Geology Reference
In-Depth Information
8πμ(λ
+
2μ)/(λ
+
μ). We take the elastic half-space to be a Poisson solid withλ
=
μ.
To reduce the solutions to those for point forces of unit magnitude, required for the
evaluation of the dip-slip and strike-slip integrals, they must then be divided by
12πμ. The point force in the solution of Mindlin problem I is in the
x
3
-direction.
That for Mindlin problem II is in the
x
1
-direction. To obtain the required solution
for a point force in the
x
2
-direction, the expressions for the first two components
are exchanged along with the exchange of variables
x
1
and
x
2
,aswellasξ
1
andξ
2
.
In addition to the distance to the field point at (
x
1
,
x
2
,
x
3
) from a source point at
(ξ
1
,ξ
2
,ξ
3
) on the fault surface, given by
(
x
1
R
=
−
ξ
1
)
2
+
(
x
2
−
ξ
2
)
2
+
(
x
3
−
ξ
3
)
2
,
(9.13)
and the distance to the field point from the point (ξ
1
,ξ
2
,
−
ξ
3
) on the image of the
fault surface in the upper half-space, given by
(
x
1
−
ξ
1
)
2
Q
=
+
(
x
2
−
ξ
2
)
2
+
(
x
3
+
ξ
3
)
2
,
(9.14)
it is convenient to introduce the abbreviations
r
2
=
x
2
sinθ
−
x
3
cosθ,
q
2
=
x
2
sinθ
+
x
3
cosθ,
(9.15)
r
3
=
x
2
cosθ
+
x
3
sinθ,
q
3
=−
x
2
cosθ
+
x
3
sinθ.
(9.16)
r
2
and
r
3
are the distances to the field point, measured normal and down-dip to
the fault, while
q
2
and
q
3
are the distances to the field point, measured normal and
up-dip to the image fault. With these abbreviations, we have
R
2
−
ξ
1
)
2
r
2
+
−
ξ)
2
=
(
x
1
+
(
r
3
,
(9.17)
Q
2
−
ξ
1
)
2
q
2
+
+
ξ)
2
=
(
x
1
+
(
q
3
(9.18)
−
ξ
1
)
2
h
2
k
2
+
ξ)
2
=
(
x
1
+
=
+
(
q
3
,
(9.19)
where
h
is the projection of
Q
on the plane
x
1
=
0and
k
is its projection on the
plane
q
3
=
0.
Expressed in indefinite integral form, the Volterra integrals for the dip-slip dis-
location (9.11) give
2
R
+
u
1
U
=
4
Q
−
4
ξ
3
x
3
3
12π
(
x
2
−
ξ
2
)sinθ
Q
3
−
Q
+
x
3
+
ξ
3
3ln
Q
+
ξ
3
+
2
x
3
−
ξ
3
R
4
x
3
−
ξ
3
Q
4
ξ
3
x
3
(
x
3
+
ξ
3
)
Q
3
−
cosθ
+
x
3
+
+
ln
Q
+
ξ
3
cosθ
+
ξ
3
−
sinθln
Q
+
+
x
3
+
q
3
6
x
3
cos
Q
−
q
2
sinθ
Q
(
Q
+
,
(9.20)
+
q
3
+
ξ)
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