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The plastic deformation of polycrystalline lead under controlled stress. Fox, Gary Wayne 1971

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THE PLASTIC DEFORMATION OF POLYCRYSTALLINE LEAD UNDER CONTROLLED STRESS by GARY WAYNE FOX B.A.Sc., U n i v e r s i t y of Toronto, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of METALLURGY We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1971 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 the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o lumbia, I agr e e t h a t the 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 agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes 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 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 . Department o f Metallurgy The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada D a t e January 5. 1972 A B S T R A C T A qualitative analysis of current creep theory has been made by studying the creep of polycrystalline lead. The behaviour of the low temp-erature stress-strain curve with prior creep history, and the strain response to decreases in creep stress were examined. The effect of temperature and creep strain on the low temperature stress-strain curve was investigated over the temperature range 0.5T to m 0.8Tm- Specimens were quenched to 77°K. after creep and strained to determine the stress-strain curve. The 77°K. yield stress was found to increase during primary creep and remain constant in steady state. Increas-ing the creep temperature drastically lowered the low temperature yield stress. The reversible flow stress ratio was found to decrease with i n -creasing temperature. These observations were in qualitative agreement with both a reaction rate theory and a rearrangement model. Stress decrease tests were carried out by reducing the creep stress after deforming the specimen varying amounts into primary and steady state in the temperature ranee 0.5T to 0.85T . The strain response m m to a stress decrease in steady state was in best agreement with the simple recovery theory. The variation In yield stress due to non-regular obstacle spacing was found to be extremely small at a l l temperatures and did not behave in accordance with the qualitative predictions of the rearrangement theory. i i TABLE OF CONTENTS Page INTRODUCTION . 1 1.1 PLASTIC FLOW AT ELEVATED TEMPERATURES 1 1.2 YIELD STRENGTH THEORY AND THE LOW TEMPERATURE STRESS-STRAIN CURVE 2 1.3 YIELD STRENGTH-RECOVERY THEORY 4 1.4 REACTION RATE THEORY 6 1.5 YIELD STRENGTH-RECOVERY-REARRANGEMENT THEORY 10 1.6 SCOPE OF THE PRESENT WORK 13 EXPERIMENTAL 15 2.1 EQUIPMENT AND MATERIALS 15 2.1.1 Specimen Preparation 15 2.1.2 Equipment . 15 2.1.3 Temperature Control 17 2.2 METHODS 17 2.2.1 Annealing In Situ 17 2.2.2 Low Temperature Stress-Strain Curves 18 2.2.3 Stress Change Experiments 20 2.2.4 Necking . . 20 RESULTS 21 3.1 LOW. TEMPERATURE YIELD STRESS MEASUREMENTS 21 3.1.1 Creep Curves 21 3.1.2 Low Temperature Stress—Strain Curves 21 3.1.3 Low Temperature Yield Stress Versus Total Strain. . 37 3.1.4 The Change i n Low. Temperature Yield Stress With. Creep Strain 43 3.1.5 Modulus Corrected Reversihle Flow Stress Ratio Versus Creep Strain . 5C 3.1.6 Reversible Flow Stress Ratio Versus Temperature . . 56 i i i i v Page 3.2 STRESS DECREASE EXPERIMENTS 58 3.2.1 Low Strain Rate Tests 58 3.2.2 High Strain Rate Tests 61 DISCUSSION 70 4.1 CREEP CURVES 70 4.2 LOW TEMPERATURE STRESS-STRAIN CURVES 71 4.3 LOW TEMPERATURE YIELD STRESS VERSUS TOTAL STRAIN 72 4.4 CHANGE IN 77°K. YIELD STRESS WITH CREEP STRAIN 73 4.5 REVERSIBLE FLOW STRESS RATIO 74 4.5.1 Reaction Rate Theory 74 4.5.2 Rearrangement Theory •• • ' 74 4.5.3 Reversible Flow Stress Ratio Versus Temperature . . . 77 4.6 STRESS DECREASE EXPERIMENTS 77 4.6.1 Low Strain Rate Results 77 4.6.2 High Strain Rate Results 78 4.6.3 Behaviour of T 80 v SUMMARY 82 CONCLUSIONS 83 APPENDICES 84 BIBLIOGRAPHY 86 LIST OF TABLES Table Page I. CLASSIFICATION OF WORK HARDENING THEORIES ' BY OBSTACLE TYPE 1 II. COMPARISON OF MEASURED AND ESTIMATED LOADING STRAINS. 45 III. VALUES OF T v CALCULATED ACCORDING TO EQUATION 16. . .67 IV. COMPARISON OF CALCULATED AND MEASURED RECOVERY TIMES. 80 v LIST OF FIGURES Figure Page 1. Shape of the L-T s t r e s s - s t r a i n curve according to the y i e l d strength theory 2 2. Response to stress changes according to recovery theory. . 5 3. Response to stress changes according to reaction r a t e theory 9 4. D i s t r i b u t i o n curve of l o c a l d i s l o c a t i o n density 10 5. Response to stress changes according to rearrangement model. 13 6. Schematic diagram of creep apparatus 16 7. Comparison of 77°K. s t r e s s - s t r a i n curves determined i n Instron and creep machine 19 8. Creep curves for p r e s t r a i n at 0.5T m 22 9. Creep curve for pres t r a i n at 0.6T 23 r m 10. Creep curves for pres t r a i n at 0.6T m 24 11. Creep curves f o r p r e s t r a i n at 0.7T^ 25 12. CreeD curves f o r p r e s t r a i n at 0.8T 26 m 13. 77°K. s t r e s s - s t r a i n curves following creep at 0.5T , a = 682 p s i m . . . . 27 c 14. 77°K. s t r e s s - s t r a i n curves following creep at 0.5T , a = 602 p s i m . . . . 28 c 15. 77°K. s t r e s s - s t r a i n curves following creep at 0.5T , a = 602 p s i m . . . . 29 c r 16. 77°K. s t r e s s - s t r a i n curves following creep at 0.6T , a = 560 p s i m . . . . 30 c 17. 77°K. s t r e s s - s t r a i n curves following creep at 0.6T , a = 400 p s i m . . . . 31 c r v i v i i Figure Page 18. 77°K. s t r e s s - s t r a i n curves following creep at 0.6T , a = 320 p s i ? . . . . 32 c 19. 77°K. s t r e s s - s t r a i n curves following creep at 0.7T , a = 215 p s i * . . . . 33 c 20. 77QK. s t r e s s - s t r a i n curves following creep at 0.7T , a = 160 p s i ? . . . . 34 c 21. 77°K. s t r e s s - s t r a i n curves following creep at 0.8T , a = 120 p s i ™ . . . . 35 c 22. 77°K. s t r e s s - s t r a i n curves following creep at 0.8Tm, a =95 p s i 36 c 23. 77°K. y i e l d stress versus t o t a l s t r a i n at 0.5T 38 J m 24. 77°K. y i e l d stress versus t o t a l s t r a i n at 0.6T , a = 560 p s i ™ . . . . . . 39 c 25. 77°K. y i e l d stress versus t o t a l s t r a i n at 0.6T 40 J m 26. 77°K. y i e l d stress versus t o t a l s t r a i n at 0.7T 41 J m 27. 77°K. y i e l d stress versus t o t a l s t r a i n at 0.8T 42 28. Change i n 77°K. y i e l d stress with creep s t r a i n at 0.5T . . 46 m 29. Change i n 77°K. y i e l d stress with creep s t r a i n at 0.61^ . . 47 30. Change i n 77°K. y i e l d stress with creep s t r a i n at 0.7Tm . . 48 31. Change i n 77°K. y i e l d stress with creep s t r a i n at 0.8Tm . . 49 32. Reversible flow stress r a t i o versus creep s t r a i n at 0.5T . 51 m 33. Reversible flow stress r a t i o versus creep s t r a i n at"*0.6T , a = 560 p s i . m. 52 c 34. Reversible flow stress r a t i o versus creep s t r a i n a t 0.6T . 53 . • m '35. Reversible flow stress r a t i o versus creep s t r a i n at 0.7T . 54 36. Reversible flow stress r a t i o versus creep s t r a i n at 0.8T . 55 m 37. Reversible flow stress r a t i o versus temperature 57 38. Chart trace of 15% stre s s decrease at 0.7T 59 m v i i i Figure Page 39. Chart trace of 5% stress decrease at 0.7T 59 m 40. Mean s t r a i n rate as a function of the time period of c a l c u l a t i o n at 0.7T 60 m 41. Mean s t r a i n rate change versus stress change at 0.7T . . 62 m 42. Chart trace of 2% stress decrease at 0.5T 63 m 43. Chart trace of 2% stress decrease at 0.6T 63 m 44. Chart trace of 5% stress decrease at 0.6T 64 m 45. Chart trace of 2% stress decrease at 0.7T 65 m 46. Chart trace of 2% stress decrease at 0.85T 65 m 47. E f f e c t of p r e s t r a i n on y i e l d i n g i n s t r e s s - s t r a i n curve. . 72 48. Relationship of y i e l d strength and creep stress to low-temperature s t r e s s - s t r a i n curve 76 49. E f f e c t of temperature on the low temperature s t r e s s -s t r a i n curve 76 50. Schematic diagram of a s t r e s s - s t r a i n curve showing a r e v e r s i b l e flow stress measurement 84 51. Temperature f l u c t u a t i o n s i n the temperature bath at 0.7T 85 m A C K N O W L E D G E M E N T S The author wishes to express hi s gratitude f o r the advice and encouragement of his research supervisor, Dr. T. H. Alden. Thanks are also extended to f a c u l t y members, fellow graduate students, and tech-n i c a l s t a f f f or h e l p f u l discussions and assistance. F i n a n c i a l assistance from the National Research Council (Grant A-4991) i s g r a t e f u l l y acknowledged. i x I N T R O D U C T I O N 1.1 P l a s t i c Flow at Elevated Temperatures Any theory of p l a s t i c flow must explain the dynamic nature of metal deformation, i . e . , the stress at which flow occurs i s a function of the rate and temperature at which deformation i s c a r r i e d out. Two dyna-mic e f f e c t s are the f a l l i n g r e v e r s i b l e i s o s t r u c t u r a l flow stress r a t i o (Appendix A) at high temperature [ C o t t r e l l and Stokes 1955] and the s t r a i n response to changes i n creep stress at constant st r u c t u r e . Resistance to p l a s t i c flow i n pure metals i s generally ascribed to an obstacle structure which impedes d i s l o c a t i o n motion. Obstacles can be regarded as thermally penetrable (thermal) when the s t r a i n process i s aided by thermal a c t i v a t i o n or thermally impenetrable (athermal) when ther-mal a c t i v a t i o n plays no role i n the s t r a i n process, P3? se. Obstacles can also be classed as thermally stable (obstacles are not l o s t with time and temperature) or thermally unstable (obstacles are l o s t with time and temp-erature) . Thus published theories of p l a s t i c deformation can be grouped into four broad categories (Table I ) . Table I) CLASSIFICATION OF WORK HARDENING THEORIES BY OBSTACLE TYPE OBSTACLE TYPE STABILITY THEORY Athermal Stable Y i e l d Strength Athermal Unstable Y i e l d Strength-Recovery Thermal Stable Simple Reaction Rate Thermal Unstable Complex Reaction Rate 1 2 Because obstacles vary in strength, (penetrability) and s t a b i l -ity with temperature, each theory makes different predictions about the way metals should behave in response to changes in temperature and stress. In the following section, the elements of three theories w i l l be presented and discussed in relation to the effects just described viz. reversible flow stress ratio and stress-strain rate response.' 