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The effects of size, temperature and strain-rate on the mechanical properties of face-centered cubic… Costanzo, Ronald Albert Joseph 1961

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THE EFFECTS OF SIZE, TEMPERATURE AND STRAIN-RATE ON THE MECHANICAL PROPERTIES OF FACE-CENTERED.CUBIC METALS ' by '. RONALD ALBERT JOSEPH COSTANZO A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE DEPARTMENT OF MINING AND METALLURGY We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF APPLIED SCIENCE Members of the Department of Mining and Metallurgy THE UNIVERSITY OF BRITISH COLUMBIA October 1961 ! In presenting 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 requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree 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 reference and study. I f u r t h e r agree that permission f o r extensive copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of 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 r k w e d without my w r i t t e n permission. Department of / / t ^ s ^ <? AV£4MXO*SL; 7f The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3, Canada. Date ABSTRACT Drawn and annealed copper wires of diameters ranging from 50 u to 900 }i were t e s t e d i n t e n s i o n and the r e s u l t s examined f o r evidence of s i z e - e f f e c t s . No s i z e - e f f e c t on y i e l d - s t r e s s or work-hardening r a t e has been d e f i n i t e l y e s t a b l i s h e d . The r e s u l t s were discussed i n terms of the f r a c t i o n of the number of grains i n the specimen which have a f r e e surface. The u l t i m a t e t e n s i l e s t r e n g t h and d u c t i l i t y decrease w i t h decreasing d i a -meter f o r diameters below 200 yu. An ex p l a n a t i o n has been put forward i n terms of void-formation during deformation. P o l y c r y s t a l s and s i n g l e c r y s t a l s of copper were t e s t e d a t room -1 -1 ~3 -1 temperature w i t h the s t r a i n - r a t e c y c l e d between 10 min and 10 mm P o l y c r y s t a l l i n e copper obeys the C o t t r e l l - S t o k e s law but shows a v a r i a t i o n i n the r a t i o of f l o w - s t r e s s e s w i t h v a r y i n g g r a i n diameter and w i t h a va r y i n g value of the f r a c t i o n of grains i n the specimen which show a f r e e surface. Copper specimens were a l s o t e s t e d w i t h the temperature c y c l e d between rj8QK and 293°K- Copper p o l y c r y s t a l s do.not obey the C o t t r e l l - S t o k e s law, the d e v i a t i o n depending on the g r a i n s i z e . These r e s u l t s are discussed i n terms of s t a c k i n g - f a u l t energy and s e v e r a l p o s s i b l e explanations are considered. Aluminum p o l y c r y s t a l s were t e s t e d w i t h the temperature c y c l e d between 78 K and 293 K. Aluminum obeys the C o t t r e l l - S t o k e s law f o r temperature v a r i a t i o n . A work-softening e f f e c t accompanies the y i e l d -drop found a t 293°K a f t e r p r i o r deformation at 78°K. This was discussed i n terms of c r o s s - s l i p and d i s l o c a t i o n climb mechanisms. v i i . ACKNOWLEDGEMENT The author i s g r a t e f u l f o r the advice and encouragement given by h i s research d i r e c t o r , Dr. E. Teghtsoonian. Thanks are extended t o Mr. R. G. B u t t e r s f o r t e c h n i c a l a s s i s t a n c e and t o f e l l o w graduate students, e s p e c i a l l y Mr. K. G. Davis, f o r many h e l p f u l d i s c u s s i o n s . S p e c i a l thanks are extended t o Miss I . Duthie who a s s i s t e d i n many of the r o u t i n e c a l c u l a t i o n s . This work was fi n a n c e d by Defence Research Board Grant No. 7 5 1 0 - 2 9 . i i . TABLE OF CONTENTS Page INTRODUCTION , 1 PREVIOUS WORK 3 1. Si z e E f f e c t s 3 a. C r y s t a l P e r f e c t i o n k-b. E f f e c t of Surface Conditions 7 c. E f f e c t s of V a r i a t i o n i n the Length of the S l i p - P l a n e 10 d. G r i p E f f e c t s 12 e. Size E f f e c t s i n P o l y c r y s t a l l i n e Wires 15 2. C o t t r e l l - S t o k e s Law . • . 15 3- Comments 20 EXPERIMENTAL PROCEDURE 22 1. M a t e r i a l s 22 2. Specimen P r e p a r a t i o n . . . 22 a. Copper Wires . 22 b. Copper Rods 23 c. Copper S i n g l e C r y s t a l s 23 d. Copper S t r i p 25 e. Aluminum S t r i p . . . 25 3- Measurements v 25 k. T e s t i n g Procedure 26 a. Normal T e n s i l e Tests 26 b. V a r i a b l e Strain-Rate Tests ' 26 c. V a r i a b l e Temperature Tests . . . 28 EXPERIMENTAL RESULTS AND OBSERVATIONS ' . . 29 1. Size E f f e c t s 29 a. Y i e l d Stress 29 b. Ul t i m a t e T e n s i l e S t r e s s 30 i i i . TABLE OF CONTENTS CONTINUED .Page c. Ductility . . . . . . 3 0 &. Work-Hardening Rate 3 2 2 . Cottrell-Stokes Law 3 8 a. Effect of Strain-Rate on Flow Stress 3 8 b. Effect of Temperature on Flow Stresses ~.kQ c. Results of Tests on Aluminum . . . • 5 8 SUMMARY OF THE RESULTS 6 3 DISCUSSION .' 6 5 1 . Size Effects 6 5 a. Yield Stress and Work-Hardening Rate 6 5 b. Ductility and Ultimate Tensile Strength 6 8 2 . Cottrell-Stokes Law . ' . - TO a. Work-Hardening . . . . . 7 1 b. Theoretical Considerations 73 c. A Possible Effect of Dislocation Density . . . . . . 7 6 d. Grain Boundaries 7 8 e. The Effect of Stacking-Fault Energy . . . . . . . . 8 l f. A Possible Effect of Dislocation Density on 6 5 ' . . 8 5 3. Work-Softening in Aluminum 8 7 SUMMARY AND CONCLUSIONS 9 1 SUGGESTED FUTURE WORK 92 BIBLIOGRAPHY 93 APPENDICES, .95 iv. FIGURES Page 1. Tensile Strengths of Silicon Rods and Whiskers 5 2. Schematic Representations of a Stress-Strain Curve for a Face-Centered Cubic Single Crystal 5 3- Strength Versus the Inverse of the Diameter for (a) Copper and (b) Iron Whiskers : • 6 k. Variation of the Strength of Silicon Whiskers with Temperature 6 5- Effect of Diameter on the Crit ical Shear Stress of Cadmium Single Crystals 9 6. Effect of an Oxide Layer on the Stress-Strain Curve for Single Crystals of Silver . . . . . . . 9 7- Variation of Flow Stress with Diameter for Copper Whiskers . . I**-8. Ratio of Flow Stresses at 293°K'and 90°K for Aluminum Single Crystals 1 Q -.5 > -K -1 9- Ratio of Flow Stresses at Strain Rates of 10 and 10 sec for Copper and Aluminum • 18 10. Ratio of Flow Stresses at 293°K and 78°K for Copper Polycrystals 19 11.. Electrolytic Cell and Power Source Used in Electropolishing Copper Wires 2k 12. Gripping System for Low-Temperature Tests on (a) Wire and Rod Specimens and (b) Strip Specimens 27 13> Yield Stress of Polycrystalline Copper Wires as a Function of the Diameter ? . . 33 lk. Ultimate Tensile Strength of Polycrystalline Copper Wires as a Function of the Diameter . . . . • . 3* 15. Ductility of Polycrystalline Copper Wires as a Function of the Diameter 35 16. Work-Hardening Rate of Polycrystalline Copper Wires as a Function of the Diameter . . . . . . . . . . . . . . . . 36 17. Ratio of Flow Stresses for Specimens 1322 and IB26 ; . . . . . . ^1 18. Ratio of Flow Stresses for Specimens IF 12, 1F16, 113, and 1113 . h r 2 19-. Ratio of Flow Stresses for Specimens 2B1, RC3, and RGk . . . . ^3 20. Ratio of Flow Stresses for Specimen SX2 ^3 FIGURES CONTINUED Page 21. Figures 17 to 20 Plotted Together for Comparison hh 22. Stress-Strain Curve for Specimen 1B22 ' U5 23. Stress-Strain Curve for Specimen RC3 • • • h6 2k. Stress-Strain Curve for Specimen SX2 . . • k-7 25. Ratio of Flow Stresses for. T = 293°K, T = 78°K for Polycrystal-line Copper . . . . . . . . . . . . . . . . . 51 26. Ratio of Flow Stresses for T-^ 293°K, T2= 78°K for Single and Polycrystals of Copper . . . 52 27. Stress Strain Curve for Specimen 1B2U '. 53 2'8. Stress-Strain Curve for Specimen RC6 5^ 29. Stress-Strain Curve for Specimen RC6 after the appearance of a visible neck . . . . . . 55 30. Stress-Strain Curve for Specimen SX3, Part 1 56 31. Stress-Strain Curve, for Specimen SX3, Part 2 57 o o 32. Ratio of Flow-Stresses for T-^ 293 K, T '= 78 K for Aluminum Polycrystals. .......... ^ ..... . 59 33- Ratio of Flow-Stresses for T ^ - 2 9 3 % Tg= 78°K for Strip Copper 60 3^. Stress-Strain Curve for Aluminum Specimen SA2 . . . . . . . . 6l 35- The Work-Softening Ratio as a Function of Strain for Various €' 62 36. Variation of the'Ratio B/A with Specimen Diameter . . . . . . 66 37- The Separation of the Flow Stress into the Elastic and Temp-erature-Dependent Components . 7^ 38. Representation of a Glide Dislocation Approaching a Row of Forest Dislocations 77 39* Stress-Concentration Factor in Front of a Dislocation Pile-up of Length L . 80 kO. A Schematic Representation of the Form pf the Stress-Strain Cugve, for Deformation at 293°K after Prior Deformation at 78 K . . . . . . . . . . ; . . . . . . 90 hi. Schematic Representation of the Load-Elongation Plot from the Instron Recorder 97 k2. Log-Log Plots of the Stress-Strain Curves of Copper Polycrystals Under Various Test Conditions 100 v i . .TABLES Page I. Ratio of Flow Stresses for T^ = 293°K, T2='78°K for Various Metal's . . . . . . . . . . . . . . . . . 17 II. Characteristics of Specimens Tested for Size Effects . 31 III. Characteristics of Specimens Tested for the Effect of Strain-Rate on Flow Stress . . 1+0 IV. Characteristics of Specimens Tested for the Effect of Temperature on Flow Stress 50 V. Stacking-Fault Energies and Cottrell-Stokes Ratios for Cu, Ag, Al and Mg 82 VI. Results of Tensile Tests on Polycrystalline Copper Wires 103 VII. Effect of Strain-Rate and Temperature on the Flow-Stress of Polycrystalline Copper ' 106 VIII. Effect of Temperature on the Flow-Stress of Polycrystalline Aluminum ' i l l INTRODUCTION Interest in the effect of diameter on the strength of crystals has, in the last ten years, grown out of whisker research. Investigations of the mechanical properties of whiskers have been quite extensive and have produced some interesting results, the one of main interest here being that the strength of a whisker depends in some manner on its diameter. Interest in whiskers developed in the early 1950's when Compton, Mendizza and Arnold"'" discovered whiskers growing in electroplated capacitors. , 2 Gait and Herring performed the first tests, a simple bend test, on tin whiskers and found a high elastic limit (as high as 2$). Interest in the effect of crystal size on the strength of metals has been aroused by Brenner's discovery that the strength of a whisker is a function of its diameter. Some investigations of the variation of strength with diameter have been carried out on single crystals produced from bulk material but have produced no firm explanation for the diameter dependence. Also, virtually a l l of the work has been'.;done using 'single crystals and the work on polycrystals has not produced particularly good results. Another line of investigation concerning the mechanism of work-hardening in face-center cubic crystals has been pursued since the early 1950's. These investigations have centered around the effects of temp-erature and strain rate on the flow stress of such metals as copper, aluminum and silver. The usual technique used for comparison of properties 1). at different strain rates and temperatures is that originated by Hollomon in 19^6. At this time the existence of a mechanical equation of state relating the flow stress to the strain, strain rate and temperature and - 2 -the exact form that i t would take was under investigation. The mechanical equation of state demands that where & = true stress C = true strain c = true strain rate T = temperature i .e . the flow stress is a function of the instantaneous values of the strain, strain rate and temperature, independent of the thermal and mechanical history of the metal. 5 In 19^8, Dorn, Goldberg and Tietz , using this technique, showed that the mechanical equation of state is not valid but that the • flow stress also depends on the thermal and mechanical history of the 6 metal. It was later realized by Cottrell and Stokes that the results of this type of test demonstrated that the difference in flow stress when a metal is deformed to a given strain at different temperatures arises from two distinct contributions: 1. The difference arising from the variation of dislocation configuration due to strain at different temperatures, and 2 . The "reversible" contribution of thermal fluctuations. The type of test mentioned show only the "reversible" part of the dependence and therefore may be used to study the form of the inter-actions between dislocations.. As the work in both of-these fields has been confined almost exclusively to single crystals, this investigation wil l be concerned with size-effects and with the temperature and strain rate dependence of the flow stress in copper polycrystals. - 3 -PREVIOUS WORK !.• Size Effects When crystals of diameter below a cr i t ical value are deformed, variations in the properties of those crystals are found. These variations, since they are a function of the diameter, are usually termed "size effects". The problem of diameter-dependence of these properties has been attacked from two directions; from tests on whiskers of various diameters and from tests on crystals produced from bulk material by various methods of diameter reduction. The f irst report of a size effect in metals was that by Taylor^ in 192U. Taylor produced fine filaments of various metals with diameters -k as small as 10 cm. by drawing out molten metal in a glass tube. Although no determination of the microstructure of these wires is mentioned, i t may be assumed that they are of a bamboo structure since the final product was formed by solidification along the length of the wire. It was found that these wires had a very high ductility: "Though britt le in bulk, bismuth and antimony are very pliable in the form of fine wires.- Antimony wire as large as 0.003 cm. diameter may be bent repeatedly without breaking". / 2 The tensile strength of antimony wire was reported to be 1800 to 2200 Kg/cm and that of bismuth about 50 Kg/cm . In comparison with this, the tensile 8 / 2 strength of bulk antimony is about 110 Kg/cm . Because bismuth is so britt le in bulk form,' no tensile strength has been reported. Taylor gave no details of testing method. No real interest in this phenomenon, however, developed until the advent of whisker research. Size effects in the strength of whiskers have been found by Brenner^ in copper and iron, and by Gyulai^ in NaCl whiskers. Single crystals of copper"^' and aluminum"^ prepared from bulk material have shown a size effect and a comparison of the strengths of silicon whiskers 13 and rods prepared from bulk crystals has shown that they are similar for similar diamters (Figure ! . )• The parameters normally measured in investigations of size-effects in face-centered cubic single crystals'are 6* - ^ 7 ^ 7 £ ^ y g ^ and 0£ which are defined in Figure 2. Generally, the following effects have been observed with decreasing diameter: 1. cfpcT' 2 increase 2. £"2 increases 3. © 1 7 Q 2 decrease A number of theories, put forward to explain these effects, wil l be discussed below. a. Crystal Perfection Brenner's results-^ for copper and iron whiskers (Figure 3-) show . that the strength is proportional to the inverse of the diameter. If, as has been postulated, the high strength of whiskers were due to the small volume resulting in a low statistical chance of the inclusion of a dis-location source, then i t would be perhaps more reasonable to expect that the strength should be dependent on the inverse of the square of the diameter, as has been found by Eder and Meyer1**' and tentatively confirmed by Moore" .^ Further, assuming that the strength depends only on the volume of the whisker and taking Eder and Meyer's results to be true, then there should also be a length dependence such that the strength is proportional to the inverse of the length. A computation of the expected length depend-ence from Moore's results on diameter dependence shows that this length dependence should be experimentally observable. No such length dependence 15 has been found 1 6 Also, according to Cottrell , the yield stress of a perfect crystal should be insensitive to temperature. The results on silicon 13 . 0 whiskers show that the yield stress at 650 C is more than double the - 5 A o 0 R O D S o W H I S K E R S - • J . . » 0 0 x 1 1 . C R O S S S E C T I O N IN S O U A R E C E N T I M E T E R S Figure 1. Tensile Strengths of Silicon Rods and Whiskers. Reproduced from Reference 12. Stress JBlope= Qx -^"TJlope^eT 4 i i 1 i i Strain Figure 2. Schematic Representation of a Stress-Strain Curve for a Face-Centered Cubic Single Crystal Showing the Parameters Mentioned in the Text. Figure 3.. Strength Versus the Inverse of the Diameter for: (a) Copper and ("b) Iron Whiskers. Reproduced from Reference 3» h f ,/ / N f RAC TURE / .A C X / " C \ \ \ \ V D0°C \ — \ 13,600 N . _i • —• 0 0.002 0.004 0.006 0.008 0.01 0.02 0.0-i 0.06 ^ - « E U . S T , C , Figure k. Variation of the Strength of Silicon Whiskers with Temperature. Reproduced from Reference 12. yield stress at 800°C (Figure k.). It was also found that small rods prepared from bulk material have the same room-temperature fracture stress as whiskers but have the same yield stress as bulk silicon crystals at 800°C. - 13 Pearson et al also found that prior deformation of.a silicon whisker at 800°C such that a dislocation density of about 10® cm~^  was introduced, has l i t t l e effect on the room-temperature fracture stress. This- also indicates that, in this case, the dislocation density has l i t t l e effect on the fracture stress. b. Effect of Surface Conditions Possibly the most reasonable source of size effects is in the surface effects. The ratio of surface area to volume, being proportional to the inverse of-;. the diameter, increases with decreasing diameter and surface effects would become more dominant. One possibility is that the strength of the oxide layer on the surface of the crystal would become more effective in smaller crystals. In a crystal of radius r, and oxide thickness > "the load at the yield point may be expressed as: fTrVy = rT{r - S f ^  + 2ff r Sol where 6^ = yield stress g^ ,= yield stress of the bulk material 6s = effective strength of the oxide layer at the yield point. Solving this equation for (f^ gives the relation: 6X = ^ + S ( £ - <v ) — V where A = 2 7 f r l V = 7 T r 2 l - 8 -12 Using this relation and = h6 gm/mm^ , Fleischer and Chalmers calculated f> 2 5^= 3(10 ) gm/mm for aluminum oxide. This value compares favourably with the value of 2-5-3 -0 (10 ) gm/mm found by Takamura from measurements on specimens with varying oxide thickness. 18 Roscoe reported that an oxide layer on the surface of cadmium crystals increases the strength and that this effect increased with increasing oxide thickness and with decreasing diameter. In contrast to this, Andrade"^ reported that Makin found that, although cadmium shows an increased strength with the addition of an oxide layer, the size effects for an oxide-coated crystal and a clean crystal do not differ appreciably (Figure 5.). This theory would also lead one' to expect, in the case of a ductile oxide, that the oxide layer should have an effect on the rest 20 of the stress-strain curve. Andrade and;Henderson found that an oxide layer on the surface of a single crystal of silver not only gave a small increase in the yield stress, but also that i t greatly increased the rate of work-hardening and the ultimate tensile stress (Figure 6 . ) . A similar effect was observed on copper crystals plated with.nickel-chromium"^. 21 Kramer and Demer tested aluminum crystals in an electrolytic cel l , electropolishing the crystals during the test. They found that, as the rate of metal removal was increased, £ 2 increased, 9-^  and 0£ decreased and 6^ 2 remained constant. , 2 2 Harper and Cottrell found that the cri t ical shear stress for single crystals of zinc was reduced by electropolishing and increased by •3 23 exposure to steam. Tests on copper whiskers-^' J , however, show no effect on the strength from an oxide layer. - 9 -0 10 . 20 30 40 50 g l i d e ( % ) Figure 6. Effect of an Oxide Layer on the Stress Strain Curve for Single Crystals of Silver. Reproduced from Reference 20. 21 I t has been suggested that the e f f e c t of an oxide l a y e r on the p l a s t i c r e g i o n of the s t r e s s - s t r a i n curve i s due to i m p u r i t y atoms being c a r r i e d t o the i n t e r i o r of the c r y s t a l by r a p i d l y t r a v e l l i n g d i s -l o c a t i o n s . This s u p p o s i t i o n i s supported by the r e s u l t s of C o t t r e l l and. 22 Harper who found that the e f f e c t of an oxide l a y e r on the p l a s t i c r e g ion diminished w i t h decreasing s t r a i n r a t e , being almost zero at very low s t r a i n r a t e s . Observations on the e f f e c t of removing the 24 surface l a y e r on the creep r a t e of cadmium s i n g l e c r y s t a l s , however, c o n t r a d i c t t h i s view. Another p o s s i b l e e x p l a n a t i o n f o r the e f f e c t of surface f i l m s and of diameter on the stre n g t h of s i n g l e c r y s t a l s i s th a t the oxide l a y e r i n h i b i t s the egress of d i s l o c a t i o n s from the c r y s t a l . This theory 25 i s supported by the observations of B a r r e t t on z i n c and m i l d s t e e l coated w i t h an oxide l a y e r . The c r y s t a l s were t w i s t e d p l a s t i c a l l y and the r a t e of u n t w i s t i n g on r e l a x i n g the loa d was measured. Soon a f t e r the l o a d was r e l a x e d , the oxide l a y e r was removed w i t h an etchant, upon which the c r y s t a l showed a s l i g h t t w i s t , contrary to the expected e f f e c t , i n the d i r e c t i o n of the o r i g i n a l deformation before resuming the u n t w i s t i n g a c t i o n . I f the theory were t r u e , then i t would be expected that the e f f e c t of an oxide f i l m on a f i n e - g r a i n e d p o l y c r y s t a l l i n e m a t e r i a l should be small and t h a t i t should increase w i t h i n c r e a s i n g g r a i n s i z e . 24,26 These c o n d i t i o n s have been reported by Andrade and R a n d a l l and by 29 Andrade and Kennedy y . c. E f f e c t s of V a r i a t i o n i n the Length of the S l i p - P l a n e The f o r e - g o i n g i n d i c a t e s t h a t the l e n g t h of the s l i p - p l a n e , measured i n the s l i p - d i r e c t i o n , could have an e f f e c t on the st r e n g t h of the c r y s t a l , an e f f e c t which can be e a s i l y demonstrated. A Frank-Read - 11 -source in a slip-plane generates dislocation loops under an applied stress. If the surface of the crystal provides a barrier strong enough to hold these loops inside the crystal, pile-ups wil l be formed. Since the stress at the head of the dislocation pile-up is proportional to the number of dislocations in the pile-up and to .the applied stress, in a large crystal, dislocations wil l break through the barrier at a lower applied stress than in a small crystal. Experimental results concerning this effect are contradictory. 12 Fleischer and Chalmers tested crystals of two orientations; one (S^ .) in which the length of"the slip-plane remained constant with crystal size and the other'(S^) in which i t varied with size. The results indicate that the size dependence of the yield stress is the same for both crystals and that the yield stress at any one diameter is the same for both orient-27 ations. Lipsett and King , in an investigation of the influence of a film of gold on the surface of single crystals of cadmium, found that the increase in cr i t ical resolved shear stress was independent of orientation. 28 In contrast to these two results, Gilman and Read found that the influence of a film of copper plated onto zinc single crystals varied with the orient-ation of the crystals, increasing as the length of the slip plane decreased. 10 It has also been suggested, by Suzuki et al that decreasing the length of the slip plane should result in a lower work-hardening rate and 12 an increase in the range of easy glide. Fleischer and Chalmers found that for crystals of orientation S , 9^  increases, 9^ remains constant and C 2 decreases with decreasing size. In contrast to this, for crystals of orientation S T T , 9 and 9 decrease and £ 0 increases with decreasing -L-L 1 2 <-size. The results of Gilman2^, who found a higher work-hardening rate for zinc single crystals with longer slip planes, agree with this result. - 12 -This effect could be attributed to the fact that dislocations have a shorter distance to travel before escaping from the surface of the crystal and, therefore, there is a smaller probability of their encountering obstacles. This would mean that a Frank-Read source would be able to produce more dislocation loops before the back-stress caused by dislocation pile-ups is high enough to produce hardening. The resolved shear stress required to operate a Frank-Read source may be expressed as: length of the pinned dislocation segment. This would indicate that, since the length 1 would be limited by the size of the slip plane, that smaller crystals should require a larger stress for slip. Brenner , however, found that i f the size effect were to be explained on this limited to the order of 10 cm. Saimoto has offered a mechanism whereby sources of this length might be introduced into a whisker but there is some question as to whether a source of this length could operate. d. Grip Effects 32 Fleischer and Chalmers , considering the effect of grip restraints on the stress-strain curve for a single crystal, have shown that the resolved shear stress on a crystal plane ( T"j) is the sum of two components; one due to the applied stress ( ^ ) and one due to the bending moment arising from the grip restraints ( Ty), and that the resultant stress on the primary system can be expressed as: ? = M b 1 where is the shear modulus, b the Burgers vector and 1 the where m. cos^C cosAp, the Schmid factor for the primary slip system a = crystal diameter L = crystal length X = the angle between the slip direction and the P specimen axis, = the angle between the slip-plane-normal and the specimen axis. This indicates that the actual stress on the primary slip-plane is lower than the measured stress resulting in a higher apparent stress, the difference depending on the ratio a/L for any given orientation and strain. The effect on the yield stress would be negligible since, at this point, the strain is very small. If the above equation is modified to take into account slip on a secondary system, the resulting expression indicates that the rate of hardening should decrease in the easy glide region and that the range of easy glide should increase with decreasing a. The authors calculated that, on this basis, a ratio of a/L which is small compared with l/lOO is required to remove the influence of grip restraints from the 12 stress-strain curve. When the work on size effects by the same authors 10 arid that by Suzuki et al was examined in this light, i t was found that the ratio was greater than the cri t ical value calculated and that the size effects were accounted for by the calculated "2^ . It would be interesting to examine the results of the diameter dependence of the flow stress of whiskers in the light of the above theory. The diameters of whiskers tested usually vary between about k / i to 20 ju and the lengths from 0-5 mm to 5 mm, with no apparent relation between diameter and lens th!5. These figures give upper and lower limits of l/25 and l/lOOO respectively to the ratio a/L. Under these circumstances, i f the variation in flow stress with diameter were due to grip effects, a statistical scatter would be expected rather than a clear- dependence as was found by Brenner^ and by Eder and Meyer"'"(Figure 7-)-- Ik Figure 7- Variation of Flow Stress with Diameter for Copper Whiskers. (Brenner and Eder and Meyer). Reproduced from Reference 15-15 e. Size Effects in Polycrystalline Wires The foregoing discussion has been concerned with single crystals only and most of the theories presented are applicable only to single crystals. The one suggestion which might be applied to polycrystals is the strengthening effect of an oxide layer.- However, as mentioned before, there is no effect from an oxide layer on polycrystalline wires until the grain diameter is of the order of the specimen diameter. 33 Shlichta tentatively reported that the results of preliminary tests on fine Taylor-process and electropolished drawn wires indicate that there is an increase in strength and a decrease in plastic deformation 3U before fracture with decreasing diameter. Later , however, he concluded that this effect was not a true size-effect but represented a "survival 35 of the fittest". McDanels, Jech and Weeton found a size effect in polycrystalline tungsten wires. In this investigation, composites of tungsten wires in a copper matrix were tested in tension. The results showed that the composites containing finer wires have a higher strength. 36 A Defence Metals Information Memorandum lists the properties of some, high-strength fine wires of various metals including high-carbon and stainless steels, nickel-base alloysy tungsten and molybdenum with diameters ranging from 0.0005" to 0.01". In general, the yield stress, ultimate stress and the total elongation increased with decreasing size. The one exception noted was a nickel base alloy which showed decreasing ductility with decreasing diameter. 2. Cottrell-Stokes Law When two specimens of a metal are deformed at the same strain rate but at two different temperatures, the stress>-strain curves diverge with increasing strain. Cottrell and Stokes^ have pointed out that the - 16 5 results of Dorn et al indicate that the difference in the two stresses at any given strain is probably due to two contributions: 1. The density and distribution of dislocations may be different in metals deformed at different temperatures to the same strain, 2. There may also be a "reversible" effect due to thermal fluctuations, an effect which wil l decrease with decreasing temperature. In .their investigations of this phenomenon in single crystals of aluminum, they also noted that the ratio of the flow stresses at. any two temperatures Is constant over the entire range of strain after an in i t ia l few percent deformation (Figure 8.). This effect has also been noted by other invest-igators working with different metals and a summary of the results is given in Table I. 37 Basinski - noted that the same effect is found i f the ratio of the flow-stresses is taken at one temperature but with two different strain rates. His results for aluminum and copper at various temperatures are shown in Figure 9- The only large deviation from Cottrell-Stokes behaviour is that found by Davis who found the ratio of flow stresses in cobalt single crystals varies with increasing strain. This was found for both variations in temperature and in strain rate. A l l of these results have been obtained, with one exception;,: by the same method, one which wil l be outlined later in this work. The one 39 exception is the work of Sylwestrowicz on polycrystalline copper who obtained each point from a different specimen. The result was that the flow stress ratio was not quite constant but varied between about O.87 to O.85 (Figure 10). - 17 -TABLE I. Ratio of Flow'Stresses Metal 4i Vf2 Reference Strain Rate Poly-crystal Cu 0.86* 5 Wot Given Poly-crystal Cu O.87-O.85 39 Not Given Single-drystal Cu 0.88 ko lO'^sec"1 Single-crystal Cu 0.88** 37 l O ^ s e c - 1 Single-crystal Al 0.79+ 6 10~5sec_1 Single and Poly-crystal Al O.78**, 37 -k - l 10 sec -1 Single-crystal Ag 0.86** 37 10"^sec~1 In a l l cases except that marked + , T = 293°K and T2= 78-K. The value marked + is for T1= 293°K, T = 90°K. The value marked * is deduced from a single experiment. The values marked ** were originally presented with reference to T2= k.2°K- These were normalized- to T2= 78°K by the method of 6 Cottrell and Stokes whereby: 6^293 . 6*\%2. = 6~293 k.2 °~~ 78 ^ 78 pcroentago e longaf ion Figure 8. Ratio of Flow Stresses at 293°K and 90°K for Aluminum Single Crystals. Reproduced from Reference 6. 9C5} . i i i -^•x 0 o.o-%fe° 0_^-Temile men 'q /m Figure 9. Ratio of Flow Stresses at Strain Rates of 10"5 and 10"^ " sec"1 for Copper and Aluminum. Reproduced from Reference 7--19 -Figure 10. Ratio of Flow Stresses at 293 K and T8°K for Copper Polycrystals. Reproduced from Reference 39-- 2.0 Sever a l t h e o r i e s have been advanced to e x p l a i n the constancy of the r a t i o of f l o w s t r e s s e s . C o t t r e l l and Adams^ have suggested t h a t kl t h i s constancy i m p l i e s t h a t , from Seeger's theory of work hardening , the d i s l o c a t i o n p a t t e r n should remain constant d u r i n g work-hardening and only the o v e r a l l d e n s i t y should change; i . e . the p r o p o r t i o n a l i t y between the d e n s i t y of f o r e s t d i s l o c a t i o n s , the d e n s i t y of jogs on screw d i s -l o c a t i o n s and the d e n s i t y of the i n t e r n a l s t r e s s e s i n the metal must hp remain constant. The theory of Seeger et, .al s t a t e s t h a t the i n t e r n a l stress, f i e l d i n a metal depends on the d i s l o c a t i o n d e n s i t y i n the same manner as the f l o w s t r e s s r e s u l t i n g from i n t e r a c t i o n s of g l i d e d i s l o c a t i o n s w i t h f o r e s t d i s l o c a t i o n s . This would imply that the hardening from these two sources should be p r o p o r t i o n a l and the r a t i o s o f the two f l o w s t r e s s e s should be constant. ' S Mott, as reported by C o t t r e l l and Stokes , has suggested t h a t "the important e l a s t i c f o r c e s on a g l i d e d i s l o c a t i o n are those from i t s immediate neighbours r a t h e r than the long-range s t r e s s e s from p i l e d up groups, f o r the e l a s t i c f o r c e s and.the f o r e s t f o r c e s a c t i n g on the d i s l o c a t i o n can both be a s c r i b e d t o the same t h i n g , the presence of nearby d i s l o c a t i o n s ! ' B a s i n s k i ^ i d e n t i f i e d the source of the e l a s t i c s t r e s s e s w i t h the f o r e s t d i s l o c a t i o n s . 3• Comments The b r i e f survey of the i n v e s t i g a t i o n s i n t o s i z e e f f e c t s given above shows t h a t , although the v a r i a t i o n of p r o p e r t i e s w i t h diameter i n s i n g l e c r y s t a l s has been w e l l e s t a b l i s h e d , the t h e o r i e s presented to account f o r t h i s v a r i a t i o n are c o n f l i c t i n g and there i s n o v u n i f i e d theory to e x p l a i n a l l s i z e e f f e c t s . Since almost a l l of the t h e o r i e s apply only to s i n g l e c r y s t a l s , any s i z e e f f e c t found i n a p o l y c r y s t a l could not be a t t r i b u t e d to any of them. I n v e s t i g a t i o n s of the C o t t r e l l - S t o k e s law have a l s o been p r i m a r i l y concerned w i t h s i n g l e c r y s t a l s . The work of Sylwestrowicz 36 on p o l y c r y s t a l s of copper., when compared t o the r e s u l t s of B a s i n s k i i n d i c a t e s t h a t there may be an anomalous e f f e c t i n copper.. This i n v e s t i g a t i o n of s i z e e f f e c t s and of the e f f e c t s of s t r a i n - r a t e and temperature on the f l o w s t r e s s i s an attempt t o c l a r i f y these two p o i n t s . - . 2 2 EXPERIMENTAL PROCEDURE 1 . Materials The copper used i n t h i s investigation was supplied by Johnson Matthey and Company and by the American Smelting and Refining Company. The aluminum used was supplied by the Aluminum Company of Canada, Ltd. A l l copper was of 99-999+$ purity.and the aluminum of 99'99+$ purity. 2 . Specimen Preparation a. Copper Wires The copper wires were prepared from Johnson-Matthey copper rods 5 mm diameter by 1 5 cm long. The wires were drawn through hardened steel dies to 0 . 1 1 5 " diameter, then through tungsten-carbide dies to O.OOV diameter, and f i n a l l y through diamond dies to a minimum diameter of 0 . 0 0 2 " . Each size was drawn from f u l l y annealed wire according to the schedule shown below: I n i t i a l F i n a l No. of Drawing Diameter ; Diameter • Steps ; 5 . mm 1 mm 3 0 2 . 5 0 . 5 3 2 1 0 . 2 3 2 0 - 5 0 . 1 ' 1 5 0 . 2 5 0 . 0 5 2 1 After drawing to the desired diameter, wire of diameter over 0-5-mm was cleaned, coiled and stored f o r future use. Wire of less than 0 - 5 mm diameter, was- immediately strung on glass U-frames and prepared f o r testing. Since a crack was found i n the middle of each rod received, random specimens of each size were sectioned and examined for i n t e r i o r cracks. No evidence of cracking was found. Specimens about 8 cm long were cut from the drawn wire and electropolished i n an ele c t r o l y t e consisting of 1 5 0 ml H^PO^, 7 5 ml each of - 23 lactic and propionic acids and 30 ml each of HgSO^  and water. A cel l potential of 7-5 volts with the specimen suspended horizontally in the cell shown in Figure 11 produced a good polish and an even, -circular cross-section. This procedure removed, a l l surface irregularities, any oxide layer accumulated during storage and the more heavily distorted surface layer, h^e specimens were then annealed in a vacuum of less o than 10-- mm of mercury for 2 hours at 250 C. Metallographic examination - 2 showed that this procedure produced a grain size of about 10 mm diameter, 6 2 an average of 10 grains per cm , and that the grain size was uniform across the cross-section. The different sizes of wire were also subjected to X-ray analysis before and after annealing to determine i f there were any wire texture•remaining. No such texture was found in the annealed wires and the as-drawn wire was found to have the normal <^ 110^  - <^L1]^ fibre texture. b. Copper Rods . Larger copper tensile specimens were machined from 10 mm diameter rod supplied by the American Smelting and Refining Company. The rods were f irst given several passes through a rolling mill in order to weld shut the intergranular cracks present in the as-received material. An approximately circular cross-section was maintained by giving the rod several passes at each reduction, turning the rod slightly between passes. Tensile specimens with a gauge length of k mm diameter were machined from the resulting rod. These were electropolished as before and then vacuum annealed at various times and temperatures to produce desired' grain sizes. Coarse grained rods were also produced by slow cooling from the melt. c. Copper Single Crystals A l l single-crystal specimens were cut from one single-crystal Figure 11. E l e c t r o l y t i c C e l l and Power Source Used i n E l e c t r o p o l i s h i n g Copper Wires. The Specimens were Suspended H o r i z o n t a l l y i n the E l e c t r o l y t e from the Frame Shown. rod about 9 mm diameter by cm long made'by the melt-solidification technique. A copper rod of the proper dimensions was cast and fitted into a hole drilled through a graphite rod. Graphite caps, the upper one .containing a reservoir of copper, were fitted over each end and the whole enclosed in a stainless-steel tube. The tube was lowered at a1 rate of 10 cm per hour through a furnace maintained at a temperature of li+00°C. d. Copper Strip The copper strip specimen was produced from 8 mm diameter Johnson-Matthey copper rod. The rod was rolled out into strips 0.015'' thick by 0.375" wide. Specimens were stamped out on the specimen punch and die described elsewhere*^, producing a gauge length 0.20" wide hy 0.75" long. These were then electropolished and annealed for o two hours at 250 C, giving a grain diameter of about 20 p.. e. Aluminum Strip The aluminum strip specimens were produced by rolling 0-5" thick bil lets of aluminum into strips 0-375" wide by O.O36" thick. Specimens were stamped from the strip as described previously. These were chemically polished for five minutes in Alcoa bright dip and annealed for 30 minutes at 450°C. 3- Measurements Diameters of 500 yx and over were measured using a Gaertner travelling microscope accurate to 1 jx. Diameters below 500 jx were measured using a Reichert microscope equipped with a Leitz micrometer eyepiece accurate to about O.h p.. From each specimen, two sets of diameter measurements were taken at right angles to each other. If the two sets of measurements agreed, a circular cross-section was - 26 . assumed. If not, the specimen was discarded. A l l diameter measurements were taken before annealing to minimize effects from specimen damage. A l l gauge length measurements were made, using the Gaertner travelling telescope, with the specimen mounted in the testing machine.. 4. Testing Procedure A l l tests, were carried out in tension on an Instron Tensile Tester equipped with an autographic recorder which produces, a load-elongation curve on a strip chart. a. Normal Tensile Tests The copper wires were tested in four batches at various strain rates and at two different temperatures. Wires of diameters 900yu, 5 0 0jx, 2 0 0 p, 1 0 0 p, and 5 0 p were tested at room temperature. For these tests, the wires were soldered to brass tabs which were held in the normal Instron vice-grips. Wires of diameters 9 0 0 y u , 5 0 0 p., o 2 0 0 p. and 1 0 0 p. were tested at 7 ° K. These wires were soldered into kk brass grips attached to a special modification to the Instron tensile tester (Figure 1 2 . ) . A l l low temperature tests were carried out in a liquid nitrogen bath contained in a wide-mouth Dewar flask. b. Variable Strain-Rate Tests Tests on the effect of varying the strain-rate on the flow stress were carried out on copper wires of,diameters 9 0 0 j x , 2 0 0 p and 5 0 ^ u and on copper rods, both single crystal and polycrystal, of diameter about h mm. The tests were done by alternating between two--strain-rates with a ratio of lOO/l. Changes in the strain-rate were accomplished by changing the gear ratio of the cross-head drive. Since the Instron,. is equipped with a gear-shift lever, this could be done almost instantaneously. (a) (*> Figure 12. Gripping System for Low-Temperature Tests on (a) Wire and Rod Specimens, and (b) Strip Specimens - 2 8 -c. Variable Temperature Tests Tests on the effect of varying temperature on the flow-stress were carried out on wires of diameters 900 Ja and 200 ji and on single and polycrystal rods of 4- mm diameter. These tests were also done on copper and aluminum strip specimens. The copper wire and rod specimens were, as before^soldered into brass grips, while the strip specimens were held 45 44 in special file-face grips screwed into a universal arrangement (Figure 12 b.). To avoid grip effects, the single crystal specimens were also mounted in the universal arrangement. The testing procedure was as follows: a. The, .specimen was deformed for a short time at room temperature, _o b. The temperature was reduced to 7 ° K and the specimen given another short deformation. c. The temperature was brought to room temperature again and the specimen given another short deformation. This procedure was repeated until the specimen fractured. Between tests, while the specimen was cooling or heating, a small load was kept on the specimen to maintain alignment of the loading system. After the specimen was immersed in the liquid nitrogen bath, and a l l but the surface boiling had ceased, five minutes was allowed in order that thermal stability could be reached. Following the low temperature extension, the liquid nitrogen bath was removed and a stream of warm air from a hair dryer was played over the specimen and grips. After a l l frost had disappeared, ten minutes was allowed to ensure temperature stability. - 29 -EXPERIMENTAL RESULTS AND OBSERVATIONS  1. Size Effects The effects of specimen diameter on the tensile properties of polycrystalline copper wires were measured at room temperature at strain-rates of 0.02 min"1 and 0.8 min"1 and at 78°K at strain rates of 0.0^ min"' and 0.2 min - 1 . The parameters measured directly from the stress-strain curve were yield stress, ultimate tensile stress, and the total elongation before fracture. In addition to these, a log-log plot of the stress-strain curve shows that, in a l l cases, the curves may be represented by . . b v an expression of the form log b = a + b l o g £ { 6~ = A£ ; a = log AJ. In some cases (see Figure ^.in Appendix I) the log-log plot shows two straight lines, each of which conforms to an expression of the type given. The parameters a and b were determined for each specimen by picking a number of points from the . stress-strain curve and flitting them to the equation by the method of least squares as is outlined in Appendix I. The above data are tabulated in Appendix II. The characteristics of the various specimens tested are shown in Table II. « a. Yield Stress The yield stress at the strain-rate of 0.8 min ^ was not measured since the autographic recorder cannot follow the load for the in i t ia l portion of the test at this rate. The stresses for 0.1$-offset (YSQ ^) and 0-5$ offset (YSQ ^ ) were measured for a l l other specimens. Since the 0.5$ offset gave a lower scatter in the yield stress and since i t is the value of offset recommended for copper^, the 0.1$ offset values were not plotted. They are, however, tabulated in Appendix,II. The values of yield stress obtained are plotted as a function of diameter in Figure 13. for both temperatures; 78 K and 293 K... The values of YSn for tests.at 78°K show no significant difference for the . 