A model for the dynamical state of dislocations in crystals during deformation

Materials Science and Engineering, 13 (1974) 269--275 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

A model for the dynamical state of dislocations in crystals during deformation KOJI S U M I N O

The Research Institute for Iron, Steel and Other Metals, Tohoku University, Sendai (Japan) (Received February 26, 1973)

Summary The equilibrium state of moving dislocations in a crystal during the constant strainrate deformation is discussed on the basis of three hypotheses. I. The configuration of immobile dislocations is such that the static free energy o f the crystal associated with existing immobile dislocations be as low as possible. II. There are certain equilibrium stationary values in the density and the velocity of moving dislocations which depend on the ratecontrolling mechanism o f dislocation motion and on the deformation condition. III. The equilibrium state of moving dislocations is determined so as to make the c o m p o n e n t o f the flow stress associated with moving dislocations minimum to maintain the given strain rate. This model is shown to give a good description of the strain-rate dependence o f the deformation behavior in germanium crystals at 6 O0° C observed experim en tally.

1. INTRODUC'rION The strain rate ~ of a crystal is generally expressed as = Nm-Ub

(1)

where N m is the density of moving dislocations, ~ the dislocation velocity averaged over all moving dislocations and b the magnitude of Burgers vector. This expression is especially useful to apply to crystals in which dislocations move in a viscous way. The velocity v of a dislocation can be discussed as a function of the applied stress T~,

the t e m p e r a t u r e T, and the dislocation configuration in the crystal if the rate-controlling process in t he dislocation m o t i o n and the m o d e of interaction between dislocations are specified. Usually such v is equated to ~ in eqn. (1) and the flow stress of the crystal is derived by solving the following equation with respect to Ta :

v(Ta, T, N* ) = 6/N m b,

(2)

where N* is a parameter specifying the densit y and the configuration of overall dislocations in the crystal. The flow stress thus obtained is a f u n c t i o n of the density of moving dislocations N m . In most theories of the flow stress so far presented the magnitude of Nm has been assumed to be i n d e p e n d e n t of or a weak funct i on of e and T. Recent experimental work on germanium crystals has shown t hat N m is a rather sensitive function of ~ and T1-3. There are infinite possible combinations of values between Nm and ~ to give any specified value of ~. However, only one specific combination among t h e m would be chosen by the crystal when the d e f o r m a t i o n condition is specified. This is substantiated by a wellk n o w n fact that the measured flow stress of crystals is reproducible for the same internal state and the same d e f o r m a t i o n condition. Thus, there must be a physical principle which governs the com bi nat i on of Nm and U to give any specified value of ~. To accomplish the t h e o r y of the flow stress of crystals it is necessary to establish the knowledge on how the c o m b i n a t i o n of N m and u is d e t e r m i n e d as a f u n c t i o n of the deformation condition. In a preceding paper 3 such

270

a combination of Nm and 0 has been clarified experimentally as a function of ~ for the deformation of germanium crystals at 600°C. The purpose of this paper is to present a model giving the combination rule of Nm and which accounts for the experimentally observed fact. 2. A MODEL ON THE EQUILIBRIUM STATE OF DISLOCATION MOTION DURING DEFORMATION

We define a crystal as in the steady state of deformation if it is being deformed in a welldefined deformation stage at a constant temperature and a constant strain rate. For example, stage I and stage II in a stress--strain curve are in the steady state of deformation while yield region and stage 0 are not so since they may be regarded as transient stages from undeformed state to stage I. It is believed that the rate-controlling mechanism in dislocation motion is identical throughout a well-defined stage of deformation. Experiments on germanium crystals1'3 have shown that the equilibrium stationary state of dislocation motion, which is characterized by constant values of N m and U against the strain, is realized in the crystal in such steady state of deformation. We deal with such equilibrium state of dislocation motion with the help of three hypotheses which will be mentioned in turn. For convenience, the discussion here is confined to stage I deformation. It has been shown that a higher value of Nm is associated with a higher level of the flow stress and longer strain-interval of each deformation stage e .

