Geoscience Reference
In-Depth Information
Z
exp .
x
0
/
x
0
dx
0
:
E
1
.x/
D
(10.20)
x
One makes sure of validity of the solution by direct substitution of Eq. (
10.19
)for
g.r/ into Eq. (
10.18
). The sought function
b
Dr
a
can be expressed through
g.x/ in the following manner
h
2
O
r
g.x/ cos
O
@
r
f
rg.x/
g
sin
i
:
1
r
b
D
(10.21)
In order to find c
1
and c
2
these equations should be supplemented by the proper
boundary conditions. Since the function
b
in Eq. (
10.21
) must go to zero at infinity,
we get that c
1
D
0. Moreover,
b
must to be limited when r
!
0. Eventually, we
obtain that
1
x
2
C
x
1
exp .
x/
x
2
C
xE
1
.x/
:
u
0
B
0
3
m
1
1
g.x/
D
(10.22)
The function g.x/ can be simplified in two extreme cases corresponding to large
and small values of argument x. Consider first the case of large distances/small
cracks, that is r
L .x
1/, when the function g.x/ simplifies to
u
0
B
0
3
m
x
2
:
v
g
(10.23)
It should be noted that if the maximal crack size satisfies the inequality l
max
r=Ǜ
where Ǜ is given by (
10.16
), this approach is valid for all the cracks.
For now, we shall be interested in the averaging over the crack size. We suppose
that the number of cracks generated per unit time with length greater than l
occurring in a specified area can be estimated from the known empirical law
obtained by Gutenberg and Richter (
1954
) for the number of EQs. According to
Turcotte (
1997
)
LJ
l
2b
;
N .l/
D
(10.24)
where b is the dimensionless empirical constant whose value varies from region to
region but is generally in the range 0:8 < b < 1:2. The constant LJ is a measure
of the regional level of seismicity. This value is measured in units of m
2b
/s. The
worldwide data correlate with (
10.24
) taking b
D
1:11 and LJ
2
10
3
m
2b
/s.
Assuming for the moment that the dependence (
10.24
) can be extrapolated down to
the crack sizes of about several meters and combining Eq. (
10.24
) and Eq. (
10.10
),
yields
Z
l
max
b
.
r
;l/
l
2bC1
dl:
h
ı
B
t
.
r
/
iD
2bLJ
(10.25)
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