Biomedical Engineering Reference
In-Depth Information
0.00
-0.05
-0.10
e
-0.15
-0.20
-0.25
0
10
20
30
40
50
t
(10
6
second)
FIGURE 3.8
Variation of
e
with time
t
for an inhomogeneous bone subjected to coupling loads (
p
= 4 MPa,
P
= 1500 N,
T
0
= 40, and φ
b
− φ
a
= 30V).
3.6.2 A Hollow, Inhomogeneous Circular Cylindrical
Bone Subjected to External Loads
The geometrical and material parameters of this problem are the same as those
used in the preceding cases, except that all material constants in Equation (3.70)
are now modified by a multiplier [1 - (1 - ξ)(
b
-
r
)/(
b
-
a
)], where 0 ≤ ξ ≤ 1 and
represents the percentage reduction of stiffness at the inner surface of the bone.
It is worth mentioning that by using the semianalytical approach, the form of
stiffness variation in the radial direction can be arbitrary. Figure 3.8 shows
the results of
e
at the outside surface of the bone for ξ = 1, 0.8, 0.6, and 0.4. The
external loads are
p
= 4 MPa,
P
= 1500 N, T
0
= 40, and φ
b
− φ
a
= 30 V. In general,
the remodeling rate declines as the initial stiffness of the inner bone surface
decreases. When time approaches infinity, it is observed that the stiffness
reduction in the radial direction has an insignificant effect on the remodeling
rate of the outside bone surface. This observation suggests that ignoring stiff-
ness reduction in the radial direction can yield satisfactory prediction of the
remodeling process occurring at the outside layer of the bone.
3.7 Extension to Thermomagnetoelectroelastic Solid
In the case of a thermomagnetoelectroelastic solid, the constitutive relation
(3.1) and the remodeling rate equation (3.4) must be augmented by some
additional terms related to magnetic field as follows [11]:
