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High-strain steady-state flow in silver at ambient and near-ambient temperatures
Scripta
METALLURGICA
Vol. 22, pp. 4 1 - 4 6 , 1 9 8 8 P r i n t e d in t h e U . S . A .
Pergamon
Journals,
Ltd.
HIGH-STRAIN STEADY-STATE FLOW IN SILVER AT AMBIENT AND NEAR-AMBIENT TEMPERATURES
M. E. Kassner and J. J. 01danl Lawrence Livermore National Laboratory (L-355) Livermore, CA 94550, USA (Received August (Revised October
17, 23,
1987) 1987)
Introduction The steady-state flow of metals has been extensively examined at high (usually above 0.5 Tm) temperatures. Much less is known about steady-state (or saturation) flow of metals at lower (0.2-0.5 Tm) temperatures, partly because large strain deformation is required. Solid specimens of hlgh-purity silver were torsionally deformed to large strains at ambient and near-ambient temperatures in this investigation to determine whether, first, a steady state could be achieved. Some earller compression tests [i], hardness tests [2], and torsion tests [3] indicate that a saturation or steady state may eventually be achieved at equivalent unlaxlal strains of roughly 3. However, other work has shown that at least some fcc metals [4,5] (and several bcc metals [6]) do not achieve a steady state over the large torsional strain range to failure at ambient temperature. Instead, it has been observed that the linear decrease in the hardening rate, 8 (= do/de), with increasing stress associated with Stage III deformation is interrupted and Stage IV deformation proceeds and, in the case of nickel [5], the new hardening is independent of stress (or strain) and a steady state is not observed. In other cases, as with copper [6,7], a discontinuity in the 8-versus-a curve is evident, although saturation is still eventually observed. Stage IV [5-7] deformation is believed to be present in this case, but it does not preclude a saturation stress that may be higher than that which would have been achieved in its absence. If our experiments reveal a steady state, the activation energy for steady-state deformation, Qc,ss, would be determined. This would provide some insight into the rate-controlllng process for steady-state flow (creep) at low and intermediate temperatures. Some have suggested [8-10] that the rate-controlling processes below about 0.6 T m are associated with dislocation pipe diffusion, and that the activation energy Qp should be constant over an appreciable temperature range. Others have suggested that the activation energy is associated with the thermal activation of dislocations over obstacles, cross-slip, or the glide of dislocations on noncompact planes [11-15], and that Qc,ss is expected to be stress-dependent and, therefore, temperature-dependent. ExDerlmental Procedure The silver used in this study was of 99.99Z purity and was provided in the form of 19-mm-diameter rod. Specimens were machined into solid torsion specimens with a 25.4-mm gage length and a 5.l-mm diameter. The specimens were annealed in vacuum for 30 min at 873 K, and the resulting grain size was about 80 pm. Torsion tests were performed on an Instron 1125, and the temperature was controlled by an Instron convection furnace to within ±i K. The shear stress, Tez, at the outer fiber was determined from torque measurements using the usual equation [16]: -
¢8z
M
(3 + k + m)
(1)
2~r 3
where M is the torque, by a = Ae k , w h e r e A i s
r is the specimen radius, k is the strain-hardening exponent (defined a constant), and m is the strain-rate sensitivity exponent. The
41 0036-9748/88 $3.00
+ .00
42
FLOW IN Ag
Vol. 22, No. 1
e q u i v a l e n t u n l a x i a l s t r e s s ~ and e q u i v a l e n t u n i a x i a l s t r a i n ¢ were c a l c u l a t e d u s i n g the von Mlses criterion. The exponent, k, was assumed equal to zero where the torque-angle of twist b e h a v i o r showed softening. Results and Discussion Specimens were deformed at equivalent uniaxlal strain rates, ~, from 5.77 x 10 -4 s -I to 5.77 x 10 -2 s -1 at 373, 296, and 233 K (0.30, 0.24, end 0.19 Tm, respectively~. Figure 1 illustrates the equivalent uniaxial stress versus equivalent strain at ~ = 5.77 x 10 -4 s -I at the three test temperatures. Steady state is not achieved until large equivalent uniaxial strains between about 1.5 and 3. This shows that steady-state behavior at ambient and near ambient temperatures can be reasonably examined by torsion tests. Fracture stresses and strains are indicated for the 296 and 373 K curves. Although the specimen at 233 K was not deformed to rupture, another specimen was twisted to failure at this temperature, but at a lower strain rate. The fracture stress at this temperature was nearly identical to the steady-state stress (to within I~). However, an apparent softening regime is observed at ambient temperature, where the flow stress decreases by about 2~ over a strain range of about 0.5 prior to fracture. More dramatically, at 373 K, a steady-state stress is maintained over a strain range of about 1.0 before there is a slow reduction in the torque by about 16~ over a relatively large strain range of 2.5. The source of this softening has not been conclusively determined, but macroscopic examination of the surfaces of specimens deformed at 373 K revealed some localized deformation coincident with softening. Examination also revealed relatively homogeneous deformation in specimens deformed at 233 and 296 K, where softening is not pronounced. This suggests the posslbility of dynamic recrystallization at 373 K. Slakhorst [17] used resistivity measurements to determine recrystallizatlon curves (~ recrys, vs time) between 328 and 413 K for 99.999~ pure silver rolled 99.5~ (~ = 3.0 at a relatively high strain rate at ambient temperature). These data indicate: sisnlflcant static recrystalllzatlon at 373 K within Just a few minutes, although extrapolation of the data to ambient temperature reveals that even a few percent static recrystalllzatlon requires over 50 hours.
