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Edge dislocations emitted along multiple inclined slip planes from a Mode I crack. II. Simultaneous emission
ELSEVIER
Mechanics of Materials 24 (19%) I 1- 17
Edge dislocations emitted along multiple inclined slip planes from a Mode I crack. II. Simultaneous emission Cai-Fu Qian i, J.C.M. Li * Department of Mechanical Engineering. University of Ro~.he.~ter. Rochester. NY 14627. USA Received 5 May 1995; revised version received 9 May 1996
Abstract In an effort to simulate dislocation distribution around a Mode I crack, edge dislocations are allowed to be emitted from the crack tip along many possible slip planes containing the crack tip. it turned out that the final distribution depends on the order of emission. So the previous paper dealt with the situation in which the emission process is one at a time, namely after one dislocation is emitted the plastic zone is re-equilibrated before the next dislocation is emitted. In this paper, many dislocations are emitted at the same time when the condition is met, namely, when the slip force exceeds the lattice friction for dislocation motion. It is found that the dislocations are more uniformly distributed than the previous case of sequential emission. The shielding effect for the same number of dislocations emitted is less when the dislocations are distributed along more slip planes. The shape of the plastic zone is different from but the size is comparable with the plastic zone based on yon Mises or Tresca criterion. A distinct dislocation-free zone appears when there is a critical stress intensity factor for dislocation emission.
1. I n t r o d u c t i o n In the preceding paper (Qian and Li, 1996) dislocations were emitted sequentially from a Mode I crack. As a result, most of the dislocations were emitted along a slip plane in which the shear stress was the largest even though 34 slip planes were available. Unless the loading is increased very slowly
• Corresponding author. Tel.: + 1-716-2754038; fax: + 1-7162562509; e-mail: [email protected]. Now at Department of Mechanical Engineering, Hong Kong University of Science of Technology, Clearwater Bay, Kowloon, Hong Kong.
so that only one dislocation can be emitted at a time and the next one will not be emitted until all the emitted dislocations have taken up their equilibrium positions, it is possible and perhaps more realistic that dislocations are emitted simultaneously along many slip planes from the crack when the loading is fast so that dislocations can be emitted along many slip planes all at once. The plastic zone around a crack tip is usually obtained analytically or numerically based on the yield stress. Examples are the well-known plastic zones using the von Mises or Tresca criterion. But since the plastic deformation of crystalline materials is due to the motion of dislocations and dislocations are likely to be emitted from a crack tip, it would be interesting to determine the shape and size of the
0167-6636/%/S15.00 Copyright © 1996 Published by Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 6 6 3 6 ( 9 6 ) 0 0 0 2 5 - 7
12
C.-F, Qian, J.C.M. l.i / Mechanics of Materials 24 (1996) 11 -.17
the dislocation will move toward the crack tip by a distance b. The dislocation will be absorbed by the crack and disappear if the distance to the crack tip is less than b or negative. These motions will take place until equilibrium prevails, namely, no more motions are possible. Then another round of emission will take place and the motions of all dislocations repeated. The computation terminates when no more dislocation can be emitted. In this study all available slip planes were placed symmetrically on both sides of the crack. But the total number of dislocations emitted refer to one side of crack only. The available slip planes for the four different cases studied here are listed in Table 1 with N, being the total number of available planes.
plastic zone around a crack tip according to the dislocation behavior. Experimental observations and computer simulations of dislocation emission from a crack tip have been reviewed in the introduction of the preceding paper (Qian and Li, 1996). In this study dislocations are allowed to be emitted simultaneously along many inclined slip planes from the tip of a Mode I crack. Equilibrium dislocation distribution around the crack tip including the plastic zone and the dislocation-free zone is studied together with the effects of the applied stress and the lattice friction stress. Increasing the applied stress further or decreasing the friction stress further will increase the number of dislocations emitted and the number of slip planes involved in dislocation emission and thereby increasing the computer time and storage beyond the present capability.
3. Results 2. Assumptions
3.1. Different number of available slip planes Most assumptions made in this study are the same as those in the preceding paper. All assumptions are presented here: (l) All dislocations are edge dislocations with their lines parallel to the tip of the crack, and all slip planes are passing through the crack tip. (2) All dislocations are emitted from the crack tip. (3) A dislocation is emitted along a slip plane as long as the slip force on the emitted dislocation (located at b from the crack tip) is larger than the friction force. After the emission of all possible dislocations, the slip forces on all the emitted dislocations are calculated according to the scheme presented in Qian and Li (1996). When the slip force is larger than the friction force, the dislocation moves forward by a distance b. If the slip force on the dislocation is negative and exceeds the friction force,
Fig. 1 shows the slip forces (in units of Ab) available for dislocation emission (placed at b from the crack tip) along each slip plane for the four cases studied with N being one half of the total number of dislocations emitted and equilibrated before the next round of emission. Since the dislocations are allowed to be emitted from all available slip planes simultaneously, all forces are decreasing with the increase of dislocations emitted and no change of slip planes is found as in the case when only one dislocation is allowed to be emitted along the slip plane with the largest slip force (Qian and Li, 1996) between equilibrium configurations. When all the slip forces for dislocation emission are finally smaller than the friction force, the dislocation emission process is terminated.
