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Deformation enhanced grain growth in a superplastic Sn - 1% Bi alloy 1971

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DEFORMATION ENHANCED GRAIN GROWTH IN A SUPERPLASTIC Sn - 1% B i ALLOY by MALCOLM ARTHUR CLARK B.A.Sc, Uni v e r s i t y of Toronto, 1966 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 stan.dard THE UNIVERSITY OF BRITISH COLUMBIA February, 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 t he 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 ee t h a t t he 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 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 pu rpo se s may be g r a n t e d by the Head o f my Department 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 1 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 . Department o f M e t a l l u r g y The U n i v e r s i t y o f B r i t i s h Co l umb i a Vancouve r 8, Canada Date A p r i l 2 3 , 1971 i ABSTRACT A Sn - 1% Bi a l l o y has been studied to determine the e f f e c t s of superplastic deformation on the g r a i n growth k i n e t i c s . Using both constant crosshead speed and creep t e s t s , the g r a i n s i z e was measured as a function of deformation time and s t r a i n over a wide range of s t r a i n rates. It was found that during deformation, considerable i n - creases i n the grain growth rates occurred when compared to s t a t i c annealing. The e f f e c t was most pronounced at intermediate s t r a i n _2 rates (»=10 /minute) i n the high s t r a i n rate s e n s i t i v i t y region. However, the grain growth rates on annealing a f t e r deformation were found to be le s s than s t a t i c rates. To a i d i n understanding the mechanism of the enhanced growth, a l t e r n a t i n g tension-compression tests were performed. The amount of grain elongation and the changes i n preferred o r i e n t a t i o n with de- formation were also measured. Grain type and g r a i n s i z e d i s t r i b u t i o n s a f t e r deformation and a f t e r annealing were established and analyzed i n terms of a grain coalescence mechanism. However, the most favourable mechanism appears to involve the production of excess vacancies i n the g r a i n boundary region leading to increased boundary mobi l i t y . ACKNOWLEDGEMENT The author washes to express h i s gratitude f o r the advice and encouragement of h i s research supervisor, Dr. T.H. Alden. Thanks are also extended to f a c u l t y members, fellow graduate students and t e c h n i c a l s t a f f f o r h e l p f u l discussions and assistance. F i n a n c i a l assistance from the Defence Research Board (Grant No.9535-41) i s g r a t e f u l l y acknowledged. i i i TABLE OF CONTENTS Page INTRODUCTION 1 1.1 S u p e r p l a s t i c i t y 1 1.2 Deformation Induced Grain Growth 3 1.3 Purpose of the Present Investigation 7 EXPERIMENTAL 8 2.1 Specimen Preparation 8 2.2 T e n s i l e Testing 9 2.3 Creep Testing 10 2.4 X-Ray Analysis 12 2.5 Metallography 13 RESULTS 15 3.1 Superplastic Properties 15 3.1.1 Log a - Log e Curves 15 3.1.2 Stres s - S t r a i n Relationship 18 3.2 Grain Growth During Annealing 23 3.3 Grain Growth During Deformation 31 3.3.1 E f f e c t of Deformation Time 3 1 3.3.2 E f f e c t of S t r a i n 3Z> 3.3.3 Grain Elongation 3 9 3.3.4 Alte r n a t i n g Tension-Compression . . . . 49 3.3.5 Grain Type D i s t r i b u t i o n 43 3.3.6 Grain Size D i s t r i b u t i o n 44 i v Page 3.4 X-Ray Analysis Results 50 3.5 Grain Growth A f t e r Deformation 57 3.5.1 Growth by Annealing 57 3.5.2 Growth Enhanced by Deformation 57 DISCUSSION 66 4.1 S u p e r p l a s t i c i t y 66 4.2 Grain Growth 67 4.3 Mechanisms of Deformation Enhanced Grain Growth •• 71 4.3.1 Grain Coalescence 71 4.3.2 Increased Driving Force 74 4.3.2.1 R e c r y s t a l l i z a t i o n 74 4.3.2.2 Grain Boundary S l i d i n g 75 4.3.2.3 Grain Elongation 78 4.3.2.4 Grain Boundary Width . . . . . . 78 4.3.3 M o b i l i t y Enhancement 79 SUMMARY AND CONCLUSION 9 2 APPENDIX A ' 93 E f f e c t of Coalescence Mechanism on Grain Type and Size D i s t r i b u t i o n 93 BIBLIOGRAPHY 104 V LIST OF FIGURES Figure No. Page 1 T y p i c a l S t r e s s - S t r a i n Rate Relationship i n Superplastic Materials 1 2 Grain Size versus True S t r a i n f o r Two I n i t i a l Grain Sizes (from data by Alden and S c h a d l e r ( 1 4 ) ) 4 3 Grip Arrangement f o r Tension-Compression Testing on Instron H 4 E f f e c t s of Processing Conditions on Stress- S t r a i n Rate Relationship ^ 5 E f f e c t s of Processing Conditions on S t r a i n Rate S e n s i t i v i t y Parameter "m" ^ 6 Comparison of S t r e s s - S t r a i n Rate Data Be- tween Single S t r a i n Rate Change Test and 19 Individual Instron Tests 7 Comparison of S t r e s s - S t r a i n Rate Data Be- tween Single S t r a i n Rate Change Test and Individual Creep Tests 2 0 8 Comparison of S t r e s s - S t r a i n Rate Data Be- tween Present and Previous Investigations •• •• 21 9 True Stress-True S t r a i n Curves at Various I n i t i a l S t r a i n Rates ' " * ' * 22 10 Creep Curves at Stresses of 500, 350 and 250 p . s . i . 24 11 Creep Curves at Stresses of 150, 100 and 40 p . s . i . 2 5 v i Figure No. Page 12 Grain Size versus Time During S t a t i c Annealing of Low Ratio. Room Temperature Extrusion M a t e r i a l 27 13 Grain Size versus Time During S t a t i c Annealing of High Ratio Room Temperature Extrusion M a t e r i a l 28 14 Grain Size versus Time During S t a t i c Annealing of Low-Ratio-0°C Temperature Extrusion M a t e r i a l 29 15 Grain Size versus Time f o r D i f f e r e n t Processing Conditions 30 16 Grain Size versus Time During Instron Deformation 32 17 Grain Size versus Time During Creep Deformation i 33 18 Relative Grain Size Change versus S t r a i n During Instron Deformation 35 19 Relative Grain Size Change versus S t r a i n During Creep Deformation 36 20 Relative Grain Size Change A f t e r 25% St r a i n versus S t r a i n Rate 38 21 Relative Grain Size Change A f t e r 25% Str a i n versus "m" 40 22 Degree of Grain Elongation A f t e r 35% Str a i n at Various S t r a i n Rates - 41 23 St r e s s - S t r a i n Relationships During Tension- Compression Testing v i i Figure No. Page 24 Grain Type D i s t r i b u t i o n f o r Annealed and Deformed Structure 45 25 Cumulative Frequency Diagram for Grain Type D i s t r i b u t i o n of Fig.24 46 26 Grain Size D i s t r i b u t i o n f or Annealed and Deformed Structures . . . . 47 27 Cumulative Frequency Diagram f o r Grain Size D i s t r i b u t i o n of Fig.26 49 28 Grain Size D i s t r i b u t i o n f o r Annealed and Deformed Structures of Equivalent Mean Grain Size 51 29 Back R e f l e c t i o n Pattern for Extruded Rod . . 5 2 30 Location of Fiber Axis on Standard Stereo- graphic Projection of T i n ^4 31 (110) Pole Figure for Extruded Rod 5 5 32 Degree of Texture Before and A f t e r Deformation 56 33 Grain Size versus Time During and A f t e r -2 Deformation at e = 10 /min. 34 E f f e c t of S t r a i n on Grain Size versus Time Curves. A f t e r Deformation ^ 35 Comparison of Grain Size versus Time Curves f o r S t a t i c Annealing and Annealing A f t e r De- fin formation of 70% S t r a i n 36 Comparison of Grain Size versus Time Curves f o r S t a t i c Annealing and Annealing A f t e r De- formation of 7% S t r a i n ^ v i i i Figure No. Page 37 Comparison of Stress-Strain Rate Data for Annealed and Deformed Material 63 38 Effect of Pre-Deformation on Grain Size In- crease During Deformation 64 39 The Grain Coalescence Mechanism 72 40 Surface Grain Boundary Migration Associ- ated with Internal Boundary Slid i n g . . .. 76 41 Comparison of Theoretical and Experi- mental Grain Size versus Time During De- formation Curves 84 42 Comparison of Theoretical and Experimental Relative Grain Size Change versus Strain Curves 86 43 Comparison of Theory and Experiment for the Relative Grain Size Change After 25% Strain 87 44 Comparison of Theory and Experiment for Post Deformation Annealing 89 45 I l l u s t r a t i o n of Grain "Encounter" During Grain Growth 90 46 Effect of a Grain Boundary Coalescence . . . . 94 47 Comparison of Calculated and Experimental Grain Type Distributions 100 48 Comparison of Calculated and Experimental Grain Size Distributions 103 i x LIST -OF TABLES Table No. Page 1 Power Law Constants f o r Grain Size versus Time Curves 26 2 Power Law Constants f o r Grain Size versus Time Curves During Deformation 34 3 Straight Line Constants f o r Relative Grain Size Change versus S t r a i n Curves .. .• 37 4 Estimated Values f o r Terms of Equation (10) ( 4 5 ) 70 5 Areas of Coalesced Grains 101 1 INTRODUCTION 1.1 - S u p e r p l a s t l c i t y Superplastic materials are characterized by large amounts of neck-free elongation i n tension and by a high value of the s t r a i n rate s e n s i t i v i t y parameter, "m", defined as ^ ? . The log a - log £ r e l a t i o n s h i p i n su p e r p l a s t i c a l l o y s i s us u a l l y an "S" shaped curve which can be divided into three stages. ( F i g . l ) . ^ S T A G E HI LOG CT . / S T A G E U - ^ S T A G E T L O G € F i g . l . T y p i c a l S t r e s s - S t r a i n Rate Relationship i n Superplastic M a t e r i a l s . Although t h i s type of curve i s t y p i c a l , not a l l a l l o y s display a Stage I region within the range of experimentally obtainable s t r a i n rates ̂ \ The largest elongations occur i n the region of highest s t r a i n rate s e n s i t i v i t y , Stage I I . The amount of extension appears (2) to increase with the value of "m" . This r e l a t i o n s h i p has been (3) explained phenomenologically by Backofen et a l . The formation 2 of an i n c i p i e n t neck i n a deforming sample causes a l o c a l increase i n s t r a i n rate. When the s t r a i n rate s e n s i t i v i t y i s high, t h i s i n - creased s t a i n rate causes the necked region to harden so that de- formation w i l l tend to continue i n the s o f t e r region away from the neck. If c e r t a i n conditions are f u l f i l l e d , s u p e r p l a s t i c i t y w i l l occur i n a wide range of a l l o y s including two phase e u t e c t i c or eutectoid systems such as P b - S n ^ and Z n - A l ^ as w e l l as phase pure systems; S n - B i ^ 5 \ N i ^ and Pb-Th^ 7\ For most a l l o y s these con- d i t i o n s are: 1) an homologous temperature greater than .5 Tm, 2) a f i n e grain s i z e and 3) phases of s i m i l a r hardness. For the purpose of forming operations i t i s necessary that the s u p e r p l a s t i c properties be displayed at high s t r a i n rates. The major requirement for such behaviour i s a f i n e grain s i z e . An increase i n grain s i z e s h i f t s the "S" curve h o r i z o n t a l l y to the l e f t so that the high "m" region occurs at lower e . During a constant s t r a i n rate or constant crosshead speed (Instron) test above 0.5 Tm, the grain s i z e may increase i n some mater- i a l s . In t h i s case the o r i g i n a l value of "m" w i l l gradually change as the s t r e s s - s t r a i n rate r e l a t i o n s h i p s h i f t s on the e axis. A sample that was o r i g i n a l l y deforming i n Stage II could, a f t e r g r a i n growth occurs, be deforming i n Stage I I I . The amount of elongation obtained would then be decreased. Experimental r e s u l t s i n d i c a t e , i n f a c t , that the highest elongations are obtained i n a l l o y s with s u b s t a n t i a l amounts of second phase which anchor the grain boundaries and retard grain coarsening. The microstructure of s up er pl as ti c a l l o y s and the changes 3 which occur during deformation have been w e l l studied. Deformation i n Stage II i s accompanied by l a r g e r amounts of g r a i n boundary (27) s l i d i n g and grain r o t a t i o n than i s observed i n the Stage I I I region The grains r e t a i n t h e i r equiaxed shape even a f t e r large amounts of elongation. No d i s l o c a t i o n substructure i s formed by the deformatio Grain boundary migration i s observed on the specimen surface and i s associated with grain s l i d i n g . 