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Creep behavior of AuNi solid solutions
Scripta
METALLURGICA
V o l . II, pp. 8 7 9 - 8 8 2 , Printed in t h e U n i t e d
1977 States
Pergamon
Press
CREEP BEHAVIOR OF Au-Ni SOLID SOLUTIONS
Farghalli A. Mohamed and Young Kyou Kim Departments of Materials Science and Mechanical Engineering University of Southern California, Los Angeles, California 90007 (n.eceived A u g u s t 8, 1 9 7 7 )
Introduction Recently, considerable attention has been focused on Creep behavior of Solid Solution alloys (i-ii). This creep behavior has been divided into two classes (I, 9): a pure metal class (stress exponent = 5) and an alloy class (stress exponent = 3). Bird et al. (2) predicted a transition in the creep behavior of a solid solution alloy, from that of the metal class to that of the alloy class, with increase of stress. This transition has been verified experimentally in AI-3% Mg (4, 5). By using the experimental transition stress for AI-3% Mg (4, 5) and assuming that the creep behavior of a solid solution alloy is governed by the sequential process of dislocation glide and climb (2-5), it has been shown that the change from the metal class to the alloy class can be accounted for by a criterion of this form (9) (~)3~ ~ _ D~ 2c = g
k2T 2 e2cb 6
(i)
where y is the stacking fault energy, G D is the diffusion coefficient for the f~r the glide process, o is the applied stant, T is the absolute temperature, e the concentration of solute atoms. The appropriate
is the shear modulus, b is the Burgers vector, climb process, D is the diffusion coefficient stress, B is a.c~nstant, k is Boltzmann's conis the solute-solvent size difference, and c is
diffusion coefficient
for climb-controlled
creep is given
(12,13) by
DAD B Dc
=
.
.
(2)
(XAD B + XBDA)f where D A and D B are the tracer diffusivities of the A and B atoms in the AB alloy, X and ~ are the atomic fractions, and f is the correlation factor. On the other hand~ the diffusivity for glide-controlled creep is the chemical interdiffusivity (14), de~ fined as *
D
~In
.
= (XAD B + XBD A)
(i +
g where ~A is the activity coefficient
~A) ~inX A
(3)
for the A species.
An examination of the validity of Eq. i, under the assumption of Dc/Dg = i, showed that the creep behavior of most of the alloys was consistent with its prediction (9). In addition, an analysis of the experimental data of the Ni-W system (i0) indicated that the variation in the value of the stress exponent, n, was in good agreement with Eq. 1 when the appropriate values of the diffusivities were incorporated, i.e., Dc/Dg ~ i. Despite this excellent agreement between prediction and experiment, the criterion seems to break down for Au-10% Ni which apparently exhibits glide behavior although the analysis suggests climb-controlled behavior. The reason for this discrepancy is not known, but it was speculated that an underestimation of the value of the stacking fault energy might be a source of error. For this alloy, 7 was estimated as 70 erg/cm 2 by assuming
879
Inc
880
CREEP
BEHAVIOR
OF
Au-Ni
SOLID
SOLUTIONS
Vol.
II,
No.
a monotonic variation (15) from YAu (50 erg/cm 2) to YNi (250 erg/cm2), while a much higher value, close to that of Ni, was found necessary to put the whole stress range in the glide regime. In this paper, creep data of the Au-Ni system are re-examined and another possibility is considered to explain the discrepancy between experiment and prediction. Analysis and Discussion As was suggested elsewhere (9, i0), the viscous glide-climb criterion may be expressed graphically by plotting, on a logarithmic scale, T2/e2cb 6 versus (y/Gb) 3(Dc/Dg)a 2. Since there is general interest in normalizing the creep behavior of metallic systems (2, 16, 17),it appears more appropriate if these two parameters are normalized and expressed as (y/Gb)3(Dc/Dg)(~/G)2 and (kT/ecl/2 b3G )2. Fig. 1 shows the plot of the criterion using the two dimensionless parameters. The line with slope of 1 represents the boundary between climb behavior on the left and glide behavior on the right. The position of this boundary, as discussed elsewhere (9), was determined from experimental data obtained in AI-3% Mg at different temperatures (4). The broken line in Fig. 1 defines a boundary obtained from data on the same alloy (5) deformed at a single temperature of 855°K. It is clear that the difference in position between the two boundaries is within the experimental error involved in such experiments. The attempt to explain the discrepancy between the criterion and creep behavior of Au-10% Ni involves three steps: a) re-examining the creep behavior of the Au-Ni system, (b) identifying another possibility to account for the discrepancy, and (c) checking the validity of this possibility. (a) Re-examining
