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Computer simulation of the glide motion of a dislocation group containing a source
Materials Science and Engineering, 49 (1980) 133 - 139
133
Computer Simulation of the Glide Motion of a Dislocation Group Containing a Source PATU, LEI CHUNG-ZI and SHIH CHANG-HSU Institute o f Metal Research, Academia Sinica, Shenyang (China)
(Received August 1, 1980; in revised form November 21, 1980)
SUMMARY
The dynamic behaviour of long straight parallel coplanar dislocations emitted from a source under a constant applied stress and a constant applied stress rate has been analysed by a few researchers. In this paper the glide motion of a dislocation group emitted from a source under a constant applied stress, the annihilation of a dislocation group under a constant applied reverse stress and the motion o f a dislocation group containing a source in the absence o f any applied stress were investigated. It was found that the glide motion o f a dislocation group containing a source is irreversible; the interaction between dislocations in a group does not affect the average velocity o f all the dislocations in the group containing a source (the average velocity is always equal to the velocity o f an isolated dislocation which has been subjected to the same applied stress) and the velocity o f the leading dislocation is only slightly larger than that o f an isolated dislocation.
group behind an obstacle. Since then, much work (in which analytical and numerical methods have been used [2 - 13] ) has been carried o u t in this field. In this paper we consider the emission, motion and annihilation of an array of long straight parallel coplanar dislocations generated from a source by means of a computer simulation method. 2. THE MODEL AND METHOD
The model and m e t h o d of computation used in this investigation start with the same procedure as used in previous papers [6, 8, 11], but in this investigation we consider the annihilation of dislocations and the application of a constant reverse stress. 2.1. The dislocation motion equation We consider a group of N identical dislocations in the slip plane Y = 0. Each dislocation is parallel to the Z axis. The dislocation positions are given by 0 ~ XN < X N - 1
1. INTRODUCTION
It is well known that plastic deformation is related to the generation, motion and annihilation of dislocations in crystals. During the process of plastic deformation, dislocations always move in a group manner. For a detailed understanding of the microscopic mechanism of flow and fracture in crystalline materials, it is necessary to study the collective and cooperative m o t i o n of dislocations. The simplest dislocation group model is an array of long straight parallel coplanar dislocations. Early in 1951, Eshelby et al. [1] gave a theoretical analysis of a pile-up of such a 0025-5416/81/0000-0000/$02.50
<
...
< X2 <
X1
where X = 0 represents the source position. In general, the jth dislocation motion equation in the group is taken to be V1 -
dt
- M I T ' e f f i l rn S i g n ( t e l f / )
(1)
where M and m are characteristic constants of the material, re~fj is the effective shear stress acting on the jth dislocation in the group and Sign is used to denote a signal function which has the following properties: Sign(tell/) = 1
reff/ > 0
= 0
reff/ = 0
= -- 1
re~ti < 0
(2)
© Elsevier Sequoia]Printed in The Netherlands
134
This means that a positive stress produced a positive velocity and vice versa. If any long-range stress in the crystal is neglected, then reffi can be written as follows: ref,j = r. +
N
A
(3)
and
XN = 0
(7)
first of all, the N t h dislocation is annihilated by the source and the number of dislocation in the group becomes N -- 1.
1=1 Xj
and A =
.b (
cos2
+
sin'a /
(4)
where j and i are the index numbers of the dislocations and ra is the applied shear stress. The second term on the right-hand side of eqn. (3) is the stress exerted on the ]th dislocation by other dislocations in the group./~ and v are the shear modulus and Poisson's ratio respectively, b is the Burgers vector and is the angle between the dislocation line and its Burgers vector.
2.4. The method o f computation At some arbitrary time t, each individual dislocation position is calculated by the R u n g e - K u t t a m e t h o d which has been programmed for the DJS-130 computer. In the present paper the calculation is made for 60 ° dislocation in silicon single crystals. The corresponding values of M, m and A are taken from our experimental data [14], as listed in Table 1. For convenience the source operation stress r, is assumed to be zero in the computation. TABLE 1 Material constants used in the c o m p u t a t i o n
2.2. The condition for source operation An initial condition is that, at t = 0, N = 1 and X1 = 0, i.e. there is one dislocation at the source. The condition for source operation is that, if an effective shear stress acting on the source is equal to or exceeds a given source operation stress r,," then a new dislocation will be emitted from the source, and the number of dislocations in the group will change from N to N + 1. In this case the condition of generation of a new dislocation is written as N A r . -- Z > r,
2.3. The condition for annihilation o f a dislocation Obviously, the Nth dislocation in the group which contains N dislocations is nearest to the source. If TeffN < 0, then the Nth dislocation moves back to the source. When A
-- < 0
]=1 X /
M (ram s-1)
A (N m - 1 )
1173 1073 973 873 773 673
1.21 1.26 1.02 1.16 1.25 1.49
3.24 1.80 3.68 7.09 4.36 8.09
4.07 4.07 4.07 4.07 4.07 4.07
X 10 -12 × 10 -13 × 10 - 1 2 × 10 - 1 5 × 10 -17 × 10 -21
3. R E S U L T S A N D D I S C U S S I O N
where the second term on the left-hand side of eqn. (5) is the stress acting on the source by dislocations (except the (N + 1)th dislocation) in the group.
