SUPERPLASTIC CREEP IN THE LEAD TIN EUTECTIC by ALBERT KEITH SURGES B.A. Sc., U n i v e r s i t y of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of METALLURGY We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r .an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e 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 p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment 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 C o l u m b i a V a n c o u v e r 8 , Canada Date O c t o b e r 2 7 . 1 9 7 0 i A b s t r a c t An extensive creep study of a s u p e r p l a s t i c m a t e r i a l has not p r e v i o u s l y been made. The present study was c a r r i e d out to determine i f there are any b a s i c d i f f e r e n c e s between the creep of coarse grained m a t e r i a l s and f i n e grained super-p l a s t i c m a t e r i a l s . The r e s u l t s give i n f o r m a t i o n about the mechanical p r o p e r t i e s of s u p e r p l a s t i c a l l o y s and are r e l e v e n t to an understanding of the mechanics of s u p e r p l a s t i c i y . At high s t r a i n r a t e s the s u p e r p l a s t i c l e a d - t i n e u t e c t i c deforms by recovery creep and a 3-stage creep curve i s observed, s i m i l a r to that found f o r coarse grained m a t e r i a l s . As the s t r a i n r a t e i s decreased, the i n i t i a l t r a n s i e n t (primary creep) disappears and the creep curve i s l i n e a r u n t i l necking occurs and t e r t i a r y creep ends i n f a i l u r e . In the p r i n c i p a l s u p e r p l a s t i c range, at medium s t r a i n r a t e s , creep curves are l i n e a r to at l e a s t 50 % s t r a i n . The recovery r a t e i s immediately equal to the s t r a i n hardening r a t e and there i s no primary creep. At low s t r a i n r a t e s the creep curve i s s l i g h t l y convex as the creep r a t e decreases w i t h time. This may be due to the s e l f e x t i n g u i s h i n g nature of d i f f u s i o n a l creep or p o s s i b l y s t r a i n induced g r a i n growth. These r e s u l t s are c o n s i s t e n t w i t h the g r a i n boundary s l i d i n g t h e o r i e s of s u p e r p l a s t i c i t y although d e t a i l s of the accommodation processes are not known. At the lowest s t r a i n r a t e s , d i f f u s i o n a l creep may operate. i i ACKNOWLEDGMENT The author i s g r a t e f u l f o r the advice of and h e l p f u l d i s c u s s i o n w i t h h i s research d i r e c t o r , Dr. T.H. Alden. Thanks are a l s o extended to R.C. Cook and K.C. Donaldson f o r t h e i r d i s c u s s i o n s and suggestions. C.B. S u l l i v a n ' s a s s i s t a n c e w i t h draghting f o r the p r e s e n t a t i o n of t h i s t h e s i s i s a l s o appreciated. I i i TABLE OF CONTENTS 1. INTRODUCTION 1 1.1 Stage II (Superplastic Stage) 3 1.1.1 Review of Experiment 3 1.1.2 Theoretical Discussion 5 1.2 Stage I II 6 1.3 Stage I 6 1.4 Previous Creep Studies on Lead and T i n Systems ... 8 2. EXPERIMENTAL 9 2.1 Material and Specimen Preparation 9 2.2 Creep Tests 10 3. RESULTS 13 3.1 Calculations 13 .3.2 Log Stress versus Log S t r a i n Ra±e Curves 13 3.3 Creep Curves 16 3.3.1 Stage II Creep Curves 16 3.3.2 Stage I Creep Curves 17 3.3.3 Stage II - Stage I II T r a n s i t i o n Creep Curves .... 17 3.3.4 Stage I II Creep Curves 23 3.4 Incremental Loading and Unloading 23 3.5 S t r a i n A f t e r - E f f e c t s 2 9 4. DISCUSSION 30 4.1 Stage I II 30 4.2 Stage I II - Stage II T r a n s i t i o n 35 4.3 Stage I II 37 4.4 Stage I • • 4 0 5. SUMMARY AND CONCLUSIONS 44 6. SUGGESTIONS FOR FUTURE WORK 46 7. APPENDICES 4 7 7.1 Computer Programme 47 7.2 Ad d i t i o n a l Creep Curves 52 i v 7.3 C a l c u l a t i o n of Theoretical Creep Curves f o r Pure Nabarro-Herring Creep and Pure Coble Creep 53 8. BIBLIOGRAPHY 64 V LIST OF FIGURES No. Page 1. C h a r a c t e r i s t i c l o g s t r e s s versus l o g s t r a i n r a t e curve. 1 2. Specimen i n g r i p s . 10 3. Constant s t r e s s creep machine. 12 4. Log s t r e s s versus l o g s t r a i n r a t e curve. (SGS, 2 microns). 14 5. Log s t r e s s versus l o g s t r a i n curve. (LGS, 8 microns). 15 6. Comparison of the S-curve data of the present and previous work. 16 7. Stage I I creep curve (SGS), 429 p s i . 18 8. Stage I I creep curve (SGS), 1716 p s i . , 19 9. Stage I I creep curve (SGS), 715 p s i . 20 10. Stage I I creep curve (LGS), 814 p s i . 21 11. Stage I creep curve (SGS), 97 p s i . 22 12. Creep curve i n stage I I - s t a g e I I t r a n s i t i o n (SGS, 3432 p s i . 24 13. Creep curve i n stage I I - s t a g e I I I t r a n s i t i o n (LGS), 3582 p s i . 25 14.. Stage I I I creep curve (5212 p s i ) . 26 15. Stage I I I creep curve (5535 p s i ) . 26 16. Stage I I I creep curve (6512 p s i ) . 28 17. S t r a i n r e q u i r e d to reach steady s t a t e versus creep s t r e s s . 27 18. Incremental lo a d i n g d u r i n g s u p e r p l a s t i c creep. 18 19. Return of an i n i t i a l t r a n s i e n t w i t h incremental loading i n stage I I I 3 1 20. T r a n s i t i o n to steady s t a t e w i t h unloading i n stage I I I . 31 21. S t r a i n r e l a x a t i o n study i n stage I I 32 22. Reloading a f t e r recovery i n stage I I I 32 23. Log s t r e s s versus l o g s t r a i n r a t e r e l a t i o n s h i p f o r Pb-2.45 wt%. v i 24. Log s t r e s s versus l o g s t r a i n r a t e r e l a t i o n s h i p f o r Pb-2.45 wt% t h a l l i u m (100 u ) . 25. Experimental, t h e o r e t i c a l N-H and t h e o r e t i c a l Coble creep curves (97 p s i ) . Appendix I I I a. Stage I I (SGS) 572 p s i . b. Stage I I (SGS), 858 p s i . c. Stage I I (SGS), 1144 p s i . d. Stage I I LGS), 407 p s i . e. Stage I I (LGS), 1221 p s i . f. Stage I (SGS), 120 p s i . g. Stage I (SGS), 143 p s i . h. Stage I (SGS), 286 p s i . i . T r a n s i t i o n (SGS), 2574 p s i . j . T r a n s i t i o n (SGS), 2860 P s i . k. T r a n s i t i o n (LGS), 1954 p s i . 1. T r a n s i t i o n (LGS), 2280 p s i . m. T r a n s i t i o n (LGS), 2606 p s i . n. T r a n s i t i o n (LGS), 2932 p s i . o. T r a n s i t i o n (LGS), 32580 p s i . p. T r a n s i t i o n (LGS), 3908 p s i . q. T r a n s i t i o n (LGS), 4234 p s i . r . T r a n s i t i o n (LGS), 4560 p s i . s. T r a n s i t i o n (LGS), 4886 p s i . t . Stage I I I , 2931 p s i . u. Stage I I I , 7326 p s i . v. Stage I I incremental l o a d i n g , w. Stage I I I unloading -1-I. I n t r o d u c t i o n S u p e r p l a s t i c behaviour has been found i n many metal systems. A l l o y s of lead and z i n c have been i n v e s t i g a t e d most f r e q u e n t l y , but systems c o n t a i n i n g n i c k e l , i r o n , aluminium, t i n , cadmium, magnesium, and copper have a l s o e x h i b i t e d s u p e r p l a s t i c p r o p e r t i e s . Studies have a l s o been made to determine deformation mechanisms which are c o n s i s t e n t w i t h experiment. Results of these s t u d i e s may a l s o be important i n the development of new metal-forming techniques. Stress versus s t r a i n r a t e r e s u l t s from t e n s i l e and creep t e s t s have been p l o t t e d as l o g s t r e s s (log a) versus l o g s t r a i n r a t e ( l o g e) to produce a c h a r a c t e r i s t i c three stage S-shaped curve. In the s u p e r p l a s t i c range, stage I I , the s t r a i n r a t e (e) i s i n s e n s i t i v e to the ap p l i e d s t r e s s (C T ) . L O G S T R A I N R A T E Figure 1. C h a r a c t e r i s t i c l o g s t r e s s - l o g s t r a i n r a t e curve. -2-Each stage of the S-curve may be described by the equation a = K e (1) where K i s a constant and m i s c a l l e d the s t r a i n r a t e s e n s i t i v i t y parameter. T y p i c a l values of m vary from l e s s than .1 f o r most metals up to 1.0 f o r hot polymers and glasses. S u p e r p l a s t i c metals have been observed to e x h i b i t m values as high as •SS''", .but are t y p i c a l l y about 0.5. In stage I I , where m i s high, propagation of a neck i s prevented by a l o c a l hardening r e s u l t i n g from an increased s t r a i n r a t e , and thus deformation w i l l proceed i n a s o f t e r p o r t i o n of the m a t e r i a l . A c c o r d i n g l y , a high value of m i s a s s o c i a t e d w i t h l a r g e elonga-2 t i o n s , r e p o r t e d l y as high as 2000 % . Mathematically, the r e l a t i o n s h i p between m and e l o n g a t i o n can be' shown more e x p l i c i t l y by f i r s t d i f f e r e n t i a t i n g equation (1) to o b t a i n I t can be seen that the l a r g e r the value of m, the more i n s e n s i t i v e s t r a i n r a t e becomes to a change i n s t r e s s . A l s o , over a c e r t a i n s t r a i n r a t e range i n s u p e r p l a s t i c m a t e r i a l s , m increases s l i g h t l y w i t h i n c r e a s i n g s t r a i n r a t e . The p o s i t i v e v a r i a t i o n of m i n s u p e r p l a s t i c m a t e r i a l s w i l l cause the f a c t o r i n brackets i n equation (2) to be reduced, and w i l l f u r t h e r reduce s e n s i t i v i t y to necking. p l a s t i c i t y i s most conveniently done by c o n s i d e r i n g the stages of the logo -loge curve s e p a r a t e l y . Of these, the most important and i n t e n s i v e l y s t u d i e d i s stage I I . (2) Further d e s c r i p t i o n of experimental and t h e o r e t i c a l s t u d i e s on super--3-1.1. Stage I I ( S u p e r p l a s t i c Stage) 1.1.1. Review of Experiment A f i n e g r a i n s i z e has been shown to be the most important m i c r o s t r u c t u r -3-11 a l requirement f o r s u p e r p l a s t i c i t y . P r o v i d i n g the phases are of comparable hardness, the composition and means by which g r a i n refinement i s achieved are 6 9 of secondary importance. ' In two phase systems, a hot or c o l d working step w i l l permit the formation of a f i n e g r a i n s i z e , w h i l e phase boundaries i n h i b i t g r a i n growth. Thus extensive s t u d i e s have been made on the e u t e c t o i d Zn - 22 wt. % A l and the Pb-Sn systems. D i l u t e a l l o y ^ ^ a n d pure metal systems,have