the Creep Data on the Au-Ni System
The creep data for Au-10% Ni, Au-25% Ni, and Ni-13% Au (taken from Fig. 7 of Ref. 18) were logarthmically replotted as ~ versus a in Fig. 2. For Ni-13% Au, experimental data fall on a straight line of slope - 3, in agreement with the glide-controlled creep behavior. On the other hand, the experimental results for Au-10% Ni and Au-25% Ni suggest a transition from slope of 5 at low stress levels to a slope of 3 at high stresses; if a single straight line is drawn through the experimental points of each alloy, an intermediate value of n, between 3 and 5, will result. A detailed analysis for data of Pb alloys (9, 19) indicated that an intermediate value of n was associated with a transition from climb to glide control. The data for these three alloys were plotted in Fig. tions: (i) Dc/Dg = 1 (ii) y varies monotonically from YAu to YNi
1 under the following assump-
(iii) G = XAuG U + ~- -~ ~ G ~ c h The length of the ~ine alloy defines the total stress range and the circles represent the points of transition from n = 5 to n = 3 taken from Fig. 2. It is apparent that, with the exception of Ni-13% Au, the agreement between experiment and prediction is rather poor. As was suggested previously (9), it is possible to account for this discrepancy by taking values of y higher than those estimated from the monotonic variation procedure (see Table I). In this case, the average values of y required to bring the transition points (marked as circles) on to the glide-climb boundary are 190 and 230 erg/cm 2 for Au-10% Ni and Au-25% Ni, respectively. Unfortunately, there are no data reported for y in the Au-Ni system to check the validity of this possibility. (b) Identifying Another Possibility
to Explain the Discrepancy
Another possible source of error, which could result in this poor correlation between the experimental results and criterion, is the incorrect assumption concerning the value of Dc/Dg. This possibility was considered previously in the Ni-W system (10), and it was shown that better correlation between creep behavior and Eq. 1 resulted when the available values of Dc/Dg were incorporated into the analysis. If it is assumed that y for the Au-Ni system is determined approximately from a monotonic variation (Table I), the values of Dc/Dg required to bring the transition points on to the boundary are 18 and i0 for Au-10% Ni and Au-25% Ni, respectively.
I0
Vol.
ii, No.
I0
CREEP B E H A V I O R OF Au-Ni
SOLID SOLUTIONS
, _ _ / /"I CLIMB
~
i/
-
/
////.~////
1.0
o
AU * 10% Ni
///~ /
t
r/"
////
A U - 25% Ni
I
i
I
. . . . . . . .
10"15
10"14
GLIDE
Ni - 13% AU
i
10-13
10"12
10-11
3
½
3 2
(kT/ec Gb ) vs. (y/Gb)3 D / D • c g are shown for the A u - N I alloys,
I
1
FIG. 1 (O/G) 2 o n a l o g a r i t h m i c scale. for D c / D g = i.
"7--
I Ni - 13%Au T: 1133K
- 25% Ni T= 1133K
AU -10~ Ni T : 1133K
AU
/
[
I
Data
,
1
,
,,
,,,,I
10
10
It
5
10
~,MNlrn 2
FIG. 2 C r e e p rate vs. stress for three A u - N i alloys.
(c) C h e c k i n g the V a l i d i t y of D c / D g > 1 for A u - N i A l l o y s The values of c h e m i c a l i n t e r d i f f u s i v i t i e s in A u - N i w e r e e x p e r i m e n t a l l y m e a s u r e d (20); D g at T = 1133 (creep t e s t i n g temperature) for A u - 1 0 % Ni and A u - 2 5 % Ni can be o b t a i n e d by i n t e r p o l a t i o n o f e x p e r i m e n t a l m e a s u r e m e n t s m a d e at I 1 2 3 ° K and I148°K. These values are also listed in T a b l e I. By u s i n g £he r e q u i r e d v a l u e s of D c / D g and the e s t i m a t e d v a l u e s o f Dg, the c l i m b - d i f f u s i v i t i e s for A u - 1 0 % Ni and A u - 2 5 % Ni w e r e c a l c u l a t e d as 6.4 x 10 -9 and 3.2 x 10 -9 cm2/sec, respectively. The lines for c l i m b - c o n t r o l l e d b e h a v i o r (n = 5) in Fig. 2 can then be e x t r a p o l a t e d to give ~ at s t r e s s e s of 4.4 and 5.4 M N / m 2 for the two alloys. T h e s e two v a l u e s o f stress w e r e s e l e c t e d to give o/G = 2 x 10 -4 . The e x t r a p o l a t e d values of ~ and the i n f e r r e d values o f D c w e r e c o m b i n e d to e s t i m a t e the n o r m a l i z e d c r e e p rates eKT/DcGb. Finally, the values of (7/Gb) w e r e d e t e r m i n e d from Fig. 1 of Ref. 9 w h i c h gives (y/Gb) as a f u n c t i o n o f ~ K T / D c G b at ~/G = 2 x 10 -4"
881
882
CREEP BEHAVIOR OF Au-Ni SOLID SOLUTIONS
Vol.
