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
 
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