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Yielding and flow in vanadium Simpson, Leonard Angus 1963

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YIELDING AND FLOW IN VANADIUM  BY  LEONARD ANGUS SIMPSON S c . , 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  i n the  Department of  METALLURGY  We accept t h i s 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 April,  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 c h a r a c t e r i s t i c s of y i e l d i n g and flow i n p o l y c r y s t a l l i n e and single c r y s t a l vanadium has been c a r r i e d out.  The  effect of grain s i z e , temperature and s t r a i n rate on these properties was studied.  It was found that there i s no s i m i l a r i t y between the mechanisms of y i e l d i n g and flow, i n vanadium which i s i n disagreement with work on iron. The r e s u l t s of t e n s i l e tests suggest that the mechanism c o n t r o l l i n g thermally activated flow i s 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 c o n t r o l l i n g thermally activated flow.  Y i e l d points were observed i n the s t r e s s - s t r a i n curve upon r e loading a t e n s i l e specimen under certain conditions and these are explained i n terms of Snoek ordering.  ii  ACKNOWLEDGEMENT The author wishes to thank Dr. J . A. Lund and Dr. E . Teghtsoonian for  their, guidance and assistance i n the interpretation of t h i s research.  Thanks are also due to Mr. E . J . Richter for technical assistance and to fellow graduate students for many h e l p f u l discussions.  F i n a n c i a l a i d for t h i s project was received i n the form of The Aluminium Laboratories Limited. Grant No. A - 1 ^ 6 3 .  Fellowship and National Research Council  This assistance i s g r a t e f u l l y acknowledged.  iii  TABLE OF CONTENTS Page  I.  II.  III.  1  INTRODUCTION AND PREVIOUS WORK A.  Introduction  1  B.  Previous Work  2  11  EXPERIMENTAL A.  Material  B.  Zone Refining  C.  Specimen Preparation  11 .  12 13  .  1.  Machining  13  2.  Electropolishing . . .  13  3*  Grain Size Determination and Control  13  D.  Tensile Testing  1^  E.  Temperature Control  17  19  EXPERIMENTAL RESULTS A.  B.  19  P o l y c r y s t a l l i n e Material 1.  Grain Growth  19  2.  Hall-Petch Equation  20  3.  S t r a i n Rate Change Tests  26  h.  Temperature Change Tests  27  5.  "Pseudo" Y i e l d Points  31  6.  Rate Equation  Single Crystals 1.  Zone Refining  . . . . .  3^ •  ^2 ^2  iv  TABLE OF CONTENTS Continued Page  C. IV.  2.  S l i p Systems  kk  3.  Strain Rate Change Tests  k6  k.  Temperature Change Tests  51  5.  Rate Equation  51  6.  Y i e l d Points  DISCUSSION  . .  VIII.  58  Hall-Petch Equation  58  B.  Thermally Activated Flow  59  C.  VII.  5$  A.  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 i n Screw Dislocations  67  Comparison Between Polycrystals and Single Crystals  69  Mechanism  70  7.  VI.  5k  Summary of Results  6.  V.  . .  Y i e l d Points  73  CONCLUSIONS  77  RECOMMENDATIONS FOR FURTHER WORK  78  APPENDICES  79  BIBLIOGRAPHY  87  V LIST OF FIGURES No.  Page  ... 1 . . Effect of the concentration of C + N i n solution on in Fe  o  Q~  after Heslop and Petch  ;  °. .  3  2.  I l l u s t r a t i o n of grain-toj-grain y i e l d i n g  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 i n t e n s i l e t e s t i n g  16  6.  Assembly for t e s t i n g at 100°C  17  7.  Inhomogeneous grain sizes after drawing, rod #2, 110X . . .  20  8.  Typical load-elongation curve for vanadium p o l y c r y s t a l  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 l d stress on grain size  2k  11.  Typical load-elongation curve for a s t r a i n rate change t e s t  25  12. 13l^*  Method of extrapolation to obtain change i n flow stress . . /S ' f increase i n £ v s . ^ for polycrystals . . . A *Y ^ increase i n £ vs. ^ v_. a and £. for polycrystals  26 28 29  f  o  r  a  n  o  r  a  n  o  r  a  n  a  15 •  /v'T ^ *" a  16.  tC\  increase i n s t r a i n rate vs g. for polycrystals  for a: decrease i n temperature vs £ and. ^  30  for  polycrystals,  32  17-  Load-elongation curves showing y i e l d points  33  18.  Method of extrapolation through y i e l d points  3^  19-  A c t i v a t i o n volume vs. flow stress  36  20.  A c t i v a t i o n volume vs. temperature .'  37  21.  Energy, H, vs. £  38 •  22.  Thermally activated component of H ° , Zs. Q, vs. £.  39  vi  LIST OF FIGURES Continued No. 23• 2ka.  Page Pre-exponential factor A v s . J for polycrystals  ko  Strain rate s e n s i t i v i t y of ^  kl  vs. temperature a  2kb.  Z^Q vs. temperature f o r single crystals  kl  25a.  Laue photograph of s u f f i c i e n t l y polished specimen  25b.  Laue photograph of i n s u f f i c i e n t l y polished specimen  . . . .  i+3  . . .  ^3  26.  Orientation of t e n s i l e axis of single c r y s t a l s  kk  27•  Gas holes i n single c r y s t a l specimens  1+5  28.  Typical s l i p traces on single c r y s t a l specimen 6A, 210X . .  29* 30.  *"  or a  n  increase i n £ vs. %  for single c r y s t a l s  / ^ Q ^ ^ f o r an increase i n £ vs. £. for single c r y s t a l s  kj  .  k^  . .  kQ  31.  Twin markings on surface of single c r y s t a l  50  32.  Laue photograph showing s l i t s i n spots due to twins . . . .  50  33«  /\ T/ /  a  for a decrease i n temperature vs ^  for. single..  a  crystals '. , A 7T o f o r a decrease i n temperature vs. 'TJ crystals  52  /  35-  36.  37•  a  f o r single 53  Strain rate s e n s i t i v i t y of the flow stress vs. temperature s e n s i t i v i t y to f i n d ln(A/jbj . (equation ( 1 5 ) )  55  Transmission electron micrograph of dislocations i n s i l i c o n i r o n , after Low and Turkalow  65  Formation of a d i s l o c a t i o n kink i n a P e i e r l s force f i e l d  66  .  vii  LIST OF TABLES  No. I. II. III. IV.  Page Summary of Previous Work  10  Analysis of Material  11  Possible S l i p Systems  k6  Summary of Results  57  I. A.  INTRODUCTION AMD PREVIOUS WORK  Introduction One of the most remarkable properties inherent i n a body centered  cubic metal i s the marked temperature dependence of i t s strength.  This has  evoked considerable interest and many investigators have t r i e d to analyze and explain t h i s property. Most of the investigations have been centered about the temperature dependence of the y i e l d strength and have not been very productive i n explaining the d e t a i l e d mechanisms of deformation.  Most of t h i s work  has been concerned with i r o n while the other body centered cubic metals have received r e l a t i v e l y l i t t l e  attention.  The adoption of rate theory to explain thermally activated flow has l e d to an i n t e n s i f i e d attack on the problem of determining the deformation mechanisms i n 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 i n a number of publications on the flow mechanisms i n i r o n , tantalum and columbium.  The problem of establishing a single mechanism as the one cont r o l l i n g thermally activated flow i s rather d i f f i c u l t and current authors are i n disagreement on many points.  However, i t must be borne i n mind  that t h i s study i s s t i l l i n the embryo stage and a vast quantity of experiments i a required before a sound theory can be devised.  It i s , therefore, the object of t h i s thesis t ° <3-<i to t h i s pool a  of knowledge a description of the temperature dependent deformation  - 2 -  Properties of vanadium.  It i s intended to make a comparison of these prop-  e r t i e s between polycrystals and single c r y s t a l s and also to investigate the role of grain size i n the y i e l d i n g and flow of vanadium.  B.  Previous work This investigation w i l l be concerned with the two experimentally  derived equations: o  where  0~LY ^  (TLY=  c±  di  C i a*  ^  s  = e  l°  -l/2  o* +  cr  +  +  +  k  L Y  *fi  (1)  d  d  "  , ,  1  /  2  ( ) 2  y i e l d stress, that i s , the lowest point on  w e r  the s t r e s s - s t r a i n curve after the beginning of p l a s t i c deformation. 0*^2. i s the flow stress, t h a t . i s ,  the stress required to main-  t a i n p l a s t i c deformation a t / any given s t r a i n . 0"i>  0~L°  a  r  a  r  e  e  athermal f r i c t i o n stresses opposing d i s l o c a t i o n motion. temperature dependent f r i c t i o n stresses opposing d i s -  location motion. . -l/2 k f 1> ^LY l°P ° ^ plots of O f 1 d Qjjf against d -1/2 d ' i s the mean grain diameter. o o* In e a r l i e r work Q T .and Q ~ have been studied as a single a  r  parameter,  r  e  s  e s  a n  /  o f ^ , and equation ( l ) has been used i n the form  QLY  =  CTo  + k  L V  d-  1 / / 2  (3)  This i s known as the Hall-Petch equation and was obtained experimentally by Petch  1  ' 2 who followed up some e a r l i e r work by H a l l .  Petch,  who worked with p o l y c r y s t a l l i n e i r o n supplemented his o r i g i n a l work with  - 3 two more papers  i n which he discussed the term (T~ . By varying the o C + N content i n solution he obtained the plots shown i n Figure 1.  Figure 1.  :  Effect of the concentration of C + N i n solution on (y° i n Fe, after Heslop and Petch.  Because the slopes were the same at a l l temperatures, Petch concluded 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 i s i n dependent of temperature.  Therefore the i n t e r s t i t i a l s would contribute t o  Q"T i n equations ( l ) and ( 2 ) .  - k -  Other factors contributing to C T \ are long range e l a s t i c f i e l d s which are created .by the d i s l o c a t i o n networks.  stress  CT* . therefore,  w i l l depend on temperature only through the e l a s t i c modulus.  A recent publication by Conrad and Schoeck^ gives evidence that the mechanisms for y i e l d i n g and flow might be the same.  A series of tests  was performed on p o l y c r y s t a l l i n e iron from which the lower y i e l d stress and -l/2 the flow stress at 5$ s t r a i n were plotted against d~ ' similar values f o r  k  i n the two p l o t s .  .  