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Theories of hot-pressing : plastic flow contribution 1971

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THEORIES OF HOT-PRESSING: PLASTIC FLOW CONTRIBUTION BY A. SADANANDA RAO B.Sc., University of Mysore, India, 1965 B.E. (Metallurgy), I.I.Sc. , Bangalore, India, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of METALLURGY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1971 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t he Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa r tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada i i - ABSTRACT The contribution of p l a s t i c flow to o v e r a l l d e n s i f i c a t i o n of a powder compact during hot-pressing has been analysed. The basis of t h i s analysis i s the incorporation of hot-working c h a r a c t e r i s t i c s of materials at elevated temperatures into an equation app l i c a b l e to hot- pressing conditions. The empirical equation r e l a t i n g steady state s t r a i n r a t e to stress i s e = Aa n and for the d e n s i f i c a t i o n of a 1 dD powder compact, the straxn rate e = — -j^- . the p a r t i c l e s are assumed to be spheres and four d i f f e r e n t packing geometric configurations: cubic, orthorhombic, rhoTabic dodecahedron and b.c.c. are considered. Taking into consideration the e f f e c t i v e stress acting at the points of contact, the equations for the s t r a i n r a t e can be combined and arranged into another equation which i s shown below: where and 6 are geometric constants and can be calculated from the packing geometry. 'A' and 'n' are material constants. D i s the r e l a t i v e density of the compact, and fR f i s the radius of sphere at any stage of deformation i n a r b i t r a r y u n i t s . Computerized plots of D vs t were obtained for lead-2% antimony, n i c k e l and alumina. Experimental v e r i f i c a t i o n of these pl o t s was c a r r i e d out using hot-pressing data for lead-2% antimony, n i c k e l and alumina spheres. The hot-compaction experiments were c a r r i e d out over a range of temperatures for each material and under d i f f e r e n t pressures. The experimental data f i t t e d w e l l with the t h e o r e t i c a l p r e d i c t i o n for the orthorhombic model. However, a deviation at the i n i t i a l stage D o o - i i i - of compaction was encountered in most cases. This deviation was explained on the basis of the contribution to densification by particle movement or rearrangement at the i n i t i a l stage, which could not be taken into account in the theoretical derivation. The stress concentration factor i.e., the effective stress acting at necks between particles has been calculated. This was found to be very much higher than that previously used by other workers. The theoretical equation for the effective stress is a ° e f f = a i ( D 2 / V / 3 R 2 - l ) This equation predicts an effective stress, which is more than an order of magnitude higher than that predicted by several empirical equations used previously. - i v - TABLE OF CONTENTS Page TITLE PAGE '. . ± ABSTRACT i i TABLE OF CONTENTS i v LIST OF FIGURES v i i LIST OF TABLES x ACKNOWLEDGEMENT x i CHAPTER I. INTRODUCTION 1 1.1 D e n s i f i c a t i o n Due to P a r t i c l e Rearrangement and Fragmentation 3 1.2 Densif i c a t i o n Due to P l a s t i c Flow 5 1.3 D e n s i f i c a t i o n Due to D i f f u s i o n a l Mass Transport. 14 1.4 Objectives of the Present Work 19 CHAPTER I I . THEORETICAL FORMULATIONS 2 1 11.1 T h e o r e t i c a l Models 2 2 I I . l . a Simple Cubic Packing 2 ^ I l . l . b Orthorhombic Packing 2 ^ I I . l . c Rhombohedral Packing ^7 I I . l . d B.C.C. Packing 2 8 11.2 General Equations 3 0 I I . 3 Ap p l i c a t i o n of the Equation 30 CHAPTER I I I . EXPERIMENTAL VERIFICATION OF THEORY 3 3 I I I . l Ma t e r i a l Selection and T h e o r e t i c a l Plots 3 3 III.2 Experimental Tests and Procedures 3 ^ I I I . 2.a Apparatus 3 ^ - v - Page III.2.a. i Lead-2% Antimony 34 I I I . 2 . a . i i N i c k e l 42 I I I . 2 . a . i i i Alumina 44 I I I . 2.b Procedures 44 III . 2 . b . i Hot-Pressing 44 I I I . 2 . b . i i Hot-Compression of Pb-2% Sb and Ni 46 I I I . 2 . b . i i . l Lead-2% Antimony 46 I I I . 2 . b . i i . 2 Nickel 48 CHAPTER IV. RESULTS 4 9 IV. 1 Metals 49 IV. 1.a Lead-2% Antimony 49 IV. l . a . i Relative Density vs. Time .. 49 I V . l . a . i i D vs. t at a Constant Tempera- ture and Under Varying Pressures 53 I V . 1 . a . i i i D vs. t at a Constant Pressure and with Varying Temperatures 53 IV.l.b Nickel 53 I V . l . b . i D vs. t at a Constant Tempera- ture and under Varying Pressures 53 I V . l . b . i i D vs. t at a Constant Pressure and with Varying Temperatures 61 IV. 2 Non-Metal 61 IV.2.a Alumina 61 IV.2.a.i D vs. t at a Constant Tempera- ture and under Varying Pressures 61 IV. 3 Strain-Rates During Hot-Pressing 61 - v i - Page CHAPTER V. DISCUSSION 64 V . l E f f e c t i v e Stress During Hot-Pressing 64 V.2 A c t i v a t i o n Energy Study 70 V.3 Packing and Deformation Geometry Inside a Die . . 73 V.3.a Packing Geometry 73 V.3.b Deformation Geometry 75 V.4 Li m i t a t i o n of the Present Analysis 80 CHAPTER VI. SUMMARY AND CONCLUSIONS 82 CHAPTER VII. SUGGESTIONS FOR FUTURE WORK 85 APPENDICES _ 87. BIBLIOGRAPHY 103 - v i i - LIST OF FIGURES Figure Page 1 A schematic d e n s i f i c a t i o n curve 2 2 (a) Plot of ln(l-D) vs. time f or fused s i l i c a at 1100°C and pressure of 1000 p s i ( a f t e r V a s i l o s 2 ^ ) . . . 7 (b) Plot of ln(l-D) vs. time for fused s i l i c a at 1150°C and pressure of 1000 p s i ( a f t e r V a s i l o s 2 5 ) . . . 8 (c) Plot of ln(l-D) vs. time f or fused s i l i c a at 1100°C and pressure of 2500 p s i (after V a s i l o s 2 5 ) . . . 9 (d) Plot of ln(l-D) vs. time for fused s i l i c a at 1500°C and pressure of 2500 p s i (after V a s i l o s 2 5 ) . . . 10 3 Temperature dependence of density of BeO while hot- pressing at 4000 p s i (after McClelland-!-7) 12 4 (a) Shrinkage p l o t s of alumina pressed at 1300°C, 2000 and 3000 p s i (af t e r F r y e r 7 ) 16 (b) Shrinkage plo t s of alumina pressed at 1300°C, 4000 and 5000 p s i (af t e r F r y e r 7 ) 17 5 Geometric r e l a t i o n s h i p of d i f f e r e n t models (af t e r Kakar 1 3) 23 6 D vs. t plot f o r Pb-2% Sb at 100°C under 1500 p s i ( t h e o r e t i c a l plots) 35 7 D vs. t plo t f o r Pb-2% Sb at 150°C under 918 p s i ( t h e o r e t i c a l plots) 36 8 D vs. t plo t for Ni at 800°C under 2162 p s i ( t h e o r e t i c a l plots) 37 9 D vs. t plot f o r Ni at 900°C under 2105 p s i (t h e o r e t i c a l plots) 38 10 D vs. t plot f o r AI2O3 at 1600°C under 5000 p s i ( t h e o r e t i c a l plots) 39 11 Hot-pressing apparatus used for Pb-2% Sb shots 41 12 Hot-pressing apparatus used for Ni and A^O^ spheres 43 13 C a l i b r a t i o n curve for the pressure-gauge used for hot-pressing of Ni and A^O^ 45. 14 D vs. t plot f o r Pb-2% Sb at 150°C under 918 p s i (solid, l i n e s represent the t h e o r e t i c a l curves for 4 d i f f e r e n t models) 50 - v i i i - Figure 15 D vs. t p l o t f o r Ni at 800°C under 2162 p s i ( s o l i d l i n e s represent the t h e o r e t i c a l curves for 4 d i f f e r e n t models) 16 D vs. t p l o t f o r Ni at 900°C under 2105 p s i ( s o l i d l i n e s represent the t h e o r e t i c a l curves for 4 d i f f e r e n t models) 17 (a) D vs. t pl o t f o r Pb-2% Sb at 100°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only) (b) D vs. t plot f o r Pb-2% Sb at 150°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model .only) 18 (a) D vs. t plot for Pb-2% Sb at 918 p s i and d i f f e r e n t temperatures ( s o l i d l i n e s represents the t h e o r e t i c a l curve for the orthorhombic model only) (b) D vs. t plot f o r Pb-2% Sb at 1010 p s i and d i f f e r e n t temperatures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only) 19 D vs. t p l o t for Ni at 800°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only) 20 D vs. t plot for Ni at 900°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only) 21 D vs. t pl o t f o r Ni at a, constant stress and d i f f e r e n t temperatures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only) 22 D vs. t plot f o r A 1 2 0 3 at 1600°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only) °eff 23 vs. D pl o t f o r d i f f e r e n t models and also of a equations used by previous workers 24 l n A vs. 1/T pl o t f o r Pb-2% Sb (The point represented by a dot with a r i n g is from the densif i c a t i o n curve at 200°C) 25 l n A vs. 1/T pl o t f o r Ni (The point represented by a dot with a r i n g is from the densif i c a t i o n curve at 700°C) - ix - Figure Page 26 Effect of container size on the efficiency of packing one-size spheres (after McGearylS) 74 27 Coordination number vs. (D - D ) for Pb-2% Sb at various temperatures (after KaSarl3) 76 28 Microstructure of Ni spheres hot-pressed at 800°C and 2162 psi 78 29 Microstructure of Ni spheres hot-pressed at 900°C and 1477 psi 79 - x - LIST OF TABLES Table Page I Predominant Mechanisms of D e n s i f i c a t i o n under Given Hot-Pressing Conditions ." 4 II D e n s i f i c a t i o n Equation Based on D i f f u s i o n a l Mass Transport 18 III Geometric Constants f o r D i f f e r e n t Models 32 IV M a t e r i a l Constants for D i f f e r e n t Materials 40 V Test Conditions f o r Hot-Pressing 47 VI Y i e l d Stress of Single C r y s t a l Sapphire (Basal S l i p ) (After Kronberg 1 5) . 