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Dynamical state of dislocations in germanium crystals during deformation
Materials Science and Engineering, 13 (1974) 263--268 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
Dynamical state of dislocations in germanium crystals during deformation KOJI SUMINO and SHOICHI KODAKA The Research Institute for Iron, Steel and Other Metals, Tohoku University, Sendai (Japan) KEN-ICHI KOJIMA Department of Physics, Yokohama City University, Yokohama (Japan) (Received February 26, 1973)
Summary Experimental results in the strain-rate change tests on germanium single crystals done in a previous paper are analysed with the use o f new velocity data for isolated dislocations. A n equilibrium stationary state o f moving dislocations appears in the deformation stage beyond the middle o f stage 0 o f the stress--strain curve. The velocity r¢ and the density Nm o f moving dislocations in the equilibrium state are observed to depend on the strain rate e a s y ~ ~o.42 and Nm cc ~o.5s, respectively, at 600°C. Transition o f the state o f dislocation motion from one equilibrium state to another by a sudden change o f the strain rate during deformation is also investigated. It is concluded that the values o f Y and N m in the equilibrium state at a certain temperature are determined completely by the strain rate and do not depend on the density o f total dislocations in the crystal nor on the values o f v and Nm which the crystal has assumed previously. A transient stage o f about 0.5% in strain is found to exist when the state o f dislocations motion is transferred from o n e equilibrium state to another by the change o f strain rate.
1. INTRODUCTION In a previous paper 1 the characteristics in the state of dislocation motion in germanium crystals during deformation were investigated
by means of the strain-rate and the temperature change experiments. Experimental data were analysed on the basis of the Sch~ifer's empirical velocity equation 2 for isolated dislocations. It was shown that the velocity and the density Nm of moving dislocations change with strain in the yield regions in a way as described by the Johnston--Gilman type model. However, they were observed to be constant against the strain in the deformation stage beyond the middle of stage 0 of the stress-strain curve. This is interpreted as that an equilibrium stationary state of dislocation motion is realized in such deformation stage. The magnitudes of 0 and Nm in such deformation stage were shown to depend on the strain rate e and the temperature T sensitively. Higher value of Nm was related to higher level of the flow stress and the longer straininterval of each deformation stage. Thus, the stress--strain curve is expanded in size maintalning the similarity of the shape by the increase in ~ or by the decrease in T. The workhardening rate in each deformation stage was observed to be almost independent of e and T. Recently, Schaumburg a has shown the Sch~er's expression for the velocity of isolated dislocations to be incorrect. Although the main conclusion reached in the previous paper is not affected by the alternation of the velocity equation of dislocations, the expressions for the dependence of ~ and N ~ on e and T reported in the previous paper suffer alternation since they depend on the expression of
264
the velocity equation of dislocations used. Data on the dislocation velocity in germanium crystals r e p o r t e d recently by two research groups (Patel and Freeland 4, Schaumburg a) have shown a reasonable agreement. This seems to suggest these new data to be reliable. Thus, the first purpose of this paper is to reanalyse the experimental results obtained in the previous paper with use of the new velocity data and to deduce the correct dependence o f 0 and Nm on the d e f o r m a t i o n condition. It is interesting to see how the equilibrium state o f dislocation motion, which is characterized by constant values of O and Nm, is affected by the internal structure of the crystal. Different dislocation structure is developed b y the d e f o r m a t i o n of different strain rate. The second purpose of this paper is, thus, to investigate the above point by observing th e behavior of D and Nm with strain when the strain rate of a crystal is changed suddenly during deformation.
2. ANALYSIS OF THE PREVIOUS EXPERIMENTAL RESULTS WITH USE OF NEW VELOCITY DATA
New data on the dislocation velocity in germanium crystals show that the activation energy o f the dislocation m o t i o n depends on the stress and tha t t he velocity--stress expon e n t depends on the temperature. Schaumburg shows th at the difference in the velocity between a screw dislocation and a 60 ° dislocation seems to diminish at temperatures above approximately 500°C for stresses lower than 1 kg/mm 2. Patel and Freeland show the velocity--stress e x p o n e n t m for a 60 ° dislocation at 580°C to be 1.6 at stresses lower than 1 ~ 2 kg/mm 2 . The flow stresses of germanium single crystals at 600°C observed in the previous paper are within this stress range. The data o f the strain-rate sensitivity of def o rmatio n behavior at 600°C observed in the previous paper are analysed here with m = 3/2. The extrapolation o f the value of m gives m = 1.6 at 600°C. However, it is shown t hat the results o f the analysis are rather insensitive to the value o f m used and t hat the difference between the magnitude o f ~ or N m deduced with m = 3/2 and that de duc e d with m = 1.6 is co mp let e l y negligible in view of the
experimental accuracy of the stress measurement. The procedure to deduce ~ and Arm by strain-rate change tests is described in the previous paper. It is assumed t hat the rate-controlling mechanism of dislocation m o t i o n is n o t affected by the interaction between dislocations. The interaction with all ot her dislocations is assumed to affect the m o t i o n of a moving dislocation through a long-range interaction stress Ti which does n o t depend on its location. Thus, the velocity of moving dislocations during d e f o r m a t i o n is given by 0 = k(T a -
703/2 = k T 3/2 eft
(1)
where Ta is the applied stress and k a constant. To determine Teff and Ti by strain-rate change tests, it is essential to assume that the magnitudes o f b o t h Nm and ri are unaltered during the strain-rate change. The validity of such an assumption may be checked by observing how the magnitudes of Tef f and ri determined depend on the value of t l / t 2 where t l is the base strain-rate and ~2 the strain rate to which the strain rate is changed from e l . If the rearrangement o f dislocations takes place during the strain-rate change test, the variation o f dislocation configuration may be a function o f the time which is spent during a strain-rate change test. This time is roughly proportional to e 2 1 . Such variation of dislocation configuration, if it exists, would naturally result in the variation of Te~f and ri with tl/~2.
