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Research Papers

Fundamental Solutions of Poroelastodynamics in Frequency Domain Based on Wave Decomposition

[+] Author and Article Information
Boyang Ding

Zhejiang University of Technology,
Hangzhou 310014, China
e-mail: dingboyang@zjut.edu.cn

Alexander H.-D. Cheng

University of Mississippi,
Oxford, MS 38677-1848
e-mail: acheng@olemiss.edu

Zhanglong Chen

Zhejiang University of Technology,
Hangzhou 310014, China
e-mail: chenzhanglong@yeah.net

1Corresponding author.

Manuscript received December 5, 2012; final manuscript received February 3, 2013; accepted manuscript posted February 19, 2013; published online August 21, 2013. Assoc. Editor: Younane Abousleiman.

J. Appl. Mech 80(6), 061021 (Aug 21, 2013) (12 pages) Paper No: JAM-12-1546; doi: 10.1115/1.4023692 History: Received December 05, 2012; Revised February 03, 2013; Accepted February 19, 2013

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.

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Topics: Fluids , Waves , Displacement
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References

Biot, M. A., 1941, “General Theory of Three-Dimensional Consolidation,” J. Appl. Phys., 12(2), pp. 155–164. [CrossRef]
Biot, M. A., and Willis, D. G., 1957, “The Elastic Coefficients of the Theory of Consolidation,” J. Appl. Phys., 24, pp. 594–601.
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.
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. [CrossRef]
Frenkel, J., 1944, “On the Theory of Seismic and Seismoelectric Phenomena in a Moist Soil,” J. Physics, 13(4), pp. 230–241.
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. [CrossRef]
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. [CrossRef]
Biot, M. A., 1962, “Generalized Theory of Acoustic Propagation in Porous Dissipative Media,” J. Acoust. Soc. Am., 34(5), pp. 1254–1264. [CrossRef]
Schanz, M., 2009, “Poroelastodynamics: Linear Models, Analytical Solutions, and Numerical Methods,” ASME Appl. Mech. Rev., 62(3), p. 030803. [CrossRef]
Burridge, R., and Vargas, C. A., 1979, “The Fundamental Solution in Dynamic Poroelasticity,” Geophys. J. Roy. Astron. Soc.58(1), pp. 61–90. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Nowacki, W., 1975, Dynamics Problems of Thermoelasticity, Noordhoff, Groningen, The Netherlands, pp. 456.
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.
Bonnet, G., 1987, “Basic Singular Solutions for a Poroelastic Medium in the Dynamic Range,” J. Acoust. Soc. Am., 82(5), pp. 1758–1762. [CrossRef]
Dominguez, J., 1991, “An Integral Formulation for Dynamic Poroelasticity,” ASME J. Appl. Mech., 58, pp. 588–591. [CrossRef]
Dominguez, J., 1992, “Boundary Element Approach for Dynamic Poroelastic Problems,” Int. J. Numer. Methods Eng., 35(2), pp. 307–324. [CrossRef]
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. [CrossRef]
Auriault, J. L., 1980, “Dynamic Behavior of a Porous Medium Saturated by a Newtonian Fluid.” Int. J. Eng. Sci.18(6), pp. 775–785. [CrossRef]
Auriault, J. L., Borne, L., and ChambonR., 1985, “Dynamics of Porous Saturated Media, Checking of the Generalized Law of Darcy,” J. Acoust. Soc. Am., 77(5), pp. 1641–1650. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Philippacopoulos, A. J., 1998, “Spectral Green's Dyadic for Point Sources in Poroelastic Media,” J. Eng. Mech., 124(1), pp. 24–31. [CrossRef]
Sahay, P. N., 2001, “Dynamic Green's Function for Homogeneous and Isotropic Porous Media,” Geophys. J. Int., 147(3), pp. 622–699. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Bear, J., and ChengA. H. D., 2010, Modeling Groundwater Flow and Contaminant Transport, Springer, New York, pp. 834.
Achenbach, J.D., 1973, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, pp. 439.
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.
Hörmander, L., 1969, Linear Partial Differential Operators, Springer, New York, pp. 285.
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. [CrossRef]
Cheng, A. H. D., and Predeleanu, M., 1987, “Transient Boundary Element Formulation for Linear Poroelasticity,” Appl. Math. Modell., 11(4), pp. 285–290. [CrossRef]
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. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Real (solid line) and imaginary (dashed line) part of x-component displacement of shear wave, u˜11s, due to an instantaneous point force in the x-direction, at f=10 Hz. (a) Along the x-axis, and (b) along the y-axis.

Grahic Jump Location
Fig. 2

Three dimensional plot of shear wave y-component displacement u˜21s on the x-y plane, due to an instantaneous point force in the x-direction, at f=10 Hz. (a) Real part, and (b) imaginary part.

Grahic Jump Location
Fig. 3

Contour plot of shear wave y-component displacement u˜21s on the x-y plane, due to an instantaneous point force in the x-direction, at f = 10 Hz. (a) Real part, and (b) imaginary part.

Grahic Jump Location
Fig. 4

Real (solid line) and imaginary (dashed line) part of x-component displacement of the first compressional wave, u˜11p1, due to an instantaneous point force in the x-direction, at f = 10 Hz. (a) Along the x-axis, and (b) along the y-axis.

Grahic Jump Location
Fig. 5

Three dimensional plot of first compressional wave y-component displacement u˜21p1 on the x-y plane, due to an instantaneous point force in the x-direction, at f = 10 Hz. (a) Real part, and (b) imaginary part.

Grahic Jump Location
Fig. 6

Real (solid line) and imaginary (dashed line) part of x-component displacement of the second compressional wave, u˜21p2, due to an instantaneous point force in the x-direction, at f = 10 Hz. (a) Along the x-axis, and (b) along the y-axis.

Grahic Jump Location
Fig. 7

Real (solid line) and imaginary (dashed line) part of x-component displacement of combined wave solution, u˜11Force, due to an instantaneous point force in the x-direction, at f = 10 Hz. (a) Along the x-axis, and (b) along the y-axis.

Grahic Jump Location
Fig. 8

Real (solid line) and imaginary (dashed line) part of x-component displacement, due to an instantaneous fluid source, at f = 10 Hz. (a) The first compressional wave, and (b) the second compressional wave.

Grahic Jump Location
Fig. 9

Real (solid line) and imaginary (dashed line) part of pressure, due to an instantaneous fluid source, at f = 10 Hz. (a) The first compressional wave, and (b) the second compressional wave.

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