I
I
I
I
I Silver
500 -
~ : 5.77 x 10-4s-1
400 A ¢0
O.
~--
FIG. 1. The e q u i v a l e n t uniaxial stress versus equivalent untaxial strain behavior of s i l v e r from t o r s i o n t e s t s a t an e q u i v a l e n t uniaxial strain rate of 5 . 7 7 x 10 - 4 s - 1 and at three temperatures.
233K
soo
,10
296K 200 373K
100
I
I
I
I
1
2
3
4
5
Strain,
It is well known that the steady-state Css = A' exp (-Qc,ss/RT)
(Oas/E) n
creep rate, ~¢ s s ,
can be d e s c r i b e d by: (2)
where A' i s a c o n s t a n t , E i s Y o u n g ' s m o d u l u s , Oss i s t h e s t e a d y - s t a t e s t r e s s , R i s t h e g a s c o n s t a n t , and T i s t h e a b s o l u t e t e m p e r a t u r e . Figure 2 is a plot of the equivalent uniaxial steady-state strain-rate C s s v e r s u s t h e m o d u l u s - c o m p e n s a t e d [18] e q u i v a l e n t u n i a x i a l steady-state stress, oss/E, at the three test temperatures. Both axes have l o g a r i t h m i c
Vol.
22,
No.
1
FLOW IN Ag
42;
scales. S t e a d y - s t a t e s t r e s s exponent v a l u e s , n , and a c t i v a t i o n e n e r g y f o r s t e a d y - s t a t e d e f o r m a t i o n v a l u e s , q c , s s , a r e e a s i l y d e t e r m i n e d f r o m t h e s e p l o t s . / The s t e a d y - s t a t e s t r e s s e x p o n e n t i s d e f i n e d f r o m Eq. 2 :
n =(8 a In~ss ~ in
(3)
(Oss/EI/i
At 296 K, n i s b e s t d e f i n e d a t a b o u t 32. The n v a l u e a t 233 K i s r o u g h l y 4 0 , and a t 373 K, n was f o u n d t o b e a b o u t 18. T h e s e v a l u e s s e e m s c o n s i s t e n t w i t h t h e t r e n d s f r o m h i g h e r t e m p e r a t u r e work by o t h e r i n v e s t i g a t o r s . Work b e t w e e n a b o u t 0 . 6 5 a n d 0 . 9 Tm i n d i c a t e t h a t n i s c o n s t a n t and a b o u t 5 [ 1 9 ] , c o n s i s t e n t w i t h p o w e r - l a w b e h a v i o r . I n t e n s i l e t e s t s below 0 . 6 5 Tm Goods and N i x [19] and Logan a n d M u k h e r J e e [20] f o u n d t h a t n i n c r e a s e s w i t h i n c r e a s i n g steady-state stress (decreasing temperature), although they did not establish steady-state conditions at temperatures within our testing range. These investigators found that n i n c r e a s e d f r o m 7 t o a b o u t 22 w i t h d e c r e a s i n g t e m p e r a t u r e . This trend is consistent with power-law-breakdown behavior. E a r l i e r work b y one o f t h e a u t h o r s f o u n d t h a t t h e c o n s t a n t structure (p) stress exponent, N = ( I ~ ! D _ ~ , for silver is over 200 [21]. This suggests In o/p,T that the relationship between n and N for silver deformed within the power-law-breakdown regime Is not as straightforward as has been suggested for metals and alloys, at least in the power-law regime [22]. It should be mentioned, that consistent with some low-temperature aluminum work [9], we expect that n should be independent of temperature for a fixed Oss/E. Therefore, the three sets of data in Fig. 2 may be better described by curves instead of lines, so that the slopes at a given ~/E on this plot are the same at the three temperatures.
~8
l(r ~
I
I
I
I
I
/
I
I
I
/=
,~ I@
•/.
. ./
.
~3K
_e o in
~ 10-4
"O
Ul
lO-S
2.0
/
•/" I 2.2
-7.:" I 2.4
I 2.6
I 2.8
I 3.0
I 3.2
I 3.4
FIG. 2. The modulus-compensated equivalent unlaxlal steady-state stress is a flmctlon of steadystate strain rate from torsion tests at three temperatures. Both axes have logarithmic scales.
3.6
~SS -3 ~-xlO
The a c t i v a t i o n
energy for steady-state
deformation,
Qc,ss,
can be c a l c u l a t e d
by
A c t i v a t i o n - e n e r g y v a l u e s were c a l c u l a t e d from t h e d a t a o f F i g . 2 a t t h e a v e r a g e t e m p e r a t u r e s o f 336 K ( 0 . 2 7 Tin) and 264 K ( 0 . 2 1 Tin) t o be 5 7 . 3 and 5 2 . 6 k J / m o l , r e s p e c t i v e l y , o r a p p r o x i m a t e l y 55 k J / m o l a t a m b i e n t t e m p e r a t u r e ( 0 . 2 4 Tm). T h e s e v a l u e s o f t h e a c t i v a t i o n
44
FLOW
IN A g
Vol.
