Civil Engineering Reference
In-Depth Information
Table 11.1
The coefficients
ij
(x
3
)
.
(a) The three first columns of [
]
ωk
t
cos
k
i
3
x
3
−
jωk
t
sin
k
i
3
x
3
ωk
t
cos
k
23
x
3
−
jωk
i
3
sin
k
i
3
x
3
ωk
i
3
cos
k
i
3
x
3
−
jωk
23
sin
k
23
x
3
−
jωk
i
3
µ
1
sin
k
i
3
x
3
ωµ
1
k
i
3
cos
k
i
3
x
3
−
jωk
23
µ
2
sin
k
23
x
3
−
D
1
cos
k
i
3
x
3
jD
1
sin
k
i
3
x
3
−
D
2
cos
k
23
x
3
2
jNk
t
k
i
3
sin
k
i
3
x
3
−
2
Nk
t
k
i
3
cos
k
i
3
x
3
2
jNk
t
k
23
sin
k
23
x
3
−
E
1
cos
k
i
3
x
3
jE
1
sin
k
i
3
x
3
−
E
2
cos
k
23
x
3
(b) The three last columns of [
]
−
jωk
t
sin
k
23
x
3
jωk
33
sin
k
33
x
3
−
ωk
33
cos
k
33
x
3
ωk
23
cos
k
23
x
3
ωk
t
cos
k
33
x
3
−
jωk
t
sin
k
33
x
3
ωµ
2
k
23
cos
k
23
x
3
ωk
t
µ
3
cos
k
33
x
3
−
jωk
t
µ
3
sin
k
33
x
3
jD
2
sin
k
23
x
3
2
jNk
33
k
t
sin
k
33
x
3
−
2
Nk
33
k
t
cos
k
33
x
3
N(k
33
−
k
t
)
cos
k
33
x
3
jN(k
33
−
k
t
)
sin
k
33
x
3
−
2
Nk
t
k
23
cos
k
23
x
3
−
jE
2
sin
k
23
x
3
0
0
weak, so that an acoustical wave propagating in the fluid phase will not exert a force
sufficient to generate vibration in the solid phase. In the rigid frame limit, the dynamic
behaviour is represented by an equivalent fluid wave equation of the form (see Section
5.7):
ρ
eq
K
eq
ω
2
p
p
+
=
0
(11.51)
K
f
/φ
is the effective bulk modulus
of the rigid frame equivalent fluid medium. Using these effective properties, the matrix
representation of the rigid frame limit is given by Equation (11.9) with the wave number
K
eq
=
where
ρ
eq
=
ρ
f
/φ
is the effective density and
given by
ω
ρ
eq
/ K
eq
.
An equivalent fluid representation can also be used for limp materials, such as aero-
nautic grade fiberglass (density of the order of 0.3 - 0.5 pcf). It is however important
to account for the inertia of the frame in the modelling of their dynamic behaviour.
The limp model can be derived from the Biot theory, assuming that the stiffness of the
frame is negligible. Various models have been proposed in the literature: Beranek (1947),
Ingard (1994), Katragada
et al
. (1995), Panneton (2007). Here, a straightforward way to
correct the equivalent fluid equation for this effect is presented. It starts from the mixed
pressure - displacement formulation derived in Appendix 6.A (Equation 6A.22):
$
div σ
s
(
u
s
)
+
ω
2
ρ
u
+
γ
grad
p
=
0
ω
2
ρ
eq
(11.52)
ω
2
ρ
eq
˜
γdiv
u
s
%
p
+
K
eq
p
−
=
0
Q/ R)
σ
s
is the stress tensor of the solid phase in vacuum,
where
γ
=
φ(ρ
12
/ ρ
22
−
ρ
eq
and K
eq
are
˜
Rin
( ρ
12
)
2
/ ρ
22
.
and
ρ
=
ρ
11
−
Coefficients
related
to
ρ
22
and