This resulted i n  It was also observed that  independent of temperature over a range from 100°K to 300°K.  k was  Data from  Petch^ also indicates a temperature independence of k^y from 19^ K to 300°K. Conrad and Schoeck interpreted the s i m i l a r i t y between k  and k p , , and the  J_iX  i d e n t i c a l temperature dependences of C T ^ ,  O^LY  an<  ^" ®~ 1'  a  s  indicating  that the same d i s l o c a t i o n mechanism i s involved i n y i e l d i n g as i n flow. The temperature independence of k^y has evbked considerable i n t e r 7  est.  Cottrell  has explained the significance of k^y i n the following way.  When a d i s l o c a t i o n source i s unpinned, i t releases an avalanche of d i s l o c a tions into the grain and these dislocations p i l e up at a grain boundary. Their stress concentration acts on sources i n the next grain and the process i s 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 p i l e up on a source a distance 1 ahead of the p i l e up i s given by  (o- - (rt) (hf LY  «  - 5-  HMYIEI DFH  ORAIN  YIELDED GRAIN  GRAIN BDY.  Figure 2.  I l l u s t r a t i o n of g r a i n - t o - g r a i n y i e l d i n g  Y i e l d i n g w i l l occur when the stress concentration at the next source the unlocking stress, that  is  is  1/2  (la) and since  2 l/d  ( &x <^  \  ^)  ^  +  =  (5)  ^  1, •1/2  :.>i/2  (6)  According to equation (6) and the C o t t r e l l - B i l b y theory, the unlocking s t r e s s ,  should decrease rapidly with increasing temperature.  This would require that k^y also decreases with temperature which i s not generally observed.. The quantity stress.  0~ i s known as the thermal component of the flow  It arises from short range forces on moving dislocations which  can be overcome by thermal a c t i v a t i o n .  -6 The use of rate theory as a means of studying thermally activated flow became firmly established after the publication of B a s i n s k i ' s c l a s s i c g paper  ,in 1959-  His analysis was. developed f o r f.  also adaptable to b . c. c.  c. c. metals but i t  is  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 d i s l o c a t i o n i s held up by an obstacle, nature of which i s s t i l l not defined.  the  This w i l l create an energy b a r r i e r  which we can depict by the force-distance curve i n Figure 3t h i s b a r r i e r a d i s l o c a t i o n must acquire an energy  To surmount  H over a length L .  the distance t r a v e l l e d by the d i s l o c a t i o n during a c t i v a t i o n i s  If  d, the  a c t i v a t i o n volume i s defined as v* where  b  (7)  = b Ld  i s theStoEgers'vector.  I f the d i s l o c a t i o n cannot overcome the b a r r i e r with the a i d of thermal energy alone i t . i s assumed that an applied stress, C , r  Q  H can be reduced with the a i d of  by an amount v* TT  to a value. A Q , given by  AQ =H - v*T  (8) a  -  Assuming the deformation process i s thermally activated one obtains the following Arrhenius type equation f o r the s t r a i n r a t e , £ :  Ae  ^  (9)  where  A = N>»>\  where  N  -  (lO)  i s the number of a c t i v a t i o n s i t e s per unit volume, i s the frequency with which the dislocations  ^  7  "try" the b a r r i e r ,  i s the average s t r a i n contributed by a d i s l o c a t i o n when i t overcomes the b a r r i e r .  £ _  i  ^  :  DISTANCE  Figure 3'.' Force-distance curve f o r thermally activated flow  - 8 Gregory remarks that the i n t e r n a l stress f i e l d due to a randomly oriented d i s l o c a t i o n network should average to zero over long distances but should give, r i s e to a l o c a l stress network i t s e l f .  within the dimensions of the  Therefore, the energy,  H, should be made up of two parts  given by H = H* + v* f  (11)  H* i s defined i n figure ( 3 ) and v* ^  where  i s the energy required  to overcome the i n t e r n a l stress f i e l d over a .distance equation ( l l )  barrier.  Substituting  into equation ( 8 ) one obtains  A Q = H* - v* Here  .  - X) ±  ='H* - v*  <\*  (12)  * i s the effective stress operating against the energy  It would be desirable to calculate H* d i r e c t l y , for t h i s would  y i e l d a good approximate value f o r the a c t i v a t i o n energy H°.  However,  there i s no d i r e c t method of doing t h i s because i t requires an evaluation of  ^ .  and therefore,  one calculates  H using  a n < a  l estimates  >  t .  by methods of extrapolation.  If the temperature dependence of the shear modulus i s and i f  A  neglected  i s not a s i g n i f i c a n t function of stress or temperature then the  11*9 following equations can be derived  :  "--(^HT  (13)  - 9 -  M  = k f  l ^> T  (15)  IE  T These equations along with equation ( 9 ) provide the means of analysis.  Several workers during the l a s t two or three years have attempted to e s t a b l i s h a mechanism for thermally activated flow by comparing measured values of v*, H ° , Z^Q and A with t h e o r e t i c a l l y calculated values. 12 Basinski and C h r i s t i a n for  observed that the change i n flow stress  a change i n s t r a i n rate decreased with increasing deformation for i r o n .  They concluded that the Peierls-Nabarro force i s the dominant factor after r u l i n g out an i n t e r s t i t i a l mechanism and a forest i n t e r s e c t i o n mechanism. However, they did- not consider any other possible mechanisms. 1 ^ , 1 5 , 1 6 , 1 9 • '. Conrad favours a P e i e r l s mechanism a l s o .  He selects  this one after r u l i n g 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 i r o n i n which f i n e l y divided precipitates are responsible for the thermal component of the flow s t r e s s .  They explain an increase i n the s t r a i n rate  sensitivity  of the flow stress with s t r a i n i n terms of a d i v i s i o n of precipitates during s t r a i n i n g due to i n t e r s e c t i o n with d i s l o c a t i o n s .  This results i n  a greater density of obstacles and hence a smaller a c t i v a t i o n volume.  - 1 0 In another p u b l i c a t i o n , Mordike, r e f e r r i n g to his work on tantalum, was unable to determine any definite mechanism although he suggests a mechanism involving the conservative motion of jogs.  However, t h i s mechanism,  20 which was f i r s t suggested by Hirsch ' for f . c c . metals,  involves the con-  s t r i c t i o n of extended jogs which are not generally observed i n b . c . c . metals. Gregory et a l " ^ favour a mechanism involving the  non-conservative  motion of jogs for columbium on the grounds that their measured a c t i v a t i o n energy was too high to be compatible with any other mechanism. A summary of the previous work i s given i n Table I . TABLE I Author  Material  H° (ev)  L (cm)  30 b  . ..  Mechanism  Ref.  Peierls  Ik  Honrad  Fe  Basinski + Christian  Fe  Did not Calculate  Did not Calculate  Peierls  12  Conrad + Frederick  Fe  Did not Calculate  : .12 b'  Peierls  19  Fe  •5  17 b  .Impurity Mechanism  10  55". b  Conservative jog mech. (not d e f i n i t e )  13  50 b  Non conservative jog.  11  Vfordike + Baasen Mordike  Gregory et a l  .5 - . 6  ' Ta  Cb  2  - 11 . Ii:. EXPERIMENTAL  A. M a t e r i a l The vanadium f o r t h i s project was supplied by Union Carbide Canada Limited, of Toronto, Ontario.  I t was i n the form of c y l i n d r i c a l  p o l y c r y s t a l l i n e rods 0.250 inches i n diameter.  The results of an  analysis f o r i n t e r s t i t i a l gases and carbon are given i n Table I I .  TABLE I I Analysis of M a t e r i a l  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 c r y s t a l s were grown i n the electron beam f l o a t i n g zone  r e f i n e r described by Snowball"^. "As-received" rods were used i n nine inch lengths.  A portion of  oneoehd,roughly 0.75 inches i n length,was machined down to a diameter of 0.125 inches to f i t the lower grip on the zone r e f i n e r .  The other end  was faced o f f squarely i n a lathe. A seed c r y s t a l , about one inch i n length, was prepared i n a similar fashion and f i t t e d t o the upper grip of the zone r e f i n e r such that there was a gap of approximately l / l 6 " between the seed and the rod. The filament was l i n e d up so that i t was just above the gap and the power was turned on. I t was found necessary to use a current of 130 ma. and a t i a l difference of 1.6 kv. to maintain a stable molten zone.  poten-  Minor ad-  justments were necessary during the pass due t o fluctuations i n the power supply, gas bursts from the rod and other minor e f f e c t s . 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 t o outgas the specimen.  The  second pass was intended to p u r i f y the rod and was c a r r i e d out at 10 cm per  hour.  The vacuum maintained was generally better than 1 0 " ^ mm of Hg. An analysis f o r i n t e r s t i t i a l s i n zone r e f i n e d material i s given i n Table I I .  - 13 C.  Specimen Preparation 1.  Machining P o l y c r y s t a l l i n e and single c r y s t a l specimens were both prepared  for t e s t i n g by the same procedure.  The p o l y c r y s t a l l i n e specimens were  manufactured from"as-received"rod.  A 1 1 / V ' length was cut from a rod and machined down to a diameter of from . 1 1 0 " to . 1 3 0 " with a gauge length of about one inch. i n i t i a l cuts were . 0 0 3 " deep.  The  At a diameter of about . 1 5 0 " the depth  was reduced after every four cuts to . 0 0 2 " ,  . 0 0 1 " and .0005".  The specimen  was then hand polished i n the lathe with 0 and 000 emery paper. 2.  Electropolishing To remove any surface deformation, a l l specimens were e l e c t r o -  polished i n a solution of the following composition:  320 cc methyl a l c o h o l , 80 cc cone, s u l f u r i c acid*  The p o l i s h i n g was c a r r i e d out at room temperature with a potent i a l difference of about 10 volts and a current of 1.5 amperes.  In a l l  cases a minimum of . 0 0 5 " was removed from the surface of the specimen. This required a p o l i s h i n g time of from f i f t e e n to twenty minutes.  3-  Grain Size Determination and Control Grain size was c o n t r o l l e d by heat treatment i n the vacuum an18  nealing furnace described by Fraser below 10~5 mm of Hg.  .  The vacuum was generally kept well  A table of temperatures, times and resultant grain  sizes i s given i n Appendix I .  - ih -  Samples were cut from the annealed specimens, mounted and polished f o r metallographic examination.  