T{°C) • IO s

IO

~ {sec 'I dNT IdE(cm 2)

o-----o

720

I~xlO

~

o--e

600 660 600

IlxPO

5

I 6x[O

IlxtO

"4

28xi09

4

38X109

D--a

n--~

I [x/O

0 86~10 ~ ~

5

O(

5

I0

15

20

(°Io)

Fig. 1. The density of total dislocations N T in stage I counted by replica electron microscopy plotted against the strain e. The temperature T and the strain rate ~ are taken as parameters.

The density of total dislocations NT in a crystal increases as the strain increases. The magnitude of NT at any specified strain depends on ~ and T rather sensitively. In Fig. 1, NT measured by replica electron microscopy is plotted against the strain e for stage I deformation of germanium crystals with an easy glide orientation, the deformation condition being taken as a parameter. Data were taken from the previous paper 2. It is seen that dNv/de is a rather sensitive function of ~ and T. On the other hand, the work-hardening rate dri/de for crystals with the same orientation has been found to be independent of T and to be a weak function of ~ in stage I. These facts lead to an important conclusion that the stress increment due to the workhardening is n o t determined by the increment of NT uniquely. This conclusion is also supported by the work-softening phenomenon found in the preceding paper 3. It has been observed that the interaction stress Ti is proportional to NT~/2 in stage 0. However, it has also been found that such relationship does not hold in stage I or in stage II1. This seems to suggest that the total dislocation density does not contribute to the interaction stress as in a usually proposed manner (ri ~ N¢/2), at least in stage I and stage II of crystals having high Peierls barrier. The configurations of immobile dislocations during deformation seen by direct observations in stage I of germanium crystals are dipoles, dipole clusters, tangled dislocations, etc. l'4'~ Such clusterings of dislocations make the sum of Burgers vectors of dislocations contained in each cluster small and reduce the energy of elastic strain field around the dislocations. It is thought that the stress field of immobile dislocations in such a configuration does not extend over a long distance. Thus, we put here the first hypothesis (Hypothesis I): The configuration o f immobile dislocations in the steady state o f deformation is determined so as to make the static free energy o f the crystal associated with existing immobile dislocations as low as possible. Direct consequences of this hypothesis are that immobile dislocations may not make the d o m i n a n t contribution to Ti and also that N w may not be related to ri simply as N~/2 cc ~i. It has been observed that the magnitude of ref~ is constant against the strain in both

271

stage I and stage II. Such magnitude of zef~ is reproducible and d e t e r m i n e d only by the def o r m a t i o n condition. Thus, we put the second hypothesis (Hypothesis II):

There are certain equilibrium stationary values in the density and in the velocity o f moving dislocations in the steady state o f deformation which depend on the rate-controlling mechanism o f dislocation motion and on the deformation condition. The interaction stress 7 i is defined by dividing the interaction force e xe r t ed on a moving dislocation segment by the magnitude of the Burgers vector. It may originate f r om longrange stress field of dislocations, resistance due to attractive junctions, line tension associated with the bowing out of dislocations around some pinning points, etc. If the main part of r~ originates from longrange stress field ar ound dislocations, it is natural to th in k that the value of r~ at any mom e n t fluctuates against the position on the active slip plane so t hat its mean value becomes zero. In the case that ri originates f r om long-range stress field of immobile dislocations, the displacement of moving dislocations does n o t affect the distribution of ri on the active slip plane. Thus, a moving dislocation feels a periodically varying interaction force during its travel and, hence, its velocity would vary every m o m e n t . However, the preceding paper has shown this model to be invalid. Here, it is shown by anot her way. Let the length of dislocations in unit volume of a crystal along which the magnitude of the interaction stress is between 7i and ri + dri be n(ri)dri. The strain rate ~ of a germanium crystal at 600°C is, then, given by

= bkfn(ri)

(T a __ Ti ) 3 / 2

= bkNm (r a - - y i )3/2 "

dT i

(3)

The meanings o f k and the e x p o n e n t are given in the preceding paper. ~i stands for some averaged interaction stress and is a f unc t i on of ra. Let the m a x i m u m value o f T i on the active slip plane be r m ~x. Then, Ti ~

