Civil Engineering Reference
In-Depth Information
Using the second equation in (6.A.13), the displacement vector of the fluid phase
u
f
is expressed in terms of the pressure
p
in the pores and in terms of the displacement
vector of the solid phase particle
u
s
:
φ
ρ
22
ω
2
grad
p
ρ
12
ρ
22
u
s
u
f
=
−
(6.A.14)
Using Equation (6.A.14), the first equation in (6.A.13) transforms into:
+
φ
ρ
12
ω
2
ρ
u
s
ρ
22
grad
p
+
div
σ
s
=
0
(6.A.15)
where the following effective density is introduced:
( ρ
12
)
2
ρ
22
ρ
=
ρ
11
−
(6.A.16)
Equation (6.A.15) is still dependent on the fluid phase displacement
u
f
because of the
dependency
σ
s
σ
s
(
u
s
,
u
f
)
. To eliminate this dependency, Equations (6.2) and (6.3),
are combined to obtain:
=
σ
ij
(
u
s
)
-
φ
Q
σ
ij
(
u
s
)
=
R
pδ
ij
(6.A.17)
with
σ
s
the stress of the frame in vacuum defined in Equation (6.A.2). Note that tilde
is used here to account for damping and possible frequency dependence of the elastic
coefficients
Q
and
R
(e.g. polymeric frame).
Equation (6.A.17) is next used to eliminate the dependency
σ
s
σ
s
(
u
s
,
u
f
)
in
Equation (6.A.15). This leads to the solid phase equation in terms of the
(
u
s
,p)
variables:
=
div
σ
s
(
u
s
)
+
ρω
2
u
s
+
γ
grad
p
=
0
(6.A.18)
with:
φ
ρ
12
Q
R
γ
=
ρ
22
−
(6.A.19)
In the case where
K
b
/K
s
1, Equation (6.A.19) reduces to (6.A.8).
Next, to derive the fluid phase equation in terms of
(
u
s
,p)
variables, the divergence
of Equation (6.A.14) is taken:
φ
ω
2
ρ
22
p
ρ
12
div
u
f
ρ
22
div
u
s
=
−
(6.A.20)
Combining this equation with the second equation in (6.A.13), the fluid phase equation
is obtained in terms of the
(
u
s
,p)
variables:
ρ
22
ρ
22
φ
2
R
ω
2
p
+
γω
2
div
u
s
p
+
=
0
(6.A.21)