Figure 1 shows Teff determined as a function o f t l / t 2 at T = 600°C for two values of t l . Te ff was det erm i ned for the same strain. It is seen t h a t t he value of Te~t has a weak dependence on tl/t2 showing t hat slight rearrangement of dislocations takes place during a strain-rate change test. The true value of the effective stress would be the value o f Teff extrapolated to e l / e 2 = 1 in Fig. 1. We determined Tef f for e l / e 2 = 10. Such values of Teft are seen t o be higher than the true values by less than 5 percent. This magnitude o f error in T~f~ is not i m p o r t a n t from the viewpoint o f t he dependence o f T~f~ on the d e f o r m a t i o n condition and is completely negligible. Figure 2 shows the behavior of ra, r~tt and Ti against the strain for the d e f o r m a t i o n at T = 600°C. The general feature observed in the
265
i
1.5
0.15 E E
e~= 4 0 X I 0"4sec-t 0.10
~
T
o
/
% E
0.05
5
T = 600°C
~O5
~1= 6 0 ]l fO-Ssec-I
F-
IJO
i
2i0
30
(1"/.) = I . I X I 0-4SgC"1
1.0
/
E E
==0.5
I
I
k~
I I
: 0,1C -II
k*
1"etf
i10
15
0.20 -- I I I L~ I
~E 0.15
~
I 20
JJ0
I
3JO
l
((%)
I
T =600ec (,= I.I X 10-4sec-'
0'05 t r
= 2, I = rO-Ssec-~ _
0t
I
I0
}, /
~2
previous paper is retained here, namely, re f( is constant against the strain in the deformation stage b e y o n d the middle o f stage 0 and workhardening proceeds entirely through ri. Figure 3 shows the stationary values of r~f( plotted against the strain rate. It is seen that r ~ f depends on the strain rate as O~= 0 . 2 8 + 0 . 0 1
Thus, from e q n . Nm depend o n ~ as cc ~ 0 . 4 2 ,
(1)
1"=
30
2~0
Fig. 1. T h e e f f e c t i v e stress reff d e t e r m i n e d for the s a m e strain p l o t t e d against ~1l¢2 for t w o values of ~1 at 600°C. e l is the base strain-rate and ~2 the strain rate to w h i c h the strain rate is changed from ~1 in d e t e r m i n i n g Tel f.
Teff cc ~c~
0.5
.
(2)
I.,.
0'~ IlO
,l5
e (%)
Fig. 2. Behavior o f the flow stress ra, the e f f e c t i v e stress Tell, and the interaction stress Ti against the strain e at 600 ° C for various strain-rates.
580°C is the same as that at T = 600°C, the c-dependence o f the absolute magnitudes of V and Nm is obtained as shown in Fig. 4. It is to be noticed that the magnitude o f 0 is lower than that estimated in the previous paper by approximately an order of magnitude and that of Nm is higher than the previous one
a n d ~ =NmOb, ~ and
Nm ~ e0.5S.
"~'ef
1.0
T = 600"C
(3)
N o w , we estimate the absolute values of and Nm. The velocity measurement of dislocations 4 has been c o n d u c t e d at T = 5 8 0 ° C and n o t at T = 600°C. Hence, the strain-rate sensitivity data at T = 6 0 0 ° C cannot be used for such a purpose. We have data o f strain-rate change tests at T = 5 8 0 ° C o n l y for ez = 1.1 × 10 - 4 sec - 1 . These data are used for the estimation o f the absolute values of ~ and Nm. Assuming that the e-dependence o f Tel f at T --
0.1 Teff o=
~o
a =0.28_+ 0.01 L
iO-S
10.4
I0-~
(s=c") Fig. 3. The variation o f the stationary value of the effective stress reff with the strain rate ~ in the deformation at 6 0 0 ° C.
266
T = 580"C
L
=
10-4
L
(5)
dx
fo le(ra -- ri(x))3/2
A
g E
i>
10-5
I0 a o E z
This Y is taken as the mean velocity of all moving dislocations in the crystal. The magnitudes of ro and zA are determined by the strain-rate-change tests for two values of e2/el at the same strain, namely, L
f
i0 7 I
10-5
I
10-4
[Ta. 1 - T 0 - 7 " A
sin (27rx/L)] 3/2
0
~1
L f dx 0 [ra. 2 - r 0 - r A sin(21rx/L)] 3/2
1
10-3
(sec-')
Fig. 4. The variation of the equilibrium values o f the density N m and the velocity b- of moving dislocations with the strain rate ~ in the d e f o r m a t i o n at 580°C.
(6) L
f also by an order of magnitude. This comes from the difference between the velocity data of S c h ~ e r and of Patel--Freeland. The above result leads to the alternation of a conclusion on the mobile fraction of dislocations reached in a previous paper5. It was concluded in the previous paper that only several percent of total dislocations are mobile at lower yield point. However, it should be concluded here that a considerable fraction (~ several tens percent} of total dislocations is mobile at lower yield point in the deformation at this temperature range. All of the above results have been derived on the basis of the model that ri is constant with respect to the location of dislocations on the active slip plane. The validity of such model can be checked by the combined strain-rate change experiments described in the previous paper. Suppose that a periodic internal stress r i which is given by
Ti= TO + ~'A Sin (2rrx/L)
dx
~2
(4)
acts on the active slip plane and makes a dominant contribution to r i. Here, ro and rA are constants, L the period of the internal stress field, x the position on the active slip plane. The time average of the velocity of a dislocation on such a slip plane is given by
dx
e3
0 [ r a . l - - r o - - r A sin(2Trx/L)] 3/2
C1
L dx f [~'a,3--T0--?'A sin (2rrx/L)] 3/2 0
w h e r e 7a, 1 , Ta, 2 and ra,a are the flow stresses for the strain rates e l , e2 and e3, respectively, in the strain-rate change tests, el being the base strain-rate. The magnitudes of ro and vA are obtained by solving simultaneous equations of eqn. (6} numerically. The magnitudes of Vo and rA determined with ~2/el = 1/10 and e 3 / q = 1/20 are given in Table 1 together with those of Ti obtained with the constant ri model for the deformation at el = 1.1 X 10 - 4 sec - 1 and T = 600°C. It is seen that the magnitude of TA is almost zero and the magnitude of ro is very close to that of ri. Since the internal state of the crystal, after a strain-rate change {~i-+~2 may be slightly different from that after el-~ea as seen in Fig. 1, the values shown in Table 1 may not be quite accurate. Nevertheless, the results in the table are effective enough to show the validity of the constant interaction stress model. Another verification of the model will be given in the paper which follows ~.
