanation of these processes can be given here because of the l i m i t e d scope of t h i s portion of the research. A large scale research programme i s required involving several d i f f e r e n t b. c. c. metals. B. . Thermally Activated Flow At present there exist the following suggested mechanisms for c o n t r o l l i n g the thermally activated flow of dislocations i n body-centered cubic metals: - 60 - •1. Intersection with a.forest 2. Impurity atoms interacting with dislocations 3. Cross sl ipping of screw dislocations k. Large Peierls stress 5. Non-conservative motion of jogs in screw dislocations 1. Intersection with a Forest The forest mechanism, although quite acceptable for f. c. c. metals, i s probably the,least acceptable for b . c. c. metals. At the beginning of deformation i t . i s unlikely that the forest Q density would be greater than 10 dislocations per square centimeter. . I f a .uniform distr ibution of forest dislocations is assumed this would lead to a spacing of 10 ^ cm. between the "trees" of the forest. A typical -21 3 value for the activation volume of vanadium is 10 cm . Now assume that d, the activation distance, . i s of the same order of magnitude as b, the Burger's vector. This is not unreasonable as any thermally activated process must involve short range forces. The distance between obstacles v* -6 i s found to be L = — ^ 1.5 x 10 cm. This is a smaller spacing than b^ one would expect for a .uniformly distributed forest. 21 It has been observed that . in f. c. c. metals the dislocations are .distributed non-uniformly,.that i s , they exist as tangles alternating with re lat ive ly dislocation free areas. I t . i s pointed out that the flow stress is determined by the spacings in these tangles. These tangles could have a smaller spacing between the "trees" of the forest and there- fore be consistent with the measurements in this research. However, this - 6 i - type of dislocation structure is not generally observed in b. c. c. metals, in fact, Gregory's electron micrographs show a relatively uniform d i s t r i - 22 bution. Also, i t has been pointed out by Wilsdorf that the spreading of these tangled regions into dislocation free regions corresponds to stage I or easy glide in f. c. c. metals, and at the beginning of stage II the distribution i s relatively uniform. A l l of the measurements in this work were taken in the later stages of deformation after heterogeneous yielding had ceased. The activation energy for a forest mechanism w i l l depend on the short range forces on the moving dislocations. Since dislocations i n b. c. c. metals are generally non-extended these forces would be related to the energy of jogs formed during intersection. This energy is generally 3 11 estimated to be b /lO which, assuming a shear modulus of k.S x 10 :"dynes/cm yields an activation energy of roughly 0.5 e.v. This is somewhat smaller.than the measured values i n this work but i t does not disagree • enough to discount the forest mechanism on this basis alone. Because of the numerous active s l i p systems in b. c. c. metals one would expect the-forest density to increase greatly with deformation, say at least by a factor of one hundred over the complete range of strain. In a uniform forest this would lead to a tenfold decrease in the activation volume with strain and a tenfold, increase in ZX^a.- The results of this research do not show this trend. If the forest i s non-uniform and spreads by the expansion of tangled regions into clear ones, then the activation volume could remain constant i f flow is controlled by the spacing in the tangles. However, i f - 62 - t h i s structure d i d e x i s t the tangled regions would soon cover the whole specimen and one might expect to see a change i n v* i n t h e . l a t t e r stages of deformation. This i s not observed. I t would seem l i k e l y by these arguments that the forest.mechanism does not.control thermally a c t i v a t e d flow i n vanadium. 2. Impurity Atoms The i n t e r a c t i o n of a uniform d i s t r i b u t i o n of i n t e r s t i t i a l impur- i t i e s with moving d i s l o c a t i o n s has been proposed as a possible mechanism responsible f o r the thermal component of the flow s t r e s s . k The work of Heslop and Petch on.iron suggests that the temperature dependent part of 0 ~ f i does not depend on the amount of C + N i n so l u t i o n . 