Table 1 Available slip planes Case
Angles of slip planes ( _+deg.)
NL
I
60
2 3 4
60 60 60
10
20 20
30
40 40
50
120
70
80 80 80
90
100 100 100
110
120 120 120
4
130
140 140
150
160 160
170
8 16 34
C.-F. Qian. J.C.M. Li / Mechanics of Materials 24 (1996) 11-17
13
2 K,/(Ab~/~)=lS"0 =
&
1.5
•
A
~
"tl/A=O.1
• $.=~o
=
1.5
=
K,/(AbWZ) =15-0 xf/A=0.1
o +
~=60 o ¢ _.=~o
•
$ =100 °
&
•
o
=j
o
o
o
B
•
=
$=lZIT
",,
°
0.5 ~.
=
•
l,
"o
o o
os
",
. F r i c t i o n t o r c . e _ l i n e . . , o..o . . . . . . . . . . .
0 ~-
•
g
----....--.--:.:_Fricti°nf°rce]ine...'..A. •
a...
0.
°°°OOOoooo
0
•
0
_
&&
-0.5
F ....
I ....
0
I ....
5
(a)
I ....
10
15
u,,,,I,,,,
20
25
....
I ....
5
35
(5)
10
X
15
I ....
20
I ....
25
•
o
x
a
A N=0
•
1.5
'tl/A=0.1
25
• •
&
:
3O•
.
..........
•
o.s
o
38
A
•
~ ........
o
~
I
,
30
(c)
I
,
,
~
60
l
,
I
i
90
,
I
•
x
150
0
180
Angles of slip planes (deg.)
(d)
x
x
x
O
O
O
O
43
l~lelI,,=,=lllelll,
-0.5
I
•
35
x
°~°°°o
,
120
• •
&
x
;::
~- ~.-~ .x.Frir_ti~'t force- lio~ . . . . . . . . . . . . . .
0
o
,
0
•
,
o
-0.5
&
X
- . . . . 0- " - b . . . . . . . .
o
40
&
• • 0.5
35
K=/(Abl/Z)=15.0
19
•
30
I ....
& A
X
1 ....
Total number of dislocations emitted, N
&
7
I ....
2
N=0 &
1.5
,,,1
-0.5 30
Total n u m b e r of dislocations emitted, N
2 =
1 ....
30
O
°°~io ,,llleple=
60
90
120
,~lOe
150
180
A n g l e s of slip p l a n e s (deg.)
Fig. 1. Slip forces on d i s l o c a t i o n s to be emitted. (a) Case I; (b) case 2; (c) case 3; (d) case 4.
Fig. 2 shows the number of dislocations emitted along each slip plane for the four different cases. The total number of dislocations emitted on all slip planes 25 •~
o
C a s e 1, N---32
•
C a s e 2, N---37
a o
C a s e 3, N = 3 8 C a s e 4, N = 4 3
20
"8 g
15
E
10
"=
5
•
&
0
0
o
0
0
0
0
0
O
o
17
o
o
0 I
I
30
60
,
o
1
,
90
,
I
,
120
B ~ ~_ _ - -,
150
--
180
A n g l e s of slip p l a n e s (deg.) Fig. 2. D i s l o c a t i o n s e m i t t e d a l o n g each slip plane for the four different cases.
is 32, 37, 38 and 43 for the four cases respectively. For some slip planes with very large or very small angles (e.g. 150 °, 160 °, 170 ° and 10° for case 4), no dislocation can be emitted. This is because the applied force is not large enough to enable dislocations to initiate and move along these slip planes, thus making the number of slip planes involved in dislocation emission (e.g. 13 for a total of 17 in case 4) less than the number of available slip planes. It is clear from Fig. 2 that the more the slip planes are available for dislocation emission, the less the number of dislocations on each slip plane, and the more the total number of dislocations on all slip planes. Fig. 3 shows the dislocation distributions for the four cases. Fig. 4 shows the plastic zone affected by the number of slip planes available for dislocation emission. When the number of slip planes increases, the number of dislocations in each slip plane decreases and hence the plastic zone decreases.
14
C.-F. Qian. J.C.M. l.i / Mechanics of Materials 24 (1996) I I - 17 1000
1000 KJ(AbW2)=15.0
'~JA=0.1
500
KJ(Ab~/2)=I5.0
.."
.•• • ..,
'Oz./A=0.1
."
500
/"
•.." :.." .o:-. ::5
o
o'. • ",
-500
-500
....
-1000
I ....
I ....
-500
-1000
I•
0 x/b
i
i
500
i
L
-1000
. . . .
1000
t
-1000
. . . .
",
,
,
0
]
. . . .
500
1000
x/b
(a) C a s e I 1000
i,
-500
(b) Case 2 800
KJ(Abl/2)=IS.0
•
~IIA=O.I 500
400
- "
•
.
o
. . .