1.2 - Deformation Induced Grain Growth A s t r u c t u r a l change, associated with s u p e r p l a s t i c i t y , which has not received d e t a i l e d attention i s an apparent enhancement (8) of grain growth during the deformation. In the Pb-Sn e u t e c t i c , the g r a i n s i z e a f t e r 600% elongation was considerably l a r g e r (4.4 u) compared to the grain s i z e (3 p) of a sample held at the t e s t i n g temperature for an equivalent length of time. The grain s i z e i n - creased l i n e a r l y with % elongation, with larger changes occurring at (9) lower s t r a i n rates. Morrison , also studied the Pb-Sn e u t e c t i c and produced micrographs displaying an increasing grain s i z e with s t r a i n . On creep t e s t i n g of the Pb-Sn e u t e c t i c , S u r g e s f o u n d a s t e a d i l y decreasing creep rate i n Stage I deformation which was t e n t a t i v e l y a t t r i b u t e d to e i t h e r d i f f u s i o n a l creep or grain growth. Other s t u d i e s ^ ' " ^ on Pb-Sn made no s p e c i f i c mention of grain growth e f f e c t s however. (14) In the eutectoid Zn-Al system, Alden and Schadler found that s i g n i f i c a n t grain growth occurred a f t e r large s t r a i n s . They deformed two samples of d i f f e r e n t i n i t i a l grain s i z e at the same s t r a i n rate i n Stage II u n t i l a neck began to form, then 4 sectioned the samples at various places along the neck and measured the g r a i n s i z e . In t h i s way specimens were obtained which had been deformed f or the same length of time but to d i f f e r e n t s t r a i n l e v e l s . The r e l a t i o n s h i p between g r a i n s i z e and s t r a i n appeared to be l i n e a r i n i t i a l l y (Fig.2). In the time taken f o r deformation the gr a i n s i z e would not have changed s i g n i f i c a n t l y with annealing only . Alden and Schadler reported no gr a i n elongation but noted increased boundary curvature with deformation which disappeared upon annealing. I 2 3 4 5 TRUE STRAIN Fig.2. Grain Size versus True S t r a i n f o r Two I n i t i a l Grain Sizes (from data by Alden and Schadler(14)), Kassawsky and B e c h t o l d a l s o studied the Zn-Al eutectoid system and t h e i r micrographs showed that growth of the z i n c - r i c h phase was occurring. Less growth appeared to take place at the lower s t r a i n rates. Micrographs and X-ray patterns revealed grain growth i n the (16) Zn-Al eutectoid i n a study by Chaudhari On the other hand, Packer and Sherby ̂ 7 >^8) r e p 0 r t e ( j l i t t l e change i n grain s i z e up to 1280% elongation although rounding of the grain boundaries did occur. (19) N u t a l l and Nicholson also reported l i t t l e g r a i n growth up to 500% elongation. In a study of the Zn-Al e u t e c t i c (95 wt.% Zn) , Packer, Johnson and S h e r b y ^ ^ found that the A l second phase l o s t i t s d i r e c t i o n a l nature produced on r o l l i n g and coarsened with super- p l a s t i c deformation. With a d i l u t e a l l o y of Zn-Al (.2 wt.% Al) C o o k ( 2 1 ) found a log-log r e l a t i o n s h i p between g r a i n s i z e and true s t r a i n up to elongations of 1000 %. In other words, growth i s rapid during the i n i t i a l stages of deformation but decreases as s t r a i n continues. (22) These r e s u l t s have been confirmed by Turner i n the 1 wt.% A l (23) a l l o y and by N a z i r i , Pearce and Williams i n the .4 wt.% A l a l l o y . (24) N a z i r i and Pearce investigated commercial p u r i t y zinc which had been r o l l e d to 90% reduction to produce a 1 to 2 micron grain s i z e . The s t r a i n rate s e n s i t i v i t y was only .2 and the maximum elongation obtained was only 200% so that the material was not con- sidered s u p e r p l a s t i c . However, a f t e r 100% elongation, the grains doubled i n s i z e , remained equiaxed and showed no evidence of s l i p , i . e . r e s u l t s s i m i l a r to those i n superplastic materials. S u p e r p l a s t i c i t y has been studied extensively i n a 45% (1 25 26) Ni, 38% Cr. 14% Fe two phase a l l o y by Brophy and co-workers ' ' In t h e i r f i r s t paper an increase i n grain s i z e with deformation was / o c Of.) noted and then studied more completely i n the two l a t e r papers ' They found that the matrix grain s i z e was dependent on the average s i z e and volume f r a c t i o n of the second phase p a r t i c l e s . The r e s u l t s were i n agreement with the Zener-McLean equation which predicts a l i m i t i n g g rain s i z e by considering the e f f e c t of second phase p a r t i c l e s on the grain boundary energy; Grain Size = ~ — V f where d i s the average p a r t i c l e diameter and i s the volume f r a c t i o n of second phase. With su p e r p l a s t i c deformation the diameter of the se- cond phase p a r t i c l e s increased roughly proportional to the power of the deformation time. The matrix grain s i z e increased accordingly so that the Zener-McLean equation was s t i l l obeyed. Lower s t r a i n rates produced a larger change i n grain s i z e f o r the same amount of s t r a i n . No theory was proposed to explain the coarsening of the second phase. Other superplastic a l l o y systems also display enhanced (27) grain growth with deformation. Alden noted that i n a Sn - 5% B i sample elongated 1000% the grain s i z e had increased to 4 microns whereas i t would have increased to only 1.6 microns on annealing (28) without deformation. In an Al-Cu e u t e c t i c a l l o y Stowell et a l found coarsening of the i n t e r m e t a l l i c CuAJ^ phase taking place with deformation. Their micrographs show evidence of s i n t e r i n g or coales- cence of some of the CuA^ p a r t i c l e s . They proposed that the coales- cence occurred as a consequence of grain boundary s l i d i n g which tended to sweep the p a r t i c l e s together. Grain growth during deformation has iport< (33) also been repo ed i n M g - A l , ( 2 9 ' 3 0 ) , T i ( 3 1 \ Mg-6Zn-.5Zr ( 3 2 ), and low a l l o y s t e e l s Some contradicatory r e s u l t s have been obtained by other work- ers however. G i f k i n s ^ observed no grain growth i n Pb-Th a l l o y s . The i n i t i a l grain s i z e was 100 u however, which i s much larger than most superplastic materials. The a l l o y s did display large elongations and a high "m" value. Despite being apparently inherent to superplastic creep, the enhancement of grain s i z e e f f e c t has been studied i n d e t a i l only 7 by Brophy et a l * . Few attempts to explain the e f f e c t have (21) been made i n the l i t e r a t u r e . Cook t e n t a t i v e l y suggested that i n d i l u t e Zn a l l o y s boundary s l i d i n g caused a differ e n c e of d i s - l o c a t i o n density on eit h e r side of the shearing boundary. The boundary would then tend to move to eliminate the areas of highest d i s l o c a t i o n density . 1.3 - Purpose of Present Investigation In view of the almost u n i v e r s a l occurrence of enhanced grain growth i n superplastic a l l o y s and i t s importance i n l i m i t i n g the amount of elongation, i t was decided to study the phenomenon i n more d e t a i l and attempt to understand the mechanism promoting the growth. For r e l a t i v e s i m p l i c i t y , a s i n g l e phase a l l o y was chosen, namely Sn - 1% B i which had been shown to be superplastic by Alden This a l l o y system offered several advantages: 1) The melting point i s low so that annealing at room temperature corresponded to an homologous temperature of .6 Tm. 2) Casting and extrusion could be done e a s i l y at,or near, room temper- ature, and 3) Normal grain growth i n d i l u t e Sn - B i a l l o y s had been extensively (34) studied and t h e o r e t i c a l l y discussed by Holmes and Winegard , and by (35) (27) Gordon . The two phase Sn - 5% B i a l l o y i s also superplastic and could provide a good comparison i f time permitted. 8 EXPERIMENTAL 2.1- Specimen Preparation The a l l o y was prepared from high p u r i t y t i n (99.999%) and bismuth (99.999%). Melting was done i n graphite c r u c i b l e s , e i t h e r i n a i r with a molten s a l t cover to lessen oxidation or i n vacuum. The melts were held at a temperature between 300 and 350°C with intermittent s t i r r i n g for at le a s t 10 minutes a f t e r a l l o y i n g was com- pleted. Casting was performed by bottom pouring into copper molds producing c y l i n d r i c a l b i l l e t s approximately 5" long and 1" i n d i a - meter. A f t e r casting a l l b i l l e t s were homogenized i n an a i r oven at 130°C ± 7°C for seven days. The ends of each b i l l e t were re- . moved and the diameter machined down. To produce a f i n e g rain s i z e s u p e r p l a s t i c material, the b i l l e t s were back extruded into rod form using high extrusion r a t i o s . Two f i n i s h e d s i z e s were produced; .150" diameter (extrusion r a t i o 40:1) and .083" diameter (extrusion r a t i o 130:1). The extrusion speed was approximately 12"/minute at 70,000 to 80,000 p s i f o r the .150" diameter rod. Speed control with the .083" diameter rod was d i f f i c u l t to a t t a i n and wide f l u c t u a t i o n s of rate were unavoidable. The mechanical properties were not affected by t h i s v a r i a t i o n . Two ex- tru s i o n temperatures were used; room temperature (22°C) and 0°C. To obtain the 0°C extrusion temperature, the b i l l e t , die blocks, ram and container were assembled and placed i n a freezer f o r 24 hours. The whole assembly was immediately transferred to the extrusion press and immersed i n an ice-water mixture during the extrusion process. For a l l extrusions, a die lubric a n t was used, ei t h e r a powdered 9 g r a p h i t e - o i l mixture or polyethylene g l y c o l . Immediately a f t e r leaving the d i e , the rod was cut into short lengths and quenched into l i q u i d nitrogen. A l l material was stored i n l i q u i d nitrogen i n a Union Carbide LD-17 container which maintained a temperature of -184°C above the l i q u i d nitrogen l e v e l so that the rods were main- tained below t h i s temperature at a l l times. For t e n s i l e or creep t e s t i n g most specimens were used "as-extruded" with no reduction i n the c r o s s - s e c t i o n a l area. To ob- t a i n high elongations i n Stage II deformation and for a l l t e s t i n g i n Stage I I I , a reduced cross-section specimen was produced. The gauge length was machined on a jewellers' lathe using a cold machining (21) technique i n which cold nitrogen gas was blown on to the s p e c i - men c o n t i n u a l l y . To ensure that no grain growth or excessive damage was introduced by t h i s procedure, a specimen was machined and ex- amined i n the o p t i c a l microscope. The o v e r a l l g r a i n s i z e was the same as i n samples annealed f o r the same length of time. The da- maged surface layer appeared to extend to a depth of only 10 y. . 2.2 - T e n s i l e Testing T e n s i l e . t e s t i n g was done on Instron floor model machines using constant crosshead speeds. The specimen was gripped using threaded s p l i t g r i p s . During tightening of the g r i p s , overloading of the specimen was prevented by automatic load c y c l i n g of the cross- head. The gauge length was taken as the distance between the grips f o r a uniform cross-section specimen and as the length having constant .diameter with the machined specimens. The s t r e s s - s t r a i n rate r e l a t i o n s h i p s were determined on s i n g l e samplesby a v a r i a b l e crosshead speed technique. A f t e r a 10 steady load was obtained at each crosshead speed the speed was changed by a factor of 2. or 2.5. True stress and s t r a i n rate values were computed using values f o r instantaneous area and length. In most cases steady stress values were obtained a f t e r approximately 1% s t r a i n at each crosshead speed. T o t a l s t r a i n during t e s t i n g through the complete speed range normally amounted to l e s s than 25%. A l - though some grain growth occurred during these tests i t s e f f e c t on the shape of the a - & curve was s l i g h t since the values obtained i n i n d i v i d u a l tests at constant crosshead speed agreed with the s t r a i n rate change t e s t s . (Results section, F i g s . 6 and 7). For grain s i z e versus s t r a i n measurements, the true s t r a i n of the specimen a f t e r testing was determined by measuring the f i n a l diameter at the section where the grain s i z e was measured. Alter n a t i n g tension-compression tests were also c a r r i e d out on an Instron. A s p e c i a l grip arrangement was made (Fig.3) to hold the specimen to minimize slack i n the linkages. Specimen ex- tension was measured with an Instron s t r a i n gauge extensometer attached to the aluminum mounts. The gauge length of the specimen was kept below .3" to minimize bending i n compression. 2.3 - Creep Testing Creep t e s t i n g was performed to obtain lower s t r a i n rates than the Instron t e s t s . The specimens were mounted i n threaded grips with the load hung d i r e c t l y on to the bottom g r i p . The stress was kept constant w i t h i n ± 5% by removing p e r i o d i c a l l y a small p o r t i o n of the load. Creep s t r a i n was measured with a t r a v e l l i n g microscope by following the movement of two marks scribed on the specimen. The error 11 Fig. 3 . Grip Arrangement for Tension-Compression Testing on Instron. 12 i n the true s t r a i n was estimated as ± 1%. For comparison with the t e n s i l e r e s u l t s , s t r a i n rate was measured from the slope of the s t r a i n versus time curve at 2% true s t r a i n . A l l creep and t e n s i l e t e s t i n g was c a r r i e d out at room temperature. 