22,
No.
energy for deformation are significantly lower than the estimated value of 81.3 kJlmol for the activation energy Qp for dlslocation plpe diffusion [23]. Figure 3 is a plot of the activation energy for steady-state deformation as a function of temperature. The graph plots six average activation energy values from this and other studies [19,24-26] calculated by us from Eq. 4 together with the range of temperature over which the values were determined. Above about 0.5 Tm, the activation energy seams to correspond to that of lattice selfdiffusion Qsd [27], as expected . Goods and Nix determined the activation energy for steady-state deformation in silver to be about ii0 kJ/mol at about 0.33 Tm, close to that of dislocation pipe diffusion. However, another interpretation is possible. While the activation energy for steady-state flow above 0.5 T m is constant over a relatively wide range of strain rate and temperature and is approximately equal to Qsd, it appears stressdependent and, therefore, temperature-dependent below 0.5 T m. This may be more supportive of a cross-slip or dlslocatlon-gllde-controlled mechanism. The activation energies for these processes are expected to be stress-dependent, while Qp would be expected to be stressindependent. The drop in the activation energy below roughly 0.5 T m in Fig. 3 suggests, to us, a change in the rate-controlllng mechanism for steady-state deformation. However, Mecking et al. [28] compared extrapolated saturatlon-stresses for silver single crystals (as well as A1, Ni, and Cu) with temperature. They found that the data can be described by a slngle formula over a wide temperature range from the intermediate range up to near the melting point, suggesting the possibility of the absence of a transition.
"~ O _=
200 / /
~="
s
-
-
i
I
/
FIG. 3. The activation energy for steady-state deformation of silver as a function of the fraction of the absolute melting temperature.
!
9I
o-~= 100 /
/
o This study • Price [26] • Schr6der et al. [25] • G o o d s and Nix [19] • M u n s o n [24]
/
//
~a C
> =0
6
0
0.5
1.0
T/Tm
An evaluation of the stress-strain curves of Fig. 1 reveals that there is an abrupt change in the rate at which hardening rate, @ (= d~Id¢), decreases wlth stress. This is shown in Fig. 4. At 373 and 233 K, @ decreases almost linearly with stress, which is characteristic of Stage Ill deformation [5-7,15]. However, at high strains between 0.8 and 1.0, there is a transition, and 8 decreases at a significantly lower rate. This behavior has been described as indicative of a transition to Stage IV deformation [5-7]. Eventually, a steady state is attained, but this stress (Oss) is 1.15 times higher at 373 K and 1.21 times higher at 233 K than a steady-state stress, Olll.ss, based on an extrapolation of Stage III e-stress data to a zero hardening rate. (Others [7] have estimated the lowtemperature steady-state stress from tensile tests, in which a steady state is not achieved, by this extrapolation.) Higher temperatures (and presumably lower strain rates) are associated with lower Stage IV deformation contributions to the total hardening. It may be more meaningful to compare n and Qc,ss values for steady-state deformation based on Olll,ss with the higher-temperature literature values. These would not include Stage IV hardening effects nor the possible effects of dynamic recrystalllzatlon at 373 K. Higher-temperature, constant-straln-rate tests of silver do not seem to be available to confirm that actual steady-state stresses are consistent wlth a linear extrapolation of @ during Stage III deformation. However, 304 stainless steel (another low-stacklng-fault-energy metal) undergoes a substantial primary creep (0.50 strain) at 1023 K under constant-straln-rate conditions, and ~ss and aIII,ss are identical [29]. If we apply Eqs. 3 and 4, then we calculate nii I = 36 and Qc,III = 47 kJ/mol at 296 K, based on extrapolated ~III,ss values at all temperatures and
1
Vol.
22,
No.
1
FLOW
IN A g
45
600
500 O.
'~1~ 400 II ~D
.m 300 O)
FIG. 4. The work-hardenlng rate as a function of stress at 233 and 373 K at a strain rate of 5.77 x 10 -4 s -I. Stage III end IV deformation regimes are indicated.
c
=
200
0
100
0
0
100 200 300 Stress, o(MPa)
400
strain rates. These calculations indicate a possible decreased rate sensitivity and activation energy for steady state, based only on Stage III deformation. It should also be mentioned that our ambient-temperature activation-energy calculation was based on tests at strain rates r o u g h l y two o r d e r s o f m a g n i t u d e h i g h e r t h a n i n t h o s e s t u d i e s r e f e r e n c e d i n F i g . 3 . If Qc,ss is recalculated based on (nonlinear) extrapolation of the data in Fig. 2 to lower strain rates, it would be roughly 5 k3/mol higher. Therefore, t h i s e n d t h e S t a g e IV a d j u s t m e n t a r e r o u g h l y compensatins, and the comparison in Fig. 3 of a 55-k3/mol value to the higher-temperature Qc,ss values is appropriate. Here, the observations that the strain corresponding to the o n s e t o f S t a g e IV d e f o r m a t i o n a n d t h e f r a c t i o n of total hardening contributed d u r i n 8 S t a g e IV appear to decrease with increasing temperature seem consistent w i t h o t h e r S t a g e IV t e m p e r a t u r e trends in nickel end aluminum [5,30]. Conclusions 1. 2.
3.
4.
