Biomedical Engineering Reference
In-Depth Information
the expressions for
u
and
U
r
are given by
r
2
2(
−
mt k
/)
uQRBImrBKmrme
=+ +
(
)(
(
)
(
))
,
r
61
71
(2.105)
2
2(
−
mt k
/)
UNAQBI mr
=− ++
(2
)(
()
+
BK mr
( )
me
r
61
71
Introducing Equation (2.105) into the boundary conditions (2.89) gives
B
6
I
1
(
ma
) +
B
7
K
1
(
ma
) = 0,
B
6
I
1
(
mb
) +
B
7
K
1
(
mb
) = 0
(2.106)
The system of equations (2.106) has a nontrivial solution only if its deter-
minant is zero:
I
1
(
ma
)
K
1
(
mb
) −
I
1
(
mb
)
K
1
(
ma
) = 0
(2.107)
Then,
B
6
= −
B
7
K
1
(
mb
)/
I
1
(
mb
)
(2.108)
Equation (2.107) can be used for determining the constant
m.
Using
Equation (2.87), the radial displacements for the solid and the fluid can be
written as
At
r
()
,
2
2(
−
mt k
/)
2
uQRBImrBKmrme
=+ +
(
)(
(
)
(
))
−
A
()
tr
−
r
61
71
1
(2.109)
At
r
()
2
2(
−
mt k
/)
2
UNAQBI mr
=− ++
(2
)(
()
+
BK mr
( )
me
−
Atr
( )
−
r
61
71
1
Making use of Equation (2.109) and the expressions for the axial
displacements,
u
z
(
t
) = −
D
1
(
t
)
z
,
U
z
(
t
) = −
D
2
(
t
)
z
(2.110)
The stress components are obtained from Equation (2.71) as
2
m
r
2
−
(
mt k
/)
σ
rr
=−
2
NQ R
(
+
)
(
BI mr
(
)
+
BK mr
(
))
e
61
71
2
(2.111)
32
−
(
mt k
/)
−
mQ
(
−
2
NR
−
AR
)(
BI mr
(
)
−
BK mr
(
))
e
60
70
2
NA
()
t
2
−−++
2
(
NAQA t
)
()
+
−+ −
(
AQDt Q
Θ
)
()
1
1
2
r
