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Stress relaxation in ionic solidsK. Tangri and D. J. Lloyd Citation: Journal of Applied Physics 45, 4268 (1974); doi: 10.1063/1.1663046 View online: http://dx.doi.org/10.1063/1.1663046 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/45/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stress-relaxation behavior in gels with ionic and covalent crosslinks J. Appl. Phys. 107, 063509 (2010); 10.1063/1.3343265 Dispersive ionic space charge relaxation in solid polymer electrolytes. II. Model and simulation J. Appl. Phys. 91, 6638 (2002); 10.1063/1.1468912 Cooperative or noncooperative dielectric relaxation in ionic solids: A discriminating experimental approach J. Appl. Phys. 85, 2821 (1999); 10.1063/1.369601 Spin-Lattice Relaxation in Ionic Solids Am. J. Phys. 35, 975 (1967); 10.1119/1.1973671 Spin-Lattice Relaxation in Ionic Solids Phys. Today 20, 88 (1967); 10.1063/1.3034407
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Stress relaxation in ionic solids K. Tangri and D. J. Lloyd
Department of Mechanical Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received 24 September 1973; in final form 20 May 1974)
The stress relaxation behavior of single-crystal MgO, single-crystal CaF" and polycrystalline AgCI has been examined to assess if the assumption of constancy of structure during relaxation is valid. It is shown that structural change occurs in all three materials, bei,ng severe at small plastic strains in the case of MgO and CaF,. In polycrystaUine AgCI, the structural change is due to recovery and increases with increasing plastic strain.
I. INTRODUCTION
In the investigation of the plastic deformation of materials, the relaxation test has been used extensively in recent years. In principle, it is an alternative form of the strain rate change test and is used to obtain information regarding the strain rate sensitivity of the flow stress and hence the activation parameters of deformation. In investigating the deformation behavior of ionic solids and ceramics, it has the great advantage that little strain is involved, which is important in inherently brittle materialS.
The relaxation of stress at constant total strain may be analyzed in several different ways. The empirical analysis of Trouton and Rankine l has been used extenSively in the literature. 2- 4 Accotding to this approach the stress decrement - Aa can be related to the relaxation time via the expression
- Aa== alog(t + k), (1)
where a and k are constants. This suggests that a plot of Aa vs log(t + k) should define a straight line with suitable choice of k. It also means that at sufficiently long times, when t» k, Aa vs log(t) is a straight line.
An alternative approach has been used by Li. 5 The plastic strain rate Ep can be expressed as 6
Ep == ¢bPm V== ¢bpmB(a*)m*, (2)
where Pm is the mobile dislocation density; V is the dislocation velocity; b is the Burgers vector; ¢, B, and m* are constants; and a* is the effective component of the flow stress aF , where aF == a* + ai' al being the athermal or internal stress component. Analyzing stress relaxation behavior on the basis of Eq. (2), Li obtained
du logdt ==log(-nK) + (-n-1)log(t+a'), (3)
where n==1/(m* -1).
Both the Trouton and Rankine, and Li analyses assume constancy of structure, ie., Pm and at during relaxation. While a considerable number of relaxation experiments are in general agreement with these assumptions, 7-9
Law and Besherslo found that the Li analysis is not applicable to the relaxation behavior of some fcc and hcp metals. Furthermore, the results on NaCI by Gupta and LF and Spracklingll indicate a strain-dependent m*, and hence Pm from Eq. (2).
The purpose of the present series of experiments is to examine the constancy of structure, i. e., Pm and a p
during relaxation tests on MgO and CaFa single crystals and polycrystalline AgCI.
4268 Journal of Applied Physics, Vol. 45, No. 10, October 1974
II. EXPERIMENTAL TECHNIQUES
. A series of tests were carried out on single crystals of MgO of (100) orientation and a total impurity content of about 250 ppm with the major impurity being Fe20 3
(10'0 ppm). The crystals were annealed at 1250 °C for 24 h prior to testing. Tests were also performed on single crystals of CaF2 of (111) orientation after an anneal of 24 h at 650 °C and on polycrystalline AgCI. The polycrystals were strained in tension at a strain rate of 4 x 10-3 Sec- l and the single crystals in compression at a strain rate of 8 x 10-6 sec-l • All tests were stopped prior to any crack formation.
