Civil Engineering Reference
In-Depth Information
This form shows that the coupling between the two phases is volumetric and of
two natures: (i) kinetic (inertial),
(φ/α)δ(u
s
i
∂p/∂x
i
)
d
, and (ii) potential (elastic),
φ(
1
+
Q/ R)δ(pu
s
i,i
)
d
. It is shown in the next section that this new formulation
couples naturally to elastic and porous media.
13.4
Coupling conditions
Considering the generic problem depicted in Figure 13.1, the coupling conditions applied
on the bounding surface
are of four types : (i) poroelastic - elastic, (ii) poroelastic -
acoustic, (iii) poroelastic - poroelastic, and (iv) poroelastic - septum. In the weak formu-
lation, Equation (13.7), the porous medium couples to other media through the following
boundary terms:
I
p
σ
ij
n
j
δu
s
i
d
S
φ(u
n
−
u
s
n
)δp
d
S
=−
−
(13.8)
This section presents the expression of the coupling conditions for various media
interfaces together with two loading conditions, imposed surface pressure and imposed
surface displacement, using the
(u,p)
formulation. Finally, the case where the poroelastic
media are inserted in a waveguide is considered.
13.4.1 Poroelastic - elastic coupling condition
The elastic medium is described in terms of its displacement vector
u
e
.Let
e
and
e
denote its volume and its boundary. Under linear elastodynamics and harmonic oscil-
lations assumptions and using displacement as the structure variable, the weak integral
form of the structure governing equation is given by
e
[
σ
ij
δe
ij
−
ω
2
˜
u
i
δu
i
]d
e
σ
ij
n
j
δu
i
d
S
δu
i
ρ
−
=
∀
0
(13.9)
where
σ
ij
and
e
ij
are the components of the structure stress and strain tensors,
ρ
e
is the
structure density,
n
i
are the components of the outward normal vector to the surface, and
δu
i
is an arbitrary admissible variation of
u
i
. The surface integral
I
e
e
σ
ij
n
j
δu
i
d
S
=−
(13.10)
represents the virtual work done by external forces applied on the surface of the structural
domain.
When the weak formulation of the poroelastic medium is combined with that of the
elastic medium, the boundary integrals of the assembly can be rewritten
I
p
I
e
σ
ij
n
j
δu
s
i
d
S
φ(u
n
−
u
s
n
)δp
d
S
σ
ij
n
j
δu
i
d
S
+
=−
−
+
(13.11)
The positive sign of the third term is due to the direction of the normal vector
n
which is inward to the elastic medium. The coupling conditions at the interface
are