N-I
m
(K)
(5)
j = l ~-1
r. -- Z
Temperature
(6)
3.1. The motion o f a dislocation group emitted from a source Calculations are carried out for various values of M, m and A, as listed in Table 1, and for different constant applied shear stresses (5.0 - 200 MN m-Z). The calculation results show that dislocation configurations are only slightly affected by r , and M. A typical dislocation distribution in the group is plotted in Fig. 1. It is shown that, in a dislocation group emitted from a source, the dislocation density decreases as the distance from the source increases. This is similar to pile-up against the source. The calculation results also show that, the larger the applied stress, the higher is the density of dislocations in the group, as shown in Fig. 2.
135
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•
°,
•
**
•
•
o
•
•
•
I
•
•
"~effj
1.5 .
~
134
i
1.0
~3 u~ i~-?
..........................
26
0.5
He,•~oe,o,,o.*o..*.o • • • 2~ e l l l * . * o e o l l i t J * * o • • • • 22 ,.. .... • . . . . . . . . . . . . 20 ° H ° ° ° o o ° * * ° o , • , • • • . • o •
•
° •
°
° °
(a)
1.5
Viso
°16
1:0
14
0.S
.....
6
0
Xj ( 10-2mm ) 1
2
Nffil0
4
3
(b)
5
Fig. 1. T h e d i s l o c a t i o n d i s t r i b u t i o n in t h e g r o u p e m i t t e d f r o m a s o u r c e for various values o f N (~'a = 3 0 M N m - 2 ; T = 1 0 7 3 K).
N,20
i
i
i
i
5
1o
Is
2o
i
J zs
I
3o
N=3S L5 3
Fig. 3. (a) T h e effective s h e a r stress a c t i n g o n e a c h d i s l o c a t i o n in t e r m s o f t h e a p p l i e d s h e a r stress ( 3 0 MN m - 2 ) ; (b) t h e v e l o c i t y o f e a c h individual dislocat i o n in t e r m s o f t h e v e l o c i t y Vis o ( 8 . 4 2 x 10 - 3 m m s - 1 ) of a n isolated d i s l o c a t i o n ( T = 1 0 7 3 K).
J 20
l
%2
",. 3
"-,.. 4
i
i
l
2.0
4.0
6.0
s h
L
i
8.0
lo.o
12.o
Xj ("lO-2mm )
Fig. 2. T h e d i s l o c a t i o n d i s t r i b u t i o n in t h e g r o u p (N = 20) f o r d i f f e r e n t a p p l i e d s h e a r stresses ( T = 1 0 7 3 K ) : c u r v e 1, 90 M N m - 2 ; c u r v e 2, 30 M N m - 2 ; c u r v e 3, 18 M N m - 2 ; c u r v e 4, 10 MN m - 2 ; c u r v e 5, 5 M N m -2.
The effective shear stress Teff,/ on each individual dislocation in the group and the velocity Vs of each dislocation are calculated from eqns. (1) and {2); they are shown in Fig. 3. It is shown that, in the dislocation group emitted from a source, each dislocation veloc-
ity differs from the others. The largest velocity is that of the leading dislocation; half of the dislocations have velocities larger than the velocity Vis o of an isolated dislocation, and the other half of the dislocations have velocities which are smaller than Viso. The result also shows that at each instant of time the average velocity V of all dislocations in the group is nearly equal to Vi, o if the isolated dislocation has been subjected to the same applied shear stress, as listed in Table 2. This result is consistent with that obtained by Arsenault and coworkers [ 1 0 , 1 1 ]. This means that the interaction between dislocations in a group cannot affect the average velocity of all dislocations. The relation between the distance X1 of the leading dislocation from the source and time t under a constant applied shear stress is shown in Fig. 4. It can be seen that the distance travelled by the leading dislocation is
TABLE 2 C o m p a r i s o n o f V a n d Vis o f o r d i f f e r e n t c o n d i t i o n s M ( r a m s- 1 )
r a (MN m - 2 )
N
Vis o ( r a m s- 1 )
V ( m m s- 1 )
1.80 1.80 1.80 1.80 1.80 1.80 4.36
30 30 30 30 30 90 30
1 5 10 20 36 20 20
8.42 8.42 8.42 8.42 8.42 3.36 1.68
8 . 4 2 X 10 - 3 8.52 x 10 - 3 8 . 5 0 x 10 - s
x 10 - i s x 10 - 1 3 x 10 - 1 3 x 10 - 1 3 X 10 - 1 3 x 10 - 1 3 x 10 - 1 7
x 10 - 3 x 10 - 3 x 10 - 3 x 10 -3
8.48 X 10 -3
x 10 - 3 x 10 - 2 X 10 - 6
8.47 X 10 - 3 3.39 x 10 - 2 1.69 × 10 - 6
136
7
6
5 t3s&
3
2
1
0
I
0"
£-. 'u l 0 "2
i,
E E >
II
fJ
10 -3 ,
0.5
i
,
i
1.0 5.0 "~a ( 10MN m°2)
10.0
Fig. 6. The dislocation velocity as measured by the indentation technique in silicon single crystals [ 15 ]
at 1073 K: o, e x p e r i m e n t a l ; - - , computational.
