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An analysis of the elastic interaction between an edge dislocation and an internal crack
Materials Science and Engineering, A I 30 (1990) 1-10
1
An Analysis of the Elastic Interaction Between an Edge Dislocation and an Internal Crack SHING-DAR WANG and SANBOH LEE Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu 30043 (Taiwan) (Received October 6, 1989; in revised form May 7, 1990)
Abstract
The elastic interaction between an edge dislocation and a finite crack has been investigated using the dislocation-modelling approach. We considered not only the applied stresses of modes I and H but also dislocations inside the crack. We obtained the stress field, the force and strain energy of the dislocation and the stress intensity factors at both crack tips. The unstable equilibrium positions of dislocation have been studied. Also, the criterion of dislocation emission from the crack tip has been analysed in order to understand the crack tip as a major source of dislocation. 1. Introduction
Dislocations around a crack tip have an intimate relationship with the ductility of crystalline materials and have therefore been studied extensively. Because of its mathematical simplicity, the elastic interaction of a screw dislocation and a crack has been solved by many people using different approaches. For example, by using dislocation modelling, Louat [1] and Lin et al. [2] solved a screw dislocation interacting with a finite crack, while Lee [3] and Juang and Lee [4] analysed the problem of a surface crack. On the other hand, by using the conformal mapping technique, Chu [5] and Majumdar and Burns [6] investigated a screw dislocation interacting with a surface crack and a semi-infinite crack respectively, while Shiue and Lee [7] studied the elastic interaction between a screw dislocation and cracks emanating from an elliptical hole. When a screw dislocation is in the vicinity of a semiinfinite crack and coplanar with the crack surface, the problem becomes the same as that of an edge dislocation. An edge dislocation near a semiinfinite crack tip but in the plane of the crack has been solved by Smith [8]. However, an edge dislo0921-5093/90/$3.50
cation is different from a screw dislocation coplanar with a finite crack. It seems very important to solve the elastic interaction between an edge dislocation and a crack. Nevertheless, the study of this topic has been very limited. Hirth and Wagoner [9], Thomson and Sinclair [10] and Lin and Thomson [11] introduced the complex potential to obtain an analytical solution for an edge dislocation interacting with a semi-infinite crack. Lakshmanan and Li [12] used dislocation modelling to obtain the stress field of an edge dislocation near a semi-infinite crack tip. Vitek [13] solved numerically the interaction between an array of edge dislocations and an elliptical hole. Riedel [14] used a dislocation model for the crack and solved numerically the dislocation distribution inside the crack and along the four inclined planes. However, none of these solutions was in closed form owing to the complicated mathematics involved. Another intriguing problem is the elastic interaction between a mixed dislocation and a semiinfinite crack. Rice and Thomson [15] and Shiue and Lee [16] studied the slip and climb forces on a mixed dislocation arising from a semi-infinite crack using a thermodynamic approach. Lee et al. [17] described graphically the image force and the potential energy of a mixed dislocation with a semi-infinite crack. However, their solutions did not address the stress field, so that many aspects of the interaction are missing. Atkinson [18] and Barnett and Asaro [19] presented procedures for solving the problem of a crack and a mixed dislocation in an anisotropic medium. Without the complete solution, a great deal of physical insight was lost. In this paper we study the elastic interaction between an edge dislocation and a finite crack in an isotropic medium using dislocation modelling. The dislocation distribution, stress field, total © Elsevier Sequoia/Printed in The Netherlands
force and strain energy of dislocation, stress intensity factor at the crack tip and dislocation emission criterion are obtained. On the basis of the force components, we find the zero-force positions of the dislocation. By analysing the dislocation emission criterion, we conclude that the crack tip can be a major source of dislocation in fracture.
/
A
f x-xfx(x')'dx' -/ +
A{(x - xo)cos(a) -
+ o. = 0 /
2. Dislocation distribution inside a finite crack
The problem is illustrated in Fig. 1. Assume that an internal crack lies along the x axis from - 1 to l with an infinite z dimension and that m dislocations are contained inside the crack. A straight edge dislocation has a sense of - z and a Burgers vector b (right-hand final start, RHFS) oriented at an angle a to the x axis. The dislocation is situated at (x0, Y0)- The medium is subject to far-field stresses crI and o H that produce mode I and II crack-opening displacements respectively as shown in Fig. 1. The crack surface is stress free so that Oyyand o~y are zero on the crack surface. Unlike the problem of a screw dislocation with a crack, which requires one array of the continuous distribution of dislocations inside the crack, two infinitesimal Burgers' vectors fx(x)dx and fr(x)dx of edge dislocations in the region between x and x + dx are constructed in order to satisfy the boundary condition. Here subscripts "x" and "y" represent the components of the Burgers vector. The governing equations for the distributions of dislocations are 01 =
(la)
!
A (fy(x)
+ [A[(x - Xo){(x --X0) 2 "1-3yo2} sin(a) -- y0{(X -- X0 )2 -- y0 2 } COS( a )]]/[{(X -- X0 )2 q- y0 2 }2]
(lb)
nt- o r i = 0
where A-
~b 2~(1 - v)
(2)
/x is the shear modulus of the medium and v is Poisson's ratio. Using the inversion theorem [20, 21], we solve the distributions from eqn. (1) as b (YoPo sin( 00 L(x)- ~rq(12-x2) 1/2 \ r,, +
yoro sin(00 -
¢0 + 6 - a)
2 6 + a)
q (m + 1)b cos(a)
+ r0 cos(a) cos(00-
- v)
+ 2(1 - /g
t I I I t t o,,.oz (x. Y.)
/~__
,
! l x - x 'dx
Gy~
IY
Y0 sin(a)}{(x - x0)2 - y02}
{(x -xo) 2+ y02}2
onx
(3a)
( l 2 --X2) 1/2
b
(YoPo
L ( x ) - ~q,.2tt - x2,1/2)
\r0
iii III _
6)) -4 3r(l2 _x2)1/:
yoro cos(00 -
cos(O0- #o+ 6 - a)
2 6 + a)
q
- r0 sin(a) cos(0o - 6)) +
-4
i i i i 1 IJI * f f y ~
Fig. 1. A n edge dislocation situated at (x., Yo) around a finite crack subjected to far-field stresses o~ and o..
2( 1 #
v)
o~x (l 2 - x 2 ) l / 2
(m + 1)b sin(a) t 2 - x2)
(3b)
where Z o = x0 + iy0 = P0 exp(i•o)
(4a)
/02-
12 = r02 e x p ( i 2 0 0 )
Z o - x = q exp(i6),
(4b)
Ix[ < l
(4c)
the definition of 00 is the same as that expressed by Lin et al. [2] and the angle 6 is shown in Fig. 1. (Note that capital Z represents a complex variable while lowercase z is the z coordinate.) The distributions are a function of x in the range between - l and l with parameters Zo, m, OI and aid When the lattice dislocation is coplanar with the crack plane, i.e. y 0 = 0 , without considering the applied stress, the distributions fx(X) with a = 0 ° and fy(X) with a = 90 ° are the same and are equivalent to the distribution of a screw dislocation coplanar with an internal crack [2]. When the crack length becomes semi-infinite, eqn. (3) reduces to the results of Li and Lakshmanan [12] if the applied stress is not considered. According to the conservation of the Burgers vector, if there is no dislocation inside the crack originally, then one of the following three situations can occur. First, when the crack tip emits a positive edge dislocation into the space, a negative edge dislocation will be created inside the crack, i.e. rn = - 1. Secondly, if the lattice dislocation comes from infinity, then the dislocation inside the crack will be zero, i.e. m = 0. Thirdly, when a dislocation is emitted from one crack tip, another dislocation will simultaneously be emitted from the other crack tip to preserve the Burgers' vector conservation. Using eqo. (3), we can obtain the stress components o~)~/arising from both the lattice dislocation and the rn dislocations inside the crack as O'y,!1)= 0"1 -{- 0"2
(5a)
O'xx(I) -~- O 1 -- 0"2
(5b)
0.xy(I)= 0.3
(5C)
where 0"1=
A(r
2r
P3 rl
sin(a
- 42 - 0 + 02)
sin(a + 4 3 - O - 01)/ !
