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The dynamic response of fluid filled porous materials is of fundamental importance to several disciplines like geophysics, acoustics and geotechnical engineering. Generally due to the presence of a pore-fluid, the dynamic response of fluid saturated poroelastic media is significantly different from that of a non-porous elastic solid. The first consistent theory concerning the dynamic response of a fluid-saturated porous medium was established by Biot [4 and 5]. In the following years the equations governing the macroscopic behavior of a porous medium also obtained by means of the homogenization (Burridge and Keller [7] and Auriault et al. [8]) and volume averaging methods (Pride et al. [9]). These methods provide a procedure for deriving macroscale dynamic equations starting from the equations governing the mechanical behavior of the medium at the microscale constituents’ level. By assumptions made regarding the permeability of the medium or the compressibility of the constituents different modified formulations have also been developed based on Biot’s original theory (Shanz and Pryl [57] and Zienkiewicz et al. [59 and 60]). In a case for example to reduce the number of unknowns involved in the Biot’s original equations a modified formulation known generally as the simplified u-p formulation is established by simply neglecting the relative inertial effects of the pore fluid flow in the medium. Despite alternative formulations proposed over the past six decades, the theory of poroelasticity founded by Biot has been the most widely accepted and employed one in description of the dynamic behavior of fluid-saturated porous materials.

An extensive literature review on different dynamic models of poroelasticity and the solutions given for their relevant problems can be found in Schanz [10]. Burridge and Vargas [11], Norris [12], Bonnet [13], Cheng et al. [14] and Philippacopoulos [15] are among the authors who investigated the fundamental solutions for a fluid saturated poroelastic medium.

Paul [16 and 17] may be the first who considered a half space problem in a fluid saturated poroelastic medium. By assuming a non-dissipative behavior, Paul [16 and 17] presented integral solutions for respectively two-dimensional plane strain and axisymmetric problems involving an impulsive line or a circular uniform load applied normally on the free surface of a fluid-saturated, porous half-space. As an extension to these works, Halpern and Christiano [18] solved the same axisymmetric problem for a time harmonic loading and a more general case incorporating a viscous fluid. Philippacopoulos [19] discussed axisymmetric wave propagation in a fluid-saturated porous half-space produced by a time harmonic concentrated load acting vertically at the free surface. Philippacopoulos [20] also considered the case of a buried vertical load and derived the solution as a superposition of the corresponding full space fundamental solution (Philippacopoulos [15]) and the relevant effects resulted from the presence of the free surface (Philippacopoulos ([19 and 21]). A comprehensive analytical treatment of two-dimensional dynamic response of a dissipative poroelastic half-plane under time-harmonic internal loads and fluid sources was given by Senjuntichai and Rajapakse [22]. General asymmetric problems in three dimensions have been considered by several authors. Based on Biot’s [4 and 5] formulation and by the method of Helmholtz decomposition separately applied to the solid and the pore fluid average displacement fields, a set of fundamental solutions for general asymmetric, buried-source problems in an isotropic poroelastic half-space was given in Ji [23].  Jin and Liu [24 and 25] also presented solutions for an isotropic fluid-saturated poroelastic half-space subjected to respectively time-harmonic horizontal surface and buried point forces. In a solution presented by Zhou et al. [26], the transient dynamic response of a fluid-saturated poroelastic half-space loaded internally by the normal and horizontal impulsive concentrated loads was investigated with the assumption of incompressible constituents and negligible inertial coupling. Also, three-dimensional time-harmonic response of a fluid saturated half space subjected to an arbitrary buried source is investigated by Chen et al. [27].