als o e x h i b i t e d s u p e r p l a s t i c i t y . The inherent problem w i t h these systems i s to produce and maintain a f i n e l y - d i v i d e d m i c r o s t r u c t u r e . For a given time and tem-perature, g r a i n s i z e i s l a r g e r i n d i l u t e a l l o y s . Large elongations during stage I I have been c o n s i s t e n t l y reported. The l a r g e s t elongations occur at s t r a i n r a t e s near that associated w i t h peak m 16 18 values ' . Maximum el o n g a t i o n may occur p r e c i s e l y at peak m but the r e s u l t i s obscurred by the decreasing s t r a i n r a t e during a t e n s i l e t e s t on a constant c r o s s -head r a t e machine such as an I n s t r o n . i T e n s i l e t e s t s i n v o l v i n g s e v e r a l g r a i n s i z e s show that the e f f e c t of g r a i n coarsening i s to s h i f t the s t r a i n r a t e at constant s t r e s s to lower v a l u e s . This s h i f t i s expressed by the r e l a t i o n s h i p k ^ 1/L ( f o r constant m) where a 1 3 5 7 8 1 2 1 3 i s u s u a l l y between 2 and A ' ' ' ' ' ' and L i s the spacing between g r a i n or phase boundaries. Temperature a l s o has an e f f e c t on the S-curve. The maximum m f o r each temperature drops w i t h decreasing temperature''". The s t r a i n r a t e correspond-i n g to peak m a l s o decreases w i t h decreasing temperature; An increase i n tempera-_4-ture s h i f t s the curve to higher s t r a i n r a t e s and to s l i g h t l y lower s t r e s s e s . G r a i n boundary s l i d i n g i s observed and i t s c o n t r i b u t i o n to t o t a l s t r a i n increases as the s t r a i n r a t e i s lowered from stage I I I i n t o stage 11^'lL. E x p e r i m e n t a l l y , the c o n t r i b u t i o n of GBS to t o t a l s t r a i n i s determined by measuring the o f f s e t of g r i d l i n e s i n s c r i b e d across g r a i n boundaries p r i o r to deformation. S u p e r p l a s t i c deformation does not cause the b u i l d up of a d i s l o c a t i o n s u b s t r u c t u r e . An Mg-Al alloy''" has been water quenched from 400°C during s u p e r p l a s t i c deformation and transmission e l e c t r o n microscopy showed no d i s l o c a t i o n t r a c e s . D i s l o c a t i o n s are present a f t e r deformation i n stage I I I . The low temperature y i e l d s t r e s s remains unchanged, r e l a t i v e to that of the 13 undeformed m a t e r i a l , a f t e r stage I I deformation . Pb-5% Cd specimens were deformed 2 % at a s e l e c t e d temperature, s t r a i n r a t e and g r a i n s i z e and immedia-t e l y quenched to -90°C. The .2% y i e l d s t r e s s was then determined. Specimens deformed i n the s u p e r p l a s t i c range showed no i n c r e a s e i n y i e l d s t r e s s w h i l e those deformed i n stage I I I showed an increase i n y i e l d s t r e s s . Grain shape remains equiaxed a f t e r as much as 1000- % e l o n g a t i o n ^ . Grain growth o c c u r s ^ ' ^ arid may be important i n the deformation process or may mask other r e l a t i o n s h i p s . 13 Recovery r a t e s are f a s t i n s u p e r p l a s t i c m a t e r i a l s and decrease w i t h i n c r e a s i n g g r a i n s i z e . Pb-5% Cd specimens were deformed 2% at -90°C, annealed f o r various times at 50°C then deformed again at -90°C. The amount of recovery, R, was determined by R = (aH - aR)/(aH - aY) , where aH i s determined a f t e r 2% s t r a i n , oR a f t e r recovery and aY on the annealed m a t e r i a l . There was 40% recovery f o r a 4.1p g r a i n s i z e a f t e r only .2 minutes while i t took 100 minutes to o b t a i n 30 % recovery i n a 15y specimen. -5-Although there i s general agreement i n the experimental observations made on s u p e r p l a s t i c m a t e r i a l s , disagreement on the r e l a t i v e importance or i n t e r p r e t a t i o n of i n d i v i d u a l obeservation has lead to a wide range of suggested mechanisms. 1.1.2. T h e o r e t i c a l D i s c u s s i o n Work on the Al^-33 wt. % Cu e u t e c t i c ^ and the Zn-Al eutectic*"'" lead to s i m i l a r proposals that the high s t r a i n r a t e s e n s i t i v i t y which c h a r a c t e r i z e s s u p e r p l a s t i c i t y i s the r e s u l t of boundary shearing and m i g r a t i o n . G r a i n boundary shear was suggested to be r a t e c o n t r o l l i n g , and mechanical o b s t r u c t -ions to s l i d i n g were removed by s t r a i n r a t e enhanced boundary m i g r a t i o n and r e c r y s t a l l i z a t i o n . At t h i s intermediate s t r a i n r a t e , boundaries become smoother and s t r e s s i s determined by viscous drag along the boundaries. Work 6 5 on the Pb-Sn and Sn-Bi systems a l s o lead to the proposal that g r a i n boundary s l i d i n g was the r a t e c o n t r o l l i n g mechanism. Experimental r e s u l t s showed^ that the g r e a t e s t c o n t r i b u t i o n of g r a i n boundary s l i d i n g occurred when the s t r a i n r a t e s e n s i t i v i t y parameter, m, reached i t s peak value. Another suggested mechanism f o r deformation of Pb-Sn*^ incorporates non-Newtonian g r a i n boundary s l i d i n g and d i f f u s i o n a l (Newtonian) creep a c t i n g together i n p a r a l l e l , and then i n s e r i e s w i t h non-Newtonian s l i p creep. Close reproduction of an ex-perimental loga - l o g ! curve, obtained by using new and p r e v i o u s l y p l o t t e d p o i n t s , could be made using semi-empirical procedures based on the model. Grai n boundary s l i d i n g was proposed to e x p l a i n the s u p e r p l a s t i c be-haviour of the Mg-Al e u t e c t i c by Lee*. GBS i s accompanied by g r a i n deformation and recovery. These cooperative processes are necessary, e s p e c i a l l y i n regions near the boundary, to permit extensive p l a s t i c deformation. No d i s l o c a t i o n s -6-were seen a f t e r s u p e r p l a s t i c d e f o r m a t i o n . T h i s i s p o s s i b l e b e c a u s e the f i n e g r a i n s i z e p e r m i t s a l l d i s l o c a t i o n s , even i n the b u l k o f a g r a i n , t o be 20 21 a t t r a c t e d t o ' and r e a c h a boundary w h i c h i s s l i d i n g by d i s l o c a t i o n 22 13 2 A movement ' o r d i f f u s i o n a l p r o c e s s e s , and be a n n i h i l a t e d . I t i s s u g g e s t e d t h a t t h i s model might e x p l a i n t h e low amounts o f GBS d u r i n g b i c r y s t a l s l i d i n g 25 26 e x p e r i m e n t s ' where t h e e f f e c t i v e g r a i n s i z e i s more o r l e s s i n f i n i t e , thus l i m i t i n g r e c o v e r y t o r e g i o n s n e a r t h e g r a i n boundary. 3 A n o t h e r model , based on e x p e r i m e n t a l work o n Pb-Sn, i n v o l v e s two competing p r o c e s s e s . These a r e N a b a r r o - H e r r i n g (N-H) c r e e p , and d i s l o c a t i o n m o t i o n . I n t h e h i g h m r e g i o n , the N-H model i s dominant and f l o w i s s t r o n g l y 27 v i s c o u s . A m o d i f i c a t i o n o f t h i s model, i n v o l v i n g the C o b l e v a r i a n t o f t h e N-H a n a l y s i s w h i c h i s based on g r a i n boundary d i f f u s i o n r a t h e r t h a n volume d i f f u s i o n , was s u g g e s t e d t o b e t t e r a c c o u n t f o r the o b s e r v e d s t r a i n r a t e s . 1.2. Stage I I I T h i s r e g i o n a t t h e h i g h s t r a i n r a t e end o f the S-curve i s n o t s u p e r -p l a s t i c . Low m v a l u e s a r e t y p i c a l . There seems to be l i t t l e controversy as t o the mode o f d e f o r m a t i o n p r e s e n t . S l i p i s t h e d e f o r m a t i o n p r o c e s s i n d i c a t e d from p h o t o m i c r o g r a p h s and e l e c t r o n m i c r o g r a p h s w h i c h show s l i p l i n e s and d i s -l o c a t i o n s t r u c t u r e s r e m a i n i n g a f t e r d e f o r m a t i o n . The c o n t r o l l i n g p r o c e s s i s p r o b a b l y r e c o v e r y by d i s l o c a t i o n c l i m b . 1.3. Stage I • ,. . _ , . . 7,28 . . 10 . . j - 29 Stage I shows g r a m e l o n g a t i o n , s t r i a t i o n s o r denuded zones a t t r a n s v e r s e b o u n d a r i e s , and r e d u c e d g r a i n b o u n d a r t y s l i d i n g ^ ' 1 1 . There has been some q u e s t i o n as t o whether t h i s s t a g e i s r e p r e s e n t a t i v e o f a s e p a r a t e -7-mechanism or i s at l e a s t p a r t i a l l y a c o n t i n u a t i o n of stage I I . Further study over a wider range of v a r i a b l e s may e x p l a i n the true r e l a t i o n s h i p . Study of deformation i n t h i s r e g i o n o f t e n r e q u i r e s long term creep t e s t s because of the low s t r a i n r a t e s i n v o l v e d . 30 Chaudhari proposes that i n stage I the d i s l o c a t i o n d e n s i t y i s small and the d i s l o c a t i o n s reaching the g r a i n boundary can be absorbed at the boundary by e i t h e r s l i d i n g or m i g r a t i o n and l o c a l d i s l o c a t i o n climb. As £he s t r e s s i s increa s e d , the f l u x of d i s l o c a t i o n s approaching a g r a i n boundary increases more r a p i d l y than does the a b i l i t y of the g r a i n boundary to absorb them. This r e s u l t s i n a d i s l o c a t i o n b u i l d u p , an i n t e r n a l s t r e s s which increases w i t h s t r a i n r a t e and f i n a l l y stage I I where s t r e s s increases r a p i d l y w i t h s t r a i n r a t e . Deformation of the Mg-Al e u t e c t i c at a low s t r a i n r a t e was st u d i e d by Lee*. He found deformation to be a combined e f f e c t of g r a i n deformation and deformation across transverse g r a i n boundaries. The l a t t e r made up 1/3 of the deformation and occurred by GBS and p o s s i b l y some d i f f u s i o n a l creep. 