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Deformation theory of hot-pressing Kakar, Ashok Kumar 1967

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The  University  of B r i t i s h  Columbia  FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of ASHOK KUMAR KAKAR B.Sc, B.Sc,  Pan jab U n i v e r s i t y ,  I n d i a , 1961  (Hons), Pan jab U n i v e r s i t y , B.E.  I n d i a , 1962  Indian I n s t i t u t e of Science, Bangalore, I n d i a , 1964  MONDAY, AUGUST 28, a t 3:30 P.M. IN ROOM 201, METALLURGY BUILDING COMMITTEE IN CHARGE Chairman: R.G. Campanella A . C D . Chaklader I. L e j a  I . McT. Cowan J.A. Lund E. P e t e r s E. T e g h t s o o n i a n  E x t e r n a l Examiner; P r o f e s s o r J . Pask Department o f M i n e r a l Technology University of C a l i f o r n i a Berkeley, C a l i f o r n i a Research S u p e r v i s o r :  A.CD.. Chaklader  DEFORMATION THEORY OF HOT  PRESSING  ABSTRACT The p o s s i b l e deformation behaviour of i n a compact has been t h e o r e t i c a l l y a n a l y z e d and mentally v e r i f i e d .  spheres experi-  The change i n c o n t a c t area r a d i u s  'a' r e l a t i v e to the p a r t i c l e r a d i u s R has been r e l a t e d to the b u l k d e n s i t y and b u l k s t r a i n f o r f o u r p o s s i b l e simple c u b i c (Z = 6 ) , orthorhombic  modes of packing:  (Z = 8), rhombohedral (Z = 12),  and body-centered  cubic  (Z = 8) . An e q u a t i o n r e l a t i n g  the above  3  2  can be r e p r e s e n t e d by D - D types of p a c k i n g s , D and D a/R  and a t a/R  = 0,  Q  q  =  D  q  parameters  (a/R)  for different  being the d e n s i t i e s a t any  respectively.  I t has been shown  e x p e r i m e n t a l l y by deforming monosized l e a d spheres at room temperature,  50 and  100°C i n a c y l i n d r i c a l d i e , t h a t  the o v e r a l l d e f o r m a t i o n i s s i m i l a r to t h a t of the o r t h o r h o m b i c a l l y packed  spheres.  A change i n the c o o r d i n a t i o n  number Z d u r i n g the deformation process was and may  p a r t i a l l y account  also  f o r the d e v i a t i o n from  u  observed the  t h e o r e t i c a l l y predicted values. S i m i l a r experiments  u s i n g s a p p h i r e and K-Monel  spheres were a l s o c a r r i e d out i n the temperature 1570  - 1700°C and 800'-. 1000°e r e s p e c t i v e l y .  showed t h a t the d e f o r m a t i o n behaviour was to t h a t o f the l e a d  spheres.  very  ranges  The  results  similar  A study of the geometry of d e f o r m a t i o n r e v e a l e d t h a t most of the spheres deformed i n a random manner, a l t h o u g h i n d i v i d u a l c o l o n i e s of o r t h o r hombic, t e t r a g o n a l and rhombohedral observed. faces,  I t was  packings were  a l s o observed t h a t the deformed  t h a t were a p p r o x i m a t e l y p e r p e n d i c u l a r to the  d i r e c t i o n of p r e s s i n g were about than those p a r a l l e l  2.2  times  larger  to the d i r e c t i o n of p r e s s i n g .  T h i s o b s e r v a t i o n has been s u b s e q u e n t l y used to modify the t h e o r e t i c a l models. and p l a s t i c  The p a r t i c l e  rearrangement  flow have been found to be the  predominant  mechanisms f o r the d e n s i f i c a t i o n of l e a d , K-Monel, and s a p p h i r e spheres under the e x p e r i m e n t a l c o n d i t i o n s used  in this  investigation. The c r i t e r i o n f o r y i e l d i n g of two hemi-  spheres of the same m a t e r i a l i n c o n t a c t was  used  i n c o r p o r a t e the y i e l d  density  equation.  s t r e n g t h i n the b a s i c  to  T h i s e q u a t i o n has been found to f i t the  d a t a o b t a i n e d d u r i n g the h o t - p r e s s i n g of the s p h e r e s . I t has been observed t h a t the d e f o r m a t i o n of  sapphire s i n g l e c r y s t a l  spheres  complex d e f o r m a t i o n p r o c e s s .  takes p l a c e by a  The presence of the  b a s a l and p r i s m a t i c s l i p has-been i d e n t i f i e d spheres deformed a t 1570 cross s l i p  and 1700°C.  Presence  i s a l s o c o n f i r m e d by the o p t i c a l  e l e c t r o n micrographs  a t these  i n the  temperatures.  and  of  GRADUATE STUDIES F i e l d of Study;.  Metallurgy  M e t a l l u r g i c a l Thermodynamics S t r u c t u r e of Metals Diffusion I Advanced P h y s i c a l Ceramics E l e c t r o n Microscopyo  C S . Samis E. T e g h t s o o n i a n L.C. Brown A . C D . Chaklader D. Tromans  R e l a t e d F i e l d s of Study Atomic and N u c l e a r P h y s i c s Analysis D i f f e r e n t i a l Equations  G.Dt.-* G r i f f i t h s " W.H. Gage W.H. Gage  PUBLICATIONS Ashok K. Kakar and A . C D . C h a k l a d e r , "Deformation Theor; of Hot P r e s s i n g " , J . App. Phys. V o l 3 8 ( J u l y ) 1967. Ashok K. Kakar and A . C D . C h a k l a d e r , "Deformation o f S i n g l e C r y s t a l Sapphire Spheres d u r i n g H o t - P r e s s i n g ' Submitted t o the J.- Am. Ceram. Soc.  DEFORMATION THEORY OF HOT-PRESSING  by  ASHOK KUMAR KAKAR B . S c , Panjab U n i v e r s i t y , I n d i a , 1961 B . S c , (Hons.), Panjab U n i v e r s i t y , I n d i a , 1962 B.E., I n d i a n I n s t i t u t e o f S c i e n c e , 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  I n t h e Department of METALLURGY We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1967  In  presenting  for  an  that  advanced  the  study, thesis  my  Department  at  in the  make  agree  partial  i t freely  that  or  representatives.  his  of  written  this  thesis  may  for  permission.  of  The University of B r i t i s h V a n c o u v e r 8, Canada.  Columbia  be  of  for  granted It  financial  of  the  British  available  permission  purposes  by  fulfilment  University  scholarly  publication  without  shall  I further for  thesis  degree  Library  Department or  this  for  Columbia,  the  Head  shall  of  of  this  my  that not  agree  and  copying  is understood gain  I  reference  extensive by  requirements  be  copying allowed  Supervisor:  Dr.  A.  C.  D.  Chaklader - i A B S T R A C T  The p o s s i b l e d e f o r m a t i o n b e h a v i o u r of spheres i n a compact has been t h e o r e t i c a l l y a n a l y z e d and e x p e r i m e n t a l l y v e r i f i e d .  The  change  i n c o n t a c t a r e a r a d i u s 'a' r e l a t i v e to the p a r t i c l e r a d i u s R has been r e l a t e d t o t h e b u l k d e n s i t y and b u l k s t r a i n f o r f o u r p o s s i b l e modes of packing:  s i m p l e c u b i c (Z = 6 ) , o r t h o r h o m b i c  (Z = 1 2 ) , and body-centered  (Z=  8 ) , rhombohedral  c u b i c (Z = 8 ) .  An e q u a t i o n r e l a t i n g the above parameters 3 by D — D  = Q  ~2  can be r e p r e s e n t e d  2 D  (/) &  0  f°  R  the d e n s i t i e s a t any a/R  r  d i f f e r e n t types of p a c k i n g s , D and D  and a t a/R  = 0, r e s p e c t i v e l y .  being  q  I t has been  shown e x p e r i m e n t a l l y by d e f o r m i n g monosized l e a d spheres a t room t e m p e r a t u r e , 50 and 100°C i n a c y l i n d r i c a l d i e , t h a t the o v e r a l l d e f o r m a t i o n i s s i m i l a r t o t h a t of the o r t h o r h o m b i c a l l y packed s p h e r e s . A change i n the c o o r d i n a t i o n number Z d u r i n g the d e f o r m a t i o n p r o c e s s a l s o observed and may  was  p a r t i a l l y account f o r the d e v i a t i o n from the  t h e o r e t i c a l l y predicted,values., S i m i l a r experiments u s i n g s a p p h i r e and K-Mpnel spheres were a l s o c a r r i e d out i n the temperature range 1570 - 1700°C and 800 - 1000°C respectively.  The r e s u l t s showed t h a t the d e f o r m a t i o n b e h a v i o u r  very s i m i l a r to t h a t of the l e a d  was  spheres.  A s t u d y o f the geometry o f d e f o r m a t i o n r e v e a l e d t h a t most of the spheres deformed I n a random manner, a l t h o u g h i n d i v i d u a l c o l o n i e s of o r t h o r h o m b i c , t e t r a g o n a l and rhombohedral p a c k i n g s were o b s e r v e d . I t was  a l s o observed t h a t the deformed f a c e s t h a t were a p p r o x i m a t e l y  p e r p e n d i c u l a r t o the d i r e c t i o n of p r e s s i n g were about 2.2  times l a r g e r  - i ithan those p a r a l l e l t o t h e d i r e c t i o n  of pressing.  This observation  has been s u b s e q u e n t l y used t o modify  the t h e o r e t i c a l  models.  The p a r t i c l e  rearrangement and p l a s t i c f l o w have been found t o be the predominant mechanisms f o r t h e d e n s i f i c a t i o n  o f l e a d , K-Monel, and s a p p h i r e  under t h e e x p e r i m e n t a l c o n d i t i o n s used i n t h i s The c r i t e r i o n f o r y i e l d i n g  spheres  investigation.  o f two hemispheres o f t h e same  m a t e r i a l i n c o n t a c t was used t o i n c o r p o r a t e t h e y i e l d s t r e n g t h i n t h e basic density equation.  T h i s e q u a t i o n has been found t o f i t t h e  data obtained d u r i n g the h o t - p r e s s i n g of the spheres. I t has been observed c r y s t a l spheres  that the deformation of sapphire  t a k e s p l a c e by a complex d e f o r m a t i o n p r o c e s s .  o f t h e b a s a l and p r i s m a t i c s l i p has been i d e n t i f i e d deformed a t 1570 and 1700°C.  Presence  by t h e o p t i c a l and e l e c t r o n micrographs  The  single presence  i n t h e spheres  of cross s l i p i s a l s o confirmed a t these  temperatures.  - iii -  A C K N O W L E D G E M E N T  The a u t h o r 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 t h e s i s d i r e c t o r , Dr. A. C. D. C h a k l a d e r .  Sincere gratitude i s also  extended t o t h e Ph.D. committee members, Dr. E. T e g h t s o o n i a n and Dean W. M. A r m s t r o n g , t o o t h e r f a c u l t y members and f e l l o w g r a d u a t e 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 .  Thanks a r e a l s o extended t o t h e members  of t h e t e c h n i c a l s t a f f whose h e l p was r e a d i l y a v a i l a b l e t o t h e a u t h o r . F i n a n c i a l a s s i s t a n c e i n t h e form o f a Welding Research C o u n c i l A s s i s t a n t s h i p , C l a y b u r n - H a r b i s o n F e l l o w s h i p , and N a t i o n a l Research C o u n c i l S t u d e n t s h i p is gratefully  acknowledged.  - ivT A B L E  OF  C O N T E N T S  Page I.  II.  INTRODUCTION .  1  A.  THEORIES BASED ON PLASTIC FLOW MECHANISM  ..  B.  THEORIES BASED ON THE/ NABARRO-HERRING CREEP MECHANISM .. .  1 3  C.  THE PRESENT CONCEPTS OF HOT-PRESSING  3  D.  SCOPE OF PRESENT INVESTIGATION  4  THEORETICAL MODELS  6  PART ONE-THEORY  6  A.  B.  GEOMETRIC RELATIONSHIPS  7  1.  C u b i c Model  8  2.  Hexagonal P r i s m a t i c Model  10  3.  Rhombic Dodecahedron Model  13  4.  T e t r a k a i d e c a h e d r o n Model  15  a)  B.C.C. Model  15  b)  T e t r a k a i d e c a h e d r o n Model  16  DEDUCED RELATIONSHIPS 1.  .  Bulk Density  , .  20 20  a)  C u b i c Model  b)  Hexagonal P r i s m a t i c Model  23  c)  Rhombic Dodecahedron Model  24  d)  T e t r a k a i d e c a h e d r o n Model  25  e)  R e l a t i o n s h i p Between D and D  27  ......  22  T a b l e o f Contents  (Cont.)  Page  2.  Bulk S t r a i n  28  3.  B u l k D e n s i t y and B u l k S t r a i n  29  4.  D, a/R and Z  .  30  PART TWO-EXPERIMENTAL VERIFICATION OF MODELS A.  PARTICLE REARRANGEMENT  B.  PLASTIC DEFORMATION OF SPHERES M a t e r i a l s Used  2.  E x p e r i m e n t a l Techniques  33  ..  ,  36  •  36  ....................  37  a)  Lead  37  b)  S i n g l e C r y s t a l Sapphire  41  c)  K-Monel  43  3.  Experimental Procedures  4.  R e s u l t s and D i s c u s s i o n a)  D v s a/R  b)  D vs Z  ..........  ...  43 48 48  .  v s a/R  52 52  d)  D vs e  •• • • • •  e)  D v s a/R v s Z  57  f)  D vs (a/R)  57  fe  2  GEOMETRY OF DEFORMATION IN THE DIE A.  33  ........  1.  c)  III.  .  INTRODUCTION  69 . .. .  69  1.  The D i e - W a l l E f f e c t  69  2.  Mode o f F i l l i n g t h e D i e  70  3.  Haphazard P a c k i n g s  72  5 7  - v iTable of Contents (Cont.) Page B.  IV.  MECHANICAL PROPERTIES A.  V.  EXPERIMENTAL TECHNIQUES AND RESULTS ....  INTRODUCTION  84 .  1.  Deformation  2.  E f f e c t o f Work-Hardening  THEORY  C.  EXPERIMENTAL VERIFICATION  MECHANISM OF —  84  Under S t a t i c C o n d i t i o n s  B.  84 87 ........  89 91  DEFORMATION OF SINGLE CRYSTAL SAPPHIRE  SHPERES BY HOT-PRESSING  VI.  75  104  A.  INTRODUCTION  104  B.  EXPERIMENTAL TECHNIQUES, RESULTS, AND DISCUSSION ...  108  1.  O p t i c a l Microscopy  108  2.  E l e c t r o n Microscopy  108  3.  Fractured Surfaces  112  DISCUSSION  118  A.  PRESSURE DISTRIBUTION IN THE DIE  B.  CORRECTED THEORETICAL MODELS  . ..  118 121  1.  C o r r e c t e d Cubic Model  121  2.  C o r r e c t e d Hexagonal P r i s m a t i c Model  122  3.  C o r r e c t e d D-o E q u a t i o n  133  a)  C o r r e c t e d Cubic Model  123  b)  C o r r e c t e d Hexagonal P r i s m a t i c Model .........  123  4.  Comparison w i t h t h e R e s u l t s  124  T a b l e o f Contents  (Cont.)  - v i i Page  C. . HOT-PRESSING—MECHANISMS  VII.  128  1.  General  ......  128  2.  H o t - P r e s s i n g o f Ceramic Oxides  3.  Creep o f M a t e r i a l s (under H o t - P r e s s i n g C o n d i t i o n s ) 133  ••••  SUMMARY AND CONCLUSIONS  VIII.  IX.  ......  OF DENSIFICATION  . 136  SUGGESTIONS FOR FUTURE WORK  139  APPENDICES APPENDIX I  131  140 SIMPLE AND SYSTEMATIC MODES OF PACKING SPHERES . .  140  APPENDIX I I  X.  BIBLIOGRAPHY  A.  THEORETICAL CALCULATIONS  144  B.  EXPERIMENTAL RESULTS  147  .  ,  . 152  - viii L I S T  OF  F I G  U R E S  No.  Page  1.  Geometry o f D e f o r m a t i o n of Two  2.  Geometric R e l a t i o n s h i p of C u b i c Model Showing the C r i t i c a l Stage  11  Geometric R e l a t i o n s h i p of Hexagonal P r i s m a t i c Model Showing the C r i t i c a l Stage  12  3.  4.  Spheres i n C o n t a c t  Geometric R e l a t i o n s h i p of Rhombic Dodecahedron Showing the C r i t i c a l Stage  9  Model ....  14  5.  Geometric R e l a t i o n s h i p of B.C.C. Model Showing the C r i t i c a l Stage 17  6.  Geometric R e l a t i o n s h i p of T e t r a k a i d e c a h e d r o n Model Showing the C r i t i c a l Stage  18  T h e o r e t i c a l R e l a t i o n s h i p of R v s a/R f o r the Proposed Models  21  T h e o r e t i c a l R e l t i o n s h i p of D v s a/R f o r the Proposed Models  26  T h e o r e t i c a l R e l a t i o n s h i p of Models  30  7.  8. 9.  -  v s a/R f o r the Proposed  10.  T h e o r e t i c a l R e l a t i o n s h i p of D v s a/R v s Z f o r the Proposed Models 31  11.  A p p a r a t u s f o r S e p a r a t i n g U n i f o r m Spheres from the I r r e g u l a r ones  34  12.  Volume S t r a i n (%) v s t h e I n i t i a l A p p l i e d Load  35  13.  Load-Compaction  38  14.  Assembly f o r H o t - P r e s s i n g Lead Spheres  15.  Schematic R e p r e s e n t a t i o n o f the D i e used f o r H o t - P r e s s i n g of Lead Spheres . 40  16.  Schematic R e p r e s e n t a t i o n of the Vacuum H o t - P r e s s used f o r S a p p h i r e and K-Monel Spheres .„  Curve i n the Case of Lead Spheres  39  42  - i xL i s t o f F i g u r e s (Cont.)  No.  *S8£.  17.  A p p a r a t u s f o r S t u d y i n g t h e Deformed Spheres  44  18.  P h o t o m i c r o g r a p h s o f Deformed Spheres Showing t h e T y p i c a l F l a t Faces  45  S t a t i s t i c a l D i s t r i b u t i o n o f t h e Number o f Spheres vs t h e C o o r d i n a t i o n Number  46  19.  20.  D v s a/R f o r Lead a t Room Temperature and S a p p h i r e at  #  1570°,  •  1600°, and  •  1700°C  ... 49  21.  D v s a/R f o r Lead a t . 5 0 and 100°C  . 50  22.  D. v s a/R f o r K-Monel  . 51  23.  ( D . - D ) v s Z f o r Lead o Change o f Z w i t h t h e I n c r e a s i n g D e f o r m a t i o n f o r K-Monel Spheres >> a / R . f o r Lead a t Room Temperature and S a p p h i r e a t  24. 25.  £  #  1600°, and M  1700°C  55  v s a/R f o r Lead a t 50 and 100°C .'  56  27.  D v s £. f o r Lead a t Room Temperature and S a p p h i r e a t  28.  H 1570°, A 1600°, and ® 1700°C D v s ^ f o r Lead a t . 50 and 100°C  29.  54  vs  Al570°, 26.  53  . .. 58 59  D v s a/R v s Z f o r Lead a t Room Temperature and S a p p h i r e a t • 1570°, ® 1600°, and B 1700°C  60  30.  D v s a/R v s Z f o r Lead a t 50 and 100°C  61  31.  2 D v s (a/R) f o r Lead a t Room Temperature, 50° and 100°C and S a p p h i r e a t ©  32.  D vs (a/R)  2  1570°,  f o r K-Monel  A  1600°, and  B  ..............  1700°C... 63 ...... 64  2 33. 34.  (D — D ) / D v s (a/R) f o r Lead a t Room Temperature ....... 66 ( D - D )/D v s ( a / R ) f o r Lead a t 50°C... ,67 o o O  q  2  L i s t o f F i g u r e s (Cont.)  x  No35. 36. 37.  38. 39. 40.  ^ 2 v s (a/R) f o r Lead a t 100°C  ( D - D )/D o o E f f e c t 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 P a c k i n g One-Size Spheres ( A f t e r McGeary39)  68 71  S t a t i s t i c a l D i s t r i b u t i o n o f t h e Number o f Spheres w i t h t h e C o o r d i n a t i o n Number (Data from J o n e s ^ l )  74  S t e r e o g r a p h i c R e p r e s e n t a t i o n o f t h e Geometry o f Def o r m a t i o n o f Spheres i n the D i e  77  P h o t o m i c r o g r a p h o f t h e Lead Sphere Deformed i n A c c o r d a n c e w i t h Hexagonal P r i s m a t i c Model  82  P h o t o m i c r o g r a p h o f t h e Lead Sphere Deformed i n A c c o r d a n c e w i t h Rhombic Dodecahedron Model  83  2 41 42.  (a/R) v s L f o r Lead a t Room Temperature (a/R)  92  v s L f o r Lead a t 50°C  2  (a/R)  44. 45. 46.  ( a / R ) v s a f o r K-Monel (D — D ) / D v s L f o r Lead a t Room Temperature ( D - D )/D„ v s L f o r Lead a t 50 and 100°C o o  95 97 98  47.  D v s a f o r K-Monel  99  48.  R e p r e s e n t a t i o n o f [1120] D i r e c t i o n i n Case o f Z i n c and S a p p h i r e C r y s t a l s ( A f t e r Kronberg63)  107  Schematic R e p r e s e n t a t i o n o f B a s a l and P r i s m a t i c S l i p Systems i n S a p p h i r e ( A f t e r Scheuplein and G i b b s ^ )  107  50.  P h o t o m i c r o g r a p h s o f t h e T y p i c a l C o n t a c t Faces  109  51.  E l e c t r o n m i c r o g r a p h s Showing E v i d e n c e o f Simple and  2  .  93  43.  49.  v s L f o r Lead a t 100°C  ...  94  2  o  o  Wavy S l i p  ..  113  52.  C r o s s - S l i p i n S a p p h i r e a t 1570°C  115  53.  C r o s s - S l i p i n S a p p h i r e a t 1700°C  116  54.  F r a c t u r e d S u r f a c e on t h e S a p p h i r e Sphere  •...  117  L i s t o f F i g u r e s (Cont.)  -  x i-  No. 55.  Approximate L i n e s o f E q u a l P r e s s u r e i n Copper Compact.Pressure 100,000 p s i ( A f t e r Duwez and Zwell )  119  D v s a/R f o r Lead (Comparison.with T h e o r e t i c a l Models);....  125  7 4  56. 57.  the Corrected  D v s a/R f o r K-Monel (Comparison w i t h t h e C o r r e c t e d T h e o r e t i c a l Models) .........  126  D e n s i f i c a t i o n Rate as a F u n c t i o n o f Time C a l c u l a t e d From t h e D e n s i f i c a t i o n p f MgO Shown a t t h e Upper R i g h t Hand Corner ( A f t e r S p r i g g s and A t t e r a a s )  129  |  58.  7 5  59.  B a s i c Systems o f S p h e r i c a l P a c k i n g s  ( A f t e r M o r g a n ^ ) . 142  60.  Shapes o f t h e Pores i n Rhombic, C u b i c , and B.C.C. P a c k i n g s ( A f t e r Morgan )  143  - xii L I S T  O:F  T A B L E S  No.  Page  I.  Deformation  Models and T h e i r C h a r a c t e r i s t i c s  II.  C o r r e l a t i o n o f C o o r d i n a t i o n Number and D e n s i t y f o r Lead Shot, 3.78 mm diam ( A f t e r J o n e s ^ l )  73  III.  Mean P r e s s u r e v s E l a s t i c L i m i t ( A f t e r T a b o r ^ )  88  IV.  R e s u l t s Showing V a l i d i t y o f E q u a t i o n P  7  ( A f t e r O ' N e i l l and G r e e n w o o d ) 52  V.  VI.  VII.  VIII.  = 3Y  m  Y i e l d S t r e n g t h V a l u e s from t h e S l o p e s o f P l o t s i n F i g s . 41 t o 47 . . .  88  100  E q u a t i o n s R e l a t i n g D e n s i t y (Volume) o f t h e Compact w i t h t h e A p p l i e d P r e s s u r e  103  C o r r e c t e d Cubic and Hexagonal P r i s m a t i c Models and T h e i r C h a r a c t e r i s t i c s  127  T h e o r e t i c a l and E x p e r i m e n t a l v s ( a / R ) and D v s ( a / R )  127  2  IX.  32  S l o p e s o f (D — D )/D  2  B a s i c Methods o f P a c k i n g and T h e i r C o n s t r u c t i o n  141  - 1 I  I N T R O D U C T I O N  H o t - p r e s s i n g of p a r t i c u l a t e compacts has r e c e n t l y gained importance s i n c e i t has an added advantage of g i v i n g g r e a t e r d e n s i f i c a t i o n at lower  temperatures  s i n t e r i n g methods. i n use s i n c e 1900,  and  s h o r t e r times  than i s p o s s i b l e by normal  D e s p i t e the f a c t t h a t h o t - p r e s s i n g methods have been i t i s o n l y d u r i n g the l a s t  have been made towards u n d e r s t a n d i n g  twenty y e a r s t h a t  efforts  the mechanisms of the hot p r e s s i n g  process.  1—8 Many e m p i r i c a l e q u a t i o n s  , first  proposed to express  the  r e l a t i o n s h i p between the d e n s i t y (volume) of the powder' compact and a p p l i e d p r e s s u r e , l e d to s p e c u l a t i o n s c o n c e r n i n g pressing.  the mechanism of  Modern t h e o r i e s of h o t - p r e s s i n g a r e d e r i v e d p r i m a r i l y  from s i n t e r i n g t h e o r i e s or from the s t r e s s - e n h a n c e d  diffusional  the  hoteither  creep  9 10 mechanism.  