Biomedical Engineering Reference
In-Depth Information
The condition that
@
p
=@
x
3
¼
0at
x
3
¼
L
requires that
@
p
=@
x
3
j
x
3
¼L
¼ bð
A
sin
2
b
e
c
b
L
þ
Bcos
b
L
Þ
¼
0 which is satisfied by setting
A
¼
0, and
cos
b
L
¼
0is
satisfied by setting
b ¼ð
1
þ
2
n
Þp=
2
L
where
n
¼
0, 1, 2,
...
. Substituting these
Z
e
c
2
t
, and summing over all
results back into
p
ð
x
3
;
t
Þ¼ð
A
cos
x
3
þ
B
sin
x
3
Þ
possible values of
n
, one obtains the representation
e
cðð
1
þ
2
nÞ
p
=
2
LÞ
1
B
n
sin
ð
þ
2
n
Þp
1
2
t
p
ð
x
3
;
t
Þ¼
x
3
:
2
L
n¼
0
p
I
C
d
K
d
then yields
The condition that
p
ð
x
3
;
0
Þ¼
¼ a
P
o
=
1
¼
a
P
o
C
d
K
d
¼
B
n
sin
ð
1
þ
2
n
Þp
p
I
p
ð
x
3
;
0
Þ¼
x
3
:
2
L
n
¼
0
If we multiply both sides of the previous equation by sin
ðð
1
þ
2
m
Þp
x
3
=
2
L
Þ
,
integrate the result from 0 to
L
, and recall the orthogonality relations
Z
L
sin
ð
dx
3
¼
sin
ð
1
þ
2
n
Þp
x
3
1
þ
2
m
Þp
x
3
L
2
d
nm
2
L
2
L
0
it follows that
dx
3
¼
C
d
K
d
Z
L
B
n
¼
a
P
o
sin
ð
1
þ
2
n
Þp
x
3
2L
a
P
o
2
L
C
d
K
d
ð
1
þ
2n
Þ
0
and the solution for the pressure field is
e
cðð
1
þ
2
nÞ
p
=
2
LÞ
C
d
K
d
1
n¼
0
2
L
a
P
o
1
sin
ð
1
þ
2
n
Þp
2
t
p
ð
x
3
;
t
Þ¼
x
3
:
ð
1
þ
2
n
Þ
2
L
Note that this result satisfies the initial conditions (Figs.
8.5
and
8.6
).
Example 8.10.3
Determine the vertical surface settlement of a layer of poroelastic material resting
on a stiff impermeable base subjected to a harmonic surface loading
T
33
¼
P
ð
t
Þ
o
t
. The layer, illustrated in Fig.
8.4
, is in the
x
1
,
x
2
plane and the
x
3
positive
coordinate direction is downward. The conditions for the drainage of the layer are
described in Example 8.10.1.
P
o
e
i
¼
o
Solution
: Substituting the surface loading
P
t
into the pressure diffusion
equation in Example 8.10.1, the diffusion takes the form
ð
t
Þ¼
P
o
e
i
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