31 Alden has r e c e n t l y proposed that s l i p at t r i p l e l i n e s i n response to s l i d i n g i s r a t e c o n t r o l l i n g . The model i n v o l v e s the viscous g l i d e of d i s -l o c a t i o n s between a source (the t r i p l e l i n e ) and a p e r f e c t s i n k (the opposite g r a i n boundary) and p r e d i c t s an m value between .33 and .5 and an a c t i v a t i o n energy of bulk d i f f u s i o n . He suggests that s u p e r p l a s t i c creep of Fe-Ni-Cr may be of t h i s type. Fe-Ni-Cr and Zn-Al (40.6 a t . % A l eutectoid) show only 2-stage loga curves. Alden suggests that stage I I i s not e n e r g e t i c a l l y f a v o r -able and only stages I and I I I are seen i n these systems. -8-Constant load creep tests on the 2-stage Zn-Al e u t e c t i c lead 32 Chaudhari to the conclusion that a d i s l o c a t i o n model was involved at temperatures above 200°C. The model involved the motion of d i s l o c a t i o n s i n an i n t e r n a l stress f i e l d generated by neighbouring d i s l o c a t i o n s . Ex-periment showed that above 200°C the s t r a i n rate i s c o n t r o l l e d by a thermally activated process with an a c t i v a t i o n energy of 35.3 k cal./g-atom: below 175°C, by a thermally activated process with an a c t i v a t i o n energy of 21k-calv/ g-atom. These values are close to those associated with bulk d i f f u s i o n i n A l and Zn r e s p e c t i v e l y . Zehr and Backofen^ represent stage I by a non-Newtonian dashpot model. They assign an m value .33 and p l o t a l i n e on the logo - logE curve but state that i t s r a t i o n a l e i s no more than speculative. 1.4 Previous Creep Studies on Lead and T i n Systems The present work was c a r r i e d out on the eutectic Pb-Sn a l l o y of f i n e grain s i z e . Constant stress creep testing was chosen to determine i f this method would show any difference between the creep of superplastic and non-superplastic materials. A l l reported creep work on lead, t i n and 33 l e a d - t i n has involved large grain s i z e materials. In lead s l i p occurred at the i n i t i a t i o n of creep and a time 1/3 law was obeyed u n t i l r e c r y s t a l l i z -34 35 ation occurred. A three stage creep curve was found. Work on the Pb-Sn eute c t i c , and on pure t i n and lead, always showed a 3-stage creep curve. I t was suggested that primary creep must always occur. 36 Creep studies on large grained t i n at stresses between 629 and 1394 p s i and 22 and 224.5°C always resulted i n 3-stage creep curves. - 9 -I f a m a t e r i a l i s t o t a l l y unloaded a f t e r deformation, i t s shape changes w i t h time beyond an i n i t i a l predominantly e l a s t i c recovery and tends to approach i t s i n i t i a l shape. This i s known as the " a n e l a s t i c a f t e r e f f e c t " . This e f f e c t i s shown by Garofalo f o r l e a d ^ 7 at 25°C where 25 % of the de-formation i s recovered. Most reported s t r e s s versus s t r a i n r a t e data has been obtained using a t e n s i l e t e s t i n g machine. This r e s u l t s i n s t r a i n r a t e decreasing and s t r e s s changing w i t h time. The r e s u l t s of these t e s t s are not u s u a l l y reported w i t h a statement i n d i c a t i n g whether c o r r e c t i o n s were made f o r these e f f e c t s inherent i n the t e n s i l e t e s t i n g machine. I t i s a l s o d i f f i c u l t to determine a "steady sate" w i t h the t e n s i l e machine. A creep t e s t should avoid these d i f f i c u l t i e s . 2. EXPERIMENTAL . i 2.1. M a t e r i a l and Specimen P r e p a r a t i o n Ingots of the Pb-Sn e u t e c t i c (61.9 % Sn) were cast i n graphite molds under argon. M a t e r i a l s of 99.99 % p u r i t y or b e t t e r were used. Ingot dimensions were 5/8 i n c h diameter and 5 inches long. The surface of the cast b i l l e t s was machined and the m a t e r i a l was • extruded at room temperature i n t o rods of.099 and .083 i n c h diameter. The f i r s t and l a s t 18 inches of the e x t r u s i o n s were discarded. The rod was then cut i n t o 3% i n c h lengths. Samples of .099 i n c h diameter rods were roughened at each end w i t h emery paper and epoxied i n t o brass g r i p s (Fig.2) w i t h Epon 828 epoxy r e s i n . The g r i p s were d r i l l e d one i n c h deep w i t h a .113 i n c h d r i l l and the hole was tapped to improve the epoxy bond. These specimens were aged -10-7 days at room temperature and then stored i n l i q u i d n i t r o g e n . The length of time of t e s t i n g was g e n e r a l l y small compared to the t o t a l time at room temperature so that g r a i n growth during t e s t i n g a t room temperature was minimized. Rods of .083 in c h diameter were sealed i n evacuated glass tubes and were annealed at 165 ± 2°C f o r 30 days'in an o i l bath to produce g r a i n growth. These rods were al s o epoxied i n t o brass g r i p s w i t h a .1015 inch tapped hole. These specimens were stored at room temperature. Comparison of l o g s t r e s s versus log s t r a i n r a t e curves p r e v i o u s l y presented f o r t h i s m a t e r i a l w i t h those r e s u l t i n g from t h i s study shows the small g r a i n s i z e m a t e r i a l (SGS) to have a g r a i n s i z e of 2 microns w h i l e the l a r g e g r a i n s i z e m a t e r i a l (LGS) has a g r a i n s i z e of 8 microns. Figure 2. Specimen i n gips 2.2. Creep Tests Creep t e s t s were performed on a constant s t r e s s machine (Figure 3). The cam design has a mechanical advantage such that the load on the specimen i s twice that i n t h e w e i g h t bucket. The constant s t r e s s cam i s designed f o r a specimen guage length of 25 mm. The a p p l i e d load was transmitted to the -11-specimen by a chromel tape which followed the contour of the cam e x a c t l y r a t h e r than tending to bow as round wires were found to do. The maximum elo n g a t i o n p o s s i b l e w i t h t h i s creep machine corresponded to 50 percent true s t r a i n . Loading and unloading operations were c a r r i e d out by lowering and r a i s i n g a s c i s s o r j a ck under the weight bucket. A l l t e s t s were done at room temperature (22±1°C). Elongation measurements were made w i t h both a t r a v e l l i n g o p t i c a l microscope and an extensometer which was connected to a modified Heathkit recorder and attached to the sample w i t h k n i f e edges. HiElongations could be measured to ± .001 cm with the microscope and to w i t h i n .0002 inches w i t h the extensometer and recorder. The microscope was used to f o l l o w the elo n g a t i o n of specimens deforming under low s t r e s s e s . Measurement was made of the displacement of a s i n g l e mark i n s c r i b e d on the specimen g r i p w i t h a razor blade. Experiment showed that there was no sli p p a g e i n the g r i p s and that i t was not necessary to observe the t r a v e l of two marks on the specimen i t s e l f . .' The extensometer was used at higher s t r e s s e s and s t r a i n r a t e s where automatic continuous recording was e s s e n t i a l . No s i g n i f i c a n t extensometer k n i f e edge i n d e n t a t i o n occurred at theses t r a i n r a t e s . The load was increased or decreased during some t e s t s to determine the e f f e c t of s t r e s s changes on the r e s u l t i n g creep curves. At high s t r e s s e s , i n v o l v i n g the l a r g e g r a i n s i z e m a t e r i a l , changes i n s t r e s s were made during the i n i t i a l t r a n s i e n t of th© creep curves. Specimens were a l s o suddenly un-loaded and the guage length was recorded w i t h the extensometer to determine i f unloading t r a n s i e n t s e x i s t e d over any s t r e s s range. -12-Figure 3. Constant s t r e s s creep machine. 3. RESULTS 3 .1. C a l c u l a t i o n s A l l c a l c u l a t i o n s were done on an IBM 360 computer. The programme i s shown i n Appendix I . Keypunched data i n c l u d e d i n i t i a l guage l e n g t h , and elongations at a s e r i e s of times (hours). A s c a l e f a c t o r was included to convert d e f l e c t i o n s on the Heathkit recorder, used whith the extensometer, to inches. The computer output included a scaled creep curve p l o t of true s t r a i n versus time and s t r a i n rateversus time. A l s o i n c l u d e d was a p r i n t e d t a b l e l i s t i n g each t r u e s t r a i n , s t r a i n r a t e and time p o i n t p l o t t e d . I n i t i a l l y , high s t r a i n r a t e values which f a l l o f f very q u i c k l y may be due to s t r a i g h t e n i n g of the specimen. This e f f e c t should be maximum when a l a r g e quage length i s used as was done w i t h t e s t s using the t r a v e l l i n g microscope. E l a s t i c s t r a i n would a l s o c o n t r i b u t e to t h i s r e s u l t . These e f f e c t s could not be i s o l a t e d and remain i n the computer output. P l o t s of s t r a i n r a t e are determined from the s t r a i n r a t e between two successive p o i n t s . This r e s u l t s i n a jagged s t r a i n r a t e curve. The s t r a i n r a t e s c a l e i s o f t e n expanded and t h i s a l s o tends to make the r e s u l t i n g curve jagged. 3.2. Log Stress versus Log S t r a i n Rate Curves S t r a i n r a t e data obtained from creep t e s t s were used to o b t a i n l o g s t r e s s versus l o g s t r a i n r a t e p l o t s f o r each g r a i n s i z e . These p l o t s are shown i n Figures 4 and 5. S t r a i n r a t e values used were the i n i t i a l s t r a i n r a t e i n stages I and I I and the steady s t a t e creep r a t e i n stage I I I . 4Y Log s t r e s s versus log s t r a i n curve (SGS, 2 microns). Figure 5. Log stress versus l o g s t r a i n rate curve (LGS, 8 microns). - 16 -The p l o t f o r the 2 micron g r a i n s i z e i s s i m i l a r to that determined by Zehr and Backofen*^. The s l o p e , or m v a l u e , i n Stage I i s .33. Stage I I shows a slope of ;5 g e n e r a l l y w i t h a peak value of about .6. The slope decreases a f t e r peak m as s t r a i n r a t e increases and approaches .10 i n stage I I I . The curve determined by Zehr and Backofen, and a l s o using C l i n e and Alden's data, i s shown i n Figure 6. Other p o i n t s are those determined i n the present work. Even i f the two s e t s of data were brought more c l o s e l y i n t o coincidence by moving one set of data to a s l i g h t l y d i f f e r e n t s t r a i n r a t e , the present work shows higher s t r e s s values i n stage I I and lower values i n stage I I I . to 10 1 ' : I I • ; 1 • 1 1 10" 7 10" 6 10"5 1 0 " 4 10~3 10" 2 10"' STRAIN RATE (SEC . - 1 ) Figure 6: Comparison of the S-curve data of the present and previous work. The p l o t f o r the 8 micron g r a i n s i z e shows only stages I I and I I I . Stage I I shows an m value of .5 which f a l l s o f f to a stage I I I value of .10. A stage I I - s t a g e t r a n s i t i o n might be present but lower s t r a i n r a t e data would be, r e q u i r e d before i t could be st a t e d that stage I has been reached. 