30 _ two strain-rates used so they are plotted together and considered as one set of points. The yield stress at 293°K shows no size effect, the averages of YS . fluctuating about an average value, of about 670 Kg/cm^. • . . . 0 . 5 At 78°K, however, the yield-stress decreases from a maximum of about IkOO Kg/cm^ at a diameter Of 500 jx to about 1000 Kg/cm2 at a diameter of 100 jx. In addition to this, i t was found that the yield stress at a diameter of 900p. is'about 1200 Kg/cm2, again lower than that at a diameter of 500 yu. -This may be an indication of a scatter higher than .that observed experimentally arid, may detract from the significance of this plot. b. Ultimate Tensile Stress Since, in every low-temperature test, the wire specimen pulled out of the solder before fracture, no ultimate tensile strengths are available for these tests. The ultimate tensile strengths for the room-temperature tests are plotted as a function of specimen diameter in Figure Ik. . The specimens strained at 0.02 min - 1 show as essentially constant ultimate stress of about 3100 Kg/cm down to a diameter of 200yu. At this point, the stress" drops off sharply with decreased diameter. The.data for specimens tested at the strain-rate of 0.8 min "^  show a very wide scatter and no trend can be determined from these. c. Ductility The values of total elongation before fracture are tabulated in Appendix II. and plotted as a function of specimen diameter in Figure 15. For the reasons outlined above, no values for the low-temperature tests are available. The data plotted in Figure 15. show that, at a.strain rate of 0.02 min""'", the ductility is essentially constant at about 38% for - 3 1 -TABLE II. Characteristics of Specimens.Tested for Size Effects Specimen Diameter A B B/A 9 0 0 /a 8 1 0 0 2 3 0 0.03 5 0 0 2 0 0 0 1 6 0 0.08 2 0 0 300 6 0 0.2 1 0 0 8 0 30 OA 50 25 15 0.6 A l l specimens have a grain diameter of 0.01 mm. A =• the number of crystals in the cross-section. B = the number of crystals in the cross-section which impinge on the surface of the specimen. - 32 -diameters between 900 p- and 500 p and thereafter appears to decrease with decreasing diameter. At the higher strain-rate (0.8 min - 1 ) , the ductility decreases approximately linearly with decreasing diameter from about 28$ at a diameter of 900 p to about YJ'fo at a diameter of 100 p. d. Work-Hardening Rate The parameters a and b in the expression log = a + b log £ are tabulated in Appendix II. The values of a are of l i t t l e significance since the log-log plots of the stress-strain curves l ie very close to-gether, making a essentially a function of b rather than an independent parameter. For the specimens showing two straight line segments, two values were calculated for each parameter (a-j_, b-]_, &2, ^2'^" However, only the parameter b^ is plotted. The values of b-^  plotted as a function of specimen diameter in Figure l6. are a measure of the rate of-, work-hardening of the specimen. Under-all test conditions, the scatter in these values of b-^  is too large to permit any definite expression of b^ as a function of diameter. Vague trends however, do show in the plots. The same holds true for the values of a-^  which are not shown. The trends observed for decreasing diameter are outlined below. Temperature Strain Rate aj_ b-^  decreases increases increases decreases decreases increases 293°K 0.02 min 293°K 0.8 " 78°K - 33 I500f I ikoo\ 1300 OJ §" 1200 bO 6 CQ 3 -noot .0) •H * 1000\ T= 78°K € = 0.05 min"1 • £ = 0.2 min"1 0 T= 293°K 0.02 min"1 O Arithmetic Averages $ 700 o 8 600 9 e 9> o o § 9 200 1+00 600 Diameter (microns) 800 Figure 13- Yield Stress of Polycrystalline Copper Wires as a Function of the Diameter - 3^ Diameter (microns) Figure 1^. Ultimate Tensile Strength of Polycrystalline Copper Wires as a Function of Diameter - 35 -ko 200 ' 400 5oo Boo Diameter (microns) Figure 15. Ductility of Polycrystalline Copper Wires as a • Function of Diameter 0.45 0 . 4 0 - 3 5 " 0 (a) T= 2 9 3 ° K O £= 0 . 0 2 min" 1 0 0 0 8 0 0 0 O 0 ooco 8 - 0 1 8 1 1 1 9 3> 0 -0 O 8 0 0 8 O O O -0 0 0 0 1 0 O (b) T = 2 9 3 ° K O £ = 0 . 8 min"1 1 1 1 2 0 0 400 6 0 0 800 Diameter (microns) Figure l 6 . (continued on page 3 6 . ) .o.k (c) T = 78°K o £ = 0.2 min"1 £ = 0.05 min"1 o o o $ o o o o o o /# o o o o £ _ o 1 • 1 200 i+oo 6oo 800 Diameter (microns) Figure 16. Work-Hardening Rate (b-, ) of Polycrystalline Copper Wires as a Function of the Diameter. - 38 -2. Cottrell-Stokes Law The data obtained from the investigation of the effect of temp-erature and strain-rate on the flow stress of copper and aluminum are tabulated in Appendix III. Included are the ratios of flow stresses for a ratio of strain-rates of 1:100 for various copper specimens and the ratios of flow, stresses for copper .and aluminum for temperatures 7" K o and 293 K. The data, displayed in Figures 17 to 20, 25 and 26, show that polycrystalline copper follows the Cottrell-Stokes law for strain-rate changes but not for temperature changes. Also, i t demonstrates that polycrystalline aluminum and copper single crystals do follow., the Cottrell-Stokes law and that the ratio of flow stresses in these cases is approximately that described in the literature (see Table I). Stress-strain curves for representative specimens of copper and aluminum are displayed in Figures 22, 23, 24, 27, 28, 29, and 30. . a. Effect of Strain-Rate on Flow Stress Copper wire and rod specimens with the characteristics listed in Table III were strained at strain rates of 10 1 min 1 and 10 ^min - 1 at a temperature of 293°K. A l l of these specimens follow the Cottrell-Stokes law as is shown in Figures 17 to 20. The ratio appears to decrease in the single crystal specimen (Figure 2l) after about 20$ deformation. Since the single crystal was damaged during preparation and no other was available for testing, no significance is attributed to this decrease. The ratios for the various specimens are plotted together for comparison in Figure 21. which shows that the ratio of flow stresses varies from O.965 for specimens 1B22 and 11B26 to O.985 for the single crystal SX2. The valuer.found for the single crystal is lower than that found by 37 Basinski (Figure 9-)J but this is for a strain-rate ratio of 1:100 whereas Basinski used a strain-rate ratio of 1:10. Also, Basinski has - 39 -shown-..that the ratio decreases as the temperature increases and the temperature used in this investigation was much .higher than that used by Basinski. Portions of the..stress-strain curves for specimens 1B22, RC3, and SX2 are shown in Figures 22, 23, and 2k. The fine-grained polycrystalline specimen (LB22) shows neither the low slope plastic flow region following yield at the high strain-rate nor serrations in the stress-strain curve. The stress-strain curves for both the coarse-grained specimen (RC3) and the single crystal (SX2) "show the low-slope plastic flow region after yield at the high strain-rate but only the single crystal shows serrations at high strain. These however, were barely noticeable. Also, only the \ single crystal, showed a yield drop on the low strain-rate portions of the stress-strain curve. TABLE I I I . C h a r a c t e r i s t i c s of Specimens Tested f o r the E f f e c t of. S t r a i n Rate on Flow Stress Specimen Diameter Gr a i n A B B/A Diameter SX2 3-5 mm 3-5 mm 1 1 1 RC4 3.8 0.3 153 40 0.26 2B1 0.9" 0.3 10 10 1 RC3 3.8 0.09 . 1500 130 0.09 113, 5 0.05 0.01 25 16 0.64 1F12, 16 0.2 0.01 400 60 0.15 1B22, 26 0.9 0.01 8100 280 0.03 A and B are as d e f i n e d i n Table I I . o 0 © . 1B22 O 1B26 || 0 • 0 • 0.965 0 . ® O 0 O O 0 0 o @ • 2 • ® 0 $ ' O O ® ® 0.960 - 0 1 1 1 1 1 5 10 15 20 25 30 35 $ Elongation -3 Figure 17. Ratio of Flow Stresses at Strain Rates of € 1 = 10 Jmia / 6 p= lO'^-min"1 for Specimens 1B22 and 1R26 _ k2 -• 0 975 £1 • • B D • • • • 113 1113 1F12 I F I 6 • O 0.970 0 0 0 • o o * — o 0 - 0 — o o # O O O » O O o o - — 0 0.965 10 15 20 io Elongation 25 Figure 18. Ratio of Flow Stresses at Strain Rates of £"]_= 10"3min~-1- and €'2- l C ^ m i n - 1 for Specimens 1F12, 1F16, 113, and 1113. 30 0.980-0-975-15 20 25 30 $ Elongation Figure 19- Ratio of Flow Stresses at Strain Rates of €j.= lO'^ min""'", € = 10~1min"-'- for Specimens RC3, 2Blf RC.ir 0.985 -O.98O -15 $ Elongation 25 30 Figure 20. Ratio of Flow Stresses at Strain Rates of €j= 10"3min , € 2 = l O ' T i i n " 1 for Specimen SX2 - kk -0 . 9 8 s 0.980 -0.975 £ 1 6 0 . 9 7 0 -0 . 9 6 5 -15 20 • $ Elongation 25 30 35 Figure 21. Figures 17 to 20 plotted together for comparison. 'fo Elongation Figure 22. Stress-Strain Curve for Specimen 1B22 , i Elongation Figure 2 3 . Stress-Strain Curve for Specimen RC3 - 48 -b. Effect of Temperature on Flow Stress Copper wires and rods with the characteristics shown i n Table IV were tested at temperatures of 78°K and 293°K at a strain-rate of about 10 ^min 1. ^he r a t i o of flow stresses at 293°K and 78°K are plotted as a function of s t r a i n i n Figure 25 and 26. The copper p o l y c r y s t a l l i n e specimens do not exhibit Cottrell-Stokes behaviour while the si n g l e . c r y s t a l specimens do. Portions of the stre s s - s t r a i n curves f o r specimens 132k, R C 6 and SX3 are plotted i n Figures 27 >to 30. I t can be seen from these that, as might be expected, the te n s i l e behaviour of the specimen approaches that of a single c r y s t a l as the grain-configuration, approaches that of a single c r y s t a l . The r a t i o of the flow stresses for the single c r y s t a l specimens shows an anomaly i n that the r a t i o , although e s s e n t i a l l y constant, shows a change i n value at one point during the deformation. Since the s t r a i n at which t h i s change i n value took place coincided with the s t r a i n at which the form of the strain-stress curve undergoes a change i n form from that seen i n Figure 30. to that seen i n Figure 31j i t was decided to treat these as two separate values rather than as one continuous curve. I t should be noted here that, although these two events coin-cided i n each of the .three single crystals tested, the s t r a i n at. which they occured was not the same f o r the three specimens. The approximate elongations at which the r a t i o changed are given below: Crystal SX1 Elongation 25$ SX3 30$ 8X4 20$ The shape of the stre s s - s t r a i n curve for specimen SX3 (Figure 30,3l) i s the same as that found by B a s i n s k i ^ i n the low-temperature portions, but not i n the high temperature.portions. The shape of the high temperature portions of the curve agree with those found by C o t t r e l l 4-tj , and Adams in that a yield drop is found after about 2.yjo elongation but the shape of the yield drop is different. In specimen R C 6 , the temperature cycling was carried on beyond the point where.the neck became visible. This resulted in a change in the shape of the low temperature portion of the stress-strain curve to that shown in A in Figure 29• and in a sharp drop in stress following a short region of plastic flow as is shown in B in Figure 29- The amount of plastic deformation before the drop in stress occured decreased and the rate and extent of the drop in stress, increased as the diameter of the neck decreased. That this phenomenon is a result of the deformation at low temperature is shown by two experiments which are also indicated o in Figure 29- In the f irst , the specimen was unloaded and held at 293 K for a short time and then reloaded. In the second, the specimen was unloaded, and held at 78°K for a time equivalent to that required for the low-temperature test. It was then brought to 293°Kand reloaded once more. In neither case was the sharp yield drop observed. TABLE IV. Characteristics of Specimens Tested for the Effect of Temperature on Flow Stress Specimen Diameter Grain Diameter A B B / A SX1 4.0 mm - 1 1 • 1 SX3 3-8 - 1 1 1 sx4 3-8 - 1 1 1 RC5 3-9 - 2 2 1 R C 6 4.0 - 5 5 1 RC2 3-5 0.2 mm 310 55 0.2 RC1 4.6 0-75 40 20 0-5 1F13 0.2 0.01 4oo 6o 0.15 1324, 27 0.9 • 0.01 8100 280 0.03 Aand-B are as defined in Table II. Specimen R C 5 is essentially a bicrystal with a few small stray grains along the grain, boundary. Specimen R C 6 consisted of five grains arranged symmetrically about the axis of the specimen and running almost the entire gauge length. - 51 -5 io 15 20 25 io Elongation Figure 25• Ratio of Flow Stresses as a Function of•Strain for T 1 = 2 9 3 % T2= 78°K 1750 1500 OJ w 1250 1000 750 2750 2500. 2250 . 2000-3 4 5 6 7 14 $ Elongation Figure 27- Stress-Strain Curve for Specimen 1B24 15 16 T 1 = 293 dK T2= 78°K 17 18 5 10 15 20 $ Elongation Figure 28. Stress-Strain- Curve for Specimen RC6 30 35 ko "jo E longat ion F igure 2 9 - S t r e s s - S t r a i n Curve f o r Specimen RC6 a f t e r the Appearance of a V i s i b l e Week 3 5 ^ 0 4 5 5 0 $ Elongation Figure 3 1 . Stress-Strain Curve for Specimen SX3, Part II. i vn —J i - 58 -c. Results of Tests on Muminum To check that the non-Cottrell-Stokes behaviour in polycrystalline copper was not due to a faulty testing procedure, a copper strip specimen was prepared and tested as described in the experimental procedure. For comparison, several aluminum strips were, prepared and tested in the same manner. The results of these tests are shown in Figures 32 and 33 along with the ratio of flow stresses found by Cottrell and Stokes^ on aluminum single crystals. The ratio of flow stresses was shown to be constant for aluminum and to have about the same value as' found by Cottrell and Stokes. The copper strip, however, was shown to behave in the same manner as the previous copper polycrystalline specimens. These results demonstrate' that the experimental procedure was not at fault. Portions of the stress-strain curve for the aluminum specimen SA2 are shown in Figure 3^. Two differences from the stress-strain curves found by Cottrell and Stokes for aluminum single crystals are evident. The f irst is that a yield drop appears at a strain of less than h°jo whereas this did not appear in the single crystals until a strain of about 6$> had been reached. Also, the yield-drop was not always evident beyond this strain whereas i t was in the single crystals. Second, the yield •stress at the low-temperature is often lower than the final stress in the previous low-temperature section of the test. This reduction in stress, which wi l l be here-in-after referred to as "work-softening", depends on.both the amount of strain at 293°K ( C* ) between the two 78°K tests and the total strain ( £ ). This is demonstrated in Figure 35- where the ratio of'the yield stress (g') to the final stress ( ) of the preceeding low-temperature test is plotted as a function of £ for different C • For small £ , the scatter is so large,that the extent of work-softening can be shown only as a band (light hatching), but the ratio is definitely less than one. 0.780 . D Aluminum Single-Crystal (Cottrell and Stokes) — I 1—: 1 1 1 I I : 5 10 15 20 25 30 35 $ Elongation Figure 32. Ratio of Flow Stresses for Aluminum Single Crystals (Reference 6) T 1 = 293°K, T2= 90°K and Aluminum Polycrystals, 0^ = 293°K, Tg= 78°K 1 vo - 60 -0.88 0.87 O 0 \ p 0.86 \ 0 L2 0.85 0 \ ° \ 0.84 ^ 0 0 1 1 1 1 1 5 10 15 20 25 30 io Elongation Figure 33. Ratio of Flow Stresses for Polycrystalline Copper Strip. T x= 293% T 2= 78°K. $ Elongation" Figure jh- Stress-Strain Curve for Aluminum Specimen SA2 fo Elongation Figure 35• Degree of Work-Softening in Aluminum as a Function of Strain for Various £ 1 • CA ro - 63 -SUMMARY OF RESULTS The results of the investigation into size-effects, are not very-revealing. The yield stress, ductility, ultimate tensile strength and rate of work-hardening were measured for copper wires with diameters ranging from 900 ^1 to 50 jx at various temperatures' and strain rates . and the results may be summarized as follows: 1. The room temperature yield-stress shows no apparent size-effect. 2. The low-temperature yield-stress decreases with decreasing diameter for wires with diameter less than 500 ^.. - 3- Both the ductility and the ultimate tensile strength decrease with decreasing diameter. k. The results on work-hardening rate show a scatter high enough that no definite size-effect can be established. A trend toward an "increasing work-hardening.rate with decreasing diameter was observed except for the high strain-rate tests where the opposite was observed. The results of the investigation of the effects of the temperature and strain-rate on the flow-stress of copper may be summarized as follows: 1. Single-crystals of copper follow the Cottrell-Stokes law for both temperature and strain-rate changes. ••"' 2. Polycrystalline copper follows the Cottrell-Stokes law for strain-rate changes but the value of the flow-stress ratio is different for different specimens being affected by: • a) the grain size, b) the fraction of the grains, in the- specimen which have a free surface. 3- Polycrystalline copper does not obey the Cottrell-Stokes law. The deviation from the expected value of the flow-stress ratio is negative and depends on strain. The results of the tests on aluminum show the following: 1. Aluminum polycrystals do obey the Cottrell-Stokes law for temperature changes. 2. The yield-drop observed by Cottrell-and Stokes^ in aluminum single crystals was found in polycrystals. 3- The yield-drop in polycrystalline aluminum is accompanied by a "work-softening" effect, not found, evidently, in single crystals. The above is a summary of the important results of this investigation and are the results which wil l be referred to in the discussion. - 65 -DISCUSSION 1. Size Effects This investigation has shown that the size-effects found in polycrystalline copper wires are, in most cases, the opposite to those found in single crystals by other investigators. The results wil l be discussed in terms of the degree to which the wires of various diameter can be referred to as "polycrystalline". Departures from the behaviour expected when this consideration is taken into account indicate that there may be size effects where none are directly observed. a..Yield Stress and Work-Hardening Rate It would be, perhaps profitable to f irst discuss the grain configuration in a polycrystal and to assess the effect of the diameter on this. The polycrystalline wires used in this investigation had a grain diameter of 0.01 mm and varied in diameter from about 0-9 mm to about 0.05 mm. E x a m i n a t i o n of Table II shows that in wires of diameter 0.9 mm, the great majority of the grains are totally enclosed in the wire and the percentage of the grains which show a free surface is negligible. In a wire of 0.05 mm diameter, a significant proportion of the grains in the wire have a free surface and less than half are totally enclosed. Figure 36. shows the relationship between the wire.^ diameter and the ratio B/A, the fraction of the grains in the wire which show a free surface. The values shown are a conservative estimate since the calculation is based on the supposition that a grain on the surface wil l have a free surface of the maximum possible area. However, i f one takes into account the relative effects of different sizes of free surfaces of the various grains on the surface, this is probably a good measure of what might be called the "effective" ratio of the number of grains showing a free surface to the total number in the specimen. The curve shown in Figure 3 6 - represents a function of the same form as the surface to volume ratio; i .e . the ratio B/A is proportional to the inverse of the wire diameter for a fixed grain size. The ratio wil l have a limiting value of one where the wire diameter is equal to the grain diameter, corresponding to a bamboo structure. The variation of the ratio • /^A may be expected to give rise to the following effects: i . The yield stress should decrease with decreasing diameter, i i . The work hardening rate should decrease with decreasing diameter. These will occur because the constraints on the grains totally enclosed in the wire wil l become less important as the diameter and thus the number of interior grains decreases. Let us now examine the results of this investigation in the light Oo of the above discussion. The yield stress at 7 ° K does in fact decrease with decreasing diameter for diameters below 0.5 mm. Above 0-5 mm diameter, where the change in the ratio B/A with diameter is very small, the yield stress decreases with increasing diameter. The room-temperature.yield-stress, however, remains approximately constant where a decrease is expected with decreasing diameter. It appears that there may be two mechanisms operating at the higher temperature; one tending to increase the yield stress and the one-, outlined above, tending to decrease the yield stress. If the oxide layer makes any contribution to the strength of the wires,, any increase in the yield stress would be proportional to the inverse of the diameter as outlined in the introduction. Since this will have the opposite effect to that of increasing the ratio B/A, these would tend to cancel. Work by previous investigators, however, indicates that the oxide-layer would have l i t t l e i f any effect in that: i . ' There appears to be no effect on the strength of 3 23 copper whiskers from the oxide layer ' i i . Wo effect on the strength of polycrystalline wires from the oxide layer is evident until the grain diameter 2k 26 29 approaches that of the specimen diameter ' ' . The smallest wires in this investigation had about 25 grains in the x-section. The::work-hardening rate, expected to decrease with decreasing diameter, tends to increase for low strain-rate tests .both at room o temperature and at 78 K. The specimens tested at the high strain rate at: room-temperature, however, do show a trend towards the expected decrease. The significance of these is very doubtful since the scatter is high enough to make the changes in hardening rate doubtful. b.. Ductility and Ultimate Tensile Strength Roberts^, studying the ductile fracture of polycrystalline copper, found that the fracture is initiated by the coalescence of voids .into a central crack in the necked region of the specimen. Under further strain, this crack grows out toward the surface of the metal. When the crack approaches the surface, void formation occurs "catastrophically" and the specimen fractures. This would appear to explain.the decrease in ductility and in the ultimate tensile strength with decreasing specimen diameter. The events outlined above, however, take place in the necked region of the specimen and.value of stress and elongation used in this investigation were' taken at the point where the stress began to decrease due to the formation of a neck. It was noted, however, that the mode of fracture changed with decreasing diameter. Wires of diameter greater - 69 -than 0.2 mm showed extensive necking and parted very slowly, the load decreasing gradually. In smaller specimens, no visible neck appeared and the specimen separated suddenly with no noticable decrease in load before fracture. From this, i t would appear that, in the small diameter specimens, the crack formation occurs at a much lower stress than in the large diameter wires and before the neck has formed visibly. There is the possibility that sub-microscopic surface-defects could act as stress-raisers and thus lower the fracture stress. The reduction in apparent ultimate strength due to a stress raiser would, of course, have a greater effect on the thin wires than on thick ones. However, the probability of a stress-raiser which is invisible under a magnification of 100 X having an appreciable effect on the fracture stress is unlikely. A more probable stress-raiser is in the voids themselves. Roberts examined specimens which, although highly strained, had not yet formed a neck. He found, that although the concentration of voids is lower than in the necked regions, there are intergranular voids. A comparison of the effect of diameter on the ultimate strength of wires found in this investigation with that found by McDanels et al35 indicates that, once again, the ratio ~B/A may be important. McDanels et al tested composites of tungsten wires in a copper matrix and found that the strength of the composite increased as the diameter of the tungsten wires decreased. Metallographic. observations indicated that neither the tungsten wires nor the copper matrix failed independently but that they fractured as a unit. For this type of test, the ratio B/A would have a very small, i f any, effect on the properties of the tungsten wire since the surface grains are constrained."by the copper matrix just as the interior grains are constrained by the surrounding tungsten. -70-• The•discussion indicates that any investigation into size effects in polycrystalline wires must be designed to offset the effect of the increasing proportion of the grains which impinge on the surface as the diameter decreases. One possibility for overcoming this is the method used by McDanels et al but this method adds a complicating factor. Another •is to use a small enough grain diameter that the ratio B/A varies negligibly over the desired range of diameters. For- diameters between 900 jx and 5 0 p., this would require a gram size of less than 1 p. 2. Cottrell-Stokes Law It is now generally accepted that the flow in a crystal may be represented by the sum of two components; one a temperature-independent and the other a temperature-dependent flow-stress. On this basis, using Seeger's notation, the flow-stress may be written: where is temperature-independent' and temperature-dependent. From this, the ratio of flow-stresses for any.crystal at two temperatures, T-j_ and can be written: # 1 = 6j,+ <£/ -pr— , „ < 1 for T > T p . . . 2 ) 6r 2 0 £ 2 + 0 J 2 1 2 If a crystal is deformed at T^'to a given strain, then deformed at Tg, the dislocation pattern and density wi l l be the same, at that strain, for the two temperatures. Thus, since 6^ is temperature-independent, 6"^ / and 6$%, wi l l be equal and the ratio of flow stresses wi l l be: (3a) ST 2 " 5* +652 the Cottrell-Stokes ratio. It may also be written: ST/ ^ 1 * £L/61 _ ( 3 b ) o r 2 1 * .6s.2/si - 71 -If Ss i s proportional to 6^  over a range of strain; i.e. 6s 1 = -C : ; 6j. 2 = C &6 6* then the flow-stress ratio becomes, from (3b): 6TI = 1 + C ' = K <i 1 + C where K i s a .'.constant, and the Cottrell-Stokes law w i l l be obeyed. If £5 is not proportional to 6^. , the Cottrell-Stokes law w i l l not be obeyed and the ratio of flow-stresses w i l l vary over a range of strain. a. Work-Hardening As was mentioned i n the introduction, there are two main theories 37 U-2 of strain-hardening, one by Basinski and one by Seeger et a l , to account for the constancy in the Cottrell-Stokes ratio. Both agree that the temperature-dependent component of the flow-stress, 65 , is the stress necessary for the glide dislocations to cut the forest dislocations This process w i l l give rise to a temperature-dependent stress in that i t is a short-range interaction involving the formation of constrictions i n extended dislocations and the formation of dislocation jogs and can there-fore be aided by thermal vibrations. The treatment of this mechanism by Basinski differs somewhat from that of Seeger et a l . Starting with a uniform distribution of obstacles, Seeger assumes that a l l of them are capable of breaking down, whereas Basinski assumes that, independent of the i n i t i a l distribution, the number breaking down w i l l depend on temperature. It i s i n the source of the temperatuie-independent component, 5^ that^these investigators show an important difference of opinion. Seeger et a l . have proposed that & i s due to an internal stress-field which impedes the motion of the glide dislocations and that this stress-field is due to the elastic stress-fields of dislocatipn pile-ups distributed randomly through the crystal. Each pile-up contains about twenty-five dislocations, the number in each being independent of strain. They • showed that the flow-stress due to this stress-field w i l l be proportional to the square-root of the dislocation density. Since 65 would depend on the dislocation density in the same manner, this would imply a proportionality between <5^  and Basinski, on the other hand, attributes the elastic component to the long-range interactions between forest and glide dislocation. Since, in this case, both <5~j a n ( i 6~£ are due to interactions between the same dislocations, the £s a n < i 6% should be proportional and the Cottrell-Stokes law obeyed. Both of these theories are based on the assumption that the rate of increase in the density of forest dislocations i s about the same as the rate of increase in the glide dislocation density. Davis has shown that i f the rate of increase of the glide dislocations is much greater than that of the forest dislocations, the proportionality between <£$ and is destroyed and the Cottrell-Stokes law is no longer obeyed, the deviation being in a positive manner. Similarly, i f the forest dislocation density increases much faster than the glide dislocation density, there w i l l be a negative deviation from the Cottrell-Stokes law. •b. Theoretical Considerations An approximate division of the flow-stress into <Sj and 6^ for a single crystal of copper was calculated by Seeger et a l as shown in Figure 3 7 * This division is based on Seeger's theory of the - 73' -flow-stress but w i l l serve as an i l l u s t r a t i o n for the following discussion. The solid curves, (S& and t\$ , represent the condition under which 6j is proportional to and the Cottrell-Stokes law is obeyed. Consider the effect on the flow-stress ratio i f 6^ were to l i e along the dotted line 6$. At a strain £, this would re-present an increase 6"6^ . over that 6^  which maintains the proportionality with 5j . Since the effect on the Cottrell-Stokes ratio i s d i f f i c u l t y to see, consider instead the ratio , where: 1 Z>6^  = 67 2 - o*r1 = ^ 2 ~ ^ 1 • 6* /? ^ •L- . since  u $ = 6/5 1 2 Simple algebra shows that: 6 7 * 1 = 1 Gr2 - 1 + ^l6r± and that an increase in represents a decrease in the Cottrell-Stokes ratio, $T 2 The increase, c7&£ , in ^ w i l l change the ratio to a new value / A T \ - 74 -Strain Figure 37- The Separation of the Flow-Stress into the Elastic and Temperature-Dependent Compounds as Calculated by Seeger et al^"2. Therefore and A reasonable assumption to make, on consideration of the two theories outlined above, would be that is proportional to In this case, although the Cottrell-Stokes ratio would have a higher value, i t would stillbbe auconstant value. Similarly, a reduction, - &6& , in 6$ would, i f were proportional to , result in a lower but s t i l l constant value for the Cottrell-Stokes ratio. It can be shown, in a similar fashion, that an increase, <S~f5^5 > in (S^5 would result in a lower Cottrell-Stokes ratio. A change in Ss which is proportional to Ss would, in effect, multiply ^ by a constant P, where P < 1 for a negative and P > 1 for a positive Thus: \ = P 6~S2 - P <fs 1 s f r j Si + P ^ X = SS2 - S~3 1 < /p + $si which means that a positive wil l cause a decrease in the Cottrell-Stokes ratio and a negative an increase. Once again, however,'if is proportional to (Sj , the ratio wil l be changed but will remain constant with strain. These effects can now be summed up as follows: Change in 6~5 or <5"$. : Change in the Cottrell-Stokes Ratio Positive - Negative Negative Positive c. A Possible Effect of Dislocation Density Since the results of this investigation have shown that the Cottrell-Stokes behaviour in polycrystals is different from that of single crystals, i t would perhaps be profitable to consider the difference between polycrystals and single crystals in the light of the above theorie One difference in the deformation, characteristics is that, due to the restraints on each grain in a polycrystal, several slip-systems are required to operate to give the required deformation. This might give an overall increase in the dislocation density over that to be found in a single crystal and may result in dislocation interactions beyond those considered by Basinski or by Seeger et a l . Let us then consider the effect of an increase in the forest dislocation density in polycrystals on the flow-stress ratio in the light of Basinski1s theory of the flow-stress. The elastic stress on the glide dislocation at point A, (Figure 38-) wil l be due to the elastic stress-fields of the forest dislocations o(-, and $2' With a low forest density, the stress at point A due to the stress fields of ftand $2 wil l be very small compared to that due to °C. With a high forest density, the forest Figure 38- Representation of a Glide Dislocation Approaching a Row of Forest Dislocations. dislocations @ and @^ wil l make a significant contribution to the stress-field at point A. If 3^ and @ are of the same sign as , the stress-field at A wil l be greater than that due to °t- alone. If ^ and @ a r e o f opposite sign to oC , the stress field at A wil l be smaller than that due to oC alone. If the sign of the forest dislocations were randomly distributed over the slip-plane, the most probable condition, then the stress field at A would be that due to oi. alone since the contributions from and @ 2 would tend to cancel. In any case, any interactions between forest dislocations, due to a high.dislocation density, would result in a proportional to the square-root of the forest density. This would mean that £6$ would be proportional to and, although the value of the Cottrell-Stokes ratio may be changed, the proportionality between a n ( i 6$ would not be upset. Thus, a mechanism of this sort could give rise to the effect shown in Figure 21. but not the deviations from the Cottrell-Stokes law shown in Figure 25-d. Grain Boundaries Another important difference between a single crystal and a polycrystal is that a polycrystal contains grain-boundaries which can act as barriers to dislocation motion. Although Basinski attributes no effect to the stress-fields of pile-ups scattered through the body of the crystal, a dislocation pile-up at a grain boundary wil l have an effect on the crystal directly ahead of i t . Consider the effect of a dislocation pile-up at a large angle . t i l t boundary. Eshelby, Frank and Nabarro^? have calculated the stress-concentration factor ahead of a dislocation pile-up to be of the form shown in Figure 39- As'suming that, the average length of a pile-up wil l be one=half of the grain-diameter, the distance over which the pile-up wil l produce a stress-concentration factor of at least two wil l be 0.25L or 0.125d, where d is the grain diameter. However, since the slip system wil l not be continuous across the grain boundary, this value of the stress concentration factor must be modified. The average angle of a high-angle boundary is about 50° to 60°. Since the angle between ( i l l ) planes is 70° and between 0-10/* directions is 60°, we can expect a slip system with the angle of the plane within ± 20° and the direction within ± 30° to the slip system from which the pile-up originated. At a distance 0.25L from the head of the pile-up, therefore, the stress-concentration factor wil l be between 2 and 2 cos 30° ^ 1.6. If we approximate the shape of the grains to be cubic, the volume affected wil l be: d 3 - 3d 3 = d 3 fl - 27 "\ I; V 517/ or slightly more than one-half the total volume. Unless 6^. <^ 6^ s , this will produce an effect of a magnitude far greater than that observed for the difference between single and polycrystals of -copper. There are two possible modifying factors: 1. The average length of a pile-up at the boundary may be less than one-half the grain diameter. 2. The grain goundary may act as a "screen", as proposed by 48 Frankel , reducing the effect of the pile-up on the adjacent grain. No quantitative calculation on these effects can be made here since there is not sufficient data on the effect of the grain diameter on the Cottrell-Stokes ratio. There is also a complicating Figure 39. Stress Concentration Factor in Front of a Dislocation Pile-up of Length L as Calculated in Reference 48. - 81 -f a c t o r i n t h a t the r a t i o appears to a l s o depend on the f r a c t i o n of the g r a i n s i n the specimen which impinge on the surface (Figure 20. ). The e f f e c t of the p i l e - u p s can, however, be assessed q u a l i t a t i v e l y . The e f f e c t , of the p i l e - u p at the g r a i n boundary w i l l be, because of the s t r e s s - c o n c e n t r a t i o n , t o reduce L\ i n the area ahead of i t . The o v e r a l l r e d u c t i o n i n 6# t,, %6$ , w i l l depend, assuming a constant p i l e - u p l e n g t h , on the number, Up, of p i l e - u p s i n the specimen: The number of p i l e - u p s , i n t u r n , w i l l depend on the d i s l o c a t i o n d e n s i t y , D , i n the specimen: Thus: £££ <^Dd . Both B a s i n s k i ' s and Seeger's t h e o r i e s give a 6^ which i s p r o p o r t i o n a l t o the square-root of the d i s l o c a t i o n d e n s i t y . This means t h a t S6Q w i l l n o t be p r o p o r t i o n a l t o o^ j and, by the previous d i s c u s s i o n , a negative d e v i a t i o n from the C o t t r e l l - S t o k e s law w i l l be expected, i n accord w i t h the experimental observations. e. The E f f e c t of Stacking F a u l t Energy on the C o t t r e l l - S t o k e s E a t i o Since the d e v i a t i o n s from the C o t t r e l l - S t o k e s law found' i n copper p o l y c r y s t a l s do not appear i n aluminum, the d i f f e r e n c e s between aluminum and copper should provide a clue t o the behaviour of copper. The p r i n c i p a l d i f f e r e n c e between copper and aluminum l i e s i n the s t a c k i n g - f a u l t energy, copper having a low and sluminum a high s t a c k i n g - f a u l t energy. Table V. shows the most probable values of s t a c k i n g - f a u l t energy as c a l c u l a t e d 50 by Thornton and H i r s c h together w i t h the C o t t r e l l - S t o k e s r a t i o s , f o r s e v e r a l temperature ranges, found by B a s i n s k i f o r the face-center cubic TI 50 metals copper, aluminum and s i l v e r and f o r the hexagonal metal magnesium - 82 TAB IE V. Stac k i n g - F a u l t Energies and C o t t r e l l - S t o k e s R a t i o s  f o r Cu, Ag, A l and Mg. Stacking F a u l t C o t t r e l l - •Stokes R a t i o , Element Energy T-L = 293°K T 2 = 78°K 200°K k.2°K 100°K k.2°K 50°K k.2°K Ag 35 ergs/cm • 0.86 * 0.95 0-97 O.98 Cu 57 0.88 * 0.93 0-95 0-97 A l 200 0.78 * 0-73 0.80 0.90 Mg (High) - 0-73 O.78 O.87 * Not c o r r e c t e d f o r the v a r i a t i o n of the e l a s t i c modulus w i t h temperature. . - 83 -There seems to he a correlation between the stacking-fault energy and the Cottrell-Stokes ratio in that the high stacking-fault . energy metals (Al and Mg) have a lower ratio than copper and silver. The values shown are corrected for the variation of the elastic shear modulus with temperature. If any other differences, in the metals are ignored and only the stacking-fault energy is considered, the theories of Basinski and Seeger et al may be tested on the basis of prediction of the effect of the stacking-fault energy on- the Cottrell-Stokes ratio. Basinski has- attributed both 6j and 6$ to interactions between forest and glide dislocations. The elastic stress, (5^ , since i t is due to long-range interactions wil l not be appreciably affected, i f at a l l , by the separation between the partial dislocations. The temperature dependent component, C% , since i t involves the actual intersection of the dislocations, wil l be profoundly affected by the separation of the partials. 51 Stroh has calculated the work required to form.a constriction in an extended dislocation from the work required to force a decrease in the separation' of the partials and the work required to bend the disr locations-against the line tension. He found that decreasing the stacking-fault energy increases the work required. Let us now examine the effect of the•stacking-fault energy on the Cottrell-Stokes ratio. By Stroh's calculation, decreasing the stacking-fault energy increases the work required to form a constriction, thus increasing 6s • As shown earlier, increasing Sj wi l l decrease the Cottrell-Stokes ratio. - Qk _ Basinski'.s theory, therefore, predicts a decrease in the Cottrell-Stokes ratio with decreasing stacking-fault energy, contrary to the results tabulated in Table V.