Civil Engineering Reference
In-Depth Information
represented by the surface domain
m
and described in terms of its displacement vector
u
m
. One side of the screen is in contact with the poroelastic domain (i.e. surface
m
−
=
)
, and the other side supports a given load (i.e. surface
m
+
)
. The hypothesis of a
thin layer supposes that
m
and
are virtually the same. The principle of virtual work
applied to the screen leads to
t
i
δu
i
d
S
t
i
δu
i
d
S
I
m
m
ω
2
mu
i
·
δu
i
d
S
=−
−
−
=
0
(13.20)
m
−
m
+
The first term corresponds to the virtual work of the inertial forces while the last two terms
correspond to the virtual work of the exterior traction forces
(
t
+
and
t
−
)
. Combining the
weak integral formulations of the poroelastic media and the limp screen, the boundary
integrals can be rewritten
I
p
I
m
σ
ij
n
j
δu
s
i
d
S
φ(u
n
−
u
s
n
)δp
d
S
+
=
−
(13.21)
t
i
δu
i
d
S
t
i
δu
i
d
S
m
ω
2
m(u
i
δu
i
)
d
S
−
−
−
m
−
m
+
The exterior forces acting on the limp screen are written as follows
t
i
σ
ij
n
j
on
m
−
=
(13.22)
t
i
on
m
+
=
F
i
where
F
is the traction force vector applied on the septum. Furthermore, since the screen
is assumed thin and impervious, the following conditions have to be verified at the
screen - poroelastic interface
m
−
u
n
u
s
n
=
−
0
(13.23)
u
s
i
=
u
i
The first relation indicates that there is no mass flux across the boundary
,and
the second relation expresses the continuity of the displacement vectors. By considering
the hypothesis of a thin layer,
m
−
=
m
+
and using these boundary conditions, the
=
coupling term can be rewritten
I
p
I
S
ω
2
m(u
s
i
δu
s
i
)
d
S
F
i
δu
s
i
d
S
+
=−
−
(13.24)
Therefore, the effect of the impervious screen on the poroelastic medium consists in an
added mass and excitation terms. Note that when the screen separates two poroelastic
media, the discontinuity of the pressure field at the screen interface should be accounted
for explicitly in the numerical implementation.
13.4.5 Case of an imposed pressure field
In the case of an imposed pressure field
p
0
applied on
, used for instance to simu-
late an acoustic plane wave (in this case the blocked pressure is used), the boundary
