10.8 Edge Dislocation in a Half-Plane
117
σ x + σ y = 2
(z) + (z)
,
σ x − σ y + 2iτ xy = 2
zz (z) + (z)
,
where the top line means a complex conjugate value and the “dash” symbol
means the differentiation operation using the variable z. By substituting into these
dependencies functions (10.36) instead of and , let us find stresses in the points
of the axis Ox after separating the real and imaginary parts:
σ y (x, 0) =
4Kδt (x 2 − t 2 )
(x 2 + t 2 ) 2 , τ xy (x, 0) = 0.
(10.37)
10.8 Edge Dislocation in a Half-Plane
Let us then consider a half-plane y 0, along the boundary of which (y = 0)
a normal pressure N is applied, which is equal to the normal stress σ y according
to formula (10.37), but of an opposite sign. For such a half-plane, Muskhelishvili
functions are represented [14] as:
3 (z) =
1
2πi
∞
−∞
Ndξ
ξ − z
, , 3 (z) = −zz
(z).
(10.38)
By substituting into the last formulas, instead of N , the formula
N(ξ ) = −
4Kδt (x 2 − t 2 )
(x 2 + t 2 ) 2 ,
we obtain as follows using formulas (10.38) the after calculation of the integral and
differentiation
3 (z) = −
2Kδt
(z + it) 2 , , 3 (z) = −
4Kδtz
(z + it) 3 .
(10.39)
Muskhelishvili functions for a half-plane y 0 with the introduced stripe δ wide
on the section −t y 0 of the negative half-axis y, by superposing respective
functions defined by formulas (10.36) and (10.39), we obtain as follows, e.g.
4 (z) = 2 (z) + 3 (z) = −
4Kδtz
(z + it) 2 (z − it)
,
4 (z) = 2 (z) + 3 (z) =
4Kδzt 2 (3iz + t)
(z + it) 3 (z − it) 2 .
(10.40)