Biomedical Engineering Reference
In-Depth Information
u
z
=
zC
(
t
) +
D
(
t
)ln(
r
/
a
),
φ =
F
(
t
)ln(
r
/
a
) + φ
a
(3.12)
where
A, B, C, D,
and
F
are unknown variables to be determined by introduc-
ing boundary conditions, and
ϖ=
λ
T
ba
10
. Substituting Equations (3.11)
2ln(
/)
and (3.12) into Equation (3.2) and later into Equation (3.1), we obtain
Bt
r
()
(
c
c
r
a
r
a
−
(3.13)
12
σ= +− −+ +ϖ
At cc
()(
)
cc cCt
)
()
ln
−
1ln
rr
11
12
11
12
13
2
11
Bt
r
()
(
c
c
r
a
r
a
12
σ= ++ −+ +ϖ −−
At c
()(
c
)
c
c
)
cCt
()
ln
ln
1
(3.14)
θθ
11
12
11
12
13
2
11
c
c
ln(/)
ln(/)
ra
ba
[
]
13
σ= +
2()
Atc
c
Ct
( )
+ ϖ
2 ln(/)1
ra
− −λ
T
(3.15)
zz
13
33
30
11
1
[
1
[
σ=
cDteFt
( )
+
( )],
D
=
eDt
( )
−κ
( )]
t
(3.16)
zr
44
15
r
15
1
r
r
e
c
ln(/)
ln(/)
ra
ba
31
(3.17)
DAte
=
2()
+
Cte
( )
+ϖ
[2ln( /) 1]
ra
−−χ
T
z
31
33
30
11
The boundary conditions (3.5) and (3.6) of stresses and electric potential
require that
c
44
D
(
t
) +
d
4
F
(
t
) = 0,
φ
b
=
F
(
t
)ln(
b
/
a
) + φ
a
(3.18)
Bt
a
()
(
c
c
12
At c
()(
+− −+ −ϖ=
c
)
c
c
)
cCt
()
0
(3.19)
11
12
11
12
13
2
11
Bt
b
()
(
c
c
b
a
−
b
a
=−
(3.20)
12
11
At cc
()(
+− −+ +ϖ
)
cc cCt
)
()
ln
−
1ln
p
11
12
11
12
13
2
2
2
*
*
π−
(
ba Atc
)[2
()
+
Ctc
()
−
FT
]
+
FT
= −
P
(3.21)
13
33
1
0
2
0
where
1
ln(/)
c
λ
−
λ
c
c
*
13
1
3
*
2
13
11
F
=
,
Fb
=π
λ−λ
(3.22)
1
2
1
3
ba
c
2
11
The unknown functions
A
(
t
),
B
(
t
),
C
(
t
),
D
(
t
), and
F
(
t
) are readily found from
Equations (3.18)-(3.21) as
*
1
cc
c
FT Pt
ba
c
+
π−
−
()
(3.23)
*
*
33
12
11
2
0
2
*
At
()
=
c
β β+ +ϖ
[
T
pt
( )]
+
FTc
33
1
2
0
13
1
013
F
*
(
2
)
3