The  s i n t e r i n g t h e o r i e s a r e based on e i t h e r p l a s t i c flow '  or v i s c o u s f l o w ^  as the mechanism of d e f o r m a t i o n  of s o l i d s .  Clark  9 and White  c o n s i d e r e d the open-pore.stage of s i n t e r i n g  i n which the m a t e r i a l  i s assumed to f l o w i n t o the l e n s between the o r i g i n a l s p h e r i c a l The  l a t e r or the c l o s e d - p o r e stage of s i n t e r i n g was  Mackenzie and  Shuttleworth^.  of c l o s e d pores  particles.  i n v e s t i g a t e d by  T h e i r theory, r e s t s on a model c o n s i s t i n g  i n a homogeneous m a t r i x which a r e reduced  i n s i z e with  time as a r e s u l t of s u r f a c e f o r c e s . A. THEORIES BASED ON PLASTIC FLOW MECHANISM Among the n o t a b l e c o n t r i b u t i o n s to the theory of h o t - p r e s s i n g ,  12 the b e s t known i s , perhaps, t h a t of Murray, Kodgers and  Williams  (also  - 2 Murray, L i v e y and W i l l i a m s  13  ) who  m o d i f i e d the s i n t e r i n g t h e o r y  Mackenzie and S h u t t l e w o r t h to e x p l a i n the observed b e h a v i o u r p r e s s i n g o f v a r i o u s o x i d e s and c a r b i d e s .  of  i n hot-  Murray i n c l u d e d the a p p l i e d  p r e s s u r e a l o n g w i t h the s u r f a c e t e n s i o n as the d r i v i n g f o r c e f o r the r a t e of d e n s i f i c a t i o n .  The  f i n a l form of t h e i r 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  where  D = the r e l a t i v e d e n s i t y a t t i m e t P = the a p p l i e d and  pressure  n = the v i s c o s i t y a t i n f i n i t e r a t e o f  shear.  T h e i r p l a s t i c f l o w t h e o r y succeeded i n e x p l a i n i n g the of r a t e o f 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 end-point  density at constant  the e f f e c t of p r e s s u r e  temperature.  Murray's  increase on  equation  has been employed t o i n t e r p r e t d a t a on a l u m i n a by Mangsen, Lambertson and Best  14  ; on s i l i c a g l a s s by V a s i l o s  15  ; i n l i q u i d - p h a s e s i n t e r i n g of  c o p p e r - b i s m u t h , sodium c h l o r i d e - w a t e r and and Charvat"*"^',  ice-menthol  by K i n g e r y , Woulbroun  and on p o t a s s i u m n i o b a t e s by J a e g e r and E g e r t o n ^ . 7  The  18 s t u d i e s on s e v e r a l c a r b i d e s by Lersmacher and  Schblz  a l s o showed t h a t  Murray's e q u a t i o n based on the p l a s t i c f l o w model can d e s c r i b e the e a r l y s t a g e s o f h o t - p r e s s i n g , but d e v i a t i o n s occur a t l o n g e r t i m e s , at higher  temperatures. 19 McClelland  BeO  particularly  and A^O^  20 '  observed t h a t the k i n e t i c s of h o t - p r e s s i n g  d i d not f o l l o w the e q u a t i o n proposed by Murray.  theory, McClelland a l s o considered  of  In h i s  t h a t the p l a s t i c f l o w mechanism i s  o p e r a t i v e d u r i n g h o t - p r e s s i n g , but he i n c l u d e d the p r e s s u r e  effective  i n c l o s i n g the pores i n s t e a d of the a p p l i e d p r e s s u r e i n the M a c k e n z i e and  - 3 Shuttleworth's equation.  H i s m o d i f i e d e q u a t i o n agreed w e l l w i t h h i s  e x p e r i m e n t a l d a t a on BeO and A^O^. B.  THEORIES BASED ON THE NABARRO-HERRING CREEP MECHANISM 21  Koval'chenko and Samsonov 22 based on t h e Nabarro  proposed a h o t - p r e s s i n g  equation  23 -Herring  creep mechanism, which was v e r i f i e d by  s t u d i e s on t u n g s t e n c a r b i d e and chromium c a r b i d e .  S c h o l z and L e r s -  24 macher  s i m p l i f i e d Koval'chenko and Samsonov's e q u a t i o n and showed t h a t  i t t o o k ' a form s i m i l a r t o t h e e q u a t i o n developed dC: = - 7 ^ Q dt  by M u r r a y , namely  4 n  where Q = p o r o s i t y . C.  THE PRESENT CONCEPTS OF HOT-PRESSING  D u r i n g t h e p a s t few y e a r s , t h e r e has been l i t t l e agreement among i n v e s t i g a t o r s r e g a r d i n g t h e mechanism o r mechanisms o p e r a t i v e during hot-pressing.  T h e i r r e s u l t s d i f f e r from those o f Murray and they  a l l suggest c e r t a i n complex mechanisms o f h o t - p r e s s i n g .  The k i n e t i c s  o f 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 a t r e l a t i v e l y low t e m p e r a t u r e s 25 was s t u d i e d by F e l t e n  u s i n g alumina powders.  d e s c r i b e d , F e l t e n concluded  Although  not e x p l i c i t e l y  t h a t t h e most p r o b a b l e i n i t i a l  mechanism was p a r t i c l e rearrangement.  densification  T h i s rearrangement may be due t o  p a r t i c l e s l i d i n g , f r a g m e n t a t i o n , p l a s t i c f l o w and d l f f u s i o n a l p r o c e s s e s . D i r e c t m i c r o s t r u c t u r a l e v i d e n c e f o r grain-boundary s l i d i n g 26 and f r a g m e n t a t i o n was o b t a i n e d by Chang and Rhodes  .  They c o n s i d e r e d  t h a t t h e d e n s i f i c a t i o n was c o n t r o l l e d by p a r t i c l e s l i d i n g , p l a s t i c f l o w and volume d i f f u s i o n .  fragmentation,  The r o l e o f f r a g m e n t a t i o n i n the 27 d e n s i f i c a t i o n p r o c e s s has been f u r t h e r r e v e a l e d by Hashimoto using calcium  _ 4 f l u o r i d e spheres o f u n i f o r m s i z e , and by S h a p i r o types of powders.  28—30  using various  U s i n g s i n g l e c r y s t a l spheres of s a p p h i r e , C o b l e and  showed t h a t f o r a l u m i n a 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 i s l i m i t e d the h o t - h a r d n e s s o f the m a t e r i a l .  T h e i r r e s u l t s show t h a t p l a s t i c  can c o n t r i b u t e to the d e n s i f i c a t i o n p r o c e s s pressing only.  Ellis  3  by flow  i n the e a r l y s t a g e s o f h o t -  F u r t h e r d e n s i f i c a t i o n i s a s c r i b e d to enhanced d i f f u s i o n  under the i n f l u e n c e of s t r e s s , e.g.,  diffusional  creep.  32 V a s i l o s and S p r i g g s  a l s o proposed d i f f u s i o n - c o n t r o l l e d creep as  the mechanism f o r d e n s i f i c a t i o n of p r e a l u m i n a and magnesia beyond u  the i n i t i a l  stages.  I n c o n t r a s t to the work o f Murray ' and  M c C l e l l a n d , they observed t h a t h i g h p u r i t y magnesia c o u l d be p r e s s u r e - s i n t e r e d c o n t i n u o u s l y t o e s s e n t i a l l y f u l l d e n s i t y over a range of t e m p e r a t u r e s . 33 R o s s i and F u l r a t h  s t u d i e d the k i n e t i c s o f the f i n a l s t a g e of  d e n s i f i c a t i o n of a l u m i n a under vacuum h o t - p r e s s i n g c o n d i t i o n s . suggested t h a t p l a s t i c f l o w 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  They a l s o 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 s t a g e , where d i f f u s i o n - c o n t r o l l e d c r e e p i s proposed as 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 . and F u l r a t h f u r t h e r concluded the f i n a l  Rossi  t h a t t h e r e i s no p a r t i c l e rearrangement i n  stage. D.  SCOPE OF PRESENT INVESTIGATION  From the p r e c e e d i n g  r e v i e w of the l i t e r a t u r e , i t a p p e a r s , t h a t  t h r e e d e n s i f i c a t i o n mechanisms have been c o n s i d e r e d hot-pressing  to be o p e r a t i v e i n the  process:  1.  P a r t i c l e rearrangement a i d e d by s l i d i n g and  2.  P l a s t i c flow  3^  Stress-enhanced d i f f u s i o n a l  and creep.  fragmentation  - 5I t appears t h a t because o f t h e c o m p l e x i t i e s o f t h e v a r i a b l e s i n v o l v e d d u r i n g h o t - p r e s s i n g , no q u a n t i t a t i v e study has been made, so f a r , to account f o r each i n d i v i d u a l s t a g e .  To t r e a t t h e d i f f e r e n t  stages  m a t h e m a t i c a l l y , i t i s n e c e s s a r y t o know o r t o c o n t r o l t h e geometry o f 34 the p a r t i c l e s and t h e p a c k i n g c o n f i g u r a t i o n s .  Pelzel  attempted t o  c o r r e l a t e t h e p e r c e n t a g e b u l k d e n s i t y o f a s p h e r i c a l compact w i t h t h e t o t a l s u r f a c e - c o n t a c t a r e a produced  by c o n t i n u o u s d e f o r m a t i o n .  He  c a l c u l a t e d h i s models on t h e b a s i s o f s i x r - f o l d and t w e l v e - f o l d c o o r d i n a t i o n s . An attempt has been made i i the p r e s e n t work t o s e p a r a t e t h e v a r i o u s s t a g e s and t o s t u d y the c o n t r i b u t i o n from t h e p l a s t i c f l o w mechanism and t h e p a r t i c l e r e o r i e n t a t i o n t o t h e d e n s i f i c a t i o n p r o c e s s during the e a r l y stages of h o t - p r e s s i n g .  T h e o r e t i c a l models have been  f o r m u l a t e d on t h e b a s i s o f t h e p l a s t i c d e f o r m a t i o n o f spheres i n a compact. The s t a r t i n g p o i n t f o r t h e models i s v a r i o u s modes o f s i m p l e and s y s t e m a t i c p a c k i n g o f u n i f o r m spheres o f e q u a l s i z e ,  .i The v a l i d i t y o f t h e s e models has been t e s t e d by deforming u n i f o r m spheres o f l e a d , K-Monel and s i n g l e c r y s t a l s a p p h i r e .  - 6II T H E O R E T T C A L M O D E L S PART ONE-THEORY A q u a n t i t a t i v e study o f t h e h o t - p r e s s i n g p r o c e s s  i sgreatly  s i m p l i f i e d by assuming t h a t t h e compact i s composed o f u n i f o r m  spheres  of e q u a l s i z e and t h a t t h e spheres a r e packed i n some r e g u l a r arrangement. The p a c k i n g o f spheres o f u n i f o r m . s i z e has been s t u d i e d by many w o r k e r s 35 in different fields.  The f i n d i n g s o f G r a t o n and F r a s e r  s i g n i f i c a n t i n t h i s respect.  a r e most  The b a s i c systems o f p a c k i n g a r e  d e s c r i b e d i n Appendix J and c a n be summarised as f o l l o w s : 1.  Simple c u b i c (Z=6)  2.  Body-centered c u b i c (Z=8)  3.  Orthorhombic (Z=8)  4.  Tetragonal  5.  Rhombohedral (Z=12)  (Z=10)  where Z i s t h e c o o r d i n a t i o n number. Of t h e s e , t h e b . c . c . p a c k i n g i s an u n s t a b l e arrangement i n a u n i d i r e c t i o n a l f i e l d o f f o r c e ( i . e . , g r a v i t a t i o n a l f o r c e ) , and i s considered here f o r t h e o r e t i c a l i n t e r e s t s only.  Furthermore, the  f o l l o w i n g assumptions have been made t o develop t h e models quantitatively: 1.  Each o f t h e above p a c k i n g s  i s s t a b l e and m a i n t a i n s i t s  symmetry upon t h e a p p l i c a t i o n o f p r e s s u r e 2.  The p r e s s u r e i s a p p l i e d e q u a l l y from a l l s i d e s  3.  The m a t e r i a l i s i s o t r o p i c and  4.  D e f o r m a t i o n t a k e s p l a c e by t h e p l a s t i c f l o w mechanism.  - 7 Under t h e s e c o n d i t i o n s , on a p p l i c a t i o n of the p r e s s u r e , p l a s t i c d e f o r m a t i o n w i l l t a k e p l a c e a t the p o i n t s o f c o n t a c t and c i r c u l a r f a c e s w i l l form on each p a r t i c l e .  flat  As the r a d i u s 'a' o f the  f a c e i n c r e a s e s , the r a d i u s 'R' o f the sphere must i n c r e a s e t o m a i n t a i n a c o n s t a n t volume.  T h i s i s t r u e o n l y i f t h e m a t e r i a l a t the f a c e s  s y m m e t r i c a l l y t o m a i n t a i n the s p h e r i c i t y o f the p a r t i c l e .  spreads  Under such  !  c o n d i t i o n s , the spheres w i l l be deformed in|to u n i f o r m a l l y shaped i p o l y h e d r a depending  upon the c o o r d i n a t i o n number.  p a c k i n g w i l l g i v e r i s e t o cubes, o r t h o r h o m b i c  Thus, s i m p l e c u b i c  packing w i l l give  hexagonal  p r i s m s , and rhombohedral p a c k i n g w i l l produce rhombic dodecahedrons on complete d e f o r m a t i o n .  I n the b . c . c . c a s e , the spheres b e g i n to deform  i n the shape of an o c t a h e d r o n , but a c r i t i c a l s t a g e i s reached when s i x spheres a t the f a c e - c e n t e r e d p o s i t i o n s b e g i n to t o u c h .  On  further  d e f o r m a t i o n , t h e p o l y h e d r o n formed i s known as a t e t r a k a i d e c a h e d r o n 36 (Z"14).  Coble  used a model (based on the t e t r a k a i d e c a h e d r o n ) i n  d e r i v i n g the s i n t e r i n g e q u a t i o n s f o r the i n t e r m e d i a t e and f i n a l s t a g e s of densification.  The  t e t r a g o n a l p a c k i n g does not seem to conform,to  r e g u l a r p o l y h e d r o n on d e f o r m a t i o n and so cannot r e s u l t i n complete d e n s i f i c a t i o n ; t h e r e f o r e i t has not been c o n s i d e r e d h e r e . A. GEOMETRIC RELATIONSHIPS  F i g u r e 1 shows the f o r m a t i o n of a f l a t face-between spheres i n c o n t a c t . c o n s i d e r e d as  The  follows.  geometry of the d e f o r m a t i o n p r o c e s s can  two be  a  - 8 The volume o f t h e p a r t i c l e a t any d e f o r m a t i o n i s g i v e n by . V = V-Z x V P s where  V = volume o f t h e sphere Z = t h e c o o r d i n a t i o n number V  = volume o f a segment o f t h e sphere.  g  I f R i s t h e r a d i u s o f t h e s p h e r e , and h i s t h e h e i g h t o f t h e segment of t h e s p h e r e , then we have V = k irR 3 and  V = irh  (3R-h)  2  ~T~  s  3  Therefore, V  = 4  3  P  TTR  - Z [ r ^ - (3R-h)]  3  3  I t c a n be seen from F i g . 1, t h a t h = R-y and  R = y +a 2  ,2  1.  (1) (2)  2  C u b i c Model  There a r e s i x p o i n t s o f c o n t a c t i n t h i s case g i v i n g r i s e to s i x f l a t f a c e s . The volume o f t h e p a r t i c l e a t any d e f o r m a t i o n w i l l be g i v e n by  2  v  =  P  4  TTR  3  -  6 [~  (3R-h)]  3  U s i n g Eq. (1) and ( 2 ) , and on s i m p l i f i c a t i o n i t c a n be  - 9 -  SPHERES IN CONTACT  Fig. 1  Geometry of Deformation of Two  Spheres  i n Contact.  - 10 shown t h a t 2 V  (R - i a )  - 2TT(2R + a ) 2  1 / 2  2  -  2  8TTR  (3)  3  3  P  3 T h i s a l s o e q u a l s the o r i g i n a l volume of the sphere ^ ^ , where P. i s 3 ° the o r i g i n a l r a d i u s of the sphere. T h e r e f o r e , from Eq. (3) 1 12 2 2 2 2 3 3 0  2TT(2R  + a )  (IT -  a )  -  8TT R  3  = 4ITP  J  0  3 = const. = 1  (e.g.)  (4)  The r a d i u s o f the f a c e i n c r e a s e s as the d e f o r m a t i o n  proceeds  and a c r i t i c a l r a d i u s i s reached when the s i x c i r c u l a r f a c e s b e g i n t o touch ( F i g . 2a).  For t h i s c o n d i t i o n , from F i g . 2b a  Crit  % r i t  =  =  1  for r  /2 S u b s t i t u t i n g t h i s i n Eq. R  C  r  l  ?  °"  =  2.  = R  2  t  (arbitrary units)  7 1 5 5 3  Hexagonal P r i s m a t i c Model  There a r e e i g h t p o i n t s o f c o n t a c t . = 4TTR 3  p  2  (4) g i v e s  Crit  V  + a  -j  3  I n t h i s case we have  (3R-h)]  1/2 =  -§^  (2R +a ) ( R - a ) 2  2  2  2  - 4TTR  3  3 =  = 1 (e.g.)  (5)  The c r i t i c a l s t a g e i s reached i n t h i s case when the s i x f a c e s i n the c l o s e - p a c k e d l a y e r b e g i n to t o u c h ( F i g . 3 a ) .  For t h i s  condition,  - 11 -  CUBIC MODEL  Fig. 2  Geometric R e l a t i o n s h i p of C u b i c Model Showing the C r i t i c a l Stage.  - 12 -  HEXAGONAL PRISMATIC MODEL  Fig. 3  Geometric R e l a t i o n s h i p o f Hexagonal P r i s m a t i c Model Showing the C r i t i c a l S t a g e .  - 13 -  from F i g . 3b 1 = — y /3 = 0.5 R c a  C  S u b s t i t u t i n g t h i s i n Eq.  C  (5) g i v e s R  c  =0.64321 ( a r b i t r a r y  3.  Units)  Rhombic Dodecahedron Model  There a r e t w e l v e p o i n t s of c o n t a c t i n t h i s c a s e . we have  = 4*r3  vp  12  J  .  =  = A ^R  3  2  =  1  1/2  - a )  N  0  (3R-h)]  J  ^ 2 .JI + a ) (R  2 4TT (2R /OT1  Therefore,  S  -  20irR -—  3  < -8-> e  ( 6 )  3 The c r i t i c a l s t a g e i s reached f a c e s b e g i n to touch ( F i g . 4 a ) .  i n t h i s case when the t w e l v e  A l s o i n t h i s model s i x f a c e s i n the  c l o s e - p a c k e d l a y e r b e g i n to touch ( F i g . 4 b ) . same as t h a t o b t a i n e d i n the hexagonal a = 0.5R c c  The second c o n d i t i o n i s the  p r i s m a t i c model, g i v i n g  A l t e r n a t e l y , from F i g . 4c a  _ p_! Cos 9 c " 2  (7)  From geometry, i t can be shown t h a t D and  = 2y  1  6 = 54°  S u b s t i t u t i n g t h e s e v a l u e s i n Eq. a  c  44'  (7), gives =..y cos 54° .c =  0.5R  c  44'  - 14 -  RHOMBIC DODECAHEDRON MODEL Fig. 4  Geometric R e l a t i o n s h i p o f Rhombic Dodecahedron Model Showing the C r i t i c a l Stage.  - 15 w h i c h on s u b s t i t u t i o n i n Eq.  Initially  (6) g i v e s  R  = 0.65560 ( a r b i t r a r y  4.  Tetrakaidecahedron  units).  Model  t h i s model has the c o n f i g u r a t i o n o f the b . c . c .  packing.  s t a g e s i n t h i s c a s e , s u b s c r i p t s 1 and 2 have  S i n c e t h e r e a r e two c r i t i c a l  been used t o d e s i g n a t e the two s t a g e s .  a)  B.C.C. Model  There a r e e i g h t p o i n t s o f c o n t a c t i n t h i s  case.  T h e r e f o r e , we have V  p  = 3 *  R  ~  <  ^  8  3  * - V ]  1/2 = j  (2R  1  2  + a )  (R -  2  2  ±  & 1  2  )  -  4TTR  3  3 = f The c r i t i c a l spheres  R  o  =  1 (e.g.)  s t a g e i s reached  (8)  i n t h i s case when the s i x  i n the f a c e - c e n t e r e d p o s i t i o n s b e g i n t o touch  i s c o n s i d e r e d as the f i r s t c r i t i c a l  stage.  ( F i g . 5b).  This  From F i g . 5b, i t can be  seen t h a t y' = 2 y s i n 9 2  (9)  1  and i t can be shown, from geometry, t h a t 9 = 35° 1 6 ' 30" Now,  at this f i r s t c r i t i c a l y  T h e r e f o r e , Eq.  2  stage,  =R  (9) becomes  1/2 R = 2 (R - a ) 2  2  x  Sin 0  - 16 w h i c h on s i m p l i f i c a t i o n  gives (a.)  = 0.5 (R ) 1  c l  1  C  C  Substituting  t h i s i n Eq.(9) g i v e s (R ) = 0.64321 ( a r b i t r a r y 1 c  b)  units)  T e t r a k a i d e c a h e d r o n Model  After  t h e f i r s t c r i t i c a l s t a g e , t h e d e f o r m a t i o n proceeds unt  the second c r i t i c a l s t a g e i s reached when a l l f o u r t e e n f a c e s come i n t o c o n t a c t ( F i g . 6 a ) . F o r t h i s c o n d i t i o n , from F i g . 6b (a )  = 1  x  c  where  2  L  2/3  L = edge l e n g t h o f the o c t a h e d r o n shown i n F i g . 5a.  I t can be shown t h a t L = /e This gives  y  ±  (a-,) = - ( ) . 2 /2 y  c  =  " /3  (R ) 2  (10)  c  A l s o , from F i g . 6c <'2>  = i  c  •  i  L  'h  /6 S i n c e from Eq. (9) y 2 = 2'y^ s i n 9  - 17 -  B.C.C. MODEL  Fig. 5  Geometric R e l a t i o n s h i p o f B.C.C. Model Showing t h e C r i t i c a l Stage.  - 18 -  Fig. 6  Geometric R e l a t i o n s h i p o f T e t r a k a i d e c a h e d r o n Model Showing t h e C r i t i c a l Stage.  - 19 therefore,  we have (a )  =  i 2/6 s i n 0  c =  ±  Alternately,  Substituting  c  2  ^  (R )  (11)  from Eq. (9) we have 1/2 2 2 (R - ap (a-.) c  V  <* >  2/2 =  (  1 / 0 1  = 2(R  2  2  - ap  sin Q  = — (R ) from Eq. (10) and s i m p l i f y i n g we /3 2  have,  C  2  (a ) ^ c  =  ± (R )  2  The volume of the p a r t i c l e i n t h i s case i s g i v e n by  V  p  =  f  -  r 3  (3R -hi)]  ~ 6[f^(3R - h ) ] 2  1/2 =  y  (2R  77  2  + a )  (R - a )  2  2  x (R  2  2  Substituting  f  R  o  1 / 2  - a ) "  - 8  0  3 =  + 2TT ( 2 R  2  TT  2  +  : a ) 2  £  3 R  1  = 1 (e.g.)  (12)  the v a l u e s of the c r i t i c a l r a d i i from Eqs. (10) and  (11)  i n t o Eq. ( 1 2 ) , we get (R ) c  = 0.66940 ( a r b i t r a r y  units)  2  I t can be seen t h a t a l l the f o r e g o i n g c a l c u l a t i o n s can be summarized  i n t o a s i n g l e e q u a t i o n , namely V  P  = a(2R  A  17  R  2 3  Q  2  + a ) (R  2  2  - a )  = 1 (e.g.)  1/2 -  BR  3  (13)  - 20 where cx,g a r e c o n s t a n t s w i t h d i f f e r e n t v a l u e s f o r each model.  These  values a r e given i n Table I . For each o f t h e models, t h e change i n R v a l u e s at v a r i o u s s t a g e s o f d e f o r m a t i o n can be c a l c u l a t e d from t h e constant-volume equation.  These v a l u e s (ALL THEORETICAL CALCULATIONS HAVE BEEN CARRIED  OUT USING AN IBM COMPUTER) a r e summarized i n t h e form o f a graph a/R and R as shown i n F i g . 7. R i s approximately equal t o R  relating  I t i s seen t h a t up t o a v a l u e o f a/R = 0.25, Q  ( t h e i n i t i a l r a d i u s ) and f o r t h i s  reason  a l l e x p e r i m e n t a l work was l i m i t e d t o a/R = 0.25. The geometry o f t h e d e f o r m a t i o n f o r a l l t h e models i s no l o n g e r t h e same a f t e r t h e c r i t i c a l s t a g e has been r e a c h e d .  Beyond t h e  c r i t i c a l s t a g e , t h e spheres a r e f u r t h e r deformed t o t h e r e s p e c t i v e p o l y h e d r a g i v i n g 100% d e n s i f i c a t i o n .  B.  DEDUCED RELATIONSHIPS  From t h e above d e r i v a t i o n s t h e f o l l o w i n g r e l a t i o n s h i p s c a n be deduced. 1.  Bulk Density  B u l k d e n s i t y i s d e f i n e d as t h e r a t i o o f t h e volume o f t h e p a r t i c l e t o t h a t o f t h e volume o f t h e u n i t c e l l , t h e volume o f t h e u n i t c e l l b e i n g t h e sum o f t h e volume o f t h e p a r t i c l e and i t s a s s o c i a t e d v o i d space volume.  