./_max i

On the o t h e r hand, as far as a macroscopic d e f o r m a t i o n takes place, Ta ~

T max i

The magnitude of the interaction stress determined by strain-rate change experiments in this case is t hat of ~i. According to the strainrate change experiments on germanium crystals, the magnitude of flow stress ra,2 immediately after the strain-rate change el -~ ~2 tends to converge to the magnitude of t-i det erm i ned as the value of ¢2 becomes smaller. max This would n o t be expect ed if ~i < r i max Thus, it may be concluded t hat ~i ~ r i • This means t hat t he width o f the distribution funct i on n(ri) is rather narrow around the experimentally det erm i ned value of ~i and that r~ is a p p r o x i m a t e l y constant with respect to the location o f moving dislocations on the active slip plane. In the case where the main part of the interaction force originates from interaction between moving dislocations themselves, the displacement of a dislocation gives rise to a change in the state of the m o t i o n of dislocations around it and, in turn, results in the variation o f the distribution of ri on its own active slip plane. It is expect ed that the interaction tends to make the dislocation velocity in the crystal uniform. This would be visualized, for example, if one recognizes a situation t hat a slowly moving dislocation exerts a force on a fast moving dislocation to decelerate the latter while the latter exerts an accelerating force on the f o r m e r as far as the densit y of moving dislocations in the crystal is constant. This means that energy and m o m e n t u m transfers take place between various parts of a crystal t hrough the interaction between moving dislocations. We have already seen that immobile dislocations do n o t determine the state of moving dislocations in the steady state of deformation. Thus, the interaction between moving dislocations such as m e n t i o n e d above is expected to play an i m p o r t a n t role in determining the state of moving dislocations themselves. It should be noticed here, however, that the velocities of all moving dislocations are never identical in the strict sense. The configuration of total dislocations is determined definitely if we give the same Tel f to all moving dislocations. When all moving dislocations displace by the same distance from such configuration, the relative configuration of the total dislocations would be altered and, consequently, the magnitude of reff at the location

272

of each moving dislocation would also be changed. Thus, the meaning of Hypothesis II is that when the dispersion in the dislocation velocity takes place a force is induced between moving dislocations so as to make their velocities uniform, and that the rearrangement of moving dislocations always takes place in this direction in the steady state of deformation. Hence, the velocities of moving dislocations are not constant even in the steady state of deformation in the strict sense but fluctuate around a certain equilibrium value with a relatively small deviation. The equilibrium stationary values of Nm and Y stated in Hypothesis II should be understood in this way. It should be emphasized here that, if such restoring force is not induced between moving dislocations, the distribution of the dislocation velocity changes with time, being controlled by some incidental factors. This must lead to the change in Nm. The reproducibility of the measured value of the flow stress would, then, be lost. It is also to be noticed that the present model gives the basis for expressing the velocity of dislocations in germanium crystals during deformation as

3. D E R I V A T I O N OF EQUILIBRIUM N m A N D ff I N T H E D E F O R M A T I O N NIUM CRYSTALS AT 600 ° C

~ = k ( r a - - r~)3/2

which is proposed in the previous papers. As a rule which governs the equilibrium values of Nm and ~ in Hypothesis II we put the third hypothesis (Hypothesis III): The equilibrium state of moving dislocations is determined so as to make the component o f the flow stress associated with moving dislocations minimum to maintain the given strain rate. For a constant strain rate the increase in the density of moving dislocations Nm leads to the decrease in the velocity and, consequently, to the decrease in the effective stress r e ~ . At the same time, the increase in Nm brings about the increase in the interaction stress between moving dislocations r o. Hence, the component of the flow stress associated with moving dislocations reff + r ° takes a minimum value for a certain value of Nm satisfying 8(~'ef ~ + 7.0 ) / O N m = 0 .