267 TABLE 1 Values of To and 7A determined by the combined strain-rate change experiments De formation stage
c (%)
TO (kg/mm 2)
TA
Ti
(kg/mm 2)
(kg/mm 2)
I I
7.4 13.0
0.4081 0.4504
0.0001 0.0003
0.404 0.449
3. TRANSITION OF THE STATE OF DISLOCATION MOTION FROM ONE EQUILIBRIUM STATE TO ANOTHER BY THE CHANGE OF STRAIN RATE T h e p r e v i o u s p a p e r a n d t h e last s e c t i o n in t h e p r e s e n t p a p e r h a v e s h o w n t h a t in a cons t a n t strain-rate d e f o r m a t i o n t h e m a g n i t u d e o f Tef f is d e t e r m i n e d e n t i r e l y b y t h e values o f t and T and does not depend on the internal structure developing with deformation once t h e m i d d l e o f stage 0 is passed. This m e a n s t h a t t h e e q u i l i b r i u m s t a t i o n a r y s t a t e o f disloc a t i o n m o t i o n , w h i c h is c h a r a c t e r i z e d b y certain c o n s t a n t values o f ~ a n d N m , is realized d u r i n g t h e c o n s t a n t strain-rate d e f o r m a t i o n a n d t h a t values o f ~ a n d Nm in such equilibr i u m s t a t e are d e t e r m i n e d o n l y b y t h e d e f o r m a t i o n c o n d i t i o n ( t , T ) . A m o d e l describing this e q u i l i b r i u m s t a t e is given in t h e n e x t paper. N o w , it is i n t e r e s t i n g t o see w h e t h e r t h e deformation history of the crystal affects such e q u i l i b r i u m s t a t e o f m o v i n g dislocations. We d e f o r m a c r y s t a l at a c e r t a i n s t r a i n - r a t e t o stage I a n d realize a c o r r e s p o n d i n g equilibr i u m s t a t e o f t h e d i s l o c a t i o n m o t i o n in t h e crystal. T h e n , w e c h a n g e t h e strain r a t e sudd e n l y t o a n o t h e r value k e e p i n g the t e m p e r a t u r e u n c h a n g e d , o b s e r v e t h e s t a t e o f t h e disloc a t i o n m o t i o n a f t e r t h e c h a n g e o f strain rate, a n d c o m p a r e it w i t h t h a t in a c r y s t a l w h i c h is d e f o r m e d at t h e latter strain r a t e f r o m t h e b e g i n n i n g o f d e f o r m a t i o n . I t is to be e m p h a sized t h a t t h e d e n s i t y o f t o t a l d i s l o c a t i o n s in t h e crystal is q u i t e d i f f e r e n t b e t w e e n t h e s e t w o crystals, o n l y t h e strain r a t e being t h e s a m e. In t h e f o l l o w i n g , t h e results o f such e x p e r i m e n t s are s h o w n . P r o c e d u r e s o f t h e s p e c i m e n preparation, deformation, determination of Teff a n d 7i, etc. are d e s c r i b e d in t h e p r e v i o u s p a p e r s 1,7, S .
1.5
T-600'C = 4.2 ll0 5 s e c ' ~
~ - 4 2 x I0-4$ec "d
',,,
,b
,'5 (
io
('/.)