•. : . . . . "
:::===.~-:
0
• •i
2 •.;" •..
-500 -400
.1000 t .... -1000
I .... -500
, .... 0
, .... 500
i
-BOO
1000
.
.
.
-400
.
.
.
'
.
0
400
. . . .
BOO
x/b
x/b
(c) Case 3
(d) Case 4
1200
Fig. 3. Dislocation distribution for each case ',,,,hen the plastic zone is fully developed. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4.
Fig. 5 shows the change of effective stress intensity factors with the emission
of dislocations• As a
r e s u l t o f s h i e l d i n g b y all e m i t t e d d i s l o c a t i o n s , e f f e c -
tive stress intensity factor decreases with the increase of dislocations emitted. The shielding effect becomes less effective for the same number of dislocations emitted when the number of available slip planes is
1000 +
*
+
t
~1.
*
-~ v
÷
500
+
14
+
+
•
x^ a
K /(Abl/~)=15.0
,., ,..!
12
't/A=0.1
~"
10E
o'+
,,.
-500 a
-1000
''
. . . . -500
,
, , t . . . . . .
0
500
, ,
1000
x ~ x ~
F
+
-I000
x
o
~
8 . -E
'~
(3 ;-
>
4
xa,
E
KJ(Abl/2)
=I"5'0
×x • xx
~ /A=(I.I ,
1500
Fig. 4. Plastic zone in each slip plane for the four different ca,~s. ( © ) Case I; ( • ) case 2; ( z', ) case 3; ( + ) case 4.
0
Case 1
• a
Car,e 2 Cac, e 3
•
Case 4
x xxA~ x×
,
,
!
. . . .
10
• ,~
x
....
x/b
x
,
20
i
t
A
i
I
30
i
A•~ i
K
,
• I
,
40
Total n u m b e r of d i s l o c a h o n s e m i t t e d F i g . 5. E f f e c d v e
stress intensity factors.
L
~..a._:
50
C.-F. Qian, J.C.M. Li / Mechanics o f Materials 24 (1996) 11 - 17 1500
3.4. Plastic zone and dislocation free zone o
1000
~ooo 500
'~
eo 0
•
O'IL
" - " - ' - - ~t."
KJ(Abt/Z)=lS.0
"
K'/(Abl/2)=IS"0
a
K /(Abl/Z)=18.0
x f/A--0.1
" 0,
lOe ~ • •
.500
•
° ° t# o
" ' " " ~ "
>"
x/A=O.06
• • r~ eo
°A o, o, o, o
k
-1000 ~-1500
15
.... "1000
I .... -SO0
I .... 0
x/A=0.1
I .... 500
I .... 1000
! .... 1500
~,,
,,
2000
2500
x/b
Fig. 6. Effects of the applied stress and friction stress on the plastic zone.
increased. For all these four cases there is a residual stress intensity factor when the crack stops emitting dislocations.
3.2. Effects of the applied stress intensity factor Applied stress is the cause for dislocation emission. Its effect on dislocation emission and distribution is studied here by increasing the applied stress intensity factor by 20% ( K a from 15 to 18Ab ~/2) as shown in Fig. 6 and Fig. 7. It is found that with increasing applied stress, the plastic zone size (Fig. 6) increases, and both the number of slip planes involved in dislocation emission and the total number of emitted dislocations also increase as shown in Fig. 7 in which the slip plane at 10° becomes a new plane to emit dislocations after K a has been increased.
3.3. Effects of the friction stress The effect of the friction stress on dislocation emission and distribution is also studied by decreasing the friction stress by 20% (rf decreased from 0.1 to 0.08A) as also shown in Figs. 6 and 7. Like increasing the applied stress, a decrease of the friction stress also increases the plastic zone size (Fig. 6) and the total number of dislocations (Fig. 7). But as shown in Fig. 7 the number of slip planes involved in dislocation emission does not increase.
Fig. 6 shows the plastic zone around a Mode I crack tip occupied by dislocations and the shape of this zone which seems not much affected by the applied stress nor the friction stress. The maximum dimension is along the 70 ° slip plane. It is seen that this shape is different from that of the well-known Mode I crack plastic zones (plane strain) based on von Mises or Tresca criterion as shown in Fig. 8 with the plastic zone obtained here also included (triangle dots) for comparison. It should be pointed out that the plastic zone obtained in this study is a result of dislocation emission from the crack tip while the von Mises or Tresca criterion uses maximum distortional energy or shear stress in any direction. However, the shape of plastic zone here is almost the same as the contour of constant shear stress o',,~ = rf as shown in Fig. 9 except that the former is larger because the farthest dislocations are repelled by all the dislocations inside. Nevertheless, with the two loops leaning forward from the crack tip, this plastic zone seems to agree more with experimental observations (Broek, 1974; Hertzberg, 1976; Knott, 1976) than the von Mises and Tresca zones.
c
6
e~
5
+
+
m
nn
~
B
m
!