1.2 Yield Strength Theory and the Low Temperature Stress-Strain Curve The simple yield strength theory has been used [Orowan 1946-7] to describe plastic deformation in metals at low temperatures where rate dependent flow is unimportant. In this description of flow, metals deform only when the applied stress equals the yield s trength of the material; deformation ceases when the yield strength rises above the applied stress due to strain hardening. The yield strength theory suggests a sharply de-fined two stage stress-strain curve with an abrupt elastic-plastic trans-it i o n (Fig. 1). strain Fig. 1. The stress-strain curve according to the yield strength theory . 3 I n t h e p l a s t i c r e g i o n t h e r e l a t i o n between s t r e s s and s t r a i n i s d e s c r i b e d by t h e e q u a t i o n da = 0 yde (1) where 0 i s t h e c o e f f i c i e n t of s t r a i n h a r d e n i n g . y M e t a l s do n o t g e n e r a l l y e x h i b i t an a b r u p t i n f l e c t i o n i n t h e s t r e s s - s t r a i n c u r v e . Kocks [1966] e x p l a i n s - t h e c u r v a t u r e o f t h e e l a s t i c -p l a s t i c t r a n s i t i o n a t 0°K by t h e n o n - r e g u l a r i t y i n a random o b s t a c l e d i s -t r i b u t i o n . " ' " The s t r e s s n e c e s s a r y f o r t h e moving d i s l o c a t i o n t o overcome a s i n g l e o b s t a c l e p a i r i s Gb a = a — (2) where G = s h e a r modulus B = b u r g e r s v e c t o r I = t h e n e a r e s t n e i g h b o u r o b s t a c l e d i s t a n c e w h i l e t h e y i e l d s t r e n g t h of t h e m a t e r i a l i s g i v e n by Gb T a — y i where £ i s t h e mean of t h e l o c a l o b s t a c l e s p a c i n g £. I t i s e v i d e n t t h a t the s p a t i a l v a r i a t i o n of s t r e n g t h i n t h e m a t e r i a l p e r m i t s some d i s l o c a t i o n s to move a t s t r e s s e s l e s s t h a n t h e m a c r o s c o p i c y i e l d s t r e n g t h ( n o t p o s s i b l e i n t h e s i m p l e y i e l d s t r e n g t h t h e o r y ) . A t low a p p l i e d s t r e s s d i s l o c a t i o n s a r e c a p a b l e of moving o n l y s h o r t d i s t a n c e s b e f o r e t h e y a r e impeded by c l o s e -l y spaced o b s t a c l e s . A c o n t i n u o u s l y i n c r e a s i n g s t r e s s however p e r m i t s g l i d e d i s l o c a t i o n s t o p e n e t r a t e p r o g r e s s i v e l y l a r g e r a r e a s of t h e s l i p p l a n e , t i l l n e a r l y t h e "'"Kocks d e f i n e s a p e r f e c t l y r e g u l a r s t r u c t u r e as one i n w h i c h the n e a r e s t n e i g h b o u r d i s t a n c e i s the same f o r a l l o b s t a c l e s . 4 whole g l i d e plane i s a c c e s s i b l e when the a p p l i e d s t r e s s equals the y i e l d s t r e n g t h . As a r e s u l t a rounded knee i n the s t r e s s - s t r a i n curve can be explained. 1.3 Y i e l d Strength-Recovery Theory The y i e l d strength-recovery theory i n c l u d e s the e f f e c t of r e -covery i n the simple y i e l d s t r e n g t h theory. Increases i n y i e l d s t r e n g t h (O yde) through s t r a i n hardening are o f f s e t by decreases due to recovery ( r y d t ) . These two counteracting e f f e c t s are described by the d i f f e r e n t i a l equation da = dr y = - r - ^ l dc + 7—^1 dt de ! t 3t 'e =0 de - r dt y y = ° (4) so e = r /0 (5) y y K J The c o e f f i c i e n t of s t r a i n hardening 0 y Ch i n the recovery l i t e r a t u r e ) i s athermal (temperature i n s e n s i t i v e ) w h i l e the r a t e of recovery, r , i s a f u n c t i o n of s t r e s s and temperature [ M i t r a and McLean 1967] . At a given temperature and s t r e s s , 0 and r are uniquely defined and a constant s t r a i n f y y r a t e i s e s t a b l i s h e d according to equation ( 5 ) . Thus the y i e l d s t r e n g t h - . recovery theory a p p l i e s only during steady s t a t e . The recovery r a t e i s determined e x p e r i m e n t a l l y by lowering the creep s t r e s s by an amount Aa ( u s u a l l y l e s s than 0.1 a c ) and r e c o r d i n g the time At to e s t a b l i s h a new steady s t a t e s t r a i n r a t e . The c o e f f i c i e n t of 5 s t r a i n hardening i s measured e i t h e r by i n c r e a s i n g the creep s t r e s s by A a at temperature and measuring the a s s o c i a t e d instantaneous s t r a i n [Evans and W i l s h i r e 1968J, or by quenching to some low temperature and determining the slope of the s t r e s s - s t r a i n curve at o_ = a [ M i t r a and McLean 1967J. REF c 2 The r e v e r s i b l e flow s t r e s s r a t i o E F aREF ET always equals one [Alden 1971b] i n the recovery theory because the y i e l d s t r e n g t h (the athermal or 0°K. flo w s t r e s s ) during p l a s t i c flow i s equal to the creep s t r e s s . This theory p r e d i c t s an asymmetric s t r a i n response to changes i n creep s t r e s s . On r a i s i n g the s t r e s s by A a there i s an i n s t a n t -aneous s t r a i n because a i s momentarily greater than T . When the creep s t r e s s equals the y i e l d s t r e n g t h through s t r a i n hardening, a new steady s t a t e s t r a i n r a t e i s e s t a b l i s h e d according.to equation (5) without any t r a n s i e n t creep. On decreasing the creep s t r e s s the theory p r e d i c t s a per-io d of zero s t r a i n r a t e because the creep s t r e s s i s l e s s than the y i e l d s t r e n g t h . When the creep s t r e s s again equals the y i e l d s t r e n g t h through recovery, a new steady s t a t e s t r a i n r a t e i s e s t a b l i s h e d immediately ( F i g . 2). time F i g . 2. Response to s t r e s s changes according to a recovery model, ^REF — i s the r a t i o of the E l a s t i c Modulus at the low temperature reference T to the Modulus at the temperature of i n t e r e s t and compensates f o r the temperature dependence of the E l a s t i c Modulus. 6 The recovery theory i s a theory of steady state lOrowan 1946-7] and as such cannot account for primary creep or trans i e n t s . Alden [1971a] has shown how the recovery theory can be modified to explain primary creep, transients and a r e v e r s i b l e flow stress r a t i o less than one. These ideas w i l l be presented i n a l a t e r s e c t i o n . 1.4 Reaction Rate Theory Reaction rate theory recognizes the dynamic or rate dependent nature of p l a s t i c flow. In these theories d i s l o c a t i o n s are aided by ther-mal a c t i v a t i o n i n overcoming obstacles to flow. These obstacles are short range because thermal a c t i v a t i o n i s capable of supplying only l i m i t e d amounts of energy at a given temperature. The temperature dependence of the defor-mation process i s described by an Arrhenius equation. i = e Qexp - [^-] (6) where i = constant o U = a c t i v a t i o n energy In the simplest type of rate theory [Becker 1925] the a c t i v a t i o n energy i s a function of s t r e s s . U = U - vo (7) o where U = a c t i v a t i o n energy v = a c t i v a t i o n volume = apparent a c t i v a t i o n energy At constant s t r e s s , U i s a constant and the energy b a r r i e r does not change with s t r a i n or time. 7 In a r e v e r s i b l e flow: stress experiment the s t r a i n rates at high and low temperature can be regarded as constant. This requires that U/kT i n equation (6) be i n v a r i a n t ( E q and U q depend on structure and are con-stant) . To maintain the s t r a i n rate constant the decrease i n s t r a i n rate accruing from the change i n temperature must be o f f s e t by an increase i n a c t i v a t i o n energy through a r i s e i n applied stress [equation C7)]. Thus a f a l l i n g flow stress r a t i o at high temperature r e s u l t s . The disadvantage of the Becker theory i s that i t predicts a constant s t r a i n rate and i s therefore unable to account for primary creep and transients. It i s possible to account for transients using r e a c t i o n rate theory. This has been done by assuming that the a c t i v a t i o n energy may vary because of s t r a i n hardening and recovery [Co ttrell--Aytekin 1950, Gasca-Neri et al. 1970]. These e f f e c t s may be incorporated into the theory by modifying equation (6) to U -va k = e Q e x p - [ — j ^ j T — ] (8) where a = e f f e c t i v e stress e = applied stress - i n t e r n a l stress = • a - a. c 1 3 In the theory of C o t t r e l l - A y t e k i n [1950J , the Arrhenius equa-t i o n i s retained as the basic equation of p l a s t i c flow while the e f f e c t s C o t t r e l l and Aytekin did not formulate t h e i r ideas i n terms of i n t e r n a l and e f f e c t i v e stresses per se, but referred to i n t e r n a l stress only i n a general way. The modern approach [Gasca-Neri et al. 1970] has been to ascribe the d r i v i n g force for s t r a i n to <j a n d that of recovery to 8 of recovery and strain hardening are introduced in the form of an auxiliary equation identical in form to the yield strength-recovery theory. do. 9a. 9a. dt S t } n + S t }i=0 (9) r=0 = 0 during s.s, 9a. , 9a. 9e 'r=0 dt 9t U U ; .'