Biomedical Engineering Reference
In-Depth Information
where [
G
], {
H
L
}, and {
H
T
} are all constant matrices.
Equation (3.47) can be solved analytically and the solution is [35]
us
()
u
(0)
h
h
r
r
∫
∫
{}
{}
[]
G
s
[ (
G
s
−τ
)
[ (
G
s
−τ
)
=
e
+
e
H
d
τ +
e
H
d
τ
(3.48)
L
T
σ
()
s
σ
(0)
0
0
r
r
where
u
r
(0) and σ
r
(0) are, respectively, displacement and stress at the bottom
surface of the layer. Rewrite Equation (3.48) as
{} {}{}{}
[
]
F
()
s
=
DF
()
s
(0)
+
DD
+
(3.49)
L
T
The exponential matrix can be calculated as follows:
[]
s
[()]
D
s
=
e
=α α
()
s
I
()[]
s
G
(3.50)
0
1
where α
0
(s) and α
1
(s) can be solved from
α αβ=
()
s
()
s
e
β
s
1
0
1
1
(3. 51)
β
s
α αβ=
()
s
()
s
e
2
0
1
2
In Equation (3.51), β
1
and β
2
are two eigenvalues of [
G
], which are given by
β
β
1
2
1
2
c
c
1
12
=− ± −
RR
54
(3.52)
2
11
Considering now
s
=
h
(i.e., the external surface of the bone layer), we obtain
{} {}{}{}
[
]
F
()
h
=
DF
()
h
(0)
+
DD
+
(3.53)
L
T
The axial stress applied at the end of the bone can be found as
1
2
c
c
u
R
c
c
ε+ σ+ λ−λ
c
c
c
c
+−
12
13
13
σ= −
c
1
c
T
(3.54)
z
13
33
z
r
1
3
11
11
11
11
The stress problem (3.54) can be solved by introducing the boundary
conditions described on the top and bottom surfaces into Equation (3.53) and
