Civil Engineering Reference
In-Depth Information
in Biot (1956) supplies relations between
ρ
o
,
ρ
1
, and these coefficients. If the frame and
the air move together at the same velocity,
u
s
u
f
=
(6.35)
there is no interaction between the frame and the air, and the macroscopic velocity
u
f
is identical to the microscopic velocity. The porous material moves as a whole, and the
kinetic energy is given by
1
u
s
2
E
c
=
2
(ρ
1
+
φρ
o
)
|
|
(6.36)
A comparison of Equations (6.36) and (6.32) yields
ρ
11
+
2
ρ
12
+
ρ
22
=
ρ
1
+
φρ
o
(6.37)
The components
q
i
of the inertial force per unit volume of material are given by
φρ
o
∂
2
u
i
∂t
2
q
i
=
(6.38)
the microscopic velocity of the air being
u
f
. A comparison of Equations (6.38) and (6.34)
yields
φρ
o
=
ρ
22
+
ρ
12
(6.39)
and thus
ρ
1
is given by
ρ
1
=
ρ
11
+
ρ
12
(6.40)
This description can be related to the case of materials having a rigid frame. If the
frame does not move, Equation (6.34) becomes
q
f
ρ
22
u
f
=
(6.41)
with
ρ
22
=
φρ
o
−
ρ
12
(6.42)
The quantity
q
f
is the inertial force acting on a mass
φρ
0
of nonviscous fluid, the
comparison of Equations (6.41) and (5.19), or of Equations (6.32) and (5.22), yields
ρ
22
u
f
α
∞
φρ
o
u
f
=
(6.43)
From Equations (6.42) and (6.43), the quantity
ρ
12
can be rewritten
ρ
12
=−
φρ
o
(α
∞
−
1
)
(6.44)
This quantity is the opposite of the inertial coupling term
ρ
a
previously defined by
Equation (4.146)
−
ρ
12
=
ρ
a
(6.45)
