Civil Engineering Reference
In-Depth Information
This provides the following power balance equation:
elas
+
iner
+
s
elas
+
s
iner
+
fs
exc
=
0
+
(13.55)
coup
where
s
elas
,
s
iner
represent the power developed by the internal and inertia forces in
the solid phase in vacuum, respectively;
elas
,
iner
represent the power developed by
the internal and inertia forces in the interstitial fluid, respectively;
fs
coup
represents the
power exchanged between the two phases; finally
exc
represents the power developed
by external loading.
The time-averaged power dissipated within the porous medium can be subdivided into
contributions from powers dissipated through structural damping in the skeleton, viscous
and thermal effects:
diss
=
t
diss
. The time-averaged power dissipated
through structural damping is obtained from the imaginary part of
s
elas
s
diss
+
diss
+
2
Im
ω
σ
ij
(
u
s
)
:
s
(
u
s
∗
)
d
1
s
diss
=
(13.56)
iner
+
The power dissipated through viscous effects is obtained from
s
iner
+
fs
coup
:
ω
3
Im
φ
2
αρ
0
ω
2
∂p
∂p
∗
∂x
i
d
1
2
)u
s
i
u
s
i
d
diss
=−
Im
(
ρ
−
∂x
i
2
Im
φ
α
Re
∂p
∂x
i
u
s
i
d
+
(13.57)
Note that in deriving the above equation, the fact that
Im
Q
R
=
0
for all materials has been used. Finally, the power dissipated through thermal effects is
obtained from
elas
2
ω
Im
φ
2
R
pp
∗
d
1
t
diss
=−
(13.58)
Section 13.9.3 presents an example illustrating the use of these expressions.
13.8
Radiation conditions
Numerical approaches based on the formulations described provide efficient tools to solve
problems where the porous material is coupled to elastic structures and finite extent acous-
tic cavities. When the porous domain is in contact with free air, the surface impedance
should be calculated accurately to account for the coupling between the porous medium
and air medium. When the air medium is bounded this is done easily through the coupling
conditions. The difficulty arises when the porous domain is in contact with an infinite
domain. The acoustic radiation of a porous medium into an unbounded fluid medium has