The polished specimens were etched  i n a solution of 15 cc. cone, l a 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 l i n e s were drawn across the photograph as i n Figure k.  The number of  grain boundaries cutting each l i n e was counted and, knowing the magnification, the mean grain diameter was calculated. D.  Tensile  Testing  Specimens were tested on.an Instron t e n s i l e t e s t i n g machine. mounting device consisted of the set of universal grip holders  The  described  17 by Snowball specimen used.  and a set of grips designed s p e c i f i c a l l y f o r the type,of A diagram of the grips i s shown i n Figure 5.  Three types of t e n s i l e tests were performed: (1)  Continuous tests on p o l y c r y s t a l s .  (2)  Strain rate change tests on single c r y s t a l s and p o l y c r y s t a l s .  (3)  Temperature change tests on single c r y s t a l s and polycrystals.  In the continuous tests the specimens were strained continuously to fracture at a constant rate of crosshead t r a v e l of .02 inches per :• minute.  Figure k.  Method of grain size measurement, specimen 9B, 210X.  - 16 -  Figure 5.  Drawing of grips used i n t e n s i l e t e s t i n g .  In the s t r a i n rate change tests the specimens were strained past i n i t i a l y i e l d i n g 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 f o r another 0.5$ s t r a i n . Then the crosshead was stopped, the load was relaxed and the crosshead was restarted at the i n i t i a l s t r a i n rate. continued u n t i l the specimen f a i l e d .  This c y c l i n g process was  These experiments were c a r r i e d out  over a temperature range of -72°C to 100°C.  - 17 The s t r a i n 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 temperature was cycled at a constant s t r a i n rate. This involved straining past the y i e l d point up to a given s t r a i n at one temperature, stopping the crosshead, relaxing the load, changing the temperature bath and reloading at the same s t r a i n rate after allowing time for thermal equilibrium to be reached.  E.  This c y c l i n g process was,: continued u n t i l the specimen f a i l e d .  Temperature Control Three p r i n c i p a l temperature baths were used.  bath was used to maintain a temperature of 100 C.  A b o i l i n g water  A photograph of t h i s  assembly i s shown i n Figure 6 .  Figure 6 .  Assembly for t e s t i n g at 100 C.  - .18  An ice water bath was used to obtain a temperature of 0°C and a mixture of acetone and s o l i d CCvj provided a temperature bath of - 7 2 ° C .  A s a l t bath of calcium chloride and ice water was used f o r two tests on single c r y s t a l s but i t was inconvenient to use and was not employed i n further t e s t s .  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.  A.  EXPERIMENTAL RESULTS  P o l y c r y s t a l l i n e Material 1.  Grain Growth Using the annealing procedures outlined i n 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 r e f i n i n g the "as-received" material.  This process y i e l d e d  a single c r y s t a l and attempts were made to make polycrystals of controlled grain size by wire-drawing and annealing  processes.  Two zone r e f i n e d bars were machined to a constant diameter and drawn down to 50$ and 30$ of the o r i g i n a l cross sectional areas.. annealing,the grain size was small at the periphery  After  and large near the  center of the rod, owing to the inhomogeneity of.deformation by drawing. A t y p i c a l structure i s shown i n Figure ( 7 ) .  Because of the non-uniformity,  t h i s procedure was abandoned and specimens were made from."as received" material.  A Laue back-reflection photograph d i d not reveal any preferred orientation.  - 20 -  HHHBHHHHBHBI  Figure 7«  2.  Inhomogeneous grain sizes after drawing, rod #2, 110X.  Hall-Petch Equation A t y p i c a l load-elongation curve for a continuous test on a  p o l y c r y s t a l i s shown i n Figure (8).  This diagram also i l l u s t r a t e s  Q £ and. fff •  method of obtaining the parameters  v  XI  ... ^  Figure (9a) shows the behavior of to d  •1/2  G~  J_)JL  T V  and  J-iJL  (y~  with respect  1 1  By comparing the two plots i t i s d i f f i c u l t to see the r e l a t i o n -  ship between ing  the  Of»i  a  n  d  ^LY ^  e  c  a  u  s  e  ° ^ ^ke scatter.  It i s more enlighten-  to study  O f i - OIy = «To • (To )  +  ( fi k  •1/2 •  k  L ) Y  d  (16)  Figure 8.  Typical load-elongation curve f o r vanadium p o l y c r y s t a l .  - 23 -• I t i s evident that i f k^y - k ^ then  Ofi "  dependent of grain s i z e .  (9b) i t i s evident that k ^ y ^ kf]_-  (V,  -  From Figure  0~Ly will'."be a constant, i n -  0* - decreases with decreasing grain s i z e . TV  This method of p l o t t i n g  the differences i s the most accurate one because the maximum error i n these -1/2 r e s u l t s l i e s i n the measurement of d  . A comparison of the difference  between the y i e l d and flow stresses f o r a given grain size i s independent of the measurement of the grain s i z e . It would have been desirable to investigate.this property further but t h i s would have required t e s t i n g a large number of specimens t o fracture at a constant s t r a i n 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 s t r a i n rate change or temperature change type.  This made i t impossible to obtain con-  sistent flow stress values at 5$ s t r a i n 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 t e s t s , however, and i s plotted i n Figure ( 1 0 ) .  There i s a trend f o r the slopes of the plots i n Figure increase with.temperature.  (lO) t o  One might question t h i s trend on the grounds  of the uncertainty of grain size measurement.  However, the specimens of  various grain sizes were made i n batches and a specimen from each batch was  tested at each temperature.  value was a l l o t t e d to each batch.  The grain size was measured and a single Therefore, i f there was an error i n the  Figure 10.  Dependence of lower y i e l d stress on grain s i z e .  - 26 -  grain size measurement i t would be the same for a l l the specimens of the given batch, and the r e l a t i v e slopes of the curves would not be  3-  affected.  Strain Rate Change Tests A t y p i c a l s t r e s s - s t r a i n curve involving s t r a i n rate changes i s  shown i n Figure ( l l ) .  The stress l e v e l at the i n i t i a t i o n of p l a s t i c  flow  was found by extrapolating the two straight portions of the curve i n Figure (12) and calculating the strain:.rate s e n s i t i v i t y  ACT  =  cr (B)  STRAIN  Figure 12.  -  cr (A).  A  CT" from the equation  (17)  £  Method of extrapolation to obtain change i n flow stress  - 27 -  the change i n the resolved shear stress, assuming a.Schmid factor of 0 . 5 . of  ^  and £  plots i s for  i n Figures ( 1 3 ) , (l*0  a  was calculated  was then plotted as a function  and ( 1 5 ) '  The general trend i n these  *d ^  to decrease s l i g h t l y with deformation i n disagreea ment with the Cottrell-Stokes Law. It i s evident that there i s no measureable dependence of on.grain size over the range studied.  ^  ^  a  This i s i n agreement with the work  of Conrad and Schoeck on p o l y c r y s t a l l i n e i r o n . Temperature, i n the range studied, has no measureable effect on the slopes of the plots i n Figures ( l 3 ) > (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 i s s u f f i c i e n t to mask a small temperature dependence of the slope.  k..  Temperature Change Tests A series of temperature change tests was performed on the poly-  c r y s t a l l i n e 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 b o i l i n g water  bath and switching.to an ice water bath.the assembly and bath came to temporary thermal equilibrium at about 8 ° C . The v a r i a b i l i t y of t h i s proo cess was r a r e l y more than + 1 C and i t was easier t o adjust the bath temperature to 8°C than to cool the system to 0°C. The specimen was allowed at least ten minutes to a t t a i n the. bath temperature at 8°C and at least f i v e minutes at 100°C.  The shorter time  was considered s u f f i c i e n t for the high temperature because the heating  .4540-  .35T=  5  .251-  •  O  O •  O  * ^.15 ^  _L  3,20  3.40  4.00 (PS./. *  3.80  3.60  4 .35-  7  .30-  4.20  • •  /0*)  &^=/00  .25-  .20-  2.80  2.60  3.00 r*  Figure 13«  /^^  a  IIC /2A I2f I3B 13 £  4.40  T=  J5_ 2.40  -72°C  3.2 0 CPS./.  P—o—o 3-60 3.40 x/o*)  f o r an increase i n £ vs.  a  for polycrystals.  0°C  O • O  IIA /IE 12 C  •  13 A  •  I3D  ro 00  .25 T = o°c  •o o  .20  If A (If /2C /3A  = /0  o  .15 ,02  .04  06  ±  .08  STRAIN  .10  .12  J4  ./6  O; 4  •  o  o  o  o  .05  0  1.60  ISO Figure lh.  2.00 A ^F-  2.20  J-  2.40 260 (PS.I*I0V  f o r an increase i n £ vs. ^  280  and £. f o r poly cry stals.  3.00  ro vo  T = -72°C IIC O 12A 12 E I3B  •  o •  10  o 7 = O  •  o  •  •  0  .02 Figure 15.  .04  .08 ,/2 .10 STftAIN S f o r an increase i n s t r a i n rate vs £. f o r polycrystals. .06  ./4  /oo°c I/O 12 b 12 F I3C I5F 10  ./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\  The trend here i s f o r A  a  against  t„  /  f „• and £ a  are shown i n Figure ( l 6 ) .  to decrease with increasing stress and s t r a i n .  It i s note able that the magnitude of A t  i s roughly twice that f o r a 61 o  s t r a i n rate change of a.factor of ten at 0 C.  Therefore, there would be  a greater l i k e l i h o o d of detecting a grain size s e n s i t i v i t y of A t these t e s t s . 5.  l n  However, such a s e n s i t i v i t y i s not observed.  "Pseudo" Y i e l d Points Under c e r t a i n conditions y i e l d points were observed upon reload-  ing the specimen a f t e r a s t r a i n rate or temperature change.  A typical  s t r e s s - s t r a i n curve showing t h i s property i s included i n Figure (l7)-  These  y i e l d points were observed;.under the following conditions: (1) (2)  i  i n s t r a i n rate change tests at 100°C, In temperature change tests involving 100°C.  In the temperature change tests the y i e l d drop was observed on both the low temperature and the high temperature.portions of the stresss t r a i n curve, but the y i e l d drops were s l i g h t l y larger on the former portion. The magnitude of the y i e l d drops was roughly the same f o r both the upper and lower s t r a i n rates i n the rate change t e s t s . drops was i n a l l cases independent of s t r a i n .  