In frequency domain, the fundamental solutions for a poroelastic half-space are re-derived in the context of Biot's theory. Based on Biot's theory, the governing field equations for the dynamic poroelasicity are established in terms of solid displacement and pore pressure. A method of potentials in cylindrical coordinate system is proposed to decouple the homogeneous Biot's wave equations into four scalar Helmholtz equations, and the general solutions to these scalar wave equations are obtained. After that, spectral Green's functions for a poroelastic full-space are found through a decomposition of solid displacement, pore pressure, and body force fields. Mirror-image technique is then applied to construct the half-space fundamental solutions.Finally, transient responses of the half-space to buried point forces are examined.
Green's functions for Biot's dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering,rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green's functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term "decoupling coefficient" for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green's functions. The correctness of the solution is demonstrated by numerically comparing the current solution with Cheng's previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green's functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method(BEM) and other applications.
By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld's identity as well as Green's functions of Stokes' solution pertaining the conventional elastic dynamic equation,the results of Green's function in an infinite space of an axisymmetric coordinate are shown in this paper.After employing a supplementary influence field and the boundary conditions in the free surface of a semi-space,the authors obtain the solutions of Green's function for Lamb's dynamic problem.Besides,the vertical displacement u_(zz) and the radial displacement u_(rz) can match Lamb's previous results,and the solutions of the linear expansion source u_(rr) and the linear torsional source u_(θθ) are also given in the paper.The authors reveal that Green's function of Stokes' solution in the semi-space is a comprehensive form of solution expressing the dynamic Lamb's problem for various situations.It may benefit the investigation of deepening and development of Lamb's problems and solution for pertinent dynamic problems conveniently.