The volume o f t h e u n i t c e l l f o r t h e v a r i o u s models  can be c a l c u l a t e d from t h e r e s p e c t i v e schematic diagrams ( F i g s . 2 t o 6) i n terms o f t h e p a r t i c l e r a d i u s R and t h e f a c e r a d i u s a a t any s t a g e o f deformation.  The b u l k d e n s i t i e s f o r t h e v a r i o u s models can be c a l c u l a t e d  as f o l l o w s . a)  - 22 -  C u b i c Model I n t h i s case t h e u n i t c e l l i s a cube.  The volume o f t h e  u n i t c e l l a t any s t a g e o f d e f o r m a t i o n can be c a l c u l a t e d from F i g . 2a.  The volume o f t h e u n i t c e l l i s g i v e n by V where  u  L  = L  3  = edge  length  = 2y 1/2 2 2 = 2(P/ - a ) Z  t h e r e f o r e , we have V  = 8(R - a ) 2  u  2  Now, the b u l k d e n s i t y  = volume o f t h e p a r t i c l e volume o f t h e u n i t c e l l  or  D=  1 ,_. o 8(R^-a?)  / 9  3 / 2  [or i n p e r c e n t ,  D-  1 8(R  2  K  100%]  (14)  - a ) 2  We know t h a t , when Eq.  a = o, then R = R . S u b s t i t u t i n g t h i s c o n d i t i o n i n Q  (4) g i v e s A * o 3 R  or therefore,  R  Q  3  „ i  = 0.62035 ( a r b i t r a r y  units)  f o r a = o, we have D = D  =  — ^ — 8R 3  0  = 52.36%  x 100  - 23 This v a l u e of the percentage  b u l k d e n s i t y agrees w i t h t h e v a l u e of  the d e n s i t y o f p a c k i n g o f . u n i f o r m spheres When a = a , then R = R £  £  i n a c u b i c a r r a y (Table I ) .  = /2 a  = 0.71553 ( a r b i t r a r y  units).  t h e r e f o r e , we have D = D  x 100  = 8 ( R c  C  2  - a  2  c  )  /  3  2  x 100 /8R  =  J  c 96.51%  The v a l u e s o f t h e b u l k d e n s i t y f o r o t h e r a/R v a l u e s have been s i m i l a r l y c a l c u l a t e d and a r e g i v e n i n Appendix I I and p l o t t e d i n F i g . 8. b)  Hexagonal P r i s m a t i c Model In  t h i s case t h e u n i t c e l l i s a h e x a g o n a l p r i s m .  The volume  o f the u n i t c e l l a t any s t a g e o f d e f o r m a t i o n can be c a l c u l a t e d from F i g . 3a. The volume o f the u n i t c e l l i s g i v e n by 3/3 8  V  D  2  H  D'^ = diameter of the base o f the hexagonal prism  where, from geometry  = 4y_  H  and  = 2y  t h e r e f o r e , we have u  =  4  =  4 /3 ( R -  /J  y 2  3/2 a ) 2  this gives, D  x 100%  = 4^3 ( R - a ) 2  2  3 / 2  (15)  - 24 We know t h a t , when  a = 0, then R = R  = 0.62035 ( a r b i t r a r y  q  units)  S u b s t i t u t i n g t h i s i n Eq. (15) g i v e s D  = 60.46%  o  and when a = a therefore,  c >  then R = R  £  = 2 a = 0.64321 ( a r b i t r a r y £  units)  from Eq. (15) D  = 83.51%  c  The v a l u e s o f t h e b u l k d e n s i t y  f o r o t h e r a/R v a l u e s have been s i m i l a r l y  c a l c u l a t e d and a r e g i v e n i n Appendix I I and p l o t t e d i n Fig.. 8. c)  Rhombic Dodecahedron Model The u n i t c e l l f o r t h i s case i s a rhombic dodecahedron.  The  volume o f t h e u n i t c e l l a t any s t a g e o f d e f o r m a t i o n can be c a l c u l a t e d from F i g . 4a. The volume o f u n i t c e l l i s g i v e n by V  = /2  U  where  = l a r g e r diagonal o f t h e rhomb  ,  1/2 2 2 = 2(R^ - a )  =  2 y  Z  t h e r e f o r e , we have  3/2 V  u  =  4/2 ( R - a ) 2  2  This gives D  =  —3L-3/2 4/2 ( R - a ) Z  Z  x l Q  °  (16)  - 25 We know t h a t , when  a = 0, then R = R  q  = 0.62035 ( a r b i t r a r y  units)  S u b s t i t u t i n g t h i s i n Eq. (16) g i v e s D  o  - 74.05%  and when therefore,  a = a , then R = R c  £  = 2 a = 0.65560 ( a r b i t r a r y £  units)  from Eq. ( 1 6 ) , D  c  =  96.40%  The v a l u e s o f t h e b u l k d e n s i t y  f o r o t h e r a/R v a l u e s have been s i m i l a r l y  c a l c u l a t e d and a r e g i v e n i n Appendix I I and p l o t t e d i n F i g . 8. d)  Tetrakaidecahedron  Model  The u n i t c e l l f o r t h i s case i s a t e t r a k a i d e c a h e d r o n .  The  volume o f t h e u n i t c e l l a t any s t a g e o f d e f o r m a t i o n can be c a l c u l a t e d from F i g . 6. The volume o f t h e u n i t c e l l i s g i v e n by v  =  & 27  u where  l  3  L = edge o f t h e o c t a h e d r o n '1 1/2 /6(R  2  -  a-p  t h e r e f o r e , we have V  This gives  u  =  32£_ 9  y{  -27 We know t h a t , When  a = 0, t h e n R = R  = 0.62035 ( a r b i t r a r y u n i t s )  Q  S u b s t i t u t i n g t h i s i n Eq. (17) g i v e s D =68.02% o and when  a = (a.) , then R = (R ) = 2 (a..) ci . c 1 l 1  = 0.64321 ( a r b i t r a r y u n i t s )  c  t h e r e f o r e , from Eq. (17) D c  = 93.95%  l  a l s o when a = (a,) , t h e n R = (R ) = /3(a.) = 3(a») 1 c 2 2 ' c  l  2  c  = 0.66490 (arbitrary units)  T h i s on s u b s t i t u t i o n i n Eq. (17) g i v e s D  = 99.45% C  2  The v a l u e s o f t h e b u l k d e n s i t y f o r o t h e r a/R v a l u e s have been s i m i l a r l y c a l c u l a t e d and a r e g i v e n i n Appendix I I and p l o t t e d i n F i g . 8. e)  R e l a t i o n s h i p Between D and D  p  I t c a n be seen from t h e f o r e g o i n g c a l c u l a t i o n s t h a t n  1 (18) 2 2^/2 A(R - a ) where A i s a c o n s t a n t h a v i n g d i f f e r e n t v a l u e s f o r d i f f e r e n t models. D  =  Z  For a = 0, R = R , Q  1  and D = D  =  -r AR o  0  (19)  3  Combining Eqs. (18).and ( 1 9 ) , D -o  2 2, [ -^ -] R o  3/2  D  =  R  (20)  a  f o r a/R = 0.25, R i s e s s e n t i a l l y a c o n s t a n t = R  Q  - 28 Therefore, D ' ... _° = [ " a D  2.3/2  3  1  or  R 2  2 (1.-- j R  D = D  _3/2  w h i c h , on e x p a n s i o n by the b i n o m i a l theorem ( n e g l e c t i n g squares and h i g h e r power), g i v e s 3 a (1 + f % ) o 2 2 2  D = D  (21)  R  " °o  D  3 2 o  =  D  a  2  (22)  F  A l s o , i t comes o u t e m p i r i c a l l y t h a t (D-D ) i s v e r y n e a r l y e q u a l t o 101.5 log  Z (a/R)  x  f o r c u b i c , h e x a g o n a l p r i s m a t i c , and rhombo4*ed*a-l models. Bulk S t r a i n  Bulk s t r a i n  can be d e f i n e d as E  where  y — E  R  (1  =  B  E )  +  1  -  3  (23)  Q  =  R o therefore,  e, b  - R  3  =  y ^R o - a  2  r  a 2 R  2  , ~ 1  3 / 2  i  - i  (2A)  o  Alternately, B u l k s t r a i n . c a n be d e f i n e d as e q u a l t o the volume o f t h e u n i t c e l l a t any d e f o r m a t i o n minus the volume of the u n i t c e l l a t no d e f o r m a t i o n d i v i d e d by t h e volume o f the u n i t c e l l a t no d e f o r m a t i o n .  From the  f o r e g o i n g c a l c u l a t i o n s , we know t h a t t h e volume of the u n i t c e l l a t any  Z / 3  2 d e f o r m a t i o n i s g i v e n by A(R  2  — a") and t h e volume o f t h e u n i t c e l l a t 3 no d e f o r m a t i o n i s g i v e n by AR , where A i s a c o n s t a n t w i t h d i f f e r e n t v a l u e s Q  f o r d i f f e r e n t models. T h e r e f o r e , we have  _ ,„ 2 • 2 A(R - a ) — AR Z  "  AR  3  3 o  2 3/2  2 Ro  w h i c h i s t h e same as.Eq. ( 2 4 ) . The b u l k s t r a i n v a l u e s f o r d i f f e r e n t a/R v a l u e s have been c a l c u l a t e d  u s i n g Eq. (24) and a r e g i v e n i n  Appendix I I and p l o t t e d i n F i g . 9. 3.  B u l k D e n s i t y and B u l k  Strain  Combining Eqs. (20) and (24) we g e t ,  £° =  1  +  e  D  b  or  D  D  D = 1 + e  (25)  fe  For s m a l l v a l u e s o f b u l k s t r a i n , Eq. (25) can be expanded t o g i v e D = D O  (1 - e, )  (26)  D  T h i s g i v e s a s t r a i g h t - l i n e r e l a t i o n s h i p between t h e b u l k d e n s i t y , and t h e b u l k s t r a i n , f o r s m a l l v a l u e s o f b u l k s t r a i n . 4. D. a/R. and Z I n o r d e r t o c o r r e l a t e a l l these parameters,  a three  d i m e n s i o n a l graph has been c o n s t r u c t e d between D, a/R, and Z. ( F i g . 1 0 ) . I t i s i n t e r e s t i n g t o n o t e t h a t the b.c.c. model does not seem t o f a l l on t h e same s u r f a c e as t h e one drawn through t h e o t h e r models.  - 30 -  a. R Fig. 9  T h e o r e t i c a l R e l a t i o n s h i p of Models.  v s a/R f o r the Proposed  - 31 -  F i g . 10  T h e o r e t i c a l R e l a t i o n s h i p of D v s a/R v s Z f o r the Proposed Models.  TABLE I  No.  Type o f p a c k i n g  D e f o r m a t i o n Models and T h e i r  D e f o r m a t i o n model  Characteristics  a*  6*  Volume o f the u n i t c e l l  - •  c  2 2 3/2 8(R - a )  52 .36  96.51  8TT/3  4TT  4/3(R -V)  2Z  60 .46  83.51  Rhombic Dodecahedron  4TT  20rr/3 4/2(R ra )  74 .05  96.41  Tetrakaidecahedron  8TT/3  4TT  , 3/2 68 .02 (R^P  (1) 93.95 (2) 99.45  Orthorhombic  Hexagonal  3.  Rhombohedral  4.  B. C. C.  Prismatic  2  3 / 2  3/2  *  R = R  8TT/3  Cubic  2.  R = R o  2 TT  Simple Cubic  1.  £ Bulk Density at  2  (32/3/9)  2  R e f . t o Eq. ( 1 3 ) .  N3  I  PART TWO  - 33 - EXPERIMENTAL VERIFICATION OF MODELS  A. In  PARTICLE REARRANGEMENT  a c t u a l p r a c t i c e , t h e i d e a l modes of p a c k i n g assumed e a r l i e r  i n the t h e o r e t i c a l models and d i s c u s s e d i n Appendix  I a r e not o b t a i n e d .  The c u b i c , o r t h o r h o m b i c , t e t r a g o n a l , o r rhombohedral p a c k i n g s , e.g.,  do  not e x i s t a c r o s s t h e d i a m e t e r of t h e d i e , because 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 of s p h e r e s . h o l d up a c e r t a i n number of s p h e r e s .  A l s o , t h e w a l l appears  to  On the a p p l i c a t i o n o f a s m a l l p r e s s u r e ,  spheres which have been l i g h t l y h e l d up by f r i c t i o n o r by b r i d g e s w i l l c o l l a p s e to lower p o s i t i o n s ; some o t h e r s w i l l f o l l o w s h o r t l y a f t e r w a r d s to fill  the v o i d space c r e a t e d by the movement.  Thus, some r e s t a c k i n g o f the  p a r t i c l e s always t a k e s p l a c e on the i n i t i a l p r e s s u r e a p p l i c a t i o n . The p a r t i c l e r e s t a c k i n g was  s t u d i e d i n a few p r e l i m i n a r y ex-  p e r i m e n t s u s i n g g l a s s spheres of 380 to 400u d i a m e t e r t o t a k e t h i s i n t o w h i l e t e s t i n g the models e x p e r i m e n t a l l y .  The g l a s s spheres were r o l l e d  down an i n c l i n e d p a n e l to s e p a r a t e the u n i f o r m and r e g u l a r spheres the i r r e g u l a r ones.  account  F i g u r e 11 shows such an arrangement.  The  from  roundest  spheres r o l l e d down the p a n e l through the s e l e c t o r and were s u b s e q u e n t l y used f o r the e x p e r i m e n t s . diam^ and 0.25  A h e a t - t r e a t e d c y l i n d r i c a l d i e (1.25 i n .  i n . h e i g h t ) made from s t e e l (Keewatin) was used i n  c o n j u n c t i o n w i t h a s p e c i a l l y d e s i g n e d s t r a i n gauge h a v i n g an a c c u r a c y of t 0.001  in.  The whole assembly was  loaded i n a floor-model t e s t i n g  machine ( I n s t r o n ) * f o r d e f o r m a t i o n p u r p o s e s .  The g l a s s spheres were  p r e s s e d t o a t o t a l l o a d of 1000 l b u s i n g a s t r a i n r a t e of 0.04 i n . / i n . / m i n . No f r a g m e n t a t i o n of t h e g l a s s spheres was o b s e r v e d .  The s t r a i n r e m a i n i n g  - 34 -  F i g . 11  Apparatus f o r S e p a r a t i n g U n i f o r m Spheres from the I r r e g u l a r Ones.  - 36 on the removal of the l o a d was r e s t a c k i n g or r e p a c k i n g .  noted and was  considered  t o be due  I n a d d i t i o n , s e p a r a t e e x p e r i m e n t s were c a r r i e d  out u s i n g a r b i t r a r y " i n i t i a l l o a d s " of 2, 5, 10, 20, and s u b s e q u e n t l y p r e s s i n g t o a f i n a l l o a d of 1000  lb.  40 l b s and  On d e c r e a s i n g  t o the i n i t i a l v a l u e s , the amount of permanent s t r a i n was " i n i t i a l load".  The  the  show t h a t a volume change of about 1.25%  the a p p l i c a t i o n of 10 p s i .  spheres by D u f f i e l d and G r o o t e n h u i s an a p p l i e d p r e s s u r e  may  be due  was  on e x t r a p o l a t i o n , The  results  can be produced  by  L u b r i c a t i n g the spheres w i t h g r a p h i t e powder  had no e f f e c t on the s t r a i n produced. 37  for  load  n o t e d f o r each  produced the v a l u e of s t r a i n a t zero i n i t i a l a p p l i e d l o a d . 12)  then  s t r a i n , r e m a i n i n g f o r each i n i t i a l a p p l i e d l o a d  t h e n p l o t t e d a g a i n s t the i n i t i a l a p p l i e d l o a d and,  (Fig.  to  of 10 p s i .  t o the mode of p a c k i n g ,  E x p e r i m e n t s c a r r i e d out u s i n g  copper  produced a volume change of 1.3The  discrepancy  4.3%  between t h e s e two r e s u l t s  the s i z e of the s p h e r e s , and  the manner  i n which s t r e s s i s a p p l i e d . B.  PLASTIC DEFORMATION OF SPHERES 1.  The  M a t e r i a l s Used  t h e o r e t i c a l models were t e s t e d u s i n g l e a d ( 2 % Sb)  because of the c o n v e n i e n c e of w o r k i n g w i t h a m e t a l h a v i n g a low point.  The  n a t u r e of the s t u d y r e q u i r e d t h a t s u f f i c i e n t l y low  s h o u l d be used so t h a t t h e r e i s no i n t e r a c t i o n a t the c o n t a c t T h i s i s p a r t i c u l a r l y i m p o r t a n t f o r a c c u r a t e measurements of contact  face radius.  I n a d d i t i o n , the range of pressure;  l a r g e enough to produce d e f o r m a t i o n w i t h i n the i n t e n d e d i.e.  up to a/R  -  0.25.  spheres melting  temperatures points.  'a',  should  the be  range of s t u d y  - 37 The study o f t h e c o n t r i b u t i o n o f p l a s t i c f l o w t o t h e mechanisms o f 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 o f c e r a m i c o x i d e s was c a r r i e d out u s i n g s i n g l e c r y s t a l s a p p h i r e spheres o f 1 mm d i a m e t e r . high cost of conducting to t h r e e o n l y .  The  each experiment l i m i t e d t h e number o f e x p e r i m e n t s  Spheres o f K-Monel were a l s o used t o t e s t models f u r t h e r .  E x t e n s i o n o f t h e models showed t h a t i t i s p o s s i b l e t o i n c o r p o r a t e i n the b a s i c d e n s i t y e q u a t i o n m a t e r i a l p r o p e r t i e s , such as t h e y i e l d The r e s u l t s i n t h i s s e c t i o n w i l l be p r e s e n t e d  strength.  and d i s c u s s e d  w i t h r e s p e c t t o l e a d and s a p p h i r e , though some r e s u l t s f o r K-Monel w i l l a l s o be g i v e n . 2.  a)  Experimental  Techniques  Lead Lead (2 wt.% Sb) s h o t s 0.120 - 0.125 i n . d i a m e t e r were used f o r  the i n v e s t i g a t i o n . The most r e g u l a r s p h e r e s ,  separated  by r o l l i n g  them  down t h e g l a s s p a n e l u s i n g t h e assembly shown i n F i g . 11^ were used f o r the experiments.  A h e a t - t r e a t e d c y l i n d r i c a l d i e (2 i n . diam  and 0.564 i n .  h e i g h t ) made from s t e e l (Keewatin) was used i n c o n j u n c t i o n w i t h t h e s t r a i n gauge d e s c r i b e d e a r l i e r . by p a c k i n g tapping.  The maximum i n i t i a l d e n s i t y was o b t a i n e d  t h e d i e i n s m a l l i n s t a l l m e n t s w i t h i n t e r m i t t e n t s h a k i n g and An i n i t i a l l o a d o f 10 l b gave a good r e f e r e n c e p o i n t . A  r a t e o f 0.35 i n . / i n . / m i n . was used f o r t h e compaction. pressed  strain  The spheres were  t o t h e v a r i o u s f i n a l l o a d s and t h e a c t u a l permanent s t r a i n was  o b t a i n e d from t h e d i f f e r e n c e i n t h e i n i t i a l and f i n a l r e a d i n g s on t h e c h a r t a t t h e same " i n i t i a l l o a d s " .  A t y p i c a l load-compaction curve obtained  the experiment i s shown i n F i g . 13.  during  - 38 -  L O A D F i g . 13  Load-Compaction Curve i n Case o f Lead Spheres.  - 39 -  F i g . 14  Assembly f o r H o t - P r e s s i n g Lead Spheres.  PRESSURE ASSEMBLY  F i g . 15  Schematic R e p r e s e n t a t i o n o f t h e D i e used f o r H o t - P r e s s i n p . o f Lead Spheres.  - 41 -. Experiments a t 50° and 100°C were c a r r i e d o u t by f i x i n g h e a t i n g c o i l s i n t h e upper and lower p l u n g e r s .  The temperature was  measured a t t h e bottom o f t h e d i e u s i n g a c h r o m e l - a l u m e l and a p o t e n t i o m e t e r .  thermocouple  A drop i n t h e temperature was observed d u r i n g  d e f o r m a t i o n and was c o u n t e r b a l a n c e d by i n c r e a s i n g t h e c u r r e n t . p o s s i b l e t o c o n t r o l t h e temperature t o w i t h i n - 3°C. the  b)  I t was  The i n i t i a l l o a d i n  h i g h temperature t e s t s was reduced t o 2 l b t o a v o i d any i n i t i a l  deformation. Fig.  resistance  F i g u r e 14 shows t h e assembly used i n t h e experiment w h i l e  15 shows t h e s c h e m a t i c r e p r e s e n t a t i o n o f t h e d i e . S i n g l e C r y s t a l Sapphire S a p p h i r e spheres o f 1 mm diam. ( s u p p l i e d by A. M i l l e r and  Co. L i b e r t y v i l l e , I l l i n o i s ) were deformed i n a g r a p h i t e d i e (0.75 i n . diam) a t 1570 , 1600  and 1700°C a t 2000 p s i i n a s p e c i a l l y d e s i g n e d  vacuum h o t - p r e s s ( t h e vacuum was 0.5 atm.). reached w i t h i n 15-20 m i n u t e s . to w i t h i n - 10°C.  The d e s i r e d temperature was  I t was p o s s i b l e t o c o n t r o l t h e temperature  The s t r a i n produced was measured u s i n g a d i a l gauge  w h i c h had a s e n s i t i v i t y o f 0.001 i n . The s c h e m a t i c diagram o f t h e assembly i s shown i n . F i g . 16. An i n s t a n t a n e o u s d e f o r m a t i o n was observed on a p p l i c a t i o n o f t h e l o a d w h i c h slowed down s i g n i f i c a n t l y w i t h i n a few seconds.  To be s u r e  t h a t d e f o r m a t i o n a t t h e p o i n t s o f c o n t a c t o c c u r r e d by p l a s t i c f l o w r a t h e r t h a n by d i f f u s i o n a l c r e e p , t h e a p p l i e d l o a d was m a i n t a i n e d f o r o n l y 30 seconds.  The permanent s t r a i n was noted from t h e d i f f e r e n c e i n the d i a l gauge f  r e a d i n g s b e f o r e a p p l y i n g t h e load.and a f t e r removing t h e l o a d .  The compact  was c o o l e d r a p i d l y , a f t e r removing t h e l o a d , by f l u s h i n g Argon gas through the  system so t h a t no s i n t e r i n g o c c u r r e d d u r i n g c o o l i n g .  - 42 -  \\ \  GRPHITE  RECORDER  F i g . 16  Schematic R e p r e s e n t a t i o n o f t h e Vacuum H o t - P r e s s Used f o r S a p p h i r e and K-Monel Spheres.  - 43 c)  K-Monel K-Monel spheres of 1/16 - 0.002 i n . d i a m e t e r were vacuum  hot-pressed i n the same assembly as used f o r s a p p h i r e .  They were deformed  i n a g r a p h i t e d i e (1 i n diam) a t 800*, 900? and 1000°C a t p r e s s u r e s of 625, 937.5, 1250, and 1562.5 p s i . The l o a d was h e l d f o r 2 minutes I t was the  only.  found t h a t the d e f o r m a t i o n was complete w i t h i n a few seconds, as  d i a l gauge d i d not r e c o r d any f u r t h e r change i n the r e a d i n g a f t e r  this period.  The d e s i r e d temperature was reached w i t h i n 5-10 3.  minutes.  Experimental Procedures  A f t e r the spheres i n the d i e were deformed by the a p p l i c a t i o n of  a c e r t a i n p r e d e t e r m i n e d l o a d , the spheres were t a k e n out s y s t e m a t i c a l l y  to measure the f l a t f a c e s formed from w h i c h an average v a l u e o f Z was obtained.  The average d i a m e t e r of the f a c e s was a l s o measured u s i n g a  s p e c i a l l y d e s i g n e d goniometer. f a c e s i s shown i n F i g . 17. are  shown i n F i g . 18.  pycnometer  The assembly used f o r s t u d y i n g the f l a t  T y p i c a l f l a t f a c e s formed d u r i n g d e f o r m a t i o n  The d e n s i t y o f the spheres was determined by the  method and the p e r c e n t a g e b u l k d e n s i t y of the compact was  c a l c u l a t e d from t h e w e i g h t and t h e volume of the compact. The p r e s s u r e w i t h i n the d i e was not the same a t a l l p o i n t s c o n t r a r y to the h y d r o s t a t i c c o n d i t i o n s assumed i n the t h e o r e t i c a l models.  I n an attempt to d e t e r m i n e the v a r i a t i o n i n a/R and Z v a l u e s  i n the whole compact, spheres from l a y e r s ( t h r e e rows o n l y ) i n two p e r p e n d i c u l a r d i r e c t i o n s were examined f o r t h e average v a l u e of a/R and the c o o r d i n a t i o n number Z. the  F i g u r e 19 shows t h e s t a t i s t i c a l d i s t r i b u t i o n o f  number of spheres v s t h e c o o r d i n a t i o n number.  I t can be seen t h a t the  - 4 4 -  F i g . 17  Apparatus f o r Studying the Deformed  Spheres.  LEAD  K-MONEL  F i g . 18 P h o t o m i c r o g r a p h s o f t h e Deformed Spheres Showing  SAPPHIRE  the T y p i c a l F l a t Faces.  40h •  32\  O  THE WHOLE COMPACT Z =7.9 mean LEAVING THE DIE WALLS Z  =8.5  CO  w ei •w w 24 CO  P *  w  161  m' s sa  8  8 C O O R D I N A T I O N ' F i g . 19  10 N U M B E R  S t a t i s t i c a l D i s t r i b u t i o n o f the Number o f Spheres v s t h e C o o r d i n a t i o n Number.  12  - 47 mean c o o r d i n a t i o n number (Z = 7.9) o f t h e s p h e r e s , i n c l u d i n g t h o s e t o u c h i n g t h e d i e - w a l l s i s about 8% s m a l l e r than t h o s e from t h e c e n t r a l p o r t i o n o f t h e compact ( l e a v i n g t h e d i e - w a l l s ) (Z=8.5).  