Civil Engineering Reference
In-Depth Information
The first equation ensures the continuity of the normal stresses on
. The second
equation ensures the continuity between the acoustic displacement and the total poroelas-
tic displacement on
. The third equation refers to the continuity of the pressure across
the boundary. Substitution of Equation (13.16) into Equation. (13.15) leads to
I
p
I
a
δ(p
a
u
s
)
d
S
+
=
(13.17)
This equation shows that the poroelastic medium will be coupled to the acoustic
medium through the classical structure - cavity coupling term. In addition, the kinematic
boundary condition
p
p
a
needs to be explicitly imposed on
. Once again, in a finite
element implementation, this latter condition is performed automatically through assem-
bling.
=
13.4.3 Poroelastic - poroelastic coupling condition
Let subscripts 1 and 2 denote the primary and secondary poroelastic media, respectively.
Both media are described in terms of their solid phase displacement vector
u
and pore
fluid pressure
p
. Combining the weak integral formulations of both poroelastic media,
the boundary integrals can be rewritten
I
1
I
2
φ
1
(u
1
n
−
σ
1
ij
n
j
δu
s
1
i
d
S
u
s
1
n
)δp
1
d
S
+
=−
−
(13.18)
φ
2
(u
2
n
−
σ
2
ij
n
j
δu
s
2
i
d
S
u
s
2
n
)δp
2
d
S
+
−
The opposite signs between the two first terms and the two last terms are due to the
direction of the normal vector
n
, chosen outwards to the primary poroelastic medium.
The coupling equations at the interface
are given by Equation (11.65)
#
$
σ
1
ij
n
j
=
σ
2
ij
n
j
φ
1
(u
1
n
−
φ
2
(u
2
n
−
u
s
1
n
)
u
s
2
n
)
=
(13.19)
%
u
s
1
i
=
u
s
2
i
p
1
=
p
2
The first condition ensures the continuity of the total normal stresses. The second
equation ensures the continuity of the relative mass flux across the boundary. The two
last equations ensure the continuity of the solid phase displacement and pore fluid pressure
fields across the boundary, respectively. Using these boundary conditions, the boundary
integral reduces to
I
p
+
I
e
=
0; this equation shows that the coupling between the two
poroelastic media is natural. Only the kinematic relations
u
s
1
=
u
s
2
and
p
1
=
p
2
will
have to be explicitly imposed on
. In a finite element implementation, this may be done
automatically through assembling.
13.4.4 Poroelastic - impervious screen coupling condition
When the stiffness of the screen is important, the latter can be modelled as a thin plate.
Here, the impervious screen is assumed thin and limp with a surface density
m
.Itis
