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Bimodal stress relaxation
Scripta METALLURGICA
Vol. 21, pp. 537-540, 1987 Printed in the U.S.A.
Pergamon Journals, Ltd All rights reserved
BIMODAL STRESSRELAXATION J. M. Galligan Department of Metallurgy and I n s t i t u t e of Materials Science University of Connecticut Storrs, CT 06268
(Received January 14, 1987) The p l a s t i c i t y of a metal results from dislocations moving through a crystal; this rate of motion is determined by barriers in the crystal, and the drag between barriers. At low temperatures, where dislocation motion is underdamped, i n e r t i a plays an important role in overcoming barriers (1,2). In contrast to this situation, at high temperatures where dislocation motion is overdamped, dislocations are held up at barriers and this determines the rate of plastic deformation of a crystal. A measure of this high temperature behavior is obtained in the stress relaxation behavior of many materials (3), where a single relaxation mode is observed. At low temperatures, however, the stress relaxation, as shown below, shows two distinct modes. Further, these modes are affected by the state of the electrons, a clear manifestation of drag processes (1,2). These observations also demonstrate an interaction between thermal processes and dislocation i n e r t i a . The experiment is carried out as follows: single crystals of lead are deformed at a temperature, T, well below where phonon damping is of importance (4). After a crystal is p l a s t i c a l l y deformed for a fixed time the tensile machine is stopped and the stress relaxation recorded; Fig. l i l l u s t r a t e s the procedure. Lead, which has a superconducting transition at ~7.18K, can be deformed in the superconducting state or in the normal state. This allows the drag on moving dislocations to be independently varied (5). In the superconducting state the electron damping is considerably less than in the normal state (5), so that below 7.18K dislocation i n e r t i a would be expected to be more important than at temperatures above 7.18K. The stress relaxation data for lead clearly shows t h i s : Fig. 2 is a plot of the log of the stress versus time for a lead crystal. These data show that for crystals deformed wholly in the superconducting state, at 6K, there are two easily distinguishable stress relaxation rates. Likewise for a crystal deformed at a temperature above 7.18K, so as to be wholly in the normal state, there are, again, two d i s t i n c t relaxation modes, Fig. 2. Additional stress relaxation data for a d i l u t e lead-tin crystal are shown in Fig. 3. These data confirm that the stress relaxation behavior is bimodal at low temperatures. A comparison of the data, given in Fig. 2., shows that the state of the electrons as well as the temperature of deformation has a profound effect on the relaxation behavior of the material. Consider, f i r s t , the differences in the slope for lead at short times, i . e . in the case of the crystal deformed at 6K, for times less than 25 seconds Fig. 2. This can be compared to the relaxation behavior for the crystal deformed at 9.5K, for times less than 15 seconds Fig. 2. This comparison shows that in this i n i t i a l stage the relaxation in rate is much faster for the crystal in the superconducting state, than in the normal state, despite the fact that the temperature is s i g n i f i c a n t l y lower in the case of the superconductor than i t is in the normal state. The more l i k e l y determinant of the stress relaxation rate at low temperatures is the difference in drag between the normal state and the superconducting state, suggesting that the f i r s t relaxation mode is mostlY related to drag processes. The second relaxation mode, Fig. 2. for times greater than 25 seconds at 6k or times greater than 15 seconds at 9.5k, is also d i f f i c u l t to reconcile with a simple barrier model. The second slope in the higher temperature deformation experiment is less than the slope at lower temperature, neglecting the question of the sign of the slope. Such behavior, however,
537 0036-9748/87 $3.00 + .00 Copyright (c) 1987 Pergamon Journals Ltd.
558
STRESS RELAXATION
Vol.
21,
No.
4
can be rationalized by recognizing that i f the drag between barriers is less, then the amount a dislocation can overshoot a barrier is greater. I f a dislocation overshoots a barrier, this w i l l exert an extra force on the mobile dislocation. As a result of this extra force a disloca. tion may mechanically overshoot the barrier, with less thermal assistance. This accounts for the slight dependence of the observed relaxation rates on temperature. These considerations suggest that i n e r t i a , at these low temperatures, is playing a role in the thermal surmounting of barriers, such as suggested by the calculations of Isaac and Granato (6). A more complete version of the present experiment is now underway, including the temperature dependence, solute concentration dependence and magnetic (7) f i e l d dependence. These results w i l l be reported elsewhere. Acknowledgements I t is a pleasure to thank a number of former students and colleagues, including J. Pellegrino, M. Suenaga and R. E. Galligan. This research has been supported by the National Science Foundation, Divison of Materials Science under contract # DMR 8603436. I t is also a pleasure to thank the A. von Humboldt Stiftung and Professor Peter Haasen. References I. 2. 3. 4. 5.