Because o f  the n a t u r e o f t h e a p p l i e d s t r e s s , w h i c h was u n i a x i a l , i n s t e a d o f h y d r o s t a t i c as assumed i n t h e models^  t h e d i f f e r e n t c o n t a c t f a c e r a d i i on t h e  i n d i v i d u a l s p h e r e s v a r i e d w i d e l y , as e x p e c t e d .  T h i s v a r i a t i o n was  p a r t i c u l a r l y e v i d e n t when t h e f a c e s formed normal t o t h e d i r e c t i o n o f t h e a p p l i e d s t r e s s were compared w i t h t h e f a c e s formed p a r a l l e l t o t h e d i r e c t i o n of t h e a p p l i e d s t r e s s .  However, t h e average v a l u e o f a/R f o r a l l t h e  spheres ( o m i t t i n g t h e s i d e s , t o p and bottom] l a y e r s ) w a s w i t h i n 3 - 4 % o f t h e mean v a l u e as d e t e r m i n e d from 20 spheres s e l e c t e d randomly from t h e c e n t r a l p o r t i o n of the d i e .  I t was a l s o found t h a t the a c c u r a c y o f  measurement c o u l d have been improved t o w i t h i n 2 - 3 % o f t h e mean v a l u e by u s i n g 50 spheres i n s t e a d o f s e l e c t i n g 20.  I n t h e case o f t h e spheres  from t h e s i d e s , t o p and bottom l a y e r s , t h e a/R v a l u e s were s l i g h t l y  larger  (about 5 - 7%) than t h e r e s t o f t h e compact, b u t t h e Z v a l u e s were r e l a t i v e l y s m a l l e r (about 1 0 % ) , so t h a t t h e o v e r a l l e r r o r i n t r o d u c e d by not c o n s i d e r i n g t h e . spheres i n c o n t a c t w i t h t h e d i e - w a l l s would be v e r y small. From a l l t h e s e c o n s i d e r a t i o n s , i t was d e c i d e d t h a t 20 spheres of l e a d ,  randomly s e l e c t e d from t h e c e n t r a l r e g i o n , c o u l d be c o n s i d e r e d  as r e p r e s e n t a t i v e o f each e x p e r i m e n t .  The e r r o r i n t r o d u c e d by s t u d y i n g  o n l y 20 spheres from t h e c e n t r a l r e g i o n o f t h e compact was c o n s i d e r e d a c c e p t a b l e i n v i e w o f t h e time saved i n n o t s t u d y i n g a l l t h e spheres (about 800) from each e x p e r i m e n t .  4.  - 48 -  R e s u l t s and D i s c u s s i o n  Twenty spheres (40 f o r s a p p h i r e and K-Monel) from t h e c e n t r a l r e g i o n o f t h e d i e were s t u d i e d f o r each compaction e x p e r i m e n t .  On each  s p h e r e , t h e f a c e s had a wide v a r i a t i o n o f 'a' v a l u e s sometimes d e v i a t i n g as much as 100 - 150% from t h e average v a l u e .  H i g h e r d e v i a t i o n s were o f t e n  observed f o r h i g h e r a/R v a l u e s , when i t was observed t h a t t h e newly formed f a c e s gave low v a l u e s o f a/R.  I n a d d i t i o n , the f o l l o w i n g  experimental  c o n d i t i o n s may have c o n t r i b u t e d t o t h i s s c a t t e r : i)  H y d r o s t a t i c c o n d i t i o n s were n o t o b t a i n e d  s t r e s s i n g o f t h e compact. perpendicular  i n the u n i a x i a l  I t was observed t h a t f a c e s t h a t were  approximately  t o t h e d i r e c t i o n o f p r e s s i n g gave l a r g e r v a l u e s than those  p a r a l l e l to the d i r e c t i o n of the p r e s s i n g . ii) iii)  D i f f e r e n c e i n t h e hardness o f d i f f e r e n t p a r t s o f t h e s p h e r e , C r y s t a l o r i e n t a t i o n i n t h e case o f s a p p h i r e .  I n t h e case o f some s p h e r e s , an i n d e n t a t i o n r a t h e r than a . f l a t f a c e was o c c a s i o n a l l y observed and was p r o b a b l y d i f f e r e n c e i n hardness a t t h e c o n t a c t p o i n t . was observed d u r i n g h o t - p r e s s i n g o f s a p p h i r e The  due t o t h e  Very l i t t l e  fragmentation  spheres.  t h e o r e t i c a l models were t e s t e d by comparing t h e f o l l o w i n g  results. a)  D v s a/R F i g u r e 20 shows t h e p l o t o f p e r c e n t a g e b u l k d e n s i t y v s  a/R f o r l e a d a t room t e m p e r a t u r e .  The e x p e r i m e n t a l  curve f o l l o w s the  general trend of the t h e o r e t i c a l curves w i t h the r i s i n g slope due  t o t h e c o o r d i n a t i o n change o b s e r v e d .  probably  The sudden s h i f t o f the c u r v e  - 49 -  Fig. 20  D vs a/R for Lead at Room Temperature and Sapphire at ® 1570°, A 1600°, and B 1700°G.  -  R F i g . 21 . D v s a/R f o r Lead a t 50 and 100°C.  50  -  - 51 -  F i g . 22  D v s a/R f o r K-Monel.  - 52 a t a/R = 0.15 c o r r e s p o n d s t o t h e s l i d i n g observed i n t h e l o a d v s compression curve.  F i g u r e 21 shows s i m i l a r p l o t s f o r t h e r e s u l t s o b t a i n e d a t 50 and  100°C.  The t h e o r e t i c a l p l o t s a r e superimposed  f o r comparison.  r e s u l t s f o r t h e s a p p h i r e spheres a r e shown i n F i g . 20.  The  F i g u r e 22 a l s o  shows t h a t t h e i n c r e a s e o f d e n s i f i c a t i o n o f t h e K-Monel spheres w i t h deformation f o l l o w s the t h e o r e t i c a l curves. b)  D  vs  Z  The d e v i a t i o n from t h e t h e o r e t i c a l c u r v e s i s a t t r i b u t e d t o t h e change o f c o o r d i n a t i o n number as t h e d e f o r m a t i o n p r o c e e d s .  A change o f  Z was observed i n a l l t h e e x p e r i m e n t s , and t h i s i s r e f l e c t e d i n t h e g e n e r a l increase i n the value of Z .  F i g u r e 23 shows t h e i n c r e a s e o f Z as a f u n c t i o n  of ( D — D ) f o r the l e a d spheres. q  The b r e a k i n t h e l i n e a t a/R = 0 . 1 5 f o r  the room temperature c u r v e c o r r e s p o n d s t o t h e p o i n t o f sudden s h i f t i n F i g . 20.  The change o f Z w i t h t h e d e f o r m a t i o n i s a l s o e v i d e n t from  F i g . 24 w h i c h shows t h e change i n t h e s t a t i s t i c a l d i s t r i b u t i o n o f t h e number o f c o n t a c t p o i n t s f o r t h e spheres from t h e c e n t r a l p o r t i o n o f t h e die. c)  v s a/R F i g u r e s 25 and 26 show t h e p l o t s o f p e r c e n t a g e b u l k s t r a i n  vs a/R f o r b o t h t h e l e a d and t h e s a p p h i r e s p h e r e s .  The l a r g e  v a l u e s o f t h e e x p e r i m e n t a l s t r a i n s a r e due to; t h e c o n t r i b u t i o n o f p a r t i c l e rearrangement and c o o r d i n a t i o n number change t o t h e e x p e r i m e n t a l s t r a i n , w h i c h has n o t been c o n s i d e r e d i n t h e t h e o r e t i c a l c u r v e s .  The c u r v e  a t 100°C shows t h a t t h e r e was a l a r g e r c o n t r i b u t i o n from t h e p a r t i c l e rearrangement.  - 53 -  •peai  aoj  z  SA  (°a-a)  ZZ '*T&  F i g . 24  Change o f Z w i t h the I n c r e a s i n g D e f o r m a t i o n f o r K-Monel Spheres.  - 55 -  F i g - 25  e, v s a/R f o r Lead a t Room Temperature and S a p p h i r e a t A 1570°, © 1600°, and" B 1700°C.  - 57 d)  D vs The s t r a i g h t l i n e r e l a t i o n s h i p between the percentage  d e n s i t y and the percentage  b u l k s t r a i n i n the t h e o r y as w e l l as  bulk experimental  o b s e r v a t i o n s (shown i n F i g s . 27 and 2 8 ) , v e r i f i e s t h a t t h e proposed models a r e o p e r a t i v e d u r i n g the d e f o r m a t i o n p r o c e s s .  The d i f f e r e n t s l o p e  i s a g a i n i n d i c a t i v e of the change i n the c o o r d i n a t i o n number.  Both  t h e o r e t i c a l and e x p e r i m e n t a l s t r a i n s a r e n e g a t i v e , c o n s e q u e n t l y ,  the  the  d a t a as p l o t t e d produced a p o s i t i v e s l o p e and t h i s i s c o n s i s t e n t w i t h Eq.  (26).  e)  D v s a/R v s Z I n o r d e r t o s u p p o r t f u r t h e r the argument t h a t the d e v i a t i o n o f  the e x p e r i m e n t a l data.from  the t h e o r e t i c a l c u r v e s was  p r i m a r i l y due  to  the g r a d u a l change of c o o r d i n a t i o n w i t h i n c r e a s e d d e f o r m a t i o n , , p l o t s were made w i t h the t h r e e v a r i a b l e s , p e r c e n t a g e as shown i n F i g s . 29 and 30.  b u l k d e n s i t y , a/R  and  Z  The e x p e r i m e n t a l d a t a as superimposed  on t h e s e p l o t s appear t o l i e on the same s u r f a c e formed by the c u b i c , orthorhombic  and rhomohedral models.  i s p r o b a b l y due  to the change i n Z.  The d e v i a t i o n of the e x p e r i m e n t a l  plots  I t i s i n t e r e s t i n g to note t h a t  the c u r v e f o r the b . c . c . model does not seem to l i e on the same s u r f a c e drawn through f)  D vs  the c u r v e s f o r o t h e r models.  (a/R)  2  2  F i g u r e 31 i s a p l o t of D vs (a/R)  f o r the l e a d and  the  s a p p h i r e spheres and F i g . 32 shows the same p l o t f o r the K-Monel s p h e r e s . I t can be seen t h a t a good f i t i s o b t a i n e d between the p r e d i c t i o n s and the e x p e r i m e n t a l o b s e r v a t i o n s .  The  theoretical  change of the s l o p e .  - 60 -  LEAD AT r i - t  0.3 F i g . 29  D v s a/R v s Z f o r Lead a t Room Temperature and Sapphire at A 1570°, © 1600°, and. • 1700°C.  - 61 -  - 62 at  100°C d u r i n g t h e l a t e r s t a g e may be due t o t h e change i n t h e n a t u r e  of t h e c o n t r i b u t i o n from t h e p a r t i c l e rearrangement t o t h e o v e r a l l densification In  process. order to normalise the d i f f e r e n c e s i n the i n i t i a l  d e n s i t i e s f o r the i n d i v i d u a l experiments,  p l o t s between  (D —  D  Q  packing  ) / D  O  2 v s (a/R) were p l o t t e d and a r e shown i n F i g s . 33 t o 35 f o r l e a d .  Again  a good f i t i s o b t a i n e d w i t h t h e t h e o r e t i c a l p r e d i c t i o n s . I t c a n be seen from Eq. (22),namely D - D  - ID  U/R)  2  <  2  2  )  AO  O  t h a t i t h o l d s f o r a f i x e d c o o r d i n a t i o n number.  T h i s does n o t i n c l u d e  any c o n t r i b u t i o n from t h e p a r t i c l e rearrangement t o t h e d e n s i f i c a t i o n I t can be seen t h a t a c c o r d i n g t o t h i s e q u a t i o n , a p l o t o f D 2 2 vs (a/R) o r between ( D — D ) / D v s (a/R) s h o u l d g i v e a s l o p e o f process.  Q  3/2 D  Q  0  o r 3/2, r e s p e c t i v e l y . B u t the c a l c u l a t e d e x p e r i m e n t a l  a r e much h i g h e r than t h e t h e o r e t i c a l l y p r e d i c t e d s l o p e s .  slopes  Such a  d i f f e r e n c e can be e x p l a i n e d on the b a s i s o f t h e e x p e r i m e n t a l  observation  t h a t t h e p l a s t i c f l o w i s accompanied by p a r t i c l e rearrangement and l i t t l e fragmentation  i n t h e case o f s a p p h i r e .  I f A D i s t h e i n c r e a s e i n t h e d e n s i t y d u r i n g h o t - p r e s s i n g tHen i t must be composed o f two p a r t s i . e . AD  =  AD  P  +  AD r  where  A D = c o n t r i b u t i o n from p l a s t i c P  and  A D ^ = c o n t r i b u t i o n from rearrangement  The c o n t r i b u t i o n from t h e p a r t i c l e rearrangement w i l l depend upon: i. ii.  the i n i t i a l packing density the applied pressure.  and  flow  F i g . 31  D v s (a/R)  f o r Lead a t Room Temperature, 50° and 100°C  and S a p p h i r e a t  ©  1570°,  A  1600°, and  E8  1700°C.  F i g . 32  D v s (a/R)  f o r K-Monel.  - 65 The complex n a t u r e o f t h i s c o n t r i b u t i o n prevented evaluation of t h i s factor.  any q u a n t i t a t i v e  But i t can be seen t h a t i t s i n c l u s i o n i n  the p l a s t i c flow equation w i l l i n c r e a s e the v a l u e of the constant.  As  37 observed  by D u f f i e l d and G r o o t e n h u i s  , a p a r t from t h e i n i t i a l r e p a c k i n g on  immediate a p p l i c a t i o n o f l o a d , a t r a n s i t i o n a l r e p a c k i n g p e r s i s t s c a u s i n g a f u r t h e r volume change o f about 1 ^ %  up t o a p r e s s u r e o f 1000 p s i .  T h i s i s s u p p o r t e d by t h e s t r a i g h t , l i n e r e l a t i o n s h i p between D v s (a/R) 2 o r (D  —  D  Q  ) / D  0  v s (a/R) , w h i c h o t h e r w i s e would n o t be so i f , t h e  c o n t r i b u t i o n from p a r t i c l e rearrangement was much d i f f e r e n t w i t h values of the load.  increasing  - 67 -  - 69 III G E O M E T R Y  OF  IN  D E F O R M A T I O N  THE  A.  D I E  INTRODUCTION  The s i m p l e and s y s t e m a t i c p a c k i n g s o f u n i f o r m spheres o f e q u a l 35  s i z e have been d i s c u s s e d i n d e t a i l by G r a t o n and F r a s e r assumed i n t h e t h e o r e t i c a l models.  and were  The r e l a t i v e s t a b i l i t y o f t h e  v a r i o u s p a c k i n g s depends upon t h e type o f p a c k i n g , t h e shape o f t h e c o n t a i n e r , t h e mode o f f i l l i n g die-wall effect etc. of  the d i e , and a few o t h e r f a c t o r s l i k e t h e  I t can be shown t h a t under the g r a v i t a t i o n a l f i e l d  f o r c e a l o n e , t h e c o n d i t i o n s f o r s t a b l e e q u i l i b r i u m a r e s a t i s f i e d by  o n l y t h e rhombphedral p a c k i n g .  The t e t r a g o n a l and o r t h o r h o m b i c  (from t h e  rhombic base o n l y ) p a c k i n g s a r e l e s s s t a b l e and have a tendency  t o change  to t h e rhombohedral p a c k i n g .  The c u b i c and o r t h o r h o m b i c  (from c u b i c base)  p a c k i n g s a r e even more u n s t a b l e . 1.  The D i e - W a l l E f f e c t  The t h e o r e t i c a l d e n s i t i e s o b t a i n e d f o r t h e s i m p l e and s y s t e m a t i c p a c k i n g s (shown i n t h e T a b l e I X , Appendix on t h e assumption  I ) have.been c a l c u l a t e d  t h a t t h e system i s i n f i n i t e l y l a r g e i n comparison w i t h  the d i a m e t e r o f t h e s p h e r e s .  I t i s evident that the systematic packings  cannot be a t t a i n e d throughout u n l e s s t h e d i e has i d e n t i c a l l y t h e same shape and a v e r y s p e c i a l e x a c t . r e l a t i o n t o t h e s i z e o f t h e u n i t c e l l o f t h e type of p a c k i n g i n v o l v e d . encountered  T h i s c o n d i t i o n i s e x t r e m e l y u n l i k e l y t o be  i n t h e d i e ; hence, t h e s y s t e m a t i c p a c k i n g i s r a r e l y  throughout t h e d i e .  found  - 70 When t h e r a t i o o f t h e c o n t a i n e r s i z e t o t h e p a r t i c l e d i a m e t e r i s l e s s than 50 i n case o f t h e o r t h o r h o m b i c , o r 100 i n t h e case o f t h e t e t r a g o n a l and rhombohedral p a c k i n g s , t h e e f f e c t o f t h e reduced d e n s i t y r e s u l t i n g from m i s a l i g n m e n t s a t the s i d e w a l l s i s o f the o r d e r o f 1% 38 or more of  .  A l s o t h e shape o f t h e c o n t a i n e r has a b e a r i n g on t h e type  p a c k i n g e.g., a r e c t a n g u l a r c o n t a i n e r w i l l f a v o u r p a c k i n g s w i t h  square  l a y e r arrangement whereas a c y l i n d r i c a l c o n t a i n e r w i l l have a s i m p l e rhombic arrangement i n t h e f i r s t  layer.  39 McGeary the p a c k i n g d e n s i t y .  has s t u d i e d t h e e f f e c t o f t h e c o n t a i n e r s i z e on H i s r e s u l t s a r e shown i n F i g . 36. He found t h a t  at D/d = 200, t h e p a c k i n g d e n s i t y reached a v a l u e o f 62.5% o f t h e t h e o r e t i c a l d e n s i t y w h i c h agreed v e r y c l o s e l y w i t h h i s e x p e r i m e n t a l maximum v a l u e s .  I t can be seen from F i g . 36 t h a t t h e p a c k i n g d e n s i t y  i s l i t t l e a f f e c t e d by an i n c r e a s e i n t h e c o n t a i n e r s i z e beyond t h e r a t i o D/d = 10. 2.  Mode o f F i l l i n g  the Die  The p a c k i n g o f spheres i s g r e a t l y i n f l u e n c e d by t h e way t h e die  isfilled.  When a p a c k i n g o f a c e r t a i n degree o f s t a b i l i t y i s  s u b j e c t e d t o s h a k i n g and t a p p i n g , t h e r e i s a tendency  f o r i t t o change o r  t r a n s l a t e i n t o a p a c k i n g o f a h i g h e r degree o f s t a b i l i t y . s u p p o r t i s adequate, t h i s tendency  i s reduced.  I f the l a t e r a l  A t t a i n m e n t o f a more  s t a b l e p a c k i n g , which i n v o l v e s l a t e r a l s p r e a d i n g o f the body , i s p r e v e n t e d by adequate s u p p o r t from t h e s i d e s .  T h e r e f o r e , t h e degree o f  s t a b i l i t y of the packing i s maintained i n s p i t e of the unstable c o n f i g u r a t i o n . O b v i o u s l y , t h e degree o f r i g i d i t y and adequacy o f t h e l a t e r a l support determine  will  t h e e x t e n t t o which t h e body w i l l conform t o a more s t a b l e p a c k i n g  - 71 -  70  F i g . 36  E f f e c t of C o n t a i n e r S i z e on the E f f i c i e n c y of P a c k i n g One-Size Spheres ( A f t e r McGeary?^).  configuration.  The v a r i o u s modes of f i l l i n g t h e d i e and v i b r a t i o n on the i n i t i a l p a c k i n g  the e f f e c t of  d e n s i t y have been s t u d i e d e x t e n s i v e l y by  40 S m i t h , F o o t e and Busang  .  They s t u d i e d the change i n the number of  contact  p o i n t s of the spheres w i t h the i n c r e a s e i n the d e n s i t y of the compact r e s u l t i n g from s h a k i n g and technique  tapping.  They employed a c o r r o s i o n  w h i c h d e v e l o p e d s m a l l r i n g s on each sphere i n d i c a t i n g the p o i n t s  of c o n t a c t .  They excluded  the spheres t o u c h i n g  observations  i n o r d e r to m i n i m i s e the d i e - w a l l e f f e c t . 41  shown i n T a b l e I I (from Jones  ).  the s i d e - w a l l s i n t h e i r Their r e s u l t s  are  I t can be seen t h a t t h e r e i s a G a u s s i a n  d i s t r i b u t i o n of the number of spheres w i t h a g i v e n c o o r d i n a t i o n number. F i g u r e 37 shows such a graph and obtained  i t can be seen t h a t the maxima a r e  around the c o o r d i n a t i o n number 8. McGeary, u s i n g ' a t r a n s p a r e n t  c o n t a i n e r and a low power  s t e r e o s c o p i c m i c r o s c o p e , showed t h a t the v e r t i c a l v i b r a t o r y t r a n s l a t i o n of a bed of spheres tends to produce the o r t h o r h o m b i c p a c k i n g extent  the t e t r a g o n a l p a c k i n g .  and  to a small  He suggested t h a t the rhombohedral  cannot o c c u r under such c o n d i t i o n s because the v e r t i c a l f o r c e w i l l t o change i t t o the t e t r a g o n a l 3.  packing tend  packing. Haphazard P a c k i n g s  There a r e v a r i o u s o t h e r schemes of arrangement w h i c h , w h i l e yet systematic ones c o n s i d e r e d  and o r d e r l y , a r e more a r b i t r a r y and l e s s s i m p l e than the u n d e r . t h e s i m p l e and  the s y s t e m a t i c modes of  packing.  The most r e a d i l y v i s u a l i s e d examples a r e the arrangements i n w h i c h the a n g l e between the rows.and s p a c i n g of the s p h e r e - c e n t e r s  d i f f e r i n some  - 73 -  TABLE I I Correlation  o f C o - o r d i n a t i o n Number and D e n s i t y f o r Lead Shot; 3.78 mm Diameter  D i s t r i b u t i o n of Numbers o f c o n t a c t s (z)  ^ . ber . , counted u m  « . . Treatment  4 5 1. Poured into glass beaker 905 2. Poured and shaken 906 3. Poured and shaken 887 4. Shaken t o maximum density 14 9 4 5. Added i n small quantities and tamped interm i t t e n t l y 1562  6  7  8  9  10 11 12  6 78 243 328 200 48  2  Mean coordination n  u  m  b  e  r  Density  0 b  " served  lated  A  0  0  6.92  0.553  0.562  3 54 173 309 233 118 14 2  0  7.34  0.560  0.575  0 14  69 182 316 212 87 7  0  8.06  0.574  0.609  0 14  8 6 1 9 2 2 33 1 93 161 226.389  1 13  77 245 322 310 208 194 192 9.14  9 . 51  0 . 628  0.641  0 . 662  0.646  The r e s u l t s a r e c a l c u l a t e d from e x p e r i e m e n t a l d a t a o f S m i t h , F o o t e , and Busang. ( A f t e r J o n e s ^ ) 1  - 74 -  4  6 C 0 ORD  F i g - 37  8  10  12  I N A.T-I'O N ' N I I M B E R  S t a t i s t i c a l D i s t r i b u t i o n o f t h e Number o f Spheres w i t h t h e C o o r d i n a t i o n Number (Data from J o n e s ) . 4 1  - 75 s y s t e m a t i c manner and degree from t h o s e i n s t a n c e s d e s c r i b e d e a r l i e r . Other t y p e s of p a c k i n g s a r e p o s s i b l e i n w h i c h no o r s y s t e m a t i c r e p e t i t i v e arrangement can be d i s c o v e r e d .  orderly  A l l these  p a c k i n g s can be grouped t o g e t h e r under the g e n e r a l term c h a o t i c o r haphazard packings.  A l s o i n a g i v e n assembly of spheres p a c k i n g i s l i k e l y to v a r y  from p l a c e to p l a c e . p a c k i n g s may  At c e r t a i n p l a c e s , one of the s i m p l e and  be developed,  a t o t h e r p l a c e s another  v a r i o u s phases o f haphazard arrangements can e x i s t . arrangements may  systematic  type, w h i l e Some of  elsewhere  these  be e n t i r e l y c h a o t i c w h i l e o t h e r s show some degree of  orderliness.  The most p r o b a b l e r e s u l t i s , t h e r e f o r e , t h a t l a y e r - c o l o n i e s of s i m p l e and s y s t e m a t i c p a c k i n g s w i l l be s e p a r a t e d by zones of d i f f e r e n t random p a c k i n g s . I n the p r e s e n t s t u d y , i t was  i n t e n d e d t o i n v e s t i g a t e the geometry  of the deformed c o n t a c t f a c e s produced by d e f o r m a t i o n d u r i n g h o t - p r e s s i n g . I t i s apparent  t h a t such a geometry w i l l be d i f f e r e n t from the  p a c k i n g because of the c o n t i n u e d r e p a c k i n g observed  during  initial  compaction.'  