.' e = r /8 (5) y y During primary creep the internal stress rises due to strain hardening [Gasca-Neri et al. 1970]. This is accompanied by a concomitant decrease in the effective stress because the creep stress is constant. The declining effective stress results in a fa l l i n g strain rate as observed. The effect of recovery in primary creep progressively increases with strain t i l l , during steady state, equation (5) can be invoked and the effects of strain hardening and recovery are self-compensating, i.e., the internal stress attains a value where any increases due to strain hardening are ex-actly offset by decreases due to recovery. Reversible flow stress ratios less than one"can be explained by this theory. The internal stress, which is the long range, athermal com-ponent of the applied stress, remains constant on quenching because the test a_^ . For simplicity an attempt has been made here to discuss the Cottrell-Aytekin theory in terms of and o . 9 i s i s o s t r u c t u r a l . The e f f e c t i v e s t r e s s , however, must increase at low temperature to maintain a constant s t r a i n rate according to equation (8). Hence, the applied stress at low temperature i s greater than i t i s at high temperature (a = a + a^) and the r e v e r s i b l e flow stress r a t i o i s less than unity. The C o t t r e l l - A y t e k i n theory i n agreement with other rate theories predicts a symmetric response to stress changes (Fig.3). On r a i s i n g the creep stress no instantaneous s t r a i n r e s u l t s , because stress and s t r a i n rate are e x p l i c i t l y r e l a t e d . Through a g i n equation (8) an increase i n stress increases the e f f e c t i v e stress and the s t r a i n rate r i s e s (Fig. 3). time Fi g . 3. Response to stress changes according to reaction rate theory. Unlike the recovery theories, a non-zero creep rate i s expected when the creep stress i s decreased. The e f f e c t i v e stress i s considered to decrease so that the s t r a i n rate f a l l s to some lower value according to equation (8). 10 1.5 Yield Strength-Recovery-Rearrangement Theory The theory types listed in Table I assume that the obstacle struc-ture is regular. Obstacle structure is describable by a single parameter; the yield strength T y in the yield strength-recovery theories, and by the activation energy U in the reaction rate theories. A recent theory [Alden 1971a] admits the possibility of non-regular obstacle structure, particu-larly at high temperature. The effect of non-regular obstacle spacing is assumed to produce a specimen of varying local yield strength (i.e., where £ is a variable). Deformation occurs locally in soft elements where obstacle spacing is large and the local yield strength is low. Thus deformation can occur when the applied stress is less than the yield strength (T is the mean of the local yield strength and is equal to aGb/l). The variation in local yield strength is described by a para-is meter designated x,^. = aGbp^ where is the width, at mid-height, of a distribution curve (Fig. 4) of the number of elements of local dis-location density N(p^) versus the local dislocation density (p ) . C L £  wl Fig. 4. Distribution curve of local dislocation density. 11 x^ i s large when obstacle structures are non-regular, and small when d i s l o c a t i o n structures are uniform. Thus at high tempera-tures where subgrains are formed [Garofalo 1965], i s expected to be large while at low temperatures where structures are regular {Livingston 1962] should be small. During primary creep d i s l o c a t i o n structures become less regular [Gupta and Strutt 1967] so that x^ i s expected to r i s e . The s o f t elements (places where deformation occurs e a s i l y ) vary strongly i n s i z e depending on the stress and structure. The degree of v a r i a t i o n i s given by the r e l a t i v e area function, A , defined as x - a A = exp - [-^ —] 0 < A < 1 (11) r x v - r — In the theory, an equation f o r the steady state behaviour of the r e l a t i v e area function i s derived by equating the derivatives of y i e l d strength and stress with respect to s t r a i n (dx /de = do/de) . The r e s u l t -y ing equation i s where A . <12) r 9 + r /t y v 9x V r = "v 9t £ At high temperatures r , determined by the thermally activated rate of rearrangement, i s presumed to be high, so that A^ i s low; at high s t r a i n rate, or low temperature (low r ) , A^ i s expected to be large. The r e l a t i v e area function and r are included i n the general v equation describing p l a s t i c flow. 12 • , r , r x - a a + v + y r y . n , 1 0 . £ = QT. ' exp - ] Q 3 ) y v A r e v e r s i b l e f l o w s t r e s s r a t i o l e s s than one i s p r e d i c t e d by t h i s e q u a t i o n b e c a u s e the r e l a t i v e a r e a f u n c t i o n i s l e s s than one a t t emp-e r a t u r e s g r e a t e r t han 0 ° K . Such a c o n d i t i o n r e q u i r e s t h a t the f l o w s t r e s s be l e s s than the y i e l d s t r e n g t h because the s t r u c t u r e v a r i a b l e s x and x y v a r e c o n s t a n t . P r i m a r y c r eep and t r a n s i e n t s a r e e x p l a i n e d i n the t h e o r y c h i e f l y by a v a r i a b l e r e l a t i v e a r e a f u n c t i o n . The i n f l u e n c e o f s t r a i n h a r d e n i n g , r e c o v e r y and r ea r r angement a r e r e f l e c t e d i n the s t r e s s - s t r a i n r e l a t i o n s h i p 0 r r de A r e e The y i e l d s t r e n g t h i s no t gove rned by r ea r r angement so t h a t dx r _ x = e (15) de y . Immedia te ly f o l l o w i n g the i n s t a n t a n e o u s s t r a i n the c r e e p s t r e s s e q u a l s the dx d o ' d T do ' y i e l d s t r e n g t h (A =1) and -r-^- >-r—. In s t e a d y s t a t e -r—^ = — so t h a t d u r -3 r , de de 3 de de dx i n g p r i m a r y c r eep — m u s t f a l l . T h i s can be a c c o m p l i s h e d by a d e c r e a s -i n g r e l a t i v e a r e a f u n c t i o n . A f a l l i n g r e l a t i v e a r e a f u n c t i o n i n p r i m a r y c r eep r e s u l t s i n a d e c r e a s i n g c r eep r a t e w i t h s t r a i n [ e q u a t i o n Q-3)]. An asymmet r i c s t r a i n r e s p o n s e to s t r e s s changes i s p r e d i c t e d ( F i g . 5). On i n c r e a s i n g the c r eep s t r e s s an i n s t a n t a n e o u s s t r a i n r e s u l t s (a > x ) f o l l o w e d by a p e r i o d of t r a n s i e n t c r e e p . The e f f e c t o f d e c r e a s i n g the s t r e s s i s to lower the r e l a t i v e a r e a f u n c t i o n t h r o u g h i n c r e a s e d x - cr y 13 so that a smaller creep rate i s immediately established. a •H time Fig. 5. Response to stress changes according to the rearrangement theory. 1.6 Scope of the Present Work The yield strength-recovery theory has the advantage that i t is easy to apply and is reasonably accurate [Gasca-Neri et al. 1971]; how-ever, i t has two serious defects which allow i t to be eliminated as an adequate explanation of the effects described here. The theory cannot explain primary creep or transients because i t applies only for a steady state condition. It is also deficient because i t predicts a reversible flow stress ratio equal to one. Both the recovery-rearrangement theory and complex reaction rate theory can deal with these phenomena in a general way. The response to a stress decrease in the recovery-rearrangement theory depends importantly on x^. To some degree, the variation of x^ with temperature and strain is known from microstructural studies. Stress decrease experiments were therefore carried out in the temperature range 0.5 T - 0.85 T . An addi-m m 14 tional set of experiments involving the determination of low temperature stress-strain curves after creep strain was performed in order to expand the knowledge of changes in micros trueture and properties during creep. From these data, reversible flow stress ratios were also calculated. E X P E R I M E N T A L 2.1 EQUIPMENT AND MATERIALS . 2.1.1 Specimen Preparation. Specimens were made from high p u r i t y lead rod (99.999%). The rod was induction melted under vacuum. The b i l -l e t thus formed was machined and subsequently cleaned i n a sodium hydrox-ide s o l u t i o n to produce a 1" diam. blank s u i t a b l e f or extrusion. Back ex-trusion to 0.083" diameter was c a r r i e d out at room temperature at a pres-sure of approximately 38,000 p s i . Polyethylene g l y c o l was used as a lub-r i c a n t . The f i r s t two feet and l a s t eight feet of the extrusion product were discarded. The wire was annealed i n a i r at room temperature for one week to produce a stable grain s i z e of 0.6 mm. Specimens were then stored at 77°K. i n a specimen storage container u n t i l t e s t i n g . Grain s i z e was not found to change s i g n i f i c a n t l y during creep. 2.1.2 Equipment. Creep tests were conducted i n a constant s t r e s s -creep machine (Fig. 6) designed for a 5.0 cm gauge length. Specimens were held i n aluminum g r i p s . The lower grip was f i x e d r i g i d l y to the machine frame; the upper grip was attached to the p u l l rod by a pin which f i t t e d snugly permitting no v e r t i c a l movement. Specimen extension was detected by a magnetic f i e l d type trans-ducer. The i r o n core of the transducer was attached i n s e r i e s to the p u l l rod of the creep machine. Changes i n current r e s u l t i n g from core movement i n the magnetic f i e l d were fed to a Daytronic Model 70 LVDT. A continuous p l o t of specimen extension versus time was provided by a Heathkit Recorder. 