Civil Engineering Reference
In-Depth Information
5.8
Summary of the semi-phenomenological models
The effective density
ρ
and the bulk modulus
K
can be written
ρ
=
ρ
0
α
∞
+
jωq
0
G(ω)
νφ
(5.50)
γ
−
1
γ
K
=
γP
0
/
−
(5.51)
ν
φ
jωq
0
G
(ω)
1
+
The expression for
G
in the Johnson
et al
. model is
1
+
1
/
2
2
α
∞
q
0
φ
2
jω
ν
G
j
(ω)
=
(5.52)
The expression for
G
in the simplified Lafarge model is
1
+
1
/
2
2
q
0
φ
2
jω
ν
G
j
(ω)
=
(5.53)
Under the hypothesis that the bulk modulus is similar to the one in circular
cross-sectional shaped pores, the following expression for
q
0
can be used in Equation
(5.53)
q
0
=
φ
2
/
8
(5.54)
leading to the Champoux - Allard model
1
1
/
2
4
2
jω
ν
G
j
(ω)
=
+
(5.55)
With a supplementary parameter,
G
j
in the Pride model and
G
j
in the full Lafarge
model can be replaced by
G
p
and
G
p
b
1
+
1
/
2
2
α
∞
q
0
bφ
2
jω
ν
G
p
(ω)
=
1
−
b
+
(5.56)
b
1
+
1
/
2
2
jω
ν
2
q
0
b
φ
G
p
(ω)
b
+
=
1
−
(5.57)
Dynamic tortuosities and permeabilities are related by
q(ω)
=
νφ/(jωα(ω))
(5.58)
q
(ω)
ν
φ/(jωα
(ω))
=
Effective densities and bulk moduli are related to dynamic tortuosities by
ρ(ω)
=
ρ
0
α(ω)
K(ω)
(5.59)
−
1
)/(γα
(ω))
]
=
P
0
/
[1
−
(γ