But the o v e r a l l geometry w i l l depend upon the type of i n i t i a l  packings,  s y s t e m a t i c o r haphazard. B.  EXPERIMENTAL TECHNIQUES AND  RESULTS  The d i s t r i b u t i o n of the c o n t a c t f a c e s on the sphere can be s t u d i e d by measuring the a n g l e s between them u s i n g a goniometer. apparatus  shown i n F i g . 17, w i t h the s p e c i a l l y designed  measure the r o t a t i o n of the sphere w i t h r e s p e c t to the a l i g n m e n t , was  used f o r the purpose.  The  goniometer t o microscope  I t s h o u l d be r e c a l l e d t h a t a l l  easily  - 76 the  t h e o r e t i c a l models assume the spheres t o be u n i f o r m and of e q u a l  size.  U n l e s s t h i s c o n d i t i o n i s s a t i s f i e d , some spheres packed i n a  c e r t a i n manner may not produce c o n t a c t p o i n t s a t low l o a d s .  This s i t u a t i o n  i s q u i t e d i f f e r e n t from measuring the c o o r d i n a t i o n number u s i n g the corrosion technique. of  show the t o t a l number  c o n t a c t p o i n t s , the former may not produce such a c o n t a c t p o i n t by  deformation. the  W h i l e the l a t t e r w i l l  Such an e f f e c t was r e a d i l y seen d u r i n g the d e f o r m a t i o n of  lead spheres.  I t was found t h a t the d e v i a t i o n from the p e r f e c t  sphericity  and the d i f f e r e n c e i n the s i z e of the l e a d spheres o c c a s i o n a l l y , caused the  deformed  sphere t o show, l e s s e r number o f c o n t a c t p o i n t s . In o r d e r t o s t u d y the geometry of the d e f o r m a t i o n , K-Monel  spheres w i t h b a l l - b e a r i n g p r e c i s i o n were used f o r the purpose.  About  160  s p h e r e s , randomly chosen from the c e n t r a l r e g i o n from f o u r d i f f e r e n t , e x p e r i m e n t s (40 from each) were used t o s t u d y the a n g u l a r r e l a t i o n s h i p between the f l a t  f a c e s formed on each s p h e r e .  The spheres were f i r s t  d i v i d e d i n t o groups a c c o r d i n g t o t h e i r c o o r d i n a t i o n numbers.  Using  a n g u l a r r e l a t i o n s h i p s , s t e r e o g r a p h i c p l o t s were made i n o r d e r t o i d e n t i f y any symmetry between the f a c e s . of  From the comparison of the d i s t r i b u t i o n  the f a c e s i n each group, i t i s c o n c l u d e d t h a t most of the spheres  deform randomly i . e . the n e i g h b o u r i n g spheres a r e randomly a l t h o u g h the c o o r d i n a t i o n number of the spheres may e v i d e n c e f o r t h i s random geometry  be the same.  The  f o r the spheres h a v i n g the same c o o r d i n a t i o n  number i s shown s c h e m a t i c a l l y i n F i g . 38. of  distributed,  The knowledge of the s i z e  t h e f a c e and the f a c t t h a t maximum c o n t a c t p o i n t s s h o u l d l i e i n the  m i d d l e l a y e r , were used t o d e t e r m i n e the o r i e n t a t i o n of the spheres w i t h r e s p e c t t o the ' d i r e c t i o n of p r e s s i n g ( F i g . 3 8 ) .  - 77 PROBABLE DIRECTION OF PRESSING  (270,90) 0(256,62) .(301,98)  (270,90) lf  (352,60)^ . ^ . _(2_08_j8^ ,  (0,0) O0-52.62)  ©(286,101)  /  ~~ ~~ ~~ -  /  i . p  (0,180)/  v  (90,90)  ^ (188,127) 0 @ -  ^(180,50). -CD" " i ©(28,50) /  C 0 , T L 3 T  O Qfo2,160)  •  i  \  % (17,155)  (73,106) ®  O  «  (0,180)  i  (120,72)  (90,90) (B)  (A)  (0,0) -  (88,18) (275,65) ^ » . ©(30*, 18) ( 0  \  O (240,j35)  \  (320,62)0 ^  (197,770 1  (270,90)  #V(86,75) O  (150,73) \ _ - - i (90,90)  / / 'I  (  A  ^  /  v  v  /  y  /  0(^60,98)  <  ®  1  5  V ° >  (65,42) !  (90,90)  (0,180) .(0,180) (O  0  0 )  (D)  (Cont.)  o  (260,47)^(0,0) /  v(270,90) v  / 1  1  7  .  5  5  >  \  (122,56) O (316,/9)@ \ / \  \ \  /  030,61) © Q l ^ 5 , 5 1 ) (230.98D  (0,0,  OA05,42)  (195,53)  Q\  O  (256,56)  /  /  /  \  O < 'H°>  /  128  r  N  (90,90)  ©  (30,115)1  vJ(19S,103) \ (312,92) N  (90,90)  (0,180)  (270,90>  (0,180)  (F)  (E)  (90,90) O  (2T5>30) X (0,0)  (io?TH) /  ' (0,0) ©<*°.*°>  /  (270,90)  ^ <183,38)^ O © (85,32) /340.65) \  1  /  v  ^ ©(45,105) (330,70W  ''O  \  /  /  (180,113)  V  /  l  O( . °) 158  V  8  p (18^0.60) ©(53,98)  0(242,105)  (0,180) 0  (135,137)  7(90,90)  (0,180)  (270,90) (G)  (H)  (Cont.)  -  F i g . 38  Stereographic Representation of the Geometry o f D e f o r m a t i o n o f Spheres i n ,the D i e .  79  -  - 80 I t can be seen from F i g . 38 t h a t not o n l y the geometry of the f l a t f a c e s i s d i f f e r e n t from sphere t o sphere but a l s o i t does not c o r r e s p o n d to the hex. spheres.  p r i s m . " model except i n the case of one  or  two  The f a c t t h a t the c o o r d i n a t i o n number v a r i e s from 4 t o 12 i n s i d e  the d i e (as a l s o observed by o t h e r s ) , s u g g e s t s t h a t the geometry of the d e f o r m a t i o n The  i s random t o a l a r g e  extent.  o t h e r o b s e r v a t i o n s made i n t h i s p a r t of the i n v e s t i g a t i o n  can be summarized as f o l l o w s : 1.  I n d i v i d u a l c o l o n i e s of the rhombohedral, the t e t r a g o n a l  the o r t h o r h o m b i c p a c k i n g s were o b s e r v e d . of the d i e , t h e r e was bohedral was  and  I n the c e n t r a l p o r t i o n  found to be a g r e a t e r p r o p o r t i o n of the rhom-  the t e t r a g o n a l p a c k i n g s .  The  o r t h o r h o m b i c arrangement  found t o be more pronounced near the w a l l s of the d i e . 2.  I t was  found t h a t though the g e n e r a l p a c k i n g  s h o u l d have been one of the above p a c k i n g s , c o n t a c t p o i n t s were m i s s i n g .  sometimes one  arrangement  or two of  T h i s w i l l be t r u e i f the spheres i n the  h o r i z o n t a l l a y e r a r e not c l o s e enough to produce c o n t a c t p o i n t s a t loads.  the  low  A l s o i f the spheres are h e l d a t one p l a c e or the o t h e r i n a  d i s o r d e r l y manner, the c o o r d i n a t i o n number w i l l be a f f e c t e d . 3.  Few  of the spheres i n c o n t a c t or n e i g h b o u r i n g  to  the  d i e - w a l l were found t o deform i n a manner s i m i l a r to the c u b i c model. 4.  I n some c a s e s , e s p e c i a l l y those near the d i e - w a l l ,  d o d e c a h e d r a l type of d e f o r m a t i o n  (Z=4)  was  observed.  and  - 81 5. at  I n o r d e r t o examine the shape of the deformed  spheres  v e r y h i g h d e f o r m a t i o n , l e a d spheres were deformed under h i g h  pressures.  On e x a m i n a t i o n , i t was  found t h a t the spheres had deformed  more i n the d i r e c t i o n of the l o a d and l a r g e v a r i a t i o n s i n a/R  values  were o b s e r v e d .  I t can  F i g u r e s 39 and 40 show the deformed s p h e r e s .  be seen t h a t the geometry of the d e f o r m a t i o n i n F i g . 39 i s s i m i l a r to t h a t of the  hexagonal p r i s m a t i c model and i n F i g . 40 i s s i m i l a r to the  rhombic dodecahedron model.  A l s o i t was  observed t h a t a t such h i g h p r e s s u r e s , most of  the spheres i n the m i d d l e of the d i e had Z=12,  c o r r e s p o n d i n g to  rhombic dodecahedron model.  6.  I n the c a s e of p a c k i n g the d i e w i t h o u t i n t e r m i t t e n t t a p p i n g  and s h a k i n g , t h e d e f o r m a t i o n produced was more d i s o r d e r l y than when the die  was  filled 7.  i n i n s t a l l m e n t s w i t h some s h a k i n g and t a p p i n g . The e f f e c t of the c o n t a i n e r s i z e was determined  d e f o r m i n g b r a s s spheres I t was  by  (1/8 i n . diam) i n a g r a p h i t e d i e (1 i n . diam).  found t h a t the d e f o r m a t i o n produced was v e r y random i n n a t u r e  and v e r y few spheres deformed a c c o r d i n g to the proposed models. c u b i c and the o r t h o r h o m b i c p a c k i n g s were o c c a s i o n a l l y 8.  The  observed.  The f a c t t h a t the i n i t i a l b u l k d e n s i t y of the compact  v a r i e s from 60% t o 62%, s u g g e s t s t h a t t h e o v e r a l l p a c k i n g arrangement i.e.  t h e mode of d e f o r m a t i o n c o r r e s p o n d s to the Z=8  the hexagonal p r i s m a t i c model.  coordination i . e .  - 82  F i g . 39  -  P h o t o m i c r o g r a p h of the Lead Sphere Deformed i n Accordance w i t h Hexagonal P r i s m a t i c Model.  - 83 -  F i g . 40  Photomicrograph of the Lead Sphere deformed i n Accordance w i t h Rhombic Dodecahedron Model.  - 84 IV M E C H A N I C A L A.  The  P R O P E R T I E S INTRODUCTION  t h e o r e t i c a l models d i s c u s s e d p r e v i o u s l y (Chapter I I )  a r e g e n e r a l i n the sense t h a t the e q u a t i o n s geometrical considerations alone.  have been developed from  The b a s i c e q u a t i o n s  should h o l d f o r  any m a t e r i a l w i t h i n the l i m i t s of the assumptions made.in t h e i r derivation.  The  f o r m a t i o n of a f l a t f a c e on any  sphere can take  place  o n l y by two mechanisms: 1.  Plastic  2.  Stress-enhanced d i f f u s i o n a l The  deformation creep.  c o n t r i b u t i o n from the d i f f u s i o n a l p r o c e s s i s c a l c u l a b l e ,  i f the d i f f u s i o n c o e f f i c i e n t s under the e x i s t i n g s t r e s s a r e knwon. from the c o n s i d e r a t i o n of the time used f o r h o t - p r e s s i n g , no c o n t r i b u t i o n from the d i f f u s i o n a l mechanism towards the f o r m a t i o n was  expected.  The  However,  significant  flat-face  a l t e r n a t i v e mechanism i . e . p l a s t i c  deformation,  i n v o l v e s the m a t e r i a l p r o p e r t i e s such as v i s c o s i t y , y i e l d s t r e n g t h , e t c . An attempt has been made to i n c l u d e the y i e l d s t r e n g t h i n t o the basic density equation,  so t h a t the knowledge of the p r o p e r t i e s of  any  m a t e r i a l can be used i n p r e d i c t i n g the d e n s i f i c a t i o n b e h a v i o u r d u r i n g hot-pressing  the  process. 1.  D e f o r m a t i o n Under S t a t i c  The mechanism of d e f o r m a t i o n  of two  Conditions spheres i n c o n t a c t under  a s t a t i c l o a d can be b e t t e r u n d e r s t o o d by f i r s t c o n s i d e r i n g the  deformation  - 85 o c c u r r i n g between a hard h e m i s p h e r i c a l i n d e n t o r and a f l a t f a c e o f a s o f t e r metal  (the r e s u l t s a r e s i m i l a r f o r the deformation  o c c u r r i n g between  a hard f l a t f a c e and a h e m i s p h e r i c a l s u r f a c e o f a s o f t e r m e t a l ) . I f the surfaces a r e pressed  t o g e t h e r w i t h a l o a d L, they w i l l , a t  f i r s t , deform e l a s t i c a l l y a c c o r d i n g t o H e r t z ' s  c l a s s i c a l equations.  The  r e g i o n o f c o n t a c t i s a c i r c l e o f r a d i u s a, where  where r = r a d i u s o f t h e hemisphere = Young's modulus o f t h e hemisphere = Young's modulus o f t h e f l a t s u r f a c e .  I t can be seen t h a t t h e a r e a o f c o n t a c t  a s L  2/3  2 (A = ira ) w i l l  increase 2 w h i l e t h e mean p r e s s u r e over t h e c i r c l e o f c o n t a c t (P =L/ua ) w i l l m 1/3  v a r y as L  .  The d e f o r m a t i o n s  i n v o l v e d i n t h i s range a r e e l a s t i c and r e v e r s i b l e .  As t h e l o a d i s i n c r e a s e d , a s t a g e i s reached  a t which the maximum shear s t r e s s  i n t h e s o f t e r m a t e r i a l exceeds i t s e l a s t i c l i m i t and p l a s t i c 43 j u s t occurs.  Timoshenko  deformation  has shown, u s i n g t h e H e r t z i a n a n a l y s i s , t h a t t h i s  r e g i o n i s s i t u a t e d a t a p o i n t about 0.5a below t h e c e n t r e o f t h e . c i r c l e of contact.  A t t h i s s t a g e , t h e mean p r e s s u r e i s g i v e n by P  - 1.1 Y m  where Y is t h e y i e l d s t r e s s o r t h e e l a s t i c l i m i t o f t h e s o f t e r m e t a l . The p l a s t i c d e f o r m a t i o n  i s l i m i t e d to a small r e g i o n .  The m a t e r i a l o u t s i d e t h i s r e g i o n has n o t reached t h e c o n d i t i o n f o r p l a s t i c i t y and i t s d e f o r m a t i o n  i s s t i l l essentially  elastic.  - 86 As t h e l o a d i s f u r t h e r i n c r e a s e d , t h e r e g i o n o f p l a s t i c i t y i n c r e a s e s u n t i l t h e whole o f t h e m a t e r i a l around the r e g i o n o f c o n t a c t 44 flows p l a s t i c a l l y .  Hencky  45 and I s h l i n s k y  have attempted t o c o r r e l a t e  the mean p r e s s u r e w i t h the e l a s t i c l i m i t o f t h e m a t e r i a l when p l a s t i c j u s t occurs.  yielding  The b a s i c e q u a t i o n used i n b o t h s t u d i e s was von M i s e s ' s  c r i t e r i o n for p l a s t i c flow.  According  to t h i s c r i t e r i o n , p l a s t i c  flow  o c c u r s when (  °1 ~ ° 2  ) 2 +  ( a  2 ~ 3 a  )  2  +  ( a  3 " °1  )2  =  where a^, o^t o"g a r e t h e p r i n c i p a l s t r e s s e s i n t h e s o l i d and Y i s the e l a s t i c l i m i t o f t h e m a t e r i a l i n pure t e n s i o n ( o r f r i c t i o n l e s s compression) experiments.  The a n a l y s i s o f b o t h Hencky and I s h l i n s k y y i e l d s  the same r e s u l t :  i . e . to a f i r s t approximation, P = c Y m  where c has a v a l u e o f a p p r o x i m a t e l y  essentially  p l a s t i c y i e l d i n g o c c u r s when  3 (Hencky, c = 2.82 and I s h l i n s k y ,  c = 2.6),. I f t h e l o a d i s f u r t h e r i n c r e a s e d , t h e a r e a over w h i c h p l a s t i c f l o w o c c u r s i n c r e a s e s and t h e mean y i e l d p r e s s u r e remains c o n s t a n t a t a v a l u e o f 3Y, p r o v i d e d , a)  approximately  that;  t h e deformed a r e a i s n o t t o o l a r g e compared w i t h t h e s i z e  o f t h e specimen.  There i s a s l i g h t i n c r e a s e i n P  m  when t h e i n d e n t a t i o n s  a r e l a r g e compared t o t h e s i z e o f t h e specimens, p r o b a b l y due t o t h e 46 i n c r e a s e d confinement o f t h e d i s p l a c e d m a t e r i a l s b) deformation  t h e m a t e r i a l does n o t work-harden as a r e s u l t o f p l a s t i c i . e . t h e y i e l d s t r e n g t h Y does n o t i n c r e a s e . However, i f t h e m a t e r i a l work-hardens d u r i n g t h e course o f  deformation,  t h e e f f e c t i v e v a l u e o f Y may be c o n s i d e r a b l y h i g h e r than t h e  - 87 v a l u e of Y a t an e a r l i e r s t a g e of d e f o r m a t i o n .  -  A close approximation  can  be reached by u s i n g m e t a l s w h i c h have been so h i g h l y worked t h a t f u r t h e r d e f o r m a t i o n produces no a p p r e c i a b l e  change i n t h e i r  elastic  limit.  Many workers have c o n f i r m e d the v a l i d i t y of the P  m  = 3Y.  Tabor  47  has  equation  g i v e n r e s u l t s f o r a hard s t e e l sphere p r e s s i n g  a g a i n s t f l a t f a c e s of o t h e r m e t a l s (Table I I I ) . The mean p r e s s u r e w h i c h i s a l s o r e f e r r e d t o as the " y i e l d p r e s s u r e "  or the  The  been f i r s t i n t r o d u c e d enlarged  term " p r e s s u r e 48  by T r e s c a  m  "pressure  of f l u i d i t y " i s a l s o found t o be independent of the i n d e n t a t i o n , t h e r e f o r e the l o a d .  P,  and  of f l u i d i t y " seems to have  to d e s i g n a t e  the t r u e s t r e s s (on  the  a r e a ) n e c e s s a r y to i n d u c e a c e r t a i n c r i t i c a l c o n d i t i o n of 49  p l a s t i c i t y . Works a l o n g t h e s e l i n e s have a l s o been c a r r i e d out by Unwin 5o 51 52 Coe , Sachs , and by o t h e r s . O ' N e i l l and Greenwood summarize the r e s u l t s f o r v a r i o u s m e t a l s and  showed t h a t the c o n d i t i o n P  ,  = 3Y i s m  satisifed  (Table  IV).' 2.  E f f e c t of Work-Hardening  I n the s i m p l e model of a h a r d s p h e r i c a l s u r f a c e the s u r f a c e of a s o f t e r m e t a l P; = 3Y i s c o n s t a n t m does not work-harden.  The  o n l y i f the m a t e r i a l  y i e l d s t r e n g t h or the e l a s t i c l i m i t  i n c r e a s e , i f the m a t e r i a l work-hardens, on the f o r m a t i o n indentation.  penetrating  T h e o r e t i c a l c o n s i d e r a t i o n s and  experimental  show t h a t the e l a s t i c l i m i t w i l l not be c o n s t a n t s i n c e the amount of d e f o r m a t i o n or s t r a i n w i l l ,  of  will  the  measurements  around the  indentation  i n g e n e r a l , v a r y from p o i n t  47 to p o i n t  .  A d e t a i l e d experimetal  the e l a s t i c l i m i t Y  e  i n v e s t i g a t i o n by Tabor showed t h a t ,  a t the edge of the i n d e n t a t i o n can be used as  an  - 88 TABLE I I I Mean P r e s s u r e Vs E l a s t i c  Work-hardened metal  Y K g  -  . / m m  P  = L/rrd 2  Kg.../mm.2  Limit  c = P /Y m  4  Tellurium-Lead  2.1  6.1  2.9  Copper  31  88  2.8  Steel  65  190  2.8 47 ( A f t e r Tabor  )  i  TABLE IV  R e s u l t s Showing V a l i d i t y o f E q u a t i o n  2 Y kg/mm Ludwick53 K o r b e r ^ & others  Metal  3Y Mean  P^Y  Mean P r e s s u r e of Fluidity kg/mm 2  Lead  1.95  1.95  5.9  6.9  Tin  A.55  4.55  13.6  12.1  Cadmium  7.75  7.75  24  31  Aluminium  14.3  14  42  41  Zinc  11.9  11.9  36  38/71  Copper  33.5  32.8  33  99  108  42.5  42.3  42  126  132  66.5  75.4  71  213  237  Iron(Krupp Nickel  soft)  13.7  ( A f t e r O ' N e i l l and Greenwood  )  - 89 average o r r e p r e s e n t a t i v e v a l u e of the e l a s t i c l i m i t w h i c h i s r e l a t e d to the mean p r e s s u r e  by a r e l a t i o n P , = m  c'Y  e  where c' has a v a l u e l y i n g between 2.7  and  3.  I t can be seen t h a t the e l a s t i c l i m i t i s independent of s i z e of the i n d e n t a t i o n and  the a r e a over w h i c h p l a s t i c f l o w o c c u r s i s  d i r e c t l y p r o p o r t i o n a l t o the l o a d f o r f u l l y worked m e t a l s . be shown t h a t the a r e a of c o n t a c t 0 8 p r o p o r t i o n a l to L ' and  I t can  f o r a work-hardening m a t e r i a l i s  f o r p a r t i a l l y work-hardened e.g.,  b r a s s or m i l d s t e e l , the a r e a of c o n t a c t (Bowden and  the  extruded 0.92  i s p r o p o r t i o n a l to L  Tabor^). O ' N e i l l " ^ has o b t a i n e d  s i m i l a r r e s u l t s f o r two h e m i s p h e r e s  of the same m e t a l p r e s s i n g a g a i n s t one  B.  The  foregoing  another.  THEORY  considerations  grow to s t a b l e s i z e s , f i x e d by a c o n s t a n t  show t h a t the c o n t a c t s t r e s s , (P  = 3Y).  areas This  f a c t w i l l be u t i l i z e d h e r e t o extend Eq. (22)to a more g e n e r a l form thus c a l c u l a t e the c o n t r i b u t i o n of the p l a s t i c f l o w mechanism to densification  and  the  process. Assuming t h a t the c o n t a c t a r e a s grow t o s t a b l e s i z e s under a 31  f i x e d s t r e s s , Coble number and  has  obtained  the a p p l i e d s t r e s s .  a r e l a t i o n s h i p between a/R,  coordination  I n h i s paper, Coble assumed t h a t i f  -  a  P  90  -  = the f i x e d s t r e s s  Z  = the c o o r d i n a t i o n number  a  = the a p p l i e d s t r e s s (assumed to be  then the l o a d t r a n s f e r r e d to each p a r t i c l e i s g i v e n 2 ira  OpZ  hydrostatic)  by  and  the l o a d a p p l i e d t o each p a r t i c l e i s g i v e n  by  2 , a 4 He  TT R  equated t h e s e two  .  loads  getting  2  Op Z ira or  2 a_ _  4  R  Za  2  =  a  4TT  R  2  ,  (27) P  2 E q u a t i n g a pZira  2 to  Q4TT  R  i s based on the assumption t h a t  the  f o r c e s o p e r a t i n g on a volume element of a medium under h y d r o s t a t i c c o n d i t i o n s can be u n e q u a l but t h a t a must be the p r e s s u r e F u r t h e r m o r e , he b e l i e v e s t h a t the e r r o r i n t r o d u c e d c o n d i t i o n s i s p r o b a b l y w i t h i n a f a c t o r of  (Coble"^).  by assuming such  2. 2  I t can be seen t h a t the s u r f a c e a r e a of the sphere w i l l no l o n g e r be the same a f t e r some d e f o r m a t i o n has s u r f a c e a r e a of the sphere can be c a l c u l a t e d a t any f o r the  s p e c i f i c models d e s c r i b e d  i n Chapter I I .  (4TTR  taken p l a c e .  )  The  s t a g e of d e f o r m a t i o n I t can be shown t h a t  f o r s m a l l d e f o r m t i o n s (a/R £ 0 . 3 ) , the s u r f a c e a r e a of the sphere e s s e n t i a l l y remains The  constant.  constant  or the f i x e d s t r e s s Op (same as P^)  i n Eq. (27),  can be equated to t h r e e times the y i e l d s t r e n g t h of the m a t e r i a l , i f i t does not work-harden, or t h r e e times the r e p r e s e n t a t i v e v a l u e of the  yield  - 91 s t r e n g t h of the m a t e r i a l , i f i t does not work-harden, or t h r e e  times  the r e p r e s e n t a t i v e v a l u e of the y i e l d s t r e n g t h , i f the m a t e r i a l work-hardens. Substituting a  p  = 3Y i n Eq. a R  Introducing  2  2  =  (27)gives 4 3YZ  a  (28)  2 2 t h i s v a l u e of a /R i n t o Eq. (22) „ 3 ^ 4 " o 2 o 3YZ ° D  or  D = D  =  o  gives  D  + ^ °  o  (29)  where D = the b u l k d e n s i t y a t any D  deformation  = the i n i t i a l b u l k .density a t a/R  Q  = 0  Y = the r e p r e s e n t a t i v e v a l u e of the y i e l d  strength  of the m a t e r i a l Z = the c o o r d i n a t i o n number of the a = the a p p l i e d h y d r o s t a t i c  C.  particles.  