Biomedical Engineering Reference
In-Depth Information
Fig. 8.7
A plot of the
function |
a
(
x
3
)| against
x
3
.
This plot has a peak of
1.06702 at
x
3
¼
3.33216 and
then it is asymptotic to the
value one as
x
3
tends to
infinity
The derivation of this solution is a problem at the end of the section. The real part
of
a
(
x
3
) is plotted in Fig.
8.7
. This plot has a peak of 1.06702 at
x
3
¼
3.33216 and
then it is asymptotic to the value one as
x
3
tends to infinity. This indicates that the
depth beyond
from the surface, where
a
(
x
3
) has an almost
constant value of one, the pore pressure
p
is in phase with the surface loading and
proportional to it by the factor
W
. The interpretation of this result is that the
departure of the pore pressure fluctuations from the undrained solution is confined
to a boundary layer of thickness
d ¼
3.33216
√
c/
o
. The semi-infinite domain solution is therefore
applicable to the finite layer problem under consideration provided
d
L
.
The desired settlement of the free surface,
u
3
(0,
t
) is calculated following the
method of Example 8.10.2 thus
d <
3
s
i
r
c
L
2
c
Þ¼
að
1
þ n
d
Þ
WLp
o
o
o
5
e
i
t
u
ð
0
;
t
tanh
:
3
K
d
L
2
ð
1
n
d
Þ
i
o
A plot against frequency of the absolute value of the function
s
i
r
c
L
2
c
o
tanh
L
2
i
o
determining the amplitude of the settlement of the free surface
u
3
(0,
t
) is shown in
Fig.
8.8
. At very large frequency the poroelastic layer behaves as if it were
undrained, that is to say
|u
(0,
t
)
|
!
1as
o !1
. At low frequencies it behaves
as if it were drained, that is to say
|u
(0,
t
)
|
!
0as
o !
0.
Problems
8.10.1. Using the assumptions of Example 8.10.1 and the equation (
8.50
) derive the
pressure diffusion equation
2
p
@
p
c
@
W
@
P
@
t
x
3
¼
t
;
@
@
Search WWH ::
Custom Search