"t-AY°(P~llCOS(a-b¢l--43-t-O-'~-Ol)]oif /90 cos(a + 4o + 4~ - 0o + 0)/ / ro +
A(rn + 1) r
ip3A |455o [ exp{ - i ( a + 4, )} 02 "1"-i 03 = - 2rpl rl \ lol --Pl
2y sin(a)
× exp{i(43- 4 1 - 0 - - 01) }
~ - ~ ° 3 [2pZpo exp{i(2~ - ~bo)}- 12p3 p~ ror x exp(i43)] exp{i( 0 o - a - 241 - 3 0)} 9
+ Aypr,~ n~ 1 2r3 Pn x exp[i{a + 4 - 30 +( - 1 )"00 - 4,,}] 9
Ayopo ~, ( - 1 ) " 2ror n=l
Pn
× exp[i{( - 1)"(a + 4o - 0 o ) - 4 . - 0}] iAr2 cos(a - 42) exp{i(0z - 2 4 2 - 0)} + 2rp2
A ( m +1) [yp +- exp{i(a + 4 - 30)} r 7 + i cos(a) exp( - i0)]
(6c)
exp(i41 )
(6d)
Z - Z 0 = P2 exp(i42)
(6e)
Z + 'go = P3 exp(i43)
(6f)
Z + Z'o = P4 exp(i44)
(6g)
Z 2 - l 2 = r 2 exp(i20)
(6h)
Zo2 - l 2 = ro2 exp(i20o)
(6i)
(Z 2 _/2)1/2 + (22 _/2)1/2 = rl exp(i01)
(6j)
(Z2-I2)l/2+(Z2-I2)I/2=r2 e x p ( i 0 2 )
(6k)
Z - Z'0 = P l
The definitions of 0 and 00 in eqns. 6(h) and 6(i) respectively are the same as those expressed in ref. 2. 01 and 02 follow from the definitions of 0 and 00. Similarly, the stress fields arising from the far-field stresses o~ and oii are expressed as 0.!v~) = O'I(K - N ) + o I I M
(6a)
(6b)
Z = x + iy = p exp(i4)
a?x ) =
sin(a - 0)
C O S ( a -~- 4 1 ) - -
0.1(K -Jr-N - 1) + an(2L - M)
a ~ = a , M + oII(K + N)
(7a)
(7b) (7c)
+ Abp__o
where
2r02 Iexp( - i200) K + i L = ~ exp{i(# - 0)}
(8a)
r
× [sin(a) exp{i(a + 4o - 200)}
2
M + iN = ~
r
exp{i(2¢ - 3 0)} - -y exp( - i0) r
(8b)
- i exp(i#0 ) + sin(9~0) exp(i2a)] + sin(a - ¢~o)exp( - i a ) - sin(2 0 o - ¢o - 2a)]
and p, ~ and r, 0 are defined in eqns. 6(c) and 6(h) respectively. When l approaches infinity and x - I remains finite, eqn. (7) can reduce to the results of Broek [22], who obtained the stress field from the Airy stress function. The total stress field components a!~ / arising from the lattice dislocation, the m dislocations inside the crack and the applied stresses are a!T/= o(!q) + a!~)
(9)
The numerical values of the stress components a!f I are checked by using personal computer. It is found that the stress components vyy(T/and Oxy(T/ vanish on the crack plane, as required by the boundary conditions, i.e. a/T/yy= a ~ I = 0 for Ixl < l. When the crack length becomes semi-infinite, the contours look like the profiles of Lakshmanan and Li [12]. When l approaches infinity, the system reduces to the case of an edge dislocation near a free surface. When the Burgers vector is perpendicular to the free surface, our equations are the same as those of Hirth and Lathe [23] and the stress contours look like the profiles of Jagannadham and Marcinkowski [24].
3. Total force and strain energy of the dislocation Using the stress field derived above and the Peach-Koehler formula, one gets the total force on the edge dislocation as
F=Fy+iFx=F1+ Fa
(10)
+ A b ( m + 1) (y0p20 exp{i(2a + 4 0 - 300)} ro
\ r0
+ i exp( - i 00 ) + sin( 00 - 2 a)} !
( 11 a)
~=~y+i~x
= olb(-P°cos(~o-Oo)exp(\
r0
ia) + cos(a))
+ o n b ( ~° [i cos(a)exp{i(¢0-00)}
- sin(a + 40 - 00)])
+(en+iol) b12 Y~°3exp{i(a-30o) }
(llb)
r0
In the above equations the subscripts x and y denote the force components along the x and y directions respectively. When both x 0 and 1 approach infinity and x 0 - I is kept constant, the above equations reduce to the result of a semiinfinite crack [15-17]. When / approaches infinity, we have the case of an edge dislocation near a free surface. Equation 1 l(a) becomes Fx + iFy = - i -
Ab
(12)
2Y0
where
F I = Fly + iFix AbyoPo 2 - -
2r0 4
exp( - i20o)
x [2 exp{i2(a + 40 - 00)} + 1]
Ab {exp( - i 2 0 o ) " 1} 4Yo --
which shows that the force on the dislocation is always towards the free surface, independent of the direction of the Burgers vector. Consider the symmetric pair of edge dislocations shown in the upper part of the inset in Fig. 5(a) (see Section 5). It is found that for a semiinfinite crack the slip force of the interaction between any two such dislocations is zero, which is in good agreement with Lin and T h o m s o n [11],
and the climb force is
Fclimb
--
ZbESsin'2'4cot(2) x{2cos(2)cos() 11]
16p
(13)
where x 0 + iy0 = p exp(ie)
14)
However, when the crack length is finite, the slip and climb components of the interaction stated above are not zero. When the crack length shrinks to zero, the interaction between the two dislocations is not zero either. The negative gradient of the strain energy is the total force on the edge dislocation. Therefore we obtain the strain energy of the edge dislocation in an infinite medium with a finite crack as
E = ~ - ira
- - - Po sin(2a + # 0 - 20o)
+ J{cos(2 0o) - 1 }+ sin(00) sin( 00 - 2 a) +4In
8(roe + p o 2 - 1 2 )
(Rr°] 2
/
+ 2(m + 1) (Y0 sin(2a - 00) \r0 - Re[ln{Zo + (Zo ° -
)
+ 2m In(l)
]
/¢
+ b[ y°A~ {al cos(a + 4 o - 0o)
+ on sin(a + 40 - 0o)}
applied stress, eqn. (15) reduces to the self-energy of an edge dislocation per unit length [23]. Secondly, when l approaches infinity, i.e. a free surface, and the slip plane a = 9 0 °, eqn. (15) becomes the result of an edge dislocation near a free surface [23] except for a constant. Thirdly, when l approaches infinity and x 0 - l remains finite, eqn. (15) reduces to the result of Lee et al. [17] except for a constant. The energy contours and trajectories of the edge dislocation in the absence of any applied stress are plotted as solid and dashed lines respectively in Fig. 2(a), where m = - 1 and a = 45 °. Here the arrow of the trajectory represents the direction of the dislocation movement. Regardless of the position of the dislocation, the edge dislocation has a tendency to be attracted either to the crack surface or to the crack tips, i.e. each trajectory following the arrow direction is terminated at the crack surface or a crack tip. Figure 2(b) and 2(c) represent the energy contours and trajectories of the edge dislocation for o I : 0.9, o n = 0 and o I = 0, Oli : 0.9 respectively, where the unit of stress is A f t . Again the solid and dashed lines represent the energy contours and trajectories respectively. In these two figures the coarsely dashed line divides the space into two regions. In one region the dislocation following the arrow direction of trajectory moves towards the crack, which is the same as in Fig. 2(a). In the other region the dislocation following the arrow direction of trajectory is repelled to infinity. It can be seen from the figures that P~ (1.76, 0.59), P2 (1.58, - 0 . 2 9 ) , ~ (1.86, 0) and P4 (1.07, - 1.03) are saddle points in unstable equilibrium. In addition, with increasing applied stress Ol and/or %, the coarsely dashed line moves towards the crack tip.
- r 0 cos(00){o I sin(a)+ Oil COS(a)} 4.
- o l y o cos(a))
(15)
where R represents the size of the crystal and ~0 is the core radius of the dislocation. Re [.] means the real part of a complex function. The strain energy is a function of the crack length/, the position (x0, Y0) of the dislocation, the Burgers vector b, its slip plane a, the m dislocations inside the crack and the external stress o I and O'i,D T h r e e special cases are worthy of mention. First, when the crack vanishes, i.e. l = O, in the absence of any
Stress intensity factor
The interaction between an edge dislocation and an internal crack is a mixture of modes I and II. The stress intensity factor at the crack tip for mode I is defined as K l./ =
lim . [{2n(x--l)}
1/2
oyy(W/,ly=0
(16a)
x~ l +
at the right-hand tip and K1_/= lim [ { 2 x l x +l[} 1/2 aye,¢Tilly=0 , x~-l-
(16b)
2!