2.4 - X-ray Analysis To determine the preferred o r i e n t a t i o n of grains i n the material,back r e f l e c t i o n photographs were taken using monochromatic (Cu-Ka) r a d i a t i o n . Since the grain s i z e was too large to produce com- ple t e Debye rings with a s t a t i c exposure, the samples were rotated i n the X-ray beam around the rod or t e n s i l e axis at a speed of 4 rpm. In t h i s way many more grains were exposed to the X-ray beam and com- plete Debye rings were produced. The o r i e n t a t i o n of the f i b e r axis was determined from the photographs To eliminate any surface e f f e c t s , a l l specimens immediately before X-raying were electropolished with a s t a i n l e s s s t e e l cathode and a voltage of 50-60 v o l t s i n a s o l u t i o n of the following composi- „. (37) t i o n : ethyl alcohol - 144 ml., water - 32 ml., n-butyl alcohol - 16 ml., aluznlr.r-a chloride - l u gn.., zinc chloride - 45 gm., In order to obtain an estimate of the degree of texture, a microdensitometer was used to scan the X-ray f i l m and measure the re- l a t i v e i n t e n s i t y around the Debye arc. 13 2.5 - Metallography Surfaces were prepared f o r etching and examination with a Porter Blum diamond kn i f e ultramicrotome, following the procedure of (38) R u s s e l l and Alden . The method i s f a s t , p a r t i c u l a r l y f o r soft metals, and can produce an extremely smooth almost d i s t o r t i o n free surface. However, only a small area, ^ x 2 mm., can be prepared. Thus, a wedge t i p of t h i s area had to be shaped on each sample with a f i n e jewellers' f i l e . Kerosene was used as a lubri c a n t to mini- mize k n i f e wear. I n i t i a l cuts were about .5 microns thick and were progressively reduced i n steps of 250A° u n t i l f i n a l s l i c e s of 500A° were taken. At each step, at l e a s t 7 s l i c e s were taken to remove the d i s t o r t e d layer from the previous cuts. A l l specimens were etched to reveal the grain boundaries i n eit h e r a 2% HC1 i n ethyl alcohol s o l u t i o n or a s o l u t i o n containing 5 c c HC1, 2 gm. f e r r i c c h l o r i d e , 30 c c . water and 60 c c . eth y l a l c o h o l . Photomicro- graphs were taken i n at le a s t three separate areas of each surface. Grain s i z e was measured on the micrographs using the i n t e r - (39) cept method with a 10 cm.circumference c i r c l e . The average — — 10 A lineal i n t e r c e p t , D, was computed from the formula: D = — — — where A = number of app l i c a t i o n s of the c i r c l e N = t o t a l number of boundary i n t e r s e c t i o n s M = magnification of the photomicrograph At l e a s t 320 in t e r s e c t i o n s were counted f o r an error i n D of better than ± 7% at the 95% confidence l e v e l . In the remainder of the text the l i n e a l intercept D w i l l be re f e r r e d to as the grain s i z e unless otherwise indicated. To determine the amount of grain elongation pro- 14 duced by deformation, some specimens were sectioned l o n g i t u d i n a l l y and the average l i n e a l intercept determined p a r a l l e l and perpendicular to the t e n s i l e axis. To determine the d i s t r i b u t i o n of grain "types" the number of sides per grain were counted f o r each g r a i n within a designated area. Grain s i z e d i s t r i b u t i o n s were determined using the method of Johnson . (41) However, the Swedish grouping method was adopted rather than the method of Johnson which i s based on the ASTM gr a i n s i z e number. If A i s the mean planar grain area i n square microns of a g r a i n class P then : A = 2. P - (1) where P takes integer values - 0. The area l i m i t s of each class are P±3̂ given by 2 2. Thus, grains having an area between 0 and 1.4 p 2 f a l l i n t o class 0,. between 1.4 and 2.8 y into class 1, between 2.8 2 and 5.8 M into class 2 etc. A serie s of squares having areas i equivalent to the cla s s l i m i t s times the photomicrograph magnifi- cation were scribed on to a sheet of clear p l a s t i c . The sheet was applied to each grain i n turn to estimate i t s c l a s s . Approximately 1000 grains were counted f or each sample. 15 RESULTS 3.1 - Superplastic Properties 3.1.1 - Log a - log e Curves Incremental s t r a i n rate change tests were performed to study the e f f e c t s of processing v a r i a b l e s on the s t r e s s - s t r a i n rate re- l a t i o n s h i p s (Fig.4). A l l tests were performed a f t e r a 20 minute anneal at room temperature. Three v a r i a b l e s are compared: 1) casting method, 2) extrusion r a t i o and 3) extrusion temperature. The reasons for studying these v a r i a b l e s were to attempt to produce agreement with published r e s u l t s and to s h i f t the high "m" region toward higher s t r a i n rates. A l l samples show at l e a s t a two stage curve with low and high "m" regions t y p i c a l of s u p e r p l a s t i c materials. A s l i g h t i n - d i c a t i o n of Stage I region i s shown by the high extrusion r a t i o material (curve 4). The e f f e c t of both a lower extrusion temperature (curve 2) and higher extrusion r a t i o (curve 4) i s to s h i f t the high "m" region to higher s t r a i n rates with extrusion r a t i o having the more powerful e f f e c t . By comparing curves 2 and 3 i t i s seen that the melting and casting conditions have no e f f e c t on the s t r e s s - s t r a i n rate curve i . e . any oxidation or contamination produced by a i r cast- ing had l i t t l e e f f e c t on the deformation c h a r a c t e r i s t i c s . The "m" values (Fig.5) of the samples of Fig.4 were ob- tained g r a p h i c a l l y by measuring the slopes on the log a - log t p l o t . Values taken from the creep data (see Fig.7) have been added to curve 4. Higher extrusion r a t i o and lower extrusion temperature have pro- duced higher peak "m" values than i n the room temperature 40:1 r a t i o 1—I I I I ) 11 "I—I I I I I 1—I—I I I 1111 1—I—I I I 111J 1 0 0 0 0 (psi) 1000 1 OAIR CAST, LOW RATIO, ROOM TEMP. 2 A AIR CAST, LOW RATIO, 0°C 3 xVACUUM CAST, LOW RATIO, 0°C 4 • VACUUM CAST,HIGH RATIO.ROOM TEMP 100 J I i i i I I I J 1—I I I I I 11 I I I 1 I l I I 1 J I 1 I I I 11 10 10 -3 ro -i € (min) 10' 1-0 Fig.4. E f f e c t s of Processing Conditions on Stress-Strain Rate Relationship. ON 17 I I i F O-AIR CAST, LOW RATIO, ROOM TEMP A AIR CAST, LOW RATIO, 0°C X VACUUM CAST, LOW RATIO, 0°C • VACUUM CAST, HIGH RATIO, ROOM TEMP, € (min"') Fig.5. E f f e c t s of Processing Conditions on S t r a i n Rate S e n s i t i v i t y Parameter "m". 18 material. The extrusion r a t i o increase produced the more pronounced e f f e c t . The peak "m" values have been s h i f t e d to higher s t r a i n rates by both v a r i a b l e s . The value of "m" i n the Stage I region of curve 4 i s approximately the same as the peak "m" of the other curves.' During the course of the incremental s t r a i n - r a t e change te s t s some grain growth occurred which could have affected the shape of the a — E curves. To e s t a b l i s h a log a - log e curve free from growth e f f e c t s , i n d i v i d u a l specimens with the same i n i t i a l grain s i z e were tested at each s t r a i n rate. Instantaneous s t r e s s - s t r a i n rate values were taken a f t e r approximately 2% s t r a i n f o r both the Instron and creep t e s t s . The p l o t t e d points (Figs.6 and 7) are a c t u a l l y the average of at l e a s t four i n d i v i d u a l t e s t s at each s t r a i n rate or stress l e v e l . Agreement between i n d i v i d u a l t e s t s and the s t r a i n rate change tests i s good i n d i c a t i n g that grain growth had l i t t l e e f f e c t i n the s t r a i n rate change t e s t . One sample of a i r cast, 40:1 extrusion r a t i o , 0°C extrusion temperature mat e r i a l , annealed to a gr a i n s i z e of 5.8 y, was used f o r an incremental s t r a i n rate change t e s t . The resultant log a - log e curve (Fig.8) could then be compared with previously published r e s u l t s ^ on s i m i l a r l y processed Sn - H B i of 5 p gr a i n s i z e . The agreement i s good considering the s l i g h t d i f f e r e n c e i n grain s i z e between the two samples. 3.1.2 - St r e s s - S t r a i n Relationships The true stress-true s t r a i n curves of the Sn - 1% B i a l l o y appear to be t y p i c a l of superplastic materials (Fig.9). At the highest s t r a i n rate (1.0"/min.), a "steady s t a t e " stress i s not maintained, but IO"4 IO"3 IO'2 ( IO'1 10 € ( m i n ) VO Fig.6 . Comparison of Stress-Strain Rate Data Between Single S t r a i n Rate Change Test and Individual Instron Tests. Fig.7. Comparison of Stress-Strain Rate Data Between Single St r a i n Rate Change Test and Individual Creep Tests. 1—I I I 1111 T 1 I I I I I T 1 I I I I I 1 1 I I I I I 0,00 0 h a (psi ) I 000 tr 00 10 -4 O F R O M A L D E N " 5= 5/i. • LOW RATIO 0"C M A T E R I A L 6=5-8/1 1 ' ' I I I 1 1 1 — I I i i i I I 1 1 1 1 i i i i I • i I , i i i . I 10" IO" 2 € (min - 1) IO- Fig.8. Comparison of Stress-Strain Rate Data Between Present and Previous Investigations. 0 10 20 3 0 40 5 0 60 7 0 8 0 TRUE € (%) Fig.9. True Stress-True S t r a i n Curves at Various I n i t i a l S t r a i n Rates. instead the flow stress c o n t i n u a l l y decreases u n t i l f a i l u r e . As the s t r a i n rate i s decreased into the s u p e r p l a s t i c region the flow stress i s maintained r e l a t i v e l y constant. At the two lower s t r a i n rates i n the Stage I I region, following an i n i t i a l decrease, the stress r i s e s with increasing s t r a i n , despite the fa c t that the s t r a i n rate i s co n t i n u a l l y decreasing during a constant crosshead speed t e s t . This increase i s l i k e l y the r e s u l t of grain growth during the t e s t . During the low stress l e v e l creep tests the s t r a i n rate usually decreases c o n t i n u a l l y with time (Figs.10 and 11). A decreas- ing s t r a i n rate with time can be i n d i c a t i v e of either primary creep or increasing g r a i n s i z e . However, i n the curves at stresses of 350 and 500 p s i a steady state creep rate p r e v a i l s a f t e r very low s t r a i n s . At these stresses then, the transient or primary creep s t r a i n i s small. (42) Usually transient s t r a i n decreases with decreasing stress , therefore, a primary region would not be expected also f o r the lower stress (< 250 p s i ) curves. The decreasing s t r a i n rate i n these curves must be due to g r a i n growth during the deformation. This e f f e c t i s s i m i l a r to the r i s i n g stress with s t r a i n i n the constant extension rate t e s t s . 3.2 - Grain Growth During Annealing The grain growth k i n e t i c s during annealing at room temperature were studied by measuring the grain s i z e on a plane perpendicular to the wire axis of the extruded rod. If the same sample was used f o r several determinations at d i f f e r e n t times i t was remicrotomed before each measurement to eliminate any surface e f f e c t s which could have affected the rate of grain growth. Grain s i z e versus time curves were determined f o r three types of material; the low r a t i o material, ex- truded at room temperature and at 0°C and the high r a t i o material ex- 30 25 20 100 40 0 2000 200 800 4000 300 1200 600 0 400 —500psi 1600 —350p«i 8000— 250psi N) Fig.10. TIME ( m i n ) Creep Curves at Stresses of 500, 350 and 250 p . s . i . 4000 8000 20000 7000 14 000 350 0 0 10000 20 000 50 000 TIME tm i n) 13000 26000 65000 16000-.150p s i 32000-1 OOps i 80000—40ps i Fig.11. Creep Curves at Stresses of 150, 100 and 40 p . s . i , truded at room temperature (Figs. 12 - 1 5 ) . On the log - log scale the grain s i z e - time curves are S-shapedwith an approximately s t r a i g h t l i n e c e n t r a l p o r t ion. The points i n the c e n t r a l region between the v e r t i c a l l i n e s were used to determine the constants, k and n, i n a power law equation; D = k t n , by a le a s t squares f i t t i n g program. The curved portions of the curves at short and long times were estimated v i s u a l l y . The con- stants, k and n, are presented i n Table 1 along with the corresponding values of r , the c o r r e l a t i o n c o e f f i c i e n t , an indication of the goodness of f i t of the data. When r = 0 there i s no c o r r e l a t i o n . When r = ± 1 there i s perfect c o r r e l a t i o n or a perfect f i t . TABLE 1 - Power Law Constants f o r Grain Size Versus Time Curves Ma t e r i a l k n r Low Ratio R.T. Extrusion .0481 .546 .981 Low Ratio 0°C Extrusion .1208 .455 .998 High Ratio R.T. Extrusion .1207 . .480 .879 In Fig.12, a " t h e o r e t i c a l " curve (dashed l i n e ) of the form — 2 — 2 2 D - Do = k t i s plo t t e d . Several values of Do and k were chosen and curves p l o t t e d u n t i l the best coincidence between the t h e o r e t i c a l and experimental curves was obtained. The best f i t was obtained f o r Do = 1.8 u and k ='.067. This t h e o r e t i c a l equation seems to represent the growth k i n e t i c s reasonably w e l l except at long times and a small portion around 1000 minutes. In Fig.15, where a l l three curves are compared without the T — THEORETICAL CURVE * - 2 - 2 2, D - CL= k f WITH D0= \-6fJL , k= 0 6 7/1/fmin) Fig.12. 100 1000 10000 100000 TIME (min) Grain Size versus Time During S t a t i c Annealing of Low Ratio Room Temperature Extrusion M a t e r i a l .  100 Fig.14. 10000 T I M E (m in ) Grain Size versus Time During S t a t i c Annealing of Low- 'Ratio-0°G Temperature -Extrusion-Material. -r • 100000 ho 2Ch i c f E X T R U S I O N C O N D I T I O N S l-LOW RAT IO - ROOM T E M P 2- HIGH R A T I O - R O O M T E M P 3- L O W R A T I O - 0°C _L to 100 Fig.15, 1 0 0 0 lOptTO 1 0 0 , 0 ^ ) 0 T I M E ( m i n ) Grain Size versus Time f or D i f f e r e n t Processing Conditions, experimental points i t i s seen that the processing v a r i a b l e s had some ef f e c t on the grain growth k i n e t i c s . The higher extrusion r a t i o had a small e f f e c t causing a s l i g h t l y lower n value and higher k. The lower extrusion temperature had a la r g e r e f f e c t producing a higher D than the other curves at the same time despite the f a c t that i t would be expected to have a smaller r e c r y s t a l l i z e d g r a i n s i z e (Do). 3.3 - Grain Growth-During Deformation. 