A hypothesis of this kind was first proposed by Takeuchi 6 in the interpretation of the lower yield stress of b.c.c, alloys. Recently, he extended such an idea to the yielding of other materials 7 due to inhomogeneous deformation. However, there seems to be no reason why such a model holds only in the yield region. It is t h o u g h t that immobile dislocations do not determine the state of moving dislocations in the steady state of deformation as demonstrated previously. Also, immobile dislocations are not expected to take a configuration which is governed by dynamical laws such as Hypothesis III since they are in a stable state. Thus, the rule governing the dynamical state of moving dislocations would be associated with only the c o m p o n e n t of the flow stress associated with moving dislocations. A complement of Hypothesis III is that in the deformation under a constant applied stress the equilibrium values of N m and ~ of moving dislocations are determined so as to make the strain rate maximum. This rule would be applicable to creep deformation.

(4)

VALUES OF OF GERMA-

We now deal with the state of dislocation motion in the steady state of deformation of germanium crystals at 600°C with the help of the three hypotheses mentioned in the preceding section. The velocity of moving dislocations in such a state of deformation is given by V'---- k ( T a - - T i ) 3 / 2 ---- k T 3e ,/~2 "

(5)

It is assumed here that ri is composed of the interaction stress due to the interaction between moving dislocations r o and that associated with the work-hardening rh, namely, 7"i = T ° + T h •

(6)

Since the values of Nm and Y are determined only by the deformation condition, r ° would be constant with respect to the strain in stage I and stage II by the first approximation. Possibly, rh comes from immobile dislocations or some other lattice defects intro-

273

duced by deformation. It is expected that T° makes a large contribution to the magnitude of ri even though the value of Nm is considerably lower than that of total dislocations since the stress field around each moving dislocation is n o t screened. The magnitude of rh may fluctuate against the position on the active slip plane. It is t h o u g h t that moving dislocations are distributed on the active slip plane so that To levels the magnitudes of r ° + rh with respect to the location of moving dislocations. For convenience, we assume r o to be given by TO

=

A ~/rNm,

It is seen from eqn. (12) that Teff cc ~0.286.

The experimental results in the previous paper show that Teff o: ~0.28±0.01.

Thus, agreement between theory and experiment is quite excellent in the strain-rate dependence of Tef f (hence, also in those of Nm and 0). It is seen from eqns. (12) and (13) that

(7)

r ° / r f f = 4/3 = 1.333,

(15)

where A is constant, and rh by r h = O(e -

%),

(8)

where @ is the work-hardening rate and eo a o constant. Here, some averaging of Ti and rh with respect to the location on the active slip plane has been done. The strain rate 4 is given by = N m b k ( T a - - 7"? - - T h ) 3 / 2 .

(9)

With the above model the flow stress ra is expressed as T a

=

(~/]~b)2/3N-m2/3 + A V~m + ®(e --eo).

(10) The sum of the first and the second terms on the right-hand side is the c o m p o n e n t of the flow stress associated with moving dislocations as mentioned in Hypothesis III. The condition of eqn. (4) gives the equilibrium value of Nm as N * = (4/3 A)6/7 (~/kb)4/7.

(11)

T e f f and T° are calculated with the help of N ~ and are given by

"ref f = ( 3 A / 4 )

4/7 ( E / k b ) 2/7 ,

TO = (4/3)3/7A4/7

( ~ / ] g b ) 217

i

(12) (13)

Accordingly, 0 is given by -U= ( 3 A / 4 ) 6 / 7

k 4/7 ( ~ / b ) 3/7 "

(14)

irrespective of the deformation condition and o the strain. The magnitude of r i can be evaluated with the help of eqn. (15) for any deformation condition since %f~ is determined experimentally. The present model requires r ° to be n o t greater than ri. Figure 2 shows the behavior of r~ff, Ti and r ° against the strain in germanium crystals for various deformation conditions, the data being taken from the preceding paper. It is seen that Ti is always higher than r o in stage I in agreement with the theoretical requirement. The magnitude of z ° coincides with that of Ti approximately at lower yield point. Possibly, the equilibrium state of the dislocation motion would be first established during stage 0. Since the magnitude o f N m ( = 4 / 0b) is evaluated from eqn. (5) by using experiment a l l y measured values of ref~, the absolute magnitude of A in eqn. (7) can be estimated from the values of r o determined. On expressing 7-0 = A x / ~ m = G b

x/N m/fl,

(16)

where G is the shear modulus, the magnitude of ~ is determined from experimental results. The data for the deformation at t = 1.1 × 10 - 4 sec - 1 and T = 580°C are used for this purpose since the data on the absolute values of ~ are available at this temperature. The equilibrium value of Teff measured for this deformation condition is found to be 0.20 kg/mm2. The corresponding values of 5 and Nm are 4.6× 1 0 - 5 cm/sec and 6.0 X 107 c m - 2 , respectively. With z o = 0.267 kg/mm 2, is obtained from eqn. (16) as

