10.6 Somigliana Dislocation in Half-Plane
107
g(l) = g 1 (l) + ig 2 (l), (i
2
= −1),
(10.10)
where l is the arch length of the curve L counted from the starting point to the
considered point.
g 1 (l) = u
− (l) − u
+ (l), g 2 (l) = v
− (l) − v
+ (l),
(10.11)
whereas u ± and v ± are the ultimate values of the components of displacement along
the axes Ox and Oy to the left (+) and right (−) from the curve L when moving
along this curve from its start.
The definition of the Somigliana dislocation shows that a ruptural deformation
(10.10)–(10.11) can be represented as a result of location of biclination in the initial
point l = 0 with the Burgers vector b(0) = g 1 (0) + ig 2 (0) and biclinations
distributed along the length of the curve L whose density will be
b(l) = g
1 (l) + ig
2 (l),
(10.12)
where derivatives are taken in the direction of the tangential line to the curve L.
Consequently, it can be deemed that biclinations are distributed on the element
dl of the curve L whose total power (total Burgers vector) is the vector with the
components
X = g
1 (l)dl, Y = g
2 (l)dl.
(10.13)
Note It can be shown that a wedge dislocation located in some point O is equivalent
to the system of biclinations evenly distributed with the density of ε along any halfstraight line coming from the point O with Burgers vectors normal to this line.
Indeed, by representing the biclination with a pair of wedge dislocations with an
opposite sign, we can make a conclusion that structural imperfections inside the area
of even distribution of biclinations are mutually destroyed (wedge-like dislocation
remains at the beginning of the half-straight line).
10.6 Somigliana Dislocation in Half-Plane
Assume that L is the beam O 1 y 1 coming from the point O 1 (Fig. 10.4a) located
at the distance H from the half-plane boundary and making an angle α with a
positive direction of the axis Ox. Moreover, the following conditions are met
indexDislocation!Somigliana!in a half plane
−
π
2
< α <
π
2
,
u
+
1 = v
+
1 = u
−
1 = 0; v
−
1 = β, (β − const),
(10.14)
107
g(l) = g 1 (l) + ig 2 (l), (i
2
= −1),
(10.10)
where l is the arch length of the curve L counted from the starting point to the
considered point.
g 1 (l) = u
− (l) − u
+ (l), g 2 (l) = v
− (l) − v
+ (l),
(10.11)
whereas u ± and v ± are the ultimate values of the components of displacement along
the axes Ox and Oy to the left (+) and right (−) from the curve L when moving
along this curve from its start.
The definition of the Somigliana dislocation shows that a ruptural deformation
(10.10)–(10.11) can be represented as a result of location of biclination in the initial
point l = 0 with the Burgers vector b(0) = g 1 (0) + ig 2 (0) and biclinations
distributed along the length of the curve L whose density will be
b(l) = g
1 (l) + ig
2 (l),
(10.12)
where derivatives are taken in the direction of the tangential line to the curve L.
Consequently, it can be deemed that biclinations are distributed on the element
dl of the curve L whose total power (total Burgers vector) is the vector with the
components
X = g
1 (l)dl, Y = g
2 (l)dl.
(10.13)
Note It can be shown that a wedge dislocation located in some point O is equivalent
to the system of biclinations evenly distributed with the density of ε along any halfstraight line coming from the point O with Burgers vectors normal to this line.
Indeed, by representing the biclination with a pair of wedge dislocations with an
opposite sign, we can make a conclusion that structural imperfections inside the area
of even distribution of biclinations are mutually destroyed (wedge-like dislocation
remains at the beginning of the half-straight line).
10.6 Somigliana Dislocation in Half-Plane
Assume that L is the beam O 1 y 1 coming from the point O 1 (Fig. 10.4a) located
at the distance H from the half-plane boundary and making an angle α with a
positive direction of the axis Ox. Moreover, the following conditions are met
indexDislocation!Somigliana!in a half plane
−
π
2
< α <
π
2
,
u
+
1 = v
+
1 = u
−
1 = 0; v
−
1 = β, (β − const),
(10.14)