0
--oL/
e't
,
0
1
i
,
2
3
40
tl3S 4 5 i
7
6
5 tss 4
3
2
1
0
Time(,e¢)
Fig. 4. The relation between the distance X 1 of the
3O
leading dislocation from the source and time t: $, +Ta; o, --r a (T and ~a have the same values as in
Fig. 1). nearly p r o p o r t i o n a l to the t i m e and t h a t the p r o p o r t i o n a l i t y f a c t o r is equal t o the distance m o v e d in u n i t t i m e b y the leading dislocation, i.e. the v e l o c i t y o f the leading dislocation u n d e r an applied shear stress. H o w e v e r , the v e l o c i t y o f the leading disloc a t i o n is strictly n o t a c o n s t a n t , as s h o w n in Fig. 5. T h e v e l o c i t y o f the leading dislocation increases sharply with N u p t o N = 6 and t h e n increases m o r e gradually. T h e v e l o c i t y decreases gradually with N f r o m N = 6 u p t o N = 36, at a c o n s t a n t rate o f change equal t o 4%. Figure 6 shows t h e leading dislocation v e l o c i t y as a f u n c t i o n o f applied shear stress. In this figure the e x p e r i m e n t a l results [15] and
2.0
Vl Vi~
Z20
i
10
0
1
2
3
4
5
Time (see)
Fig. 7. The relation between the number N of dislocations in the group and time t: e, +Ta; O, --T a ( T and Ta have the same values as in Fig. 1). o u r p r e s e n t c o m p u t a t i o n results are compared; t h e y agree well. T h e n u m b e r o f dislocations e m i t t e d f r o m the s o u r c e is p l o t t e d against t i m e in Fig. 7. It can b e seen t h a t t h e emission rate d N / d t o f dislocations f r o m the source u n d e r a c o n s t a n t applied shear stress decreases as the n u m b e r o f dislocations in the g r o u p increases.
1,5
1.0 ~ 05
o
, 5
-
e , 10
-
.
o , 15
.
, , 20
o
, 25
o i 30
, 35
40
N
Fig. 5. The number of dislocations in the group plotted against the velocity of the leading dislocation: o, +Ta; o, --T a (T and r a have the same values as in Fig. 1).
3.2. T h e irreversibility o f the glide m o t i o n o f a dislocation g r o u p c o n t a i n i n g a source T h e following q u e s t i o n is v e r y interesting. T h e r e are N dislocations in a group which are e m i t t e d f r o m a source u n d e r a c o n s t a n t applied shear stress ra during the time t = 0 t o t = t N. If at t = t N we s u b s t i t u t e +% f o r - - % , w h a t will be the m o t i o n o f each dislocation in such a group in t i m e ?
137
Obviously, dislocations in the group will move back to the source and will continuously be annihilated by the source. Under the applied shear stress --% the relation between the leading dislocation velocity 171 and the number N of dislocations in the group is shown in Fig. 5. The result obtained for +% is also included for comparison. In contrast with the case when +%, the leading dislocation velocity is always smaller than that of an isolated dislocation subjected to the same applied shear stress --% and slowly increasing with decreasing number of dislocations in the group. This can possibly be explained because the sign of stress exerted on the leading dislocation by other dislocations in the group is contrary to the sign of the applied shear stress --%. The relation between N, X1 and t is given in Figs. 7 and 4. During the time t3s, all dislocations in the group are n o t annihilated by the source. There are still AN dislocations in the group and the leading dislocation is still at a distance AX 1 from the source as long as, at time t36 + A t, all dislocations can be annihilated by the source. On close scrutiny of the curves in Figs. 5, 7 and 4, it can be seen that a dislocation group containing a source cannot recover to its initial state during a stress cycle of +% and - - % , i.e. the glide motion of a dislocation group containing a source is irreversible. Such a feature m a y be used to explain some properties, such as fatigue, in crystalline materials. We also investigated the effect of %, N and M on AN, AX1 and A t. We f o u n d that AN is proportional to N. The velocity of each dislocation in the group as it moves to the source is shown in Fig. 8. From a simple computation we conclude that at any time the average velocity of all the dislocations in the group is equal to the velocity of an isolated dislocation which has been subjected to the same applied shear stress
--T a .