stress.  EXPERIMENTAL VERIFICATION  E q u a t i o n s (28) and  (29) a r e based on the assumptions  1.  the a p p l i e d 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  2.  Z remains  constant  There i s c o n s i d e r a b l e  experimental  the h y d r o s t a t i c c o n d i t i o n s a r e not o b t a i n e d hot-pressing  process.  e v i d e n c e w h i c h shows t h a t  d u r i n g the  conventional  Under t h e s e c o n d i t i o n s , as e x p e c t e d , t h e r e  found t o be a l a r g e v a r i a t i o n i n the c o n t a c t . a r e a s  was  on each p a r t i c l e .  An average v a l u e o f the c o n t a c t a r e a s has been assumed t o be  representative  - 94 -  -95-  0  400  800 o  2 F i g . 44  (a/R)  v s a f o r K-Monel.  1200  1600  - 96 of the e x p e r i m e n t .  F i g u r e s 41-44  show the p l o t of (a/R)  l o a d ) f o r l e a d a t room t e m p e r a t u r e , 50° 800 , (D  —  900, D  Q  ) / D  According  and 1000°C. o  and  2  vs a (or a p p l i e d  100°C,and K-Monel a t  S i m i l a r l y , F i g s . 45 t o 47 show the p l o t of  vs a (or the a p p l i e d l o a d ) f o r the l e a d and K-Monel  t o Eqs.  (28) and  ( 2 9 ) , t h e s e r e l a t i o n s h i p s s h o u l d be  p r o v i d e d Z remains c o n s t a n t .  I t can be seen t h a t the  the i n i t i a l s t a g e s o f h o t - p r e s s i n g . the o b s e r v a t i o n s o f F e l t e n  25  , Shapiro  be due  linear  experimental  r e s u l t s g i v e a good f i t w i t h the t h e o r e t i c a l p r e d i c t i o n s . from the l i n e a r i t y a t low l o a d v a l u e s may  spheres.  The d e v i a t i o n  to the change of Z d u r i n g  Such a c o n c l u s i o n i s c o n s i s t e n t w i t h 28—30  , and D u f f i e l d and G r o o t e n h u i s  namely t h a t the c o n t r i b u t i o n from p a r i c l e r e o r i e n t a t i o n i s mostly d u r i n g  37  ,  the  i n i t i a l a p p l i c a t i o n of l o a d s . I t i s t h e o r e t i c a l l y p o s s i b l e t o c a l c u l a t e the y i e l d o f the m a t e r i a l from the m o d i f i e d d e n s i t y e q u a t i o n - Eq. Z remains c o n s t a n t .  However, e x p e r i m e n t a l l y i t was  strength  (29), provided  observed t h a t Z  changes c o n t i n u o u s l y w i t h the p r o g r e s s i v e d e f o r m a t i o n  during  hot-pressing  w h i c h makes i t d i f f i c u l t t o c a l c u l a t e the y i e l d s t r e n g t h from the d a t a . But the change i n Z, as e x p e r i m e n t a l l y o b s e r v e d , was 10%.  Under t h e s e c i r c u m s t a n c e s ,  o n l y about 8 t o  the v a l u e s of the y i e l d s t r e n g t h were  computed from the s l o p e s of the F i g s . 41 to 47 u s i n g the mean v a l u e of The  c a l c u l a t e d y i e l d s t r e n g t h v a l u e s f o r l e a d , K-Monel and  sapphire  Z.  are  g i v e n i n T a b l e V.  I t can be seen from T a b l e V t h a t the c a l c u l a t e d y i e l d v a l u e s from the  2 p l o t s (a/R)" vs a (or L) and D — D D  o  Q  strength  (or D) vs L (or a )  - 97 -  F i g . 47  D v s a For K-Monel.  TABLE V  Y i e l d S t r e n g t h V a l u e s from t h e S l o p e s o f P l o t s i n F i g s . 41 t o 47  Y i e l d Strength ( p s i ) Relationship  Lead (2%Sb) Room Temp.  (D-Do) u  K-Monel  Sapphire  50°C  100°C  800°C  900°C  1000°C  1570°C  1600°C  1700°C  3842  4070  2095  8800  5320  4245  16,570  15,520  9635  7210  7470  4010  14,280  10,090  6,910  35,600  35,420  22,210  vs L o  or D vs a  o o r L vs ( a / R )  2  - 101 d i f f e r n e a r l y by a f a c t o r o f 2 i n a l l c a s e s .  -  The v a l u e s o b t a i n e d  2 from the p l o t s o f a vs (a/R)  appear to be about two times the v a l u e 62  o f t h e y i e l d s t r e n g t h r e p o r t e d by Kronberg range.  I t i s b e l i e v e d t h a t Eq.  f o r sapphire i n t h i s  temperature  (27) i s a c c u r a t e p o s s i b l y w i t h i n a f a c t o r  o f 2 (as a l s o s t a t e d by C o b l e ^ ) ; t h e r e f o r e the c a l c u l a t i o n s , b a s e d on equation give larger y i e l d strength values.  this  That means the r e s u l t s o b t a i n e d  from the p l o t s (D — D ) VS L would have t o be c o r r e s p o n d i n g l y reduced.  But  q  Do i n the l a t e r c a s e , the v a l u e s o b t a i n e d s h o u l d be i n c r e a s e d by a f a c t o r o f about 1.5 i n (D — D ) / D q  because of the c o n t r i b u t i o n from p a r t i c l e rearrangement c o n t a i n e d o  values.  I t i s a l s o p o s s i b l e t h a t the d i f f e r e n c e between  the two r e s u l t s i s a f f e c t e d by i ) (D — D ) / D 2 Q  b u l k of the compact and i i ) (a/R) 20 o r 40 spheres  o  values which represent  the  v a l u e s w h i c h r e p r e s e n t the mean o f  only.  I n o r d e r to compare the y i e l d s t r e n g t h v a l u e o b t a i n e d from these c a l c u l a t i o n s f o r the l e a d spheres a t room temperature of the a c t u a l y i e l d s t r e n g t h , m i c r o h a r d n e s s lead spheres.  with that  measurements were made on  F o l l o w i n g Tabor's o b s e r v a t i o n s , the y i e l d s t r e n g t h of the  m a t e r i a l i s a p p r o x i m a t e l y 2.9 (same as M i c r o h a r d n e s s  to 3 times the V i c k e r ' s hardness number  number).  A l t h o u g h no c o n c l u s i o n s can be drawn  from t h i s a n a l y s i s w i t h o u t d o i n g an e x t e n s i v e s t u d y , t h e y i e l d s t r e n g t h v a l u e determined  from the m i c r o h a r d n e s s  t e s t s (about 6000 p s i a t 23°C) f o r  t h e . l e a d ( 2 % Sb) spheres l i e s between those v a l u e s c a l c u l a t e d from the e x p e r i m e n t a l p l o t s (Table V ) . I t i s i n t e r e s t i n g to compare Eq.  (29) w i t h the o t h e r e m p i r i c a l  and semi-feempirical r e l a t i o n s h i p s g i v e n i n the l i t e r a t u r e .  These a r e  - 102 summarized i n T a b l e V I .  Equations  -  ( i ) t o ( v i ) seem t o t a k e i n t o account  only  one mechanism of compaction i . e . p l a s t i c d e f o r m a t i o n and have been supported by o b s e r v a t i o n s on m e t a l powders.  Equation  ( V i i ) comprises  of  two p a r t s a)  F i l l i n g of l a r g e s i z e h o l e s by p a r t i c l e s l i d i n g  b)  F i l l i n g of h o l e s w h i c h a r e s m a l l e r than the p a r t i c l e s i z e  p r o b a b l y by p l a s t i c f l o w o r by  fragmentation.  A l l these e q u a t i o n s seem t o be a p p l i c a b l e f o r the e n t i r e range of d e n s i f i c a t i o n .  But t h e Eq.  (29) o b t a i n e d i n the p r e s e n t  investigation  i s a p p l i c a b l e f o r l i m i t e d range of d e n s i f i c a t i o n p r o c e s s i . e . e a r l y o f compaction;  and a d e v i a t i o n from l i n e a r i t y i s expected  s t a g e s of d e n s i f i c a t i o n w i t h the h i g h e r v a l u e s of a/R  ( >  stages  a t the l a t e r 0.3)  and a l s o  from o t h e r mechanism of d e n s i f i c a t i o n , such as s t r e s s enhanced d i f f u s i o n , w h i c h may  be r a t e d e t e r m i n i n g a t a l a t e r s t a g e .  I t i s p r e f e r r e d , however,  to l e a v e any f o r m a l comparison of the r e l a t i o n s h i p w i t h the o t h e r s g i v e n i n the l i t e r a t u r e , as the r e s u l t s h e r e i n a r e not s u f f i c i e n t l y d e t a i l e d to w a r r a n t  c r i t i c a l assessment of the d e n s i t y - p r e s s u r e r e l a t i o n s h i p .  - 103 TABLE V I  E q u a t i o n s R e l a t i n g D e n s i t y ( o r Volume)  of t h e Compact w i t h the A p p l i e d P r e s s u r e  No.  Equation  w =» k^ — k  (i) (ii)  fdw  (iii)  Pw  Reference  log P  2  2  = -kP lr  3  = constant  P « lc 1 n  (iv)  (v)  D-D  (vi)  D - D  0  0  1  1 E  ~ o 1 - D  - kP  6,8  D  1/3  - kP  7 58  2  V •» a^ exp — ( k ^ P )  (vii)  + a  2  D = D  (viii)  exp -  (k /P) 2  + kP  0  where w = the apparent p  volume  the a p p l i e d p r e s s u r e  D = the d e n s i t y at pressure P D  o = the d e n s i t y a t z e r o p r e s s u r e V = f r a c t i o n a l volume  ^1'^2'^ a  l' 2 a  constants = fractional  coefficients.  59  P r e s e n t work  - 104 V M E C H A N I S M SXNG'LE  OF  —  C R Y S T A L HOT  In  S A P P H I R E -  A.  D E F O R M A T I O N  OF  S P H E R E S B Y  P R E S S I N G INTRODUCTION  the p r e s e n t s t u d y i t has been observed t h a t permanent s t r a i n  can be produced d u r i n g the h o t - p r e s s i n g of s i n g l e c r y s t a l s a p p h i r e spheres i n the temperature range 1570°-1700°C.  The d e n s i f i c a t i o n p r o c e s s has been  i n t e r p r e t e d i n terms of the p a r t i c l e movements g i v i n g r i s e t o r e p a c k i n g , and the  p l a s t i c f l o w mechanism.  A l t h o u g h the c o n t r i b u t i o n from the  p a r t i c l e rearrangment c a n n o t , be q u a n t i t a t i v e l y d e t e r m i n e d , the o v e r a l l d e n s i f i c a t i o n b e h a v i o u r of the s a p p h i r e spheres i s v e r y s i m i l a r to t h a t of  the l e a d s p h e r e s , where b o t h the p a r t i c l e rearrangement  and the f l a t  f a c e f o r m a t i o n by p l a s t i c f l o w c o n t r i b u t e d to the d e n s i f i c a t i o n p r o c e s s . The c o n t r i b u t i o n from p l a s t i c f l o w i s c a l c u l a b l e from the proposed g e o m e t r i c models, knowing  t h e average c o o r d i n a t i o n number Z and the average  f a c e r a d i u s 'a'. 31 Coble  has s t u d i e d the f l a t f a c e f o r m a t i o n between two  c r y s t a l s a p p h i r e spheres (1 mm 1530°C.  diam) i n c o n t a c t under a s t a t i c l o a d , a t  He s p e c u l a t e d t h a t the d e f o r m a t i o n might be due to b o t h b a s a l and  n o n - b a s a l s l i p , t w i n n i n g o r some o t h e r complex mechanism. the  single  The n a t u r e of  d e f o r m a t i o n of the s a p p h i r e spheres has been i n v e s t i g a t e d  i n the p r e s e n t work by examining t h e deformed  qualitatively  spheres under o p t i c a l and  e l e c t r o n m i c r o s c o p e s and r e l a t i n g the s l i p l i n e s t o the c r y s t a l  orientation.  - 105 I n t h i s r e s p e c t , i t would be d e s i r a b l e t o make a s u r v e y of  t h e p r e s e n t day knowledge o f t h e d e f o r m a t i o n c h a r a c t e r i s t i c s o f s i n g l e  c r y s t a l sapphire. 60 McCandless and Y e n n i  were t h e f i r s t t o o b s e r v e b a s a l  slip  i n s i n g l e c r y s t a l s a p p h i r e r o d s deformed by bending around an a x i s normal to  t h e h e x a g o n a l a x i s a t 1300-1400°C.  Wachtman and M a x w e l l ^ s u b s e q u e n t l y  s t u d i e d t h e . t e n s i l e c r e e p o f s i n g l e c r y s t a l rods o f L i n d e s y n t h e t i c s a p p h i r e . They showed t h a t s a p p h i r e c o u l d be deformed p l a s t i c a l l y above 900°C.  The  o n s e t o f c r e e p near t h i s temperature was found t o be p r e c e d e d by a pronounced i n c u b a t i o n p e r i o d .  W i t h i n c r e a s i n g temperature and s t r e s s ,  however, t h e l e n g t h o f t h e d e l a y p e r i o d was found t o d e c r e a s e s h a r p l y . From t h e o r i e n t a t i o n s t u d y , Wachtman and M a x w e l l d e t e r m i n e d t h a t t h e s l i p system (0001) [1120] was o p e r a t i v e .  Kronberg  confirmed both t h i s  slip  p l a n e and t h e s l i p d i r e c t i o n i n t e n s i l e d e f o r m a t i o n o f s a p p h i r e between 1200° and 1700°C. the  He a l s o s t u d i e d t h e d e f o r m a t i o n as a f u n c t i o n o f -3 -1  s t r a i n r a t e between 10  t o 10  i n . / I n . / m i n . H i s work showed t h e  e x i s t e n c e o f a r e l a t i v e l y sharp t r a n s i t i o n between complete f r a c t u r e and m a s s i v e p l a s t i c f l o w , t h e s p e c i f i c t r a n s i t i o n being s e n s t i v i e to the s t r a i n r a t e . the  brittle temperature  F o r t h e range o f t h e r a t e s used,  t r a n s i t i o n temperature i n c r e a s e d from a p p r o x i m a t e l y 1270  t o 1520^0.  He a l s o found t h a t w i t h o n l y a few degrees d i f f e r e n c e i n t h e t r a n s i t i o n t e m p e r a t u r e , i t was p o s s i b l e t o o b t a i n e i t h e r no p l a s t i c f l o w on t h e low temperature s i d e o r a . r e l a t i v e l y l a r g e amount o f e x t e n t i o n (about 100%) on the  h i g h temperature s i d e .  d r o p s . i . e . t h e subsequent the  He a l s o observed pronounced y i e l d - p o i n t f l o w s t r e s s e s were found t o be approxmately  s t r e s s e s r e q u i r e d to i n i t i a t e macroscopic  flow.  half  - 106 Kronberg  63  had  e a r l i e r presented  i n the form of a model the  j u s t i f i c a t i o n f o r the e x p e r i m e n t a l l y observed elements of the system (0001) [1120] and  -  slip  t w i n n i n g i n the s a p p h i r e s i n g l e c r y s t a l s .  i n t e r e s t i n g t o n o t e t h a t [1120] d i r e c t i o n i n s a p p h i r e i s a t 30°  It is  to the  row  o f c l o s e s t packed oxygen i o n s as d i f f e r e n t from [1120] d i r e c t i o n i n z i n c p a r a l l e l t o the c l o s e s t packed row of atoms.  F i g u r e 48 shows the  two  cases f o r a comparative study. 64 Nye  has a l s o observed b a s a l s l i p i n corundum c r y s t a l s bent i n t o  the form o f a U-shaped l o o p .  The P r i s m a t i c s l i p system  (1120)  [lOlO]  65 has been observed by K l a s s e n - N e k l y u n d o v a on corundum c r y s t a l s above 2000°C.  The  during deformation r e l a t i o n between the  studies two  s l i p systems o b s e r v e d i n corundum i s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g . 49. S c h e u p l e i n and G i b b s ^ ^ have s t u d i e d the b a s a l and p r i s m a t i c d i s l o c a t i o n s u s i n g an e t c h p i t t e c h n i q u e .  They found t h a t b a s a l s l i p c o u l d  more r e a d i l y than p r i s m a t i c .  The  l a t t e r was  found to be induced more  r e a d i l y a t t e m p e r a t u r e s near 2000°G i n c r y s t a l s o r i e n t e d 67 for  basal glide.  I n t h e i r l a t e r paper  unfavourably  , they showed t h a t b o t h  d i s l o c a t i o n s c o u l d be a c t i v a t e d c o n c u r r e n t l y and between them c o u l d  occur  that important i n t e r a c t i o n s  occur.  68 Lommel and K r o n b e r g observed d i s l o c a t i o n s i n s a p p h i r e and ruby s i n g l e c r y s t a l s by means of a h i g h r e s o l u t i o n B e r g - B a r r e t t x-ray technique.  Their thermal  e t c h i n g d a t a were found to be i n agreement 69  w i t h the d a t a o f S c h e u p l e i n and G i b b s . c h e m i c a l p o l i s h i n g and  Stephens and A l f o r d  etching techniques  used  to r e v e a l the d i s l o c a t i o n  s t r u c u t r e s of s a p p h i r e and ruby , c r y s t a l s grown by the f l a m e - f u s i o n They o b s e r v e d b o t h the b a s a l s l i p and  the  techniques.  the p r i s m a t i c s l i p around 1800°C.  PRISMATIC SLIP  <IOlO>  (•) TOad efnles rcpraeant sine stoma to an umWlylug «ael plan* and open abelee leproaenl the atoma in the piano jovo. The vector describee the smallest diaptooamont that •eh atom to a given plane moat undergo to order to rotors • normal atruntme. I t ia equal to A r  F i g . 48  _  I  BASAL SLIP  <K>af»  (b) t^rajB nusd circles lepaeaaot odds ion* to an oaatartjssg baaal peine and large open eawaaa n y i m i l l oridstanatothe plane above, rkneil fUW sod open <en^dodsji^sl™JBeen tone and boles, respectively, lvfag on the nwdtom plane between the oxide abaeta. The voter baa the —ree oigssetoonae ee for part (a) of the figure.  R e p r e s e n t a t i o n o f [11?0] D i r e c t i o n i n Case of Z i n c and S a p p h i r e Crystals (After Kronberg^).  49  Schematic R e p r e s e n t a t i o n o f B a s a l and P r i s m a t i c S l i p Systems i n ^ S a p p h i r e ( A f t e r S c h e u p l e i n and Gibbs . ) .  - 108 A l s o b a s a l t w i n n i n g was  B.  -  observed i n a l l f l u x grown c r y s t a l s .  EXPERIMENTAL TECHNIQUES, RESULTS, AND 1.  DISCUSSION  O p t i c a l Microscopy  The c o n t a c t f a c e s formed on d e f o r m a t i o n of the s i n g l e  crystal  s a p p h i r e spheres t o g e t h e r w i t h the undeformed s u r f a c e s o f the deformed spheres were examined f o r any s l i p l i n e s , d e f o r m a t i o n o r f r a c t u r e I t was noted t h a t t h e d e f o r m a t i o n markings observed on a l l the f a c e s . more than o n e . s l i p system  markings.  shown i n F i g . 50 were not  The s e t s of markings  suggest the presence of  under the p r e s e n t d e f o r m a t i o n c o n d i t i o n s .  F i g u r e s 50(c) and 50(d) shows t h a t e x t e n s i v e s l i p has p l a c e i n the f r e e s u r f a c e near the deformed f a c e .  taken  The two s e t s of  slip  l i n e s shown i n F i g . 50(c) were i d e n t i f i e d from the o r i e n t a t i o n of t h e c r y s t a l determined by the b a c k - r e f l e c t i o n Laue t e c h n i q u e u s i n g the s t a n d a r d p r o j e c t i o n f o r the corundum c r y s t a l ^ .  S i m i l a r technqiues  h e l p e d i n a n a l y z i n g one or b o t h of t h e s e s l i p p l a n e s r e s p o n s i b l e f o r d e f o r m a t i o n of s a p p h i r e . F i g u r e 50(e) shows t h a t inhomogeneous d e f o r m a t i o n has p l a c e ( l e f t of the f l a t f a c e ) .  taken  The c i r c u l a r marking on the r i g h t of t h e  c o n t a c t f a c e seems t o be p r e s e n t i n the undeformed spheres produced  by  grinding. 2. I t was  E l e c t r o n Microscopy  found t h a t the o p t i c a l m i c r o s c o p y h e l p e d v e r y  i n s t u d y i n g the d e t a i l s of the f l a t f a c e formed.  little  For t h i s r e a s o n , i t was  d e c i d e d t o s t u d y the d e f o r m a t i o n c h a r a c t e r i s t i c s of t h e s i n g l e  crystal  - 109 -  - 110 -  (d)  X400  (Cont.)  - Ill -  (e) F i g . 50  X600  Photomicrographs of the T y p i c a l C o n t a c t Faces.  - 112 spheres u s i n g r e p l i c a t e c h n i q u e s .  Because o f t h e v e r y s m a l l s i z e o f t h e  s u r f a c e t o be s t u d i e d , r e p l i c a s were made w i t h about one t h i r d t o one h a l f o f t h e s u r f a c e o f t h e sphere making an i m p r e s s i o n acetate f i l m .  i n the c e l l u l o s e  The r e p l i c a was expected t o c o n t a i n a t l e a s t one f l a t  face. F i g u r e s 51 t o 53 p r o v i d e enough e v i d e n c e f o r t h e presence of t h e p l a s t i c f l o w mode.of d e f o r m a t i o n s a p p h i r e s i n g l e c r y s t a l spheres.  during hot-pressing of the  F i g u r e 51(a)  shows t h e p r e s e n c e o f  s l i p l i n e s p o s s i b l y representing the b a s a l s l i p . i n F i g s . 51(b)  and 51(c) s t r o n g l y s u p p o r t s  s l i p has taken  place.  P r e s e n c e o f wavy s l i p  t h e c o n c l u s i o n t h a t complex  F i g u r e s 52 and 53 show a stepped s t r u c t u r e s i m i l a r t o one 71 observed i n a - b r a s s  72 and copper  crystals.  Following the i n t e r p r e t a t i o n  used f o r t h e s e m a t e r i a l s , c r o s s s l i p i s thought t o be t h e o p e r a t i n g 67 mechanism.  S c h e u p l e i n and Gibbs  have advanced a mechanism f o r c r o s s 63  s l i p i n corundum based on t h e e a r l i e r model o f K r o n b e r g i n t e r a c t i o n o f b a s a l and p r i s m a t i c s l i p s .  d e s c r i b i n g the  They p o s t u l a t e t h a t t h e screw  segments o f p r i s m a t i c d i s l o c a t i o n s can decompose i n t o b a s a l d i s l o c a t i o n s . Further decomposition  o f b a s a l d i s l o c a t i o n i n t o q u a r t e r p a r t i a l s and s t a c k i n g  f a u l t s r e s t r i c t s the motion of p r i s m a t i c d i s l o c a t i o n e n t e r i n g the b a s a l 3. F r a c t u r e d S u r f a c e s At.some p l a c e s on t h e f r e e s u r f a c e o f t h e s a p p h i r e c h i p s were seen t o be f l a k e d - o f f . s u r f a c e on a s a p p h i r e sphere,.. place.  sphere,  F i g u r e s 54 shows such a c h i p p e d - o f f  The m a r k i n g s show t h a t f r a c t u r e has taken  No attempt was made t o study t h e n a t u r e o r t h e c h a r a c t e r i s t i c s  of t h e f r a c t u r e s u r f a c e any f u r t h e r .  plane.  - 113 -  (b) X5000  (Cont.)  - 114  (c)  F i g . 51  -  X 10,000  E l e c t r o n m i c r o g r a p h s Showing E v i d e n c e of Simple Wavy S l i p .  and  - 115 -  - 116 -  F i g . 53  Cross S l i p i n Sapphire a t 1700°C.  X5000  - 117  F i g . 54  F r a c t u r e d S u r f a c e on the S a p p h i r e  -  Sphere.  X400  VI  - 118 -  D I S C U S S I O N  The t h e o r e t i c a l models have been d e r i v e d on the assumption t h a t the p r e s s u r e  i s a p p l i e d e q u a l l y from a l l d i r e c t i o n s .  