The size of these y i e l d  Because of the temperature  dependence of the y i e l d drops i t i s u n l i k e l y that "th^y could be due to any c h a r a c t e r i s t i c of the t e s t i n g apparatus.  °  •  .3*  o o I  V30L  .04  .02  :  .06 STRAIN  _L_  08  10  14  .12.  E T = 8°C~~I00 IIB  O O  .50-  •  CO  MF I2D  40[  .301 1.60  _L  1.80  2.00  2.20  240 (P.S.I. x  Figure 1 6 .  A t  ^  o r  a  2.60  2.80  |0 )  decrease i n temperature vs. 6- and  4  f o r polycrystals  3.00  C  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 l a t e r the  method of extrapolation i l l u s t r a t e d i n Figure (l8) was used.  Figure l 8 .  6.  Method of extrapolation through y i e l d points.  Rate Equation The a c t i v a t i o n volume was calculated f o r each specimen  plotted as a function of stress.  To do t h i s , the s t r a i n rate  was found from the points i n Figures (13)  and  sensitivity  and (lk-) and used i n equation  (13)-  - 35 -  Results of these calculations are plotted i n Figure (19).  There appears  to he a s l i g h t increase i n v* with stress at 0°C and -72°C.  This increase  i s small, however, and therefore an average value of v* was taken plotted against temperature.  and  This i s shown i n Figure ( 2 0 ) .  The energy, H, was calculated using data from the temperature change tests and equation ( l k ) . that f o r the mean temperature  The value of v* used i n t h i s equation was  of the t e s t , 327°K.  Since no s t r a i n rate  change test was doner at 327°K, v* was found by i n t e r p o l a t i o n on Figure ( 2 0 ) . H i s plotted as a function of s t r a i n i n Figure ( 2 1 ) .  I t i s seen that H  increases somewhat with s t r a i n . Using equation (15), the thermally activated component of the a c t i v a t i o n energy, A Q, was calculated f o r a temperature 10$ s t r a i n .  The value obtained was .60 e.v.  of 327°K and at  A Q was also calculated from  equation ( 8 ) and plotted as a function of s t r a i n i n Figure ( 2 2 ) . The value f o r - t . .in t h i s equation was taken as the average of the applied stresses at the two t e s t i n g temperatures. 10$ s t r a i n i s 0 . 7 2 e.v.  From Figure (22<),  A Q at  This i s somewhat d i f f e r e n t from the value obtained  i n equation (15) and serves as a rough measure of the accuracy of t h i s type of analysis.  The factor. A i n the rate equation was determined.from (9)  equation  and i s plotted i n Figure ( 2 3 ) as a function of s t r a i n . o A rough estimate of the a c t i v a t i o n energy, H , may be made with  the a i d of Figure  (2*4-8).  When the s t r a i n rate s e n s i t i v i t y i s equal to zero  100°c/40  *  I  /oo° c  120  •  o  SINGLE  CftVVT^  o  loa-  80  60f•7Z°C  40 -72 C  20  /.o  JL  2.0  3.0 (PS.I. v  Figure 19•  40 IO )  A c t i v a t i o n volume vs. flow stress  4  OA  Figure 20.  A c t i v a t i o n volume vs. temperature.  POLYCR Y.ST A L S  2.5  32 7 ° K  2.0  SINGLE  CRYSTALS  287°K > 15  32 7 ° K 356°K  237°K .0  0.5 L  o  0  .02  £)4  1 .06  .08 STRAW  Figure 21. Energy, H, vs. £  JO  ,12  .14  Figure 23»  Pre-exponential factor A vs. £. f o r p o l y c r y s t a l s .  5.0  100  200  300  T Figure 2ha..  400  (°K)  Strain rate s e n s i t i v i t y 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) i s extrapolated to the temperature, T , at which  T  ~2Tir\ir~  =  °> "t^sn H  can be calculated from equation H° = kT  c  (9). (18)  m(A/^)  Taking a value f o r ln(A/g) of 25 from Figure (23) T  Q  of klO°K  and a value f o r  from Figure (2^),H° i s found to be approximately 0.9 e.v.  Since H = 2.3 e.v. at 10$ s t r a i n , t h i s suggests that about l.k goes into overcoming the i n t e r n a l stress B.  e.v  t^.  Single Crystals 1. ' Zone Refining Laue back r e f l e c t i o n photographs were taken of each single  c r y s t a l 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 f o r a s u f f i c i e n t l y  polished and an I n s u f f i c i e n t l y polished specimen. remove about .005"  I t was necessary to  to 'obtain sharply defined spots.  The orientation of each specimen was plotted and a l l were found to be of the same orientation to within 2 ° .  specimens  The position of the  t e n s i l e axis with.respect to the standard st.ereographic t r i a n g l e i s shown i n Figure  (26).  Figure 25b.  Laue photograph of an i n s u f f i c i e n t l y polished specimen.  - kk  The single c r y s t a l rods were a l l radiographed a number of small gas holes.  -  and found to contain  These were also observed while the specimens  were being polished during which the holes would appear as small surface A photograph of a p i t t e d specimen i s shown i n Figure ( 2 7 ) -  pits. 2.  S l i p Systems Metallographic examination of the.deformed single c r y s t a l s  revealed that at l e a s t two s l i p systems were operative during  deformation.  Near the necked region evidence of three or more systems was frequently observed.  T y p i c a l s l i p traces are shown.in Figure ( 2 8 ) .  - 45 -  Figure 27.  Figure 2 8 .  Gas holes i n single c r y s t a l specimens,  Typical s l i p traces on single c r y s t a l , specimen 6A, 2 1 0 X .  The s l i p d i r e c t i o n was found to be  <Clll> .  Due to the complex-  i t y of separating the s l i p markings of the systems involved, an attempt to determine the s l i p planes was abandoned.  It i s already known that the s l i p planes i n a b . c. c. metal are {lio} ,  {ll2} and [123} •  Using the known orientation and a stereographic  projection, the angles between the t e n s i l e axis and a l l of the possible s l i p planes were measured.  From these angles Schmid factors were calculated  and the largest of these are tabulated i n Table I I I .  - 46 -  TABLE III Possible S l i p Systems S l i p Direction  S l i p Plane  Schmid Factor  •{ml  (101)  .468  [In]  (101)  .482  [111]  (112)  .468  [In]  (ll2)  .456  [in]  (213)  .483  [In]  (213)  • 500  The  Schmid factors are roughly the same f o r each type of s l i p  Because of t h i s , an average factor of 0.48  plane.  was  used to calculate  the resolved shear stress i n the single c r y s t a l s . 3.  Strain Rate Change Tests These tests were c a r r i e d out using the same s t r a i n rates as f o r The r e s u l t s are plotted i n Figures (29)  the p o l y c r y s t a l s .  Z\ Tj .is /  a  o  (30).  e s s e n t i a l l y independent of s t r a i n and stress at  0°C  0  and 100 C.  At -72 C  A  'r u  increase  and  of  A  °£  with  to be regarded as d e f i n i t e .  appears to decrease with deformation. a a  and £ shown by specimen 5A.is too small  The  o  O  T=  .14  -72°C  O  SB  O.  7£  ./3l  /.80  /.6 0  2.20  200  T= IOO°C • 6A O SD  .10  2  o  .05  «  6  .80  0  -o */.oo  I.2Q  <j ./5L 7A . 0 ° C  66, -/9°C 5A,  ./o  25°C •<D-  o  -o-  .051 .90 Figure 2 9 .  /\^T ,a  10  -50 1-30 or* (PS./. x 10*) f o r an increase i n £ vs. t f o r single c r y s t a l s .  J.  a  ,25L  SB  O  O  7E 7A  <D • 6B  .20L  ©  •  5A  e  x  6A  5D  0°C  o-  v. cc  =fl3=  I9°C  .10  _0_  •  •  o  -G  0  G-  25°C  • /oo°c  -05L  0  .02  .04  .06  .08  S T R A I N  Figure 30 •  A^Co for 3.  .12  E  an increase i n £ vs. £.for single c r y s t a l s .  .14  J6 4=-  cx>  It i s evident that the r e s u l t s f o r single c r y s t a l s are not as reproduceable as those f o r p o l y c r y s t a l s .  The.reason f o r t h i s may he the  gas holes i n the single c r y s t a l specimens. o Due to the l i m i t e d d u c t i l i t y of vanadium below 200 K most of the  tests were performed above t h i s temperature.  However, one c r y s t a l  was tested at 78°K to determine the s t r a i n rate s e n s i t i v i t y of the flow stress at, t h i s temperature.  I t was possible to cycle the s t r a i n 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 t h i s temperature. Sudden reductions of stress occurred i n the e l a s t i c region by a sharp cracking sound.  accompanied  Metallographic examination revealed twin l i k e  markings on the surface of the specimen. (Figure (3l))-  Also a Laue back  r e f l e c t i o n photograph of the supposed twinned region was taken using a beam i n the form of a long narrow s l i t (Figure 3 2 ) . f i l m i s s l i t i n an i d e n t i c a l manner.  Each spot on the  Small spots corresponding to the  gaps were not d e f i n i t e l y observed but they could have been masked by the background r a d i a t i o n .  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  not  observe twinning down to 78°K«  ent  orientation, which could explain t h i s  i n h i s work on vanadium, d i d  However, h i s specimens v/eosEof a d i f f e r 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 r e s u l t s of these tests are plotted i n Figures (33) and  In a l l cases, except f o r specimen 6D, / \ ^ decreases s l i g h t l y ... a  (34).  with deformation.  Since only four points could be obtained f o r 6D, i t can-  not be d e f i n i t e l y concluded that A ^ 5.  increases with  r  C  Q  .for t h i s specimen.  Rate Equation • The a c t i v a t i o n volume f o r each specimen which was subjected t o  a s t r a i n rate change t e s t was calculated and plotted as a function of stress and temperature i n Figures ('19) and (20).  As f o r the polycrystals,  an average value f o r v* was used i n the plot of v* against temperature. The a c t i v a t i o n volume i s generally larger f o r the single crystals than f o r the polycrystals at any given temperature.  This difference de-  creases as the temperature increases to 100°C where v* i s the same f o r both types of specimen. The energy, H, was calculated from equation ( l 4 ) and i s plotted as a function of s t r a i n i n Figure ( 2 l ) . A Q at 10$ s t r a i n was calculated from equation (15) f o r the various mean t e s t i n g temperatures and i s plotted as a function of temperature i n Figure (24b) assuming  = 0 at 0°K.  A Q was also calculated from equation (8) and i s plotted as a function of s t r a i n i n Figure (22).  A Q again shows a small tendency to  decrease with s t r a i n although.this decrease i s not as marked as f o r the polycrystals.  