Saturated Half-space Research Articles

3D dynamic responses of a multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure

Dynamic Green's function plays an important role in the study of various wave radiation, scattering and soil-structure interaction problems. However, little research has been done on the response of transversely isotropic saturated layered media. In this paper, the 3D dynamic responses of a multi-layered transversely isotropic saturated half-space subjected to concentrated forces and pore pressure are investigated. First, utilizing Fourier expansion in circumferential direction accompanied by Hankel integral transform in radial direction, the wave equations for transversely isotropic saturated medium in cylindrical coordinate system are solved. Next, with the aid of the exact dynamic stiffness matrix for in-plane and out-of-plane motions, the solutions for multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure are obtained by direct stiffness method. A FORTRAN computer code is developed to achieve numerical evaluation of the proposed method, and its accuracy is validated through comparison with existing solutions that are special cases of the more general problems addressed. In addition, selected numerical results for a homogeneous and a layered material model are performed to illustrate the effects of material anisotropy, load frequency, drainage condition and layering on the dynamic responses. The presented solutions form a complete set of Green's functions for concentrated forces (including horizontal load in x(y)-direction, vertical load in z-direction) as well as pore pressure, which lays the foundation for further exploring wave propagation of complex local site in a layered transversely isotropic saturated half-space by using the BEMs.

Three-dimensional dynamic Green's functions for transversely isotropic saturated half-space subjected to buried loads

Based on Biot's theory of saturated porous medium, three-dimensional (3D) dynamic Green's functions for a transversely isotropic (TI) saturated half-space is solved. The 3D Green's functions correspond to solutions of buried uniformly distributed loads acting on a circular disk of a homogeneous TI saturated half-space. First, a set of scalar potential functions are adopted to solve the governing equations of TI saturated medium in the cylindrical coordinate system. Next, by employing the Fourier series expansion and Hankel transform, the solutions for the potential functions are obtained in transformed domain, and then the displacements, stresses and pore fluid pressure are expressed via the relationships between these functions. Finally, the boundary conditions of problem are introduced, the solutions are derived for three different cases of vertical load in z-direction, horizontal load in x(y)-direction and pore fluid pressure in z-direction. The accuracy of the proposed method is validated by comparison with existing solutions for isotropic elastic and saturated as well as TI saturated (surface load) media that are special cases of the more general problems addressed. Parametric studies in both frequency and time domain are performed to investigate the influences of the material anisotropy, buried depth load frequency and drainage condition on the dynamic responses of the TI saturated half-space. Numerical results indicate that material anisotropy and surface drainage condition are significant important for the accurate assessment of the dynamic responses subjected to buried sources. In addition, The obtained Green's functions form a complete set of fundamental solutions of the TI saturated medium, by which the key to applying the BEMs to a variety of radiation, scattering and interaction problems associated with wave propagation in TI saturated half-space is solved.

Free-field response of a transversely isotropic saturated half-space subjected to incident plane qP1- and qSV-waves

Many natural soils and rocks are not only saturated but are also anisotropic. Due to the fundamental importance of having an understanding of the free-field response of an anisotropic half-space subjected to incident seismic waves, the surface displacements, surface strain, rocking, and pore pressure due to reflection of plane waves in a transversely isotropic (TI) saturated half-space are discussed. The governing equations in u-w formulation are first derived and the plane harmonic wave solution of these equations is then proposed. The free-field response is finally determined by specifying the stress free and surface drainage conditions at the boundary. The half-space surface can be either completely permeable or impermeable, and the excitation includes incident plane qP1- and qSV-waves. The proposed formulations are verified by comparing our results with those for the isotropic half-space. A parametric study is performed to illustrate the influences of the material anisotropy, surface drainage condition and permeability on the surface motions. Numerical results show that the surface motions of the TI half-space are significantly different from those of the isotropic case. The surface motions of the TI half-space are much more dependent on the TI parameters when the incident waves are qSV-waves than when they are qP1-waves because the dynamic responses change sharply as the incident angles approach the critical angles. In contrast with the isotropic case, with an identical angle of 45°, the total reflection angle of the TI half-space varies with the TI parameters. In general, the surface motions of the permeable surface are smaller than those of the impermeable surface, particularly in response to incident qSV-waves.