274 flow stress at some strain which is thought to correspond to the lower yield point. Haasen 9 has extended the idea to the deformation of covalent crystals. He assumes that total dislocations in a crystal are moving and also that the stress component associated with work hardening comes entirely from the interaction between such total dislocations. The condition of the lower yield point for any deformation condition is given by

P5

,

T =EO0°C

E =4.0x

10-4sec t

To EI0

E E

"~05

o/~

T? Te ff

Io

T = 600°C

APO

E

~o

(%}

3o

~= I Ix1 0 - 4 s e c - a ~

OT a

OT a

=

E

~e

bN T

-bN T

=0.

(17)

~e

~05 •

o---o1"elf

0

I0

20 E

30

Since N T is an increasing function of e and NT = Nm, eqn. (17) leads to

(%)

aTalaN m = 0 T =600°C,

E = 2 I xlO-Ssec "l

O~

To

E E

I" i

v F~ °-~.'~o__

.--o .....

o-----,,

. . . . .

I0

5

E

•. . . . .

Teff 15

(%)

Fig. 2. B e h a v i o r o f t h e f l o w s t r e s s Ta, t h e e f f e c t i v e s t r e s s r e f f , t h e t o t a l i n t e r a c t i o n s t r e s s Ti a n d t h e i n t e r . • • o action stress due to mobile dislocations r i against the strain e at 600 ° C for various strain rates.

fi= 6.8. This result is satisfactory in the view that the theoretically expected value of /3 is approximately 2~.

4. D I S C U S S I O N

4.1. Comparison of the present model with Haasen's theory of lower yield stress According to the Johnston--Gilman type model of yielding, the lower yield point appears as a result of the balance of the stress decrease due to the increase in Nm with the stress increase due to the work hardening. The magnitude of Nm is assumed to increase with e in such a model even in the deformation stage beyond the lower yield point. The numerical calculation by Johnston s has actually shown that there appears a minimum in the

(18)

which is essentially identical with eqn. (4) in form. However, the physical idea from which eqn. (4) originates is different from that from which eqn. (18) originates. In the model proposed in this paper the magnitude of Nm satisfying eqn. (4) gives the equilibrium density of moving dislocations in any steady state of deformation. On the other hand, in Haasen's model it means the magnitude of Nm or N w at the lower yield point and differs from the values of Nm in any deformation stages other than the lower yield point. The experiments in the previous papers show that though the value of Te f f is still decreasing against the strain at lower yield point the value there is rather close to the equilibrium value of Te ~f in the steady state of deformation. Also, NT is rather close to Nm at the lower yield point. Thus, it is probable that the lower yield stress would have the apparently same dependence on the deformation condition as the equilibrium value of r e ~ has. It should be emphasized here that the present model does not conflict with the Johnston--Gilman type model of yielding. From the viewpoint of the present model, the yield phenomenon is interpreted as a transient effect appearing in the period during which the magnitude of Nm increases from the value in the undeformed state to the equilibrium value of the given deformation condition. Thus, the idea of the Johnston-Gilman model is applied to the deformation stage before the steady

275 state is r e a c h e d , n a m e l y , t o the yield region. 4.2. Possible m e c h a n i s m for e - d e p e n d e n c e o f Teff

