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Theories of hot-pressing : plastic flow contribution Rao, A. Sadananda 1971

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THEORIES OF HOT-PRESSING:  PLASTIC FLOW CONTRIBUTION  BY  A. SADANANDA RAO B.Sc., U n i v e r s i t y o f Mysore, I n d i a , 1965 B.E. ( M e t a l l u r g y ) , I . I . S c . , B a n g a l o r e , I n d i a , 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department of  METALLURGY  We accept t h i s t h e s i s as conforming required  t o the  standard  THE  UNIVERSITY OF BRITISH COLUMBIA August, 1971  In  presenting  this  thesis  an a d v a n c e d d e g r e e the L i b r a r y  shall  I f u r t h e r agree for  scholarly  by h i s of  this  written  at  the U n i v e r s i t y  make  it  It  by  permission.  Depa r t m e n t Columbia  shall  the  requirements  B r i t i s h Columbia, for  I agree  r e f e r e n c e and copying of  this  that  not  copying  or  for  that  study. thesis  t h e Head o f my D e p a r t m e n t  is understood  financial gain  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of  for extensive  p u r p o s e s may be g r a n t e d  for  fulfilment of  freely available  that permission  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  ii  -  ABSTRACT  The  c o n t r i b u t i o n of  p l a s t i c f l o w to o v e r a l l d e n s i f i c a t i o n of a  powder compact d u r i n g h o t - p r e s s i n g has been a n a l y s e d . t h i s a n a l y s i s i s the i n c o r p o r a t i o n of hot-working m a t e r i a l s a t e l e v a t e d temperatures pressing conditions.  The  c h a r a c t e r i s t i c s of  i n t o an e q u a t i o n a p p l i c a b l e to h o t -  empirical equation r e l a t i n g  s t r a i n r a t e to s t r e s s i s e = A a  n  The b a s i s of  steady  state  and f o r the d e n s i f i c a t i o n o f a  1 dD powder compact, the s t r a x n r a t e e = — -j^- . the p a r t i c l e s a r e assumed to be spheres and geometric b.c.c. acting  c o n f i g u r a t i o n s : c u b i c , orthorhombic,  are  considered.  D  rhoTabic dodecahedron and  T a k i n g i n t o c o n s i d e r a t i o n the e f f e c t i v e  and  i n t o another  6 a r e geometric c o n s t a n t s and can be c a l c u l a t e d from 'A' and  deformation i n a r b i t r a r y Computerized  'n' a r e m a t e r i a l c o n s t a n t s . f  R  f  the  D i s the r e l a t i v e  i s the r a d i u s of sphere a t any  stage of  units.  p l o t s of D vs t were o b t a i n e d f o r l e a d - 2 %  n i c k e l and alumina.  antimony,  E x p e r i m e n t a l v e r i f i c a t i o n of these p l o t s was  out u s i n g h o t - p r e s s i n g d a t a f o r l e a d - 2 % antimony, n i c k e l and The hot-compaction  of temperatures  can  o  d e n s i t y o f the compact, and  The  stress  e q u a t i o n which i s shown below:  o  packing geometry.  spheres.  packing  a t the p o i n t s of c o n t a c t , the e q u a t i o n s f o r the s t r a i n r a t e  be combined and arranged  where  four d i f f e r e n t  experiments  carried  alumina  were c a r r i e d out over a  range  f o r each m a t e r i a l and under d i f f e r e n t p r e s s u r e s .  e x p e r i m e n t a l data f i t t e d w e l l w i t h the t h e o r e t i c a l  f o r the orthorhombic  prediction  model. However, a d e v i a t i o n a t the i n i t i a l  stage  - iii  of compaction was  -  encountered i n most cases.  This deviation  was  explained on the basis of the contribution to d e n s i f i c a t i o n by p a r t i c l e movement or rearrangement at the i n i t i a l stage, which could not be taken into account i n the t h e o r e t i c a l derivation. The stress concentration  factor i . e . , the e f f e c t i v e stress acting  at necks between p a r t i c l e s has been calculated.  This was  found to be  very much higher than that previously used by other workers.  The  t h e o r e t i c a l equation for the e f f e c t i v e stress i s  a  °  e f f  =  (D /V 2  a i  / 3  R -l) 2  This equation predicts an e f f e c t i v e s t r e s s , which i s more than an order of magnitude higher than that predicted by several empirical equations used previously.  - iv-  TABLE OF CONTENTS Page TITLE PAGE  '. .  ABSTRACT  ± i i  TABLE OF CONTENTS  iv  LIST OF FIGURES  v i i  LIST OF TABLES  x  ACKNOWLEDGEMENT  CHAPTER I . 1.1  xi  INTRODUCTION  1  D e n s i f i c a t i o n Due t o P a r t i c l e Rearrangement and Fragmentation  3  1.2  D e n s i f i c a t i o n Due t o P l a s t i c Flow  1.3  D e n s i f i c a t i o n Due t o D i f f u s i o n a l Mass T r a n s p o r t .  14  1.4  O b j e c t i v e s of the P r e s e n t Work  19  CHAPTER I I . 11.1  5  THEORETICAL FORMULATIONS T h e o r e t i c a l Models  2  1  2  2  II.l.a  Simple Cubic P a c k i n g  2  ^  Il.l.b  Orthorhombic  2  ^  II.l.c  Rhombohedral P a c k i n g  ^7  II. l.d  B.C.C. P a c k i n g  2  8  0  Packing  11.2  General Equations  3  II. 3  Application  30  of the E q u a t i o n  CHAPTER I I I . EXPERIMENTAL VERIFICATION OF THEORY S e l e c t i o n and T h e o r e t i c a l P l o t s  3  3  3  3  III. l  Material  III.2  E x p e r i m e n t a l T e s t s and Procedures  3  ^  I I I . 2.a Apparatus  3  ^  - v -  Page III.2.a. i  Lead-2% Antimony  34  III.2.a.ii  Nickel  42  I I I . 2 . a . i i i Alumina  44  I I I . 2.b Procedures  CHAPTER IV. IV. 1  44  III.2.b.i  Hot-Pressing  44  III. 2.b.ii  Hot-Compression o f Pb-2% Sb and N i 46  III.2.b.ii.l  Lead-2% Antimony  46  III.2.b.ii.2  Nickel  48  RESULTS  4  Metals IV. 1.a  49 Lead-2% Antimony  49  IV. l . a . i  R e l a t i v e D e n s i t y v s . Time ..  49  IV.l.a.ii  D v s . t a t a Constant Temperat u r e and Under V a r y i n g P r e s s u r e s  53  D v s . t a t a Constant P r e s s u r e and w i t h V a r y i n g Temperatures  53  IV.1.a.iii  IV.l.b  Nickel IV.l.b.i  IV.l.b.ii  53 D v s . t a t a Constant Temperat u r e and under V a r y i n g P r e s s u r e s D v s . t a t a Constant  Non-Metal IV.2.a  IV. 3  61 61  Alumina IV.2.a.i  53  Pressure  and w i t h V a r y i n g Temperatures IV. 2  9  61 D v s . t a t a Constant Temperat u r e and under V a r y i n g P r e s s u r e s  Strain-Rates During Hot-Pressing  61 61  - vi-  Page  CHAPTER V.  DISCUSSION  64  V.l  E f f e c t i v e S t r e s s During Hot-Pressing  64  V.2  A c t i v a t i o n Energy  70  V.3  Packing and Deformation  Study Geometry I n s i d e a D i e . .  73  V.3.a  P a c k i n g Geometry  73  V.3.b  Deformation  75  V.4 CHAPTER V I .  Geometry  L i m i t a t i o n o f the P r e s e n t A n a l y s i s SUMMARY AND CONCLUSIONS  80 82  CHAPTER V I I . SUGGESTIONS FOR FUTURE WORK  85  APPENDICES _  87.  BIBLIOGRAPHY  103  - vii -  LIST OF FIGURES Figure 1 2  Page A schematic d e n s i f i c a t i o n (a) P l o t  of l n ( l - D )  1100°C and p r e s s u r e (b) P l o t of l n ( l - D ) 1150°C and p r e s s u r e (c) P l o t o f l n ( l - D ) 1100°C and p r e s s u r e (d) P l o t o f l n ( l - D ) 1500°C and p r e s s u r e 3  curve  2  v s . time f o r f u s e d s i l i c a a t o f 1000 p s i ( a f t e r v s . time f o r f u s e d o f 1000 p s i ( a f t e r v s . time f o r f u s e d o f 2500 p s i ( a f t e r v s . time f o r f u s e d o f 2500 p s i ( a f t e r  Vasilos ^)... s i l i c a at Vasilos )... s i l i c a at Vasilos )... s i l i c a at Vasilos )...  Temperature dependence o f d e n s i t y o f BeO w h i l e h o t p r e s s i n g a t 4000 p s i ( a f t e r McClelland-!- )  12  2  2 5  8  2 5  9  2 5  10  7  4  5  (a) S h r i n k a g e 2000 and 3000 (b) Shrinkage 4000 and 5000  p l o t s o f alumina p s i (after Fryer p l o t s of alumina p s i (after Fryer  Geometric r e l a t i o n s h i p Kakar )  7  7  p r e s s e d a t 1300°C, ) p r e s s e d a t 1300°C, )  of d i f f e r e n t models  16 17  (after 23  1 3  6  7  D v s . t p l o t f o r Pb-2% Sb at 100°C under 1500 p s i (theoretical plots)  35  D v s . t p l o t f o r Pb-2% Sb a t 150°C under 918 p s i (theoretical plots)  36  D v s . t p l o t f o r N i at 800°C under 2162 p s i (theoretical plots)  37  D v s . t p l o t f o r N i a t 900°C under 2105 p s i (theoretical plots)  38  D v s . t p l o t f o r AI2O3 a t 1600°C under 5000 p s i (theoretical plots)  39  11  H o t - p r e s s i n g apparatus used f o r Pb-2% Sb shots  41  12  H o t - p r e s s i n g apparatus used f o r N i and A ^ O ^  43  13  C a l i b r a t i o n curve f o r t h e pressure-gauge used f o r h o t - p r e s s i n g of N i and A ^ O ^  45.  D v s . t p l o t f o r Pb-2% Sb at 150°C under 918 p s i ( s o l i d , l i n e s r e p r e s e n t the t h e o r e t i c a l curves f o r 4 d i f f e r e n t models)  50  7  8  9  10  14  spheres  - viii  -  Figure 15  D v s . t p l o t f o r N i a t 800°C under 2162 p s i ( s o l i d l i n e s r e p r e s e n t the t h e o r e t i c a l curves f o r 4 d i f f e r e n t models)  16  D v s . t p l o t f o r N i a t 900°C under 2105 p s i ( s o l i d l i n e s r e p r e s e n t the t h e o r e t i c a l curves f o r 4 d i f f e r e n t models)  17  (a) D v s . pressures curve f o r (b) D v s . pressures curve f o r  18  (a) D v s . t p l o t f o r Pb-2% Sb a t 918 p s i and d i f f e r e n t temperatures ( s o l i d l i n e s r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model o n l y ) (b) D v s . t p l o t f o r Pb-2% Sb at 1010 p s i and d i f f e r e n t temperatures ( s o l i d l i n e r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model o n l y )  19  D vs. t p l o t f o r N i a t 800°C under d i f f e r e n t p r e s s u r e s ( s o l i d l i n e r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model o n l y )  20  D v s . t p l o t f o r N i a t 900°C under d i f f e r e n t p r e s s u r e s ( s o l i d l i n e r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model o n l y )  21  D v s . t p l o t f o r N i a t a, c o n s t a n t s t r e s s and d i f f e r e n t temperatures ( s o l i d l i n e r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model o n l y )  22  D v s . t p l o t f o r A 1 0 a t 1600°C under d i f f e r e n t p r e s s u r e s ( s o l i d l i n e r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model o n l y )  23  t p l o t f o r Pb-2% Sb at ( s o l i d l i n e represents the orthorhombic model t p l o t f o r Pb-2% Sb at ( s o l i d l i n e represents the orthorhombic model  2  100°C under d i f f e r e n t the t h e o r e t i c a l only) 150°C under d i f f e r e n t the t h e o r e t i c a l .only)  3  °eff  vs. D plot f o r d i f f e r e n t models and a l s o o f a e q u a t i o n s used by p r e v i o u s workers  24  l n A v s . 1/T p l o t f o r Pb-2% Sb (The p o i n t r e p r e s e n t e d by a dot w i t h a r i n g i s from the d e n s i f i c a t i o n curve at 200°C)  25  l n A v s . 1/T p l o t f o r N i (The p o i n t r e p r e s e n t e d by dot w i t h a r i n g i s from the d e n s i f i c a t i o n curve at 700°C)  a  - ix -  Figure 26  27 28  29  Page E f f e c t of container size on the e f f i c i e n c y of packing one-size spheres (after McGearylS)  74  Coordination number vs. (D - D ) for Pb-2% various temperatures (after KaSarl3)  76  Sb at  Microstructure of Ni spheres hot-pressed and 2162 p s i  at 800°C  Microstructure of Ni spheres hot-pressed and 1477 p s i  at 900°C  78 79  - x LIST OF TABLES Table I  II  Page Predominant Mechanisms of D e n s i f i c a t i o n under Given H o t - P r e s s i n g C o n d i t i o n s ."  4  D e n s i f i c a t i o n E q u a t i o n Based on D i f f u s i o n a l Mass Transport  18  Geometric Constants f o r D i f f e r e n t Models  32  IV V  M a t e r i a l Constants f o r D i f f e r e n t M a t e r i a l s Test Conditions f o r Hot-Pressing  40 47  VI  Y i e l d S t r e s s o f S i n g l e C r y s t a l Sapphire Slip) (After Kronberg )  69  III  1 5  (Basal .  - x i-  ACKNOWLEDGEMENT  The author i s g r a t e f u l f o r t h e a d v i c e and encouragement g i v e n by h i s r e s e a r c h d i r e c t o r Dr. A . C D . Chaklader.  Thanks a r e a l s o  extended t o o t h e r f a c u l t y members and f e l l o w graduate s t u d e n t s f o r many h e l p f u l d i s c u s s i o n s . Research C o u n c i l  F i n a n c i a l a s s i s t a n c e from the N a t i o n a l  (NRC Grant No. A-2461) i s g r a t e f u l l y acknowledged.  CHAPTER I INTRODUCTION  Hot-pressing refers to d e n s i f i c a t i o n of a powder compact under pressure at elevated temperatures.  