23 Calculations using l i n e a r e l a s t i c i t y theory ind i c a t e that .the i n t e r a c t i o n energy between an i n t e r s t i t i a l carbon atom and a d i s l o c a t i o n i n i r o n i s about 1 e.v. However, i t . i s pointed Out that the use of l i n e a r e l a s t i c i t y theory almost.certainly would y i e l d an over-estimation. This energy i s ca l c u l a t e d from equation (19). //b 1 + >> sinQ A T , , , E = fff ~ A v ^9) Where E i s a maximum f o r 9 = and :p = b, and A"V i s the volume change due to one i n t e r s t i t i a l per un i t c e l l . The shear m o d u l u s , , f o r vanadium i s s l i g h t l y more than one h a l f of that f o r i r o n . Also, since the l a t t i c e parameter i s la r g e r f o r vanadium than f o r i r o n i t i s l i k e l y that A V w i l l also be smaller. Hence, a maximum value f o r E i n vanadium would be roughly ; 0.5 e.v. - 63 - We would expect the activation volume for an i n t e r s t i t i a l mech- anism to decrease slightly with strain. The number of i n t e r s t i t i a l s w i l l not change with strain and therefore the activation length, L, w i l l not change. The activation distance, d, should decrease somewhat since as the flow stress increases with work hardening the dislocation w i l l rise to a higher level on the force-distance curve. The observed activation volume i s relatively constant with strain or in some cases increases slightly in disagreement with an i n t e r s t i t i a l mechanism. can calculate the spacing L in the activation volume. In the polycrys- ta l l i n e material there, is roughly one i n t e r s t i t i a l per 1000 atoms. This is equivalent.to an i n t e r s t i t i a l in every 500 unit cells and an average spacing of 3/500 a ^ 8 a where a is the lattice parameter. Thus we have a spacing between the i n t e r s t i t i a l s of roughly 2.5 x 10 cm. This is too small to agree with the measured value for L in this work. For -7 the single crystals L would be 3 x 10 cm. which.is also too small to agree with this work. It might be argued that the•interstitials may be segregated due to the presence of dislocations and tend to be closer together along dis- location lines before straining. However,, i t . i s l i k e l y that saturation of dislocations would occur, before a large fraction of the i n t e r s t i t i a l s could segregate and therefore the average spacing should not be too far from that for a uniform distribution. If the i n t e r s t i t i a l s are assumed to be uniformly distributed one -7 - 6k - 3- Cross Slipping of Screw. Dislocations Observation of wavy s l ip l ines in b. c. c. metals has been inter- preted as evidence of cross sl ipping of screw\ dislocations. This process 2k has been.described by many authors . It involves a screw . segment of an obstructed dislocation, loop, cross sl ipping to a para l l e l s l ip plane. The screw segment.is locked in the new s l ip plane by an edge dipole which is formed.in the cross s l ip process. Mult ipl icat ion occurs when the locked screw:, segment acts as a Frank-Read source. 25 Johnston and .Gilman have shown that the ease with which such a source operates is inversely proportional to the length of edge dipole or jogs formed.in the cross s l ip process. The stress to operate a source is given by ^ = 8 H ' f i \ p . ) JT . (20) Where d' is the spacing,between the new and old s l ip planes. Assuming Jy = 0 .3 (Poisson's ratio) and taking = k x 10^ ps i . which i s as . -8 high an applied stress as any specimen received one finds d = 30 x 10 cm. or dJ";= l i b . This . i s a minimum value for d J since in most tests the applied stress was less than k x 10^ ps i . This distance is too great to be effectively overcome by thermal fluctuations and the energy of the edge dipoles produced in the process would be prohibit ive. k. Peierls-Nabarro Force The idea of the Peierls-Nabarro force as the mechanism controll ing thermally activated flow is a popular one. However, the choice of this - 65 - 4 mechanism has in most cases been somewhat arbitrary. Heslop and Petch chose i t for lack of a better one rather than as a result of any intensive study. Actually a high Peierls stress might be expected for b . c. c. metals since departure from close packing should favour the narrowing of d is lo- cations . It is d i f f