Biomedical Engineering Reference
In-Depth Information
The solutions of these equations for
T
11
,
T
22
,tr
T,
and
T
33
are
n
d
Þ
ð
n
d
Þ
1
2
T
11
¼
T
22
¼
P
ð
t
Þ
a
p
;
ð
1
n
d
Þ
ð
1
n
d
E
d
d
d
@
u
3
¼
ð
1
þ n
Þ
2
ð
1
2
n
Þ
tr
T
¼
x
3
3
a
p
P
ð
t
Þ
a
p
ð
1
2
n
d
Þ
@
ð
1
n
d
Þ
ð
1
n
d
Þ
and
3
K
d
ð
n
d
Þ
@
u
3
1
T
33
¼
P
ð
t
Þ¼
x
3
a
p
ð
1
þ n
d
Þ
@
and the single strain component
E
33
, is given by
d
E
33
¼
@
u
3
ð
1
þ n
Þ
x
3
¼
Þ
ð
P
ð
t
Þþa
p
Þ:
@
3
K
d
ð
1
n
d
The stress equations of motion ((3.37) or (3.38)), or equilibrium in this case,
reduce to the condition that the derivative of the
T
33
stress component with respect
to
x
3
must vanish, thus from the equation directly above,
d
@
@
3
K
d
ð
1
n
Þ
@
u
3
x
3
a
p
¼
0
:
x
3
ð
1
þ n
d
Þ
@
E
33
, substitution of
E
33
into the pressure diffusion equation (
8.60
),
and use of (
8.38
) for both material and drained constants yields
Since tr
E
¼
2
p
@
p
c
@
W
dP
dt
t
x
3
¼
;
(8.65)
@
@
where
c
I
W
a
að
þ n
d
Þ
1
c
¼
;
W
¼
K
d
3
L
ð
1
n
d
Þþa
2
ð
1
þ n
d
ÞÞ
and
c
I
represents the value of the constant
c
when the matrix material and the pore
fluid are incompressible,
K
11
3
K
d
d
ð
1
n
Þ
c
I
¼
:
mð
1
þ n
d
Þ
In the special case when the matrix material and the fluid are incompressible,
c
1. The two following examples examine special solutions of these
equations that may then be specialized to these more special assumptions by the
appropriate selection of
c
and
W
.
¼
c
I
and
W
¼
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