Steady-state or saturation conditions can be achieved in silver at temperatures between 0 . 1 9 a n d 0 . 3 0 Tm b y t o r s i o n a l deformation. The activation energy for steady-state d e f o r m a t i o n o v e r t h i s r a n g e i s r o u g h l y 55 k 3 / m o l and is significantly lower then that for lattice self-diffusion or dislocation ptpe diffusion. This suggests that although the rate-controlling process for steady-state d e f o r m a t i o n a b o v e , r o u g h l y , 0 . 5 Xm a p p e a r s t o b e a s s o c i a t e d w i t h v o l u m e d i f f u s i o n , other processes such as a cross-slip or dislocation-glide-controlled m e c h a n i s m may operate at lower temperatures. At equivalent uniaxial strains between 0.8 end about 1.0, there is an abrupt change in the rate at which the hardening rate decreases with stress, a n d S t a g e IV h a r d e n i n g a p p e a r s t o c o m m e n c e . T h e s t r a i n c o r r e s p o n d i n g t o t h e o n s e t o f t h i s S t a g e IV a n d t h e f r a c t i o n o f t h e t o t a l h a r d e n i n g a c c u m u l a t e d d u r i n g S t a g e IV d e c r e a s e w i t h i n c r e a s i n g temperature; An a c t i v a t i o n energy ten be calculated based on steady-state stress ~IIIpss values t h a t do n o t i n c l u d e S t a g e IV d e f o r m a t i o n t h a t w e r e d e t e r m i n e d b y l i n e a r e x t r a p o l a t i o n of the hardentn8 rate over Stage III deformation to a zero value. This activation energy is slightly l e s s t h e n t h e m e a s u r e d v a l u e w h i c h i n c l u d e s S t a g e IV d e f o r m a t i o n .
46
FLOW
IN Ag
Vol.
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i
Acknowledgments This work was supported by Lawrence Livermore National Laboratory under the auspices of the U. S. Department of Energy, Contract W-7405-Eng-48. The author appreciates the help with the mechanical testing by M. Stratman and L. Allison. References i. 2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17 18 19 20 21 22 23 24 25 26 27 28 29 30.
H. G. Grewe, E. Kappler, H. Pflugmacher, E. Schneider, and W. Welser, Z. Physik 224, 85 (1969). I. Saunders and J. Nutting, Metal Sci. 18, 571 (1984). H. P. StUwe and H. Turck, Z. Metallktmde 55, 699 (1964). G. R. Johnson, J. M. Hoegfeldt, U. S. Lindholme, and A. Nagy, J. Eng. Mater. Tech. 105, 42 (1983). D. A. Hughes, J. C. Glbellng, and W. D. Nix, Strength of Metals and Alloys, H. J. Mcqueen, J.-P. Bailon, J. I. Dickson, J. J. Jonas, and M. G. Akben, eds., p. 51, Pergamon, Oxford (1985). J . Gil S e v i l l a n o , P. van H o u t t e , and E. A e r n o u d t , P r o g . M a t e r . S c i . 25, 69 (1981). H. Mecking and A. G r i n b e r g , S t r e n g t h o f M e t a l s and A l l o y s , P. Haasen, V. G e r o l d , and G. K o s t o r z , e d s . , p . 289, Pergamon, Oxford (1979). O. D. Sherby and J . Weertman, Acta M e t a l l . 27, 387 (1979). H. Luthy, A. K. M i l l e r , and 0. D. S h e r b y , Acta M e t a l l . 28, 169 ( 1 9 8 0 ) . H. E. Evans and G. Knowles, A c t a M e t a l l . 25, 963 (1977). W. D. Nix and B. I l s c h n e r , S t r e n g t h o f M e t a l s and A l l o y s , P. Haasen, V. G e r o l d , and G. K o s t o r z , e d s . , p. 1503, Pergamon, Oxford (1979). A. Arlell and A. K. Mukherjee, Creep and Fracture of Engineering Materials and Structures, B. Wilshlre and D. R. J. 0wen, eds., p. 77, Pineridge, Swansea (1981). D. Caillard and J. L. Martin, Rev. Phys. Appl. 22, 169 (1987). J. P. Porrler, Rev. Phys. Appl. ii, 731 (1976). H. Mecking, Work Hardening in Tension and Fatigue, A. W. Thompson, ed., p. 67, AIME, New York (1977). D. S. Fields and W. A. Backofen, Proc. ASTM 57, 1259 (1957). J. W. H. G. Slakhorst, Acta Metall. 23, 301 (1975). W. KSster, Z. Metallkunde 39, 1 (1948). S. H. Goods and W. D. Nix, Acta Metall. 26, 753 (1978). R. W. Logan and A. K. MukherJee, Scripta Metall. 17, 63 (1983). M. E. Kassner and A. K. MukherJee, Scripta Metall. 17, 741 (1983). O. D. Sherby, R. H. Klundt, and A. K. Miller, Metall. Trans. 8A, 843 (1977). D. Turnbull and R. Hoffman, Acta Metall. 2, 419 (1954). D. E. Munson, Ph.D. Dissertation, Stanford Univ., Stanford, CA (1960). K. Schroder, A. S. Grannuzzi, and G. Gorska, Acta Metall. 16, 469 (1968). C. E. Price, Acta Metall. 14, 1781 (1966). R. E. Hoffman and D. Turnbull, J. AppI. Phys. 22, 634 (1951). H. Mecklng, B. Nicklas, N. Zarubova, and U. F. Kooks, Acta Metall. 34, 527 (1986). M. E. Kassner, A. K. Miller, and A. A. Ziaai-Moayyed, Metall. Trans. 16A, 1069 (1985). W. Roberts, Deformation, Processing, and Structure, G. Krauss ed., p. 109, American Society for Metals (1984).