i c u l t to support this mechanism on the basis of numeri- cal results . The Peierls force w i l l be large only for dislocations lying in close packed directions, that i s , <̂ 111̂ > in b. c. c. There is no reason for assuming that most of the dislocations are of this type a l - 26 though electron microscopy studies on s i l i con iron do- support this (Figure (36)). Figure 36. Transmission electron micrograph of dislocations in s i l i con iron, after Low and Turkalow. - 66 - However, this .is a particular case. One must keep in mind the work of Stein and Low 2 ^ who showed that the mobility of edge dislocations .is approximately twenty times that for screw.i dislocations in s i l i c o n iron. Since the s l i p direction i s <̂ 111> one would expect to observe loops with extended screw segments. The deformation takes place mainly by edge dis- locations which are not oriented to experience a large Peierls force. This behavior has not been observed.in any pure b. c. c. metal. The Peierls mechanism is based on a .series of energy maxima and minima parallel to the dislocation line as in Figure (37). PROPAGATION DIRECTION < M I > Figure 37* Formation of a .dislocation kink in a Peierls force f i e l d . - 67 - The Peierls mechanism would operate in the following manner. .In order- that a dislocation propagates, a segment of a . c r i t i c a l length, L c r r must overcome the barrier . . L c r i s determined by the attractive force between two kinks of opposite sign and the applied stress. . If the loop segment is longer than L then the kinks w i l l propagate in opposite direc- cr tions and the dislocations w i l l move forward. Thus L is determined by cr the applied stress • At lower temperatures ^ is larger and therefore L c r can be smaller than at high temperatures. This would be helpful in explaining the dependence of v* on temperature. 2 8 Read has pointed out that the controll ing factor in this pro- cess is the formation of a.kink. The sideways motion of a kink occurs with relative, ease compared with the forward motion of the dislocation against the high.Peierls stress. 2 9 Seeger , has calculated the kink energy, in a publication on an internal f r i c t i o n peak due to kink formation in copper. Because of the many "a pr ior i" assumptions in this calculation a re l iable estimate for vanadium cannot be made. To obtain a value for the kink energy i t would be necessary to do internal f r i c t i o n measurements on vanadium. It may be d i f f i c u l t to observe a peak due to kink formation because of the many other internal f r i c t i o n sources in vanadium. 5• Non-Conservative Motion of Jogs in Screw Dislocation When a screw dislocation acquires a jog, i t . i s of the edge orientation and therefore can glide conservatively only in the direction of i t s Burgers vector which is para l l e l to the screw, dislocation,'/ l ine . - 68 - I t will.therefore cause a drag on the motion of a screw di s l o c a t i o n . This i s also true for jogs i n mixed dislocations, that - i s , any dis l o c a t i o n with a screw component. I f these jogs are to move with the dislocation'they must move non-conservatively, leaving "behind a t r a i l of vacancies or i n t e r s t i t i a l s . The creation of i n t e r s t i t i a l s i s energetically unfavourable. The vacancy mechanism would be governed by the energy of formation of vacancies which i s of the order 2 to 3 e.v. There.is some argument against the vacancy mechanism. I t has been suggested by F r i e d e l that vacancy jogs might move conservatively along the disl o c a t i o n loop u n t i l they can glide conservatively with the loop or they 34 may combine with jogs of opposite sign and annihilate one another. Frank has pointed out that the former p o s s i b i l i t y i s u n l i k e l y as i t requires a stationary shape of the disloc a t i o n loop. This would require a higher di s l o c a t i o n v e l o c i t y i n regions of convex curvature of the dislocation. I t i s more l i k e l y that the jog w i l l lag behind and either form an edge dipole or move i n non-conservative jumps. The annihilation of jogs would r e s u l t . i n an increase i n activation volume with s t r a i n . Also, even i f the jogs do not annihilate each other the spreading of the disloc a t i o n loop; would resu l t i n an increase i n the jog spacing. However, new