Biomedical Engineering Reference
In-Depth Information
on the periosteal surface and
0
UB
=σ−
C
(2.20)
e
Ze z
e
on the endosteal surface, where the subscripts
p
and
e
refer to periosteal
and endosteal, respectively. Since
U
e
and
U
p
are the velocities normal to
the inner and outer surfaces of the cylinders, respectively, they are calcu-
lated as
da
at
db
at
U
=−
,
U
=
(2.21)
e
p
where the minus sign appearing in the expression for
U
e
denotes that
the outward normal of the endosteal surface is in the negative coordi-
nate direction. The stress in the axial direction of the hollow cylinder is
given by
σ=
−
π−
P
ba
(2.22)
zz
2
2
(
)
Substituting Equations (2.21) and (2.22) into Equations (2.19) and (2.20),
we have
0
1
db
dt
−
π−
−
P
ba
C
B
p
=
(2.23)
2
2
B
(
)
Zp
Zp
0
−
1
da
dt
−
π−
−
P
ba
C
B
e
Ze
=
(2.24)
2
2
B
(
)
Ze
Subtracting Equation (2.24) from Equation (2.23), we have the differential
equation
0
0
C
B
1
da
dt
1
db
dt
C
B
p
e
Ze
+
=
−
(2.25)
B
B
Ze
Zp
Zp
which has the solution
++
0
C
B
0
at
B
()
bt
B
()
C
B
a
B
b
B
p
e
0
0
+= −
t
(2.26)
Ze
Zp
Ze
Zp
Ze
Zp
where the initial size of the hollow cylinder has been used as an initial
condition. So that the size of the cylinder does not become unbounded in