Reflection and transmission of elastic waves by a fluid-filled crack located in a poroelastic formation

Elastic wave propagation in the system: porous saturated half-space – plane-parallel layer (crack) filled with a viscous fluid – porous saturated half-space is considered in this paper. Reflection and transmission coefficients of compressional and shear waves propagating in such system are calculated. Within the frame of validity of the single-scattering approximation, an expression for the complex wave number of the compressional wave propagating in a rock containing a system of thin parallel layers (cracks) is obtained. The performed calculations have shown that this wave presents strong velocity dispersion, especially at low frequencies corresponding to seismic measurements. The reflection coefficient of the elastic compressional wave incident at the boundary between the elastic half-space and porous fractured medium is calculated. It was shown that the reflection coefficient of this wave strongly depends on the frequency, especially in the low frequency range. This frequency dependence is determined by the intensive hydrodynamic flows (mesoscopic flow) generated at the boundary between the cracks and the porous matrix. The importance of the effects associated with these mesoscopic flows is illustrated by calculations of synthetic seismograms.

An Integral Equation Model for a Pile in a Layered Transversely Isotropic Saturated Soil

Objective:In this paper, an integral equation model is established to predict the time-dependent response of a vertically loaded pile embedded in a layered Transversely Isotropic Saturated Soil (TISS).Methods:Based on the fictitious pile method, the pile-soil system is decomposed into an extended saturated half-space and a fictitious pile. The extended half-space is treated as a layered TISS, while the fictitious pile is considered as a 1D bar. The pile-soil compatibility is accomplished by requiring that the axial strain of the fictitious pile be equal to the vertical strain of the extended layered TISS along the axis of the pile. The second kind Fredholm integral equation of the pile is then derived by using the aforementioned compatibility condition and the fundamental solution of the layered TISS, which is equivalent to the solution of the layered TISS subjected to a uniformly-distributed load acting vertically over a circular area with the radius equal to that of the pile. The fundamental solution of the layered TISS is obtainedviathe Reflection-Transmission Matrix (RTM) method for the layered TISS. Applying the Laplace transform to the Fredholm integral equation, and solving the resulting integral equation, the transformed solution is obtained. The time domain solution of the pile-soil system is retrievedviathe inverse Laplace transform.Results and Conclusion:Numerical results of this paper agree with existing solutions very well, validating the proposed pile-soil interaction model. A parametric study is carried out to examine the influence of some parameters on the response of the pile-soil system.

Dynamic Responses of Saturated Half-Spaces under Distributed Buried Loads

AbstractIn this paper, three-dimensional dynamic responses of a saturated half-space under arbitrary distributed buried time-harmonic load are investigated through analytical developments. In the d...

Seepage Consolidation under Plane Deformation of Elastic Half-Space

The process of seepage consolidation of elastic saturated half-space under the action of an arbitrary normal surface load is investigated in the case of plane deformation. The fluid and skeleton grains are assumed to be incompressible. A new mathematical model of consolidation is proposed with the use of the compatibility equation. Analytic dependences for the sum of effective normal stresses and fluid pressure are found on the basis of this model. The total normal stresses are explicitly expressed in terms of these dependences.

The influence of pore-fluid in the soil on ground vibrations from a tunnel embedded in a layered half-space

A computationally efficient semi-analytical solution for ground-borne vibrations from underground railways is proposed and used to investigate the influence of hydraulic boundary conditions at the scattering surfaces and the moving ground water table on ground vibrations. The arrangement of a dry soil layer with varying thickness resting on a saturated poroelastic half-space, which includes a circular tunnel subject to a harmonic load at the tunnel invert, creates the scenario of a moving water table for research purposes in this paper. The tunnel is modelled as a hollow cylinder, which is made of viscoelastic material and buried in the half-space below the ground water table. The wave field in the dry soil layer consists of up-going and down-going waves while the wave field in the tunnel wall consists of outgoing and regular cylindrical waves. The complete solution for the saturated half-space with a cylindrical hole is composed of down-going plane waves and outgoing cylindrical waves. By adopting traction-free boundary conditions on the ground surface and continuity conditions at the interfaces of the two soil layers and of the tunnel and the surrounding soil, a set of algebraic equations can be obtained and solved in the transformed domain. Numerical results show that the moving ground water table can cause an uncertainty of up to 20 dB for surface vibrations.