METALLURGICA
Vol. 22, pp. 4 1 - 4 6 , 1 9 8 8 P r i n t e d in t h e U . S . A .
Pergamon
Journals,
Ltd.
HIGH-STRAIN STEADY-STATE FLOW IN SILVER AT AMBIENT AND NEAR-AMBIENT TEMPERATURES
M. E. Kassner and J. J. 01danl Lawrence Livermore National Laboratory (L-355) Livermore, CA 94550, USA (Received August (Revised October
17, 23,
1987) 1987)
Introduction The steady-state flow of metals has been extensively examined at high (usually above 0.5 Tm) temperatures. Much less is known about steady-state (or saturation) flow of metals at lower (0.2-0.5 Tm) temperatures, partly because large strain deformation is required. Solid specimens of hlgh-purity silver were torsionally deformed to large strains at ambient and near-ambient temperatures in this investigation to determine whether, first, a steady state could be achieved. Some earller compression tests [i], hardness tests [2], and torsion tests [3] indicate that a saturation or steady state may eventually be achieved at equivalent unlaxlal strains of roughly 3. However, other work has shown that at least some fcc metals [4,5] (and several bcc metals [6]) do not achieve a steady state over the large torsional strain range to failure at ambient temperature. Instead, it has been observed that the linear decrease in the hardening rate, 8 (= do/de), with increasing stress associated with Stage III deformation is interrupted and Stage IV deformation proceeds and, in the case of nickel [5], the new hardening is independent of stress (or strain) and a steady state is not observed. In other cases, as with copper [6,7], a discontinuity in the 8-versus-a curve is evident, although saturation is still eventually observed. Stage IV [5-7] deformation is believed to be present in this case, but it does not preclude a saturation stress that may be higher than that which would have been achieved in its absence. If our experiments reveal a steady state, the activation energy for steady-state deformation, Qc,ss, would be determined. This would provide some insight into the rate-controlllng process for steady-state flow (creep) at low and intermediate temperatures. Some have suggested [8-10] that the rate-controlling processes below about 0.6 T m are associated with dislocation pipe diffusion, and that the activation energy Qp should be constant over an appreciable temperature range. Others have suggested that the activation energy is associated with the thermal activation of dislocations over obstacles, cross-slip, or the glide of dislocations on noncompact planes [11-15], and that Qc,ss is expected to be stress-dependent and, therefore, temperature-dependent. ExDerlmental Procedure The silver used in this study was of 99.99Z purity and was provided in the form of 19-mm-diameter rod. Specimens were machined into solid torsion specimens with a 25.4-mm gage length and a 5.l-mm diameter. The specimens were annealed in vacuum for 30 min at 873 K, and the resulting grain size was about 80 pm. Torsion tests were performed on an Instron 1125, and the temperature was controlled by an Instron convection furnace to within ±i K. The shear stress, Tez, at the outer fiber was determined from torque measurements using the usual equation [16]: -
¢8z
M
(3 + k + m)
(1)
2~r 3
where M is the torque, by a = Ae k , w h e r e A i s
r is the specimen radius, k is the strain-hardening exponent (defined a constant), and m is the strain-rate sensitivity exponent. The
41 0036-9748/88 $3.00
+ .00
42
FLOW IN Ag
Vol. 22, No. 1
e q u i v a l e n t u n l a x i a l s t r e s s ~ and e q u i v a l e n t u n i a x i a l s t r a i n ¢ were c a l c u l a t e d u s i n g the von Mlses criterion. The exponent, k, was assumed equal to zero where the torque-angle of twist b e h a v i o r showed softening. Results and Discussion Specimens were deformed at equivalent uniaxlal strain rates, ~, from 5.77 x 10 -4 s -I to 5.77 x 10 -2 s -1 at 373, 296, and 233 K (0.30, 0.24, end 0.19 Tm, respectively~. Figure 1 illustrates the equivalent uniaxial stress versus equivalent strain at ~ = 5.77 x 10 -4 s -I at the three test temperatures. Steady state is not achieved until large equivalent uniaxial strains between about 1.5 and 3. This shows that steady-state behavior at ambient and near ambient temperatures can be reasonably examined by torsion tests. Fracture stresses and strains are indicated for the 296 and 373 K curves. Although the specimen at 233 K was not deformed to rupture, another specimen was twisted to failure at this temperature, but at a lower strain rate. The fracture stress at this temperature was nearly identical to the steady-state stress (to within I~). However, an apparent softening regime is observed at ambient temperature, where the flow stress decreases by about 2~ over a strain range of about 0.5 prior to fracture. More dramatically, at 373 K, a steady-state stress is maintained over a strain range of about 1.0 before there is a slow reduction in the torque by about 16~ over a relatively large strain range of 2.5. The source of this softening has not been conclusively determined, but macroscopic examination of the surfaces of specimens deformed at 373 K revealed some localized deformation coincident with softening. Examination also revealed relatively homogeneous deformation in specimens deformed at 233 and 296 K, where softening is not pronounced. This suggests the posslbility of dynamic recrystallization at 373 K. Slakhorst [17] used resistivity measurements to determine recrystallizatlon curves (~ recrys, vs time) between 328 and 413 K for 99.999~ pure silver rolled 99.5~ (~ = 3.0 at a relatively high strain rate at ambient temperature). These data indicate: sisnlflcant static recrystalllzatlon at 373 K within Just a few minutes, although extrapolation of the data to ambient temperature reveals that even a few percent static recrystalllzatlon requires over 50 hours.