In analyzing stress relaxation tests, it is assumed that there is no change in structure during the relaxation and that the mobile dislocation density remains constant. One way of checking these assumptions, as suggested by Lloyd and Embury, 8 is to carry out a relaxation test from a suitable stress level and then reload to the same stress and repeat the test. If the relaxation curves superimpose, constancy of structure is established. Furthermore, if instead of relaxing the specimen from the same stress level, the specimen is unloaded to some intermediate stress and then allowed to relax, the resulting relaxation curve should again superimpose on the master relaxation curve. This type of approach has been used in the present investigation and three specific testing procedures, illustrated diagrammatically in Fig. 1, have been employed.
0...
'0 o o ...J
A
Time t
FIG. 1. Schematic of testing procedure.
Copyright © 1974 American Institute of Physics
8 c
4268
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E 'jf --(L
" 0 CJ -1
,
4b
40
Ep' 0.65%
Ep~ 0.6 %
Ep= 0.15%
Ep' 0.05%
j--------'----Ep '0.005% .0;' 85%0;
'--- Ep::: O,O"r ::.20%0;.
11- 2kgm,
Ll __ Imln
FIG. 2. Stress relaxation curves for a MgO crystal after various plastic strains.
Procedure A involves a master relaxation followed by a second stress relaxation from the same stress level.
Procedure B is similar to A but instead of relaxing from an equivalent stress level the specimen is unloaded to some intermediate stress level and allowed to relax.
Procedure C modifies that of A so that the second relaxation occurs from a stress higher than that of the master relaxation.
III. RESULTS AND DISCUSSION A. Stress relaxation in MgO
Figure 2 shows some typical relaxation curves for MgO crystal deformed at room temperature. Curves 1 and 2 are relaxations carried out from stresses of 20 and 85%, respectively, of the yield stress (ly. Curve 1 was reproducible from specimen to specimen and is a reflection of the machine relaxation. Curve 2, on the other hand, represents relaxation behavior after some plastic flow in the premacroyield region has occurred. Curves 3, 4a, and.5 are relaxations after plastic strains of 0.05, 0.15, and 0.6%, respectively. I~ is apparent that the extent of relaxation increases with increasing strain. However, beyond about 0.6 (experimental values vary from 0.6 to 1% plastic strain from specimen to specimen), the relaxation curves effectively become constant. Curves 4a and 4b refer to the master and second relaxation from the same stress level following testing procedure A. It is apparent that the second relaxation test shows a slower rate of relaxation at an equivalent time and an over-all reduction in the stress relaxed when compared with the master relaxation. This was a characteristic feature of all repeat relaxations regardless of the extent of plastic strain and was, in fact, maintained up to fracture. The detailed profile of the second relaxation was a function of strain. Incremental unloading following procedure B for the second relaxation also showed a decreased relaxation rate and
4269 J. Appl. Phys., Vol. 45, No. 10, October 1974
30
°
\ 20
" E 0"-.
10
o 2 3 4
Ep (%l
FIG. 3. Variation of the velocity stress exponent m* with plastic strain Ep for MgO deformed at room temperature.
a reduction in the relaxed stress for the second relaxation.
If testing procuedure C is followed and the second relaxation is carried out at a stress where the work-hardening rate is equal to that prior to the first master relaxation, the second relaxation (curve 6) superimposes on the master curve (curve 5). Furthermore, if the second relaxation is unloaded to some intermediate stress after reaching an equivalent work-hardening rate the subsequent relaxation curve (curve 7) also superimposes on the master relaxation plot. The relaxation results, therefore, show that the same work-hardening rate has to be achieved on restraining after relaxation to obtain a subsequent relaxation curve which superimposes on the master curve. This indicates that a structural change-a change in (II or/and Pm-iS occurring during the relaxation test. On subsequent straining, some plastic strain is needed to develop the equivalent structure existing prior to the master relaxation. Development of an equivalent work-hardening rate reflects the existence of a similar dislocation structure.
