Civil Engineering Reference
In-Depth Information
Equations (6.52) and (6.53) become
−
ω
2
( ρ
11
u
s
ρ
12
u
f
)
=
(P
−
N)
∇∇
·
u
s
2
u
s
+
Q
∇∇
·
u
f
+
+
N
∇
(6.54)
ω
2
( ρ
22
u
f
ρ
12
u
s
)
∇∇
·
u
f
∇∇
·
u
s
−
+
=
R
+
Q
(6.55)
where
jσφ
2
G(ω)
ω
ρ
12
=−
ρ
a
+
jσφ
2
G(ω)
ω
ρ
11
=
ρ
1
+
ρ
a
−
(6.56)
jσφ
2
G(ω)
ω
Three other formalisms of the Biot theory are presented in Appendix 6.A: (i) Biot's
second formulation (Biot 1962), (ii) the Dazel representation (Dazel
et al
. 2007) and (iii)
the mixed displacement pressure formulation (Atalla
et al
. 1998). The latter will be used
in Chapter 13 to illustrate the finite element implementation of the Biot theory.
ρ
22
=
φρ
o
+
ρ
a
−
6.5
The two compressional waves and the shear wave
6.5.1 The two compressional waves
As in the case for an elastic solid, the wave equations of the dilatational and the rotational
waves can be obtained by using scalar and vector displacement potentials, respectively.
Velocity potentials are used in Chapter 8. Two scalar potentials for the frame and the air,
ϕ
s
and
ϕ
f
, are defined for the compressional waves, giving
u
s
ϕ
s
=
∇
(6.57)
u
f
ϕ
f
=
∇
(6.58)
By using the relation
2
ϕ
2
∇∇
=
∇
∇
ϕ
(6.59)
in Equations (6.54) and (6.55), it can be shown that
ϕ
s
and
ϕ
s
are related as follows:
ω
2
( ρ
11
ϕ
s
ρ
12
ϕ
f
)
2
ϕ
s
2
ϕ
f
−
+
=
P
∇
+
Q
∇
(6.60)
−
ω
2
( ρ
22
ϕ
f
ρ
12
ϕ
s
)
=
R
∇
2
ϕ
f
2
ϕ
s
+
+
Q
∇
(6.61)
Let us denote by [
ϕ
] the vector
[
ϕ
s
,ϕ
f
]
T
[
ϕ
]
=
(6.62)
Equations (6.60) and (6.61) can then be reformulated as
ω
2
[
ρ
][
ϕ
]
=
[
M
]
∇
2
[
ϕ
]
−
(6.63)
