Geology Reference
In-Depth Information
•Ÿ
2
C
O
•Ÿ
2
force couple in the
x
2
direction with an arm in the
x
1
direction. It results:
@
u
i
@x
2
u
i
.Ÿ; r/
D
(10.37)
h
M
8
•
i2
.2
/
x
1
r
3
C
•
i1
x
2
To obtain the static field of displacement as-
sociated with a force couple, we take the limit
as •Ÿ
2
!
0and
f
!1
, so that we can always
consider a finite moment
f
•Ÿ
2
!
M
. Therefore,
combining (
10.31
)and(
10.37
) we see that the
displacement will be given by:
u
i
.r/
D
r
3
i
3
x
1
x
2
x
i
r
5
(10.39)
Therefore, a double couple like that in
Fig.
10.10
would generate the following dis-
placement field:
h
M
8
•
i1
.2
/
x
2
r
3
C
•
i2
x
1
u
j
.
r
/
D
r
3
i
3
x
1
x
2
x
i
r
5
h
.
1/
•
i1
x
2
r
3
M
4
r
3
C
•
i2
x
1
(10.38)
u
i
.r/
D
i
3
x
1
x
2
x
i
r
5
A similar calculation can be performed to
determine the displacement field generated by a
(10.40)
Finally, in spherical coordinates the double-
couple displacement field assumes the form:
8
<
4 r
2
1
C
2
sin
2
™ sin 2¥
M
u
r
D
M
u
™
D
8 r
2
.1
/sin 2™ sin 2¥
(10.41)
:
M
4 r
2
.1
/sin ™ cos 2¥
u
¥
D
A comparison of this solution with (
10.34
)
shows that the displacement field associated with
a double couple decreases with distance more
rapidly (as 1/
r
2
) than that generated by a point
force. In the
x
1
x
2
planewehavethat™
D
/2,
thereby
u
™
D
0 and:
(
u
r
D
4 r
2
1
C
2
sin 2¥
M
(10.42)
M
u
¥
D
4 r
2
.1
/cos 2¥
Fig. 10.9
A
force couple in the
x
1
direction is applied at
location Ÿ D (Ÿ
1
,Ÿ
2
,Ÿ
3
) with an arm •Ÿ
2
in the
x
1
x
2
plane
Fig. 10.10
A double force couple in the
x
1
x
2
plane and the corresponding pattern of displacement for
u
r
and
u
¥
