Dynamic response of a multi-layered poroelastic medium

An exact stiffness matrix method is presented to evaluate the dynamic response of a multi-layered poroelastic medium due to time-harmonic loads and fluid sources applied in the interior of the layered medium. The system under consideration consists of N layers of different properties and thickness overlying a homogeneous half-plane or a rigid base. Fourier integral transform is used with respect to the x-co-ordinate and the formulation is presented in the frequency domain. Fourier transforms of average displacements of the solid matrix and pore pressure at layer interfaces are considered as the basic unknowns. Exact stiffness (impedance) matrices describing the relationship between generalized displacement and force vectors of a layer of finite thickness and a half-plane are derived explicitly in the Fourier-frequency space by using rigorous analytical solutions for Biot's elastodynamic theory for porous media. The global stiffness matrix and the force vector of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces. The numerical solution of the global equation system for discrete values of Fourier transform parameter together with the application of numerical quadrature to evaluate inverse Fourier transform integrals yield the solutions for poroelastic fields. Numerical results for displacements and stresses of a few layered systems and vertical impedance of a rigid strip bonded to layered poroelastic media are presented. The advantages of the present method when compared to existing approximate stiffness methods and other methods based on the determination of layer arbitrary coefficients are discussed.

[1]  M. Biot.  MECHANICS OF DEFORMATION AND ACOUSTIC PROPAGATION IN POROUS MEDIA , 1962 .

[2]  N. A. Haskell.  The Dispersion of Surface Waves on Multilayered Media , 1953 .

[3]  H. Deresiewicz.  The effect of boundaries on wave propagation in a liquid-filled porous solid: II. Love waves in a porous layer , 1961 .

[4]  R. J. Apsel,et al.  On the Green's functions for a layered half-space. Part II , 1983 .

[5]  Paul Christiano,et al.  Response of poroelastic halfspace to steady‐state harmonic surface tractions , 1986 .

[6]  Eduardo Kausel,et al.  Point Loads in Cross‐Anisotropic, Layered Halfspaces , 1989 .

[7]  R. Rajapakse,et al.  Elastodynamic Green's functions of orthotropic half plane , 1991 .

[8]  A. Skempton.  THE PORE-PRESSURE COEFFICIENTS A AND B , 1954 .

[9]  T. H. Watson.  A note on fast computation of Rayleigh wave dispersion in the multilayered elastic half-space , 1970, Bulletin of the Seismological Society of America.

[10]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[11]  J. Rice,et al.  Some basic stress diffusion solutions for fluid‐saturated elastic porous media with compressible constituents , 1976 .

[12]  J. Z. Zhu,et al.  The finite element method , 1977 .

[13]  M. Kassir,et al.  Interaction functions of a rigid strip bonded to saturated elastic half-space , 1988 .

[14]  Paul Christiano,et al.  Steady‐state harmonic response of a rigid plate bearing on a liquid‐saturated poroelastic halfspace , 1986 .

[15]  J. E. Luco,et al.  Dynamic response of rigid foundations of arbitrary shape , 1976 .

[16]  N. A. Haskell.  Crustal reflection of plane P and SV waves , 1962 .

[17]  J. H. Wilkinson,et al.  AN ESTIMATE FOR THE CONDITION NUMBER OF A MATRIX , 1979 .

[18]  H. Poulos,et al.  Elastic solutions for soil and rock mechanics , 1973 .

[19]  E. Kausel,et al.  Dynamic loads in the interior of a layered stratum: An explicit solution , 1982 .

[20]  W. Thomson,et al.  Transmission of Elastic Waves through a Stratified Solid Medium , 1950 .

[21]  N. A. Haskell,et al.  Crustal Reflection of Plane SH Waves , 1960 .

[22]  Dimitri E. Beskos,et al.  Integral Equation for Dynamic Poroelasticity in Frequency Domain with BEM Solution , 1991 .

[23]  M. Biot.  Theory of Propagation of Elastic Waves in a Fluid‐Saturated Porous Solid. I. Low‐Frequency Range , 1956 .

[24]  Stanley B. Dong,et al.  On Natural Vibrations and Waves in Laminated Orthotropic Plates , 1972 .

[25]  R. Fateman,et al.  A System for Doing Mathematics by Computer. , 1992 .

[26]  R. Skalak,et al.  On uniqueness in dynamic poroelasticity , 1963 .

[27]  A. J. Philippacopoulos,et al.  Axisymmetric Vibration of Disk Resting on Saturated Layered Half-Space , 1989 .

[28]  John L. Tassoulas,et al.  Analysis of Foundations on Fluid‐Filled poroelastic stratum , 1993 .

[29]  M. Biot.  Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. II. Higher Frequency Range , 1956 .

[30]  F. Schwab.  Surfa ce-wave dispersion computations: Knopoff's method , 1970, Bulletin of the Seismological Society of America.

[31]  A. Philippacopoulos.  Waves in partially saturated medium due to surface loads , 1988 .

[32]  Freeman Gilbert,et al.  Propagator matrices in elastic wave and vibration problems , 1966 .

[33]  Analysis of in-plane waves in layered half-space by global-local finite element method , 1988 .

[34]  L. Knopoff.  A matrix method for elastic wave problems , 1964 .

[35]  E. Kausel,et al.  Stiffness matrices for layered soils , 1981 .

[36]  B. Kennett,et al.  Reflections, rays, and reverberations , 1974, Bulletin of the Seismological Society of America.

[37]  John L. Tassoulas,et al.  DYNAMIC STIFFNESS OF FOUNDATIONS ON FLUID-FILLED POROELASTIC STRATUM , 1993 .

[38]  H. Deresiewicz.  The effect of boundaries on wave propagation in a liquid-filled porous solid: IV. Surface waves in a half-space , 1962 .

[39]  H. Deresiewicz.  The effect of boundaries on wave propagation in a liquid-filled porous solid: III. Reflection of plane waves at a free plane boundary (general case) , 1960 .