X"\
1
\\
fl
/'\
I
l y
I\
I
I
at the left-hand tip. The definition of the stress intensity factor at the crack tip for mode II is the same as that for mode I except that a Ix/in eqns. 16(a) and 16(b)is replaced by ox~Xt~We • obtain the stress intensity factors at the two tips as
I
",,,~/,,~~ ~ , ~ ; ~ + - +. ,-~
KLl + iKII, t
I1~.
-----r
',
I-o2 I
.X///'/JJ.~ - ~, I/-~", 'l j ~i~. / / -
'
r
I I i ~,~;,
-0.5
I
I
,
~x
,
_
= -A(~I)
1/2 Y-~°3 [l e x p { i ( a -
3 0o)}
ro
, ', 7; , ~ - - ~ / / / / / 2
+P0 exp{i(a + ¢ 0 - 300)}] -1
..~.._ ,' i ,2_~-~'~-~, ', ,,/,~./.. -1.5
~ ~ o-7 , ~, -1,5
-2
~-.,~ ",, , , / ~ - . / / . .
-I
-0.5
0.5
0
x/~ 2
I ~,++,,]
\
\
1.5
/-..//,/,,o~,
.,9 ~.. / ' . . / .... I
-7./~ ~ ~ ' X ~ ",.'\-Z~7/÷J
1
/ '/'//.-V(_.+--~'
,
m + l - - - cos(0o) ro
x e x p ( - i a ) + ( q + ialt)(~l) m
,>Z/o:/.///1
,,
t)+(
tarocos(0~-~o))
1.5
Y'-C"/Q
P~,~,.'~'Z"!++ ~',."lIL"
-..
+ iA
(17a)
KI, - l + iKii, - ~
= A(nl) 1/2 Y-~° 3 [l exp{i(a - 300)}
'
r0
O5
~
o.yl//~p-r-//TR, lg ~ i I I/Itt'~
-05
~ I
I
I
;-~s ; 0[]3-- I ( l l I
I
ftt...~j,
- P0 exp{i(a + ¢0 - 300)}]
I
i-o,s
', ',+i~
-iA ¢,'/'f/ \./
I
/ X -1.! -/
I I
/
""~,-.._
b'~
u[ -1.5
-2
I / -I
I
I I I
!
l! -0.5
I
I
IL 0
I 1.2 ,
t
I
1
I /
i I
/
i
/
/
. \X 3
\
r
X
1,5
2
2.5
I \
'A-"- % " , I \
~
I
.,/->~,, ~, ~ ~/"~v
--T
0
I
i
4m
-0.5
/
/ /
/
/ /
-I
t-4. / -2 -1.5
V\
/'-. /,. \ / \
'~t -1
-(15
0
0.5 x/$
+,
r0
cos(00)
cos(00-¢0)/!
"-.
~',XF\ I
1.5
1/2
(17b)
Generally, the stress intensity factor at the crack tip is a function of the position of the edge dislocation and its slip plane. Therefore an edge dislocation influences both mode I and mode II stress intensity factors. When the edge dislocation is located on the x axis and its slip plane is perpendicular to the crack surface, the stress intensity factor at the crack tip is pure mode I and its magnitude decreases with increasing distance from the crack tip. On the other hand, if the situation is the same as the above except that the slip plane of the dislocation is parallel to the crack surface, the stress intensity factor at the crack tip is pure mode II. Furthermore, the contours of K I arising
x~s X,
/ "/C , X "/ ' ,, ., ~$ I °" -/~. ",~,
-1.5
m+l
xexp(-ia)-(oi+krn)(~l)
I I I
! ilia 0.5
xll
2
P0 r0
t)+(
2
"
2.5
Fig. 2. E n e r g y contours (solid lines) and trajectories (dashed lines) of an e d g e dislocation of Burgers' vector (b/2 I/:, b/2 llZ) near a crack of length 21 in w h i c h m = - 1: (a) o I = a . = 0; (b) only ol = 0.9; (c) only %1 = 0.9. T h e units of energy and stress are Ab and A/I respectively.
from the dislocation of slip plane ct = 0 ° are the same as those of Ktx arising from the dislocation of slip plane a = 90 °. Owing to the geometric symmetry and assuming m = - 1, we find the stress intensity factor at the crack tip related to the position of the dislocation and its slip plane ct as follows: the values of KI, I (x n, Yo, a), KI_ ! ( - x0, Y0, - a), - KL - l ( - x0, Y0, ~r - a), KI, _ l (-xo, -YII, a - s t ) , -KI,_ l (--Xo, -Yo, a) and KL/ (x0, --Y0, ~r-- a) are the same; the values of KiLt (x0, y0, a), - K H _ / (-xo, y o , - a ) , Ku. /
(-xo, yo, ;r-a),
K,I_ I
2
I
5. Dislocation emission criterion
It can be seen from experiment that edge dislocations are emitted from the crack tip [25]. If w e
I
I
I 415
1.5
-0.3
[ 0,7
"-02
KI 1
0.5
c
/
-0.5
(-xo, -Yo, a-er),
- K I I , - / ( - x 0 , -Y0, a) and Kit,/ (x0, -Y0, - c t ) are also the same. We give some examples to illustrate the physical picture. Figures 3(a) and 3(b) show the positions of the edge dislocation of slip plane a = 45 ° to generate the same KL/and KIL/ respectively. It can be seen that the values of K U and KIL/ are negative over the whole space except for K1./ near the upper crack surface. This implies that the dislocation has a shielding effect on the fracture. However, the dislocation is often emitted from the crack tip. Thus the plane containing the dislocation and the crack tip is parallel to the slip plane of the dislocation. The contours of this kind of dislocation to generate the same stress intensity factor at the crack tip are shown in Figs. 4(a) and 4(b) for modes I and II respectively. It can be seen from Fig. 4(a) that the curves are antisymmetric with the plane y = 0. It is known that when a dislocation is emitted from the crack tip, the stress intensity factor at the crack tip decreases. Therefore the upper plane of Fig. 4(a) is favourable for the existence of the dislocation. If the sign of the Burgers vector is changed, the lower plane of Fig. 4(a) is also favourable for the existence of such a dislocation. On the other hand, in Fig. 4(b) the stress intensity factor at the crack tip is symmetric with the plane y = 0. The value of Kn at the right-hand-side crack tip is negative in front of the tip and positive in its wake. Thus the dislocation emitted from the right-hand-side crack tip is favourable for small ct in order to lower the K u level at that tip. In addition, when ! approaches infinity and x o - I is kept constant, eqn. 17(a) reduces to the result of Rice and T h o m s o n [15] and Shiue and Lee [16].
I
-0.1 -1
-1.5
-2 -2
-1.5
-1
-05
0
0.5
1
1.5
2
x/,#
?=
,/0001.0 KII
0.5
c0.7
_~.~.-o~
-.i~ -~
I
-,.~
/I -i
/ -o.~
h o
I I o.~
-;~-
i.~
\
x/2
Fig. 3. Contours of dislocation of Burgers" vector (b/2 ~/2, bl2~/2) to generate the same stress intensity factor at the right-hand crack tip, (a) Kt,t(b) Knj. The unit of stress intensity factor is A(zr/I)1/2,and m = - 1. assume no dislocation inside the crack before emission, a negative (m = - 1) edge dislocation will appear inside the crack after the emission of a positive edge dislocation because of the conservation of the Burgers' vector. We are interested in finding out the criterion of dislocation emission from the crack tip?According to Rice and Thomson [15], when the distance between the crack tip and the zero-slip-force position of a dislocation is less than the core radius ~0 of the dislocation, a dislocation is spontaneously emitted from the crack tip. The slip force on the dislocation is the
-(m+ 1)~j ~0 3 cos 41
o
[3 +cos
(19a)
o
-0.5
+ 2 ( - ~ ) 1/2 c o s ( a ) - ~ - / ° c o s ( ~ ) }
-1
(19b)
1.