II, No. 1O
TABLE 1 Creep Data, Chemical Interdiffusivities, and Stacking Fault Energies for Three Alloys of Au-Ni System Creep Data y (erg/cm 2) Dg Monotonic Variation
Creep Data
Alloy
T(OK)
Au-10% Ni
1133
0.97-8.6
3.6 x i0 -I0
70
62
IAu-25% Ni
1133
0.97-13.8
3.2 x i0 -I0
i00
95
Ni-13% Ni
1133
8.6 - 28.4
1.5 x i0 -II
224
--
(cm2/sec)
(MN/m 2)
i
As can be seen from Table I, the values of y estimated from this procedure (creep data pnocedure) agree reasonably well with the assumed values Of y (monotonic variation procedure). This provides favorable support for the possibility that D_/D may be responsible for the discrepancy between creep behavior of gold-nickel alloys andUth~ viscous glideclimb criterion. Conclusions I.
Two possibilities can explain the marked disagreement between the glide-climb criterion and the creep behavior of Au-10% Ni and Au-25% Ni: an underestimation of y and/or an incorrect assumption of Dc/Dg = i.
2.
It has been shown the possibility of Dc/Dg > 1 yields values of y which agree reasonably well with those estimated from the monotonic variation procedure. Acknowledgements
The authors wish to thank Professors S. M. Copley and M. Gerstein for the departmental facilities made available. Thanks are also due to Dr. H. Oikawa of the Tohoku University for some discussions, and Mr. R. Lee for typing the manuscript.
References
I, 2, 3. 4, 5, 6. 7. 8. 9. lO. II. 12. 13. 14. 15. 16. 17 18. 19. 20.
O.D, Sherby and P.M. Burke, Prog, Mat, Sci, 13, 325 (1968). J.E. Bird, A.K. Mukherjee and J,E. Dorn, Quantitative Relation Between Properties and Microstructure, edited by D.G, Brandon and A.Rosen, p. 255, Isreal University Press (1969). W.R. Cannon and O.D. Sherby, Met. Trans. 5, I179 (1970). K.L. Murty, F,A. Mohamed and J.E. Dorn, Acta Met. 20, 465 (1972). F,A. Mohamed,Ph.D, University of California, BerkeT6y (1972)~~ W.R. Johnson, C.R. Barrett and W.D. Nix, Met. Trans. 3, 963 (1972). H. Oikawa an_d S. Karashima, Met. Trans. 5, I179 (1974T. K. Kurachova, I. Saxl and J. Cadek, Acta Met. 22, 465 (1974). F.A. Mohamed and T.G. Langdon, Acta Met. 22, 7 ~ (1974). F.A. Mohamed and T.G. Langdon, Met. Trans. 6A, 927 (1975). S. Takeuchi and A.S. Argon, Acta Met. 24, 883 (1976). C. Herring, J. Appl. Phys. 21_i, 437 (19~). J. Weertman, Trans. ASM. 61, 681 (1968). L.S. Darken, Trans. AIME,-T75, 184 (1948). P.C.J. Gallagher, Met. Trans~ 1, 2429 (1970). A.K. Mukherjee, JoE. Bird and J.E. Dorn, The Interaction Between Dislocations and Point Defects, Harwell, p. 431 (1968). F.A. Mohamed, K.L. Murty and J.W. Morris, Trans. AIME. 4, 935 (1973). C.M. Sellars and A.G. Quarrell, J. Inst. Metals. 90, 329 (1961/1962). J. Weertman, Trans. Am. Inst. Min. Engrs. 218, 203F-(1960). J.E. Reynolds, B.L. Averbach and M. Cohen, Acta Met. 5, 29 (1957).