10
.0 "-
<!
O.IL... _____ .l-_____ --L..~ ____ _'
10 100 1000
Time t ( sees)
FIG. 4. Log(do/ dt) vs logt for a stress relaxation after 0.35% plastic strain.
K. Tangri and D.J. Lloyd 4269
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Ep~ 0.80%
Ep ~ 0.23%
Imin Ep~ 0.05%
Time t (mins)
FIG. 5. stress relaxation curves for a CaF2 crystal after various plastic strains.
Values of m* obtained via Li analysis, which is based on the assumption that up Pm' and m* are constant, are shown in Fig. 3. It is seen that m* decreases drastically with strain up to about 0.6%. However, in the strain range 1-3%, considerably less but unsystematic variation in m*, with values from a low of 7 to a high of 15, is observed. Recent etch-pit investigation on MgO by Singh and Coble12 also shows a similar variation in m* values.
For a given instant in the relaxation process, Eq. (3) may be rewritten
(4a)
and
loliy = 10gA + logpm
+ m * logO'* , (4b)
where the constant A incorporates cp, b, and B. From Eq. (4a) , it is seen that a variation in m*, when m* has to be a constant, is in reality a consequence of the variation in 17* (due to a change in ut ) and/or P'" during relaxation. Furthermore, Eq. (4b) shows that a small variation in m* will demand a large compensatory variation in a* for a constant Pm or in Pm for a constant a*. Assuming that Pm is not changed during relaxation, the experimentally observed variation of 4 around a mean value of 11 in m* requires, via Eq. (4b) , that a* and as such 17, during relaxation must vary by at least a factor of 20. However, a maximum increase of about 10% in the applied stress during C-type tests was found necessary to produce relaxation curves which superimpose on the master relaxation curve. In view of the above, it may be concluded that during relaxation, apart from an increase in u/' significant changes in Pm are to be expected. Thus, in the strain range of 1-3%, where m* . changes and as such, 17/ and Pm changes are less marked, reasonable plots for Eq. (3) are expected as observed. However, in the lower strain region (Ep < 0.6) where m* values show a large variation indicating drastic structural changes, Li analysis is not expected to be applicable. This is confirmed by the 10gdO'/dt-vs-logt plot as shown in Fig. 4, which yields a slope of less than unity, an obviously impossible result.
4270 J. Appl. Phys., Vol. 45, No. 10, October 1974
20~
E 10~
I o 2
Ep (%l
3
FIG. 6. Variation of the velocity stress exponent m* with plastic strain Ep for CaF2 deformed at 150°C.
B. Stress relaxation in CaF2
The relaxation behavior of CaF2 single crystals at 150°C is very similar to that of MgO. As shown in Fig. 5, the relaxation prOfile changes with strain in the early stages of deformation and then becomes constant after a plastic strain of approximately O. 5%. A second relaxation from the same stress level resulted in a decreased relaxation rate, regardless of the strain involved. The decrease in relaxation rate tends to be less in CaF2 than in MgO. The Li analysis was not applicable at small strains, giving a gradient < 1; however, beyond about 0.2% strain, reasonable results were obtained. The variation of m* with strain is shown in Fig. 6. The average m* = 10 compares with the values of 6.5 and 8± 2 for edge and screw dislocations, respectively, obtained by Evans and Pratt13 from etch-pit experiments.
C. Stress relaxation in AgCI
The stress relaxation curves for AgCI change with strain as shown in Fig. 7; the extent of stress relaxation in any given time interval increases with increaSing strain. Furhtermore, there is a significant decrease in the yield stress on subsequent testing after stress re-
Ep ~ 6.8%
[-o2k9ms E = 1.5 '%
I min
FIG. 7. Stress relaxation curves for polycrystalline Agel after various plastic strains.