-1.5
Al'slip=i~00 ]
-21~M*f I -2
-1.5
-1
I
/
I
-0.5
/
I
0
II
0.5
I x, 1
2
I
1.5
+ 31J6/{5 cos ( ~ ) - cos (2)1 ]
x/,#
(19c)
2
the subscript "c" corresponds to the force on the dislocation induced by the crack, and i, representing the far-field stresses, can be I or II. The first term on the left-hand side of eqn. (1 8) corresponds to the force on the dislocation arising from far-field stresses. Therefore the stress intensity factors due to the applied load at the crack tip for dislocation emission can be written
1.5
as
Ki D =
-1
'
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x/2
Fig. 4. As Fig. 3 except that the direction of the Burgers vector is the same as the direction of emission instead of the constant direction (b/2 l/~,b/2m).
sum of Fxcos(a) and Fysin(a) expressed in eqn. (10). Assume that ~0 is smaller than l; then the zero slip force on the dislocation, where the dislocation is ~0 away from the right-hand crack tip, is expressed as
oif~slip +
fc,slip = 0
( 1 S)
where
Ab I
~o
fc,slip= 2~0 1+161 x {3 + 2 cos(a)+ cos(2a)+ 2 cos(3a)}
- - ( ~ l ) 1 / 2 fc,slip
(20)
f, slip
where i is I or II. Here the crack-blunting parameter influencing the zero slip force on the dislocation is negligible. It is worthwhile to mention that the value of rn + 1 in eqn. 19(a) is the total number of dislocations inside the crack before dislocation emission from the crack tip. Figure 5(a) shows the effect of crack length on the stress intensity factors due to an applied load for dislocation emission, KI, D and KII,D , for m = 0. Note that the emission direction from the crack tip is the same as the slip direction a of the emitted dislocation. Since KI,D and Kn,D are always positive, and if we assume that KI,D and Kn,D of the top set in Fig. 5(a) are positive, which is proportional to b, then the counterparts of the bottom set are positive only when b is negative. This means that if an edge dislocation is emitted from the crack tip corresponding to KI.D and KII,Dof the bottom set, its Burgers' vector direction has to switch sign. It can be seen from Fig. 5(a) that for constant a,
20
i
p
r
I
I ////
16
\
12
TI
I
m
~o
.L' .(3 :=L -4
2
/ -8
-12
-'16
-2O o [ 0
I 30
30
\~
\
25
Jr
I 60
I"Jl
I
90
I
IJll ,:::
/ [I
120
/ / i/ /
150
////
'
;t ]
20
-7-4
-,
1
15 '~0
;
5
.a :1
0
2
m:]
-5
//// / //
l////
-10
/t,'/;/
-15
--4 -7
~\ \\ "'. "'~',
"%,',
i~I IIII
\ \ \
till
\
///
-20
ill
',, \
-2~
-3(
~
I
3o
I Ihl i
6o
9o
I
~2o
I
~so
180
Fig. 5. Relation between stress intensity factors due to applied load for dislocation emission and angle a. Solid and dashed lines represent K~,D and Kn, D respectively. (a) Effect of crack length with rn = 0. T h e inset describes the geometric symmetry related to dislocation emission from the crack tip. (b) Effect of m.
both KI. D and Kn.D decrease with decreasing crack length, i.e. a lower applied stress intensity factor is needed to emit a dislocation from a smaller crack. This result is in agreement with ref. 6, where it was shown that the stress intensity factor induced by a screw dislocation decreased with decreasing crack length. This implies that if the stress intensity factor for crack propagation is a material constant, a specimen with a smaller crack is more ductile. The minimum magnitude of Kt, D and its corresponding angle G~min for dislocation emission decrease with decreasing crack length, i.e. the easiest direction GCmi n for dislocation emission due to a far-field stress at decreases with decreasing crack length. However, the minimum magnitude of Ku. D for any crack length is associated with a = 0. This means that the direction a = 0 is easiest for edge dislocation emission under a far-field stress o., independent of the crack length. Figure 5(b) shows the effect of m on the stress intensity factors due to an applied load for dislocation emission, KLD and Kn.D, for a given crack length l = 40~0. It can be seen that for a given a and crack length, the magnitudes of both Kt.D and K,. D increase with decreasing m, i.e. a lower applied stress intensity factor is needed to emit dislocations from a crack with larger m. The minimum magnitude of K~,D and its corresponding angle amin for dislocation emission decrease with increasing m, i.e. the easiest direction am~. for dislocation emission due to a far-field stress ot increases with decreasing m. However, the minimum magnitude of KIt,D for any m corresponds to the emission angle ct = 0. The implication is that the direction a = 0 is the easiest one for edge dislocation emission due to a far-field stress o., independent of m. The Burgers vector of the emitted edge dislocation is antisymmetric with respect to the x axis for mode I and symmetric for mode II because the crack-opening displacement is along the y direction for mode I and along the x direction for mode II. Assuming that no dislocation is inside the crack before dislocation emission, it is found that the stress intensity factor at the right-hand crack tip for the case of a positive edge dislocation (Burgers' vector b) emitted from the righthand crack tip and a negative edge dislocation (Burgers' vector - b ) simultaneously emitted from the left-hand crack tip is lower than that for the case of only positive edge dislocations emitted from the right-hand crack tip. Finally, when the
10
crack length is equal to or larger than 104 ~0, it can be treated as a semi-infinite crack, so that the effects of both l and rn disappear and the result is reduced to that of Lin and Thomson [11]. An example is shown in Fig. 5(a).
Acknowledgments This work was supported by the National Science Council, Taiwan. References
6. Summary and conclusions A complete solution of the elastic interaction between an edge dislocation and a finite crack has been obtained using the dislocation-modelling method. Two continuous distributions fx(X) and fy(X) of dislocations with Burgers' vector in the x and y directions respectively are used to simulate the internal crack. Generally, the interaction between an edge dislocation and a crack is of mixed mode type. An exception is when the dislocation is situated along the x axis and its slip plane is either parallel with or perpendicular to the crack surface. With no applied stress, the dislocation is always attracted to the crack surface or to the two crack tips. However, with an applied stress, the entire space is divided into two regions. In one region the dislocation moves towards the crack and in the other it is repelled to infinity. For a constant angle a, the stress intensity factor due to an applied load at the crack tip for dislocation emission increases with increasing crack length and decreasing m. Some special cases have been considered and the results are consistent with those reported in other studies. Finally, because of the stress field of a single dislocation is solved, our results can be easily extended to systems with multiple dislocations.
1 N. P. Louat, Proc. First Int. Conf. on Fracture, Sendai, 1965, Japan Society for the Strength and Fracture of Materials, pp. 117-132. 2 K. M. Lin, C. T. Hu and S. Lee, Mater. Sci. Eng., 95 (1987) 167. 3 S, Lee, Eng. Fract. Mech., 22(1985) 429. 4 R. R. Juang and S. Lee, J. Appl. Phys., 59 (1986) 3421. 5 S.N.G. Chu, J. Appl. Phys., 53(1982) 8678. 6 B. S. Majumdar and S. J. Burns, Acta Metall., 29 (1981) 579. 7 S.T. Shiue and S. Lee, J. Appl. Phys., 64 (1987) 129. 8 E. Smith, Acta Metall., 14(1966) 556. 9 J. P. Hirth and R. H. Wagoner, Int. J. Solids Struct., 12 (1976) 117. 10 R. Thomson and J. Sinclair, Acta Metall., 30 (1982) 1325. 11 I.H. Lin and R. Thomson, Acta Metall., 34 (1986) 187. 12 V. Lakshmanan and J. C. M. Li, Mater. Sci. Eng., A104 (1988) 95. 13 V. Vitek, J. Mech. Phys. Solids, 24(1975) 67. 14 H. Riedel, J. Mech. Phys. Solids, 24(1976) 277. 15 J.R. Rice and R. Thomson, Philos. Mag., 29(1974) 73. 16 S.T. Shiue and S. Lee, Eng. Fract. Mech., 22 (1985) 1105. 17 S. Lee, S. J. Burns and J. C. M. Li, Mater. Sci. Eng., 83 (1986)65. 18 C. Atkinson, Int. J. Fract. Mech., 2 (1966) 567. 19 D. M. Barnett and R. J. Asaro, J. Mech. Phys. Solids, 20 (1972) 353. 20 A. K. Head and N. Louat, Aust. J. Phys., 8(1965) I. 21 N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Leyden, 1977. 22 D. Brock, Elementary Engineering Fracture Mechanics, Martinus Nijhoff, The Hague, 3rd edn., 1982. 23 J.P. Hirth and J. Lothe, Theory of Dislocations, McGrawHill, New York, 2nd edn., 1983. 24 K. Jagannadham and M. J. Marcinkowski, Phys. Status Solidi A, 50 (1978) 293. 25 S.M. Ohr, Mater. Sci. Eng., 72(1985) 1.
1
An Analysis of the Elastic Interaction Between an Edge Dislocation and an Internal Crack SHING-DAR WANG and SANBOH LEE Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu 30043 (Taiwan) (Received October 6, 1989; in revised form May 7, 1990)
Abstract
The elastic interaction between an edge dislocation and a finite crack has been investigated using the dislocation-modelling approach. We considered not only the applied stresses of modes I and H but also dislocations inside the crack. We obtained the stress field, the force and strain energy of the dislocation and the stress intensity factors at both crack tips. The unstable equilibrium positions of dislocation have been studied. Also, the criterion of dislocation emission from the crack tip has been analysed in order to understand the crack tip as a major source of dislocation. 1. Introduction
Dislocations around a crack tip have an intimate relationship with the ductility of crystalline materials and have therefore been studied extensively. Because of its mathematical simplicity, the elastic interaction of a screw dislocation and a crack has been solved by many people using different approaches. For example, by using dislocation modelling, Louat [1] and Lin et al. [2] solved a screw dislocation interacting with a finite crack, while Lee [3] and Juang and Lee [4] analysed the problem of a surface crack. On the other hand, by using the conformal mapping technique, Chu [5] and Majumdar and Burns [6] investigated a screw dislocation interacting with a surface crack and a semi-infinite crack respectively, while Shiue and Lee [7] studied the elastic interaction between a screw dislocation and cracks emanating from an elliptical hole. When a screw dislocation is in the vicinity of a semiinfinite crack and coplanar with the crack surface, the problem becomes the same as that of an edge dislocation. An edge dislocation near a semiinfinite crack tip but in the plane of the crack has been solved by Smith [8]. However, an edge dislo0921-5093/90/$3.50
cation is different from a screw dislocation coplanar with a finite crack. It seems very important to solve the elastic interaction between an edge dislocation and a crack. Nevertheless, the study of this topic has been very limited. Hirth and Wagoner [9], Thomson and Sinclair [10] and Lin and Thomson [11] introduced the complex potential to obtain an analytical solution for an edge dislocation interacting with a semi-infinite crack. Lakshmanan and Li [12] used dislocation modelling to obtain the stress field of an edge dislocation near a semi-infinite crack tip. Vitek [13] solved numerically the interaction between an array of edge dislocations and an elliptical hole. Riedel [14] used a dislocation model for the crack and solved numerically the dislocation distribution inside the crack and along the four inclined planes. However, none of these solutions was in closed form owing to the complicated mathematics involved. Another intriguing problem is the elastic interaction between a mixed dislocation and a semiinfinite crack. Rice and Thomson [15] and Shiue and Lee [16] studied the slip and climb forces on a mixed dislocation arising from a semi-infinite crack using a thermodynamic approach. Lee et al. [17] described graphically the image force and the potential energy of a mixed dislocation with a semi-infinite crack. However, their solutions did not address the stress field, so that many aspects of the interaction are missing. Atkinson [18] and Barnett and Asaro [19] presented procedures for solving the problem of a crack and a mixed dislocation in an anisotropic medium. Without the complete solution, a great deal of physical insight was lost. In this paper we study the elastic interaction between an edge dislocation and a finite crack in an isotropic medium using dislocation modelling. The dislocation distribution, stress field, total © Elsevier Sequoia/Printed in The Netherlands
force and strain energy of dislocation, stress intensity factor at the crack tip and dislocation emission criterion are obtained. On the basis of the force components, we find the zero-force positions of the dislocation. By analysing the dislocation emission criterion, we conclude that the crack tip can be a major source of dislocation in fracture.
/
A
f x-xfx(x')'dx' -/ +
A{(x - xo)cos(a) -
+ o. = 0 /
2. Dislocation distribution inside a finite crack
The problem is illustrated in Fig. 1. Assume that an internal crack lies along the x axis from - 1 to l with an infinite z dimension and that m dislocations are contained inside the crack. A straight edge dislocation has a sense of - z and a Burgers vector b (right-hand final start, RHFS) oriented at an angle a to the x axis. The dislocation is situated at (x0, Y0)- The medium is subject to far-field stresses crI and o H that produce mode I and II crack-opening displacements respectively as shown in Fig. 1. The crack surface is stress free so that Oyyand o~y are zero on the crack surface. Unlike the problem of a screw dislocation with a crack, which requires one array of the continuous distribution of dislocations inside the crack, two infinitesimal Burgers' vectors fx(x)dx and fr(x)dx of edge dislocations in the region between x and x + dx are constructed in order to satisfy the boundary condition. Here subscripts "x" and "y" represent the components of the Burgers vector. The governing equations for the distributions of dislocations are 01 =
(la)
!
A (fy(x)
+ [A[(x - Xo){(x --X0) 2 "1-3yo2} sin(a) -- y0{(X -- X0 )2 -- y0 2 } COS( a )]]/[{(X -- X0 )2 q- y0 2 }2]
(lb)
nt- o r i = 0
where A-
~b 2~(1 - v)
(2)
/x is the shear modulus of the medium and v is Poisson's ratio. Using the inversion theorem [20, 21], we solve the distributions from eqn. (1) as b (YoPo sin( 00 L(x)- ~rq(12-x2) 1/2 \ r,, +
yoro sin(00 -
¢0 + 6 - a)
2 6 + a)
q (m + 1)b cos(a)
+ r0 cos(a) cos(00-
- v)
+ 2(1 - /g
t I I I t t o,,.oz (x. Y.)
/~__
,
! l x - x 'dx
Gy~
IY
Y0 sin(a)}{(x - x0)2 - y02}
{(x -xo) 2+ y02}2
onx
(3a)
( l 2 --X2) 1/2
b
(YoPo
L ( x ) - ~q,.2tt - x2,1/2)
\r0
iii III _
6)) -4 3r(l2 _x2)1/:
yoro cos(00 -
cos(O0- #o+ 6 - a)
2 6 + a)
q
- r0 sin(a) cos(0o - 6)) +
-4
i i i i 1 IJI * f f y ~
Fig. 1. A n edge dislocation situated at (x., Yo) around a finite crack subjected to far-field stresses o~ and o..
2( 1 #
v)
o~x (l 2 - x 2 ) l / 2
(m + 1)b sin(a) t 2 - x2)
(3b)
where Z o = x0 + iy0 = P0 exp(i•o)
(4a)
/02-
12 = r02 e x p ( i 2 0 0 )
Z o - x = q exp(i6),
(4b)
Ix[ < l
(4c)
the definition of 00 is the same as that expressed by Lin et al. [2] and the angle 6 is shown in Fig. 1. (Note that capital Z represents a complex variable while lowercase z is the z coordinate.) The distributions are a function of x in the range between - l and l with parameters Zo, m, OI and aid When the lattice dislocation is coplanar with the crack plane, i.e. y 0 = 0 , without considering the applied stress, the distributions fx(X) with a = 0 ° and fy(X) with a = 90 ° are the same and are equivalent to the distribution of a screw dislocation coplanar with an internal crack [2]. When the crack length becomes semi-infinite, eqn. (3) reduces to the results of Li and Lakshmanan [12] if the applied stress is not considered. According to the conservation of the Burgers vector, if there is no dislocation inside the crack originally, then one of the following three situations can occur. First, when the crack tip emits a positive edge dislocation into the space, a negative edge dislocation will be created inside the crack, i.e. rn = - 1. Secondly, if the lattice dislocation comes from infinity, then the dislocation inside the crack will be zero, i.e. m = 0. Thirdly, when a dislocation is emitted from one crack tip, another dislocation will simultaneously be emitted from the other crack tip to preserve the Burgers' vector conservation. Using eqo. (3), we can obtain the stress components o~)~/arising from both the lattice dislocation and the rn dislocations inside the crack as O'y,!1)= 0"1 -{- 0"2
(5a)
O'xx(I) -~- O 1 -- 0"2
(5b)
0.xy(I)= 0.3
(5C)
where 0"1=
A(r
2r
P3 rl
sin(a
- 42 - 0 + 02)
sin(a + 4 3 - O - 01)/ !