METALLURGICA
V o l . II, pp. 8 7 9 - 8 8 2 , Printed in t h e U n i t e d
1977 States
Pergamon
Press
CREEP BEHAVIOR OF Au-Ni SOLID SOLUTIONS
Farghalli A. Mohamed and Young Kyou Kim Departments of Materials Science and Mechanical Engineering University of Southern California, Los Angeles, California 90007 (n.eceived A u g u s t 8, 1 9 7 7 )
Introduction Recently, considerable attention has been focused on Creep behavior of Solid Solution alloys (i-ii). This creep behavior has been divided into two classes (I, 9): a pure metal class (stress exponent = 5) and an alloy class (stress exponent = 3). Bird et al. (2) predicted a transition in the creep behavior of a solid solution alloy, from that of the metal class to that of the alloy class, with increase of stress. This transition has been verified experimentally in AI-3% Mg (4, 5). By using the experimental transition stress for AI-3% Mg (4, 5) and assuming that the creep behavior of a solid solution alloy is governed by the sequential process of dislocation glide and climb (2-5), it has been shown that the change from the metal class to the alloy class can be accounted for by a criterion of this form (9) (~)3~ ~ _ D~ 2c = g
k2T 2 e2cb 6
(i)
where y is the stacking fault energy, G D is the diffusion coefficient for the f~r the glide process, o is the applied stant, T is the absolute temperature, e the concentration of solute atoms. The appropriate
is the shear modulus, b is the Burgers vector, climb process, D is the diffusion coefficient stress, B is a.c~nstant, k is Boltzmann's conis the solute-solvent size difference, and c is
diffusion coefficient
for climb-controlled
creep is given
(12,13) by
DAD B Dc
=
.
.
(2)
(XAD B + XBDA)f where D A and D B are the tracer diffusivities of the A and B atoms in the AB alloy, X and ~ are the atomic fractions, and f is the correlation factor. On the other hand~ the diffusivity for glide-controlled creep is the chemical interdiffusivity (14), de~ fined as *
D
~In
.
= (XAD B + XBD A)
(i +
g where ~A is the activity coefficient
~A) ~inX A
(3)
for the A species.
An examination of the validity of Eq. i, under the assumption of Dc/Dg = i, showed that the creep behavior of most of the alloys was consistent with its prediction (9). In addition, an analysis of the experimental data of the Ni-W system (i0) indicated that the variation in the value of the stress exponent, n, was in good agreement with Eq. 1 when the appropriate values of the diffusivities were incorporated, i.e., Dc/Dg ~ i. Despite this excellent agreement between prediction and experiment, the criterion seems to break down for Au-10% Ni which apparently exhibits glide behavior although the analysis suggests climb-controlled behavior. The reason for this discrepancy is not known, but it was speculated that an underestimation of the value of the stacking fault energy might be a source of error. For this alloy, 7 was estimated as 70 erg/cm 2 by assuming
879
Inc
880
CREEP
BEHAVIOR
OF
Au-Ni
SOLID
SOLUTIONS
Vol.
II,
No.
a monotonic variation (15) from YAu (50 erg/cm 2) to YNi (250 erg/cm2), while a much higher value, close to that of Ni, was found necessary to put the whole stress range in the glide regime. In this paper, creep data of the Au-Ni system are re-examined and another possibility is considered to explain the discrepancy between experiment and prediction. Analysis and Discussion As was suggested elsewhere (9, i0), the viscous glide-climb criterion may be expressed graphically by plotting, on a logarithmic scale, T2/e2cb 6 versus (y/Gb) 3(Dc/Dg)a 2. Since there is general interest in normalizing the creep behavior of metallic systems (2, 16, 17),it appears more appropriate if these two parameters are normalized and expressed as (y/Gb)3(Dc/Dg)(~/G)2 and (kT/ecl/2 b3G )2. Fig. 1 shows the plot of the criterion using the two dimensionless parameters. The line with slope of 1 represents the boundary between climb behavior on the left and glide behavior on the right. The position of this boundary, as discussed elsewhere (9), was determined from experimental data obtained in AI-3% Mg at different temperatures (4). The broken line in Fig. 1 defines a boundary obtained from data on the same alloy (5) deformed at a single temperature of 855°K. It is clear that the difference in position between the two boundaries is within the experimental error involved in such experiments. The attempt to explain the discrepancy between the criterion and creep behavior of Au-10% Ni involves three steps: a) re-examining the creep behavior of the Au-Ni system, (b) identifying another possibility to account for the discrepancy, and (c) checking the validity of this possibility. (a) Re-examining