3.3. Creep Curves 3.3.1. Stage I I Creep Curves For specimens of both g r a i n s i z e s the true s t r a i n versus time curves were s t r a i g h t l i n e s . No primary creep was observed. Steady s t a t e creep was -17-observed from the s t a r t of each t e s t . No necking was present during deforma-tion. For SGS specimens, l i n e a r creep curves were observed from near 300 p s i -2 -1 with a s t r a i n rate of 5 x 10 hr , to 2400 p s i which i s approximately the stress at maximum m. Figures 7 and 8 show curves near the extremes of this stress range. Another curve i s shown i n Figure 9. Above t h i s stress range m decreased as stage III was approached. Below 300 p s i , stage I behaviour was found. More curves t y p i c a l of stage II are shown i n Appendix I I , Figures a to c. Creep tests on LGS specimens did not extend to low enough s t r a i n rates to determine a t r a n s i t i o n stress and s t r a i n rate between stages I and I I . Straight l i n e creep curves were evident up to a stress of 1500 p s i where m began to decrease. Examples are shown i n Figure 10 and Appendix I I , Figures d and e. 3.3.2. Stage I Creep Curves Creep tests c a r r i e d out i n the stress range associated with stage I showed a creep curve with ever decreasing s t r a i n rate. At a stress of 97 p s i , a SGS specimen was extended to a true s t r a i n of .446 a f t e r 696.5 hours. Figure 11 shows this creep curve. LGS creep tests at low stresses were only taken to a few percent elongation and only the creep rate rather than the creep curve shape was determined. Figures f,g, and h i n Appendix II show more SGS creep curves for stage I. 3.3.3. Stage II -Stage III T r a n s i t i o n Creep Curves As s t r a i n rate i s increased above that associated with peak m, the creep curves change from the s t r a i g h t l i n e s of stage I I . Creep curves i n t h i s t r a n s i t i o n range a l l had an i n i t i a l l i n e a r region but the curve increased i n slope, as i n t e r t i a r y creep, a f t e r a s t r a i n of a few percent. A l l specimens, -19-o Figure 11. Stage I creep curve (SGS), 97 p s i . -23-i n t h i s region of decreasing m, exhibited neck formation and f a i l u r e occurred at l e s s than .50 true s t r a i n . The t r a n s i t i o n region continued to 5000 p s i , i n the LGS, where m approached a constant value of .10. High s t r a i n rates pre-vented the determination of the end of the stage II - stage I II t r a n s i t i o n . Figures 12 and 13 show creep curves f o r the stage II - stage II t r a n s i t i o n for both grain s i z e s . More plots are shown i n Figures i to s i n Appendix I I . 3.3.4. Stage I II Creep Curves Creep pl o t s i n this stress range showed 3-stage creep curves. There was an i n i t i a l transient. Primary creep was followed by steady state or second-ary creep as s t r a i n increased. Fracture generally occurred at lower and lower s t r a i n s as the creep stress was increased. Creep curves were obtained only for LGS specimens because the very high s t r a i n rates involved to a t t a i n stage III behaviour i n the SGS material made recording of the deformation impossible with the experimental equipment used. Figures 14,15 and 16 demonstrate creep curves, obtained from LGS specimens, over a range of s t r e s s . Two more curves are shown i n Figures t and u i n Appendix I I . Figure 17 i s a p l o t of the s t r a i n at which steady state creep appears versus the testing stress for. the LGS material. The beginning of stage I I I i s thus near 2500 p s i . ii 3.4. Incremental Loading and Unloading Incremental loading and unloading tests were made during creep tests on specimens of both grain sizes i n stage II and I I I . In stage I I , where creep curves are l i n e a r , increasing and decreasing the load r e s u l t s i n an abrupt t r a n s i t i o n to l i n e a r curves of increasing or decreasing slope. Figure 18 i s an example of incremental loading of a SGS specimen. The short s t r a i n rate peak, Figure 12. Creep curve i n stage IT - stage I I t r a n s i t i o n (SGS), 3432 p s i . Figure 13. Creep curve i n stage I I - stage I I I t r a n s i t i o n (LGS), 3582 p s i . -26-T I : T "I 1 — 1 1 1 1 1 Tc 0.0 0.0B 0.16 0.24 0.32 . 0.4 0.48 0.56 0.64 0.72 D.8 TIME HOUR mo - 1 ) Figure 1A. Stage I I I creep curve (5212 p s i ) . - C S ' _ = 6 8 0 3 GM S - 5 5 3 5 P S I O = . 0 B 3 I N . E X T . . i 1 1 1 1 1 1 1 1 1 -|"--; 3.0 O.OB • 0.16 0.24 • 0.32 0.4 0.4B 0.56 0.64 0 72 OS TIME HOUR 1 X 1 0 " 1 ) Figure 15. Stage I I I creep curve (5535 p s i ) S T R A I N T O S T E A D Y S T A T E -28-Figure 18. Incremental loading during superplastic creep. -29-when the s t r e s s reaches 832 p s i , i s due to an e r r o r i n one reading e x a c t l y at t r a n s i t i o n . Another incremental l o a d i n g example i s shown i n F i g u r e v Appendix I I . Curves have the same slope at a l l s t r e s s e s i n stage I I , during incremental l o a d i n g , as when they are i n i t i a l l y loaded to that s t r e s s . S t r a i n r a t e i s independent of p r i o r h i s t o r y i n stage I I . Incremental l o a d i n g during primary creep i n stage I I I r e s u l t s i n a change i n the i n i t i a l t r a n s i e n t . The slope of the curve i s i n s t a n t l y i n -creased and a t r a n s i e n t remains to a higher s t r a i n than would have occurred at the lower s t r e s s . Figure 19 shows the r e s u l t of i n c r e a s i n g the load during steady s t a t e creep i n stage I I I . Here, a t r a n s i e n t reappears. Figure 20 shows the immediate t r a n s i t i o n to steady s t a t e creep when the load i s decreased during primary creep. This r e s u l t i s a l s o shown i n Figure w, Appendix I I . The purpose of these t e s t s was to compare the behaviour of a s u p e r p l a s t i c m a t e r i a l , during l o a d i n g and unloading, to the behaviour of i c n - s u p e r p l a s t i c m a t e r i a l s under s i m i l a r t e s t i n g c o n d i t i o n s . 3.5. S t r a i n A f t e r - E f f e c t s Specimens deforming i n stage I I and stage I I I were unloaded. These specimens were then continuously measured w i t h an extensometer to look f o r s t r a i n r e l a x a t i o n . At no time was there any s i g n of c o n t r a c t i o n during stage I I . An example of t h i s r e s u l t i s shown i n Figure 21. ReusIts of t e s t s on LGS samples i n stage I I I were not con c l u s i v e but always suggested some s t r a i n r e l a x a t i o n . Relaxations of .003 to .004 inches were found on a guage length of .500 inches a f t e r times of 45 minutes to 60 minutes a f t e r s t r a i n s ^ o f .10 ^30-to .30. The return of primary creep a f t e r recovery was looked f o r . Once again the e f f e c t i s small as shown by Figure 22 a f t e r 5 minutes recovery time. I t may be that recovery rates are too slow at room temperature. Recovery at higher temperatures forced the removal of the extensometer and specimen d i s t o r t i o n occurred. These problems once again lead to inconclusive r e s u l t s . . 4. DISCUSSION Creep studies of superplastic materials have seldom been made. Pre-33-35 vious work on lead, t i n and the l e a d - t i n eutectic has not involved stresses, s t r a i n rates and grain sizes necessary to obtain superplastic be-38 haviour. Packer, Johnson and Sherby used constant stress creep tests i n the i r study of eutectic Zn-Al and state that they found n e g l i g i b l e s t r a i n hardening during superplastic creep and that the creep rate remained constant under constant stress and temperature. Zehr and Backofen*^ used a creep test 32 to obtain a low s t r a i n rate value on an S-curve for Pb-Sn. Chaudhari has also done some creep work. The present study has been the only one s p e c i f i c a l l y designed to gain more i n s i g h t i n t o the superplastic phenomena through the study of creep curve d e t a i l s over a wide stress range. Discussion w i l l cover mainly the merits of recently suggested mechanisms of s u p e r p l a s t i c i t y , considered i n view of new creep r e s u l t s and other consistently reported observations. 4.1. Stage III Deformation i n stage III i s not superplastic. I t i s s i m i l a r to the creep of coarse grained materials where deformation i s by recovery creep. This conclusion follows from the current observations of primary creep, a -31-Figure 20. T r a n s i t i o n to steady s t a t e w i t h unloading i n stage I I I . -32-Figure 22. Reloading a f t e r recovery i n stage I I I . -33-3-stage c r e e p c u r v e and a low m v a l u e n e a r .10. A d i s l o c a t i o n s u b s t r u c t u r e i s a l s o p r e s e n t a f t e r s t a g e I I I c r e e p 1>6,7,39^ a g ^ n t ^ e c r e e p Qf c o a r s e 37 g r a i n e d m a t e r i a l s . R e s u l t s ( F i g u r e 21) a l s o seem to show t h e r e t u r n o f an i n i t i a l t r a n s i e n t a f t e r r e c o v e r y . However, s i g n i f i c a n t l y l a r g e r t r a n s i e n t s were n o t seen a t l a r g e r r e c o v e r y t i m e s . A s t r a i n a f t e r - e f f e c t , whereby the specimens s h o r t e n e d s l i g h t l y when the l o a d was removed a f t e r e l o n g a t i o n , has 37 a l s o been o b s e r v e d i n c o a r s e g r a i n e d m a t e r i a l s a f t e r r e c o v e r y c r e e p . P r e s e n t r e s u l t s seem t o show t h a t t h i s d i d o c c u r a f t e r s t a g e I I I d e f o r m a t i o n . N e c k i n g was a s s o c i a t e d w i t h t e r t i a r y c r e e p . R e c o v e r y c r e e p t h e o r i e s i n v o l v e a g e n e r a l e q u a t i o n f o r s t e a d y s t a t e s t r a i n r a t e , E = r / h (3) where r i s t h e r e c o v e r y r a t e , d a / d t , and h i s s t r a i n h a r d e n i n g dh/de. S t r a i n h a r d e n i n g i s a s s o c i a t e d w i t h an i n c r e a s e i n d i s l o c a t i o n d e n s i t y (p) w h i l e r e c o v e r y r e p r e s e n t s a decrease,. P r i m a r y c r e e p r e s u l t s i f the i n i t i a l r e c o v e r y r a t e i s s m a l l e r t h a n t h e h a r d e n i n g r a t e . The v a l u e of r 1 i n an a n n e a l e d specimen i s c l o s e t o z e r o . Steady s t a t e c r e e p r e p r e s e n t s a b a l a n c e between h a r d e n i n g and r e c o v e r y . Such a b a l a n c e i s e x p e c t e d o n l y a f t e r some i n c r e a s e i n d i s l o c a t i o n d e n s i t y , r e l a t i v e t o t h e a n n e a l e d c o n d i t i o n , w h i c h o c c u r s d u r i n g p r i m a r y c r e e p . 