Civil Engineering Reference
In-Depth Information
This equation is the classical equivalent fluid equation for absorbing media with a
source term. The first two terms of this equation may be obtained directly from Biot's
equations in the limit of a rigid skeleton.
Grouping Equations (6.A.18) and (6.A.21), the Biot poroelasticity equations in terms
of
(
u
s
,p)
variables are given by:
$
div
σ
s
(
u
s
)
ρω
2
u
s
+
+
γ
grad
p
=
0
ρ
22
R
ω
2
p
+
ρ
22
φ
2
(6.A.22)
γω
2
div
u
s
%
p
+
=
0
This system exhibits the classical form of a fluid-structure coupled equation. However,
the coupling is of a volume nature since the poroelastic material is a superposition in space
and time of the elastic and fluid phases. The first two terms of the structure equation
represent the dynamic behaviour of the material in vacuum, while the first two terms
of the fluid equation represent the dynamic behaviour of the fluid when the frame is
supposed motionless. The third terms in both equations couple the dynamics of the two
phases. It is shown in Chapter 13 that this formalism leads to a simple weak formulation
for finite element based numerical implementations. An example of the application of
this formalism to the optimization of the surface impedance of a porous material is given
in Kanfoud and Hamdi (2009).
References
Allard, J.F., Depollier, C., Guignouard, P. and Rebillard, P. (1991) Effect of a resonance of the
frame on the surface impedance of glass wool of high density and stiffness.
J. Acoust. Soc.
Amer
.,
89
, 999 - 1001.
Atalla N., Panneton, R. and Debergue, P. (1998) A mixed displacement pressure formulation for
poroelastic materials.
J. Acoust. Soc. Amer
.,
104
, 1444 - 1452.
Beranek, L. (1947) Acoustical properties of homogeneous isotropic rigid tiles and flexible blankets.
J. Acoust. Soc. Amer
.,
19
, 556 - 68.
Biot, M.A. (1956) The theory of propagation of elastic waves in a fluid-saturated porous solid. I.
Low frequency range. II. Higher frequency range.
J. Acoust. Soc. Amer
.,
28
, 168 - 91.
Biot, M.A. and Willis, D.G. (1957) The elastic coefficients of the theory of consolidation.
J. Appl.
Mechanics
,
24
, 594 - 601.
Biot, M.A. (1962) Generalized theory of acoustic propagation in porous dissipative media.
J.
Acoust. Soc. Amer
.,
34
, 1254 -1264.
Brown, R.J.S. and Korringa, J. (1975), On the dependence of the elastic properties of a porous
rock on the compressibility of the pore fluid.
Geophysics
,
40
, 608 -16.
Cheng, A.H.D. (1997) Material coefficients of anisotropic poroelasticity.
Int. J. Rock Mech. Min.
Sci
.
34
, 199 -205.
Dahl, M.D., Rice, E.J. and Groesbeck, D.E. (1990) Effects of fiber motion on the acoustical
behaviour of an anisotropic, flexible fibrous material.
J. Acoust. Soc. Amer
.,
87
, 54 - 66.
Dazel, O., Brouard, B., Depollier., C. and Griffiths. S. (2007) An alternative Biot's displacement
formulation for porous materials.
J. Acoust. Soc. Amer
.,
121
, 3509 - 3516.
Depollier, C., Allard, J.F. and Lauriks, W. (1988) Biot theory and stress - strain equations in porous
sound absorbing materials.
J. Acoust. Soc. Amer
.,
84
, 2277 -9.
Gorog S., Panneton, R. and Atalla, N. (1997) Mixed displacement - pressure formulation for acous-
tic anisotropic open porous media.
J. Applied Physics
,
82
(9), 4192 - 4196.