- Any assessment of the effect of the stacking-f ault energy in light of Seeger's theory is .more complex since 64 wil l also depend on the stacking-fault energy. The number of dislocations in a pile-up wil l be a function of the stacking-fault energy and thus the magnitude of the stress-field due to the pile-up will depend on the stacking-fault energy. This comes about because the extended dislocations in a pile-up must be constricted before they can escape through cross-slip or through dislocation climb. The stress on the leading dislocation is a function of the number of dislocations in the pile-up and of the applied stress. Consequently, the number of dislocations in a pile-up wil l be"higher in a low stacking-fault energy metal. By Seeger's theory, then, we could expect an increase in both 6s and 6 5 with a decrease in the stacking-fault energy. The effect of this on the Cottrell-Stokes ratio is uncertain but i t is possible that ittcould give an increase in the ratio. This discussion ignores, of course, any difference other than the stacking-fault energy between high .and low stacking-fault energy metals. Based on the limited data available, however, there is a rough correlation between the Cottrell-Stokes ratio and the stacking fault which seems to be independent of other differences. It also appears to be qualitatively independent of the temperature range considered. More experiments, however, on metals of different stacking fault energies must be carried out to definitely establish the existence of the correlation. - Q 5 -Another useful experiment in an investigation of this type would be to alter the stacking-fault energy of a metal by the additions of small amounts of a suitable impurity and to examine the effect on the Cottrell-Stokes ratio. f. A Possible Effect of Dislocation Density on In an earlier part of this discussion, we considered the possible effect of the dislocation density on the elastic interactions between glide and forest dislocations. Let us now consider the effect of increasing the dislocation density on the temperature-dependent component of the flow stress. With a high enough dislocation density, the interactions between parallel extended dislocations wil l decrease the average separation of the partial dislocations, resulting, effectively, in a higher stacking-fault energy and a lower stress required to form a constriction in the extended dislocation. This wil l reduce 65 and, according to Basinski"s theory, result in a higher value of the Cottrell-Stokes ratio- The actual, results on polycrystalline copper (Figure 2l) show that the ratio is lower than that for single crystals. This reduction in the ratio, however, on the basis of the proposed mechanism, fits in with the observed correlation between the stacking-fault; energy and the Cottrell-Stokes ratio. The dislocation density required to produce an appreciable effect on the temperature-dependent component of the flow-stress can 5 3 be estimated quite easily. Cottrell' has caluclated the separation of the two partial dislocations bounding a stacking fault to be: r .= M (bj.bp) 2/7-<r where €. is the stacking-fault energy, b-^  and b 2 the Burgers vectors of the partials andja the elastic shear modulus. In a face center cubic - 86 -.crystal.;, a. normal dissociation mechanism i s : | <io5,-^! < 2 n> + g < l l 5> For t h i s pair of p a r t i a l s : b r b 2 = a and r = u a' 12 2 2hTT6 In copper, ergs/cm 2 and a = 3-6 X. In aluminum 200 ergs/cm 2 and a = 4 . 0 £. Takingyi = 4 X 1011dyne/cm^, r = 10"^ cm f o r copper and r -- 10" cm f o r aluminum. I f the centers of the extended dislocations approached to within 10 r, the separation between the p a r t i a l dislocations would be reduced by about 10$. This distance between dislocations corresponds to d i s l o c a t i o n 12 -2 14 -2 . . . densities of about 10 cm and 10 cm i n copper and aluminum respectively. 8 1 2 - 2 The normal di s l o c a t i o n density i s about 10 to 10 cm i n a cold-worked single c r y s t a l . I f the di s l o c a t i o n density i n a copper polycrystal i s at a l l higher than that i n a single c r y s t a l , we can expect an effect on' from the extra reduction i n the width of the stacking-fault. In aluminum, since a much higher dislo c a t i o n density i s required to s i g n i f i c a n t l y decrease the width of the stacking-fault, the effect would be reduced by a factor of about 100 and would probably be neg l i g i b l e . The foregoing discussion indicated that the process of work-hardening and the origin of the temperature-dependent and temperature-independent components of the flow stress may be more complex than those o 7 42 proposed by Basinski-" or by Seeger et a l . I t has been shown that i t i s possible to explain the v a r i a t i o n of the Cottrell-Stokes r a t i o between single and polycrystals of copper and the deviations from the C o t t r e l l -Stokes r a t i o i n terms of these theories only by proposing an extra mechanism which was not considered i n the o r i g i n a l theory. - 87 -Examination of the Cottrell'-Stokes ratios for metals of different stacking-fault;.energies has indicated a possible variation in the ratio with varying stacking-fault which is opposite to that predicted by Basinski's theory. It should be stressed here, however, that this correlation is far from being definitely established and requires further investigations. 3- Work-Softening in Aluminum Cottrell and "Stokes^  observed a yield drop in aluminum crystals deformed after prior deformation at a lower temperature. They proposed that this yield drop indicates the operation of a mechanism by which the plastically deformed crystal can rid itself of some of the work-hardening introduced through the low-temperature deformation. This proposal is supported by the work-softening observed in this investigation on aluminum polycrystals. As can be seen'in Figure 3^. the work-softening does not appear until the strain at which the yield drop f irst appears. After a few percent deformation, the mganitude of the work-softening for small is approximate y constant over the range of strain studied (Figure 35*)• Also, for large C', the work-hardening introduced by the plastic de-formation after the yield drop compensates to some extent for the work-softening occuring at the yield drop. . It appears that the dislocation' configuration produced by deformation at allow temperature is unstable, at the same strain, at higher temperatures. The fact that the work-softening is recoverable with further deformation at high temperature indicates that the difference in the stable configuration is one of degree rather than kind. Also, the fact that the ratio 6? , for small £ ' , settles down to a constant value indicates that a definite percentage of the obstacles formed at - 8 8 -low temperatures break down at the higher temperature. That this effect is absent in copper means that the mechanism responsible for the work-softening in aluminum is one which is affected by the stacking-fault energy such as cross-slip or dislocation climb. By Basinski' s theory, this process would presumably be a dislocations-annihilation or polyganization through cross-slip or climb. From Seeger's theory, the same effect might be expected through the escape of dislocations from the head of a'.pile-up by cross-slip or climb. Since this type of process occurs more easily at higher temperatures, the equilibrium number of dislocations would be lower than at low temperatures. Examination of Figure 3 ^ . also shows that the results on work-softening in aluminum support the explanation offered by Cottrell and Stokes for the observed difference in flow-stress ratios on going from high to low and from low to high temperatures. Figure kO shows a schematic representation of the form of the yield drop observed in aluminum deformation at room temperature after prior deformation at 78 ° K . A yield stress of the value A would be expected assuming that the change in flow stress is reversible; that is , that the change in the flow stress at this strain should be the same for both directions of change of temperature. The point B represents the actual observed yield stresses than would be found on going from 2 9 3 ° K to 78 ° K . Evidence for this form of the curve is presented by Cottrell and 6 Stokes as follows: 1. The difference in the ratios of.flow stresses does not become appreciable until the strain at which the yield drop appears. 2 . If a specimen is strained to a stress represented by point C, and held at this stress, a delayed yield drop preceeded by creep at stress C is observed. Added to this is the evidence shown in Figure 3^- Here, i t can be seen that, although the work-softening does not appear until the strain at which the first yeild drop is observed, work-softening occurs even in tests where there is not observed yield drop and the magnitude of the work-softening does not depend on the size of the yield drop. This indicates that the dislocation configuration present at 78°K is unstable at 293°K and wil l break down under an applied stress lower than that expected for plastic flow. In some cases, the cr i t ica l stress appears to have been low enough that no yield drop was observed in conjunction with the work-softening. That thermal agitation alone, is not sufficient to cause the breakdown in the dislocation con-figuration is indicated in Figure 3^- which shows the stress-strain curve for a specimen which was strained at 78°K, unloaded and held at room-temperature for several hours, then restrained at 78°K. Np work-softening was observed in this case. - 90 -Strain Figure-:40. A Schematic Representation of the Form of the Stress-Strain Curve for Deformation at 2 9 3 ° K after Prior Deformation at 7 8 ° K . - 91 -SUMMARY AND CONCLUSIONS 1. The important results have been summarized in a previous section and wi l l not be repeated here. 2. In the-tests on the effect of temperature on the flow-stress, the shape of the stress-strain curve changes for specimens of different characteristics. In contrast to this, the shape of the stress-strain curve for polycrystalline aluminum is essentially the same as found in the literature^ for single crystals of aluminum. 5. The deviations from the Cottrell-Stokes law for temperature changes and the variation in the Cottrell-Stokes ratio found for strain-rate changes are probably due to dislocation mechanisms beyond those considered by either ^6 kl Basinski-^ or Seeger et a l . . These mechanisms wi l l be ones which exert a minor influence on.the flow-stress. ' k. The rough correlation between the stacking fault energy and the Cottrell-Stokes ratio' noted in the discussion is the opposite to that * 36 predicted by Basinski s theory. 5. The yield-drop found in aluminum at high temperature after prior deformation at a lower temperature is probably due to the release of dis-locations from a barrier by a thermally activated process such as cross-slip or dislocation climb. - 92 -SUGGESTED FUTURE WORK This work has been,primarily of an exploratory nature and the results have suggested directions of future work which might be profitably-explored. 1. Any future work on size effects in polycrystalline materials should be done using a procedure which wil l eliminate the effect of the ratio B'/A. Some possible methods have been suggested in the discussion. 2. If the Cottrell-Stokes behaviour, for both temperature and strain-rate changes, in polycrystalline copper were studied over different temperature ranges, the mechanism responsible for the deviations from the Cottrell-Stokes law may become more evident. 3- The work-softening found in aluminum polycrystals may also provide significant information i f studied over a series of temperature ranges. It may also be profitable to determine i f the same effects can be produced by strain-rate changes. The effects of different increments of low-temperature strain on the magnitude of the work-softening may also be studied. k. The correlation between the stacking-fault energy and the Cottrell-Stokes ratio noted in the discussion may be investigated by two types of experiment: a. Tests on more metals of high and low stacking-fault; energies and on nickel which has an intermediate stacking-fault energy (about 100 ergs/cm ). . b. Tests on metals doped with appropriate impurities to change the stacking-fault energy. Two alloys which might provide information i f studied over a wide range of compositions are nickel-cobalt and copper-zinc. - 93 -BIBLIOGRAPHY I f K. G. Compton, A. Mendizza and S. M. Arnold, Corrosion, Vol. 7 ( 1 9 5 1 ) , 3 2 7 -2 . J . K. Gait and C. Herring, Phys. Rev., Vol. 8 5 ( 1 9 5 2 ) , 1 0 6 0 . 3- S. S. Brenner, J . App. Phys., Vol. 2 7 ( 1 9 5 6 ' ) , 1 4 8 4 . 4. J . H. Hollomon, Trans. A.I.M .E., Vol. 1 7 1 ( 1 9 4 7 ) , 5 3 5 -5- J . E. Dorn, A. Goldberg and T. E. Tietz, Trans. A.I.M.E., Vol. 1 8 0 ( 1 9 4 9 ) 2 0 5 = 6. A. H. Cottrell and R. J . Stokes, Proc. Roy. Soc. Vo l . ' 2 3 ^ ( 1 9 5 5 / 5 6 ) 1 7 -7- G. F. Taylor, Phys. Rev., V o l . 2 3 ( l 9 g 4 ) , 6 5 5 . 8. A.S.M. Metals Handbook, 1 9 6 1 / II98. 9- Z. Gyulai; Z. Physik, V o l . 1 3 8 ( 1 9 5 4 ) , 3 1 7 . 1 0 . H. Suzuki, S. Ikeda and S. Takeuchi, Phys. Soc. Japan, J.,_Vol. 1 1 ( 1 9 5 6 ) 3 8 2 . l i . J . Garstone, R. W. K. Honeycombe and G. Greetham, Acta. Met., V o l . 4 ( 1 9 5 6 ) 4 8 5 1 2 . R. L. Fleipcher and B. Chalmers, Trans. A.I.M.E., Vol. 2 1 2 ( 1 9 5 8 ) , 2 6 5 . 13. G. L. Pearson, W.'T. Read Jr. and W. L. Feldmann, Acta.Met. V o l . 5 ( l 9 5 7 ) l 8 l . 14. F. Y. Eder and V. Meyer, Naturwissenschaften, V o l . 4 7 ( 1 9 6 0 ) , 3 5 2 . 1 5 - . z.. E. Moore, Private Communication. 1 6 . A. H. Cottrell, "Dislocations and Plastic Flow in Crystals", Clarendon Press, Oxford, p. 5 3 -1 7 - J . Takamura,, Mem. Faculty of Eng. Kyoto U. , Vol 1 8 ( 1 9 5 6 ) , 2 5 5 . 1 8 . R. Roscoe, Phil Mag., Vol. 2 1 ( 1 9 3 6 ) , 3 9 9 . 1 9 - E. W. da C. Andrade, "Properties of Metallic Surfaces", Inst, of Metals, ' London, Monograph and Report Series, n. 1 3 , 1 9 5 2 . 2 0 . E. N. da C. Andrade and C. Henderson, Phil . Trans. Roy. S o c , V o l 2 4 4 A ( 1 9 5 1 ) , 1 7 7 . 2 1 . I . R. Kramer and J . Demer, Trans. A.I.M.E., V o l . 2 2 1 No. 4 ( 1 9 6 1 ) 7 8 0 . 2 2 . . S. Harper and H. Cottrell', Proc. Phys. Soc, Vol. 6 3 B ( l 9 5 0 ) , 3 3 1 . 2 3 - S. Saimoto, 'M.A.Sc Thesis submitted in the Department of Metallurgy, University of British Columbia, June i 9 6 0 . 2k. E. N. da C. Andrade and R. F. Y. Randall, Nature, Vol. 1 6 2 ( 1 9 4 8 ) , 8 9 0 . 2 5 . C. S. Barrett, Acta.Met., Vol. l ( l 9 5 3 ) , 2 . 2 6 . E. N. da C. Andrade and R. F. Y. Randall, Proc Phys. S o c , V o l . 6 5 B ( l 9 5 2 ) 4 4 5 2 7 - F. R. Li.pse.tt and R. King, Proc Phys. Soc, Vol. 7 0 B ( l 9 5 7 ) , 6 0 8 . - 9k -28. J . J . Gilman and T. A. Read, Trans. A.I.M.E., V o l 1 9 l ( l 9 5 l ) , 792. 29. J . J . Gilman, Trans. A.I.M.E., V o l . 197(1953), 1217-30. E. N. da C. Andrade and A. J . Kennedy, Proc. Phys. S o c , V o l . 2 4 4 A(l95l), 177-31. S. S. Brenner, J . App.' Phys., V o l . 28(1957), 1023. 32. R. L. F l e i s c h e r and B. Chalmers, J . Mech. and Phys. of Solids.Vol.6(1958) 307-33. P. J . S c h l i c h t a , "Growth and P e r f e c t i o n of C r y s t a l s " , Wiley, (1958), 214. 34. P. J . S c h l i c h t a , P r i v a t e Communication. 35. D. L. McDanels, R. W. Jech and J . W. Weeton, Metals Prog. Vol.78(1960)6, 118. 36. P h y s i c a l and Mechanical P r o p e r t i e s of Some High-Strength F i n e Wires, Memorandum No. 80, Defense Metals Information Center, Jan. 1961. 37. Z. S. B a s i n s k i , P h i l . Mag., V o l . 4(1959), 393-38. K. G. Davis, P r i v a t e Communication. 39- W. D. Sylwestrowicz, Trans. A.I.M.E., V o l . 212(1958), 617• 40. M. A. Adams and A. H. C o t t r e l l , P h i l . Mag., V o l . 46(1955), I I 8 7 . 41. A. Seeger, " D i s l o c a t i o n s and Mechanical P r o p e r t i e s of Crystals',' Wiley, 24-3• 42. A. Seeger, J . D i e h l , S. Mader and H. Rebstock, P h i l Mag. V o l . 2(1957), 323-43. V. B. Lawson, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, A p r i l 1961. 44. R. F. Snowball, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, October i960. 45- R- W. Fraser, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, November i960. 46. A.S.M. Metals Handbook, 1961, 962:.. 47. H. C. Rogers, Trans. A.I.M.E., V o l . 218(1960), 498. 48. J . D. Eshelby, F. C. Frank and F. R. N. Nabarro, P h i l . Mag., Vol- 42(1951) 351' 49. J'. P. F r a n k e l , Acta. Met., V o l . 6(1958), 215-50. P. R. Thornton and P. B. H i r s c h , P h i l . Mag., V o l . 3(1958), 738. 51. Z. S. B a s i n s k i , Aust. J . Phys., V o l . 13(1960), 284. 52. A. N. Stroh, Proc. Phys. S o c , V o l . 67B(l954), 427-53. Reference l 6 . Page 74. APPENDICES APPENDIX I. APPENDIX I.  Calculation of Parameters a and b The parameters a and be represent the intercept and the slope respectively of the log-log plot of the stress-strain curve. The values for each specimen were found by calculating the stress at each of a number of points picked from the Instron plot of load versus elongation and fitting these to an expression of the form . log £ = a + b log £ by the methods of least squares. Figure Ul. shows a schematic representation of the Instron plot together with a correction curve which compensates for slackness in the grips and for the inertia of the recording system. A l l elongation measurements are taken with reference to this correction curve. The stress (5 and the strain £ at point A are calculated as follows: V c L V-g = cross-head speed V c = chart speed L = gauge length (measured between the grips) . 6 r = — (1 +C) P = load a = i n i t i a l cross sectional area o The factor ( l +£•) takes into account the reduction in cross-sectional area, due to strain and is derived as follows: The in i t i a l volume of the specimen is VQ = aQ L. When the specimen is extended an amount ^ L , the volume becomes V l = a i (L +A L). where and where / Correction / Curve f Instron t Plot / / P r • 1 Elongation Figure kl.. Schematic Representation of the Load-Elongation plot from the Instron Recorder. Assuming V-^  = YQ: a n (L + & L) = a L l o §1 = 1 + ^ = 1 + £ a o L a l = a o 1 +£ The l o g and l o g ^  are determined f o r each p o i n t picked from the I n s t r o n p l o t . Since the r e l a t i o n s h i p •' i s of the form l o g 6"* = a + b l o g € , the values of a and b which s a t i s f y the l e a s t squares s o l u t i o n are those which s a t i s f y the equation: n a + b £ log£ = ^logS* a£log € + b5"(log€ f = £log£ log^ For specimen 1324, whose l o g - l o g p l o t forms two s t r a i g h t l i n e s , the f o l l o w i n g sets of equations were found: 18 a± + I9.9U3 b x = 59-760 19-9^3 a x + . 22.693 b 1 = 66A70 which gives a l = 2.836. b x = 0.1+37 and 10 a 2 + 14.566 b 2 = 3U.527 14.566 a + 21.274 b 2 = 50.310 which gives • aJ = 3.004 b 2 = 0.308 where a 1 and a 1 are c a l c u l a t e d f o r £ expressed as a percentage. These are r e f f e r r e d to £ expressed i n i n / i n by a 1 = a 1 - 2 b 1 = I.962 a 2 = a 1 - 2 b 2 = 2.388 The correlation coefficient, r, was calculated for randomly selected specimens as shown below: log £ logC - log logC 6*log6* ^ logc 2 where p For the above specimen, the values of r found by this method were r^ s 1 .000 and r^ = 0.995- The values of r found for other specimens were also a l l very close to 1 .0 indicating a close f i t of the points to the equation since a perfect f i t corresponds to r = 1 .0 . - 1 0 0 -0 . 6 0 . 8 l . o 1 . 2 l.k 1 . 6 log °jo Elongation Figure k2. a. 1D1 log °jo Elongation Figure 42. b. Figure U-2. c. Log-Log Plots of the Stress-Strain Curves for Representative Specimens. APPENDIX II. _ 103 -TABLE VI. The Results of Tensile Tests of Polycrystalline Copper Wires a. Temperature = 293°K Strain Rate =0.02 min"1 D e c i m e n Diameter (microns) Total Strain UTS (Kg/cm2) Yield Stress YS , YS TKg/cm2) ° a i b l &2 h 2 B 9 926 37-1 3080 56O 680 1.966 0.435 2. .261 0.345 BIO 926 44.8 3280 590 700 1.898 0.458 2.425 0.299 B l l 930 41.6 3200 520 660 1.960 0.438 2. •993 0.317 B12 922 38.3 3100 560 680 1.954 0.443 2, •371 0.315 B13 932 37-3 3060 490 630 1-971 0.430 2. .220 0-324 Bl4 932 37-9 3100 540 650 1.962 0-437 2.388 0.308 B 1 5 917 36.5 3100 560 690 1.911 0.459 2. •373 0.316 Bl6 930 37-3 3070 570 690 1.941 0.444 2.360 0.316 BIT 930 38.0 3080 54o 680 1.893 0 . 46 i 2, .988 0.318 C 3 518 36.9 3070 530 650 1.924 0.450 2.438 0.294 c 4 518 38.2 3270 560 700 1.842 0.483 2. .444 0.298 C 5 518 38.2 3200 560 680 1.799 O.495 2.405 O.308 C 6 518 36.7 2880 - - 1.902 0.449 2. •357 0.310 C 7 518 38.4 2900 490 600' I.889 0-453 2. •977 0.309 C .8 518 38.3 2900 .490 610 1.880 0.456 2. .406 0.296 Cl4 518 37-9 3100 500 64o I.891 0.462 2.366 0.318 F 1 203 29.0 .3150 610 720 1.951 0.455 2-346 0-333 F 2 203 31-5 3200 570 710 I.926 O.458 2. 270 0-355 F 3 203 33-2 3280 570 710 1.867 0.482 2, .220 0.370 F 5 203 31-9 3250 590 710 I.909 0.468 2. • 322 0.340 F 7 203 29-3 3120 550 700 I.918 0.463 2, • 340 0-333 F 8 203 31-3 3170 550 - 700 1.746 0.515 2. •337 0.333 F 9 203 33-9 3240 590 720 1.923 0.461 2, .297 0.345 F10 203 31-6 3200 590 700 1.880 O.477 2. • 357 0.328 G 5 100 18.8 2540 _ _ 1.863 0.473 _ G 6 . 100 19.5 2560 460 610 1.820 0.486 -G 8 100 19.8 2600 480 670 1.867 O.472 -G 9 101 21.3 264o 470 600 1.894 o.46i G10 101 16.9 2400 420 600 1.825 0.483 -I 1 48 13-6 2460 570 730 1.884 0.482 _ I 2 48 14.9 246o 550 700 1.956 0.450 -I 3 48 24.2 2800 590 700 1.765 0.498 -- 104 -b. Temperature = 2 9 3 ° K Strain Rate = 0 . 8 min ^ Diameter Total Strain UTSg Specimen (microns) ($) . (Kg/cm ) a - j _ • a^  b^ IB 1 946 2 8 . 1 2 6 5 0 2 .074 0 . 3 9 8 2 . 3 3 1 0 . 3 1 7 IB 2 914 2 5 . 2 2 6 2 0 2 . 1 9 0 O .368 2 . 451 0 . 2 8 6 IB 9 9 5 0 2 8 . 8 2770 2 . 0 2 9 0 .418 2-^73 0 . 2 8 1 1B10 • 9^7 • 2 9 . 4 2800 2 . 1 3 0 0 . 3 9 1 2 .461 0 . 2 8 6 1B11 952 2 9 . 1 2 7 8 0 2 . 0 5 2 0 . 4 1 5 2 . 5 0 8 0 . 2 7 1 i c 6 502 2 3 - 0 2570 2 \174 0 . 3 7 2 2 . 3 1 8 0 . 3 2 6 1C 7 495 24 . 2 2 7 0 0 2 . 1 3 3 0 . 3 9 1 2 - 3 7 5 0 . 3 1 3 1C 8 493 2 3 . 4 2670 2 .249 0 . 3 5 7 2 . 6 0 1 0 .245 i c 9 495 2 3 - 6 2 6 6 0 2 .174 0 . 3 7 7 2 .444 0 . 2 9 1 1C10 4 8 8 2 2 . 3 . : 2570 2 . 1 6 0 - 0 . 3 7 9 2 . 401 0 . 3 0 2 1C12 512 ' 2 3 . 7 2760 2 . 2 1 2 0 . 3 6 6 2-577 0 . 2 5 0 1C14 500 2 3 - 5 •2860 2 . 2 6 8 0 . 3 5 4 2 . 6 7 0 ' 0 . 2 2 7 IC15 503 2 2 . 3 2 6 9 0 2 . 2 5 7 0 . 3 5 0 2 . 5 4 8 0 . 2 5 0 IF 1 '• 211 1 7 . 8 2 5 0 0 2 . 1 7 5 0 . 3 7 7 - -IF 3 215 . 1 7 - 8 2 5 8 0 2 . 3 0 3 0 . 3 4 3 - -IF 4 257 1 5 . 8 2 3 0 0 2 . 420 0 . 2 9 9 - -1G 2 . . 95 1 7 . 3 2 6 6 0 2 . 3 2 6 0 . 3 4 0 -1G 3 100 1 5 - 3 2590 2 . 4 6 7 0 . 2 9 6 •- -1G 4 9 8 24 . 2 2850 2 . 9 7 2 0 . 3 5 3 - -1G 5. 99 1 7 . 8 2600 2-324 0 . 3 3 7 - -c. Temperature = 78°K .Sp< )ecimen Diameter (microns) Y i e l d YS 1 (Kg/. Stress YS r-cm2) - 5 . a l b l L1BI3A 903 1150 1250 1.996 O.487 L1B13B 903 1150 1250 I . 9 4 4 0.503 L1B14A 928 1060 1190. 2.032 0 . 1+68 L1315A 917 - - 1.920 0.503 L1B16A • 928 850 1150 I . 9 6 7 0.488 L1317A 897 1060 1190 2.110 0.455 L1B17B 897 • 1050 1190 2.024 0.480 L1C22A 515- _ _ I .96O 0.500 L1C24A 501 1210 1370 2.028 0.479' L1C24B 501 1180 13^0 2.043 0.475 L1C25A 475 1290 ' 1500 2.096 0.473 L1C25B 475 1240 1500 1.992 0.506 L1C26A 490 l l 6 0 1410 2.591 0 A 6 5 L1C26B 490 l l 6 0 1410 - -L1C27A 474 . - - 2.095 0.476 L1F 6A 215- 920 1100 I . 6 7 6 0.482 L1F 6B 215 890 1080 1.954 0.493 L1F 7A 204 960 1210 1-916 . 0.514 L1F 7B .204 96Q 1180 -L1F 9A 218 • 1020 1140 1.936 0.576 L1F 9B 218 1010 1150 1-9^3 O.500 L1F11A 218 1030 1200 1.880 • O.521 L1G 6 99 810 1080 2.085 O.456 L1G 7 102 780 970 1.865 O.508 L1G 8 98 830 960 1.847 O.520 L1G 9 100 850 960 I . 8 5 6 0.515 L1G10 97 - - 1.802 0.536 L1G11 101 - - 1.468 O.619 APPENDIX III. TABLE -VII. Effect of Strain Rate, and Temperature on" the  Flow-Stress of Polycrystalline Copper a. Strain Rate' • , . £:/= l O ' ^ m i n " 1 , ^ -10"1min"1 ' ' Ratio of Specimen Diameter Strain Flow-Stresses . mm •- SX2 3-5 RC3 3-8 2B1 0.9 RC4 3.8 113 0.05 0.068 0.986 0.098 0.985 0.126 0.987 0.164 0.985 0.206 0.985 0.251 0.982 0.292 0.979 0.3W 0.975 0.005 ' 0.935 0.012 0.972 0.024 0.977 0.051 0.978 0.080 . 0.978 0.106 "0.978 0.138 0.977 0.171 0.977 0.203 0.976 0.238 0.978 0.270 0.978 0.308 0.977 0.366 0.975 0.048 0.980 0.130 0.979 0.164 0.978 0.048 ,0.980 0.102 0.980 0.139 0.978 0.226 0.980 0.268 0.979 0.174' 0.979 0.005 0.974 0.024 • 0.973 0.036 0.974 0.056 0.974 0.075 0.975 Specimen Diameter mm Strain Ratio of Flow-Stresses 115 0.05 0.003 0.931 0.022 0.974 0.043 0.974 0.062 0.975 O.O83 0.975 0.111 0.975 0.122 O.976 0.139 0.973 0.152 0.974 0.166 0.974 0 . l 8 l 0.974 1F12 0.2 0.009 O.96I 0.025 O.969 • 0.042 0.970 - O.O58 O.968 0.077 0.971 0.094 0-970 0.110 0.969 0.131 . O.968 O.I5I O.969 0.171 .0.970 O.I89 O.969 0.205 O.969 0.221 0.970 0.239 O.968 0.257 O.968 0.275 0.970 0.295 0.970 O.316 0.970 1F16 . 0.2 . 0.055 0.970 0.068 0.970 0.081 O.969 O.O96 O.968 0.109 0.970 0.122 O.968 O.I36 O.967 0.149 O.968 0.162 O.969 , 0.177 O.967 0.202 O.968 0.215 O.968 0.229 0.970 0.243 O.968 0.257 O.969 0.272 O.969 0.285 O.967 Ratio of Specimen Diameter Strain Flow-Stresses mm 1B22 0.9 IB.26 0.9 Temperature T 1 = 2 9 3 % T 2 SXl 4.0 0.007 O.96O 0.031 O.966 0.048 0.965 0.070 0.967 0.091 O.968 0.110 O.966 0.130 0.964 0.151 0.964 0.168 O.965 0.191 O.963 0.217 0.964 0.246 0.963 0.274 0.963 0.302 O.965 0.325 0.965 0.348 O.965 0.014 0.938 0.039 0-957 0.059 O.966 0.094 O.967 0.110 O.965 0.132 0.964 0.157 0.964 0.180 O.966 0.204 0.963 0.228 0.9.64 0.253 O.965 0.281 O.963 0.309 O.966 0.332 O.963 0.358 O.960 O.387 O.961 78°K 0.020 0.881:. 0.055 0.886 O.O87 0.888 0.124 0.890 0.158 0.889 0.202 0.894 0.252 0.894 0.312 0.881 0.346 0.882 Specimen Diameter ' mm Strain Ratio of Flow-Stresses SX3 3.8 O.O38 O.87O 0.077 O.893 0.122 O.889 0.179 0.886 0.228 0.888 O.287 0.886 0.340 0.880 0.4l4 0.881 0A98 0.880 SX4 3.8 . 0.031 0.880 0.120 0.886 0.183 0.885 0.255 0.881 RC1 4.6 0.054 O.885 0.074 0.884 0.094 0.879 0.115 . 0.881 0'. 136 0.877 0.156 0.874 0.178 0.871 0.197 0.873 0.216 0.867 0.226 0.851 0.273, . 0.853 RC2 3.5 0.012 O.885 0.031 0.884 0.051 0.886 0.093 0.875 0.091 0.879 0.105 • 0.876 0.120 O.874 0.137 0.874 0.153 0.874 0.167 0.866 RC5 3.9 0.028 0.877 • 0.072 0.887 0.129 0.890 0.172 0.884 0.214 0.888 0.258 0.884 O.303 0.884 O.3U8 0.885 0.392 0.887 Specimen Diameter Strain Ratio of Flow-Stresses - 110 RC6 4 . 0 0-022 . 0 . 8 8 0 " 0 . 0 4 8 0 . 8 8 5 0 . 0 7 5 0 . 8 8 8 " 0 . 1 0 2 0 . 8 8 6 0 . 1 2 9 0 . 8 8 6 0 . 1 5 2 0 . 8 8 3 0 . 1 7 9 0 . 8 8 3 0 . 2 0 5 0 . 8 8 0 "' 0 . 2 5 2 • 0.882 "1F13 0.2 0 . 0 1 6 O . 8 7 9 0 . 0 3 8 0 . 8 8 0 0 . 0 5 8 0 . 8 7 3 0 . 0 7 5 0 . 8 7 2 0.100 ' 0 . 8 6 5 .0.124 0 . 8 6 3 0 . 1 5 0 0 . 8 6 0 . 0 . 1 7 9 0.849 0.212 O . 8 5 5 0.245 0.848 0.278 0.846 LB24 . 0-9 0.010 0.846 0.024 O . 8 7 6 " C.OUO 0 . 8 7 9 O.O56 O . 8 7 7 " - '0.071 O . 8 6 9 0 . 0 9 4 " 0 . 8 6 8 0 . 1 1 5 0.864 O . 1 3 5 0 . 8 6 2 O . 1 5 5 O . 8 5 9 , O..176 O . 8 5 7 0.204 0 . 8 5 1 0 . 2 2 3 0.846 0.249 0.848 LB27 0.9 0 . 0 1 7 0 . 8 5 6 0 . o 4 l 0 . 8 8 2 . O . 0 6 7 0.868 . 0 . 0 9 8 0 . 8 6 2 0 . 1 3 2 0 . 8 5 9 0 . 1 7 1 0 . 8 5 6 0 . 2 1 3 0.848 0 . 2 4 9 0.845 0 . 2 9 4 0 . 8 3 7 APPENDIX IV. TABLE VIII. Effect of Temperature on.the Flow-Stress of Aluminum a. Cottrell-Stokes Law Tl = 2 9 3 % T 2 = T8°K Ratio of Specimen Strain Flow-Stresses SA5 0.138 0-TT6 0.184. 0.771 0.212 0-TT3 0.243 0.TT2 0.2T4 O.T69 0.300 0.TT4 0.334 0.T68 O.362 0.TT4 SA6 0.043 0-TT3 0.053 0-TT6 0.068 0.TT4 0.08T 0.TT2 0.112 0.771 o.i4o 0.771 0.150 0.771 0.153 0.774 0.169 0.770 0.205 0.771 0.24T 0.772 0.304 0.774 0.363 0-773 SAT 0.138 0.772 0.169 • 0.777 0.208 0.770 0.280 0.776 0.305 0.770 b. Work-Softening T l = 293°K T 2 = 78°K Specimen €' Strain SA8 0.026 0.080 1.131 0.118 1.034 0.156 0-975 0.193 0-955 0.231 0.971 0.271 0-955 0.308 0-979 0-346 0-975 Specimen £•' S t r a i n SA9 0.050 0 . 0 9 2 1 . 4 7 1 0 . 1 5 7 1 . 0 8 2 0 . 3 1 4 1.021 O . 3 7 8 1 . 0 1 5 0 . 4 4 1 1 . 0 0 6 0 . 5 0 5 O . 9 8 6 SA10 0 . 0 1 3 0 . 0 5 3 1 . 1 4 1 0 . 0 7 7 1 - 0 3 3 0.101 1 . 0 0 3 0 . 1 2 5 0 . 9 9 0 0 . 1 4 9 0 . 9 7 8 -0 . 1 7 3 0 . 9 7 9 ' 0 . 1 9 7 O . 9 6 3 0 . 2 2 1 0 . 9 7 0 0 . 2 7 0 0 . 9 6 4 0 . 2 9 4 O . 9 8 3 0 . 3 1 8 0 . 9 7 2 0 . 3 4 2 0 . 9 7 1 O . 3 6 6 0 . 9 5 1 0 . 3 9 0 O . 9 6 2 0 , 415 O . 9 5 8 SA11 0 . 0 0 5 0 . 0 4 1 I . 0 6 5 0 . 0 8 7 0 . 9 8 8 0 . 1 3 0 0 . 984 O.I78 O . 9 6 9 0 . 2 2 5 O . 9 6 2 0 . 2 7 1 O . 9 6 7 0 . 3 1 7 0 - 9 5 9 .0 .364 O . 9 6 3 0.020 . 0 . 0 7 1 - I.O96 0 . 1 1 6 O . 9 8 8 0 . 1 6 2 0 . 9 7 1 0 . 2 0 9 O . 9 3 8 0 . 2 5 6 O . 9 6 2 0 . 3 0 2 O . 9 5 8 0 . 3 4 7 O.95O SA12 0.040 0 . 0 8 0 1 . 3 0 0 0 . 1 3 0 1 . 0 2 8 0 . l 8 l 1.020 0 . 2 3 2 0 . 997 0 . 2 8 2 O . 9 8 8 SA13' 0 . 0 0 3 0 . 0 4 2 1 . 0 0 3 0 . 1 3 3 O . 9 6 9 0 . 2 2 4 0 . 9 5 2 0 . 3 3 9 0 . 9 7 3 0 . 4 2 9 O . 9 6 7 0 . 0 1 3 O . 3 2 6 O . 9 6 2 O . O 6 7 0.120 1 .542 0 . 2 1 1 1 . 0 6 0 0 . 3 0 1 1 . 0 1 5 0 . 4 i 6 0 . 9 9 0 APPENDIX V. Estimated Errors 1. Size Effects The estimated error in the yield stress, ultimate tensile strength and elongation before fracture are calculated for a representative specimen, Clk. The diameter was measured to ± lyu but difficulties in measuring a true diameter reducedthe accuracy to about ± 3 Diameter = d = 518 ± 3 fx = 518 u ± 0.6$ Area = a = TT d 2 = 2.11 X 10~3cm2 ± 1.2$ ~k~' The Instron tensile tester is rated by the manufacturer at an accuracy of better than ± 1 $ of full-scale load when the slope of the load elongation curve is such that more than three seconds is required for the pen'..to travel the f u l l width of the chart. Load' at yield, P = 2.86 * 0.05 Ih = 2.86 lb ± 1-7$ Since £ is very small at the yield point, errors in the (l + € ) term wil l have a negligible effect. ¥ S 0 . = o.k^k P (1 * 0.017) ( 1 + € ) J a (1 * 0.012) - v ; = 6k0 Kg/cm2 ± 3$ = 640 ± 20 Kg/cm2. The gauge-length was measured to ± 0.01 mm but difficulties in measurement reduced the accuracy to about t 0.05 mm. Gauge length = 62.41* 0.05 mm = 62.kl mm i 0.1$ The strain was .-calculated to be 0.001-(l * 0.00l)(x ± l ) where x is as defined in Appendix I. The elongation before fracture: £= o.ooi ( i * o.ooi) 379 ( l * 0.003) = 0.379 ( l * 0.004) = 0.379 ± 0.002 The ultimate tensile strength is calculated by the same method as the yield stress. p. = 10.6 ± 0.2 lb = 10.6 lb ± 2 $ 1 + € = 1-379 ± 0.002 - = 1-379 (1 ± 0-002) UTS = 3100 (1 ± 0.034) Kg/cm2 = 3100 ±"100 Kg/cm The limits of accuracy found for the other specimens tested for size effects were found to be of the same order as those calculated for specimen C14. 2. Effect of Temperature and Strain Rate on the Flow-Stress Since only a ratio of flow-stresses was required, only the loads were measured and the only errors are those due to the limits of accuracy of the machine. The Instron tensile tester is equipped with a zero suppression by means of which the area of the load-elongation curve to be studied can be magnified, thus increasing the accuracy. The max-imum estimated error for a l l specimens varied from about 0.8$ at low strains to about 0-1 -$ at high strains.  THE EFFECTS OF SIZE, TEMPERATURE AND STRAIN-RATE ON THE MECHANICAL PROPERTIES OF FACE-CENTERED.CUBIC METALS ' by '. RONALD ALBERT JOSEPH COSTANZO A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE DEPARTMENT OF MINING AND METALLURGY We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF APPLIED SCIENCE Members of the Department of Mining and Metallurgy THE UNIVERSITY OF BRITISH COLUMBIA October 1961 ! In presenting 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 requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree 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 reference and study. I f u r t h e r agree that permission f o r extensive copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of 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 r k w e d without my w r i t t e n permission. Department of / / t ^ s ^ <? AV£4MXO*SL; 7f The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3, Canada. Date ABSTRACT Drawn and annealed copper wires of diameters ranging from 50 u to 900 }i were t e s t e d i n t e n s i o n and the r e s u l t s examined f o r evidence of s i z e - e f f e c t s . No s i z e - e f f e c t on y i e l d - s t r e s s or work-hardening r a t e has been d e f i n i t e l y e s t a b l i s h e d . The r e s u l t s were discussed i n terms of the f r a c t i o n of the number of grains i n the specimen which have a f r e e surface. The u l t i m a t e t e n s i l e s t r e n g t h and d u c t i l i t y decrease w i t h decreasing d i a -meter f o r diameters below 200 yu. An ex p l a n a t i o n has been put forward i n terms of void-formation during deformation. P o l y c r y s t a l s and s i n g l e c r y s t a l s of copper were t e s t e d a t room -1 -1 ~3 -1 temperature w i t h the s t r a i n - r a t e c y c l e d between 10 min and 10 mm P o l y c r y s t a l l i n e copper obeys the C o t t r e l l - S t o k e s law but shows a v a r i a t i o n i n the r a t i o of f l o w - s t r e s s e s w i t h v a r y i n g g r a i n diameter and w i t h a va r y i n g value of the f r a c t i o n of grains i n the specimen which show a f r e e surface. Copper specimens were a l s o t e s t e d w i t h the temperature c y c l e d between rj8QK and 293°K- Copper p o l y c r y s t a l s do.not obey the C o t t r e l l - S t o k e s law, the d e v i a t i o n depending on the g r a i n s i z e . These r e s u l t s are discussed i n terms of s t a c k i n g - f a u l t energy and s e v e r a l p o s s i b l e explanations are considered. Aluminum p o l y c r y s t a l s were t e s t e d w i t h the temperature c y c l e d between 78 K and 293 K. Aluminum obeys the C o t t r e l l - S t o k e s law f o r temperature v a r i a t i o n . A work-softening e f f e c t accompanies the y i e l d -drop found a t 293°K a f t e r p r i o r deformation at 78°K. This was discussed i n terms of c r o s s - s l i p and d i s l o c a t i o n climb mechanisms. v i i . ACKNOWLEDGEMENT The author i s g r a t e f u l f o r the advice and encouragement given by h i s research d i r e c t o r , Dr. E. Teghtsoonian. Thanks are extended t o Mr. R. G. B u t t e r s f o r t e c h n i c a l a s s i s t a n c e and t o f e l l o w graduate students, e s p e c i a l l y Mr. K. G. Davis, f o r many h e l p f u l d i s c u s s i o n s . S p e c i a l thanks are extended t o Miss I . Duthie who a s s i s t e d i n many of the r o u t i n e c a l c u l a t i o n s . This work was fi n a n c e d by Defence Research Board Grant No. 7 5 1 0 - 2 9 . i i . TABLE OF CONTENTS Page INTRODUCTION , 1 PREVIOUS WORK 3 1. Si z e E f f e c t s 3 a. C r y s t a l P e r f e c t i o n k-b. E f f e c t of Surface Conditions 7 c. E f f e c t s of V a r i a t i o n i n the Length of the S l i p - P l a n e 10 d. G r i p E f f e c t s 12 e. Size E f f e c t s i n P o l y c r y s t a l l i n e Wires 15 2. C o t t r e l l - S t o k e s Law . • . 15 3- Comments 20 EXPERIMENTAL PROCEDURE 22 1. M a t e r i a l s 22 2. Specimen P r e p a r a t i o n . . . 22 a. Copper Wires . 22 b. Copper Rods 23 c. Copper S i n g l e C r y s t a l s 23 d. Copper S t r i p 25 e. Aluminum S t r i p . . . 25 3- Measurements v 25 k. T e s t i n g Procedure 26 a. Normal T e n s i l e Tests 26 b. V a r i a b l e Strain-Rate Tests ' 26 c. V a r i a b l e Temperature Tests . . . 28 EXPERIMENTAL RESULTS AND OBSERVATIONS ' . . 29 1. Size E f f e c t s 29 a. Y i e l d Stress 29 b. Ul t i m a t e T e n s i l e S t r e s s 30 i i i . TABLE OF CONTENTS CONTINUED .Page c. Ductility . . . . . . 3 0 &. Work-Hardening Rate 3 2 2 . Cottrell-Stokes Law 3 8 a. Effect of Strain-Rate on Flow Stress 3 8 b. Effect of Temperature on Flow Stresses ~.kQ c. Results of Tests on Aluminum . . . • 5 8 SUMMARY OF THE RESULTS 6 3 DISCUSSION .' 6 5 1 . Size Effects 6 5 a. Yield Stress and Work-Hardening Rate 6 5 b. Ductility and Ultimate Tensile Strength 6 8 2 . Cottrell-Stokes Law . ' . - TO a. Work-Hardening . . . . . 7 1 b. Theoretical Considerations 73 c. A Possible Effect of Dislocation Density . . . . . . 7 6 d. Grain Boundaries 7 8 e. The Effect of Stacking-Fault Energy . . . . . . . . 8 l f. A Possible Effect of Dislocation Density on 6 5 ' . . 8 5 3. Work-Softening in Aluminum 8 7 SUMMARY AND CONCLUSIONS 9 1 SUGGESTED FUTURE WORK 92 BIBLIOGRAPHY 93 APPENDICES, .95 iv. FIGURES Page 1. Tensile Strengths of Silicon Rods and Whiskers 5 2. Schematic Representations of a Stress-Strain Curve for a Face-Centered Cubic Single Crystal 5 3- Strength Versus the Inverse of the Diameter for (a) Copper and (b) Iron Whiskers : • 6 k. Variation of the Strength of Silicon Whiskers with Temperature 6 5- Effect of Diameter on the Crit ical Shear Stress of Cadmium Single Crystals 9 6. Effect of an Oxide Layer on the Stress-Strain Curve for Single Crystals of Silver . . . . . . . 9 7- Variation of Flow Stress with Diameter for Copper Whiskers . . I**-8. Ratio of Flow Stresses at 293°K'and 90°K for Aluminum Single Crystals 1 Q -.5 > -K -1 9- Ratio of Flow Stresses at Strain Rates of 10 and 10 sec for Copper and Aluminum • 18 10. Ratio of Flow Stresses at 293°K and 78°K for Copper Polycrystals 19 11.. Electrolytic Cell and Power Source Used in Electropolishing Copper Wires 2k 12. Gripping System for Low-Temperature Tests on (a) Wire and Rod Specimens and (b) Strip Specimens 27 13> Yield Stress of Polycrystalline Copper Wires as a Function of the Diameter ? . . 33 lk. Ultimate Tensile Strength of Polycrystalline Copper Wires as a Function of the Diameter . . . . • . 3* 15. Ductility of Polycrystalline Copper Wires as a Function of the Diameter 35 16. Work-Hardening Rate of Polycrystalline Copper Wires as a Function of the Diameter . . . . . . . . . . . . . . . . 36 17. Ratio of Flow Stresses for Specimens 1322 and IB26 ; . . . . . . ^1 18. Ratio of Flow Stresses for Specimens IF 12, 1F16, 113, and 1113 . h r 2 19-. Ratio of Flow Stresses for Specimens 2B1, RC3, and RGk . . . . ^3 20. Ratio of Flow Stresses for Specimen SX2 ^3 FIGURES CONTINUED Page 21. Figures 17 to 20 Plotted Together for Comparison hh 22. Stress-Strain Curve for Specimen 1B22 ' U5 23. Stress-Strain Curve for Specimen RC3 • • • h6 2k. Stress-Strain Curve for Specimen SX2 . . • k-7 25. Ratio of Flow Stresses for. T = 293°K, T = 78°K for Polycrystal-line Copper . . . . . . . . . . . . . . . . . 51 26. Ratio of Flow Stresses for T-^ 293°K, T2= 78°K for Single and Polycrystals of Copper . . . 52 27. Stress Strain Curve for Specimen 1B2U '. 53 2'8. Stress-Strain Curve for Specimen RC6 5^ 29. Stress-Strain Curve for Specimen RC6 after the appearance of a visible neck . . . . . . 55 30. Stress-Strain Curve for Specimen SX3, Part 1 56 31. Stress-Strain Curve, for Specimen SX3, Part 2 57 o o 32. Ratio of Flow-Stresses for T-^ 293 K, T '= 78 K for Aluminum Polycrystals. .......... ^ ..... . 59 33- Ratio of Flow-Stresses for T ^ - 2 9 3 % Tg= 78°K for Strip Copper 60 3^. Stress-Strain Curve for Aluminum Specimen SA2 . . . . . . . . 6l 35- The Work-Softening Ratio as a Function of Strain for Various €' 62 36. Variation of the'Ratio B/A with Specimen Diameter . . . . . . 66 37- The Separation of the Flow Stress into the Elastic and Temp-erature-Dependent Components . 7^ 38. Representation of a Glide Dislocation Approaching a Row of Forest Dislocations 77 39* Stress-Concentration Factor in Front of a Dislocation Pile-up of Length L . 80 kO. A Schematic Representation of the Form pf the Stress-Strain Cugve, for Deformation at 293°K after Prior Deformation at 78 K . . . . . . . . . . ; . . . . . . 90 hi. Schematic Representation of the Load-Elongation Plot from the Instron Recorder 97 k2. Log-Log Plots of the Stress-Strain Curves of Copper Polycrystals Under Various Test Conditions 100 v i . .TABLES Page I. Ratio of Flow Stresses for T^ = 293°K, T2='78°K for Various Metal's . . . . . . . . . . . . . . . . . 17 II. Characteristics of Specimens Tested for Size Effects . 31 III. Characteristics of Specimens Tested for the Effect of Strain-Rate on Flow Stress . . 1+0 IV. Characteristics of Specimens Tested for the Effect of Temperature on Flow Stress 50 V. Stacking-Fault Energies and Cottrell-Stokes Ratios for Cu, Ag, Al and Mg 82 VI. Results of Tensile Tests on Polycrystalline Copper Wires 103 VII. Effect of Strain-Rate and Temperature on the Flow-Stress of Polycrystalline Copper ' 106 VIII. Effect of Temperature on the Flow-Stress of Polycrystalline Aluminum ' i l l INTRODUCTION Interest in the effect of diameter on the strength of crystals has, in the last ten years, grown out of whisker research. Investigations of the mechanical properties of whiskers have been quite extensive and have produced some interesting results, the one of main interest here being that the strength of a whisker depends in some manner on its diameter. Interest in whiskers developed in the early 1950's when Compton, Mendizza and Arnold"'" discovered whiskers growing in electroplated capacitors. , 2 Gait and Herring performed the first tests, a simple bend test, on tin whiskers and found a high elastic limit (as high as 2$). Interest in the effect of crystal size on the strength of metals has been aroused by Brenner's discovery that the strength of a whisker is a function of its diameter. Some investigations of the variation of strength with diameter have been carried out on single crystals produced from bulk material but have produced no firm explanation for the diameter dependence. Also, virtually a l l of the work has been'.;done using 'single crystals and the work on polycrystals has not produced particularly good results. Another line of investigation concerning the mechanism of work-hardening in face-center cubic crystals has been pursued since the early 1950's. These investigations have centered around the effects of temp-erature and strain rate on the flow stress of such metals as copper, aluminum and silver. The usual technique used for comparison of properties 1). at different strain rates and temperatures is that originated by Hollomon in 19^6. At this time the existence of a mechanical equation of state relating the flow stress to the strain, strain rate and temperature and - 2 -the exact form that i t would take was under investigation. The mechanical equation of state demands that where & = true stress C = true strain c = true strain rate T = temperature i .e . the flow stress is a function of the instantaneous values of the strain, strain rate and temperature, independent of the thermal and mechanical history of the metal. 5 In 19^8, Dorn, Goldberg and Tietz , using this technique, showed that the mechanical equation of state is not valid but that the • flow stress also depends on the thermal and mechanical history of the 6 metal. It was later realized by Cottrell and Stokes that the results of this type of test demonstrated that the difference in flow stress when a metal is deformed to a given strain at different temperatures arises from two distinct contributions: 1. The difference arising from the variation of dislocation configuration due to strain at different temperatures, and 2 . The "reversible" contribution of thermal fluctuations. The type of test mentioned show only the "reversible" part of the dependence and therefore may be used to study the form of the inter-actions between dislocations.. As the work in both of-these fields has been confined almost exclusively to single crystals, this investigation wil l be concerned with size-effects and with the temperature and strain rate dependence of the flow stress in copper polycrystals. - 3 -PREVIOUS WORK !.• Size Effects When crystals of diameter below a cr i t ical value are deformed, variations in the properties of those crystals are found. These variations, since they are a function of the diameter, are usually termed "size effects". The problem of diameter-dependence of these properties has been attacked from two directions; from tests on whiskers of various diameters and from tests on crystals produced from bulk material by various methods of diameter reduction. The f irst report of a size effect in metals was that by Taylor^ in 192U. Taylor produced fine filaments of various metals with diameters -k as small as 10 cm. by drawing out molten metal in a glass tube. Although no determination of the microstructure of these wires is mentioned, i t may be assumed that they are of a bamboo structure since the final product was formed by solidification along the length of the wire. It was found that these wires had a very high ductility: "Though britt le in bulk, bismuth and antimony are very pliable in the form of fine wires.