60  fOO°C —  -26°C  -50  .401 .60  70  80  .90  1.00  110  2.901  6C  /00°C  >-66C 0  a; - -70[__ & 60  .30  BO  TO  40k\00°C~B°C  5C -©-  .301  1  -251 .60  -70 Figure 33-  .80  A f  .90  |.00  ro  1.10  f o r a decrease i n temperature vs tT f o r single crystals. /  a  - 54 An average value f o r A at.10$ s t r a i n was calculated by p l o t t i n g against ln^A/g)  =p  |  i n Figure '(-35) •  This y i e l d e d a value f o r  of 2 5 . 9 which i s quite comparable to that f o r the p o l y c r y s t a l l i n e  material. .  6.  Y i e l d Points Y i e l d points similar to those observed during the tests on poly-  crystals also appeared in.the single c r y s t a l t e s t s .  They appeared under  the same temperature conditions as f o r the p o l y c r y s t a l s but.in t h i s case they d i d not show up u n t i l a f t e r about 5$ s t r a i n .  They increased i n size  somewhat with increasing s t r a i n but they were always smaller than those observed during the p o l y c r y s t a l t e s t s . One t e s t was performed by straining continuously to 20$ s t r a i n at the basic s t r a i n rate and then the rate was increased by a factor of ten;  This was done to determine whether the size of the y i 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 s t r a i n i n a normal t e s t .  This i s shown.in Figure  (17). A summary of the r e s u l t s i s included i n Table IV.  - 55 -  Figure 35»  Strain rate s e n s i t i v i t y of the flow stress vs. temperature s e n s i t i v i t y to f i n d ln(A/g) . (equation  (15))•  - 56 -  C.  1.  Summary of Results  The lover y i e l d stress and the flow stress of vanadium both obey H a l l Petch equations. -1/2  2.  The slopes of the y i e l d stress and flow stress versus d •'  plots are  not i d e n t i c a l f o r vanadium. 3.  The slope, k^y* increases with temperature. .  k.  Vanadium does not obey the Cottrell-Stokes Law. A, f or decreases s l i g h t l y with ' t  5' 6.  As t' r  a  a  i s constant  »  i s Independent of grain size over the range tested.  The a c t i v a t i o n volume f o r single c r y s t a l specimens diverges from that f o r polycrystals at low temperatures.  7.  The a c t i v a t i o n energy, H°, i s roughly 1 e. v. f o r thermally  activated  flow i n vanadium.  8.  Y i e l d points are observed upon reloading a specimen i n t e s t s involving a . temperature of 100°C.  9.  Vanadium deforms by twinning at -196°C when the orientation i s as i n Figure ( 2 6 ) .  - 57 TABLE IV  Summary of Results  Specimen  Single Crystals  T  H(e.v.),£ =0.1  287  1-7  .76  327  1-5  .81  310  1.5  •75  1.1*.  • 76  237  1.3  . .hi  327  2.3 •  356  Polycrystals  •  Q(ew), £ = 0 . 1  •72  H^e.v.^  0.9  25.9 '  0.9  25.2  - 58 IV. A.  DISCUSSION  Hall-Fetch. Equation From the preceding r e s u l t s come two i n t e r e s t i n g observations: 1.  k^y increases w i t h 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  b a s i s of C o t t r e l l ' s ' t h e o r y , k^y should depend on the stress to unlock d i s l o c a t i o n sources pinned by impurity atoms. should decrease w i t h . i n c r e a s i n g temperature. i s observed.  Hence, one expects that k  LY  However, j u s t the opposite  This suggests that the C o t t r e l l theory, i s an o v e r - s i m p l i f i c a t i o n .  These observations d i f f e r from those of other workers  5,6  i n that  5 their 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. I f one attempts t o use t h i s idea t o e x p l a i n 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 w i t h temperature.  Such  a s i 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 f o r the behavior.of k  ,. i s worthy of , LY  d i s c u s s i o n . . I f the sources are pinned d i s l o c a t i o n s , t h e n those  specimens  annealed at a higher temperature ( l a r g e r g r a i n s ) might be expected t o have fewer sources.  This would r e s u l t i n a l a r g e r value f o r 1  (equation  (6))  - 59 than i n f i n e grained specimens.  However, f o r any given g r a i n size  1  should be the same and while d i f f e r i n g heat treatments may a f f e c t the a c t u a l values f o r k^y* i t . i s hard t o see how i t could a f f e c t the r e l a t i v e values, over a range of temperature. The d i f f e r e n c e between k-^y and k ^ i s not s u r p r i s i n g . assumes .that because k ^ = k^y i  n  Conrad  h i s measurements the same deformation  mechanism i s i n v o l v e d i n y i e l d i n g as i n flow. s t r a i n of .05 as i t was i n t h i s work.  His k was measured a t a fl  In a p o l y c r y s t a l l i n e specimen considerable work hardening occurs near the g r a i n boundaries.  Hence, one might expect that the f l o w stress  at give percent s t r a i n would depend on the g r a i n s i z e through the rate of work hardening i n the grains and t h i s i s not n e c e s s a r i l y r e l a t e d t o the lower y i e l d s t r e s s .  An equivalence between y i e l d i n g and flow mechanisms  would then suggest an equivalence between the a c t i v a t i o n and stopping of sources and there i s no- sound reason t o make t h i s assumption. The r e s u l t s of t h i s work are i n disagreement w i t h the work of Conrad and i t appears as i f h i s conclusions were premature.  No a l t e r n a t i v e  explanation of these processes can be given here because of the l i m i t e d scope of t h i s  p o r t i o n of the research. A large scale research programme  i s r e q u i r e d i n v o l v i n g several d i f f e r e n t b. c. c. metals. B. . Thermally A c t i v a t e d Flow At present there e x i s t the f o l l o w i n g suggested mechanisms f o r c o n t r o l l i n g the thermally a c t i v a t e d f l o w of d i s l o c a t i o n s i n body-centered cubic metals:  - 60 -  1.  •1.  Intersection with a.forest  2.  Impurity atoms i n t e r a c t i n g with dislocations  3.  Cross s l i p p i n g of screw dislocations  k.  Large P e i e r l s stress  5.  Non-conservative motion of jogs i n screw dislocations  Intersection with a Forest The forest mechanism, although quite acceptable for f.  c. c.  metals, i s probably t h e , l e a s t acceptable for b . c. c. metals. At the beginning of deformation i t . i s u n l i k e l y that the forest Q  density would be greater than 10  dislocations per square centimeter. . I f  a .uniform d i s t r i b u t i o n of forest dislocations i s assumed t h i s would lead to a spacing of 10 ^ cm. between the "trees" of the f o r e s t . -21 3 value for the a c t i v a t i o n volume of vanadium i s 10  cm .  A typical Now assume  that d, the a c t i v a t i o n d i s t a n c e , . i s of the same order of magnitude as b , the Burger's vector.  This i s 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 i s a smaller spacing than b^ one would expect for a .uniformly d i s t r i b u t e d f o r e s t . 21 It has been observed  t h a t . i n f.  c. c. metals the dislocations  are .distributed non-uniformly,.that i s , they e x i s t as tangles alternating with r e l a t i v e l y d i s l o c a t i o n free areas.  I t . i s pointed out that the flow  stress i s determined by the spacings i n these tangles.  These tangles  could have a smaller spacing between the "trees" of the forest and therefore be consistent with the measurements i n t h i s research.  However, t h i s  - 6itype of d i s l o c a t i o n structure i s not generally observed i n b. c. c. metals, i n f a c t , Gregory's electron micrographs show a r e l a t i v e l y 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 d i s l o c a t i o n free regions corresponds to stage I or easy glide i n f . c. c. metals, and at the beginning of stage I I the d i s t r i b u t i o n i s r e l a t i v e l y uniform.  A l l of the measurements i n t h i s  work were taken i n the l a t e r stages of deformation a f t e r  heterogeneous  y i e l d i n g had ceased. The a c t i v a t i o n energy f o r a forest mechanism w i l l depend on the short range forces on the moving d i s l o c a t i o n s .  Since dislocations i n  b. c. c. metals are generally non-extended these forces would be related to the energy of jogs formed during i n t e r s e c t i o n . estimated to be :"dynes/cm  This energy i s  generally  3  b /lO which, assuming a shear modulus of k.S x 10  y i e l d s an a c t i v a t i o n energy of roughly 0.5 e.v.  11  This i s somewhat  smaller.than the measured values i n t h i s work but i t does not disagree • enough to discount the forest mechanism on t h i s basis alone.  Because of the numerous active s l i p systems i n 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 s t r a i n . In a uniform forest t h i s would lead to a tenfold decrease i n the a c t i v a t i o n volume with s t r a i n and a tenfold, increase i n  ZX^a.-  The r e s u l t s of t h i s  research do not show t h i s trend.  If the forest i s non-uniform and spreads by the expansion of tangled regions into clear ones, then the a c t i v a t i o n volume could remain constant i f flow i s controlled by the spacing i n the tangles.  However, i f  - 62 -  t h i s s t r u c t u r e d i d e x i s t t h e t a n g l e d r e g i o n s would soon cover t h e whole specimen  and one might  of d e f o r m a t i o n .  expect t o see a change i n v * i n t h e . l a t t e r  stages  T h i s i s n o t observed.  I t would seem l i k e l y by these arguments t h a t t h e forest.mechanism does n o t . c o n t r o l t h e r m a l l y a c t i v a t e d f l o w i n vanadium.  2.  Impurity Atoms The  i n t e r a c t i o n o f a u n i f o r m d i s t r i b u t i o n o f i n t e r s t i t i a l impur-  i t i e s w i t h moving d i s l o c a t i o n s has been proposed as a p o s s i b l e mechanism r e s p o n s i b l e f o r t h e t h e r m a l component o f t h e f l o w s t r e s s . k The work o f Heslop and P e t c h  o n . i r o n suggests t h a t t h e temperature  dependent p a r t o f 0 ~ f i does n o t depend on t h e amount o f C + N i n s o l u t i o n . 23 Calculations  using linear e l a s t i c i t y  t h e o r y i n d i c a t e t h a t .