40 A modern t h e o r y f o r s t a g e I I I i s t h a t o f McLean . D e f o r m a t i o n a t h i g h t e m p e r a t u r e s i n v o l v e s the b e h a v i o u r o f a t h r e e - d i m e n s i o n a l d i s l o c a t i o n n e t -work. Three i m p o r t a n t a s p e c t s of t h i s b e h a v i o u r a r e t h e t e m p e r a t u r e i n s e n s -i t i v i t y o f the f l o w s t r e s s of a g i v e n n e t w o r k , the n etwork's tendency t o c o a r s e n on h e a t i n g w h i c h c o r r e s p o n d s t o a d e c r e a s e i n p and i t s r e f i n e m e n t -34-on s t r a i n i n g w h i c h corresponds to an i c r e a s e . McLean shows that the recovery r a t e , r , v a r i e s w i t h s t r e s s by 3 r a a (4) The recovery process i n v o l v e s d i f f u s i o n c o n t r o l l e d d i s l o c a t i o n climb and m i g r a t i o n of jogs i n screws. Therefore, i f D i s the d i f f u s i o n c o e f f i c i e n t 3 r a a D (5) P l a s t i c deformation r e f i n e s the d i s l o c a t i o n network. During deformation, moving d i s l o c a t i o n s are h e l d up at p o i n t s where the network i s f i n e and bow out to increase the average d i s l o c a t i o n d e n s i t y . M u l t i p l e s l i p permits network geometry to remain constant as the meshes become smaller . The s t r a i n hardening c o e f f i c i e n t h = 8c/3e i s a measure of the r e f i n i n g a c t i o n . I f -3/2 expression f o r r from eqn.(5) and h from the e m p i r i c a l r e l a t i o n s h i p haa are used i n eqn(3) the r e s u l t i s 4-5 (6) or i f K = ( V B D ) 1 7 4 ' 5 e = BDa .22 , a = Ke (7) which i s the same as equation (1) w i t h m = .22. This value of m i s not q u i t e as low as that found experimentally. Theories of t h i s type are, of c o u r s e / r a t h e r s p e c u l a t i v e and e v i d e n t l y approximate i n nature. I t i s not known to what extent the p h y s i c a l model of McLean corresponds to the de-t a i l e d deformation and recovery processes i n stage I I I . - 35 -4.2. Stage II-Stage I I I T r a n s i t i o n In the stage I I - s t a g e I I I t r a n s i t i o n , on i n c r e a s i n g s t r a i n r a t e , m f a l l s from a maximum to a sm a l l e s s e n t i a l l y constant value a s s o c i a t e d w i t h stage I I I . The t r a n s i t i o n s t r a i n r a t e 9 -1 -1 range was from 2 x 1 0 - z hr. to 2 hr. f o r the 8 micron g r a i n s i z e and above 2.5 hr 1 f o r the 2 micron g r a i n s i z e . The high s t r a i n r a t e s i n v o l v e d prevented d i f f e r e n t i a t i o n between the end of the t r a n s i t i o n range and the s t a r t of stage I I I i n the sm a l l g r a i n s i z e specimens. Creep curves i n the t r a n s i t i o n r e g ion are convex upward ( t e r t i a r y creep) (Figures 12 and 13). There i s no primary creep. F a i l u r e occurs at s t r a i n s of l e s s than 41 .50 and f a i l u r e occurs at a neck. Chaudhari may have explained t h i s r e s u l t mathematically. D i f f e r e n t i a t i o n of equation (1) leads to dt 1 j-da - l o g ^ dm j ^ ) ^ m a K m The e f f e c t of a la r g e m value i s to decrease the tendency f o r a neck to grow. Conversely, an increase i n the bracketed term increases the s e n s i t i v i t y to necking. I f m i s decreasing w i t h i n c r e a s i n g s t r a i n r a t e , as i t i s i n the t r a n s i t i o n range, dm i s negative and the second term w i l l be added to the f i r s t . The bracketed term becomes l a r g e r and s e n s i t i v i t y to necking i s more pronounced. Observations of neck formation and growth a f t e r only a few percent s t r a i n were made during the present 16 work (Figures 12 and 13). Previous observations of l a r g e s u p e r p l a s t i c elongations using i n i t i a l s t r a i n r a t e s i n t h i s r e g i o n were made w i t h I n s t r o n t e n s i l e machines. These machines have a constant cross head" speed and the s t r a i n r a t e imposed on a specimen i s continuously decreasing. The e f f e c t of a -36-decreasing s t r a i n r a t e i n the t r a n s i t i o n range i s to give a r i s i n g m value as t e s t i n g proceeds. A convex upward creep curve has p r e v i o u s l y been r e p o r t e d , i n the l i t e r -36 ature but no comment or e x p l a n a t i o n was attempted. Constant s t r e s s creep t e s t s were performed on a number of pure t i n specimens. Four d i f f e r e n t g r a i n s i z e s were used but only the f i n e s t , 37 microns, e x h i b i t e d t h i s behaviour. Others showed primary creep. Figure 23 shows a l o g a r i t h m i c p l o t of s t r e s s versus s t r a i n r a t e using Breen and Weertman's creep r a t e data f o r the s m a l l e s t g r a i n s i z e . This curve i s somewhat comparable to th a t of Alden and d i f f e r e n c e s may be due to g r a i n growth and the f a c t that Alden's curve was obtained using an I n s t r o n t e n s i l e machine, m v a r i e s from .3 to .12. The increase i n ui w i t h decreasing strain.r=te i s due to a slow increase i n sub-grain s i z e v i t h decreasing s t r a i n r a t e ' - . Stage II i s reached when sub-grain 1 13 • s i z e has reached the g r a i n s i z e ' . As t h i s c o n d i t i o n i s approached, accommodation fo r GBS by s l i p becomes e a s i e r and m increases to i t s maximum value. The stage I I I -stage I I t r a n s i t i o n i s a s s o c i a t e d w i t h increased amounts of GBS"\ S u p e r p l a s t i c i t y may a c t u a l l y be p o s s i b l e i n l a r g e r grained m a t e r i a l s i f , at low enough s t r a i n r a t e s , the sub g r a i n s i z e approaches the g r a i n s i z e . Creep 43 t e s t s by G i f k i n s may be i n d i c a t i v e of t h i s . The g r a i n s i z e of extruded Pb-2.45 wt% t h a l l i u m was 100 microns. Stresses of 300,500,1000,1500, and 2000 p s i were used on creep specimens. These creep r e s u l t s are p l o t t e d i n Figure 24. The m value increases from .15 to .45. Micros t r u e t u r a l observations showed that s l i p l i n e s became l e s s r e g u l a r and" more widely spread as the s t r e s s was decreased and at the lowest s t r a i n r a t e s deformation proceeded by "boundary micro-flow" and grains were unde-formed. At 500 p s i e l o n g a t i o n to f a i l u r e was 208 %. S i m i l a r observations weremade -37-by Wood et a l on aluminum w i t h a g r a i n s i z e of 100 to 200 microns. S l i p was prominent during deformation at lower temperatures and higher r a t e s of s t r a i n . As the temperature was increased or the s t r a i n r a t e decreased, s l i p l i n e s g r a d u a l l y vanished and the elements of the ass o c i a t e d substructure showed an increase i n s i z e . This l e f t a coarse s u b s t r u c t u r e which increased i n s i z e w i t h a f u r t h e r i n -crease i n temperature or decrease i n s t r a i n r a t e u n t i l i t was the s i z e of the g r a i n i t s e l f . This c o i n c i d e d w i t h the onset of prominent "boundary micro flow" or GBS. They found that l a t t i c e s t r u c t u r e was unchanged and no s t r a i n hardening occurred during deformation. The observations of Wood et a l and G i f k i n s seem to be very s i m i l a r to the observations made on f i n e grained s u p e r p l a s t i c m a t e r i a l s where GBS becomes 13 more dominant as the s t r a i n r a t e i s decreased from that of stage I I I 4.3. Stage I I The new observations a s s o c i a t e d w i t h stage I I are that the creep curves are l i n e a r and that there are no unloading t r a n s i e n t s . The l i n e a r creep curves are shown i n Figures 8,11. There are no signs of primary or t e r t i a r y creep, only steady s t a t e . The r e s u l t s of unloading t e s t s are shown i n Figure 21. No c o n t r a c t -ions due to recovery has occurred. These r e s u l t s show that recovery t h e o r i e s , which may be v a l i d f o r normal creep, and stage I I I . a r e u n s a t i s f a c t o r y f o r super-45 p l a s t i c creep. Alden has demonstrated that many proposed models f o r super-p l a s t i c i t y are r e a l l y models based on recovery creep and are thus unacceptable. An 32 i n t e r n a l s t r e s s model which incorporates an accumulation of d i s l o c a t i o n s near the g r a i n boundary seems to be unacceptable because d i s l o c a t i o n s are not seen d u r i n g s u p e r p l a s t i c deformation even when specimens are quenched under l o a d * . Figure 23. Log s t r e s s versus l o g s t r a i n r a t e r e l a t i o n s h i p f o r pure tin,(37u) _1000 _ 10-2 STRAIN RATE 10" I ( D A Y " 1 ) Figure 24. Log s t r e s s versus l o g s t r a i n - r a t e r e l a t i o n s h i p f o r Pb-2.45 wt.