"t-AY°(P~llCOS(a-b¢l--43-t-O-'~-Ol)]oif /90 cos(a + 4o + 4~ - 0o + 0)/ / ro +
A(rn + 1) r
ip3A |455o [ exp{ - i ( a + 4, )} 02 "1"-i 03 = - 2rpl rl \ lol --Pl
2y sin(a)
× exp{i(43- 4 1 - 0 - - 01) }
~ - ~ ° 3 [2pZpo exp{i(2~ - ~bo)}- 12p3 p~ ror x exp(i43)] exp{i( 0 o - a - 241 - 3 0)} 9
+ Aypr,~ n~ 1 2r3 Pn x exp[i{a + 4 - 30 +( - 1 )"00 - 4,,}] 9
Ayopo ~, ( - 1 ) " 2ror n=l
Pn
× exp[i{( - 1)"(a + 4o - 0 o ) - 4 . - 0}] iAr2 cos(a - 42) exp{i(0z - 2 4 2 - 0)} + 2rp2
A ( m +1) [yp +- exp{i(a + 4 - 30)} r 7 + i cos(a) exp( - i0)]
(6c)
exp(i41 )
(6d)
Z - Z 0 = P2 exp(i42)
(6e)
Z + 'go = P3 exp(i43)
(6f)
Z + Z'o = P4 exp(i44)
(6g)
Z 2 - l 2 = r 2 exp(i20)
(6h)
Zo2 - l 2 = ro2 exp(i20o)
(6i)
(Z 2 _/2)1/2 + (22 _/2)1/2 = rl exp(i01)
(6j)
(Z2-I2)l/2+(Z2-I2)I/2=r2 e x p ( i 0 2 )
(6k)
Z - Z'0 = P l
The definitions of 0 and 00 in eqns. 6(h) and 6(i) respectively are the same as those expressed in ref. 2. 01 and 02 follow from the definitions of 0 and 00. Similarly, the stress fields arising from the far-field stresses o~ and oii are expressed as 0.!v~) = O'I(K - N ) + o I I M
(6a)
(6b)
Z = x + iy = p exp(i4)
a?x ) =
sin(a - 0)
C O S ( a -~- 4 1 ) - -
0.1(K -Jr-N - 1) + an(2L - M)
a ~ = a , M + oII(K + N)
(7a)
(7b) (7c)
+ Abp__o
where
2r02 Iexp( - i200) K + i L = ~ exp{i(# - 0)}
(8a)
r
× [sin(a) exp{i(a + 4o - 200)}
2
M + iN = ~
r
exp{i(2¢ - 3 0)} - -y exp( - i0) r
(8b)
- i exp(i#0 ) + sin(9~0) exp(i2a)] + sin(a - ¢~o)exp( - i a ) - sin(2 0 o - ¢o - 2a)]
and p, ~ and r, 0 are defined in eqns. 6(c) and 6(h) respectively. When l approaches infinity and x - I remains finite, eqn. (7) can reduce to the results of Broek [22], who obtained the stress field from the Airy stress function. The total stress field components a!~ / arising from the lattice dislocation, the m dislocations inside the crack and the applied stresses are a!T/= o(!q) + a!~)
(9)
The numerical values of the stress components a!f I are checked by using personal computer. It is found that the stress components vyy(T/and Oxy(T/ vanish on the crack plane, as required by the boundary conditions, i.e. a/T/yy= a ~ I = 0 for Ixl < l. When the crack length becomes semi-infinite, the contours look like the profiles of Lakshmanan and Li [12]. When l approaches infinity, the system reduces to the case of an edge dislocation near a free surface. When the Burgers vector is perpendicular to the free surface, our equations are the same as those of Hirth and Lathe [23] and the stress contours look like the profiles of Jagannadham and Marcinkowski [24].
3. Total force and strain energy of the dislocation Using the stress field derived above and the Peach-Koehler formula, one gets the total force on the edge dislocation as
F=Fy+iFx=F1+ Fa
(10)
+ A b ( m + 1) (y0p20 exp{i(2a + 4 0 - 300)} ro
\ r0
+ i exp( - i 00 ) + sin( 00 - 2 a)} !
( 11 a)
~=~y+i~x
= olb(-P°cos(~o-Oo)exp(\
r0
ia) + cos(a))
+ o n b ( ~° [i cos(a)exp{i(¢0-00)}
- sin(a + 40 - 00)])
+(en+iol) b12 Y~°3exp{i(a-30o) }
(llb)
r0
In the above equations the subscripts x and y denote the force components along the x and y directions respectively. When both x 0 and 1 approach infinity and x 0 - I is kept constant, the above equations reduce to the result of a semiinfinite crack [15-17]. When / approaches infinity, we have the case of an edge dislocation near a free surface. Equation 1 l(a) becomes Fx + iFy = - i -
Ab
(12)
2Y0
where
F I = Fly + iFix AbyoPo 2 - -
2r0 4
exp( - i20o)
x [2 exp{i2(a + 40 - 00)} + 1]
Ab {exp( - i 2 0 o ) " 1} 4Yo --
which shows that the force on the dislocation is always towards the free surface, independent of the direction of the Burgers vector. Consider the symmetric pair of edge dislocations shown in the upper part of the inset in Fig. 5(a) (see Section 5). It is found that for a semiinfinite crack the slip force of the interaction between any two such dislocations is zero, which is in good agreement with Lin and T h o m s o n [11],
and the climb force is
Fclimb
--
ZbESsin'2'4cot(2) x{2cos(2)cos() 11]
16p
(13)
where x 0 + iy0 = p exp(ie)
14)
However, when the crack length is finite, the slip and climb components of the interaction stated above are not zero. When the crack length shrinks to zero, the interaction between the two dislocations is not zero either. The negative gradient of the strain energy is the total force on the edge dislocation. Therefore we obtain the strain energy of the edge dislocation in an infinite medium with a finite crack as
E = ~ - ira
- - - Po sin(2a + # 0 - 20o)
+ J{cos(2 0o) - 1 }+ sin(00) sin( 00 - 2 a) +4In
8(roe + p o 2 - 1 2 )
(Rr°] 2
/
+ 2(m + 1) (Y0 sin(2a - 00) \r0 - Re[ln{Zo + (Zo ° -
)
+ 2m In(l)
]
/¢
+ b[ y°A~ {al cos(a + 4 o - 0o)
+ on sin(a + 40 - 0o)}
applied stress, eqn. (15) reduces to the self-energy of an edge dislocation per unit length [23]. Secondly, when l approaches infinity, i.e. a free surface, and the slip plane a = 9 0 °, eqn. (15) becomes the result of an edge dislocation near a free surface [23] except for a constant. Thirdly, when l approaches infinity and x 0 - l remains finite, eqn. (15) reduces to the result of Lee et al. [17] except for a constant. The energy contours and trajectories of the edge dislocation in the absence of any applied stress are plotted as solid and dashed lines respectively in Fig. 2(a), where m = - 1 and a = 45 °. Here the arrow of the trajectory represents the direction of the dislocation movement. Regardless of the position of the dislocation, the edge dislocation has a tendency to be attracted either to the crack surface or to the crack tips, i.e. each trajectory following the arrow direction is terminated at the crack surface or a crack tip. Figure 2(b) and 2(c) represent the energy contours and trajectories of the edge dislocation for o I : 0.9, o n = 0 and o I = 0, Oli : 0.9 respectively, where the unit of stress is A f t . Again the solid and dashed lines represent the energy contours and trajectories respectively. In these two figures the coarsely dashed line divides the space into two regions. In one region the dislocation following the arrow direction of trajectory moves towards the crack, which is the same as in Fig. 2(a). In the other region the dislocation following the arrow direction of trajectory is repelled to infinity. It can be seen from the figures that P~ (1.76, 0.59), P2 (1.58, - 0 . 2 9 ) , ~ (1.86, 0) and P4 (1.07, - 1.03) are saddle points in unstable equilibrium. In addition, with increasing applied stress Ol and/or %, the coarsely dashed line moves towards the crack tip.
- r 0 cos(00){o I sin(a)+ Oil COS(a)} 4.
- o l y o cos(a))
(15)
where R represents the size of the crystal and ~0 is the core radius of the dislocation. Re [.] means the real part of a complex function. The strain energy is a function of the crack length/, the position (x0, Y0) of the dislocation, the Burgers vector b, its slip plane a, the m dislocations inside the crack and the external stress o I and O'i,D T h r e e special cases are worthy of mention. First, when the crack vanishes, i.e. l = O, in the absence of any
Stress intensity factor
The interaction between an edge dislocation and an internal crack is a mixture of modes I and II. The stress intensity factor at the crack tip for mode I is defined as K l./ =
lim . [{2n(x--l)}
1/2
oyy(W/,ly=0
(16a)
x~ l +
at the right-hand tip and K1_/= lim [ { 2 x l x +l[} 1/2 aye,¢Tilly=0 , x~-l-
(16b)
2!