In some materials such as CaF 2 l o and InSb 11 r ~ has been observed to increase with strain in stage I. In the view o f t h e p r e s e n t model one possible interpretation for such a p h e n o m e n o n is as follows. It has b e e n r e p o r t e d t h a t laminar regions into which p r i m a r y dislocations c a n n o t p e n e t r a t e d e v e l o p as the strain increases during stage I d e f o r m a tion o f g e r m a n i u m c r y s t a l s 4 ' 5 ' 1 2 ' l 3 T h e d e n s i t y o f f o r e s t dislocations inside such laminar regions is very high and almost comparable with t h e density o f p r i m a r y dislocations outside t h e laminar regions. T h e laminar regions are localized along the p r i m a r y slip planes and o c c u p y a relatively small f r a c t i o n o f t h e crystal volume. T h e y are revealed o n the specimen s u r f a c e as i n h o m o g e n e i t i e s in the slip b a n d d i s t r i b u t i o n and in t h e e t c h pit d i s t r i b u t i o n . These are k n o w n as bands o f seco n d a r y slip and stripes, respectively. T h e crystal v o l u m e t h r o u g h which p r i m a r y dislocations m o v e f r e e l y decreases as such laminar regions d e v e l o p inside the crystal. Thus, t o k e e p t h e strain rate f o r a w h o l e crystal cons t a n t t h e local d e n s i t y o f m o v i n g dislocations s h o u l d increase a c c o r d i n g l y even if t h e t o t a l length o f m o v i n g dislocations in the crystal is c o n s t a n t p r o v i d e d t h a t n o d e f o r m a t i o n takes place inside t h e laminar regions. T h e q u a n t i t y A in eqn. (7) for such a case m a y be given b y A =Ao(1--f)-l/2

(19)

w h e r e Ao is a c o n s t a n t characterizing the int e r a c t i o n b e t w e e n m o v i n g dislocations, f is the v o l u m e f r a c t i o n o f the laminar regions and is an increasing f u n c t i o n o f the strain. Thus, t h e Magnitudes o f N m, r ~ and r °

given b y eqns. (11) t o (13) b e c o m e to d e p e n d o n t h e strain. It s h o u l d be n o t e d t h a t Tel f b e c o m e s an increasing f u n c t i o n o f t h e strain. It is t h o u g h t t h a t t h e m a g n i t u d e o f f is small c o m p a r e d with 1 or d e f o r m a t i o n due t o s e c o n d a r y slip takes place in the laminar regions in materials such as Ge, a - F e 14, NaC11 o, etc. f o r which T e f f is observed to be i n d e p e n d e n t o f t h e strain. T h e variation o f ref~ with the strain in such materials w o u l d n o t be det e c t e d within t h e a c c u r a c y o f e x p e r i m e n t s . On t h e o t h e r h a n d , the m a g n i t u d e o f f w o u l d be r a t h e r large and n o d e f o r m a t i o n takes place inside the laminar regions in materials such as CaF 2 and InSb resulting in the dep e n d e n c e o f reef on the strain. This w o u l d possibly be associated with characteristics o f dislocations in such materials.

REFERENCES

1 K. Sumino and K. Kojima, Crystal Lattice Defects, 2 (1971) 159. 2 K. Kojirna and K. Sumino, Crystal Lattice Defects, 2 (1971) 147. 3 K. Sumino, S. Kodaka and K. Kojima, Mater. Sci. Eng., 13 (1974) 263. 4 H. Alexander, Phys. Status Solidi, 27 (1968) 391. 5 K. Kojima and K. Sumino, J. Phys. Soc. Japan, 26 (1969) 1213. 6 S. Takeuchi, J. Phys. Soc. Japan, 27 (1969) 929. 7 S. Takeuchi, J. Phys. Soc. Japan, 35 (1973) 188. 8 W.G. Johnston, J. Appl. Phys., 33 (1962) 2716. 9 P. Haasen, Z. Physik, 167 (1962) 461. 10 A.G. Evans and P.L. Pratt, Phil. Mag., 21 (1970) 951. 11 H. Shimizu and K. Sumino, to be published. 12 H. Alexander and P. Haasen, Can. J. Phys., 45 (1967) 1209. 13 K. Kojima and K. Sumino, J. Phys. Soc. Japan, 31 (1971) 171. 14 W.A. Spitig and A.S. Keh, Met. Trans., 1 (1970) 3325.