Biomedical Engineering Reference
In-Depth Information
time, it is necessary for the first term on the right-hand side of Equation (2.26)
to vanish; thus, a constant, say σ
0
, can be defined as
0
0
C
B
C
B
p
e
σ= =
(2.27)
0
Ze
Zp
To obtain an explicit and readily understood answer, Cowin and Van
Buskirk assumed that changes in the radii are small in terms of initial radius
a
0
.
Specifically, they assumed that the squares in the quantities ε and η, where
a
a
b
b
ε= − =−
1,
1
(2.28)
0
0
are negligible. The definitions (2.28) and the definition (2.27) are substituted
into Equation (2.26), leading to
a
B
b
B
0
0
ε
()
t
+ η
()
t
=
0
(2.29)
Ze
Zp
When Equations (2.28), (2.29), and the assumed smallness in the change of
radii are employed in Equation (2.23), it takes the form
d
dt
η
=α−β η
b
B
b
(2.30)
0
Zp
0
where
P
ba
α=
π−
+σ
0
(2. 31)
2
2
(
)
0
0
β=
−
2(
PB aBb
ba
+
π−
)
Ze
0
Zp
0
(2.32)
2
22
(
)
0
0
The solutions to Equations (2.23) and (2.24) are easily obtained with this
approximation as
=−
α
β
+
α
β
−β
t
−β
t
at
()
a
B e
(1
−
),
b tb
()
=
B
(1
−
e
)
(2.33)
0
Ze
0
Zp
The final radii of the cylinder are determined from Equation (2.33) as the
limiting values of
a
and
b,
as
t
tends to infinity,
=−
α
β
=+
α
β
aa
Bb
,
b
B
(2.34)
∞
0
Ze
∞
0
Zp