the Creep Data on the Au-Ni System
The creep data for Au-10% Ni, Au-25% Ni, and Ni-13% Au (taken from Fig. 7 of Ref. 18) were logarthmically replotted as ~ versus a in Fig. 2. For Ni-13% Au, experimental data fall on a straight line of slope - 3, in agreement with the glide-controlled creep behavior. On the other hand, the experimental results for Au-10% Ni and Au-25% Ni suggest a transition from slope of 5 at low stress levels to a slope of 3 at high stresses; if a single straight line is drawn through the experimental points of each alloy, an intermediate value of n, between 3 and 5, will result. A detailed analysis for data of Pb alloys (9, 19) indicated that an intermediate value of n was associated with a transition from climb to glide control. The data for these three alloys were plotted in Fig. tions: (i) Dc/Dg = 1 (ii) y varies monotonically from YAu to YNi
1 under the following assump-
(iii) G = XAuG U + ~- -~ ~ G ~ c h The length of the ~ine alloy defines the total stress range and the circles represent the points of transition from n = 5 to n = 3 taken from Fig. 2. It is apparent that, with the exception of Ni-13% Au, the agreement between experiment and prediction is rather poor. As was suggested previously (9), it is possible to account for this discrepancy by taking values of y higher than those estimated from the monotonic variation procedure (see Table I). In this case, the average values of y required to bring the transition points (marked as circles) on to the glide-climb boundary are 190 and 230 erg/cm 2 for Au-10% Ni and Au-25% Ni, respectively. Unfortunately, there are no data reported for y in the Au-Ni system to check the validity of this possibility. (b) Identifying Another Possibility
to Explain the Discrepancy
Another possible source of error, which could result in this poor correlation between the experimental results and criterion, is the incorrect assumption concerning the value of Dc/Dg. This possibility was considered previously in the Ni-W system (10), and it was shown that better correlation between creep behavior and Eq. 1 resulted when the available values of Dc/Dg were incorporated into the analysis. If it is assumed that y for the Au-Ni system is determined approximately from a monotonic variation (Table I), the values of Dc/Dg required to bring the transition points on to the boundary are 18 and i0 for Au-10% Ni and Au-25% Ni, respectively.
I0
Vol.
ii, No.
I0
CREEP B E H A V I O R OF Au-Ni
SOLID SOLUTIONS
, _ _ / /"I CLIMB
~
i/
-
/
////.~////
1.0
o
AU * 10% Ni
///~ /
t
r/"
////
A U - 25% Ni
I
i
I
. . . . . . . .
10"15
10"14
GLIDE
Ni - 13% AU
i
10-13
10"12
10-11
3
½
3 2
(kT/ec Gb ) vs. (y/Gb)3 D / D • c g are shown for the A u - N I alloys,
I
1
FIG. 1 (O/G) 2 o n a l o g a r i t h m i c scale. for D c / D g = i.
"7--
I Ni - 13%Au T: 1133K
- 25% Ni T= 1133K
AU -10~ Ni T : 1133K
AU
/
[
I
Data
,
1
,
,,
,,,,I
10
10
It
5
10
~,MNlrn 2
FIG. 2 C r e e p rate vs. stress for three A u - N i alloys.
(c) C h e c k i n g the V a l i d i t y of D c / D g > 1 for A u - N i A l l o y s The values of c h e m i c a l i n t e r d i f f u s i v i t i e s in A u - N i w e r e e x p e r i m e n t a l l y m e a s u r e d (20); D g at T = 1133 (creep t e s t i n g temperature) for A u - 1 0 % Ni and A u - 2 5 % Ni can be o b t a i n e d by i n t e r p o l a t i o n o f e x p e r i m e n t a l m e a s u r e m e n t s m a d e at I 1 2 3 ° K and I148°K. These values are also listed in T a b l e I. By u s i n g £he r e q u i r e d v a l u e s of D c / D g and the e s t i m a t e d v a l u e s o f Dg, the c l i m b - d i f f u s i v i t i e s for A u - 1 0 % Ni and A u - 2 5 % Ni w e r e c a l c u l a t e d as 6.4 x 10 -9 and 3.2 x 10 -9 cm2/sec, respectively. The lines for c l i m b - c o n t r o l l e d b e h a v i o r (n = 5) in Fig. 2 can then be e x t r a p o l a t e d to give ~ at s t r e s s e s of 4.4 and 5.4 M N / m 2 for the two alloys. T h e s e two v a l u e s o f stress w e r e s e l e c t e d to give o/G = 2 x 10 -4 . The e x t r a p o l a t e d values of ~ and the i n f e r r e d values o f D c w e r e c o m b i n e d to e s t i m a t e the n o r m a l i z e d c r e e p rates eKT/DcGb. Finally, the values of (7/Gb) w e r e d e t e r m i n e d from Fig. 1 of Ref. 9 w h i c h gives (y/Gb) as a f u n c t i o n o f ~ K T / D c G b at ~/G = 2 x 10 -4"
881
882
CREEP BEHAVIOR OF Au-Ni SOLID SOLUTIONS
Vol.