- Antimony wire as large as 0.003 cm. diameter may be bent repeatedly without breaking". / 2 The tensile strength of antimony wire was reported to be 1800 to 2200 Kg/cm and that of bismuth about 50 Kg/cm . In comparison with this, the tensile 8 / 2 strength of bulk antimony is about 110 Kg/cm . Because bismuth is so britt le in bulk form,' no tensile strength has been reported. Taylor gave no details of testing method. No real interest in this phenomenon, however, developed until the advent of whisker research. Size effects in the strength of whiskers have been found by Brenner^ in copper and iron, and by Gyulai^ in NaCl whiskers. Single crystals of copper"^' and aluminum"^ prepared from bulk material have shown a size effect and a comparison of the strengths of silicon whiskers 13 and rods prepared from bulk crystals has shown that they are similar for similar diamters (Figure ! . )• The parameters normally measured in investigations of size-effects in face-centered cubic single crystals'are 6* - ^ 7 ^ 7 £ ^ y g ^ and 0£ which are defined in Figure 2. Generally, the following effects have been observed with decreasing diameter: 1. cfpcT' 2 increase 2. £"2 increases 3. © 1 7 Q 2 decrease A number of theories, put forward to explain these effects, wil l be discussed below. a. Crystal Perfection Brenner's results-^ for copper and iron whiskers (Figure 3-) show . that the strength is proportional to the inverse of the diameter. If, as has been postulated, the high strength of whiskers were due to the small volume resulting in a low statistical chance of the inclusion of a dis-location source, then i t would be perhaps more reasonable to expect that the strength should be dependent on the inverse of the square of the diameter, as has been found by Eder and Meyer1**' and tentatively confirmed by Moore" .^ Further, assuming that the strength depends only on the volume of the whisker and taking Eder and Meyer's results to be true, then there should also be a length dependence such that the strength is proportional to the inverse of the length. A computation of the expected length depend-ence from Moore's results on diameter dependence shows that this length dependence should be experimentally observable. No such length dependence 15 has been found 1 6 Also, according to Cottrell , the yield stress of a perfect crystal should be insensitive to temperature. The results on silicon 13 . 0 whiskers show that the yield stress at 650 C is more than double the - 5 A o 0 R O D S o W H I S K E R S - • J . . » 0 0 x 1 1 . C R O S S S E C T I O N IN S O U A R E C E N T I M E T E R S Figure 1. Tensile Strengths of Silicon Rods and Whiskers. Reproduced from Reference 12. Stress JBlope= Qx -^"TJlope^eT 4 i i 1 i i Strain Figure 2. Schematic Representation of a Stress-Strain Curve for a Face-Centered Cubic Single Crystal Showing the Parameters Mentioned in the Text. Figure 3.. Strength Versus the Inverse of the Diameter for: (a) Copper and ("b) Iron Whiskers. Reproduced from Reference 3» h f ,/ / N f RAC TURE / .A C X / " C \ \ \ \ V D0°C \ — \ 13,600 N . _i • —• 0 0.002 0.004 0.006 0.008 0.01 0.02 0.0-i 0.06 ^ - « E U . S T , C , Figure k. Variation of the Strength of Silicon Whiskers with Temperature. Reproduced from Reference 12. yield stress at 800°C (Figure k.). It was also found that small rods prepared from bulk material have the same room-temperature fracture stress as whiskers but have the same yield stress as bulk silicon crystals at 800°C. - 13 Pearson et al also found that prior deformation of.a silicon whisker at 800°C such that a dislocation density of about 10® cm~^  was introduced, has l i t t l e effect on the room-temperature fracture stress. This- also indicates that, in this case, the dislocation density has l i t t l e effect on the fracture stress. b. Effect of Surface Conditions Possibly the most reasonable source of size effects is in the surface effects. The ratio of surface area to volume, being proportional to the inverse of-;. the diameter, increases with decreasing diameter and surface effects would become more dominant. One possibility is that the strength of the oxide layer on the surface of the crystal would become more effective in smaller crystals. In a crystal of radius r, and oxide thickness > "the load at the yield point may be expressed as: fTrVy = rT{r - S f ^  + 2ff r Sol where 6^ = yield stress g^ ,= yield stress of the bulk material 6s = effective strength of the oxide layer at the yield point. Solving this equation for (f^ gives the relation: 6X = ^ + S ( £ - <v ) — V where A = 2 7 f r l V = 7 T r 2 l - 8 -12 Using this relation and = h6 gm/mm^ , Fleischer and Chalmers calculated f> 2 5^= 3(10 ) gm/mm for aluminum oxide. This value compares favourably with the value of 2-5-3 -0 (10 ) gm/mm found by Takamura from measurements on specimens with varying oxide thickness. 18 Roscoe reported that an oxide layer on the surface of cadmium crystals increases the strength and that this effect increased with increasing oxide thickness and with decreasing diameter. In contrast to this, Andrade"^ reported that Makin found that, although cadmium shows an increased strength with the addition of an oxide layer, the size effects for an oxide-coated crystal and a clean crystal do not differ appreciably (Figure 5.). This theory would also lead one' to expect, in the case of a ductile oxide, that the oxide layer should have an effect on the rest 20 of the stress-strain curve. Andrade and;Henderson found that an oxide layer on the surface of a single crystal of silver not only gave a small increase in the yield stress, but also that i t greatly increased the rate of work-hardening and the ultimate tensile stress (Figure 6 . ) . A similar effect was observed on copper crystals plated with.nickel-chromium"^. 21 Kramer and Demer tested aluminum crystals in an electrolytic cel l , electropolishing the crystals during the test. They found that, as the rate of metal removal was increased, £ 2 increased, 9-^  and 0£ decreased and 6^ 2 remained constant. , 2 2 Harper and Cottrell found that the cri t ical shear stress for single crystals of zinc was reduced by electropolishing and increased by •3 23 exposure to steam. Tests on copper whiskers-^' J , however, show no effect on the strength from an oxide layer. - 9 -0 10 . 20 30 40 50 g l i d e ( % ) Figure 6. Effect of an Oxide Layer on the Stress Strain Curve for Single Crystals of Silver. Reproduced from Reference 20. 21 I t has been suggested that the e f f e c t of an oxide l a y e r on the p l a s t i c r e g i o n of the s t r e s s - s t r a i n curve i s due to i m p u r i t y atoms being c a r r i e d t o the i n t e r i o r of the c r y s t a l by r a p i d l y t r a v e l l i n g d i s -l o c a t i o n s . This s u p p o s i t i o n i s supported by the r e s u l t s of C o t t r e l l and. 22 Harper who found that the e f f e c t of an oxide l a y e r on the p l a s t i c r e g ion diminished w i t h decreasing s t r a i n r a t e , being almost zero at very low s t r a i n r a t e s . Observations on the e f f e c t of removing the 24 surface l a y e r on the creep r a t e of cadmium s i n g l e c r y s t a l s , however, c o n t r a d i c t t h i s view. Another p o s s i b l e e x p l a n a t i o n f o r the e f f e c t of surface f i l m s and of diameter on the stre n g t h of s i n g l e c r y s t a l s i s th a t the oxide l a y e r i n h i b i t s the egress of d i s l o c a t i o n s from the c r y s t a l . This theory 25 i s supported by the observations of B a r r e t t on z i n c and m i l d s t e e l coated w i t h an oxide l a y e r . The c r y s t a l s were t w i s t e d p l a s t i c a l l y and the r a t e of u n t w i s t i n g on r e l a x i n g the loa d was measured. Soon a f t e r the l o a d was r e l a x e d , the oxide l a y e r was removed w i t h an etchant, upon which the c r y s t a l showed a s l i g h t t w i s t , contrary to the expected e f f e c t , i n the d i r e c t i o n of the o r i g i n a l deformation before resuming the u n t w i s t i n g a c t i o n . I f the theory were t r u e , then i t would be expected that the e f f e c t of an oxide f i l m on a f i n e - g r a i n e d p o l y c r y s t a l l i n e m a t e r i a l should be small and t h a t i t should increase w i t h i n c r e a s i n g g r a i n s i z e . 24,26 These c o n d i t i o n s have been reported by Andrade and R a n d a l l and by 29 Andrade and Kennedy y . c. E f f e c t s of V a r i a t i o n i n the Length of the S l i p - P l a n e The f o r e - g o i n g i n d i c a t e s t h a t the l e n g t h of the s l i p - p l a n e , measured i n the s l i p - d i r e c t i o n , could have an e f f e c t on the st r e n g t h of the c r y s t a l , an e f f e c t which can be e a s i l y demonstrated. A Frank-Read - 11 -source in a slip-plane generates dislocation loops under an applied stress. If the surface of the crystal provides a barrier strong enough to hold these loops inside the crystal, pile-ups wil l be formed. Since the stress at the head of the dislocation pile-up is proportional to the number of dislocations in the pile-up and to .the applied stress, in a large crystal, dislocations wil l break through the barrier at a lower applied stress than in a small crystal. Experimental results concerning this effect are contradictory. 12 Fleischer and Chalmers tested crystals of two orientations; one (S^ .) in which the length of"the slip-plane remained constant with crystal size and the other'(S^) in which i t varied with size. The results indicate that the size dependence of the yield stress is the same for both crystals and that the yield stress at any one diameter is the same for both orient-27 ations. Lipsett and King , in an investigation of the influence of a film of gold on the surface of single crystals of cadmium, found that the increase in cr i t ical resolved shear stress was independent of orientation. 28 In contrast to these two results, Gilman and Read found that the influence of a film of copper plated onto zinc single crystals varied with the orient-ation of the crystals, increasing as the length of the slip plane decreased. 10 It has also been suggested, by Suzuki et al that decreasing the length of the slip plane should result in a lower work-hardening rate and 12 an increase in the range of easy glide. Fleischer and Chalmers found that for crystals of orientation S , 9^  increases, 9^ remains constant and C 2 decreases with decreasing size. In contrast to this, for crystals of orientation S T T , 9 and 9 decrease and £ 0 increases with decreasing -L-L 1 2 <-size. The results of Gilman2^, who found a higher work-hardening rate for zinc single crystals with longer slip planes, agree with this result. - 12 -This effect could be attributed to the fact that dislocations have a shorter distance to travel before escaping from the surface of the crystal and, therefore, there is a smaller probability of their encountering obstacles. This would mean that a Frank-Read source would be able to produce more dislocation loops before the back-stress caused by dislocation pile-ups is high enough to produce hardening. The resolved shear stress required to operate a Frank-Read source may be expressed as: length of the pinned dislocation segment. This would indicate that, since the length 1 would be limited by the size of the slip plane, that smaller crystals should require a larger stress for slip. Brenner , however, found that i f the size effect were to be explained on this limited to the order of 10 cm. Saimoto has offered a mechanism whereby sources of this length might be introduced into a whisker but there is some question as to whether a source of this length could operate. d. Grip Effects 32 Fleischer and Chalmers , considering the effect of grip restraints on the stress-strain curve for a single crystal, have shown that the resolved shear stress on a crystal plane ( T"j) is the sum of two components; one due to the applied stress ( ^ ) and one due to the bending moment arising from the grip restraints ( Ty), and that the resultant stress on the primary system can be expressed as: ? = M b 1 where is the shear modulus, b the Burgers vector and 1 the where m. cos^C cosAp, the Schmid factor for the primary slip system a = crystal diameter L = crystal length X = the angle between the slip direction and the P specimen axis, = the angle between the slip-plane-normal and the specimen axis. This indicates that the actual stress on the primary slip-plane is lower than the measured stress resulting in a higher apparent stress, the difference depending on the ratio a/L for any given orientation and strain. The effect on the yield stress would be negligible since, at this point, the strain is very small. If the above equation is modified to take into account slip on a secondary system, the resulting expression indicates that the rate of hardening should decrease in the easy glide region and that the range of easy glide should increase with decreasing a. The authors calculated that, on this basis, a ratio of a/L which is small compared with l/lOO is required to remove the influence of grip restraints from the 12 stress-strain curve. When the work on size effects by the same authors 10 arid that by Suzuki et al was examined in this light, i t was found that the ratio was greater than the cri t ical value calculated and that the size effects were accounted for by the calculated "2^ . It would be interesting to examine the results of the diameter dependence of the flow stress of whiskers in the light of the above theory. The diameters of whiskers tested usually vary between about k / i to 20 ju and the lengths from 0-5 mm to 5 mm, with no apparent relation between diameter and lens th!5. These figures give upper and lower limits of l/25 and l/lOOO respectively to the ratio a/L. Under these circumstances, i f the variation in flow stress with diameter were due to grip effects, a statistical scatter would be expected rather than a clear- dependence as was found by Brenner^ and by Eder and Meyer"'"(Figure 7-)-- Ik Figure 7- Variation of Flow Stress with Diameter for Copper Whiskers. (Brenner and Eder and Meyer). Reproduced from Reference 15-15 e. Size Effects in Polycrystalline Wires The foregoing discussion has been concerned with single crystals only and most of the theories presented are applicable only to single crystals. The one suggestion which might be applied to polycrystals is the strengthening effect of an oxide layer.- However, as mentioned before, there is no effect from an oxide layer on polycrystalline wires until the grain diameter is of the order of the specimen diameter. 33 Shlichta tentatively reported that the results of preliminary tests on fine Taylor-process and electropolished drawn wires indicate that there is an increase in strength and a decrease in plastic deformation 3U before fracture with decreasing diameter. Later , however, he concluded that this effect was not a true size-effect but represented a "survival 35 of the fittest". McDanels, Jech and Weeton found a size effect in polycrystalline tungsten wires. In this investigation, composites of tungsten wires in a copper matrix were tested in tension. The results showed that the composites containing finer wires have a higher strength. 36 A Defence Metals Information Memorandum lists the properties of some, high-strength fine wires of various metals including high-carbon and stainless steels, nickel-base alloysy tungsten and molybdenum with diameters ranging from 0.0005" to 0.01". In general, the yield stress, ultimate stress and the total elongation increased with decreasing size. The one exception noted was a nickel base alloy which showed decreasing ductility with decreasing diameter. 2. Cottrell-Stokes Law When two specimens of a metal are deformed at the same strain rate but at two different temperatures, the stress>-strain curves diverge with increasing strain. Cottrell and Stokes^ have pointed out that the - 16 5 results of Dorn et al indicate that the difference in the two stresses at any given strain is probably due to two contributions: 1. The density and distribution of dislocations may be different in metals deformed at different temperatures to the same strain, 2. There may also be a "reversible" effect due to thermal fluctuations, an effect which wil l decrease with decreasing temperature. In .their investigations of this phenomenon in single crystals of aluminum, they also noted that the ratio of the flow stresses at. any two temperatures Is constant over the entire range of strain after an in i t ia l few percent deformation (Figure 8.). This effect has also been noted by other invest-igators working with different metals and a summary of the results is given in Table I. 37 Basinski - noted that the same effect is found i f the ratio of the flow-stresses is taken at one temperature but with two different strain rates. His results for aluminum and copper at various temperatures are shown in Figure 9- The only large deviation from Cottrell-Stokes behaviour is that found by Davis who found the ratio of flow stresses in cobalt single crystals varies with increasing strain. This was found for both variations in temperature and in strain rate. A l l of these results have been obtained, with one exception;,: by the same method, one which wil l be outlined later in this work. The one 39 exception is the work of Sylwestrowicz on polycrystalline copper who obtained each point from a different specimen. The result was that the flow stress ratio was not quite constant but varied between about O.87 to O.85 (Figure 10). - 17 -TABLE I. Ratio of Flow'Stresses Metal 4i Vf2 Reference Strain Rate Poly-crystal Cu 0.86* 5 Wot Given Poly-crystal Cu O.87-O.85 39 Not Given Single-drystal Cu 0.88 ko lO'^sec"1 Single-crystal Cu 0.88** 37 l O ^ s e c - 1 Single-crystal Al 0.79+ 6 10~5sec_1 Single and Poly-crystal Al O.78**, 37 -k - l 10 sec -1 Single-crystal Ag 0.86** 37 10"^sec~1 In a l l cases except that marked + , T = 293°K and T2= 78-K. The value marked + is for T1= 293°K, T = 90°K. The value marked * is deduced from a single experiment. The values marked ** were originally presented with reference to T2= k.2°K- These were normalized- to T2= 78°K by the method of 6 Cottrell and Stokes whereby: 6^293 . 6*\%2. = 6~293 k.2 °~~ 78 ^ 78 pcroentago e longaf ion Figure 8. Ratio of Flow Stresses at 293°K and 90°K for Aluminum Single Crystals. Reproduced from Reference 6. 9C5} . i i i -^•x 0 o.o-%fe° 0_^-Temile men 'q /m Figure 9. Ratio of Flow Stresses at Strain Rates of 10"5 and 10"^ " sec"1 for Copper and Aluminum. Reproduced from Reference 7--19 -Figure 10. Ratio of Flow Stresses at 293 K and T8°K for Copper Polycrystals. Reproduced from Reference 39-- 2.0 Sever a l t h e o r i e s have been advanced to e x p l a i n the constancy of the r a t i o of f l o w s t r e s s e s . C o t t r e l l and Adams^ have suggested t h a t kl t h i s constancy i m p l i e s t h a t , from Seeger's theory of work hardening , the d i s l o c a t i o n p a t t e r n should remain constant d u r i n g work-hardening and only the o v e r a l l d e n s i t y should change; i . e . the p r o p o r t i o n a l i t y between the d e n s i t y of f o r e s t d i s l o c a t i o n s , the d e n s i t y of jogs on screw d i s -l o c a t i o n s and the d e n s i t y of the i n t e r n a l s t r e s s e s i n the metal must hp remain constant. The theory of Seeger et, .al s t a t e s t h a t the i n t e r n a l stress, f i e l d i n a metal depends on the d i s l o c a t i o n d e n s i t y i n the same manner as the f l o w s t r e s s r e s u l t i n g from i n t e r a c t i o n s of g l i d e d i s l o c a t i o n s w i t h f o r e s t d i s l o c a t i o n s . This would imply that the hardening from these two sources should be p r o p o r t i o n a l and the r a t i o s o f the two f l o w s t r e s s e s should be constant. ' S Mott, as reported by C o t t r e l l and Stokes , has suggested t h a t "the important e l a s t i c f o r c e s on a g l i d e d i s l o c a t i o n are those from i t s immediate neighbours r a t h e r than the long-range s t r e s s e s from p i l e d up groups, f o r the e l a s t i c f o r c e s and.the f o r e s t f o r c e s a c t i n g on the d i s l o c a t i o n can both be a s c r i b e d t o the same t h i n g , the presence of nearby d i s l o c a t i o n s ! ' B a s i n s k i ^ i d e n t i f i e d the source of the e l a s t i c s t r e s s e s w i t h the f o r e s t d i s l o c a t i o n s . 3• Comments The b r i e f survey of the i n v e s t i g a t i o n s i n t o s i z e e f f e c t s given above shows t h a t , although the v a r i a t i o n of p r o p e r t i e s w i t h diameter i n s i n g l e c r y s t a l s has been w e l l e s t a b l i s h e d , the t h e o r i e s presented to account f o r t h i s v a r i a t i o n are c o n f l i c t i n g and there i s n o v u n i f i e d theory to e x p l a i n a l l s i z e e f f e c t s . Since almost a l l of the t h e o r i e s apply only to s i n g l e c r y s t a l s , any s i z e e f f e c t found i n a p o l y c r y s t a l could not be a t t r i b u t e d to any of them. I n v e s t i g a t i o n s of the C o t t r e l l - S t o k e s law have a l s o been p r i m a r i l y concerned w i t h s i n g l e c r y s t a l s . The work of Sylwestrowicz 36 on p o l y c r y s t a l s of copper., when compared t o the r e s u l t s of B a s i n s k i i n d i c a t e s t h a t there may be an anomalous e f f e c t i n copper.. This i n v e s t i g a t i o n of s i z e e f f e c t s and of the e f f e c t s of s t r a i n - r a t e and temperature on the f l o w s t r e s s i s an attempt t o c l a r i f y these two p o i n t s . - . 2 2 EXPERIMENTAL PROCEDURE 1 . Materials The copper used i n t h i s investigation was supplied by Johnson Matthey and Company and by the American Smelting and Refining Company. The aluminum used was supplied by the Aluminum Company of Canada, Ltd. A l l copper was of 99-999+$ purity.and the aluminum of 99'99+$ purity. 2 . Specimen Preparation a. Copper Wires The copper wires were prepared from Johnson-Matthey copper rods 5 mm diameter by 1 5 cm long. The wires were drawn through hardened steel dies to 0 . 1 1 5 " diameter, then through tungsten-carbide dies to O.OOV diameter, and f i n a l l y through diamond dies to a minimum diameter of 0 . 0 0 2 " . Each size was drawn from f u l l y annealed wire according to the schedule shown below: I n i t i a l F i n a l No. of Drawing Diameter ; Diameter • Steps ; 5 . mm 1 mm 3 0 2 . 5 0 . 5 3 2 1 0 . 2 3 2 0 - 5 0 . 1 ' 1 5 0 . 2 5 0 . 0 5 2 1 After drawing to the desired diameter, wire of diameter over 0-5-mm was cleaned, coiled and stored f o r future use. Wire of less than 0 - 5 mm diameter, was- immediately strung on glass U-frames and prepared f o r testing. Since a crack was found i n the middle of each rod received, random specimens of each size were sectioned and examined for i n t e r i o r cracks. No evidence of cracking was found. Specimens about 8 cm long were cut from the drawn wire and electropolished i n an ele c t r o l y t e consisting of 1 5 0 ml H^PO^, 7 5 ml each of - 23 lactic and propionic acids and 30 ml each of HgSO^  and water. A cel l potential of 7-5 volts with the specimen suspended horizontally in the cell shown in Figure 11 produced a good polish and an even, -circular cross-section. This procedure removed, a l l surface irregularities, any oxide layer accumulated during storage and the more heavily distorted surface layer, h^e specimens were then annealed in a vacuum of less o than 10-- mm of mercury for 2 hours at 250 C. Metallographic examination - 2 showed that this procedure produced a grain size of about 10 mm diameter, 6 2 an average of 10 grains per cm , and that the grain size was uniform across the cross-section. The different sizes of wire were also subjected to X-ray analysis before and after annealing to determine i f there were any wire texture•remaining. No such texture was found in the annealed wires and the as-drawn wire was found to have the normal <^ 110^  - <^L1]^ fibre texture. b. Copper Rods . Larger copper tensile specimens were machined from 10 mm diameter rod supplied by the American Smelting and Refining Company. The rods were f irst given several passes through a rolling mill in order to weld shut the intergranular cracks present in the as-received material. An approximately circular cross-section was maintained by giving the rod several passes at each reduction, turning the rod slightly between passes. Tensile specimens with a gauge length of k mm diameter were machined from the resulting rod. These were electropolished as before and then vacuum annealed at various times and temperatures to produce desired' grain sizes. Coarse grained rods were also produced by slow cooling from the melt. c. Copper Single Crystals A l l single-crystal specimens were cut from one single-crystal Figure 11. E l e c t r o l y t i c C e l l and Power Source Used i n E l e c t r o p o l i s h i n g Copper Wires. The Specimens were Suspended H o r i z o n t a l l y i n the E l e c t r o l y t e from the Frame Shown. rod about 9 mm diameter by cm long made'by the melt-solidification technique. A copper rod of the proper dimensions was cast and fitted into a hole drilled through a graphite rod. Graphite caps, the upper one .containing a reservoir of copper, were fitted over each end and the whole enclosed in a stainless-steel tube. The tube was lowered at a1 rate of 10 cm per hour through a furnace maintained at a temperature of li+00°C. d. Copper Strip The copper strip specimen was produced from 8 mm diameter Johnson-Matthey copper rod. The rod was rolled out into strips 0.015'' thick by 0.375" wide. Specimens were stamped out on the specimen punch and die described elsewhere*^, producing a gauge length 0.20" wide hy 0.75" long. These were then electropolished and annealed for o two hours at 250 C, giving a grain diameter of about 20 p.. e. Aluminum Strip The aluminum strip specimens were produced by rolling 0-5" thick bil lets of aluminum into strips 0-375" wide by O.O36" thick. Specimens were stamped from the strip as described previously. These were chemically polished for five minutes in Alcoa bright dip and annealed for 30 minutes at 450°C. 3- Measurements Diameters of 500 yx and over were measured using a Gaertner travelling microscope accurate to 1 jx. Diameters below 500 jx were measured using a Reichert microscope equipped with a Leitz micrometer eyepiece accurate to about O.h p.. From each specimen, two sets of diameter measurements were taken at right angles to each other. If the two sets of measurements agreed, a circular cross-section was - 26 . assumed. If not, the specimen was discarded. A l l diameter measurements were taken before annealing to minimize effects from specimen damage. A l l gauge length measurements were made, using the Gaertner travelling telescope, with the specimen mounted in the testing machine.. 4. Testing Procedure A l l tests, were carried out in tension on an Instron Tensile Tester equipped with an autographic recorder which produces, a load-elongation curve on a strip chart. a. Normal Tensile Tests The copper wires were tested in four batches at various strain rates and at two different temperatures. Wires of diameters 900yu, 5 0 0jx, 2 0 0 p, 1 0 0 p, and 5 0 p were tested at room temperature. For these tests, the wires were soldered to brass tabs which were held in the normal Instron vice-grips. Wires of diameters 9 0 0 y u , 5 0 0 p., o 2 0 0 p. and 1 0 0 p. were tested at 7 ° K. These wires were soldered into kk brass grips attached to a special modification to the Instron tensile tester (Figure 1 2 . ) . A l l low temperature tests were carried out in a liquid nitrogen bath contained in a wide-mouth Dewar flask. b. Variable Strain-Rate Tests Tests on the effect of varying the strain-rate on the flow stress were carried out on copper wires of,diameters 9 0 0 j x , 2 0 0 p and 5 0 ^ u and on copper rods, both single crystal and polycrystal, of diameter about h mm. The tests were done by alternating between two--strain-rates with a ratio of lOO/l. Changes in the strain-rate were accomplished by changing the gear ratio of the cross-head drive. Since the Instron,. is equipped with a gear-shift lever, this could be done almost instantaneously. (a) (*> Figure 12. Gripping System for Low-Temperature Tests on (a) Wire and Rod Specimens, and (b) Strip Specimens - 2 8 -c. Variable Temperature Tests Tests on the effect of varying temperature on the flow-stress were carried out on wires of diameters 900 Ja and 200 ji and on single and polycrystal rods of 4- mm diameter. These tests were also done on copper and aluminum strip specimens. The copper wire and rod specimens were, as before^soldered into brass grips, while the strip specimens were held 45 44 in special file-face grips screwed into a universal arrangement (Figure 12 b.). To avoid grip effects, the single crystal specimens were also mounted in the universal arrangement. The testing procedure was as follows: a. The, .specimen was deformed for a short time at room temperature, _o b. The temperature was reduced to 7 ° K and the specimen given another short deformation. c. The temperature was brought to room temperature again and the specimen given another short deformation. This procedure was repeated until the specimen fractured. Between tests, while the specimen was cooling or heating, a small load was kept on the specimen to maintain alignment of the loading system. After the specimen was immersed in the liquid nitrogen bath, and a l l but the surface boiling had ceased, five minutes was allowed in order that thermal stability could be reached. Following the low temperature extension, the liquid nitrogen bath was removed and a stream of warm air from a hair dryer was played over the specimen and grips. After a l l frost had disappeared, ten minutes was allowed to ensure temperature stability. - 29 -EXPERIMENTAL RESULTS AND OBSERVATIONS  1. Size Effects The effects of specimen diameter on the tensile properties of polycrystalline copper wires were measured at room temperature at strain-rates of 0.02 min"1 and 0.8 min"1 and at 78°K at strain rates of 0.0^ min"' and 0.2 min - 1 . The parameters measured directly from the stress-strain curve were yield stress, ultimate tensile stress, and the total elongation before fracture. In addition to these, a log-log plot of the stress-strain curve shows that, in a l l cases, the curves may be represented by . . b v an expression of the form log b = a + b l o g £ { 6~ = A£ ; a = log AJ. In some cases (see Figure ^.in Appendix I) the log-log plot shows two straight lines, each of which conforms to an expression of the type given. The parameters a and b were determined for each specimen by picking a number of points from the . stress-strain curve and flitting them to the equation by the method of least squares as is outlined in Appendix I. The above data are tabulated in Appendix II. The characteristics of the various specimens tested are shown in Table II. « a. Yield Stress The yield stress at the strain-rate of 0.8 min ^ was not measured since the autographic recorder cannot follow the load for the in i t ia l portion of the test at this rate. The stresses for 0.1$-offset (YSQ ^) and 0-5$ offset (YSQ ^ ) were measured for a l l other specimens. Since the 0.5$ offset gave a lower scatter in the yield stress and since i t is the value of offset recommended for copper^, the 0.1$ offset values were not plotted. They are, however, tabulated in Appendix,II. The values of yield stress obtained are plotted as a function of diameter in Figure 13. for both temperatures; 78 K and 293 K... The values of YSn for tests.at 78°K show no significant difference for the . 30 _ two strain-rates used so they are plotted together and considered as one set of points. The yield stress at 293°K shows no size effect, the averages of YS . fluctuating about an average value, of about 670 Kg/cm^. • . . . 0 . 5 At 78°K, however, the yield-stress decreases from a maximum of about IkOO Kg/cm^ at a diameter Of 500 jx to about 1000 Kg/cm2 at a diameter of 100 jx. In addition to this, i t was found that the yield stress at a diameter of 900p. is'about 1200 Kg/cm2, again lower than that at a diameter of 500 yu. -This may be an indication of a scatter higher than .that observed experimentally arid, may detract from the significance of this plot. b. Ultimate Tensile Stress Since, in every low-temperature test, the wire specimen pulled out of the solder before fracture, no ultimate tensile strengths are available for these tests. The ultimate tensile strengths for the room-temperature tests are plotted as a function of specimen diameter in Figure Ik. . The specimens strained at 0.02 min - 1 show as essentially constant ultimate stress of about 3100 Kg/cm down to a diameter of 200yu. At this point, the stress" drops off sharply with decreased diameter. The.data for specimens tested at the strain-rate of 0.8 min "^  show a very wide scatter and no trend can be determined from these. c. Ductility The values of total elongation before fracture are tabulated in Appendix II. and plotted as a function of specimen diameter in Figure 15. For the reasons outlined above, no values for the low-temperature tests are available. The data plotted in Figure 15. show that, at a.strain rate of 0.02 min""'", the ductility is essentially constant at about 38% for - 3 1 -TABLE II. Characteristics of Specimens.Tested for Size Effects Specimen Diameter A B B/A 9 0 0 /a 8 1 0 0 2 3 0 0.03 5 0 0 2 0 0 0 1 6 0 0.08 2 0 0 300 6 0 0.2 1 0 0 8 0 30 OA 50 25 15 0.6 A l l specimens have a grain diameter of 0.01 mm. A =• the number of crystals in the cross-section. B = the number of crystals in the cross-section which impinge on the surface of the specimen. - 32 -diameters between 900 p- and 500 p and thereafter appears to decrease with decreasing diameter. At the higher strain-rate (0.8 min - 1 ) , the ductility decreases approximately linearly with decreasing diameter from about 28$ at a diameter of 900 p to about YJ'fo at a diameter of 100 p. d. Work-Hardening Rate The parameters a and b in the expression log = a + b log £ are tabulated in Appendix II. The values of a are of l i t t l e significance since the log-log plots of the stress-strain curves l ie very close to-gether, making a essentially a function of b rather than an independent parameter. For the specimens showing two straight line segments, two values were calculated for each parameter (a-j_, b-]_, &2, ^2'^" However, only the parameter b^ is plotted. The values of b-^  plotted as a function of specimen diameter in Figure l6. are a measure of the rate of-, work-hardening of the specimen. Under-all test conditions, the scatter in these values of b-^  is too large to permit any definite expression of b^ as a function of diameter. Vague trends however, do show in the plots. The same holds true for the values of a-^  which are not shown. The trends observed for decreasing diameter are outlined below. Temperature Strain Rate aj_ b-^  decreases increases increases decreases decreases increases 293°K 0.02 min 293°K 0.8 " 78°K - 33 I500f I ikoo\ 1300 OJ §" 1200 bO 6 CQ 3 -noot .0) •H * 1000\ T= 78°K € = 0.05 min"1 • £ = 0.2 min"1 0 T= 293°K 0.02 min"1 O Arithmetic Averages $ 700 o 8 600 9 e 9> o o § 9 200 1+00 600 Diameter (microns) 800 Figure 13- Yield Stress of Polycrystalline Copper Wires as a Function of the Diameter - 3^ Diameter (microns) Figure 1^. Ultimate Tensile Strength of Polycrystalline Copper Wires as a Function of Diameter - 35 -ko 200 ' 400 5oo Boo Diameter (microns) Figure 15. Ductility of Polycrystalline Copper Wires as a • Function of Diameter 0.45 0 . 4 0 - 3 5 " 0 (a) T= 2 9 3 ° K O £= 0 . 0 2 min" 1 0 0 0 8 0 0 0 O 0 ooco 8 - 0 1 8 1 1 1 9 3> 0 -0 O 8 0 0 8 O O O -0 0 0 0 1 0 O (b) T = 2 9 3 ° K O £ = 0 . 8 min"1 1 1 1 2 0 0 400 6 0 0 800 Diameter (microns) Figure l 6 . (continued on page 3 6 . ) .o.k (c) T = 78°K o £ = 0.2 min"1 £ = 0.05 min"1 o o o $ o o o o o o /# o o o o £ _ o 1 • 1 200 i+oo 6oo 800 Diameter (microns) Figure 16. Work-Hardening Rate (b-, ) of Polycrystalline Copper Wires as a Function of the Diameter. - 38 -2. Cottrell-Stokes Law The data obtained from the investigation of the effect of temp-erature and strain-rate on the flow stress of copper and aluminum are tabulated in Appendix III. Included are the ratios of flow stresses for a ratio of strain-rates of 1:100 for various copper specimens and the ratios of flow, stresses for copper .and aluminum for temperatures 7" K o and 293 K. The data, displayed in Figures 17 to 20, 25 and 26, show that polycrystalline copper follows the Cottrell-Stokes law for strain-rate changes but not for temperature changes. Also, i t demonstrates that polycrystalline aluminum and copper single crystals do follow., the Cottrell-Stokes law and that the ratio of flow stresses in these cases is approximately that described in the literature (see Table I). Stress-strain curves for representative specimens of copper and aluminum are displayed in Figures 22, 23, 24, 27, 28, 29, and 30. . a. Effect of Strain-Rate on Flow Stress Copper wire and rod specimens with the characteristics listed in Table III were strained at strain rates of 10 1 min 1 and 10 ^min - 1 at a temperature of 293°K. A l l of these specimens follow the Cottrell-Stokes law as is shown in Figures 17 to 20. The ratio appears to decrease in the single crystal specimen (Figure 2l) after about 20$ deformation. Since the single crystal was damaged during preparation and no other was available for testing, no significance is attributed to this decrease. The ratios for the various specimens are plotted together for comparison in Figure 21. which shows that the ratio of flow stresses varies from O.965 for specimens 1B22 and 11B26 to O.985 for the single crystal SX2. The valuer.found for the single crystal is lower than that found by 37 Basinski (Figure 9-)J but this is for a strain-rate ratio of 1:100 whereas Basinski used a strain-rate ratio of 1:10. Also, Basinski has - 39 -shown-..that the ratio decreases as the temperature increases and the temperature used in this investigation was much .higher than that used by Basinski. Portions of the..stress-strain curves for specimens 1B22, RC3, and SX2 are shown in Figures 22, 23, and 2k. The fine-grained polycrystalline specimen (LB22) shows neither the low slope plastic flow region following yield at the high strain-rate nor serrations in the stress-strain curve. The stress-strain curves for both the coarse-grained specimen (RC3) and the single crystal (SX2) "show the low-slope plastic flow region after yield at the high strain-rate but only the single crystal shows serrations at high strain. These however, were barely noticeable. Also, only the \ single crystal, showed a yield drop on the low strain-rate portions of the stress-strain curve. TABLE I I I . C h a r a c t e r i s t i c s of Specimens Tested f o r the E f f e c t of. S t r a i n Rate on Flow Stress Specimen Diameter Gr a i n A B B/A Diameter SX2 3-5 mm 3-5 mm 1 1 1 RC4 3.8 0.3 153 40 0.26 2B1 0.9" 0.3 10 10 1 RC3 3.8 0.09 . 1500 130 0.09 113, 5 0.05 0.01 25 16 0.64 1F12, 16 0.2 0.01 400 60 0.15 1B22, 26 0.9 0.01 8100 280 0.03 A and B are as d e f i n e d i n Table I I . o 0 © . 1B22 O 1B26 || 0 • 0 • 0.965 0 . ® O 0 O O 0 0 o @ • 2 • ® 0 $ ' O O ® ® 0.960 - 0 1 1 1 1 1 5 10 15 20 25 30 35 $ Elongation -3 Figure 17. Ratio of Flow Stresses at Strain Rates of € 1 = 10 Jmia / 6 p= lO'^-min"1 for Specimens 1B22 and 1R26 _ k2 -• 0 975 £1 • • B D • • • • 113 1113 1F12 I F I 6 • O 0.970 0 0 0 • o o * — o 0 - 0 — o o # O O O » O O o o - — 0 0.965 10 15 20 io Elongation 25 Figure 18. Ratio of Flow Stresses at Strain Rates of £"]_= 10"3min~-1- and €'2- l C ^ m i n - 1 for Specimens 1F12, 1F16, 113, and 1113. 30 0.980-0-975-15 20 25 30 $ Elongation Figure 19- Ratio of Flow Stresses at Strain Rates of €j.= lO'^ min""'", € = 10~1min"-'- for Specimens RC3, 2Blf RC.ir 0.985 -O.98O -15 $ Elongation 25 30 Figure 20. Ratio of Flow Stresses at Strain Rates of €j= 10"3min , € 2 = l O ' T i i n " 1 for Specimen SX2 - kk -0 . 9 8 s 0.980 -0.975 £ 1 6 0 . 9 7 0 -0 . 9 6 5 -15 20 • $ Elongation 25 30 35 Figure 21. Figures 17 to 20 plotted together for comparison. 'fo Elongation Figure 22. Stress-Strain Curve for Specimen 1B22 , i Elongation Figure 2 3 . Stress-Strain Curve for Specimen RC3 - 48 -b. Effect of Temperature on Flow Stress Copper wires and rods with the characteristics shown i n Table IV were tested at temperatures of 78°K and 293°K at a strain-rate of about 10 ^min 1. ^he r a t i o of flow stresses at 293°K and 78°K are plotted as a function of s t r a i n i n Figure 25 and 26. The copper p o l y c r y s t a l l i n e specimens do not exhibit Cottrell-Stokes behaviour while the si n g l e . c r y s t a l specimens do. Portions of the stre s s - s t r a i n curves f o r specimens 132k, R C 6 and SX3 are plotted i n Figures 27 >to 30. I t can be seen from these that, as might be expected, the te n s i l e behaviour of the specimen approaches that of a single c r y s t a l as the grain-configuration, approaches that of a single c r y s t a l . The r a t i o of the flow stresses for the single c r y s t a l specimens shows an anomaly i n that the r a t i o , although e s s e n t i a l l y constant, shows a change i n value at one point during the deformation. Since the s t r a i n at which t h i s change i n value took place coincided with the s t r a i n at which the form of the strain-stress curve undergoes a change i n form from that seen i n Figure 30. to that seen i n Figure 31j i t was decided to treat these as two separate values rather than as one continuous curve. I t should be noted here that, although these two events coin-cided i n each of the .three single crystals tested, the s t r a i n at. which they occured was not the same f o r the three specimens. The approximate elongations at which the r a t i o changed are given below: Crystal SX1 Elongation 25$ SX3 30$ 8X4 20$ The shape of the stre s s - s t r a i n curve for specimen SX3 (Figure 30,3l) i s the same as that found by B a s i n s k i ^ i n the low-temperature portions, but not i n the high temperature.portions. The shape of the high temperature portions of the curve agree with those found by C o t t r e l l 4-tj , and Adams in that a yield drop is found after about 2.yjo elongation but the shape of the yield drop is different. In specimen R C 6 , the temperature cycling was carried on beyond the point where.the neck became visible. This resulted in a change in the shape of the low temperature portion of the stress-strain curve to that shown in A in Figure 29• and in a sharp drop in stress following a short region of plastic flow as is shown in B in Figure 29- The amount of plastic deformation before the drop in stress occured decreased and the rate and extent of the drop in stress, increased as the diameter of the neck decreased. That this phenomenon is a result of the deformation at low temperature is shown by two experiments which are also indicated o in Figure 29- In the f irst , the specimen was unloaded and held at 293 K for a short time and then reloaded. In the second, the specimen was unloaded, and held at 78°K for a time equivalent to that required for the low-temperature test. It was then brought to 293°Kand reloaded once more. In neither case was the sharp yield drop observed. TABLE IV. Characteristics of Specimens Tested for the Effect of Temperature on Flow Stress Specimen Diameter Grain Diameter A B B / A SX1 4.0 mm - 1 1 • 1 SX3 3-8 - 1 1 1 sx4 3-8 - 1 1 1 RC5 3-9 - 2 2 1 R C 6 4.0 - 5 5 1 RC2 3-5 0.2 mm 310 55 0.2 RC1 4.6 0-75 40 20 0-5 1F13 0.2 0.01 4oo 6o 0.15 1324, 27 0.9 • 0.01 8100 280 0.03 Aand-B are as defined in Table II. Specimen R C 5 is essentially a bicrystal with a few small stray grains along the grain, boundary. Specimen R C 6 consisted of five grains arranged symmetrically about the axis of the specimen and running almost the entire gauge length. - 51 -5 io 15 20 25 io Elongation Figure 25• Ratio of Flow Stresses as a Function of•Strain for T 1 = 2 9 3 % T2= 78°K 1750 1500 OJ w 1250 1000 750 2750 2500. 