Biomedical Engineering Reference
In-Depth Information
and
P
ba
∫
σ=⇒σ=
π−
0
ds
P
(2.59)
zz
0
zz
2
2
(
)
s
The remodeling rate equation of this problem is defined by Equation (2.7).
The corresponding stress-stress relations, strain-displacement relations,
and the equilibrium equation are, respectively, defined by Equations (2.11),
(2.13), and (2.14). Tsili assumed the solution to the preceding problem to be
in the form
u
r
=
A
(
t
)
r
+
B
(
t
)/
r
,
u
θ
= 0,
u
z
=
C
(
t
)
z
(2.60)
where
A
(
t
),
B
(
t
), and
C
(
t
) are unknown functions to be determined from the
boundary conditions (2.57)-(2.59). Substituting Equation (2.60) into Equation
(2.13), we have
ε= −
At
()
Bt
()/,
r
2
ε =+
A tBtr
()
()/,
2
rr
θθ
(2.61)
ε=
Ct
(),
ε=ε=ε=
0
zz
r
θ
rz
z
θ
The assumed strain (2.61) is substituted into the stress-strain relations
(2.11), giving
2
σ=λ+
2(
GAt
)()2
−
GBt
()/
r
+ λ
Ct
( ),
rr
2
T
T
1
2
σ=λ+
2(
GAt
)()2
+
GBt
()/
r
+ λ
Ct
( ),
θθ
2
T
T
1
(2.62)
σ=λ
2( )(
At
+ λ+
2)(),
GCt
zz
1
1
1
σ=σ=σ=
0
r
θ
θ
z
zr
Applying the boundary conditions (2.57)-(2.59), the functions
A
(
t
),
B
(
t
), and
C
(
t
) can be given by
2
πλ+ + λ
π− λ+ λ+ −λ
[
b
(
2)
G
PP
]
1
1
1
1
0
At
()
=−
]
,
2
2
2
2(
ba
)[(
G
)(
2)
G
2
T
1
1
1
22
abP
Gb
1
Bt
()
=−
)
,
(2.63)
2(
2
−
a
2
T
πλ +λ+
π− λ+ λ+ −λ
[
bP
2
(
GP
)]
11
2
T
0
Ct
()
=
2
2
2
(
ba
)[(
G
)(
2)
G
]
2
T
1
1
1
Substituting the solution of
A
(
t
),
B
(
t
), and
C
(
t
) into Equation (2.60), we obtain
the expressions of the displacement in terms of the loading
P
0
and
P
1
as