I
I
I
I
I Silver
500 -
~ : 5.77 x 10-4s-1
400 A ¢0
O.
~--
FIG. 1. The e q u i v a l e n t uniaxial stress versus equivalent untaxial strain behavior of s i l v e r from t o r s i o n t e s t s a t an e q u i v a l e n t uniaxial strain rate of 5 . 7 7 x 10 - 4 s - 1 and at three temperatures.
233K
soo
,10
296K 200 373K
100
I
I
I
I
1
2
3
4
5
Strain,
It is well known that the steady-state Css = A' exp (-Qc,ss/RT)
(Oas/E) n
creep rate, ~¢ s s ,
can be d e s c r i b e d by: (2)
where A' i s a c o n s t a n t , E i s Y o u n g ' s m o d u l u s , Oss i s t h e s t e a d y - s t a t e s t r e s s , R i s t h e g a s c o n s t a n t , and T i s t h e a b s o l u t e t e m p e r a t u r e . Figure 2 is a plot of the equivalent uniaxial steady-state strain-rate C s s v e r s u s t h e m o d u l u s - c o m p e n s a t e d [18] e q u i v a l e n t u n i a x i a l steady-state stress, oss/E, at the three test temperatures. Both axes have l o g a r i t h m i c
Vol.
22,
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1
FLOW IN Ag
42;
scales. S t e a d y - s t a t e s t r e s s exponent v a l u e s , n , and a c t i v a t i o n e n e r g y f o r s t e a d y - s t a t e d e f o r m a t i o n v a l u e s , q c , s s , a r e e a s i l y d e t e r m i n e d f r o m t h e s e p l o t s . / The s t e a d y - s t a t e s t r e s s e x p o n e n t i s d e f i n e d f r o m Eq. 2 :
n =(8 a In~ss ~ in
(3)
(Oss/EI/i
At 296 K, n i s b e s t d e f i n e d a t a b o u t 32. The n v a l u e a t 233 K i s r o u g h l y 4 0 , and a t 373 K, n was f o u n d t o b e a b o u t 18. T h e s e v a l u e s s e e m s c o n s i s t e n t w i t h t h e t r e n d s f r o m h i g h e r t e m p e r a t u r e work by o t h e r i n v e s t i g a t o r s . Work b e t w e e n a b o u t 0 . 6 5 a n d 0 . 9 Tm i n d i c a t e t h a t n i s c o n s t a n t and a b o u t 5 [ 1 9 ] , c o n s i s t e n t w i t h p o w e r - l a w b e h a v i o r . I n t e n s i l e t e s t s below 0 . 6 5 Tm Goods and N i x [19] and Logan a n d M u k h e r J e e [20] f o u n d t h a t n i n c r e a s e s w i t h i n c r e a s i n g steady-state stress (decreasing temperature), although they did not establish steady-state conditions at temperatures within our testing range. These investigators found that n i n c r e a s e d f r o m 7 t o a b o u t 22 w i t h d e c r e a s i n g t e m p e r a t u r e . This trend is consistent with power-law-breakdown behavior. E a r l i e r work b y one o f t h e a u t h o r s f o u n d t h a t t h e c o n s t a n t structure (p) stress exponent, N = ( I ~ ! D _ ~ , for silver is over 200 [21]. This suggests In o/p,T that the relationship between n and N for silver deformed within the power-law-breakdown regime Is not as straightforward as has been suggested for metals and alloys, at least in the power-law regime [22]. It should be mentioned, that consistent with some low-temperature aluminum work [9], we expect that n should be independent of temperature for a fixed Oss/E. Therefore, the three sets of data in Fig. 2 may be better described by curves instead of lines, so that the slopes at a given ~/E on this plot are the same at the three temperatures.
~8
l(r ~
I
I
I
I
I
/
I
I
I
/=
,~ I@
•/.
. ./
.
~3K
_e o in
~ 10-4
"O
Ul
lO-S
2.0
/
•/" I 2.2
-7.:" I 2.4
I 2.6
I 2.8
I 3.0
I 3.2
I 3.4
FIG. 2. The modulus-compensated equivalent unlaxlal steady-state stress is a flmctlon of steadystate strain rate from torsion tests at three temperatures. Both axes have logarithmic scales.
3.6
~SS -3 ~-xlO
The a c t i v a t i o n
energy for steady-state
deformation,
Qc,ss,
can be c a l c u l a t e d
by
A c t i v a t i o n - e n e r g y v a l u e s were c a l c u l a t e d from t h e d a t a o f F i g . 2 a t t h e a v e r a g e t e m p e r a t u r e s o f 336 K ( 0 . 2 7 Tin) and 264 K ( 0 . 2 1 Tin) t o be 5 7 . 3 and 5 2 . 6 k J / m o l , r e s p e c t i v e l y , o r a p p r o x i m a t e l y 55 k J / m o l a t a m b i e n t t e m p e r a t u r e ( 0 . 2 4 Tm). T h e s e v a l u e s o f t h e a c t i v a t i o n
44
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Vol.
