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S
=
A
1
+
A
2
, where
A
1
is a drained surface (pore pressure is constant
u
= 0) and
A
2
is
closed surface (no flow condition). Adopting a volumetric approach, i.e. volume
strain
and mean effective stress
'
are related through a bulk modulus
K
:
' = K
,
the general storage equation, expressed by
58
'
u
=
(1/
c
v
)
(
u
)/
t
with
c
v
= kK/
w
(16.14)
Two cases are considered: a cyclic pore pressure
u
at boundary
A
1
and a cyclic
loading
on the entire volume.
The first case implies that the loading
is constant, and at the drained boundary
A
1
a harmonic loading is applied
u
1
=
U
1
cos(
t
). The harmonic solution, employing
complex algebra, seeks the real part of the complex response, subjected to the
complex extension of the loading
u
1
=
U
1
exp(
i
2
t
). Since a harmonic response is
considered, everywhere the pore pressure has identical frequency. Thus,
u
=
U
exp(
i
2
t
), where the amplitude
U
is only a function of space. Equation (16.14)
becomes
2
2
U = i
c
v
2
U
(16.15)
A simple solution is sought by averaging over space (see Chapter 6
E
)
,
rendering
(16.15) into,
2
UdV = i
c
v
2
UdV
(16.16)
Introduction of average field pressures
U
=
UdV / V
and
u
=
udV / V
(16.17)
and adopting an approximate two-dimensional parabolic distribution of the field
pressure, shown in Fig 16.8, so that
2
UdV =
UdA =
3(
U
1
U
)
A
1
/
L
1
(16.18)
where,
A
1
is the drained surface and
L
1
the relevant flow path, renders (16.16) into
2
U = U
1
(
1
i
)/(1+
)
(16.19)
with
L
1
L
2
/
3
c
v
and
L
2
=
V/A
1
, the hydraulic radius (drainage capacity). The
total harmonic response is therefore
=
2
2
)
u = U
exp(
i
2
t
)
= U
1
exp(
i
2
t
)
(
1
i
)/(1+
(16.20)
The real part of this expression, the real response, becomes after some
elaboration
58
The compressibility of the pore fluid
)
can be included by adopting
c
v
= kK/
(
w
+n
)
K
)
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