Biomedical Engineering Reference
In-Depth Information
directions in bone, they are oriented primarily in the direction parallel to the
axis of bone diaphysis [42,43] or the axis of the samples. As a force acting on
a collagen fiber in any direction can be divided into normal force and shear
force, it can be assumed, without loss of generality, that the mineral colla-
gen in the model is parallel to the sample axis. The model can be analyzed
using “shear lag” theory [41,44]. Based on this theory, it is assumed that the
mineral phase carries normal load only and the collagen phase carries shear
load only (Figure 9.20b). The external normal stresses σ
0
and σ
1
can be deter-
mined using traditional beam theory. According to the reciprocal theorem of
shear stress, the shear stresses τ
0
and τ
1
are equal to the corresponding shear
stresses on the cross section. The shear stresses at any point on the cross sec-
tion can also be determined using traditional beam theory.
Let
h
and
w
be the widths of the collagen and mineral components,
respectively, and the model thickness be one unit. The force equilibria of
the isolated elements (Figure 9.20c) are derived using the analysis method
reported in the literature [41]:
w
d
dx
σ
0
+τ +τ=
0
(9.23)
0
w
d
dx
σ
1
−τ−τ =
0
(9.24)
1
Let
u
0
and
u
1
be displacements of the two mineral components, respec-
tively, and assume the mineral component is linear elastic. Then,
E
du
dx
E
du
dx
0
1
σ=
,
σ=
,
τ= γ
G
(9.25)
0
1
where
E
is the elastic modulus of the mineral phase,
G
is the shear elastic
modulus, and γ is the shear strain in the collagen layer; γ is also equal to
uu
h
1
0
γ=
(9.26)
Substituting the preceding three equations into Equations (9.23) and (9.24)
and then subtracting Equation (9.23) from Equation (9.24) yields
2
wE h
d
dx
γ
−γ=τ +τ
2
1
(9.27)
m
0
2
with the following boundary conditions:
γ|
x
=∞
= a bounded value, and also
γ|
x
=0
= a bounded value.