jogs should be introduced continuously by i n t e r - section with the forest and i t i s conceivable that an equilibrium could be set up such that the spacing of jogs remains roughly constant. • , • ' - 69 - . It is d i f f i c u l t to believe that such an equilibrium .could be set up instantaneously and therefore one might expect to see some evidence of this in the early stages of deformation. In fact,, in some of the tests, the largest scatter in the plots of 2\ ^ a against ^ a w a s observed for the f i r s t point. The main objection to the vacancy mechanism here is that the jmsasured activation energy is small. The energy of formation of vacancies in b. c. c. metals is roughly 2 to 3 e - v - which,is not very compatible ..with the measured value, of H 0 ' * 1 e. v. for vanadium. There is some ques- t ion, however, as to the val id i ty of assuming the energy to form a vacancy at a jog i s the same as that for forming one in a perfect l a t t i c e . It is known for instance that the energy to form a vacancy in an alloy i s much less than that for the pure.metal.. Unfortunately a detailed analysis of this problem would require sophisticated quantum mechanical calculations which are beyond the scope of this work. 6. Comparison Between Polycrystals and Single Crystals The divergence between polycrystals and single crystals in the plots of v* against temperature is d i f f i c u l t to explain. On the basis of an i n t e r s t i t i a l mechanism, the polycrystal curve should l i e below that for the single crystals since the polycrystals have a higher i n t e r s t i t i a l content. This would result in a smaller activation length, L , and hence a.smaller activation volume. However this would not explain the convergence of the two curves with increasing temperature. It is possible that the difference in v* l i es in the activation distance, d. This might be brought about by a difference in the shapes - 70 - of the force-distance curves for the two materials. It i s d i f f i c u l t to explain why this should occur i f the same mechanism is operative i n both types of specimens. On the basis of a vacancy jog mechanism the energy to form a vacancy may depend on the internal stress f i e l d . From the measurements of H° and H i t appears that the internal stress f i e l d i s larger for the polycrystalline specimens. This could be responsible for a difference in the shapes of the force-distance curves. The value of the factor A i n equation (9) is roughly the same for the two materials. The trend for .A to decrease by 10 to 10 with strain might be explained by a decrease in the number of mobile dislocations. One would not expect A.or to decrease by more than a factor of five with strain. A decrease in the number of mobile dislocations means those remaining must move at a higher velocity which would result.in an observed work hardening. 7. Mechanism It i s rather d i f f i c u l t to pick out a single mechanism as the controlling one in thermally activated flow. The factcthat the opinions of so many authors dif f e r i s evidence enough for that. What is generally done is to select a mechanism which fits'the data best and declare this to be the most probable one. On the basis.of the preceding arguments i t i s reasonably safe to rule out the forest,, cross s l i p and i n t e r s t i t i a l mechanisms on the - 71 - grounds of severe disagreement between theory and experiment. The Peierls mechanism can not be compared with, the data because i t i s not possible to o calculate v*, H and A accurately from theory. Certainly the important point here is whether the dislocations do l i e predominantly in the ^111^ directions. There is not sufficient evidence available at this time to answer this question. The jog mechanism is the best one for explaining the increase in activation volume with strain which was observed in some specimens. The jogs are the only obstacles which could spread apart with deformation. An argument against the jog mechanism is that.it cannot operate at low temperatures. This is because the vacancy formed must diffuse away im- mediately or i t will.'