It i s usually conducted  at a tempera-  ture several hundred degrees below the temperature at which s i n t e r i n g i s carried out (without any external pressure).  Hot-pressing has gained  importance as a commercial process since one can achieve greater density at lower temperatures  and shorter times than i n conventional  s i n t e r i n g , and with much more microstructural control. The d e n s i f i c a t i o n of powder compacts during hot-pressing has been generally studied as a function of temperature,  pressure and time.  For  k i n e t i c analysis, the change of r e l a t i v e density (or bulk density) as a function of time under isothermal conditions and at a constant pressure i s usually determined.  A t y p i c a l d e n s i f i c a t i o n curve i s shown  schematically i n Figure 1, where the r e l a t i v e density vs time i s plotted at a constant temperature and pressure. recognised on the curve.  Three d i s t i n c t regions are  The extent of the contribution from each of  the three mechanisms of mass transport, i . e . p a r t i c l e rearrangement with or without fragmentation, p l a s t i c flow, and d i f f u s i o n , depends on the type of material, the temperature and the stress l e v e l used during hot-pressing.  For example, b r i t t l e solids i n a compact at r e l a t i v e l y  0-5  Time (arbitrary scale) Figure 1.  A schematic d e n s i f i c a t i o n curve.  - 3 low  temperature  hot-pressing.  tend t o f r a c t u r e as soon as a l o a d i s a p p l i e d d u r i n g The  d e n s i f i c a t i o n i s p r i m a r i l y a c h i e v e d by  and p a r t i c l e movement. the temperature  T h i s w i l l be  f o l l o w e d by p l a s t i c  fragmentation flow, only i f  and s t r e s s are h i g h enough t h a t d i s l o c a t i o n s  can move.  I f no d i f f u s i o n a l p r o c e s s occurs a f t e r t h i s s t a g e , the compact reach an "end-point  d e n s i t y " which i s below i t s t h e o r e t i c a l d e n s i t y .  M e t a l powder compacts can be expected  t o have a much g r e a t e r  to d e n s i f i c a t i o n from p l a s t i c flow than ceramic flow would i n g e n e r a l be a s i g n i f i c a n t t i o n at h i g h temperatures The expected  5-10  Also, plastic  c o n t r i b u t i n g f a c t o r to d e n s i f i c a -  stresses. be  t o be predominant i n v a r i o u s m a t e r i a l s and i n wide ranges and p r e s s u r e are summarized i n T a b l e  D e n s i f i c a t i o n Due Felten  Al^O^  and h i g h  oxides.  contribution  d i f f e r e n t mechanisms of m a t e r i a l t r a n s p o r t which can  temperature  1.1  will  y).  He  1.  to P a r t i c l e Rearrangement and  Fragmentation  c a r r i e d out a number of h o t - p r e s s i n g i n v e s t i g a t i o n s  h a v i n g t h r e e d i f f e r e n t average conducted  1300°C) to minimize densification.  particle sizes  a l l h i s experiments  of  (0.05  u, 0.3  at low temperatures  the c o n t r i b u t i o n from p l a s t i c flow and  with u and  (750-  diffusion  to  H i s d a t a f i t t e d w e l l w i t h a p l a s t i c flox^j e q u a t i o n , 19  developed  by Murray, L i v e y and W i l l i a m s ,  of d e n s i f i c a t i o n .  There was  but o n l y at the l a t e r  a l a r g e d e v i a t i o n at the i n i t i a l  d e n s i f i c a t i o n from the t h e o r e t i c a l p r e d i c t i o n . that t h i s  deviation  a t  He  concluded  the v e r y e a r l y stage must be  to another mode of d e n s i f i c a t i o n which he c a l l e d p a r t i c l e T h i s rearrangement may  a l s o be a s s o c i a t e d w i t h  stages  stage of that attributed  rearrangement.  fragmentation.  - 4 -  Table 1.  Predominant Mechanisms of D e n s i f i c a t i o n under Given HotPressing Conditions.  Low temperature  Fracture  High pressure P l a s t i c flow  Large p a r t i c l e s i z e  Boundary d i f f u s i o n  High temperature ^.  Lattice diffusion  Low pressure Small p a r t i c l e s i z e  - 5 -  S i m i l a r l y Chang and Rhodes,"*" who uranium-carbide range 500  powder compacts a f t e r h o t - p r e s s i n g i n the  to 1500°C under p r e s s u r e s v a r y i n g from  atmospheres, concluded a significant  I.2  s t u d i e d the m i c r o s t r u c t u r e s of  that p a r t i c l e s l i d i n g  r o l e i n the i n i t i a l  D e n s i f i c a t i o n Due  and  temperature  10,000 t o 46,000 fragmentation  played  stage of d e n s i f i c a t i o n .  to P l a s t i c Flow  Among the c o n t r i b u t i o n s to the t h e o r y of h o t - p r e s s i n g , the b e s t known i s perhaps t h a t of Murray, Rodger and W i l l i a m s ( a l s o Murray, 19 L i v e y and W i l l i a m s )  who  m o d i f i e d the s i n t e r i n g  and S h u t t l e w o r t h t o e x p l a i n the observed v a r i o u s o x i d e s and  carbides.  The  theory of Mackenzie  behaviour  i n h o t - p r e s s i n g , of  f i n a l form of the e q u a t i o n , n e g l e c t i n g  the c o n t r i b u t i o n from s i n t e r i n g , i s  dF  =  M  4n" <1-D>  where D i s the r e l a t i v e d e n s i t y at time t , P i s the a p p l i e d p r e s s u r e  and  Q  n i s the v i s c o s i t y .  I t was  assumed by Mackenzie and  t h a t a l l s o l i d s can be d i v i d e d i n t o two As metals  succeeded  groups - Newtonian and Bingham.  and o x i d e s can o n l y f l o w above a c r i t i c a l s t r e s s ,  approximated equation  Shuttleworth  the c r y s t a l l i n e s o l i d s as the "Bingham s o l i d  (la) contains a v i s c o s i t y  term.  ".  they Hence,  T h e i r p l a s t i c f l o w theory  i n e x p l a i n i n g the i n c r e a s e d r a t e of d e n s i f i c a t i o n w i t h  p r e s s u r e and  the e f f e c t  of p r e s s u r e on end p o i n t d e n s i t y at a c o n s t a n t  temperature. Mangsen, Lambertson and Best  16  s t u d i e d the h o t - p r e s s i n g c h a r a c t e r i s t i c s  - 6 -  of  aluminum-oxide t o understand  t i o n process.  They observed  the mechanisms i n v o l v e d i n the  that t h e i r experimental data  densifica-  fitted  19  v e r y w e l l w i t h the Murray, L i v e y and W i l l i a m s v a l u e s were c a l c u l a t e d from these d a t a c l o s e l y agreed Murray's interpret used  viscosity which  literature.  e q u a t i o n has a l s o been employed by V a s i l o s  25  to  the d a t a o b t a i n e d d u r i n g h o t - p r e s s i n g of f u s e d s i l i c a .  He  the i n t e g r a t e d form of e q u a t i o n l a , as f o l l o w s :  =  t = 0; D = D  constant  observed  q  |^ t  Q  The  +  c  (the i n i t i a l  c equals-ln(l-D ).  Figure 2.  or  The  as a f u n c t i o n of temperature  w i t h the v a l u e s r e p o r t e d i n the  19  -In(l-D)  when  equation.  (lb)  pressed d e n s i t y ) .  So the  integration  He p l o t t e d - l n ( l - D ) vs time, as shown i n  l i n e a r r e l a t i o n between these two  quantities i s essentially  r e g a r d l e s s of d i f f e r e n c e s i n p a r t i c l e s i z e , a p p l i e d p r e s s u r e ,  temperature.  curves agreed  V i s c o s i t y v a l u e s computed from  r e a s o n a b l y w e l l w i t h those  the s l o p e s of  r e p o r t e d i n the  these  literature.  McClelland"'"''' m o d i f i e d Murray's e q u a t i o n to f i t h i s h o t - p r e s s i n g d a t a f o r BeO  and A^O^,  t a k i n g i n t o account  p r e s s u r e which accompanies c l o s i n g used  the p o r e s .  effective  E q u a t i o n l a can  be  to d e s c r i b e the v a r i a t i o n of d e n s i t y w i t h time o n l y i f  pressure, is  the change of  remains  constant during h o t - p r e s s i n g .  The p r e s s u r e which  e f f e c t i v e i n c l o s i n g the p o r e s , however, i s not e q u a l to the a p p l i e d  pressure.  Due  to the presence  of the v o i d s , whose s i z e changes d u r i n g  the h o t - p r e s s i n g o p e r a t i o n , the a r e a over which the a p p l i e d l o a d i s transmitted increases with increasing density.  Thus, the p r e s s u r e  - 7 -  F i g u r e 2(a)..  P l o t of l n ( l - D ) vs. 1100°C and  time f o r f u s e d s i l i c a  p r e s s u r e o f 1000  psi  (after  at  Vasilos  2 5  ).  - 8 -  10  20  30 TIME  Figure 2(b).  40  50  (MIN)  Plot of ln(l-D) vs. time for fused s i l i c a at 1150°C and pressure of 1000 p s i (after V a s i l o s ) . 2 5  -  9  -  Figure 2 ( c ) . P l o t of l n ( l - D ) vs. time f o r fused s i l i c a at 25 1100°C and pressure of 2500 p s i ( a f t e r V a s i l o s  ).  - 10  Figure 2(d).  -  P l o t of l n ( l - D ) vs. time f o r fused s i l i c a at 1150°c and pressure of 2500 p s i (after V a s i l o s ) . 2 5  e f f e c t i v e i n c l o s i n g the pores i s a f u n c t i o n of p o r o s i t y . final  form of equation  X  McClelland's  i s as f o l l o w s :  D (dD/x{ln[l/(l-x  2 / 3  ) ] + a l n x})  =  Kt  (lc)  o  where x i s (1-D), D i s t h e d e n s i t y D  T  o  = the i n i t i a l  a  =  K  =  dens  /P  T  c  3P —-  4n  =  yield  P  =  pressure  n  =  viscosity.  c  a t time t  stress  The l e f t h a n d d i f f e r e n t values  side of equation o f D.  l c can be i n t e g r a t e d g r a p i c a l l y f o r  McClelland"'"^  t h e o r e t i c a l c u r v e s and e x p e r i m e n t a l  found good agreement between t h e data  as  is  shown i n F i g u r e  3.  12 K a k a r and C h a k l a d e r  developed a mathematical model f o r i n t e r p r e t a -  t i o n s o f d e n s i f i c a t i o n due t o p l a s t i c f l o w . the  p a r t i c l e and p a c k i n g c o n f i g u r a t i o n i n a d i e , t h e y c a l c u l a t e d t h e  change o f d e n s i t y the p o i n t s  o f a compact o f s p h e r e s p r o d u c e d by d e f o r m a t i o n a t  of contact  of the spheres.  c o n f i g u r a t i o n s were c o n s i d e r e d ; r h o m b i c d o d e c a h e d r o n (z is  Knowing t h e geometry of  = 12)  t h e c o o r d i n a t i o n number. Due  to i n d e n t a t i o n  as shown i n F i g u r e space and t h i s w i l l  cubic  calculation, several  (z  hexagonal prism  = 6),  and t e t r a k a i d e c a h e d r o n  (Z = 14)  T h e s e a r e shown i n F i g u r e  at the point  5a, m a t e r i a l w i l l increase  For t h i s  of contact  be t r a n s f e r r e d t o f i l l  the density  where  5.  b e t w e e n two  o f t h e compact.  spheres the void  The  8),  (Z =  final  Z  - 12 -  50 •  40  1  —  0  1  30  ' 60  '—• 90  1  TIME  Figure 3.  •  120  ,  150  180  210  240  (min)  Temperature dependence of density of BeO while hot-pressing at 4000 p s i (after M c C l e l l a n d ) . 17  - 13 -  d e n s i t y due  to these i n d e n t a t i o n s at s e v e r a l p o i n t s of c o n t a c t  upon Z) i s g i v e n as  (depending  follows:  D  -  D  Q  =  3/2  D  D  -  D  q  =  101.5  Q  ( | )  (at  2  |  <  0.3)  (2a)  or  where Z = 6,  8 and  12:  compact at a > 0 and flat  log Z  D and  D o  (|)  (2b)  2  are the r e l a t i v e d e n s i t i e s of  at a = 0 r e s p e c t i v e l y , a i s the r a d i u s of  f a c e produced by  i n d e n t a t i o n , and  the the  R i s the r a d i u s of the sphere at 12  any  stage of d e n s i f i c a t i o n .  Kakar and  Chaklader  tested this  u s i n g hot-compaction data for l e a d , K-monel and A ^ O ^ obtained  a v e r y good f i t w i t h  equation  spheres and  the t h e o r e t i c a l l y p r e d i c t e d v a l u e s  they of  the hexagonal p r i s m a t i c model. They i n t r o d u c e d e q u a t i o n s 2a and  2b by  c o n t a c t between two  Using  pressure  and  temperature dependent terms i n t o  considering  particles  t h a t d e f o r m a t i o n at the p o i n t s  i s e s s e n t i a l l y a s e l f - i n d e n t a t i o n process. 9 10  the y i e l d i n g c r i t e r i o n of Hencky  and  Ishlinsky  i . e . , the s t r e s s (o^) n e c e s s a r y f o r y i e l d i n g d u r i n g three  for indentation  indentation i s  times' the y i e l d s t r e s s (Y) of the m a t e r i a l , the e f f e c t i v e s t r e s s  a c t i n g on each f a c e to 0.60 to  of  (cr ) and  the  c o o r d i n a t i o n number 8  r e l a t i v e d e n s i t y of green compact), e q u a t i o n  (corresponding  (2a) was  transformed  give D - D  Q  -  D £  a '  (  2  C  )  - 14  -  An exponential temperature dependence of the y i e l d strength modifies equation 2c i n t o o exp(Q/RT) _  D D - D  o  =  -2  (2d)  In t h i s equation a i s the applied pressure, A i s a pre-exponential constant and Q i s the a c t i v a t i o n energy f o r y i e l d i n g . 