Biomedical Engineering Reference
In-Depth Information
Fig. 8.6 Illustration of the vertical settlement of a layer of poroelastic material resting on a stiff
impermeable base subjected to a constant surface loading. This is a plot of the function g ( t ) against
time. The curve underneath the top curve is for a layer twice as thick as the layer of the top curve.
The two curves below are for layers that are five and ten times as thick, respectively, as the layer
associated with the top curve. Compare with Fig. 1.11c
c d 2 f
i
o
f
ð
x 3 Þ
dx 3 ¼
i
o
Wp o
o
after dividing through by e i
t . The solution to this ordinary differential equation is
s
i
s
i
L 2
c
L 2
c
A sinh x 3
L
o
B cosh x 3
L
o
f
ð
x 3 Þ¼
Wp o þ
þ
:
The boundary conditions of the previous example, namely that the pressure is
zero at x 3 ¼
L yield the following
solution to the original partial differential equation above, thus
0 and that the pressure gradient is zero at x 3 ¼
2
4
s
i
s
i
s
i
3
5 e i
L 2
c
L 2
c
L 2
c
o
sinh x 3
L
o
cosh x 3
L
o
o
t
p
ð
x 3 ;
t
Þ¼
Wp o 1
þ
:
tanh
The solution to the problem of semi-infinite domain of poroelastic material
subjected to a harmonic surface loading T 33 ¼
o
t may also be
obtained without difficulty. This problem is also illustrated in Fig. 8.4 if the stiff
impermeable base is removed. The solution to the original differential equation
above with the boundary conditions that the pressure is zero at x 3 ¼
p o e i
P
ð
t
Þ¼
0 and that the
pressure gradient tends to zero as x 3 becomes large is
e x 3
p
o
o = c
e i
t
i
p
ð
x 3 ;
t
Þ¼
Wp o a
ð
x 3 Þ
;
where a
ð
x 3 Þ¼½
1
:
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