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energy for deformation are significantly lower than the estimated value of 81.3 kJlmol for the activation energy Qp for dlslocation plpe diffusion [23]. Figure 3 is a plot of the activation energy for steady-state deformation as a function of temperature. The graph plots six average activation energy values from this and other studies [19,24-26] calculated by us from Eq. 4 together with the range of temperature over which the values were determined. Above about 0.5 Tm, the activation energy seams to correspond to that of lattice selfdiffusion Qsd [27], as expected . Goods and Nix determined the activation energy for steady-state deformation in silver to be about ii0 kJ/mol at about 0.33 Tm, close to that of dislocation pipe diffusion. However, another interpretation is possible. While the activation energy for steady-state flow above 0.5 T m is constant over a relatively wide range of strain rate and temperature and is approximately equal to Qsd, it appears stressdependent and, therefore, temperature-dependent below 0.5 T m. This may be more supportive of a cross-slip or dlslocatlon-gllde-controlled mechanism. The activation energies for these processes are expected to be stress-dependent, while Qp would be expected to be stressindependent. The drop in the activation energy below roughly 0.5 T m in Fig. 3 suggests, to us, a change in the rate-controlllng mechanism for steady-state deformation. However, Mecking et al. [28] compared extrapolated saturatlon-stresses for silver single crystals (as well as A1, Ni, and Cu) with temperature. They found that the data can be described by a slngle formula over a wide temperature range from the intermediate range up to near the melting point, suggesting the possibility of the absence of a transition.
"~ O _=
200 / /
~="
s
-
-
i
I
/
FIG. 3. The activation energy for steady-state deformation of silver as a function of the fraction of the absolute melting temperature.
!
9I
o-~= 100 /
/
o This study • Price [26] • Schr6der et al. [25] • G o o d s and Nix [19] • M u n s o n [24]
/
//
~a C
> =0
6
0
0.5
1.0
T/Tm
An evaluation of the stress-strain curves of Fig. 1 reveals that there is an abrupt change in the rate at which hardening rate, @ (= d~Id¢), decreases wlth stress. This is shown in Fig. 4. At 373 and 233 K, @ decreases almost linearly with stress, which is characteristic of Stage Ill deformation [5-7,15]. However, at high strains between 0.8 and 1.0, there is a transition, and 8 decreases at a significantly lower rate. This behavior has been described as indicative of a transition to Stage IV deformation [5-7]. Eventually, a steady state is attained, but this stress (Oss) is 1.15 times higher at 373 K and 1.21 times higher at 233 K than a steady-state stress, Olll.ss, based on an extrapolation of Stage III e-stress data to a zero hardening rate. (Others [7] have estimated the lowtemperature steady-state stress from tensile tests, in which a steady state is not achieved, by this extrapolation.) Higher temperatures (and presumably lower strain rates) are associated with lower Stage IV deformation contributions to the total hardening. It may be more meaningful to compare n and Qc,ss values for steady-state deformation based on Olll,ss with the higher-temperature literature values. These would not include Stage IV hardening effects nor the possible effects of dynamic recrystalllzatlon at 373 K. Higher-temperature, constant-straln-rate tests of silver do not seem to be available to confirm that actual steady-state stresses are consistent wlth a linear extrapolation of @ during Stage III deformation. However, 304 stainless steel (another low-stacklng-fault-energy metal) undergoes a substantial primary creep (0.50 strain) at 1023 K under constant-straln-rate conditions, and ~ss and aIII,ss are identical [29]. If we apply Eqs. 3 and 4, then we calculate nii I = 36 and Qc,III = 47 kJ/mol at 296 K, based on extrapolated ~III,ss values at all temperatures and
1
Vol.
22,
No.
1
FLOW
IN A g
45
600
500 O.
'~1~ 400 II ~D
.m 300 O)
FIG. 4. The work-hardenlng rate as a function of stress at 233 and 373 K at a strain rate of 5.77 x 10 -4 s -I. Stage III end IV deformation regimes are indicated.
c
=
200
0
100
0
0
100 200 300 Stress, o(MPa)
400
strain rates. These calculations indicate a possible decreased rate sensitivity and activation energy for steady state, based only on Stage III deformation. It should also be mentioned that our ambient-temperature activation-energy calculation was based on tests at strain rates r o u g h l y two o r d e r s o f m a g n i t u d e h i g h e r t h a n i n t h o s e s t u d i e s r e f e r e n c e d i n F i g . 3 . If Qc,ss is recalculated based on (nonlinear) extrapolation of the data in Fig. 2 to lower strain rates, it would be roughly 5 k3/mol higher. Therefore, t h i s e n d t h e S t a g e IV a d j u s t m e n t a r e r o u g h l y compensatins, and the comparison in Fig. 3 of a 55-k3/mol value to the higher-temperature Qc,ss values is appropriate. Here, the observations that the strain corresponding to the o n s e t o f S t a g e IV d e f o r m a t i o n a n d t h e f r a c t i o n of total hardening contributed d u r i n 8 S t a g e IV appear to decrease with increasing temperature seem consistent w i t h o t h e r S t a g e IV t e m p e r a t u r e trends in nickel end aluminum [5,30]. Conclusions 1. 2.
3.
4.