2250 . 2000-3 4 5 6 7 14 $ Elongation Figure 27- Stress-Strain Curve for Specimen 1B24 15 16 T 1 = 293 dK T2= 78°K 17 18 5 10 15 20 $ Elongation Figure 28. Stress-Strain- Curve for Specimen RC6 30 35 ko "jo E longat ion F igure 2 9 - S t r e s s - S t r a i n Curve f o r Specimen RC6 a f t e r the Appearance of a V i s i b l e Week 3 5 ^ 0 4 5 5 0 $ Elongation Figure 3 1 . Stress-Strain Curve for Specimen SX3, Part II. i vn —J i - 58 -c. Results of Tests on Muminum To check that the non-Cottrell-Stokes behaviour in polycrystalline copper was not due to a faulty testing procedure, a copper strip specimen was prepared and tested as described in the experimental procedure. For comparison, several aluminum strips were, prepared and tested in the same manner. The results of these tests are shown in Figures 32 and 33 along with the ratio of flow stresses found by Cottrell and Stokes^ on aluminum single crystals. The ratio of flow stresses was shown to be constant for aluminum and to have about the same value as' found by Cottrell and Stokes. The copper strip, however, was shown to behave in the same manner as the previous copper polycrystalline specimens. These results demonstrate' that the experimental procedure was not at fault. Portions of the stress-strain curve for the aluminum specimen SA2 are shown in Figure 3^. Two differences from the stress-strain curves found by Cottrell and Stokes for aluminum single crystals are evident. The f irst is that a yield drop appears at a strain of less than h°jo whereas this did not appear in the single crystals until a strain of about 6$> had been reached. Also, the yield-drop was not always evident beyond this strain whereas i t was in the single crystals. Second, the yield •stress at the low-temperature is often lower than the final stress in the previous low-temperature section of the test. This reduction in stress, which wi l l be here-in-after referred to as "work-softening", depends on.both the amount of strain at 293°K ( C* ) between the two 78°K tests and the total strain ( £ ). This is demonstrated in Figure 35- where the ratio of'the yield stress (g') to the final stress ( ) of the preceeding low-temperature test is plotted as a function of £ for different C • For small £ , the scatter is so large,that the extent of work-softening can be shown only as a band (light hatching), but the ratio is definitely less than one. 0.780 . D Aluminum Single-Crystal (Cottrell and Stokes) — I 1—: 1 1 1 I I : 5 10 15 20 25 30 35 $ Elongation Figure 32. Ratio of Flow Stresses for Aluminum Single Crystals (Reference 6) T 1 = 293°K, T2= 90°K and Aluminum Polycrystals, 0^ = 293°K, Tg= 78°K 1 vo - 60 -0.88 0.87 O 0 \ p 0.86 \ 0 L2 0.85 0 \ ° \ 0.84 ^ 0 0 1 1 1 1 1 5 10 15 20 25 30 io Elongation Figure 33. Ratio of Flow Stresses for Polycrystalline Copper Strip. T x= 293% T 2= 78°K. $ Elongation" Figure jh- Stress-Strain Curve for Aluminum Specimen SA2 fo Elongation Figure 35• Degree of Work-Softening in Aluminum as a Function of Strain for Various £ 1 • CA ro - 63 -SUMMARY OF RESULTS The results of the investigation into size-effects, are not very-revealing. The yield stress, ductility, ultimate tensile strength and rate of work-hardening were measured for copper wires with diameters ranging from 900 ^1 to 50 jx at various temperatures' and strain rates . and the results may be summarized as follows: 1. The room temperature yield-stress shows no apparent size-effect. 2. The low-temperature yield-stress decreases with decreasing diameter for wires with diameter less than 500 ^.. - 3- Both the ductility and the ultimate tensile strength decrease with decreasing diameter. k. The results on work-hardening rate show a scatter high enough that no definite size-effect can be established. A trend toward an "increasing work-hardening.rate with decreasing diameter was observed except for the high strain-rate tests where the opposite was observed. The results of the investigation of the effects of the temperature and strain-rate on the flow-stress of copper may be summarized as follows: 1. Single-crystals of copper follow the Cottrell-Stokes law for both temperature and strain-rate changes. ••"' 2. Polycrystalline copper follows the Cottrell-Stokes law for strain-rate changes but the value of the flow-stress ratio is different for different specimens being affected by: • a) the grain size, b) the fraction of the grains, in the- specimen which have a free surface. 3- Polycrystalline copper does not obey the Cottrell-Stokes law. The deviation from the expected value of the flow-stress ratio is negative and depends on strain. The results of the tests on aluminum show the following: 1. Aluminum polycrystals do obey the Cottrell-Stokes law for temperature changes. 2. The yield-drop observed by Cottrell-and Stokes^ in aluminum single crystals was found in polycrystals. 3- The yield-drop in polycrystalline aluminum is accompanied by a "work-softening" effect, not found, evidently, in single crystals. The above is a summary of the important results of this investigation and are the results which wil l be referred to in the discussion. - 65 -DISCUSSION 1. Size Effects This investigation has shown that the size-effects found in polycrystalline copper wires are, in most cases, the opposite to those found in single crystals by other investigators. The results wil l be discussed in terms of the degree to which the wires of various diameter can be referred to as "polycrystalline". Departures from the behaviour expected when this consideration is taken into account indicate that there may be size effects where none are directly observed. a..Yield Stress and Work-Hardening Rate It would be, perhaps profitable to f irst discuss the grain configuration in a polycrystal and to assess the effect of the diameter on this. The polycrystalline wires used in this investigation had a grain diameter of 0.01 mm and varied in diameter from about 0-9 mm to about 0.05 mm. E x a m i n a t i o n of Table II shows that in wires of diameter 0.9 mm, the great majority of the grains are totally enclosed in the wire and the percentage of the grains which show a free surface is negligible. In a wire of 0.05 mm diameter, a significant proportion of the grains in the wire have a free surface and less than half are totally enclosed. Figure 36. shows the relationship between the wire.^ diameter and the ratio B/A, the fraction of the grains in the wire which show a free surface. The values shown are a conservative estimate since the calculation is based on the supposition that a grain on the surface wil l have a free surface of the maximum possible area. However, i f one takes into account the relative effects of different sizes of free surfaces of the various grains on the surface, this is probably a good measure of what might be called the "effective" ratio of the number of grains showing a free surface to the total number in the specimen. The curve shown in Figure 3 6 - represents a function of the same form as the surface to volume ratio; i .e . the ratio B/A is proportional to the inverse of the wire diameter for a fixed grain size. The ratio wil l have a limiting value of one where the wire diameter is equal to the grain diameter, corresponding to a bamboo structure. The variation of the ratio • /^A may be expected to give rise to the following effects: i . The yield stress should decrease with decreasing diameter, i i . The work hardening rate should decrease with decreasing diameter. These will occur because the constraints on the grains totally enclosed in the wire wil l become less important as the diameter and thus the number of interior grains decreases. Let us now examine the results of this investigation in the light Oo of the above discussion. The yield stress at 7 ° K does in fact decrease with decreasing diameter for diameters below 0.5 mm. Above 0-5 mm diameter, where the change in the ratio B/A with diameter is very small, the yield stress decreases with increasing diameter. The room-temperature.yield-stress, however, remains approximately constant where a decrease is expected with decreasing diameter. It appears that there may be two mechanisms operating at the higher temperature; one tending to increase the yield stress and the one-, outlined above, tending to decrease the yield stress. If the oxide layer makes any contribution to the strength of the wires,, any increase in the yield stress would be proportional to the inverse of the diameter as outlined in the introduction. Since this will have the opposite effect to that of increasing the ratio B/A, these would tend to cancel. Work by previous investigators, however, indicates that the oxide-layer would have l i t t l e i f any effect in that: i . ' There appears to be no effect on the strength of 3 23 copper whiskers from the oxide layer ' i i . Wo effect on the strength of polycrystalline wires from the oxide layer is evident until the grain diameter 2k 26 29 approaches that of the specimen diameter ' ' . The smallest wires in this investigation had about 25 grains in the x-section. The::work-hardening rate, expected to decrease with decreasing diameter, tends to increase for low strain-rate tests .both at room o temperature and at 78 K. The specimens tested at the high strain rate at: room-temperature, however, do show a trend towards the expected decrease. The significance of these is very doubtful since the scatter is high enough to make the changes in hardening rate doubtful. b.. Ductility and Ultimate Tensile Strength Roberts^, studying the ductile fracture of polycrystalline copper, found that the fracture is initiated by the coalescence of voids .into a central crack in the necked region of the specimen. Under further strain, this crack grows out toward the surface of the metal. When the crack approaches the surface, void formation occurs "catastrophically" and the specimen fractures. This would appear to explain.the decrease in ductility and in the ultimate tensile strength with decreasing specimen diameter. The events outlined above, however, take place in the necked region of the specimen and.value of stress and elongation used in this investigation were' taken at the point where the stress began to decrease due to the formation of a neck. It was noted, however, that the mode of fracture changed with decreasing diameter. Wires of diameter greater - 69 -than 0.2 mm showed extensive necking and parted very slowly, the load decreasing gradually. In smaller specimens, no visible neck appeared and the specimen separated suddenly with no noticable decrease in load before fracture. From this, i t would appear that, in the small diameter specimens, the crack formation occurs at a much lower stress than in the large diameter wires and before the neck has formed visibly. There is the possibility that sub-microscopic surface-defects could act as stress-raisers and thus lower the fracture stress. The reduction in apparent ultimate strength due to a stress raiser would, of course, have a greater effect on the thin wires than on thick ones. However, the probability of a stress-raiser which is invisible under a magnification of 100 X having an appreciable effect on the fracture stress is unlikely. A more probable stress-raiser is in the voids themselves. Roberts examined specimens which, although highly strained, had not yet formed a neck. He found, that although the concentration of voids is lower than in the necked regions, there are intergranular voids. A comparison of the effect of diameter on the ultimate strength of wires found in this investigation with that found by McDanels et al35 indicates that, once again, the ratio ~B/A may be important. McDanels et al tested composites of tungsten wires in a copper matrix and found that the strength of the composite increased as the diameter of the tungsten wires decreased. Metallographic. observations indicated that neither the tungsten wires nor the copper matrix failed independently but that they fractured as a unit. For this type of test, the ratio B/A would have a very small, i f any, effect on the properties of the tungsten wire since the surface grains are constrained."by the copper matrix just as the interior grains are constrained by the surrounding tungsten. -70-• The•discussion indicates that any investigation into size effects in polycrystalline wires must be designed to offset the effect of the increasing proportion of the grains which impinge on the surface as the diameter decreases. One possibility for overcoming this is the method used by McDanels et al but this method adds a complicating factor. Another •is to use a small enough grain diameter that the ratio B/A varies negligibly over the desired range of diameters. For- diameters between 900 jx and 5 0 p., this would require a gram size of less than 1 p. 2. Cottrell-Stokes Law It is now generally accepted that the flow in a crystal may be represented by the sum of two components; one a temperature-independent and the other a temperature-dependent flow-stress. On this basis, using Seeger's notation, the flow-stress may be written: where is temperature-independent' and temperature-dependent. From this, the ratio of flow-stresses for any.crystal at two temperatures, T-j_ and can be written: # 1 = 6j,+ <£/ -pr— , „ < 1 for T > T p . . . 2 ) 6r 2 0 £ 2 + 0 J 2 1 2 If a crystal is deformed at T^'to a given strain, then deformed at Tg, the dislocation pattern and density wi l l be the same, at that strain, for the two temperatures. Thus, since 6^ is temperature-independent, 6"^ / and 6$%, wi l l be equal and the ratio of flow stresses wi l l be: (3a) ST 2 " 5* +652 the Cottrell-Stokes ratio. It may also be written: ST/ ^ 1 * £L/61 _ ( 3 b ) o r 2 1 * .6s.2/si - 71 -If Ss i s proportional to 6^  over a range of strain; i.e. 6s 1 = -C : ; 6j. 2 = C &6 6* then the flow-stress ratio becomes, from (3b): 6TI = 1 + C ' = K <i 1 + C where K i s a .'.constant, and the Cottrell-Stokes law w i l l be obeyed. If £5 is not proportional to 6^. , the Cottrell-Stokes law w i l l not be obeyed and the ratio of flow-stresses w i l l vary over a range of strain. a. Work-Hardening As was mentioned i n the introduction, there are two main theories 37 U-2 of strain-hardening, one by Basinski and one by Seeger et a l , to account for the constancy in the Cottrell-Stokes ratio. Both agree that the temperature-dependent component of the flow-stress, 65 , is the stress necessary for the glide dislocations to cut the forest dislocations This process w i l l give rise to a temperature-dependent stress in that i t is a short-range interaction involving the formation of constrictions i n extended dislocations and the formation of dislocation jogs and can there-fore be aided by thermal vibrations. The treatment of this mechanism by Basinski differs somewhat from that of Seeger et a l . Starting with a uniform distribution of obstacles, Seeger assumes that a l l of them are capable of breaking down, whereas Basinski assumes that, independent of the i n i t i a l distribution, the number breaking down w i l l depend on temperature. It i s i n the source of the temperatuie-independent component, 5^ that^these investigators show an important difference of opinion. Seeger et a l . have proposed that & i s due to an internal stress-field which impedes the motion of the glide dislocations and that this stress-field is due to the elastic stress-fields of dislocatipn pile-ups distributed randomly through the crystal. Each pile-up contains about twenty-five dislocations, the number in each being independent of strain. They • showed that the flow-stress due to this stress-field w i l l be proportional to the square-root of the dislocation density. Since 65 would depend on the dislocation density in the same manner, this would imply a proportionality between <5^  and Basinski, on the other hand, attributes the elastic component to the long-range interactions between forest and glide dislocation. Since, in this case, both <5~j a n ( i 6~£ are due to interactions between the same dislocations, the £s a n < i 6% should be proportional and the Cottrell-Stokes law obeyed. Both of these theories are based on the assumption that the rate of increase in the density of forest dislocations i s about the same as the rate of increase in the glide dislocation density. Davis has shown that i f the rate of increase of the glide dislocations is much greater than that of the forest dislocations, the proportionality between <£$ and is destroyed and the Cottrell-Stokes law is no longer obeyed, the deviation being in a positive manner. Similarly, i f the forest dislocation density increases much faster than the glide dislocation density, there w i l l be a negative deviation from the Cottrell-Stokes law. •b. Theoretical Considerations An approximate division of the flow-stress into <Sj and 6^ for a single crystal of copper was calculated by Seeger et a l as shown in Figure 3 7 * This division is based on Seeger's theory of the - 73' -flow-stress but w i l l serve as an i l l u s t r a t i o n for the following discussion. The solid curves, (S& and t\$ , represent the condition under which 6j is proportional to and the Cottrell-Stokes law is obeyed. Consider the effect on the flow-stress ratio i f 6^ were to l i e along the dotted line 6$. At a strain £, this would re-present an increase 6"6^ . over that 6^  which maintains the proportionality with 5j . Since the effect on the Cottrell-Stokes ratio i s d i f f i c u l t y to see, consider instead the ratio , where: 1 Z>6^  = 67 2 - o*r1 = ^ 2 ~ ^ 1 • 6* /? ^ •L- . since  u $ = 6/5 1 2 Simple algebra shows that: 6 7 * 1 = 1 Gr2 - 1 + ^l6r± and that an increase in represents a decrease in the Cottrell-Stokes ratio, $T 2 The increase, c7&£ , in ^ w i l l change the ratio to a new value / A T \ - 74 -Strain Figure 37- The Separation of the Flow-Stress into the Elastic and Temperature-Dependent Compounds as Calculated by Seeger et al^"2. Therefore and A reasonable assumption to make, on consideration of the two theories outlined above, would be that is proportional to In this case, although the Cottrell-Stokes ratio would have a higher value, i t would stillbbe auconstant value. Similarly, a reduction, - &6& , in 6$ would, i f were proportional to , result in a lower but s t i l l constant value for the Cottrell-Stokes ratio. It can be shown, in a similar fashion, that an increase, <S~f5^5 > in (S^5 would result in a lower Cottrell-Stokes ratio. A change in Ss which is proportional to Ss would, in effect, multiply ^ by a constant P, where P < 1 for a negative and P > 1 for a positive Thus: \ = P 6~S2 - P <fs 1 s f r j Si + P ^ X = SS2 - S~3 1 < /p + $si which means that a positive wil l cause a decrease in the Cottrell-Stokes ratio and a negative an increase. Once again, however,'if is proportional to (Sj , the ratio wil l be changed but will remain constant with strain. These effects can now be summed up as follows: Change in 6~5 or <5"$. : Change in the Cottrell-Stokes Ratio Positive - Negative Negative Positive c. A Possible Effect of Dislocation Density Since the results of this investigation have shown that the Cottrell-Stokes behaviour in polycrystals is different from that of single crystals, i t would perhaps be profitable to consider the difference between polycrystals and single crystals in the light of the above theorie One difference in the deformation, characteristics is that, due to the restraints on each grain in a polycrystal, several slip-systems are required to operate to give the required deformation. This might give an overall increase in the dislocation density over that to be found in a single crystal and may result in dislocation interactions beyond those considered by Basinski or by Seeger et a l . Let us then consider the effect of an increase in the forest dislocation density in polycrystals on the flow-stress ratio in the light of Basinski1s theory of the flow-stress. The elastic stress on the glide dislocation at point A, (Figure 38-) wil l be due to the elastic stress-fields of the forest dislocations o(-, and $2' With a low forest density, the stress at point A due to the stress fields of ftand $2 wil l be very small compared to that due to °C. With a high forest density, the forest Figure 38- Representation of a Glide Dislocation Approaching a Row of Forest Dislocations. dislocations @ and @^ wil l make a significant contribution to the stress-field at point A. If 3^ and @ are of the same sign as , the stress-field at A wil l be greater than that due to °t- alone. If ^ and @ a r e o f opposite sign to oC , the stress field at A wil l be smaller than that due to oC alone. If the sign of the forest dislocations were randomly distributed over the slip-plane, the most probable condition, then the stress field at A would be that due to oi. alone since the contributions from and @ 2 would tend to cancel. In any case, any interactions between forest dislocations, due to a high.dislocation density, would result in a proportional to the square-root of the forest density. This would mean that £6$ would be proportional to and, although the value of the Cottrell-Stokes ratio may be changed, the proportionality between a n ( i 6$ would not be upset. Thus, a mechanism of this sort could give rise to the effect shown in Figure 21. but not the deviations from the Cottrell-Stokes law shown in Figure 25-d. Grain Boundaries Another important difference between a single crystal and a polycrystal is that a polycrystal contains grain-boundaries which can act as barriers to dislocation motion. Although Basinski attributes no effect to the stress-fields of pile-ups scattered through the body of the crystal, a dislocation pile-up at a grain boundary wil l have an effect on the crystal directly ahead of i t . Consider the effect of a dislocation pile-up at a large angle . t i l t boundary. Eshelby, Frank and Nabarro^? have calculated the stress-concentration factor ahead of a dislocation pile-up to be of the form shown in Figure 39- As'suming that, the average length of a pile-up wil l be one=half of the grain-diameter, the distance over which the pile-up wil l produce a stress-concentration factor of at least two wil l be 0.25L or 0.125d, where d is the grain diameter. However, since the slip system wil l not be continuous across the grain boundary, this value of the stress concentration factor must be modified. The average angle of a high-angle boundary is about 50° to 60°. Since the angle between ( i l l ) planes is 70° and between 0-10/* directions is 60°, we can expect a slip system with the angle of the plane within ± 20° and the direction within ± 30° to the slip system from which the pile-up originated. At a distance 0.25L from the head of the pile-up, therefore, the stress-concentration factor wil l be between 2 and 2 cos 30° ^ 1.6. If we approximate the shape of the grains to be cubic, the volume affected wil l be: d 3 - 3d 3 = d 3 fl - 27 "\ I; V 517/ or slightly more than one-half the total volume. Unless 6^. <^ 6^ s , this will produce an effect of a magnitude far greater than that observed for the difference between single and polycrystals of -copper. There are two possible modifying factors: 1. The average length of a pile-up at the boundary may be less than one-half the grain diameter. 2. The grain goundary may act as a "screen", as proposed by 48 Frankel , reducing the effect of the pile-up on the adjacent grain. No quantitative calculation on these effects can be made here since there is not sufficient data on the effect of the grain diameter on the Cottrell-Stokes ratio. There is also a complicating Figure 39. Stress Concentration Factor in Front of a Dislocation Pile-up of Length L as Calculated in Reference 48. - 81 -f a c t o r i n t h a t the r a t i o appears to a l s o depend on the f r a c t i o n of the g r a i n s i n the specimen which impinge on the surface (Figure 20. ). The e f f e c t of the p i l e - u p s can, however, be assessed q u a l i t a t i v e l y . The e f f e c t , of the p i l e - u p at the g r a i n boundary w i l l be, because of the s t r e s s - c o n c e n t r a t i o n , t o reduce L\ i n the area ahead of i t . The o v e r a l l r e d u c t i o n i n 6# t,, %6$ , w i l l depend, assuming a constant p i l e - u p l e n g t h , on the number, Up, of p i l e - u p s i n the specimen: The number of p i l e - u p s , i n t u r n , w i l l depend on the d i s l o c a t i o n d e n s i t y , D , i n the specimen: Thus: £££ <^Dd . Both B a s i n s k i ' s and Seeger's t h e o r i e s give a 6^ which i s p r o p o r t i o n a l t o the square-root of the d i s l o c a t i o n d e n s i t y . This means t h a t S6Q w i l l n o t be p r o p o r t i o n a l t o o^ j and, by the previous d i s c u s s i o n , a negative d e v i a t i o n from the C o t t r e l l - S t o k e s law w i l l be expected, i n accord w i t h the experimental observations. e. The E f f e c t of Stacking F a u l t Energy on the C o t t r e l l - S t o k e s E a t i o Since the d e v i a t i o n s from the C o t t r e l l - S t o k e s law found' i n copper p o l y c r y s t a l s do not appear i n aluminum, the d i f f e r e n c e s between aluminum and copper should provide a clue t o the behaviour of copper. The p r i n c i p a l d i f f e r e n c e between copper and aluminum l i e s i n the s t a c k i n g - f a u l t energy, copper having a low and sluminum a high s t a c k i n g - f a u l t energy. Table V. shows the most probable values of s t a c k i n g - f a u l t energy as c a l c u l a t e d 50 by Thornton and H i r s c h together w i t h the C o t t r e l l - S t o k e s r a t i o s , f o r s e v e r a l temperature ranges, found by B a s i n s k i f o r the face-center cubic TI 50 metals copper, aluminum and s i l v e r and f o r the hexagonal metal magnesium - 82 TAB IE V. Stac k i n g - F a u l t Energies and C o t t r e l l - S t o k e s R a t i o s  f o r Cu, Ag, A l and Mg. Stacking F a u l t C o t t r e l l - •Stokes R a t i o , Element Energy T-L = 293°K T 2 = 78°K 200°K k.2°K 100°K k.2°K 50°K k.2°K Ag 35 ergs/cm • 0.86 * 0.95 0-97 O.98 Cu 57 0.88 * 0.93 0-95 0-97 A l 200 0.78 * 0-73 0.80 0.90 Mg (High) - 0-73 O.78 O.87 * Not c o r r e c t e d f o r the v a r i a t i o n of the e l a s t i c modulus w i t h temperature. . - 83 -There seems to he a correlation between the stacking-fault energy and the Cottrell-Stokes ratio in that the high stacking-fault . energy metals (Al and Mg) have a lower ratio than copper and silver. The values shown are corrected for the variation of the elastic shear modulus with temperature. If any other differences, in the metals are ignored and only the stacking-fault energy is considered, the theories of Basinski and Seeger et al may be tested on the basis of prediction of the effect of the stacking-fault energy on- the Cottrell-Stokes ratio. Basinski has- attributed both 6j and 6$ to interactions between forest and glide dislocations. The elastic stress, (5^ , since i t is due to long-range interactions wil l not be appreciably affected, i f at a l l , by the separation between the partial dislocations. The temperature dependent component, C% , since i t involves the actual intersection of the dislocations, wil l be profoundly affected by the separation of the partials. 51 Stroh has calculated the work required to form.a constriction in an extended dislocation from the work required to force a decrease in the separation' of the partials and the work required to bend the disr locations-against the line tension. He found that decreasing the stacking-fault energy increases the work required. Let us now examine the effect of the•stacking-fault energy on the Cottrell-Stokes ratio. By Stroh's calculation, decreasing the stacking-fault energy increases the work required to form a constriction, thus increasing 6s • As shown earlier, increasing Sj wi l l decrease the Cottrell-Stokes ratio. - Qk _ Basinski'.s theory, therefore, predicts a decrease in the Cottrell-Stokes ratio with decreasing stacking-fault energy, contrary to the results tabulated in Table V.- Any assessment of the effect of the stacking-f ault energy in light of Seeger's theory is .more complex since 64 wil l also depend on the stacking-fault energy. The number of dislocations in a pile-up wil l be a function of the stacking-fault energy and thus the magnitude of the stress-field due to the pile-up will depend on the stacking-fault energy. This comes about because the extended dislocations in a pile-up must be constricted before they can escape through cross-slip or through dislocation climb. The stress on the leading dislocation is a function of the number of dislocations in the pile-up and of the applied stress. Consequently, the number of dislocations in a pile-up wil l be"higher in a low stacking-fault energy metal. By Seeger's theory, then, we could expect an increase in both 6s and 6 5 with a decrease in the stacking-fault energy. The effect of this on the Cottrell-Stokes ratio is uncertain but i t is possible that ittcould give an increase in the ratio. This discussion ignores, of course, any difference other than the stacking-fault energy between high .and low stacking-fault energy metals. Based on the limited data available, however, there is a rough correlation between the Cottrell-Stokes ratio and the stacking fault which seems to be independent of other differences. It also appears to be qualitatively independent of the temperature range considered. More experiments, however, on metals of different stacking fault energies must be carried out to definitely establish the existence of the correlation. - Q 5 -Another useful experiment in an investigation of this type would be to alter the stacking-fault energy of a metal by the additions of small amounts of a suitable impurity and to examine the effect on the Cottrell-Stokes ratio. f. A Possible Effect of Dislocation Density on In an earlier part of this discussion, we considered the possible effect of the dislocation density on the elastic interactions between glide and forest dislocations. Let us now consider the effect of increasing the dislocation density on the temperature-dependent component of the flow stress. With a high enough dislocation density, the interactions between parallel extended dislocations wil l decrease the average separation of the partial dislocations, resulting, effectively, in a higher stacking-fault energy and a lower stress required to form a constriction in the extended dislocation. This wil l reduce 65 and, according to Basinski"s theory, result in a higher value of the Cottrell-Stokes ratio- The actual, results on polycrystalline copper (Figure 2l) show that the ratio is lower than that for single crystals. This reduction in the ratio, however, on the basis of the proposed mechanism, fits in with the observed correlation between the stacking-fault; energy and the Cottrell-Stokes ratio. The dislocation density required to produce an appreciable effect on the temperature-dependent component of the flow-stress can 5 3 be estimated quite easily. Cottrell' has caluclated the separation of the two partial dislocations bounding a stacking fault to be: r .= M (bj.bp) 2/7-<r where €. is the stacking-fault energy, b-^  and b 2 the Burgers vectors of the partials andja the elastic shear modulus. In a face center cubic - 86 -.crystal.;, a. normal dissociation mechanism i s : | <io5,-^! < 2 n> + g < l l 5> For t h i s pair of p a r t i a l s : b r b 2 = a and r = u a' 12 2 2hTT6 In copper, ergs/cm 2 and a = 3-6 X. In aluminum 200 ergs/cm 2 and a = 4 . 0 £. Takingyi = 4 X 1011dyne/cm^, r = 10"^ cm f o r copper and r -- 10" cm f o r aluminum. I f the centers of the extended dislocations approached to within 10 r, the separation between the p a r t i a l dislocations would be reduced by about 10$. This distance between dislocations corresponds to d i s l o c a t i o n 12 -2 14 -2 . . . densities of about 10 cm and 10 cm i n copper and aluminum respectively. 8 1 2 - 2 The normal di s l o c a t i o n density i s about 10 to 10 cm i n a cold-worked single c r y s t a l . I f the di s l o c a t i o n density i n a copper polycrystal i s at a l l higher than that i n a single c r y s t a l , we can expect an effect on' from the extra reduction i n the width of the stacking-fault. In aluminum, since a much higher dislo c a t i o n density i s required to s i g n i f i c a n t l y decrease the width of the stacking-fault, the effect would be reduced by a factor of about 100 and would probably be neg l i g i b l e . The foregoing discussion indicated that the process of work-hardening and the origin of the temperature-dependent and temperature-independent components of the flow stress may be more complex than those o 7 42 proposed by Basinski-" or by Seeger et a l . I t has been shown that i t i s possible to explain the v a r i a t i o n of the Cottrell-Stokes r a t i o between single and polycrystals of copper and the deviations from the C o t t r e l l -Stokes r a t i o i n terms of these theories only by proposing an extra mechanism which was not considered i n the o r i g i n a l theory. - 87 -Examination of the Cottrell'-Stokes ratios for metals of different stacking-fault;.energies has indicated a possible variation in the ratio with varying stacking-fault which is opposite to that predicted by Basinski's theory. It should be stressed here, however, that this correlation is far from being definitely established and requires further investigations. 3- Work-Softening in Aluminum Cottrell and "Stokes^  observed a yield drop in aluminum crystals deformed after prior deformation at a lower temperature. They proposed that this yield drop indicates the operation of a mechanism by which the plastically deformed crystal can rid itself of some of the work-hardening introduced through the low-temperature deformation. This proposal is supported by the work-softening observed in this investigation on aluminum polycrystals. As can be seen'in Figure 3^. the work-softening does not appear until the strain at which the yield drop f irst appears. After a few percent deformation, the mganitude of the work-softening for small is approximate y constant over the range of strain studied (Figure 35*)• Also, for large C', the work-hardening introduced by the plastic de-formation after the yield drop compensates to some extent for the work-softening occuring at the yield drop. . It appears that the dislocation' configuration produced by deformation at allow temperature is unstable, at the same strain, at higher temperatures. The fact that the work-softening is recoverable with further deformation at high temperature indicates that the difference in the stable configuration is one of degree rather than kind. Also, the fact that the ratio 6? , for small £ ' , settles down to a constant value indicates that a definite percentage of the obstacles formed at - 8 8 -low temperatures break down at the higher temperature. That this effect is absent in copper means that the mechanism responsible for the work-softening in aluminum is one which is affected by the stacking-fault energy such as cross-slip or dislocation climb. By Basinski' s theory, this process would presumably be a dislocations-annihilation or polyganization through cross-slip or climb. From Seeger's theory, the same effect might be expected through the escape of dislocations from the head of a'.pile-up by cross-slip or climb. Since this type of process occurs more easily at higher temperatures, the equilibrium number of dislocations would be lower than at low temperatures. Examination of Figure 3 ^ . also shows that the results on work-softening in aluminum support the explanation offered by Cottrell and Stokes for the observed difference in flow-stress ratios on going from high to low and from low to high temperatures. Figure kO shows a schematic representation of the form of the yield drop observed in aluminum deformation at room temperature after prior deformation at 78 ° K . A yield stress of the value A would be expected assuming that the change in flow stress is reversible; that is , that the change in the flow stress at this strain should be the same for both directions of change of temperature. The point B represents the actual observed yield stresses than would be found on going from 2 9 3 ° K to 78 ° K . Evidence for this form of the curve is presented by Cottrell and 6 Stokes as follows: 1. The difference in the ratios of.flow stresses does not become appreciable until the strain at which the yield drop appears. 2 . If a specimen is strained to a stress represented by point C, and held at this stress, a delayed yield drop preceeded by creep at stress C is observed. Added to this is the evidence shown in Figure 3^- Here, i t can be seen that, although the work-softening does not appear until the strain at which the first yeild drop is observed, work-softening occurs even in tests where there is not observed yield drop and the magnitude of the work-softening does not depend on the size of the yield drop. This indicates that the dislocation configuration present at 78°K is unstable at 293°K and wil l break down under an applied stress lower than that expected for plastic flow. In some cases, the cr i t ica l stress appears to have been low enough that no yield drop was observed in conjunction with the work-softening. That thermal agitation alone, is not sufficient to cause the breakdown in the dislocation con-figuration is indicated in Figure 3^- which shows the stress-strain curve for a specimen which was strained at 78°K, unloaded and held at room-temperature for several hours, then restrained at 78°K. Np work-softening was observed in this case. - 90 -Strain Figure-:40. A Schematic Representation of the Form of the Stress-Strain Curve for Deformation at 2 9 3 ° K after Prior Deformation at 7 8 ° K . - 91 -SUMMARY AND CONCLUSIONS 1. The important results have been summarized in a previous section and wi l l not be repeated here. 2. In the-tests on the effect of temperature on the flow-stress, the shape of the stress-strain curve changes for specimens of different characteristics. In contrast to this, the shape of the stress-strain curve for polycrystalline aluminum is essentially the same as found in the literature^ for single crystals of aluminum. 5. The deviations from the Cottrell-Stokes law for temperature changes and the variation in the Cottrell-Stokes ratio found for strain-rate changes are probably due to dislocation mechanisms beyond those considered by either ^6 kl Basinski-^ or Seeger et a l . . These mechanisms wi l l be ones which exert a minor influence on.the flow-stress. ' k. The rough correlation between the stacking fault energy and the Cottrell-Stokes ratio' noted in the discussion is the opposite to that * 36 predicted by Basinski s theory. 5. The yield-drop found in aluminum at high temperature after prior deformation at a lower temperature is probably due to the release of dis-locations from a barrier by a thermally activated process such as cross-slip or dislocation climb. - 92 -SUGGESTED FUTURE WORK This work has been,primarily of an exploratory nature and the results have suggested directions of future work which might be profitably-explored. 1. Any future work on size effects in polycrystalline materials should be done using a procedure which wil l eliminate the effect of the ratio B'/A. Some possible methods have been suggested in the discussion. 2. If the Cottrell-Stokes behaviour, for both temperature and strain-rate changes, in polycrystalline copper were studied over different temperature ranges, the mechanism responsible for the deviations from the Cottrell-Stokes law may become more evident. 3- The work-softening found in aluminum polycrystals may also provide significant information i f studied over a series of temperature ranges. It may also be profitable to determine i f the same effects can be produced by strain-rate changes. The effects of different increments of low-temperature strain on the magnitude of the work-softening may also be studied. k. The correlation between the stacking-fault energy and the Cottrell-Stokes ratio noted in the discussion may be investigated by two types of experiment: a. Tests on more metals of high and low stacking-fault; energies and on nickel which has an intermediate stacking-fault energy (about 100 ergs/cm ). . b. Tests on metals doped with appropriate impurities to change the stacking-fault energy. Two alloys which might provide information i f studied over a wide range of compositions are nickel-cobalt and copper-zinc. - 93 -BIBLIOGRAPHY I f K. G. Compton, A. Mendizza and S. M. Arnold, Corrosion, Vol. 7 ( 1 9 5 1 ) , 3 2 7 -2 . J . K. Gait and C. Herring, Phys. Rev., Vol. 8 5 ( 1 9 5 2 ) , 1 0 6 0 . 3- S. S. Brenner, J . App. Phys., Vol. 2 7 ( 1 9 5 6 ' ) , 1 4 8 4 . 4. J . H. Hollomon, Trans. A.I.M .E., Vol. 1 7 1 ( 1 9 4 7 ) , 5 3 5 -5- J . E. Dorn, A. Goldberg and T. E. Tietz, Trans. A.I.M.E., Vol. 1 8 0 ( 1 9 4 9 ) 2 0 5 = 6. A. H. Cottrell and R. J . Stokes, Proc. Roy. Soc. Vo l . ' 2 3 ^ ( 1 9 5 5 / 5 6 ) 1 7 -7- G. F. Taylor, Phys. Rev., V o l . 2 3 ( l 9 g 4 ) , 6 5 5 . 8. A.S.M. Metals Handbook, 1 9 6 1 / II98. 9- Z. Gyulai; Z. 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F l e i s c h e r and B. Chalmers, J . Mech. and Phys. of Solids.Vol.6(1958) 307-33. P. J . S c h l i c h t a , "Growth and P e r f e c t i o n of C r y s t a l s " , Wiley, (1958), 214. 34. P. J . S c h l i c h t a , P r i v a t e Communication. 35. D. L. McDanels, R. W. Jech and J . W. Weeton, Metals Prog. Vol.78(1960)6, 118. 36. P h y s i c a l and Mechanical P r o p e r t i e s of Some High-Strength F i n e Wires, Memorandum No. 80, Defense Metals Information Center, Jan. 1961. 37. Z. S. B a s i n s k i , P h i l . Mag., V o l . 4(1959), 393-38. K. G. Davis, P r i v a t e Communication. 39- W. D. Sylwestrowicz, Trans. A.I.M.E., V o l . 212(1958), 617• 40. M. A. Adams and A. H. C o t t r e l l , P h i l . Mag., V o l . 46(1955), I I 8 7 . 41. A. Seeger, " D i s l o c a t i o n s and Mechanical P r o p e r t i e s of Crystals',' Wiley, 24-3• 42. A. Seeger, J . D i e h l , S. Mader and H. Rebstock, P h i l Mag. V o l . 2(1957), 323-43. V. B. Lawson, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, A p r i l 1961. 44. R. F. Snowball, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, October i960. 45- R- W. Fraser, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, November i960. 46. A.S.M. Metals Handbook, 1961, 962:.. 47. H. C. Rogers, Trans. A.I.M.E., V o l . 218(1960), 498. 48. J . D. Eshelby, F. C. Frank and F. R. N. Nabarro, P h i l . Mag., Vol- 42(1951) 351' 49. J'. P. F r a n k e l , Acta. Met., V o l . 6(1958), 215-50. P. R. Thornton and P. B. H i r s c h , P h i l . Mag., V o l . 3(1958), 738. 51. Z. S. B a s i n s k i , Aust. J . Phys., V o l . 13(1960), 284. 52. A. N. Stroh, Proc. Phys. S o c , V o l . 67B(l954), 427-53. Reference l 6 . Page 74. APPENDICES APPENDIX I. APPENDIX I.  