Fundamental solutions of poroelastodynamics in the frequency domain have been derived by Cheng et al. (1991, “Integral Equation for Dynamic Poroelasticity in Frequency Domain With BEM Solution,” J. Eng. Mech., 117(5), pp. 1136–1157) for the point force and fluid source singularities in 2D and 3D, using an analogy between poroelasticity and thermoelasticity. In this paper, a formal derivation is presented based on the decomposition of a Dirac δ function into a rotational and a dilatational part. The decomposition allows the derived fundamental solutions to be separated into a shear and two compressional wave components, before they are combined. For the point force solution, each of the isolated wave components contains a term that is not present in the combined wave field; hence can be observable only if the present approach is taken. These isolated wave fields may be useful in applications where it is desirable to separate the shear and compressional wave effects. These wave fields are evaluated and plotted.

References

References
1.
Biot
,
M. A.
,
1941
, “
General Theory of Three-Dimensional Consolidation
,”
J. Appl. Phys.
,
12
(
2
), pp.
155
164
.10.1063/1.1712886
2.
Biot
,
M. A.
, and
Willis
,
D. G.
,
1957
, “
The Elastic Coefficients of the Theory of Consolidation
,”
J. Appl. Phys.
,
24
, pp.
594
601
.
3.
Detournay
,
E.
, and
Cheng
,
A. H. D.
,
1993
, “
Fundamentals of Poroelasticity
,”
Comprehensive Rock Engineering: Principles, Practice And Projects, Vol. II, Analysis and Design Method
,
C.
Fairhurst
, ed.,
Pergamon
,
New York
, pp.
113
171
.
4.
Rice
,
J. R.
, and
Cleary
,
M. P.
,
1976
, “
Some Basic Stress Diffusion Solutions for Fluid-Saturated Elastic Porous Media With Compressible Constituents
,”
Rev. Geophys.
,
14
(
2
), pp.
227
241
.10.1029/RG014i002p00227
5.
Frenkel
,
J.
,
1944
, “
On the Theory of Seismic and Seismoelectric Phenomena in a Moist Soil
,”
J. Physics
,
13
(
4
), pp.
230
241
.
6.
Biot
,
M. A.
,
1956a
, “
Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. 1. Low-Frequency Range
,”
J. Acoust. Soc. Am.
,
28
, pp.
168
178
.10.1121/1.1908239
7.
Biot
,
M. A.
,
1956b
, “
Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. 2. Higher Frequency Range
,”
J. Acoust. Soc. Am.
,
28
, pp.
179
191
.10.1121/1.1908241
8.
Biot
,
M. A.
,
1962
, “
Generalized Theory of Acoustic Propagation in Porous Dissipative Media
,”
J. Acoust. Soc. Am.
,
34
(
5
), pp.
1254
1264
.10.1121/1.1918315
9.
Schanz
,
M.
,
2009
, “
Poroelastodynamics: Linear Models, Analytical Solutions, and Numerical Methods
,”
ASME Appl. Mech. Rev.
,
62
(
3
), p.
030803
.10.1115/1.3090831
10.
Burridge
,
R.
, and
Vargas
,
C. A.
,
1979
, “
The Fundamental Solution in Dynamic Poroelasticity
,”
Geophys. J. Roy. Astron. Soc.
58
(
1
), pp.
61
90
.10.1111/j.1365-246X.1979.tb01010.x
11.
Norris
,
A. N.
,
1985
, “
Radiation From a Point Source and Scattering Theory in a Fluid Saturated Porous Solid
,”
J. Acoust. Soc. Am.
,
77
, pp.
2012
2023
.10.1121/1.391773
12.
Manolis
,
G. D.
, and
Beskos
,
D. E.
(
1989
), “
Integral Formulation and Fundamental Solutions of Dynamic Poroelasticity and Thermoelasticity
,”
Acta Mech.
,
76
(
1–2
), pp.
89
104
.10.1007/BF01175798
13.
Manolis
,
G. D.
, and
Beskos
,
D. E.
,
1990
, “
Errata in Integral Formulation and Fundamental Solutions of Dynamic Poroelasticity and Thermoelasticity
,”
Acta Mech.
,
83
(
3–4
), pp.
223
226
.10.1007/BF01172983
14.
Cheng
,
A. H. D.
,
Badmus
,
T.
, and
Beskos
,
D. E.
,
1991
, “
Integral Equation for Dynamic Poroelasticity in Frequency Domain With BEM Solution
,”
J. Eng. Mech.
,
117
(
5
), pp.
1136
1157
.10.1061/(ASCE)0733-9399(1991)117:5(1136)
15.
Nowacki
,
W.
,
1975
,
Dynamics Problems of Thermoelasticity
,
Noordhoff
,
Groningen
, The Netherlands, pp.
456
.
16.
Kupradze
,
V. D.
,
Gezelia
,
T. G.
,
Basheleishvili
,
M. O.
, and
Burchuladze
,
T. V.
,
1979
,
Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity
,
North-Holland
,
Amsterdam
.
17.
Bonnet
,
G.
,
1987
, “
Basic Singular Solutions for a Poroelastic Medium in the Dynamic Range
,”
J. Acoust. Soc. Am.
,
82
(
5
), pp.
1758
1762
.10.1121/1.395169
18.
Dominguez
,
J.
,
1991
, “
An Integral Formulation for Dynamic Poroelasticity
,”
ASME J. Appl. Mech.
,
58
, pp.
588
591
.10.1115/1.2897229
19.
Dominguez
,
J.
,
1992
, “
Boundary Element Approach for Dynamic Poroelastic Problems
,”
Int. J. Numer. Methods Eng.