X"\
1
\\
fl
/'\
I
l y
I\
I
I
at the left-hand tip. The definition of the stress intensity factor at the crack tip for mode II is the same as that for mode I except that a Ix/in eqns. 16(a) and 16(b)is replaced by ox~Xt~We • obtain the stress intensity factors at the two tips as
I
",,,~/,,~~ ~ , ~ ; ~ + - +. ,-~
KLl + iKII, t
I1~.
-----r
',
I-o2 I
.X///'/JJ.~ - ~, I/-~", 'l j ~i~. / / -
'
r
I I i ~,~;,
-0.5
I
I
,
~x
,
_
= -A(~I)
1/2 Y-~°3 [l e x p { i ( a -
3 0o)}
ro
, ', 7; , ~ - - ~ / / / / / 2
+P0 exp{i(a + ¢ 0 - 300)}] -1
..~.._ ,' i ,2_~-~'~-~, ', ,,/,~./.. -1.5
~ ~ o-7 , ~, -1,5
-2
~-.,~ ",, , , / ~ - . / / . .
-I
-0.5
0.5
0
x/~ 2
I ~,++,,]
\
\
1.5
/-..//,/,,o~,
.,9 ~.. / ' . . / .... I
-7./~ ~ ~ ' X ~ ",.'\-Z~7/÷J
1
/ '/'//.-V(_.+--~'
,
m + l - - - cos(0o) ro
x e x p ( - i a ) + ( q + ialt)(~l) m
,>Z/o:/.///1
,,
t)+(
tarocos(0~-~o))
1.5
Y'-C"/Q
P~,~,.'~'Z"!++ ~',."lIL"
-..
+ iA
(17a)
KI, - l + iKii, - ~
= A(nl) 1/2 Y-~° 3 [l exp{i(a - 300)}
'
r0
O5
~
o.yl//~p-r-//TR, lg ~ i I I/Itt'~
-05
~ I
I
I
;-~s ; 0[]3-- I ( l l I
I
ftt...~j,
- P0 exp{i(a + ¢0 - 300)}]
I
i-o,s
', ',+i~
-iA ¢,'/'f/ \./
I
/ X -1.! -/
I I
/
""~,-.._
b'~
u[ -1.5
-2
I / -I
I
I I I
!
l! -0.5
I
I
IL 0
I 1.2 ,
t
I
1
I /
i I
/
i
/
/
. \X 3
\
r
X
1,5
2
2.5
I \
'A-"- % " , I \
~
I
.,/->~,, ~, ~ ~/"~v
--T
0
I
i
4m
-0.5
/
/ /
/
/ /
-I
t-4. / -2 -1.5
V\
/'-. /,. \ / \
'~t -1
-(15
0
0.5 x/$
+,
r0
cos(00)
cos(00-¢0)/!
"-.
~',XF\ I
1.5
1/2
(17b)
Generally, the stress intensity factor at the crack tip is a function of the position of the edge dislocation and its slip plane. Therefore an edge dislocation influences both mode I and mode II stress intensity factors. When the edge dislocation is located on the x axis and its slip plane is perpendicular to the crack surface, the stress intensity factor at the crack tip is pure mode I and its magnitude decreases with increasing distance from the crack tip. On the other hand, if the situation is the same as the above except that the slip plane of the dislocation is parallel to the crack surface, the stress intensity factor at the crack tip is pure mode II. Furthermore, the contours of K I arising
x~s X,
/ "/C , X "/ ' ,, ., ~$ I °" -/~. ",~,
-1.5
m+l
xexp(-ia)-(oi+krn)(~l)
I I I
! ilia 0.5
xll
2
P0 r0
t)+(
2
"
2.5
Fig. 2. E n e r g y contours (solid lines) and trajectories (dashed lines) of an e d g e dislocation of Burgers' vector (b/2 I/:, b/2 llZ) near a crack of length 21 in w h i c h m = - 1: (a) o I = a . = 0; (b) only ol = 0.9; (c) only %1 = 0.9. T h e units of energy and stress are Ab and A/I respectively.
from the dislocation of slip plane ct = 0 ° are the same as those of Ktx arising from the dislocation of slip plane a = 90 °. Owing to the geometric symmetry and assuming m = - 1, we find the stress intensity factor at the crack tip related to the position of the dislocation and its slip plane ct as follows: the values of KI, I (x n, Yo, a), KI_ ! ( - x0, Y0, - a), - KL - l ( - x0, Y0, ~r - a), KI, _ l (-xo, -YII, a - s t ) , -KI,_ l (--Xo, -Yo, a) and KL/ (x0, --Y0, ~r-- a) are the same; the values of KiLt (x0, y0, a), - K H _ / (-xo, y o , - a ) , Ku. /
(-xo, yo, ;r-a),
K,I_ I
2
I
5. Dislocation emission criterion
It can be seen from experiment that edge dislocations are emitted from the crack tip [25]. If w e
I
I
I 415
1.5
-0.3
[ 0,7
"-02
KI 1
0.5
c
/
-0.5
(-xo, -Yo, a-er),
- K I I , - / ( - x 0 , -Y0, a) and Kit,/ (x0, -Y0, - c t ) are also the same. We give some examples to illustrate the physical picture. Figures 3(a) and 3(b) show the positions of the edge dislocation of slip plane a = 45 ° to generate the same KL/and KIL/ respectively. It can be seen that the values of K U and KIL/ are negative over the whole space except for K1./ near the upper crack surface. This implies that the dislocation has a shielding effect on the fracture. However, the dislocation is often emitted from the crack tip. Thus the plane containing the dislocation and the crack tip is parallel to the slip plane of the dislocation. The contours of this kind of dislocation to generate the same stress intensity factor at the crack tip are shown in Figs. 4(a) and 4(b) for modes I and II respectively. It can be seen from Fig. 4(a) that the curves are antisymmetric with the plane y = 0. It is known that when a dislocation is emitted from the crack tip, the stress intensity factor at the crack tip decreases. Therefore the upper plane of Fig. 4(a) is favourable for the existence of the dislocation. If the sign of the Burgers vector is changed, the lower plane of Fig. 4(a) is also favourable for the existence of such a dislocation. On the other hand, in Fig. 4(b) the stress intensity factor at the crack tip is symmetric with the plane y = 0. The value of Kn at the right-hand-side crack tip is negative in front of the tip and positive in its wake. Thus the dislocation emitted from the right-hand-side crack tip is favourable for small ct in order to lower the K u level at that tip. In addition, when ! approaches infinity and x o - I is kept constant, eqn. 17(a) reduces to the result of Rice and T h o m s o n [15] and Shiue and Lee [16].
I
-0.1 -1
-1.5
-2 -2
-1.5
-1
-05
0
0.5
1
1.5
2
x/,#
?=
,/0001.0 KII
0.5
c0.7
_~.~.-o~
-.i~ -~
I
-,.~
/I -i
/ -o.~
h o
I I o.~
-;~-
i.~
\
x/2
Fig. 3. Contours of dislocation of Burgers" vector (b/2 ~/2, bl2~/2) to generate the same stress intensity factor at the right-hand crack tip, (a) Kt,t(b) Knj. The unit of stress intensity factor is A(zr/I)1/2,and m = - 1. assume no dislocation inside the crack before emission, a negative (m = - 1) edge dislocation will appear inside the crack after the emission of a positive edge dislocation because of the conservation of the Burgers' vector. We are interested in finding out the criterion of dislocation emission from the crack tip?According to Rice and Thomson [15], when the distance between the crack tip and the zero-slip-force position of a dislocation is less than the core radius ~0 of the dislocation, a dislocation is spontaneously emitted from the crack tip. The slip force on the dislocation is the
-(m+ 1)~j ~0 3 cos 41
o
[3 +cos
(19a)
o
-0.5
+ 2 ( - ~ ) 1/2 c o s ( a ) - ~ - / ° c o s ( ~ ) }
-1
(19b)
1.
