Biomedical Engineering Reference
In-Depth Information
4.2.4 Analytical Solution of Surface Remodeling
An analytical solution of Equation (4.26) can be obtained if a homogeneous
property is assumed for bone materials. In that case, the inhomogeneous lin-
ear differential equations system (4.26) can be converted into the following
homogeneous one:
d
dt
ε
′
d
dt
η
= ′ε+ ′η
=
BB
ε+ η
,
BB
(4.33)
1
2
1
2
by introducing two new variables such that
ε′ = ε − ε
∞
,
η′ = η − η
∞
(4.34)
1
det
1
det
ε=
(
BB
′
−
BB
′ η=
),
(
BB
′ − ′
BB
31
)
(4.35)
∞
32
32
∞
31
M
M
BB
1
2
M
=
(4.36)
BB
′
′
1
2
det
M
=
BB
′ − ′
BB
12
(4.37)
12
The solution of Equation (4.33), subject to the initial conditions that ε(0) = 0
and η(0) = 0, can be expressed in four possible forms that fulfill the physics
of the problem (i.e., when
t
→ ∞, ε and η must be limited quantities,
a
<
b,
and
the solution must be stable). The form of the solution depends on the roots of
the following quadratic equation:
2
strs
MM
−
+
det
=
0
(4.38)
where
tr
M
=+′ =+
BB ss
2
(4.39)
1
2
1
All the theoretically possible solutions are shown as follows:
Case A:
When
(
)
2
12 21
,
Equation (4.38) has two different roots,
s
1
and
s
2
, both of which are real and
distinct. Then the solutions of the equations are
BB
− ′
+
4
BB
′ >
0,
BB
+ ′ <
0
, and
BB
′ −
BB
′ >
0
1
2
21
1
2
1
(
)
(
)
−
st
−
st
ε=
−
ss
s
ε− +−ε
Be
B
s
e
,
1
2
2
∞
3
3
1
∞
1
2
(4.40)
1
(
)
(
)
−
st
−
st
η=
−
ss
s
η−′
B
e
+
B
′ −η
s
e
1
2
2
∞
3
3
1
∞
1
2