69 - x i - ACKNOWLEDGEMENT The author i s g r a t e f u l f o r the advice and encouragement given by his research d i r e c t o r Dr. A.CD. Chaklader. Thanks are also extended to other f a c u l t y members and fellow graduate students f o r many h e l p f u l discussions. F i n a n c i a l assistance from the National Research Council (NRC Grant No. A-2461) i s g r a t e f u l l y acknowledged. CHAPTER I INTRODUCTION Hot-pressing refers to densification of a powder compact under pressure at elevated temperatures. It is usually conducted at a tempera- ture several hundred degrees below the temperature at which sintering is carried out (without any external pressure). Hot-pressing has gained importance as a commercial process since one can achieve greater density at lower temperatures and shorter times than in conventional sintering, and with much more microstructural control. The densification of powder compacts during hot-pressing has been generally studied as a function of temperature, pressure and time. For kinetic analysis, the change of relative density (or bulk density) as a function of time under isothermal conditions and at a constant pressure is usually determined. A typical densification curve is shown schematically in Figure 1, where the relative density vs time is plotted at a constant temperature and pressure. Three distinct regions are recognised on the curve. The extent of the contribution from each of the three mechanisms of mass transport, i.e. particle rearrangement with or without fragmentation, plastic flow, and diffusion, depends on the type of material, the temperature and the stress level used during hot-pressing. For example, b r i t t l e solids i n a compact at relatively 0-5 Time (arbitrary scale) Figure 1. A schematic densification curve. - 3 - low temperature tend to fracture as soon as a load i s applied during hot-pressing. The d e n s i f i c a t i o n i s p r i m a r i l y achieved by fragmentation and p a r t i c l e movement. This w i l l be followed by p l a s t i c flow, only i f the temperature and stress are high enough that d i s l o c a t i o n s can move. If no d i f f u s i o n a l process occurs a f t e r t h i s stage, the compact w i l l reach an "end-point density" which i s below i t s t h e o r e t i c a l density. Metal powder compacts can be expected to have a much greater contribution to d e n s i f i c a t i o n from p l a s t i c flow than ceramic oxides. Also, p l a s t i c flow would i n general be a s i g n i f i c a n t contributing factor to d e n s i f i c a - t i o n at high temperatures and high stresses. The d i f f e r e n t mechanisms of material transport which can be expected to be predominant i n various materials and i n wide ranges of temperature and pressure are summarized i n Table 1. 1.1 D e n s i f i c a t i o n Due to P a r t i c l e Rearrangement and Fragmentation Felten c a r r i e d out a number of hot-pressing in v e s t i g a t i o n s with Al^O^ having three d i f f e r e n t average p a r t i c l e sizes (0.05 u, 0.3 u and 5-10 y). He conducted a l l h i s experiments at low temperatures (750- 1300°C) to minimize the contribution from p l a s t i c flow and d i f f u s i o n to d e n s i f i c a t i o n . His data f i t t e d w e l l with a p l a s t i c flox^j equation, 19 developed by Murray, Livey and Williams, but only at the l a t e r stages of d e n s i f i c a t i o n . There was a large deviation at the i n i t i a l stage of d e n s i f i c a t i o n from the t h e o r e t i c a l p r e d i c t i o n . He concluded that that t h i s deviation a t the very e a r l y stage must be a t t r i b u t e d to another mode of d e n s i f i c a t i o n which he c a l l e d p a r t i c l e rearrangement. This rearrangement may also be associated with fragmentation. - 4 - Table 1. Predominant Mechanisms of Densification under Given Hot- Pressing Conditions. Fracture P l a s t i c flow Low temperature High pressure Large p a r t i c l e size Boundary dif f u s i o n Lattice d i f f u s i o n High temperature .̂ Low pressure Small p a r t i c l e size - 5 - S i m i l a r l y Chang and Rhodes,"*" who studied the microstructures of uranium-carbide powder compacts a f t e r hot-pressing i n the temperature range 500 to 1500°C under pressures varying from 10,000 to 46,000 atmospheres, concluded that p a r t i c l e s l i d i n g and fragmentation played a s i g n i f i c a n t r o l e i n the i n i t i a l stage of d e n s i f i c a t i o n . I.2 D e n s i f i c a t i o n Due to P l a s t i c Flow Among the contributions to the theory of hot-pressing, the best known i s perhaps that of Murray, Rodger and Williams (also Murray, 19 Livey and Williams) who modified the s i n t e r i n g theory of Mackenzie and Shuttleworth to explain the observed behaviour i n hot-pressing, of various oxides and carbides. The f i n a l form of the equation, neglecting the contribution from s i n t e r i n g , i s dF = 4n" <1-D> M where D i s the r e l a t i v e density at time t, P i s the applied pressure and Q n i s the v i s c o s i t y . It was assumed by Mackenzie and Shuttleworth that a l l s o l i d s can be divided i n t o two groups - Newtonian and Bingham. As metals and oxides can only flow above a c r i t i c a l s t r e s s , they approximated the c r y s t a l l i n e s o l i d s as the "Bingham s o l i d ". Hence, equation (la) contains a v i s c o s i t y term. Their p l a s t i c flow theory succeeded i n explaining the increased rate of d e n s i f i c a t i o n with pressure and the e f f e c t of pressure on end point density at a constant temperature. 16 Mangsen, Lambertson and Best studied the hot-pressing c h a r a c t e r i s t i c s - 6 - of aluminum-oxide to understand the mechanisms involved i n the d e n s i f i c a - t i o n process. They observed that t h e i r experimental data f i t t e d 1 9 very well with the Murray, Livey and Williams equation. The v i s c o s i t y values were calculated from these data as a function of temperature which c l o s e l y agreed with the values reported i n the l i t e r a t u r e . 1 9 2 5 Murray's equation has also been employed by V a s i l o s to i n t e r p r e t the data obtained during hot-pressing of fused s i l i c a . He used the integrated form of equation l a , as follows: -In(l-D) = | ^ t + c (lb) when t = 0 ; D = D q (the i n i t i a l pressed density). So the i n t e g r a t i o n constant c e q u a l s - l n ( l - D Q ) . He pl o t t e d - l n ( l - D ) vs time, as shown i n Figure 2 . The l i n e a r r e l a t i o n between these two quantities i s e s s e n t i a l l y observed regardless of differences i n p a r t i c l e s i z e , applied pressure, or temperature. V i s c o s i t y values computed from the slopes of these curves agreed reasonably w e l l with those reported i n the l i t e r a t u r e . McClelland"'"''' modified Murray's equation to f i t h i s hot-pressing data for BeO and A^O^, taking into account the change of e f f e c t i v e pressure which accompanies closing the pores. Equation l a can be used to describe the v a r i a t i o n of density with time only i f pressure, remains constant during hot-pressing. The pressure which i s e f f e c t i v e i n c l o s i n g the pores, however, i s not equal to the applied pressure. Due to the presence of the voids, whose s i z e changes during the hot-pressing operation, the area over which the applied load i s transmitted increases with increasing density. Thus, the pressure - 7 - Figure 2(a).. Plot of ln(l-D) vs. time for fused s i l i c a at 1100°C and pressure of 1000 p s i (after V a s i l o s 2 5 ) . - 8 - 10 2 0 30 4 0 5 0 T I M E (MIN) Figure 2(b). Plot of ln(l-D) vs. time for fused s i l i c a at 1150°C and pressure of 1000 psi (after V a s i l o s 2 5 ) . - 9 - Figure 2(c). Plot of ln(l-D) vs. time for fused s i l i c a at 25 1100°C and pressure of 2500 p s i (after Vasilos ). - 10 - Figure 2(d). Plot of ln(l-D) vs. time for fused s i l i c a at 1150°c and pressure of 2500 p s i (after V a s i l o s 2 5 ) . e f f e c t i v e i n c l o s i n g t h e p o r e s i s a f u n c t i o n o f p o r o s i t y . M c C l e l l a n d ' s f i n a l form o f e q u a t i o n i s as f o l l o w s : X o D ( d D / x { l n [ l / ( l - x 2 / 3 ) ] + a l n x}) = K t ( l c ) where x i s (1-D), D i s t h e d e n s i t y a t t i m e t D = the i n i t i a l dens o a = T /P c 3P K = — - 4n T = y i e l d s t r e s s c P = p r e s s u r e n = v i s c o s i t y . The l e f t h a n d s i d e o f e q u a t i o n l c can be i n t e g r a t e d g r a p i c a l l y f o r d i f f e r e n t v a l u e s o f D. McClelland"'"^ f ound good agreement between t h e t h e o r e t i c a l c u r v e s and e x p e r i m e n t a l d a t a as i s shown i n F i g u r e 3. 12 K a k a r and C h a k l a d e r d e v e l o p e d a m a t h e m a t i c a l model f o r i n t e r p r e t a - t i o n s o f d e n s i f i c a t i o n due t o p l a s t i c f l o w . Knowing t h e geometry of the p a r t i c l e and p a c k i n g c o n f i g u r a t i o n i n a d i e , t h e y c a l c u l a t e d t h e change o f d e n s i t y o f a compact o f s p h e r e s p r o d u c e d by d e f o r m a t i o n a t the p o i n t s of c o n t a c t o f t h e s p h e r e s . F o r t h i s c a l c u l a t i o n , s e v e r a l c o n f i g u r a t i o n s were c o n s i d e r e d ; c u b i c (z = 6 ) , h e x a g o n a l p r i s m (Z = 8 ) , rhombic dodecahedron (z = 12) and t e t r a k a i d e c a h e d r o n (Z = 14) where Z i s t h e c o o r d i n a t i o n number. These a r e shown i n F i g u r e 5. Due t o i n d e n t a t i o n a t t h e p o i n t o f c o n t a c t between two s p h e r e s as shown i n F i g u r e 5a, m a t e r i a l w i l l be t r a n s f e r r e d t o f i l l t h e v o i d space and t h i s w i l l i n c r e a s e t h e d e n s i t y o f t h e compact. The f i n a l - 12 - 5 0 • 4 0 1 — 1 ' '—• 1 • , 0 3 0 6 0 9 0 120 150 180 210 2 4 0 TIME (min) Figure 3. Temperature dependence of density of BeO while hot-pressing at 4000 psi (after McClelland 1 7). - 13 - density due to these indentations at several points of contact (depending upon Z) i s given as follows: D - D Q = 3/2 D Q ( | ) 2 (at | < 0.3) (2a) or D - D q = 101.5 log Z (|) 2 (2b) where Z = 6, 8 and 12: D and D are the r e l a t i v e d e n s i t i e s of the o compact at a > 0 and at a = 0 r e s p e c t i v e l y , a i s the radius of the f l a t face produced by indentation, and R i s the radius of the sphere at 12 any stage of d e n s i f i c a t i o n . Kakar and Chaklader tested t h i s equation using hot-compaction data for lead, K-monel and A^O^ spheres and they obtained a very good f i t with the t h e o r e t i c a l l y predicted values of the hexagonal prismatic model. They introduced pressure and temperature dependent terms into equations 2a and 2b by considering that deformation at the points of contact between two p a r t i c l e s i s e s s e n t i a l l y a s e l f - i n d e n t a t i o n process. 9 10 Using the y i e l d i n g c r i t e r i o n of Hencky and Ishlinsky for indentation i . e . , the stress (o^) necessary f o r y i e l d i n g during indentation i s three times' the y i e l d stress (Y) of the material, the e f f e c t i v e stress acting on each face (cr ) and the coordination number 8 (corresponding to 0.60 r e l a t i v e density of green compact), equation (2a) was transformed to give D a D - D Q - £ ' ( 2 C ) - 14 - An exponential temperature dependence of the y i e l d strength modifies equation 2c into D o exp(Q/RT) D - Do = -2 _ (2d) In this equation a i s the applied pressure, A i s a pre-exponential constant and Q i s the activation energy for y i e l d i n g . 