3.3. The m o t i o n o f a dislocation group w i t h o u t any applied stress Head and Wood [3] have studied the motion of a dislocation group in the absence of any applied stress. Their treatment, however, requires that the number of dislocations in the group is held constant during motion. In this section we shall consider the
ZO V.j
N=36
,.ot
1.5 [-
i
0.5
5
N=20
i
I0
j
L
15
20
25
~
30
35
40
J Fig. 8. The velocity o f each individual dislocation in the group in terms o f the velocity (--8.42 x 10 - a m m s- l ) o f the isolated dislocation ( T and r a have the same values as in Fig. 1).
motion of a dislocation group containing a source in the absence of any applied stress, 10 3 I
10-I
3 j 33 29 2s 21 i?
r.:.:,7.:.¢:.:.;
13 g
5
.
i
; • ~
xj~,o'~
I 14
Fig. 9. The arrangement of dislocations in the group in the absence of any applied stress for different times (T = 1073 K).
motion of a dislocation group containing a source in the absence of any applied stress, i.e. the number of dislocations is changed during the motion. In this case the mutual repulsion of dislocations is the only driving force and thus causes the motion of the group. The shape of the dislocation distribution as it moves is plotted in Fig. 9. Initially, there are 36 dislocations in the group. Under the mutual repulsion of dislocations, each dislocation moves in its own manner. At one end of the group which is far away from the source, the dislocations expand. However, at the other end near to the source, the dislocations move to and consequently are annihilated by the source. There are some dislocations in the group which move in a complicated manner. Initially, they move away from the source and their velocity is positive but, after some time, the velocity becomes zero. Finally, the velocity becomes negative and
138
they move to and are annihilated by the source. In principle, after a sufficiently long time there exists only one dislocation (i.e. the leading dislocation) in the group; all other dislocations are annihilated by the source. From Fig. 9 it is possible to obtain the motion of the zero stress point of the group. The relation between N, X1, V1 and t is shown in Fig. 10. The velocity of each dislocation in the group for different N values is given in Fig. 11.
N X1~1O-2mm. )
3o.3 25
.
.
.
15
'.,X, \\
11
// 1.
,o.
E
20
~ 5
15 10
'5
N V1 i
i
i
t
i
,o"
I
Io
Id
Time(see
0
Io'
)
Fig. 10. The relation between N, V1, X 1 and time t (T = 1073 K).
It can be seen that in the absence of any applied stress the algebraic average velocity of all dislocations in the group is equal to zero.
4. CONCLUSIONS
Computer simulation methods were used to study several characteristics of the glide motion of a dislocation group containing a source. The following conclusions were reached. (1) A dislocation group containing a source cannot recover to its initial state during a stress cycle of +% and --%, i.e. the glide motion of a dislocation group containing a source is irreversible. Such a feature may be used to explain some properties, such as fatigue, of crystalline materials. (2) The interaction between dislocations in a group does n o t affect the average velocity of all dislocations in the group containing a source (the average velocity is always equal to the velocity of an isolated dislocation which has been subjected to the same applied stress). (3) The velocity V1 of the leading dislocation is only slightly larger than the velocity Viso of an isolated dislocation under a constant applied stress.
ACKNOWLEDGMENT 1.5
,
V} ( Io'3mm
,
~ c "1 )
The authors are deeply indebted to Professor R. J. Arsenault, University of Maryland, for his critical reading of the manuscript.
1.o
o5
REFERENCES
-0.5 20 -1.0 -1.5 -2.0
'
'
'
'
5
10
15
20
' 25
30
J Fig. 11. The velocity of each dislocation in the group for different N values in the absence of any applied stress (T = 1073 K).
1 J. D. Eshelby, F. C. Frank and F. R. Nabarro, Philos. Mag., 42 (1951) 351. 2 A. K. Head, Philos. Mag., 26 (1972) 43, 6 5 ; 2 7 (1973) 531. 3 A. K. Head and W. W. Wood, Philos. Mag., 27 (1973) 505, 519. 4 J. J. Weertman, J. Appl. Phys., 28 (1957) 1185. 5 A. R. Rosenfield and G. T. Hahn, in A. R. Rosenfield, G. T. Hahn, A. L. Bement, Jr., and R. I. Jaffee (eds.), Dislocation Dynamics, McGraw-Hill, New York, 1968, p. 255. 6 M. F. Kanninen and A. R. Rosenfield, Philos. Mag., 20 (1969) 569. 7 A. R. Rosenfield and M. F. Kanninen, Philos. Mag., 22 (1970) 143.
139 8 T. Yokobori, A. T. Yokobori, Jr., and A. Kamei, Philos. Mag., 30 (1974) 367. 9 A. T. Yokobori, Jr., T. Yokobori and A. Kamei, J. Appl. Phys., 46 (1975) 3720. 10 R.J. Arsenault and T. W. Cadman, Scr. Metall., 12 (1978) 633. 11 R. J. Arsenault and C. J. K. Kuo, MetaU. Trans. A, 9 (1978) 459.
12 Patu and L. Chung-zi, Internal Rep. 79-154, October 1979 (Institute of Metal Research, Academia Sinica). 13 A. T. Yokobori, Jr., T. Yokobori and A. Kamei, Mater. Sci. Eng., 40 (1979) 111. 14 Patu and H. Yi-zhen, Acta Phys. Sin., 29 (1980) 698. 15 Patu and H. Yi-zhen, Phys. Status Solidi A, 59 (1980) 195.
