10.6 Somigliana Dislocation in Half-Plane
109
where 2
λ =
Eν
(1 + ν)(1 − 2ν)
, μ =
E
2(1 + ν)
;
as previously (p. 9), E is the Young modulus, and ν is the Poisson coefficient.
By making passages to the limit in formula (10.15) at R 1 → 0, R 2 → ∞,
we will obtain Muskhelishvili functions for an infinite plane with a dislocation
introduced along the axis O 1 y 1 :
1 (z 1 ) = 1 (z 1 ) =
k
z 1
,
(10.16)
where the following designation is introduced for short
k =
(λ + μ)
2π(λ + 2μ)
.
(10.17)
Using the known [14] formulas of conversion of Muskhelishvili functions when
replacing rectangular coordinate axes, let us write functions (10.16) in the axes xOy
(Fig. 10.4a). To do it, let us at first turn the axes x 1 O 1 y 1 by the angle α so that the
new Y-axis takes a vertical position and is directed from bottom to top. Then we will
make a parallel transfer of axes by the value of H (Fig. 10.4a) vertically upwards.
As a result, for an infinite plane with dislocation, we obtain
2 (z) =
k
(z + iH )e iα , (z = x + iy),
2 (z) =
k[(z + 2iH ) cos α + iz sin α]
(z + iH ) 2
.
(10.18)
10.6.2 Functions , , for a Half-Plane with Dislocation
The components of stresses corresponding to functions (10.18) are designated as
X x ,
Y y , and
X y . They are related to functions (10.18) by dependencies (9.15)
Y y +
X x = 2
2 (z) + 2 (z)
,
Y y −
X x + 2i
X y = 2
zz
2 (z) + 2 (z)
,
(10.19)
where the straight line above still (p. 90) means a complex conjugate value, and the
dash means the differentiation operation under the argument z.
2 Considered plane strain.