Civil Engineering Reference
In-Depth Information
φ
2
K
s
1
−
φ
−
K
b
/K
s
+
φK
s
/K
f
R
=
(6.20)
Equations (6.2) - (6.3) were replaced by Biot in a latter work (Biot 1962) by equivalent
stress - strain relations involving new definitions of stresses and strains. This second rep-
resentation of the Biot theory is given in Appendix 6.A and is used in Chapter 10 for
transversally isotropic media.
6.2.3 A simple example
Let us consider a glass wool. The glass is very stiff, compared to the glass wool itself. In
a first approximation, the volume of glass can be assumed to be constant in the second
gedanken experiment, represented in Figure 6.4. The material the frame is made of is
not compressible. This property remains valid for most of the frames of sound absorbing
porous media. A unit volume of material contains a volume (1
−
φ
)offrameat
p
1
=
0.
The same volume of frame, at
p
1
different from zero, is in a volume of porous material
equal to 1
+
θ
1
. The porosity
φ
is given by
φ
)(
1
θ
1
)
1
−
φ
=
(
1
−
+
(6.21)
The dilatation of the air in the material is due to the variation of porosity, and
θ
1
is
given by
θ
1
)
φ
(
1
+
=
φ
(6.22)
For this material, the description of the second 'gedanken experiment' is simpler than
in the general case. By using Equations (6.21) and (6.22), the following relation relating
θ
1
to
θ
1
can be obtained in the case of small dilatations
(
1
−
φ)
θ
1
θ
1
=
(6.23)
φ
Equations (6.8) and (6.9) can be rewritten
(P
−
K
b
=
1
3
N)
−
Q
(
1
−
φ)
4
(6.24)
φ
Q
=
R(
1
−
φ)/φ
(6.25)
Figure 6.4
The second 'gedanken experiment' with a glass wool. The fibres are dis-
placed, but the volume of glass is constant.
