Biomedical Engineering Reference
In-Depth Information
2
//)
2
3
−
(
mt
k
σ
zz
=− −+ −
(
2
QN
AR
QBImrBKmrme
)(
(
)
(
))
60
70
(2.112)
−+ −++
2
(
AQAt
)
() (
2
NAQD tQ
)
()
−
Θ
1
1
2
2
3(
−
mt k
/)
σ=
(
Q
−
2
NARBImrBKmrme
−
)(
()
−
( )
60
70
(2.113)
−+ −+ −Θ
2(
RQAt
)()(
RQDt
)()
R
1
1
where
I
0
(
mr
) and
K
0
(
mr
) are the modified Bessel functions of the first and
second kind of zero order, respectively.
The solution of the system of equations (2.84)-(2.86) leads to the expres-
sions for the unknowns,
A
1
(
t
),
A
2
(
t
), and
D
1
(
t
) as
=
+++π++++π− +Θ
−
(
AQRP aA NQRp Nb aQR
abNANQR
2
)
2
(
22 )
2(
2
2
)(
)
A
,
1
2
2
2(
)
π +++
(32 63)
22
abp
Nb
−
A
=
)
,
2
2
2
2(
−
a
2
2
2
=
−+++ −π+++π −
(
AQRNPa AQRp Nb aQR
ab ANQR
2
)
(
2
)
(
)(
+ Θ
)
D
(2.114)
1
2
2
(
−
)(3
+ ++
263)
Taking [29]
Θ=
+++
+
32 63
ANQR
QR
and
B
=
1
(2.115)
7
and substituting Equation (2.114) into Equations (2.109)-(2.113), the displace-
ments and stresses in the hollow cylinder can be obtained.
The formulations discussed in this chapter provide an initial insight into
bone remodeling theory. Extensions to multifield bone remodeling analysis
are discussed in the next two chapters.
References
1. Wolff J.
Das gesetz der transformation der knochen.
Berlin: A. Hirschwald (1892).
2. Bauman W. A., Spungen A. M., Wang J., Pierson R. N., Schwartz E. Continuous
loss of bone during chronic immobilization: A monozygotic twin study.
Osteoporosis International
10 (2): 123-127 (1999).
3. Zerwekh J. E., Ruml L. A., Gottschalk F., Pak C. Y. C. The effects of twelve
weeks of bed rest on bone histology, biochemical markers of bone turnover, and