3.3.1 f E f f e c t of Deformation Time The grain s i z e , as determined immediately a f t e r deformation w i l l subsequently be referred to as "deformed g r a i n s i z e " and w i l l be compared to the "annealed grain s i z e " taken from the curves i n Fig s . 12 to 14 at the time required f o r the deformation. One serie s of tests was performed on the room temperature ex- truded low r a t i o material using Instron tests at four d i f f e r e n t s t r a i n rates. A second s e r i e s was c a r r i e d out on the high r a t i o material using creep te s t s at several stress l e v e l s . In both s e r i e s , at each stress l e v e l or s t r a i n rate, samples were tested to d i f f e r e n t amounts of s t r a i n and the grain s i z e measured immediately a f t e r t e s t i n g . In F i g s . 16 and 17 the deformation grain s i z e i s p l o t t e d against the t e s t i n g time f o r both s e r i e s of tests along with the annealed grain s i z e curves. A l l curves through the experimental points were drawn using regression analysis to the power law equation, except f o r the two lowest s t r a i n rates i n Fig.17. The constants, k and n, and the c o r r e l a t i o n c o e f f i c i - ent r are presented i n Table 2. 20 10 9 8 7 6 D 5 (LL) "STATIC ANNEAL ING A 0' I -2 X |0 O 2x l0~ 1 •10 I 0 0 0 1 0 0 0 T E S T I N G T I M E (min) Fig.16. Grain Size versus Time During Instron Deformation •000 10000 100000 T E S T I N G T I M E (min) Fig.17. Grain Size versus Time During Creep Deformation. u> 34 TABLE 2 - Power Law Constants f o r Grain Size Versus Time Curves During Deformation S t r a i n Rate /min. n 1.0 2.127 -.0162 -.211 .0 2.82 .0196 .152 10~ 2 .640 .433 .985 2 x 10~ 3 .677 .308 .967 7 x 10~ 4 .353 .386 .868 2 x 10~ 4 .296 .374 .987 1 x 10~ 4 .108 .496 .998 2.7 x 10~ 5 .138 .454 .998 A l l s t r a i n rates.produced an increase i n the grain s i z e . Except f o r the two highest s t r a i n rates, the "n" values remain between.3 and .5 -4 regardless of s t r a i n rate. At s t r a i n rates below 10 /min., the de- formation causes an upward s h i f t of the annealed curve independent of the s t r a i n rate and "k" remains constant. At the higher rates (> 2x10 4/min.) the value of "k" increases with s t r a i n rate. 3.3.2. E f f e c t of S t r a i n The change i n grain s i z e with s t r a i n was also studied using these t e s t s . In order to eliminate the e f f e c t of d i f f e r e n t amounts of annealed grain growth occurring at the d i f f e r e n t s t r a i n rates a value was defined and plotted against true s t r a i n (Figs.18 and 19). DA Fig.18. Relative Grain Size Change versus S t r a i n During Instron Deformation. Fig.19. Relative Grain Size Change versus S t r a i n During Creep Deformation. AD i s defined as the difference between the deformation grain s i z e , Dp, and the annealed grain s i z e ^ D ^ a f t e r the time required f o r deform- ation. In other words i s the r e l a t i v e a d d i t i o n a l increase i n D A grain s i z e due to the deformation. Regression analysis was used to determine the best f i t s t r a i g h t l i n e s at each s t r a i n rate. The constants, M and b, i n the equation: # = M E + b and the c o r r e l a t i o n c o e f f i c i e n t , r, are given i n Table 3. TABLE 3 - Straight Line Constants f o r Relative Grain Size Change Versus S t r a i n Curves • f . -1» E (mm ) M(% _ 1) b r 1.0 .004 .086 .537 .1 .0094 .100 .946 IO" 2 .020 .015 .992 2 x 10" 3 .021 .012 .994 -4 7 x 10 .028 -.059 .959 -4 2 x 10 .015 .020 .879 -4 1 x 10 4 .009 .035 .944 2.8 x 1 0 - 5 .007 .032 .876 The amount of grain s i z e enhancement at a given value of s t r a i n (25%) f i r s t increases, then decreases with continually decreasing s t r a i n rate (Fig,20). The error bars i n Fig.20 i n d i c a t e the standard error of Fig.20. Relative Grain Size Change A f t e r 25% Strain versus S t r a i n Rate. estimate of the regression l i n e s at the 95% confidence l e v e l . There i s l i t t l e c o r r e l a t i o n between the degree of g r a i n s i z e enhancement and "m" value, (Fig.21) i n the high "m" regions. However, AD/D^ does increase with "m" i n the t r a n s i t i o n region between Stage II and Stage I I I . 3.3.3. - Grain Elongation The grain s i z e i n the l o n g i t u d i n a l d i r e c t i o n (D^) and the transverse d i r e c t i o n (D^) were determined for a serie s of specimen's deformed approximately 35% at d i f f e r e n t s t r a i n rates. Using (43) Rachinger's a n a l y s i s , the amount of s t r a i n due to s l i p (or other grain elongation processes) can be calculated from the rate D-̂ /D̂  by the formula: e s l i p = ( ̂  ) 3 - 1 D T The percentage of the deformation due to elongation pro- cesses can then be calculated (Fig.22). The contribution of grain elongation to the s t r a i n appears to increase with decreasing s t r a i n -2 rate from the high m region (e = 10 /min.) to the lower m Stage I region. At the higher s t r a i n rates (Stage II - Stage III t r a n s i t i o n region) the contribution may also increase s l i g h t l y . 3.3.4. - Al t e r n a t i n g Tension - Compression Tension-compression t e s t s were c a r r i e d out on the low extrusion r a t i o , 0°C extrusion temperature material. The specimens were annealed for 4 hours before t e s t i n g while the epoxy used f o r gripping hardened. The s t r e s s - s t r a i n cycle used f o r the test i s i l l u s t r a t e d i n Fig.23. The stress at 2% s t r a i n on each h a l f cycle (Fig.23) remained f a i r l y constant m Fig.21. Relative Grain Size Change A f t e r 25% S t r a i n versus "m". Fig.22. Degree of Grain Elongation After 35% Strain at Various S t r a i n Rates. 2000 2 0 0 0»' a) TYPICAL CYCLE - — i 1 1 1 r 2000 cr ( p s i ) 1900 1800 -COMPRESSION TENSION -I 1 L__b I 0 8 16 24 32 40 ACCUMULATED € (%) 48 b) FLOW STRESS AT END OF EACH CYCLE Fig.23. S t r e s s - S t r a i n Relationships During Tension-Compression Testing. 43 with the accumulated s t r a i n u n t i l the l a s t cycle when the specimen seemed to s t a r t s l i p p i n g from the g r i p s . The t e s t was then terminated, grain s i z e measured and X-rays taken. Because several minutes were required to change the d i r e c t i o n of the stress due to slack i n the grips and t e s t i n g machine, a control sample was needed f o r comparison. A f t e r the 4 hour anneal, the con- t r o l sample was strained to 2% s t r a i n , i n tension, unloaded to zero stress and annealed f o r the equivalent length of time needed to re- verse the machine to the compression cycle. The 2% s t r a i n tension cycle was then repeated a number of times u n t i l the t o t a l time of t e s t i n g and the s t r a i n cycles were the same as i n the tension-compression te s t . The control sample had a t o t a l t e n s i l e s t r a i n of 38% whereas the, tension-compression sample had a net t e n s i l e s t r a i n of only 2%. The measured grain s i z e a f t e r t e s t i n g was 3.70 microns f o r the tension- compression specimen compared to 3.30 microns f o r the co n t r o l sample. Thus, i t would appear that reversing the d i r e c t i o n of the s t r a i n has l i t t l e e f f e c t on the grain s i z e enhancement. 3.3.5 - Grain Type D i s t r i b u t i o n The grain "type" was defined as the number of sides possessed by an i n d i v i d u a l grain i n the plane of examination. The d i s t r i b u t i o n of types was determined by counting the number of sides of each grain i n - side a c e r t a i n area. A sample was deformed i n creep at an i n i t i a l s t r a i n rate of 2.8 x 10 "Vmin. to a s t r a i n of 14% and micrographs taken of the deformed section and the undeformed g r i p section. The g r i p section represents a s t a t i c anneal of equivalent time (10,000 minutes). The grain s i z e s of the two sections were 9.34 u f o r the deformed and 7.9 u 44 fo r the undeformed. Both d i s t r i b u t i o n s are of s i m i l a r form (Fig.24) except f o r an increase i n r e l a t i v e frequency of 8, 9 and 10 sided grains i n the deformed specimen compared to the undeformed. Grains having le s s than 6 sides appeared s l i g h t l y l e s s frequently i n the de- formed material. The cumulative frequency d i s t r i b u t i o n s (Fig.25) are plo t t e d with the grain type on a logarthmic s c a l e . Both curves are close to a s t r a i g h t l i n e p l o t t e d i n t h i s fashion, although the deformed d i s - t r i b u t i o n departs s l i g h t l y from the s t r a i g h t l i n e at the upper end. 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 on these scales indicates a "log-normal" (44) d i s t r i b u t i o n i n agreement with r e s u l t s of Feltham on annealed < aluminum. This form of the d i s t r i b u t i o n appears to be t y p i c a l of annealed materials where the dr i v i n g force f o r gr a i n growth i s surface (45) energy A Chi-square test f o r comparing d i s t r i b u t i o n s was performed on these two d i s t r i b u t i o n s and showed that they d i f f e r e d s i g n i f i c a n t l y at the 97.5% confidence l e v e l . A difference of means t e s t , performed on the two d i s t r i b u t i o n s showed that the d i s t r i b u t i o n means d i f f e r e d at only the 57% confidence l e v e l . 3,3.6 - Grain Size D i s t r i b u t i o n s Grain s i z e d i s t r i b u t i o n s were performed on the same samples used i n Section 3.3.5 using the s i z e grouping previously defined. The r e l a t i v e frequency d i s t r i b u t i o n s , p l o t t e d i n Fig.26 show that the de- formation has produced a s h i f t of the curve to the r i g h t and an increase i n the mean grain c l a s s . Calculations show a s l i g h t l y greater d i s - persion i n the d i s t r i b u t i o n a f t e r deformation. 45 1 I I 1 1 1 1 1 1 1 STAN. MEAN DEV 5 6 7 8 9 1 0 II |2 1 3 14 G R A I N TYPE Fig.24. Grain Type D i s t r i b u t i o n for Annealed and Deformed Structure. Fig.25. Cumulative Frequency Diagram f o r Grain Type D i s t r i b u t i o n of Eig.24. 47 ig.26. Grain Size D i s t r i b u t i o n f o r Annealed and Deformed Structures. 48 The means of these d i s t r i b u t i o n s can be rela t e d to the average l i n e a l intercept which has been used as a measure of the grain (41) s i z e previously. The mean grain area i n two dimensions , A, = exp [In A_ + ( l n S G A ) 2 ] (2) 2~~ where A^ i s the geometric mean of the grain area and S ̂  i s the geometric standard deviation. Also l n A G = iii In 2 (3) and l n = Sm l n 2 (4) where m and Sm are the mean and standard deviation r e s p e c t i v e l y of the grain classes. Thus, a value of A can be calculated using the values of m and Sm computed for Fig.26. The square root of A can then be compared to D. The calculated values of /A were 10.1 and 7.4 microns f o r the deformed and annealed samples compared to 9.34 and 7.9 f o r D. Agreement between the two grain s i z e measuring methods i s thus s a t i s f a c t o r y . The cumulative frequency d i s t r i b u t i o n s (Fig.27) f o r two samples again show that the s i z e d i s t r i b u t i o n s are e s s e n t i a l l y log-normal. In order to see i f the shapes of the s i z e d i s t r i b u t i o n s had been changed by the deformation, d i s t r i b u t i o n s of an annealed and deformed sample of equivalent mean grain s i z e should be compared. By s h i f t i n g the annealed curve to the ri g h t to a p o s i t i o n where i t produced an average grain area or l i n e a l intercept equivalent to the deformed sample such a comparison could be made. Although the disp e r s i o n of gr a i n sizes i s thought to increase somewhat with annealing t i t w i l l be assumed that the standard deviation would not change appreciably f o r t h i s change i n Fig.27. Cumulative Frequency Diagram f o r Grain Size D i s t r i b u t i o n of Fig.26. 50 average grain s i z e ( 7 .4 to 10 .1 u ) . From e q u a t i o n 2 ! new value of m, the mean grain c l a s s , was calculated to give a /K of 10 .1 u . The di f f e r e n c e between t h i s new m of 5.76 and the o l d m f o r the annealed sample ( 4 ,88 ) was the amount that each point on the annealed cumulative frequency pl o t was s h i f t e d to the r i g h t . This procedure produced a new annealed d i s t r i b u t i o n with a mean grain class of 5.74 and a standard deviation of 1 .63 . From the points of t h i s s h i f t e d curve a r e l a t i v e frequency d i s t r i b u t i o n was c a l c u l a t e d and compared to the deformed frequency d i s t r i b u t i o n curve (Fig, 2 8 ) . These two curves then re- present two samples with the same l i n e a l intercept of 10.1,one having been deformed and the other annealed. Although the deformed sample seems to possess a higher pro- portion of class 6 sizes no other trends are evident and both curves are b a s i c a l l y the same shape. The grain s i z e d i s t r i b u t i o n appears to be unchanged by the deformation process. 