15 16 F i g . 6. Schematic diagram of creep apparatus. 17 Specimens were loaded manually using a laboratory model screw jack. The load pan was aligned under the cam on the jack table before lowering the stage. With care i t was possible under most conditions to load the specimens r a p i d l y without undue shock to the system. 2.1.3 Temperature Control. The sample temperature was con-t r o l l e d by means of a s i l i c o n e o i l bath with magnetic s t i r r e r . I t was possible to maintain temperature within ±1 C° using an immersion heater i n conjunction with a YSI Model 71 temperature c o n t r o l l e r . A tempering s a l t pot without s t i r r i n g was used for the 0.8 T^ tests and for annealing the specimens in situ. Temperature was maintained within ±2 C° by a constant s e t t i n g of the power source. Manual adjust-ments were made i f necessary. In order to determine the low temperature s t r e s s - s t r a i n curves and the 77°K. y i e l d s t r e s s , specimens were quenched under load by removing the temperature bath and immersing the specimen d i r e c t l y i n l i q u i d nitrogen. An estimate of the time to cool the specimen can be obtained by observing the length of the pen trace on quenching. The i n i t i a l cooling rate was approximately 80 C°/sec. and the time to reach an approximately steady value was less than 30 sees. This estimate i s considered to be conservative because i t includes the e f f e c t of the machine frame. The system was allowed at l e a s t a further three minutes i n the l i q u i d n i t r o -gen before the s t r e s s - s t r a i n t e s t . 2.2 METHODS 2.2.1 Annealing In Situ. Specimens were usually deformed while 18 placing them in the creep machine. Such damage was removed by annealing the specimens in situ at 200°C. for 90 sees, before testing to provide uniform properties from, specimen to specimen. A minor load was placed on the specimen during the f i r s t 30 sees, to straighten the specimen. The sample was then air cooled to room temperature. 2.2.2 Low Temperature Stress-Strain Curves. The low tempera-ture stress-strain test for each specimen was carried out i n the creep machine after the sample had been deformed varying amounts into primary creep (e.g., 1%, 2%, 3%, and 4% total strain). This procedure was carried out for different creep stresses in the temperature range 0.51^-0.8T . m The stress-strain curve at 77°K. was determined by removing the creep load to establish a zero strain point, and adding lead shot to the load pan at regular intervals. The extension associated with each addi-tion was recorded. With this information a true stress-true strain curve was calculated and plotted. The 77°K. yield stress, was determined by the extrapolation of the elastic and plastic regions of the curve. Values of the 77°K. yield stress determined in this way differed only slightly from those obtained by a 0.2% offset method and were less l i k e -ly to be influenced by changes in the shape of the elastic-plastic trans-it i o n . The flow stresses determined in the creep machine were in good agreement with Instron tests performed at approximately the same strain rate. The stress-strain curves of four annealed specimens tested at 77°K. are shown in Fig. 7. Two were conducted in the creep machine by the method described and two were tested in an Instron machine. Differences in true s t r a i n (%) F i g . 7. Comparison of 77 PK. s t r e s s - s t r a i n curves of annealed specimens deter-mined i n Instron and creep machine. 20 the curves are ascribed to preloads associated with cooling the Instron specimens i n l i q u i d nitrogen. At t h i s low temperature C0.13T^) minor differences i n s t r a i n rate are not expected to a f f e c t the s t r e s s - s t r a i n curve to any appreciable degree. Load additions to the specimen i n a l l the creep machine tests were made at regular time i n t e r v a l s so that the s t r a i n rate among these samples was constant. 2.2.3 Stress Change Experiments. Stress decreases were made at i n t e r v a l s along the creep curve by manually applying the necessary loads to a counterweight arm on the creep machine. Stress decrements of 0.02o"c and 0.05c?c were performed at the temperatures 0.5, 0.6, 0.7, and 0.851^ -3 -1 at £ g =10 min . Another set of stress change experiments were con-ducted at a lower s t r a i n r ate, (e - 10 4 min at 0.7T . Stress s.s. m decreases of 2%, 5%, 10% and 15% of the creep stress were made. 2.2.4 Necking. Necking was found to be a problem i n the samples used. Consequently experiments were r e s t r i c t e d to t o t a l creep s t r a i n s of less than about 7%. Alden [ p r i v . comm.] and Weinberg [1968] using larger s i z e samples have experienced no d i f f i c u l t i e s up to 10% s t r a i n . Necking i n this work at such low s t r a i n s was a t t r i b u t e d to a s i z e e f f e c t . R E S U L T S 3.1 LOW TEMPERATURE YIELD STRESS MEASUREMENTS 3.1.1 Creep Curves. Creep curves depicting the p r e s t r a i n at a given stress and temperature have been plotted i n Fig s . 8-12. The creep curves exhibit the following features common to most creep tests [Chalmers 1959]: an "instantaneous" s t r a i n e^, c a l l e d here the loading s t r a i n (OA i n F i g . 9), which i s associated with the a p p l i c a t i o n of the creep s t r e s s ; a primary or transient creep region. (AB) of decreasing creep rate; and a "steady-state" creep where the creep rate assumes a nearly constant value CBC) . Examination of these curves reveals that the shape of the creep curve i s influenced by temperature and creep s t r e s s . At a given tempera-ture increasing the creep stress increases the loading s t r a i n and raises the creep rate at any given s t r a i n . The amount of primary creep at any temperature i s decreased by r a i s i n g the creep s t r e s s . 1 At low temperature and low s t r e s s , primary creep occupies a large part of the creep curve on a time scale, while at high temperature the extent of primary creep i s li m i t e d . • * 3.1.2i Low Temperature S t r e s s - S t r a i n Curves. The low temperature s t r e s s - s t r a i n curves obtained by deformation at 77°K. following creep are reproduced i n Fig s . 13-22. Y i e l d i n g i s gradual rather than abrupt with an e l a s t i c - p l a s t i c s t r a i n of about 0.5%, and a l i n e a r p l a s t i c region following The e f f e c t of temperature on the shape of the creep curve i s d i f f i c u l t to see. i n the res u l t s presented here. Changes i n the curve with temperature are more dramatic over wide temperature ranges [Garofalo 1965]. 21 23 24 (%) UTBjas ana^ 28 T I — ! ; 1 r 1.0 2.0 true s t r a i n (%) F i g . 14. 77°K. s t r e s s - s t r a i n curves following creep at 0.5T a = 602 p s i . 1 c r 29 1.0 2.0 true s t r a i n (%) F i g . 15. 77°K. s t r e s s - s t r a i n curves following creep at 0.5T , a = 602 p s i . m c V 6% prestrain O 9% p r e s t r a i n 1 . 0 2 . 0 true s t r a i n (%) 3 . 0 UJ o F i g . 16. 77°K. s t r e s s - s t r a i n curves following creep at 0.6T , a - 560 p s i , m c 31 ; • 3% p r e s t r a i n O 4% p r e s t r a i n I I i. 0 true s t r a i n (%) F i g . 17. 77°K, s t r e s s - s t r a i n curves following creep at 0.6T , a = 400 p s i . m 32 • • • • • • ..... . • 0 1.0 2.0 true s t r a i n (%) i g . 18. 77°K. s t r e s s - s t r a i n curves following creep at 0.6T a =, 320 p s i . 1 33 i I I i i Ol ' • * • • ' 0 1.0 2.0 true s t r a i n (%) F i g . 19. 77°K. s t r e s s - s t r a i n curves following creep at 0.7T a = 215 p s i . r 34 3 5 T 1 T " 37 y i e l d i n g . The flow stress i s higher a f t e r creep, r e l a t i v e to the annealed samples,and increases with increasing t o t a l s t r a i n (creep s t r a i n + load-ing s t r a i n ) . For a given t o t a l s t r a i n , the 77°K. s t r e s s - s t r a i n curve i s lowered at high temperatures. The extent of y i e l d i n g , i . e . , the e l a s t i c -p l a s t i c t r a n s i t i o n does not appear to change with temperature or t o t a l s t r a i n . S p e c i f i c a l l y , the t r a n s i t i o n i s not sharper following low temp-erature creep than a f t e r high temperature creep and i s unaffected by the amount of creep s t r a i n . To e s t a b l i s h the existence of such d i f f e r e n c e s , i f any, more c a r e f u l measurement of the s t r e s s - s t r a i n curve i n the neigh-bourhood of the e l a s t i c - p l a s t i c t r a n s i t i o n would be required. Some small v a r i a t i o n s i n the slope of the s t r e s s - s t r a i n curve occurred from specimen to specimen. This may be the r e s u l t of a cumula-tiv e e f f e c t of s t r a i n while recording data at high str e s s e s . Since the e f f e c t was small i n most cases, the 77°K. flow stress was always deter-mined by the i n t e r s e c t i o n of the extrapolated e l a s t i c and l i n e a r p l a s t i c parts of the s t r e s s - s t r a i n curve. 3.1.3 Low Temperature Y i e l d Stress Versus Total S t r a i n . Graphs of 77°K. y i e l d s t r e s s , a 0 versus t o t a l s t r a i n e (loading s t r a i n + / / K.. , t o t . creep s t r a i n ) are presented i n Fi g s . 23-27. A value of the low tempera-ture y i e l d stress was determined as described, from a s i n g l e specimen aft e r a given amount of t o t a l s t r a i n i n the creep machine. From the s t r e s s - s t r a i n curves ( F i g s . 13-22) a serie s of composite plots of the low temperature flow stress versus t o t a l s t r a i n were constructed. 