22 Rummler and Palmour studied the densification k i n e t i c s of magnesium-aluminate spinel during vacuum hot-pressing. They observed that below 1350°C, the densification k i n e t i c s of the spinel compacts were i n agreement with Murray's expression which i n i t s i n t e g r a l form predicts a linear r e l a t i o n between log porosity and time. Hence they concluded that the densification of magnesium-aluminate spinel below 1350°C was mainly due to p l a s t i c flow. I.3 Densification Due to Dif f u s i o n a l Mass Transport Koval'Chenko and Samsonov"'"4 proposed a hot-pressing equation based 20 on the Nabarro-Herring creep mechanism, which was v e r i f i e d by studies 23 on tungsten-carbide and chromium-carbide. Scholz and Lersmacher sim p l i f i e d Koval'Chenko and Samsonov's"*"^ equation and showed that i t 19 took a form s i m i l a r to the equation developed by Murray et a l . v i z _ dQ = 3P dt 4q yi where Q = porosity. 2 Coble and E l l i s carried out hot-pressing experiments on aluminum single c r y s t a l spheres at 1530°C. They measured the effect of load on - 15 - i n i t i a l neck growth between sin g l e c r y s t a l spheres and observed that the neck areas were larger than those of sintered spheres and that they were, constant for a l l times between 10 and 480 minutes. They calculated the p l a s t i c flow contribution to the d e n s i f i c a t i o n during hot-pressing. This c a l c u l a t i o n was based on the hydrostatic nature of stress and the geometric r e l a t i o n s between p a r t i c l e s . From t h i s they found that for aluminum oxide the contribution of p l a s t i c flow to d e n s i f i c a t i o n at the pressures normally used i n hot-pressing was small. Hence they concluded that the f i n a l stage of d e n s i f i c a t i o n of alumina occurs by enhanced d i f f u s i o n under the influence of s t r e s s . 21 Rossi and Fulrath also studied the k i n e t i c s of the f i n a l stage of d e n s i f i c a t i o n of alumina under vacuum hot-pressing conditions. They suggested that p l a s t i c flow may be operative at an intermediate stage but d e f i n i t e l y not during the f i n a l stage where d i f f u s i o n - c o n t r o l l e d creep was proposed to be the mechanism responsible f o r d e n s i f i c a t i o n . 26 Vasilos and Spriggs calculated the apparent bulk d i f f u s i o n c o e f f i c i e n t s from the d e n s i f i c a t i o n data for alumina and magnesia during hot-pressing and obtained an order of magnitude higher values for pressureless s i n t e r i n g . When the pressure c o r r e c t i o n term due to changing porosity was introduced into t h e i r c a l c u l a t i o n s , t h e i r calculated d i f f u s i o n - c o e f f i c i e n t s were i n better agreement with those obtained from the s i n t e r i n g data. From these observations they concluded that the d e n s i f i c a t i o n of a hot-pressed compact, beyond the i n i t i a l stage i s a d i f f u s i o n c o n t r o l l e d process. A new treatment, proposed by F r y e r 7 f o r the f i n a l d e n s i f i c a t i o n step during hot-pressing (was based on a model involving the bulk d i f f u s i o n - 16 - 003h (h-o 0-02 OOI O 3000 lb. in:1 • 2000 ib.iri? OOI 0 0 2 003 Figure 4(a) Shrinkage plots of alumina pressed at 1300°C, 2000 and > -• 3000 psi (after Fryer 7). - 17 - Figure 4(b). Shrinkage plots of alumina pressed at 1300°C, 4000, and 5000 psi (after Fryer 7). - 18 - Table I I . D e n s i f i c a t i o n Equations Based on D i f f u s i o n a l Mass Transport. 2 , AL. 5/2 1 L ' 4 o 2?rd KT a - L/ TT x x • = neck radius Coble and E l l i s 40 D.nba dD 1 dt (1-D) KT d N-H creep model Rossi and Fulrath 21 D = • 2 eR KT a 4fi c a = a ( l + 2P) c e = dP dt V a s i l o s and Spriggs 26 i d v 1 s_ v s dt 7  i L VA5/3 A ,2 KT V Fryer I d D 4 0 D l " ( a , 2Y. D dt - 3 D 3 K T D̂ r l a t t i c e d i f f u s i o n Coble I d D 7 ' 5 V . a , 2Y, D dt " d^ KT r boundary d i f f u s i o n Coble L load R = grain radius D = r e l a t i v e density K = Boltzmann constant D l = bulk d i f f u s i o n c o e f f i c i e n t T = temperature i n °Kelvin Db = boundary d i f f u s i o n c o e f f i c i e n t V = s = volume of s o l i d n = vacancy volume Z = a numerical constant a = applied stress P = porosity = ( 1 - D ) a c = e f f e c t i v e stress e = s t r a i n rate b = a s t r e s s - i n t e n s i t y factor Y = surface energy d = grain diameter W = grain boundary width - 19 - of vacancies. The f i n a l form of h i s equation i s shown i n Table I I . He tested h i s equation with hot-pressing data for alumina powder at 1300°C. The experimental data f i t t e d w e ll with the t h e o r e t i c a l p r e d i c t i o n , as shown i n Figures 4a&4b• 4 Coble has also developed a model f o r the f i n a l stage of d e n s i f i c a - t i o n of a powder compact, which e x p l i c i t l y includes both the surface energy and applied pressure as the d r i v i n g force. This model i s based on a steady state d i f f u s i v e flow of material between concentric s p h e r i c a l s h e l l s . The d r i v i n g force i s expressed as follows P D.F = ^ + = 2x . a D where P^ = applied force y = surface energy D = r e l a t i v e density r = radius of closed pore. 4 The r e s u l t i n g equations are also shown i n Table I I . Coble concluded that t h i s new approach should be able to predict d e n s i f i c a t i o n up to the t h e o r e t i c a l density of a powder compact. 1.4 Objectives of the Present Work The contribution of p l a s t i c flow to d e n s i f i c a t i o n during hot- pressing has been a subject of controversy. Although i t has been accepted that p l a s t i c flow occurs at a c e r t a i n stage during hot-pressing, which p r i m a r i l y depends on the temperature used, the extent of d e n s i f i c a - tion a r i s i n g from any p l a s t i c flow mechanism has not been unambiguously determined, - 20 - The aim of t h i s work was to study q u a n t i t a t i v e l y the extent of d e n s i f i c a t i o n by p l a s t i c flow during hot-pressing. Generally hot- pressing i s c a r r i e d out ^ 0.6 T (where T i s the melting point i n °K) m m of the material concerned. This i s also approximately the hot-working temperature of the material. With t h i s observation, an attempt has been made to derive an equation based on steady-state hot-deformation and subsequently test the equation with hot-compaction data for both m e t a l l i c and non-metallic materials. CHAPTER II THEORETICAL FORMULATIONS It has been found experimentally 1"'" that f o r a large number of materials, the steady-state s t r a i n rate i s related to stress by a power law as follows e = A a * (3) and the s t r a i n rate during d e n s i f i c a t i o n of a powder compact ( i n a die) i s approximately given by e = — = - 5̂. (4) hdt D dt where "D" i s the r e l a t i v e density, "h" i s the instantaneous compact height; "A" i s a constant at constant temperature; "n" i s a material constant and i s the st r e s s . For a compact t h i s stress changes with d e n s i f i c a t i o n during hot-pressing. In the development of the theory, i t i s assumed that the p a r t i c l e s i n a compact are monosized spheres and that they are arranged i n a regular three-dimensional array. During hot-pressing, they deform p l a s t i c a l l y , at the points of contact (necks) and form f l a t faces. The compacts density change, as a r e s u l t of th i s deformation, with respect to neck radius, has been derived by Kakar and Chaklader and i s given by D = 2 \ 3/2 ( 5 ) 3 ( R Z - a V ^ D i s the r e l a t i v e density of compact at neck radius "a", "R" i s the instantaneous radius of the p a r t i c l e at neck radius "a" and g i s a geometric constant which depends on the packing configuration. The e f f e c t i v e stress acting on a compact w i l l be affected by the packing geometry. II.1 T h e o r e t i c a l Models The deformation geometries of two spheres i n contact and other simple geometric configurations are shown i n Figure 5. Only four deforma- tio n models were considered - (1) simple cubic (Z = 6), (2) orthorhombic (Z = 8), (3) body-centred cubic (Z = 8), and (4) rhombohedral (Z = 12), where Z i s the coordination number. The b.c.c. packing i s an unstable arrangement i n a u n i d i r e c t i o n a l f i e l d of force ( i . e . g r a v i t a t i o n a l f o r c e ) . For t h e o r e t i c a l purposes, i t i s assumed that each type of packing i s stable and maintains i t s symmetry on the a p p l i c a t i o n of pressure, and that the material at the points of contact spreads symmetrically during deformation to maintain the s p h e r i c i t y of the p a r t i c l e . The applied load can be considered to be acting on the whole unit c e l l where the unit c e l l i s defined as a s p a c e - f i l l i n g polyhedron with a deformed sphere situated i n s i d e the polyhedron ( a l t e r n a t i v e l y , the - 23 - Figure 5. Geometric r e l a t i o n s h i p of..different models (After 13 Kakar ). - 24 - unit c e l l i s composed of the deformed sphere with i t s associated void space). II.1.a. Simple Cubic Packing Consider a cubic array of spheres deformed under hydrostatic pressure. Each sphere w i l l have s i x f l a t faces formed as shown i n Figures 5b ^and 2 2 1/2 5 ^ . The unit c e l l i n t h i s case i s a cube of side 2y where y = (R -a ) 2 The c r o s s - s e c t i o n a l area of the unit c e l l i s 4y . If there i s a 2 load ' £' on each face, then the stress on each face i s £/4y and the t o t a l pressure acting on each sphere i s — y . This has to be equal to 4y 2 the t o t a l load on the system under hydrostatic conditions. In conventional hot-pressing generally a u n i d i r e c t i o n a l load i s applied, but because of the existence of back stress from the die-wall and other plunger, the stress at the c e n t r a l core of a die i s approximately i s o t r o p i c . This 13 was experimentally observed by Kakar, while measuring the contact face r a d i i of hot-pressed lead spheres. The load acting on each face of the unit c e l l can be divided i n t o two components; one part acting on the sphere through i t s indented f l a t face and the other part on the void space, i . e . 