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 p o i n t e d Out t h a t t h e use o f l i n e a r elasticity energy  t h e o r y a l m o s t . c e r t a i n l y would y i e l d  This  i s c a l c u l a t e d from e q u a t i o n (19).  E  =  //b fff  1 + >>  Where E i s a maximum f o r 9 =  sinQ ~  slightly  parameter  T  ,  ,  ,  ^9)  v  i s t h e volume change  The shear m o d u l u s , ,  more than one h a l f o f t h a t f o r i r o n .  f o r vanadium  Also, since the l a t t i c e  i s l a 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  a l s o be s m a l l e r . 0.5 e.v.  A  A  and :p = b , and A"V  due t o one i n t e r s t i t i a l p e r u n i t c e l l . is  an o v e r - e s t i m a t i o n .  that A V  will  Hence, a maximum v a l u e f o r E i n vanadium would be r o u g h l y ;  - 63 We would expect the a c t i v a t i o n volume f o r an i n t e r s t i t i a l mechanism to decrease s l i g h t l y with s t r a i n .  The number of i n t e r s t i t i a l s w i l l  not change with s t r a i n and therefore the a c t i v a t i o n 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 d i s l o c a t i o n w i l l r i s e to a higher l e v e l on the force-distance curve.  The observed a c t i v a t i o n  volume i s r e l a t i v e l y constant with s t r a i n or i n some cases increases s l i g h t l y i n disagreement  with an i n t e r s t i t i a l mechanism.  If the i n t e r s t i t i a l s are assumed to be uniformly d i s t r i b u t e d one can calculate the spacing  L  i n the a c t i v a t i o n volume.  In the polycrys-  t a l l i n e material there, i s roughly one i n t e r s t i t i a l per 1000 i s equivalent.to an i n t e r s t i t i a l i n every 500 spacing of  3/500  a ^  8  atoms.  This  unit c e l l s and an average  a where a i s the l a t t i c e parameter.  have a spacing between the i n t e r s t i t i a l s of roughly 2.5 i s too small to agree with the measured value f o r  L  x 10  -7  Thus we cm.  i n this work.  This For  -7  the single c r y s t a l s  L  would be 3 x 10  cm. which.is also too small to  agree with t h i s work. It might be argued that t h e • i n t e r s t i t i a l s may be segregated due to the presence of dislocations and tend to be closer together along d i s location l i n e s before s t r a i n i n g .  However,, i t . i s l i k e l y that saturation of  dislocations would occur, before a large f r a c t i o n 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 f a r from that for a uniform d i s t r i b u t i o n .  - 6k 3-  Cross Slipping of Screw. Dislocations Observation of wavy s l i p l i n e s i n b . c. c. metals has been i n t e r -  preted as evidence of cross s l i p p i n g of screw\ d i s l o c a t i o n s .  This process  2k has been.described by many authors  .  It involves a screw . segment of an  obstructed dislocation, loop, cross s l i p p i n g to a p a r a l l e l s l i p plane.  The  screw segment.is locked i n the new s l i p plane by an edge dipole which i s formed.in the cross s l i p process.  M u l t i p l i c a t i o n 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 i s inversely proportional to the length of edge dipole or jogs formed.in the cross s l i p process.  The stress to operate a source  i s given by ^  =  8H'fi\ p. )  JT  (20)  .  Where d' i s the spacing,between the new and o l d s l i p planes. Jy = 0 . 3 (Poisson's r a t i o ) and taking  =  Assuming  k x 10^ p s i . which i s as  .  -8  high an applied stress as any specimen received one finds d = 30 x 10 or d ";= J  cm.  lib. T h i s . i s a minimum value for d  stress was less than k x 10^ p s i . effectively  J  since i n most tests the applied  This distance i s too great to be  overcome by thermal fluctuations and the energy of the edge  dipoles produced i n the process would be p r o h i b i t i v e .  k.  Peierls-Nabarro Force The idea of the Peierls-Nabarro force as the mechanism c o n t r o l l i n g  thermally activated flow is a popular one.  However, the choice of t h i s  - 65 -  4 mechanism has i n most cases been somewhat a r b i t r a r y .  Heslop and Petch  chose i t for lack of a better one rather than as a result of any intensive study.  Actually a high P e i e r l s stress might be expected for b . c. c. metals  since departure from close packing should favour the narrowing of d i s l o cations . It i s d i f f i c u l t to support t h i s mechanism on the basis of numerical results.  The P e i e r l s force w i l l be large only for dislocations  i n close packed d i r e c t i o n s , that i s , reason for assuming that most  <^111^> i n b . c. c.  lying  There i s no  of the dislocations are of t h i s type a l -  26 though electron microscopy studies on s i l i c o n iron (Figure  do-  support t h i s  (36)).  Figure 36.  Transmission electron micrograph of dislocations i r o n , after Low and Turkalow.  in silicon  - 66 However, t h i s .is a p a r t i c u l a r case. Stein and Low ^ who 2  One must keep i n mind the work of  showed that the mobility  of edge dislocations .is  approximately twenty times that f o r screw.i dislocations Since the s l i p d i r e c t i o n i s extended screw segments.  <^111>  i n s i l i c o n iron.  one would expect to observe loops with  The deformation takes place mainly by edge d i s -  locations which are not oriented to experience a large P e i e r l s force.  This  behavior has not been observed.in any pure b. c. c. metal.  The P e i e r l s mechanism i s based on a .series of energy maxima and minima p a r a l l e l to the d i s l o c a t i o n l i n e as i n Figure (37).  PROPAGATION DIRECTION  <MI>  Figure 37*  Formation of a .dislocation kink i n a P e i e r l s force f i e l d .  - 67 The P e i e r l s mechanism would operate i n the following manner. .In order- that a d i s l o c a t i o n propagates, a segment of a . c r i t i c a l length, must overcome the b a r r i e r . .  L  c  r  is  determined by the a t t r a c t i v e  L  c  r  r  force  between two kinks of opposite sign and the applied s t r e s s . . I f the loop segment i s longer than L then the kinks w i l l propagate i n opposite d i r e c cr tions and the dislocations w i l l move forward. Thus L i s determined by cr the applied stress L  c r  •  At lower temperatures ^  can be smaller than at high temperatures.  i s larger and therefore  This would be h e l p f u l i n  explaining the dependence of v* on temperature. 28  Read  has pointed out that the c o n t r o l l i n g factor i n t h i s pro-  cess i s the formation of a.kink.  The sideways motion of a kink occurs  with r e l a t i v e , ease compared with the forward motion of the d i s l o c a t i o n against the h i g h . P e i e r l s s t r e s s . 29  Seeger  , has calculated the kink energy, i n a publication on an  i n t e r n a l f r i c t i o n peak due to kink formation i n copper.  Because of the  many "a p r i o r i " assumptions i n t h i s c a l c u l a t i o n a r e l i a b l e vanadium cannot be made.  estimate for  To obtain a value for the kink energy i t would  be necessary to do i n t e r n a l 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 i n t e r n a l f r i c t i o n sources i n vanadium. 5•  Non-Conservative Motion of Jogs i n Screw Dislocation When a screw d i s l o c a t i o n acquires a jog, i t . i s of the edge  orientation and therefore can glide conservatively only i n the d i r e c t i o n of i t s Burgers vector which i s p a r a l l e l to the screw, dislocation,'/ l i n e .  - 68 I t w i l l . t h e r e f o r e cause a drag on the motion of a screw d i s l o c a t i o n .  This  i s a l s o true f o r jogs i n mixed d i s l o c a t i o n s , that - i s , any d i s l o c a t i o n w i t h a screw component. I f these jogs are t o move w i t h the d i s l o c a t i o n ' t h e y must move non-conservatively, l e a v i n g "behind a t r a i l of vacancies or i n t e r s t i t i a l s . The c r e a t i o n of i n t e r s t i t i a l s i s e n e r g e t i c a l l y unfavourable.  The vacancy  mechanism would be governed by the energy of formation of vacancies which i s of the order 2 t o 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 c o n s e r v a t i v e l y along the d i s l o c a t i o n loop u n t i l they can g l i d e c o n s e r v a t i v e l y w i t h the loop or they 34  may combine with jogs of opposite sign and a n n i h i l a t e 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 d i s l o c a t i o n loop.  This would require a higher  d i 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 d i s l o c a t i o n . I t i s more l i k e l y that the jog w i l l l a g behind and e i t h e r form an edge dipole or move i n non-conservative jumps. The a n n i h i l a t i o n of jogs would r e s u l t . i n an increase i n a c t i v a t i o n volume w i t h s t r a i n .  A l s o , even i f the jogs do not a n n i h i l a t e each other  the spreading of the d i s l o c a t i o n loop; would r e s u 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 w i t h the f o r e s t and i t i s conceivable that an e q u i l i b r i u m could be set up such that the spacing of jogs remains roughly constant.  • , •  '  - 69 -  . It i s 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 t h i s i n the early stages of deformation. the largest scatter i n the plots of  2\ ^  a  In fact,, i n some of the t e s t s , against  ^  w  a  s  a  observed for  the f i r s t point.  The main objection to the vacancy mechanism here i s that the jmsasured a c t i v a t i o n energy i s small.  The energy of formation of vacancies  i n b . c. c. metals i s roughly 2 to 3 e  v  - which,is not very compatible  ..with the measured value, of H ' * 1 e. v. for vanadium. 0  There i s some ques-  t i o n , however, as to the v a l i d i t y of assuming the energy to form a vacancy at a jog i s the same as that for forming one i n a perfect l a t t i c e .  It  is  known f o r instance that the energy to form a vacancy i n an a l l o y i s much less than that f o r the pure.metal..  Unfortunately a d e t a i l e d analysis of  t h i s problem would require sophisticated quantum mechanical calculations which are beyond the scope of t h i s work.  6.  Comparison Between Polycrystals and Single Crystals The divergence between polycrystals and single crystals i n the  plots of v* against temperature i s 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 p o l y c r y s t a l 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 r e s u l t i n a smaller a c t i v a t i o n length, L , and hence a.smaller activation volume.  