Geology Reference
In-Depth Information
where
V
D
awh
is the volume of the region.
Similarly, the shear strain will be given by:
11.2
The Asymmetric Strain
Tensor
w
D
M
0
sin
2
™
•
u
sin ™
©
xy
D
(11.8)
Molnar (
1983
) proposed the following solution to
the problem of estimating the average strain rate
of a deforming region from its seismicity. Let us
consider a rectangular region of thickness
h
and
dimensions
w
and
a
, crossed from side to side by
a vertical fault of length
L
and strike ™ (Fig.
11.2
).
Let us assume that a small (compared to
w
,
a
,
and
L
) strike-slip displacement •
u
occurs along
the fault during an earthquake. In this instance,
the average strain in the
y
direction will be given
by:
V
Let us consider now the components ©
xx
and
©
yx
. The points along the left side of the region
have displacement zero in the
x
direction for a
length
w
2
,and•
u
sin™ for a length
w
1
.There-
fore, the average displacement of the left side
in the
x
direction is (
w
1
/
w
)•
u
sin™. Similarly, the
mean displacement of the right side in the
x
direction is (
w
3
/
w
)•
u
sin™. Therefore, the average
change in length is •
a
D
(
w
3
-
w
1
)•
u
sin ™/
w
D
-
L
•
u
sin ™ cos™/
w
.
Then, the average strain in the
x
direction will
be given by:
•
u
cos ™
w
©
yy
D
(11.5)
•a
a
D
L•
u
sin ™ cos
™
a
w
D
M
0
sin ™ cos
™
V
(11.9)
©
xx
D
M
0
D
Lh•
u
(11.6)
A similar calculation leads to the following ex-
pression for ©
yx
:
Therefore, (
11.5
) can be rewritten as follows:
L•
u
cos
2
™
a
w
D
M
0
cos
2
™
©
yy
D
M
0
sin ™ cos ™
V
©
yx
D
(11.10)
(11.7)
V
Fig. 11.2
A rectangular
region cut by a vertical
fault with strike ™