22 Rummler and Palmour  studied the d e n s i f i c a t i o n k i n e t i c s of  magnesium-aluminate s p i n e l during vacuum hot-pressing. that below 1350°C, the  They observed  d e n s i f i c a t i o n k i n e t i c s of the s p i n e l compacts  were i n agreement w i t h Murray's expression which i n i t s i n t e g r a l form p r e d i c t s a l i n e a r r e l a t i o n between l o g p o r o s i t y and time.  Hence they  concluded that the d e n s i f i c a t i o n of magnesium-aluminate s p i n e l below 1350°C was mainly due to p l a s t i c flow. I.3  D e n s i f i c a t i o n Due  to D i f f u s i o n a l Mass Transport  Koval'Chenko and Samsonov"'" proposed a hot-pressing equation based 4  20 on the Nabarro-Herring  creep mechanism, which was v e r i f i e d by studies 23 on tungsten-carbide and chromium-carbide. Scholz and Lersmacher s i m p l i f i e d Koval'Chenko and Samsonov's"*"^ equation and showed that i t 19 took a form s i m i l a r to the equation developed by Murray et a l . v i z _ dQ dt  =  3P 4q  yi  where Q = p o r o s i t y . 2 Coble and E l l i s  c a r r i e d out hot-pressing experiments on aluminum  single c r y s t a l spheres at 1530°C.  They measured the e f f e c t of load on  - 15 -  i n i t i a l neck growth between s i n g l e c r y s t a l spheres  and observed  the neck areas were l a r g e r than those o f s i n t e r e d spheres were, c o n s t a n t f o r a l l times between 10 and 480 minutes.  and They  that  that  they  calculated  the p l a s t i c flow c o n t r i b u t i o n t o the d e n s i f i c a t i o n d u r i n g h o t - p r e s s i n g . T h i s c a l c u l a t i o n was  based  on the h y d r o s t a t i c n a t u r e of s t r e s s  the geometric r e l a t i o n s between p a r t i c l e s . for  and  From t h i s they found  aluminum o x i d e the c o n t r i b u t i o n o f p l a s t i c  the p r e s s u r e s n o r m a l l y used i n h o t - p r e s s i n g was  that  flow t o d e n s i f i c a t i o n at small.  Hence they  concluded that the f i n a l stage o f d e n s i f i c a t i o n of alumina occurs by enhanced d i f f u s i o n under the i n f l u e n c e o f s t r e s s . 21 R o s s i and F u l r a t h of  a l s o s t u d i e d the k i n e t i c s of the f i n a l  stage  d e n s i f i c a t i o n of alumina under vacuum h o t - p r e s s i n g c o n d i t i o n s .  suggested  that p l a s t i c  flow may  be o p e r a t i v e at an i n t e r m e d i a t e stage but  d e f i n i t e l y not d u r i n g the f i n a l stage where d i f f u s i o n - c o n t r o l l e d was  proposed  creep  t o be the mechanism r e s p o n s i b l e f o r d e n s i f i c a t i o n . 26  V a s i l o s and S p r i g g s coefficients  They  c a l c u l a t e d the apparent b u l k  diffusion  from the d e n s i f i c a t i o n d a t a f o r alumina and magnesia  d u r i n g h o t - p r e s s i n g and o b t a i n e d an o r d e r of magnitude h i g h e r values term due  for  pressureless sintering.  to changing p o r o s i t y was  When the p r e s s u r e  introduced into their  correction calculations,  t h e i r c a l c u l a t e d d i f f u s i o n - c o e f f i c i e n t s were i n b e t t e r agreement w i t h o b t a i n e d from the s i n t e r i n g d a t a .  From these o b s e r v a t i o n s they  that the d e n s i f i c a t i o n o f a h o t - p r e s s e d compact, beyond the  those  concluded  initial  stage i s a d i f f u s i o n c o n t r o l l e d p r o c e s s . A new  treatment, proposed  step d u r i n g h o t - p r e s s i n g w a s (  by F r y e r based  7  f o r the f i n a l  densification  on a model i n v o l v i n g the b u l k  diffusion  - 16 -  O  3000 lb. in:  •  2000 ib.iri?  1  003h  (h-o 0-02  OOI  OOI  0 0 2  003  Figure 4(a) Shrinkage plots of alumina pressed at 1300°C, 2000 and >  3000 p s i (after F r y e r ) . 7  -•  - 17 -  Figure 4(b).  Shrinkage plots of alumina pressed at 1300°C, 4000, and 5000 p s i (after F r y e r ) . 7  - 18 -  Table I I .  D e n s i f i c a t i o n E q u a t i o n s Based on D i f f u s i o n a l Mass T r a n s p o r t .  , AL. 5/2 L o  '  2?rd  4  KT  2 Coble and E l l i s radius  1  N-H creep model  (1-D)  Rossi  and F u l r a t h  21  KT d  i 1 s  L/ TT x  x •• = neck  • 2 D = eR KT a 4fi c  v  -  D.nba  40  dD dt  a  1  d  a e  s_  7  dt  IdD  ,2  A  4  -  0  3 D  D  3  l " K  = a ( l + 2P) =  V a s i l o s and S p r i g g s  dP dt  i L VA5/3  v  IdD D dt  c  a  , 2Y.  ^D  r  (  T  Fryer  V  KT  ' V . a , 2Y,  lattice diffusion  Coble  boundary d i f f u s i o n  Coble  5  7  D dt "  d^ KT  L  r  load  R = grain  = relative  D  density  radius  K = Boltzmann c o n s t a n t  = bulk d i f f u s i o n c o e f f i c i e n t  T = temperature i n °Kelvin  = boundary d i f f u s i o n c o e f f i c i e n t  V == volume of s o l i d s  n  = vacancy volume  Z = a  a  =  P = porosity =  D  l  D  b  a  c  applied  stress  = effective  stress  b  = a stress-intensity factor  d  = grain  W  = g r a i n boundary w i d t h  diameter  numerical  e = strain Y =  surface  constant (1-D)  rate energy  26  - 19 -  of v a c a n c i e s . He  The  f i n a l form of h i s e q u a t i o n i s shown i n T a b l e I I .  t e s t e d h i s e q u a t i o n w i t h h o t - p r e s s i n g d a t a f o r alumina powder at  1300°C.  The  e x p e r i m e n t a l data f i t t e d w e l l w i t h the  theoretical  p r e d i c t i o n , as shown i n F i g u r e s 4a&4b• 4 Coble  has  a l s o developed  a model f o r the f i n a l stage of d e n s i f i c a -  t i o n of a powder compact, which e x p l i c i t l y energy  and  i n c l u d e s both  a p p l i e d p r e s s u r e as the d r i v i n g f o r c e .  the s u r f a c e  T h i s model i s based  on a steady s t a t e d i f f u s i v e flow of m a t e r i a l between c o n c e n t r i c spherical shells. D.F  where P^  = =  The  2x  ^  driving . +  P  f o r c e i s expressed  as f o l l o w s  a  D  =  applied force  y  =  surface  D  =  relative density  r  =  r a d i u s of c l o s e d pore.  energy  4 The  r e s u l t i n g e q u a t i o n s are a l s o shown i n T a b l e I I .  that t h i s new  Coble  concluded  approach should be a b l e to p r e d i c t d e n s i f i c a t i o n up to the  t h e o r e t i c a l d e n s i t y of a powder compact.  1.4  O b j e c t i v e s of the P r e s e n t Work The  c o n t r i b u t i o n of p l a s t i c flow t o d e n s i f i c a t i o n d u r i n g h o t -  p r e s s i n g has been a s u b j e c t of c o n t r o v e r s y . accepted  Although  i t has been  t h a t p l a s t i c f l o w occurs at a c e r t a i n stage d u r i n g h o t - p r e s s i n g ,  which p r i m a r i l y depends on the temperature  used,  the e x t e n t of d e n s i f i c a -  t i o n a r i s i n g from any p l a s t i c flow mechanism has not been unambiguously determined,  - 20  The  aim of t h i s work was  -  to study q u a n t i t a t i v e l y  d e n s i f i c a t i o n by p l a s t i c f l o w d u r i n g h o t - p r e s s i n g . pressing i s carried  out ^ 0.6  T m  of the m a t e r i a l concerned. temperature of the m a t e r i a l .  (where T m  Generally  subsequently  metallic  and  test  T h i s i s a l s o approximately  hot-  the hot-working  With t h i s o b s e r v a t i o n , an attempt  has  hot-deformation  the e q u a t i o n w i t h hot-compaction data  non-metallic materials.  of  i s the m e l t i n g p o i n t i n °K)  been made to d e r i v e an e q u a t i o n based on s t e a d y - s t a t e and  the e x t e n t  f o r both  CHAPTER I I THEORETICAL FORMULATIONS  I t has been found experimentally "'" t h a t f o r a l a r g e number o f 1  m a t e r i a l s , the s t e a d y - s t a t e  s t r a i n r a t e i s r e l a t e d t o s t r e s s by a  power law as f o l l o w s  e  and  =  A  (3)  a *  the s t r a i n r a t e d u r i n g d e n s i f i c a t i o n o f a powder compact  d i e ) i s approximately  e  =  — hdt  g i v e n by  =  -D dt ^5.  (4)  where "D" i s the r e l a t i v e d e n s i t y , "h" i s the i n s t a n t a n e o u s h e i g h t ; "A" i s a c o n s t a n t constant  and  at constant  i s the s t r e s s .  d e n s i f i c a t i o n during  compact  temperature; "n" i s a m a t e r i a l  F o r a compact t h i s s t r e s s changes w i t h  hot-pressing.  In the development of the t h e o r y ,  i t i s assumed t h a t the p a r t i c l e s  i n a compact are monosized spheres and t h a t they regular three-dimensional  array.  During  p l a s t i c a l l y , a t the p o i n t s o f c o n t a c t The  (in a  are arranged  hot-pressing,  ina  they  deform  (necks) and form f l a t  faces.  compacts d e n s i t y change, as a r e s u l t o f t h i s d e f o r m a t i o n ,  with  respect  to neck r a d i u s , has been d e r i v e d by Kakar and  i s given  Chaklader  and  by  D  2 \  =  3/2  ( 5 )  3(R -aV^ Z  D i s the r e l a t i v e d e n s i t y of compact at neck r a d i u s " a " , "R" instantaneous  r a d i u s of the p a r t i c l e at neck r a d i u s  geometric constant  which depends on the packing  e f f e c t i v e s t r e s s a c t i n g on a compact w i l l  be  " a " and  i s the g is a  configuration.  a f f e c t e d by  the  The  packing  geometry.  II.1  T h e o r e t i c a l Models The  simple  d e f o r m a t i o n geometries of two  spheres i n c o n t a c t  geometric c o n f i g u r a t i o n s are shown i n F i g u r e 5.  t i o n models were c o n s i d e r e d (Z = 8 ) ,  -  (1) simple  (3) body-centred c u b i c  cubic  (Z = 8 ) , and  where Z i s the c o o r d i n a t i o n number. arrangement i n a u n i d i r e c t i o n a l f i e l d  The  (Z = 6 ) ,  and  other  Only f o u r deforma(2)  orthorhombic  (4) rhombohedral (Z =  b.c.c.  of f o r c e  packing  i s an  unstable  (i.e. gravitational  force).  For t h e o r e t i c a l purposes, i t i s assumed that each type of  packing  i s s t a b l e and m a i n t a i n s i t s symmetry on the a p p l i c a t i o n of  pressure,  and  symmetrically  t h a t the m a t e r i a l at the p o i n t s of c o n t a c t d u r i n g d e f o r m a t i o n to m a i n t a i n  12),  spreads  the s p h e r i c i t y of  the  particle. The  a p p l i e d l o a d can be  c e l l where the u n i t c e l l  considered  to be  a c t i n g on  the whole u n i t  i s d e f i n e d as a s p a c e - f i l l i n g p o l y h e d r o n  a deformed sphere s i t u a t e d i n s i d e the polyhedron  (alternatively,  with the  - 23 -  F i g u r e 5.  Geometric r e l a t i o n s h i p o f . . d i f f e r e n t models ( A f t e r 13 Kakar  ).  - 24  unit  cell  -  i s composed of the deformed sphere w i t h i t s a s s o c i a t e d  void  space).  II.1.a.  Simple Cubic P a c k i n g  C o n s i d e r a c u b i c a r r a y of spheres deformed under h y d r o s t a t i c Each sphere w i l l have s i x f l a t 5^.  The The  load  faces  formed as shown i n F i g u r e s  pressure.  5b ^and  2 2 1/2 case i s a cube o f s i d e 2y where y = (R -a ) 2 c r o s s - s e c t i o n a l area of the u n i t c e l l i s 4y . I f there i s a unit c e l l  in this  ' £' on each f a c e , then the s t r e s s on each f a c e i s £/4y  t o t a l pressure  a c t i n g on each sphere i s — y 4y  .  T h i s has  2  and  to be  the  e q u a l to  2  the t o t a l l o a d on hot-pressing  the system under h y d r o s t a t i c c o n d i t i o n s .  In  conventional  g e n e r a l l y a u n i d i r e c t i o n a l l o a d i s a p p l i e d , but because of  the e x i s t e n c e  o f back s t r e s s from the d i e - w a l l and  other p l u n g e r ,  s t r e s s at the c e n t r a l core of a d i e i s a p p r o x i m a t e l y i s o t r o p i c . 13 was  experimentally  observed by Kakar,  w h i l e measuring the  the  This  contact  face  r a d i i of h o t - p r e s s e d l e a d spheres. The two  l o a d a c t i n g on each f a c e of the u n i t c e l l  components; one  flat  f a c e and  p a r t a c t i n g on the sphere through i t s  divided into indented  the other p a r t on the v o i d space, i . e .  2 4y a  where  can be  i s the  =  2 ira  +  ca^  s t r e s s on the neck,  •  i s the s t r e s s on the v o i d  (= 0 ) , a i s the a p p l i e d s t r e s s , c i s the c r o s s - s e c t i o n area of and  a i s the  radius  of the neck.  (6)  space  porosity  - 25 4y  °l  2  T  =  ira  or 2  2 C/;  o  a-,  1  TT  a  2  From e q u a t i o n (5)  ;(R -a ) 2  2  3 / 2  v/here g = 8 f o r the simple c u b i c packing and t h i s e q u a t i o n can be r e w r i t t e n as  R -a  S u b s t i t u t i n g equation  0 1  * (D  =  S u b s t i t u t i n g equation  (8) i n e q u a t i o n (7)  2 / 3  B  2 / 3  R -D  °  2  (9) i n e q u a t i o n (3)  1 £  =  A  ^  Combining equations  1 dD D dt  n  (D  V  (10) and  . 1 \/4 r  A  2 /  ( D  / 3  R -1) 2  }  (  1  0  (4)  a 2/3 2/3 2_ 3  R  -,  n  1 1 }  U  1  ;  )  - 26 This  on i n t e g r a t i o n y i e l d s D D {D 1  II.l.b.  p  2 / 3  R -l} 2  dD  = /  A (-TT-)  dt  d e f o r m a t i o n w i l l have e i g h t  the deformed sphere and the u n i t  faces.  The c r o s s - s e c t i o n a l  2 2 1/2 (R -a ) .  Computing  2/3 y o  =  2  0  O2  =  a r e a of the u n i t  the l o a d  T r a  ^o^  +  =  applied  =  s t r e s s on the neck  The model  c e l l are shown i n F i g u r e s 5c^ and r-  5c2«  (12)  Orthorhombic P a c k i n g  Each sphere a f t e r for  2 / 3  cell  f o r the top f a c e  co^  i s 2/3 y  2 where y =  of the u n i t  cell,  (13)  stress  s t r e s s on p o r o s i t y = 0  2/3  a1  TT  2 ^  2  a  (14)  or 2  a, 1  Substituting  =  ^  R -a — ~ — I 2  TT  equation  a  ±  .  TT/2/3  n  (8) i n e q u a t i o n  1 0  ,. (15;  2  a [D  2  /  V  (15)  (16) /  3  R -1] 2  - 27 -  where 3  =  4/3  f o r the orthorhombic  Combining e q u a t i o n s  ( 3 ) , (4) and (16)  ( D The  packing.  2  /  W - l )  i n t e g r a l form o f e q u a t i o n (17) i s  D max  /o  _ D  „.„ „  1  {D  $  R -1}  n  t dt  o  II.I.e.  =/ Jo  n Af.— —)  dt  5  (18)  TT/2/3  Rhombohedral Packing  Each sphere has twelve p o i n t s o f c o n t a c t and the r e s u l t a n t  unit  i s a rhombic dodecahedron as shown i n F i g u r e s 5d^ and 5d^.  cell  2 The  cross-sectional  2 2 1/2 (R -a ) .  =  CT^ = a^  =  where y =  Computing the l o a d s f o r the top f a c e o f the u n i t  2 2/3 y a  a  area of the u n i t c e l l i s 2 / J y  2 = 3ira cos Qo-^ + ca^  applies  cell,  (19)  stress  s t r e s s on the neck stress  on p o r o s i t y  = 0  From geometry cos 6 = /2/3  o  1  - " ^ — " ^ a /6TT a  (20)  or, a  1  =  2 /  3  R -a —£ 2  2  a  ,o-n (21)  - 28 -  S u b s t i t u t i n g equation  (8) i n e q u a t i o n  (21)  (22)  u//T  1  where 3  [D  2 / 3  3  2 / 3  R -l] 2  4/2~ f o r the rhombohedral  =  Combining equations  ( 3 ) , (4) and (22)  1. dD D  d  packing.  , 1  o  \r//2  t  T h i s , on i n t e g r a t i o n ,  (D  2 / 3  3  ,  2 / 3  R -D 2  yields  r^vv'v'v-i/'dD</  Il.l.d. The  f w - i - , " dt Jo  D  o  "/^  shape of deformed sphere  i s schematically represented i n  There are e i g h t p o i n t s of c o n t a c t d u r i n g the  stage o f deformation. 16  The (R -a ) 2  2  (24)  B.C.C. P a c k i n g  F i g u r e s 5e^ and Se.^initial  n  c r o s s - s e c t i o n a l a r e a o f the u n i t c e l l 1 / 2  i s —— y  2 where y =  .  Computing the loads f o r the top f a c e o f the u n i t  y a 2  = 47ra  2  cos 6a^ +  co^  cell  (25)  - 29 -  a  =  applied  stress  =  s t r e s s on neck  ~ s t r e s s on p o r o s i t y = 0 From geometry,  cos8  =  /3  16 3  1  — /3 y o a^ 2  4TT  or tR>  /4  °1  =  —  a  /3TT  Substituting  equation  dD d  2  a  (8) i n e q u a t i o n  2 /  Combining e q u a t i o n s  D  -a 2~  (27)  ,/3/4 [ D V R - 1 ]  1  I  2  ( 3 ) , (4) and  .  T h i s , on i n t e g r a t i o n g i v e s  o  2  (28)  1  \ m  t  / 3  a  (D V R -1) 2/  /3  2  - 30 II.2  General  Equations  From the above, i t appears  one  can w r i t e g e n e r a l e q u a t i o n s  f o r the s t r e s s c o n c e n t r a t i o n f a c t o r and  f o r the d e n s i f i c a t i o n  both  as  follows:  a  =  «  ( 3 1 )  a D V/ R -l) 2/  3  2  l(  where  and 3 are c o n s t a n t s which vary w i t h p a c k i n g geometry, and  hence  max  p  rTW V 7  D  / 3  R -l} 2  n  dD J  A ( / ) dt o  o  I t s h o u l d be noted  1  t h a t the d e r i v a t i o n i s based  the s t r e s s i s h y d r o s t a t i c i n n a t u r e .  (32)  n  on a c o n d i t i o n where  I t i s a l s o assumed t h a t the  strain  r a t e a t the p o i n t s of c o n t a c t and i n the whole system approaches a steadys t a t e c o n d i t i o n as soon as a l o a d i s a p p l i e d .  I I . 3 A p p l i c a t i o n of the E q u a t i o n The  a n a l y t i c a l s o l u t i o n of the above e q u a t i o n i n a c l o s e d form i s  not p o s s i b l e . .835  The  l i m i t s of i n t e g r a t i o n D  m a x  have v a l u e s v a r y i n g from  f o r the hexagonal p r i s m a t i c model to 0.995 f o r the t e t r a k a i d e c a h e d r o n  model.  In e q u a t i o n  calculable.  (32) g,  constants and  they  The v a l u e s of these c o n s t a n t s are shown i n T a b l e I I I .  n are m a t e r i a l constants and working d a t a .  and ct^ are geometric  can be determined  from  A  and  steady-state hot-  Knowing a l l the c o n s t a n t s , the change of D,  d e n s i t y of a compact, as a f u n c t i o n of time  are  the r e l a t i v e ,  ( t ) at a constant  temperature  -31-  and p r e s s u r e can be o b t a i n e d by s o l v i n g the e q u a t i o n i n a computer.  The  computer programme i s g i v e n i n Appendix 5. For each of the t h e o r e t i c a l d e f o r m a t i o n models, R changes as f o r m a t i o n proceeds;  i.e.,R i s a f u n c t i o n of D.  i n R a t v a r i o u s stages of d e f o r m a t i o n  de-  However, the change  can be c a l c u l a t e d from the c o n s t a n t 13  volume e q u a t i o n as d e s c r i b e d by Kakar Appendix  .  T a b l e s of R vs D are g i v e n i n  1.  1  - 32 -  Table I I I . Geometric Constants o f D i f f e r e n t  Model  cubic hexagonal p r i s m rhombic  dodecahedron  tetrakaidecahedron  a,  Models  3  1  D  D o  max  TT/4  8  .523  0.965  TT/2/3  4/3  .604  0.835  4/2 32/3"  ,740  0.964  .680  0.995  TT/2  TT/3/4  CHAPTER I I I EXPERIMENTAL VERIFICATION OF THEORY  III.l  M a t e r i a l S e l e c t i o n and T h e o r e t i c a l P l o t s To compare the compaction  data f o r h o t - p r e s s i n g w i t h t h a t of  t h e o r e t i c a l l y p r e d i c t e d b e h a v i o u r , both m e t a l l i c and n o n - m e t a l l i c m a t e r i a l s were chosen.  The s e l e c t i o n of lead-2% antimony, n i c k e l  alumina f o r e x p e r i m e n t a l work was  based  upon the a v a i l a b i l i t y  and  of  spherical particles for hot-pressing. As d i s c u s s e d p r e v i o u s l y , the t h e o r e t i c a l e q u a t i o n can only be t e s t e d if  the v a l u e s of the m a t e r i a l c o n s t a n t s A and n f o r any m a t e r i a l are known  o r , a r e e x p e r i m e n t a l l y determined  by hot-working  data.  The v a l u e s of  these constants f o r a l a r g e number of m e t a l l i c and n o n - m e t a l l i c m a t e r i a l s have been r e p o r t e d i n the l i t e r a t u r e . (spheres of Pb-2%  The p u r i t i e s of the  Sb and Ni) used i n t h i s i n v e s t i g a t i o n are not  same f o r which the v a l u e s of A and n are a v a i l a b l e i n the For t h i s reason, the v a l u e s of A and n f o r Pb-2% determined machine.  from s t e a d y - s t a t e hot-compression The  the  literature.  Sb and N i were  experiments  d e t a i l s of the e x p e r i m e n t a l procedure  later section.  metals  i n an I n s t r o n  are g i v e n i n a  The v a l u e s of s t r e s s and s t r a i n r a t e are shown i n  Appendix 2.  The v a l u e s of A and n f o r alumina  literature.  The v a l u e of the constants A and n f o r a l l the m a t e r i a l s are  shown i n Table  IV.  are o b t a i n e d from  the  - 34 -  Using the v a l u e s from the t a b l e and the t h e o r e t i c a l e q u a t i o n ( e q u a t i o n 32), a s e r i e s of p l o t s of r e l a t i v e d e n s i t y of time ( t ) were generated f o r d i f f e r e n t pressures. (a)  under 2105 (c)  temperatures  and  These t h e o r e t i c a l p l o t s are shown i n the f o l l o w i n g F o r l e a d - 2 % antimony  l e a d - 2 % antimony (b)  (D) as a f u n c t i o n  figures.  at 100°C under 1500 p s i i n F i g . 6,  at 150°C under 918 p s i i n F i g . 7.  For n i c k e l at 800°C under  2162 p s i i n F i g . 8, n i c k e l at 900°C  p s i i n F i g . 9. For alumina a t 1600°C under 5000 p s i i n F i g . 10.  For these p l o t s a l l f o u r geometric models were employed.  III.2  E x p e r i m e n t a l T e s t s and Procedures  III.2.a. (i)  Apparatus Lead-2%  antimony:  For these h o t - p r e s s i n g experiments 1.5  average diameter l e a d - 2 % antimony was  s h o t s were used.  s u p p l i e d by the Lead Shot I n d u s t r i e s L i m i t e d .  f o r h o t - p r e s s i n g of l e a d - 2 % antimony u n i a x i a l compressive l o a d was bar.  A stainless steel  This material The apparatus used  shots i s shown i n F i g u r e 11.  A  a p p l i e d t o the system through a simple  (316) rod a c t i v a t e d by a l e v e r arm was  i n s i d e a s t a i n l e s s s t e e l tube.  mm  The  compact was  the s t e e l rod i n a d i e and the s t r e s s was between the rod and a p l u g welded  placed  p o s i t i o n e d on top of  a p p l i e d t o the  specimen  i n t o the c e n t r e of the tube.  To  ensure a u n i f o r m s t r e s s a c r o s s the compact a h e m i s p h e r i c a l p l u n g e r and cup were used t o t r a n s m i t the l o a d t o the compact. f u r n a c e w i t h a u n i f o r m hot zone 3 i n c h e s l o n g was s t a i n l e s s s t e e l tube.  A 400 watt  tube  p l a c e d around the  The temperature of the f u r n a c e was  raised  and  Figure 6.  D vs. t plot f o r Pb-2% Sb at 100°C under 1500 p s i (theoretical p l o t s ) .  TIME(min)  F i g u r e 7.  D vs.  t p l o t f o r Pb-2%  Sb  at 150°Cunder 918  psi  (theoretical  plots).  - 37 -  F i g u r e 9.  D vs. t plot  f o r N i at 900°C under 2105 p s i ( t h e o r e t i c a l  plots).  - 40 -  T a b l e IV.  M a t e r i a l Constants f o r D i f f e r e n t  Materials  Temperature  Lead-2%  Nickel  Alumina  antimony  (°C)  Materials  100  4.11  150  3.519  800  4.6  x 10  900  3.4  x 10  1600  Values o b t a i n e d from Warshaw e t a l .  n  A (sec ^)  x 10  -21  x 10  2.0 x 10  -19  -23  4.2  4.6  -22  -22  4.2  4.6  +  4.0  +  - 41 -  Furnace Thermocouple  Hell  Welded  Plug  Steel  Pressure Boss  Transmitting  Die Specimen  Stainless  Stei^l  Furnace Stainless  Tube Steel Rod  Bench Lever  Am  Tranducer  Load  F i g u r e 11.  6 Hot-pressing  - Dial  Gauge  S u p p o r t i n g Rod for Transducer & D i a l Gauge  apparatus used f o r Pb-2% Sb s h o t s .  - 42 m a i n t a i n e d at the d e s i r e d temperature. s i t u a t e d 1/2  A chromel-alumel  thermocouple  i n c h from the specimen operated a p r o p o r t i o n a l  temperature  c o n t r o l l e r which m a i n t a i n e d the specimen temperature w i t h i n + 5°C. The compaction was  determined by the d i s p l a c e m e n t of the e x t e n s i o n  of the s t a i n l e s s s t e e l r o d , which was  o b t a i n e d by the output from an  e l e c t r o m a g n e t i c t r a n s d u c e r a t t a c h e d to the e x t e n s i o n r o d . output was which was The  The t r a n s d u c e r  a n a l y s e d by a P h i l l i p s PR 19300, d i r e c t r e a d i n g b r i d g e connected to a "Heath K i t " Servo Recorder.  l o a d i n g system was  c a l i b r a t e d by hanging weights at the p o i n t  of c o n t a c t of the b a r and the s t a i n l e s s s t e e l r o d .  The l o a d was v e r y  c l o s e t o the p r o d u c t o f the weight on the bar and the r a t i o o f the arms.  The weight of the b a r was  compensated f o r by l o a d i n g a s m a l l  weight at the o t h e r end.  III.2.a.ii Nickel 0.65  mm  (99% N i and 0.7%  Co):  For h o t - p r e s s i n g of n i c k e l ,  average diameter n i c k e l b a l l s were used.  These were s u p p l i e d  by the S h e r r i t t Gordon Mines L i m i t e d , F o r t Saskatchewan, h o t - p r e s s i n g was  The  c a r r i e d out i n a molybdenum d i e w i t h an e x t e r n a l  graphite sleeve, using induction heating. s l e e v e a c t e d as s u s c e p t o r s . F i g u r e 12.  Alberta.  Both the d i e and  The e x p e r i m e n t a l s e t up i s shown i n  The temperature of the compact was  p l u n g e r at a d i s t a n c e 1/6 the same thermocouple.  