Biomedical Engineering Reference
In-Depth Information
The stress in the axial direction is obtained by substituting Equation (2.33)
back into Equation (2.22) as
σ=
−
P
ba
e
−β
t
−β
t
π−
+σ −
(1
e
)
(2.35)
zz
0
2
2
(
)
0
0
The final stress is obtained by taking the value of the stress as
t
tends to
infinity:
σ
zz
|
t
→∞
= σ
0
(2.36)
This value of the stress indicates that the strain has returned to its reference
value
E
z
0
and that the speed of the surface remodeling has approached zero
for both endosteal and periosteal surfaces.
2.3.3 Extension to Poroelastic Bone with Fluid
On the basis of Biot's theory of consolidation, Papathanasopoulou et al. [24]
modified the theory described in Section 2.3.1 by incorporating fluid flow in
the description of the process of bone remodeling. In the modified theory,
bone was assumed to be a porous isotropic solid through the pores of
which a viscous compressible fluid flows, and the formulation described in
Section 2.3 was revised using the theory of consolidation introduced by Biot
[25]. The system of solid plus fluid is assumed to be a system with conser-
vation properties. The solid part is considered to have compressibility and
shearing rigidity and the fluid to be compressible. This section presents a
brief review of this modified surface bone remodeling theory [24].
Considering the effect of fluid flow, the stress tensor in a porous material
can be expressed as
σ=σ+δσ
ij
(2.37)
ij
ij
where δ
ij
are Kronecker's deltas, σ
ij
are the stress components applied to the
solid part, and σ represents the total normal force applied to the fluid part
of the faces of a cube of the bulk material with unit size (here, bone matrix
plus fluid). Sigma is related to the hydrostatic pressure
p
of the fluid in the
pores by
σ = −
fp
(2.38)
where
f
is the porosity defined as
f
=
V
p
/
V
b
(2.39)
