Biomedical Engineering Reference
In-Depth Information
e
rc
(
ϕ−ϕ
)
ln(/)
−
αψ−ψ
(
)
ln(/)
15
44
b
a
15
44
b
a
ε=−
(3.85)
rz
ba
rc
ba
=−
ϕ−ϕ
(
)
ln(/)
,
=−
ψ−ψ
(
)
ln(/)
b
a
b
a
E
H
(3.86)
r
r
rba
rba
Substituting Equation (3.82)-(3.86) yields
r
s
2
*
2
A
F
cc
c
FT Pt
ba
c
+
π−
−
()
=
33
12
0
2
eAe
*
()
c
*
[
*
T
pt
( )]
FTc
*
+
ββ +
+ϖ+
33
1
2
0
13
1
013
*
2
(
)
3
11
s
s
*
A
ϖ
[2 ln(/) ]
r
a
−
A
F
FT Pt
ba
+
π−
( )
rr
zz
*
2
0
2
+
+
FT
−
(
c
+
c
)
1
0
11
12
*
2
c
(
)
11
3
(3.87)
2
cc
c
ϖ
−
φ−φ
e
c
13
12
b
a
15
*
*
E
s
−ββ+ −
2
c
[
T
pt
()]
ba
A
+
A
13
1
2
0
r
zr
r
ln(/)
11
44
−
ψ−ψ
+
α
b
a
E
15
s
G
A
r
zr
rba
ln(/)
c
44
In comparison with Equation (3.37), the only difference is due to the last term
(underlined term).
As numerical illustration of the bone devolution process, the results pre-
sented in Qu and Qin [11] are summarized here. In these authors' work, a
femur with
a
= 25 mm and
b
= 35 mm is considered. The material properties
assumed for the bone are
c
11
= 15(1 +
e
)GPa,
c
12
=
c
13
= 6.6(1 +
e
)GPa,
c
33
= 12(1 =
e
)GPa,
c
44
= 4.4(1 +
e
)GPa, λ
1
= 0.621(1 +
e
) × 10
5
NK
−1
m
−2
,
λ
3
= 0.55(1 +
e
) × 10
5
NK
−1
m
−2
,
χ
3
= 0.133(1 +
e
)CK
−1
m
−2
(3.88)
e
31
= −0.435(1 +
e
)C/m
2
,
e
33
= 1.75(1 +
e
)C/m
2
,
e
15
= 1.14(1 +
e
)C/m
2
,
κ
1
= 111.5(1 +
e
)κ
0
,
α
15
= 550(1 +
e
) N/A m
κ
3
= 126(1 +
e
)κ
0
, κ
0
= 8.85 × 10
−12
C
2
/Nm
2
= permittivity of free space
The remodeling rate coefficients are assumed to be
C
0
= 3.09 × 10
−9
sec
−1
,
C
1
= 2 × 10
−7
sec
−1
,
C
2
= 10
−6
sec
−1