.draw the jog ;bac.k again. At low temperatures this diffusion might be d i f f i c u l t . On the basis of preceding arguments i t appears as i f the Peierls mechanism is.the most likely, one. However, this, may be just because there is insufficient evidence available to dispute i t . What one should .inquire at this juncture is just how valid is 'the- application of rate theory to this process. . It has been assumed that there is a single thermally activated mechanism operative. •In fact, i t is possible that the average effects of two or more mechanisms were, measured. Since two mechanisms-;:wauld l i k e l y have different temperature dependences, the predominance of one could.change with respect to the other over a .range of temperatures. - 72 - With the disagreement between theory and experiment in a l l the proposed mechanisms i t . i s conceivable that.more than one mechanism could well be the controlling ones. What, is needed at this time i s a reliable mathematical analysis of the Peierls mechanism so that. i t sr.'.importance in b, c. c. metals can be definitely established. The assumption used.in the derivation of equations 13, lh, and 15, that A i s not significantly dependent on stress, i s worthy of discus- sion. This assumption means that the number of mobile dislocations does 30 not change greatly during a strain rate change. Martinson contends that,in LiF the number of mobile dislocations i s a sensitive function of the mean stress prevailing during a stress increase. He used this hypothesis to explain his i r r e v e r s i b i l i t y of /^H^. He found that A ^ o - f o r a strain rate increase was not equal to that for a decrease and also that upon unloading and reloading at the same strain rate the flow stress showed a change. . This type of i r r e v e r s i b i l i t y was not observed in this work. • The flow stress was completely reversible to within the limits of exper- imental accuracy. Also, the strain rate change tests of 100 times yielded a A ^ * . approximately twice as large as that for a factor of 10 change from the same basic strain rate. If the number of mobile dislocations were to increase with the mean stress prevailing then we would expect /^^«.for a change of 100 times to be less than twice that for the factor of 10 change. C. Yield Points The y i e l d points obtained upon reloading after cycling have been observed by many' experimenters but have not been satisfactorily explained by any of them. It is important to understand this process however, in order to justify the method of extrapolation used to determine the flow stress in a strain rate or temperature change test. Some authors have attempted to explain this effect as being a type of relaxation phenomenon. Upon unloading the specimen or even just stopping.the crosshead, the dislocations relax into a configuration of lower energy.- Hence, energy must be supplied to bring the dislocations from their relaxed configuration back to the pattern which they were in just before the crosshead was stopped. This hypothesis i s strengthened by the ir r e v e r s i b i l i t y of plastic flow. When a stress is removed from a crystal ' one might expect the dislocations i n pile ups to run back along their slip planes causing large reverse plastic flow. . In fact, this does not .occur and therefore there must be some obstacles to this;;process. Makin^Jias described a process for f. c. c. crystals in which Cottrell-Lomer sessile dislocations are formed on unloading and these prevent large scale reverse plasticity.. Since energy is released upon •formation of these sessiles a higher stress than the flow stress i s required to dissociate them. This would result .in an observed y i e l d drop. Makin' ' observed y i e l d drops at temperatures ranging from - 195°C to 100°C. The magnitude of the drops appeared to be independent of temp- erature and was proportional to the reduction in stress on unloading. The yiel d drops in this thesis however were observed only in tests involving - 7^ - the temperature 100 C, and they showed no dependence .on the amount of un- loading. These f a c t s , combined with .the f a c t that a mechanism of s e s s i l e d i s l o c a t i o n formation s i m i l a r to the Cottrell-Lomer mechanism i s not known f o r b. c. c. metals suggest that Makin's theory does not apply to vanadium. 