108
10 Mathematical Structural Imperfections
a
b
Fig. 10.4 Somigliana dislocation in half-plane (a) and in annulus (b)
where u
±
1 , v
±
1 mean the ultimate values of the components of displacement along
the axes Ox and Oy to the left (+) and right (−) from the beam O 1 y 1 while moving
in the positive direction of the axis O 1 y 1 .
There is a problem: find elastic equilibrium of the half-plane y 0 (Fig. 10.4b)
in the conditions of (10.14)
10.6.1 Functions , for the Plane with Dislocation
Let us use the solution to the problem for an annulus with an internal radius R 1 and
external radius R 2 (Fig. 10.4b) where a cut is made along the axis O 1 y 1 and the right
edge of the cut is moved relative to the left secured edge by the value of β in the
direction indicated by the arrow in Fig. 10.4b. After the shift, cut edges are soldered
together and the left edge is released from securing.
Muskhelishvili functions for this ring are given [14] by the formulas 1
1 ) = −
(λ + μ)
2π(λ + 2μ)
2z 1
R 2
1 + R 2
2
−
1
z 1
,
(z 1 ) =
(λ + μ)
2π(λ + 2μ)
1
z 1
+
2R 2
1 R 2
2
R 2
1 + R 2
2
·
1
z 3
1
,
(z 1 = x 1 + iy 1 ),
(10.15)
1 In the fifth edition of the monograph [14] the formula for the function (z 1 ) contains misprints.