Biomedical Engineering Reference
In-Depth Information
QR
NAQ
+
++
ε
σ+δσ=
Ce
()
ξε+−ξδ
(1
)
(2.68)
ij
ij
ijkm
km
km
233
in which the bulk volume of the poroelastic medium has been assumed to
remain constant throughout the remodeling process, which is denoted by
the sum of coefficients ξ and 1 - ξ being equal to unity. Papathanasopoulou
et al. [29] further assumed the remodeling rate equation to be in the form of
QR
NAQ
+
++
ε
eAeAeE
=
()
+
()
+
(2.69)
1
2
233
where
A
1
(
e
) and
A
2
(
e
) are material constants dependent upon the change in
the volume fraction
e.
The system of Equations (2.13), (2.40), (2.46), (2.47), (2.65)-(2.67), (2.68), and
(2.69), together with the proper initial conditions, constitute a well-posed
initial boundary value problem. Papathanasopoulou et al. proposed the
following solution, which satisfies the equations mentioned:
uurt
=
(,),
u
=
0,
u
=
Dtz
() ,
r
r
θ
z
1
(2.70)
UUrt
=
(,),
U
=
0,
U
=
Dtz
()
r
r
θ
z
2
Then, the constitutive equations (2.46) become
NA
u
r
∂
∂
u
r
+
∂
∂
u
z
+
∂
∂
++
∂
U
r
U
r
U
z
r
r
z
r
r
z
σ= +
(2
)
+
A
Q
,
rr
∂
NA
u
r
∂
∂
u
r
+
∂
∂
u
z
∂
∂
++
∂
U
r
U
r
U
z
+
r
r
z
r
r
z
σ= +
+
(2
)
A
Q
,
θθ
∂
(2.71)
NA
u
z
∂
∂
∂
∂
u
r
u
r
+
∂
∂
++
∂
U
r
U
r
U
z
z
r
r
r
r
z
σ= +
(2
)
+
A
+
Q
,
zz
∂
σ=
∂
∂
++
∂
u
r
u
r
u
z
∂
∂
++
∂
U
r
U
r
U
z
+
r
r
z
r
r
z
Q
R
,
∂
∂
σ=σ=σ=
0
r
θ
θ
z
rz
The equilibrium equation (2.46), in this case, is simplified as
∂σ +σ
∂
(
)
+
σ−σ
rr
rr
θθ
=
0,
r
r
(2.72)
∂σ +σ
∂
(
)
zz
=
0
z