% t h a l l i u m (lOOy). i O J C O I -39-A l s o , the low temperature y i e l d s t r e s s is unchanged a f t e r stage I I deformation I f d i s l o c a t i o n s are accumulated during creep, t h i s y i e l d s t r e s s should i n c r e a s e . Mechanisms and accommodation processes which would show no t r a n s i e n t s are those which do not i n v o l v e a s t r u c t u r a l change Grain boundary s l i d i n g w i t h some s p e c i a l accommodation processes, Nabarro-Herring or Coble creep and g r a i n boundary m i g r a t i o n are examples. Grain boundary m i g r a t i o n , i n a s s o c i a t i o n w i t h GBS, has been observed 46 47 i n s u p e r p l a s t i c m a t e r i a l s ' . Opposition to the idea that m i g r a t i o n i s r a t e -c o n t r o l l i n g i s based on doubt that m i g r a t i o n can be e f f e c t i v e when many phase boundaries are present i n two phase systems. At the same time, i t must be con-si d e r e d that i f there are equal amounts of two phases, each phase might s t i l l be i n contact w i t h 50 % of the l i k e phase. Thus there i s a reasonable chance f o r mi g r a t i o n to be an e f f e c t i v e means of accommodation f o r GBS i n a s u p e r p l a s t i c m a t e r i a l . With only one phase present, d i l u t e a l l o y systems do not, of course, present t h i s problem. Grain e l o n g a t i o n should r e s u l t during d i f f u s i o n a l creep. This i s not observed i n stage 11*^ but Zehr and Bachofen's explanation i s that elongated grains w i l l experience a shape r e l a x a t i o n during s t r a i n i n g through d i r e c t migra-t i o n and r e c r y s t a l l i z a t i o n . They c l a i m that the presence of striated bands on a 2 micron Pb-Sn e u t e c t i c specimen p u l l e d at 3.3 x 10 ^ sec. * supports d i f f u s i o n a l creep. This s t r a i n r a t e i s very near the t r a n s i t i o n to stage I . S i m i l a r s t r i a t i o n s 47 have been oserved i n Cd-5 % Pb over s t r a i n r a t e s a s s o c i a t e d w i t h stages I and I I . 47 Donaldson suggests that s t r i a t i o n s may i n d i c a t e that boundary s l i d i n g occurs on p r e f e r r e d c r y s t a l l o g r a p h i c planes. The t e s t of t h i s i s to c o r r e l a t e g r a i n o r i e n t -a t i o n and the planar o r i e n t a t i o n of the s l i p p e d boundaries w i t h s t r i a t i o n spacing. -40-Narrow s t r i a t i o n spacing would i n d i c a t e l a r g e m i s - o r i e n t a t i o n . The absence of s t r i a t i o n s would i n d i c a t e that the s l i d i n g plane i s a p r e f e r r e d plane. In a Mg-Al a l l o y , the s t r a i n c o n t r i b u t i o n of g r a i n boundary s l i d i n g i n stage I I reached 65 %*. Alden suggests that the g r a i n boundaries may act as " p e r f e c t " s i n k s and that s l i d i n g i s r a t e - c o n t r o l l e d by d i f f u s i o n . This model demands that the absorption r a t e by boundaries of d i s l o c a t i o n s generated at t r i p l e l i n e s be so high that i t i s not r a t e c o n t r o l l i n g . Fast recovery r a t e s i n 13 s u p e r p l a s t i c m a t e r i a l s support t h i s but l i t t l e or no climb can be inv o l v e d or t h i s becomes a recovery creep model. Absorption may be enhanced by GBS by the a t t r a c t i o n f o r c e between s l i d i n g boundaries and d i s l o c a t i o n s * . The r a t e of s l i d i n g i s determined by the e f f e c t i v e v i s c o s i t y of the boundary. The boundaries are rough and the s l i d i n g r a t e i s determined by d i f f u s i o n around these rough areas. The s c a l e of roughness w i l l u s u a l l y i n c r ease w i t h g r a i n s i z e . The model p r e d i c t s the s t r a i n r a t e semi q u a n t i t a t i v e l y i n agreement w i t h observed e f f e c t s of s t r e s s , g r a i n s i z e , and temperature. 4.4. Stage I Stage I , as shown i n Figure 11, i s s i m i l a r to stage I I i n that there are no t r a n s i e n t s s i m i l a r to those found i n stage I I I . The creep curve i s not l i n e a r , however. There i s a d e f i n i t e decrease i n s t r a i n r a t e w i t h time. This decreasing s t r a i n r a t e i s not b e l i e v e d to be i n any way r e p r e s e n t a t i v e of a r e -covery creep model where a decreasing creep r a t e i s found during primary creep. Stage I may represent a change i n the r a t e c o n t r o l l i n g process f o r GBS from d i f f u s i o n around bumps on boundaries to s l i p at t r i p l e l i n e s . I f d i s l o c a -t i o n s are emitted from a source, proceed through a m a t e r i a l without b a r r i e r s and are then absorbed at a " p e r f e c t " s i n k , no s t r a i n hardening w i l l r e s u l t . ' I f the -41-sources of d i s l o c a t i o n s are the edges of s l i d i n g g r a i n s , and the si n k s are g r a i n 30 boundaries, Alden suggests that s t r a i n r a t e w i l l depend on s t r e s s to the power 2 to 3. The slope depends on whether accommodation d i s l o c a t i o n s move on a few s l i p planes or throughout the g r a i n volume. A l i n e a r array i s expected i n a l l o y s w i t h a low s t a c k i n g f a u l t energy such as Fe-Ni-Cr and m should be .5. Motion normal to the s l i p plane i s e a s i e r through cross s l i p i n high s t a c k i n g f a u l t energy ma t e r i a l : and the slope should be .33 as was observed. Lee* found the behaviour i n stage I to be a combined e f f e c t of g r a i n deformation and deformation across transverse g r a i n boundaries, accounting roughly f o r 1/3 and 2/3 of the t o t a l deformation, r e s p e c t i v e l y . The l a t t e r c o n s i s t e d , at l e a s t i n p a r t , of GBS but whether the remaining f r a c t i o n was due to GBS or Coble creep could not be determined. Several experimental observations suggest that d i f f u s i o n a l creep may 7 28 10 co n t r i b u t e to deformation i n stage I . These are g r a i n e l o n g a t i o n ' , s t r i a t i o n s 29 and denunded zones , and creep curves of decreasing slope. Creep curves of decreasing slope could occur by the s e l f - e x t i n g u i s h i n g nature of d i f f u s i o n a l creep. A t h e o r e t i c a l a n a l y s i s of N-H and Coble creep has been made and d e t a i l s are shown i n Appendix I I I . Figure 25 shows the shape of creep curves expected f o r pure N-H and Coble creep. A stage I experimental curve i s a l s o shown and a l l are put on a time s c a l e r e presenting the f r a c t i o n of t o t a l time to leach .40 true s t r a i n . Times in v o l v e d f o r .40 s t r a i n are .87 h r s . f o r Coble creep and 5.39 x 10^ hrs. f o r N-H creep. The experimental curve represents 700 hours and could be p a r t i a l l y r e p r e s e n t a t i v e of some combination of N-H and Coble creep. I t has been reported that there i s no g r a i n e l o n g a t i o n i n the Pb-Sn e u t e c t i c * ^ i n stage I ; There i s g r a i n growth during deformation*^ and a l s o g r a i n 16 s t r a i n . Grain r o t a t i o n has been observed . Lack of g r a i n e l o n g a t i o n might not -42-exclude d i f f u s i o n a l creep as an important p a r t of the deformation mechanism of stage I . A g r a i n may elongate, by d i f f u s i o n , p a r a l l e l to the t e n s i l e a x i s . G r a i n boundary m i g r a t i o n could account f o r the r e t u r n of an equiaxed shape and an increased g r a i n s i z e . Grain r o t a t i o n would change the t e n s i l e a x i s and permit e l o n g a t i o n i n a l l d i r e c t i o n s . I f g r a i n growth does not i n v o l v e d i f f u s i o n a l creep, g r a i n s t r a i n would not be expected.Lee* measured the t r a v e l of two marker wholly w i t h i n a g r a i n . In stage I I I no movement was found. In stage I I t h i s was equal to .21 of the t o t a l s t r a i n and i n stage I t h i s increased to .30. Growth of one g r a i n at the expense of another would not c o n t r i b u t e to an i n t e r n a l t r a v e l of markers but only to the g r a i n volume. Gra i n growth by any means, however, could r e s u l t i n a creep curve of decreasing slope because the s t r a i n r a t e does decrease w i t h i n c r e a s i n g g r a i n s i z e . Stage I deformation i n v o l v e s GBS and probably some combination of N-H and Coble creep. 4> FRACTION OF TIME TO REACH -40 STRAIN Figure 25. Experimental, t h e o r e t i c a l N-H and t h e o r e t i c a l Coble creep curves (97 p s i ) . -4A-5. SUMMARY AND CONCLUSIONS In the s u p e r p l a s t i c range, m a t e r i a l s d i s p l a y unusual creep p r o p e r t i e s . S t r a i n r a t e i s dependent on g r a i n s i z e and i n s e n s i t i v e to s t r e s s . Creep curves are l i n e a r w i t h no i n i t i a l or f i n a l t r a n s i e n t s . A m a t e r i a l i s considered " s u p e r p l a s t i c " i f s u p e r p l a s t i c p r o p e r t i e s are observed at experimentally reason-able s t r a i n r a t e s . There i s evidence that s u p e r p l a s t i c p r o p e r t i e s may occur at low enough s t r a i n r a t e s i n "normal" m a t e r i a l s . Conversely, a " s u p e r p l a s t i c " m a t e r i a l may d i s p l a y normal creep p r o p e r t i e s at s u f f i c i e n t l y high s t r a i n r a t e s . In stage IT, a l i n e a r creep curve suggests that the deformation mechanism accounting f o r s u p e r p l a s t i c behaviour must be one which i n v o l v e s no s i g n i f i c a n t s t r u c t u r a l change. Creep curves i n stage I I show no t r a n s i e n t s , a f t e r l o a d i n g or unloading and the s t r a i n r a t e , at any s t r e s s i n the s u p e r p l a s t i c range, i s independent of p r i o r s u p e r p l a s t i c deformation h i s t o r y . P r i o r evidence i n d i c a t e s that most of the deformation i n stage I I i s accomplished by g r a i n boundary s l i d i n g . There must be accommodation f o r g r a i n boundary s l i d i n g to operate.Accommodation could i n v o l v e one or more of g r a i n boundary m i g r a t i o n , d i f f u s i o n , or s l i p at t r i p l e p o i n t s . The m value f o r stage I I i n the- l e a d - t i n e u t e c t i c i s near .5 and has a maximum value near .6. At low s t r a i n r a t e s creep curves show an ever decreasing sl o p e , t h i s could be due