+
R
L~
&
L~
A
L~
&
0
~
4
+
r'1
o
3
D
A
2
A
o
1
-~
0
m
+
A
,,m
A
+
+
0
i
0
30
60
90
120
150
-
-
180
Angles of slip planes (deg.) -,
Ka/(Abt/2)=15.0
xf/A=0.1
+
Ka/(AbZ/2)=18.0
xr/A=0-1
o
Ka/(AbZ/2)=IS.0
xf/A=0.08
Fig. 7. Effects of the applied stress and friction stress on dislocation emission.
16
C.-F. Qian. J.C.M. Li / Mechanics o f Materials 24 (1996) 11-17
1500 K =18Ab
1000
//
i
Oy=2~I
500 o
(
-500 -I000 ....
K /(Abln)=50.0 ,t/A=0.1
~/2
'tt=0"lA
i ....
I ....
i ....
-1500 - 1 5 0 0 - 1 0 0 0 -500
I ....
0
(a)
500
4
/ :::"'-i""
-
"."
;:.."
"-__~ ~ •
-4
"
'°t--
.
"...
"5~
~
.
.
.
;'
.
'
~'",~o-",~"--~o
.,,b
: : . . *.. "..
\ ,, -.. I ....
-5
0
5 x/1000b
1000 1500
x/b
10
15
Fig. 10. Dislocation-free zone when the plastic zone is not fully developed.
1500 K'=18Abt/2
1000
xl=0.1A
•A
(3y=2"t f
500
/
• •
a
-500
~
,
i
•
-1000 -1500 ' - 1 5 0 0 - 1 0 0 0 -500 .
.
.
.
.
.
.
.
.
(b)
.
.
'
.
0
×/b
"
"
'
.
500
.
.
.
'
.
.
.
.
1000 1500
Fig. 8. Plastic zone shapes according to von Mises and Tresca yield criteria. (a) Von Mises criterion; (b) Tresca criterion.
Fig. 3 shows the dislocation distribution along each slip plane involved in dislocation emission.
Strictly speaking there is no dislocation-free zone (DFZ) because there is no sudden change of dislocation density around the crack tip. This is because the plastic zone is fully developed, namely, there is no critical stress intensity factor needed for dislocation emission. If there is such a critical stress intensity factor and the plastic zone is not fully developed, there will be a DFZ. Fig. I0 shows the dislocation distribution at a high applied stress intensity factor (50Ab ~/2) and a large critical stress intensity factor (Kc, = 41 Ab I/2) for dislocation emission. The crack tip region is magnified and also included in Fig. 10 to show the dislocation distribution along the slip planes at + 10 and + 150 °. Since the plastic zone has not fully developed, a DFZ clearly exists around the crack tip. Fig. 11 shows the ratio of DFZ over the
1500 1000 500
K =18Abi n xl=0.1A °Y=2"tl
2O •
I
• 1
: ~
-1000
• •
'If/
5
A=O.
1
,
1
=**
NI fa.
-500
K /(Abln)=50.0
•
1 0 ~ -iP
t-
F -1 500 . . . . ' . . . . ' . . . . ' . . . . ' . . . . " . . . . -1500-1000-500 0 500 1000 1500 x/b Fig. 9. Radial shear stress contour (curve) and the computed plastic zone (points).
oF, 0
• !
30
i
L
]
60
,
I
I
90 ¢ (deg.)
120
,
150
180
Fig. I1. The relative magnitude of DFZ in each slip ptane involved in dislocation emission.
C.-F. Qian. J.C.M. Li/Mechanics of Materials 24 (1996) 11-17
spacing between the first and second dislocations from the crack. Since the critical stress intensity factor for dislocation emission may be material dependent, it could be the reason why some people observed a DFZ around a Mode I crack, but others did not.
4. Conclusions (I) For the same applied stress intensity factor and the same friction stress for dislocation motion, increasing the number of available slip planes increases the total number of dislocations emitted at saturation, reduces the number of dislocations in each slip plane and produces less shielding for the same number dislocations emitted (but slightly more at saturation). (2) Increasing the applied stress intensity factor or decreasing the lattice friction stress for dislocation motion increases the total number dislocations emitted at saturation, the size of the plastic zone and the fraction of occupied available slip planes. (3) The shape of the plastic zone is different from but the size is comparable with the plastic zone based on von Mises or Tresca criterion. It is similar
17
to the shape but larger than the contour of constant polar (r, tk) shear stress equal to the lattice friction stress for dislocation motion. (4) A distinct dislocation-free zone develops only if there is a critical stress intensity factor for dislocation emission.
Acknowledgements This work was supported partly by an Allied-Signal fellowship and partly by NSF through DMR9221326 monitored by Dr. Bruce MacDonald.
References Broek, D. (1974), Elementary Engineering Fracture Mechanics, Noordhoff International Publishing, Leiden. Hertzberg, R.W. (1976), Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons, Inc., New York, pp. 273-277. Knott, J.E. (1976), Fundamentals" of Fracture Mechanics, John Wiley & Sons, Inc., New York. Qian, C.-F. and J.C.M. Li (1996), this issue, Mech. Mater. 24, 1-10.