Fig. 5. The stress--strain behavior of a specimen deformed first at a strain rate of 4.2 x 10 - 5 s e c - 1 to the middle of stage I and, then, subjected to the change of strain rate to a strain rate of 4.2 × 10 - 4 sec - 1 at 600 °C. Full circles and full triangles represent 7ef f and ri for the specimen, respectively. The behavior of a specimen deformed at a strain rate of 4.2 x 10 - 4 sec - 1 throughout all deformation stages is shown by broken lines. Open circles and open triangles show Tel f and zi of the latter specimen, respectively. Figure 5 s h o w s stress--strain b e h a v i o r o f a s p e c i m e n d e f o r m e d first at ~ = 4.2 X 10 - 5 sec- 1 and then subjected to the change of the strain r a t e to t = 4.2 X 10 - 4 sec - 1 at t h e m i d d l e o f stage I. B e h a v i o r o f a s p e c i m e n def o r m e d at e = 4.2 X 10 - 4 sec - 1 f r o m t h e b e g i n n i n g o f d e f o r m a t i o n is also s h o w n b y b r o k e n lines. It is seen t h a t t h e value o f zef~ a f t e r t h e c h a n g e o f strain r a t e c o i n c i d e s w i t h t h a t o f t h e s p e c i m e n d e f o r m e d at t h e s e c o n d strain r a t e f r o m t h e beginning. T h u s , it is conc l u d e d t h a t t h e values o f ~ a n d Nm d o n o t dep e n d o n t h e d e n s i t y o f t o t a l d i s l o c a t i o n s in the c r y s t a l or o n t h e values o f ~ a n d N m w h i c h t h e c r y s t a l has a s s u m e d p r e v i o u s l y b u t are d e t e r m i n e d c o m p l e t e l y b y t h e value o f t h e strain r a t e o f t h e crystal. T h e d i f f e r e n c e in t h e f l o w stress b e t w e e n t w o s t r e s s - - s t r a i n curves in Fig. 5 a f t e r t h e c h a n g e o f strain r a t e is seen to be e n t i r e l y d u e t o t h e d i f f e r e n c e in Ti. Figure 6 s h o w s t h e b e h a v i o r d u e to t h e c h a n g e o f strain r a t e f r o m ~ = 4.2 X 10 - 4 sec - 1 to ~ = 4.2 X 10 - 5 sec - 1 . B r o k e n lines show the stress~strain characteristics of a s p e c i m e n d e f o r m e d at t = 4.2 X 1 0 - 5 s e c - 1 f r o m t h e b e g i n n i n g o f d e f o r m a t i o n . As to Te~f essentially t h e s a m e result as t h a t in Fig. 5 is o b t a i n e d . T h u s , also in this case, it is concluded that the equilibrium state of moving d i s l o c a t i o n s is c o n t r o l l e d o n l y b y t h e deform a t i o n c o n d i t i o n . As first o b s e r v e d in t h e pre-
268
J.5 I0
T - 600"C -4.2 110-"$e¢-'~
T -600'C - 4 2 x lO-4se¢ -' -~--~- 4.2 z
JO~e¢-'
~--4.2 ~ I0" ~$e¢ -'
~ - hC
j"j 0.5
.~-~ 0.5
Tiff • ~-o--__.o..o--o-..r-o---o-_-.~.~.~--.o-
~: = - .
=
T,.
.,. Tiff
(%]
06
~
,b
,'5 E
Fig. 6. T h e stress---strain b e h a v i o r o f a s p e c i m e n def o r m e d first at a s t r a i n rate o f 4.2 × 10 _ 4 sec - 1 to t h e m i d d l e o f stage I and, t h e n s u b j e c t e d t o t h e c h a n g e o f strain rate to a strain rate o f 4.2 × 10 - 5 sec - 1 at 600°C. Full circles a n d full triangles repres e n t 7ef f a n d Ti for t h e s p e c i m e n , respectively. T h e b e h a v i o r o f a s p e c i m e n d e f o r m e d at a strain rate o f 4.2 × 10 - 5 sec - 1 t h r o u g h o u t all d e f o r m a t i o n stages is s h o w n b y b r o k e n lines. O p e n circles s h o w r e f f o f the latter specimen.
vious paper 5 work-softening takes place after the lowering of the strain rate. Figure 6 shows clearly such work-softening to proceed through Ti. This phenomenon may be interpreted in the following way. A considerable fraction of moving dislocations is immobilized by the lowering of the strain rate. Such immobilized dislocations may not be able to take the most stable configuration instantaneously. Thus, unstable configuration of immobilized dislocations would be realized in the crystal. The long-range internal stress fields around such immobilized dislocations are n o t released efficiently. Thus, they would make a rather large contribution to 7i. This point will be discussed in more detail in the next paper. As the deformation proceeds the immobilized dislocations would settle into a more stable configuration by means of the trapping of moving dislocations and relaxing the long-range internal stress fields around them, thus, ri becoming smaller as the strain increases. This is thought to be the cause for the work-softening occurring after the lowering of the strain rate. On deforming an undeformed crystal at a certain strain-rate, a fairly large strain is needed before the equilibrium state of moving dislocations is established. Namely, yield region and approximately a half of stage 0 where %f~ varies with strain precede the region of constant r~ff. Similarly, a transient stage is
t0
(%)
Fig. 7. T r a n s i e n t b e h a v i o r o f tee f o b s e r v e d for t h e c h a n g e o f s t r a i n rate f r o m 4.2 × 10 - 4 sec - 1 t o 4.2 × 10 - 5 sec - 1 at 6 0 0 ° C .
observed to exist when the state of dislocation motion changes from one equilibrium state to another. Such transient stage has been omitted to show in Figs. 5 and 6. The transient phenomenon is demonstrated in Fig. 7. A specimen deformed first at e = 4.2 × 10 - 4 sec-1 at 600°C is subjected to the change of strain rate to ~ = 4.2 × 10 - 5 sec - 1 in the middle of stage I and then, after a strain of about 1.3%, subjected again to the change of strain rate back to e = 4.2 × 10 -4 sec-~. The transient behavior in 7elf is shown for the first change of strain rate. It is seen that a transient stage of approximately 0.5% in strain appears before Teff reaches a new equilibrium value from an old one. If the equilibrium state of dislocation motion could change instantaneously from one to another, then it is impossible to determine the values of ~i and ~ by strain-rate change tests. It is emphasized here that the existence of the transient stage seen in Fig. 7 makes it possible to determine T~f~ and ~i by strain-rate change tests.
REFERENCES 1 K. S u m i n o a n d K. Kojima, Crystal Lattice Defects, 2 ( 1 9 7 1 ) 159. 2 S. Sch~ifer, Phys. Status Solidi, 19 ( 1 9 6 7 ) 297. 3 H. S c h a u m b u r g , Phil. Mag., 25 ( 1 9 7 2 ) 1429. 4 J.R. Patel a n d P.E. F r e e l a n d , J. Appl. Phys., 42 (1971) 3298. 5 K. K o j i m a a n d K. S u m i n o , Crystal Lattice Defects, 2 ( 1 9 7 1 ) 147. 6 K. S u m i n o , Mater. Sci. Eng., this issue. 7 K. K o j i m a a n d K. S u m i n o , J. Phys. Soc. Japan, 26 ( 1 9 6 9 ) 1213. 8 K. K o j i m a a n d K. S u m i n o , J. Phys. Soc. Japan, 31 ( 1 9 7 1 ) 171.