133
Computer Simulation of the Glide Motion of a Dislocation Group Containing a Source PATU, LEI CHUNG-ZI and SHIH CHANG-HSU Institute o f Metal Research, Academia Sinica, Shenyang (China)
(Received August 1, 1980; in revised form November 21, 1980)
SUMMARY
The dynamic behaviour of long straight parallel coplanar dislocations emitted from a source under a constant applied stress and a constant applied stress rate has been analysed by a few researchers. In this paper the glide motion of a dislocation group emitted from a source under a constant applied stress, the annihilation of a dislocation group under a constant applied reverse stress and the motion o f a dislocation group containing a source in the absence o f any applied stress were investigated. It was found that the glide motion o f a dislocation group containing a source is irreversible; the interaction between dislocations in a group does not affect the average velocity o f all the dislocations in the group containing a source (the average velocity is always equal to the velocity o f an isolated dislocation which has been subjected to the same applied stress) and the velocity o f the leading dislocation is only slightly larger than that o f an isolated dislocation.
group behind an obstacle. Since then, much work (in which analytical and numerical methods have been used [2 - 13] ) has been carried o u t in this field. In this paper we consider the emission, motion and annihilation of an array of long straight parallel coplanar dislocations generated from a source by means of a computer simulation method. 2. THE MODEL AND METHOD
The model and m e t h o d of computation used in this investigation start with the same procedure as used in previous papers [6, 8, 11], but in this investigation we consider the annihilation of dislocations and the application of a constant reverse stress. 2.1. The dislocation motion equation We consider a group of N identical dislocations in the slip plane Y = 0. Each dislocation is parallel to the Z axis. The dislocation positions are given by 0 ~ XN < X N - 1
1. INTRODUCTION
It is well known that plastic deformation is related to the generation, motion and annihilation of dislocations in crystals. During the process of plastic deformation, dislocations always move in a group manner. For a detailed understanding of the microscopic mechanism of flow and fracture in crystalline materials, it is necessary to study the collective and cooperative m o t i o n of dislocations. The simplest dislocation group model is an array of long straight parallel coplanar dislocations. Early in 1951, Eshelby et al. [1] gave a theoretical analysis of a pile-up of such a 0025-5416/81/0000-0000/$02.50
<
...
< X2 <
X1
where X = 0 represents the source position. In general, the jth dislocation motion equation in the group is taken to be V1 -
dt
- M I T ' e f f i l rn S i g n ( t e l f / )
(1)
where M and m are characteristic constants of the material, re~fj is the effective shear stress acting on the jth dislocation in the group and Sign is used to denote a signal function which has the following properties: Sign(tell/) = 1
reff/ > 0
= 0
reff/ = 0
= -- 1
re~ti < 0
(2)
© Elsevier Sequoia]Printed in The Netherlands
134
This means that a positive stress produced a positive velocity and vice versa. If any long-range stress in the crystal is neglected, then reffi can be written as follows: ref,j = r. +
N
A
(3)
and
XN = 0
(7)
first of all, the N t h dislocation is annihilated by the source and the number of dislocation in the group becomes N -- 1.
1=1 Xj
and A =
.b (
cos2
+
sin'a /
(4)
where j and i are the index numbers of the dislocations and ra is the applied shear stress. The second term on the right-hand side of eqn. (3) is the stress exerted on the ]th dislocation by other dislocations in the group./~ and v are the shear modulus and Poisson's ratio respectively, b is the Burgers vector and is the angle between the dislocation line and its Burgers vector.
2.4. The method o f computation At some arbitrary time t, each individual dislocation position is calculated by the R u n g e - K u t t a m e t h o d which has been programmed for the DJS-130 computer. In the present paper the calculation is made for 60 ° dislocation in silicon single crystals. The corresponding values of M, m and A are taken from our experimental data [14], as listed in Table 1. For convenience the source operation stress r, is assumed to be zero in the computation. TABLE 1 Material constants used in the c o m p u t a t i o n
2.2. The condition for source operation An initial condition is that, at t = 0, N = 1 and X1 = 0, i.e. there is one dislocation at the source. The condition for source operation is that, if an effective shear stress acting on the source is equal to or exceeds a given source operation stress r,," then a new dislocation will be emitted from the source, and the number of dislocations in the group will change from N to N + 1. In this case the condition of generation of a new dislocation is written as N A r . -- Z > r,
2.3. The condition for annihilation o f a dislocation Obviously, the Nth dislocation in the group which contains N dislocations is nearest to the source. If TeffN < 0, then the Nth dislocation moves back to the source. When A
-- < 0
]=1 X /
M (ram s-1)
A (N m - 1 )
1173 1073 973 873 773 673
1.21 1.26 1.02 1.16 1.25 1.49
3.24 1.80 3.68 7.09 4.36 8.09
4.07 4.07 4.07 4.07 4.07 4.07
X 10 -12 × 10 -13 × 10 - 1 2 × 10 - 1 5 × 10 -17 × 10 -21
3. R E S U L T S A N D D I S C U S S I O N
where the second term on the left-hand side of eqn. (5) is the stress acting on the source by dislocations (except the (N + 1)th dislocation) in the group.