2 2 4y a = ira + ca^ • (6) where i s the stress on the neck, i s the stress on the void space (= 0), a i s the applied s t r e s s , c i s the cross-section area of porosity and a i s the radius of the neck. - 25 - 2 or 4y °l = T i r a 2 2 a-, o C/; 1 TT 2 a From equation (5) ; ( R 2 - a 2 ) 3 / 2 v/here g = 8 for the simple cubic packing and th i s equation can be rewritten as R -a Substituting equation (8) i n equation (7) 0 1 = * ( D 2 / 3 B 2 / 3 R 2 - D ° Substituting equation (9) i n equation (3) 1 n £ = A ^ ( D 2 / V / 3 R 2 - 1 ) } ( 1 0 ) Combining equations (10) and (4) 1 dD . r 1 a -,n D dt A \ / 4 ( D2/3 32/3 R2_ 1 } 1 U 1 ; - 26 - This on i n t e g r a t i o n y i e l d s D D 1 { D 2 / 3 p 2 / 3 R 2 - l } dD = / A (-TT-) dt (12) I I . l . b . Orthorhombic Packing Each sphere af t e r deformation w i l l have eight faces. The model for the deformed sphere and the unit c e l l are shown i n Figures 5c^ and r- 2 5c2« The cro s s - s e c t i o n a l area of the unit c e l l i s 2/3 y where y = 2 2 1/2 (R -a ) . Computing the load for the top face of the unit c e l l , 2/3 y 2o = T r a^o^ + co^ (13) 0 = applied stress = stress on the neck O2 = stress on porosity = 0 a 2/3 2 ^ a (14) or 1 TT 2 2 ^ R 2-a 2 n , . a, = — ~ — a (15; 1 TT I Substituting equation (8) i n equation (15) 1 a 0 ± . TT/2/3 [ D 2 / V / 3 R 2 - 1 ] (16) - 27 - where 3 = 4/3 for the orthorhombic packing. Combining equations (3), (4) and (16) ( D 2 / W - l ) The i n t e g r a l form of equation (17) i s D /o max _ 1 „.„ „ n t n D {D $ R -1} dt =/ Af.—5—) dt (18) o Jo TT/2/3 II.I.e. Rhombohedral Packing Each sphere has twelve points of contact and the resultant unit c e l l i s a rhombic dodecahedron as shown i n Figures 5d^ and 5d^. 2 The c r o s s - s e c t i o n a l area of the unit c e l l i s 2/J y where y = 2 2 1/2 (R -a ) . Computing the loads f o r the top face of the unit c e l l , 2 2 2/3 y a = 3ira cos Qo-^ + ca^ (19) a = applies stress CT^ = stress on the neck a^ = stress on porosity = 0 From geometry cos 6 = /2/3 o 1 - " ^ — " ^ a (20) /6TT a 2 / 3 R 2-a 2 ,o-n a 1 = —£ a (21) or, - 28 - Substituting equation (8) i n equation (21) r ^ v v'v'v-i/'dD- f w - i - , " </D Jo "/^ o (22) 1 u//T [ D 2 / 3 3 2 / 3 R 2 - l ] where 3 = 4/2~ for the rhombohedral packing. Combining equations (3), (4) and (22) 1. dD , 1 o , n D d t \ r//2 ( D 2 / 3 3 2 / 3 R 2 - D This, on i n t e g r a t i o n , y i e l d s dt (24) I l . l . d . B.C.C. Packing The shape of deformed sphere i s schematically represented i n Figures 5e^ and Se.^- There are eight points of contact during the i n i t i a l stage of deformation. 16 2 The c r o s s - s e c t i o n a l area of the unit c e l l i s —— y where y = ( R 2 - a 2 ) 1 / 2 . Computing the loads f o r the top face of the unit c e l l y 2 a = 47ra2 cos 6a^ + co^ (25) - 29 - a = applied stress = stress on neck ~ stress on porosity = 0 From geometry, cos8 = — /3 16 /3 y 2 1 3 4TT o a^ or / t> 2 2 4 R -a °1 = — 2 ~ a /3TT a Substituting equation (8) i n equation (27) 1 ,/3/4 [D 2 /V / 3R 2-1] Combining equations (3), (4) and (28) I dD . 1 a D d t \ m (D 2 /V / 3R 2-1) This, on i n t e g r a t i o n gives o - 30 - II.2 General Equations From the above, i t appears one can write general equations both f o r the stress concentration factor and for the d e n s i f i c a t i o n as follows: a = « ( 3 1 ) a l (D 2 /V/ 3R 2-l) where and 3 are constants which vary with packing geometry, and hence max p r T W 7 V / 3 R 2 - l } n dD J A ( / ) n dt (32) D o 1 o It should be noted that the d e r i v a t i o n i s based on a condition where the stress i s hydrostatic i n nature. I t i s also assumed that the s t r a i n rate at the points of contact and i n the whole system approaches a steady- state condition as soon as a load i s applied. II.3 A p p l i c a t i o n of the Equation The a n a l y t i c a l s o l u t i o n of the above equation i n a closed form i s not possible. The l i m i t s of i n t e g r a t i o n D m a x have values varying from .835 for the hexagonal prismatic model to 0.995 for the tetrakaidecahedron model. In equation (32) g, and ct^ are geometric constants and they are c a l c u l a b l e . The values of these constants are shown i n Table I I I . A and n are material constants and can be determined from steady-state hot- working data. Knowing a l l the constants, the change of D, the r e l a t i v e , density of a compact, as a function of time (t) at a constant temperature -31- and pressure can be obtained by solving the equation i n a computer. The computer programme i s given i n Appendix 5. For each of the t h e o r e t i c a l deformation models, R changes as de- formation proceeds; i.e.,R i s a function of D. However, the change i n R at various stages of deformation can be calculated from the constant 13 volume equation as described by Kakar . Tables of R vs D are given i n Appendix 1. 1 - 32 - Table III. Geometric Constants of D i f f e r e n t Models Model a, 3 D D 1 o max cubic hexagonal prism rhombic dodecahedron tetrakaidecahedron TT/4 TT/2/3 TT/2 TT/3/4 8 4/3 4/2 32/3" .523 .604 ,740 .680 0.965 0.835 0.964 0.995 CHAPTER I I I EXPERIMENTAL VERIFICATION OF THEORY I I I . l M a t erial Selection and T h e o r e t i c a l Plots To compare the compaction data for hot-pressing with that of t h e o r e t i c a l l y predicted behaviour, both m e t a l l i c and non-metallic materials were chosen. The s e l e c t i o n of lead-2% antimony, n i c k e l and alumina for experimental work was based upon the a v a i l a b i l i t y of s p h e r i c a l p a r t i c l e s for hot-pressing. As discussed previously, the t h e o r e t i c a l equation can only be tested i f the values of the material constants A and n for any material are known or,are experimentally determined by hot-working data. The values of these constants f o r a large number of m e t a l l i c and non-metallic materials have been reported i n the l i t e r a t u r e . The p u r i t i e s of the metals (spheres of Pb-2% Sb and Ni) used i n t h i s i n v e s t i g a t i o n are not the same for which the values of A and n are a v a i l a b l e i n the l i t e r a t u r e . For this reason, the values of A and n for Pb-2% Sb and Ni were determined from steady-state hot-compression experiments i n an Instron machine. The d e t a i l s of the experimental procedure are given i n a l a t e r section. The values of stress and s t r a i n rate are shown i n Appendix 2. The values of A and n for alumina are obtained from the l i t e r a t u r e . The value of the constants A and n for a l l the materials are shown i n Table IV. - 34 - Using the values from the table and the t h e o r e t i c a l equation (equation 32), a seri e s of pl o t s of r e l a t i v e density (D) as a function of time (t) were generated for d i f f e r e n t temperatures and pressures. These t h e o r e t i c a l plots are shown i n the following f i g u r e s . (a) For lead-2% antimony at 100°C under 1500 p s i i n F i g . 6, lead-2% antimony at 150°C under 918 p s i i n F i g . 7. (b) For n i c k e l at 800°C under 2162 p s i i n F i g . 8, n i c k e l at 900°C under 2105 p s i i n F i g . 9. (c) For alumina at 1600°C under 5000 p s i i n F i g . 10. For these plots a l l four geometric models were employed. III.2 Experimental Tests and Procedures III.2.a. Apparatus (i) Lead-2% antimony: For these hot-pressing experiments 1.5 mm average diameter lead-2% antimony shots were used. This material was supplied by the Lead Shot Industries Limited. The apparatus used for hot-pressing of lead-2% antimony shots i s shown i n Figure 11. A u n i a x i a l compressive load was applied to the system through a simple bar. A s t a i n l e s s s t e e l (316) rod activated by a lever arm was placed i n s i d e a s t a i n l e s s s t e e l tube. The compact was positioned on top of the s t e e l rod i n a die and the stress was applied to the specimen between the rod and a plug welded i n t o the centre of the tube. To ensure a uniform stress across the compact a hemispherical plunger and cup were used to transmit the load to the compact. A 400 watt tube furnace with a uniform hot zone 3 inches long was placed around the s t a i n l e s s s t e e l tube. The temperature of the furnace was raised and Figure 6. D vs. t plot for Pb-2% Sb at 100°C under 1500 psi (theoretical plots). TIME(min) Figure 7. D vs. t plot for Pb-2% Sb at 150°Cunder 918 p s i ( t h e o r e t i c a l p l o t s ) . - 37 - Figure 9. D vs. t plot for Ni at 900°C under 2105 p s i ( t h e o r e t i c a l p l o t s ) .  - 40 - Table IV. M a t e r i a l Constants for D i f f e r e n t Materials Materials Temperature (°C) A (sec ^) n Lead-2% antimony 100 -21 4.11 x 10 4.2 150 -19 3.519 x 10 4.2 Nickel 800 -23 4.6 x 10 4.6 900 -22 3.4 x 10 4.6 Alumina 1600 -22 + 2.0 x 10 4.0 + Values obtained from Warshaw et a l . - 41 - F u r n a c e T h e r m o c o u p l e H e l l W e l d e d S t e e l P l u g P r e s s u r e T r a n s m i t t i n g B o s s D i e S p e c i m e n S t a i n l e s s S t e i ^ l F u r n a c e T u b e S t a i n l e s s S t e e l R o d Bench L e v e r Am T r a n d u c e r Load 6 - D i a l Gauge S u p p o r t i n g Rod f o r T r a n s d u c e r & D i a l Gauge Figure 11. Hot-pressing apparatus used f o r Pb-2% Sb shots. - 42 - maintained at the desired temperature. A chromel-alumel thermocouple situated 1/2 inch from the specimen operated a proportional temperature c o n t r o l l e r which maintained the specimen temperature within + 5°C. The compaction was determined by the displacement of the extension of the s t a i n l e s s s t e e l rod, which was obtained by the output from an electromagnetic transducer attached to the extension rod. The transducer output was analysed by a P h i l l i p s PR 19300, d i r e c t reading bridge which was connected to a "Heath K i t " Servo Recorder. The loading system was c a l i b r a t e d by hanging weights at the point of contact of the bar and the s t a i n l e s s s t e e l rod. The load was very close to the product of the weight on the bar and the r a t i o of the arms. The weight of the bar was compensated f o r by loading a small weight at the other end. I I I . 2 . a . i i Nickel (99% Ni and 0.7% Co): For hot-pressing of n i c k e l , 0.65 mm average diameter n i c k e l b a l l s were used. These were supplied by the S h e r r i t t Gordon Mines Limited, Fort Saskatchewan, Alberta. The hot-pressing was c a r r i e d out i n a molybdenum die with an external graphite sleeve, using induction heating. Both the die and graphite sleeve acted as susceptors. The experimental set up i s shown i n Figure 12. The temperature of the compact was recorded through the top plunger at a distance 1/6 inch from the specimen and was controlled by the same thermocouple. A Pt-Pt-10% Rh thermocouple was used. The measurement of the displacment of the moveable bottom ram (the top ram with the thermocouple was r i g i d l y attached to the press) was obtained by the output from an electromagnetic transducer attached to the ram. The transducer output was analysed by a P h i l l i p s d i r e c t reading measuring - 43 - Figure 12. Hot-pressing apparatus used for Ni and A l 0 spheres. - 44 - bridge, which was connected to a "Heath K i t " servo-recorder, as used previously f o r the compaction experiments with lead-2% antimony spheres. ( i i i ) Alumina: The same apparatus (hot-pressing induction unit) which was used f o r n i c k e l experiments was also used for the compaction studies of alumina. In t h i s case,however, a graphite die was used instead of a molybdenum die-graphite sleeve combination as used previously. 