32 Birnbaum- has also discussed t h i s e f f e c t , i n f . c. c. metals. He suggests that upon unloading, the d i s l o c a t i o n s r e l a x u n t i l they react e l a s t i c a l l y with f o r e s t d i s l o c a t i o n s and are held. The y i e l d point i s a r e s u l t of r e l e a s i n g the d i s l o c a t i o n from i t s bound configuration. This type of mechanism could c e r t a i n l y be applicable to b. c. c. metals since with the numerous s l i p planes a high f o r e s t density i s l i k e l y . However, i t would be d i f f i c u l t to explain the temperature dependence-, of the y i e l d drops on t h i s b a s i s . Birnbaum observes that-the magnitude of the y i e l d drops are independent of temperature over a range from 72°K to 293°K. I t therefore appears that the y i e l d point effect, i n vanadium i s not c o n t r o l l e d by the same: mechanism as f o r f . c. c. metals. The f a c t that i t occurs only at high temperatures or at a low temperature a f t e r cooling from 100°C suggests a d i f f u s i o n c o n t r o l l e d mechanism. One might expect that the d i s l o c a t i o n becomes locked by i n t e r s t i t i a l s when they stop and an unlocking stress i s required to r e i n i t i a t e flow. Now the times of r e l a x a t i o n are f a r too short f o r the normal d i f f u s i o n mechanisms. In the s t r a i n - r a t e change tests the crosshead was stopped f o r l e s s than one minute before reloading while i n the temperature change t e s t s the.load was relaxed f o r no more than twenty minutes. , In spite of the longer r e s t i n g periods during temperature change t e s t s the - 75. - y i e l d drops showed no significant difference in magnitude from those in the strain-rate change tests. The i n t e r s t i t i a l content, in the single crystals was about one half of that for the polycrystals and. the observed y i e l d drops were smaller for the single crystals. This relationship suggests that .interstitials might be involved. Thus we are looking for a diffusion controlled.interstitial locking mechanism which can operate in a very short time ..relative to normal diffusion times. The only mechanism which seems to satisfy this is Snoek ordering. A brief description of this mechanism w i l l be given. It is well known that the i n t e r s t i t i a l s in b. c. c. metals cause tetragonal distortion when they occupy the (OOg-) (cube edge) position in the unit c e l l . There are three types of these sites corresponding to the three directions of tetragonality or the three space axes. Under no applied stress the. i n t e r s t i t i a l s should be present in these three sites .in equal fractions. However, under'an applied stress some sites w i l l be preferable energetically to others and at temperatures where diffusion can take place the i n t e r s t i t i a l s should redistribute themselves so as to in- crease the population of the sites of lower energy. This, is known as the Snoek effect. 33 Schoeck andftB&eger state that the stress f i e l d of a dislocation should have a similar effect on the i n t e r s t i t i a l s . • Hence we expect a redistribution which.would lower the energy of the dislocation and there- fore lock i t . This involves the movement of atoms a distance of only - 76 - one atomic jump.and. could occur in a very short.time. This process appears to explain the observed y i e ld drops very adequately. The actual intent of Schoeck and Seeger was not to explain the y i e l d drops but to explain a .temperature independent region on the flow stress versus temperature curve f o r i r o n v . They show that the Snoek effect w i l l creat a f r i c t i o n force on dislocations which i s inversely proportional to their velocity. On the basis of this explanation i t would be correct to extra- polate through the y i e ld drop, as was done in this thesis, to obtain the correct value for the flow stress. - 77 - V. CONCLUSIONS , 1. A s imilarity between the mechanisms of yielding and of flow is not observed for vanadium. 2. k increases with temperature for vanadium. 3' Vanadium does not obey the Cottrell-Stokes Law. it-.. It is unlikely that the forest, cross s l ip and i n t e r s t i t i a l mechanisms control thermally activated flow in vanadium. 5. The controll ing mechanism for thermally activated flow is probably either the Peierls-Nabarro force or the non-conservative motion of vacancy jogs. However, there is also some evidence against each of these. 6. It is possible that there i s not a unique mechanism controll ing thermally activated flow. 7- Vanadium deforms by twinning .at 78°K when oriented in certain directions. 