3.4 - X-Ray Analysis In the back r e f l e c t i o n photographs, a v a r i a t i o n i n i n t e n s i t y around the circumference of some of the Debye rings was observed (Fig.29 ) i n d i c a t i n g that a texture was present i n the as-extruded material. In order to determine the o r i e n t a t i o n of the f i b e r axis, the X-ray pattern was f i r s t indexed using Bunn charts. I t was found that the rings with the most prominent evidence of texture corresponded to r e f l e c t i o n s from the (442) and (640) planes. By measuring the angle a (Fig.29 ) at the centre of the arcs on these ri n g s , the angle p was (36) calculated from the formula : cos p = cos 0 cos a 51 Fig.28. Grain Size D i s t r i b u t i o n f or Annealed and Deformed Structures of Equivalent Mean Grain Size. Fig.29. Back Refle c t i o n Pattern for Extruded Rod. 53 where p i s the angle between the normal to the plane i n question, and the f i b e r axis. By a graphical method (Fig.30), the l o c a t i o n at the f i b e r axis was then found to be within 14° of a [110] pole of the c r y s t a l (Fig.30). In other words, a large proportion of the c r y s t a l s were aligned with a [110] pole almost p a r a l l e l to the d i r e c t i o n of the wire axis. A [110] pole f i g u r e was p l o t t e d i n Fig.31, with the s o l i d l i n e s representing the l o c a t i o n of two of the [110] poles calculated from the centre of the Debye arcs. The dotted l i n e s are the locations of the poles at the point where the i n t e n s i t y of the Debye arc f a l l s to one h a l f of the peak i n t e n s i t y . These dotted l i n e s then i n d i c a t e the strength or degree of p e r f e c t i o n of the texture. The strength of the texture was measured with the micro- densitometer by determining the r e l a t i v e i n t e n s i t y of the arc of the (640) r e f l e c t i o n as a function of the angle a, with a = 0° at the peak i n t e n s i t y p o s i t i o n (Fig.32). The extrusion r a t i o had l i t t l e e f f e c t on the degree of the texture (Fig.32(a)). Annealing of the extruded rod also had l i t t l e e f f e c t (not shown). Deformation did appear to influence the texture however, (Fig.32(a)). Only the highest s t r a i n rate used (1.0/minute) produced a strengthening of the texture (Fig.32(b)). A l l other s t r a i n rates produced a weakening of the texture to about the same degree except for t = .1/minute which produced a s l i g h t l y greater e f f e c t . These r e s u l t s i n d i c a t e that during superplastic deformation random grain r o t a t i o n i s taking place as a r e s u l t of grain boundary s l i d i n g causing a decrease i n the texture. 54 Fig.30. Location of Fiber Axis on Standard Stereographic Projection of Tin. 55 I N D I C A T E S " A N G U L A R S P R E A D OF P O L E S Fig.31. (110) Pole Figure for Extruded Rod. 0 15 30 45 60 75 A N G L E a ( DEGREES) a) E F F E C T OF STRA IN ON D E G R E E OF T E X T U R E 0 15 30 45 60 75 A N G L E CT (DEGREES) b) E F F E C T OF STRA IN RATE ON D E G R E E OF T E X T U R E Fig.32. Degree of Texture Before and Aft e r Deformation. 57 3.5 - Grain Growth A f t e r Deformation 3.5.1 - Growth by Annealing To determine i f any fa c t o r s a f f e c t i n g g r a i n growth were changed by deformation, samples were deformed then annealed at room temperature. The grain s i z e v a r i a t i o n with time during and a f t e r -2 deformation at a s t r a i n rate of 10 /min. i s plo t t e d i n Fig.33 along with the s t a t i c annealing curve f o r t h i s m a t e r i a l . Immediately a f t e r the stress i s removed, the growth rate i s lower. I t i s i n t e r e s t - ing to note that a f t e r deformation and considerable post deformation annealing, the grain s i z e i s a c t u a l l y less than that of a sample s t a t i c a l l y annealed f o r the equivalent length of time. Similar post deformation annealing curves (Fig.34) f o r d i f f e r e n t amounts of s t r a i n suggest that the rate of growth a f t e r deformation i s a function of s t r a i n ; small s t r a i n s are followed by a f a s t e r growth rate than large s t r a i n s , Further i l l u s t r a t i o n i s afforded by Figs.35 and 36 where two curves of Fig.34, f o r 70% s t r a i n and 6.9% s t r a i n are replotted along with sections of the s t a t i c annealing curve with the time axis s h i f t e d ; the time to reach an annealed grain s i z e equivalent to the deformation grain s i z e i s taken as zero time. In t h i s way, i t i s possible to compare growth rates of two samples having reached the same grain s i z e by two routes; annealing and deformation. These curves con- fir m that the post deformation growth rate i s less than the s t a t i c annealing rate but only a f t e r large amounts of s t r a i n i n g . 3.5.2 - Growth Enhanced by Deformation The deformation seemed to be producing some change i n the material structure which, i n turn, caused a reduction i n the grain O O A F T E R DEFORMATION 100 I 000 I 100 00 I 00.0 00 00 T I M E (min) Fig.33. Grain Size versus Time During and A f t e r Deformation at i 10~ 2/min.  20T 9 8 7 6 / / D iLL) 5 A S T A T I C A N N E A L I N G A N N E A L I N G A F T E R D E F O R M A T I O N 10 _ L 1 0 0 Fig.35. 1 0 0 0 M E ( m n) 0,0 0 0 Comparison of Grain Size versus Time Curves f o r St Annealing and Annealing A f t e r Deformation of 7DA S I 00,000 o a t i c t r a i n . 20r / / 10 9. 8 7 6 D 5 / / / / / S T A T f C A N N E A L I N G -L JL A N N E A L I N G A F T E R D E F O R M A T I O N _L 10 100 1000 0 IO0 0 T I M E ( m i n ) Fig.36. Comparison of Grain Size versus Time Curves f o r S t a t i c - --v. r v . - p . Annealing and Annealing-After Deformation-of 7% S t r a i n . I O O C T D O 62 growth rate. To determine i f t h i s s t r u c t u r a l change aff e c t e d the s t r e s s - s t r a i n rate r e l a t i o n s h i p s , the following experiment was performed. A sample of the low r a t i o 0°C extrusion temperature material was de- formed i n the Instron to 85% true s t r a i n at a s t r a i n rate of 10 /min. The grain s i z e a f t e r deformation was 5.5 microns. A piece of the same material was annealed to a grain s i z e of 5.8 microns and the two samples used to perform an incremental s t r a i n rate change te s t (Fig.37). The deformed material behaved very s i m i l a r l y to the annealed material although the deformed material had a s l i g h t l y higher "m" value at the low s t r a i n rates. The s t r u c t u r a l changes produced by the p r i o r deformation had l i t t l e e f f e c t on the s t r e s s - s t r a i n rate r e l a t i o n - ships. To see i f the change had any e f f e c t on the growth k i n e t i c s during further deformation, another set of experiments was performed. _2 Two samples were again deformed to 85% s t r a i n (at e = 10 /min.) and a grain s i z e of 5.5 microns. One of these was then annealed to a grain s i z e of 6.2 microns to see i f the s t r u c t u r a l changes might be alt e r e d by an annealing treatment. Two other samples were annealed to s i m i l a r grain sizes (5.8 and 6.5 microns) f o r comparison. A l l four samples were;then deformed at t = 6.7 x 10 ^/min. to a s t r a i n of approximately 15% and the grain s i z e s measured. The i n i t i a l and f i n a l g r ain sizes are presented i n Fig.38 as a function of the deformation time at e = 6.7 x 10 Vmin. The annealed samples had a much greater increase i n gr a i n s i z e than the previously deformed samples; the annealed sample had a r e l a t i v e en- hancement (^ ) of 60% compared to only 30% f o r the deformed sample. A The s t r u c t u r a l change produced by deformation had the same 1 1 1 1 « ' M I i i ! i M M ; 1 — i — i n u n 1 — r T T 10,000 cr ( p s i ) 1 0 0 0 O O ANNEALED 0 = 5-8/1 A — — A DEFORMED D=5'5/£ IQQ! 1 — ' — ' i i i 111 1 — i — i M i n i 1 — i ' i i i 111 i i i i i i 111 10 -4 > o i o . • 10 1-0 6 ( m i n 1 ) Fig.37. Comparison of Stress-Strain Rate Data for Annealed and Deformed M a t e r i a l . 64 i r X A N N E A L E D 0 100 200 300 400 5 00 -4 - i DEFORMATION TIME A T € = 6 - 7 n l O (min) (mirr) Fig.38. Effect of Pre-Deformation on Grain Size Increase During Deformation. 65 e f f e c t i n t h i s case as i t d i d on post deformation annealing, i . e . i t reduced the grain growth k i n e t i c s as compared to undeformed material. In other words, deformation i n the su p e r p l a s t i c region causes a reduction i n subsequent grain growth rates exclusive of whether t h i s growth i s produced by annealing or by further deformation. 66 DISCUSSION 4.1 - S u p e r p l a s t i c i t y Evidence has been presented to suggest that the Sn-1% B i a l l o y used i n t h i s study i s su p e r p l a s t i c . The phenomenological basis of s u p e r p l a s t i c i t y requires a strong dependence of the flow stress o n s t r a i n rate. The Sn-Bi a l l o y does posseses high "m" regions (Figs. 4 and 5). The c h a r a c t e r i s t i c three-stage curve t y p i c a l of most super- p l a s t i c a l l o y s was not obtained, despite the use of very low stress creep tests (Fig,7). A s l i g h t reduction i n "m" can be seen i n Fig.7 but mainly i n the creep test region. The decreased "m" could be due to the change i n t e s t i n g procedure rather than a d i f f e r e n t mechanism c h a r a c t e r i s t i c of Stage I deformation. Elongation values from a l l o y s regarded as superplastic range from 300 to 2000%. The maximum value obtained f o r t h i s a l l o y was s l i g h t l y greater than 300%. Elongation values reported by Alden on the same a l l o y with s i m i l a r s t r a i n rates ranged from 450 to 500%. A possible reason f o r t h i s d i f f e r e n c e could be differences i n impurity (12) content . A higher maximum value of "m" was obtained by Alden (Fig.8), lending support to t h i s supposition. A f t e r large amounts of deformation the grains of the Sn-Bi a l l o y remained equiaxed, a prominent feature of superplastic a l l o y s . The r a t i o of the average l i n e a r intercepts measured i n the l o n g i t u d i n a l and transverse d i r e c t i o n s (D^/D^) was found to be 1.15 a f t e r 300% elongation. On the basis of these r e s u l t s i t i s concluded that the Sn - 1% Bi a l l o y under study i s supe r p l a s t i c . 67 4.2 - Grain Growth An equation can be derived f o r the growth of c e l l s i n a soap f r o t h where the surface energy of the f i l m i s the d r i v i n g f o r c e . A pressure d i f f e r e n c e e x i s t s across each curved c e l l w a l l , the pressure being greater on the concave side and proportional to the curvature of the w a l l . The pressure d i f f e r e n c e Causes d i f f u s i o n from the high to the low pressure side r e s u l t i n g i n movement of the walls toward the centre of curvature. The rate of motion of the walls and hence the rate of increase of the average c e l l diameter (—) w i l l be proportional dt to the curvature, C. In an equiaxed structure the curvature can be assumed pro- p o r t i o n a l to the inverse of the average c e l l diameter so that: = K'C = ^ (5) dt D Integration of t h i s equation and evaluation of the constant of i n t e - g ration y i e l d s ; D 2 - D 2 = 2 Kt (6) o where D q i s the average c e l l s i z e at t = 0. Experimental observations have confirmed that t h i s equation (47) adequately describes the growth of c e l l s i n a soap f r o t h The growth of grains i n a metal can be represented by a s i m i l a r type of equation except that the exponent i s usually found to be greater than 2 and can change with temperature, p u r i t y of the metal, and the presence of impurities. I t was shown (Eig.12) that the room temperature grain growth of the Sn-Bi could be approximated by an equation s i m i l a r to (6). Therefore one may conclude that grain growth during s t a t i c annealing 68 i s driven by the reduction of surface energy of the grain boundaries. s i m i l a r atomistic theories to explain the migration rates of grain boundaries i n which the segregation of impurity atoms to the boundary control the movement. Both theories depend on the fac t that the equilibrium concentration of the impurity i n the region of the grain boundary w i l l be d i f f e r e n t from the concentration i n the bulk of the material. The l a t t i c e d i s t o r t i o n caused by the impurity atom i s less i f i t i s placed i n the already d i s t o r t e d boundary region instead of i n the i n t e r i o r c r y s t a l l a t t i c e . Thus, i f a force acts on the grain boundary tending to separate i t from the array of excess impurity atoms, the free energy of the system w i l l be increased. The boundary w i l l have to climb an energy h i l l which i s equivalent to a r e s t r a i n i n g force exerted by the impurities on the boundary. overcome the maximum r e s t r a i n i n g f o r c e , the boundary w i l l break away from the impurity array and migrate at a rate that depends only on the net rate of atom transfer across the boundary. If however, the d r i v i n g force i s not s u f f i c i e n t to cause t h i s breakway the boundary w i l l tend to move away from the impurity array a short distance u n t i l the r e s t r a i n i n g force j u s t equals.the d r i v i n g force. The boundary w i l l then move with a constant v e l o c i t y with the impurity array t r a i l i n g behind. The rate of migration w i l l then be controlled by the d i f f u s i o n of the impurity atoms i n the d i s t o r t e d region adjacent to the boundary. Lucke and Detert (48) and Gordon and Vandermeer (49) proposed I f the force acting to move the boundary i s large enough to The rate of grain boundary migration can be wri t t e n as: G = (7) 69 where M i s the mobility of the rate determining step. F i s the free energy per atom and x i s the distance co-ordinate i n the d i r e c t i o n of boundary mi- gration. The m o b i l i t y i s r e l a t e d to a d i f f u s i o n c o e f f i c i e n t by the equation: M - £ (8) where i s the d i f f u s i o n c o e f f i c i e n t f o r the impurity atoms i n the re- gion adj acent to the boundary when the rate i s impurity dependent. Lucke and Detert consider that the impurity atoms w i l l t r a i l so f a r behind the boundary that they can be considered as d i f f u s i n g i n the un- d i s t o r t e d bulk material and D 1 w i l l be equal to that f o r l a t t i c e d i f f u s i o n of the impurity. In contrast Gordon and Vandermeer suggest that the impurity atoms w i l l be d i f f u s i n g c l o s e r to the boundary i n a much more d i s t o r t e d area. D 1 then could be s i g n i f i c a n t l y greater than that f o r l a t t i c e d i f f u s i o n . dF Vandermeer and Gordon developed an equation f o r by assuming that i n grain growth the surface energy i s the d r i v i n g f o r ce: (9) dF 2 ^ s Vm dx N X D C_ where Y g i s the s p e c i f i c g r a i n boundary energy V i s the molar volume of the parent material m r N i s Avogadro's number X i s the width of the grain boundary D i s the average grain diameter C_ i s the equilibrium concentration of impurity i n the grain boundary. 