Biomedical Engineering Reference
In-Depth Information
p
(
t
)
p
1
(
t
)
p
1
(
t
)
FIGURE 3.2
Decomposition of the medullar pin problem into two separate subproblems.
Let
a
and
b
denote the inner and outer radii, respectively, of the hollow
bone cylinder at the instant after the solid isotropic cylinder has been
forced into the hollow cylinder. Although the radii of the hollow cylinder
will actually change during the adaptation process, the deviation of these
quantities from
a
and
b
will be a small quantity negligible in small strain
t h e or y.
At an arbitrary instant in time after the two cylinders have been forced
together, the pressure of the interaction is
p
1
(
t
). The radial displacement of
the solid cylinder at its surface is
=
−µ+λ
µλ+µ
(2 )()
2(32)
pta
1
u
(3.61)
1
Using expression (3.29), the radial displacement of the bone at its inner
surface is obtained as
*
a
F
cc
c
FT Pt
ba
c
+
π−
−
()
33
12
2
0
2
*
*
*
u
=
c
β β− +
[
T
pt
( )
pt
( )]
+ϖ+
FTc
2
33
1
2
0
1
13
1
013
*
2
(
)
3
11
*
*
+
ββ −
a
[
Tpt
( )
+
pt
( )]
−
ϖ
a
0
1
(3.62)
(
c
−
c
)
c
11
12
11
As it is assumed that the two surfaces have perfect contact, the two dis-
placements have the relationship
+
δ
+=+
a
u
au
2
(3.63)
0
1
0
2
Hence, we find
δ = 2(
u
2
−
u
1
)
(3.64)
