Biomedical Engineering Reference
In-Depth Information
2
ε+ σ+ λ−λ
c
c
u
R
c
c
c
c
c
c
+−
∫∫
12
13
13
13
c
1
c
TdSPt
=−
( )
(3.55)
13
33
z
r
1
3
S
11
11
11
11
For a thick-walled bone section or a section with variable volume fraction in
the radial direction, we can divide the bone into a number of sublayers, each
of which is sufficiently thin and is assumed to be composed of a homogeneous
material. Within a layer we take the mean value of the volume fraction of
the layer as the layer's volume fraction. As a consequence, the analysis
described previously for a thin and homogeneous bone can be applied here
for each sublayer in a straightforward manner. For instance, for the j th layer,
Equation (3.53) becomes
{
}
{
}
{}{}
()
j
=
()
j
()
j
()
j
()
j
F
()
h
D
()
h
F
(0)
+
D
+
D
(3.56)
j
j
L
T
j
where h j denotes the thickness of the j th sublayer.
Considering the continuity of displacements and transverse stresses across
the interfaces between these fictitious sublayers, we have
{
} {
}
()
j
( )
j
+
F
()
h
=
F
(0)
(3.57)
j
After establishing Equation (3.56) for all sublayers, the following equation
can be obtained by using Equations (3.56) and (3.57) recursively:
()
N
()
N
()
N
{( )}
F
h
=
[
D
(
h
)]{ (
F
h
)}{
+
D
}{
+
D
}
N
N
N
−1
L
T
()
N
(
N
1
)
(
N
1
)
(
N
1
)
(
N
)
(
N
)
=
[
D
(
h
)]{[
D
(
h
)]{ (
F
h
)}{
+
D
}{
+
D
}}
+
{
D
}
+
{
D
}
N
N
1
N
2
L
T
L
T
()
N
(
N
1
)
=
[
D
(
h
)]
[
D
(
h
)]{ (
F
h
)}
N
1
N
2
()
N
(
N
1
)
(
N
1
)
(
N
)
(
N
)
+
[
D
(
h
)]{{
D
}{
+
D
}}
+
{
D
}
+
{
D
}
N
L
T
L
T
=
(
Nj
)
()
N
(
N
1
)
(
N
2
)
=
[
D
(
h
)][
D
(
h
)][
D
(
h
)][
D
(
h
)]{ (
F
h
)}
N
1
N
2
N
j
N
−−
j
1
()
N
(
N
1
)
Nj
−+
1
(
Nj
)
(
Nj
)
+
[
D
(
h
)][
D
(
h
1 D
)][
(
h
)]{{
D
}{
+
D
}}
N
N
Nj
−+
1
L
T
()
N
(
N
1
)
Nj
−+
2
(
NNj
−+
1
)
(
Nj
−+
1
)
+
[
D
(
h
)][
D
(
h
)][
D
(
h
)]{{
D
}{
+
D
}}
+
N
N
1
Nj
−+
2
L
T
()
N
(
N
1
)
(
N
1
)
(
N
)
(
N
)
+
[
D
(
h
)]{{
D
}{
+
D
}}
+
{
D
}
+
{
D
}
N
L
T
L
T
=
[ {(0)}{}
(3.58)
F
+
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