8. Yie ld drops obtained upon reloading are attributed to Snoek ordering. - 78 - VI. RECOMMENDATIONS FOR FURTHER WORK The results of this research indicate the need for an extensive study into the importance of the Peierls-Nabarro force in b . c. c. metals. This would, include internal f r i c t i o n work to determine dislocation kink energies and electron microscopy studies to study the orientation of dis- location l ines . Before this i s done i t i s unlikely that any further pro- gress w i l l be made in the work hardening theory. It would also be interesting,to conduct an extensive investigation of the y i e l d point effects. This would involve work at higher temperatures than were used here and a thorough investigation of the dependence of this effect on.impurity content. - 79 - VII. APPENDICES - 8o - APPENDIX I A. Annealing Data for Grain Growth Specimen Annealing Temp. Annealing Time Grain Size °C hr. microns 8A 870 1 14.5 8B . 94o 1 16.6 8C 1020 1 26 .5 8D 1090 1 33-4 9A 870 1 • 17.8 9B 9ko 1 24 9C 1020' 1 35-6 9D 1090 1 52.8 10A * 1090 4 Too large and nonuniform LOB 1050 4 55 IOC 995 4 43 10D 900 1-5 Not measured 10E 820 1 12.4 L1A,B,C & D 870 l 19.4 I1E,F, 12A|& B 1020 2 54.6 L2C,D,E & F 1090 8 69 L3A,B & C 920 l 26.1 L3D, E & F 970 2 35 Rod 10 deformed 4$ before annealing. APPENDIX II A. Polycrystals Specimen Area of Gauge Testing Crosshead Approx. Cross section length LY Temp. Speed Uniform 2 i n . k o„ . -1 Deformation Remarks i n . P . S . I . x 10 C in.mm. 1o 8A .0lk5 1.01+5 1+.1+9 25 .02 21+ 8B .011+5 1.028 I+.69 25 .02 25 8C .0137 0.925 1+.52 25 .02 25 8D .0126 1.015 1+.32 25 .02 20 9A .0109 1.057 1+.1+8 25 . 0 0 2 - . 0 2 13 9B .0102 0.99k k .63 25 . 0 1 - 0 . 1 15 9C .00800 0.906 Not Observed 100 . 0 1 - 0 . 1 12 Loaded non- axia l ly . 9D .OO693 O.962 3-k5 100 . 0 1 - 0 . 1 13 10A .00916 1.01+8 Not tested due to undesirable grain size. 10B .011+1 0.995 3.96 25 .02 18 IOC .011+1 O.9I+2 k .36 25 .02 18 10D .0132 0 . 9  - - - Damaged during mounting. 10E .0137 0.950 1+.76 25 .02 20 11A .OO967 O.9I+I 5-09 0 . 0 1 - 0 . 1 16 11B .0111 0.978 1+.07 100, -8 .01 13 11C .0115 0.95^ 6.1+0 -72 . 0 1 - 0 . 1 17 11D .0098k O.986 4 . 1 1 100 . 0 1 - 0 . 1 lk H E ?0098l+ 0 .923 k .86 0 . 0 1 - 0 . 1 18 I 11F .00816 0 .937 3-55 100 ,8 .01 12 OO H 12A .0117 0 .973 6.1+0 -72 . 0 1 - 0 . 1 16 12B .0111 0.956 3.52 100 . 0 1 - 0 . 1 17 1 Specimen Area of Gauge Testing Crosshead Approx. Cross Section length LY Temp. Speed Uniform . 2 xn. i n . 4 P.S.I. x 10 . °c . :-.i i n . mm. Deformation * Remarks 12C .0115 0-951 4.61 0 .01-0.1 14 12D .0115 O.965 3.19 100,8 .01 14 12E .0107 0.912 6.18 -72 .01-0.1 15 12F . O O 9 6 7 0.956 3.26 100 .01-0.1 15 13A .0100 O.928 4.80 0 .01-0.1 15 133 .0109 1.027 6.36 -72 .01-0.1 13 13C .0104 0-954 3.65 100 .01-0.1 15 13D .0115 0.940 4.54 . 0 .01-1.0 14 13E .0106 0.917 6.27 -72 .01-1.0 13 13F .00984 0.915 3.57 100 .01-6.1 13 I CD ro B. Single Crystals Specimen Area of Gauge Temperature Crosshead Approx. Remarks Cross Section Length Speed Uniform . 2 i n . 0 -1 Elongation in. C in.mm. 4A .00709 .972 - - - Damaged during mounting. 4B .00785 0.991 100,8 .01 12 he .00818 0.936 25 .002-.02 10 non axial loading 5A .0106 0.937 25 .01-0.1 19 5B .0104 0-933 -72 .01-0.1 19 5C .OO833 O.967 100,8 .01 15 5D .OO968 0.950 100 .01-0.1 10 6A .00664 0.972 100 .01-0.1 26 6B .00566 0.910 100 .01-0.1 23 6c .00738 0.955 100,66 .01 12 6D .00849 O.917 100,-26 .01-0.1 14 6E .00832 0.931 -19 .01-0.1 14 7A •00753 0.953 0 .01-0.1 10 7B .OO899 0.975 100,-72 .01 11- 7C .OO967 O.987 -195 .01-0.1 -4 Twinning observed 7-D .0119 O.947 0,-72 .01 8 7E -.0111 a 9i4 -72 -.01-0.1 17 I 00 1 - 81+ - APPENDIX III A. Precision of Measurements 1. Grain Size. Measurements Taking the batch of specimens 13 A, B and C as S>!typic.al:...example, the precision of the measurements of the mean grain diameter can be estimated. Define: d° = dn- = calculated mean grain diameter th diameter from the i . measurement the residual (d Q - d^). Measurement 1 2 3 h 5 6 d° (microns) 26.1 26 .1 26 .1 26 .1 26 .1 26 .1 d^ (microns) ] | ( m i c r o n s ) 23-3 30.0 22.6 27.2 27.5 26.1+ 2 .8 3-9 3-5 1.1 1.1+ 0.3 U v j j = 13.0 ula: The probable error of the mean is calculated from Peter's form- .8U5 n P. E . = /n(^-i) / 6"x 5 Hence the mean grain diameter is i = 1 i = 1 ,2 , n x 13-0 = 2 .0 d Q = 2 6 . 