N-I
m
(K)
(5)
j = l ~-1
r. -- Z
Temperature
(6)
3.1. The motion o f a dislocation group emitted from a source Calculations are carried out for various values of M, m and A, as listed in Table 1, and for different constant applied shear stresses (5.0 - 200 MN m-Z). The calculation results show that dislocation configurations are only slightly affected by r , and M. A typical dislocation distribution in the group is plotted in Fig. 1. It is shown that, in a dislocation group emitted from a source, the dislocation density decreases as the distance from the source increases. This is similar to pile-up against the source. The calculation results also show that, the larger the applied stress, the higher is the density of dislocations in the group, as shown in Fig. 2.
135
*36 * * * 1 , ° ° 1 ° 1 1 . , * * * * °
•
°,
•
**
•
•
o
•
•
•
I
•
•
"~effj
1.5 .
~
134
i
1.0
~3 u~ i~-?
..........................
26
0.5
He,•~oe,o,,o.*o..*.o • • • 2~ e l l l * . * o e o l l i t J * * o • • • • 22 ,.. .... • . . . . . . . . . . . . 20 ° H ° ° ° o o ° * * ° o , • , • • • . • o •
•
° •
°
° °
(a)
1.5
Viso
°16
1:0
14
0.S
.....
6
0
Xj ( 10-2mm ) 1
2
Nffil0
4
3
(b)
5
Fig. 1. T h e d i s l o c a t i o n d i s t r i b u t i o n in t h e g r o u p e m i t t e d f r o m a s o u r c e for various values o f N (~'a = 3 0 M N m - 2 ; T = 1 0 7 3 K).
N,20
i
i
i
i
5
1o
Is
2o
i
J zs
I
3o
N=3S L5 3
Fig. 3. (a) T h e effective s h e a r stress a c t i n g o n e a c h d i s l o c a t i o n in t e r m s o f t h e a p p l i e d s h e a r stress ( 3 0 MN m - 2 ) ; (b) t h e v e l o c i t y o f e a c h individual dislocat i o n in t e r m s o f t h e v e l o c i t y Vis o ( 8 . 4 2 x 10 - 3 m m s - 1 ) of a n isolated d i s l o c a t i o n ( T = 1 0 7 3 K).
J 20
l
%2
",. 3
"-,.. 4
i
i
l
2.0
4.0
6.0
s h
L
i
8.0
lo.o
12.o
Xj ("lO-2mm )
Fig. 2. T h e d i s l o c a t i o n d i s t r i b u t i o n in t h e g r o u p (N = 20) f o r d i f f e r e n t a p p l i e d s h e a r stresses ( T = 1 0 7 3 K ) : c u r v e 1, 90 M N m - 2 ; c u r v e 2, 30 M N m - 2 ; c u r v e 3, 18 M N m - 2 ; c u r v e 4, 10 MN m - 2 ; c u r v e 5, 5 M N m -2.
The effective shear stress Teff,/ on each individual dislocation in the group and the velocity Vs of each dislocation are calculated from eqns. (1) and {2); they are shown in Fig. 3. It is shown that, in the dislocation group emitted from a source, each dislocation veloc-
ity differs from the others. The largest velocity is that of the leading dislocation; half of the dislocations have velocities larger than the velocity Vis o of an isolated dislocation, and the other half of the dislocations have velocities which are smaller than Viso. The result also shows that at each instant of time the average velocity V of all dislocations in the group is nearly equal to Vi, o if the isolated dislocation has been subjected to the same applied shear stress, as listed in Table 2. This result is consistent with that obtained by Arsenault and coworkers [ 1 0 , 1 1 ]. This means that the interaction between dislocations in a group cannot affect the average velocity of all dislocations. The relation between the distance X1 of the leading dislocation from the source and time t under a constant applied shear stress is shown in Fig. 4. It can be seen that the distance travelled by the leading dislocation is
TABLE 2 C o m p a r i s o n o f V a n d Vis o f o r d i f f e r e n t c o n d i t i o n s M ( r a m s- 1 )
r a (MN m - 2 )
N
Vis o ( r a m s- 1 )
V ( m m s- 1 )
1.80 1.80 1.80 1.80 1.80 1.80 4.36
30 30 30 30 30 90 30
1 5 10 20 36 20 20
8.42 8.42 8.42 8.42 8.42 3.36 1.68
8 . 4 2 X 10 - 3 8.52 x 10 - 3 8 . 5 0 x 10 - s
x 10 - i s x 10 - 1 3 x 10 - 1 3 x 10 - 1 3 X 10 - 1 3 x 10 - 1 3 x 10 - 1 7
x 10 - 3 x 10 - 3 x 10 - 3 x 10 -3
8.48 X 10 -3
x 10 - 3 x 10 - 2 X 10 - 6
8.47 X 10 - 3 3.39 x 10 - 2 1.69 × 10 - 6
136
7
6
5 t3s&
3
2
1
0
I
0"
£-. 'u l 0 "2
i,
E E >
II
fJ
10 -3 ,
0.5
i
,
i
1.0 5.0 "~a ( 10MN m°2)
10.0
Fig. 6. The dislocation velocity as measured by the indentation technique in silicon single crystals [ 15 ]
at 1073 K: o, e x p e r i m e n t a l ; - - , computational.
