Biomedical Engineering Reference
In-Depth Information
where
2
QAR
NA Q
−
+−
=
q
QAR
q
=
,
s
(2.45)
2
2
(2
3) 3
−
The total stress field of the bulk material, in the absence of body forces,
satisfies the equilibrium equations (2.14) provided that σ
rr
,
σ
θθ
, and σ
zz
in
Equation (2.14) are, respectively, replaced by (σ
rr
+ σ), (σ
θθ
+ σ), and (σ
zz
+ σ);
that is,
∂σ +σ
∂
(
)
1
∂σ
∂θ
+
∂σ
∂
+
σ−σ
rr
r
θ
rz
rr
θθ
+
=
0,
r
r
z
r
∂σ
∂
1
(
∂σ +σ
∂θ
)
+
∂σ
2
r
θ
θθ
θ
z
+
∂
+σ=
0,
(2.46)
r
θ
r
r
zr
∂σ
∂
1
∂σ
∂θ
+
∂σ +σ
∂
(
)1
rz
θ
z
zz
+
+σ=
0
rz
r
r
z
r
Darcy's law governing the flow of a fluid is
∂σ
∂
∂
∂
∂σ
∂θ
=
∂
∂
∂σ
∂
∂
∂
=
C
(
Uu
−
),
C
(
Uu
−
),
=
C
(
Uu
−
)
(2.47)
r
r
θ
θ
z
z
r
t
r
t
z
t
where
u
i
are the average displacements in the solid,
U
i
denote the aver-
age displacements in the fluid, and
C
is a constant that depends on the
permeability κ
m
,
the porosity
f
of the medium, and the viscosity η of the
fluid [26]:
2
=
η
κ
f
(2.48)
C
m
Since the constitutive equation (2.6) is not suitable for the poroelastic mate-
rial under consideration, Papathanasopoulou et al. rewrote Equation (2.40)
as follows:
σ+σ
σ+σ
σ+σ
σ
σ
σ
Ex
Ex
Ex
E
E
E
+ε
+ε
+ε
2
NX
+
+
X
X
000
rr
rr
1
AQ
2
NX
+
X
000
θθ
θθ
2
X
X
2
NX
+
000
zz
zz
3
=
(2.49)
0
0
0
N
0
0
θ
z
θ
z
0
0
0
0
N
0
zr
zr
0
0
0
0
0
N
r
r
θ
θ
