Biomedical Engineering Reference
In-Depth Information
where
X
=
A
+
Q.
It follows from Equation (2.49) that
(2
NXxXxx QR
NXxXxx QR
NXxXxx QR
+
)
+ +=+
(
)
1
2
3
⇒===
+
++
QR
NAQ
(2
+
)
+ +=+
(
)
xxx
(2.50)
2
3
1
1
2
3
233
(2
+
)
+ +=+
(
)
3
1
2
where
x
1
,
x
2
, and
x
3
represent parameters related to the fraction of the fluid
dilatation in the
r,
θ, and
z
directions, respectively.
Making use of Equations (2.6) and (2.49), the stress-strain relation for isotropic
poroelastic bone can be written as
QR
NAQ
+
++
ε
σ+δσ=
C
ε +δ
(2. 51)
ij
ij
ijkmkmkm
233
Using an analogous approach to that of Cowin and Van Buskirk [23] and
Equation (2.51), the constitutive equation for the speed of the remodeling
surface can be written as
QR
NAQ
+
++
ε−ε
(
)
0
0
UC
=
(,)
SS S
ε
( )
−ε
( )
+δ
()
S
()
S
(2.52)
ij
ij
ij
ij
233
where
S
is used (instead of
Q
in the previous section) to represent a surface
point, to avoid confusion with constant
Q
used in this section. In cylindrical
coordinates, this expression can be written as
QR
NAQ
+
++
ε
+ε +
QR
NAQ
+
++
ε
UC
=ε +
C
Rrr
Z
zz
233
2
33
(2.53)
QR
NAQ
+
++
ε
+ε+ε+ε−
0
+ε +
C
C
C
C
C
θ θ
RZ
rz
Rr
θ
θ
θ
Zz
θ
233
where
QR
NAQ
+
++
ε
+ε +
QR
NAQ
+
++
ε
CC
0
=ε +
0
0
C
0
0
Rrr
Z
zz
233
2
33
(2.54)
+ε+ε+ε
QR
NAQ
+
++
ε
0
0
0
0
0
+ε +
C
C
C
C
θ θ
RZ
rz
Rr
θ
θ
θ
Zz
θ
233
The constitutive equation for the speed of the remodeling surface can be
written in terms of stresses as
U
=
B
R
σ
rr
+
B
Z
σ
zz
+
B
θ
σ
θθ
+
B
RZ
σ
rz
+
B
R
θ
σ
r
θ
+
B
θ
Z
σ
θ
z
+
B
σ −
C
0
(2.55)