r e c o r d e d through the top  i n c h from the specimen and was  A P t - P t - 1 0 % Rh thermocouple was  c o n t r o l l e d by used.  measurement of the d i s p l a c m e n t o f the moveable bottom ram w i t h the thermocouple was  graphite  The  (the top ram  r i g i d l y a t t a c h e d to the p r e s s ) was o b t a i n e d  by the output from an e l e c t r o m a g n e t i c t r a n s d u c e r a t t a c h e d to the The t r a n s d u c e r output was  ram.  a n a l y s e d by a P h i l l i p s d i r e c t r e a d i n g measuring  - 43 -  Figure 1 2 .  H o t - p r e s s i n g apparatus used f o r N i and A l 0  spheres.  - 44 -  b r i d g e , which was connected t o a "Heath K i t " s e r v o - r e c o r d e r , as used p r e v i o u s l y f o r the compaction experiments w i t h l e a d - 2 % antimony  (iii)  Alumina:  The same apparatus ( h o t - p r e s s i n g  spheres.  induction  u n i t ) which was used f o r n i c k e l experiments was a l s o used for  the compaction  s t u d i e s o f alumina.  die  was used i n s t e a d o f a molybdenum d i e - g r a p h i t e s l e e v e  as used p r e v i o u s l y .  1 mm  In t h i s case,however, a g r a p h i t e combination  s a p p h i r e spheres were used i n t h i s case and  these were s u p p l i e d by A. M i l l e r and Company, L i b e r t y v i l l e ,  III.2.b. (i) die,  Illinois.  Procedures Hot-pressing:  A weighed  amount o f spheres was poured i n t o a  tapped and w e l l - s h a k e n i n o r d e r t o o b t a i n u n i f o r m p a c k i n g and  maximum d e n s i t y .  The i n i t i a l h e i g h t o f the d i e w i t h p l u n g e r s was  measured and s u b s e q u e n t l y the h e i g h t o f d i e , w i t h p l u n g e r s and specimen, b e f o r e h o t - p r e s s i n g was a l s o measured. the  specimen.  I t took 15-20 minutes  T h i s gave t h e i n i t i a l h e i g h t of  f o r the specimen  to r e a c h the  f u r n a c e temperature a f t e r the d i e assembly was i n t r o d u c e d i n the f u r n a c e . During t h i s h e a t i n g up p e r i o d , the p l u n g e r d i s p l a c e m e n t r e c o r d i n g was connected.  No a p p r e c i a b l e s h r i n k a g e o r expansion was r e c o r d e d  d u r i n g the h e a t i n g up p e r i o d . temperature the pre-weighed In  system  the case of n i c k e l  Once the specimen a t t a i n e d the e q u i l i b r i u m  l o a d was p l a c e d on the l e v e r arm f o r l e a d . and alumina the l o a d was a p p l i e d  a h y d r a u l i c j a c k , which a c t i v a t e d the p r e s s . by an I n s t r o n machine u s i n g an 'F' c e l l .  The p r e s s was  through  calibrated  The c a l i b r a t i o n curve i s  shown i n F i g u r e 13. A hydrogen  atmosphere  was used d u r i n g h o t - p r e s s i n g o f l e a d - 2 % antimony  - 45 -  8000 "  sooo -  I  5 <  o  4 0 0 0  .  < U <  2 0 0 0 -  1400  M E T E R READING (psi)  Figure 13.  Calibration curve for the pressure-gauge used for hot-pressing of Ni and A l , ^ .  - 46  and n i t r o g e n was  used  -  i n the case of n i c k e l and  a p p l i c a t i o n of p r e s s u r e , the s h r i n k a g e was time.  When the compaction  curve had  r a t e was  alumina.  d r a s t i c a l l y reduced  c o o l e d i n the f u r n a c e .  The  c a l c u l a t e d b e f o r e and  p r e s s i n g u s i n g the h e i g h t of the compact and which remained c o n s t a n t .  compaction the  The weight of the compact was  experimental V.  after  the diameter  From these measurements the i n i t i a l  d e n s i t i e s were c a l c u l a t e d .  the  The h e i g h t of the specimen i n  d u r i n g h o t - p r e s s i n g are shown i n T a b l e  The volume of the compact was  remained c o n s t a n t .  and  shut o f f and  the d i e w i t h p l u n g e r s a f t e r h o t - p r e s s i n g was. measured. t e s t c o n d i t i o n s used  the  r e c o r d e d as a f u n c t i o n of  l e v e l l e d o f f , the power to the f u r n a c e was  specimen was  After  hot-  of the d i e ,  known and  also  and  bulk-  final  From the r e c o r d e r c h a r t the compact h e i g h t  at any i n s t a n t d u r i n g compaction  c o u l d be found.  Using  these c h a r t s ,  the change i n b u l k - d e n s i t y as a f u n c t i o n of time under i s o t h e r m a l c o n d i t i o n s was  obtained.  Afterwards  the v a l u e s of the b u l k d e n s i t y  were converted i n t o r e l a t i v e d e n s i t y by d i v i d i n g t h e o r e t i c a l d e n s i t y of the  solid.  I I I . 2 . b . i i Hot-compression  o f l e a d - 2 % antimony  1.  Lead-2% antimony:  The Pb-2%  the form of s m a l l i n g o t s (0.5 x 0.5 these i n g o t s w i t h a j e w e l l e r s saw, 0.2  x 1 inch.  The  these v a l u e s by  and  nickel  Sb shots were melted  x 3 inches).  and  cast i n  Specimens were cut  the m a j o r i t y of dimensions  specimens were annealed  the  a t 100°C f o r 5-6  from  were 0.2  hours.  P r i o r to t e s t i n g , the ends of each specimen were ground p e r p e n d i c u l a r to the l o n g i t u d i n a l a x i s .  A l l specimens were t e s t e d i n a f u r n a c e a t t a c h e d  to an I n s t r o n t e n s i l e t e s t i n g machine under c o n d i t i o n s of c o n s t a n t  strain  x  - 47 -  Table  V.  Material  Lead2% antimony  Nickel  Alumina  Test Conditions f o r Hot-Pressing  Sphere diameter i n mm  Temperature i n °C  Pressure i n psi  Atmosphere  1.5  100  918,1010,1285  hydrogen  1.5  150  918,1010  hydrogen  0.65  800  2162,3208  nitrogen  0.65  900  1477,2105  nitrogen  1600  3200,4000  nitrogen  1.0  - 48 -  r a t e s . F o r t e s t i n g , the specimens were p l a c e d buttons  (  constant  0.7" d i a . , 0.2" t h i c k n e s s ) .  value  the t e s t was stopped.  u p r i g h t between alumina  When the l o a d a t t a i n e d a From t h i s l o a d and the c r o s s -  s e c t i o n a l area of the specimen, the s t r e s s was c a l c u l a t e d .  The e x p e r i -  ments were done a t 100 and 150°C and at s t r a i n r a t e s of 0.002, 0.005, 0.01 and 0.02 in/min.  2.  Nickel:  These r e s u l t s are shown i n Appendix 2.  N i c k e l b a l l s were m e l t e d by i n d u c t i o n h e a t i n g  i n the form of s m a l l i n g o t s  (0.5" d i a . , 3" l e n g t h ) .  Cylindrical  specimens (0.25" d i a . , 0.9" l e n g t h ) were p r e p a r e d by t u r n i n g i n g o t s on The  a lathe.  these  The specimens were annealed a t 1000°C f o r 5 h o u r s .  ends of each specimen were ground p e r p e n d i c u l a r  axis.  and c a s t  to the l o n g i t u d i n a l  Experiments were conducted at 800 and 900°C and 0.002, 0.005,  0.01, and 0.02 in/min s t r a i n r a t e s . Appendix 2.  These r e s u l t s a r e shown i n  CHAPTER IV RESULTS IV,1  Metals  IV.1.a. (i)  Lead-2% Antimony Relative Density (D) vs. Time (t)  I n i t i a l experiments with Pb-2%  Sb were made to test which of  the four d i f f e r e n t geometrical models f i t t e d the experimental data. For t h i s , hot-pressing was done at 150°C under 918 p s i and curve was obtained.  compaction  Both theoretical plots and the experimental  data are shown i n Figure 14.  I t i s apparent from the figure that the  data coincided with the orthorhombic model i n the l a t e r stages of hotpressing, with a deviation at an early stage of compaction.  The same  observation was made for n i c k e l compacts at 800°C and 900°C under 2162 p s i and 2105 p s i pressures respectively, as shown i n Figures 15 and 16. An o v e r a l l comparison of compaction data with the t h e o r e t i c a l curves indicated  that the experimental data followed closely the t h e o r e t i c a l l y  predicted curves for the orthorhombic model. theoretical plots i n subsequent  For this reason a l l the  figures were computed with geometric  constants for the orthorhombic model only.  The reason for this  agreement between the data and the curves for the orthorhombic model i s discussed i n a l a t e r section.  In some cases, the experimental  compaction data obtained were beyond the upper l i m i t of the density, predicted by the theoretical orthorhombic model.  However, these data  were excluded from the figures, as the theory of compaction i s not v a l i d over .835 r e l a t i v e density.  TIME (min)  F i g u r e 14.  D vs. t plot the  f o r Pb-2% Sb at 150°C under 918 p s i ( s o l i d  theoretical  curves f o r 4 d i f f e r e n t m o d e l s ) .  lines  represent  I-OT  Ni 800°C o 2162 psi 0-9  B-C-C 0  RHOMB  ORTHO r, i—  1  S • CUBIC  3  4  5  _l_ 9  10  TIME (min)  Figure 15. D vs. t plot f o r Ni at 800°C under 2162 p s i ( s o l i d l i n e s represent the theoretical curves for 4 different models).  r  l-O  T  1  i  1  i  ~1  1  1  Ni 9 0 0 ° C o 2105 psi  >-  0-9  *!  "  =  "B-C-C  —  5  CO  UJ Q  a  0-8 /  _ O _  ^)RTH0  ^•rnnir  .  °  0-7  -  0-6  -  0-5 0  L 1  1 2  l 3  1  >  4  5  i  1  i  l  6  7  8  9  TIME(min)  Figure 16.  D.vs. t plot for Ni at 900°C under 2105 p s i ( s o l i d l i n e s represent the theoretical curves for 4 different models).  10  -  (ii)  53  -  D v s . t at a c o n s t a n t temperature  and under v a r y i n g p r e s s u r e s :  For t h i s , h o t - p r e s s i n g experiments  were c a r r i e d out at 100  and  1010  150°C (2 s t r e s s e s ) under 918,  and  t h e o r e t i c a l curves f o r the orthorhombic the computer.  The  psi.  A s e r i e s of  model o n l y were o b t a i n e d  e x p e r i m e n t a l d a t a and  shown i n F i g u r e s 17a and 17b.  1285  (3 s t r e s s e s )  the t h e o r e t i c a l curves  from are  A good f i t of e x p e r i m e n t a l p o i n t s w i t h  the t h e o r e t i c a l p l o t s can be seen p a r t i c u l a r l y at the l a t e r stages  of  compaction.  ( i i i ) D v s . t at a c o n s t a n t s t r e s s and w i t h v a r y i n g  temperatures:  For t h i s , e x p e r i m e n t s were made at 100°and 150°C at a s t r e s s of 918 p s i . In a d d i t i o n , o t h e r experiments under 1010  p s i stress.  The  were a l s o c a r r i e d out at 100  e x p e r i m e n t a l d a t a and  are compared i n F i g u r e s 18a and 18b Again a good f i t i s apparent theoretical  IV.l.b.  150°C  the t h e o r e t i c a l s e t s of  plots  experiments.  between e x p e r i m e n t a l p o i n t s and  the  prediction.  Nickel  (i) The  f o r these two  and  D vs. t a t a constant temperature  e x p e r i m e n t a l c o n d i t i o n s used  and v a r y i n g p r e s s u r e s :  to f o l l o w the change of r e l a t i v e d e n s i t y  as a f u n c t i o n of time f o r n i c k e l are 800°C a t 2162 experimental  compaction  d a t a and  are compared i n F i g u r e 19.  3208 p s i .  Another s e t of h o t - p r e s s i n g  the c o r r e s p o n d i n g  The  the t h e o r e t i c a l p l o t u s i n g e q u a t i o n  were c a r r i e d out at 900°C under p r e s s u r e s of 1477 r e s u l t s and  and  and  (32)  experiments  2105  psi.  These  t h e o r e t i c a l p l o t s are shown i n F i g u r e  20.  0-9  Pb-Sb 100° C O  918 p.s.i.  •  1010 p.s.i.  O  1285 p.s.i.  O-E-  z  UI Q UI > UI  4>  0-7  tr  o-6  5  6~  TIME (min)  Figure 17(a).  D vs. t plot for Pb-2% Sb at 100°C under d i f f e r e n t pressures (solid l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only).  1  09h  C  to  1  I50°C © - lOIOp.s.i. o = 918 p.s.i.  Pb-Sb=  JR _ .^aax  0-8  z  LU Q  LU > Ul LL)  0-7I-  0-6L  4  5  10  6  T I M E (min)  F i g u r e 17(b).  D v s . t p l o t f o r Pb-2% Sb at 150°C under d i f f e r e n t (solid model  pressures  l i n e r e p r e s e n t s the t h e o r e t i c a l curve f o r the only).  orthorhombic  TIME (min) ure 18(a).  D vs. t p l o t (solid model  f o r Pb-2%  Sb at 918 p s i and d i f f e r e n t  l i n e s r e p r e s e n t s the t h e o r e t i c a l only).  temperatures  curve f o r the  orthorhombic  r  1  0-9  l_ P b - S b lOIOpsi • I50°C o IOO°C  CO  UJ Q  o-8  UJ  >  UI  ar  0-6L 10  TIME (min)  F i g u r e 18(b).  D vs. t plot (solid model  line only).  f o r Pb-2% Sb at 1010 p s i and d i f f e r e n t r e p r e s e n t s the t h e o r e t i c a l  temperatures  curve f o r the orthorhombic  0-9  Ni  800°C  9  3 2 0 8 psi  o  2162  psi  >to LU 0-8rQ  UJ  > Ln 00  UJ  0-7  0-6  _L 5  TIME  F i g u r e 19.  D vs. t plot  6 (min)  f o r N i at 800°C under d i f f e r e n t p r e s s u r e s  r e p r e s e n t s the t h e o r e t i c a l  (solid 1  curve f o r the orthorhombic model  me  only).  'igure 20.  