Civil Engineering Reference
In-Depth Information
where
∇
·
u
denotes the divergence of
u
∂u
1
∂x
1
+
∂u
2
∂x
2
+
∂u
3
∂x
3
∇
·
u
=
(6.5)
The 'gedanken experiments' suggested by Biot provide an evaluation of the elasticity
coefficients
P
,
N
,
Q
and
R
. These experiments are static, but they give a description
which remains valid for wavelengths large compared with the characteristic dimension
of the representative elementary volume. There are three gedanken experiments which
are described in Biot and Willis (1957).
First, the material is subjected to a pure shear
(θ
s
θ
f
=
=
0
)
.Wethenhave
σ
ij
σ
ij
=
2
Ne
s
ij
and
=
0
(6.6)
It is clear that
N
is the shear modulus of the material, and consequently the shear
modulus of the frame, since the air does not contribute to the shear restoring force.
In the second experiment, the material is surrounded by a flexible jacket that is
subjected to a hydrostatic pressure
p
1
. As shown in Figure 6.2, the pressure of the air
inside the jacket remains constant and equal to
p
o
.
This experiment provides a definition for the bulk modulus
K
b
of the frame at constant
pressure in air
K
b
=−
p
1
/θ
1
(6.7)
θ
1
being the dilatation of the frame, and
σ
11
,
σ
22
,
σ
33
being equal to
−
p
1
. For the case
of the materials studied in this topic,
K
b
is the bulk modulus of the frame in vacuum.
Equations (6.2) and (6.3) can be rewritten
Qθ
1
4
3
N)θ
1
+
−
p
1
=
(P
−
(6.8)
Rθ
1
Qθ
1
+
0
=
(6.9)
In these equations,
θ
1
is the dilatation of the air in the material, and is generally
unknown
apriori
. This dilatation is due to the variation in the porosity of the frame,
which is not directly predictable, the microscopic field of stresses in the frame for this
experiment being very complicated.
p
1
P
0
Figure 6.2
The frame of the jacketed material is subjected to a hydrostatic pressure p
1
while the pressure in the air in the jacket is equal to p
o
.
