World Journal of Engineering
The interaction between a screw dislocation and a semi-infinite
wedge crack with a nano circular inclusion at tip
Song Hao-Peng Gao Cun-Fa
College of Aerospace Engineering
Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China
The interaction between a screw dislocation
and a surface nano semi-cylindrical inclusion
in half-plane is investigated. Utilizing the
image dislocations method, the closed form
solutions of the stress fields in the matrix and
the inclusion region are derived. The results
were used to study the interaction of a screw
dislocation and a semi-infinite wedge crack
with a nano circular inclusion at tip by means
of conformal mapping. The present solutions
include previous results as special cases.
For a simple screw dislocation of Burgers
inclusion of radius R whose axis is in the
surface. The complex potentials of a screw
dislocation in region 2 near a surface
semi-cylindrical inclusion of radius R are:
Bk 
(4)
bz
b
1
1
1
(

 ) z
*
2 i w  w0 w  w w 2 i
22
R2
( )k 1 w k 2
k  0 1   2  (1  k )( 0 / R) w0


(1)
k 0
F2 ( w) 
k 0

inclusion, the stress fields of the dislocation
are, respectively, (Fang 2006)
w R
k 0
22
bz
1 k 1
(
)
1  2  (1  k )( 0 / R) 2 i  w0
F2 ( w) 
through the point w0 near a nano circular
F1 ( w)   Bk wk

Where
vector b , parallel to the z axis and passing


F1 ( w)   Bk wk   Bk wk
bz
b
1
1
1
(

 ) z
*
2 i w  w0 w  w w 2 i
22
R 2 k 1  k 2
(
) w
k  0 1   2  (1  k )( 0 / R)  w0

bz
b
1
1
1
(

 ) z
*
2 i w  w0 w  w w 2 i

22
R2
( )k 1 w k 2
k  0 1   2  (1  k )( 0 / R) w0


(5)
The following transform function
(2)
w  ( z )  z1/ m , z  G(w)  wm , 0  m  2
where
Bk 
2 2
bz 1 k 1
( ) ,
1  2  (1  k )( 0 / R) 2 i w0
and m 
2  

(6)
which can map the half-plane with a
w  R 2 / w0 and the superbar stands for the
half-circular inclusion (with radius R ) in the
w -plane on to a circular inclusion (with
complex conjugate.
Now consider a screw dislocation in
medium A near a surface semi-cylindrical
radius R  R ) with a semi-infinite wedge
crack (with radian  )in the z -plane (Fig.
1).
m
1037
World Journal of Engineering
y
 Re{ 2 1
z0
2
Fig.3
x
8
stresses
at
Eqs. (4-5).With the aid of the mapping
function (6), the stress field at z due to a
2 / 1  2 , 0 / 1  105 ,   0 ,
  1.2 105 . It is seen that the shielding
screw dislocation at z0 near a nano circular
effect increases with the increase of  and
R.
inclusion with a semi-infinite wedge crack can
be obtained as:
1df1 ( z ) dw
dw
0 / 1  2 106
dz

1 ( m1 1) 
z
( Bk z k   Bk z k ) (7)
m
k 0
k 0
 yz 2 ( w)  i xz 2 ( w) 
0 / 1  5 107
0 / 1  2 107
2 df 2 ( z ) dw
dw
dz
2bz 1 ( m1 1) 1
1
1
z
[


*
2 i m
z  z0 z  z z
Fig.3.
K III 0 (normalized SIF) versus 
(dislocation azimuth angle)
22
R2

( )k 1 z  k 2
k  0 1   2  (1  k )( 0 / R ) z0

  /5
22
R 2 k 1  k 2

(
) z
]
k  0 1   2  (1  k )( 0 / R)  z0

1
1
1
(

 )
*
z  z0 z  z
z
 and 0 .
Fig.4 shows the normalized SIF versus 
Let the screw dislocation be in region 2, the
complex potential in the w -plane is given by
 yz1 ( z )  i xz1 ( z ) 
 / R  1.2 . It is seen
decreases with the increase of
inclusion with a semi-infinite wedge crack
  2 / 5
(8)
  /2
The stress intensity factor (SIF) at the right
tip of the wedge crack can be obtained by
z 0
normalized
that the stresses intensity factor is always
negative, which means positive screw
dislocations can reduce the SIF of the wedge
crack (shielding effect). The shielding effect
Fig 1 A screw dislocation near a nano circular
K III  lim 2
the
   / 9 , R  10 ,
1

shows
(14)
intensity factors (SIF) versus  at 2 / 1  2 ,

 1

1 
[  Bk z k   Bk z k ]}
m k 0
k 0
Fig.4.
1
(1 )
z m
 yz1
K III 0 (normalized SIF) versus R
(radius)
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