1 mm sapphire spheres were used i n t h i s case and these were supplied by A. M i l l e r and Company, L i b e r t y v i l l e , I l l i n o i s . III.2.b. Procedures (i) Hot-pressing: A weighed amount of spheres was poured i n t o a die, tapped and well-shaken i n order to obtain uniform packing and maximum density. The i n i t i a l height of the die with plungers was measured and subsequently the height of die, with plungers and specimen, before hot-pressing was also measured. This gave the i n i t i a l height of the specimen. I t took 15-20 minutes f o r the specimen to reach the furnace temperature a f t e r the die assembly was introduced i n the furnace. During t h i s heating up period, the plunger displacement recording system was connected. No appreciable shrinkage or expansion was recorded during the heating up period. Once the specimen attained the equilibrium temperature the pre-weighed load was placed on the lever arm for lead. In the case of n i c k e l and alumina the load was applied through a hydraulic jack, which activated the press. The press was c a l i b r a t e d by an Instron machine using an 'F' c e l l . The c a l i b r a t i o n curve i s shown i n Figure 13. A hydrogen atmosphere was used during hot-pressing of lead-2% antimony - 45 - 8000 " sooo - I 5 < o 4 0 0 0 . < U < 2 0 0 0 - 1400 METER READING (psi) Figure 13. Calibration curve for the pressure-gauge used for hot-pressing of Ni and A l , ^ . - 46 - and nitrogen was used i n the case of n i c k e l and alumina. A f t e r the a p p l i c a t i o n of pressure, the shrinkage was recorded as a function of time. When the compaction rate was d r a s t i c a l l y reduced and the compaction curve had l e v e l l e d o f f , the power to the furnace was shut off and the specimen was cooled i n the furnace. The height of the specimen i n the die with plungers a f t e r hot-pressing was. measured. The experimental test conditions used during hot-pressing are shown i n Table V. The volume of the compact was calculated before and a f t e r hot- pressing using the height of the compact and the diameter of the d i e , which remained constant. The weight of the compact was known and also remained constant. From these measurements the i n i t i a l and f i n a l bulk- den s i t i e s were calculated. From the recorder chart the compact height at any instant during compaction could be found. Using these charts, the change i n bulk-density as a function of time under isothermal conditions was obtained. Afterwards the values of the bulk density were converted into r e l a t i v e density by d i v i d i n g these values by the t h e o r e t i c a l density of the s o l i d . I I I . 2 . b . i i Hot-compression of lead-2% antimony and n i c k e l 1. Lead-2% antimony: The Pb-2% Sb shots were melted and cast i n the form of small ingots (0.5 x 0.5 x 3 inches). Specimens were cut from these ingots with a jewellers saw, the majority of dimensions were 0.2 x 0.2 x 1 inch. The specimens were annealed at 100°C for 5-6 hours. P r i o r to t e s t i n g , the ends of each specimen were ground perpendicular to the l o n g i t u d i n a l axis. A l l specimens were tested i n a furnace attached to an Instron t e n s i l e t e s t i n g machine under conditions of constant s t r a i n - 47 - Table V. Test Conditions f o r Hot-Pressing Material Sphere diameter Temperature Pressure i n Atmosphere i n mm i n °C p s i Lead- 1.5 100 918,1010,1285 hydrogen 2% antimony 1.5 150 918,1010 hydrogen Nickel 0.65 800 2162,3208 nitrogen 0.65 900 1477,2105 nitrogen Alumina 1.0 1600 3200,4000 nitrogen - 48 - rates. For t e s t i n g , the specimens were placed upright between alumina buttons ( 0.7" d i a . , 0.2" thickness). When the load attained a constant value the test was stopped. From t h i s load and the cross- s e c t i o n a l area of the specimen, the stress was calculated. The experi- ments were done at 100 and 150°C and at s t r a i n rates of 0.002, 0.005, 0.01 and 0.02 in/min. These r e s u l t s are shown i n Appendix 2. 2. N i c k e l : N i c k e l b a l l s were melted by induction heating and cast i n the form of small ingots (0.5" d i a . , 3" length). C y l i n d r i c a l specimens (0.25" d i a . , 0.9" length) were prepared by turning these ingots on a lathe. The specimens were annealed at 1000°C for 5 hours. The ends of each specimen were ground perpendicular to the l o n g i t u d i n a l axis. Experiments were conducted at 800 and 900°C and 0.002, 0.005, 0.01, and 0.02 in/min s t r a i n rates. These r e s u l t s are shown i n Appendix 2. CHAPTER IV RESULTS IV,1 Metals IV.1.a. Lead-2% Antimony (i) Relative Density (D) vs. Time (t) I n i t i a l experiments with Pb-2% Sb were made to test which of the four different geometrical models fitt e d the experimental data. For this, hot-pressing was done at 150°C under 918 psi and compaction curve was obtained. Both theoretical plots and the experimental data are shown in Figure 14. It is apparent from the figure that the data coincided with the orthorhombic model in the later stages of hot- pressing, with a deviation at an early stage of compaction. The same observation was made for nickel compacts at 800°C and 900°C under 2162 psi and 2105 psi pressures respectively, as shown in Figures 15 and 16. An overall comparison of compaction data with the theoretical curves indicated that the experimental data followed closely the theoretically predicted curves for the orthorhombic model. For this reason a l l the theoretical plots in subsequent figures were computed with geometric constants for the orthorhombic model only. The reason for this agreement between the data and the curves for the orthorhombic model is discussed in a later section. In some cases, the experimental compaction data obtained were beyond the upper limit of the density, predicted by the theoretical orthorhombic model. However, these data were excluded from the figures, as the theory of compaction is not valid over .835 relative density. TIME (min) Figure 14. D vs. t plot f o r Pb-2% Sb at 150°C under 918 p s i ( s o l i d l i n e s represent the t h e o r e t i c a l curves for 4 d i f f e r e n t models). I-OT 0-9 Ni 800°C o 2162 psi B-C-C RHOMB 0 ORTHO r, S • CUBIC 3 4 5 TIME (min) _ l _ 9 10 i—1 Figure 15. D vs. t plot for Ni at 800°C under 2162 psi (solid lines represent the theoretical curves for 4 different models). >- CO UJ Q l-O 0-9 Ni o r T 1 900°C 2105 psi 1 i i 1 ~1 1 " * ! — 5 = "B-C-C 0-8 / ° a _ O _ ^ ) R T H 0 ^•rnnir . 0-7 - 0-6 - 0-5 L 1 l 1 > i 1 i l 0 1 2 3 4 5 6 7 8 9 10 TIME(min) Figure 16. D.vs. t plot for Ni at 900°C under 2105 psi (solid lines represent the theoretical curves for 4 different models). - 53 - ( i i ) D vs. t at a constant temperature and under varying pressures: For this, hot-pressing experiments were c a r r i e d out at 100 (3 stresses) and 150°C (2 stresses) under 918, 1010 and 1285 p s i . A series of t h e o r e t i c a l curves f o r the orthorhombic model only were obtained from the computer. The experimental data and the t h e o r e t i c a l curves are shown i n Figures 17a and 17b. A good f i t of experimental points with the t h e o r e t i c a l p l o t s can be seen p a r t i c u l a r l y at the l a t e r stages of compaction. ( i i i ) D vs. t at a constant stress and with varying temperatures: For this,experiments were made at 100°and 150°C at a stress of 918 p s i . In addition,other experiments were also c a r r i e d out at 100 and 150°C under 1010 p s i s t r e s s . The experimental data and the t h e o r e t i c a l p l o t s are compared i n Figures 18a and 18b for these two sets of experiments. Again a good f i t i s apparent between experimental points and the t h e o r e t i c a l p r e d i c t i o n . IV.l.b. Nickel (i ) D vs. t a t a constant temperature and varying pressures: The experimental conditions used to follow the change of r e l a t i v e density as a function of time for n i c k e l are 800°C at 2162 and 3208 p s i . The experimental compaction data and the t h e o r e t i c a l p l o t using equation (32) are compared i n Figure 19. Another set of hot-pressing experiments were c a r r i e d out at 900°C under pressures of 1477 and 2105 p s i . These res u l t s and the corresponding t h e o r e t i c a l p l o t s are shown i n Figure 20. 0-9 O-E- z UI Q UI > 0 -7 UI tr o-6 Pb-Sb 100° C O 918 p.s.i. • 1010 p.s.i. O 1285 p.s.i. 5 6~ TIME (min) 4> Figure 17(a). D vs. t plot for Pb-2% Sb at 100°C under different pressures (solid line represents the theoretical curve for the orthorhombic model only). 09h LL) 1 1 Pb-Sb= I50°C © - lOIOp.s.i. o = 918 p.s.i. C 0-8 to z LU Q LU > 0-7I- JR _ .^aax U l 0 - 6 L 4 5 6 TIME (min) 10 Figure 17(b). D vs. t pl o t f o r Pb-2% Sb at 150°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve f or the orthorhombic model only). TIME (min) ure 18(a). D vs. t plot for Pb-2% Sb at 918 p s i and d i f f e r e n t temperatures ( s o l i d l i n e s represents the t h e o r e t i c a l curve f o r the orthorhombic model only). 0-9 1 r l_ Pb-Sb lOIOpsi • I50°C o IOO°C CO UJ Q UJ > UI ar o-8 0 - 6 L 10 TIME (min) Figure 18(b). D vs. t plot for Pb-2% Sb at 1010 p s i and d i f f e r e n t temperatures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only). 0-9 >- to LU Q 0-8r- UJ > U J 0-7 0-6 Ni 8 0 0 ° C 9 3 2 0 8 psi o 2162 psi _L 5 6 T I M E (min) Ln 00 me Figure 19. D vs. t plot f o r Ni at 800°C under d i f f e r e n t pressures ( s o l i d 1 represents the t h e o r e t i c a l curve for the orthorhombic model only). 