Geoscience Reference
In-Depth Information
2
Re(
u
)
= U
1
cos(
2
t
) /
(1+
)
(16.21)
where
= arctan(
). Solution (16.21) represents a response with a decay of
2
) and a delay of
1/
(1+
/
2
, as shown in Fig 16.9.
U
U
U
L
L
L
U
1
U
1
U
1
U
1
U
U
1
U
U
1
U
A
1
A
1
A
1
U
1
U
1
U
1
V
V
V
A
1
A
1
A
1
U
U
U
U
U
U
Figure 16.8
H2
u
t
U
1
u
u
1
u
1
U
1
t
H2
Figure 16.9 Harmonic average response to cyclic boundary loading
The second case defines at the drained boundary
A
1
a pore pressure
u
1
= 0, and
on the entire volume an harmonic loading
t
). The harmonic solution,
employing complex algebra, seeks the real part of the complex response, subjected
to the complex extension of the loading
=
S
1
cos(
2
t
). Since a harmonic response
is considered, everywhere the pore pressure has identical frequency. Thus,
u
=
U
exp[
i
=
S
1
exp(
i
2
t
], where the amplitude
U
is a function of space. Then, the field response
(16.14) is described by
2
2
U = i
c
v
2
(
U - S
1
)
(16.22)
Volume average yields
2
UdV = i
c
v
2
(
U
-
S
1
)
dV
(16.23)
Introduction of average field pressures according to (16.17), and adopting an
approximate two-dimensional parabolic distribution of the field pressure, shown in
Fig 16.10, gives
2
UdV =
UdA =
-3
UA
1
/L
1
(16.24)
This renders (16.22) into
Search WWH ::
Custom Search