104
10 Mathematical Structural Imperfections
Fig. 10.3 Model of an edge
dislocation
u =
b
2π
arctan
y
x
+
1
2(1 − ν)
xy
x 2 + y 2
,
v = −
b
8π(1 − ν)
(1 − 2ν) ln(x
2
+ y
2 ) +
x 2 − y 2
x 2 + y 2
,
w = 0,
(10.1)
or in polar coordinates (r, ϕ):
u =
b
2π
ϕ +
sin
2 ϕ
4(1 − ν)
,
v = −
b
2π
1 − 2ν
2(1 − ν)
ln r +
cos 2ϕ
4(1 − 2ν)
,
w = 0,
(10.2)
where b is the module of the burgers vector and
ϕ = arctg
y
x
.
Since there is no movement in the direction of the dislocation line, there is
a flat deformation. For known displacements by the formulas (2.2)–(2.3), the
deformations ε x , ε y , and γ xy are determined, and then by Hooke’s law (1.20)–
(1.21)—stress components
σ x = −
Gb
2π(1 − ν)
y(3x 2 + y 2 )
(x 2 + y 2 ) 2 ,
σ y =
Gb
2π(1 − ν)
y(x 2 − y 2 )
(x 2 + y 2 ) 2 ,
τ xy =
Gb
2π(1 − ν)
x(x 2 − y 2 )
(x 2 + y 2 ) 2 .
(10.3)