Geoscience Reference
In-Depth Information
1.5.4 Wave modes in the pure undrained
regime
Before we get into the realmof the dynamic Biot
-
Frenkel
theory, it is instructive to review how the two wave
modes (pressure or primary (P-)waves and shear or sec-
ondary (S-)waves) are obtained in the elastic case. In
elastic media, Hooke
'
s law can be written in two different
equivalent forms using the elements of the tensors or the
tensors themselves:
T
ij
,
j
=
λ
u
+
G u
j
,
ij
+
Gu
i
,
jj
1 160
T
ij j
=
λ
u
+
G u
i
,
jj
+
Gu
j
,
ji
1 161
After straightforward algebraic manipulations using
the properties of the Kronecker delta and the fact that
we can change the order of the derivatives, we obtain
λ
u
+
G
ρ
G
ρ
u
j
,
ji
+
u
i
,
jj
=
u
i
1 162
2
3
G
T
ij
=
K
u
−
ε
kk
δ
ij
+2
G
ε
ij
1 151
Equation (1.162) can be written in vectorial nota-
tions as
T
=
λ
u
Tr
ε
I
3
+2
G
ε
1 152
λ
u
+
G
ρ
G
ρ
2
u
=
u
∇∇
u
+
∇
1 163
where the Lamé coefficient
λ
u
is given by
2
3
G
Using the general property
λ
u
≡
K
u
−
1 153
2
u
=
∇
∇∇
u
−∇
×
∇
×
u
1 164
and
I
3
denotes the 3 × 3 identity matrix (
T
and
denote
the stress and deformation tensors). If we define the bulk
deformation as
ε
we obtain
λ
u
+2
G
ρ
G
ρ
∇∇
u
−
∇
×
∇
×
u
=
u
1 165
θ
≡∇
u
=
ε
kk
1 154
Hooke
'
s law can be written as
Now, we define the following two parameters:
T
=
λ
θ
I
+2
G
ε
1 155
c
p
=
λ
u
+2
G
ρ
u
1 166
In order to find the field equation for the displacement
of the solid phase, we need to combine Hooke
c
S
=
G
ρ
s law
(which is a constitutive equation) with a continuity equa-
tion, actually the momentum conservation equation
applied to the elastic material. This equation corresponds
to Newton
'
1 167
From Equations (1.165)
-
(1.167), we have
'
s law:
c
p
∇∇
c
S
∇
u
−
×
∇
×
u
=
u
1 168
2
u
∂
ρ
∂
∇
T
=
1 156
Equation (1.168) corresponds to the wave equation of
elasticity. To find the different wave modes, we need to
use a Helmholtz decomposition of the displacement field:
t
2
T
ij
,
j
=
ρ
u
i
1 157
where
ρ
denotes the bulk density of the material.
The divergence of the stress tensor can be computed as
follows:
u
=
∇
φ
+
∇
×
Ψ
1 169
∇ Ψ
=0
1 170
T
ij
=
λ
u
u
k
,
k
δ
ij
+
Gu
i
,
j
+
u
j
,
i
1 158
where
is a vectorial potential.
Equation (1.170) corresponds to the so-called coulomb
gauge. The Helmholtz decomposition of a vector field
φ
is a scalar potential and
Ψ
T
ij
,
j
=
λ
u
u
k
,
ki
+
Gu
i
,
jj
+
u
j
,
ij
1 159