10.3 Mathematical Wedge-Shaped Dislocation
105
In the polar coordinates of the formula (10.3) take the form
σ x = −
Gb
2π(1 − ν)
b
r
(2 + cos 2ϕ) sin ϕ,
σ y =
Gb
2π(1 − ν)
sin ϕ cos 2ϕ
r
,
τ xy =
Gb
2π(1 − ν)
cos ϕ cos 2ϕ
r
.
(10.4)
The largest normal stress σ x acts along the Ox axis and is tensile at y < 0 and
compressive at y > 0. This is also seen in the model of crystal lattice distortions in
the vicinity of the dislocation core, shown in Fig. 10.3.
10.3 Mathematical Wedge-Shaped Dislocation
We cut out a wedge from an unbounded elastic plane with apex at the point (x 0 ; 0).
The banks of the formed section are spaced from each other at a distance of δ =
ε(x − x 0 ), (ε − const). Now let us combine the points of the opposite edges of the
cut, equidistant from the top of the wedge, and glue the plane material. The resulting
structural imperfection is called a [9] wedge-shaped (wedge) dislocation. In this
case, a plane stress state arises, axisymmetric with respect to the point (x 0 ; 0).
Let us take this point as the origin of the polar coordinate system (r, α). Using
the solution of the Lame problem (p. 18), we write down the radial displacement
component
u r =
1 − ν
4π
εr ln Ar +
B
r
,
(10.5)
where A and B are constants, and in this case B = 0 should be set from the condition
of bounded displacement at the origin, and the parameter ε is called the power of
the wedge dislocation.
The constant A corresponds to uniform axially symmetric tension. Next, we put
A = 1.
We find the tangential component of the displacement (u α ) from the condition
that any circle centered at the origin of coordinates deforms uniformly along its
length (axisymmetric stress state). Assuming that there is no rotation of the cut, we
find
u α = −
εr
2
1 −
α
π
, (0 < α < 2π) ,
(10.6)
105
In the polar coordinates of the formula (10.3) take the form
σ x = −
Gb
2π(1 − ν)
b
r
(2 + cos 2ϕ) sin ϕ,
σ y =
Gb
2π(1 − ν)
sin ϕ cos 2ϕ
r
,
τ xy =
Gb
2π(1 − ν)
cos ϕ cos 2ϕ
r
.
(10.4)
The largest normal stress σ x acts along the Ox axis and is tensile at y < 0 and
compressive at y > 0. This is also seen in the model of crystal lattice distortions in
the vicinity of the dislocation core, shown in Fig. 10.3.
10.3 Mathematical Wedge-Shaped Dislocation
We cut out a wedge from an unbounded elastic plane with apex at the point (x 0 ; 0).
The banks of the formed section are spaced from each other at a distance of δ =
ε(x − x 0 ), (ε − const). Now let us combine the points of the opposite edges of the
cut, equidistant from the top of the wedge, and glue the plane material. The resulting
structural imperfection is called a [9] wedge-shaped (wedge) dislocation. In this
case, a plane stress state arises, axisymmetric with respect to the point (x 0 ; 0).
Let us take this point as the origin of the polar coordinate system (r, α). Using
the solution of the Lame problem (p. 18), we write down the radial displacement
component
u r =
1 − ν
4π
εr ln Ar +
B
r
,
(10.5)
where A and B are constants, and in this case B = 0 should be set from the condition
of bounded displacement at the origin, and the parameter ε is called the power of
the wedge dislocation.
The constant A corresponds to uniform axially symmetric tension. Next, we put
A = 1.
We find the tangential component of the displacement (u α ) from the condition
that any circle centered at the origin of coordinates deforms uniformly along its
length (axisymmetric stress state). Assuming that there is no rotation of the cut, we
find
u α = −
εr
2
1 −
α
π
, (0 < α < 2π) ,
(10.6)