0
--oL/
e't
,
0
1
i
,
2
3
40
tl3S 4 5 i
7
6
5 tss 4
3
2
1
0
Time(,e¢)
Fig. 4. The relation between the distance X 1 of the
3O
leading dislocation from the source and time t: $, +Ta; o, --r a (T and ~a have the same values as in
Fig. 1). nearly p r o p o r t i o n a l to the t i m e and t h a t the p r o p o r t i o n a l i t y f a c t o r is equal t o the distance m o v e d in u n i t t i m e b y the leading dislocation, i.e. the v e l o c i t y o f the leading dislocation u n d e r an applied shear stress. H o w e v e r , the v e l o c i t y o f the leading disloc a t i o n is strictly n o t a c o n s t a n t , as s h o w n in Fig. 5. T h e v e l o c i t y o f the leading dislocation increases sharply with N u p t o N = 6 and t h e n increases m o r e gradually. T h e v e l o c i t y decreases gradually with N f r o m N = 6 u p t o N = 36, at a c o n s t a n t rate o f change equal t o 4%. Figure 6 shows t h e leading dislocation v e l o c i t y as a f u n c t i o n o f applied shear stress. In this figure the e x p e r i m e n t a l results [15] and
2.0
Vl Vi~
Z20
i
10
0
1
2
3
4
5
Time (see)
Fig. 7. The relation between the number N of dislocations in the group and time t: e, +Ta; O, --T a ( T and Ta have the same values as in Fig. 1). o u r p r e s e n t c o m p u t a t i o n results are compared; t h e y agree well. T h e n u m b e r o f dislocations e m i t t e d f r o m the s o u r c e is p l o t t e d against t i m e in Fig. 7. It can b e seen t h a t t h e emission rate d N / d t o f dislocations f r o m the source u n d e r a c o n s t a n t applied shear stress decreases as the n u m b e r o f dislocations in the g r o u p increases.
1,5
1.0 ~ 05
o
, 5
-
e , 10
-
.
o , 15
.
, , 20
o
, 25
o i 30
, 35
40
N
Fig. 5. The number of dislocations in the group plotted against the velocity of the leading dislocation: o, +Ta; o, --T a (T and r a have the same values as in Fig. 1).
3.2. T h e irreversibility o f the glide m o t i o n o f a dislocation g r o u p c o n t a i n i n g a source T h e following q u e s t i o n is v e r y interesting. T h e r e are N dislocations in a group which are e m i t t e d f r o m a source u n d e r a c o n s t a n t applied shear stress ra during the time t = 0 t o t = t N. If at t = t N we s u b s t i t u t e +% f o r - - % , w h a t will be the m o t i o n o f each dislocation in such a group in t i m e ?
137
Obviously, dislocations in the group will move back to the source and will continuously be annihilated by the source. Under the applied shear stress --% the relation between the leading dislocation velocity 171 and the number N of dislocations in the group is shown in Fig. 5. The result obtained for +% is also included for comparison. In contrast with the case when +%, the leading dislocation velocity is always smaller than that of an isolated dislocation subjected to the same applied shear stress --% and slowly increasing with decreasing number of dislocations in the group. This can possibly be explained because the sign of stress exerted on the leading dislocation by other dislocations in the group is contrary to the sign of the applied shear stress --%. The relation between N, X1 and t is given in Figs. 7 and 4. During the time t3s, all dislocations in the group are n o t annihilated by the source. There are still AN dislocations in the group and the leading dislocation is still at a distance AX 1 from the source as long as, at time t36 + A t, all dislocations can be annihilated by the source. On close scrutiny of the curves in Figs. 5, 7 and 4, it can be seen that a dislocation group containing a source cannot recover to its initial state during a stress cycle of +% and - - % , i.e. the glide motion of a dislocation group containing a source is irreversible. Such a feature m a y be used to explain some properties, such as fatigue, in crystalline materials. We also investigated the effect of %, N and M on AN, AX1 and A t. We f o u n d that AN is proportional to N. The velocity of each dislocation in the group as it moves to the source is shown in Fig. 8. From a simple computation we conclude that at any time the average velocity of all the dislocations in the group is equal to the velocity of an isolated dislocation which has been subjected to the same applied shear stress
--T a .