Though i t  i s w e l l e s t a b l i s h e d t h a t such an assumption i s h a r d l y met i n p r a c t i c e during hot-pressing, problem.  i t seems t o s i m p l i f y t h e q u a n t i t a t i v e a n a l y s i s o f t h e  Thus b e f o r e any d i s c u s s i o n o f t h e p r e s e n t r e s u l t s i s a t t e m p t e d ,  i t i s d e s i r a b l e t o see how f a r t h i s a s s u m p t i o n i s v a l i d under t h e p r e s e n t experimental  conditions. A.  PRESSURE DISTRIBUTION IN THE DIE  I t i s a w e l l known f a c t t h a t t h e d i s t r i b u t i o n o f d e n s i t y w i t h i n cold-pressed  powder compacts i s i n f l u e n c e d by the t y p e o f powder,  shape o f t h e d i e , l u b r i c a t i o n d u r i n g  the p r o c e s s ,  e t c . The a n a l y s i s o f  the d e n s i t y and t h e s t r a i n d i s t r i b u t i o n i n t h e compact i s v e r y h e l p f u l i n o b t a i n i n g i n f o r m a t i o n about t h e s t r e s s d i s t r i b u t i o n w i t h i n a powder compact. 73  Kamm, S t e i n b e r g  and W u l f f  used a d e f o r m a b l e g r i d w i t h i n  a powder compact t o s t u d y the d e n s i t y d i s t r i b u t i o n i n the compact. From t h e i r measurements, they found t h a t t h e d e n s e s t p a r t o f a compact p r e s s e d from one s i d e i s a t t h e top o u t e r c i r c u m f e r e n c e and t h e l e a s t dense r e g i o n a t t h e bottom c i r c u m f e r e n c e near the s t a t i o n a r y p l u n g e r . density gradients are a t t r i b u t e d to the d i e - w a l l f r i c t i o n .  The  They found  t h a t as the r a t i o o f the d i a m e t e r t o t h e h e i g h t o f the compact i s i n c r e a s e d the d i e - w a l l s p l a y a c o r r e s p o n d i n g l y  smaller  role.  74  Duwez and Z w e l l based on d e n s i t y measurements.  have a l s o s t u d i e d t h e p r e s s u r e They r e p r e s e n t  distribution  the d i s t r i b u t i o n o f  pressure  - 119 w i t h i n t h e compact by t r a c i n g l i n e s o f e q u a l p r e s s u r e i n a l o n g i t u d i n a l direction.  F i g u r e 55 shows t h e approximate  l o c a t i o n of the l i n e s of  e q u a l p r e s s u r e i n a copper compact o f t h i c k n e s s t o d i a m e t e r r a t i o o f 0.42.  The v a r i a t i o n o f d i e - w a l l p r e s s u r e w i t h h e i g h t observed by them  i s s u b s t a n t i a t e d by t h e r e s u l t s o f many o t h e r w o r k e r s .  R e s u l t s on t h e  s t u d i e s o f t h e p r e s s u r e d i s t r i b u t i o n i n t h e d i e on t h e s e l i n e s l e a d t o a most i m p o r t a n t p r a c t i c a l consequence o f t h e d i e - w a l l f r a i c t i o n i . e . u n i f o r m d i s t r i b u t i o n s o f d e n s i t y w i t h i n a powder compact a r e t h e f u n c t i o n of t h e h e i g h t o f t h e compact.  F i g - 55  Approximate L i n e s o f E q u a l P r e s s u r e i n Copper Compact. P r e s s u r e 100,000 p s i ( A f t e r Duwez and.Zwell ). 7 4  - 120 The unequal d i s t r i b u t i o n o f p r e s s u r e t h r o u g h o u t . t h e  compact  a r i s e s from t h e w a l l f r i c t i o n a l f o r c e w h i c h opposes t r a n s m i s s i o n o f the a p p l i e d p r e s s u r e i n i t s neighbourhood.  Such p r e s s u r e v a r i a t i o n s a t  l o c a l r e g i o n s w i t h i n t h e compact can l e a d t o r e l a t i v e s l i d i n g o f p a r t i c l e s . Thus c o n s i d e r a b l e movements o f p a r t i c l e s can be expected near t h e w a l l s o f t h e d i e .  to take place  T h i s r e l a t i v e movement, p a r t i c u l a r l y a t h i g h  p r e s s u r e s , can cause d i s t o r t i o n o f t h e p a r t i c l e s , produce  fragmentation  by s h e a r i n g f o r c e s i n these a r e a s , and may cause d i s l o d g e m e n t o f s u r f a c e f i l m s i n metals.  I t can be concluded  from t h e works o f s e v e r a l i n v e s t i g a t o r s  t h a t t h e maximum u n i f o r m d i s t r i b u t i o n o f p r e s s u r e w i t h i n a powder compact can be o b t a i n e d by f i x i n g t h e r a t i o o f t h e t h i c k n e s s t o t h e diameter was  o f t h e d i e t o about 0.5.  :  I n t h e p r e s e n t work t h i s  ratio  r e s p e c t i v e l y , 0.25 f o r l e a d , 0.40 f o r K-Monel and 0.30 f o r s a p p h i r e . I t can be seen t h a t above c o n c l u s i o n s , a r e based on measurements  of t h e d e n s i t y o f v e r y s m a l l p o r t i o n s o f t h e compact.  These may n o t  n e c e s s a r i l y a p p l y t o t h e d i s t r i b u t i o n o f p r e s s u r e a t each c o n t a c t p o i n t o f t h e powder p a r t i c l e .  I t i s observed  i n t h e p r e s e n t work t h a t t h e r e  i s a l a r g e v a r i a t i o n i n c o n t a c t f a c e r a d i i f o r each p a r t i c l e . deforming  By  l e a d spheres t o l a r g e r . v a l u e s o f s t r a i n i t was observed  that  the the deformed f a c e s t h a t were normal (or w i t h i n 30°) t o t h e d i r e c t i o n of p r e s s i n g were about 2.2 times l a r g e r than those to t h e d i r e c t i o n o f ^ p r e s s i n g .  parallel  Since the deformation  t a k e s p l a c e under a  f i x e d s t r e s s (as d i s c u s s e d i n Chapter I V ) , t h e v a r i a t i o n i n 'a' v a l u e s suggests  t h a t t h e f o r c e s a c t i n g a t each c o n t a c t p o i n t a r e u n e q u a l c o n t r a r y  t o t h e a s s u m p t i o n made i n t h e t h e o r e t i c a l models.  This observation w i l l  - 121 be used t o m o d i f y t h e t h e o r e t i c a l models. B.  CORRECTED THEORETICAL MODELS BASED ON EXPERIMENTAL RESULTS E x p e r i m e n t a l l y i t was observed &2  2.2 a^  =  that  (approximately)  where &2 ~ r a d i u s o f t h e f a c e s normal t o t h e d i r e c t i o n o f p r e s s i n g a^ = r a d i u s o f t h e f a c e s p a r a l l e l t o t h e d i r e c t i o n o f pressing. U s i n g t h i s o b s e r v a t i o n , t h e o r e t i c a l models have been m o d i f i e d t o t a k e i n t o account t h e a l g e b r i c mean r a d i u s o f t h e c o n t a c t f a c e s . 1.  C o r r e c t e d Cubic Model  I n t h i s case t h e r e w i l l be 4 a^'s and 2 the average o r t h e mean r a d i u s a a Substituting  a  2  m m  Therefore,  w i l l be g i v e n by  m  = (4a-, + 2a.)/6 J-  A  = 2.2 a^, we g e t  or  s.  = 1.4 a^,  a, = 0.71 a 1 m  and  a  0  =1.57 a  2  m  F u r t h e r m o r e , t h e c o n s t a n t volume e q u a t i o n can be r e w r i t t e n as 4  3  i l 3  TTR  ( 2 R  3  2  —  +  4Trh-,  2 (3R -  2  2 _  1  1  ,  N  —  1/2 2 )  +  2 L-  .  2-rrho  £r  ( 2 R  3 3  =1  2  2  +  a  2  }  ( R  2 _ 2  8TTR  -  (3R — h )  ,  (e.g.) N  1/2 &  2  )  2  . 1  (30)  - 122 S u b s t i t u t i n g t h e v a l u e s o f a l and a„ i n terms o f a., t h e l Z m r a d i u s o f t h e p a r t i c l e a t any d e f o r m a t i o n can be o b t a i n e d from Eq. (30) f o r t h e d i f f e r e n t v a l u e s o f a^. rectangular parallelepiped.  The u n i t c e l l i n t h i s case w i l l be a  The volume o f t h e u n i t c e l l w i l l be g i v e n  by  c '  V  ^ 2  8  2  2  = 8 (R Z  a  ±  ) or 2  1/2 2  - a  )  2  and t h e b u l k d e n s i t y w i l l be g i v e n by D = 1/V . 100% c T a b l e V I I summarises t h e v a l u e s o f R, V , D, and (D — D ) c o f o r t h e v a l u e s o f a up t o 0.3R. I t can be c a l c u l a t e d from t h e d a t a i n m Table V I I that r  2 D-D  = 1.78 D  o  2.  (a /R) o m  (31)  C o r r e c t e d Hexagonal P r i s m a t i c Wockl  I n t h i s case t h e r e w i l l be 6a^'s and 2a2's.  Therefore, the  average o r t h e mean r a d i u s w i l l be g i v e n by a Substituting a  2  - (6a.. + 2 a ) / 8 l  m  0  = 2.2 a-p we g e t a ^ - 1.3 a ^  or  a, = 0.77 a 1 m a„ = 1.69 a 2 m  and  The c o n s t a n t volume e q u a t i o n , i n t h i s c a s e , becomes 4_TT  3  or  2TT(2R  2  +  a  2 x  )  R _  6TTh^  3  ( W  3 (R  2  -  a  ±  3  R  _  "  h  i  _  )  2TTh  "1'  3  1/2 )  +  j  1  (2R  2  +  2 2  (  3  W  "  R  _  )  h  =  (  e  .  g  <  )  "2  2 (R  2  -  a ) 2  -  4TTR  3  =  1  - 123  -  S u b s t i t u t i n g the v a l u e s of a-, and a_ i n terms o f a R the 1 2 m ? 6  r a d i u s of the p a r t i c l e a t any d e f o r m a t i o n can be c a l c u l a t e d from Eq. for  d i f f e r e n t v a l u e s of' a . m  The u n i t c e l l i n t h i s case w i l l be  a hexagonal prism w i t h d i f f e r e n t h e i g h t .  (32)  again  The volume of t h e u n i t  cell  w i l l be g i v e n by v  c  =  4 / 3  "yi y  1/2  2  = 4/3  (R  2  2  - _)  2  (R  a]  - a  2  )  2  and t h e b u l k d e n s i t y w i l l be g i v e n by = 1/V  D  . 100%  c  T a b l e V I I summarizes t h e v a l u e s o f R, \#C, for  the v a l u e s o f a D - D  m  up to 0.3R. = 2.03  o  D  o  D,  and  (D — D Q )  The f o l l o w i n g m o d i f i e d d e n s i t y e q u a t i o n  (a/R) m  (33)  2  can a l s o be c a l c u l a t e d from the d a t a g i v e n i n T a b l e V I I .  C o r r e c t e d D-o  3. Equation of a's.  Equation  (28) w i l l a l s o be a f f e c t e d because of the two  I n t h i s case we can w r i t e the new  values  e q u a l i t y e q u a t i o n f o r the  loads as: a)  C o r r e c t e d Cubic Model 3 Y  (4Tia  2 1  +  2Tra  2  2  ) =  2  4TT  R  o r i n terms o f a , t h i s can be s i m p l i f i e d t o m 2 a  m  a  5.21 b)  (34)  a  Y  ~  0.87  Z Y  C o r r e c t e d Hexagonal P r i s m a t i c Model 3  Y  (eva^  +  2  Tra  2  2  )  =  4TT  R  2  two  124  o r i n terms o f  -  t h i s c a n be s i m p l i f i e d t o 2  -ED  2  4.  —2 6.96  =  Y  =  0.87  2  ( Z Y  35) '  v  Comparison w i t h t h e R e s u l t s  These m o d i f i e d e q u a t i o n s f o r t h e c u b i c and t h e h e x a g o n a l p r i s m a t i c models were used t o r e p l o t t h e d a t a g i v e n i n F i g s . 2 0 t o 2 2 and a r e shown i n F i g s . 5 6 and 5 7 . shown f o r comparison.  The o r i g i n a l t h e o r e t i c a l p l o t s a r e a l s o  I t can be seen t h a t t h e m o d i f i e d  theoretical  c u r v e f o r t h e h e x a g o n a l p r i s m a t i c model g i v e s a b e t t e r f i t w i t h t h e experimental data. constant 1.5 D 2.03 D  q  q  I n a d d i t i o n , i t can be seen t h a t t h e v a l u e o f t h e  i n t h e Eq. ( 2 2 ) changes t o 1 . 7 8 D  f o r t h e h e x a g o n a l p r i s m a t i c models.  q  f o r t h e c u b i c and  A comparison o f these  c o r r e c t e d t h e o r e t i c a l slopes w i t h the experimental slopes of the curves 2  2  ( D — D ) / D v s (a/R) o r D v s (a/R) shows t h a t t h e c o r r e c t e d v a l u e s o o o f t h e t h e o r e t i c a l s l o p e s a r e much c l o s e r t o t h e e x p e r i m e n t a l v a l u e s . T a b l e V I I I summarizes t h e s e r e s u l t s . A l s o t h e Eq. ( 3 3 ) changes t o D  -  D  + l^ - 3  Q  3  D  (for Z =  o  8)  (3>6)  and t h i s c o r r e c t e d e q u a t i o n w i l l g i v e d i f f e r e n t v a l u e s o f t h e p r e d i c t e d y i e l d s t r e n g t h (about 1 6 % l a r g e r ) . I t can be concluded  from t h e good agreement between t h e  c o r r e c t e d t h e o r e t i c a l p l o t s and t h e e x p e r i m e n t a l r e s u l t s t h a t t h e c o n t r i b u t i o n t o t h e d e n s i f i c a t i o n may be c h i e f l y from t h e p l a s t i c f l o w mechanism i n the l a t e r stages of the experimental o b s e r v a t i o n s .  As t h e maximum  initial  p a c k i n g d e n s i t y o f t h e spheres was o b t a i n e d b e f o r e any a p p l i c a t i o n o f t h e  -.125  F i g . 56  -  D v s a/R f o r Lead (Comparison w i t h t h e C o r r e c t e d T h e o r e t i c a l -Models).  - 126 -  F i g . 57  D vs a/R f o r K-Monel (Comparison w i t h the C o r r e c t e d T h e o r e t i c a l Models).  - 127 TABLE V I I C o r r e c t e d C u b i c and Hexagonal P r i s m a t i c Models and T h e i r Characterisitcs  a/R m  Model  V  (arb. U n i t s )  D-D (%) '  Corrected  0.1  0.6203  7.843R  53.30  0.94  Cubic  0.2  0.6213  7.365R  56.02  3.66  Model  0.3  0.6248  6.556R  60.81  8.45  Corrected  0.1  0.6204  6.787R  61.70  1.24  Hex.. P r i s m .  0.2  0.6219  6.363R  56.36  4.90  Model  0.3  0.6267  5.654R  71.66  11.20  TABLE V I I I  T h e o r e t i c a l and E x p e r i m e n t a l S l o p e s o f P l o t s ~  ( D  D  Material  Temp  D  o  )  vs (a/R)  Room Temp. 50 100  800 900 1000  2  D v s (a/R)'  (D ~ D ) v s (a/R)' 0  3/2 modified to 2.03 f o r  K-Monel  and D v s ( a / R )  o  Theoretical  Lead  2  Hex. Model  Prism.  Experimental  2.75,2.20 2.57 4.8,2.3  Theoretical  3 D/ modified 0  to  2  Experimental 2.15D , 2.69D 2.92D 3.68D° 2.28D D  a  2.03D 2.72D o 2.73D£ for Hex. P r i s m . 1.99D Model n  Q  - 128 l o a d t h e c o n t r i b u t i o n from p a r t i c l e rearrangement was expected  t o be  s m a l l d u r i n g t h e l a t e r s t a g e s o f t h e experiment;.  Ci.  HOT-PRESSING-MECHANISMS OF DENSIFICATION  1.  General  Many t h e o r i e s have been put f o r w a r d i n r e c e n t y e a r s t o d e s c r i b e t h e d e n s i f i c a t i o n mechanism d u r i n g h o t - p r e s s i n g p r o c e s s . Most o f t h e s e t h e o r i e s a r e s u p p o r t e d by works on ceramic o x i d e s . p r o c e s s as a whole has been suggested a complex mechanism.  The d e n s i f i c a t i o n  by t h e r e c e n t i n v e s t i g a t o r s t o be  I t has been shown t h a t t h e e n t i r e p r o c e s s o f  d e n s i f i c a t i o n cannot.-, be r e p r e s e n t e d by a s i n g l e e q u a t i o n .  While  f r a g m e n t a t i o n and p a r t i c l e rearrangement a r e c o n s i d e r e d t o be r e s p o n s i b l e for  the d e n s i f i c a t i o n i n the i n i t i a l stages  s t a g e s o f h o t - p r e s s i n g have been concluded  X6 25 30 33 ' ' ; the f i n a l t o be c o n t r o l l e d by t h e s t r e s s  31-33 enhanced d i f f u s i o n a l creep  .  The c o n t r i b u t i o n s from t h e d i f f e r e n t  mechanisms a t d i f f e r e n t s t a g e s o f d e n s i f i c a t i o n have been proposed ( h y p o t h e t i c a l l y ) by S p r i g g s and A t t e r a a s ^ (shown i n F i g . 5 8 ) . A c c o r d i n g 7  to  them, t h e predominant mechanisms o f d e n s i f i c a t i o n d u r i n g t h e v e r y  e a r l y s t a g e s o f t h e h o t - p r e s s i n g p r o c e s s a r e p a r t i c l e rearrangement and p l a s t i c flow.  And t h e maximum c o n t r i b u t i o n from t h e s e two mechanisms  o c c u r s w i t h i n t h e f i r s t 30 seconds o f t h e h o t - p r e s s i n g p r o c e s s . O b v i o u s l y , t h e s e maxima o r peak c o n t r i b u t i o n s e i t h e r from t h e p a r t i c l e rearrangement o r from t h e p l a s t i c f l o w w i l l v a r y w i t h d i f f e r e n t m a t e r i a l s and t h e i n i t i a l p a c k i n g c o n f i g u r a t i o n . Whether t h e d e n s i f i c a t i o n  will  t a k e p l a c e by f r a g m e n t a t i o n , p a r t i c l e rearrangement, p l a s t i c f l o w , o r stress-enhanced  d i f f u s i o n a l creep w i l l depend upon t h e t y p e o f m a t e r i a l  - 129 -  TIME F i g . 58  [MIN]  Densif i c a t i o n Rate as A Function of Time C a l c u l a t e d From the D e n s i f i c a t i o n of MgO shown a t the Upper Right Hand Corner. ( A f t e r Spriggs and A t t e r a a s ) . 7 5  - 130 the t e m p e r a t u r e and t h e s t r e s s l e v e l used d u r i n g t h e h o t - p r e s s i n g experiments.  M e t a l compacts w i l l have.a much g r e a t e r c o n t r i b u t i o n  from p l a s t i c f l o w than ceramic o x i d e s .  A l s o p l a s t i c f l o w i s enhanced by  h i g h t e m p e r a t u r e s and h i g h s t r e s s e s . I t i s e v i d e n t from t h e above d i s c u s s i o n t h a t 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 i s a n e t r e s u l t o f s e v e r a l mechanisms.  The c o n -  t r i b u t i o n from one mechanism a l o n e , i . e . p a r t i c l e rearrangement, p l a s t i c f l o w o r stress-enhanced  d i f f u s i o n a l creep i s thus n o t e a s i l y measured.  One o f the e a s i e s t methods i s t o c o n s i d e r models based on some s i m p l e and ' systematic packings of one-sized s p h e r i c a l p a r t i c l e s .  The p r e s e n t work  e s s e n t i a l l y d e s c r i b e s t h e d e n s i f i c a t i o n o f such compacts by t h e p l a s t i c f l o w mechanism a l o n e .  Because o f t h e d i f f i c u l t y  i n s t a r t i n g t h e compaction  e x p e r i m e n t s w i t h one o f t h e i d e a l p a c k i n g s , p a r t i c l e rearrangement i s a l s o a c t i v e a l t h o u g h t h e p a r t i c l e rearrangement was kept t o a minimum by s h a k i n g and t a p p i n g t h e spheres as they were packed i n the d i e , t h e c o n t r i b u t i o n from p a r t i c l e rearrangement w i t h i n c r e a s i n g def o r m a t i o n has been w e l l demonstrated i n t h e p r e s e n t work.  The good agreement  between t h e t h e o r e t i c a l p r e d i c t i o n s (based- on i d e a l p a c k i n g ) and t h e e x p e r i m e n t a l r e s u l t s (random p a c k i n g )  leads to the c o n c l u s i o n that the  d e n s i f i c a t i o n i s t a k i n g p l a c e c h i e f l y by p l a s t i c f l o w i n t h e e a r l y stages of the h o t - p r e s s i n g process.  I n t h e i n t e r m e d i a t e and f i n a l  stages, c a l c u l a t i o n s of the c o n t r i b u t i o n of p l a s t i c flow to the d e n s i f i c a t i o n p r o c e s s a r e c o m p l i c a t e d by t h e s i m u l t a n e o u s  occurrence  of  stress-enhanced  d i f f u s i o n a l c r e e p ; t h e r e f o r e , was n o t attempted i n t h e p r e s e n t work.  - 131 2.  H o t - P r e s s i n g of Ceramic O x i d e s —  C o n t r i b u t i o n from P l a s t i c  von M i s e s ^ and T a y l o r ^ have shown t h a t f i v e  Flow  independent  s l i p systems a r e n e c e s s a r y f o r p l a s t i c f l o w w i t h i n p o l y c r y s t a l l i n e m a t e r i a l s , von M i s e s ' s c r i t e r i o n a l s o assumes s l i p t o be homogeneous and c a p a b l e of the  o c c u r r i n g on a l l the p o s s i b l e s l i p systems i n any g i v e n volume of material.  However, i n m a t e r i a l s where a v a i l a b l e independent  slip  systems a r e l i m i t e d , d e f o r m a t i o n by t w i n n i n g , bending and k i n k i n g can reduce t h e number o f d i s t i n c t systems r e q u i r e d . In case of c e r a m i c o x i d e m a t e r i a l s l i k e MgO, independent s l i p systems  {110}  <110>  are  only  two  a v a i l a b l e a t low  78 temperatures  .  But a t t e m p e r a t u r e s above 600°C, i t has been shown  79 by H u l s e , C o p l e y , and Pask {100>:  <110>  t h a t t h r e e o t h e r s l i p systems of the form  b ecome o p e r a t i v e i n c o m p r e s s i o n .  Though the  a v a i l a b i l i t y o f t h e s e f i v e independent s l i p systems s a t i s f i e s t h e c r i t e r i o n for  p l a s t i c f l o w , s l i p i n s i n g l e c r y s t a l s o f MgO  has been found t o be 80 inhomogeneous up t o about 1700°C ( i n t e n s i o n e x p e r i m e n t s ) . In s a p p h i r e s i n g l e c r y s t a l s , o n l y b a s a l s l i p has been observed  by o t h e r i n v e s t i g a t o r s up t o a temperature of 1800°C.  E v i d e n c e f o r the  p r e s e n c e of b o t h b a s a l s l i p and p r i s m a t i c s l i p and p o s s i b l y t w i n n i n g has been demonstrated i n the p r e s e n t i n v e s t i g a t i o n . I t i s i m p o r t a n t t o keep i n mind the c h i e f d i f f e r e n c e between the  p l a s t i c f l o w as i t a p p l i e s t o h o t - p r e s s i n g and p l a s t i c f l o w d u r i n g  t e n s i l e o r c o m p r e s s i v e d e f o r m a t i o n of s i n g l e c r y s t a l s .  I n case o f h o t -  p r e s s i n g , t h e compact i s composed of s i n g l e c r y s t a l spheres w i t h an i n i t i a l b u l k d e n s i t y o f about.58%.  Thus t h e spheres can r e l i e v e .  - 132 themselves o f h i g h s t r e s s e s by p a r t i c l e s l i d i n g o r r o t a t i o n of t h e spheres.  A l s o t h e a v a i l a b l e f r e e s u r f a c e a l l o w s the spheres t o deform  w i t h l i t t l e c o n s t r a i n t from the n e i g h b o u r i n g d e n s i f i c a t i o n proceeds,  spheres.  However, as t h e  the spheres w i l l be l e s s l i a b l e t o o r i e n t and  s l i d e because o f t h e c o n s t r a i n t imposed by denser p a c k i n g geometry.  Thus i n m a t e r i a l s , p a r t i c u l a r l y ceramic o x i d e s , where t h e number o f a v a i l a b l e a c t i v e s l i p systems a r e few a t a p a r t i c u l a r temperature,  c o n t r i b u t i o n from d i s l o c a t i o n g l i d e w i l l be l i m i t e d .  This  i s b e l i e v e d t o be t h e o n l y c o n t r i b u t i o n from p l a s t i c f l o w i n such m a t e r i a l F u r t h e r d e n s i f i c a t i o n s h o u l d t a k e p l a c e by s t r e s s - e n h a n c e d creep.  diffusional  That t h e c o n t r i b u t i o n from p l a s t i c f l o w i n ceramic o x i d e s i s  l i m i t e d by such a f a c t o r , has been s u b s t a n t i a t e d by t h e work o f C o b l e 31 and E l l i s  They s t u d i e d the , i n c r e a s e i n the c o n t a c t a r e a between  two s i n g l e c r y s t a l s a p p h i r e spheres (1 mm diam) as a f u n c t i o n o f time at  1530°C.  