Dynamical state of dislocations in germanium crystals during deformation KOJI SUMINO and SHOICHI KODAKA The Research Institute for Iron, Steel and Other Metals, Tohoku University, Sendai (Japan) KEN-ICHI KOJIMA Department of Physics, Yokohama City University, Yokohama (Japan) (Received February 26, 1973)
Summary Experimental results in the strain-rate change tests on germanium single crystals done in a previous paper are analysed with the use o f new velocity data for isolated dislocations. A n equilibrium stationary state o f moving dislocations appears in the deformation stage beyond the middle o f stage 0 o f the stress--strain curve. The velocity r¢ and the density Nm o f moving dislocations in the equilibrium state are observed to depend on the strain rate e a s y ~ ~o.42 and Nm cc ~o.5s, respectively, at 600°C. Transition o f the state o f dislocation motion from one equilibrium state to another by a sudden change o f the strain rate during deformation is also investigated. It is concluded that the values o f Y and N m in the equilibrium state at a certain temperature are determined completely by the strain rate and do not depend on the density o f total dislocations in the crystal nor on the values o f v and Nm which the crystal has assumed previously. A transient stage o f about 0.5% in strain is found to exist when the state o f dislocations motion is transferred from o n e equilibrium state to another by the change o f strain rate.
1. INTRODUCTION In a previous paper 1 the characteristics in the state of dislocation motion in germanium crystals during deformation were investigated
by means of the strain-rate and the temperature change experiments. Experimental data were analysed on the basis of the Sch~ifer's empirical velocity equation 2 for isolated dislocations. It was shown that the velocity and the density Nm of moving dislocations change with strain in the yield regions in a way as described by the Johnston--Gilman type model. However, they were observed to be constant against the strain in the deformation stage beyond the middle of stage 0 of the stress-strain curve. This is interpreted as that an equilibrium stationary state of dislocation motion is realized in such deformation stage. The magnitudes of 0 and Nm in such deformation stage were shown to depend on the strain rate e and the temperature T sensitively. Higher value of Nm was related to higher level of the flow stress and the longer straininterval of each deformation stage. Thus, the stress--strain curve is expanded in size maintalning the similarity of the shape by the increase in ~ or by the decrease in T. The workhardening rate in each deformation stage was observed to be almost independent of e and T. Recently, Schaumburg a has shown the Sch~er's expression for the velocity of isolated dislocations to be incorrect. Although the main conclusion reached in the previous paper is not affected by the alternation of the velocity equation of dislocations, the expressions for the dependence of ~ and N ~ on e and T reported in the previous paper suffer alternation since they depend on the expression of
264
the velocity equation of dislocations used. Data on the dislocation velocity in germanium crystals r e p o r t e d recently by two research groups (Patel and Freeland 4, Schaumburg a) have shown a reasonable agreement. This seems to suggest these new data to be reliable. Thus, the first purpose of this paper is to reanalyse the experimental results obtained in the previous paper with use of the new velocity data and to deduce the correct dependence o f 0 and Nm on the d e f o r m a t i o n condition. It is interesting to see how the equilibrium state o f dislocation motion, which is characterized by constant values of O and Nm, is affected by the internal structure of the crystal. Different dislocation structure is developed b y the d e f o r m a t i o n of different strain rate. The second purpose of this paper is, thus, to investigate the above point by observing th e behavior of D and Nm with strain when the strain rate of a crystal is changed suddenly during deformation.
2. ANALYSIS OF THE PREVIOUS EXPERIMENTAL RESULTS WITH USE OF NEW VELOCITY DATA
New data on the dislocation velocity in germanium crystals show that the activation energy o f the dislocation m o t i o n depends on the stress and tha t t he velocity--stress expon e n t depends on the temperature. Schaumburg shows th at the difference in the velocity between a screw dislocation and a 60 ° dislocation seems to diminish at temperatures above approximately 500°C for stresses lower than 1 kg/mm 2. Patel and Freeland show the velocity--stress e x p o n e n t m for a 60 ° dislocation at 580°C to be 1.6 at stresses lower than 1 ~ 2 kg/mm 2 . The flow stresses of germanium single crystals at 600°C observed in the previous paper are within this stress range. The data o f the strain-rate sensitivity of def o rmatio n behavior at 600°C observed in the previous paper are analysed here with m = 3/2. The extrapolation o f the value of m gives m = 1.6 at 600°C. However, it is shown t hat the results o f the analysis are rather insensitive to the value o f m used and t hat the difference between the magnitude o f ~ or N m deduced with m = 3/2 and that de duc e d with m = 1.6 is co mp let e l y negligible in view of the
experimental accuracy of the stress measurement. The procedure to deduce ~ and Arm by strain-rate change tests is described in the previous paper. It is assumed t hat the rate-controlling mechanism of dislocation m o t i o n is n o t affected by the interaction between dislocations. The interaction with all ot her dislocations is assumed to affect the m o t i o n of a moving dislocation through a long-range interaction stress Ti which does n o t depend on its location. Thus, the velocity of moving dislocations during d e f o r m a t i o n is given by 0 = k(T a -
703/2 = k T 3/2 eft
(1)
where Ta is the applied stress and k a constant. To determine Teff and Ti by strain-rate change tests, it is essential to assume that the magnitudes o f b o t h Nm and ri are unaltered during the strain-rate change. The validity of such an assumption may be checked by observing how the magnitudes of Tef f and ri determined depend on the value of t l / t 2 where t l is the base strain-rate and ~2 the strain rate to which the strain rate is changed from e l . If the rearrangement o f dislocations takes place during the strain-rate change test, the variation o f dislocation configuration may be a function o f the time which is spent during a strain-rate change test. This time is roughly proportional to e 2 1 . Such variation of dislocation configuration, if it exists, would naturally result in the variation of Te~f and ri with tl/~2.