3.3. The m o t i o n o f a dislocation group w i t h o u t any applied stress Head and Wood [3] have studied the motion of a dislocation group in the absence of any applied stress. Their treatment, however, requires that the number of dislocations in the group is held constant during motion. In this section we shall consider the
ZO V.j
N=36
,.ot
1.5 [-
i
0.5
5
N=20
i
I0
j
L
15
20
25
~
30
35
40
J Fig. 8. The velocity o f each individual dislocation in the group in terms o f the velocity (--8.42 x 10 - a m m s- l ) o f the isolated dislocation ( T and r a have the same values as in Fig. 1).
motion of a dislocation group containing a source in the absence of any applied stress, 10 3 I
10-I
3 j 33 29 2s 21 i?
r.:.:,7.:.¢:.:.;
13 g
5
.
i
; • ~
xj~,o'~
I 14
Fig. 9. The arrangement of dislocations in the group in the absence of any applied stress for different times (T = 1073 K).
motion of a dislocation group containing a source in the absence of any applied stress, i.e. the number of dislocations is changed during the motion. In this case the mutual repulsion of dislocations is the only driving force and thus causes the motion of the group. The shape of the dislocation distribution as it moves is plotted in Fig. 9. Initially, there are 36 dislocations in the group. Under the mutual repulsion of dislocations, each dislocation moves in its own manner. At one end of the group which is far away from the source, the dislocations expand. However, at the other end near to the source, the dislocations move to and consequently are annihilated by the source. There are some dislocations in the group which move in a complicated manner. Initially, they move away from the source and their velocity is positive but, after some time, the velocity becomes zero. Finally, the velocity becomes negative and
138
they move to and are annihilated by the source. In principle, after a sufficiently long time there exists only one dislocation (i.e. the leading dislocation) in the group; all other dislocations are annihilated by the source. From Fig. 9 it is possible to obtain the motion of the zero stress point of the group. The relation between N, X1, V1 and t is shown in Fig. 10. The velocity of each dislocation in the group for different N values is given in Fig. 11.
N X1~1O-2mm. )
3o.3 25
.
.
.
15
'.,X, \\
11
// 1.
,o.
E
20
~ 5
15 10
'5
N V1 i
i
i
t
i
,o"
I
Io
Id
Time(see
0
Io'
)
Fig. 10. The relation between N, V1, X 1 and time t (T = 1073 K).
It can be seen that in the absence of any applied stress the algebraic average velocity of all dislocations in the group is equal to zero.
4. CONCLUSIONS
Computer simulation methods were used to study several characteristics of the glide motion of a dislocation group containing a source. The following conclusions were reached. (1) A dislocation group containing a source cannot recover to its initial state during a stress cycle of +% and --%, i.e. the glide motion of a dislocation group containing a source is irreversible. Such a feature may be used to explain some properties, such as fatigue, of crystalline materials. (2) The interaction between dislocations in a group does n o t affect the average velocity of all dislocations in the group containing a source (the average velocity is always equal to the velocity of an isolated dislocation which has been subjected to the same applied stress). (3) The velocity V1 of the leading dislocation is only slightly larger than the velocity Viso of an isolated dislocation under a constant applied stress.
ACKNOWLEDGMENT 1.5
,
V} ( Io'3mm
,
~ c "1 )
The authors are deeply indebted to Professor R. J. Arsenault, University of Maryland, for his critical reading of the manuscript.
1.o
o5
REFERENCES
-0.5 20 -1.0 -1.5 -2.0
'
'
'
'
5
10
15
20
' 25
30
J Fig. 11. The velocity of each dislocation in the group for different N values in the absence of any applied stress (T = 1073 K).
1 J. D. Eshelby, F. C. Frank and F. R. Nabarro, Philos. Mag., 42 (1951) 351. 2 A. K. Head, Philos. Mag., 26 (1972) 43, 6 5 ; 2 7 (1973) 531. 3 A. K. Head and W. W. Wood, Philos. Mag., 27 (1973) 505, 519. 4 J. J. Weertman, J. Appl. Phys., 28 (1957) 1185. 5 A. R. Rosenfield and G. T. Hahn, in A. R. Rosenfield, G. T. Hahn, A. L. Bement, Jr., and R. I. Jaffee (eds.), Dislocation Dynamics, McGraw-Hill, New York, 1968, p. 255. 6 M. F. Kanninen and A. R. Rosenfield, Philos. Mag., 20 (1969) 569. 7 A. R. Rosenfield and M. F. Kanninen, Philos. Mag., 22 (1970) 143.
139 8 T. Yokobori, A. T. Yokobori, Jr., and A. Kamei, Philos. Mag., 30 (1974) 367. 9 A. T. Yokobori, Jr., T. Yokobori and A. Kamei, J. Appl. Phys., 46 (1975) 3720. 10 R.J. Arsenault and T. W. Cadman, Scr. Metall., 12 (1978) 633. 11 R. J. Arsenault and C. J. K. Kuo, MetaU. Trans. A, 9 (1978) 459.
12 Patu and L. Chung-zi, Internal Rep. 79-154, October 1979 (Institute of Metal Research, Academia Sinica). 13 A. T. Yokobori, Jr., T. Yokobori and A. Kamei, Mater. Sci. Eng., 40 (1979) 111. 14 Patu and H. Yi-zhen, Acta Phys. Sin., 29 (1980) 698. 15 Patu and H. Yi-zhen, Phys. Status Solidi A, 59 (1980) 195.