Civil Engineering Reference
In-Depth Information
Figure 6.3
A nonjacketed material subjected to an increase of pressure.
In the third experiment, represented in Figure 6.3, the material is nonjacketed, and is
subjected to an increase of pressure
p
1
in air. This variation in pressure is transmitted to
the frame, and the components of the stress tensor for the frame become
τ
ij
=−
p
f
(
1
−
φ)δ
ij
(6.10)
Equations (6.2) and (6.3) can be rewritten
Qθ
2
4
3
N)θ
2
+
−
p
f
(
1
−
φ)
=
(P
−
(6.11)
Rθ
2
Qθ
2
+
−
φp
f
=
(6.12)
In these equations,
θ
2
and
θ
2
are the dilatations of the frame and the air, respectively.
The quantity
−
p
f
/θ
2
, which will be denoted by
K
s
, is the bulk modulus of the elastic
solid from which the frame is made
p
f
/θ
2
K
s
=−
(6.13)
In this last experiment there is no variation of the porosity, the deformation of the
frame is the same as if the material were not porous, and can be associated with a simple
change of scale.
The quantity
−
p
f
/θ
2
is the bulk modulus
K
f
of the air
p
f
/θ
2
K
f
=−
(6.14)
From Equations (6.7) -(6.14), a system of three equations containing the three
unknown parameters
P
,
Q
and
R
can be written
Q/K
s
+
R/K
f
=
φ
(6.15)
4
(P
−
3
N)/K
s
+
Q/K
f
=
1
−
φ
(6.16)
(P
K
b
=
1
Q
2
R
4
3
N)
−
−
(6.17)
The elastic coefficients
P
,
Q
and
R
, calculated by using Equations (6.15) - (6.17) are
given by
(
1
−
φ)
1
−
φ
−
K
s
+
φ
K
s
K
b
K
s
K
f
K
b
4
3
N
P
=
+
(6.18)
1
−
φ
−
K
b
/K
s
+
φK
s
/K
f
[1
−
φ
−
K
b
/K
s
]
φK
s
Q
=
(6.19)
1
−
φ
−
K
b
/K
s
+
φK
s
/K
f