70 They also assumed that the boundary migration rate was equal to the rate of change of the average grain diameter. By combining equation 7, 8 and 9 and s u b s t i t u t i n g f o r and D 1 the following equation was developed: 2y V D 1 exp -( $ ~ ) dD _ f 's m o ^ RT , 1_ . dt _ 1 RT X A C J * - U U J o D where D 1 and Q 1 are the pre-exponential f a c t o r and a c t i v a t i o n energy per mole f o r d i f f u s i o n of solute i n the region near the boundary A i s a v i b r a t i o n a l entropy f a c t o r C i s the atomic f r a c t i o n of solute o E i s the i n t e r n a l energy gain per mole of solute atoms trans- fe r r e d to the boundary from the bulk material. This equation i s i d e n t i c a l to equation (5): dD _ K dt D with the bracketed term equal to K. (45) Holmes and Winegard made estimates of the values of the terms i n equation (10) for d i l u t e Sn-Bi a l l o y s (Table 4). TABLE 4 - Estimated Values f o r Terms of Equation (10)v J Term Value Y„ I O - 5 cals/cm 2 s V m 3 16.3 cm n 1 2 D Q 72 cm /sec. 5 x 10 ^ cm. 71 They were then able to c a l c u l a t e a value of 22,000 c a l f o r Q 1 + E from experimentally determined grain growth curves. Using the values of Table 4 c a l c u l a t i o n s f o r the three grain growth curves (Fig.15) yielded an average value f o r Q 1 + E of 24,000 c a l . which compares favourably with Winegard's r e s u l t . Since Winegard's data was shown to be consistent with the impurity dependent (35) migration model , i t can be concluded that the boundary migration i n the Sn-1% B i a l l o y i s also c o n t r o l l e d by the d i f f u s i o n of the solute B i atoms. Various mechanisms by which deformation can increase the average g r a i n s i z e at a given time w i l l now be discussed. 4.3 - Mechanisms of Deformation Enhanced Grain Growth 4.3.1 - Grain Coalescence Grain boundaries i n a s i n g l e phase metal e x i s t because of d i f f e r e n t c r y s t a l l o g r a p h i c o r i e n t a t i o n s of the grains (Fig.39a). If one of the adjacent grains rotates (Fig.39b) i n a d i r e c t i o n so that the misorientation i s eliminated (Fig.39c)„then the boundary i t s e l f w i l l have been eliminated and the two o r i g i n a l grains w i l l have coalesced into one. A f t e r the coalescence some l o c a l migration of the surround- ing boundaries w i l l probably occur (Fig.39d). I t i s obvious that operation of t h i s mechanism w i l l produce an increased grain s i z e . L i ^ " ^ proposed that a s i m i l a r phenomenon occurred with subgrains and used the mechanism as an explanation f o r the o r i g i n of r e c r y s t a l l i z a t i o n n u c l e i i . During superplastic or creep deformation grain boundary s l i d i n g i s accompanied by a r o t a t i o n of the grains with respect to \ \ 73 each other. The amount of r o t a t i o n can be quite large i . e . 25-30° a f t e r 45% e x t e n s i o n ^ ^ . Thus, stress-induced r o t a t i o n could produce coalescences i n a s i m i l a r fashion to L i ' s mechanism. In creep deformation i t i s generally f o u n d t h a t a l i n e a r r e l a t i o n s h i p e x i s t s between the amount of s l i d i n g and the t o t a l s t r a i n . The amount of g r a i n r o t a t i o n and the number of coalescence reactions should also increase c o n t i n u a l l y with s t r a i n . I f these r e - l a t i o n s h i p s are assumed true f o r superplasti c materials, the g r a i n coalescence mechanism predicts q u a l i t a t i v e l y the curves of Figs.18 and 19 where the grain s i z e enhancement increaseswith s t r a i n . The e f f e c t s of s t r a i n rate are also consistent with the mechanism. The grain elongation measurements show that as the s t r a i n rate i s lowered from the high "m" region the percentage of the t o t a l deformation due to grain boundary s l i d i n g decreases. Thus, f o r a given amount of s t r a i n , the lower s t r a i n rates would have experienced less g r a i n r o t a t i o n and should show a smaller rate of grain growth. In f a c t , the r e s u l t s show that, i n the superplas ti c region, the lower the s t r a i n rate the lower the r e l a t i v e enhancement f o r a given amount of s t r a i n (Fig.19). However, the r e s u l t s of the a l t e r n a t i n g tension-compression tests are not explained. The a l t e r n a t i n g s t r e s s - s t r a i n cycles would be expected to approximately reverse the d i r e c t i o n of the grain rotations so that the large net rotations needed to produce a s i g n i f i c a n t number of coalescences would not occur. Thus, t h i s type of deformation should produce very l i t t l e g r a in enhancement. In f a c t , the a l t e r n a t i n g de- formation produced s l i g h t l y more gr a i n growth than the corresponding tension only control sample. 74 In order to attempt to t h e o r e t i c a l l y predict the e f f e c t s of the coalescence mechanism on the grain type and s i z e d i s t r i b u t i o n s the analysis of Appendix I was c a r r i e d out. The c a l c u l a t i o n s show that the s l i g h t changes i n the grain type d i s t r i b u t i o n curves can be accomplished i f 6% of the o r i g i n a l grains coalesced during deformation. However, t h i s 6% coalescence was not s u f f i c i e n t to account for the observed changes i n the grain s i z e d i s t r i b u t i o n . I t i s therefore concluded that the d i s t r i b u t i o n c a l c u l a t i o n s and the tension-compression experiements show that the grain coalescence mechanism cannot be the cause of the grain s i z e enhancement during deformation. 4.3.2 - Increased Driving Force It appears more l i k e l y that the g r a i n s i z e enhancement i s a r e s u l t of a "speeding up" of the normal g r a i n growth processes caused by increased boundary migration rates. Deformation could i n - crease e i t h e r the mobility or the d r i v i n g f o r c e . Mechanisms a f f e c t i n g the d r i v i n g force w i l l be considered i n t h i s section. 4.3.2.1 - R e c r y s t a l l i z a t i o n R e c r y s t a l l i z a t i o n taking place continuously during the de- formation could produce an increased g r a i n s i z e . Continuous re- , , _ , , (17,18,20,26,52) „ c r y s t a l l i z a t i o n has been suggested by some authors to be associated with superplastic deformation. The r e c r y s t a l l i z e d grains would be equiaxed thus explaining the observed lack of d i r e c t i o n a l i t y i n g r ain shape a f t e r deformation. R e c r y s t a l l i z a t i o n would be i n i t i a t e d by stress concentrations at t r i p l e l i n e s or at l o c a l d i s t o r t i o n s around 75 s l i d i n g grain boundaries. However, i t does not seem l i k e l y that r e c r y s t a l l i z a t i o n during deformation could be the cause of the increased grain s i z e i n super- p l a s t i c metals. Studies of r e c r y s t a l l i z a t i o n during creep deformation of large grained metals ( P b ^ 3 ' " ^ , N i ^ ~ ^ and Mg^~^) show that the grain s i z e a f t e r r e c r y s t a l l i z a t i o n occurs can be e i t h e r l a r g e r or smaller than the o r i g i n a l s i z e . Theories(53,55) q ^ t ^ e o r i g i n Q f re- c r y s t a l l i z a t i o n during creep require a d i s l o c a t i o n or subboundary net- work to provide the n u c l e i i and the d r i v i n g force f o r growth. However, one of the main c h a r a c t e r i s t i c s of superplastic deformation i s the lack of d i s l o c a t i o n networks during deformation. R e c r y s t a l l i z a t i o n i s also d i f f u c l t to imagine i n a two phase superplastic system since simultaneous r e c r y s t a l l i z a t i o n and growth of both phases would be r e - quired. 4.3.2.2 - Grain Boundary S l i d i n g Since grain boundary s l i d i n g i s observed i n a l l superplastic a l l o y s i t s e f f e c t s on grain boundary migration must be considered. Studies of grain boundary s l i d i n g i n p o l y c r y s t a l s and b i c r y s t a l s have shown that boundary migration i s often connected with s l i d i n g . However, an important conclusion of the b i and t r i c r y s t a l studies (57-59) i s that migration does not necess a r i l y accompany s l i d i n g . Migration depends on the configuration of the sample (^0,61) a n ( j t ^ e f 62 ) r e l a t i o n s h i p between the surface examined and the d i r e c t i o n of s l i d i n g . There appear to be two types of migration associated with b i c r y s t a l s l i d i n g . The f i r s t t y p e ^ 2 ' * ^ occurs when s l i d i n g produces a step onthe c r y s t a l surface and the boundary migrates to the low 76 energy p o s i t i o n with respect to t h i s new surface (Fig.40). The second type appears to be caused by the b u i l d up of d i s l o c a t i o n s at the boundary as a r e s u l t of the s l i d i n g p r o c e s s ^ 2 ' . S U R F A C E I N I T I A L G R A I N B O U N D A R Y B O U N D A R Y A F T E R S L I D I N G A N D M I G R A T I O N Fig.40 Surface Grain Boundary Migration Associated with Internal Boundary S l i d i n g . ( 63—67 ^ Studies on p o l y c r y s t a l s also suggest that migration i s not a necessary r e s u l t of g r a i n boundary s l i d i n g . Almost a l l ob- servations were made on the surface of the specimens so that the mi- gration may be a t t r i b u t e d to the surface step e f f e c t or to another surface e f f e c t investigated by B e l l and Langdon and G i t t i n s and G i f k i n s ^ 4 \ They observed that g r a i n boundaries i n t e r s e c t i n g the surface tended to orient themselves at 90° to the surface during annealing and during creep. If samples were annealed p r i o r to creep so that most of the angles approached 90° the amount of migration during the deformation was greatly reduced. Surface observations also do not suggest a mechanism f o r a grain s i z e increase since the migration occurs both away from and toward the centre of curvature of the b o u n d a r i e s ^ 1 However, Ishida, et a l , using an i n t e r n a l marker method, found simultaneous migration and s l i d i n g i n the i n t e r i o r of samples and concluded that some migration could be driven by s t r a i n energy. The s t r a i n energy could be present i n two forms; as d i s l o c a t i o n arrays adjacent to the s l i d i n g boundary and produced d i r e c t l y by the shearing process^or as arrays at t r i p l e points ̂ "^'^^ produced by l o c a l deformation to r e l i e v e stress concentrations. A model of stress induced grain growth based on the f i r s t ( 2 1 ) type of array was suggested by Cook . A di f f e r e n c e i n d i s l o c a t i o n density across a shearing boundary was postulated with the larger grains containing the higher density. The boundaries would then migrate i n such a way as to decrease the s i z e of the la r g e r grains and increase the s i z e of the smaller. In normal g r a i n growth the l a r g e r grains grow at the expense of the smaller. The operation of t h i s model i s thus i n d i r e c t opposition to the normal growth process and would only decrease the s i z e dispersion of the grain structure and not produce an increase i n average grain s i z e . Changes i n the grain s i z e dispersion were not observed experimentally i n the Sn-1% B i a l l o y . If the d i s l o c a t i o n structure was arranged so that migration toward the centre of curvature tended to decrease the stored s t r a i n energy, the normal growth process could be enhanced. The r e s u l t s of the alt e r n a t i n g tension-compression test appear to disprove t h i s type of model. The d i s l o c a t i o n pile-ups produced i n the tension cycle would probably be counteracted by s i m i l a r pile-ups on the opposite side of the boundary when the stress reverses i n the compression c y c l e . Thus, a large net b u i l d up of d i s l o c a t i o n s on one side of a shearing boundary would be u n l i k e l y i n th i s type of experiment. Hence, a large 78 net increase i n the grain growth rate would not be expected, contrary to experiment. The l o c a l deformation structure at t r i p l e points also seems an u n l i k e l y mechanism to promote growth since the induced mi- gration i s equally l i k e l y to be toward or away from the centre of curvature of the blocking grain 4.3.2.3 - Grain Elongation An elongated g r a i n must possess greater boundary curvature than one of equivalent volume but with an equiaxed shape. An increase i n l o c a l curvature should produce an increase i n boundary migration. Some grain elongation did take place i n the a l l o y studied and could have been caused by e i t h e r s l i p processes or Nabarro-Herring d i f f u s i o n . However, i t i s d i f f i c u l t to see how g r a i n elongation can lead to a net increase i n grain s i z e . The increased curvature causes migration i n a d i r e c t i o n to restore the o r i g i n a l equiaxed shape without an increase i n g r a i n volume. Also, i f an elongated grain structure was responsible f o r the grain s i z e enhancement, the accelerated rate of growth during deformation would be expected to continue a f t e r the de- formation ceased u n t i l the elongation was removed. Measurements re - vealed that very long post deformation annealing times were required before the equiaxed shape was restored. Figure 32, however, reveals a sharp decrease i n growth rate as the deformation ends. 4.3.2.4 - Grain Boundary Width An increase i n the d r i v i n g force term could be effected by a decrease i n the boundary width (Equation (9)). Since accurate measurements of grain boundary width are not a v a i l a b l e and con- troversy s t i l l e x i s t s as to the exact nature of the high angle bound- ary i t i s d i f f i c u l t to assess the e f f e c t s of deformation on the boundary width. If grain boundary s l i d i n g tended to smooth the boundary, the width could be reduced, However, some experimental evidence ' suggests that s l i d i n g boundaries become corrugated. 