Civil Engineering Reference
In-Depth Information
frame stress tensor by the transmission of the pressure into the frame. The components
[1
φ)/φ
]
σ
ij
pδ
ij
are added to the stress tensor of the frame in vacuum to
obtain the total stress tensor. The frame displacement vector
u
s
+
(
1
−
=−
and the fluid-discharge
φ(
u
f
u
s
)
where
u
f
displacement vector
w
=
−
is the fluid displacement vector are used
instead of
u
s
and
u
f
.Using
ζ
=−
∇
.
w
, the total stress components and the pressure are
given by
θ
s
−
ζ
σ
xx
=
(
2
G
+
A)e
xx
+
Ae
yy
+
Fe
zz
+
K
f
(10.14)
φ
θ
s
−
ζ
σ
yy
=
Ae
xx
+
(
2
G
+
A)e
yy
+
Fe
zz
+
K
f
(10.15)
φ
θ
s
−
ζ
σ
zz
=
Fe
xx
+
Fe
yy
+
Ce
zz
+
K
f
(10.16)
φ
σ
yz
=
2
G
e
yz
(10.17)
σ
xz
=
2
G
e
xz
(10.18)
σ
xy
=
2Ge
xy
(10.19)
K
f
(θ
s
−
ζ)
φ
−
p
=
(10.20)
where
K
f
is the bulk modulus of the saturating air. Equations (10.14) - (10.20) can be
rewritten
σ
xx
=
(
2
B
1
+
B
2
)e
xx
+
B
2
e
yy
+
B
3
e
zz
+
B
6
ζ
(10.21)
σ
yy
=
B
2
e
xx
+
(
2
B
1
+
B
2
)e
yy
+
B
3
e
zz
+
B
6
ζ
(10.22)
σ
zz
=
B
3
e
xx
+
B
3
e
yy
+
B
4
e
zz
+
B
7
ζ
(10.23)
σ
xz
=
2
B
5
e
xz
(10.24)
σ
yz
=
2
B
5
e
yz
(10.25)
σ
xy
=
2
B
1
e
xy
(10.26)
p
=
B
6
e
xx
+
B
6
e
yy
+
B
7
e
zz
+
B
8
ζ
(10.27)
where
B
1
=
G,B
2
=
A
+
K
f
/φ,B
3
=
F
+
K
f
/φ,B
4
=
C
+
K
f
/φ,
B
5
=
G
,B
6
=
B
7
=−
B
8
=−
K
f
/φ.
10.3.2 Wave equations
It has been shown by Sanchez-Palencia (1980) that, for anisotropic porous media, the
dynamic viscous permeability (or the dynamic tortuosity) are symmetrical tensors of the
second order. This is also true for the different parameters that describe the visco-inertial
interactions between both phases. These tensors are diagonal in three orthogonal directions.
