Biomedical Engineering Reference
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2 p
@x 1
2 p
@x 2
2 p
@x 3
K eff @
1
p
@t
1
m
K 11 @
1
m
K 22 @
1
m
K 33 @
@
@
@
d
12
d
31
E 3
d
12
d
23
E 2
d
23
d
31
E 3
E 1 n
1
E 1 n
T 11
@t þ
E 2 n
1
E 1 n
T 22
@t þ
E 3 n
1
E 2 n
T 33
@t
¼
:
(8.64)
The boundary conditions on the pore pressure field are coincident with those
described at the end of the previous section.
8.10 Some Example Isotropic Poroelastic Problems
Example 8.10.1
Formulate the differential equations governing the problem of determining the
vertical surface settlement of a layer of poroelastic material resting on a stiff
impermeable base subjected to a constant surface loading. The layer, illustrated
in Fig. 8.4 , is in the x 1 , x 2 plane and the x 3 positive coordinate is in the thickness
direction and it is pointed downward in Fig. 8.4 . The surface is subjected to an
applied compressive stress T 33 ¼
P ( t ), the only nonzero strain component is E 33 .
The free surface of the layer permits the passage of fluid out of the layer.
Solution : First, since the free surface of the layer permits the passage of fluid and the
supporting base of the layer is impermeable, the boundary conditions on the pore
pressure field are p
¼
0at x 3 ¼
0,
@
p
=@
x 3 ¼
0at x 3 ¼
L . Next, using the fact that
the only nonzero strain component is E 33 ( E 33 ¼ @
u 3 =@
x 3 from (3.52)) and that the
applied compressive stress T 33 ¼
P ( t ) is uniform throughout
the layer,
the
strain-stress-pressure relations ( 8.1 ) specialize to the following:
1
E d
d
d tr T
d
0
¼
1
þ n
Þ
T 11 n
þð
1
2
n
Þa
p
g;
1
E d
d
d tr T
d
0
¼
1
þ n
Þ
T 22 n
þð
1
2
n
Þa
p
g
@
u 3
1
E d
d
d tr T
d
x 3 ¼
1
þ n
Þ
P
ð
t
Þn
þð
1
2
n
Þa
p
g:
@
Fig. 8.4 Illustration of a
layer of poroelastic material
resting on a stiff impermeable
base subjected to a uniform
time varying surface loading
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