to d i f f u s i o n or g r a i n growth. GBS occurs i n stage I, but i s not as dominating as i n stage I I . Gra i n s t r a i n increases i n stage I and i s notably i n d i c a t i v e of a d i f f u s i o n a l process. Some combination of N-H and Coble d i f f u s i o n a l creep i s l i k e l y s i n c e the experimental creep r a t e s ; a r e much f a s t e r than those expected f o r N-H creep and much slower than could be -45-a t t r i b u t e d to Coble creep. The r a t e c o n t r o l l i n g process i s not known. Grai n growth, g r a i n r o t a t i o n , g r a i n boundary m i g r a t i o n , and s l i p and t r i p l e l i n e s may a l l have some e f f e c t on deformation and the r e s u l t i n g creep curve. The m value f o r stage I i s .33 and i s constant over the s t r e s s range s t u d i e d . In the stage I I - stage I I t r a n s i t i o n r e g i o n , s u p e r p l a s t i c p r o p e r t i e s depend on the t e s t i n g method. In a constant s t r e s s creep t e s t t e r t i a r y creep begins, as necks propagate and grow, a f t e r only a few percent s t r a i n and f a i l u r e occurs at l e s s than.50 true s t r a i n i n the l e a d - t i n e u t e c t i c . Testing under c o n d i t i o n s where the s t r a i n r a t e decreases as the t e s t proceeds reduces the tendency to neck and e l o n g a t i o n to f a i l u r e i s much greater. -46-SUGGESTIONS FOR FUTURE WORK There are s e v e r a l l i n e s of i n v e s t i g a t i o n which could extend from the present work. These i n c l u d e : (1) A determination of creep curves and creep r a t e s over a wide s t r e s s range i n stage I f o r s e v e r a l g r a i n s i z e s . (2) A micrographic study of the v a r i a t i o n of the c o n t r i b u t i o n of g r a i n boundary s l i d i n g and g r a i n s t r a i n w i t h s t r e s s over stage I . A study of g r a i n e l o n g a t i o n , g r a i n boundary m i g r a t i o n and g r a i n r o t a t i o n might prove h e l p f u l at low s t r e s s e s . (3) A study of a c t i v a t i o n energy i n stage I over a wide s t r e s s and temperature range and a l s o i n t o stage I I . (4) An e v a l u a t i o n of f a c t o r s , such as time temperature and e l o n g a t i o n , c o n t r i b u t -i n g to g r a i n growth during stage I I deformation. (5) An i n v e s t i g a t i o n of e l o n g a t i o n at v a r i o u s constant s t r e s s e s over the stage I I -stage I I I t r a n s i t i o n to determine i f there i s a r e l a t i o n s h i p between'elongation to f a i l u r e and m. (6) A more c a r e f u l study of the r e l a t i o n s h i p between s t r a i n r a t e and g r a i n s i z e . 1 2 R e l a t i o n s h i p s between /L and 1/4^ 5 have been reported i n the l i t e r a t u r e 1,3,10,13,20^ j-^ggg r e i a t i o n s i p s to be v a l i d , each value taken must be i n stage I I . Some r e l a t i o n s h i p s i n the l i t e r a t u r e have i n v o l v e d a' s t r e s s which i s i n two stages of the S-curve and are t h e r e f o r e i n v a l i d . The present work 1 3 shows a /L dependence although there i s some doubt i n the g r a i n s i z e s . A 13 1 •• 3 p l o t of v a l i d p o i n t s f o r Pb-5 %Cd and e u t e c t i c Pb-Sn shows a /L r e l a t i o n -1 3 s h i p . The r e l a t i o n s h i p s show s c a t t e r between -2.2 and -3.9 f o r Pb-Sn and -2 and -3 f o r a Mg-Al a l l o y . -47-APPENDIX I Computer Programme •--•i-T!.•/<•; in r, C v n i L ^ r MAIM Q^-17-6<1 1 Ai?lt04 PAGE 0001 r AM/ll.vcjc; n r T H F CRFFP TEST CUPVFS PHB004 P l M c . M s i r i M T [Tl c ( 14) , INOI20.20 ), TIMFf?0,?0),GP.AD(20,?0), PHB005 l S T " A l N(?'.-0), TTMry(200),RATF(200),INI 200),DIAL(20,20),XC(5),YC(5) PH8006 s '»0 0 ? R F A O f , NFxrr.i.! PHB007 /> T i m q FC'P^ATI 15X.I3) PHB008 0 0 4 'IV f-0 1. = 1 , NFXFCU PHB009 ooo^ RFAOfS, DT1TLF PHB010 i FriPM&T('14A4) 000 7 PrAO(5,4) SCAFAC»MSHIFT,ELNOT '.' ? C " 4 riRMATI FI 0.5, 5X.I 3,Fa.3) r.ro / K = 0 PHB014 001 0 f = o .. •'.v..:• •. ' PHB015 001 1 s u « s m = 0 . ., •. -.. PHB016 001 ? OP 10 I = l.NSHIFT PHB017 R F AO(5,5) SHIFT, NMEASW -V'. PHB018 0014 rpRMAT(F10.5,5X, 13) ' " ' PHB0L9 0 015 SUMSHI = SUMSHI • SHIFT • . . . PHB020 001*1 0 T MF NS I ON COMNTSI 9,50,7) , IMPORT (20 ,50 )', T IHEXHt 9i50 )t STRA IH (9 , IMTRANS(O),CRISIS! 20,20,7) 50).PHB021 PHB022 0017 OP 20 J = 1, NMFASU PHB023 coin PFAO (5,ft) TIMF(l , j),GRAD(I , J ) , 0 I A L t t i a i * t l»P0RT ( I , J J , ( C R I S I S < I , J,PHB024 1 I J ) , IJ = 1,7) rt • PHB025 0010 FPRMAT(F9.2,1X,F5.1,3X,F8.5,2X,I2,7A4) , , 00 ?0 K = K + 1 . \ PHB027 00 21 I N ( K ) = K y i^^'i^M^'Vy^:.):['."' :'. PHB028 00?? IF ( OTALtl.J) .GT.O.) GO TO 1? « - PHB029 00? 3 OFITAL = SCAFAC * GRAO(I.J) * SUM'HI r , PHB030 00?4 GH TO 11 PHB031 ^ 0025 12 DFLTAL = 01AL(I,J) " r PHB032 00?f> On TP 11 PHB033 0027 11., TSTRAI = ALOG ( 1. • DEL TAL / ELNOT) * * / , C SFLECTION P.F COMMENTS ' v/Vt'Si'.^'^fey'' ^^i^fi^.^.,;:-^,.' ' PHB035 O0?3 IF ( IMPORT (I , J) . GT.O ) GO TO 13 . ' * _' PHB036 0079 GO TO 14 • " ' - •' " PHB037 00 30 13 M = M + i . / • .. . • . .• .•.'•:v.tv , jv^;^\ :v : i ;K^^ PHB038 , 0031 00 120 IJ = 1,7 PHB039 r 0032 _ 120. . CPMNTS(L,M,I J)..- CRIStS . U , J t I J > PH8040 0033 I»P0RT(L,M) = IMP0RT(1,J) , ' PHB041 0034 TIMEXMIL ,M) : = T I K E ( I . J ) PHB042 0035 STRAIM(L.M) = TSTRAI PHB043 0036 MTRANSID = M , PHB044 0037 14 STRAIN(K) • . = TSTRAI t , PHB045 003S .'. TIMEX 002 o r, -i , N S T A R T = 1 P H H 0 5 9 I-. "j r. 1 " F J M 1 S = 4 0 P H B 0 6 0 0 0 «. 7 • I P A C F = K / 4 0 + 1 PHB061 N P A O T F = K - ( . H ' P A G F - 1 ) * t,n P H B .16.2 IT ( N P A O ! F . l - ' O . O ) O P T O 41 P H 3 0 6 3 ,, ; )r,c on T O 4 ? . , PHB064 0 0 ^ 6 . 41 .NPAGP ;=. NPAOF. - 1 PHn065 " 0 ^ 7 42 • ' " D O 40 J = 1,N»AGF . .PH8066 ons a / N U M F R O = ' N I I M C R 0 + 1 • "'• ' ' ' PHfl067 •; WRITFI6,?) NUMFRO , T I T L E '.V PHB068 "060 ? FORMAT(lOX ,33HANA|.YSI .S PF THE C R F F P T F S T CURVES. 1 2/. 5HPAGE ,!?.4X ,PHR069 1 IHS/66X, 1H*/1X, 1 4 A 4 . 2 ! /66X.1H* 1/7H NI.IMRFR , 10X.4HTIME, 10X.6HSTRAIM ,PHB070 . J' : 2 9 X , UHSTRAIN R A T F ,9X , 1H*/16X , 5HH0URS , 2 3 X , 10H H O U R * * -.... I F ( K > L T > N p I N , S ) R ; 0 T O 43^-.'I?I4J.; . V ' 1 . :~;SV' . i , 12 x, i H * i . ; .PHB071 0 0 6 1 '.'..''. ' • " "PH3072" , 0 0 6 2 G O TO 46 .' -.• , ••'':jr;$:ie~'i,•:'• "• PHB07 3 0 0 6 3 4 3 ' ' N F I N I S = 'K '" ;'V;'v«i^;>' ' •'' .-'Or.' •..''•' PHR074 0 0 6 4 46 . , . 00.130 I = NSTART ,NFINI$/: . ' . ••.•.•',..; ;,r^"; •>',..,;,• PHB075 0 0 6 5 V . N S P A = INf-T) - ( !VU »/10n»:-l6 v '= ' ' : . ; ' :^ . :i,V : \ x : PHB076 • . 0 0 6 6 h. IF ( N S P A.EQ.l) GO T O 44' '-'-'^^^C •?> \v ?.' "• \vA;•/,=''.'; •:•' PHB077 0 0 6 7 ' ...... ..... . . ' G O - T O 45 ;:• •' ; ; ' ' - . ^ ^ ^ ^ ^ ^ f ^ ^ ^ Y ^ PHB078 0 0 6 3 • • 44 .• • WRITER,7).M; . ' . ' , . . ' • , •'• ' , ; ; 5 S S 3 ^ .'' .; ':' . ' PH8079' • 0 0 6 9 7 . ••'•• FORMAT { 66X> l.H*')'•'•'• '•'••' "'''>'• PHP080 0 0 7 0 • 0 0 7 1 .•• 4 5 3 •''" .; WRITE!*,3).-IMUt.. TI,MEX! iI ; , lSST ,«A , lfH M , RATE (11 •'• FORMAT ( 3 X , 1 .3 , 6X , F 9 . 3 ,7xV'EiiQ'^3.'»^ feiE,tO=. 3 ,'.11 X^TH*!-'*!'-if. i 'V PHB081 ' PIIB082 0 0 7 2 . . . . 1 3 0 . .:f ..NSTART .= ",' NSTART ;*-.*Oj; | " ' i . ' , v • ' . . '• "PHB084" 0 0 7 4 /V'NFINIS = j.NF IN! S' +;j ^ O W ^ 0 l ^ m ^ ^ ^ ^ r < P . '••\")>"}i~'' •'• • i ,PHB08 5 0 0 7 5 ••• W R I T E t6,9 ) -r; 'rtji.v-i '•.''' • PHBC86 0 0 7 6 9 . FORMAT! 1H11 . l V ' ; - 1 i " : ^ ^ l ^ | ^ M f e i ; ^ .-, • PHB 0 (3 7 • v- 0077 . •.'• V"r;.40^ '; >:vCONTINUE^,:^f X^^^^S^^^M^^^^^ ; A TR ANSF'E R^/tQi^THEPL'OTT-NGi^'OU^ v^'X": '.C:'.- " -PHB088 :.l.'i.'.C_.'J CH^ ::y"V'.": J"' '' , 1 .PHB0S9 " •' ooV no' 70 i ^ K : ^ : * i i a ^ W w € l ^ PHB091 '• 00«0 TiMtt.n .,= T t M F x m l 1 . / . l?:':^'/'' • •1 '• ''-'^ PHB092 . . 00«l RAT!L,.IK.i»:j..RATEm.;;.'^ )rt,'i/:-;^'v'. . t\- . PHB093 • . 008? • . STOAd.Il = STRAINm?©^«P^ # § S ^ ;.*'~iv^ .'jv/-''' v}-.^;; .PHB094 :'±Y 0O83._ , 0 0 / 8 4 •' ,'...•; I.kti/7P4= ; ! i.C.ONTTNllfi^B -v i^ 0O;;,10d'. Xtfyktr-lt-ff&'Wti^ CS-'«'^'-\ . PHB095. .'PHB096 . '.';V- 0 0 8 5 : . . . V T I T (I..T1U: TITLE (,i:)?'\V'<::-'ti;VffMf»?€?^ J'!,- ••'-;'^ ''7j^ .V^ 'V.'.'t"''%>i1| ;PHB097 • 00P6 ' • 100 ' CONT INUE.VM" • '•'' '•'•kCij/feSHli&Sv »'-^S'^%te:KiJW'*'J.':-'..'r'-vtV'"' ' ' 5i:''-:< '^ 'PHB098 . . . . •; 0 0 8 7 KTRANS(L) = K > > , • ' t , , 1 PHB099 ',. 0 0 8 8 ' •• ' " 6 0 ' / ..V.'CONTINUF ,» ' • , i , , ' " , 1 * P H B 1 0 0 0 0 8 9 . .... ... .- .. NA7'A.i^c.,yc. v;5,;;.i!...5,.i.^ ./,..'-.';::•. 5 P H B 1 0 1 . "V"... "c'".' ''^jCiyV* • P H B 1 0 2 •.'•" -'''fit - PLOTTING ROUTINE ^ * ' » 1 r\r-f^V-'.-j'':- "'•'.•''V' ;,PHB103 ',r'"c ''A ;!.;;:-'i: :C.'.>'"..''' •'' P H B 1 0 4 • ; 0090 C A I L PLOTS , > :;i>on- ao L - i,NEXECir ..i;',.-^..j.,-..... PHB105. : . . 0 0 9 1 :-"i:^A"; •" '" - P H B 1 0 6 • i::'^ 0 0 9 ? ' K. = KTP.ANS!L) . P H B 1 0 7 . . ... 0 0 9 3 " ..on'90 I = i , K . . ' - • P H B 1 0 8 • ' : ' : . V . . O 0 9 4 -'.-"i>-7 TIMEX! I ) : = . : T I M ( L ' , 11 ' - .i..v!''-„..; ;/-'''.;c'.'.;rv-.^'''' P H B 1 0 9 • ' • 0 0 9 5 R A T F ( I 1 = RAT 1 1 . , t ) -•".•:V''-.-"':'' ;' -.V •'• - : ^ " ' V ' •• . ' ;.v'"' ' •.. •• P H B 1 1 0 0 0 9 6 ., ..... ... S T P A ! \ ( I) - S T R A ( 1 , I) .., ..• .•;••/< ..... . •PHB112 ' J : 0 0 9 7 9 0 " :•• CONT'TNMF:'-'-.-. > ' - . ' : . : - 'V . ! ' ' : •'' •:''.• '• • ':'.V • ;e .'': "'' :'' -PHB113 • . : 009s ;.. on up. i... f i ' » ' . 4 v. .... . .' ','••. • ' i ' . . ' " ' . • v PHB114 1 -49-0 W P I I F P MAIN 03-17-C9 18:31: 0 4 PAC.P 0003 pnqn TITLF (1 ) = TIT(Ltl) PHBI lb 0 100 110 CONTIVUF PHB116 0 to 1 CALL 1 IMF (XC.,YC,5,l) PHB117 010? WRITF(ft.lOOO) X C . Y C 0 1 0 ' 1 000 FORMAT(12F6.2) 0 104 CALL PLOT ( l.,0.,-3) PHB118 0 105 CALL SC AL F (TIMFX,K,10.,XMIN, DX , I) PHB119 in of. CALL SCAtE (STRAIN,K,7.,YMIN,DY,1) PHB120 0107 CALL SCALE PHB121 CIO" nn 50 I = 1,K PH8122 oioo RATE (I) = RATE(I) f I. PHB123 Oil:) STRAIN! I) = STRAIN!! 