Mechanics of Materials 24 (19%) I 1- 17
Edge dislocations emitted along multiple inclined slip planes from a Mode I crack. II. Simultaneous emission Cai-Fu Qian i, J.C.M. Li * Department of Mechanical Engineering. University of Ro~.he.~ter. Rochester. NY 14627. USA Received 5 May 1995; revised version received 9 May 1996
Abstract In an effort to simulate dislocation distribution around a Mode I crack, edge dislocations are allowed to be emitted from the crack tip along many possible slip planes containing the crack tip. it turned out that the final distribution depends on the order of emission. So the previous paper dealt with the situation in which the emission process is one at a time, namely after one dislocation is emitted the plastic zone is re-equilibrated before the next dislocation is emitted. In this paper, many dislocations are emitted at the same time when the condition is met, namely, when the slip force exceeds the lattice friction for dislocation motion. It is found that the dislocations are more uniformly distributed than the previous case of sequential emission. The shielding effect for the same number of dislocations emitted is less when the dislocations are distributed along more slip planes. The shape of the plastic zone is different from but the size is comparable with the plastic zone based on yon Mises or Tresca criterion. A distinct dislocation-free zone appears when there is a critical stress intensity factor for dislocation emission.
1. I n t r o d u c t i o n In the preceding paper (Qian and Li, 1996) dislocations were emitted sequentially from a Mode I crack. As a result, most of the dislocations were emitted along a slip plane in which the shear stress was the largest even though 34 slip planes were available. Unless the loading is increased very slowly
• Corresponding author. Tel.: + 1-716-2754038; fax: + 1-7162562509; e-mail: [email protected]. Now at Department of Mechanical Engineering, Hong Kong University of Science of Technology, Clearwater Bay, Kowloon, Hong Kong.
so that only one dislocation can be emitted at a time and the next one will not be emitted until all the emitted dislocations have taken up their equilibrium positions, it is possible and perhaps more realistic that dislocations are emitted simultaneously along many slip planes from the crack when the loading is fast so that dislocations can be emitted along many slip planes all at once. The plastic zone around a crack tip is usually obtained analytically or numerically based on the yield stress. Examples are the well-known plastic zones using the von Mises or Tresca criterion. But since the plastic deformation of crystalline materials is due to the motion of dislocations and dislocations are likely to be emitted from a crack tip, it would be interesting to determine the shape and size of the
0167-6636/%/S15.00 Copyright © 1996 Published by Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 6 6 3 6 ( 9 6 ) 0 0 0 2 5 - 7
12
C.-F, Qian, J.C.M. l.i / Mechanics of Materials 24 (1996) 11 -.17
the dislocation will move toward the crack tip by a distance b. The dislocation will be absorbed by the crack and disappear if the distance to the crack tip is less than b or negative. These motions will take place until equilibrium prevails, namely, no more motions are possible. Then another round of emission will take place and the motions of all dislocations repeated. The computation terminates when no more dislocation can be emitted. In this study all available slip planes were placed symmetrically on both sides of the crack. But the total number of dislocations emitted refer to one side of crack only. The available slip planes for the four different cases studied here are listed in Table 1 with N, being the total number of available planes.
plastic zone around a crack tip according to the dislocation behavior. Experimental observations and computer simulations of dislocation emission from a crack tip have been reviewed in the introduction of the preceding paper (Qian and Li, 1996). In this study dislocations are allowed to be emitted simultaneously along many inclined slip planes from the tip of a Mode I crack. Equilibrium dislocation distribution around the crack tip including the plastic zone and the dislocation-free zone is studied together with the effects of the applied stress and the lattice friction stress. Increasing the applied stress further or decreasing the friction stress further will increase the number of dislocations emitted and the number of slip planes involved in dislocation emission and thereby increasing the computer time and storage beyond the present capability.
3. Results 2. Assumptions
3.1. Different number of available slip planes Most assumptions made in this study are the same as those in the preceding paper. All assumptions are presented here: (l) All dislocations are edge dislocations with their lines parallel to the tip of the crack, and all slip planes are passing through the crack tip. (2) All dislocations are emitted from the crack tip. (3) A dislocation is emitted along a slip plane as long as the slip force on the emitted dislocation (located at b from the crack tip) is larger than the friction force. After the emission of all possible dislocations, the slip forces on all the emitted dislocations are calculated according to the scheme presented in Qian and Li (1996). When the slip force is larger than the friction force, the dislocation moves forward by a distance b. If the slip force on the dislocation is negative and exceeds the friction force,
Fig. 1 shows the slip forces (in units of Ab) available for dislocation emission (placed at b from the crack tip) along each slip plane for the four cases studied with N being one half of the total number of dislocations emitted and equilibrated before the next round of emission. Since the dislocations are allowed to be emitted from all available slip planes simultaneously, all forces are decreasing with the increase of dislocations emitted and no change of slip planes is found as in the case when only one dislocation is allowed to be emitted along the slip plane with the largest slip force (Qian and Li, 1996) between equilibrium configurations. When all the slip forces for dislocation emission are finally smaller than the friction force, the dislocation emission process is terminated.