'igure 20. D vs. t plot for Ni at 900°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only). T 1 1 1 1 1 1 I r Ni at ~ constant cr T = 9 0 0 ° C T I M E (minK Figure 21. D vs. t plot f o r Ni at ^constant stress and d i f f e r e n t temperatures ( s o l i d l i n e represents the t h e o r e t i c a l curve for the orthorhombic model only). - 61 - ( i i ) D vs . t at a constant pressure and varying temperatures: To study the e f f e c t of changing the temperature at a constant stress on the compaction behaviour of n i c k e l spheres, the curves i n Figure 19 and 20 f o r the conditions 2105 p s i at 900°C and 2165 p s i at 800°C are replotted i n Figure 21. I t shows that an increase i n temperature of hot-pressing by 100°C s h i f t s the r e l a t i v e density curve by about 10%. IV.2. Non-Metal IV.2.a. Alumina (i) D vs. t a t a constant temperature and varying stresses: As only • very few alumina spheres (single c r y s t a l sapphire spheres) were a v a i l a b l e f or the hot-compaction studies, experiments were done at one temperature (1600°C) and at two d i f f e r e n t stresses (3200 and 4000 p s i ) . The t h e o r e t i c a l curves f o r D vs. t were also generated under these experimental conditions using equation (32). The experimental data and the t h e o r e t i c a l p l o t s are shown i n Figure 22. In t h i s case a good f i t between the experimental points and the t h e o r e t i c a l p r e d i c t i o n can be seen during the intermediate stage of hot-pressing. A deviation between the t h e o r e t i c a l p r e d i c t i o n and experimental r e s u l t s at the very eary stage and during the l a s t stage of compaction was encountered. IV.3. Strain-Rates During Hot-Pressing For the c a l c u l a t i o n of s t r a i n - r a t e s from the hot-pressing data, the following equation was used • = 1 dD £ D dt os - AlgOj 1600°C • 3200 psi > ° 4000 psi to LU Q 0-8 - TIME (min) Figure 22. D vs. t plot for A l ^ at 1600°C under d i f f e r e n t pressures ( s o l i d l i n e represents the t h e o r e t i c a l curve f o r the orthorhombic model only). - 63 - From the knowledge of the relative density at any stage of compaction and from the slope of the curve D vs. t at that point, the values of strain-rates were calculated and these were tabulated in Appendix 4. It can be seen that the strain rates used for hot compression experiments of Pb-2% Sb and Ni are similar to the values of strain rates at the intermediate stage of densification, although at the i n i t i a l stage of densification, the strain rates were very much higher than that used for hot-working experiments. It is also at the i n i t i a l stage, a deviation between the experimental data and the theoretical plots was encountered. CHAPTER V DISCUSSION V. 1. E f f e c t i v e Stress During Hot-Pressing 19 Since the f i r s t t h e o r e t i c a l work of Murray, Livey and Williams, on the p l a s t i c flow theory of hot-pressing, the proponents of s t r e s s - 20 enhanced d i f f usional creep (the Nabarro-Herring creep ) have contended that any contribution by p l a s t i c flow to the o v e r a l l d e n s i f i c a t i o n must be small since, the stress on the system during hot-pressing i s generally low. It has been recognized l a t e l y by most workers, that the e f f e c t i v e stress i n hot-pressing i s d i f f e r e n t from the applied pressure and that t h i s i s a function of porosity. M c C l e l l a n d , 1 7 the f i r s t person to recognise t h i s and to take into account porosity, introduced the following form of e f f e c t i v e stress P a a e f f 2/3 P where P q i s the applied pressure and V i s the volume f r a c t i o n porosity. 26 Another form of e f f e c t i v e stress was introduced by V a s i l o s et a l . for applying to c r y s t a l l i n e materials. a r c = P (1 + 2V ) e f f a p - 65 - This was adopted from the change of e l a s t i c modulus with porosity as predicte< t h e o r e t i c a l l y and observed experimentally. Rossi and Fulrath generalised 26 the form used by Vasilos et a l . as follows a = P (1 + b V ) e f f a p' where b i s an empirical constant C - 2 ) . Subsequently a large number of workers (Fairnsworth and Coble,^, Coble, Fryer, and Koval'Chenko and Samsonov ) used the following form of e f f e c t i v e stress equation P a e f f = r f where D i s the r e l a t i v e density. Some of these r e l a t i o n s are plo t t e d i n Figure 23. I t can be seen from t h i s f i g u r e that the maximum e f f e c t i v e s t r e s s i s no more than a factor of two larger than the applied stress on a compact even at a r e l a t i v e density of 0.5. On the basis of th i s evidence i t has been suggested that the c o n t r i - bution to d e n s i f i c a t i o n by p l a s t i c flow during hot-pressing i s quite small, as the contact stress i s not large enough to cause the material to deform. Moreover, from the theory and experiments of s e l f - indentation, i t i s well-known that a ma t e r i a l can only deform i f the contact stress i s about 3 times the y i e l d stress of the material ("yield c r i t e r i o n " ) . For th i s reason, workers i n t h i s f i e l d tend to assume that any contribution by p l a s t i c flow has to be very small.  - 67 - However, a l l the equations for the e f f e c t i v e s t r e s s , r e f e r r e d to above, have not been rigorously derived. A l l of them are eit h e r empirical or obtained from f i t t i n g experimental data into an a r b i t r a r y equation. Attempts to formulate a rigorous equation have not been very successful as the l o c a l i s e d stresses w i l l vary both with respect to the volume f r a c t i o n of porosity and to the pore shape, which i s not known. The whole problem w i l l be further complicated i f the p a r t i c l e s i n a si n g l e system are random-shaped and vary i n s i z e . On the other hand, the problem can be s i m p l i f i e d by assuming the p a r t i c l e s are monosized spheres and i f t h e i r geometry of packing i s known. The stress i s hydrostatic i n nature and each shpere i s being deformed at i t s points of contact uniformly. Under these conditions,the e f f e c t i v e stress (a ,-,-) with respect to the r e l a t i v e density of the err compact i s given by equation (31) a ° e f f = , n2/32/32 c^CD 3 R -1) The values of ct^ and 3 (geometric constants) have already been given i n Table III for four d i f f e r e n t geometric models. The r e l a t i o n s h i p between D and R was calculated using the constant volume equation and i s tabulated i n Appendix 1. From these the e f f e c t i v e stress with respect 0" cc ef f to the applied stress (——) as a function of r e l a t i v e density (D) has been calculated. The values are given i n Appendix 3 and are p l o t t e d i n Figure 23. It i s quite apparent from t h i s f i g u r e that the stress e f f e c t i v e at the points of contact i s very much la r g e r than that assumed by previous workers. For example,below 0.79 r e l a t i v e density the - 68 - e f f e c t i v e stress i s more than 3 times the applied s t r e s s f o r a l l the geometric models. The instantaneous stress at the contact points of spheres i s extremely high and drops very r a p i d l y to about 10 times with an increase i n r e l a t i v e density of les s than 0.10. From above,it i s apparent that the contact stresses with respect to the applied stress i n a compact are more than an order of magnitude larger than o r i g i n a l l y thought. This implies that depending upon the magnitude of the applied s t r e s s , the e f f e c t i v e stress i s reasonably high,upto•• a compact density of 0.85 to 0.90 (depending upon the packing geometry). I f i t i s accepted that the packing configuration of spheres i n a die approximates to the orthorhombic packing, then the e f f e c t i v e stress i s more than three times the applied stress up to 0.85 r e l a t i v e density of a compact. It w i l l be informative at t h i s stage to analyse the magnitude and e f f e c t of contact stress i n an oxide compact. Compacts of alumina are generally hot-pressed i n the temperature range 1400-1700°C under a pressure of 2000 to 6000 p s i . The y i e l d stress of alumina as a function of temperature has been reported by Kronberg"^ and i s shown i n Table VI. It i f i s considered that the hot-pressing of alumina spheres was done at 1600°C under 4000 p s i and i f i t i s considered that the packing geometry i s that of the orthorhombic model, the contact stress w i l l be i n the order of 16,000 p s i at 0.80 r e l a t i v e density of the compact, which i s much more than three times the y i e l d stress f o r basal s l i p i n alumina. In the present i n v e s t i g a t i o n however, an end-point r e l a t i v e density of 0.75 was obtained at 1600°C under 4000 p s i . This discrepancy may. be explained from the fac t that the resolved y i e l d stress - 69 - Table V i Y i e l d Stress of Single C r y s t a l Sapphire (Basal S l i p ) Temperature (°C) 1400 1500 1600 1700 S t r a i n Rate (in/in/min) 0.05 0.05 0.05 0.05 Y i e l d Stress (psi) .. 8500 6000 4500 3000 - 70 - for the rhombohedral s l i p and the rhombohedral twinning are much higher than the resolved y i e l d stress for basal s l i p and only 3 basal s l i p systems are av a i l a b l e for the deformation of alumina under these experimental conditions. It should be mentioned here that the e f f e c t i v e stress calculated t h e o r e t i c a l l y using equation (31) i s most l i k e l y the upper l i m i t of the stress acting on the contact faces. "In p r a c t i c e , some of the applied stress w i l l be l o s t i n die-wall f r i c t i o n and also i n creating the large f l a t face (indentation) on the top and bottom layer of spheres immediately a f t e r the a p p l i c a t i o n of load, as has been observed by 13 Kakar. Thus, i t i s possible that the e f f e c t i v e s t r e s s i n any p a r t i c u l a r system may not be as high as that predicted by equation (31). V.2. A c t i v a t i o n Energy Study An Arrhenius type of equation can be wr i t t e n f o r the constants A i n equation (32) as follows A = A' exp(-Q/RT) where A' i s a temperature independent constant and Q i s the a c t i v a t i o n energy f o r the process, R and T have t h e i r usual meaning. From the slope of a l n A vs. 1/T p l o t , the a c t i v a t i o n energy for the process can be obtained. These plo t s f o r lead and n i c k e l are shown i n Figures 24 and 25, re s p e c t i v e l y . For these p l o t s , the value 'A' for lead at 200°C and that of n i c k e l at 700°C were obtained from the experimental d e n s i f i c a t i o n curve. As can be seen from these figures, these values of 'A' (back calculated from the d e n s i f i c a t i o n data) l i e on the stra i g h t l i n e drawn through the other points. The - 71 - Figure 24. ln A vs. 1/T plot for Pb-2% Sb (the point represented by a dot with a ring is from the densification curve at 200°C). - 72 - Figure 25. l n A vs. 1/T p l o