4.3.3 - M o b i l i t y Enhancement The only way the m o b i l i t y of the g r a i n boundaries can be increased by deformation i s through the d i f f u s i o n c o e f f i c i e n t (Equation (8)). S t r a i n enhanced d i f f u s i o n has been a subject of considerable controversy i n the l i t e r a t u r e . Most experimental work has been con- cerned with e i t h e r s e l f d i f f u s i o n c o e f f i c i e n t s or bulk c o e f f i c i e n t s of a major a l l o y i n g element. Several i n v e s t i g a t o r s ^ ^ have reported large increases i n the s e l f d i f f u s i v i t y of Ag a f t e r deformation. In- (7273) (7A) creases by factors of 10 to 15 were found i n Ag, Fe ' and Cu These r e s u l t s have been questioned however, on t h e o r e t i c a l and ex- perimental grounds by a number of w r i t e r s C o n t r a d i c t o r y ex- (78) perimental evidence has also been presented f o r Ag , i n which maximum enhancements of only 2 times were obtained. Other systems, such as Z n - A l ^ 7 9 \ S - N i ^ 8 ° \ Cu-Ag^ 8 2^ and N i - F e ^ 8 1 ^ showed l i t t l e increase with deformation. Excess vacancies produced by the motion of d i s l o c a t i o n s during deformation have been used to explain the apparent enhancement. However, t h e o r e t i c a l arguments showed that t h i s mechanism was i n - s u f f i c i e n t to account f o r the larger enhancements. A d i s l o c a t i o n pipe model appeared more p l a u s i b l e , the increased d i s l o c a t i o n density pro- v i d i n g f a s t e r paths f o r the d i f f u s i n g atoms. Despite the contro- v e r s i a l nature of t h i s subject, i t seems reasonable to conclude that some increase i n the bulk d i f f u s i o n c o e f f i c i e n t i s p o s s i b l e . The e f f e c t i s highly dependent on conditions of temperature, and s t r a i n r a t e. The e f f e c t s cf deformation on the grain boundary d i f f u s i v i t y ( 83) have not received widespread atte n t i o n . Bhat and Vitovec studied the d i f f u s i o n of zinc i n copper with superimposed fatigue s t r a i n i n g and found no s i g n i f i c a n t enhancement of the volume d i f f u s i o n c o e f f i c i e n t . However, they did f i n d increased d i f f u s i o n along grain boundaries which, they postulated, might be due to damage r e s u l t i n g from grain boundary s l i d i n g . Blackburn and Brown measured the grain boundary d i f f u s i o n c o e f f i c i e n t s of Ag d i f f u s i n g i n copper b i c r y s t a l s . Tests on s t a t i c and s l i d i n g boundaries showed a s l i g h t increase (30%) i n grain boundary d i f f u s i v i t y when the boundaries were s l i d i n g . The r e s u l t s cannot be taken as conclusive since the number of measurements was small and the observed increase was w i t h i n the experimental e r r o r . Thus, there i s some suggestion that grain boundary d i f f u s i o n c o e f f i c i e n t s can also be increased by deformation and i n p a r t i c u l a r by grain boundary s l i d i n g . Processes of grain boundary s l i d i n g , such as climb-glide of d i s l o c a t i o n s or Nabarro-Herring migration of grain boundary bumps, which involve motion of vacancies, could conceivably produce an excess of vacancies i n the grain boundary region and i n - crease the d i f f u s i o n c o e f f i c i e n t . Since the c o n t r o l l i n g f a c t o r i n grain boundary migration i s the d i f f u s i o n of the B i atoms i n a d i s - torted region close to the boundary a l o c a l increase i n the vacancy concentration i n t h i s region would lead to f a s t e r grain boundary mi- gration. If the production rate of vacancies i s proportional to the s t r a i n rate and the annealing rate of vacancies i s proportional to the excess vacancy concentration, then the equation governing the vacancy concentration i s : dn = Ki e dt - n K 0 dt (11) X -L X 2 where n = atomic f r a c t i o n s of vacancies due to s t r a i n x = constant such that e i s the rate of vacancy production e = s t r a i n rate K2 = constant equal to — where x i s the average l i f e t i m e of a vacancy. Integration of t h i s equation y i e l d s : K e n x = ^ — (1 - exp (-K 2t)) (12) where t i s the time to reach a value of s t r a i n at the given s t r a i n rate. If d i f f u s i o n takes place by a vacancy mechanism, the d i f f u s i v i t y i s pro- p o r t i o n a l to the t o t a l atomic f r a c t i o n of vacancies. Thus, the d i f f u s i - v i t y i n a strained system w i l l be proportional to ( n x + n v) where n v i s the equilibrium vacancy concentration (mole f r a c t i o n ) . In preceding sections i t was shown that during annealing: dD _ ^ dF dt ix dF where M was proportional to the d i f f u s i v i t y and - j ^ was proportional to — . Combining these f a c t o r s y i e l d s : D k' n ^ - (13) dt D 82 when the vacancy concentration i s i n thermal equilibrium and: ,- k' (n + n ) ^ = — - 5 2 - (14) dt D when excess vacancies are generated. Substituting f o r n from x equation (12) gives: - k' ( KT~ (1 - exp (-K t ) ) + n ) — = (15) dt D Integration of t h i s equation gives the grain s i z e v a r i a t i o n with time during deformation: -2 2 / K 1 \ K l f1 ~ exp (-K t) \ D 2 - D o 2 = 2k' [ ( ^ + n v ) t - ) ] (16) This equation can be compared to the integrated form of equation (13): D 2 - D 2 = 2 k ' n t (17) o v If the values of the various constants can be estimated, t h e o r e t i c a l grain s i z e versus time curves during deformation can be calculated. An approximate value of n^ of 10 9 was obtained from^^^ : n v = exp - AH f/ R T (18) where AH^, the a c t i v a t i o n energy f o r formation of vacancy, = 11.8 k c a l / (87) mole . Then k' was determined from the k value f o r the t h e o r e t i c a l unstrained grain growth curve (Fig.12). The constant w i l l depend upon the nature of the vacancy sinks i n the material. The most e f f e c t i v e sink i n t h i s case w i l l pro- (85) bably be the grain boundary. G i r i f a l c o and Grimes have shown that 83 for a p l a t e - l i k e sink: 2D 2D K 2 = ~~ = ~2~ e X p (~ A V R T ) ( 1 9 ) L L where D^ i s the vacancy d i f f u s i o n c o e f f i c i e n t (OJ) AH^ i s the a c t i v a t i o n energy for vacancy motion = 15.7 kcal/mole 2 (88) D q i s the frequency f a c t o r f o r vacancy d i f f u s i o n = 9.7 cm /sec. and L i s the perpendicular distance from the sink. An accurate value for the distance L cannot be determined since i t depends on the boundary shear process that produces the vacancies. A value of L = 100A° :was taken as a reasonable estimate which yi e l d e d K 2 ^ 10 3/min. To determine K^, values of D, t and t f o r one experimental point were inserted i n equation (16) along with the values of the other constants. The t h e o r e t i c a l curves were thus constraindd to pass through only t h i s one point and could be used f o r comparison with experimental points at other s t r a i n rates. The point chosen was: — _2 D = 4 . 7 p a t t = 100 minutes and i = 10 /min. —3 (89) —3 which gave = 4.1 x 10 . Barry and Buown estimated = 10 f o r s t r a i n enhanced bulk d i f f u s i o n . Equation (16) was now used to c a l c u l a t e a s e r i e s of g r a i n s i z e versus time curves f o r d i f f e r e n t s t r a i n rates (Fig.41). The experimental points from F i g s . 16 and 17 and the unstrained annealing curve are shown i n Fig.41 f o r comparison. Agreement of the experimental points with the t h e o r e t i c a l curves i s s a t i s f a c t o r y from s t r a i n rates from 1.0/minute to 7 x 10 ^/minute. The lower s t r a i n rate curves do not agree quite so w e l l . I t should be remembered that the points for these low s t r a i n rates were determined by creep t e s t i n g the high ex- 20- 10 9 8 1\ 6 D 5 l > ) 4 I L_ 0-1 STAT IC ANNEAL ING 1 1 EXPERIMENTAL POINTS € (m in) • 1 0 A 0- X I o"2, © 2x10* A 7XI0"* V 2x10 / • 1x10 1000 Fig.41, 10 100 TEST ING TIME (min) Comparison of Theoretical and Experimental Grain Size versus Time During Deformation Curves. 10,000 00 85 trusion r a t i o material which seemed to possess s l i g h t l y d i f f e r e n t grain growth c h a r a c t e r i s t i c s from the lower extrusion r a t i o material (Fig.15). The same equation (16) can be used to c a l c u l a t e t h e o r e t i c a l AD . , — versus s t r a i n curves by computing the time required to reach a \ c e r t a i n value of s t r a i n and i n s e r t i n g i t i n the equation (Fig.42). Again agreement i s good f o r the high s t c a i n rates shown. The trend to lower ^p- values with decreasing s t r a i n rate i n the superplastic region i s also predicted. T h e o r e t i c a l ^ values at 25% true s t r a i n DA f o r a l l s t r a i n rates were calculated and compared to the experimental points from Fig.21 (Fig.43). The experimental trend i s again pre- dicted by the theory. Predictions of the ainealing behaviour a f t e r deformation can also be made. When excess vacancies are no longer produced equals zero. The excess vacancies that were present at the end of the deformation w i l l continue to influence the grain growth u n t i l they anneal out. I f the deformation ends at time t ^ , the excess vacancies present , n^ , w i l l be given by: K. X l n x = ^ e (1 - exp (- K 2 t ^ ) (20) The loss rate of the excess vacancies i s given by: d n dt Integration of t h i s equation y i e l d s ^ = - n x K 2 (21) K2 ( t l ~ t ) n = n e (22) x x^ T R U E STRAIN 1%) Fig.42. Comparison of Theoretical and Experimental Rel a t i v e Grain • v , . - . Size Change versus S t r a i n Curves.--— -..v.. Fig.43. Comparison of Theory and Experiment f o r the Relative Grain Size Change Aft e r 25% S t r a i n . 