Dynamic Green’s Functions For Homogeneous And Stratified Poroelastic Half-spaces

Many materials encountered in engineering can be considered as porous mediaconsisting of an assemblage of solid particles of a pore space. The pore space may befilled with air, a viscous fluid or both. The theory of elastic wave propagation in thefluid saturated porous media was developed by Biot in a series of papers. The wavepropagation in a homogeneous and stratified poroelastic half-space is a subject offundamental interest in applied mechanics and civil engineering because of itsrelevance to dynamic soil-structure interaction, geotechnical earthquake engineering,foundation vibration, and seismology.In the present work, based on Biot’s theory, governing equations formulated withfield variables u w (solid displacement and relative solid-fluid displacement) andu p(solid displacement and pore pressure), respectively, in frequency domain arere-derived. In a cylindrical coordinate system, a method of displacement potentials isdeveloped for the u w formulation to reduce the homogeneous wave equations tofour scalar Helmholtz equations and thus the general solutions are found.For the u pformulation, four potentials are introduced to decouple thehomogeneous wave equations. It can be shown that those potentials stratify fourscalar Helmholtz equations representing the motions of P1-, P2-, SV-, SH-waves inthe porous media, respectively. By the methods of separation of variables, the generalsolutions to those Helmholtz equations are found.The steady-state and transient responses of a homogeneous poroelastic half-spacesubjected to surface tractions are investigated by using the general solutions tohomogeneous wave equations. By virtue of a method of displacement potentials, the fundamental singularsolutions for a point force and fluid source in an infinite poroelastic medium arederived, respectively, in frequency domain. The mirror image technique is thenapplied to construct the dynamic Green’s functions for a homogeneous poroelastichalf-space. Explicit analytical solutions for displacement fields and pore pressure areobtained in terms of semi-infinite Hankel type integrals with respect to horizontalwavenumber. In two limiting cases, the solutions presented in this study are shown toreduce to known counterparts of elastodynamics and those of Lame’s problem, thusensuring the validity of our result. Numerical results show that in the frequencydomain full-space and half-space solutions agree well for large depths. At the freesurface, however, the difference between the two solutions reaches a maximum. Thesingular behavior of displacement fields at the source point can be clearly observedfrom displacement curves. In addition, it is shown that the arrivals of direct S andreflected and converted SP1, SS waves could not be observed from the waveformsinduced by fluid injection.In cylindrical coordinates, a three vector baseR mk，S mk，T mkis introduced toexpress the foregoing general solutions in terms of them, and then, utilize thecontinuity condition between any two adjacent layers and the boundary condition atthe surface to form the propagator matrix formulation. After that, we introduce thegeneralized reflection and transmission matrices for individual interface and layers,and then, develop the recursive produce to combine these matrices into globalreflection and transmission matrices for a stack of homogeneous layers. The pointforce and fluid source are introduced in terms of discontinuities in displacement,traction, fluid flux, and pore pressure across the source plane. In order to provide ameans to check the validity of the present approach, we apply our formulation tocalculate the displacement field in a homogeneous poroelastic half-space which isassumed to be composed of two layers, each with the same set of material parameters.It is shown that the numerical results computed with the proposed method agree well with those computed with analytical solution in frequency domain for a homogeneousporoelastic half-space. Numerical results show that it is impossible to observe thearrivals of transmitted S and transmitted and converted SP waves from the waveformsinduced by fluid injection.