10.6 Somigliana Dislocation in Half-Plane
113
X 0
x =
2k{2x(y − H )V 1 − V 2 [x 2 − (y − H ) 2 ]}
[x 2 + (y − H ) 2 ] 2
−
−
2ky{x[3(y − H ) 2 − x 2 ]V 3 + (y + H )[(y − H ) 2 − 3x 2 ]V 4 }
[x 2 + (y − H ) 2 ] 3
;
Y 0
y = −
2k{V 5 [x 2 − (y − H ) 2 ] + 2x(y − H )(y − 2H ) cos α}
[x 2 + (y − H ) 2 ] 2
+
+
2ky{x[3(y − H ) 2 − x 2 ]V 3 + (y + H )[(y − H ) 2 − 3x 2 ]V 6}
[x 2 + (y − H ) 2 ] 3
;
X 0
y = −
2k{(H cos α + x sin α)[x 2 − (y − H ) 2 ] + 2xy(y − H ) sin α}
[x 2 + (y − H ) 2 ] 2
−
−
2ky{(y − H )[(y − H ) 2 − 3x 2 ]V 8 + x[x 2 − 3(y − H ) 2 ]V 7 }
[x 2 + (y − H ) 2 ] 3
(10.32)
where
V 1 = (4H − y) cos α + 2x sin α, V 2 = x cos α + (H + 2y) sin α,
V 3 = (5H − y) cos α + x sin α, V 4 = x cos α + (y + H ) sin α,
V 5 = x cos α + H sin α,
V 6 = (y + H ) sin α + x cos α,
V 7 = x cos α + (3H + y) sin α, V 8 = x sin α + (5H − y) cos α.
(10.33)
By summing the corresponding components of stresses in formulas (10.20)
and 10.32), we obtain the solution to the problem:
X x =
X x + X
0
x ; Y y =
Y y + Y
0
y ; X y =
X y + X
0
y .
(10.34)
10.6.5 Addition to Geomechanics
It is known [4] that one of the manifestations of geodynamic processes in the Earth’s
crust is the formation of faults called disjunctive dislocations. Schematically,
such dislocation is shown in Fig. 10.6 where the digits 1 and 2 designate a fixed
and shifted crust blocks, respectively. This type of fracture is called [4] upcast in
geology. An arrow in Fig. 10.6 indicates the direction of block 2 shift. Stresses
caused by such movements of the crust are called tectonic.
If a rock massif in the vicinity of the dislocation is conditionally considered as
homogeneous, isotropic, and ideally elastic, it is obvious that the problem solved
in this paragraph serves an almost adequate mathematical model of a rock massif
with upcast dislocation. In other conditions, the answer to the question about the
applicability of the solution (10.34) to the assessment of tectonic stresses in the
vicinity of a disjunctive dislocation can be obtained by specifying the model and
comparing the results of field and numerical experiments.