38 800 I i i • L 0 2 4 6 8 t o t a l s t r a i n (%) F i g . 2 4 . 7 7 ° K . y i e l d stress versus t o t a l s t r a i n at 0 . 6 T . m 40 900 tn tn cn g 700 cn T J i H 0) • H Ni o 500 30(T O ' a = 400 p s i c A cr = 320 p s i c J L 2 .3 t o t a l s t r a i n (%) F i g . 25. 77°K. y i e l d stress versus t o t a l s t r a i n at 0.6T m 41 900 •rl cn a cn cn cu u cn T 3 rH O) •rl >> O 700 500 h 300 F i g . 26. 1 2 3 4 t o t a l s t r a i n (%) 77°K. y i e l d stress versus t o t a l s t r a i n at 0.7T m F i g . 27. 77°K. y i e l d stress versus t o t a l s t r a i n at 0. 43 The results show a large i n i t i a l increase in the 77°K. yield stress at small total strain. This is attributable to the loading strain. At high temperature where the creep stress is low the abrupt rise in the 77°K. yield stress is small because the loading strain is small (compare Figs. 23 and 27). The graphs also show a gradually i n -creasing low temperature yield stress with total strain during primary creep. Although there is only limited evidence, i t appears that the 77°K. yield stress, Oyy> is constant during steady state (Fig. 24). The effect of increasing creep stress at constant temperature is to raise the 77°K. yield stress at a given strain. 3.1.4 The Change in Low Temperature Yield Stress With Creep  Strain. The large i n i t i a l rise in the 77°K. yield stress, with total strain particularly at low temperature, is somewhat misleading because i t includes a large component of stress due to the loading strain. An attempt has been made to distinguish between the effect of loading strain and creep strain on the low temperature yield stress. This has been done by calculating the increase in the 77°K. yield stress above the modu-lus corrected creep stress, Ac , and plotting i t against the creep, strain (total strain - loading strain). The change in 77°K. yield stress ha a v equals // K. E77 : °77°K. " a c 17 The loading strain is d i f f i c u l t to measure accurately. Strain rates are so high at the beginning of creep that i t is d i f f i c u l t to dis-tinguish between the loading strain and the onset of creep. Furthermore 44 the loading s t r a i n i s not accurately p l o t t e d because tLe recorder response i s slower than the actual loading rate; hence estimates of trie loading s t r a i n have been used. These values were determined by assuming that load-ing i s indeed instantaneous and therefore athermal, so that the r e s u l t i n g s t r a i n i s given by the 77°K. s t r e s s - s t r a i n curve of an unstrained annealed specimen. The stress value employed i n this determination was the creep stress, modulus corrected to 77°K. In Table II a comparison of measured and estimated values of the loading s t r a i n has been made. Measured loading s t r a i n s were obtained by averaging the observed loading s t r a i n s from the chart traces. Agree-ment between measured and estimated values appears to be reasonably good considering the problems attendant i n obtaining both sets of numbers. The diffe r e n c e between the 77°K. y i e l d stress and the modulus corrected creep stress i d „ has been p l o t t e d versus creep s t r a i n c 77 . r creep i n F i g s . 28-31. The re s u l t s show a large i n i t i a l increase i n the low temperature y i e l d stress at low creep s t r a i n s and a gradually increasing Aa^y during primary creep. At any given temperature the high creep stress curve usually l i e s below the low creep stress curve. The curves suggest a roughly constant increase of the 77°K. y i e l d stress above the modulus corrected creep stress at a l l temperatures, so that the e f f e c t of hardening during creep comprises a much larger part of the t o t a l hardening at high temperatures than at low temperatures. TABLE II COMPARISON OF MEASURED AND ESTIMATED LOADING STRAINS T/T Stress e measured e estimated (psi) ° (%) ° (%) 0.5 68 2 6 0 2 1.6 1.2 1.4 1.1 0.6 560 4 0 0 3 2 0 1.5 0.80 0.40 1.1 0.60 0.37 0.7 2 1 5 1 6 0 0.35 0 . 1 5 0.20 0.08 0.8 1 2 0 9 5 0.10 0.0 0.04 0.01 46 F i g . -28. Change i n 77°K. y i e l d stress with creep s t r a i n at 0.5T . m %1 ' t — ( . . . . . . 1 500 A ^ — A —'o -400 a. A -(psi) 300 ^8 r~-r -O b 1 200 7°K. o a = c 400 p s i b 100 • A • cr = c 320 p s i • -0 l 2 creep (%) 3 . 29. Change i n 77°K. y i e l d stress with creep s t r a i n at F i g . 30. Change i n 77°K. y i e l d s t r e s s w i t h creep s t r a i n at 0.7T . m Fig. 31. Change in 77°K. yield stress with creep strain at 0.8T m 50 3.1.5 Modulus Corrected Reversible Flow Stress Ratio Versus  Creep S t r a i n . The modulus corrected i s o s t r u c t u r a l r e v e r s i b l e flow s t r e s s r a t i o s g T E 7 7 °77 Et [ C o t t r e l l and Stokes 1955] were calculated from the s t r e s s - s t r a i n curve data a f t e r creep. The r a t i o i s defined as the high temperature flow stress divided by the modulus corrected low temperature flow s t r e s s , evaluated at constant structure and s t r a i n rate. These values were plotted against the creep s t r a i n at which the sample was quenched and de-formed at 77°K. (Figs. 32-36). The data of Koster 1948 was used for the 2 modulus co r r e c t i o n . The importance of this flow stress r a t i o i s that i t gives a comparison of the strength of a material at constant structure at high and low temperature, and thus information about the nature of the obstacles to flow. Moreover, there i s comparable r e v e r s i b l e flow stress information a v a i l a b l e following t e n s i l e tests i n lead [Weinberg 1968] and other f . c . c . metals [ C o t t r e l l and Stokes 1955]. The r e v e r s i b l e flow stress r a t i o decreases with increasing creep s t r a i n ( F i g . 32) and appears to be constant during steady state (Fig. 33). At zero creep s t r a i n a r e v e r s i b l e flow stress r a t i o equal to one i s not observed as would be expected i f the loading s t r a i n were t r u l y instantaneous (and thus athermal) and i f the creep s t r a i n were p r e c i s e l y known. These conditions do not hold i n these experiments (See discussion), Shear Modulus and E l a s t i c Modulus are re l a t e d by a constant i f the material i s i s o t r o p i c so that the r a t i o of the moduli has the same temperature dependence. T T .500 0 2 4 £ (%) c r e e p F i g . 33. R e v e r s i b l e f l o w s t r e s s r a t i o v e r s u s creep s t r a i n a t 0.6T , a = 5 6 0 p s i m c K3 Cn 56 F i n a l l y , at any given temperature, a lower creep stress i s associated with a smaller r e v e r s i b l e flow stress r a t i o ( F i g . 36) at constant s t r a i n . 3.1.6 Reversible Flow Stress Ratio Versus Temperature. The e f f e c t of temperature on the r e v e r s i b l e flow s t r e s s r a t i o i s depicted i n F i g . 37. The r e v e r s i b l e flow stress r a t i o s i n the f i g u r e were determined -3 - i at a constant s t r a i n rate of 5.56 x 10 min . In the same p l o t , the data 3 of Weinberg [1968] for <100> s i n g l e c r y s t a l lead i s shown. His data apply - 2 - 1 at a s t r a i n r a t e of 1.02 x 10 min . Creep data from the present study are not r e l i a b l e at the s t r a i n rate used by Weinberg. The creep rate i s -2 ~1 changing too r a p i d l y i n the v i c i n i t y of e =10 min to allow accurate c o r r e l a t i o n of s t r a i n and s t r a i n r a t e for the determination of the rever-s i b l e flow stress r a t i o . Also the r e v e r s i b l e flow stress r a t i o data i s most inaccurate at these s t r a i n rates because they occur i n the region where specimen loading takes place. In most cases flow stress r a t i o s were not a v a i l a b l e at these high s t r a i n rates. The r e v e r s i b l e flow stress r a t i o versus temperature p l o t shows a continuously decreasing r a t i o with temperature. There i s no plateau evident near 0.5T i n contrast to the data of Weinberg and others [ C o t t r e l l and m Stokes 1955] . The r e v e r s i b l e flow stress r a t i o s l i e w e l l below those of Weinberg. - • ' -The degree of s c a t t e r i n the r e s u l t s may i n d i c a t e a s t r a i n de-pendence of the r a t i o although the r a t i o i s normally s t r a i n independent Weinberg found the r e s u l t s of p o l y c r y s t a l l i n e lead to be s i m i l a r to those of <100> s i n g l e c r y s t a l s . 57 0.90 0.70 o H W W o H D 0.5CT 0.30 0.4 1 r Weinberg Ov E=1.02 x 10~ 2min _ 1 <100> s i n g l e • c r y s t a l 0.6 0.8 1.0 T/T m Weinberg, 1968, Trans. A.I.M.E., 242, 2115. "measurements on large-g r a i n e d p o l y c r y s t a l l i n e specimens of 59 lead were s i m i l a r to r a t i o s obtained f o r <100> o r i e n t e d specimens." F i g . 37. R e v e r s i b l e flow s t r e s s r a t i o versus homologous temperature 58 [ C o t t r e l l and Stokes 1955] . Therefore, more probably the d i s p a r i t y r e s u l t s from inaccuracy i n i n t e r p r e t i n g the r e v e r s i b l e flow stress r a t i o s from the data i t s e l f . Error may r e s u l t at two steps i n the procedure: i n the corre-l a t i o n of the s t r a i n rate and s t r a i n from the creep curves, and i n the value loading s t r a i n used to c a l c u l a t e the creep s t r a i n . These two e f f e c t s could lead to an error of ±0.25 i n the r a t i o . It i s concluded that the observed r e s u l t s give a true i n d i c a t i o n of the temperature dependence of the rever-s i b l e flow stress r a t i o . 