Calculation of Parameters a and b The parameters a and be represent the intercept and the slope respectively of the log-log plot of the stress-strain curve. The values for each specimen were found by calculating the stress at each of a number of points picked from the Instron plot of load versus elongation and fitting these to an expression of the form . log £ = a + b log £ by the methods of least squares. Figure Ul. shows a schematic representation of the Instron plot together with a correction curve which compensates for slackness in the grips and for the inertia of the recording system. A l l elongation measurements are taken with reference to this correction curve. The stress (5 and the strain £ at point A are calculated as follows: V c L V-g = cross-head speed V c = chart speed L = gauge length (measured between the grips) . 6 r = — (1 +C) P = load a = i n i t i a l cross sectional area o The factor ( l +£•) takes into account the reduction in cross-sectional area, due to strain and is derived as follows: The in i t i a l volume of the specimen is VQ = aQ L. When the specimen is extended an amount ^ L , the volume becomes V l = a i (L +A L). where and where / Correction / Curve f Instron t Plot / / P r • 1 Elongation Figure kl.. Schematic Representation of the Load-Elongation plot from the Instron Recorder. Assuming V-^  = YQ: a n (L + & L) = a L l o §1 = 1 + ^ = 1 + £ a o L a l = a o 1 +£ The l o g and l o g ^  are determined f o r each p o i n t picked from the I n s t r o n p l o t . Since the r e l a t i o n s h i p •' i s of the form l o g 6"* = a + b l o g € , the values of a and b which s a t i s f y the l e a s t squares s o l u t i o n are those which s a t i s f y the equation: n a + b £ log£ = ^logS* a£log € + b5"(log€ f = £log£ log^ For specimen 1324, whose l o g - l o g p l o t forms two s t r a i g h t l i n e s , the f o l l o w i n g sets of equations were found: 18 a± + I9.9U3 b x = 59-760 19-9^3 a x + . 22.693 b 1 = 66A70 which gives a l = 2.836. b x = 0.1+37 and 10 a 2 + 14.566 b 2 = 3U.527 14.566 a + 21.274 b 2 = 50.310 which gives • aJ = 3.004 b 2 = 0.308 where a 1 and a 1 are c a l c u l a t e d f o r £ expressed as a percentage. These are r e f f e r r e d to £ expressed i n i n / i n by a 1 = a 1 - 2 b 1 = I.962 a 2 = a 1 - 2 b 2 = 2.388 The correlation coefficient, r, was calculated for randomly selected specimens as shown below: log £ logC - log logC 6*log6* ^ logc 2 where p For the above specimen, the values of r found by this method were r^ s 1 .000 and r^ = 0.995- The values of r found for other specimens were also a l l very close to 1 .0 indicating a close f i t of the points to the equation since a perfect f i t corresponds to r = 1 .0 . - 1 0 0 -0 . 6 0 . 8 l . o 1 . 2 l.k 1 . 6 log °jo Elongation Figure k2. a. 1D1 log °jo Elongation Figure 42. b. Figure U-2. c. Log-Log Plots of the Stress-Strain Curves for Representative Specimens. APPENDIX II. _ 103 -TABLE VI. The Results of Tensile Tests of Polycrystalline Copper Wires a. Temperature = 293°K Strain Rate =0.02 min"1 D e c i m e n Diameter (microns) Total Strain UTS (Kg/cm2) Yield Stress YS , YS TKg/cm2) ° a i b l &2 h 2 B 9 926 37-1 3080 56O 680 1.966 0.435 2. .261 0.345 BIO 926 44.8 3280 590 700 1.898 0.458 2.425 0.299 B l l 930 41.6 3200 520 660 1.960 0.438 2. •993 0.317 B12 922 38.3 3100 560 680 1.954 0.443 2, •371 0.315 B13 932 37-3 3060 490 630 1-971 0.430 2. .220 0-324 Bl4 932 37-9 3100 540 650 1.962 0-437 2.388 0.308 B 1 5 917 36.5 3100 560 690 1.911 0.459 2. •373 0.316 Bl6 930 37-3 3070 570 690 1.941 0.444 2.360 0.316 BIT 930 38.0 3080 54o 680 1.893 0 . 46 i 2, .988 0.318 C 3 518 36.9 3070 530 650 1.924 0.450 2.438 0.294 c 4 518 38.2 3270 560 700 1.842 0.483 2. .444 0.298 C 5 518 38.2 3200 560 680 1.799 O.495 2.405 O.308 C 6 518 36.7 2880 - - 1.902 0.449 2. •357 0.310 C 7 518 38.4 2900 490 600' I.889 0-453 2. •977 0.309 C .8 518 38.3 2900 .490 610 1.880 0.456 2. .406 0.296 Cl4 518 37-9 3100 500 64o I.891 0.462 2.366 0.318 F 1 203 29.0 .3150 610 720 1.951 0.455 2-346 0-333 F 2 203 31-5 3200 570 710 I.926 O.458 2. 270 0-355 F 3 203 33-2 3280 570 710 1.867 0.482 2, .220 0.370 F 5 203 31-9 3250 590 710 I.909 0.468 2. • 322 0.340 F 7 203 29-3 3120 550 700 I.918 0.463 2, • 340 0-333 F 8 203 31-3 3170 550 - 700 1.746 0.515 2. •337 0.333 F 9 203 33-9 3240 590 720 1.923 0.461 2, .297 0.345 F10 203 31-6 3200 590 700 1.880 O.477 2. • 357 0.328 G 5 100 18.8 2540 _ _ 1.863 0.473 _ G 6 . 100 19.5 2560 460 610 1.820 0.486 -G 8 100 19.8 2600 480 670 1.867 O.472 -G 9 101 21.3 264o 470 600 1.894 o.46i G10 101 16.9 2400 420 600 1.825 0.483 -I 1 48 13-6 2460 570 730 1.884 0.482 _ I 2 48 14.9 246o 550 700 1.956 0.450 -I 3 48 24.2 2800 590 700 1.765 0.498 -- 104 -b. Temperature = 2 9 3 ° K Strain Rate = 0 . 8 min ^ Diameter Total Strain UTSg Specimen (microns) ($) . (Kg/cm ) a - j _ • a^  b^ IB 1 946 2 8 . 1 2 6 5 0 2 .074 0 . 3 9 8 2 . 3 3 1 0 . 3 1 7 IB 2 914 2 5 . 2 2 6 2 0 2 . 1 9 0 O .368 2 . 451 0 . 2 8 6 IB 9 9 5 0 2 8 . 8 2770 2 . 0 2 9 0 .418 2-^73 0 . 2 8 1 1B10 • 9^7 • 2 9 . 4 2800 2 . 1 3 0 0 . 3 9 1 2 .461 0 . 2 8 6 1B11 952 2 9 . 1 2 7 8 0 2 . 0 5 2 0 . 4 1 5 2 . 5 0 8 0 . 2 7 1 i c 6 502 2 3 - 0 2570 2 \174 0 . 3 7 2 2 . 3 1 8 0 . 3 2 6 1C 7 495 24 . 2 2 7 0 0 2 . 1 3 3 0 . 3 9 1 2 - 3 7 5 0 . 3 1 3 1C 8 493 2 3 . 4 2670 2 .249 0 . 3 5 7 2 . 6 0 1 0 .245 i c 9 495 2 3 - 6 2 6 6 0 2 .174 0 . 3 7 7 2 .444 0 . 2 9 1 1C10 4 8 8 2 2 . 3 . : 2570 2 . 1 6 0 - 0 . 3 7 9 2 . 401 0 . 3 0 2 1C12 512 ' 2 3 . 7 2760 2 . 2 1 2 0 . 3 6 6 2-577 0 . 2 5 0 1C14 500 2 3 - 5 •2860 2 . 2 6 8 0 . 3 5 4 2 . 6 7 0 ' 0 . 2 2 7 IC15 503 2 2 . 3 2 6 9 0 2 . 2 5 7 0 . 3 5 0 2 . 5 4 8 0 . 2 5 0 IF 1 '• 211 1 7 . 8 2 5 0 0 2 . 1 7 5 0 . 3 7 7 - -IF 3 215 . 1 7 - 8 2 5 8 0 2 . 3 0 3 0 . 3 4 3 - -IF 4 257 1 5 . 8 2 3 0 0 2 . 420 0 . 2 9 9 - -1G 2 . . 95 1 7 . 3 2 6 6 0 2 . 3 2 6 0 . 3 4 0 -1G 3 100 1 5 - 3 2590 2 . 4 6 7 0 . 2 9 6 •- -1G 4 9 8 24 . 2 2850 2 . 9 7 2 0 . 3 5 3 - -1G 5. 99 1 7 . 8 2600 2-324 0 . 3 3 7 - -c. Temperature = 78°K .Sp< )ecimen Diameter (microns) Y i e l d YS 1 (Kg/. Stress YS r-cm2) - 5 . a l b l L1BI3A 903 1150 1250 1.996 O.487 L1B13B 903 1150 1250 I . 9 4 4 0.503 L1B14A 928 1060 1190. 2.032 0 . 1+68 L1315A 917 - - 1.920 0.503 L1B16A • 928 850 1150 I . 9 6 7 0.488 L1317A 897 1060 1190 2.110 0.455 L1B17B 897 • 1050 1190 2.024 0.480 L1C22A 515- _ _ I .96O 0.500 L1C24A 501 1210 1370 2.028 0.479' L1C24B 501 1180 13^0 2.043 0.475 L1C25A 475 1290 ' 1500 2.096 0.473 L1C25B 475 1240 1500 1.992 0.506 L1C26A 490 l l 6 0 1410 2.591 0 A 6 5 L1C26B 490 l l 6 0 1410 - -L1C27A 474 . - - 2.095 0.476 L1F 6A 215- 920 1100 I . 6 7 6 0.482 L1F 6B 215 890 1080 1.954 0.493 L1F 7A 204 960 1210 1-916 . 0.514 L1F 7B .204 96Q 1180 -L1F 9A 218 • 1020 1140 1.936 0.576 L1F 9B 218 1010 1150 1-9^3 O.500 L1F11A 218 1030 1200 1.880 • O.521 L1G 6 99 810 1080 2.085 O.456 L1G 7 102 780 970 1.865 O.508 L1G 8 98 830 960 1.847 O.520 L1G 9 100 850 960 I . 8 5 6 0.515 L1G10 97 - - 1.802 0.536 L1G11 101 - - 1.468 O.619 APPENDIX III. TABLE -VII. Effect of Strain Rate, and Temperature on" the  Flow-Stress of Polycrystalline Copper a. Strain Rate' • , . £:/= l O ' ^ m i n " 1 , ^ -10"1min"1 ' ' Ratio of Specimen Diameter Strain Flow-Stresses . mm •- SX2 3-5 RC3 3-8 2B1 0.9 RC4 3.8 113 0.05 0.068 0.986 0.098 0.985 0.126 0.987 0.164 0.985 0.206 0.985 0.251 0.982 0.292 0.979 0.3W 0.975 0.005 ' 0.935 0.012 0.972 0.024 0.977 0.051 0.978 0.080 . 0.978 0.106 "0.978 0.138 0.977 0.171 0.977 0.203 0.976 0.238 0.978 0.270 0.978 0.308 0.977 0.366 0.975 0.048 0.980 0.130 0.979 0.164 0.978 0.048 ,0.980 0.102 0.980 0.139 0.978 0.226 0.980 0.268 0.979 0.174' 0.979 0.005 0.974 0.024 • 0.973 0.036 0.974 0.056 0.974 0.075 0.975 Specimen Diameter mm Strain Ratio of Flow-Stresses 115 0.05 0.003 0.931 0.022 0.974 0.043 0.974 0.062 0.975 O.O83 0.975 0.111 0.975 0.122 O.976 0.139 0.973 0.152 0.974 0.166 0.974 0 . l 8 l 0.974 1F12 0.2 0.009 O.96I 0.025 O.969 • 0.042 0.970 - O.O58 O.968 0.077 0.971 0.094 0-970 0.110 0.969 0.131 . O.968 O.I5I O.969 0.171 .0.970 O.I89 O.969 0.205 O.969 0.221 0.970 0.239 O.968 0.257 O.968 0.275 0.970 0.295 0.970 O.316 0.970 1F16 . 0.2 . 0.055 0.970 0.068 0.970 0.081 O.969 O.O96 O.968 0.109 0.970 0.122 O.968 O.I36 O.967 0.149 O.968 0.162 O.969 , 0.177 O.967 0.202 O.968 0.215 O.968 0.229 0.970 0.243 O.968 0.257 O.969 0.272 O.969 0.285 O.967 Ratio of Specimen Diameter Strain Flow-Stresses mm 1B22 0.9 IB.26 0.9 Temperature T 1 = 2 9 3 % T 2 SXl 4.0 0.007 O.96O 0.031 O.966 0.048 0.965 0.070 0.967 0.091 O.968 0.110 O.966 0.130 0.964 0.151 0.964 0.168 O.965 0.191 O.963 0.217 0.964 0.246 0.963 0.274 0.963 0.302 O.965 0.325 0.965 0.348 O.965 0.014 0.938 0.039 0-957 0.059 O.966 0.094 O.967 0.110 O.965 0.132 0.964 0.157 0.964 0.180 O.966 0.204 0.963 0.228 0.9.64 0.253 O.965 0.281 O.963 0.309 O.966 0.332 O.963 0.358 O.960 O.387 O.961 78°K 0.020 0.881:. 0.055 0.886 O.O87 0.888 0.124 0.890 0.158 0.889 0.202 0.894 0.252 0.894 0.312 0.881 0.346 0.882 Specimen Diameter ' mm Strain Ratio of Flow-Stresses SX3 3.8 O.O38 O.87O 0.077 O.893 0.122 O.889 0.179 0.886 0.228 0.888 O.287 0.886 0.340 0.880 0.4l4 0.881 0A98 0.880 SX4 3.8 . 0.031 0.880 0.120 0.886 0.183 0.885 0.255 0.881 RC1 4.6 0.054 O.885 0.074 0.884 0.094 0.879 0.115 . 0.881 0'. 136 0.877 0.156 0.874 0.178 0.871 0.197 0.873 0.216 0.867 0.226 0.851 0.273, . 0.853 RC2 3.5 0.012 O.885 0.031 0.884 0.051 0.886 0.093 0.875 0.091 0.879 0.105 • 0.876 0.120 O.874 0.137 0.874 0.153 0.874 0.167 0.866 RC5 3.9 0.028 0.877 • 0.072 0.887 0.129 0.890 0.172 0.884 0.214 0.888 0.258 0.884 O.303 0.884 O.3U8 0.885 0.392 0.887 Specimen Diameter Strain Ratio of Flow-Stresses - 110 RC6 4 . 0 0-022 . 0 . 8 8 0 " 0 . 0 4 8 0 . 8 8 5 0 . 0 7 5 0 . 8 8 8 " 0 . 1 0 2 0 . 8 8 6 0 . 1 2 9 0 . 8 8 6 0 . 1 5 2 0 . 8 8 3 0 . 1 7 9 0 . 8 8 3 0 . 2 0 5 0 . 8 8 0 "' 0 . 2 5 2 • 0.882 "1F13 0.2 0 . 0 1 6 O . 8 7 9 0 . 0 3 8 0 . 8 8 0 0 . 0 5 8 0 . 8 7 3 0 . 0 7 5 0 . 8 7 2 0.100 ' 0 . 8 6 5 .0.124 0 . 8 6 3 0 . 1 5 0 0 . 8 6 0 . 0 . 1 7 9 0.849 0.212 O . 8 5 5 0.245 0.848 0.278 0.846 LB24 . 0-9 0.010 0.846 0.024 O . 8 7 6 " C.OUO 0 . 8 7 9 O.O56 O . 8 7 7 " - '0.071 O . 8 6 9 0 . 0 9 4 " 0 . 8 6 8 0 . 1 1 5 0.864 O . 1 3 5 0 . 8 6 2 O . 1 5 5 O . 8 5 9 , O..176 O . 8 5 7 0.204 0 . 8 5 1 0 . 2 2 3 0.846 0.249 0.848 LB27 0.9 0 . 0 1 7 0 . 8 5 6 0 . o 4 l 0 . 8 8 2 . O . 0 6 7 0.868 . 0 . 0 9 8 0 . 8 6 2 0 . 1 3 2 0 . 8 5 9 0 . 1 7 1 0 . 8 5 6 0 . 2 1 3 0.848 0 . 2 4 9 0.845 0 . 2 9 4 0 . 8 3 7 APPENDIX IV. TABLE VIII. Effect of Temperature on.the Flow-Stress of Aluminum a. Cottrell-Stokes Law Tl = 2 9 3 % T 2 = T8°K Ratio of Specimen Strain Flow-Stresses SA5 0.138 0-TT6 0.184. 0.771 0.212 0-TT3 0.243 0.TT2 0.2T4 O.T69 0.300 0.TT4 0.334 0.T68 O.362 0.TT4 SA6 0.043 0-TT3 0.053 0-TT6 0.068 0.TT4 0.08T 0.TT2 0.112 0.771 o.i4o 0.771 0.150 0.771 0.153 0.774 0.169 0.770 0.205 0.771 0.24T 0.772 0.304 0.774 0.363 0-773 SAT 0.138 0.772 0.169 • 0.777 0.208 0.770 0.280 0.776 0.305 0.770 b. Work-Softening T l = 293°K T 2 = 78°K Specimen €' Strain SA8 0.026 0.080 1.131 0.118 1.034 0.156 0-975 0.193 0-955 0.231 0.971 0.271 0-955 0.308 0-979 0-346 0-975 Specimen £•' S t r a i n SA9 0.050 0 . 0 9 2 1 . 4 7 1 0 . 1 5 7 1 . 0 8 2 0 . 3 1 4 1.021 O . 3 7 8 1 . 0 1 5 0 . 4 4 1 1 . 0 0 6 0 . 5 0 5 O . 9 8 6 SA10 0 . 0 1 3 0 . 0 5 3 1 . 1 4 1 0 . 0 7 7 1 - 0 3 3 0.101 1 . 0 0 3 0 . 1 2 5 0 . 9 9 0 0 . 1 4 9 0 . 9 7 8 -0 . 1 7 3 0 . 9 7 9 ' 0 . 1 9 7 O . 9 6 3 0 . 2 2 1 0 . 9 7 0 0 . 2 7 0 0 . 9 6 4 0 . 2 9 4 O . 9 8 3 0 . 3 1 8 0 . 9 7 2 0 . 3 4 2 0 . 9 7 1 O . 3 6 6 0 . 9 5 1 0 . 3 9 0 O . 9 6 2 0 , 415 O . 9 5 8 SA11 0 . 0 0 5 0 . 0 4 1 I . 0 6 5 0 . 0 8 7 0 . 9 8 8 0 . 1 3 0 0 . 984 O.I78 O . 9 6 9 0 . 2 2 5 O . 9 6 2 0 . 2 7 1 O . 9 6 7 0 . 3 1 7 0 - 9 5 9 .0 .364 O . 9 6 3 0.020 . 0 . 0 7 1 - I.O96 0 . 1 1 6 O . 9 8 8 0 . 1 6 2 0 . 9 7 1 0 . 2 0 9 O . 9 3 8 0 . 2 5 6 O . 9 6 2 0 . 3 0 2 O . 9 5 8 0 . 3 4 7 O.95O SA12 0.040 0 . 0 8 0 1 . 3 0 0 0 . 1 3 0 1 . 0 2 8 0 . l 8 l 1.020 0 . 2 3 2 0 . 997 0 . 2 8 2 O . 9 8 8 SA13' 0 . 0 0 3 0 . 0 4 2 1 . 0 0 3 0 . 1 3 3 O . 9 6 9 0 . 2 2 4 0 . 9 5 2 0 . 3 3 9 0 . 9 7 3 0 . 4 2 9 O . 9 6 7 0 . 0 1 3 O . 3 2 6 O . 9 6 2 O . O 6 7 0.120 1 .542 0 . 2 1 1 1 . 0 6 0 0 . 3 0 1 1 . 0 1 5 0 . 4 i 6 0 . 9 9 0 APPENDIX V. Estimated Errors 1. Size Effects The estimated error in the yield stress, ultimate tensile strength and elongation before fracture are calculated for a representative specimen, Clk. The diameter was measured to ± lyu but difficulties in measuring a true diameter reducedthe accuracy to about ± 3 Diameter = d = 518 ± 3 fx = 518 u ± 0.6$ Area = a = TT d 2 = 2.11 X 10~3cm2 ± 1.2$ ~k~' The Instron tensile tester is rated by the manufacturer at an accuracy of better than ± 1 $ of full-scale load when the slope of the load elongation curve is such that more than three seconds is required for the pen'..to travel the f u l l width of the chart. Load' at yield, P = 2.86 * 0.05 Ih = 2.86 lb ± 1-7$ Since £ is very small at the yield point, errors in the (l + € ) term wil l have a negligible effect. ¥ S 0 . = o.k^k P (1 * 0.017) ( 1 + € ) J a (1 * 0.012) - v ; = 6k0 Kg/cm2 ± 3$ = 640 ± 20 Kg/cm2. The gauge-length was measured to ± 0.01 mm but difficulties in measurement reduced the accuracy to about t 0.05 mm. Gauge length = 62.41* 0.05 mm = 62.kl mm i 0.1$ The strain was .-calculated to be 0.001-(l * 0.00l)(x ± l ) where x is as defined in Appendix I. The elongation before fracture: £= o.ooi ( i * o.ooi) 379 ( l * 0.003) = 0.379 ( l * 0.004) = 0.379 ± 0.002 The ultimate tensile strength is calculated by the same method as the yield stress. p. = 10.6 ± 0.2 lb = 10.6 lb ± 2 $ 1 + € = 1-379 ± 0.002 - = 1-379 (1 ± 0-002) UTS = 3100 (1 ± 0.034) Kg/cm2 = 3100 ±"100 Kg/cm The limits of accuracy found for the other specimens tested for size effects were found to be of the same order as those calculated for specimen C14. 2. Effect of Temperature and Strain Rate on the Flow-Stress Since only a ratio of flow-stresses was required, only the loads were measured and the only errors are those due to the limits of accuracy of the machine. The Instron tensile tester is equipped with a zero suppression by means of which the area of the load-elongation curve to be studied can be magnified, thus increasing the accuracy. The max-imum estimated error for a l l specimens varied from about 0.8$ at low strains to about 0-1 -$ at high strains.  THE EFFECTS OF SIZE, TEMPERATURE AND STRAIN-RATE ON THE MECHANICAL PROPERTIES OF FACE-CENTERED.CUBIC METALS ' by '. RONALD ALBERT JOSEPH COSTANZO A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE DEPARTMENT OF MINING AND METALLURGY We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF APPLIED SCIENCE Members of the Department of Mining and Metallurgy THE UNIVERSITY OF BRITISH COLUMBIA October 1961 ! In presenting 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 requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree 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 reference and study. I f u r t h e r agree that permission f o r extensive copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of 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 r k w e d without my w r i t t e n permission. Department of / / t ^ s ^ <? AV£4MXO*SL; 7f The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3, Canada. Date ABSTRACT Drawn and annealed copper wires of diameters ranging from 50 u to 900 }i were t e s t e d i n t e n s i o n and the r e s u l t s examined f o r evidence of s i z e - e f f e c t s . No s i z e - e f f e c t on y i e l d - s t r e s s or work-hardening r a t e has been d e f i n i t e l y e s t a b l i s h e d . The r e s u l t s were discussed i n terms of the f r a c t i o n of the number of grains i n the specimen which have a f r e e surface. The u l t i m a t e t e n s i l e s t r e n g t h and d u c t i l i t y decrease w i t h decreasing d i a -meter f o r diameters below 200 yu. An ex p l a n a t i o n has been put forward i n terms of void-formation during deformation. P o l y c r y s t a l s and s i n g l e c r y s t a l s of copper were t e s t e d a t room -1 -1 ~3 -1 temperature w i t h the s t r a i n - r a t e c y c l e d between 10 min and 10 mm P o l y c r y s t a l l i n e copper obeys the C o t t r e l l - S t o k e s law but shows a v a r i a t i o n i n the r a t i o of f l o w - s t r e s s e s w i t h v a r y i n g g r a i n diameter and w i t h a va r y i n g value of the f r a c t i o n of grains i n the specimen which show a f r e e surface. Copper specimens were a l s o t e s t e d w i t h the temperature c y c l e d between rj8QK and 293°K- Copper p o l y c r y s t a l s do.not obey the C o t t r e l l - S t o k e s law, the d e v i a t i o n depending on the g r a i n s i z e . These r e s u l t s are discussed i n terms of s t a c k i n g - f a u l t energy and s e v e r a l p o s s i b l e explanations are considered. Aluminum p o l y c r y s t a l s were t e s t e d w i t h the temperature c y c l e d between 78 K and 293 K. Aluminum obeys the C o t t r e l l - S t o k e s law f o r temperature v a r i a t i o n . A work-softening e f f e c t accompanies the y i e l d -drop found a t 293°K a f t e r p r i o r deformation at 78°K. This was discussed i n terms of c r o s s - s l i p and d i s l o c a t i o n climb mechanisms. v i i . ACKNOWLEDGEMENT The author i s g r a t e f u l f o r the advice and encouragement given by h i s research d i r e c t o r , Dr. E. Teghtsoonian. Thanks are extended t o Mr. R. G. B u t t e r s f o r t e c h n i c a l a s s i s t a n c e and t o f e l l o w graduate students, e s p e c i a l l y Mr. K. G. Davis, f o r many h e l p f u l d i s c u s s i o n s . S p e c i a l thanks are extended t o Miss I . Duthie who a s s i s t e d i n many of the r o u t i n e c a l c u l a t i o n s . This work was fi n a n c e d by Defence Research Board Grant No. 7 5 1 0 - 2 9 . i i . TABLE OF CONTENTS Page INTRODUCTION , 1 PREVIOUS WORK 3 1. Si z e E f f e c t s 3 a. C r y s t a l P e r f e c t i o n k-b. E f f e c t of Surface Conditions 7 c. E f f e c t s of V a r i a t i o n i n the Length of the S l i p - P l a n e 10 d. G r i p E f f e c t s 12 e. Size E f f e c t s i n P o l y c r y s t a l l i n e Wires 15 2. C o t t r e l l - S t o k e s Law . • . 15 3- Comments 20 EXPERIMENTAL PROCEDURE 22 1. M a t e r i a l s 22 2. Specimen P r e p a r a t i o n . . . 22 a. Copper Wires . 22 b. Copper Rods 23 c. Copper S i n g l e C r y s t a l s 23 d. Copper S t r i p 25 e. Aluminum S t r i p . . . 25 3- Measurements v 25 k. T e s t i n g Procedure 26 a. Normal T e n s i l e Tests 26 b. V a r i a b l e Strain-Rate Tests ' 26 c. V a r i a b l e Temperature Tests . . . 28 EXPERIMENTAL RESULTS AND OBSERVATIONS ' . . 29 1. Size E f f e c t s 29 a. Y i e l d Stress 29 b. Ul t i m a t e T e n s i l e S t r e s s 30 i i i . TABLE OF CONTENTS CONTINUED .Page c. Ductility . . . . . . 3 0 &. Work-Hardening Rate 3 2 2 . Cottrell-Stokes Law 3 8 a. Effect of Strain-Rate on Flow Stress 3 8 b. Effect of Temperature on Flow Stresses ~.kQ c. Results of Tests on Aluminum . . . • 5 8 SUMMARY OF THE RESULTS 6 3 DISCUSSION .' 6 5 1 . Size Effects 6 5 a. Yield Stress and Work-Hardening Rate 6 5 b. Ductility and Ultimate Tensile Strength 6 8 2 . Cottrell-Stokes Law . ' . - TO a. Work-Hardening . . . . . 7 1 b. Theoretical Considerations 73 c. A Possible Effect of Dislocation Density . . . . . . 7 6 d. Grain Boundaries 7 8 e. The Effect of Stacking-Fault Energy . . . . . . . . 8 l f. A Possible Effect of Dislocation Density on 6 5 ' . . 8 5 3. Work-Softening in Aluminum 8 7 SUMMARY AND CONCLUSIONS 9 1 SUGGESTED FUTURE WORK 92 BIBLIOGRAPHY 93 APPENDICES, .95 iv. FIGURES Page 1. Tensile Strengths of Silicon Rods and Whiskers 5 2. Schematic Representations of a Stress-Strain Curve for a Face-Centered Cubic Single Crystal 5 3- Strength Versus the Inverse of the Diameter for (a) Copper and (b) Iron Whiskers : • 6 k. Variation of the Strength of Silicon Whiskers with Temperature 6 5- Effect of Diameter on the Crit ical Shear Stress of Cadmium Single Crystals 9 6. Effect of an Oxide Layer on the Stress-Strain Curve for Single Crystals of Silver . . . . . . . 9 7- Variation of Flow Stress with Diameter for Copper Whiskers . . I**-8. Ratio of Flow Stresses at 293°K'and 90°K for Aluminum Single Crystals 1 Q -.5 > -K -1 9- Ratio of Flow Stresses at Strain Rates of 10 and 10 sec for Copper and Aluminum • 18 10. Ratio of Flow Stresses at 293°K and 78°K for Copper Polycrystals 19 11.. Electrolytic Cell and Power Source Used in Electropolishing Copper Wires 2k 12. Gripping System for Low-Temperature Tests on (a) Wire and Rod Specimens and (b) Strip Specimens 27 13> Yield Stress of Polycrystalline Copper Wires as a Function of the Diameter ? . . 33 lk. Ultimate Tensile Strength of Polycrystalline Copper Wires as a Function of the Diameter . . . . • . 3* 15. Ductility of Polycrystalline Copper Wires as a Function of the Diameter 35 16. Work-Hardening Rate of Polycrystalline Copper Wires as a Function of the Diameter . . . . . . . . . . . . . . . . 36 17. Ratio of Flow Stresses for Specimens 1322 and IB26 ; . . . . . . ^1 18. Ratio of Flow Stresses for Specimens IF 12, 1F16, 113, and 1113 . h r 2 19-. Ratio of Flow Stresses for Specimens 2B1, RC3, and RGk . . . . ^3 20. Ratio of Flow Stresses for Specimen SX2 ^3 FIGURES CONTINUED Page 21. Figures 17 to 20 Plotted Together for Comparison hh 22. Stress-Strain Curve for Specimen 1B22 ' U5 23. Stress-Strain Curve for Specimen RC3 • • • h6 2k. Stress-Strain Curve for Specimen SX2 . . • k-7 25. Ratio of Flow Stresses for. T = 293°K, T = 78°K for Polycrystal-line Copper . . . . . . . . . . . . . . . . . 51 26. Ratio of Flow Stresses for T-^ 293°K, T2= 78°K for Single and Polycrystals of Copper . . . 52 27. Stress Strain Curve for Specimen 1B2U '. 53 2'8. Stress-Strain Curve for Specimen RC6 5^ 29. Stress-Strain Curve for Specimen RC6 after the appearance of a visible neck . . . . . . 55 30. Stress-Strain Curve for Specimen SX3, Part 1 56 31. Stress-Strain Curve, for Specimen SX3, Part 2 57 o o 32. Ratio of Flow-Stresses for T-^ 293 K, T '= 78 K for Aluminum Polycrystals. .......... ^ ..... . 59 33- Ratio of Flow-Stresses for T ^ - 2 9 3 % Tg= 78°K for Strip Copper 60 3^. Stress-Strain Curve for Aluminum Specimen SA2 . . . . . . . . 6l 35- The Work-Softening Ratio as a Function of Strain for Various €' 62 36. Variation of the'Ratio B/A with Specimen Diameter . . . . . . 66 37- The Separation of the Flow Stress into the Elastic and Temp-erature-Dependent Components . 7^ 38. Representation of a Glide Dislocation Approaching a Row of Forest Dislocations 77 39* Stress-Concentration Factor in Front of a Dislocation Pile-up of Length L . 80 kO. A Schematic Representation of the Form pf the Stress-Strain Cugve, for Deformation at 293°K after Prior Deformation at 78 K . . . . . . . . . . ; . . . . . . 90 hi. Schematic Representation of the Load-Elongation Plot from the Instron Recorder 97 k2. Log-Log Plots of the Stress-Strain Curves of Copper Polycrystals Under Various Test Conditions 100 v i . .TABLES Page I. Ratio of Flow Stresses for T^ = 293°K, T2='78°K for Various Metal's . . . . . . . . . . . . . . . . . 17 II. Characteristics of Specimens Tested for Size Effects . 31 III. Characteristics of Specimens Tested for the Effect of Strain-Rate on Flow Stress . . 1+0 IV. Characteristics of Specimens Tested for the Effect of Temperature on Flow Stress 50 V. Stacking-Fault Energies and Cottrell-Stokes Ratios for Cu, Ag, Al and Mg 82 VI. Results of Tensile Tests on Polycrystalline Copper Wires 103 VII. Effect of Strain-Rate and Temperature on the Flow-Stress of Polycrystalline Copper ' 106 VIII. Effect of Temperature on the Flow-Stress of Polycrystalline Aluminum ' i l l INTRODUCTION Interest in the effect of diameter on the strength of crystals has, in the last ten years, grown out of whisker research. Investigations of the mechanical properties of whiskers have been quite extensive and have produced some interesting results, the one of main interest here being that the strength of a whisker depends in some manner on its diameter. Interest in whiskers developed in the early 1950's when Compton, Mendizza and Arnold"'" discovered whiskers growing in electroplated capacitors. , 2 Gait and Herring performed the first tests, a simple bend test, on tin whiskers and found a high elastic limit (as high as 2$). Interest in the effect of crystal size on the strength of metals has been aroused by Brenner's discovery that the strength of a whisker is a function of its diameter. Some investigations of the variation of strength with diameter have been carried out on single crystals produced from bulk material but have produced no firm explanation for the diameter dependence. Also, virtually a l l of the work has been'.;done using 'single crystals and the work on polycrystals has not produced particularly good results. Another line of investigation concerning the mechanism of work-hardening in face-center cubic crystals has been pursued since the early 1950's. These investigations have centered around the effects of temp-erature and strain rate on the flow stress of such metals as copper, aluminum and silver. The usual technique used for comparison of properties 1). at different strain rates and temperatures is that originated by Hollomon in 19^6. At this time the existence of a mechanical equation of state relating the flow stress to the strain, strain rate and temperature and - 2 -the exact form that i t would take was under investigation. The mechanical equation of state demands that where & = true stress C = true strain c = true strain rate T = temperature i .e . the flow stress is a function of the instantaneous values of the strain, strain rate and temperature, independent of the thermal and mechanical history of the metal. 5 In 19^8, Dorn, Goldberg and Tietz , using this technique, showed that the mechanical equation of state is not valid but that the • flow stress also depends on the thermal and mechanical history of the 6 metal. It was later realized by Cottrell and Stokes that the results of this type of test demonstrated that the difference in flow stress when a metal is deformed to a given strain at different temperatures arises from two distinct contributions: 1. The difference arising from the variation of dislocation configuration due to strain at different temperatures, and 2 . The "reversible" contribution of thermal fluctuations. The type of test mentioned show only the "reversible" part of the dependence and therefore may be used to study the form of the inter-actions between dislocations.. As the work in both of-these fields has been confined almost exclusively to single crystals, this investigation wil l be concerned with size-effects and with the temperature and strain rate dependence of the flow stress in copper polycrystals. - 3 -PREVIOUS WORK !.• Size Effects When crystals of diameter below a cr i t ical value are deformed, variations in the properties of those crystals are found. These variations, since they are a function of the diameter, are usually termed "size effects". The problem of diameter-dependence of these properties has been attacked from two directions; from tests on whiskers of various diameters and from tests on crystals produced from bulk material by various methods of diameter reduction. The f irst report of a size effect in metals was that by Taylor^ in 192U. Taylor produced fine filaments of various metals with diameters -k as small as 10 cm. by drawing out molten metal in a glass tube. Although no determination of the microstructure of these wires is mentioned, i t may be assumed that they are of a bamboo structure since the final product was formed by solidification along the length of the wire. It was found that these wires had a very high ductility: "Though britt le in bulk, bismuth and antimony are very pliable in the form of fine wires.- Antimony wire as large as 0.003 cm. diameter may be bent repeatedly without breaking". / 2 The tensile strength of antimony wire was reported to be 1800 to 2200 Kg/cm and that of bismuth about 50 Kg/cm . In comparison with this, the tensile 8 / 2 strength of bulk antimony is about 110 Kg/cm . Because bismuth is so britt le in bulk form,' no tensile strength has been reported. Taylor gave no details of testing method. No real interest in this phenomenon, however, developed until the advent of whisker research. Size effects in the strength of whiskers have been found by Brenner^ in copper and iron, and by Gyulai^ in NaCl whiskers. Single crystals of copper"^' and aluminum"^ prepared from bulk material have shown a size effect and a comparison of the strengths of silicon whiskers 13 and rods prepared from bulk crystals has shown that they are similar for similar diamters (Figure ! . )• The parameters normally measured in investigations of size-effects in face-centered cubic single crystals'are 6* - ^ 7 ^ 7 £ ^ y g ^ and 0£ which are defined in Figure 2. Generally, the following effects have been observed with decreasing diameter: 1. cfpcT' 2 increase 2. £"2 increases 3. © 1 7 Q 2 decrease A number of theories, put forward to explain these effects, wil l be discussed below. a. Crystal Perfection Brenner's results-^ for copper and iron whiskers (Figure 3-) show . that the strength is proportional to the inverse of the diameter. If, as has been postulated, the high strength of whiskers were due to the small volume resulting in a low statistical chance of the inclusion of a dis-location source, then i t would be perhaps more reasonable to expect that the strength should be dependent on the inverse of the square of the diameter, as has been found by Eder and Meyer1**' and tentatively confirmed by Moore" .^ Further, assuming that the strength depends only on the volume of the whisker and taking Eder and Meyer's results to be true, then there should also be a length dependence such that the strength is proportional to the inverse of the length. A computation of the expected length depend-ence from Moore's results on diameter dependence shows that this length dependence should be experimentally observable. No such length dependence 15 has been found 1 6 Also, according to Cottrell , the yield stress of a perfect crystal should be insensitive to temperature. The results on silicon 13 . 0 whiskers show that the yield stress at 650 C is more than double the - 5 A o 0 R O D S o W H I S K E R S - • J . . » 0 0 x 1 1 . C R O S S S E C T I O N IN S O U A R E C E N T I M E T E R S Figure 1. Tensile Strengths of Silicon Rods and Whiskers. Reproduced from Reference 12. Stress JBlope= Qx -^"TJlope^eT 4 i i 1 i i Strain Figure 2. Schematic Representation of a Stress-Strain Curve for a Face-Centered Cubic Single Crystal Showing the Parameters Mentioned in the Text. Figure 3.. Strength Versus the Inverse of the Diameter for: (a) Copper and ("b) Iron Whiskers. Reproduced from Reference 3» h f ,/ / N f RAC TURE / .A C X / " C \ \ \ \ V D0°C \ — \ 13,600 N . _i • —• 0 0.002 0.004 0.006 0.008 0.01 0.02 0.0-i 0.06 ^ - « E U . S T , C , Figure k. Variation of the Strength of Silicon Whiskers with Temperature. Reproduced from Reference 12. yield stress at 800°C (Figure k.). It was also found that small rods prepared from bulk material have the same room-temperature fracture stress as whiskers but have the same yield stress as bulk silicon crystals at 800°C. - 13 Pearson et al also found that prior deformation of.a silicon whisker at 800°C such that a dislocation density of about 10® cm~^  was introduced, has l i t t l e effect on the room-temperature fracture stress. This- also indicates that, in this case, the dislocation density has l i t t l e effect on the fracture stress. b. Effect of Surface Conditions Possibly the most reasonable source of size effects is in the surface effects. The ratio of surface area to volume, being proportional to the inverse of-;. the diameter, increases with decreasing diameter and surface effects would become more dominant. One possibility is that the strength of the oxide layer on the surface of the crystal would become more effective in smaller crystals. In a crystal of radius r, and oxide thickness > "the load at the yield point may be expressed as: fTrVy = rT{r - S f ^  + 2ff r Sol where 6^ = yield stress g^ ,= yield stress of the bulk material 6s = effective strength of the oxide layer at the yield point. Solving this equation for (f^ gives the relation: 6X = ^ + S ( £ - <v ) — V where A = 2 7 f r l V = 7 T r 2 l - 8 -12 Using this relation and = h6 gm/mm^ , Fleischer and Chalmers calculated f> 2 5^= 3(10 ) gm/mm for aluminum oxide. This value compares favourably with the value of 2-5-3 -0 (10 ) gm/mm found by Takamura from measurements on specimens with varying oxide thickness. 18 Roscoe reported that an oxide layer on the surface of cadmium crystals increases the strength and that this effect increased with increasing oxide thickness and with decreasing diameter. In contrast to this, Andrade"^ reported that Makin found that, although cadmium shows an increased strength with the addition of an oxide layer, the size effects for an oxide-coated crystal and a clean crystal do not differ appreciably (Figure 5.). This theory would also lead one' to expect, in the case of a ductile oxide, that the oxide layer should have an effect on the rest 20 of the stress-strain curve. Andrade and;Henderson found that an oxide layer on the surface of a single crystal of silver not only gave a small increase in the yield stress, but also that i t greatly increased the rate of work-hardening and the ultimate tensile stress (Figure 6 . ) . A similar effect was observed on copper crystals plated with.nickel-chromium"^. 21 Kramer and Demer tested aluminum crystals in an electrolytic cel l , electropolishing the crystals during the test. They found that, as the rate of metal removal was increased, £ 2 increased, 9-^  and 0£ decreased and 6^ 2 remained constant. , 2 2 Harper and Cottrell found that the cri t ical shear stress for single crystals of zinc was reduced by electropolishing and increased by •3 23 exposure to steam. Tests on copper whiskers-^' J , however, show no effect on the strength from an oxide layer. - 9 -0 10 . 20 30 40 50 g l i d e ( % ) Figure 6. Effect of an Oxide Layer on the Stress Strain Curve for Single Crystals of Silver. Reproduced from Reference 20. 21 I t has been suggested that the e f f e c t of an oxide l a y e r on the p l a s t i c r e g i o n of the s t r e s s - s t r a i n curve i s due to i m p u r i t y atoms being c a r r i e d t o the i n t e r i o r of the c r y s t a l by r a p i d l y t r a v e l l i n g d i s -l o c a t i o n s . This s u p p o s i t i o n i s supported by the r e s u l t s of C o t t r e l l and. 22 Harper who found that the e f f e c t of an oxide l a y e r on the p l a s t i c r e g ion diminished w i t h decreasing s t r a i n r a t e , being almost zero at very low s t r a i n r a t e s . Observations on the e f f e c t of removing the 24 surface l a y e r on the creep r a t e of cadmium s i n g l e c r y s t a l s , however, c o n t r a d i c t t h i s view. Another p o s s i b l e e x p l a n a t i o n f o r the e f f e c t of surface f i l m s and of diameter on the stre n g t h of s i n g l e c r y s t a l s i s th a t the oxide l a y e r i n h i b i t s the egress of d i s l o c a t i o n s from the c r y s t a l . This theory 25 i s supported by the observations of B a r r e t t on z i n c and m i l d s t e e l coated w i t h an oxide l a y e r . The c r y s t a l s were t w i s t e d p l a s t i c a l l y and the r a t e of u n t w i s t i n g on r e l a x i n g the loa d was measured. Soon a f t e r the l o a d was r e l a x e d , the oxide l a y e r was removed w i t h an etchant, upon which the c r y s t a l showed a s l i g h t t w i s t , contrary to the expected e f f e c t , i n the d i r e c t i o n of the o r i g i n a l deformation before resuming the u n t w i s t i n g a c t i o n . I f the theory were t r u e , then i t would be expected that the e f f e c t of an oxide f i l m on a f i n e - g r a i n e d p o l y c r y s t a l l i n e m a t e r i a l should be small and t h a t i t should increase w i t h i n c r e a s i n g g r a i n s i z e . 24,26 These c o n d i t i o n s have been reported by Andrade and R a n d a l l and by 29 Andrade and Kennedy y . c. E f f e c t s of V a r i a t i o n i n the Length of the S l i p - P l a n e The f o r e - g o i n g i n d i c a t e s t h a t the l e n g t h of the s l i p - p l a n e , measured i n the s l i p - d i r e c t i o n , could have an e f f e c t on the st r e n g t h of the c r y s t a l , an e f f e c t which can be e a s i l y demonstrated. A Frank-Read - 11 -source in a slip-plane generates dislocation loops under an applied stress. If the surface of the crystal provides a barrier strong enough to hold these loops inside the crystal, pile-ups wil l be formed. Since the stress at the head of the dislocation pile-up is proportional to the number of dislocations in the pile-up and to .the applied stress, in a large crystal, dislocations wil l break through the barrier at a lower applied stress than in a small crystal. Experimental results concerning this effect are contradictory. 12 Fleischer and Chalmers tested crystals of two orientations; one (S^ .) in which the length of"the slip-plane remained constant with crystal size and the other'(S^) in which i t varied with size. The results indicate that the size dependence of the yield stress is the same for both crystals and that the yield stress at any one diameter is the same for both orient-27 ations. Lipsett and King , in an investigation of the influence of a film of gold on the surface of single crystals of cadmium, found that the increase in cr i t ical resolved shear stress was independent of orientation. 28 In contrast to these two results, Gilman and Read found that the influence of a film of copper plated onto zinc single crystals varied with the orient-ation of the crystals, increasing as the length of the slip plane decreased. 10 It has also been suggested, by Suzuki et al that decreasing the length of the slip plane should result in a lower work-hardening rate and 12 an increase in the range of easy glide. Fleischer and Chalmers found that for crystals of orientation S , 9^  increases, 9^ remains constant and C 2 decreases with decreasing size. In contrast to this, for crystals of orientation S T T , 9 and 9 decrease and £ 0 increases with decreasing -L-L 1 2 <-size. The results of Gilman2^, who found a higher work-hardening rate for zinc single crystals with longer slip planes, agree with this result. - 12 -This effect could be attributed to the fact that dislocations have a shorter distance to travel before escaping from the surface of the crystal and, therefore, there is a smaller probability of their encountering obstacles. This would mean that a Frank-Read source would be able to produce more dislocation loops before the back-stress caused by dislocation pile-ups is high enough to produce hardening. The resolved shear stress required to operate a Frank-Read source may be expressed as: length of the pinned dislocation segment. This would indicate that, since the length 1 would be limited by the size of the slip plane, that smaller crystals should require a larger stress for slip. Brenner , however, found that i f the size effect were to be explained on this limited to the order of 10 cm. Saimoto has offered a mechanism whereby sources of this length might be introduced into a whisker but there is some question as to whether a source of this length could operate. d. Grip Effects 32 Fleischer and Chalmers , considering the effect of grip restraints on the stress-strain curve for a single crystal, have shown that the resolved shear stress on a crystal plane ( T"j) is the sum of two components; one due to the applied stress ( ^ ) and one due to the bending moment arising from the grip restraints ( Ty), and that the resultant stress on the primary system can be expressed as: ? = M b 1 where is the shear modulus, b the Burgers vector and 1 the where m. cos^C cosAp, the Schmid factor for the primary slip system a = crystal diameter L = crystal length X = the angle between the slip direction and the P specimen axis, = the angle between the slip-plane-normal and the specimen axis. This indicates that the actual stress on the primary slip-plane is lower than the measured stress resulting in a higher apparent stress, the difference depending on the ratio a/L for any given orientation and strain. The effect on the yield stress would be negligible since, at this point, the strain is very small. If the above equation is modified to take into account slip on a secondary system, the resulting expression indicates that the rate of hardening should decrease in the easy glide region and that the range of easy glide should increase with decreasing a. The authors calculated that, on this basis, a ratio of a/L which is small compared with l/lOO is required to remove the influence of grip restraints from the 12 stress-strain curve. When the work on size effects by the same authors 10 arid that by Suzuki et al was examined in this light, i t was found that the ratio was greater than the cri t ical value calculated and that the size effects were accounted for by the calculated "2^ . It would be interesting to examine the results of the diameter dependence of the flow stress of whiskers in the light of the above theory. The diameters of whiskers tested usually vary between about k / i to 20 ju and the lengths from 0-5 mm to 5 mm, with no apparent relation between diameter and lens th!5. These figures give upper and lower limits of l/25 and l/lOOO respectively to the ratio a/L. Under these circumstances, i f the variation in flow stress with diameter were due to grip effects, a statistical scatter would be expected rather than a clear- dependence as was found by Brenner^ and by Eder and Meyer"'"(Figure 7-)-- Ik Figure 7- Variation of Flow Stress with Diameter for Copper Whiskers. (Brenner and Eder and Meyer). Reproduced from Reference 15-15 e. Size Effects in Polycrystalline Wires The foregoing discussion has been concerned with single crystals only and most of the theories presented are applicable only to single crystals. The one suggestion which might be applied to polycrystals is the strengthening effect of an oxide layer.- However, as mentioned before, there is no effect from an oxide layer on polycrystalline wires until the grain diameter is of the order of the specimen diameter. 33 Shlichta tentatively reported that the results of preliminary tests on fine Taylor-process and electropolished drawn wires indicate that there is an increase in strength and a decrease in plastic deformation 3U before fracture with decreasing diameter. Later , however, he concluded that this effect was not a true size-effect but represented a "survival 35 of the fittest". McDanels, Jech and Weeton found a size effect in polycrystalline tungsten wires. In this investigation, composites of tungsten wires in a copper matrix were tested in tension. The results showed that the composites containing finer wires have a higher strength. 36 A Defence Metals Information Memorandum lists the properties of some, high-strength fine wires of various metals including high-carbon and stainless steels, nickel-base alloysy tungsten and molybdenum with diameters ranging from 0.0005" to 0.01". In general, the yield stress, ultimate stress and the total elongation increased with decreasing size. The one exception noted was a nickel base alloy which showed decreasing ductility with decreasing diameter. 