They found t h a t a t each l o a d , t h e v a l u e o f r a d i u s o f the  c o n t a c t f a c e formed remained c o n s t a n t a f t e r about 10 m i n u t e s . The spheres i n t h e d i e w i l l have random c r y s t a l o r i e n t a t i o n w i t h respect to t h e i r neighbours. at  The e f f e c t i v e s t r e s s e s  t h e c o n t a c t p o i n t s w h i c h a r e q u i t e l a r g e compared t o t h e f l o w  s t r e s s can cause f r a c t u r e , b r i t t l e places.  or d u c t i l e , at unfavourably  oriented  I t has been r e p o r t e d e a r l i e r t h a t a s m a l l amount of f r a g m e n t a t i o n  was a l s o observed  d u r i n g the h o t - p r e s s i n g o f s i n g l e c r y s t a l s a p p h i r e spher  Thus t h e d e n s i f i c a t i o n o f s a p p h i r e spheres i s m a i n l y a t t r i b u t e d t o p l a s t i c f l o w and p a r t i c l e rearrangement.  - 133 3.  Creep o f M a t e r i a l s (Under H o t - P r e s s i n g C o n d i t i o n s )  The mode o f d e f o r m a t i o n o f m a t e r i a l s d u r i n g h o t - p r e s s i n g c a n be e a s i l y r e l a t e d w i t h t h e mechanisms o f h i g h temperature creep d e f o r m a t i o n of  m a t e r i a l s under c o n s t a n t c o m p r e s s i v e s t r e s s .  Most o f t h e experiments  on t h e c r e e p o f m a t e r i a l s a r e conducted under s i m p l e t e n s i o n and as a r e s u l t l i t t l e i n f o r m a t i o n i s a v a i l a b l e on t h e c r e e p o f m a t e r i a l s under s i m p l e shear, t o r s i o n o r compression.  I t has been shown from t h e l i m i t e d  i n f o r m a t i o n a v a i l a b l e t h a t t h e c r e e p r a t e s may be s e n s i t i v e t o t h e nature of stress applied.  I t has been found t h a t t h e creep r a t e i n 81  t e n s i o n i s g r e a t e r t h a n t h a t i n compression f o r s t a i n l e s s s t e e l and l e a d 82 81 and n i c k e l , whereas t h e r e v e r s e i s t r u e f o r t h e Woods m e t a l  ,  The h o t - p r e s s i n g experiments a r e m o s t l y c a r r i e d o u t a t t e m p e r a t u r e s above 0.5T « m  of  The t o t a l c r e e p s t r a i n produced w i l l be composed  t h e s t r a i n s produced i n t h e i n d i v i d u a l s t a g e s i . e . ,  t r a n s i e n t f l o w and s t e a d y s t a t e c r e e p . i s o b t a i n e d on immediate l o a d i n g .  The i n i t i a l i n s t a n t a n e o u s s t r a i n  The h i g h temperature creep (above 0.5T )  r e s u l t s on m a t e r i a l s such as s t a i n l e s s s t e e l 86 brass  the i n i t i a l extension  m  83  , 8 4 , Fe-MrCr-Mn a l l o y  8 , copper  87 , lead  e t c . have shown t h a t t h e i n s t a n t a n e o u s s t r a i n i s due t o  p l a s t i c deformation.', I t c a n be expected t h a t t h e l a r g e c o n t a c t s t r e s s e s d u r i n g t h e h o t - p r e s s i n g o f t h e s p h e r e s , f o r s h o r t t i m e s , would account f o r most o f t h e d e f o r m a t i o n t a k i n g p l a c e . The t r a n s i e n t c r e e p i s c h a r a c t e r i z e d by t h e d e c r e a s i n g r a t e of  c r e e p s t r a i n w i t h t i m e . I n t h i s s t a g e , t h e t h e o r y o f work-hardening go proposed by Mott has been found t o be i n agreement w i t h  - 134 r e s u l t s of many w o r k e r s .  The  -  a c t i v a t i o n energy f o r t h i s s t a g e 87  was  found t o be 7.7  k c a l / g atom f o r l e a d by F e l t h a m  compared t o  the a c t i v a t i o n energy f o r volume d i f f u s i o n i n l e a d about 25.7 atom.  I t has been observed t h a t the s t r a i n r a t e i s a c c e l e r a t e d  the end of the t r a n s i e n t c r e e p . been d e s c r i b e d explained  kcal/g  towards  T h i s a c c e l e r a t i o n i n the creep r a t e 89  t o mark the s t a r t of r e c r y s t a l l i z a t i o n .  Gifkins  has  has  t h i s a c c e l e r a t i o n of creep r a t e to be dependent upon the  c r e e p of new  g r a i n s formed d u r i n g r e c r y s t a l l i z a t i o n and not t h e i r c r e a t i o n .  T h i s has been e x p l a i n e d and  - 27  p r i m a r y c r e e p of new  s e m i q u a n t i t a t i v e l y r e s u l t i n g from i n i t i a l grains.  r e a d i l y i s shown by the new a f t e r the f o r m a t i o n  That the s o f t new  g r a i n s do deform  c o a r s e s l i p l i n e s t h a t appear v e r y  of the g r a i n s and which have been t a k e n as 90 91  i n d i c a t i o n of the o n s e t of r e c r y s t a l l i z a t i o n  .  extension  Gifkins  has  soon an also  shown t h a t the i n s t a n t a n e o u s s t r a i n has an e f f e c t on the c r e e p s t r a i n to i n i t i a t e r e c r y s t a l l i z a t i o n d u r i n g h i s s t u d i e s on the e f f e c t of p r i o r s t r a i n on the r e c r y s t a l l i z a t i o n of  lead.  Steady s t a t e c r e e p , i s c h a r a c t e r i z e d by the c o n s t a n c y of the s t r a i n r a t e under c o n d t i o n s .of c o n s t a n t  stress.  Recovery and  strain 92  h a r d e n i n g can o c c u r s i m u l t a n e o u s l y g i v i n g a constant  creep r a t e .  h e r e , as shown by C o t t r e l l and A y t e k i n  Experimental evidence s t r o n g l y i n d i c a t e s  t h a t the h i g h temperature creep i s d i f f u s i o n - c o n t r o l l e d . I n the p r e s e n t s e t of e x p e r i m e n t s , the s t r e s s was f o r o n l y a s h o r t p e r i o d of time. t e m p e r a t u r e and obtained  100°C, i t was  I n case of l e a d s p h e r e s , b o t h a t room  found t h a t p r a c t i c a l l y the same s t r a i n s were  i r r e s p e c t i v e of whether the r e q u i r e d l o a d was  or 5 seconds.  applied  a p p l i e d i n 5 minutes  I n case o f . s a p p h i r e and K-Monel s p h e r e s , the t o t a l  load  ,  - 135 was a p p l i e d w i t h i n about 30 seconds. instantaneous a few seconds.  I t was found t h a t t h e  s t r a i n observed on l o a d i n g slowed down q u i c k l y w i t h i n These o b s e r v a t i o n s  suggest t h a t most p a r t o f t h e  observed s t r a i n d u r i n g h o t - p r e s s i n g , i s due t o t h e i n s t a n t a n e o u s and  transient strains.  Although  no c o n c l u s i o n s can be drawn r e g a r d i n g  the b a s i c mechanisms i n v o l v e d d u r i n g h o t p r e s s i n g o f l e a d and K-Monel spheres,  i t c a n be assumed t h a t t h e s t r a i n produced d u r i n g a v e r y  s h o r t p e r i o d o f h o t - p r e s s i n g i s due t o p l a s t i c f l o w mechanism. S i n c e t h e h o t - p r e s s i n g e x p e r i m e n t s a r e b e i n g c a r r i e d o u t above 0.5T , m  r e c o v e r y and r e c r y s t a l l i z a t i o n can be assumed t o o c c u r c o n c u r r e n t l y i n the l a t e r p a r t s o f t h e e x p e r i m e n t s .  -  136 -  VII S U M M A R Y  AND  The d e f o r m a t i o n b e h a v i o u r  C O N C L U S I O N S  o f spheres i n a compact can be p r e d i c t e d  q u a n t i t a t i v e l y by assuming t h e spheres t o be packed i n s i m p l e and s y s t e m a t i c modes o f p a c k i n g .  On a p p l i c a t i o n o f  h y d r o s t a t i c pressure, p l a s t i c deformation  takes place a t the  p o i n t s o f c o n t a c t and t h e spheres a r e deformed t o r e g u l a r polyhedr.a depending upon the c o o r d i n a t i o n number. The d e n s i f i c a t i o n produced d u r i n g compaction c a n be d e s c r i b e d i n terms o f an e q u a t i o n r e l a t i n g t h e c o n t a c t f a c e r a d i u s a, p a r t i c l e r a d i u s R, i n i t i a l p a c k i n g d e n s i t y D , and t h e d e n s i t y D, q  a t any d e f o r m a t i o n .  The f a c t t h a t t h e r a d i u s o f t h e sphere  remains p r a c t i c a l l y c o n s t a n t f o r s m a l l d e f o r m a t i o n s  ( f o r a/R <_  0 . 3 ) l e a d s t o t h e f i n a l d e n s i t y e q u a t i o n g i v e n by D = D + ir D (a/R) . o z o 2  The b u l k s t r a i n can bei r e l a t e d t o t h e sphere parameters by the relationship  ^ e.  = [ ( R - a ) / R '] 2  2  D  2  - 1  O  and t h e b u l k d e n s i t y and t h e b u l k s t r a i n can be r e l a t e d by t h e relationship D= 1  which on e x p a n s i o n  + £  b  gives, to a f i r s t D  = D  o  (1  - e. ) b  approximation,  - 137  -  I t has been shown by d e f o r m i n g u n i f o r m monosized spheres of l e a d , K-Monel and  sapphire  t h a t the d e f o r m a t i o n of spheres i n a compact  i s s i m i l a r to t h a t of the o r t h o r h o m b i c a l l y The  i n i t i a l stages,of  packed s p h e r e s .  d e n s i f i c a t i o n a r e a t t r i b u t e d t o the combined  e f f e c t of the p a r t i c l e rearrangement and p l a s t i c f l o w mechanisms. Very l i t t l e f r a g m e n t a t i o n  was  observed i n the case of the  sapphire  spheres. A l a r g e v a r i a t i o n i n the c o n t a c t pressing.  I t was  f a c e r a d i i was  observed d u r i n g  hot-  found t h a t the r a d i i of the f a c e s t h a t were  approximately perpendicular of p r e s s i n g a r e 2.2  (or w i t h i n :30 ) a  to the d i r e c t i o n  times the r a d i i of the f a c e s formed i n the  d i r e c t i o n p a r a l l e l to the d i r e c t i o n of  pressing.  A s t u d y of the geometry of the d e f o r m a t i o n showed t h a t most pf spheres deformed i n a random manner. o r t h o r h o m b i c , t e t r a g o n a l and  the  I n d i v i d u a l c o l o n i e s of  rhombohedral p a c k i n g s were  o c c a s i o n a l l y observed. On  the b a s i s of the f a c t t h a t the c o n t a c t a r e a s grow to a s t a b l e  s i z e f i x e d by a c o n s t a n t  s t r e s s . e q u a l to 3 times the y i e l d  the d e n s i t y e q u a t i o n was  modified  t o i n c l u d e the y i e l d  A l i n e a r r e l a t i o n s h i p between d e n s i t y and e s t a b l i s h e d and  experimentally  strength.  was q u a n t i t a t i v e l y  tested.  The  d e n s i f i c a t i o n of s a p p h i r e  was  e s t a b l i s h e d u s i n g o p t i c a l and  of the b a s a l , and  pressure  strength,  s i n g l e c r y s t a l s by p l a s t i c d e f o r m a t i o n e l e c t r o n microscopy.  p r i s m a t i c s l i p and  d e f o r m a t i o n haS4 been demonstrated. f l o w t o the d e n s i f i c a t i o n of s a p p h i r e  The  presence  a l s o o t h e r complex modes of The  c o n t r i b u t i o n of  plastic  ( l i k e other ceramic oxides)  - 138 i s l i m i t e d by the s t a t e of s t r e s s , temperature used, and t h e number a v a i l a b l e s l i p systems under the e x p e r i m e n t a l  conditions.  - 139  -  VIII S U G G E S T I O N S  FOR  F U T U R E  WORK  A study i n w h i c h the d i a m e t e r of the spheres i s i n t e g r a l l y r e l a t e d t o the d i a m e t e r of the d i e may  be u s e f u l as such an arrangement  w i l l r e s t r i c t l a r g e s c a l e p a r t i c l e movement. A p a r t i c u l a r p a c k i n g can be o b t a i n e d by f i x i n g the geometry of the d i e thus d e n s i f i c a t i o n of t h i s p a c k i n g may  and  t e s t the t h e o r e t i c a l models  more a c c u r a t e l y .  The d e f o r m a t i o n  of s p h e r e s under i s o s t a t i c c o n d i t i o n s may  elucidate  the mechanisms o f h o t - p r e s s i n g . A study under f i x e d geometry t o i d e n t i f y the predominent mechanism o f h o t - p r e s s i n g d u r i n g the l a t e r s t a g e s of h o t - p r e s s i n g . The  e x t e n s i o n of the d e f o r m a t i o n  range beyond a/R  e q u a l t o 0.3  show the l i m i t of 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 and  should  i n i t i a t i o n of  d i f f u s i o n - c o n t r o l l e d d e n s i f i c a t i o n d u r i n g the h o t - p r e s s i n g  process.  IX  - 140 -  A P P E N D I X  I  THE PACKING OF SPHERES OF UNIFORM SIZE  The p a c k i n g o f spheres o f u n i f o r m s i z e has been s t u d i e d by 37 many workers  in.different fields.  G r a t o n and F r a s e r  have d i s c u s s e d  t h i s s u b j e c t e x t e n s i v e l y and t h e i r f i n d i n g s a r e most s i g n i f i c a n t i n t h i s respect.  There a r e f i v e s i m p l e and s y s t e m a t i c modes of p a c k i n g of u n i f o r m 93  spheres.  These a r e shown i n F i g . 59 (from Morgan  ) and can be d e s c r i b e d  as f o l l o w s . 1.  Cubic P a c k i n g  T h i s p a c k i n g i s c o n s t r u c t e d by p l a c i n g spheres i n square (Fig.  60b).  formation  Spheres i n second and subsequent l a y e r s - a r e p l a c e d v e r t i c a l l y  over those i n the p r e c e e d i n g l a y e r s . p a c k i n g most open and l e a s t 2.  Such an arrangement makes t h i s  stable. Orthorhombic  Packing  F i g u r e 59 shows t h a t t h i s t y p e of p a c k i n g can be o b t a i n e d e i t h e r by s t a c k i n g spheres i n second to  layer h o r i z o n t a l l y o f f s e t with respect  t h o s e i n the f i r s t l a y e r by a d i s t a n c e R (sphere r a d i u s ) a l o n g the  d i r e c t i o n of one s e t of rows, or by s t a c k i n g v e r t i c a l l y over those i n f i r s t s i m p l e rhombic l a y e r .  I t t u r n s out t h a t t h e s e two ways of p a c k i n g  a r e i d e n t i c a l i n n a t u r e though of d i f f e r e n t o r i e n t a t i o n i n space. 3.  Body-Centered-Cubic  Packing  F i g u r e s 59b and 60c show t h i s type of p a c k i n g .  I t can  seen t h a t i f the spheres i n the t h i r d l a y e r have to l i e v e r t i c a l l y  be over  those i n the f i r s t , i t forms a v e r y u n s t a b l e arrangement under g r a v i t a t i o n a l  - 141  -  force alone. 4.  Tetragonal Packing  T h i s i s c o n s t r u c t e d by p l a c i n g spheres  i n second  h o r i z o n t a l l y o f f s e t w i t h r e s p e c t to those of the f i r s t R a l o n g the d i r e c t i o n of one of the s e t s of rows. s i m p l e rhombic i n t h i s  l a y e r , by a d i s t a n c e  A l l the l a y e r s  Rhombohedral P a c k i n g  i n case of orthorhombic  p a c k i n g , t h i s can be c o n s t r u c t e d  e i t h e r from square l a y e r type base or from simple rhombic l a y e r base.  But  i n t h i s case the spheres  i n second  o f f s e t w i t h r e s p e c t to those i n the f i r s t bisecting of  the a n g l e between two  type  l a y e r are h o r i z o n t a l l y  layer  in a direction  s e t s of rows by a d i s t a n c e of R^2  square l a y e r f o r m a t i o n , and o f 2R//3in  formation.  are  case. 5.  As  layer  case of simple rhombic  i n case, layer  These two ways of p a c k i n g a r e i d e n t i c a l i n n a t u r e though of  d i f f e r e n t o r i e n t a t i o n i n space. T a b l e ' IX  summarises the b a s i c methods of simple  s y s t e m a t i c p a c k i n g of u n i f o r m TABLE IX  :  spheres.  B a s i c Methods of P a c k i n g and  Method of Packing  and  Coordination number  their Construction  D e n s i t y of the Unit C e l l  Density  %  6  TT/6  52.36  Orthorhombic  8  TT/3/9  60.46  Body-Centered-Cubic  8  TT/3/8  68.02  Simple  Cubic  Tetragonal  10  2*/9  69.81  Rhombohedral  12  TT/2/6  74.05  - 142  (a) (fc) (e) (g)  Cubic Orthorhombic (from C u b i c Base) Rhombohedral (from C u b i c B a s e ) . Face-Centred Cubic Rhombohedral (from Rhombic  Fift. 59  B a s i c Systems of Spher  (b) (d) (f) (fa) e).  Body-centred Orthorhombic Tetragonal Rhombohedral Close-packed Face-centred  -  cubic (from Rhombic baseO (from Rhombic base) hexagonal cubic.  93. . P a c k i n g s ( A f t e r Morgan]*  )•  thi «M|tMP> " f t i l l * '  F i g . 60  (rj Hndy-rrtiiml rtMr  Shapes o f t h e P o r e s i n Rhombic, C u b i c , and B.C.C. P a c k i n g s (After Morgan^) ^  A P P E N D I X A.  THEORETICAL 1.  a/R  R  - 144 -  II  CALCULATIONS  C u b i c Model  Bulk Density  Bulk S t r a i n  (%)  (%)  0.00  0.620350  52.360  0  0.05  0.620352  52.556  0.37  0.10  0.620374  53.149  1.48  0.15  0.620469  54.145  3.30  0.20  0.620728  55.565  5.77  0.25  0.621282  57.423  8.82  0.30  0.622307  59.749  12.37  0.35  0.624038  62.576  16.32  0.40  0.626787  65.937  20.59  0.45  0.630972  69.869  25.06  0.50  0.637180  74.393  29.62  0,55  0.646268  79.498  34.14  0.60  0.659573  85.085  38.46  0.65  0.679356  90.843  42.36  0.70  0.709891  95.936  45.42  0.7071  0.715532  96.506  45.74  - 145 2.  a/R  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  R  Bulk Density (%)  Bulk Density (%)  Hex. P r i s m .  B. C. C. Model  Models  Bulk Strain (%)  0.00  0.620350  60.460  68.017  0  0.05  0.620352  60.687  68.273  0.37  0.10  0.620382  61.369  69.040  1.48  0.15  0.620509  62.511  70.325  3.28  0.20  0.620854  64.121  72.137  5.71  0.25  0.621593  66.207  74.483  8.68  0.30  0.622964  68.774  77.371  12.09  0.35  0.625287  71.824  80.802  15.82  0.40  0.628993  75.340  84.757  19.75  0.45  0.634679  79.272  89.181  23.73  0.50  0.643213  83.507  93.946  27.60  a  l/R  0.577350  a  2/R  0.333333  R  0.669404  Bulk Density (%)[Z = 14]  99.450  - 146 3.  a/R  R  Rhombic Dodecahedron Model  B u l k D e n s i t y (%)  Bulk S t r a i n  0.00  0.620350  74.048  0  0.05  0.620353  74.325  0.37  0.10  0.620397  75.156  1.47  0.15  0.620588  76.531  3.24  0.20  0.621107  78.437  5.95  0.25  0.622218  80.842  8.40  0.30  0.624288  83.697  11.53  0.35  0.627815'  86.908  14.80  0.40  0.633499  90.317  18.01  0.45  0.642361  93.646  20.93  0.50  0.655600  96.410  23.19  (%)  B. 1.  Load Lbs.  %  D %  - 147 -  EXPERIMENTAL RESULTS Lead a t room temp.  D-D, %  Z  a  R  V  1000  60.77  62.43  .022  2.65  7.67  0.0993  0.0099  1000  60.67  62.29  .021  2.56  7.55  0.1100  0.0121  1000  60.96  62.84  ,031  3.00  7.65  0.1180  0.0139  1500  60.67  63.02  ,039  3.70  7.80  0.1350  0.0182  2000  60.40  63.23  .047  4.45  7.60  0.1419  0.0201  2000  60.19  63.19  .050  4.73  7.56  0.1468  0.0216  2000  60.85  63.93  .051  4.81  7.56  0.1491  0.0220  3000  60.73  63.80  .051  4.80  8.40  0.1487  0.0221  3000  60.71  64.59  .064  6.00  8.42  0.1495  0.0224  3000  60.85  64.89  .067  6.22  8.26  0.1597  0.0256  4000  60.43  65.67  .087  7.95  8.72  0.1583  0.0251  4000  60.18  65.19  .083  7.82  8.61  0.1640  0.0269  4000  60.75  66.13  .089  8.12  9.06  0.1695  0.0287  4500  61.41  66.60  .084  7.82  8.39  0.1850  0.0342  5000  61.20  67.96  .110  9.93  8.30  0.2030  0.0412  5000  60.95  67.48  .107  9.67  8.73  0.2030  0.0412  5000  61.03  68.41  .121  10.79  8.80  0.2100  0.0441  6000  60.75  68.50  .128  11.30  8.87  0.2112  0.0446  6000  61.31  69.50  .134  11.80  9.03  0.2276  0.0518  2.  D %  Load lbs.  D  Lead a t 50°C  - 148 -  a_ R  - c D  (")  600  61.10  63.39  .036  3.59  8.60  0.1176  0.0138  750  61.28  63.52  .037  3.51  8.65  0.1178  0.0139  1000  61.08  62.87  .029  2.85  8.63  0.1139  0.0130  1000  61.27  63.76  .041  3.92  8.70  0.1267  0.0160  2000  60.81  63.85  .050  4.73  9.05  0.1446  0.0209  2000  61.25  64.31  .050  4.76  8.35  0.1469  0.0216  2000  60.59  63.62  ,050  4.76  8.72  0.1547  0.0234  3000  61.13  66.12  ,082  7.54  9.25  0.1629  0.0266  3000  60.53  65.61  ,084  7.73  8.95  0.1675  0.0280  3000  61.30  65.82  ,074  6.85  8.70  0.1797  0.0323  4000  61.29  66.74  ,089  8.15  8.85  0.1915  0.0366  4000  61.42  67.26  ,095  8.70  9.25  0.1983  0.0393  4000  60.96  67.45  .106  9.61  9.50  0.2030  0.0412  5000  60.75  67.47  ,111  9.96  8.75  0.1979  0.0392  5000  61.31  68.19  .112  10.09  9.30  0.1979  0.0392  5000  60.98  68.32  ,120  10.73  9.50  0.1984  0.0394  5000  61.79  68.84  .114  9.75  0.2063  0.0426  6000  61.66  68.08  .104  9.35  0.2000  0.0400  7000  62.40  70.50  .130  9.35  0.2242  0.0523  10.25 9.41 11.46  - 149 -  3. Lead at 100°C  Load lbs.  Br  D %  D-D,  500  61.75  63.66 .031  3.01  8.30  0.1011  0.0102  500  61.53  63.72 .036  3.41  8.33  0.1059  0.0112  500  61.25  63.96  .044  4 ,.23  8.43  0.1114  0.0124  1000  60.99  65.16  .068  6.39  8.75  0.1308  0.0171  1000  61.12  65.18  .066  6.22  8.74  0.1319  0.0174  1000  61.01  65.23 .069  6.45  8.60  0.1348  0.0182  2000  61.30  66.63 .087  8.02  9.05  0.1432  0.0205  2000  61.44  66.73 .088  7.90  9.35  0.1485  0.0220  2000  60.67  66.61  .098  8.92  8.85  0.1650  0.0273  3000  61.12  68.25 .117  10.44  9.75  0.1767  0.0312  3000  61.06  68.68 .125  11.07  9.40  0.1893  0.0358  3500  61.59  69.95 .136  11.96  9.40  0.2166  0.0468  3500  61.05  69.82 .144  12.54  9.65  0.2227  0.0496  4000  61.01  70.84 .161  13.87  10.05 0.2232  0.04'98  4000  61.35  70.22 .145  12.62  9.95  0.2262  0.0512  4000  60.91  70.03 .150  13.02  10.35 0.2302  0.0530  4000  61.12  70.96 .161  13.85  9.43  0.0545  a_  (-)  R  0.2334  - 150 4.  D  D  o  b  e  Sapphire  ,  z  a/R  (a/R)  Temp  2  0  c  58.05  59.92  3.11  7.50  0.0999  0.00998  1570  57.68  59.67  3.34  7.48  0.0997  0.00994  1600  57.80  60.98  5.23  7.55  0.1261  0.0159  1700  5.  a/R  Z  „  p  s  K-Monel a t 800°C  .  (a/R)  2  D  %  0.099  8.2  625  0.0098  62.50  0.110  8.8  937.5  0.0121  63.14  0.121  9.9  1250  0.0146  63.50  0.124  9.1  1250  0.0154  63.20  0.133  9.6  1562.5  0.0178  64.00  0.138  9.5  1562.5  0.0190  64.13  6.  a/R  Z  - 151 -  R-Monel a t 900°C  (a/R)  a  D  psi  %  0.113  8.8  625  0.0128  63.02  0.125  9.6  937.5  0.0156  63.70  0.137  9.6  1250  0.0188  64.80  0.139  9.8  1250  0.0193  64.43  0.151  9.8  1562.5  0.0228  65.10  0.160  9.9  1562.5  0.0256  65.31  7. 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