3.2 STRESS DECREASE EXPERIMENTS 3.2.1 Low S t r a i n Rate Tests, Measurements were made of the change in s t r a i n rate following a stress decrease - A number of experiments c were conducted a f t e r f i r s t e s t a b l i s h i n g a moderately f a s t steady state -4 -1 s t r a i n rate of 10 min. . Stress decrements of 2%, 5%, 10% and 15% of the creep stress a were used. The s t r a i n rate that ensues from the stress c decrease ought s t r i c t l y to be measured immediately a f t e r the t e s t . In these experiments this was impossible because at the high s t r a i n s e n s i t i v i t y used -4 -1 (5.07 x 10 i n . of chart) e r r a t i c extension versus time curves were r e -corded (Figs. 38-39). I t was concluded that the i r r e g u l a r i t i e s i n the ex-tension-time curve resulted from small temperature f l u c t u a t i o n s i n the o i l bath (Appendix B). To minimize this problem mean s t r a i n rates i were calculated following the stress decreases using d i f f e r e n t time periods up to two minutes a f t e r the creep stress was reduced. Values of these mean s t r a i n rates e have been plotted i n F i g . 40 against the time period over which T 1 ' T time (min) F i g . 38. C h a r t t r a c e o f 15% s t r e s s d e c r e a s e a t 0.7T . m F i g . 39. C h a r t t r a c e of 5% s t r e s s d e c r e a s e a t 0.7T . m T T o o A o O Aa = -0.02 a c A Aa = -0.05 a c • Aa = -0.10 a c O Aa = -0.15 a_ time (min) F i e . 40. Mean s t r a i n rate as a function of the time period of c a l c u l a t i o n at 0.7T m 61 the averaging was done. From these data a time period of 1.6 min f o r the c a l c u l a t i o n of the mean s t r a i n rate was chosen on the basis of two c r i -t e r i a : Cl) the largest stress decrement should produce the smallest mean s t r a i n rate; and C2) the time should be as short as possible so that the s t r a i n rate relates to the structure of the material at the time of the stress reduction. Stress change experiments were then c a r r i e d out at 0.71^ using stress decrements of 2 % , 5%, 10% and 15% of the creep s t r e s s . The mean s t r a i n rate, evaluated over the time period 1.6 min, was used to compute the response to stress changes. In F i g . 41 the r e s u l t s of repeated tests on four specimens are presented. The logarithm of the change i n s t r a i n rate, Ae = i - , ss 1.6 mxn, following a stress decrease i s plotted against the stress decrement - ~-. o c There appears to be a general increase i n the logarithm of the s t r a i n rate change with decreasing — ; however, there Is too much scatter f o r the r e -c sui t s to be meaningful. 3.2.2 High S t r a i n Rate Tests. The e f f e c t of temperature f l u c -tuations becomes less important at higher s t r a i n rates CFig. 42-43). When the creep stress i s reduced at these high s t r a i n rates the new s t r a i n rate can be measured d i r e c t l y . Stress decrease tests were performed throughout primary creep and into steady s t a t e over the temperature range 0.5T - 0.85T m m Stress decreases of 2% and 5% of the creep stress were made. Chart traces for a number of these /tests i n steady state are pre-sented i n Figs. 42-46. The nature of the response to a stress decrease changes with temperature. At 0,5T the s t r a i n rate f a l l s to zero when the 5 xlO 2x IO 0.05 0.10 0.15 -ha/a c F i g . 41. Mean s t r a i n r a t e change versus s t r e s s change at 0.7T . m 63 i 1 0.142 - > (in) / on o f f extension 0.141 0.140 0 0.25 0.50 time (min) F i g . 43. C h a r t t r a c e o f 2% s t r e s s d e c r e a s e a t 0.6T . m 0.123 Fig,. 44. Chart trace of a 5% s t r e s s decrease at 0.6T . m 0.114h 0.113P 0.112 0 0.1 0.2 time (min) 0.3 F i g . 45. Chart traces of 2% stress decrease at 0.7T . m 1 r " / — T — 0 .092 / Df f -0 091 / on -0. 090 | 0 0.1 time 0.2 (min) 0.3 Fig 46. Chart trace of 2% stress decrease at 0.85T . m 66 creep stress i s decreased. The s t r a i n rate also becomes zero f o r a time at - — =5% at 0.6T ( F i g . 44). At 0.7T and 0.85T a non-zero s t r a i n a . m m m c rate r e s u l t s when the creep rate i s diminished. I f the general s t r a i n r a t e equation (equation 13) of the rearrange ment theory i s d i f f e r e n t i a t e d with respect to s t r e s s , a simple equation i s obtained r e l a t i n g x to the stress decrease -Aa and the s t r a i n r a t e change v Ae, de 1 x - a —••! = ——- [b + r + r ]exp - [—^ " da 1 x 0 y v • x -x , x , T v y v v y x = & i (16) v de The information obtained from these high s t r a i n rate experiments (£• ~ -3 -1 10 min. ) was used to c a l c u l a t e x^ at d i f f e r e n t t o t a l s t r a i n s along the creep curve. The r e s u l t s are l i s t e d i n Table I I I . The value of x appears to be larger at 0.5T than at 0.85T v m m (for - — = 2% x i s 30 p s i at 0.5T and 4.0 p s i at 0.85T ). Also, x a v m m v c appears to be influenced more by the magnitude of the stress decrease than either of the other factors which are expected to strongly govern T v (temp-erature and s t r a i n ) . From examination of these r e s u l t s the value of r does v not change with s t r a i n during creep. This can be seen f o r example at 0.85T m with a stress decrease of 1.21 p s i . The value of x^ i s 3.10 p s i ea r l y i n creep and 2.90 p s i l a t e r i n steady s t a t e . This same pattern exists at other temperatures. TABLE III VALUES OF x CALCULATED ACCORDING TO EQUATION (16) T/T Ao p s i spec # z. . % z min. x p s i m tot v 0.5 30.2 1 2.76 5.49 x 10 33.1 3.79 5.51 x 10~ 4 30.2 . 4.42 4.40 x 1 0 - 4 30.2 5.20 2.89 x 1 0 _ 4 30.2 6.20 3.11 x I O - 4 30.2 0.5 84.5 2 2.66 1.15 x 10 84.5 5.63 5.45 x 10~ 4 84.5 6.16 . 4.09 x 10~ 4 84.5 0.5 84.5 3 3.95 6.03 x 10 84.5 4.23 8.69 x IO" 4 84.5 5.15 4.25 x 10~ 4 84.5 0.6 30.2 4 2.11 1.42 x 10 33.5 3.30 3.15 x 10~ 2 30.2 4.52 2.84 x 1 0 " 3 30.2 0.6 30.2 5 2.21 7.25 x 10 35.3 3.64 3.54 x 1 0 " 3 30.2 6.58 1.51. x 10~ 3 30.2 67 68 TABLE I I I (Con t inued ) T/T ACT psi spec # e. .% e min.^ " T psi m ^ r tot v 0.6 12.1 6 1.91 1.99 X i o "2 15.0 3.54 3.92 X i o "3 21.5 5.48 3.84 X -3 10 14.2 6.96 2.79 X i o "3 15.2 0.6 12.1 7 2.03 1.79 x 10~"2 16.2 3.60 4.50 x IO - 3. 19.9 5.77 3.83 x I O - 3 17.9 0.7 14.5 1.66 3.44 5.38 6.96 4.9 x 10 9.67 x 10 2.92 x 10 -3 -3 2.01 x 10 -3 16.7 17.3 27.4 19.2 0.7 14.5 9 3.25 1.97 x 10~2 16.8 6.88 7.31 x I O - 3 17.6 7.12 1.34 x I O - 3 18.7 0.7 14.5 10 2.41 5.47 x -2 10 15.3 3.93 1.30 x i o " 2 18.4 5.87 3.14 X i o "3 17.0 7.39 1.27 X H f3 16.3 0.7 6.54 11 2.21 1.23 X i o "2 •11.9 3.84 3.52 X i o "3 14.7 5.48 1.30 X 10 " 8.4 69 TABLE III (Continued) T/Tm Aa p s i spec // e . . % tot • e . -1 mxn. T p s i V 0.7 6 .54 12 2.71 4.08 5.68 1.50 x 10~ 2 6.44 x 10" 3 1.96 x 10~ 3 11.3 11.7 8.9 0.85 3.02 13 4.91 9.23 x 6.30 3.63 x 6.82 3.60 x 7.48 2.64 x 7.97 2.34 x 10" 3 4.05 10~ 3 4.78 10~ 3 5.22 I O - 3 7.03 10~ 3 5.03 4.08 7.99 x 10 3 3.10 4.76 4.17 x 10" 3 4.33 5.26 3.77 -3 x 10 2.44 5.92 3.06 x 10~ 3 2.15 6.67 2.66 x 10~ 3 2.82 7.37 2.70 x 10~ 3 2.90 D I S C U S S I O N 4.1 CREEP CURVES Both, the reaction rate theory and the rearrangement theory can explain the general shape of the creep curve i n primary and steady state. The theories are distinguished by the kind and s p a t i a l arrangement of the obstacles. In a reaction rate theory of the type proposed by C o t t r e l l and Aytekin [1950] and modified by Gasca-Neri et dl. [1970] the obstacles to d i s l o c a t i o n flow are thermally penetrable and r e g u l a r l y spaced. The d r i v i n g force for g l i d e i s the e f f e c t i v e stress (applied stress minus i n -ternal s t r e s s ) . Deformation occurs e a s i l y where the e f f e c t i v e stress com-prises a large part of the applied s t r e s s . The i n t e r n a l stress i s increased by s t r a i n hardening, which diminishes the e f f e c t i v e s t r e s s , and decreased by recovery which increases the e f f e c t i v e s t r e s s . In primary creep the s t r a i n rate f a l l s because s t r a i n harden-ing reduces the e f f e c t i v e stress and diminishes the s t r a i n rate according to the Arrhenius equation [equation ( 8 ) ] . The d r i v i n g force for recovery i s the i n t e r n a l s t r e s s , so that as deformation proceeds the recovery rate increases. Then, i n steady s t a t e the e f f e c t s of s t r a i n hardening and recovery are balanced so that the i n t e r n a l and e f f e c t i v e stresses do not change with time. The s t r a i n r a t e i s constant. In the rearrangement theory obstacles to flow are thermal and may be non-regularlv arrayed. The theory explains many of the creep pheno-mena by changes i n the non-regularity of the obstacle structure. 70 71 The r e s u l t s of Gupta and S t r u t t £1967] and Garofalo [1965] show, that obstacles are r e g u l a r l y spaced e a r l y i n creep and tend to become l e s s r e g u l a r as creep progresses. The s t r a i n r a t e f a l l s d uring creep according to equation 0-3) because the r a t e of change of-'x w i t h time r ^ i s expected to d i m i n i s h w i t h s t r a i n . One a l s o a n t i c i p a t e s that the r e l a t i v e area func-t i o n decreases w i t h s t r a i n due to s t r a i n hardening, which increases the x - a y i e l d s t r e n g t h x , and a l s o through x which may i n c r e a s e (A v In steady s t a t e the s t r u c t u r e does not change ap p r e c i a b l y so that a n e a r l y constant creep r a t e r e s u l t s . 