2. Cottrell-Stokes Law When two specimens of a metal are deformed at the same strain rate but at two different temperatures, the stress>-strain curves diverge with increasing strain. Cottrell and Stokes^ have pointed out that the - 16 5 results of Dorn et al indicate that the difference in the two stresses at any given strain is probably due to two contributions: 1. The density and distribution of dislocations may be different in metals deformed at different temperatures to the same strain, 2. There may also be a "reversible" effect due to thermal fluctuations, an effect which wil l decrease with decreasing temperature. In .their investigations of this phenomenon in single crystals of aluminum, they also noted that the ratio of the flow stresses at. any two temperatures Is constant over the entire range of strain after an in i t ia l few percent deformation (Figure 8.). This effect has also been noted by other invest-igators working with different metals and a summary of the results is given in Table I. 37 Basinski - noted that the same effect is found i f the ratio of the flow-stresses is taken at one temperature but with two different strain rates. His results for aluminum and copper at various temperatures are shown in Figure 9- The only large deviation from Cottrell-Stokes behaviour is that found by Davis who found the ratio of flow stresses in cobalt single crystals varies with increasing strain. This was found for both variations in temperature and in strain rate. A l l of these results have been obtained, with one exception;,: by the same method, one which wil l be outlined later in this work. The one 39 exception is the work of Sylwestrowicz on polycrystalline copper who obtained each point from a different specimen. The result was that the flow stress ratio was not quite constant but varied between about O.87 to O.85 (Figure 10). - 17 -TABLE I. Ratio of Flow'Stresses Metal 4i Vf2 Reference Strain Rate Poly-crystal Cu 0.86* 5 Wot Given Poly-crystal Cu O.87-O.85 39 Not Given Single-drystal Cu 0.88 ko lO'^sec"1 Single-crystal Cu 0.88** 37 l O ^ s e c - 1 Single-crystal Al 0.79+ 6 10~5sec_1 Single and Poly-crystal Al O.78**, 37 -k - l 10 sec -1 Single-crystal Ag 0.86** 37 10"^sec~1 In a l l cases except that marked + , T = 293°K and T2= 78-K. The value marked + is for T1= 293°K, T = 90°K. The value marked * is deduced from a single experiment. The values marked ** were originally presented with reference to T2= k.2°K- These were normalized- to T2= 78°K by the method of 6 Cottrell and Stokes whereby: 6^293 . 6*\%2. = 6~293 k.2 °~~ 78 ^ 78 pcroentago e longaf ion Figure 8. Ratio of Flow Stresses at 293°K and 90°K for Aluminum Single Crystals. Reproduced from Reference 6. 9C5} . i i i -^•x 0 o.o-%fe° 0_^-Temile men 'q /m Figure 9. Ratio of Flow Stresses at Strain Rates of 10"5 and 10"^ " sec"1 for Copper and Aluminum. Reproduced from Reference 7--19 -Figure 10. Ratio of Flow Stresses at 293 K and T8°K for Copper Polycrystals. Reproduced from Reference 39-- 2.0 Sever a l t h e o r i e s have been advanced to e x p l a i n the constancy of the r a t i o of f l o w s t r e s s e s . C o t t r e l l and Adams^ have suggested t h a t kl t h i s constancy i m p l i e s t h a t , from Seeger's theory of work hardening , the d i s l o c a t i o n p a t t e r n should remain constant d u r i n g work-hardening and only the o v e r a l l d e n s i t y should change; i . e . the p r o p o r t i o n a l i t y between the d e n s i t y of f o r e s t d i s l o c a t i o n s , the d e n s i t y of jogs on screw d i s -l o c a t i o n s and the d e n s i t y of the i n t e r n a l s t r e s s e s i n the metal must hp remain constant. The theory of Seeger et, .al s t a t e s t h a t the i n t e r n a l stress, f i e l d i n a metal depends on the d i s l o c a t i o n d e n s i t y i n the same manner as the f l o w s t r e s s r e s u l t i n g from i n t e r a c t i o n s of g l i d e d i s l o c a t i o n s w i t h f o r e s t d i s l o c a t i o n s . This would imply that the hardening from these two sources should be p r o p o r t i o n a l and the r a t i o s o f the two f l o w s t r e s s e s should be constant. ' S Mott, as reported by C o t t r e l l and Stokes , has suggested t h a t "the important e l a s t i c f o r c e s on a g l i d e d i s l o c a t i o n are those from i t s immediate neighbours r a t h e r than the long-range s t r e s s e s from p i l e d up groups, f o r the e l a s t i c f o r c e s and.the f o r e s t f o r c e s a c t i n g on the d i s l o c a t i o n can both be a s c r i b e d t o the same t h i n g , the presence of nearby d i s l o c a t i o n s ! ' B a s i n s k i ^ i d e n t i f i e d the source of the e l a s t i c s t r e s s e s w i t h the f o r e s t d i s l o c a t i o n s . 3• Comments The b r i e f survey of the i n v e s t i g a t i o n s i n t o s i z e e f f e c t s given above shows t h a t , although the v a r i a t i o n of p r o p e r t i e s w i t h diameter i n s i n g l e c r y s t a l s has been w e l l e s t a b l i s h e d , the t h e o r i e s presented to account f o r t h i s v a r i a t i o n are c o n f l i c t i n g and there i s n o v u n i f i e d theory to e x p l a i n a l l s i z e e f f e c t s . Since almost a l l of the t h e o r i e s apply only to s i n g l e c r y s t a l s , any s i z e e f f e c t found i n a p o l y c r y s t a l could not be a t t r i b u t e d to any of them. I n v e s t i g a t i o n s of the C o t t r e l l - S t o k e s law have a l s o been p r i m a r i l y concerned w i t h s i n g l e c r y s t a l s . The work of Sylwestrowicz 36 on p o l y c r y s t a l s of copper., when compared t o the r e s u l t s of B a s i n s k i i n d i c a t e s t h a t there may be an anomalous e f f e c t i n copper.. This i n v e s t i g a t i o n of s i z e e f f e c t s and of the e f f e c t s of s t r a i n - r a t e and temperature on the f l o w s t r e s s i s an attempt t o c l a r i f y these two p o i n t s . - . 2 2 EXPERIMENTAL PROCEDURE 1 . Materials The copper used i n t h i s investigation was supplied by Johnson Matthey and Company and by the American Smelting and Refining Company. The aluminum used was supplied by the Aluminum Company of Canada, Ltd. A l l copper was of 99-999+$ purity.and the aluminum of 99'99+$ purity. 2 . Specimen Preparation a. Copper Wires The copper wires were prepared from Johnson-Matthey copper rods 5 mm diameter by 1 5 cm long. The wires were drawn through hardened steel dies to 0 . 1 1 5 " diameter, then through tungsten-carbide dies to O.OOV diameter, and f i n a l l y through diamond dies to a minimum diameter of 0 . 0 0 2 " . Each size was drawn from f u l l y annealed wire according to the schedule shown below: I n i t i a l F i n a l No. of Drawing Diameter ; Diameter • Steps ; 5 . mm 1 mm 3 0 2 . 5 0 . 5 3 2 1 0 . 2 3 2 0 - 5 0 . 1 ' 1 5 0 . 2 5 0 . 0 5 2 1 After drawing to the desired diameter, wire of diameter over 0-5-mm was cleaned, coiled and stored f o r future use. Wire of less than 0 - 5 mm diameter, was- immediately strung on glass U-frames and prepared f o r testing. Since a crack was found i n the middle of each rod received, random specimens of each size were sectioned and examined for i n t e r i o r cracks. No evidence of cracking was found. Specimens about 8 cm long were cut from the drawn wire and electropolished i n an ele c t r o l y t e consisting of 1 5 0 ml H^PO^, 7 5 ml each of - 23 lactic and propionic acids and 30 ml each of HgSO^  and water. A cel l potential of 7-5 volts with the specimen suspended horizontally in the cell shown in Figure 11 produced a good polish and an even, -circular cross-section. This procedure removed, a l l surface irregularities, any oxide layer accumulated during storage and the more heavily distorted surface layer, h^e specimens were then annealed in a vacuum of less o than 10-- mm of mercury for 2 hours at 250 C. Metallographic examination - 2 showed that this procedure produced a grain size of about 10 mm diameter, 6 2 an average of 10 grains per cm , and that the grain size was uniform across the cross-section. The different sizes of wire were also subjected to X-ray analysis before and after annealing to determine i f there were any wire texture•remaining. No such texture was found in the annealed wires and the as-drawn wire was found to have the normal <^ 110^  - <^L1]^ fibre texture. b. Copper Rods . Larger copper tensile specimens were machined from 10 mm diameter rod supplied by the American Smelting and Refining Company. The rods were f irst given several passes through a rolling mill in order to weld shut the intergranular cracks present in the as-received material. An approximately circular cross-section was maintained by giving the rod several passes at each reduction, turning the rod slightly between passes. Tensile specimens with a gauge length of k mm diameter were machined from the resulting rod. These were electropolished as before and then vacuum annealed at various times and temperatures to produce desired' grain sizes. Coarse grained rods were also produced by slow cooling from the melt. c. Copper Single Crystals A l l single-crystal specimens were cut from one single-crystal Figure 11. E l e c t r o l y t i c C e l l and Power Source Used i n E l e c t r o p o l i s h i n g Copper Wires. The Specimens were Suspended H o r i z o n t a l l y i n the E l e c t r o l y t e from the Frame Shown. rod about 9 mm diameter by cm long made'by the melt-solidification technique. A copper rod of the proper dimensions was cast and fitted into a hole drilled through a graphite rod. Graphite caps, the upper one .containing a reservoir of copper, were fitted over each end and the whole enclosed in a stainless-steel tube. The tube was lowered at a1 rate of 10 cm per hour through a furnace maintained at a temperature of li+00°C. d. Copper Strip The copper strip specimen was produced from 8 mm diameter Johnson-Matthey copper rod. The rod was rolled out into strips 0.015'' thick by 0.375" wide. Specimens were stamped out on the specimen punch and die described elsewhere*^, producing a gauge length 0.20" wide hy 0.75" long. These were then electropolished and annealed for o two hours at 250 C, giving a grain diameter of about 20 p.. e. Aluminum Strip The aluminum strip specimens were produced by rolling 0-5" thick bil lets of aluminum into strips 0-375" wide by O.O36" thick. Specimens were stamped from the strip as described previously. These were chemically polished for five minutes in Alcoa bright dip and annealed for 30 minutes at 450°C. 3- Measurements Diameters of 500 yx and over were measured using a Gaertner travelling microscope accurate to 1 jx. Diameters below 500 jx were measured using a Reichert microscope equipped with a Leitz micrometer eyepiece accurate to about O.h p.. From each specimen, two sets of diameter measurements were taken at right angles to each other. If the two sets of measurements agreed, a circular cross-section was - 26 . assumed. If not, the specimen was discarded. A l l diameter measurements were taken before annealing to minimize effects from specimen damage. A l l gauge length measurements were made, using the Gaertner travelling telescope, with the specimen mounted in the testing machine.. 4. Testing Procedure A l l tests, were carried out in tension on an Instron Tensile Tester equipped with an autographic recorder which produces, a load-elongation curve on a strip chart. a. Normal Tensile Tests The copper wires were tested in four batches at various strain rates and at two different temperatures. Wires of diameters 900yu, 5 0 0jx, 2 0 0 p, 1 0 0 p, and 5 0 p were tested at room temperature. For these tests, the wires were soldered to brass tabs which were held in the normal Instron vice-grips. Wires of diameters 9 0 0 y u , 5 0 0 p., o 2 0 0 p. and 1 0 0 p. were tested at 7 ° K. These wires were soldered into kk brass grips attached to a special modification to the Instron tensile tester (Figure 1 2 . ) . A l l low temperature tests were carried out in a liquid nitrogen bath contained in a wide-mouth Dewar flask. b. Variable Strain-Rate Tests Tests on the effect of varying the strain-rate on the flow stress were carried out on copper wires of,diameters 9 0 0 j x , 2 0 0 p and 5 0 ^ u and on copper rods, both single crystal and polycrystal, of diameter about h mm. The tests were done by alternating between two--strain-rates with a ratio of lOO/l. Changes in the strain-rate were accomplished by changing the gear ratio of the cross-head drive. Since the Instron,. is equipped with a gear-shift lever, this could be done almost instantaneously. (a) (*> Figure 12. Gripping System for Low-Temperature Tests on (a) Wire and Rod Specimens, and (b) Strip Specimens - 2 8 -c. Variable Temperature Tests Tests on the effect of varying temperature on the flow-stress were carried out on wires of diameters 900 Ja and 200 ji and on single and polycrystal rods of 4- mm diameter. These tests were also done on copper and aluminum strip specimens. The copper wire and rod specimens were, as before^soldered into brass grips, while the strip specimens were held 45 44 in special file-face grips screwed into a universal arrangement (Figure 12 b.). To avoid grip effects, the single crystal specimens were also mounted in the universal arrangement. The testing procedure was as follows: a. The, .specimen was deformed for a short time at room temperature, _o b. The temperature was reduced to 7 ° K and the specimen given another short deformation. c. The temperature was brought to room temperature again and the specimen given another short deformation. This procedure was repeated until the specimen fractured. Between tests, while the specimen was cooling or heating, a small load was kept on the specimen to maintain alignment of the loading system. After the specimen was immersed in the liquid nitrogen bath, and a l l but the surface boiling had ceased, five minutes was allowed in order that thermal stability could be reached. Following the low temperature extension, the liquid nitrogen bath was removed and a stream of warm air from a hair dryer was played over the specimen and grips. After a l l frost had disappeared, ten minutes was allowed to ensure temperature stability. - 29 -EXPERIMENTAL RESULTS AND OBSERVATIONS  1. Size Effects The effects of specimen diameter on the tensile properties of polycrystalline copper wires were measured at room temperature at strain-rates of 0.02 min"1 and 0.8 min"1 and at 78°K at strain rates of 0.0^ min"' and 0.2 min - 1 . The parameters measured directly from the stress-strain curve were yield stress, ultimate tensile stress, and the total elongation before fracture. In addition to these, a log-log plot of the stress-strain curve shows that, in a l l cases, the curves may be represented by . . b v an expression of the form log b = a + b l o g £ { 6~ = A£ ; a = log AJ. In some cases (see Figure ^.in Appendix I) the log-log plot shows two straight lines, each of which conforms to an expression of the type given. The parameters a and b were determined for each specimen by picking a number of points from the . stress-strain curve and flitting them to the equation by the method of least squares as is outlined in Appendix I. The above data are tabulated in Appendix II. The characteristics of the various specimens tested are shown in Table II. « a. Yield Stress The yield stress at the strain-rate of 0.8 min ^ was not measured since the autographic recorder cannot follow the load for the in i t ia l portion of the test at this rate. The stresses for 0.1$-offset (YSQ ^) and 0-5$ offset (YSQ ^ ) were measured for a l l other specimens. Since the 0.5$ offset gave a lower scatter in the yield stress and since i t is the value of offset recommended for copper^, the 0.1$ offset values were not plotted. They are, however, tabulated in Appendix,II. The values of yield stress obtained are plotted as a function of diameter in Figure 13. for both temperatures; 78 K and 293 K... The values of YSn for tests.at 78°K show no significant difference for the . 30 _ two strain-rates used so they are plotted together and considered as one set of points. The yield stress at 293°K shows no size effect, the averages of YS . fluctuating about an average value, of about 670 Kg/cm^. • . . . 0 . 5 At 78°K, however, the yield-stress decreases from a maximum of about IkOO Kg/cm^ at a diameter Of 500 jx to about 1000 Kg/cm2 at a diameter of 100 jx. In addition to this, i t was found that the yield stress at a diameter of 900p. is'about 1200 Kg/cm2, again lower than that at a diameter of 500 yu. -This may be an indication of a scatter higher than .that observed experimentally arid, may detract from the significance of this plot. b. Ultimate Tensile Stress Since, in every low-temperature test, the wire specimen pulled out of the solder before fracture, no ultimate tensile strengths are available for these tests. The ultimate tensile strengths for the room-temperature tests are plotted as a function of specimen diameter in Figure Ik. . The specimens strained at 0.02 min - 1 show as essentially constant ultimate stress of about 3100 Kg/cm down to a diameter of 200yu. At this point, the stress" drops off sharply with decreased diameter. The.data for specimens tested at the strain-rate of 0.8 min "^  show a very wide scatter and no trend can be determined from these. c. Ductility The values of total elongation before fracture are tabulated in Appendix II. and plotted as a function of specimen diameter in Figure 15. For the reasons outlined above, no values for the low-temperature tests are available. The data plotted in Figure 15. show that, at a.strain rate of 0.02 min""'", the ductility is essentially constant at about 38% for - 3 1 -TABLE II. Characteristics of Specimens.Tested for Size Effects Specimen Diameter A B B/A 9 0 0 /a 8 1 0 0 2 3 0 0.03 5 0 0 2 0 0 0 1 6 0 0.08 2 0 0 300 6 0 0.2 1 0 0 8 0 30 OA 50 25 15 0.6 A l l specimens have a grain diameter of 0.01 mm. A =• the number of crystals in the cross-section. B = the number of crystals in the cross-section which impinge on the surface of the specimen. - 32 -diameters between 900 p- and 500 p and thereafter appears to decrease with decreasing diameter. At the higher strain-rate (0.8 min - 1 ) , the ductility decreases approximately linearly with decreasing diameter from about 28$ at a diameter of 900 p to about YJ'fo at a diameter of 100 p. d. Work-Hardening Rate The parameters a and b in the expression log = a + b log £ are tabulated in Appendix II. The values of a are of l i t t l e significance since the log-log plots of the stress-strain curves l ie very close to-gether, making a essentially a function of b rather than an independent parameter. For the specimens showing two straight line segments, two values were calculated for each parameter (a-j_, b-]_, &2, ^2'^" However, only the parameter b^ is plotted. The values of b-^  plotted as a function of specimen diameter in Figure l6. are a measure of the rate of-, work-hardening of the specimen. Under-all test conditions, the scatter in these values of b-^  is too large to permit any definite expression of b^ as a function of diameter. Vague trends however, do show in the plots. The same holds true for the values of a-^  which are not shown. The trends observed for decreasing diameter are outlined below. Temperature Strain Rate aj_ b-^  decreases increases increases decreases decreases increases 293°K 0.02 min 293°K 0.8 " 78°K - 33 I500f I ikoo\ 1300 OJ §" 1200 bO 6 CQ 3 -noot .0) •H * 1000\ T= 78°K € = 0.05 min"1 • £ = 0.2 min"1 0 T= 293°K 0.02 min"1 O Arithmetic Averages $ 700 o 8 600 9 e 9> o o § 9 200 1+00 600 Diameter (microns) 800 Figure 13- Yield Stress of Polycrystalline Copper Wires as a Function of the Diameter - 3^ Diameter (microns) Figure 1^. Ultimate Tensile Strength of Polycrystalline Copper Wires as a Function of Diameter - 35 -ko 200 ' 400 5oo Boo Diameter (microns) Figure 15. Ductility of Polycrystalline Copper Wires as a • Function of Diameter 0.45 0 . 4 0 - 3 5 " 0 (a) T= 2 9 3 ° K O £= 0 . 0 2 min" 1 0 0 0 8 0 0 0 O 0 ooco 8 - 0 1 8 1 1 1 9 3> 0 -0 O 8 0 0 8 O O O -0 0 0 0 1 0 O (b) T = 2 9 3 ° K O £ = 0 . 8 min"1 1 1 1 2 0 0 400 6 0 0 800 Diameter (microns) Figure l 6 . (continued on page 3 6 . ) .o.k (c) T = 78°K o £ = 0.2 min"1 £ = 0.05 min"1 o o o $ o o o o o o /# o o o o £ _ o 1 • 1 200 i+oo 6oo 800 Diameter (microns) Figure 16. Work-Hardening Rate (b-, ) of Polycrystalline Copper Wires as a Function of the Diameter. - 38 -2. Cottrell-Stokes Law The data obtained from the investigation of the effect of temp-erature and strain-rate on the flow stress of copper and aluminum are tabulated in Appendix III. Included are the ratios of flow stresses for a ratio of strain-rates of 1:100 for various copper specimens and the ratios of flow, stresses for copper .and aluminum for temperatures 7" K o and 293 K. The data, displayed in Figures 17 to 20, 25 and 26, show that polycrystalline copper follows the Cottrell-Stokes law for strain-rate changes but not for temperature changes. Also, i t demonstrates that polycrystalline aluminum and copper single crystals do follow., the Cottrell-Stokes law and that the ratio of flow stresses in these cases is approximately that described in the literature (see Table I). Stress-strain curves for representative specimens of copper and aluminum are displayed in Figures 22, 23, 24, 27, 28, 29, and 30. . a. Effect of Strain-Rate on Flow Stress Copper wire and rod specimens with the characteristics listed in Table III were strained at strain rates of 10 1 min 1 and 10 ^min - 1 at a temperature of 293°K. A l l of these specimens follow the Cottrell-Stokes law as is shown in Figures 17 to 20. The ratio appears to decrease in the single crystal specimen (Figure 2l) after about 20$ deformation. Since the single crystal was damaged during preparation and no other was available for testing, no significance is attributed to this decrease. The ratios for the various specimens are plotted together for comparison in Figure 21. which shows that the ratio of flow stresses varies from O.965 for specimens 1B22 and 11B26 to O.985 for the single crystal SX2. The valuer.found for the single crystal is lower than that found by 37 Basinski (Figure 9-)J but this is for a strain-rate ratio of 1:100 whereas Basinski used a strain-rate ratio of 1:10. Also, Basinski has - 39 -shown-..that the ratio decreases as the temperature increases and the temperature used in this investigation was much .higher than that used by Basinski. Portions of the..stress-strain curves for specimens 1B22, RC3, and SX2 are shown in Figures 22, 23, and 2k. The fine-grained polycrystalline specimen (LB22) shows neither the low slope plastic flow region following yield at the high strain-rate nor serrations in the stress-strain curve. The stress-strain curves for both the coarse-grained specimen (RC3) and the single crystal (SX2) "show the low-slope plastic flow region after yield at the high strain-rate but only the single crystal shows serrations at high strain. These however, were barely noticeable. Also, only the \ single crystal, showed a yield drop on the low strain-rate portions of the stress-strain curve. TABLE I I I . C h a r a c t e r i s t i c s of Specimens Tested f o r the E f f e c t of. S t r a i n Rate on Flow Stress Specimen Diameter Gr a i n A B B/A Diameter SX2 3-5 mm 3-5 mm 1 1 1 RC4 3.8 0.3 153 40 0.26 2B1 0.9" 0.3 10 10 1 RC3 3.8 0.09 . 1500 130 0.09 113, 5 0.05 0.01 25 16 0.64 1F12, 16 0.2 0.01 400 60 0.15 1B22, 26 0.9 0.01 8100 280 0.03 A and B are as d e f i n e d i n Table I I . o 0 © . 1B22 O 1B26 || 0 • 0 • 0.965 0 . ® O 0 O O 0 0 o @ • 2 • ® 0 $ ' O O ® ® 0.960 - 0 1 1 1 1 1 5 10 15 20 25 30 35 $ Elongation -3 Figure 17. Ratio of Flow Stresses at Strain Rates of € 1 = 10 Jmia / 6 p= lO'^-min"1 for Specimens 1B22 and 1R26 _ k2 -• 0 975 £1 • • B D • • • • 113 1113 1F12 I F I 6 • O 0.970 0 0 0 • o o * — o 0 - 0 — o o # O O O » O O o o - — 0 0.965 10 15 20 io Elongation 25 Figure 18. Ratio of Flow Stresses at Strain Rates of £"]_= 10"3min~-1- and €'2- l C ^ m i n - 1 for Specimens 1F12, 1F16, 113, and 1113. 30 0.980-0-975-15 20 25 30 $ Elongation Figure 19- Ratio of Flow Stresses at Strain Rates of €j.= lO'^ min""'", € = 10~1min"-'- for Specimens RC3, 2Blf RC.ir 0.985 -O.98O -15 $ Elongation 25 30 Figure 20. Ratio of Flow Stresses at Strain Rates of €j= 10"3min , € 2 = l O ' T i i n " 1 for Specimen SX2 - kk -0 . 9 8 s 0.980 -0.975 £ 1 6 0 . 9 7 0 -0 . 9 6 5 -15 20 • $ Elongation 25 30 35 Figure 21. Figures 17 to 20 plotted together for comparison. 'fo Elongation Figure 22. Stress-Strain Curve for Specimen 1B22 , i Elongation Figure 2 3 . Stress-Strain Curve for Specimen RC3 - 48 -b. Effect of Temperature on Flow Stress Copper wires and rods with the characteristics shown i n Table IV were tested at temperatures of 78°K and 293°K at a strain-rate of about 10 ^min 1. ^he r a t i o of flow stresses at 293°K and 78°K are plotted as a function of s t r a i n i n Figure 25 and 26. The copper p o l y c r y s t a l l i n e specimens do not exhibit Cottrell-Stokes behaviour while the si n g l e . c r y s t a l specimens do. Portions of the stre s s - s t r a i n curves f o r specimens 132k, R C 6 and SX3 are plotted i n Figures 27 >to 30. I t can be seen from these that, as might be expected, the te n s i l e behaviour of the specimen approaches that of a single c r y s t a l as the grain-configuration, approaches that of a single c r y s t a l . The r a t i o of the flow stresses for the single c r y s t a l specimens shows an anomaly i n that the r a t i o , although e s s e n t i a l l y constant, shows a change i n value at one point during the deformation. Since the s t r a i n at which t h i s change i n value took place coincided with the s t r a i n at which the form of the strain-stress curve undergoes a change i n form from that seen i n Figure 30. to that seen i n Figure 31j i t was decided to treat these as two separate values rather than as one continuous curve. I t should be noted here that, although these two events coin-cided i n each of the .three single crystals tested, the s t r a i n at. which they occured was not the same f o r the three specimens. The approximate elongations at which the r a t i o changed are given below: Crystal SX1 Elongation 25$ SX3 30$ 8X4 20$ The shape of the stre s s - s t r a i n curve for specimen SX3 (Figure 30,3l) i s the same as that found by B a s i n s k i ^ i n the low-temperature portions, but not i n the high temperature.portions. The shape of the high temperature portions of the curve agree with those found by C o t t r e l l 4-tj , and Adams in that a yield drop is found after about 2.yjo elongation but the shape of the yield drop is different. In specimen R C 6 , the temperature cycling was carried on beyond the point where.the neck became visible. This resulted in a change in the shape of the low temperature portion of the stress-strain curve to that shown in A in Figure 29• and in a sharp drop in stress following a short region of plastic flow as is shown in B in Figure 29- The amount of plastic deformation before the drop in stress occured decreased and the rate and extent of the drop in stress, increased as the diameter of the neck decreased. That this phenomenon is a result of the deformation at low temperature is shown by two experiments which are also indicated o in Figure 29- In the f irst , the specimen was unloaded and held at 293 K for a short time and then reloaded. In the second, the specimen was unloaded, and held at 78°K for a time equivalent to that required for the low-temperature test. It was then brought to 293°Kand reloaded once more. In neither case was the sharp yield drop observed. TABLE IV. Characteristics of Specimens Tested for the Effect of Temperature on Flow Stress Specimen Diameter Grain Diameter A B B / A SX1 4.0 mm - 1 1 • 1 SX3 3-8 - 1 1 1 sx4 3-8 - 1 1 1 RC5 3-9 - 2 2 1 R C 6 4.0 - 5 5 1 RC2 3-5 0.2 mm 310 55 0.2 RC1 4.6 0-75 40 20 0-5 1F13 0.2 0.01 4oo 6o 0.15 1324, 27 0.9 • 0.01 8100 280 0.03 Aand-B are as defined in Table II. Specimen R C 5 is essentially a bicrystal with a few small stray grains along the grain, boundary. Specimen R C 6 consisted of five grains arranged symmetrically about the axis of the specimen and running almost the entire gauge length. - 51 -5 io 15 20 25 io Elongation Figure 25• Ratio of Flow Stresses as a Function of•Strain for T 1 = 2 9 3 % T2= 78°K 1750 1500 OJ w 1250 1000 750 2750 2500. 2250 . 2000-3 4 5 6 7 14 $ Elongation Figure 27- Stress-Strain Curve for Specimen 1B24 15 16 T 1 = 293 dK T2= 78°K 17 18 5 10 15 20 $ Elongation Figure 28. Stress-Strain- Curve for Specimen RC6 30 35 ko "jo E longat ion F igure 2 9 - S t r e s s - S t r a i n Curve f o r Specimen RC6 a f t e r the Appearance of a V i s i b l e Week 3 5 ^ 0 4 5 5 0 $ Elongation Figure 3 1 . Stress-Strain Curve for Specimen SX3, Part II. i vn —J i - 58 -c. Results of Tests on Muminum To check that the non-Cottrell-Stokes behaviour in polycrystalline copper was not due to a faulty testing procedure, a copper strip specimen was prepared and tested as described in the experimental procedure. For comparison, several aluminum strips were, prepared and tested in the same manner. The results of these tests are shown in Figures 32 and 33 along with the ratio of flow stresses found by Cottrell and Stokes^ on aluminum single crystals. The ratio of flow stresses was shown to be constant for aluminum and to have about the same value as' found by Cottrell and Stokes. The copper strip, however, was shown to behave in the same manner as the previous copper polycrystalline specimens. These results demonstrate' that the experimental procedure was not at fault. Portions of the stress-strain curve for the aluminum specimen SA2 are shown in Figure 3^. Two differences from the stress-strain curves found by Cottrell and Stokes for aluminum single crystals are evident. The f irst is that a yield drop appears at a strain of less than h°jo whereas this did not appear in the single crystals until a strain of about 6$> had been reached. Also, the yield-drop was not always evident beyond this strain whereas i t was in the single crystals. Second, the yield •stress at the low-temperature is often lower than the final stress in the previous low-temperature section of the test. This reduction in stress, which wi l l be here-in-after referred to as "work-softening", depends on.both the amount of strain at 293°K ( C* ) between the two 78°K tests and the total strain ( £ ). This is demonstrated in Figure 35- where the ratio of'the yield stress (g') to the final stress ( ) of the preceeding low-temperature test is plotted as a function of £ for different C • For small £ , the scatter is so large,that the extent of work-softening can be shown only as a band (light hatching), but the ratio is definitely less than one. 0.780 . D Aluminum Single-Crystal (Cottrell and Stokes) — I 1—: 1 1 1 I I : 5 10 15 20 25 30 35 $ Elongation Figure 32. Ratio of Flow Stresses for Aluminum Single Crystals (Reference 6) T 1 = 293°K, T2= 90°K and Aluminum Polycrystals, 0^ = 293°K, Tg= 78°K 1 vo - 60 -0.88 0.87 O 0 \ p 0.86 \ 0 L2 0.85 0 \ ° \ 0.84 ^ 0 0 1 1 1 1 1 5 10 15 20 25 30 io Elongation Figure 33. Ratio of Flow Stresses for Polycrystalline Copper Strip. T x= 293% T 2= 78°K. $ Elongation" Figure jh- Stress-Strain Curve for Aluminum Specimen SA2 fo Elongation Figure 35• Degree of Work-Softening in Aluminum as a Function of Strain for Various £ 1 • CA ro - 63 -SUMMARY OF RESULTS The results of the investigation into size-effects, are not very-revealing. The yield stress, ductility, ultimate tensile strength and rate of work-hardening were measured for copper wires with diameters ranging from 900 ^1 to 50 jx at various temperatures' and strain rates . and the results may be summarized as follows: 1. The room temperature yield-stress shows no apparent size-effect. 2. The low-temperature yield-stress decreases with decreasing diameter for wires with diameter less than 500 ^.. - 3- Both the ductility and the ultimate tensile strength decrease with decreasing diameter. k. The results on work-hardening rate show a scatter high enough that no definite size-effect can be established. A trend toward an "increasing work-hardening.rate with decreasing diameter was observed except for the high strain-rate tests where the opposite was observed. The results of the investigation of the effects of the temperature and strain-rate on the flow-stress of copper may be summarized as follows: 1. Single-crystals of copper follow the Cottrell-Stokes law for both temperature and strain-rate changes. ••"' 2. Polycrystalline copper follows the Cottrell-Stokes law for strain-rate changes but the value of the flow-stress ratio is different for different specimens being affected by: • a) the grain size, b) the fraction of the grains, in the- specimen which have a free surface. 3- Polycrystalline copper does not obey the Cottrell-Stokes law. The deviation from the expected value of the flow-stress ratio is negative and depends on strain. The results of the tests on aluminum show the following: 1. Aluminum polycrystals do obey the Cottrell-Stokes law for temperature changes. 2. The yield-drop observed by Cottrell-and Stokes^ in aluminum single crystals was found in polycrystals. 3- The yield-drop in polycrystalline aluminum is accompanied by a "work-softening" effect, not found, evidently, in single crystals. The above is a summary of the important results of this investigation and are the results which wil l be referred to in the discussion. - 65 -DISCUSSION 1. Size Effects This investigation has shown that the size-effects found in polycrystalline copper wires are, in most cases, the opposite to those found in single crystals by other investigators. The results wil l be discussed in terms of the degree to which the wires of various diameter can be referred to as "polycrystalline". Departures from the behaviour expected when this consideration is taken into account indicate that there may be size effects where none are directly observed. a..Yield Stress and Work-Hardening Rate It would be, perhaps profitable to f irst discuss the grain configuration in a polycrystal and to assess the effect of the diameter on this. The polycrystalline wires used in this investigation had a grain diameter of 0.01 mm and varied in diameter from about 0-9 mm to about 0.05 mm. E x a m i n a t i o n of Table II shows that in wires of diameter 0.9 mm, the great majority of the grains are totally enclosed in the wire and the percentage of the grains which show a free surface is negligible. In a wire of 0.05 mm diameter, a significant proportion of the grains in the wire have a free surface and less than half are totally enclosed. Figure 36. shows the relationship between the wire.^ diameter and the ratio B/A, the fraction of the grains in the wire which show a free surface. The values shown are a conservative estimate since the calculation is based on the supposition that a grain on the surface wil l have a free surface of the maximum possible area. However, i f one takes into account the relative effects of different sizes of free surfaces of the various grains on the surface, this is probably a good measure of what might be called the "effective" ratio of the number of grains showing a free surface to the total number in the specimen. The curve shown in Figure 3 6 - represents a function of the same form as the surface to volume ratio; i .e . the ratio B/A is proportional to the inverse of the wire diameter for a fixed grain size. The ratio wil l have a limiting value of one where the wire diameter is equal to the grain diameter, corresponding to a bamboo structure. The variation of the ratio • /^A may be expected to give rise to the following effects: i . The yield stress should decrease with decreasing diameter, i i . The work hardening rate should decrease with decreasing diameter. These will occur because the constraints on the grains totally enclosed in the wire wil l become less important as the diameter and thus the number of interior grains decreases. Let us now examine the results of this investigation in the light Oo of the above discussion. The yield stress at 7 ° K does in fact decrease with decreasing diameter for diameters below 0.5 mm. Above 0-5 mm diameter, where the change in the ratio B/A with diameter is very small, the yield stress decreases with increasing diameter. The room-temperature.yield-stress, however, remains approximately constant where a decrease is expected with decreasing diameter. It appears that there may be two mechanisms operating at the higher temperature; one tending to increase the yield stress and the one-, outlined above, tending to decrease the yield stress. If the oxide layer makes any contribution to the strength of the wires,, any increase in the yield stress would be proportional to the inverse of the diameter as outlined in the introduction. Since this will have the opposite effect to that of increasing the ratio B/A, these would tend to cancel. Work by previous investigators, however, indicates that the oxide-layer would have l i t t l e i f any effect in that: i . ' There appears to be no effect on the strength of 3 23 copper whiskers from the oxide layer ' i i . Wo effect on the strength of polycrystalline wires from the oxide layer is evident until the grain diameter 2k 26 29 approaches that of the specimen diameter ' ' . The smallest wires in this investigation had about 25 grains in the x-section. The::work-hardening rate, expected to decrease with decreasing diameter, tends to increase for low strain-rate tests .both at room o temperature and at 78 K. The specimens tested at the high strain rate at: room-temperature, however, do show a trend towards the expected decrease. The significance of these is very doubtful since the scatter is high enough to make the changes in hardening rate doubtful. b.. Ductility and Ultimate Tensile Strength Roberts^, studying the ductile fracture of polycrystalline copper, found that the fracture is initiated by the coalescence of voids .into a central crack in the necked region of the specimen. Under further strain, this crack grows out toward the surface of the metal. When the crack approaches the surface, void formation occurs "catastrophically" and the specimen fractures. This would appear to explain.the decrease in ductility and in the ultimate tensile strength with decreasing specimen diameter. The events outlined above, however, take place in the necked region of the specimen and.value of stress and elongation used in this investigation were' taken at the point where the stress began to decrease due to the formation of a neck. It was noted, however, that the mode of fracture changed with decreasing diameter. Wires of diameter greater - 69 -than 0.2 mm showed extensive necking and parted very slowly, the load decreasing gradually. In smaller specimens, no visible neck appeared and the specimen separated suddenly with no noticable decrease in load before fracture. From this, i t would appear that, in the small diameter specimens, the crack formation occurs at a much lower stress than in the large diameter wires and before the neck has formed visibly. There is the possibility that sub-microscopic surface-defects could act as stress-raisers and thus lower the fracture stress. The reduction in apparent ultimate strength due to a stress raiser would, of course, have a greater effect on the thin wires than on thick ones. However, the probability of a stress-raiser which is invisible under a magnification of 100 X having an appreciable effect on the fracture stress is unlikely. A more probable stress-raiser is in the voids themselves. Roberts examined specimens which, although highly strained, had not yet formed a neck. He found, that although the concentration of voids is lower than in the necked regions, there are intergranular voids. A comparison of the effect of diameter on the ultimate strength of wires found in this investigation with that found by McDanels et al35 indicates that, once again, the ratio ~B/A may be important. McDanels et al tested composites of tungsten wires in a copper matrix and found that the strength of the composite increased as the diameter of the tungsten wires decreased. Metallographic. observations indicated that neither the tungsten wires nor the copper matrix failed independently but that they fractured as a unit. For this type of test, the ratio B/A would have a very small, i f any, effect on the properties of the tungsten wire since the surface grains are constrained."by the copper matrix just as the interior grains are constrained by the surrounding tungsten. -70-• The•discussion indicates that any investigation into size effects in polycrystalline wires must be designed to offset the effect of the increasing proportion of the grains which impinge on the surface as the diameter decreases. One possibility for overcoming this is the method used by McDanels et al but this method adds a complicating factor. Another •is to use a small enough grain diameter that the ratio B/A varies negligibly over the desired range of diameters. For- diameters between 900 jx and 5 0 p., this would require a gram size of less than 1 p. 2. Cottrell-Stokes Law It is now generally accepted that the flow in a crystal may be represented by the sum of two components; one a temperature-independent and the other a temperature-dependent flow-stress. On this basis, using Seeger's notation, the flow-stress may be written: where is temperature-independent' and temperature-dependent. From this, the ratio of flow-stresses for any.crystal at two temperatures, T-j_ and can be written: # 1 = 6j,+ <£/ -pr— , „ < 1 for T > T p . . . 2 ) 6r 2 0 £ 2 + 0 J 2 1 2 If a crystal is deformed at T^'to a given strain, then deformed at Tg, the dislocation pattern and density wi l l be the same, at that strain, for the two temperatures. Thus, since 6^ is temperature-independent, 6"^ / and 6$%, wi l l be equal and the ratio of flow stresses wi l l be: (3a) ST 2 " 5* +652 the Cottrell-Stokes ratio. It may also be written: ST/ ^ 1 * £L/61 _ ( 3 b ) o r 2 1 * .6s.2/si - 71 -If Ss i s proportional to 6^  over a range of strain; i.e. 6s 1 = -C : ; 6j. 2 = C &6 6* then the flow-stress ratio becomes, from (3b): 6TI = 1 + C ' = K <i 1 + C where K i s a .'