K. Tangri and D.J. Lloyd 4270
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b <l
o
0.02
003
~ ----0(0)
(b)
-~- .---~----'-10 20 30 40
Time,! (mlns)
FIG. 8. Decrease in the yield stress after (a) time at zero stress and (b) time undergoing stress relaxation.
laxation, the extent of the decrease increases with increasing stress relaxation time. The decrease in the flow stress after relaxation indicates that the specimen is undergoing recovery and that the internal stress is decreasing during the stress relaxation test. This is in agreement with the increasing extent of relaxation with increasing strain, since the driving force for recovery, i . e., the stored energy which is reflected in the internal stress, is increasing with strain. It is also in agreement with the extent of relaxation increasing with increasing relaxation time since recovery would be thermally activated and hence time dependent. The recovery probably involves both thermal recovery in the usual sense-zero applied stress and dynamic recovery since plastic flow is also occurring. It is, therefore, of interest to compare recovery during relaxation with static recovery, i. e., recovery at the same temperature but with zero applied load. The results in Fig. 8 compare the recovery behavior of a specimen undergoing relaxation with that of a speCimen strained to the same stress level and recovered at room temperature under zero stress. It can be seen that recovery occurs more rapidly under stress relaxation. This would be expected since in the relaxation situation dislocations are still overcoming barriers and hence relieving local back stress without travelling large distances to interact with other barriers and increase the back stresses in these areas. Thus, if barriers can be overcome without any extensive subsequent work hardening, therma'l recovery will be enhanced by an applied stress or stress relaxation.
4271 J. Appl. Phys., Vol. 45, No. 10, October 1974
Thornton and Cahn14 came to a similar conclusion in terms of stress-assisted thermal recovery in copper.
In view of the structural changes occurring due to recovery during a room temperature test on AgCI, Li analysis is not expected to be applicable. This is confirmed by a slope of less than unity for plots of Eq. (3) in both the low- and high-strain regions.
IV. SUMMARY
The stress relaxation behavior of MgO, CaF2 and AgCl have been investigated in terms of the Li analysis. It is shown that the assumption of constant structure is invalid in all three materials.
In MgO and CaF2, structural change during relaxation is particularly severe at small strains and renders the analysis invalid, giving a velocity stress exponent value approaching infinity. However, at strains greater than about 1%, structural change is less severe and the Li analysis can be applied to the results giving m* == 7 -15 for MgO and 5-12 for CaF2 •
In the case of AgCI, the structural change occurring during relaxation is a significant decrease in the internal stress through a recovery mechanism. As a result the Li analysis is not applicable in any strain range.
ACKNOWLEDGMENT The authors are grateful to the Atomic Energy of
Canada Ltd. for financial support of this work.
1F. T. Trouton and A. O. Rankine, Philos. Mag. 8, 538 (1904). 2p. Feltham, J. Inst. Met. 89, 210 (1960-61). 3F. Guiu and P. L. Pratt, Phys. Status Solidi 6, 111 (1964). 4F. W. Noble and D. Hull, Acta Metall. 12, 1089 (1964). 5J.C.M. Li, Can. J. Phys. 46, 493 (1967). 6W.G. Johnston andJ.J. Gilman, J. Applo Phys. 30, 129 (1959).
71. Gupta andJ.C.M. Li, Metall. Trans. 1, 2323 (1970). 8D. J. Lloyd and J. D. Embury, Phys. status Solidi B 43, 393 (1971).
91. Gupta and J. C. M. Li, Mater. Sci. Eng. 6, 20 (1970). 10C.C. Law and D.N. Beshers, Scr. Metallo 6, 635 (1972). 11M. T. Sprackling, Philos. Mag. 27, 265 (1973). 12R. N. Singh and R. L. Coble, J. Appl, Phys. 46, 981 (1974), 13A. G. Evans and P. L. Pratt, Philos. Mag. 20, 1213 (1969). 14p. H. Thornton and R. W. Cahn, J. Inst. Met. 89, 455
(1960-61).
K. Tangri and D.J. Lloyd 4271
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