Figure 1 shows Teff determined as a function o f t l / t 2 at T = 600°C for two values of t l . Te ff was det erm i ned for the same strain. It is seen t h a t t he value of Te~t has a weak dependence on tl/t2 showing t hat slight rearrangement of dislocations takes place during a strain-rate change test. The true value of the effective stress would be the value o f Teff extrapolated to e l / e 2 = 1 in Fig. 1. We determined Tef f for e l / e 2 = 10. Such values of Teft are seen t o be higher than the true values by less than 5 percent. This magnitude o f error in T~f~ is not i m p o r t a n t from the viewpoint o f t he dependence o f T~f~ on the d e f o r m a t i o n condition and is completely negligible. Figure 2 shows the behavior of ra, r~tt and Ti against the strain for the d e f o r m a t i o n at T = 600°C. The general feature observed in the
265
i
1.5
0.15 E E
e~= 4 0 X I 0"4sec-t 0.10
~
T
o
/
% E
0.05
5
T = 600°C
~O5
~1= 6 0 ]l fO-Ssec-I
F-
IJO
i
2i0
30
(1"/.) = I . I X I 0-4SgC"1
1.0
/
E E
==0.5
I
I
k~
I I
: 0,1C -II
k*
1"etf
i10
15
0.20 -- I I I L~ I
~E 0.15
~
I 20
JJ0
I
3JO
l
((%)
I
T =600ec (,= I.I X 10-4sec-'
0'05 t r
= 2, I = rO-Ssec-~ _
0t
I
I0
}, /
~2
previous paper is retained here, namely, re f( is constant against the strain in the deformation stage b e y o n d the middle o f stage 0 and workhardening proceeds entirely through ri. Figure 3 shows the stationary values of r~f( plotted against the strain rate. It is seen that r ~ f depends on the strain rate as O~= 0 . 2 8 + 0 . 0 1
Thus, from e q n . Nm depend o n ~ as cc ~ 0 . 4 2 ,
(1)
1"=
30
2~0
Fig. 1. T h e e f f e c t i v e stress reff d e t e r m i n e d for the s a m e strain p l o t t e d against ~1l¢2 for t w o values of ~1 at 600°C. e l is the base strain-rate and ~2 the strain rate to w h i c h the strain rate is changed from ~1 in d e t e r m i n i n g Tel f.
Teff cc ~c~
0.5
.
(2)
I.,.
0'~ IlO
,l5
e (%)
Fig. 2. Behavior o f the flow stress ra, the e f f e c t i v e stress Tell, and the interaction stress Ti against the strain e at 600 ° C for various strain-rates.
580°C is the same as that at T = 600°C, the c-dependence o f the absolute magnitudes of V and Nm is obtained as shown in Fig. 4. It is to be noticed that the magnitude o f 0 is lower than that estimated in the previous paper by approximately an order of magnitude and that of Nm is higher than the previous one
a n d ~ =NmOb, ~ and
Nm ~ e0.5S.
"~'ef
1.0
T = 600"C
(3)
N o w , we estimate the absolute values of and Nm. The velocity measurement of dislocations 4 has been c o n d u c t e d at T = 5 8 0 ° C and n o t at T = 600°C. Hence, the strain-rate sensitivity data at T = 6 0 0 ° C cannot be used for such a purpose. We have data o f strain-rate change tests at T = 5 8 0 ° C o n l y for ez = 1.1 × 10 - 4 sec - 1 . These data are used for the estimation o f the absolute values of ~ and Nm. Assuming that the e-dependence o f Tel f at T --
0.1 Teff o=
~o
a =0.28_+ 0.01 L
iO-S
10.4
I0-~
(s=c") Fig. 3. The variation o f the stationary value of the effective stress reff with the strain rate ~ in the deformation at 6 0 0 ° C.
266
T = 580"C
L
=
10-4
L
(5)
dx
fo le(ra -- ri(x))3/2
A
g E
i>
10-5
I0 a o E z
This Y is taken as the mean velocity of all moving dislocations in the crystal. The magnitudes of ro and zA are determined by the strain-rate-change tests for two values of e2/el at the same strain, namely, L
f
i0 7 I
10-5
I
10-4
[Ta. 1 - T 0 - 7 " A
sin (27rx/L)] 3/2
0
~1
L f dx 0 [ra. 2 - r 0 - r A sin(21rx/L)] 3/2
1
10-3
(sec-')
Fig. 4. The variation of the equilibrium values o f the density N m and the velocity b- of moving dislocations with the strain rate ~ in the d e f o r m a t i o n at 580°C.
(6) L
f also by an order of magnitude. This comes from the difference between the velocity data of S c h ~ e r and of Patel--Freeland. The above result leads to the alternation of a conclusion on the mobile fraction of dislocations reached in a previous paper5. It was concluded in the previous paper that only several percent of total dislocations are mobile at lower yield point. However, it should be concluded here that a considerable fraction (~ several tens percent} of total dislocations is mobile at lower yield point in the deformation at this temperature range. All of the above results have been derived on the basis of the model that ri is constant with respect to the location of dislocations on the active slip plane. The validity of such model can be checked by the combined strain-rate change experiments described in the previous paper. Suppose that a periodic internal stress r i which is given by
Ti= TO + ~'A Sin (2rrx/L)
dx
~2
(4)
acts on the active slip plane and makes a dominant contribution to r i. Here, ro and rA are constants, L the period of the internal stress field, x the position on the active slip plane. The time average of the velocity of a dislocation on such a slip plane is given by
dx
e3
0 [ r a . l - - r o - - r A sin(2Trx/L)] 3/2
C1
L dx f [~'a,3--T0--?'A sin (2rrx/L)] 3/2 0
w h e r e 7a, 1 , Ta, 2 and ra,a are the flow stresses for the strain rates e l , e2 and e3, respectively, in the strain-rate change tests, el being the base strain-rate. The magnitudes of ro and vA are obtained by solving simultaneous equations of eqn. (6} numerically. The magnitudes of Vo and rA determined with ~2/el = 1/10 and e 3 / q = 1/20 are given in Table 1 together with those of Ti obtained with the constant ri model for the deformation at el = 1.1 X 10 - 4 sec - 1 and T = 600°C. It is seen that the magnitude of TA is almost zero and the magnitude of ro is very close to that of ri. Since the internal state of the crystal, after a strain-rate change {~i-+~2 may be slightly different from that after el-~ea as seen in Fig. 1, the values shown in Table 1 may not be quite accurate. Nevertheless, the results in the table are effective enough to show the validity of the constant interaction stress model. Another verification of the model will be given in the paper which follows ~.