1+1. PHB124 0111 l-IR?TF<6,1000) TIMEX( I ), STRAIN (I I.RATFd ) 0112 50 CONTINUE PHB125 0 1 1 1 CALL AXIS (0..I..18H TIME HOUR,-18,10.,0 .,XMIN ,DX) PHB126 0114 CALL AXIS (0..1..12H STRAIN , 1 2, 7. , 90. , YMI N, D Y ) PHB127 0 11 = CALL IINF (TIMEX,STRAIN,K,l) PHB128 011ft XO = TIMFX(K) - .4 , PHB129 -.0117 ..YO =. STRAIN (K) .1. • : .. ;' ' • " • PHB130 CUR CALL SYMBOL (XO,Y0,.07,6HSTRAIN,0.,ft) PHB131 Olio CALL AXIS (10.,1.,25H STRAIN RATE 1/HOUR,-25,7., 90.,7MIN ,DZ)PHB132 0120 CALL LINE ( TTMEX.P.ATE.K, tl • PHB133 0121 ZP = RATF(K) • .5 ...... PHB134 . 0 122 CALL SYMBOL 1X0,Z0, .07, UHSTRATN RATE.O. , 111 ' • PHB135 . "12 3 CALL. SYMROL tl. , 8. , . 1 4, T I TLF , 0. , 80 ), .«.sj •***.> PH8136 "c PHB137 c PRINTING.(IF COMMENTS .. ,•'•',)[ "li /' "?S'0i 'idM :., :• PHB138 . c ' PHB139 012'. M = MTRANS(L) . . ,•,:•;:..:'.-••,..'. PHB140 • 0125 DT MENS T ON TI MM<50),STRAM(501.XFf6),YF(6)i XE(2) ,YE(2) ,C0M1I4) .C0MPHB141 1213),LfAP(50) . PHB142 012ft .. 00 9999 t = l,6 ' • "i. • •', .• ' 0127 XF(1)=0. 1 (' , 012B 9909 YFII) = 0. •> * 0129 XE(1)=O. :v . . . . .-i. : . . . . . . v , - « • . 0130 XE(2)=0. 0131 YE(1)=0. 013? YE(2)=0. 0 133 SPACIN = 0. PHB143 0134 ALTITU - 0. PHB144 0 135 01 STAN = 1. • .. ... .T.-.•.••:. . J ' -'•.>:.. . PHB 150 01.36 KOUNT=0 '. •'''^•^•''^'s'ti'^'V'?" 0137 WRITF(ft,1002) KOUNT 'S^til ••"••Ji". 0138 1002 FORMAT(16) < 1 0139 • on ' 140 J - 1, M I'-; : -V'- v : PHB151 0140 TIMM(J) = (TTMEXM(L.J) - XHIN) / DX ' .,:kV- PHB152 0141 STR AM ( J ) .'= (STRAIM(L.J) - YMIN >/ OY • .1 ^ • PHB153 014? XF(1) •= TTMM(J) .-.< . PHB154 ... 0143 XF(2) - XF(1) + .0? PHB155 0144 XF(3) = XF( 1 ) PHB156 0145 XF(4) = XF( 1 ) • - .02 PHB157 014ft XF(5) = X F(1) '. PHB158 01 47 XF(6) = XF(1) PHB159 014 8 LOGICAL MIDDLE, PLACE,CLFAR PHB160 014 0 .' .MIDDLE. =. STR AM (J) .GT.5. •• ' PHB161 -50-rp(} T(; ••, rv i ; rr'MM i. "A IN 0 3 - 1 7 - 6 ' ) ' ' 18:31:04 PAGF 0Q04 0 1 so PLAC.F = IMPORTU.J) .LT.? PHB162 0 1 S 1 I F ( J.GT.l) G O Tp 1 V 3 PHB163 0 1 R 7 G O T'l 1 4 4 --> PHB164 < 0 1 5 3 1 4 3 0 1ST AN = TIMM(J) - TIMOUT PHB165 0 1 s 4 I F ( PISTAM . L T . . 2 ) GO TO 1 4 1 ! "' PHB166 0 1 5 R~ 1 4 4 x r n i = T I M « I J I . . V PHB167 0 1 5 6 I F ( X F ( 1 ) .LT. .1) T I M M I J ) = .1 PHB168 0 1 5 7 I F { x t r m . G T . 9.35 i T I M M I J I = 9.85 PHB169 0 15 8 X F ( 2 ) = TTMMU) PH8170 r> i '.T xm = X F ( ? I PHB171 "Iff X 0 2 = X F I ? ) * .1 PHB172 1^61 1 4 1 IF ( MIPPLF) GO TO 146 PHB173 0 1 6 ? " YF( 1) = 5 . P PHB174 " 1 6 3 YF(?) = 6. PHB175 " 1 6 4 YP1 = YE ( ? ) PHB176 C I 6 5 V F ( T'l = STRAM(J) •' .08 PHB17 7 0 1 6 6 v r ( 6 ) = STRAMI J I + .?5 PH8178 0 1 6 7 K0UNT=K0UNT+1 • 0 1 6 8 W ? I T F ( 6 , 1 9 0 ? ) K O U N T 0 16? GO T O 147 PHB179 0170 1 . 4 6 YF( 1) = 3.7 *' : •• . PHB180 0171 Y F ( 2 ) = 3.5 PHB181 017? Y01 = 2.06 PHB182 0173 YM1) = S T R AM IJ 1 - .08 PHB183 0 1 7 4 Y F ( 6 ) = STBAMUI - . 2 5 . . . . . ' ',' ... , • PHB184 Ot7S 147 Y P 2 = Y01 '•'''',' PHB185 0176 ANGLF = 9 0 . PHB186 0 1 7 7 LFAP ( J ) - 0 PHB187 0175 K O U N T = K C U N T + l 0179 WITE(6,I002) K O U N T 0 1 8 0 I F ( OISTAN .LT. . 2 ... AND. PL AC E ; ) GO TO 142 ' PHB188 0181 G O T O 148 ..... .,.,.''•••.•• :, • ••'.'••' •PHB189 01"? 145 I F I PLACE I G O T O ,141 d ' ' '•' ' PHB190 0183 I F ( TIMMIJ) . G T . .72 ) GO T O 151 PHB191 0184 X01 = .1 ' , '. . ;j PHB192 0185 G O T O 1 5 2 ' • ' • . ' - • • , / ]•'• , " ••: •'' PHB194 0186 • . . 151 .. I F (..TIMMIJ) . G T . 9.27 ). GO.TO 153 ;.'•'"_.-.. . .• '• ,r-.'. •' ' . ''.. . . PHB195 0187 X01 = TIMMIJ) - .72 .". .• ;•.«;.'» PHB196 0188 G O T O 152 .". . 1 PHB198 0189 153 xm =8.55 ' PHB 199 0190 152 X02 = X01 . •. •• PHB201 0 1 9 1 XE(1) = TIMMIJ) . : "'•"'•;' •..'•''.' ~ • PHB202 019? . XE ( 2 I =. TIMMIJ) .... •.'".'. . ,.1 •• PHB203 0193 ROOM = TIMMIJ) - SPACIN . ' •". .'•''? '• '•" ' ',. PHB204 0194 CLEAR '= ROOM . G T . 1.44 . O R . VCLEAR . G T . .2 . PHB205 0195 VCl.FAR = STRAM(J) - A L T I T U ' PHB206 01O6 SPACIN = TIMMIJ) PHB207 0 1 9 7 ALTITU = STRAMI J) ' . "• PHB208 0198 K O U N T = K O U N T + I . . . ' : . : . . . . . , . ; • ' . ' . ' ' . . • ' • . : • ' . 0199 WRITEI6,10021 KOUNT ' 0?00 I F ( MIOOLF) G O T O 154 PHB209 0 7 0 1 I F ( .NOT. CLEAR ) G O T O 157 PHB210 0?0? 156 I F ( IFAP ( J-l) . F O . 1 ) G O T O 15R PHB211 0 7 0 3 159 Y F(1) = STRAMIJ) + .08 PHB212 . . . 0 7 0 4 LFAPt J 1 = 1 J PHB213 -51-r tf'L'I'-'A' I 1/ [ II c" MAPI 03-17-69 1P:31:04 °*GF 0005 YF(6)'= STRAM(J) + .75 PHB214 0 7 0 (• YF( 1 > - YFIM + .? PHB215 " 7 0 7 Y>-(?) = YC(6I * .4 PHB216 vni = YF(?) + .1 PHB217 vn? = YF(?) PHB218 0 7 1 KPU'NT-KOIINT + 1 C > 1 I WPITF(6,1007) K flUNT 071 ? 00 TO 155 PHB220 o?l < 1 54 IF ( .MOT. CLEAR ) GO TO 156 ; PHB221 0 714 157 IF ( LEAP(J-l) .EQ. ? ) GO T O 159 PHB222 0715 153 YF(l) = STRAM(J) - .OB PHR223 0 ? 1 L cA°(j) = ? PHB224 r? \ i YF ( 6 ) = S TR AM t J I .- .75 PHB225 0 71 1 YF( 1 ) = VF(6) - .2 ••. , PHB226 0 7 I n YE(?) = YF(6) - .4 ' \ .. PHB227 r:7?o YD 1 = YF(7) - .1 PHB228 <~??1 155 ANGI. F = 0. PHB230 07?? YP2 = YEI?) - .? • ' ., , PHB231 i??t KOUNT=KOUNT*l 0 ? 7 /, WRITE (6,100?) KOUNT 0 ? 7 c 149 DO 160 1 = 1,4 PHB232 0??6 160 CPM1(I) = COMNTSIL,J, I) PHB233 0??7 in 170 I = 5,7 PH8234 0 ? 7 o IA = I - 4 -. • . ' . • •• ' , PHB235 0??° 170 CPM?(IA) = COMNTSIL, J, I ) . •.. PHB236 "O?30 CALL LINF ( XE«YE,2,1) ; . , PHB237 o ? 31 WRITF(6,1000) XE,YE 0 71? CALL SYMBOL ( X01 ,Y01 , .07 ,C0M1 , ANGLE, 24) PHB238 r, ? 3 3 CALL SYMpnt (X02,YD2,.07,COM2,ANGLE,18) PHB239 0 7 3/, WPITF(6,1000) X01,X02,YDl,Y02 , V: 0 ? ? r i TTMOUT = TIMM(J) PHB240 n?3f. 14? YF(?) =.YF(1) PHB241 0737 YF(3) = STRAM(J) PHB242 ^7 3? YFI4) = YF ( 11 • • - ' •'• PHB243 0?3i YF(5) = YFf 1) .. ... PHB244 0?40 CALL LINE (XF, Y F , 6,1) PHB245 o?4i WRITEI6.1000) . X F . Y F ; 0?47 1 40 CONTINUE ., PHB251 c THE END O F COMMENTS' PHB252 0?4"< CALL PLOT (12.,0..-3) PH8253 0?',4 no CONTINUE ... , , .. PHB256 P?45 CALL P L O T N O . , > . ' PHB257 0246 .. S T O P _ • PHB258 0 74 7 END •• , '. •. V ' - . ' ... PHB259 -52-APPENDIX I I A d d i t i o n a l Creep Curves Figure a. Stage I I (SGS), 572 p s i . F i g u r e b. Stage I I (SGS), 858 p s i . Figure e. Stage I I (LGS), 1221 p s i Figure f. Stage I (SGS), 120 p s i . Figure i . T r a n s i t i o n (SGS), 2574 p s i . Figure k. T r a n s i t i o n (LGS). 1954 p s i . Figure m. T r a n s i t i o n (LGS),.2606 p s i . Figure 1. T r a n s i t i o n (LGS), 2280 p s i . Figure n. T r a n s i t i o n (LGS,), 2932 p s i . Figure o. T r a n s i t i o n (LGS), 3258 p s i . Figure p. T r a n s i t i o n (LGS), 3908 p s i . Figure r . T r a n s i t i o n (LGS), 4560 p s i . Figure s. T r a n s i t i o n (LGS), 4886 p s i . Figure u. Stage I I I , 7326 p s i Figure v. Stage II incremental loading. -58-Figure w. Stage I I I unloading. -59-APPENDIX I I I C a l c u l a t i o n o f T h e o r e t i c a l Creep Curves f o r P u r e N a b a r r o - H e r r i n g Creep and P u r e C o b l e Creep. N-H. Assume t h a t a g r a i n deforms o n l y by N-H d i f f u s i o n a l c r e e p . I n i t i a l l e n g t h = l o I n i t i a l w i d t h s = w6 I n c r e a s e s i n l e n g t h = A l I n c r e a s e i n w i d t h = Aw F i n a l l e n g t h 1 = l o + o l F i n a l w i d t h w = wo + Aw E n g i n e e r i n g s t r a i n = = E True s t r a i n de = d l / 1 ; e = l n ^ = I n 1 + ^ = I n (1 4- E) l o l o . F, + 1 = e e o r E = e £ - 1 F o r c o n s t a n t volume e. + + £g = 0 Assume E £ = £ 3 = - Hz\ w i t h a P o i s s o n s r a t i o o f % f o r p l a s t i c d e f o r m a t i o n . l n ( l + f , . - % l „ ( l + | i , 1. ( - i „ ( 1 ( 1 + E T ) = ( 1 + % L ) ~ h Assume i n i t i a l g r a i n d i m e n s i o n s L o , Wo A f t e r e n g i n e e r i n g s t r a i n E -60-L = Lo(1 + E ) = Loe E J_i _ \ , _ G / 9 W = Wo(l + E 2) 2 = Woe ' 2 - E 2 2 2 2 2c Define g r a i n s i z e L where L = ^(L rw ) = %Lo e + ^ Wo e I f the g r a i n i s i n i t i a l l y equiaxed -2 ! 2 j. 2e , _£-, L = %Lo {e + e } From Zehr and B a c k o f e n ^ e " « V D L where a i s a geometrical constant - 10 -23 3 v i s the atomic volume (1.43 x 10 cm f o r Sn) D L i s the c o e f f i c i e n t f o r l a t t i c e d i f f u s i o n — 16 o k i s Boltzman's constant = 1.38 x 10 erg/ K T i s the absolute temperature (°K) Pure N-H creep i s much slower than that found exp e r i m e n t a l l y . Values of D f o r Sn w i l l give the f a s t e s t r a t e . I f the g r a i n s i z e i s 2 microns and T = 26°C. J{ N-H = 2.74 x 1 0 1 7 d y n e S I s e c = 3.96 x 1 0 1 2 l b - s e c / i n 2 crn^ de = dt 71m T2 , f v c raD 1** -f mvD T J • , T 2 a\ L) t = h Lo kT e L d t 2 2e kT o ( e 2 e + e e ) d e - \ Lo {^- + e £ + Jg} t = Z L N ^ { e 2 e _ 2 e - e 4s -61-a = 97 p s i 71 N-H = 3.96 x 1 0 1 2 ^ 5 f £ " x * r = 1.1 x 10 9 v 3600 sec i n ,9 _ 1-1 x 10 • , 2e o -e , * ""4 x 97 { e " 2 e + 1 } t = 2.85 x 1 0 5 { e 2 e - 2e" e + 1} hrs N-H Creep F r a c t i o n of time True S t r a i n Time (hrs) to reach .40 s t r a i n 0 0 0 .01 1.14 x 10 4 2.1 .02 2.28 x 10 4 4.2 .05 5.70 x 10 4 10.6 .10 . 1.16 x 10 5 21.5 .20 2.42 x 10 5 45.0 .30 3.82 x 10 5 71.0 .40 5.39 x 10 5 100.0 Coble The a n a l y s i s i s s i m i l a r to that f o r N-H creep. Grain s i z e i s i d e n t i f i e d as L 3 = h ( L o 3 + Wo3) = h L o 3 e 3 £ + %Wo3e 3 / 2 e f o r equiaxed grains Lo = Wo .3. L J = % L o J ( e " " + e '") 7 3 _ , T_3,_3e , _ - J / 2 E > a rn L 3kT I = 7 l c ° ~~ BvwF u gb -62-where D ^ i s the grain boundary d i f f u s i o n c o e f f i c i e n t i s a constant - 150 ,-7 w i s grain boundary width 10 cm) T i s 26° C L i s 2 microns a i s 97 p s i 77 co = = 3.88 x 1 0 ^ dynes sec/cm 2 since coble creep i s much f a s t e r than the experimental rate and 7 ^ gives a slower rate of creep than7?Sn. L 3de agvwD kT dt T 3 f , 3e . - /2e, , ,aBvwD„>. , Lo (e + e )de = (____££ ) 2 J k T t = L o \ t 3vwDgk v 6 a r 3e . - 3/2 . . 1 e - 2e + 1 J t = Lo kT 3vwD gb f^)Le3e - 2e •3/2e + l ] , 7 f 3e „ - 3/2e . .1 t = . 271 e - 2e I hrs. - 6 3 -Coble Creep F r a c t i o n of time True S t r a i n Time (hrs) to reach .40 s t r a i n 0 0 0 .01 1.62 x 10~ 2 .0187 •02 3.24'x 10" 2 , .0374 .05 8.4 x 10~ 2 .096 .10 1.7 x 10" 1 .196 .20 ,. 3.6 x 1 0 - 1 .415 .30 5.9 x 10" 1 ' .680 .40 • * 8.7 x IO" 1 1.00 -64-BIBLIOGRAPHY 1. D. Lee, G.E. Res. and Dev. 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