Civil Engineering Reference
In-Depth Information
lateral boundary conditions of the tube on the impedance curve has been conducted by
Pilon
et al
. (2003) using the finite element method (see Chapter 13).
Appendix 6.A: Other representations of the Biot theory
The Biot second representation (Biot 1962)
The total stress components
σ
ij
=
σ
ij
σ
ij
+
σ
ij
−
=
φpδ
ij
and the pressure
p
are used
instead of
σ
ij
and
σ
ij
. The displacements
u
s
−
u
s
)
are used instead of the
couple
u
s
,
u
f
. The medium the frame is made of is not compressible. The stress - strain
Equation (6.3) can be replaced by
and
w
=
φ(
u
f
−
φp
=
K
f
(
div
u
s
−
ζ)
(6.A.1)
where
ζ
=−
div
w
. The stress elements of the frame in vacuum are given by
σ
ij
=
δ
ij
K
b
−
3
N
div
u
s
2
+
2
Ne
s
ij
(6.A.2)
The stress elements of the saturated frame are given by
σ
ij
=
δ
ij
K
b
−
3
N
div
u
s
−
(
1
−
φ)p
2
+
2
Ne
s
ij
(6.A.3)
and the total stress components are given by
δ
ij
K
b
−
3
N
div
u
s
p
2
σ
ij
=
+
2
Ne
s
ij
−
(6.A.4)
Equations (6.A.1) and (6.A.4) provide a simple description of the stress in a porous
medium when the bulk modulus
K
s
of the elastic solid from which the frame is made
is much larger than the other coefficients of rigidity. For instance, the third 'gedanken
experiment' can be described as follows. A variation
dθ
f
creates a variation
ξ
=
K
f
d
θ
f
. This variation is related to a variation of the
diagonal elements of the total stress which is the sum of the variation in air and in the
frame of these elements
−
φ
d
θ
f
and a variation d
p
−
=−
=−
d
p
. A general description of the second
representation, when
K
s
is not very large compared with the other rigidity coefficients, and
when the porous structure is anisotropic, was performed by Cheng (1997). A description
of the different waves with the second representation is performed in Chapter 10 for
transversally isotropic porous media.
φ
d
p
−
(
1
−
φ)
d
p
The Dazel representation (Dazel
et al.
2007)
Only the simple case where the medium the frame is made of is not compressible is
considered. The total displacement
u
t
φ
u
f
and
u
s
are used instead of
the couple
u
s
,
u
f
. The normal velocity in air at an air porous layer interface is equal to
the normal component of the total displacement. With
ς
t
φ)
u
s
=
(
1
−
+
=
∇
.
u
t
and
K
eq
=
K
f
/φ
the
pressure is given by
K
eq
ς
t
p
=−
(6.A.5)
