Biomedical Engineering Reference
In-Depth Information
where
V
p
is the volume of the pores contained in a sample of bulk volume
V
b
.
Thus,
f
represents the fraction of the volume of the porous material occupied
by the pores.
The stress-strain equations (2.11) now become
σ= ε+ +ε σ=ε
2
N
EQ
,
N
,
rr
rr
r
θ
r
θ
σ= ε+ +εσ=ε
2
N
EQ
,
N
,
(2.40)
θθ
θθ
θ
z
θ
z
σ= ε+ +ε σ=ε σ=+ε
2
N
EQ
,
N
,
QER
zz
zz
rz
rz
for an isotropic poroelastic material, where
A, N, R,
and
Q
are the elastic
constants of the material, in accordance with Biot's formulation [25], and
E
and ε are the dilatations of the solid and fluid:
E
= ε
rr
+ ε
θθ
+ ε
zz
(2.41)
for solid and
()
f
()
f
()
f
ε=ε+ε+ε
θθ
(2.42)
rr
zz
for fluid, where the superscript
f
represents the related variable being
associated with fluid.
The inverse of Equation (2.40) gives the strain-stress relations as
1
Q
R
ε= +σ+σ+σ −
(1
q
)
q
(
)
(3 ),
q
+ σ
rr
rr
θθ
zz
2
N
1
2
Q
R
ε= +σ+σ+σ −
(1
q
)
q
(
)
(3 ),
q
+ σ
θθ
θθ
rr
zz
N
(2.43)
1
Q
R
ε= +σ+σ+σ −
(1
q
)
q
(
)
(3 ),
q
+ σ
zz
zz
θθ
rr
2
N
ε=σ
/,
N
ε =σ
/,
N
ε=σ
/
N
θ
z
θ
z
rz
rz
r
θ
r
θ
and
31
2
q
+
σ+σ+σ−
σ
3
Q
R
E
=
,
rr
θθ
zz
N
(2.44)
σ
(
)
2
ε=−σ+σ +σ ++
Qs
(
)
13
Qs
R
rr
θθ
zz