t for Ni (the point represented by a dot with a r i n g i s from the d e n s i f i c a t i o n curve at 700°C). a c t i v a t i o n energy obtained from Figure 24 for lead was 28.7 Kcal/mole which i s close to the a c t i v a t i o n energy for s e l f - d i f f u s i o n of lead (26.9 Kcal/mole). S i m i l a r l y the a c t i v a t i o n energy calculated from Figure 25 for n i c k e l was 57.5 Kcal/mole. This value i s also close to the a c t i v a t i o n energy for hot-working of n i c k e l (71 Kcal/mole)."'""'" V.3. Packing and Deformation Geometry Inside a Die V.3.a. Packing Geometry It i s seen from a l l the Figures "14., 15 and -̂ 1 that the experi- mental data follows c l o s e l y the t h e o r e t i c a l l y derived equation for the hexagonal prism model. This agreement indicates that the o v e r a l l packing geometry of monosized spheres i n a die may be s i m i l a r to the orthqrhombic packing. When a die i s randomly f i l l e d with monosized spheres, with intermittent shaking and tapping i n order to achieve a uniform packing the spheres tend to spread l a t e r a l l y to achieve the most stable configuration. However, the die-wall o f f e r s resistance to l a t e r a l spreading. As a r e s u l t , a c e r t a i n degree of s t a b i l i t y i n packing i s 18 maintained i n spite of an unstable configuration. McGeary studied the various modes of f i l l i n g the die and the e f f e c t of container s i z e on packing density. His r e s u l t s are shown i n Figure 26. When the r a t i o of the die diameter to the sphere diameter i s greater than 10, the packing density of the compact reaches a maximum of 62.5% of the t h e o r e t i c a l density. This value i s close to the as-compacted density for the orthorhombic packing. 24 Smith, Foote and Busang studied the coordination number of spheres - 74 - 70 D / container diameter d \ sphere diameter Figure 26. E f f e c t of container s i z e on the e f f i c i e n c y of 18 packing one-size spheres (after McGeary ). - 75 - i n a die a f t e r shaking and tapping. Their r e s u l t s showed a Gaussian d i s t r i b u t i o n of the number of sphere with a given coordination number. The average coordination number of the sphere was close to 8. These r e s u l t s further confirm that the o v e r a l l packing geometry i n a die approximates to that of the orthorhombic model. V.3.b. Deformation Geometry The width of the die does not permit an i n t e g r a l number of spheres across the diameter; hence the i d e a l packing discussed i n the above section does not e x i s t across the diameter of the die since a cer t a i n number of spheres are l i g h t l y held against the die-wall. The loosely held spheres would rearrange as soon as a load i s applied, r e s u l t i n g i n a higher r e l a t i v e density. As no fragmentation was observed i n metal compacts t h i s increase i n density can be a t t r i b u t e d to p a r t i c l e rearrangement. However, i n the case of sapphire (Al^O^) some fragmentation was noted a f t e r the experiments. This fragmentation associated with rearrangement might have contributed to d e n s i f i c a t i o n 13 at the i n i t i a l stage. In addition, Kakar observed an increase i n the average coordination number of deformed spheres of lead, as the bulk density of a compact of lead spheres was increased during hot-pressing as shown i n Figure 27. This indicates that p a r t i c l e rearrangement occurs as d e n s i f i c a t i o n of a powder-compact proceeds, e s p e c i a l l y at the i n i t i a l stage of compaction. On further loading, the p a r t i c l e s begin to deform. Although i n d i v i d u a l colonies of rhombohedral or tetragonal deformation were observed, the majority of the spheres showed a hexagonal prism model of deformation. 7 ' ' " 1 1 U 0 2.0 4.0 6.0 8.0 10.0 ( D - Do) Figure 27. Coordination number vs. (D - D ) for Pb-2% Sb at various temperatures (after K a k a r 1 3 ) . This i s reavealed i n Figures 28 and 29 which represent n i c k e l spheres deformed at 800 and 900°C. As can be seen from the f i g u r e s , the coordination i n one,plane during deformation i s 6 and the deformed sphere took the form of a hexagonal prism. It has already been noted i n the density vs. time curves that there i s a disagreement between the t h e o r e t i c a l curves and experimental data, p a r t i c u l a r l y at the i n i t i a l stage of compaction. This disagree- ment can be explained from the arguments above. In the t h e o r e t i c a l d e r i v a t i o n , no consideration was given to account f o r any increase i n density due to the change of coordination number. There i s experimental evidence i n t h i s study and' also 12 reported by Kakar and Chaklader, that p a r t i c l e rearrangement and fragmentation (for oxides) are s i g n i f i c a n t contributing factors to d e n s i f i c a t i o n at the i n i t i a l stage of hot-pressing. This i s e s p e c i a l l y true immediately after the a p p l i c a t i o n of load when the e f f e c t i v e stress at the points of contact may be very high. The experimentally observed increase i n density, i n the f i r s t few minutes of compaction which was always greater than the t h e o r e t i c a l l y predicted increase, may be explained as follows: In the i n i t i a l stage of hot-pressing, the increase i n density i s due to two factors ( i ) indentation at the points of contact and ( i i ) p a r t i c l e rearrangement leading to a higher average coordination number per sphere. I t has not been possible at present to incorporate the increase i n density due to the second e f f e c t , into the t h e o r e t i c a l equations. In addition, to account f o r the deviation,the e f f e c t of the existence of very high s t r a i n rates, at the i n i t i a l stage of hot pressing,' (as seen experimentally), should be taken into consideration. _ 78 _ 2162 p s i . (80x) _79- Figure 29. Microstructure of Ni spheres hot-pressed at 900°C and 1477 p s i . (95x) -80- V.4 Limitations of the Present Analysis 1. In the t h e o r e t i c a l consideration, the stress i s considered to be hydrostatic, but i n p r a c t i c e a u n i a x i a l stress i s applied i n hot- pressing. However, due to the existence of back stress from the die- vall and the other plunger, the stress may be assumed to be i s o t r o p i c However, this assumption may not be a good one i n p r a c t i c e . 2. Although the same heat-treatments were given for the b a l l s used i n hot-pressing and the specimens used i n hot-compression, the structure of these may not be the same. Since the constant 'A' i s a function of structure, the values are subjected to unknown er r o r . 3. In the hot-compression of Pb-Sb and Ni specimens, the f r i c t i o n between specimen and the alumina buttons was not taken into consideration. This may give r i s e to some errors i n evaluating the value of steady- state stress and hence the values of 'A' and 'n'. 4. The compostion of the Pb-Sb a l l o y was o r i g i n a l l y considered to be Pb-2% Sb (manufacturer's composition) but i t was found l a t e r that i t was a c t u a l l y Pb-15% Sb. Since t h i s i s a hyper-eutectic alloy,the power law. dependence of steady-state s t r a i n rate on s t r e s s , giving the values of *A' and 'n' shown i n Table IV may not be completely v a l i d . However, i n the temperature and s t r a i n rate range used i n t h i s i n v e s t i g a t i o n , super- p l a s t i c behaviour would not be expected, and i t has been assumed the values of 'A' and 'n' are r e l i a b l e from the compression t e s t s . 5. In the theory, the bulk s t r a i n - r a t e w i t h i n the compact i s used, based on the instaneous density. I t would be more correct to consider -81- the strain-rate localised i n the contact areas, If i t -were possible to characterise It. The error introduced i n this way i s unknown, but i s not believed to be appreciable beyond the early stages of densification. CHAPTER VI SUMMARY AND CONCLUSIONS Isothermal densification curves of a powder compact during hot- pressing have been theoretically calculated using the geometry of deformation of particles and hot-compression data. The particles are assumed to be monosized spheres. Four different deformation models were considered: Cubic (Z = 6), orthorhombic (Z = 8), b.c.c. (Z = 8), and rhombohedral (Z = 12) where Z is the coordination of the sphere. For hot-compression an equation of type e = Aa was used. The f i n a l densification equation relating the relative density and time for different ideal packing arrangements has been derived which is The equation was solved i n a computer to obtain the theoretical plots. However, in order to use this equation the values for material constants (A and n) were necessary and were determined by hot-compression tests in an Instron machine. Three different materials were used for this purpose - these are Pb-2% Sb, Ni and A^O^. Theoretical curves for a l l four different geometric models were generated by the computer. o -83 - These t h e o r e t i c a l curves were compared with hot-pressing data of spheres of the same materials at d i f f e r e n t temperatures and pressures. The following conclusions can be made (1) The general t h e o r e t i c a l equation proposed i s found to be obeyed by the s p h e r i c a l p a r t i c l e s during the intermediate stage of hot-pressing. (2) The experimental points follow c l o s e l y the t h e o r e t i c a l l y predicted curve for the hexagonal prismatic deformation model. (3) This indicates the o v e r a l l packing geometry of sphere i n s i d e the die coincides with the orthorhombic packing i n agreement with the observation of previous workers. (4) A deviation was encountered at the i n i t i a l stage of d e n s i f i c a - t i o n , which could be explained from p a r t i c l e rearrangement at the beginning of hot-pressing as was observed i n th i s study and by other wo rker s p r e v i ously. A t h e o r e t i c a l equation f o r c a l c u l a t i n g the e f f e c t i v e stress acting on the contact faces i n a compact of spheres has been derived. This i s : a apjj ° e f f ~ a l ( D 2 / V / 3 R 2 - l ) Theoretical plots of r e l a t i v e e f f e c t i v e stress as a function r e l a t i v e density were computed f o r 4 d i f f e r e n t geometric models. When these are compared with the e f f e c t i v e stress plots used by previous workers, i t was observed that the actual e f f e c t i v e stress i s very much higher than that considered so f a r . - 84 - From above, i t is concluded that the contribution to densification during hot-pressing by plastic flow is more than that considered by previous investigators. CHAPTER V I I SUGGESTIONS FOR FUTURE WORK 1. M e t a l l o g r a p h i c s t u d y o f h o t - p r e s s e d compacts s h o u l d be done f o r c h e c k i n g t h e change o f R w i t h r e s p e c t t o change i n D i n a d i e as R i s a v e r y c r i t i c a l f u n c t i o n f o r e f f e c t i v e s t r e s s . The e x p e r i m e n t a l l y d e t e r m i n e d R s h o u l d be compared w i t h t h e t h e o r e t i c a l R and n e c e s s a r y c o r r e c t i o n t o the d e n s i f i c a t i o n e q u a t i o n s h o u l d be made f o r b e t t e r u n d e r s t a n d i n g . 