II, No. 1O
TABLE 1 Creep Data, Chemical Interdiffusivities, and Stacking Fault Energies for Three Alloys of Au-Ni System Creep Data y (erg/cm 2) Dg Monotonic Variation
Creep Data
Alloy
T(OK)
Au-10% Ni
1133
0.97-8.6
3.6 x i0 -I0
70
62
IAu-25% Ni
1133
0.97-13.8
3.2 x i0 -I0
i00
95
Ni-13% Ni
1133
8.6 - 28.4
1.5 x i0 -II
224
--
(cm2/sec)
(MN/m 2)
i
As can be seen from Table I, the values of y estimated from this procedure (creep data pnocedure) agree reasonably well with the assumed values Of y (monotonic variation procedure). This provides favorable support for the possibility that D_/D may be responsible for the discrepancy between creep behavior of gold-nickel alloys andUth~ viscous glideclimb criterion. Conclusions I.
Two possibilities can explain the marked disagreement between the glide-climb criterion and the creep behavior of Au-10% Ni and Au-25% Ni: an underestimation of y and/or an incorrect assumption of Dc/Dg = i.
2.
It has been shown the possibility of Dc/Dg > 1 yields values of y which agree reasonably well with those estimated from the monotonic variation procedure. Acknowledgements
The authors wish to thank Professors S. M. Copley and M. Gerstein for the departmental facilities made available. Thanks are also due to Dr. H. Oikawa of the Tohoku University for some discussions, and Mr. R. Lee for typing the manuscript.
References
I, 2, 3. 4, 5, 6. 7. 8. 9. lO. II. 12. 13. 14. 15. 16. 17 18. 19. 20.
O.D, Sherby and P.M. Burke, Prog, Mat, Sci, 13, 325 (1968). J.E. Bird, A.K. Mukherjee and J,E. Dorn, Quantitative Relation Between Properties and Microstructure, edited by D.G, Brandon and A.Rosen, p. 255, Isreal University Press (1969). W.R. Cannon and O.D. Sherby, Met. Trans. 5, I179 (1970). K.L. Murty, F,A. Mohamed and J.E. Dorn, Acta Met. 20, 465 (1972). F,A. Mohamed,Ph.D, University of California, BerkeT6y (1972)~~ W.R. Johnson, C.R. Barrett and W.D. Nix, Met. Trans. 3, 963 (1972). H. Oikawa an_d S. Karashima, Met. Trans. 5, I179 (1974T. K. Kurachova, I. Saxl and J. Cadek, Acta Met. 22, 465 (1974). F.A. Mohamed and T.G. Langdon, Acta Met. 22, 7 ~ (1974). F.A. Mohamed and T.G. Langdon, Met. Trans. 6A, 927 (1975). S. Takeuchi and A.S. Argon, Acta Met. 24, 883 (1976). C. Herring, J. Appl. Phys. 21_i, 437 (19~). J. Weertman, Trans. ASM. 61, 681 (1968). L.S. Darken, Trans. AIME,-T75, 184 (1948). P.C.J. Gallagher, Met. Trans~ 1, 2429 (1970). A.K. Mukherjee, JoE. Bird and J.E. Dorn, The Interaction Between Dislocations and Point Defects, Harwell, p. 431 (1968). F.A. Mohamed, K.L. Murty and J.W. Morris, Trans. AIME. 4, 935 (1973). C.M. Sellars and A.G. Quarrell, J. Inst. Metals. 90, 329 (1961/1962). J. Weertman, Trans. Am. Inst. Min. Engrs. 218, 203F-(1960). J.E. Reynolds, B.L. Averbach and M. Cohen, Acta Met. 5, 29 (1957).