All of the above solutions are based on the assumption that the solid skeleton of the medium is completely isotropic and the permeability at any single point in the medium does not vary with the direction. While isotropy is an appealing material assumption, most soils and geomaterials exhibit some degrees of anisotropy in both their stiffness and permeability due to their original deposition, compaction or microstructural characteristics. The first basic formulation for propagation of the elastic waves in an anisotropic fluid-saturated poroelastic medium was presented by Biot [28 and 29], as an extension to his original theory in isotropic medium (Biot [4 and 5]). This formulation has been employed by many authors to study the propagation characteristics of the stress waves in anisotropic porous media. Such studies mostly discuss the existence of different interfacial wave modes and fundamental features of the wave front curves propagating in such media (Schmit [30], Sharma and Gogna [31], Carcione [32 and 33], Liu et al. [34], Sharma [35, 36 and 37]). A review of literature indicates that only few attempts have been made for analytical solution of dynamic boundary-value problems in anisotropic porous media. As one notable contribution, one can mention the work of Kazi-Aoual et al. [38], who obtained a set of time-harmonic fundamental solutions for an infinite transversely isotropic porous medium. The solution of Kazi-Aoual et al. [38] was an extension to their solution (Kazi-Aoual et al. [39]) for the same problem in the corresponding non-porous elastic medium, where Kupradze's [40] method employed to obtain the fundamental solutions for an infinite transversely isotropic medium. However, they could not obtain the solutions in an explicit form. Also more recently, Kumar et al. [41 and 42] investigated the elastodynamic responses of respectively axisymmetric and plane-strain full-space transversely isotropic porous media loaded by concentrated point forces in the frequency domain.

Unlike solutions for porous media, solutions for boundary-value problems in anisotropic non-porous elastic media have been considered by many authors. Payton [43] gives a comprehensive summery on studies concerning dynamic problems in transversely isotropic elastic medium, including solutions for a selected set of loadings applied inside or on the boundary of a half-space. Rajapakse and Wang [44 and 45] also presented analytical solutions for two and three dimensional responses of respectively transversely isotropic and orthotropic semi-infinite elastic media subjected to arbitrary time-harmonic loads acting at an interior plane of loading. They applied the Fourier expansion and Hankel integral transforms to derive general solutions for equations of motion expressed in terms of displacements. In a more recent study, Rahimian et al. [46] presented a new and efficient elastodynamic potential method for determination of the displacement and stress fields of a transversely isotropic elastic half-space subjected to time-harmonic loads on its free surface. This solution employs the displacement-potential functions introduced by Eskandari-Ghadi [47] to uncouple the equations of motion in a transversely isotropic medium. Many recent studies in dynamic response problems of transversely isotropic elastic media (Khojasteh et al. [48, 49 and 50], Gharahi et al. [51], Ai et al. [52 and 53] and Kalantari et al. [54]) have been founded on the basis of the method proposed by Rahimian et al. [46].

In what follows, by extending the work undertaken by Rahimian et al. [46], an elastodynamic potential method is presented for treatment of the corresponding problems in fluid-saturated poroelastic media. The problem is considered in the framework of Biot's theory [28 and 29] presented for transversely isotropic fluid-saturated poroelastic media. The solution is given for the equations of motion in a cylindrical coordinate system written in terms of the displacement components of the solid skeleton and the pore fluid pressure field (i.e., the u-p formulation) in a frequency domain. It is worth noting that since such a formulation is achieved in a frequency domain, there is no need to consider any additional assumption or simplification regarding Biot’s original equations and this means that there will be no constraint or limitation on the ranges of validity of the solutions.

By application of the potential representations introduced in terms of two scalar potential functions for the displacement components of the solid skeleton and the pore fluid pressure field it is shown that the coupled set of the governing differential equations of motion are reduced to a pair of uncoupled equations in terms of the scalar potential functions. By appropriate use of Fourier expansions and Hankel integral transforms these equations can be written as a pair of ordinary differential equations in a spatial and time transformed domain. By solution of these equations one can achieve the general solution for the displacements, stresses and the fluid discharge components as well as the pore fluid pressure of a transversely isotropic fluid-saturated poroelastic medium. By the help of the general solution provided, three dimensional response of a transversely isotropic poroelastic half-space subjected to an arbitrarily distribution of a time-harmonic mechanical loading on its free surface is investigated. By application of the relevant boundary conditions, a set of fundamental solutions for the displacements and the pore fluid pressure are presented in the transformed domain. These solutions are presented for two different hydraulic boundary conditions; fully permeable and completely impermeable free surfaces of the half-space. To retrieve the solutions in a physical domain numerical inversion is employed with a proper account of possible presence of singularities on the path of integration. As a specific solution, Green’s functions corresponding to the case where the loading is distributed uniformly over a circular area on the surface of the half space is presented. To verify the validity of the analytical solution presented in this paper, both analytical and numerical results have been compared with existing solutions for a transversely isotropic single phase elastic medium. Some illustrative numerical examples are also presented to clarify the influence of the degree of the material anisotropy and the frequency of excitation on the response.

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