However t h i s would not explain the convergence of the  two curves with increasing temperature.  It i s possible that the difference i n v* l i e s i n the a c t i v a t i o n distance, d.  This might be brought about by a difference i n the shapes  - 70 -  of the force-distance curves f o r the two materials.  I t i s d i f f i c u l t to  explain why t h i s should occur i f the same mechanism i s 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 i n t e r n a l stress f i e l d .  From the measurements  of H° and H i t appears that the i n t e r n a l stress f i e l d i s larger f o r the p o l y c r y s t a l l i n e specimens.  This could be responsible f o r a difference i n  the shapes of the force-distance curves.  The value of the f a c t o r for the two materials.  A  i n equation (9) i s roughly the same  The trend f o r .A  to decrease by 10  to 10  with  s t r a i n might be explained by a decrease i n the number of mobile d i s l o c a t i o n s . One would not expect with s t r a i n .  A.or  to decrease by more than a factor of f i v e  A decrease i n the number of mobile dislocations means those  remaining must move at a higher v e l o c i t y which would r e s u l t . i n an observed work hardening.  7.  Mechanism I t i s rather d i f f i c u l t to pick out a single mechanism as the  c o n t r o l l i n g one i n thermally activated flow.  The factcthat the opinions  of so many authors d i f f e r i s evidence enough f o r that.  What i s generally  done i s to select a mechanism which f i t s ' t h e data best and declare t h i s to be the most probable  one.  On the b a s i s . o f 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 P e i e r l s  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 i s whether the d i s l o c a t i o n s do l i e predominantly  ^111^  i n the  d i r e c t i o n s . There i s not s u f f i c i e n t evidence available at t h i s time to answer t h i s question. The jog mechanism i s the best one f o r explaining the increase i n a c t i v a t i o n volume with s t r a i n which was observed i n some specimens.  The  jogs are the only obstacles which could spread apart with deformation.  An  argument against the jog mechanism i s t h a t . i t cannot operate at low temperatures.  This i s because the vacancy formed must d i f f u s e away im-  mediately or i t will.'.draw the jog ;bac.k again.  At low temperatures  this  d i f f u s i o n might be d i f f i c u l t . On the basis of preceding arguments i t appears as i f the P e i e r l s mechanism is.the most likely, one.  However, this, may be just because there  i s i n s u f f i c i e n t evidence available to dispute i t .  What one should .inquire at t h i s juncture i s just  how v a l i d i s  'the- application of rate theory to t h i s process. . I t has been assumed that there i s a single thermally activated mechanism operative. •In f a c t , i t i s possible that the average e f f e c t s of two or more mechanisms were, measured. Since two mechanisms-;:wauld l i k e l y have d i f f e r e n t 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 i n a l l the proposed mechanisms i t . i s conceivable that.more than one mechanism could well be the c o n t r o l l i n g ones.  What, i s needed at t h i s time i s a r e l i a b l e  mathematical analysis of the P e i e r l s mechanism so that. i t sr.'.importance i n b, c. c. metals can be d e f i n i t e l y established. The assumption used.in the derivation of equations 13, lh,  and  15, that A i s not s i g n i f i c a n t l y dependent on stress, i s worthy of discussion.  This assumption means that the number of mobile dislocations does 30  not change greatly during a s t r a i n rate change.  Martinson  contends  t h a t , i n L i F the number of mobile dislocations i s a sensitive function of the mean stress p r e v a i l i n g during a stress increase. to explain his i r r e v e r s i b i l i t y of /^H^.  He used t h i s hypothesis  He found that A ^ o - f o r a s t r a i n  rate increase was not equal to that f o r a decrease and also that upon unloading and reloading at the same s t r a i n 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 i n t h i s work. • The flow stress was completely reversible to within the l i m i t s  of exper-  imental accuracy. Also, the s t r a i n rate change tests of 100 times yielded a  A^*.  approximately twice as large as that f o r a factor of 10 change from the same basic s t r a i n rate.  I f the number of mobile dislocations were to  increase with the mean stress p r e v a i l i n g then we would expect /^^«.for a change of 100 times to be less than twice that f o r the factor of 10 change.  C.  Y i e l d Points The y i e l d points obtained upon reloading a f t e r c y c l i n g have been  observed by many' experimenters but have not been s a t i s f a c t o r i l y explained by any of them.  I t i s important to understand t h i s process however, i n  order to j u s t i f y the method of extrapolation used t o determine the flow stress i n a s t r a i n rate or temperature change t e s t . Some authors have attempted to explain t h i s e f f e c t 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 t h e i r relaxed configuration back to the pattern which they were i n just before the crosshead was stopped.  This hypothesis i s strengthened by the  i r r e v e r s i b i l i t y of p l a s t i c flow.  When a stress i s removed from a c r y s t a l '  one might expect the dislocations i n p i l e ups to run back along t h e i r slip planes causing large reverse p l a s t i c flow. . In f a c t , t h i s does not .occur and therefore there must be some obstacles to this;;process.  Makin^Jias described a process f o r f . c. c. c r y s t a l s i n which Cottrell-Lomer s e s s i l e dislocations are formed on unloading and these prevent large scale reverse p l a s t i c i t y . .  Since energy i s released upon  •formation of these s e s s i l e s a higher stress than the flow stress i s required t o dissociate them.  This would r e s u l t .in an observed y i e l d drop.  Makin' ' observed y i e l d drops at temperatures to 100°C.  ranging from - 195°C  The magnitude of the drops appeared to be independent of temp-  erature and was proportional to the reduction i n stress on unloading.  The  y i e l d drops i n t h i s thesis however were observed only i n tests involving  - 7^ -  the temperature 100 loading.  C, and they showed no dependence .on the amount o f un-  These f a c t s , combined w i t h .the f a c t t h a t a mechanism o f s e s s i l e  d i s l o c a t i o n f o r m a t i o n s i m i l a r t o the C o t t r e l l - L o m e r mechanism i s not known f o r b . c. c. metals suggest t h a t Makin's  t h e o r y does not a p p l y t o vanadium.  32 Birnbaum-  has a l s o d i s c u s s e d t h i s e f f e c t , i n f . c. c. m e t a l s .  He  suggests t h a t upon u n l o a d i n g , the d i s l o c a t i o n s r e l a x u n t i l they r e a c t e l a s t i c a l l y w i t h f o r e s t d i s l o c a t i o n s and are h e l d .  The y i e l d p o i n t i s a  r e s u l t o f 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 c o n f i g u r a t i o n .  This  type o f mechanism c o u l d c e r t a i n l y be a p p l i c a b l e t o b . c. c. metals  since  w i t h the numerous s l i p p l a n e s a h i g h f o r e s t d e n s i t y i s l i k e l y .  However,  i t would be d i f f i c u l t t o e x p l a i n the temperature dependence-, of the y i e l d drops on t h i s b a s i s .  Birnbaum  observes t h a t - t h e magnitude  of the y i e l d  drops are independent o f temperature over a range from 72°K t o 293°K. I t t h e r e f o r e appears t h a t the y i e l d p o i n t e f f e c t , 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. m e t a l s .  The  t h a t i t o c c u r s o n l y a t h i g h temperatures or a t a low temperature c o o l i n g 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.  fact after  One  might  expect t h a t the d i s l o c a t i o n becomes l o c k e d by i n t e r s t i t i a l s when they stop and an u n l o c k i n g s t r e s s i s r e q u i r e d t o r e i n i t i a t e f l o w .  Now  the times o f r e l a x a t i o n are f a r t o o s h o r t 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 t e s t s the c r o s s h e a d was  stopped f o r l e s s t h a n one minute b e f o r e r e l o a d i n g w h i l e i n the temperature change t e s t s t h e . l o a d was  r e l a x e d f o r no more t h a n twenty minutes. , I n  s p i t e of the l o n g e r r e s t i n g p e r i o d s d u r i n g temperature change t e s t s the  - 75. y i e l d drops showed no s i g n i f i c a n t difference i n magnitude from those i n the strain-rate change t e s t s .  The  i n t e r s t i t i a l content, i n the single c r y s t a l s was  about  one  h a l f of that f o r the polycrystals and. the observed y i e l d drops were smaller for the single c r y s t a l s . might be  This r e l a t i o n s h i p suggests that . i n t e r s t i t i a l s  involved.  Thus we are looking f o r a d i f f u s i o n c o n t r o l l e d . i n t e r s t i t i a l locking mechanism which can operate i n a very short time ..relative to normal d i f f u s i o n times. i s Snoek ordering.  The  only mechanism which seems to s a t i s f y t h i s  A b r i e f description of t h i s mechanism w i l l be given.  It i s well known that the i n t e r s t i t i a l s i n b. c. c. metals cause tetragonal d i s t o r t i o n when they occupy the (OOg-) (cube edge) p o s i t i o n i n the unit c e l l .  There are three types of these s i t e s corresponding to  the three directions of tetragonality or the three space axes. applied stress the. i n t e r s t i t i a l s .in equal f r a c t i o n s . preferable  Under no  should be present i n these three s i t e s  However, under'an applied stress some s i t e s w i l l be  energetically to others and at temperatures where d i f f u s i o n can  take place the i n t e r s t i t i a l s  should r e d i s t r i b u t e themselves so as to i n -  crease the population of the s i t e s of lower energy.  