Civil Engineering Reference
In-Depth Information
(a)
(b)
Figure 6.1
(a) Material set on a rigid floor; (b) material set between two elastic plates.
and stresses are defined as forces acting on the frame or the air per unit area of porous
material. As a consequence, the stress tensor components for air are
σ
ij
=−
φpδ
ij
(6.1)
p
being the pressure and
φ
the porosity.
The stress tensor
σ
s
at a point
M
for the frame is an average of the different local
tensors in the frame in the neighbourhood of
M
.
6.2.2 Stress - strain relations in the Biot theory: The potential
coupling term
The displacement vectors for the frame will be denoted by
u
s
. The macroscopic average
dispacement of air will be denoted by
u
f
, while the corresponding strain tensors have
elements represented by
e
s
ij
and
e
ij
.
Biot developed an elegant Lagrangian model where the stress - strain relations are
derived from a potential energy of deformation. A detailed description of the model is
given by Johnson (1986). It has been shown by Pride and Berryman (1998) that, as for
the description of the fluid-rigid frame interaction in Chapter 5, the validity of the Biot
stress - strain relations is restricted to the case where the wavelengths are much larger
than the dimensions of the volume of homogenization. The stress - strain relations in the
Biot theory are
σ
ij
=
[
(P
−
2
N)θ
s
+
Qθ
f
]
δ
ij
+
2
Ne
s
ij
(6.2)
σ
ij
(Qθ
s
Rθ
f
)δ
ij
=
(
−
φp)δ
ij
=
+
(6.3)
In these equations,
θ
s
and
θ
f
are the dilatations of the frame and of the air,
respectively. If
Q
=
0, Equation (6.2) is identical to Equation (1.78) and becomes the
stress - strain relation in elastic solids. Equation (6.3) becomes the stress - strain relation
in elastic fluids. The coefficient
Q
is a potential coupling coefficient. The two terms
Qθ
f
and
Qθ
s
give the contributions of the air dilatation to the stress in the frame, and
of the frame dilatation to the pressure variation in the air in the porous material. The
same coefficient
Q
appears in both Equations (6.2) and (6.3), because
Qθ
s
and
Qθ
f
are obtained by the derivation of a potential energy of interaction per unit volume of
material
E
PI
, which must be given in the context of a linear model by
∇
·
u
s
)(
∇
·
u
f
)
E
PI
=
Q(
(6.4)