D vs. t plot  f o r N i at 900°C under d i f f e r e n t p r e s s u r e s  r e p r e s e n t s the t h e o r e t i c a l  curve f o r the orthorhombic  (solid model  line  only).  T  1  1  1  1  1  1  I  r  Ni at ~ constant cr T = 900°  C  TIME  F i g u r e 21.  (minK  D v s . t p l o t f o r N i at ^constant s t r e s s  and d i f f e r e n t  temperatures  r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model o n l y ) .  (solid  line  - 61 -  (ii)  D vs . t at a c o n s t a n t p r e s s u r e and v a r y i n g temperatures:  study the e f f e c t of changing the compaction  behaviour  of n i c k e l spheres,  20 f o r the c o n d i t i o n s 2105 r e p l o t t e d i n F i g u r e 21.  the temperature  p s i a t 900°C and  at a c o n s t a n t s t r e s s  2165  and  p s i a t 800°C are  I t shows t h a t an i n c r e a s e i n temperature  of 10%.  Non-Metal  IV.2.a.  Alumina  (i) As o n l y  D vs. t a t a constant temperature • v e r y few alumina  spheres  temperature  4000 p s i ) .  The  (1600°C) and at two  and v a r y i n g s t r e s s e s :  (single c r y s t a l sapphire  were a v a i l a b l e f o r the hot-compaction one  on  the curves i n F i g u r e 19  h o t - p r e s s i n g by 100°C s h i f t s the r e l a t i v e d e n s i t y curve by about  IV.2.  To  s t u d i e s , experiments  were done at  d i f f e r e n t s t r e s s e s (3200  and  t h e o r e t i c a l curves f o r D v s . t were a l s o generated  these e x p e r i m e n t a l c o n d i t i o n s u s i n g e q u a t i o n d a t a and  spheres)  (32).  The  the t h e o r e t i c a l p l o t s are shown i n F i g u r e 22.  good f i t between the e x p e r i m e n t a l p o i n t s and  under  experimental In t h i s  the t h e o r e t i c a l  can be seen d u r i n g the i n t e r m e d i a t e stage of h o t - p r e s s i n g .  case a  prediction A deviation  between the t h e o r e t i c a l p r e d i c t i o n and e x p e r i m e n t a l r e s u l t s at the v e r y eary stage and  IV.3.  d u r i n g the l a s t  S t r a i n - R a t e s During  stage of compaction  was  encountered.  Hot-Pressing  For the c a l c u l a t i o n of s t r a i n - r a t e s from the h o t - p r e s s i n g d a t a , f o l l o w i n g e q u a t i o n was  used  • £  =  1 dD D dt  the  os > to  AlgOj 1600°C •  3200 psi  °  4000 psi  LU Q  0-8 -  TIME (min) F i g u r e 22.  D vs. t plot  f o r A l ^ a t 1600°C under d i f f e r e n t p r e s s u r e s ( s o l i d  r e p r e s e n t s the t h e o r e t i c a l curve f o r the orthorhombic model  only).  line  - 63 -  From the knowledge of the r e l a t i v e density at any stage of compaction and from the slope of the curve D vs. t at that point, the values of strain-rates were calculated and these were tabulated i n Appendix 4 . It can be seen that the s t r a i n rates used for hot compression experiments of Pb-2% Sb and Ni are similar to the values of s t r a i n rates at the intermediate stage of d e n s i f i c a t i o n , although at the i n i t i a l stage of d e n s i f i c a t i o n , the s t r a i n rates were very much higher than that used f o r hot-working experiments.  I t i s also at the i n i t i a l  stage, a deviation between the experimental data and the theoretical plots was encountered.  CHAPTER V DISCUSSION  V. 1.  E f f e c t i v e Stress During Since the f i r s t  Hot-Pressing  t h e o r e t i c a l work of Murray, L i v e y and W i l l i a m s ,  on the p l a s t i c flow t h e o r y of h o t - p r e s s i n g ,  the proponents of  19  stress-  20 enhanced d i f f u s i o n a l creep (the N a b a r r o - H e r r i n g that any  creep  c o n t r i b u t i o n by p l a s t i c flow to the  ) have contended  overall  densification  must be s m a l l since, the s t r e s s on the system d u r i n g h o t - p r e s s i n g i s g e n e r a l l y low.  I t has been r e c o g n i z e d l a t e l y by most workers,  that the e f f e c t i v e s t r e s s i n h o t - p r e s s i n g i s d i f f e r e n t from the a p p l i e d p r e s s u r e and  t h a t t h i s i s a f u n c t i o n of p o r o s i t y .  first  to r e c o g n i s e t h i s and  person  to take i n t o account  i n t r o d u c e d the f o l l o w i n g form of e f f e c t i v e P  a  eff  McClelland,  the  1 7  porosity,  stress  a 2/3  P where P  q  i s the a p p l i e d p r e s s u r e and V  i s the volume f r a c t i o n p o r o s i t y . 26  Another form of e f f e c t i v e s t r e s s was a p p l y i n g to c r y s t a l l i n e m a t e r i a l s .  a  eff r c  =  P  a  (1 + 2V ) p  i n t r o d u c e d by V a s i l o s et a l .  for  - 65 -  T h i s was adopted from t h e change o f e l a s t i c modulus w i t h p o r o s i t y as predicte< t h e o r e t i c a l l y and observed e x p e r i m e n t a l l y .  26  the form used by V a s i l o s e t a l .  a  =  eff  as f o l l o w s  C-2).  Subsequently a l a r g e number o f workers Fryer,  generalised  P (1 + b V ) a p'  where b i s an e m p i r i c a l constant  Coble,  R o s s i and F u l r a t h  (Fairnsworth  and C o b l e , ^ ,  and Koval'Chenko and Samsonov ) used t h e f o l l o w i n g  form of e f f e c t i v e s t r e s s e q u a t i o n  P a  eff  =  rf  where D i s the r e l a t i v e  density.  Some o f these r e l a t i o n s a r e p l o t t e d i n F i g u r e from t h i s f i g u r e  23.  I t can be seen  t h a t the maximum e f f e c t i v e s t r e s s i s no more than a  f a c t o r o f two l a r g e r than the a p p l i e d s t r e s s on a compact even a t a r e l a t i v e d e n s i t y of 0.5. On bution  the b a s i s o f t h i s evidence i t has been s u g g e s t e d t h a t the c o n t r i to d e n s i f i c a t i o n by p l a s t i c f l o w d u r i n g h o t - p r e s s i n g  s m a l l , as the c o n t a c t to deform.  i s quite  s t r e s s i s n o t l a r g e enough t o cause t h e m a t e r i a l  Moreover, from the theory  and experiments o f s e l f -  i n d e n t a t i o n , i t i s well-known t h a t a m a t e r i a l can o n l y deform i f the contact  s t r e s s i s about 3 times the y i e l d s t r e s s o f t h e m a t e r i a l  criterion").  F o r t h i s r e a s o n , workers i n t h i s f i e l d  t h a t any c o n t r i b u t i o n by p l a s t i c flow has t o be v e r y  ("yield  tend t o assume small.  -  67 -  However, a l l t h e e q u a t i o n s f o r the e f f e c t i v e s t r e s s , r e f e r r e d t o above, have not been r i g o r o u s l y d e r i v e d . or  A l l o f them a r e e i t h e r e m p i r i c a l  o b t a i n e d from f i t t i n g e x p e r i m e n t a l d a t a i n t o an a r b i t r a r y  equation.  Attempts to f o r m u l a t e a r i g o r o u s e q u a t i o n have not been v e r y s u c c e s s f u l as the l o c a l i s e d s t r e s s e s w i l l v a r y both w i t h r e s p e c t to t h e volume f r a c t i o n o f p o r o s i t y and t o the pore  shape, which i s not known.  whole problem w i l l be f u r t h e r c o m p l i c a t e d i f the p a r t i c l e s system a r e random-shaped and  vary  The  i n a single  i n size.  On the o t h e r hand, the problem can be s i m p l i f i e d by assuming the p a r t i c l e s a r e monosized spheres known.  and i f t h e i r geometry of p a c k i n g i s  The s t r e s s i s h y d r o s t a t i c i n n a t u r e and each shpere  deformed at i t s p o i n t s o f c o n t a c t u n i f o r m l y .  i s being  Under these c o n d i t i o n s , t h e  effective stress  (a ,-,-) w i t h r e s p e c t t o the r e l a t i v e d e n s i t y o f the err compact i s g i v e n by e q u a t i o n (31)  a °  e f f  =  , 2/32/32 n  3  c^CD  R -1)  The v a l u e s o f ct^ and 3 (geometric c o n s t a n t s ) have a l r e a d y been g i v e n i n T a b l e III f o r f o u r d i f f e r e n t  geometric models.  The r e l a t i o n s h i p between  D and R was c a l c u l a t e d u s i n g the c o n s t a n t volume e q u a t i o n and i s t a b u l a t e d i n Appendix 1.  From these the e f f e c t i v e s t r e s s w i t h r e s p e c t  0" cc eff to  the a p p l i e d s t r e s s  been c a l c u l a t e d . F i g u r e 23.  (——)  as a f u n c t i o n of r e l a t i v e d e n s i t y (D) has  The v a l u e s a r e g i v e n i n Appendix 3 and a r e p l o t t e d i n  I t i s q u i t e apparent  from  t h i s f i g u r e t h a t the s t r e s s  e f f e c t i v e a t the p o i n t s of c o n t a c t i s v e r y much l a r g e r than assumed by p r e v i o u s workers.  that  F o r example,below 0.79 r e l a t i v e d e n s i t y the  - 68 e f f e c t i v e s t r e s s i s more than 3 times the a p p l i e d s t r e s s f o r a l l the geometric models.  The i n s t a n t a n e o u s s t r e s s a t the c o n t a c t p o i n t s o f  spheres i s extremely h i g h and drops v e r y r a p i d l y t o about  10 times w i t h  an i n c r e a s e i n r e l a t i v e d e n s i t y o f l e s s than 0.10. From a b o v e , i t i s apparent t h a t the c o n t a c t s t r e s s e s w i t h r e s p e c t to the a p p l i e d s t r e s s i n a compact are more than an o r d e r o f magnitude l a r g e r than o r i g i n a l l y  thought.  T h i s i m p l i e s t h a t depending  upon the  magnitude o f the a p p l i e d s t r e s s , the e f f e c t i v e s t r e s s i s r e a s o n a b l y high,upto•• a compact d e n s i t y o f 0.85 t o 0.90 (depending upon the p a c k i n g geometry).  I f i t i s accepted t h a t the p a c k i n g c o n f i g u r a t i o n o f spheres  i n a d i e approximates  to t h e orthorhombic p a c k i n g , then the e f f e c t i v e  s t r e s s i s more than t h r e e times the a p p l i e d s t r e s s up to 0.85  relative  d e n s i t y of a compact. I t w i l l be i n f o r m a t i v e at t h i s stage to a n a l y s e the magnitude and e f f e c t o f c o n t a c t s t r e s s i n an oxide compact. g e n e r a l l y h o t - p r e s s e d i n the temperature p r e s s u r e of 2000 t o 6000 p s i .  Compacts o f alumina are  range 1400-1700°C under a  The y i e l d s t r e s s o f alumina as a f u n c t i o n  of temperature has been r e p o r t e d by Kronberg"^ and i s shown i n T a b l e V I . I t i f i s c o n s i d e r e d t h a t the h o t - p r e s s i n g o f alumina spheres was done a t  1600°C under  4000 p s i and i f i t i s c o n s i d e r e d t h a t the p a c k i n g  geometry i s t h a t of the orthorhombic model, the c o n t a c t s t r e s s w i l l be i n the o r d e r o f 16,000 p s i a t 0.80 r e l a t i v e d e n s i t y o f the compact, which i s much more than t h r e e times the y i e l d s t r e s s f o r b a s a l s l i p i n alumina.  In the p r e s e n t i n v e s t i g a t i o n however, an end-point  d e n s i t y o f 0.75 was o b t a i n e d at 1600°C under d i s c r e p a n c y may. be e x p l a i n e d from the f a c t  4000 p s i .  relative  This  that the r e s o l v e d y i e l d  stress  - 69 -  Table V i  Y i e l d S t r e s s o f S i n g l e C r y s t a l Sapphire  Temperature (°C)  S t r a i n Rate  (in/in/min)  (Basal  Slip)  Y i e l d Stress (psi)  1400  0.05  .. 8500  1500  0.05  6000  1600  0.05  4500  1700  0.05  3000  - 70 -  for  the rhombohedral s l i p  and the rhombohedral twinning a r e much h i g h e r  than  the r e s o l v e d y i e l d s t r e s s f o r b a s a l s l i p  and o n l y 3 b a s a l s l i p  a v a i l a b l e f o r the d e f o r m a t i o n o f alumina  under these e x p e r i m e n t a l c o n d i t i o n s .  It  s h o u l d be mentioned here  t h e o r e t i c a l l y using equation  t h a t the e f f e c t i v e s t r e s s  (31) i s most l i k e l y  the s t r e s s a c t i n g on the c o n t a c t f a c e s . applied stress w i l l the l a r g e f l a t immediately  be l o s t  systems a r e  calculated  the upper l i m i t o f  "In p r a c t i c e , some of the  i n die-wall friction  and a l s o i n c r e a t i n g  f a c e ( i n d e n t a t i o n ) on the top and bottom l a y e r o f spheres  a f t e r the a p p l i c a t i o n of l o a d , as has been observed by  13 Kakar.  Thus, i t i s p o s s i b l e t h a t the e f f e c t i v e s t r e s s i n any  p a r t i c u l a r system may not be as h i g h as t h a t p r e d i c t e d by e q u a t i o n ( 3 1 ) . V.2.  A c t i v a t i o n Energy Study An A r r h e n i u s type o f e q u a t i o n can be w r i t t e n f o r t h e c o n s t a n t s A i n  equation  (32) as f o l l o w s  A  =  A' exp(-Q/RT)  where A' i s a temperature energy  f o r the p r o c e s s , R and T have  From the s l o p e o f a for  independent  constant and Q i s the a c t i v a t i o n t h e i r u s u a l meaning.  