88 where n i s the atomic f r a c t i o n of excess vacancies at time t a f t e r x deformation ceases. Substituting equation(20)into equation (14) gives: k' (n e K 2 ( t l _ ° + n ) ft X l V (23) D If i s the grain s i z e at time t ^ i n t e g r a t i o n of t h i s equation gives the time dependence of the grain s i z e a f t e r deformation: 2k' n D 2 - S^2 = 2k' n v (t - t 1 ) + - i r (1 - exp (K 2 ( t ^ t ) ) (24) A curve was calculated from t h i s equation with t ^ = 120(min.) and = 4.6 u and compared to the corresponding experimental curve (Fig. 44). The theory c o r r e c t l y p redicts the sharp t r a n s i t i o n i n the curve a f t e r deformation, but does not predict the actual grain s i z e a f t e r long periods of post deformation annealing. The t h e o r e t i c a l curve coincides with the s t a t i c annealing curve of Fig.35 rather than the actual post deformation curve. These decreased growth rates a f t e r deformation can be explained by a mechanism involving the e f f e c t s of de- formation on texture. N e i l s e n ^ ^ has discussed a coalescence-type mechanism which occurs as a consequence of g r a i n "encounters" during normal grain growth. Normal grain growth occurs by the disappearance of the small grains of few sides. In a t y p i c a l sequence a four, sided g r a i n (Fig.45a No.5) shrinks u n t i l the upper two t r i p l e points meet (b) 89 O O 90 (a) (b) (c) Fig.45. Illustration of Grain "Encounter" During Grain Growth. This unstable configuration changes to the more stable one (c) and grain 5 has lost a side. In the process grains 2 and 4 have "encountered" or formed a new boundary between them. Nielsen proposed that i f grains 2 and 4 were of like orientation or slightly misoriented a grain bound- ary would not form between them. They would have coalesced to form a larger grain and increased the average grain size. The frequency of encounters that lead to a coalescence will depend on the relative number of like-oriented grains present in the structure. Compared to a random structure, a textured material will have a higher proportion of these like orientations and consequently a higher grain growth rate. Since superplastic deformation causes a randomization of the original texture the growth rate after deformation should be less than a statically annealed sample as is observed. The grain size distribution analysis and the tension-com- p»Ession experiment are also consistent with the diffusivity enhancement model. Since the deformation ceases only a speeding up of the normal growth processes, equivalent grain size samples should possess the same size distributions regardless of the path taken to reach that grain s i z e . The curves of Fig.28 show that the g r a i n s i z e d i s t r i b u t i o n s are indeed i d e n t i c a l . In the tension-compression experiment r e - v e r s a l of the stress d i r e c t i o n should s t i l l produce a vacancy ex- cess i n the grain boundary region. As long as the boundaries are s l i d i n g , vacancies w i l l be generated and the mobility increased r e - gardless of the s l i d i n g d i r e c t i o n . Experimentally the stress d i r e c t i o n had l i t t l e e f f e c t on the grain s i z e enhancement produced. Since a l l of the experimental r e s u l t s are consistent with the d i f f u s i v i t y enhancement model, i t seems the most l i k e l y cause of the deformation induced grain growth. SUMMARY AND CONCLUSIONS 1) Superplastic deformation of Sn - 1% B i was found to pro- duce considerable increases i n grain growth rates. During annealing the g r a i n s i z e k i n e t i c s could be described by an equation of the —2 2 2 form D - D = k t. The e f f e c t of deformation was to increase the o value of the constant k. 2) The gr a i n growth enhancement was most pronounced at high s t r a i n rates i n the superplastic region. Reduction of s t r a i n rate produced a lessening of the growth ra t e s . 3) The r e l a t i v e grain s i z e change, ̂ p- , was found to depend l i n e a r l y on s t r a i n . 4) The grain growth rates on annealing a f t e r deformation were les s than s t a t i c rates. 5) Grain type and grain s i z e d i s t r i b u t i o n s were s i m i l a r i n deformed and annealed structures. 6 ) A mechanism involving the production of excess vacancies i n the grain boundary region leading to increased boundary mobility was found to be consistent with a l l experimental observations. 93 APPENDIX A E f f e c t of Coalescence Mechanism on Grain Type and Size D i s t r i b u t i o n s If the coalescence mechanism i s responsible f o r the grain s i z e increase during deformation, changes i n the grain type and s i z e d i s t r i b u t i o n s would be expected. In t h i s section an attempt w i l l be made to c a l c u l a t e the expected d i s t r i b u t i o n s a f t e r deformation from the s t a t i c annealing d i s t r i b u t i o n s . It w i l l be assumed that the d i s t r i b u t i o n curves (Figs. 24 and 26) f o r s t a t i c annealing are representative of the actual d i s - t r i b u t i o n s at any time during annealing. This assumption i s usually (44) made i n t h e o r e t i c a l considerations of grain growth although some changes i n the s i z e d i s t r i b u t i o n may occur on extended annealing Since i t has been shown that the d i s t r i b u t i o n curves taken from a random 2 dimensional section and the actual s p a t i a l d i s t r i b u t i o n are (41 44) s i m i l a r i n form ' the analysis w i l l be done i n 2 dimensions. When a boundary coalesces, the l o c a l d i s t r i b u t i o n of g r a i n types i s a f f e c t e d . In a perfect 6 sided network (Fig.46), when grains 1 and 2 undergo a coalescence so that the indicated boundary disappears, the combined 1-2 grain w i l l have 8 sides. The adjacent grains (5 and 9) have become 5 sided. 94 Fig.46. E f f e c t of a Grain Boundary Coalescence. In general when a boundary separating grains having m and n sides disappears, a combined grain i s produced with sides numbering (m + n) - 4 and the two adjacent grains have each l o s t a side. In addi t i o n , the total number of grains has been reduced by one. The p r o b a b i l i t y of any one grain taking part i n a coalescence w i l l depend on i t s number of sides (m) and also on i t s r e l a t i v e frequency i n the structure (f ). The frequency f ^ w i l l be defined as X_/Xi. where X i s the number of m sided grains i n a given area and m t m X t i s the t o t a l number of grains i n the same area. The p r o b a b i l i t y of an (m + n) coalescence taking place w i l l vary as i . n f^. The p r o b a b i l i t y of producing an adjacent grain having p-1 sides w i l l equal the p r o b a b i l i t y of fin d i n g a p sided grain i n an adjacent s i t e . The 95 p r o b a b i l i t y of f i n d i n g a p sided g r a i n i n any s i t e i n the structure w i l l vary as i t s r e l a t i v e frequency ( f ^ ) . Therefore,for each coalescence that occurs there w i l l be produced i n the adjacent p o s i t i o n s : 2 f grains of p-1 sides + 2 f grains of m-1 sides m + 2 f grains of n-1 sides n To s i m p l i f y the c a l c u l a t i o n s , coalescences w i l l be res- t r i c t e d to the most frequently occurring grains, the 4, 5, 6 and 7 sided grains. In addi t i o n m w i l l be r e s t r i c t e d to values from 3 to 13. If a number of coalesced grains are produced by a given amount of s t r a i n , the t o t a l w i l l be composed of the number formed from each combination i . e . X c = a + b + c + d + e + f + g + h + i + j (25) a - number of 4-4 coalescence b = I I I I 4-5 I I c I I I I 4-6 I I d I I I I 4-7 I I e = I I I I 5-5 I I f - n: I I 5-6 I I g - I I I I 5-7 I I h — I I . I I 6-6 I I i 5 ". I I 6-7 i t 3 = I I it 7-7 M The r a t i o a:b:c: ... w i l l be proportional to the p r o b a b i l i t y of occurrence of each combination (m f . n f ) i . e . m n 96 a:b:c: ... = 4 i^.h ; 4 tfy 5 . f 5 ; 4 6 . f g : ... (26) If i s known and i s small compared to Xfc then values of a, b, c .... can be calculated from the o r i g i n a l frequency d i s t r i b u t i o n . The number of grains of side m a f t e r a number of coalescences have occurred w i l l be defined as X'm. For one (4-4) coalescence X^ w i l l be reduced by two X'. = X. - 2 4 4 However, a coalesced grain has been produced having sides equal to m + n - 4 (=4) X'. = X. - 2 + 1 4 4 Also 2 f j . , 4 sided grains have been produced i n the adjacent p o s i t i o n X X' = X. - 2 + 1 + 2 ^r- 4 4 X. F i n a l l y , 2 f^»4 sided grains have l o s t a side i n the adjacent p o s i t i o n X X, X' 4 = X 4 - 2 + 1 + 2 ^ - 2 ̂  (27) This equation w i l l apply f o r every 4-4 coalescence that occurs, therefore, f o r "a" 4-4 coalescences: X' = X. - a + 4 4 / X X. K (2irt ~ 2rt ) \ <28> The frequency of the other grains w i l l be affected only by the adjacent grain reactions: X' - X. + [ 2 ^ - 2 ̂  1 a X ' 6 - X 6 , + ( 2 ^ - a 97 The 13 sided grains w i l l only loae by the adjacent grains since 14 sided grains were not permitted o r i g i n a l l y : X13 X ' l 3 = X13 ~ 2 7~ 3 X t A 2 sided grain i s possible by the adjacent mechanism but i s unstable and w i l l disappear very quickly. I t w i l l be included i n the c a l - culations however: X 3 X' 2 = 2 ^ a For each of these "a" coalescences the t o t a l number of grains has been reduced by one: X - t - X t " 3 This analysis can be extended to the other combinations. For example for "a" 4-4 and "b" 4-5 coalescences the equations w i l l be: X 3 X' 2 = 2 ^ (a + b) X' 3 = X 3 + ( 2 ^ - 2 ̂  ) (a + b) 0 X' 4 = X 4 + ^ 2 j ~ - 2 Y- \ (a + b) - (a + b) x ' : c = x r + ^ ~ 2 ) (a + b) + b - b ( 2 T t ~ 2 T t ) ( a + b ) •5 5 X ' 6 = X 6 + X ' l 3 = X13 + ( 2 If »(a + b ) X» + = Xfc - (a + b) The equations obtained after a l l possible combinations are considered are: X'2 = + \ 2 ^ ) (a + b + c ...) X t / X'3 = X 3 + ( 2 Y~ - 2 ) (a + b + c...) X'4 = X 4 + ( 2 ^ - 2 ^ J (a + b + c ...) - (a + b + c + d) X'5 = X 5 + ( 2 ^ - 2 ^ ) (a + b + c ...) - (2e + f + g) X'6 = X & + ^2 Y~ - 2 Y~ j (a + b + c .. .) - (f + 2h + i ) + e x, x 6 x„ x. X', = X, + 12 ~ - 2 ̂ - J (a + b + c . . . . ) - (g + i + 2j) + f 7 - 7 r x t - x t / X X \ X ' 8 = X g + ( 2 ^ - 2 ^ 1 ( a + b + c . . . ) + g + h X'9 = X 9 + [2 - 2 ^ ) (a + b + c...) + !' ' I _ Y L 10 10 X10 x t 2 x t i x l l 2 — — x t t L X12 <\ X t f 2 X 1 3 I X t ( 7 \ " 2 x f ' (a + b + c ...) + j X ' u = X n + ( 2 ^ - 2 ^ ) (a + b + c...) X » 1 2 = X 1 2 + (2 ̂  - 2 ̂  J (a + b + c...) ) X ' l 3 = X13 + ( " 2 F ^ l (a+-b + c...) X' = X t - (a + b + c + ...) 99 By taking successive increments f o r X £ and knowing the o r i g i n a l d i s t r i b u t i o n , the new d i s t r i b u t i o n can be c a l c u l a t e d from the equations. Three increments of.X .were taken, each increment being 2% of the t o t a l number of grains (X f c). The c a l c u l a t e d d i s - t r i b u t i o n a f t e r the 6% coalscence could then be compared with the deformed d i s t r i b u t i o n (Fig.4 7 ) . The theory p r e d i c t s the observed decrease i n the frequency of the 4, 5 and 6 sided grains as compared to the annealed s t a t e . I t also p r e d i c t s the observed frequency increase i n the 8, 9 and 10 sided grains. However, disagreement i s not ob- tained f o r the 2, 3 , 7, 11, 12 and 13 sided grains. I t seems reason- able to conclude however, that the theory predicts the general trend of the d i s t r i b u t i o n changes and that only approximately 6% of the grains need coalesce to produce the degree of change observed. In order to determine i f t h i s 6% coalescence could also produce the observed changes i n the grain s i z e d i s t r i b u t i o n a s i m i l a r analysis can be performed. F i r s t , i t i s necessary to estimate the (44) most probable area of each type of gr a i n . Feltham has shown that a one-one r e l a t i o n s h i p e x i s t s between the type and s i z e d i s t r i b u t i o n s i . e . the smaller grains would, i n general, be those with few sides and the l a r g e r grains those with many sides. This r e l a t i o n s h i p was suggested by the log-normal d i s t r i b u t i o n s of the grain s i z e s and types. To prove the r e l a t i o n s h i p , the actual s i z e d i s t r i b u t i o n s f o r each type of grain were measured. The mean diameter of each type varied l i n e a r l y with the number of sides. ,It has been shown that the s i z e and type d i s t r i b u t i o n s of the Sn - 1% B i are also log-normal so i t w i l l be assumed that the same l i n e a r r e l a t i o n s h i p i s v a l i d . By oomparing the two d i s t r i b u t i o n curves f o r the annealed 1 1 1 1 1 1 r A A A N N E A L E D 2 3 4 5 6 7 8 9 10 II 12 13 14 GRAIN T Y P E Fig.47. Comparison of Calculated and Experimental Grain Type D i s t r i b u t i o n s . 101 material i t seemed that the three sided grains should have mean diameters corresponding to grain classes 1 and 2, 4 sided grains to 3 and 4 classes, 5 sided grains to cl a s s 5, 6 sided to cl a s s 6 and 7 sided to class 7. The most probable diameters of the 3, 4, 5, 6 and 7 sided grains were thus taken to be 1.9, 3.8, 6.4, 9, and 12.8 u r e s p e c t i v e l y . The areas corresponding to these diameters are 2.8, 11, 2 32, 64 and 128 (u ) r e s p e c t i v e l y . Again only the coalescences between the 4, 5, 6 and 7 sided grains w i l l be considered. For one m + n coalescence a new grain i s formed having an area which i s the sum of the areas of the combining grains. At the same time the number of grains i n the m sided g r a i n class has been reduced by one. S i m i l a r l y the number of grains i n the class of the n-sided g r a i n has been depleted by one. The possible combinations are l i s t e d i n Table 5, along with the area of the coalesced grains. TABLE 5 - Areas of Coalesced Grains Combination No. of Area of the Grain Class Combinations Coalesced to which Grain (ji ) Coalesced Grain belongs 4 - 4 a 22 4 4 - 5 b 43 5 4 - 6 c 75 6 4 - 7 d 139 7 5 - 5 e 64 6 .5 - 6 f 96 7 5 - 7 8 160 7 6 - 6 h 128 7 6 - 7 i 192 8 - 7 - 7 j 256 8 102 When a l l these combinations take place the number of gr a i n i n clas s 6 has been increased by c + e but at the same time has been decreased by a l l those reactions i n v o l v i n g a 6 sided grain i . e . c, f , 2h and i grains have been l o s t . I f Np i s the number of grains o r i g i n a l l y i n grain c l a s s p and N' p i s the number a f t e r (a + b + c ....) coalescences have occurred then the following equation can be written: g + b i + c + e 2 j + d + f + g + h It w i l l be assumed that the grains l o s t from grain classes 3 and 4 by the reactions involving 4 sided grains w i l l be equally divided between the two clas s e s : N.. 4 = N 4 _ ( 2 a _ £ W _ c _ t ^ ) + a N ' = N (2a + b + c + d) 3 3 2 Using the experimental values f o r N^, etc. (Fig.26) and taking X c i n steps of 2% of the t o t a l number of grains the s i z e d i s t r i b u t i o n s a f t e r coalescence can be calculated. When the r e s u l t s f o r a t o t a l of 6% coalescence are compared to the actual as deformed distribution (Fig.48), i t i s seen that the predicted changes are very small compared to the actual measured r e s u l t s . 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