267 TABLE 1 Values of To and 7A determined by the combined strain-rate change experiments De formation stage
c (%)
TO (kg/mm 2)
TA
Ti
(kg/mm 2)
(kg/mm 2)
I I
7.4 13.0
0.4081 0.4504
0.0001 0.0003
0.404 0.449
3. TRANSITION OF THE STATE OF DISLOCATION MOTION FROM ONE EQUILIBRIUM STATE TO ANOTHER BY THE CHANGE OF STRAIN RATE T h e p r e v i o u s p a p e r a n d t h e last s e c t i o n in t h e p r e s e n t p a p e r h a v e s h o w n t h a t in a cons t a n t strain-rate d e f o r m a t i o n t h e m a g n i t u d e o f Tef f is d e t e r m i n e d e n t i r e l y b y t h e values o f t and T and does not depend on the internal structure developing with deformation once t h e m i d d l e o f stage 0 is passed. This m e a n s t h a t t h e e q u i l i b r i u m s t a t i o n a r y s t a t e o f disloc a t i o n m o t i o n , w h i c h is c h a r a c t e r i z e d b y certain c o n s t a n t values o f ~ a n d N m , is realized d u r i n g t h e c o n s t a n t strain-rate d e f o r m a t i o n a n d t h a t values o f ~ a n d Nm in such equilibr i u m s t a t e are d e t e r m i n e d o n l y b y t h e d e f o r m a t i o n c o n d i t i o n ( t , T ) . A m o d e l describing this e q u i l i b r i u m s t a t e is given in t h e n e x t paper. N o w , it is i n t e r e s t i n g t o see w h e t h e r t h e deformation history of the crystal affects such e q u i l i b r i u m s t a t e o f m o v i n g dislocations. We d e f o r m a c r y s t a l at a c e r t a i n s t r a i n - r a t e t o stage I a n d realize a c o r r e s p o n d i n g equilibr i u m s t a t e o f t h e d i s l o c a t i o n m o t i o n in t h e crystal. T h e n , w e c h a n g e t h e strain r a t e sudd e n l y t o a n o t h e r value k e e p i n g the t e m p e r a t u r e u n c h a n g e d , o b s e r v e t h e s t a t e o f t h e disloc a t i o n m o t i o n a f t e r t h e c h a n g e o f strain rate, a n d c o m p a r e it w i t h t h a t in a c r y s t a l w h i c h is d e f o r m e d at t h e latter strain r a t e f r o m t h e b e g i n n i n g o f d e f o r m a t i o n . I t is to be e m p h a sized t h a t t h e d e n s i t y o f t o t a l d i s l o c a t i o n s in t h e crystal is q u i t e d i f f e r e n t b e t w e e n t h e s e t w o crystals, o n l y t h e strain r a t e being t h e s a m e. In t h e f o l l o w i n g , t h e results o f such e x p e r i m e n t s are s h o w n . P r o c e d u r e s o f t h e s p e c i m e n preparation, deformation, determination of Teff a n d 7i, etc. are d e s c r i b e d in t h e p r e v i o u s p a p e r s 1,7, S .
1.5
T-600'C = 4.2 ll0 5 s e c ' ~
~ - 4 2 x I0-4$ec "d
',,,
,b
,'5 (
io
('/.)
Fig. 5. The stress--strain behavior of a specimen deformed first at a strain rate of 4.2 x 10 - 5 s e c - 1 to the middle of stage I and, then, subjected to the change of strain rate to a strain rate of 4.2 × 10 - 4 sec - 1 at 600 °C. Full circles and full triangles represent 7ef f and ri for the specimen, respectively. The behavior of a specimen deformed at a strain rate of 4.2 x 10 - 4 sec - 1 throughout all deformation stages is shown by broken lines. Open circles and open triangles show Tel f and zi of the latter specimen, respectively. Figure 5 s h o w s stress--strain b e h a v i o r o f a s p e c i m e n d e f o r m e d first at ~ = 4.2 X 10 - 5 sec- 1 and then subjected to the change of the strain r a t e to t = 4.2 X 10 - 4 sec - 1 at t h e m i d d l e o f stage I. B e h a v i o r o f a s p e c i m e n def o r m e d at e = 4.2 X 10 - 4 sec - 1 f r o m t h e b e g i n n i n g o f d e f o r m a t i o n is also s h o w n b y b r o k e n lines. It is seen t h a t t h e value o f zef~ a f t e r t h e c h a n g e o f strain r a t e c o i n c i d e s w i t h t h a t o f t h e s p e c i m e n d e f o r m e d at t h e s e c o n d strain r a t e f r o m t h e beginning. T h u s , it is conc l u d e d t h a t t h e values o f ~ a n d Nm d o n o t dep e n d o n t h e d e n s i t y o f t o t a l d i s l o c a t i o n s in the c r y s t a l or o n t h e values o f ~ a n d N m w h i c h t h e c r y s t a l has a s s u m e d p r e v i o u s l y b u t are d e t e r m i n e d c o m p l e t e l y b y t h e value o f t h e strain r a t e o f t h e crystal. T h e d i f f e r e n c e in t h e f l o w stress b e t w e e n t w o s t r e s s - - s t r a i n curves in Fig. 5 a f t e r t h e c h a n g e o f strain r a t e is seen to be e n t i r e l y d u e t o t h e d i f f e r e n c e in Ti. Figure 6 s h o w s t h e b e h a v i o r d u e to t h e c h a n g e o f strain r a t e f r o m ~ = 4.2 X 10 - 4 sec - 1 to ~ = 4.2 X 10 - 5 sec - 1 . B r o k e n lines show the stress~strain characteristics of a s p e c i m e n d e f o r m e d at t = 4.2 X 1 0 - 5 s e c - 1 f r o m t h e b e g i n n i n g o f d e f o r m a t i o n . As to Te~f essentially t h e s a m e result as t h a t in Fig. 5 is o b t a i n e d . T h u s , also in this case, it is concluded that the equilibrium state of moving d i s l o c a t i o n s is c o n t r o l l e d o n l y b y t h e deform a t i o n c o n d i t i o n . As first o b s e r v e d in t h e pre-
268
J.5 I0
T - 600"C -4.2 110-"$e¢-'~
T -600'C - 4 2 x lO-4se¢ -' -~--~- 4.2 z
JO~e¢-'
~--4.2 ~ I0" ~$e¢ -'
~ - hC
j"j 0.5
.~-~ 0.5
Tiff • ~-o--__.o..o--o-..r-o---o-_-.~.~.~--.o-
~: = - .