Table 1 Available slip planes Case
Angles of slip planes ( _+deg.)
NL
I
60
2 3 4
60 60 60
10
20 20
30
40 40
50
120
70
80 80 80
90
100 100 100
110
120 120 120
4
130
140 140
150
160 160
170
8 16 34
C.-F. Qian. J.C.M. Li / Mechanics of Materials 24 (1996) 11-17
13
2 K,/(Ab~/~)=lS"0 =
&
1.5
•
A
~
"tl/A=O.1
• $.=~o
=
1.5
=
K,/(AbWZ) =15-0 xf/A=0.1
o +
~=60 o ¢ _.=~o
•
$ =100 °
&
•
o
=j
o
o
o
B
•
=
$=lZIT
",,
°
0.5 ~.
=
•
l,
"o
o o
os
",
. F r i c t i o n t o r c . e _ l i n e . . , o..o . . . . . . . . . . .
0 ~-
•
g
----....--.--:.:_Fricti°nf°rce]ine...'..A. •
a...
0.
°°°OOOoooo
0
•
0
_
&&
-0.5
F ....
I ....
0
I ....
5
(a)
I ....
10
15
u,,,,I,,,,
20
25
....
I ....
5
35
(5)
10
X
15
I ....
20
I ....
25
•
o
x
a
A N=0
•
1.5
'tl/A=0.1
25
• •
&
:
3O•
.
..........
•
o.s
o
38
A
•
~ ........
o
~
I
,
30
(c)
I
,
,
~
60
l
,
I
i
90
,
I
•
x
150
0
180
Angles of slip planes (deg.)
(d)
x
x
x
O
O
O
O
43
l~lelI,,=,=lllelll,
-0.5
I
•
35
x
°~°°°o
,
120
• •
&
x
;::
~- ~.-~ .x.Frir_ti~'t force- lio~ . . . . . . . . . . . . . .
0
o
,
0
•
,
o
-0.5
&
X
- . . . . 0- " - b . . . . . . . .
o
40
&
• • 0.5
35
K=/(Abl/Z)=15.0
19
•
30
I ....
& A
X
1 ....
Total number of dislocations emitted, N
&
7
I ....
2
N=0 &
1.5
,,,1
-0.5 30
Total n u m b e r of dislocations emitted, N
2 =
1 ....
30
O
°°~io ,,llleple=
60
90
120
,~lOe
150
180
A n g l e s of slip p l a n e s (deg.)
Fig. 1. Slip forces on d i s l o c a t i o n s to be emitted. (a) Case I; (b) case 2; (c) case 3; (d) case 4.
Fig. 2 shows the number of dislocations emitted along each slip plane for the four different cases. The total number of dislocations emitted on all slip planes 25 •~
o
C a s e 1, N---32
•
C a s e 2, N---37
a o
C a s e 3, N = 3 8 C a s e 4, N = 4 3
20
"8 g
15
E
10
"=
5
•
&
0
0
o
0
0
0
0
0
O
o
17
o
o
0 I
I
30
60
,
o
1
,
90
,
I
,
120
B ~ ~_ _ - -,
150
--
180
A n g l e s of slip p l a n e s (deg.) Fig. 2. D i s l o c a t i o n s e m i t t e d a l o n g each slip plane for the four different cases.
is 32, 37, 38 and 43 for the four cases respectively. For some slip planes with very large or very small angles (e.g. 150 °, 160 °, 170 ° and 10° for case 4), no dislocation can be emitted. This is because the applied force is not large enough to enable dislocations to initiate and move along these slip planes, thus making the number of slip planes involved in dislocation emission (e.g. 13 for a total of 17 in case 4) less than the number of available slip planes. It is clear from Fig. 2 that the more the slip planes are available for dislocation emission, the less the number of dislocations on each slip plane, and the more the total number of dislocations on all slip planes. Fig. 3 shows the dislocation distributions for the four cases. Fig. 4 shows the plastic zone affected by the number of slip planes available for dislocation emission. When the number of slip planes increases, the number of dislocations in each slip plane decreases and hence the plastic zone decreases.
14
C.-F. Qian. J.C.M. l.i / Mechanics of Materials 24 (1996) I I - 17 1000
1000 KJ(AbW2)=15.0
'~JA=0.1
500
KJ(Ab~/2)=I5.0
.."
.•• • ..,
'Oz./A=0.1
."
500
/"
•.." :.." .o:-. ::5
o
o'. • ",
-500
-500
....
-1000
I ....
I ....
-500
-1000
I•
0 x/b
i
i
500
i
L
-1000
. . . .
1000
t
-1000
. . . .
",
,
,
0
]
. . . .
500
1000
x/b
(a) C a s e I 1000
i,
-500
(b) Case 2 800
KJ(Abl/2)=IS.0
•
~IIA=O.I 500
400
- "
•
.
o
. . .