l n A v s . 1/T p l o t , the a c t i v a t i o n  the p r o c e s s can be o b t a i n e d .  energy  These p l o t s f o r l e a d and n i c k e l a r e  shown i n F i g u r e s 24 and 25, r e s p e c t i v e l y .  F o r these p l o t s , the v a l u e  'A' f o r l e a d a t 200°C and t h a t o f n i c k e l a t 700°C were o b t a i n e d the e x p e r i m e n t a l d e n s i f i c a t i o n curve.  As can be seen from  f i g u r e s , these v a l u e s of 'A' (back c a l c u l a t e d data) l i e on the s t r a i g h t l i n e  drawn through  from  these  from the d e n s i f i c a t i o n the o t h e r p o i n t s .  The  - 71 -  Figure 24.  l n A vs. 1/T plot for Pb-2%  Sb (the point represented by a  dot with a ring i s from the d e n s i f i c a t i o n curve at 200°C).  -  Figure  25.  l n A v s . 1/T  plot  72 -  for Ni  (the p o i n t r e p r e s e n t e d by a dot  w i t h a r i n g i s from the d e n s i f i c a t i o n  curve at  700°C).  a c t i v a t i o n energy  o b t a i n e d from F i g u r e 24 f o r l e a d was  which i s c l o s e t o the a c t i v a t i o n energy Kcal/mole). for  Similarly  n i c k e l was  57.5  a c t i v a t i o n energy  V.3.  Packing  V.3.a. It  (26.9  c a l c u l a t e d from F i g u r e  25  T h i s v a l u e i s a l s o c l o s e to the  f o r hot-working  and Deformation  Kcal/mole  f o r s e l f - d i f f u s i o n of l e a d  the a c t i v a t i o n energy  Kcal/mole.  28.7  of n i c k e l  (71 Kcal/mole)."'""'"  Geometry I n s i d e a Die  Packing Geometry i s seen from a l l the F i g u r e s "14.,  15  mental d a t a f o l l o w s c l o s e l y the t h e o r e t i c a l l y the hexagonal p r i s m model.  derived equation f o r  T h i s agreement i n d i c a t e s t h a t  packing geometry of monosized spheres orthqrhombic  and -^1 t h a t the e x p e r i -  i n a d i e may  the o v e r a l l  be s i m i l a r t o the  packing.  When a d i e i s randomly f i l l e d  w i t h monosized spheres,  with  i n t e r m i t t e n t shaking and t a p p i n g i n o r d e r to a c h i e v e a uniform the spheres  tend to spread l a t e r a l l y  configuration. spreading.  to a c h i e v e the  most  stable  However, the d i e - w a l l o f f e r s r e s i s t a n c e to  As a r e s u l t , a c e r t a i n degree of s t a b i l i t y  packing  lateral  i n packing i s 18  maintained  i n spite of an u n s t a b l e c o n f i g u r a t i o n . McGeary  v a r i o u s modes of f i l l i n g packing d e n s i t y . the d i e diameter  the d i e and  density.  the e f f e c t of c o n t a i n e r s i z e  H i s r e s u l t s are shown i n F i g u r e 26. to the sphere diameter  d e n s i t y of the compact reaches  s t u d i e d the  When the r a t i o of  i s g r e a t e r than 10,  a maximum of 62.5%  on  of the  the p a c k i n g  theoretical  T h i s v a l u e i s c l o s e t o the as-compacted d e n s i t y f o r the  orthorhombic  packing.  Smith, Foote and Busang  24  s t u d i e d the c o o r d i n a t i o n number of  spheres  - 74 -  70  D d  F i g u r e 26.  Effect  / c o n t a i n e r diameter \ sphere diameter  o f c o n t a i n e r s i z e on t h e e f f i c i e n c y o f  18 packing  one-size  spheres  ( a f t e r McGeary  ).  - 75 -  in  a d i e a f t e r shaking and t a p p i n g .  T h e i r r e s u l t s showed a Gaussian  d i s t r i b u t i o n of the number o f sphere w i t h a g i v e n c o o r d i n a t i o n number. The average c o o r d i n a t i o n number of the sphere was  c l o s e to 8.  These  r e s u l t s f u r t h e r c o n f i r m t h a t the o v e r a l l p a c k i n g geometry i n a d i e approximates  V.3.b.  t o t h a t of the orthorhombic model.  Deformation Geometry  The w i d t h of the d i e does not p e r m i t an i n t e g r a l number o f spheres a c r o s s the diameter; hence the i d e a l p a c k i n g d i s c u s s e d i n the above s e c t i o n does not e x i s t  a c r o s s the diameter o f the d i e s i n c e a  c e r t a i n number of spheres are l i g h t l y h e l d a g a i n s t l o o s e l y h e l d spheres  the d i e - w a l l .  would r e a r r a n g e as soon as a l o a d i s a p p l i e d ,  r e s u l t i n g i n a higher r e l a t i v e density.  As no f r a g m e n t a t i o n was  observed i n m e t a l compacts t h i s i n c r e a s e i n d e n s i t y to  The  p a r t i c l e rearrangement.  some f r a g m e n t a t i o n was  can be  attributed  However, i n the case o f s a p p h i r e (Al^O^)  noted a f t e r  a s s o c i a t e d w i t h rearrangement  the experiments.  This fragmentation  might have c o n t r i b u t e d to d e n s i f i c a t i o n 13  at  the i n i t i a l  average  stage.  In a d d i t i o n , Kakar  c o o r d i n a t i o n number of deformed  d e n s i t y of a compact of l e a d spheres was as shown i n F i g u r e 27.  observed an i n c r e a s e i n the  spheres of l e a d , as the  bulk  increased during hot-pressing  This i n d i c a t e s that p a r t i c l e  rearrangement  occurs as d e n s i f i c a t i o n of a powder-compact p r o c e e d s , e s p e c i a l l y at the  initial  stage o f  compaction.  On f u r t h e r l o a d i n g , the p a r t i c l e s b e g i n t o deform. c o l o n i e s of rhombohedral  Although  individual  or t e t r a g o n a l d e f o r m a t i o n were observed, the  m a j o r i t y of the spheres showed a hexagonal p r i s m model of d e f o r m a t i o n .  7 ' 0  ' 2.0  " 4.0  1 6.0 (D -  F i g u r e 27.  C o o r d i n a t i o n number v s . (after  Kakar ). 1 3  1  8.0  U  10.0  Do)  (D - D ) f o r Pb-2%  Sb at v a r i o u s  temperatures  This i s reavealed  i n Figures  deformed a t 800 and 900°C.  28  and 29 which r e p r e s e n t  As can be seen from the f i g u r e s , the  c o o r d i n a t i o n i n one,plane d u r i n g  d e f o r m a t i o n i s 6 and the deformed  sphere took the form o f a hexagonal It  n i c k e l spheres  prism.  has a l r e a d y been noted i n t h e d e n s i t y v s . time curves  t h e r e i s a disagreement between the t h e o r e t i c a l curves and data, p a r t i c u l a r l y at t h e i n i t i a l ment can be e x p l a i n e d  stage o f compaction.  This  that  experimental disagree-  from t h e arguments above.  In the t h e o r e t i c a l d e r i v a t i o n , no c o n s i d e r a t i o n was g i v e n t o account f o r any i n c r e a s e i n d e n s i t y due t o the change of c o o r d i n a t i o n number.  There i s e x p e r i m e n t a l  e v i d e n c e i n t h i s study and'  also  12 r e p o r t e d by Kakar and C h a k l a d e r , fragmentation  t h a t p a r t i c l e rearrangement and  ( f o r oxides) are s i g n i f i c a n t  d e n s i f i c a t i o n at the i n i t i a l  c o n t r i b u t i n g f a c t o r s to  stage o f h o t - p r e s s i n g .  This i s e s p e c i a l l y  t r u e immediately a f t e r the a p p l i c a t i o n o f l o a d when the e f f e c t i v e s t r e s s a t the p o i n t s of contact may be v e r y The  experimentally  high.  observed i n c r e a s e i n d e n s i t y , i n the f i r s t  few minutes o f compaction which was always g r e a t e r p r e d i c t e d i n c r e a s e , may be e x p l a i n e d hot-pressing, at  at  to i n c o r p o r a t e  I t has n o t been p o s s i b l e  the i n c r e a s e i n d e n s i t y due t o t h e second  e f f e c t , i n t o the t h e o r e t i c a l e q u a t i o n s . d e v i a t i o n , t h e e f f e c t o f the e x i s t e n c e  In a d d i t i o n , to account f o r the  o f very high  s t r a i n r a t e s , a t the  i n i t i a l stage o f h o t p r e s s i n g , ' (as seen e x p e r i m e n t a l l y ) , into  stage o f  and ( i i ) p a r t i c l e rearrangement l e a d i n g to a  average c o o r d i n a t i o n number p e r sphere.  present  In the i n i t i a l  the i n c r e a s e i n d e n s i t y i s due t o two f a c t o r s ( i ) i n d e n t a t i o n  the p o i n t s o f c o n t a c t  higher  as f o l l o w s :  than the t h e o r e t i c a l l y  consideration.  s h o u l d be taken  _ 78 _  2162 p s i .  (80x)  _79-  Figure 29.  Microstructure of Ni spheres hot-pressed at 900°C and 1477 p s i . (95x)  -80V.4  L i m i t a t i o n s of the P r e s e n t A n a l y s i s 1.  In the t h e o r e t i c a l c o n s i d e r a t i o n , the s t r e s s i s c o n s i d e r e d to be  h y d r o s t a t i c , but i n p r a c t i c e a u n i a x i a l s t r e s s i s a p p l i e d i n hotpressing.  However, due  to the e x i s t e n c e of back s t r e s s from the d i e -  v a l l and the o t h e r p l u n g e r , the s t r e s s may However, t h i s assumption may  2.  be assumed to be  isotropic  not be a good one i n p r a c t i c e .  Although the same h e a t - t r e a t m e n t s were g i v e n f o r the b a l l s  i n h o t - p r e s s i n g and the specimens used i n hot-compression, of these may  not be the same.  used  the s t r u c t u r e  S i n c e the constant 'A' i s a f u n c t i o n of  s t r u c t u r e , the v a l u e s are s u b j e c t e d to unknown e r r o r .  3.  I n the hot-compression  between specimen T h i s may  of Pb-Sb and N i specimens,  and the alumina buttons was  Pb-2%  Sb  not taken i n t o  friction  consideration.  g i v e r i s e t o some e r r o r s i n e v a l u a t i n g the v a l u e o f steady-  s t a t e s t r e s s and hence the v a l u e s of 'A'  4.  the  The  compostion  (manufacturer's  a c t u a l l y Pb-15% Sb.  and  o f the Pb-Sb a l l o y was  'n'.  originally  composition) but i t was  c o n s i d e r e d t o be  found l a t e r t h a t i t was  S i n c e t h i s i s a h y p e r - e u t e c t i c a l l o y , t h e power law.  dependence of s t e a d y - s t a t e s t r a i n r a t e on s t r e s s , g i v i n g the v a l u e s of *A'  and  'n' shown i n T a b l e IV may  the temperature  not be completely v a l i d .  However, i n  and s t r a i n r a t e range used i n t h i s i n v e s t i g a t i o n ,  super-  p l a s t i c b e h a v i o u r would not be expected, and i t has been assumed the v a l u e s of 'A' and  5.  'n' are r e l i a b l e from the compression  tests.  In the t h e o r y , the b u l k s t r a i n - r a t e w i t h i n the compact i s used,  based on the i n s t a n e o u s d e n s i t y .  I t would be more c o r r e c t to c o n s i d e r  -81the s t r a i n - r a t e l o c a l i s e d i n the contact areas, I f i t -were possible to characterise I t .  The error introduced i n this way i s unknown, but i s  not believed to be appreciable beyond the early stages of d e n s i f i c a t i o n .  CHAPTER VI SUMMARY AND CONCLUSIONS  Isothermal densification curves of a powder compact during hotpressing have been t h e o r e t i c a l l y calculated using the geometry of deformation of p a r t i c l e s and hot-compression are assumed to be monosized spheres. models were considered:  data.  The p a r t i c l e s  Four d i f f e r e n t deformation  Cubic (Z = 6), orthorhombic  (Z = 8), b.c.c.  (Z = 8), and rhombohedral (Z = 12) where Z i s the coordination of the sphere. used.  For hot-compression  an equation of type  e = Aa  was  The f i n a l d e n s i f i c a t i o n equation r e l a t i n g the r e l a t i v e density  and time for d i f f e r e n t i d e a l packing arrangements has been derived which i s  o The equation was solved i n a computer to obtain the theoretical plots. However, i n order to use this equation the values for material constants (A and n) were necessary and were determined by hot-compression an Instron machine.  tests i n  Three different materials were used f o r this  purpose - these are Pb-2% Sb, Ni and A^O^.  Theoretical curves f o r  a l l four different geometric models were generated by the computer.  -83  -  These t h e o r e t i c a l curves were compared w i t h h o t - p r e s s i n g data o f spheres The (1)  of the same m a t e r i a l s a t d i f f e r e n t temperatures  and p r e s s u r e s .  