=
T,.
.,. Tiff
(%]
06
~
,b
,'5 E
Fig. 6. T h e stress---strain b e h a v i o r o f a s p e c i m e n def o r m e d first at a s t r a i n rate o f 4.2 × 10 _ 4 sec - 1 to t h e m i d d l e o f stage I and, t h e n s u b j e c t e d t o t h e c h a n g e o f strain rate to a strain rate o f 4.2 × 10 - 5 sec - 1 at 600°C. Full circles a n d full triangles repres e n t 7ef f a n d Ti for t h e s p e c i m e n , respectively. T h e b e h a v i o r o f a s p e c i m e n d e f o r m e d at a strain rate o f 4.2 × 10 - 5 sec - 1 t h r o u g h o u t all d e f o r m a t i o n stages is s h o w n b y b r o k e n lines. O p e n circles s h o w r e f f o f the latter specimen.
vious paper 5 work-softening takes place after the lowering of the strain rate. Figure 6 shows clearly such work-softening to proceed through Ti. This phenomenon may be interpreted in the following way. A considerable fraction of moving dislocations is immobilized by the lowering of the strain rate. Such immobilized dislocations may not be able to take the most stable configuration instantaneously. Thus, unstable configuration of immobilized dislocations would be realized in the crystal. The long-range internal stress fields around such immobilized dislocations are n o t released efficiently. Thus, they would make a rather large contribution to 7i. This point will be discussed in more detail in the next paper. As the deformation proceeds the immobilized dislocations would settle into a more stable configuration by means of the trapping of moving dislocations and relaxing the long-range internal stress fields around them, thus, ri becoming smaller as the strain increases. This is thought to be the cause for the work-softening occurring after the lowering of the strain rate. On deforming an undeformed crystal at a certain strain-rate, a fairly large strain is needed before the equilibrium state of moving dislocations is established. Namely, yield region and approximately a half of stage 0 where %f~ varies with strain precede the region of constant r~ff. Similarly, a transient stage is
t0
(%)
Fig. 7. T r a n s i e n t b e h a v i o r o f tee f o b s e r v e d for t h e c h a n g e o f s t r a i n rate f r o m 4.2 × 10 - 4 sec - 1 t o 4.2 × 10 - 5 sec - 1 at 6 0 0 ° C .
observed to exist when the state of dislocation motion changes from one equilibrium state to another. Such transient stage has been omitted to show in Figs. 5 and 6. The transient phenomenon is demonstrated in Fig. 7. A specimen deformed first at e = 4.2 × 10 - 4 sec-1 at 600°C is subjected to the change of strain rate to ~ = 4.2 × 10 - 5 sec - 1 in the middle of stage I and then, after a strain of about 1.3%, subjected again to the change of strain rate back to e = 4.2 × 10 -4 sec-~. The transient behavior in 7elf is shown for the first change of strain rate. It is seen that a transient stage of approximately 0.5% in strain appears before Teff reaches a new equilibrium value from an old one. If the equilibrium state of dislocation motion could change instantaneously from one to another, then it is impossible to determine the values of ~i and ~ by strain-rate change tests. It is emphasized here that the existence of the transient stage seen in Fig. 7 makes it possible to determine T~f~ and ~i by strain-rate change tests.
REFERENCES 1 K. S u m i n o a n d K. Kojima, Crystal Lattice Defects, 2 ( 1 9 7 1 ) 159. 2 S. Sch~ifer, Phys. Status Solidi, 19 ( 1 9 6 7 ) 297. 3 H. S c h a u m b u r g , Phil. Mag., 25 ( 1 9 7 2 ) 1429. 4 J.R. Patel a n d P.E. F r e e l a n d , J. Appl. Phys., 42 (1971) 3298. 5 K. K o j i m a a n d K. S u m i n o , Crystal Lattice Defects, 2 ( 1 9 7 1 ) 147. 6 K. S u m i n o , Mater. Sci. Eng., this issue. 7 K. K o j i m a a n d K. S u m i n o , J. Phys. Soc. Japan, 26 ( 1 9 6 9 ) 1213. 8 K. K o j i m a a n d K. S u m i n o , J. Phys. Soc. Japan, 31 ( 1 9 7 1 ) 171.