•. : . . . . "
:::===.~-:
0
• •i
2 •.;" •..
-500 -400
.1000 t .... -1000
I .... -500
, .... 0
, .... 500
i
-BOO
1000
.
.
.
-400
.
.
.
'
.
0
400
. . . .
BOO
x/b
x/b
(c) Case 3
(d) Case 4
1200
Fig. 3. Dislocation distribution for each case ',,,,hen the plastic zone is fully developed. (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4.
Fig. 5 shows the change of effective stress intensity factors with the emission
of dislocations• As a
r e s u l t o f s h i e l d i n g b y all e m i t t e d d i s l o c a t i o n s , e f f e c -
tive stress intensity factor decreases with the increase of dislocations emitted. The shielding effect becomes less effective for the same number of dislocations emitted when the number of available slip planes is
1000 +
*
+
t
~1.
*
-~ v
÷
500
+
14
+
+
•
x^ a
K /(Abl/~)=15.0
,., ,..!
12
't/A=0.1
~"
10E
o'+
,,.
-500 a
-1000
''
. . . . -500
,
, , t . . . . . .
0
500
, ,
1000
x ~ x ~
F
+
-I000
x
o
~
8 . -E
'~
(3 ;-
>
4
xa,
E
KJ(Abl/2)
=I"5'0
×x • xx
~ /A=(I.I ,
1500
Fig. 4. Plastic zone in each slip plane for the four different ca,~s. ( © ) Case I; ( • ) case 2; ( z', ) case 3; ( + ) case 4.
0
Case 1
• a
Car,e 2 Cac, e 3
•
Case 4
x xxA~ x×
,
,
!
. . . .
10
• ,~
x
....
x/b
x
,
20
i
t
A
i
I
30
i
A•~ i
K
,
• I
,
40
Total n u m b e r of d i s l o c a h o n s e m i t t e d F i g . 5. E f f e c d v e
stress intensity factors.
L
~..a._:
50
C.-F. Qian, J.C.M. Li / Mechanics o f Materials 24 (1996) 11 - 17 1500
3.4. Plastic zone and dislocation free zone o
1000
~ooo 500
'~
eo 0
•
O'IL
" - " - ' - - ~t."
KJ(Abt/Z)=lS.0
"
K'/(Abl/2)=IS"0
a
K /(Abl/Z)=18.0
x f/A--0.1
" 0,
lOe ~ • •
.500
•
° ° t# o
" ' " " ~ "
>"
x/A=O.06
• • r~ eo
°A o, o, o, o
k
-1000 ~-1500
15
.... "1000
I .... -SO0
I .... 0
x/A=0.1
I .... 500
I .... 1000
! .... 1500
~,,
,,
2000
2500
x/b
Fig. 6. Effects of the applied stress and friction stress on the plastic zone.
increased. For all these four cases there is a residual stress intensity factor when the crack stops emitting dislocations.
3.2. Effects of the applied stress intensity factor Applied stress is the cause for dislocation emission. Its effect on dislocation emission and distribution is studied here by increasing the applied stress intensity factor by 20% ( K a from 15 to 18Ab ~/2) as shown in Fig. 6 and Fig. 7. It is found that with increasing applied stress, the plastic zone size (Fig. 6) increases, and both the number of slip planes involved in dislocation emission and the total number of emitted dislocations also increase as shown in Fig. 7 in which the slip plane at 10° becomes a new plane to emit dislocations after K a has been increased.
3.3. Effects of the friction stress The effect of the friction stress on dislocation emission and distribution is also studied by decreasing the friction stress by 20% (rf decreased from 0.1 to 0.08A) as also shown in Figs. 6 and 7. Like increasing the applied stress, a decrease of the friction stress also increases the plastic zone size (Fig. 6) and the total number of dislocations (Fig. 7). But as shown in Fig. 7 the number of slip planes involved in dislocation emission does not increase.
Fig. 6 shows the plastic zone around a Mode I crack tip occupied by dislocations and the shape of this zone which seems not much affected by the applied stress nor the friction stress. The maximum dimension is along the 70 ° slip plane. It is seen that this shape is different from that of the well-known Mode I crack plastic zones (plane strain) based on von Mises or Tresca criterion as shown in Fig. 8 with the plastic zone obtained here also included (triangle dots) for comparison. It should be pointed out that the plastic zone obtained in this study is a result of dislocation emission from the crack tip while the von Mises or Tresca criterion uses maximum distortional energy or shear stress in any direction. However, the shape of plastic zone here is almost the same as the contour of constant shear stress o',,~ = rf as shown in Fig. 9 except that the former is larger because the farthest dislocations are repelled by all the dislocations inside. Nevertheless, with the two loops leaning forward from the crack tip, this plastic zone seems to agree more with experimental observations (Broek, 1974; Hertzberg, 1976; Knott, 1976) than the von Mises and Tresca zones.
c
6
e~
5
+
+
m
nn
~
B
m
!