.constant, and the Cottrell-Stokes law w i l l be obeyed. If £5 is not proportional to 6^. , the Cottrell-Stokes law w i l l not be obeyed and the ratio of flow-stresses w i l l vary over a range of strain. a. Work-Hardening As was mentioned i n the introduction, there are two main theories 37 U-2 of strain-hardening, one by Basinski and one by Seeger et a l , to account for the constancy in the Cottrell-Stokes ratio. Both agree that the temperature-dependent component of the flow-stress, 65 , is the stress necessary for the glide dislocations to cut the forest dislocations This process w i l l give rise to a temperature-dependent stress in that i t is a short-range interaction involving the formation of constrictions i n extended dislocations and the formation of dislocation jogs and can there-fore be aided by thermal vibrations. The treatment of this mechanism by Basinski differs somewhat from that of Seeger et a l . Starting with a uniform distribution of obstacles, Seeger assumes that a l l of them are capable of breaking down, whereas Basinski assumes that, independent of the i n i t i a l distribution, the number breaking down w i l l depend on temperature. It i s i n the source of the temperatuie-independent component, 5^ that^these investigators show an important difference of opinion. Seeger et a l . have proposed that & i s due to an internal stress-field which impedes the motion of the glide dislocations and that this stress-field is due to the elastic stress-fields of dislocatipn pile-ups distributed randomly through the crystal. Each pile-up contains about twenty-five dislocations, the number in each being independent of strain. They • showed that the flow-stress due to this stress-field w i l l be proportional to the square-root of the dislocation density. Since 65 would depend on the dislocation density in the same manner, this would imply a proportionality between <5^  and Basinski, on the other hand, attributes the elastic component to the long-range interactions between forest and glide dislocation. Since, in this case, both <5~j a n ( i 6~£ are due to interactions between the same dislocations, the £s a n < i 6% should be proportional and the Cottrell-Stokes law obeyed. Both of these theories are based on the assumption that the rate of increase in the density of forest dislocations i s about the same as the rate of increase in the glide dislocation density. Davis has shown that i f the rate of increase of the glide dislocations is much greater than that of the forest dislocations, the proportionality between <£$ and is destroyed and the Cottrell-Stokes law is no longer obeyed, the deviation being in a positive manner. Similarly, i f the forest dislocation density increases much faster than the glide dislocation density, there w i l l be a negative deviation from the Cottrell-Stokes law. •b. Theoretical Considerations An approximate division of the flow-stress into <Sj and 6^ for a single crystal of copper was calculated by Seeger et a l as shown in Figure 3 7 * This division is based on Seeger's theory of the - 73' -flow-stress but w i l l serve as an i l l u s t r a t i o n for the following discussion. The solid curves, (S& and t\$ , represent the condition under which 6j is proportional to and the Cottrell-Stokes law is obeyed. Consider the effect on the flow-stress ratio i f 6^ were to l i e along the dotted line 6$. At a strain £, this would re-present an increase 6"6^ . over that 6^  which maintains the proportionality with 5j . Since the effect on the Cottrell-Stokes ratio i s d i f f i c u l t y to see, consider instead the ratio , where: 1 Z>6^  = 67 2 - o*r1 = ^ 2 ~ ^ 1 • 6* /? ^ •L- . since  u $ = 6/5 1 2 Simple algebra shows that: 6 7 * 1 = 1 Gr2 - 1 + ^l6r± and that an increase in represents a decrease in the Cottrell-Stokes ratio, $T 2 The increase, c7&£ , in ^ w i l l change the ratio to a new value / A T \ - 74 -Strain Figure 37- The Separation of the Flow-Stress into the Elastic and Temperature-Dependent Compounds as Calculated by Seeger et al^"2. Therefore and A reasonable assumption to make, on consideration of the two theories outlined above, would be that is proportional to In this case, although the Cottrell-Stokes ratio would have a higher value, i t would stillbbe auconstant value. Similarly, a reduction, - &6& , in 6$ would, i f were proportional to , result in a lower but s t i l l constant value for the Cottrell-Stokes ratio. It can be shown, in a similar fashion, that an increase, <S~f5^5 > in (S^5 would result in a lower Cottrell-Stokes ratio. A change in Ss which is proportional to Ss would, in effect, multiply ^ by a constant P, where P < 1 for a negative and P > 1 for a positive Thus: \ = P 6~S2 - P <fs 1 s f r j Si + P ^ X = SS2 - S~3 1 < /p + $si which means that a positive wil l cause a decrease in the Cottrell-Stokes ratio and a negative an increase. Once again, however,'if is proportional to (Sj , the ratio wil l be changed but will remain constant with strain. These effects can now be summed up as follows: Change in 6~5 or <5"$. : Change in the Cottrell-Stokes Ratio Positive - Negative Negative Positive c. A Possible Effect of Dislocation Density Since the results of this investigation have shown that the Cottrell-Stokes behaviour in polycrystals is different from that of single crystals, i t would perhaps be profitable to consider the difference between polycrystals and single crystals in the light of the above theorie One difference in the deformation, characteristics is that, due to the restraints on each grain in a polycrystal, several slip-systems are required to operate to give the required deformation. This might give an overall increase in the dislocation density over that to be found in a single crystal and may result in dislocation interactions beyond those considered by Basinski or by Seeger et a l . Let us then consider the effect of an increase in the forest dislocation density in polycrystals on the flow-stress ratio in the light of Basinski1s theory of the flow-stress. The elastic stress on the glide dislocation at point A, (Figure 38-) wil l be due to the elastic stress-fields of the forest dislocations o(-, and $2' With a low forest density, the stress at point A due to the stress fields of ftand $2 wil l be very small compared to that due to °C. With a high forest density, the forest Figure 38- Representation of a Glide Dislocation Approaching a Row of Forest Dislocations. dislocations @ and @^ wil l make a significant contribution to the stress-field at point A. If 3^ and @ are of the same sign as , the stress-field at A wil l be greater than that due to °t- alone. If ^ and @ a r e o f opposite sign to oC , the stress field at A wil l be smaller than that due to oC alone. If the sign of the forest dislocations were randomly distributed over the slip-plane, the most probable condition, then the stress field at A would be that due to oi. alone since the contributions from and @ 2 would tend to cancel. In any case, any interactions between forest dislocations, due to a high.dislocation density, would result in a proportional to the square-root of the forest density. This would mean that £6$ would be proportional to and, although the value of the Cottrell-Stokes ratio may be changed, the proportionality between a n ( i 6$ would not be upset. Thus, a mechanism of this sort could give rise to the effect shown in Figure 21. but not the deviations from the Cottrell-Stokes law shown in Figure 25-d. Grain Boundaries Another important difference between a single crystal and a polycrystal is that a polycrystal contains grain-boundaries which can act as barriers to dislocation motion. Although Basinski attributes no effect to the stress-fields of pile-ups scattered through the body of the crystal, a dislocation pile-up at a grain boundary wil l have an effect on the crystal directly ahead of i t . Consider the effect of a dislocation pile-up at a large angle . t i l t boundary. Eshelby, Frank and Nabarro^? have calculated the stress-concentration factor ahead of a dislocation pile-up to be of the form shown in Figure 39- As'suming that, the average length of a pile-up wil l be one=half of the grain-diameter, the distance over which the pile-up wil l produce a stress-concentration factor of at least two wil l be 0.25L or 0.125d, where d is the grain diameter. However, since the slip system wil l not be continuous across the grain boundary, this value of the stress concentration factor must be modified. The average angle of a high-angle boundary is about 50° to 60°. Since the angle between ( i l l ) planes is 70° and between 0-10/* directions is 60°, we can expect a slip system with the angle of the plane within ± 20° and the direction within ± 30° to the slip system from which the pile-up originated. At a distance 0.25L from the head of the pile-up, therefore, the stress-concentration factor wil l be between 2 and 2 cos 30° ^ 1.6. If we approximate the shape of the grains to be cubic, the volume affected wil l be: d 3 - 3d 3 = d 3 fl - 27 "\ I; V 517/ or slightly more than one-half the total volume. Unless 6^. <^ 6^ s , this will produce an effect of a magnitude far greater than that observed for the difference between single and polycrystals of -copper. There are two possible modifying factors: 1. The average length of a pile-up at the boundary may be less than one-half the grain diameter. 2. The grain goundary may act as a "screen", as proposed by 48 Frankel , reducing the effect of the pile-up on the adjacent grain. No quantitative calculation on these effects can be made here since there is not sufficient data on the effect of the grain diameter on the Cottrell-Stokes ratio. There is also a complicating Figure 39. Stress Concentration Factor in Front of a Dislocation Pile-up of Length L as Calculated in Reference 48. - 81 -f a c t o r i n t h a t the r a t i o appears to a l s o depend on the f r a c t i o n of the g r a i n s i n the specimen which impinge on the surface (Figure 20. ). The e f f e c t of the p i l e - u p s can, however, be assessed q u a l i t a t i v e l y . The e f f e c t , of the p i l e - u p at the g r a i n boundary w i l l be, because of the s t r e s s - c o n c e n t r a t i o n , t o reduce L\ i n the area ahead of i t . The o v e r a l l r e d u c t i o n i n 6# t,, %6$ , w i l l depend, assuming a constant p i l e - u p l e n g t h , on the number, Up, of p i l e - u p s i n the specimen: The number of p i l e - u p s , i n t u r n , w i l l depend on the d i s l o c a t i o n d e n s i t y , D , i n the specimen: Thus: £££ <^Dd . Both B a s i n s k i ' s and Seeger's t h e o r i e s give a 6^ which i s p r o p o r t i o n a l t o the square-root of the d i s l o c a t i o n d e n s i t y . This means t h a t S6Q w i l l n o t be p r o p o r t i o n a l t o o^ j and, by the previous d i s c u s s i o n , a negative d e v i a t i o n from the C o t t r e l l - S t o k e s law w i l l be expected, i n accord w i t h the experimental observations. e. The E f f e c t of Stacking F a u l t Energy on the C o t t r e l l - S t o k e s E a t i o Since the d e v i a t i o n s from the C o t t r e l l - S t o k e s law found' i n copper p o l y c r y s t a l s do not appear i n aluminum, the d i f f e r e n c e s between aluminum and copper should provide a clue t o the behaviour of copper. The p r i n c i p a l d i f f e r e n c e between copper and aluminum l i e s i n the s t a c k i n g - f a u l t energy, copper having a low and sluminum a high s t a c k i n g - f a u l t energy. Table V. shows the most probable values of s t a c k i n g - f a u l t energy as c a l c u l a t e d 50 by Thornton and H i r s c h together w i t h the C o t t r e l l - S t o k e s r a t i o s , f o r s e v e r a l temperature ranges, found by B a s i n s k i f o r the face-center cubic TI 50 metals copper, aluminum and s i l v e r and f o r the hexagonal metal magnesium - 82 TAB IE V. Stac k i n g - F a u l t Energies and C o t t r e l l - S t o k e s R a t i o s  f o r Cu, Ag, A l and Mg. Stacking F a u l t C o t t r e l l - •Stokes R a t i o , Element Energy T-L = 293°K T 2 = 78°K 200°K k.2°K 100°K k.2°K 50°K k.2°K Ag 35 ergs/cm • 0.86 * 0.95 0-97 O.98 Cu 57 0.88 * 0.93 0-95 0-97 A l 200 0.78 * 0-73 0.80 0.90 Mg (High) - 0-73 O.78 O.87 * Not c o r r e c t e d f o r the v a r i a t i o n of the e l a s t i c modulus w i t h temperature. . - 83 -There seems to he a correlation between the stacking-fault energy and the Cottrell-Stokes ratio in that the high stacking-fault . energy metals (Al and Mg) have a lower ratio than copper and silver. The values shown are corrected for the variation of the elastic shear modulus with temperature. If any other differences, in the metals are ignored and only the stacking-fault energy is considered, the theories of Basinski and Seeger et al may be tested on the basis of prediction of the effect of the stacking-fault energy on- the Cottrell-Stokes ratio. Basinski has- attributed both 6j and 6$ to interactions between forest and glide dislocations. The elastic stress, (5^ , since i t is due to long-range interactions wil l not be appreciably affected, i f at a l l , by the separation between the partial dislocations. The temperature dependent component, C% , since i t involves the actual intersection of the dislocations, wil l be profoundly affected by the separation of the partials. 51 Stroh has calculated the work required to form.a constriction in an extended dislocation from the work required to force a decrease in the separation' of the partials and the work required to bend the disr locations-against the line tension. He found that decreasing the stacking-fault energy increases the work required. Let us now examine the effect of the•stacking-fault energy on the Cottrell-Stokes ratio. By Stroh's calculation, decreasing the stacking-fault energy increases the work required to form a constriction, thus increasing 6s • As shown earlier, increasing Sj wi l l decrease the Cottrell-Stokes ratio. - Qk _ Basinski'.s theory, therefore, predicts a decrease in the Cottrell-Stokes ratio with decreasing stacking-fault energy, contrary to the results tabulated in Table V.- Any assessment of the effect of the stacking-f ault energy in light of Seeger's theory is .more complex since 64 wil l also depend on the stacking-fault energy. The number of dislocations in a pile-up wil l be a function of the stacking-fault energy and thus the magnitude of the stress-field due to the pile-up will depend on the stacking-fault energy. This comes about because the extended dislocations in a pile-up must be constricted before they can escape through cross-slip or through dislocation climb. The stress on the leading dislocation is a function of the number of dislocations in the pile-up and of the applied stress. Consequently, the number of dislocations in a pile-up wil l be"higher in a low stacking-fault energy metal. By Seeger's theory, then, we could expect an increase in both 6s and 6 5 with a decrease in the stacking-fault energy. The effect of this on the Cottrell-Stokes ratio is uncertain but i t is possible that ittcould give an increase in the ratio. This discussion ignores, of course, any difference other than the stacking-fault energy between high .and low stacking-fault energy metals. Based on the limited data available, however, there is a rough correlation between the Cottrell-Stokes ratio and the stacking fault which seems to be independent of other differences. It also appears to be qualitatively independent of the temperature range considered. More experiments, however, on metals of different stacking fault energies must be carried out to definitely establish the existence of the correlation. - Q 5 -Another useful experiment in an investigation of this type would be to alter the stacking-fault energy of a metal by the additions of small amounts of a suitable impurity and to examine the effect on the Cottrell-Stokes ratio. f. A Possible Effect of Dislocation Density on In an earlier part of this discussion, we considered the possible effect of the dislocation density on the elastic interactions between glide and forest dislocations. Let us now consider the effect of increasing the dislocation density on the temperature-dependent component of the flow stress. With a high enough dislocation density, the interactions between parallel extended dislocations wil l decrease the average separation of the partial dislocations, resulting, effectively, in a higher stacking-fault energy and a lower stress required to form a constriction in the extended dislocation. This wil l reduce 65 and, according to Basinski"s theory, result in a higher value of the Cottrell-Stokes ratio- The actual, results on polycrystalline copper (Figure 2l) show that the ratio is lower than that for single crystals. This reduction in the ratio, however, on the basis of the proposed mechanism, fits in with the observed correlation between the stacking-fault; energy and the Cottrell-Stokes ratio. The dislocation density required to produce an appreciable effect on the temperature-dependent component of the flow-stress can 5 3 be estimated quite easily. Cottrell' has caluclated the separation of the two partial dislocations bounding a stacking fault to be: r .= M (bj.bp) 2/7-<r where €. is the stacking-fault energy, b-^  and b 2 the Burgers vectors of the partials andja the elastic shear modulus. In a face center cubic - 86 -.crystal.;, a. normal dissociation mechanism i s : | <io5,-^! < 2 n> + g < l l 5> For t h i s pair of p a r t i a l s : b r b 2 = a and r = u a' 12 2 2hTT6 In copper, ergs/cm 2 and a = 3-6 X. In aluminum 200 ergs/cm 2 and a = 4 . 0 £. Takingyi = 4 X 1011dyne/cm^, r = 10"^ cm f o r copper and r -- 10" cm f o r aluminum. I f the centers of the extended dislocations approached to within 10 r, the separation between the p a r t i a l dislocations would be reduced by about 10$. This distance between dislocations corresponds to d i s l o c a t i o n 12 -2 14 -2 . . . densities of about 10 cm and 10 cm i n copper and aluminum respectively. 8 1 2 - 2 The normal di s l o c a t i o n density i s about 10 to 10 cm i n a cold-worked single c r y s t a l . I f the di s l o c a t i o n density i n a copper polycrystal i s at a l l higher than that i n a single c r y s t a l , we can expect an effect on' from the extra reduction i n the width of the stacking-fault. In aluminum, since a much higher dislo c a t i o n density i s required to s i g n i f i c a n t l y decrease the width of the stacking-fault, the effect would be reduced by a factor of about 100 and would probably be neg l i g i b l e . The foregoing discussion indicated that the process of work-hardening and the origin of the temperature-dependent and temperature-independent components of the flow stress may be more complex than those o 7 42 proposed by Basinski-" or by Seeger et a l . I t has been shown that i t i s possible to explain the v a r i a t i o n of the Cottrell-Stokes r a t i o between single and polycrystals of copper and the deviations from the C o t t r e l l -Stokes r a t i o i n terms of these theories only by proposing an extra mechanism which was not considered i n the o r i g i n a l theory. - 87 -Examination of the Cottrell'-Stokes ratios for metals of different stacking-fault;.energies has indicated a possible variation in the ratio with varying stacking-fault which is opposite to that predicted by Basinski's theory. It should be stressed here, however, that this correlation is far from being definitely established and requires further investigations. 3- Work-Softening in Aluminum Cottrell and "Stokes^  observed a yield drop in aluminum crystals deformed after prior deformation at a lower temperature. They proposed that this yield drop indicates the operation of a mechanism by which the plastically deformed crystal can rid itself of some of the work-hardening introduced through the low-temperature deformation. This proposal is supported by the work-softening observed in this investigation on aluminum polycrystals. As can be seen'in Figure 3^. the work-softening does not appear until the strain at which the yield drop f irst appears. After a few percent deformation, the mganitude of the work-softening for small is approximate y constant over the range of strain studied (Figure 35*)• Also, for large C', the work-hardening introduced by the plastic de-formation after the yield drop compensates to some extent for the work-softening occuring at the yield drop. . It appears that the dislocation' configuration produced by deformation at allow temperature is unstable, at the same strain, at higher temperatures. The fact that the work-softening is recoverable with further deformation at high temperature indicates that the difference in the stable configuration is one of degree rather than kind. Also, the fact that the ratio 6? , for small £ ' , settles down to a constant value indicates that a definite percentage of the obstacles formed at - 8 8 -low temperatures break down at the higher temperature. That this effect is absent in copper means that the mechanism responsible for the work-softening in aluminum is one which is affected by the stacking-fault energy such as cross-slip or dislocation climb. By Basinski' s theory, this process would presumably be a dislocations-annihilation or polyganization through cross-slip or climb. From Seeger's theory, the same effect might be expected through the escape of dislocations from the head of a'.pile-up by cross-slip or climb. Since this type of process occurs more easily at higher temperatures, the equilibrium number of dislocations would be lower than at low temperatures. Examination of Figure 3 ^ . also shows that the results on work-softening in aluminum support the explanation offered by Cottrell and Stokes for the observed difference in flow-stress ratios on going from high to low and from low to high temperatures. Figure kO shows a schematic representation of the form of the yield drop observed in aluminum deformation at room temperature after prior deformation at 78 ° K . A yield stress of the value A would be expected assuming that the change in flow stress is reversible; that is , that the change in the flow stress at this strain should be the same for both directions of change of temperature. The point B represents the actual observed yield stresses than would be found on going from 2 9 3 ° K to 78 ° K . Evidence for this form of the curve is presented by Cottrell and 6 Stokes as follows: 1. The difference in the ratios of.flow stresses does not become appreciable until the strain at which the yield drop appears. 2 . If a specimen is strained to a stress represented by point C, and held at this stress, a delayed yield drop preceeded by creep at stress C is observed. Added to this is the evidence shown in Figure 3^- Here, i t can be seen that, although the work-softening does not appear until the strain at which the first yeild drop is observed, work-softening occurs even in tests where there is not observed yield drop and the magnitude of the work-softening does not depend on the size of the yield drop. This indicates that the dislocation configuration present at 78°K is unstable at 293°K and wil l break down under an applied stress lower than that expected for plastic flow. In some cases, the cr i t ica l stress appears to have been low enough that no yield drop was observed in conjunction with the work-softening. That thermal agitation alone, is not sufficient to cause the breakdown in the dislocation con-figuration is indicated in Figure 3^- which shows the stress-strain curve for a specimen which was strained at 78°K, unloaded and held at room-temperature for several hours, then restrained at 78°K. Np work-softening was observed in this case. - 90 -Strain Figure-:40. A Schematic Representation of the Form of the Stress-Strain Curve for Deformation at 2 9 3 ° K after Prior Deformation at 7 8 ° K . - 91 -SUMMARY AND CONCLUSIONS 1. The important results have been summarized in a previous section and wi l l not be repeated here. 2. In the-tests on the effect of temperature on the flow-stress, the shape of the stress-strain curve changes for specimens of different characteristics. In contrast to this, the shape of the stress-strain curve for polycrystalline aluminum is essentially the same as found in the literature^ for single crystals of aluminum. 5. The deviations from the Cottrell-Stokes law for temperature changes and the variation in the Cottrell-Stokes ratio found for strain-rate changes are probably due to dislocation mechanisms beyond those considered by either ^6 kl Basinski-^ or Seeger et a l . . These mechanisms wi l l be ones which exert a minor influence on.the flow-stress. ' k. The rough correlation between the stacking fault energy and the Cottrell-Stokes ratio' noted in the discussion is the opposite to that * 36 predicted by Basinski s theory. 5. The yield-drop found in aluminum at high temperature after prior deformation at a lower temperature is probably due to the release of dis-locations from a barrier by a thermally activated process such as cross-slip or dislocation climb. - 92 -SUGGESTED FUTURE WORK This work has been,primarily of an exploratory nature and the results have suggested directions of future work which might be profitably-explored. 1. Any future work on size effects in polycrystalline materials should be done using a procedure which wil l eliminate the effect of the ratio B'/A. Some possible methods have been suggested in the discussion. 2. If the Cottrell-Stokes behaviour, for both temperature and strain-rate changes, in polycrystalline copper were studied over different temperature ranges, the mechanism responsible for the deviations from the Cottrell-Stokes law may become more evident. 3- The work-softening found in aluminum polycrystals may also provide significant information i f studied over a series of temperature ranges. It may also be profitable to determine i f the same effects can be produced by strain-rate changes. The effects of different increments of low-temperature strain on the magnitude of the work-softening may also be studied. k. The correlation between the stacking-fault energy and the Cottrell-Stokes ratio noted in the discussion may be investigated by two types of experiment: a. Tests on more metals of high and low stacking-fault; energies and on nickel which has an intermediate stacking-fault energy (about 100 ergs/cm ). . b. Tests on metals doped with appropriate impurities to change the stacking-fault energy. Two alloys which might provide information i f studied over a wide range of compositions are nickel-cobalt and copper-zinc. - 93 -BIBLIOGRAPHY I f K. G. Compton, A. Mendizza and S. M. Arnold, Corrosion, Vol. 7 ( 1 9 5 1 ) , 3 2 7 -2 . J . K. Gait and C. Herring, Phys. Rev., Vol. 8 5 ( 1 9 5 2 ) , 1 0 6 0 . 3- S. S. Brenner, J . App. Phys., Vol. 2 7 ( 1 9 5 6 ' ) , 1 4 8 4 . 4. J . H. Hollomon, Trans. A.I.M .E., Vol. 1 7 1 ( 1 9 4 7 ) , 5 3 5 -5- J . E. Dorn, A. Goldberg and T. E. Tietz, Trans. A.I.M.E., Vol. 1 8 0 ( 1 9 4 9 ) 2 0 5 = 6. A. H. Cottrell and R. J . Stokes, Proc. Roy. Soc. Vo l . ' 2 3 ^ ( 1 9 5 5 / 5 6 ) 1 7 -7- G. F. Taylor, Phys. Rev., V o l . 2 3 ( l 9 g 4 ) , 6 5 5 . 8. A.S.M. Metals Handbook, 1 9 6 1 / II98. 9- Z. Gyulai; Z. Physik, V o l . 1 3 8 ( 1 9 5 4 ) , 3 1 7 . 1 0 . H. Suzuki, S. Ikeda and S. Takeuchi, Phys. Soc. Japan, J.,_Vol. 1 1 ( 1 9 5 6 ) 3 8 2 . l i . J . Garstone, R. W. K. Honeycombe and G. Greetham, Acta. Met., V o l . 4 ( 1 9 5 6 ) 4 8 5 1 2 . R. L. Fleipcher and B. Chalmers, Trans. A.I.M.E., Vol. 2 1 2 ( 1 9 5 8 ) , 2 6 5 . 13. G. L. Pearson, W.'T. Read Jr. and W. L. Feldmann, Acta.Met. V o l . 5 ( l 9 5 7 ) l 8 l . 14. F. Y. Eder and V. Meyer, Naturwissenschaften, V o l . 4 7 ( 1 9 6 0 ) , 3 5 2 . 1 5 - . z.. E. Moore, Private Communication. 1 6 . A. H. Cottrell, "Dislocations and Plastic Flow in Crystals", Clarendon Press, Oxford, p. 5 3 -1 7 - J . Takamura,, Mem. Faculty of Eng. Kyoto U. , Vol 1 8 ( 1 9 5 6 ) , 2 5 5 . 1 8 . R. Roscoe, Phil Mag., Vol. 2 1 ( 1 9 3 6 ) , 3 9 9 . 1 9 - E. W. da C. Andrade, "Properties of Metallic Surfaces", Inst, of Metals, ' London, Monograph and Report Series, n. 1 3 , 1 9 5 2 . 2 0 . E. N. da C. Andrade and C. Henderson, Phil . Trans. Roy. S o c , V o l 2 4 4 A ( 1 9 5 1 ) , 1 7 7 . 2 1 . I . R. Kramer and J . Demer, Trans. A.I.M.E., V o l . 2 2 1 No. 4 ( 1 9 6 1 ) 7 8 0 . 2 2 . . S. Harper and H. Cottrell', Proc. Phys. Soc, Vol. 6 3 B ( l 9 5 0 ) , 3 3 1 . 2 3 - S. Saimoto, 'M.A.Sc Thesis submitted in the Department of Metallurgy, University of British Columbia, June i 9 6 0 . 2k. E. N. da C. Andrade and R. F. Y. Randall, Nature, Vol. 1 6 2 ( 1 9 4 8 ) , 8 9 0 . 2 5 . C. S. Barrett, Acta.Met., Vol. l ( l 9 5 3 ) , 2 . 2 6 . E. N. da C. Andrade and R. F. Y. Randall, Proc Phys. S o c , V o l . 6 5 B ( l 9 5 2 ) 4 4 5 2 7 - F. R. Li.pse.tt and R. King, Proc Phys. Soc, Vol. 7 0 B ( l 9 5 7 ) , 6 0 8 . - 9k -28. J . J . Gilman and T. A. Read, Trans. A.I.M.E., V o l 1 9 l ( l 9 5 l ) , 792. 29. J . J . Gilman, Trans. A.I.M.E., V o l . 197(1953), 1217-30. E. N. da C. Andrade and A. J . Kennedy, Proc. Phys. S o c , V o l . 2 4 4 A(l95l), 177-31. S. S. Brenner, J . App.' Phys., V o l . 28(1957), 1023. 32. R. L. F l e i s c h e r and B. Chalmers, J . Mech. and Phys. of Solids.Vol.6(1958) 307-33. P. J . S c h l i c h t a , "Growth and P e r f e c t i o n of C r y s t a l s " , Wiley, (1958), 214. 34. P. J . S c h l i c h t a , P r i v a t e Communication. 35. D. L. McDanels, R. W. Jech and J . W. Weeton, Metals Prog. Vol.78(1960)6, 118. 36. P h y s i c a l and Mechanical P r o p e r t i e s of Some High-Strength F i n e Wires, Memorandum No. 80, Defense Metals Information Center, Jan. 1961. 37. Z. S. B a s i n s k i , P h i l . Mag., V o l . 4(1959), 393-38. K. G. Davis, P r i v a t e Communication. 39- W. D. Sylwestrowicz, Trans. A.I.M.E., V o l . 212(1958), 617• 40. M. A. Adams and A. H. C o t t r e l l , P h i l . Mag., V o l . 46(1955), I I 8 7 . 41. A. Seeger, " D i s l o c a t i o n s and Mechanical P r o p e r t i e s of Crystals',' Wiley, 24-3• 42. A. Seeger, J . D i e h l , S. Mader and H. Rebstock, P h i l Mag. V o l . 2(1957), 323-43. V. B. Lawson, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, A p r i l 1961. 44. R. F. Snowball, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, October i960. 45- R- W. Fraser, M.A.Sc. Thesis submitted i n the Department of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, November i960. 46. A.S.M. Metals Handbook, 1961, 962:.. 47. H. C. Rogers, Trans. A.I.M.E., V o l . 218(1960), 498. 48. J . D. Eshelby, F. C. Frank and F. R. N. Nabarro, P h i l . Mag., Vol- 42(1951) 351' 49. J'. P. F r a n k e l , Acta. Met., V o l . 6(1958), 215-50. P. R. Thornton and P. B. H i r s c h , P h i l . Mag., V o l . 3(1958), 738. 51. Z. S. B a s i n s k i , Aust. J . Phys., V o l . 13(1960), 284. 52. A. N. Stroh, Proc. Phys. S o c , V o l . 67B(l954), 427-53. Reference l 6 . Page 74. APPENDICES APPENDIX I. APPENDIX I.  Calculation of Parameters a and b The parameters a and be represent the intercept and the slope respectively of the log-log plot of the stress-strain curve. The values for each specimen were found by calculating the stress at each of a number of points picked from the Instron plot of load versus elongation and fitting these to an expression of the form . log £ = a + b log £ by the methods of least squares. Figure Ul. shows a schematic representation of the Instron plot together with a correction curve which compensates for slackness in the grips and for the inertia of the recording system. A l l elongation measurements are taken with reference to this correction curve. The stress (5 and the strain £ at point A are calculated as follows: V c L V-g = cross-head speed V c = chart speed L = gauge length (measured between the grips) . 6 r = — (1 +C) P = load a = i n i t i a l cross sectional area o The factor ( l +£•) takes into account the reduction in cross-sectional area, due to strain and is derived as follows: The in i t i a l volume of the specimen is VQ = aQ L. When the specimen is extended an amount ^ L , the volume becomes V l = a i (L +A L). where and where / Correction / Curve f Instron t Plot / / P r • 1 Elongation Figure kl.. Schematic Representation of the Load-Elongation plot from the Instron Recorder. Assuming V-^  = YQ: a n (L + & L) = a L l o §1 = 1 + ^ = 1 + £ a o L a l = a o 1 +£ The l o g and l o g ^  are determined f o r each p o i n t picked from the I n s t r o n p l o t . Since the r e l a t i o n s h i p •' i s of the form l o g 6"* = a + b l o g € , the values of a and b which s a t i s f y the l e a s t squares s o l u t i o n are those which s a t i s f y the equation: n a + b £ log£ = ^logS* a£log € + b5"(log€ f = £log£ log^ For specimen 1324, whose l o g - l o g p l o t forms two s t r a i g h t l i n e s , the f o l l o w i n g sets of equations were found: 18 a± + I9.9U3 b x = 59-760 19-9^3 a x + . 22.693 b 1 = 66A70 which gives a l = 2.836. b x = 0.1+37 and 10 a 2 + 14.566 b 2 = 3U.527 14.566 a + 21.274 b 2 = 50.310 which gives • aJ = 3.004 b 2 = 0.308 where a 1 and a 1 are c a l c u l a t e d f o r £ expressed as a percentage. These are r e f f e r r e d to £ expressed i n i n / i n by a 1 = a 1 - 2 b 1 = I.962 a 2 = a 1 - 2 b 2 = 2.388 The correlation coefficient, r, was calculated for randomly selected specimens as shown below: log £ logC - log logC 6*log6* ^ logc 2 where p For the above specimen, the values of r found by this method were r^ s 1 .000 and r^ = 0.995- The values of r found for other specimens were also a l l very close to 1 .0 indicating a close f i t of the points to the equation since a perfect f i t corresponds to r = 1 .0 . - 1 0 0 -0 . 6 0 . 8 l . o 1 . 2 l.k 1 . 6 log °jo Elongation Figure k2. a. 1D1 log °jo Elongation Figure 42. b. Figure U-2. c. Log-Log Plots of the Stress-Strain Curves for Representative Specimens. APPENDIX II. _ 103 -TABLE VI. The Results of Tensile Tests of Polycrystalline Copper Wires a. Temperature = 293°K Strain Rate =0.02 min"1 D e c i m e n Diameter (microns) Total Strain UTS (Kg/cm2) Yield Stress YS , YS TKg/cm2) ° a i b l &2 h 2 B 9 926 37-1 3080 56O 680 1.966 0.435 2. .261 0.345 BIO 926 44.8 3280 590 700 1.898 0.458 2.425 0.299 B l l 930 41.6 3200 520 660 1.960 0.438 2. •993 0.317 B12 922 38.3 3100 560 680 1.954 0.443 2, •371 0.315 B13 932 37-3 3060 490 630 1-971 0.430 2. .220 0-324 Bl4 932 37-9 3100 540 650 1.962 0-437 2.388 0.308 B 1 5 917 36.5 3100 560 690 1.911 0.459 2. •373 0.316 Bl6 930 37-3 3070 570 690 1.941 0.444 2.360 0.316 BIT 930 38.0 3080 54o 680 1.893 0 . 46 i 2, .988 0.318 C 3 518 36.9 3070 530 650 1.924 0.450 2.438 0.294 c 4 518 38.2 3270 560 700 1.842 0.483 2. .444 0.298 C 5 518 38.2 3200 560 680 1.799 O.495 2.405 O.308 C 6 518 36.7 2880 - - 1.902 0.449 2. •357 0.310 C 7 518 38.4 2900 490 600' I.889 0-453 2. •977 0.309 C .8 518 38.3 2900 .490 610 1.880 0.456 2. .406 0.296 Cl4 518 37-9 3100 500 64o I.891 0.462 2.366 0.318 F 1 203 29.0 .3150 610 720 1.951 0.455 2-346 0-333 F 2 203 31-5 3200 570 710 I.926 O.458 2. 270 0-355 F 3 203 33-2 3280 570 710 1.867 0.482 2, .220 0.370 F 5 203 31-9 3250 590 710 I.909 0.468 2. • 322 0.340 F 7 203 29-3 3120 550 700 I.918 0.463 2, • 340 0-333 F 8 203 31-3 3170 550 - 700 1.746 0.515 2. •337 0.333 F 9 203 33-9 3240 590 720 1.923 0.461 2, .297 0.345 F10 203 31-6 3200 590 700 1.880 O.477 2. • 357 0.328 G 5 100 18.8 2540 _ _ 1.863 0.473 _ G 6 . 100 19.5 2560 460 610 1.820 0.486 -G 8 100 19.8 2600 480 670 1.867 O.472 -G 9 101 21.3 264o 470 600 1.894 o.46i G10 101 16.9 2400 420 600 1.825 0.483 -I 1 48 13-6 2460 570 730 1.884 0.482 _ I 2 48 14.9 246o 550 700 1.956 0.450 -I 3 48 24.2 2800 590 700 1.765 0.498 -- 104 -b. Temperature = 2 9 3 ° K Strain Rate = 0 . 8 min ^ Diameter Total Strain UTSg Specimen (microns) ($) . (Kg/cm ) a - j _ • a^  b^ IB 1 946 2 8 . 1 2 6 5 0 2 .074 0 . 3 9 8 2 . 3 3 1 0 . 3 1 7 IB 2 914 2 5 . 2 2 6 2 0 2 . 1 9 0 O .368 2 . 451 0 . 2 8 6 IB 9 9 5 0 2 8 . 8 2770 2 . 0 2 9 0 .418 2-^73 0 . 2 8 1 1B10 • 9^7 • 2 9 . 4 2800 2 . 1 3 0 0 . 3 9 1 2 .461 0 . 2 8 6 1B11 952 2 9 . 1 2 7 8 0 2 . 0 5 2 0 . 4 1 5 2 . 5 0 8 0 . 2 7 1 i c 6 502 2 3 - 0 2570 2 \174 0 . 3 7 2 2 . 3 1 8 0 . 3 2 6 1C 7 495 24 . 2 2 7 0 0 2 . 1 3 3 0 . 3 9 1 2 - 3 7 5 0 . 3 1 3 1C 8 493 2 3 . 4 2670 2 .249 0 . 3 5 7 2 . 6 0 1 0 .245 i c 9 495 2 3 - 6 2 6 6 0 2 .174 0 . 3 7 7 2 .444 0 . 2 9 1 1C10 4 8 8 2 2 . 3 . : 2570 2 . 1 6 0 - 0 . 3 7 9 2 . 401 0 . 3 0 2 1C12 512 ' 2 3 . 7 2760 2 . 2 1 2 0 . 3 6 6 2-577 0 . 2 5 0 1C14 500 2 3 - 5 •2860 2 . 2 6 8 0 . 3 5 4 2 . 6 7 0 ' 0 . 2 2 7 IC15 503 2 2 . 3 2 6 9 0 2 . 2 5 7 0 . 3 5 0 2 . 5 4 8 0 . 2 5 0 IF 1 '• 211 1 7 . 8 2 5 0 0 2 . 1 7 5 0 . 3 7 7 - -IF 3 215 . 1 7 - 8 2 5 8 0 2 . 3 0 3 0 . 3 4 3 - -IF 4 257 1 5 . 8 2 3 0 0 2 . 420 0 . 2 9 9 - -1G 2 . . 95 1 7 . 3 2 6 6 0 2 . 3 2 6 0 . 3 4 0 -1G 3 100 1 5 - 3 2590 2 . 4 6 7 0 . 2 9 6 •- -1G 4 9 8 24 . 2 2850 2 . 9 7 2 0 . 3 5 3 - -1G 5. 99 1 7 . 8 2600 2-324 0 . 3 3 7 - -c. Temperature = 78°K .Sp< )ecimen Diameter (microns) Y i e l d YS 1 (Kg/. Stress YS r-cm2) - 5 . a l b l L1BI3A 903 1150 1250 1.996 O.487 L1B13B 903 1150 1250 I . 9 4 4 0.503 L1B14A 928 1060 1190. 2.032 0 . 1+68 L1315A 917 - - 1.920 0.503 L1B16A • 928 850 1150 I . 9 6 7 0.488 L1317A 897 1060 1190 2.110 0.455 L1B17B 897 • 1050 1190 2.024 0.480 L1C22A 515- _ _ I .96O 0.500 L1C24A 501 1210 1370 2.028 0.479' L1C24B 501 1180 13^0 2.043 0.475 L1C25A 475 1290 ' 1500 2.096 0.473 L1C25B 475 1240 1500 1.992 0.506 L1C26A 490 l l 6 0 1410 2.591 0 A 6 5 L1C26B 490 l l 6 0 1410 - -L1C27A 474 . - - 2.095 0.476 L1F 6A 215- 920 1100 I . 6 7 6 0.482 L1F 6B 215 890 1080 1.954 0.493 L1F 7A 204 960 1210 1-916 . 0.514 L1F 7B .204 96Q 1180 -L1F 9A 218 • 1020 1140 1.936 0.576 L1F 9B 218 1010 1150 1-9^3 O.500 L1F11A 218 1030 1200 1.880 • O.521 L1G 6 99 810 1080 2.085 O.456 L1G 7 102 780 970 1.865 O.508 L1G 8 98 830 960 1.847 O.520 L1G 9 100 850 960 I . 8 5 6 0.515 L1G10 97 - - 1.802 0.536 L1G11 101 - - 1.468 O.619 APPENDIX III. TABLE -VII. Effect of Strain Rate, and Temperature on" the  Flow-Stress of Polycrystalline Copper a. Strain Rate' • , . £:/= l O ' ^ m i n " 1 , ^ -10"1min"1 ' ' Ratio of Specimen Diameter Strain Flow-Stresses . mm •- SX2 3-5 RC3 3-8 2B1 0.9 RC4 3.8 113 0.05 0.068 0.986 0.098 0.985 0.126 0.987 0.164 0.985 0.206 0.985 0.251 0.982 0.292 0.979 0.3W 0.975 0.005 ' 0.935 0.012 0.972 0.024 0.977 0.051 0.978 0.080 . 0.978 0.106 "0.978 0.138 0.977 0.171 0.977 0.203 0.976 0.238 0.978 0.270 0.978 0.308 0.977 0.366 0.975 0.048 0.980 0.130 0.979 0.164 0.978 0.048 ,0.980 0.102 0.980 0.139 0.978 0.226 0.980 0.268 0.979 0.174' 0.979 0.005 0.974 0.024 • 0.973 0.036 0.974 0.056 0.974 0.075 0.975 Specimen Diameter mm Strain Ratio of Flow-Stresses 115 0.05 0.003 0.931 0.022 0.974 0.043 0.974 0.062 0.975 O.O83 0.975 0.111 0.975 0.122 O.976 0.139 0.973 0.152 0.974 0.166 0.974 0 . l 8 l 0.974 1F12 0.2 0.009 O.96I 0.025 O.969 • 0.042 0.970 - O.O58 O.968 0.077 0.971 0.094 0-970 0.110 0.969 0.131 . O.968 O.I5I O.969 0.171 .0.970 O.I89 O.969 0.205 O.969 0.221 0.970 0.239 O.968 0.257 O.968 0.275 0.970 0.295 0.970 O.316 0.970 1F16 . 0.2 . 0.055 0.970 0.068 0.970 0.081 O.969 O.O96 O.968 0.109 0.970 0.122 O.968 O.I36 O.967 0.149 O.968 0.162 O.969 , 0.177 O.967 0.202 O.968 0.215 O.968 0.229 0.970 0.243 O.968 0.257 O.969 0.272 O.969 0.285 O.967 Ratio of Specimen Diameter Strain Flow-Stresses mm 1B22 0.9 IB.26 0.9 Temperature T 1 = 2 9 3 % T 2 SXl 4.0 0.007 O.96O 0.031 O.966 0.048 0.965 0.070 0.967 0.091 O.968 0.110 O.966 0.130 0.964 0.151 0.964 0.168 O.965 0.191 O.963 0.217 0.964 0.246 0.963 0.274 0.963 0.302 O.965 0.325 0.965 0.348 O.965 0.014 0.938 0.039 0-957 0.059 O.966 0.094 O.967 0.110 O.965 0.132 0.964 0.157 0.964 0.180 O.966 0.204 0.963 0.228 0.9.64 0.253 O.965 0.281 O.963 0.309 O.966 0.332 O.963 0.358 O.960 O.387 O.961 78°K 0.020 0.881:. 0.055 0.886 O.O87 0.888 0.124 0.890 0.158 0.889 0.202 0.894 0.252 0.894 0.312 0.881 0.346 0.882 Specimen Diameter ' mm Strain Ratio of Flow-Stresses SX3 3.8 O.O38 O.87O 0.077 O.893 0.122 O.889 0.179 0.886 0.228 0.888 O.287 0.886 0.340 0.880 0.4l4 0.881 0A98 0.880 SX4 3.8 . 0.031 0.880 0.120 0.886 0.183 0.885 0.255 0.881 RC1 4.6 0.054 O.885 0.074 0.884 0.094 0.879 0.115 . 0.881 0'. 136 0.877 0.156 0.874 0.178 0.871 0.197 0.873 0.216 0.867 0.226 0.851 0.273, . 0.853 RC2 3.5 0.012 O.885 0.031 0.884 0.051 0.886 0.093 0.875 0.091 0.879 0.105 • 0.876 0.120 O.874 0.137 0.874 0.153 0.874 0.167 0.866 RC5 3.9 0.028 0.877 • 0.072 0.887 0.129 0.890 0.172 0.884 0.214 0.888 0.258 0.884 O.303 0.884 O.3U8 0.885 0.392 0.887 Specimen Diameter Strain Ratio of Flow-Stresses - 110 RC6 4 . 0 0-022 . 0 . 8 8 0 " 0 . 0 4 8 0 . 8 8 5 0 . 0 7 5 0 . 8 8 8 " 0 . 1 0 2 0 . 8 8 6 0 . 1 2 9 0 . 8 8 6 0 . 1 5 2 0 . 8 8 3 0 . 1 7 9 0 . 8 8 3 0 . 2 0 5 0 . 8 8 0 "' 0 . 2 5 2 • 0.882 "1F13 0.2 0 . 0 1 6 O . 8 7 9 0 . 0 3 8 0 . 8 8 0 0 . 0 5 8 0 . 8 7 3 0 . 0 7 5 0 . 8 7 2 0.100 ' 0 . 8 6 5 .0.124 0 . 8 6 3 0 . 1 5 0 0 . 8 6 0 . 0 . 1 7 9 0.849 0.212 O . 8 5 5 0.245 0.848 0.278 0.846 LB24 . 0-9 0.010 0.846 0.024 O . 8 7 6 " C.OUO 0 . 8 7 9 O.O56 O . 8 7 7 " - '0.071 O . 8 6 9 0 . 0 9 4 " 0 . 8 6 8 0 . 1 1 5 0.864 O . 1 3 5 0 . 8 6 2 O . 1 5 5 O . 8 5 9 , O..176 O . 8 5 7 0.204 0 . 8 5 1 0 . 2 2 3 0.846 0.249 0.848 LB27 0.9 0 . 0 1 7 0 . 8 5 6 0 . o 4 l 0 . 8 8 2 . O . 0 6 7 0.868 . 0 . 0 9 8 0 . 8 6 2 0 . 1 3 2 0 . 8 5 9 0 . 1 7 1 0 . 8 5 6 0 . 2 1 3 0.848 0 . 2 4 9 0.845 0 . 2 9 4 0 . 8 3 7 APPENDIX IV. TABLE VIII. Effect of Temperature on.the Flow-Stress of Aluminum a. Cottrell-Stokes Law Tl = 2 9 3 % T 2 = T8°K Ratio of Specimen Strain Flow-Stresses SA5 0.138 0-TT6 0.184. 0.771 0.212 0-TT3 0.243 0.TT2 0.2T4 O.T69 0.300 0.TT4 0.334 0.T68 O.362 0.TT4 SA6 0.043 0-TT3 0.053 0-TT6 0.068 0.TT4 0.08T 0.TT2 0.112 0.771 o.i4o 0.771 0.150 0.771 0.153 0.774 0.169 0.770 0.205 0.771 0.24T 0.772 0.304 0.774 0.363 0-773 SAT 0.138 0.772 0.169 • 0.777 0.208 0.770 0.280 0.776 0.305 0.770 b. Work-Softening T l = 293°K T 2 = 78°K Specimen €' Strain SA8 0.026 0.080 1.131 0.118 1.034 0.156 0-975 0.193 0-955 0.231 0.971 0.271 0-955 0.308 0-979 0-346 0-975 Specimen £•' S t r a i n SA9 0.050 0 . 0 9 2 1 . 4 7 1 0 . 1 5 7 1 . 0 8 2 0 . 3 1 4 1.021 O . 3 7 8 1 . 0 1 5 0 . 4 4 1 1 . 0 0 6 0 . 5 0 5 O . 9 8 6 SA10 0 . 0 1 3 0 . 0 5 3 1 . 1 4 1 0 . 0 7 7 1 - 0 3 3 0.101 1 . 0 0 3 0 . 1 2 5 0 . 9 9 0 0 . 1 4 9 0 . 9 7 8 -0 . 1 7 3 0 . 9 7 9 ' 0 . 1 9 7 O . 9 6 3 0 . 2 2 1 0 . 9 7 0 0 . 2 7 0 0 . 9 6 4 0 . 2 9 4 O . 9 8 3 0 . 3 1 8 0 . 9 7 2 0 . 3 4 2 0 . 9 7 1 O . 3 6 6 0 . 9 5 1 0 . 3 9 0 O . 9 6 2 0 , 415 O . 9 5 8 SA11 0 . 0 0 5 0 . 0 4 1 I . 0 6 5 0 . 0 8 7 0 . 9 8 8 0 . 1 3 0 0 . 984 O.I78 O . 9 6 9 0 . 2 2 5 O . 9 6 2 0 . 2 7 1 O . 9 6 7 0 . 3 1 7 0 - 9 5 9 .0 .364 O . 9 6 3 0.020 . 0 . 0 7 1 - I.O96 0 . 1 1 6 O . 9 8 8 0 . 1 6 2 0 . 9 7 1 0 . 2 0 9 O . 9 3 8 0 . 2 5 6 O . 9 6 2 0 . 3 0 2 O . 9 5 8 0 . 3 4 7 O.95O SA12 0.040 0 . 0 8 0 1 . 3 0 0 0 . 1 3 0 1 . 0 2 8 0 . l 8 l 1.020 0 . 2 3 2 0 . 997 0 . 2 8 2 O . 9 8 8 SA13' 0 . 0 0 3 0 . 0 4 2 1 . 0 0 3 0 . 1 3 3 O . 9 6 9 0 . 2 2 4 0 . 9 5 2 0 . 3 3 9 0 . 9 7 3 0 . 4 2 9 O . 9 6 7 0 . 0 1 3 O . 3 2 6 O . 9 6 2 O . O 6 7 0.120 1 .542 0 . 2 1 1 1 . 0 6 0 0 . 3 0 1 1 . 0 1 5 0 . 4 i 6 0 . 9 9 0 APPENDIX V. Estimated Errors 1. Size Effects The estimated error in the yield stress, ultimate tensile strength and elongation before fracture are calculated for a representative specimen, Clk. The diameter was measured to ± lyu but difficulties in measuring a true diameter reducedthe accuracy to about ± 3 Diameter = d = 518 ± 3 fx = 518 u ± 0.6$ Area = a = TT d 2 = 2.11 X 10~3cm2 ± 1.2$ ~k~' The Instron tensile tester is rated by the manufacturer at an accuracy of better than ± 1 $ of full-scale load when the slope of the load elongation curve is such that more than three seconds is required for the pen'..to travel the f u l l width of the chart. Load' at yield, P = 2.86 * 0.05 Ih = 2.86 lb ± 1-7$ Since £ is very small at the yield point, errors in the (l + € ) term wil l have a negligible effect. ¥ S 0 . = o.k^k P (1 * 0.017) ( 1 + € ) J a (1 * 0.012) - v ; = 6k0 Kg/cm2 ± 3$ = 640 ± 20 Kg/cm2. The gauge-length was measured to ± 0.01 mm but difficulties in measurement reduced the accuracy to about t 0.05 mm. Gauge length = 62.41* 0.05 mm = 62.kl mm i 0.1$ The strain was .-calculated to be 0.001-(l * 0.00l)(x ± l ) where x is as defined in Appendix I. The elongation before fracture: £= o.ooi ( i * o.ooi) 379 ( l * 0.003) = 0.379 ( l * 0.004) = 0.379 ± 0.002 The ultimate tensile strength is calculated by the same method as the yield stress. p. = 10.6 ± 0.2 lb = 10.6 lb ± 2 $ 1 + € = 1-379 ± 0.002 - = 1-379 (1 ± 0-002) UTS = 3100 (1 ± 0.034) Kg/cm2 = 3100 ±"100 Kg/cm The limits of accuracy found for the other specimens tested for size effects were found to be of the same order as those calculated for specimen C14. 2. Effect of Temperature and Strain Rate on the Flow-Stress Since only a ratio of flow-stresses was required, only the loads were measured and the only errors are those due to the limits of accuracy of the machine. The Instron tensile tester is equipped with a zero suppression by means of which the area of the load-elongation curve to be studied can be magnified, thus increasing the accuracy. The max-imum estimated error for a l l specimens varied from about 0.8$ at low strains to about 0-1 -$ at high strains. 

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