1 + 2 . 0 microns - 85 - From these calculations i t i s apparentt that the precision of the grain size measurements is about 8$. 2. Tensile Testing The Instron Operating Instructions state that the accuracy of the load -weighing system.is better than 0.5$ irregardless of the range in use. This is about the same as the accurary in reading the chart which is about •+ 0.2 of the smallest scale div is ion. j Sample calculation for the lower y i e l d stress Assume the following typical values: F = kOO l b . (load at the lower y ie ld point) XJJL d = Q . l i n . (diameter of specimen) F_ LY = TT <r Differentiating logarithmically: I crLY , £ j ^ y X (TLY " F LY - . d Accuracy of F^y = +.."2 l b . Accuracy of d = + .001" The maximum error i s : S ai LY 2 , _ .001 „ c d -gr— = m + 2 — = -025 or 2'& ^ LY The accuracy of the measurements of Z^'tg^ i s of pr inc ipal con- cern. The smaller Zi rt a> "the more d i f f i c u l t a precise measurement w i l l be. - 86 - This is evident in the plots in Figure (19) where the high temperature plots show:;.the-'greatest scatter. The accuracy i s estimated by: (f ( A ^ a ) = S ( ^ a 2 ) + S ( ^af l A ^ a A ^ a At 100°C the specimens were generally tested at 500 l b . f u l l scale deflection on the Instron.. Hence $ = $ ^ a l = +1 l b . AT4 ~ 13 l b . Atr a 13 ~ This, of course, is a maximum estimate as the errors could compen- sate' for each other. Errors in the calculations of the rate parameters cannot be easily estimated without knowing the exact shape of the force - distance curve. The acatter in the plots of AQ against temperature (Figure 2^b) and in ca l - culations of A Q.by different methods serves as a rough measure of the accuracy. From these comparisons one might expect our estimations of A Q o „, • and H to be out as much as 25yo» However, t h i s . i s s t i l l adequate for deter- mining a mechanism of flow since the theoretical calculations are probably not much more precise than this . •• - 8? - VIII. BIBLIOGRAPHY 1. H a l l , E . 0 . , Proc. Phys. Soc. Lond. B 64 747, (1951) . 2 . Petch, N. J . , . J . Iron and Steel Inst . , Y[k 25, (1953) . 3- Cracknell , A . , Petch, N. J . , Acta Met. 3 186, (1955) . k. Heslop, J . , . Petch, N. J . , . P h i l . Mag. 1 866, (1956) . 5- Conrad, H . , Schoeck, G . , Acta Met. 8 791 , ( i 9 6 0 ) . 6. Petch, ET.' J.,. P h i l Mag. 3 IO89, (1958) . 7- C o t t r e l l , A... H . , Trans. A.. I . M . E . , 212 192, (1958) . ..8i . Bas inski , .Z . S., P h i l . Mag. 4 393, (1959) . 9- Conrad, H . , Wiedersich, H . , Acta.Met. 8 128, ( i 9 6 0 ) . . 10. Mordike, B. L . , Haasen, P . , P h i l . Mag. 7 459, (1962) . 11 . Gregory, D.. P . , Stroh, A. N . , Rowe, G. H . , to be published. 12. Basinski, Z. S., Christ ian, J . W., Aust. J . Phys. 13_ 299, ( i 9 6 0 ) . 13 . Moraike, B. L . , Z. Metallkunde 9 586, (1962) . Ik. Conrad, H . , J . Iron and Steel Inst. 198 364, (1961) . 1 5 . , Conrad,, II., Phi l . . Mag. 5 745, ( i 9 6 0 ) . 16 . Conrad, H . , To be published. 17- Snowball, R. F . , M . A . Sc. Thesis, University of B r i t i s h Columbia, (1961) 18 . Fraser,. R. W., M. A. Sc. Thesis , .University of B r i t i s h Columbia,' ( i 960 ) 19 . Conrad, H . , Frederick, S., Acta Met. 10 1013, (1962) . 20. Hirsch, P. B . , P h i l . Mag. 7 67 , (1962) . 21 . Mott, N. F . , . Trans. A. I . M. E . 2 l 8 962, ( i 9 6 0 ) . 22 . Kuhlmann-Wilsdorf, D . , Trans. A. I . M. E . , 224 1047, (1962) . 23 . Cottre l l , . A. H . , Dislocations and Plast ic Flow in Crystals, Oxford University Press, ( l953)> page 134. 24. ' Low, J . R . , Guard, R. ¥ . , Acta Met.., 7 171, (1959) . - 88 - BIBLIOGRAPHY Continued 25 . Johnston, W. G . , Oilman, J . J . , J . Appl. Phys. _3_1 632 ( i 9 6 0 ) . 26. Low, J . R. , Turkalow, A. M. Acta Met., 10 362, (1962) . 27- Stein, D . F . , Low, J . R.-, J . Appl. Phys. 31 362, ( i 9 6 0 ) . 28 . Read, W. T . , Dislocations in Crystals , McGraw H i l l , N. Y . , (1963) , Page 46. 29 . Seeger, A . , P h i l . Mag.. 46, 1 1 9 4 , ( 1 9 5 5 ) . 30. Martinson, R. H . , M. A. Sc. Thesis, University of B r i t i s h Columbia, (1963) . 31 . Makin, M. J . , P h i l . Mag. 3 287, (1958) . 32. Birnbaum, H. K . , Acta Met. 9_ 320, (1961) . 33- Schoeck, G.,Seeger, A . , Acta Met. 7 469, (1959)- 34. Frank, F . C , II Nuovo Cimento 7 Supp. 1 -2 , 386, (1958).


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