+
R
L~
&
L~
A
L~
&
0
~
4
+
r'1
o
3
D
A
2
A
o
1
-~
0
m
+
A
,,m
A
+
+
0
i
0
30
60
90
120
150
-
-
180
Angles of slip planes (deg.) -,
Ka/(Abt/2)=15.0
xf/A=0.1
+
Ka/(AbZ/2)=18.0
xr/A=0-1
o
Ka/(AbZ/2)=IS.0
xf/A=0.08
Fig. 7. Effects of the applied stress and friction stress on dislocation emission.
16
C.-F. Qian. J.C.M. Li / Mechanics o f Materials 24 (1996) 11-17
1500 K =18Ab
1000
//
i
Oy=2~I
500 o
(
-500 -I000 ....
K /(Abln)=50.0 ,t/A=0.1
~/2
'tt=0"lA
i ....
I ....
i ....
-1500 - 1 5 0 0 - 1 0 0 0 -500
I ....
0
(a)
500
4
/ :::"'-i""
-
"."
;:.."
"-__~ ~ •
-4
"
'°t--
.
"...
"5~
~
.
.
.
;'
.
'
~'",~o-",~"--~o
.,,b
: : . . *.. "..
\ ,, -.. I ....
-5
0
5 x/1000b
1000 1500
x/b
10
15
Fig. 10. Dislocation-free zone when the plastic zone is not fully developed.
1500 K'=18Abt/2
1000
xl=0.1A
•A
(3y=2"t f
500
/
• •
a
-500
~
,
i
•
-1000 -1500 ' - 1 5 0 0 - 1 0 0 0 -500 .
.
.
.
.
.
.
.
.
(b)
.
.
'
.
0
×/b
"
"
'
.
500
.
.
.
'
.
.
.
.
1000 1500
Fig. 8. Plastic zone shapes according to von Mises and Tresca yield criteria. (a) Von Mises criterion; (b) Tresca criterion.
Fig. 3 shows the dislocation distribution along each slip plane involved in dislocation emission.
Strictly speaking there is no dislocation-free zone (DFZ) because there is no sudden change of dislocation density around the crack tip. This is because the plastic zone is fully developed, namely, there is no critical stress intensity factor needed for dislocation emission. If there is such a critical stress intensity factor and the plastic zone is not fully developed, there will be a DFZ. Fig. I0 shows the dislocation distribution at a high applied stress intensity factor (50Ab ~/2) and a large critical stress intensity factor (Kc, = 41 Ab I/2) for dislocation emission. The crack tip region is magnified and also included in Fig. 10 to show the dislocation distribution along the slip planes at + 10 and + 150 °. Since the plastic zone has not fully developed, a DFZ clearly exists around the crack tip. Fig. 11 shows the ratio of DFZ over the
1500 1000 500
K =18Abi n xl=0.1A °Y=2"tl
2O •
I
• 1
: ~
-1000
• •
'If/
5
A=O.
1
,
1
=**
NI fa.
-500
K /(Abln)=50.0
•
1 0 ~ -iP
t-
F -1 500 . . . . ' . . . . ' . . . . ' . . . . ' . . . . " . . . . -1500-1000-500 0 500 1000 1500 x/b Fig. 9. Radial shear stress contour (curve) and the computed plastic zone (points).
oF, 0
• !
30
i
L
]
60
,
I
I
90 ¢ (deg.)
120
,
150
180
Fig. I1. The relative magnitude of DFZ in each slip ptane involved in dislocation emission.
C.-F. Qian. J.C.M. Li/Mechanics of Materials 24 (1996) 11-17
spacing between the first and second dislocations from the crack. Since the critical stress intensity factor for dislocation emission may be material dependent, it could be the reason why some people observed a DFZ around a Mode I crack, but others did not.
4. Conclusions (I) For the same applied stress intensity factor and the same friction stress for dislocation motion, increasing the number of available slip planes increases the total number of dislocations emitted at saturation, reduces the number of dislocations in each slip plane and produces less shielding for the same number dislocations emitted (but slightly more at saturation). (2) Increasing the applied stress intensity factor or decreasing the lattice friction stress for dislocation motion increases the total number dislocations emitted at saturation, the size of the plastic zone and the fraction of occupied available slip planes. (3) The shape of the plastic zone is different from but the size is comparable with the plastic zone based on von Mises or Tresca criterion. It is similar
17
to the shape but larger than the contour of constant polar (r, tk) shear stress equal to the lattice friction stress for dislocation motion. (4) A distinct dislocation-free zone develops only if there is a critical stress intensity factor for dislocation emission.
Acknowledgements This work was supported partly by an Allied-Signal fellowship and partly by NSF through DMR9221326 monitored by Dr. Bruce MacDonald.
References Broek, D. (1974), Elementary Engineering Fracture Mechanics, Noordhoff International Publishing, Leiden. Hertzberg, R.W. (1976), Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons, Inc., New York, pp. 273-277. Knott, J.E. (1976), Fundamentals" of Fracture Mechanics, John Wiley & Sons, Inc., New York. Qian, C.-F. and J.C.M. Li (1996), this issue, Mech. Mater. 24, 1-10.