Biomedical Engineering Reference
In-Depth Information
where
1
∏
()
j
[]
ψ=
[
D
(
h
)]
j
jN
=
N
i
∑
∏
()
j
(
N
−
)
(
N
−
1)
()
N
()
N
(3.59)
[]
Ω
=
[
D
(
h
)] {{{
D
}
+
{
D
}} {
+
D
}
+
{
D
}}
j
L
T
L
T
i
=
2
jN
=
It can be seen that Equation (3.58) has the same structure and dimension
as in Equation (3.53). After introducing the boundary condition imposed
on the two transverse surfaces and considering Equation (3.55), the surface
displacements and/or stresses can be obtained. Introducing these solutions
back into the equations at sublayer level, the displacements, stresses, and
then strains within each sublayer can be further calculated.
3.5 Internal Surface Pressure Induced by a Medullar Pin
Prosthetic devices often employ metallic pins fitted into the medulla of a
long bone as a means of attachment. These medullar pins cause the bone in
the vicinity of the pin to change its internal structure and external shape. In
this section we introduce the model presented in references 4, 9, and 14 for
external changes in bone shape. The theory is applied here to the problem
of determining the changes in external bone shape that result from a pin
force-fitted into the medulla. The diaphysial region of a long bone is modeled
here as a hollow circular cylinder, and external changes in shape are changes
in the external and internal radii of the hollow circular cylinder.
The solution of this problem can be obtained by decomposing the problem
into two separate subproblems: the problem of the remodeling of a hollow
circular cylinder of adaptive bone material subjected to external loads, and
the problem of an isotropic solid elastic cylinder subjected to an external
pressure. These two problems are illustrated in Figure 3.2.
For an isotropic solid elastic cylinder subjected to an external pressure
p
(
t
),
the displacement in the radial direction is given by
=
−µ+λ
µλ+µ
(2 )()
2(32)
ptr
u
(3.60)
where λ and μ are Lamé's constants for an isotropic solid elastic cylinder.
In this problem we calculate the pressure of interaction
p
(
t
), which occurs
when an isotropic solid cylinder of radius
a
0
+ δ/2 is forced into a hollow
adaptive bone cylinder of radius
a
0
.