M. Suenaga and J. M. Galligan, Scripta Met. 5, 829 (1971). A.V. Granato, Phys. Rev. B4, 2196 (1971). F.W. Noble and D. Hull, Acta Met. 12, I089 (1964). J. Pellegrino, B. Zhou and J. M. Gall---igan, Scripta Met. 16, 985 (1982). C.S. Pan~, T. H. Lin and J. M. Galligan, Jnl. of Applied Physics, 41, 333 (1978) R.D. Isaac and A. V. Granato, Proc. 5th I n t ' l . Conference on the Strength of Metals and Alloys, Edited by Peter Haasen e t . a l . Pergamon Press, Oxford, p425 (1979). J. M. Galligan, T. H. Lin and C. S. Pang, Phys, Rev. Letters. 38 405 (1977).
6. 7.
Tensile Machine ed
(/) (/)
(n
¢D ,._
(n (]3
03
03
,0--
Stroin Fig. 1
Time
In the l e f t hand diagram above a schematic of the stress-strain curve for a lead crystal at low temperature is shown. In the right hand diagram a schematic of the stress-relaxatlon curve is illustrated.
Vol.
21, No. 4
STRESS
RELAXATION
539
e9.5K • 6.OK
5
2 C
V
I¢1 L_
.2
I
I
I
I
A
I
10
15
20
25
30
35
time (sec.) Fig. 2
The stress decrement, plotted on a logarithmic scale, as a function of time. Note that the intercept on the ordinate for the data from the two temperatures has been displaced for c l a r i t y , The applied stress is roughly the same for both measurements.
540
STRESS RELAXATION
Vol.
4 Lead-Tin crystal T= 42K Superconducting State
3
1
i 2
=
l
o 01
w o
,2 l
I 15
I
30
Time (sec) Fig.
3.
S t r e s s R ~ l a x a t i o n data f o r a l p a d - t i n c r y s t a l , p l o t t ~ d on a l o g a r i t h m i c s c a l e in s t r e s s decrement versus t i m e .
21, No. 4
Vol. 21, pp. 537-540, 1987 Printed in the U.S.A.
Pergamon Journals, Ltd All rights reserved
BIMODAL STRESSRELAXATION J. M. Galligan Department of Metallurgy and I n s t i t u t e of Materials Science University of Connecticut Storrs, CT 06268
(Received January 14, 1987) The p l a s t i c i t y of a metal results from dislocations moving through a crystal; this rate of motion is determined by barriers in the crystal, and the drag between barriers. At low temperatures, where dislocation motion is underdamped, i n e r t i a plays an important role in overcoming barriers (1,2). In contrast to this situation, at high temperatures where dislocation motion is overdamped, dislocations are held up at barriers and this determines the rate of plastic deformation of a crystal. A measure of this high temperature behavior is obtained in the stress relaxation behavior of many materials (3), where a single relaxation mode is observed. At low temperatures, however, the stress relaxation, as shown below, shows two distinct modes. Further, these modes are affected by the state of the electrons, a clear manifestation of drag processes (1,2). These observations also demonstrate an interaction between thermal processes and dislocation i n e r t i a . The experiment is carried out as follows: single crystals of lead are deformed at a temperature, T, well below where phonon damping is of importance (4). After a crystal is p l a s t i c a l l y deformed for a fixed time the tensile machine is stopped and the stress relaxation recorded; Fig. l i l l u s t r a t e s the procedure. Lead, which has a superconducting transition at ~7.18K, can be deformed in the superconducting state or in the normal state. This allows the drag on moving dislocations to be independently varied (5). In the superconducting state the electron damping is considerably less than in the normal state (5), so that below 7.18K dislocation i n e r t i a would be expected to be more important than at temperatures above 7.18K. The stress relaxation data for lead clearly shows t h i s : Fig. 2 is a plot of the log of the stress versus time for a lead crystal. These data show that for crystals deformed wholly in the superconducting state, at 6K, there are two easily distinguishable stress relaxation rates. Likewise for a crystal deformed at a temperature above 7.18K, so as to be wholly in the normal state, there are, again, two d i s t i n c t relaxation modes, Fig. 2. Additional stress relaxation data for a d i l u t e lead-tin crystal are shown in Fig. 3. These data confirm that the stress relaxation behavior is bimodal at low temperatures. A comparison of the data, given in Fig. 2., shows that the state of the electrons as well as the temperature of deformation has a profound effect on the relaxation behavior of the material. Consider, f i r s t , the differences in the slope for lead at short times, i . e . in the case of the crystal deformed at 6K, for times less than 25 seconds Fig. 2. This can be compared to the relaxation behavior for the crystal deformed at 9.5K, for times less than 15 seconds Fig. 2. This comparison shows that in this i n i t i a l stage the relaxation in rate is much faster for the crystal in the superconducting state, than in the normal state, despite the fact that the temperature is s i g n i f i c a n t l y lower in the case of the superconductor than i t is in the normal state. The more l i k e l y determinant of the stress relaxation rate at low temperatures is the difference in drag between the normal state and the superconducting state, suggesting that the f i r s t relaxation mode is mostlY related to drag processes. The second relaxation mode, Fig. 2. for times greater than 25 seconds at 6k or times greater than 15 seconds at 9.5k, is also d i f f i c u l t to reconcile with a simple barrier model. The second slope in the higher temperature deformation experiment is less than the slope at lower temperature, neglecting the question of the sign of the slope. Such behavior, however,
537 0036-9748/87 $3.00 + .00 Copyright (c) 1987 Pergamon Journals Ltd.