This, i s known as  the  Snoek e f f e c t .  33 Schoeck andftB&eger  state that the stress f i e l d of a d i s l o c a t i o n  should have a similar e f f e c t on the i n t e r s t i t i a l s . • Hence we expect a r e d i s t r i b u t i o n which.would lower the energy of the d i s l o c a t i o n and therefore lock i t .  This involves the movement of atoms a distance  of only  - 76 one atomic jump.and. could occur i n a very short.time.  This process appears  to explain the observed y i e l d 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 t h e i r v e l o c i t y .  On the basis of t h i s explanation i t would be correct to extrapolate through the y i e l d drop, as was done i n t h i s t h e s i s , to obtain the correct value for the flow s t r e s s .  - 77 V.  1.  CONCLUSIONS  ,  A s i m i l a r i t y between the mechanisms of y i e l d i n g and of flow i s not observed for vanadium.  2.  k  increases with temperature for vanadium.  3'  Vanadium does not obey the Cottrell-Stokes Law.  it-..  It i s u n l i k e l y that the f o r e s t ,  cross s l i p and i n t e r s t i t i a l mechanisms  control thermally activated flow i n vanadium. 5.  The c o n t r o l l i n g mechanism for thermally activated flow i s probably either the Peierls-Nabarro force or the non-conservative motion of vacancy jogs.  However, there i s also some evidence against each of  these. 6.  It i s possible that there i s not a unique mechanism c o n t r o l l i n g thermally activated flow.  7-  Vanadium deforms by twinning .at 78°K when oriented i n c e r t a i n d i r e c t i o n s .  8.  Y i e l d drops obtained upon reloading are a t t r i b u t e d to Snoek ordering.  - 8 7  V I . RECOMMENDATIONS FOR FURTHER WORK  The results of t h i s research indicate the need for an extensive study into the importance of the Peierls-Nabarro force i n b . c. c.  metals.  This would, include i n t e r n a l f r i c t i o n work to determine d i s l o c a t i o n kink energies and electron microscopy studies to study the orientation of d i s location lines.  Before t h i s i s done i t i s u n l i k e l y that any further pro-  gress w i l l be made i n the work hardening theory.  It would also be i n t e r e s t i n g , t o of the y i e l d point e f f e c t s .  conduct an extensive  investigation  This would involve work at higher temperatures  than were used here and a thorough investigation of the dependence of t h i s effect on.impurity content.  - 79 -  VII.  APPENDICES  - 8o APPENDIX I A.  Annealing Data for Grain Growth  Specimen  Annealing Temp. °C  Annealing Time hr.  microns  8A 8B 8C 8D 9A 9B 9C 9D 10A *  870 . 94o 1020 1090 870 9ko 1020' 1090 1090  1 1 1 1 1• 1 1 1 4  LOB IOC 10D 10E L1A,B,C & D I1E,F, 12A|& B L2C,D,E & F L3A,B & C L3D, E & F  1050 995 900 820 870 1020 1090 920 970  4 4 1-5 1 l 2 8 l 2  Rod 10 deformed 4$ before  annealing.  Grain Size  14.5 16.6 26.5 33-4 17.8 24 35-6 52.8 Too large and nonuniform 55 43 Not measured 12.4 19.4 54.6 69 26.1 35  APPENDIX II A.  Polycrystals  Specimen  Area of Cross section in.  2  8A  .0lk5  8B  .011+5 .0137 .0126 .0109 .0102 .00800  8C 8D  9A 9B 9C  Gauge length in. 1.01+5 1.028 0.925 1.015 1.057 0.99k 0.906  9D 10A  .OO693  .00916  O.962 1.01+8  10B IOC 10D  .011+1 .011+1 .0132  0.995  10E 11A 11B 11C 11D HE 11F 12A 12B  .0137  0.950 O.9I+I 0.978 0.95^ O.986 0.923 0.937 0.973 0.956  .OO967  .0111 .0115 .0098k ?0098l+ .00816 .0117 .0111  O.9I+2  0.992  LY k P . S . I . x 10 1+.1+9 I+.69 1+.52 1+.32 1+.1+8 k.63 Not Observed 3-k5  3.96 k.36  1+.76 5-09 1+.07 6.1+0 4.11 k.86 3-55 6.1+0 3.52  Testing Temp. o„ C  Crosshead Speed .  in.mm.  -1  Approx. Uniform Deformation  25 25 25 25 25 25 100  .02 .02 .02 .02 .002-.02 .01-0.1 .01-0.1  21+ 25 25 20 13 15 12  100  .01-0.1  13  25 25  .02 .02  -  25 0 100, -8 -72 100 0 100,8 -72 100  Loaded nonaxially. Not tested due to undesirable grain s i z e .  18 18  .02 .01-0.1 .01 .01-0.1 .01-0.1 .01-0.1 .01 .01-0.1 .01-0.1  Remarks  1o  Damaged during mounting.  20 16 13 17 lk 18 12 16 17  I  OO  H 1  Specimen  Area of Cross Section  .  xn.  2  12C 12D 12E 12F  .0115  13A 133 13C 13D 13E 13F  .0100 .0109 .0104 .0115 .0106 .00984  .0115 .0107 .OO967  Gauge length in.  0-951 O.965 0.912 0.956 O.928 1.027 0-954 0.940 0.917 0.915  Testing Temp.  LY  4 P.S.I. x 10  4.61 3.19 6.18 3.26 4.80 6.36 3.65 4.54 6.27 3.57  .  °c 0 100,8 -72 100 0 -72 100 .0 -72 100  Crosshead Speed . :-.i i n . mm.  .01-0.1 .01 .01-0.1 .01-0.1 .01-0.1 .01-0.1 .01-0.1 .01-1.0 .01-1.0 .01-6.1  Approx. Uniform Deformation  Remarks  *  14  14 15 15 15 13 15 14 13 13  I  CD  ro  B.  Single Crystals  Specimen  Area of Cross Section . 2 in.  Gauge Length in.  4A  .00709  .972  4B he  .00785 .00818  0.991 0.936  5A 5B 5C 5D 6A 6B 6c 6D 6E 7A 7B 7C  .0106 .0104 .OO833 .OO968 .00664 .00566 .00738 .00849 .00832 •00753 .OO899 .OO967  0.937 0-933 O.967 0.950 0.972 0.910 0.955 O.917 0.931 0.953 0.975 O.987  7-D 7E  .0119 -.0111  O.947 a 9i4  Temperature 0  C  100,8 25  Crosshead Speed -1 in.mm.  -  Approx. Uniform Elongation  -  .01 .002-.02  12 10  25 -72 100,8 100 100 100 100,66 100,-26 -19 0 100,-72 -195  .01-0.1 .01-0.1 .01 .01-0.1 .01-0.1 .01-0.1 .01 .01-0.1 .01-0.1 .01-0.1 .01 .01-0.1  19 19 15 10 26 23 12 14 14 10 11-4  0,-72 -72  .01 -.01-0.1  8 17  Remarks  Damaged during mounting. non a x i a l loading  Twinning observed  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°  =  d-  =  n  calculated mean grain diameter th diameter from the i . the r e s i d u a l ( d  Q  measurement  - d^).  Measurement  d ° (microns)  d^ (microns)  1 2 3  26.1 26.1 26.1 26.1 26.1 26.1  23-3 30.0 22.6 27.2 27.5 26.1+  h 5 6  ]|(microns) 2.8 3-9 3-5 1.1 1.1+  0.3  U v j j = 13.0 The probable error of the mean i s calculated from Peter's formula: P. E . =  .8U5  /n(^-i) / 6"x 5  n i = 1 x 13-0 = 2.0  Hence the mean grain diameter i s  d  Q  =  2 6 . 1 + 2 . 0 microns  i = 1,2,  n  - 85 From these calculations i t i s apparentt that the precision of the grain size measurements i s 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 i n use. This i s about the same as the accurary i n reading the chart which i s about •+ 0.2 of the smallest scale d i v i s i o n .  j Sample c a l c u l a t i o n for the lower y i e l d stress Assume the following t y p i c a l values: F  = kOO l b . (load at the lower y i e l d point) XJJL  d = Q . l i n . (diameter of specimen) F_ LY  =  TT <r  Differentiating logarithmically:  I crLY , £ j ^ y (TLY  "  F  LY  X -  .  d  Accuracy of F^y = +.."2 l b . Accuracy of  d  = + .001"  The maximum error i s :  S aiLY  -gr—  „ m2 ,—_ .001 = - '& + 2  =  ^ LY  The accuracy of the measurements of cern.  The smaller  Zi t > r  a  025 2 or  Z^'tg^  c d  i s of p r i n c i p a l con-  "the more d i f f i c u l t a precise measurement w i l l be.  - 86 This i s evident i n the plots i n Figure (19) where the high temperature plots show: .the-'greatest ;  scatter.  The accuracy i s estimated by:  (f ( A ^ a )  S (^a2) +  =  ^afl  A ^a  A^a At  S (  100°C the specimens were generally tested at 500 l b . f u l l  scale deflection on the Instron..  Hence  $  = $ ^  AT4 Atr  a  ~  l  =  +1  lb.  13 l b .  13 ~  a  T h i s , of course, i s a maximum estimate as the errors could compensate' for each other. Errors i n the calculations of the rate parameters cannot be e a s i l y estimated without knowing the exact shape of the force - distance curve. acatter i n the plots of culations of accuracy.  The  AQ against temperature (Figure 2^b) and i n c a l -  A Q.by d i f f e r e n t methods serves as a rough measure of the  From these comparisons one might expect our estimations of  o and H to be out as much as  „,  25yo»  AQ  • However, t h i s . i s s t i l l adequate for deter-  mining a mechanism of flow since the t h e o r e t i c a l calculations are probably not much more precise than t h i s . ••  - 8? VIII. BIBLIOGRAPHY 1.  H a l l , E . 0 . , Proc. Phys. Soc. Lond. B 64 7 4 7 , ( 1 9 5 1 ) .  2.  Petch, N. J . , . J . Iron and Steel I n s t . , Y[k 25, ( 1 9 5 3 ) .  3-  Cracknell, A . , Petch, N. J . , Acta Met. 3 1 8 6 , ( 1 9 5 5 ) .  k.  Heslop, J . , . Petch, N. J . , . P h i l . Mag. 1 8 6 6 , ( 1 9 5 6 ) .  5-  Conrad, H . , Schoeck, G . , Acta Met. 8 7 9 1 , ( i 9 6 0 ) .  6.  Petch, ET.' J.,. P h i l Mag. 3 I O 8 9 , ( 1 9 5 8 ) .  7-  C o t t r e l l , A... H . , Trans. A.. I . M . E . , 212 1 9 2 , ( 1 9 5 8 ) .  ..8i . B a s i n s k i , . Z . 9-  S., P h i l . Mag. 4 3 9 3 , ( 1 9 5 9 ) .  Conrad, H . , Wiedersich,  H . , Acta.Met. 8 1 2 8 , ( i 9 6 0 ) .  .  10.  Mordike, B . L . , Haasen, P . , P h i l . Mag. 7 4 5 9 , ( 1 9 6 2 ) .  11.  Gregory, D.. P . , Stroh, A. N . , Rowe, G. H . , to be published.  12.  Basinski,  13.  Moraike, B . L . , Z. Metallkunde  Ik.  Conrad, H . , J . Iron and Steel Inst. 198 364, ( 1 9 6 1 ) .  Z . S.,  C h r i s t i a n , J . W., Aust. J . Phys. 13_ 2 9 9 , ( i 9 6 0 ) . 9 586, (1962).  1 5 . , Conrad,, II., P h i l . . Mag. 5 7 4 5 , ( 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,  18.  Fraser,. R. W., M. A. Sc. T h e s i s , . U n i v e r s i t y of B r i t i s h Columbia,' ( i 9 6 0 )  19.  Conrad, H . , Frederick,  20.  H i r s c h , P. B . , P h i l . Mag. 7 6 7 , ( 1 9 6 2 ) .  21.  Mott, N. F . , . Trans. A. I . M. E . 2 l 8 9 6 2 , ( i 9 6 0 ) .  22.  Kuhlmann-Wilsdorf,  23.  C o t t r e l l , . A. H . , Dislocations and P l a s t i c Flow i n C r y s t a l s , University Press, (l953)> page 1 3 4 .  (1961)  S., Acta Met. 10 1 0 1 3 , ( 1 9 6 2 ) .  D . , Trans. A. I . M. E . , 224 1 0 4 7 , ( 1 9 6 2 ) .  24. ' Low, J . R . , Guard, R. ¥ . , Acta Met.., 7 1 7 1 , ( 1 9 5 9 ) .  Oxford  - 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 3 6 2 , ( 1 9 6 2 ) .  27-  S t e i n , D . F . , Low, J . R.-, J . Appl. Phys. 31 3 6 2 , ( i 9 6 0 ) .  28.  Read, W . T . , Dislocations i n C r y s t a l s , 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 2 8 7 , ( 1 9 5 8 ) .  32.  Birnbaum, H. K . , Acta Met. 9_ 3 2 0 , ( 1 9 6 1 ) .  33-  Schoeck, G.,Seeger, A . , Acta Met. 7 4 6 9 , ( 1 9 5 9 ) -  McGraw H i l l , N. Y . , ( 1 9 6 3 ) ,  34. Frank, F . C , II Nuovo Cimento 7 Supp. 1 - 2 , 3 8 6 , (1958).  

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