-1.5
Al'slip=i~00 ]
-21~M*f I -2
-1.5
-1
I
/
I
-0.5
/
I
0
II
0.5
I x, 1
2
I
1.5
+ 31J6/{5 cos ( ~ ) - cos (2)1 ]
x/,#
(19c)
2
the subscript "c" corresponds to the force on the dislocation induced by the crack, and i, representing the far-field stresses, can be I or II. The first term on the left-hand side of eqn. (1 8) corresponds to the force on the dislocation arising from far-field stresses. Therefore the stress intensity factors due to the applied load at the crack tip for dislocation emission can be written
1.5
as
Ki D =
-1
'
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x/2
Fig. 4. As Fig. 3 except that the direction of the Burgers vector is the same as the direction of emission instead of the constant direction (b/2 l/~,b/2m).
sum of Fxcos(a) and Fysin(a) expressed in eqn. (10). Assume that ~0 is smaller than l; then the zero slip force on the dislocation, where the dislocation is ~0 away from the right-hand crack tip, is expressed as
oif~slip +
fc,slip = 0
( 1 S)
where
Ab I
~o
fc,slip= 2~0 1+161 x {3 + 2 cos(a)+ cos(2a)+ 2 cos(3a)}
- - ( ~ l ) 1 / 2 fc,slip
(20)
f, slip
where i is I or II. Here the crack-blunting parameter influencing the zero slip force on the dislocation is negligible. It is worthwhile to mention that the value of rn + 1 in eqn. 19(a) is the total number of dislocations inside the crack before dislocation emission from the crack tip. Figure 5(a) shows the effect of crack length on the stress intensity factors due to an applied load for dislocation emission, KI, D and KII,D , for m = 0. Note that the emission direction from the crack tip is the same as the slip direction a of the emitted dislocation. Since KI,D and Kn,D are always positive, and if we assume that KI,D and Kn,D of the top set in Fig. 5(a) are positive, which is proportional to b, then the counterparts of the bottom set are positive only when b is negative. This means that if an edge dislocation is emitted from the crack tip corresponding to KI.D and KII,Dof the bottom set, its Burgers' vector direction has to switch sign. It can be seen from Fig. 5(a) that for constant a,
20
i
p
r
I
I ////
16
\
12
TI
I
m
~o
.L' .(3 :=L -4
2
/ -8
-12
-'16
-2O o [ 0
I 30
30
\~
\
25
Jr
I 60
I"Jl
I
90
I
IJll ,:::
/ [I
120
/ / i/ /
150
////
'
;t ]
20
-7-4
-,
1
15 '~0
;
5
.a :1
0
2
m:]
-5
//// / //
l////
-10
/t,'/;/
-15
--4 -7
~\ \\ "'. "'~',
"%,',
i~I IIII
\ \ \
till
\
///
-20
ill
',, \
-2~
-3(
~
I
3o
I Ihl i
6o
9o
I
~2o
I
~so
180
Fig. 5. Relation between stress intensity factors due to applied load for dislocation emission and angle a. Solid and dashed lines represent K~,D and Kn, D respectively. (a) Effect of crack length with rn = 0. T h e inset describes the geometric symmetry related to dislocation emission from the crack tip. (b) Effect of m.
both KI. D and Kn.D decrease with decreasing crack length, i.e. a lower applied stress intensity factor is needed to emit a dislocation from a smaller crack. This result is in agreement with ref. 6, where it was shown that the stress intensity factor induced by a screw dislocation decreased with decreasing crack length. This implies that if the stress intensity factor for crack propagation is a material constant, a specimen with a smaller crack is more ductile. The minimum magnitude of Kt, D and its corresponding angle G~min for dislocation emission decrease with decreasing crack length, i.e. the easiest direction GCmi n for dislocation emission due to a far-field stress at decreases with decreasing crack length. However, the minimum magnitude of Ku. D for any crack length is associated with a = 0. This means that the direction a = 0 is easiest for edge dislocation emission under a far-field stress o., independent of the crack length. Figure 5(b) shows the effect of m on the stress intensity factors due to an applied load for dislocation emission, KLD and Kn.D, for a given crack length l = 40~0. It can be seen that for a given a and crack length, the magnitudes of both Kt.D and K,. D increase with decreasing m, i.e. a lower applied stress intensity factor is needed to emit dislocations from a crack with larger m. The minimum magnitude of K~,D and its corresponding angle amin for dislocation emission decrease with increasing m, i.e. the easiest direction am~. for dislocation emission due to a far-field stress ot increases with decreasing m. However, the minimum magnitude of KIt,D for any m corresponds to the emission angle ct = 0. The implication is that the direction a = 0 is the easiest one for edge dislocation emission due to a far-field stress o., independent of m. The Burgers vector of the emitted edge dislocation is antisymmetric with respect to the x axis for mode I and symmetric for mode II because the crack-opening displacement is along the y direction for mode I and along the x direction for mode II. Assuming that no dislocation is inside the crack before dislocation emission, it is found that the stress intensity factor at the right-hand crack tip for the case of a positive edge dislocation (Burgers' vector b) emitted from the righthand crack tip and a negative edge dislocation (Burgers' vector - b ) simultaneously emitted from the left-hand crack tip is lower than that for the case of only positive edge dislocations emitted from the right-hand crack tip. Finally, when the
10
crack length is equal to or larger than 104 ~0, it can be treated as a semi-infinite crack, so that the effects of both l and rn disappear and the result is reduced to that of Lin and Thomson [11]. An example is shown in Fig. 5(a).
Acknowledgments This work was supported by the National Science Council, Taiwan. References
6. Summary and conclusions A complete solution of the elastic interaction between an edge dislocation and a finite crack has been obtained using the dislocation-modelling method. Two continuous distributions fx(X) and fy(X) of dislocations with Burgers' vector in the x and y directions respectively are used to simulate the internal crack. Generally, the interaction between an edge dislocation and a crack is of mixed mode type. An exception is when the dislocation is situated along the x axis and its slip plane is either parallel with or perpendicular to the crack surface. With no applied stress, the dislocation is always attracted to the crack surface or to the two crack tips. However, with an applied stress, the entire space is divided into two regions. In one region the dislocation moves towards the crack and in the other it is repelled to infinity. For a constant angle a, the stress intensity factor due to an applied load at the crack tip for dislocation emission increases with increasing crack length and decreasing m. Some special cases have been considered and the results are consistent with those reported in other studies. Finally, because of the stress field of a single dislocation is solved, our results can be easily extended to systems with multiple dislocations.
1 N. P. Louat, Proc. First Int. Conf. on Fracture, Sendai, 1965, Japan Society for the Strength and Fracture of Materials, pp. 117-132. 2 K. M. Lin, C. T. Hu and S. Lee, Mater. Sci. Eng., 95 (1987) 167. 3 S, Lee, Eng. Fract. Mech., 22(1985) 429. 4 R. R. Juang and S. Lee, J. Appl. Phys., 59 (1986) 3421. 5 S.N.G. Chu, J. Appl. Phys., 53(1982) 8678. 6 B. S. Majumdar and S. J. Burns, Acta Metall., 29 (1981) 579. 7 S.T. Shiue and S. Lee, J. Appl. Phys., 64 (1987) 129. 8 E. Smith, Acta Metall., 14(1966) 556. 9 J. P. Hirth and R. H. Wagoner, Int. J. Solids Struct., 12 (1976) 117. 10 R. Thomson and J. Sinclair, Acta Metall., 30 (1982) 1325. 11 I.H. Lin and R. Thomson, Acta Metall., 34 (1986) 187. 12 V. Lakshmanan and J. C. M. Li, Mater. Sci. Eng., A104 (1988) 95. 13 V. Vitek, J. Mech. Phys. Solids, 24(1975) 67. 14 H. Riedel, J. Mech. Phys. Solids, 24(1976) 277. 15 J.R. Rice and R. Thomson, Philos. Mag., 29(1974) 73. 16 S.T. Shiue and S. Lee, Eng. Fract. Mech., 22 (1985) 1105. 17 S. Lee, S. J. Burns and J. C. M. Li, Mater. Sci. Eng., 83 (1986)65. 18 C. Atkinson, Int. J. Fract. Mech., 2 (1966) 567. 19 D. M. Barnett and R. J. Asaro, J. Mech. Phys. Solids, 20 (1972) 353. 20 A. K. Head and N. Louat, Aust. J. Phys., 8(1965) I. 21 N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Leyden, 1977. 22 D. Brock, Elementary Engineering Fracture Mechanics, Martinus Nijhoff, The Hague, 3rd edn., 1982. 23 J.P. Hirth and J. Lothe, Theory of Dislocations, McGrawHill, New York, 2nd edn., 1983. 24 K. Jagannadham and M. J. Marcinkowski, Phys. Status Solidi A, 50 (1978) 293. 25 S.M. Ohr, Mater. Sci. Eng., 72(1985) 1.