Steady-state or saturation conditions can be achieved in silver at temperatures between 0 . 1 9 a n d 0 . 3 0 Tm b y t o r s i o n a l deformation. The activation energy for steady-state d e f o r m a t i o n o v e r t h i s r a n g e i s r o u g h l y 55 k 3 / m o l and is significantly lower then that for lattice self-diffusion or dislocation ptpe diffusion. This suggests that although the rate-controlling process for steady-state d e f o r m a t i o n a b o v e , r o u g h l y , 0 . 5 Xm a p p e a r s t o b e a s s o c i a t e d w i t h v o l u m e d i f f u s i o n , other processes such as a cross-slip or dislocation-glide-controlled m e c h a n i s m may operate at lower temperatures. At equivalent uniaxial strains between 0.8 end about 1.0, there is an abrupt change in the rate at which the hardening rate decreases with stress, a n d S t a g e IV h a r d e n i n g a p p e a r s t o c o m m e n c e . T h e s t r a i n c o r r e s p o n d i n g t o t h e o n s e t o f t h i s S t a g e IV a n d t h e f r a c t i o n o f t h e t o t a l h a r d e n i n g a c c u m u l a t e d d u r i n g S t a g e IV d e c r e a s e w i t h i n c r e a s i n g temperature; An a c t i v a t i o n energy ten be calculated based on steady-state stress ~IIIpss values t h a t do n o t i n c l u d e S t a g e IV d e f o r m a t i o n t h a t w e r e d e t e r m i n e d b y l i n e a r e x t r a p o l a t i o n of the hardentn8 rate over Stage III deformation to a zero value. This activation energy is slightly l e s s t h e n t h e m e a s u r e d v a l u e w h i c h i n c l u d e s S t a g e IV d e f o r m a t i o n .
46
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i
Acknowledgments This work was supported by Lawrence Livermore National Laboratory under the auspices of the U. S. Department of Energy, Contract W-7405-Eng-48. The author appreciates the help with the mechanical testing by M. Stratman and L. Allison. References i. 2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17 18 19 20 21 22 23 24 25 26 27 28 29 30.
H. G. Grewe, E. Kappler, H. Pflugmacher, E. Schneider, and W. Welser, Z. Physik 224, 85 (1969). I. Saunders and J. Nutting, Metal Sci. 18, 571 (1984). H. P. StUwe and H. Turck, Z. Metallktmde 55, 699 (1964). G. R. Johnson, J. M. Hoegfeldt, U. S. Lindholme, and A. Nagy, J. Eng. Mater. Tech. 105, 42 (1983). D. A. Hughes, J. C. Glbellng, and W. D. Nix, Strength of Metals and Alloys, H. J. Mcqueen, J.-P. Bailon, J. I. Dickson, J. J. Jonas, and M. G. Akben, eds., p. 51, Pergamon, Oxford (1985). J . Gil S e v i l l a n o , P. van H o u t t e , and E. A e r n o u d t , P r o g . M a t e r . S c i . 25, 69 (1981). H. Mecking and A. G r i n b e r g , S t r e n g t h o f M e t a l s and A l l o y s , P. Haasen, V. G e r o l d , and G. K o s t o r z , e d s . , p . 289, Pergamon, Oxford (1979). O. D. Sherby and J . Weertman, Acta M e t a l l . 27, 387 (1979). H. Luthy, A. K. M i l l e r , and 0. D. S h e r b y , Acta M e t a l l . 28, 169 ( 1 9 8 0 ) . H. E. Evans and G. Knowles, A c t a M e t a l l . 25, 963 (1977). W. D. Nix and B. I l s c h n e r , S t r e n g t h o f M e t a l s and A l l o y s , P. Haasen, V. G e r o l d , and G. K o s t o r z , e d s . , p. 1503, Pergamon, Oxford (1979). A. Arlell and A. K. Mukherjee, Creep and Fracture of Engineering Materials and Structures, B. Wilshlre and D. R. J. 0wen, eds., p. 77, Pineridge, Swansea (1981). D. Caillard and J. L. Martin, Rev. Phys. Appl. 22, 169 (1987). J. P. Porrler, Rev. Phys. Appl. ii, 731 (1976). H. Mecking, Work Hardening in Tension and Fatigue, A. W. Thompson, ed., p. 67, AIME, New York (1977). D. S. Fields and W. A. Backofen, Proc. ASTM 57, 1259 (1957). J. W. H. G. Slakhorst, Acta Metall. 23, 301 (1975). W. KSster, Z. Metallkunde 39, 1 (1948). S. H. Goods and W. D. Nix, Acta Metall. 26, 753 (1978). R. W. Logan and A. K. MukherJee, Scripta Metall. 17, 63 (1983). M. E. Kassner and A. K. MukherJee, Scripta Metall. 17, 741 (1983). O. D. Sherby, R. H. Klundt, and A. K. Miller, Metall. Trans. 8A, 843 (1977). D. Turnbull and R. Hoffman, Acta Metall. 2, 419 (1954). D. E. Munson, Ph.D. Dissertation, Stanford Univ., Stanford, CA (1960). K. Schroder, A. S. Grannuzzi, and G. Gorska, Acta Metall. 16, 469 (1968). C. E. Price, Acta Metall. 14, 1781 (1966). R. E. Hoffman and D. Turnbull, J. AppI. Phys. 22, 634 (1951). H. Mecklng, B. Nicklas, N. Zarubova, and U. F. Kooks, Acta Metall. 34, 527 (1986). M. E. Kassner, A. K. Miller, and A. A. Ziaai-Moayyed, Metall. Trans. 16A, 1069 (1985). W. Roberts, Deformation, Processing, and Structure, G. Krauss ed., p. 109, American Society for Metals (1984).