4.2 LOW TEMPERATURE STRESS-STRAIN CURVES The experimental r e s u l t s do not agree w i t h the q u a l i t a t i v e p r e -d i c t i o n s of the r e a c t i o n r a t e theory. I t i s reasonable to assume that the obstacles to flow at high temperature are present i n roughly the same num-bers at low temperatures i f the quench i s e f f e c t i v e . I f then the o b s t a c l e s are r e g u l a r l y spaced, the low temperature s t r e s s - s t r a i n curve ought to show the same abrupt y i e l d i n g r e g a r d l e s s of the thermal-mechanical h i s t o r y of the specimen. Abrupt y i e l d i n g was not observed. The rearrangement theory i s more s p e c i f i c . Gradual y i e l d i n g i s ex-pected where the s t r u c t u r e i s non-regular and an abrupt e l a s t i c - p l a s t i c trans i t i o n r e s u l t s when the s t r u c t u r e i s r e g u l a r . Y i e l d i n g here was gradual, a l -though i t d i d not vary markedly w i t h p r i o r creep h i s t o r y . As discussed pre-v i o u s l y , t h i s may have r e s u l t e d from the l a c k of d e t a i l e d measurements i n the r e g i o n of the e l a s t i c - p l a s t i c t r a n s i t i o n . Yet, i t i s w e l l known, f o r example, that samples p r e s t r a i n e d at low homologous temperatures e x h i b i t 72 abrupt y i e l d i n g while annealed specimens y i e l d gradually CFlg. 47). strain Fig. 47. Effect of prestrain on yielding in stress-strain curve, Prior to testing i t had been anticipated that the low temperature stress-strain curves would show some variation in the nature of the yield-ing depending on the prestrain. The fact that yielding did not vary may be ax explained by the rate change of x^ with strain, 0^ 3e 't Strain tends to decrease the value of x . Evidently the precise shape of the stress-strain curve following creep cannot be predicted from the theory. 4.3 LOW; TEMPERATURE YIELD STRESS VERSUS TOTAL STRAIN These results are most effective In distinguishing between harden-ing due to loading and that due to actual creep. Large values of 77°K. yield 73 stress are obtained i n specimens after creep at 0.5T^ because the creep stress required to deform the specimen at this temperature was large. By contrast, creep at 0-ST^ required a low: creep stress and resulted i n a small value of the low temperature y i e l d stress. The increasing 77°K. y i e l d stress with s t r a i n i s compatible with both the reaction rate theory and the rearrangement theory. In the reaction rate theory the i n t e r n a l stress r i s e s during creep. On quenching to low temperature the long range i n t e r n a l stress remains constant. The increase i n 77°K. y i e l d stress i s primarily a r e f l e c t i o n of the r i s i n g athermal in t e r n a l stress during creep. In the rearrangement theory there i s an equation which describes the change In y i e l d strength with s t r a i n [equation ( 1 7 ) ] . dx r _ £ = 9 [ i _ i _ ] + ( 1 7) de y A e r dx Immediately after loading A^ = 1 and i s po s i t i v e . The rate change of y i e l d strength with s t r a i n w i l l be positive provided that decreases i n the value of the r e l a t i v e area function with s t r a i n are counteracted by changes i n r and e. v 4.4 CHANGE I N 77°K. YIELD STFESS WITH CREEP STRAIN The 77°K. y i e l d stress and the modulus corrected creep stress should be equal immediately after loading. The curves of Figs. 2 8 - 3 2 ought then to pass through the o r i g i n . The fact that they do not may demonstrate the d i f f i -culty of loading the specimen rapidly but without shock to the system. If the specimen i s loaded too rap i d l y , the sample i s overstrained by Ine r t i a effects 74 and an anomalously high 77 K. y i e l d stress r e s u l t s m a large o o v '' ' 77°K - cr . Nevertheless, i t i s thought that the 77°K. y i e l d stress ° T does indeed increase very r a p i d l y at small creep s t r a i n s because even at where the creep stress i s small a large value of 77 °K.. y i e l d stress i s observed. The e f f e c t , however, may be exaggerated by the load-ing problems. 4.5 REVERSIBLE FLOW STRESS RATIO 4.5.1 Reaction Rate Theory. In the reaction rate theory the tendency for the s t r a i n rate to decrease on lowering the temperature at constant structure i n the flow stress measurement must be o f f s e t by an increase i n the e f f e c t i v e stress [equation (8)]. Thus a flow stress r a t i o l e s s than one i s observed. As the creep temperature i s increased the e f f e c t i v e stress must also increase to compensate and therefore a f a l l i n g r a t i o with temperature i s achieved. Changes i n the flow stress r a t i o with s t r a i n and creep stress are governed by the e f f e c t of these variables on the e f f e c t i v e s t r e s s . These r e s u l t s cannot be explained without d e t a i l e d knowledge of the i n t e r -nal s t r e s s . 4.5.2 Rearrangement Theory. The magnitude of x^ i n r e l a t i o n to the y i e l d strength T y of a material dictates the l e v e l of the applied stress under any conditions and also the nature of the e l a s t i c - p l a s t i c t r a n s i t i o n i n the low temperature s t r e s s - s t r a i n curve Jequation (11)]. Generally the larger the value of x^ the greater the d i s p a r i t y between the flow stress and the y i e l d strength and the more gradual the y i e l d i n g . 75 Indirect m i c r o s t r u c t u r a l evidence on the r e g u l a r i t y of obstacle spacing i s u s e f u l i n explaining the behaviour of the r a t i o with, temperature, creep stress and.strain. Here, tKe f a l l i n g r e v e r s i b l e flow stress r a t i o with, increasing temperature i s explained by an Increasing value of x^ r e l a -t i v e to the y i e l d strength., which permits increasing amounts of s t r a i n at stresses below the y i e l d strength, as the temperature i s r a i s e d . A decreas-ing r a t i o during creep suggests that the creep stress and the y i e l d strength grow f a r t h e r apart. Such a r e s u l t may be explained by an increasing x with v s t r a i n . Raising the creep rate would be expected to increase the r a t i o by decreasing x^; the structure i s more regular. These q u a l i t a t i v e observations are i n agreement with the general behaviour of the microstructure with s t r e s s , s t r a i n and temperature. High temperature favours the formation of subgrains which tend to increase x^ [Garofalo 1965]. In this work the nature of y i e l d i n g i n the low temperature s t r e s s -s t r a i n curve did not change but only s h i f t e d along the stress a x i s . This implies that x^ may not be a strong v a r i a b l e except i n the cases mentioned previously. Alden,[1971c] has advanced an argument which admits a f a l l i n g flow stress r a t i o with temperature without large v a r i a t i o n In x . Assuming that a l l obstacles to flow are athermal, the point A i n F i g . 48 i s the e l a s t i c l i m i t and equals the modulus corrected creep s t r e s s . Point B corresponds to the extrapolated y i e l d strength of the material and AB i s a measure of x . The r e v e r s i b l e flow, stress r a t i o i s v given by A/B,. 76 CO s t r a i n F i g . 48. Relationship of y i e l d strength and creep stress to the L-T s t r e s s - s t r a i n curve. In F i g . 49 two hypothetical s t r e s s - s t r a i n curves with the same x are presented. A decrease i n r e v e r s i b l e flow stress r a t i o with increas-v ing temperature r e s u l t s even when T y - a I s constant because the y i e l d strength i s d r a s t i c a l l y lower a f t e r high temperature creep. s t r a i n F i g . 49. E f f e c t of creep temperature on L-T s t r e s s - s t r a i n curve. 77 4.5.3 Reversible Flow Stress Ratio Versus Temperature. In agreement with the data of Weinberg [1968] a f a l l i n g r e v e r s i b l e flow stress r a t i o with temperature i s observed ( F i g . 37). The r e s u l t s d i f f e r i n two respects from those of Weinberg: (1) no plateau i s recorded at 0.51^; and (2) the values of the r a t i o at a given temperature are much lower than those of Weinberg. The present r e s u l t s , while showing no plateau, are i n agreement with the s t r a i n rate s e n s i t i v i t y measurements of Weinberg which display a smoothly increasing "m" with temperature. Weinberg i n explaining t h i s con-t r a d i c t i o n , i n his own r e s u l t s suggested that the plateau was probably the r e s u l t of recovery. The present r e s u l t s i n F i g . 37 were determined at a s t r a i n rate -3 -1 of 5.56 x 10 min compared to Weinberg's which were obtained at i = 1.02 -2 -1 x 10 min . It i s f e l t that the diff e r e n c e i n the s t r a i n rate i s unable to account f o r the d i s p a r i t y i n the r e s u l t s . The ratio,determined at e = -2 -1 1.0 x 10 min at 0.61^ s t i l l l i e s w e l l below the Weinberg data ( F i g . 37). The lower flow stress r a t i o s measured here are a t t r i b u t e d to the effectiveness of the present technique i n eliminating recovery during the quench. Weinberg used a two stage process to quench his specimens; fan cooling to room temperature followed by a l i q u i d nitrogen quench. During the quench the crosshead was manually adjusted to maintain the stress at less than 0.250,, . In this work specimens were quenched under load d i r e c t l flow into l i q u i d nitrogen. Since the thermal mass was also small by comparison, quench times were less than 30 sees. 4.6 STRESS DECREASE EXPERIMENTS 4.6.1 Low S t r a i n Results. Stress change experiments appear to be 78 simple tests of the stress-strain rate response of metals at constant struc-ture and elevated temperatures. Such experiments, however, can be d i f f i -cult to conduct and the results a problem to analyse. Experiments per-formed at low strain rates are desirable because a larger number of tests can be conducted before an ins t a b i l i t y occurs; however, at low: strain rates even minor temperature fluctuations affect the creep curve (Figs. 38-39). The idea of using a mean strain rate was not successful i n obviating this problem. A high strain sensitivity was used when recording extension-time curves in order to accurately measure strain rates immediately following the stress decrease. The same sensitivity was used for a l l stress decrease tests to provide the same basis of comparison for a l l specimens and for a l l experimental conditions. The results of these experiments showed a period of accelerating creep rate after some stress changes. This means that the strain rate that is measured depends to a certain extent on the strain and time scale used in the tests. Strain rate changes in Instron machines do not appear to offer any advantages over constant stress creep machines. Weinberg 11968] observed complicated transients following fixed strain rate changes making the measurement of the stress change Aa uncertain. The stress decrease experiments at low strain rate i l l u s t r a t e the d i f f i c u l t i e s involved in making such measurements, and were unsuccess-f u l because temperature fluctuations made measurement of the new. strain rate uncertain. 