f o l l o w i n g c o n c l u s i o n s can be made The g e n e r a l t h e o r e t i c a l e q u a t i o n proposed  i s found t o be  obeyed by the s p h e r i c a l p a r t i c l e s d u r i n g the i n t e r m e d i a t e stage o f hot-pressing. (2)  The e x p e r i m e n t a l p o i n t s f o l l o w c l o s e l y the t h e o r e t i c a l l y  p r e d i c t e d curve f o r t h e hexagonal p r i s m a t i c d e f o r m a t i o n model. (3)  T h i s i n d i c a t e s t h e o v e r a l l p a c k i n g geometry of sphere  i n s i d e the d i e c o i n c i d e s w i t h  the orthorhombic  p a c k i n g i n agreement w i t h  the o b s e r v a t i o n of p r e v i o u s workers. (4)  A d e v i a t i o n was encountered  a t the i n i t i a l  stage o f d e n s i f i c a -  t i o n , which c o u l d be e x p l a i n e d from p a r t i c l e rearrangement a t the b e g i n n i n g o f h o t - p r e s s i n g as was observed  i n t h i s study and by o t h e r  wo r k e r s p r e v i o u s l y . A t h e o r e t i c a l equation f o r c a l c u l a t i n g  the e f f e c t i v e  stress  a c t i n g on the c o n t a c t f a c e s i n a compact o f spheres has been d e r i v e d . This i s : a °  e  f  f  ~  a  l (  D  apjj  2 /  V  / 3  R -l) 2  T h e o r e t i c a l p l o t s of r e l a t i v e e f f e c t i v e s t r e s s as a f u n c t i o n r e l a t i v e d e n s i t y were computed f o r 4 d i f f e r e n t geometric  models.  When these a r e compared w i t h the e f f e c t i v e s t r e s s p l o t s used by p r e v i o u s workers, i t was observed  t h a t the a c t u a l  s t r e s s i s v e r y much h i g h e r than t h a t c o n s i d e r e d so f a r .  effective  - 84  -  From above, i t i s concluded that the contribution to d e n s i f i c a t i o n during hot-pressing  by p l a s t i c flow i s more than that considered  by previous investigators.  CHAPTER V I I SUGGESTIONS FOR FUTURE WORK  1.  M e t a l l o g r a p h i c study  checking a very  of hot-pressed  c o m p a c t s s h o u l d b e done f o r  t h e change o f R w i t h r e s p e c t t o change i n D i n a d i e as R i s  critical  function f o reffective stress.  The e x p e r i m e n t a l l y  d e t e r m i n e d R s h o u l d be compared w i t h t h e t h e o r e t i c a l R and n e c e s s a r y c o r r e c t i o n to the d e n s i f i c a t i o n equation  s h o u l d b e made f o r b e t t e r  understanding.  2.  The v a l u e s o f e f f e c t i v e s t r e s s a t d i f f e r e n t s t a g e s  s h o u l d a l s o be e v a l u a t e d hemispherical  3.  experimentally.  T h i s c a n b e done b y u s i n g  specimens.  From t h e above e x p e r i m e n t t h e y i e l d  c r i t e r i o n a t an e l e v a t e d  t e m p e r a t u r e s h o u l d be c h e c k e d . ( Y i e l d c r i t e r i o n r e f e r s required  4.  of densification  t o deform by s e l f - i n d e n t a t i o n i s 3 times  I f s t r a i n r a t e dependence  t = A { S i n h(a^a^)} , n  t h e same t r e a t m e n t  to "Stress  the y i e l d stress".)  o n . s t r e s s i s g i v e n by  one c a n w r i t e a d e n s i f i c a t i o n e q u a t i o n b y a s d o n e i n C h a p t e r I I , as f o l l o w s  following  I  -86D max  t  D"  1  /  Sin h [ ~ . a  i  (D  2 /  V  / 3  R -D 2  ]  dD =|  A dt  J  A computer programme for this i s given in Appendix 5 .  By knowing  the constants A, 0 2 and n for different materials, the validity of this equation should be determined.  APPENDICES  APPENDIX 1  Theoretical 1.  Calculations  Cubic Model  R  Bulk Density (%)  0.620350  52.360  0.620352  52.556  0.620374  53.149  0.620469  54.145  0.620728  55.565  0.621282  57.423  0.622307  59.749  0.624038  62.576  0.626787  65.937  0.630972  69.869  0.637180  74.393  0.646268  79.498  0.659573  85.085  0.679356  90.843  0.709891  95.936  0.715532  96.506  S 9  -  2.  "  Hexagonal P r i s m a t i c and T e t r a k a i d e c a h e d r o n Models  R  Bulk D e n s i t y  (%)  Bulk D e n s i t y  (%)  Hex. P r i s m .  b . c . c . model  0.620350  60.460  68.017  0.620352  60.687  68.273  0.620382  61.369  69.040  0.620509  62.511  70.325  0.620854  64.121  72.137  0.621593  66.207  74.483  0.622964  68.774  77.371  0.625287  71.824  80.802  0.628993  75.340  84.757  0.634679  79.272  89.181  0.643213  83.507  93.946  R  Bulk D e n s i t y (%)[Z = 14]  0.669404  99.450  -90-  Rhombic Dodecahedron Model  R  Bulk Density (%)  0.620350  74.048  0.620353  74.325  0.620397  75.156  0.620588  76.531  0.621107  78.437  0.622218  80.842  0.624288  83.697  0.627815  86.908  0.633499  90.317  0.642361  93.646  0.655600  96.410  - 91 -  APPENDIX 2 Hot-Compression Data 1.  Lead-2% Antimony - 100°C  e (sec "*") 3.9401 x I O  (psi) 5257  - 5  9.8502 x 10~  5  6507  1.9700 x 10~  4  7646  3.9401 x I O  2.  a  8986  - 4  Lead-2% Antimony - 150"C  e (sec ) 1  a  3.4972 x 10" 8.7443 x I O  5  1934  - 5  2398  1.6666 x 10~ 2.9682 x I O  (psi)  4  2797  - 4  3195  - 92 -  3.  Nickel - 800°C  • -1 e (sec )  a (psi)  -5 3.4904 X 10  6387  -5 7.4904 x 10  7461  -4 1.5873 X 10  8817  2.9629 X IO"  10070  4  4.  4  Nickel - 900°C  e (sec "*")  a (psi)  2.6666 x 10  3914  9.8619 x 10'  5171  1.7543 x 10'  5845  2.7777 x 10'  6446  - 93 -  APPENDIX 3  E f f e c t i v e Stress f o r Different Models  Cubic Model effective  CT  ^applied  0.52  2666  0.53  120  0.55  30.22  0.59  12.80  0.62  9.08  0.65  6.66  0.69  5.0  0.74  3.81  0.79  2.93  0.85  2.25  - 94 -  Hexagonal Prism Model  D  0.60  a a  effective applied 2414  0.61  104.3  0.62  46.9  0.64  26.1  0.66  16.41  0.68  11.68  0.71  7.8  0.75  5.769  0.79  4.33  0.83  3.3  - 95 -  3.  B.C.C. Model  "effective °applied  0.68  1557  0.69  69  0.70  31.2  0.72  17.42  0.74  10.93  0.77  7.39  0.80  5.2  0.84  3.8  0.89  2.8  0.93  2.2  0.99  1.46  - 96  -  Rhombic Dodecahedron Model  effective applied  0.74  959.5  0.75  42  0.76  19.14  0.78  9.882  0.80  6.69  0.83  4.52  0.86  3.210  0.90  2.355  0.93  1.774  0.96  1.354  - 97 -  APPENDIX 4 STRAIN RATES AT DIFFERENT STAGES OF DENSIFICATION  1.  (a) Lead-2% Antimony - 100°C, 918 p s i Relative Density  (b)  Strain Rate  (D)  (sec )  0.64  1.017 x 10"  2  0.66  1.449 x 10"  3  0.68  2.785 x 10~  4  0.71  6.376 x I O  0.75  1.796 x 10~  -1  - 5  5  Lead-2% Antimony - 150°C, 918 p s i Relative Density  Strain Rate  (D)  (sec" )  0.66  1.241 x IO"  0.68  2.384 x 1 0  0.71  5.459 x IO"  0.75  1.538 x I O  0.79  4.609 x 10"  4  0.83  1.472 x 10"  4  1  1  _ 2  3  - 3  - 98 -  2.  (a) Nickel - 800°C, 2162 p s i  Cb)  Relative Density  Strain Rate  (D)  (sec ) -1  0.66  3.913 x I O  0.68  0.426 x 10"  3  0.71  1.278 x 10~  3  0.75  3.192 x I O  0.79  8.59  0.83  2.444 x I O  - 2  - 4  x IO"  5  - 5  Nickel - 900° C, 2105 p s i  Relative Density CD)  Strain Rate (sec ) -1  0.66  2.551 x 1 0  0.68  4.206 x 10~  0.71  8.357 x I O  - 3  0.75  2.086 x I O  - 3  0.79  5.575 x 1 0  _ 4  0.83  1.598 x IO"  _ 1  2  4  - 99 -  3.  Alumina - 1600°C, 4000 p s i  Relative Density  Strain Rate  (D)  (sec ) -1  0.64  2.375 x 1 0  0.66  3.703 x IO"  0.68  7.716 x I O  0.71  1.895 x IO"  4  0.75  5.635 x 10~  5  0.79  1.799 x I O  0.83  6.071 x 10~  _ 2  3  - 4  - 5  6  APPENDIX 5 COMPUTER PROGRAMME  RFS NU^ 010678  >  UNIVERSITY OF 6 C COMPUTING CENTRE  MTS(AN1 20)  1 1 :32:01  08-18-71  ******************** PLEASE RETURN TU ELECTRICAL ENGINEERING ******************** SSIG XRSA P=8 T=1M PRIO=V **LAST SIGNUN WAS: 11 :33 :35 07-13-71 USER "XRSA" SIGNED ON AT 11:32:07 UN 08-1B-71 JUbl H L b l READY. SL1S F I L E ! 1 INTEGER ST0RE1, STORE 2, CASE 2 REAL N 2.26 ~3 PAT A T AKE 1 .TAX E2 , T AKE3,TA*E 4 / ' G EN E ~,^ R A T T STgC » 1AL" J : 4 READI5.1) N t BETA, A, SIGMA, ALPHA, DELTAT, ALPHA 2 5 1 FORMAT <2F10.4,F10.5,4F10.4) 6 dRITE ( 6 , DN, BETA, A, SIGMA, ALPHA, DELTAT, ALPHA2 7 CASE=2 8 IFIABSIALPHA2).GT.0.00005) CAsE=l ~~9 r A L U t l = UUfcSI 1( IAKb1, IAKE3,CASE) 10 VALUE2 = QUEST2( TAK.E2, TAKE4.CASE) 10. 1 10.2 11 WRITE(6,102) VALUE1, VALUE2 12 102 F0RMAT(///40X'THIS CORRESPONDS TO THE ',2A4, 'FORMULA CASE') Ti C NUW KtAU IH-E T7ABTF"PF R"~V5—DEVALUES 14 DIMENSION X( 16) ,H 16) 15 DATA X , Y / 1 6 * 0 . 0 , 16*0.0 / 16 WRITE(6,32) 17 32 FORMATI5X,'TABLE OF D VERSUS R VALUES') 18 DO 2 12=1,16 T9 REAU(5,3) U,R,Z : 20 3 F0RMATI3F10.4) 21 WRITEI6,31)D,R 22 31 FORMAT(5X,E14.7,5X,E14.7) 23 X(I2) = D 24 Y(I2)=R r  I F I Z . 6 T . 0 . 1 )  -tt  26 27 28 29 30 "31 32 33 34 35 36 "37 38 39 40 41 42  2 4  5  r  1  GO TU 4  T  :  CONTINUE I2MINS=I2-1 JTIMES=I2MINS*2 ISUM=12 OMAX=D DTSrOT=XTT) DELTAD=IDMAX-DNOT)/JTIMES W R I T E ( 6 , 5 ) DMAX,OELTAD,ISUM FORMAT(/5X,'DMAX = ' , F 10 . 4 , 5X , ' D ELT AD = , F 1 0 . 5 , 5 X , • T 0 T A L 1CARDS OF D VS R READ = ' , I 4 ) SUM1=0.0 U=UNU1 JT IMES=JT IMES-1 DO 6 I6=1,JTIMES FACT3=D*BETA FACT4=FACT3**0.667 FACT 1 = 1(0*BETAI**0.667 I*R*R-1.0 1  "ST  bALTZ=IFACTl**'N77D  43 .1 44 45 46 46. 1  IFICASE.E0.2) GO TO 150 VALUE=1.0/(SINH((ALPHA2/ALPHA)*SIG MA/FACT 1) ) ** N FACT2=VALUE/D  ' ,_, O FI 1  v  NUMBER OF  —  >  <rtrr2— / 150 48 49 50 51 —52 53 54 55 56 7 57 5B 9 59 10 60 61 6Z 11 63 tm 65 109 66 12 67 6 68 13 69 7-t) 71 72 73 74 75 Tt 77 78 152 79 151 80 81 8-2 83 84 85 86 87 16 8-8 1-4 89 90 91 91.1 92 ^ 94 95 96 96.1 97 -9^ 99 100 END UF F I L E  CONTINUE SUM1 = SUM1 + F A C T 2 * D E L T A 0 D=D+DELTAD ST0RE1=1 M=0 DO 7 I7=U,ISUM IF ( A B S U l 1 7 1 - 0 ) . L E . 0 . 0 0 0 2 ) M= 1 7 IF(X(17).LT.D) 5T0RE1 = 17 I F ( X ( I 7 ) . G T . U I GO TO 9 CONTINUE GO TO 10 SIUKE 2=I I I F ( H . E U . O ) GO TO 11 R=YIM) GO TO 12 DELT = D-XI S T O R E D R=Y< ST0KE11 + I Y I S T 0 R E 2 ) - Y I S T O R E l I )*OELT / ( X( S T 0 R E 2 ) - X ( S T ORE 1 ) ) WRITEI6.109) CJ,R FURMAT!/'D='F10.7.5X,'R=',F10.7) WRITE(6,13) SUM1 CONTINUE FORMAT!'SUM1=»E16.9) CONS=A*(<SIGMA/ALPHA)**NI*(10.**(-20)) IF ( C A S E . EQ. 1) CONS=A<- ( 1 0 . * « ! - 8 ) ) SUM2=0.0 FACT = SUMl/(CONS*OELrAT) ' NCOUNT=0 00 152 1152=1,100 I F I F A C T . L T . 1 1 0 0 0 . 0 ) ) GO TO 151 FACT = F A C T / 1 0 . 0 NCUUNT=NCOUNT+1 CONTINUE NLIM = F ACT I F I N L I M . L T . 4 ) NLIM=4 NL1 = N L l M / 4 r F T m l . L I .2T~NL1=1 " DU 14 I 1 4 = 1 , N L I M , N L 1 TIME= I 14*C)ELTAT * ( 1 0 . **NCOUN r) SUM2=TIME*CONS WRITE16,16)TIME,SUM2 FORMAT I / ' T I M E = ' E 1 2 . 5 , 5 X , ' S U M 2 = ' , E 1 2 . 5) CUNT I NU6 STOP END FUNCTION QUEST 1( TAKE 1, TAKE3 , CAS E) INTEGER CASE QUE ST1 = T AKE1 IF 1 C A S E . E Q V 2 ) QUEST 1-TAKE3 RETURN END FUNCTION Q U E S T 2 ( T A K E 2 , TAKE 4,CAS El INTEGER CASE QUEST2= TAKE 2 T F 1 t W S t iT:Q-; 2T-(TOTrSTZ=TAKE4 RETURN END  •  ^ ,_, O ^» ;  :  "  :  —  :  - 103 -  BIBLIOGRAPHY  1.  R. Chang and C.G. Rhodes, J . Am.  2.  R.L. Coble and J.S. E l l i s , J . Am.  3.  R.L. Coble, i n "Sintering and Related Phenomena", Ed., Kuezynski, N.A.  Ceram. Soc. , 4-5, 379 (1962). Ceram. Soc. , 46,  438-41 (1963) G.C.  Hooton and C.F. Gibbon (Gordon and Breach,  Science Publishers, New York), (1967), p. 329. 4.  R.L. Coble, J . App. Phys., 41, 4798 (1970).  5.  P.L. Farnsworth and R.L. Coble, J . Am.  6.  E.J. Felten, J . Am.  7.  G.M.  Ceram. Soc., 49, 264 (1966).  Ceram. Soc., 44, 381, (1961).  Fryer, Trans. B r i t . Ceram. Soc., 66, 127 (1967) and 68,  181-185 (1969). 8.  J.K. Mackenzie and R. Shuttleworth, Proc. Phys. Soc. (London), B, 62, 833-852 (1949).  9.  H.  HeiErky,  Z. Angew. Math. Mech., _3> 241 (1923).  10.  J . Ishlinsky, J . App. Math. Mech. U.S.S.R. , {5, 233 (1944).  11.  J . J . Jonas, CM.  Sellars and W.J. McG. Tegart, i n "Metallurgical  Reviews", v o l . 14, Review 130, 1 (1969). 12.  A.K. Kakar and A.CD.  Chaklader, J . App. Phys. , J38, 3223-30 (1967).  13.  A.K. Kakar "Deformat ion Theory of Hot—Pressing", Ph.D.  Thesis,  Department of Metallurgy, University of B r i t i s h Columbia 14.  M.S.  (1967).  Koval'Chenko and G.V.Samsonov, Poroshkovaya Met., 1, 3  (1961).  .  15.  M.L. Kronberg, J . Am.  16.  G.E. Mangsen, W.A. 43, 55 (1960).  Ceram. Soc., 45, 6, 274 (1962).  Lambertson  and B. Best, J . Am.  Ceram. Soc.,  -104-  17.  J.D. McClelland, In "Powder Metallurgy Proceedings of International Conference, New York, 1960, W. Leszynski, Ed. (Interscience Publishers, New York, 1961).  18.  R.K. McGeary, J . Am. Ceram. Soc., 44, 513 (1961).  19.  P. Murray, D.T. Livey and J . Williams, in"Ceramic Fabrication Process", W.D. Kingery, Ed. (Technology Press Cambridge, Mass., and John Wiley and Sons, Inc., New York, 1958).  20.  F.R.N. Nabarro, i n the "Strength of Solids", London (Physical Society),  21.  75 (1948).  R.C. Rossi and R.M. Fulrath, J . Am. Ceram. Soc., 48,  558-64  (1965). 22.  D.R. Rummler and H. Palmour, J . Am. Ceram. Soc. , _51, no. 6, 320 (1968).  23.  S. Scholz and B. Lersmacher, Arch. Eisenhuettenw,  41, 98 (1964).  24.  W.O.  25.  T. Vasilos, J . Am. Ceram. Soc. , 4_3, no. 10, 517 (1960).  26.  T. Vasilos and R.M. Spriggs, J . Am. Ceram. Soc. , 46_,  Smith, P.D. Foote and P.F. Busang, Phys. Rev., 34, 1272 (1929).  493-96  (1963). 27.  S.I. Warshaw and F.H. Norton, J . Am. Ceram. Soc., 45, 479 (1962).  

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