2. The v a l u e s o f e f f e c t i v e s t r e s s a t d i f f e r e n t s t a g e s o f d e n s i f i c a t i o n s h o u l d a l s o be e v a l u a t e d e x p e r i m e n t a l l y . T h i s can be done by u s i n g h e m i s p h e r i c a l specimens. 3. From the above e x p e r i m e n t t h e y i e l d c r i t e r i o n a t an e l e v a t e d t e m p e r a t u r e s h o u l d be c h e c k e d . ( Y i e l d c r i t e r i o n r e f e r s t o " S t r e s s r e q u i r e d t o deform by s e l f - i n d e n t a t i o n i s 3 t i m e s t h e y i e l d s t r e s s " . ) 4. I f s t r a i n r a t e dependence on . s t r e s s i s g i v e n by t = A { S i n h(a^a^)}n, one can w r i t e a d e n s i f i c a t i o n e q u a t i o n by f o l l o w i n g the same t r e a t m e n t as done i n C h a p t e r I I , as f o l l o w s I -86- D t max / D"1 Sin h [ ~ . - ] dD =| a i ( D 2 / V / 3 R 2 - D J A dt A computer programme for this is given in Appendix 5 . By knowing the constants A, 0 2 and n for different materials, the validity of this equation should be determined. APPENDICES APPENDIX 1 Theoretical Calculations 1. Cubic Model R Bulk Density (%) 0.620350 52.360 0.620352 52.556 0.620374 53.149 0.620469 54.145 0.620728 55.565 0.621282 57.423 0.622307 59.749 0.624038 62.576 0.626787 65.937 0.630972 69.869 0.637180 74.393 0.646268 79.498 0.659573 85.085 0.679356 90.843 0.709891 95.936 0.715532 96.506 - S 9 " 2. Hexagonal Prismatic and Tetrakaidecahedron Models R Bulk Density (%) Bulk Density (%) Hex. Prism. b.c.c. model 0.620350 60.460 68.017 0.620352 60.687 68.273 0.620382 61.369 69.040 0.620509 62.511 70.325 0.620854 64.121 72.137 0.621593 66.207 74.483 0.622964 68.774 77.371 0.625287 71.824 80.802 0.628993 75.340 84.757 0.634679 79.272 89.181 0.643213 83.507 93.946 R Bulk Density (%)[Z = 14] 0.669404 99.450 -90- Rhombic Dodecahedron Model R Bulk Density (%) 0.620350 74.048 0.620353 74.325 0.620397 75.156 0.620588 76.531 0.621107 78.437 0.622218 80.842 0.624288 83.697 0.627815 86.908 0.633499 90.317 0.642361 93.646 0.655600 96.410 - 91 - APPENDIX 2 Hot-Compression Data 1. Lead-2% Antimony - 100°C e (sec "*") a (psi) 3.9401 x IO - 5 5257 9.8502 x 10~5 6507 1.9700 x 10~4 7646 3.9401 x IO - 4 8986 2. Lead-2% Antimony - 150"C e (sec 1) a (psi) 3.4972 x 10"5 1934 8.7443 x IO - 5 2398 1.6666 x 10~4 2797 2.9682 x IO - 4 3195 - 92 - 3. Nickel - 800°C • -1 e (sec ) a (psi) -5 3.4904 X 10 6387 -5 7.4904 x 10 7461 -4 1.5873 X 10 4 8817 2.9629 X IO"4 10070 4. Nickel - 900°C e (sec "*") a (psi) 3914 5171 5845 6446 2.6666 x 10 9.8619 x 10' 1.7543 x 10' 2.7777 x 10' - 93 - APPENDIX 3 Effective Stress for Different Models Cubic Model CTeffective ^applied 0.52 0.53 0.55 0.59 0.62 0.65 0.69 0.74 0.79 0.85 2666 120 30.22 12.80 9.08 6.66 5.0 3.81 2.93 2.25 - 94 - Hexagonal Prism Model a D effective a applied 0.60 2414 0.61 104.3 0.62 46.9 0.64 26.1 0.66 16.41 0.68 11.68 0.71 7.8 0.75 5.769 0.79 4.33 0.83 3.3 - 95 - 3. B.C.C. Model " e f f e c t i v e °applied 0.68 1557 0.69 69 0.70 31.2 0.72 17.42 0.74 10.93 0.77 7.39 0.80 5.2 0.84 3.8 0.89 2.8 0.93 2.2 0.99 1.46 - 96 - Rhombic Dodecahedron Model e f f e c t i v e a p p l i e d 0.74 0.75 0.76 0.78 0.80 0.83 0.86 0.90 0.93 0.96 959.5 42 19.14 9.882 6.69 4.52 3.210 2.355 1.774 1.354 - 97 - APPENDIX 4 STRAIN RATES AT DIFFERENT STAGES OF DENSIFICATION 1. (a) Lead-2% Antimony - 100°C, 918 psi Relative Density Strain Rate (D) (sec - 1) 0.64 1.017 x 10"2 0.66 1.449 x 10"3 0.68 2.785 x 10~4 0.71 6.376 x IO - 5 0.75 1.796 x 10~5 (b) Lead-2% Antimony - 150°C, 918 psi Relative Density Strain Rate (D) (sec" 1) 0.66 1.241 x IO" 1 0.68 2.384 x 10 _ 2 0.71 5.459 x IO" 3 0.75 1.538 x IO - 3 0.79 4.609 x 10"4 0.83 1.472 x 10"4 - 98 - 2. (a) Nickel - 800°C, 2162 psi Relative Density Strain Rate (D) (sec - 1) 0.66 3.913 x IO - 2 0.68 0.426 x 10"3 0.71 1.278 x 10~3 0.75 3.192 x IO - 4 0.79 8.59 x IO"5 0.83 2.444 x IO - 5 Cb) Nickel - 900° C, 2105 psi Relative Density Strain Rate CD) (sec - 1) 0.66 2.551 x 10 _ 1 0.68 4.206 x 10~2 0.71 8.357 x IO - 3 0.75 2.086 x IO - 3 0.79 5.575 x 10 _ 4 0.83 1.598 x IO"4 - 99 - 3. Alumina - 1600°C, 4000 psi Relative Density Strain Rate (D) (sec - 1) 0.64 2.375 x 10 _ 2 0.66 3.703 x IO" 3 0.68 7.716 x IO - 4 0.71 1.895 x IO"4 0.75 5.635 x 10~5 0.79 1.799 x IO - 5 0.83 6.071 x 10~6 APPENDIX 5 COMPUTER PROGRAMME RFS NU^ 010678 UNIVERSITY OF 6 C COMPUTING CENTRE MTS(AN1 20) 1 1 :32:01 08-18-71 ******************** PLEASE RETURN TU ELECTRICAL ENGINEERING ******************** SSIG XRSA P=8 T=1M PRIO=V **LAST SIGNUN WAS: 11 :33 :35 07-13-71 USER "XRSA" SIGNED ON AT 11:32:07 UN 08-1B-71 > JUb l H L b l READY. SL1S FILE! 1 INTEGER ST0RE1, STORE 2, CASE 2 REAL N 2.26 ~ 3 PAT A T AKE 1 .TAX E2 , T AKE3,TA*E 4 /'G EN E r~,^rR A T T 1 STgC » 1 A L " TJ : 4 READI5.1) N t BETA, A, SIGMA, ALPHA, DELTAT, ALPHA 2 5 1 FORMAT <2F10.4,F10.5,4F10.4) 6 dRITE (6 , DN, BETA, A, SIGMA, ALPHA, DELTAT, ALPHA2 7 CASE=2 8 IFIABSIALPHA2).GT.0.00005) CAsE=l ~~9 rALUtl = UUfcSI 1( IAKb1, IAKE3,CASE) 10 VALUE2 = QUEST2( TAK.E2, TAKE4.CASE) 10. 1 10.2 11 WRITE(6,102) VALUE1, VALUE2 12 102 F0RMAT(///40X'THIS CORRESPONDS TO THE ',2A4, 'FORMULA CASE') Ti C NUW KtAU IH-E T7ABTF"PF R"~V5—DEVALUES 14 DIMENSION X( 16) ,H 16) ' 15 DATA X,Y/16*0.0, 16*0.0 / ,_, 16 WRITE(6,32) O 17 32 FORMATI5X,'TABLE OF D VERSUS R VALUES') F-1 18 DO 2 12=1,16 I T 9 REAU(5,3) U,R,Z : 20 3 F0RMATI3F10.4) 21 WRITEI6,31)D,R 22 31 FORMAT(5X,E14.7,5X,E14.7) 23 X(I2) = D 24 Y(I2)=R - t t I F I Z . 6 T . 0 . 1 ) G O T U 4 : v 26 2 CONTINUE 27 4 I2MINS=I2-1 28 JTIMES=I2MINS*2 29 ISUM=12 30 OMAX=D "31 DTSrOT=XTT) 32 DELTAD=IDMAX-DNOT)/JTIMES 33 W R I T E ( 6 , 5 ) DMAX,OELTAD,ISUM 34 5 FORMAT(/5X,'DMAX = ' , F 10 . 4 , 5X , ' D ELT AD = 1 , F 1 0 . 5 , 5 X , • T 0 T A L NUMBER OF 35 1CARDS OF D VS R READ = ' , I 4 ) 36 SUM1=0.0 "37 U=UNU1 38 JT IMES=JT IMES-1 39 DO 6 I6=1,JTIMES 40 FACT3=D*BETA 41 FACT4=FACT3**0.667 42 FACT 1 = 1(0*BETAI**0.667 I*R*R-1.0 " S T b A L T Z = I F A C T l * * ' N 7 7 D 43 .1 4 4 IFICASE.E0.2) GO TO 150 4 5 VALUE=1.0/(SINH((ALPHA2/ALPHA)*SIG MA/FACT 1) ) ** N 46 FACT2=VALUE/D 46. 1 — <rtrr2— / 150 CONTINUE 48 SUM1 = SUM1 + FACT2*DELTA0 49 D=D+DELTAD 50 ST0RE1=1 51 M=0 > — 5 2 DO 7 I7=U,ISUM • 53 IF ( A B S U l 171-0) . L E . 0.0002) M= 1 7 54 I F ( X ( 1 7 ) . L T . D ) 5T0RE1 = 17 55 I F ( X ( I 7 ) . G T . U I GO TO 9 56 7 CONTINUE 57 GO TO 10 5B 9 SIUKE 2=I I 59 10 IF (H.EU.O) GO TO 11 60 R=YIM) 61 GO TO 12 6Z 11 DELT = D-XI STORED 63 R=Y< ST0KE11 + I Y I ST0RE2) -Y I STORE l I )*OELT / ( X( S T0RE2) -X( S T ORE 1 ) ) tm WRITEI6.109) CJ,R 65 109 F U R M A T ! / ' D = ' F 1 0 . 7 . 5 X , ' R = ' , F 1 0 . 7 ) 66 12 WRITE(6,13) SUM1 67 6 CONTINUE 68 13 F O R M A T ! ' S U M 1 = » E 1 6 . 9 ) 69 CONS=A* (<S IGMA/ALPHA)**N I * (10. * * ( -20) ) 7-t) IF (CASE. EQ. 1) CONS=A<- ( 10. * « ! - 8 ) ) 71 SUM2=0.0 72 FACT = SUMl/(CONS*OELrAT) ' ^ 73 NCOUNT=0 ,_, 74 00 152 1152=1,100 O 75 I F I F A C T . L T . 1 1 0 0 0 . 0 ) ) GO TO 151 ^» Tt FACT = FACT/10 .0 ; 77 NCUUNT=NCOUNT+1 78 152 CONTINUE 79 151 NLIM = F ACT 80 IF INL IM.LT.4 ) NLIM=4 81 NL1 =NLlM/4 8-2 r F T m l . L I .2T~NL1=1 " : 83 DU 14 I 14=1,NLIM,NL1 84 TIME= I 14*C)ELTAT * ( 10. **NCOUN r) 85 SUM2=TIME*CONS 86 WRITE16,16)TIME,SUM2 87 16 FORMAT I / 'T IME = ' E 1 2 . 5,5X, 'SUM2= ' , E 1 2 . 5) 8-8 1-4 CUNT I NU6 : " 89 STOP 90 END 91 FUNCTION QUEST 1( TAKE 1, TAKE3 , CAS E) 91 .1 INTEGER CASE 92 QUE ST1 = T AKE1 ^ IF 1CASE.EQV2) QUEST 1-TAKE3 — 94 RETURN 95 END 96 FUNCTION QUEST2(TAKE2, TAKE 4,CAS El 96.1 INTEGER CASE 97 QUEST2= TAKE 2 -9^ TF1 tWSt iT:Q-; 2T-(TOTrSTZ=TAKE4 : 99 RETURN 100 END END UF F ILE - 103 - BIBLIOGRAPHY 1. R. Chang and C.G. Rhodes, J. Am. Ceram. Soc. , 4-5, 379 (1962). 2. R.L. Coble and J.S. E l l i s , J. Am. Ceram. Soc. , 46, 438-41 (1963) 3. R.L. Coble, in "Sintering and Related Phenomena", Ed., G.C. Kuezynski, N.A. Hooton and C.F. Gibbon (Gordon and Breach, Science Publishers, New York), (1967), p. 329. 4. R.L. Coble, J. App. Phys., 41, 4798 (1970). 5. P.L. Farnsworth and R.L. Coble, J. Am. Ceram. Soc., 49, 264 (1966). 6. E.J. Felten, J. Am. Ceram. Soc., 44, 381, (1961). 7. G.M. Fryer, Trans. Brit. Ceram. Soc., 66, 127 (1967) and 68, 181-185 (1969). 8. J.K. Mackenzie and R. Shuttleworth, Proc. Phys. Soc. (London), B, 62, 833-852 (1949). 9. H. HeiErky, Z. Angew. Math. Mech., _3> 241 (1923). 10. J. Ishlinsky, J. App. Math. Mech. U.S.S.R. , {5, 233 (1944). 11. J.J. Jonas, CM. Sellars and W.J. McG. Tegart, in "Metallurgical Reviews", vol. 14, Review 130, 1 (1969). 12. A.K. Kakar and A.CD. Chaklader, J. App. Phys. , J38, 3223-30 (1967). 13. A.K. Kakar "Deformat ion Theory of Hot—Pressing", Ph.D. Thesis, Department of Metallurgy, University of British Columbia (1967). 14. M.S. Koval'Chenko and G.V.Samsonov, Poroshkovaya Met., 1, 3 (1961). . 15. M.L. Kronberg, J. Am. Ceram. Soc., 45, 6, 274 (1962). 16. G.E. Mangsen, W.A. Lambertson and B. Best, J. Am. Ceram. Soc., 43, 55 (1960). -104- 17. J.D. McClelland, In "Powder Metallurgy Proceedings of International Conference, New York, 1960, W. Leszynski, Ed. (Interscience Publishers, New York, 1961). 18. R.K. McGeary, J. Am. Ceram. Soc., 44, 513 (1961). 19. P. Murray, D.T. Livey and J. Williams, in"Ceramic Fabrication Process", W.D. Kingery, Ed. (Technology Press Cambridge, Mass., and John Wiley and Sons, Inc., New York, 1958). 20. F.R.N. Nabarro, in the "Strength of Solids", London (Physical Society), 75 (1948). 21. R.C. Rossi and R.M. Fulrath, J. Am. Ceram. Soc., 48, 558-64 (1965). 22. D.R. Rummler and H. Palmour, J. Am. Ceram. Soc. , _51, no. 6, 320 (1968). 23. S. Scholz and B. Lersmacher, Arch. Eisenhuettenw, 41, 98 (1964). 24. W.O. Smith, P.D. Foote and P.F. Busang, Phys. Rev., 34, 1272 (1929). 25. T. Vasilos, J. Am. Ceram. Soc. , 4_3, no. 10, 517 (1960). 26. T. Vasilos and R.M. Spriggs, J. Am. Ceram. Soc. , 46_, 493-96 (1963). 27. S.I. Warshaw and F.H. Norton, J. Am. Ceram. Soc., 45, 479 (1962).

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