4.6.2 High. Strain Rate Results. The results of the stress change -3 -1 experiments carried out at the higher strain rate .(e-„ ~ 10 min ) showed a 79 period of zero s t r a i n rate at 0.5T and at 0.6T for A.o = -0.05a . At m m c higher temperatures non-zero s t r a i n rates were observed when- the creep stress was decreased; however, even at the high temperatures, the s t r a i n rate f e l l to about 10% of the o r i g i n a l creep r a t e when the stress was decreased by only 2% to 5%. Such d r a s t i c changes i n s t r a i n r a t e are not i n agreement with a reaction rate or rearrangement model. In the recovery theory the s t r a i n rate during steady state i s determined by the recovery rate and the s t r a i n hardening c o e f f i c i e n t . y Aa 1_ At * 0, y At - -f-'^- as) £ 0 ss y A large s t r a i n rate and a small stress decrease favour a short period of zero s t r a i n rate. In these r e s u l t s the highest p r a c t i c a l steady s t a t e -4 -1 creep rate attainable at CL^T^ was about 3 x 10 min . At temperatures -3 -1 greater than 0*5^ the steady state creep rate was 3 x 10 min . A com-parison of experimentally observed recovery times At ( F i g . 42) and times calculated according to equation (18), presented i n Table IV, demonstrates that recovery times were probably not observed at the higher temperatures because they were of too short duration to be detected by the recorder."*" The r e s u l t s of stress decrease experiments are i n best agreement with a "^"Calculated recovery times i n the table disagree with measured values by about an order of magnitude. Equation (J.8) does not pr e d i c t exact quantitative agreement unless 0 i s about 10 p s l . 80 TABLE IV A COMPARISON OF CALCULATED AND MEASURED RECOVERY TIMES r l ss (psi) ,calc. (mm) ,meas. (minj 0.5 3.11 x 10~ 4 30.24 2.31 0.35 0.6 3.84 x 10" 3 12.07 0.08 0.0 0.7 3.52 x 10~ 3 6.54 0.04 0.0 0.85 3.06 x 10~ 3 1.21 0.01 0.0 0.5 4.09 X i o " 4 84.53 4.91 0.6 0.6 3.54 X i o " 3 30.23 0.20 0.04 0.7 3.14 X i o " 3 14.49 0.11 0.0 0.85 3.60 X -3 10 3.02 0.02 0.0 simple recovery model. 4.6.3 Behaviour of x . Values of x calculated according to _ v v o equation (16) remained very close to the value of the stress decrement Aa regardless of the temperature or s t r a i n the sample experienced, x v which was expected to increase with, temperature a c t u a l l y appeared to decrease. The .most s t r i k i n g observation one notes i s that the values of x are extremely small, e.g. at 0.8T the value of x i s about 5 p s i . where v m v the y i e l d stress of the. material i s probably about 700 p s i . Values of x are calculated to be small because the s t r a i n rate drops so d r a s t i c a l l y when the creep stress i s reduced. This makes £/Ac i n equation 0.6) nearly one 81 2 and l i t t l e variation in is registered. On these grounds the.theory cannot be discounted, but equation. (16) may not accurately describe the stress-strain rate relationship in f . c c . metals. 2 The possibility of a spontaneous contraction when reducing the creep stress i s not incorporated i n the theory and may have an effect on the observed strain rate. S U M M A R Y Current theories of p l a s t i c deformation at elevated temperatures make c e r t a i n q u a l i t a t i v e predictions about the low temperature s t r e s s -s t r a i n curves a f t e r creep and about the shape of the creep curve on chang-ing the creep s t r e s s . The f i r s t part of t h i s study was concerned with the measurement of the low temperature s t r e s s e s t r a i n curves of specimens deformed i n t o p r i -mary creep i n the temperature range 0>5Tm to 0.85T m. The 77°K. s t r e s s -s t r a i n curves were determined i n the creep machine by noting the extension associated with successive load additions to the deformed specimen. The 77°K. y i e l d stress was found to increase during primary creep and to decrease strongly with increasing temperature of p r e s t r a i n . Reversible flow stress r a t i o s decreased with increasing temperature i n agreement with e i t h e r a reac-t i o n rate or rearrangement model. Part two of the thesis dealt with the s t r a i n response of the s p e c i -men to decreases i n creep stress and measurement of the parameter x i n the v rearrangement theory. Samples were deformed varying amounts into primary and steady state where the creep stress was decreased by 2% to 5%. The nature of the s t r a i n response to the stress decrease was noted. In a d d i t i o n , x^ was calculated according to equation (16)• The shape of the creep curve following the stress decrease agreed most c l o s e l y w i t h the simple recovery model. Values of x .were very small and not In general agreement with the rearrangement theory; the p o s s i b i l i t y of a spon-taneous contraction on reducing the, creep .stress was not ruled out. 82 C O N C L U S I O N S 1. The low temperature s t r e s s - s t r a i n curve a f t e r creep exhibited gradual y i e l d i n g and a l i n e a r p l a s t i c region over the f i r s t 3% s t r a i n . A reaction theory cannot account for gradual y i e l d i n g while the rearrange-ment model incorporates this feature of the stress s t r a i n curve as an i n -t r i n s i c part of the theory. The nature of the y i e l d i n g was unaffected by thermal-mechanical h i s t o r y . 2. The r e v e r s i b l e flow stress r a t i o at constant s t r a i n rate was found to decrease l i n e a r l y with temperature over the temperature region 0.5T - 0.8T . Both a reaction rate model and a rearrangement model can ex-m m ° p l a i n t h i s r e s u l t . 3. The s t r a i n rate i n steady state was observed to f a l l d r a s t i -c a l l y l a response to only minor decreases i n the creep s t r e s s . The s t r a i n response to a stress decrease agreed most c l o s e l y with the simple recovery theory. 4. Values of calculated according to the rearrangement theory were generally very small and did not agree vrith the q u a l i t a t i v e predictions of the theory regarding s t r a i n and p a r t i c u l a r l y temperature. 83 APPENDIX A DETERMINATION OF THE REVERSIBLE FLOW STRESS RATIO The r e v e r s i b l e flow stress r a t i o determined by the usual s t r e s s -s t r a i n method i s the r a t i o of the flow stress at high temperature to the flow stress at low temperature (A/B i n F i g . 50). The specimen i s strained into the p l a s t i c region where i t i s quenched and the low temperature flow stress i s measured. s t r a i n F i g . 50. Schematic of a s t r e s s - s t r a i n curve showing a r e v e r s i b l e flow stress measurement (T n > T 0 ) . In these creep experiments the creep stress was used i n place of the high temperature flow stress i n the c a l c u l a t i o n of the r e v e r s i b l e flow stress r a t i o . 84 APPENDIX B THE EFFECT OF MINOR TEMPERATURE FLUCTUATIONS ON THE OBSERVED CREEP RATE Minor changes i n temperature were recorded i n the temperature bath ( F i g . 51). 1 5 sec H time F i g . 51. Temperature f l u c t u a t i o n s i n the temperature bath at 0.7T . m The s t r a i n rate r e s u l t i n g from one of the most severe temperature fluctuations was calculated and found to be greater at times than the steady state creep rate i n the low s t r a i n rate experiments. change i n temperature c . , . s t r a i n rate = — — f : — • x coefr. of thermal expansion change i n time 1C° 60 sec. _ Q _ . .-6 "J" z - . — x 29.0 x 10 -^5-0.6 sec. min. C = 2.90 x lO^minT1 85 B I B L I O G R A P H Y Alden, T.H.,1971a, submitted to P h i l . Mag. Alden, T.H.,1971b, submitted to S c r i p t a M e t . Alden, T.H.,1971c, submitted to Met. Trans. Becker, R., 1925, Z. Phyzik, 26, 919. Chalmers, B., 1959, Physical Metallurgy. Wiley, New York. C o t t r e l l , A.H., and Aytekin, V., 1950, J. Inst, of Metals, 77, 389. C o t t r e l l , A.H., and Stokes, R.J. , 1955, Proc. Roy. S o c , 233A, 17. Evans, W.J., and Wilshire, B., 1968, Trans A.I.M.E., 242. 2514. Garofalo, F., 1965, Fundamentals of Creep and Creep-Rupture i n Metals, MacMillan, New York. Gasca-Neri, R. , Ahlquist, C.N.', and Nix, W.D. , 1970, Acta Met. 18, 655. Gasca-Neri, R. , Ahlquist, C.N. , and Nix, W.D., 1971, S c r i p t a M e t . , 5_, 733. Gupta, V.P., and S t r u t t , P.R., 1967, Can. J. Phys., 45, 1213. Kocks, V.F., 1966, P h i l . Mag., 13, 541. Koster, W., 1948, Z. Metallk., 39, 1. Livingston, J.D., 1962, Acta Met., 10, 229. Mitra, S.K., and McLean, D., 1967, Mat. S c i . J., 1, 192. Orowan, E., 1946-7, J.W. Scot. Iron and Steel Inst., J54, 45. Weinberg, F., 1968, Trans. A.I.M.E., 242, 2111. 86 

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