,
35
(
2
), pp.
307
324
.10.1002/nme.1620350206
20.
Boutin
,
C.
,
Bonnet
,
G.
, and
Bard
,
P. Y.
,
1987
, “
Green Functions and Associated Sources in Infinite and Stratified Poroelastic Media
,”
Geophys. J. R. Astron. Soc.
,
90
(
3
), pp.
521
550
.10.1111/j.1365-246X.1987.tb00741.x
21.
Auriault
,
J. L.
,
1980
, “
Dynamic Behavior of a Porous Medium Saturated by a Newtonian Fluid
.”
Int. J. Eng. Sci.
18
(
6
), pp.
775
785
.10.1016/0020-7225(80)90025-7
22.
Auriault
,
J. L.
,
Borne
,
L.
, and
Chambon
R.
,
1985
, “
Dynamics of Porous Saturated Media, Checking of the Generalized Law of Darcy
,”
J. Acoust. Soc. Am.
,
77
(
5
), pp.
1641
1650
.10.1121/1.391962
23.
Chen
,
J.
,
1994a
, “
Time Domain Fundamental Solution to Biot's Complete Equations of Dynamic Poroelasticity. Part I: Two-Dimensional Solution
.”
Int. J. Solids Struct.
,
31
(
10
), pp.
1447
1490
.10.1016/0020-7683(94)90186-4
24.
Chen.
J.
,
1994b
, “
Time Domain Fundamental Solution to Biot's Complete Equations of Dynamic Poroelasticity. Part II: Three-Dimensional Solution
,”
Int. J. Solids Struct.
,
31
(
2
), pp.
169
202
.10.1016/0020-7683(94)90049-3
25.
Kaynia
,
A. M.
, and
Banerjee
,
P. K.
,
1993
, “
Fundamental Solutions of Biot's Equation of Dynamic Poroelasticity
,”
Int. J. Eng. Sci.
,
31
(
5
), pp.
817
830
.10.1016/0020-7225(93)90126-F
26.
Zimmerman
,
C.
, and
Stern
,
M.
,
1993
, “
Boundary Element Solution of 3-D Wave Scatter Problems in a Poroelastic Medium
,”
Eng. Anal. Boundary Elements
,
12
(
4
), pp.
223
240
.10.1016/0955-7997(93)90050-U
27.
Philippacopoulos
,
A. J.
,
1998
, “
Spectral Green's Dyadic for Point Sources in Poroelastic Media
,”
J. Eng. Mech.
,
124
(
1
), pp.
24
31
.10.1061/(ASCE)0733-9399(1998)124:1(24)
28.
Sahay
,
P. N.
,
2001
, “
Dynamic Green's Function for Homogeneous and Isotropic Porous Media
,”
Geophys. J. Int.
,
147
(
3
), pp.
622
699
.10.1046/j.1365-246x.2001.01562.x
29.
Ding
,
B. Y.
, and
Yuan
,
J. H.
,
2011
, “
Dynamic Green's Functions of a Two-Phase Saturated Medium Subjected to a Concentrated Force
,”
Int. J. Solids Struct.
,
48
, pp.
2288
2303
.10.1016/j.ijsolstr.2011.04.006
30.
Halpern
,
M. R.
, and
Christiano
,
P.
,
1986
, “
Response of Poroelastic Halfspace to Steady-State Harmonic Surface Tractions
,”
Int. J. Numer. Analyt. Meth. Geomech.
,
10
(
6
), pp.
609
632
.10.1002/nag.1610100605
31.
Senjuntichai
,
T.
, and
Rajapakse
,
R. K. N. D.
,
1994
, “
Dynamic Green's Functions of Homogeneous Poroelastic Half-Plane
,”
J. Eng. Mech.
,
120
(
11
), pp.
2381
2464
.10.1061/(ASCE)0733-9399(1994)120:11(2381)
32.
Bear
,
J.
, and
Cheng
A. H. D.
,
2010
,
Modeling Groundwater Flow and Contaminant Transport
,
Springer
,
New York
, pp.
834
.
33.
Achenbach
,
J.D.
,
1973
,
Wave Propagation in Elastic Solids
,
North-Holland
,
Amsterdam
, pp.
439
.
34.
Bonnet
,
G.
, and
Auriault
,
J. L.
,
1985
, “
Dynamics of Saturated and Deformable Porous Media: Homogenization Theory and Determination of the Solid-Liquid Coupling Coefficients
,”
Physics of Finely Divided Matter, Proc. Winter School
,
N.
Boccara
and
Z. M.
Daoud
, eds.,
Springer-Verlag
,
New York
, pp.
306
316
.
35.
Hörmander
,
L.
,
1969
,
Linear Partial Differential Operators
,
Springer
,
New York
, pp.
285
.
36.
Yew
,
C. H.
, and
Jogi
,
P. N.
,
1978
, “
Determination of Biot's Parameters for Sandstones, 1. Static Tests
,”
Exp. Mech.
,
18
(
5
), pp.
167
172
.10.1007/BF02324137
37.
Cheng
,
A. H. D.
, and
Predeleanu
,
M.
,
1987
, “
Transient Boundary Element Formulation for Linear Poroelasticity
,”
Appl. Math. Modell.
,
11
(
4
), pp.
285
290
.10.1016/0307-904X(87)90144-2
38.
Cheng
,
A. H. D.
, and
Detournay
,
E.
,
1998
, “
On Singular Integral Equations and Fundamental Solutions of Poroelasticity
,”
Int. J. Solids Struct.
,
35
(
34-35
), pp.
4521
4555
.10.1016/S0020-7683(98)00082-1
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