106
10 Mathematical Structural Imperfections
where
α = arctg
y
x − x 0
, r =
(x − x 0 ) 2 + y 2 .
(10.7)
10.4 Mathematical Biclination
Let, as in the previous paragraph, at the point (x 0 , 0) of an unbounded elastic plane,
there is a wedge dislocation of power ε. We place at the point (x 0 + , 0) a
wedge dislocation of power −ε. Then the components (u, v) of displacement in
the direction of the axes Ox and Oy from two dislocations at → 0 will be
u =
∂u
∂x
, v =
∂v
∂x
,
where u and v are the components of movement in the direction of the Ox and Oy
axes for a wedge dislocation, defined by the formulas
u = u r cos α − u α sin α , v = u r sin α + u α cos α.
(10.8)
Denoting further εε = b, we get
u =
b
2π
1 − ν
2
ln r + cos
2 α
+ sin
2 α
,
v = −
1 + ν
8π
b sin 2α −
b
2
1 −
α
π
.
(10.9)
From the formula (10.9), it follows that the displacement component v for α =
2π exceeds by b its value for α = 0, i.e. there is a gap of displacements on the semidirect x > 0, y = 0. The latter means that at the point (0, 0) the edge dislocation
considered above (p. 103) is created with the burgers vector b parallel to the Oy
axis.
10.5 Flat Dislocation of Somigliana
Assume a section is made along some curve L in an unlimited elastic plane. The
section edges can be split and material is inserted into the slit, or material can
be cut off from the edges of the cut. If the edges of such a slit are than glued by
preliminary compressing and shifting them relative to each other, this structural
imperfection is referred to as [2, 9, 12] the flat dislocation of Somigliana. Let
us set the displacement difference on the curve L after gluing as
10 Mathematical Structural Imperfections
where
α = arctg
y
x − x 0
, r =
(x − x 0 ) 2 + y 2 .
(10.7)
10.4 Mathematical Biclination
Let, as in the previous paragraph, at the point (x 0 , 0) of an unbounded elastic plane,
there is a wedge dislocation of power ε. We place at the point (x 0 + , 0) a
wedge dislocation of power −ε. Then the components (u, v) of displacement in
the direction of the axes Ox and Oy from two dislocations at → 0 will be
u =
∂u
∂x
, v =
∂v
∂x
,
where u and v are the components of movement in the direction of the Ox and Oy
axes for a wedge dislocation, defined by the formulas
u = u r cos α − u α sin α , v = u r sin α + u α cos α.
(10.8)
Denoting further εε = b, we get
u =
b
2π
1 − ν
2
ln r + cos
2 α
+ sin
2 α
,
v = −
1 + ν
8π
b sin 2α −
b
2
1 −
α
π
.
(10.9)
From the formula (10.9), it follows that the displacement component v for α =
2π exceeds by b its value for α = 0, i.e. there is a gap of displacements on the semidirect x > 0, y = 0. The latter means that at the point (0, 0) the edge dislocation
considered above (p. 103) is created with the burgers vector b parallel to the Oy
axis.
10.5 Flat Dislocation of Somigliana
Assume a section is made along some curve L in an unlimited elastic plane. The
section edges can be split and material is inserted into the slit, or material can
be cut off from the edges of the cut. If the edges of such a slit are than glued by
preliminary compressing and shifting them relative to each other, this structural
imperfection is referred to as [2, 9, 12] the flat dislocation of Somigliana. Let
us set the displacement difference on the curve L after gluing as