558
STRESS RELAXATION
Vol.
21,
No.
4
can be rationalized by recognizing that i f the drag between barriers is less, then the amount a dislocation can overshoot a barrier is greater. I f a dislocation overshoots a barrier, this w i l l exert an extra force on the mobile dislocation. As a result of this extra force a disloca. tion may mechanically overshoot the barrier, with less thermal assistance. This accounts for the slight dependence of the observed relaxation rates on temperature. These considerations suggest that i n e r t i a , at these low temperatures, is playing a role in the thermal surmounting of barriers, such as suggested by the calculations of Isaac and Granato (6). A more complete version of the present experiment is now underway, including the temperature dependence, solute concentration dependence and magnetic (7) f i e l d dependence. These results w i l l be reported elsewhere. Acknowledgements I t is a pleasure to thank a number of former students and colleagues, including J. Pellegrino, M. Suenaga and R. E. Galligan. This research has been supported by the National Science Foundation, Divison of Materials Science under contract # DMR 8603436. I t is also a pleasure to thank the A. von Humboldt Stiftung and Professor Peter Haasen. References I. 2. 3. 4. 5.
M. Suenaga and J. M. Galligan, Scripta Met. 5, 829 (1971). A.V. Granato, Phys. Rev. B4, 2196 (1971). F.W. Noble and D. Hull, Acta Met. 12, I089 (1964). J. Pellegrino, B. Zhou and J. M. Gall---igan, Scripta Met. 16, 985 (1982). C.S. Pan~, T. H. Lin and J. M. Galligan, Jnl. of Applied Physics, 41, 333 (1978) R.D. Isaac and A. V. Granato, Proc. 5th I n t ' l . Conference on the Strength of Metals and Alloys, Edited by Peter Haasen e t . a l . Pergamon Press, Oxford, p425 (1979). J. M. Galligan, T. H. Lin and C. S. Pang, Phys, Rev. Letters. 38 405 (1977).
6. 7.
Tensile Machine ed
(/) (/)
(n
¢D ,._
(n (]3
03
03
,0--
Stroin Fig. 1
Time
In the l e f t hand diagram above a schematic of the stress-strain curve for a lead crystal at low temperature is shown. In the right hand diagram a schematic of the stress-relaxatlon curve is illustrated.
Vol.
21, No. 4
STRESS
RELAXATION
539
e9.5K • 6.OK
5
2 C
V
I¢1 L_
.2
I
I
I
I
A
I
10
15
20
25
30
35
time (sec.) Fig. 2
The stress decrement, plotted on a logarithmic scale, as a function of time. Note that the intercept on the ordinate for the data from the two temperatures has been displaced for c l a r i t y , The applied stress is roughly the same for both measurements.
540
STRESS RELAXATION
Vol.
4 Lead-Tin crystal T= 42K Superconducting State
3
1
i 2
=
l
o 01
w o
,2 l
I 15
I
30
Time (sec) Fig.
3.
S t r e s s R ~ l a x a t i o n data f o r a l p a d - t i n c r y s t a l , p l o t t ~ d on a l o g a r i t h m i c s c a l e in s t r e s s decrement versus t i m e .
21, No. 4