Civil Engineering Reference
In-Depth Information
poroelastic wave propagation are assumed. Also, the air contained in the porous medium
is at rest. The presentation will be limited to time harmonic behaviour
(
exp
(jωt))
.
13.2
Displacement based formulations
The coupled poroelasticity equations (6.54) - (6.56) can be rewritten in the compact form
div σ
s
(
u
s
,
u
f
)
ω
2
ρ
11
u
s
ω
2
ρ
12
u
f
+
+
=
0
(13.1)
div σ
f
(
u
s
,
u
f
)
ω
2
ρ
22
u
f
ω
2
ρ
12
u
s
+
+
=
0
Using the solid and fluid displacement vectors
(
u
s
,
u
f
)
as primary variables, the weak
integral form of the poroelasticity equations, reads
ρ
12
ω
2
u
i
δu
s
i
)
d
(σ
ij
(
u
s
,
u
f
)e
s
ij
(δ
u
s
)
ρ
11
ω
2
u
s
i
δu
s
i
−
−
−
σ
ij
(
u
s
,
u
f
)n
j
δu
s
i
d
S
=
0
(σ
ij
(
u
s
,
u
f
)e
ij
(δ
u
f
)
ρ
22
ω
2
u
i
δu
i
−
ρ
12
ω
2
u
s
i
δu
i
)
d
−
−
(13.2)
σ
ij
(
u
s
,
u
f
)n
j
δu
i
d
S
=
0
(δ
u
s
,δ
u
f
)
∀
where
δ
u
s
and
δ
u
f
denote admissible variations of
u
s
and
u
f
, respectively, and where
and
denote the medium domain and its boundary (Figure. 13.1). For the finite
element implementation of Equation (13.2), an analogy with three-dimensional elastic
solid elements is used; however, this time, six degrees of freedom per node are necessary:
three displacement components of the solid phase and three displacement components of
the fluid phase. Moreover, because of the viscous and thermal dissipation mechanisms,
the system matrices are frequency dependent. In consequence, for large 3-D multilayer
Elastic
domain
Supports
Excitations
u
i
or p
i
Γ
Ω
Primary
poroelastic domain
Septum
Acoustic
domain
Secondary
poroelastic
domain
Figure 13.1
A poroelastic domain with typical boundary and loading conditions.