The equivalent stiffness of a saturated poroelastic halfspace interacting with an infinite beam under a moving point load

The equivalent stiffness of a saturated poroelastic halfspace that supports an infinite Euler-Bernoulli beam under a moving point load has been investigated in this paper. The smooth contact condition is assumed for the interface of beam and halfspace, however, with an improved continuity condition, i.e. the continuities of contact normal stress and contact vertical displacements are imposed across the width of the beam. The equivalent stiffness of the halfspace has been evaluated using a contour integration approach such that contributions of the Rayleigh pole and the dispersion branches can be explicitly taken into account. Influences of the continuity condition and the permeability on the equivalent stiffness have been studied. It is found that the imaginary part of the equivalent stiffness is nonzero since the viscous coupling between two phases of the saturated halfspace has been considered. This observation fundamentally differentiates the present paper to the work by Shi and Selvadurai (2016) where an infinite permeability has been assumed for the halfspace, i.e. null viscous coupling.

Dynamic impedance functions for a rigid strip footing resting on a multi-layered transversely isotropic saturated half-space

This paper is concerned with the dynamic impedance functions (force-displacement relationships) of a surface rigid strip footing resting on a multi-layered transversely isotropic (TI) saturated half-space. The rigid footing is perfectly bonded to the layered half-space and is subjected to time-harmonic vertical, horizontal and moment loadings. The half-space under consideration consists of a number of horizontal layers with different thicknesses and an underlying half-space, which are all governed by the Biot's poroelastodynamic theory. The surface of the half-space can be either fully permeable or impermeable. The dynamic interaction problem is solved by employing an indirect boundary element method (IBEM), which uses Green's functions for uniform strip loads acting on the surface of a multi-layered TI saturated half-space. The discretization of the method is restricted to the footing-subsoil interface because of the layered half-space kernel functions, and the accuracy of the method would not be affected by the thickness of the discrete layers because of the exact dynamic stiffness matrix. Comparison with the existing solutions for the TI elastic and isotropic saturated media is conducted to verify the method, which are special cases of the more general problems addressed. Selected numerical solutions are presented to portray the influence of material anisotropy, frequency of excitation, surfaced drainage condition and layering on the dynamic impedance functions. Numerical results show that the dynamic impedance functions for the TI material can be significantly different from those of the isotropic material. The variation of the TI parameters alters the resonant frequencies of the layer and further alters the dynamic interaction between the layer and the footing.

Plane strain dynamic responses of a multi-layered transversely isotropic saturated half-space

An exact dynamic stiffness matrix method is proposed to evaluate the plane strain responses due to time-harmonic loads and pore fluid pressure applied in the interior or on the surface of a multi-layered transversely isotropic (TI) saturated half-space. First, the governing equations of the TI saturated medium are solved analytically by using the Fourier transform and the exact global dynamic stiffness matrix in the wavenumber domain is established describing the relationship between the generalized displacement and force vectors. Then, solutions of the multi-layered system for discrete wavenumbers are obtained by using the dynamic stiffness matrix method, which are then synthesized to retrieve the responses in the physical domain. The accuracy of the method is confirmed by comparison with existing solutions for the TI elastic as well as isotropic saturated media that are special cases of the more general problems addressed. Selected numerical results are presented to investigate the effects of material anisotropy, layering, surface drainage condition, buried depth of the source and permeability on the responses. It is found that material anisotropy is very important for the accurate assessment of the dynamic responses subjected to time-harmonic sources. In addition, the presented solutions form a complete set of Green's functions which is required in the application of plane strain boundary methods for multi-layered TI saturated medium.

Fundamental solutions of a multi-layered transversely isotropic saturated half-space subjected to moving point forces and pore pressure

The steady-state dynamic response of a multi-layered transversely isotropic (TI) saturated half-space due to point forces and pore pressure moving with a constant speed is investigated in this paper. To solve this problem, the dynamic stiffness method combined with the inverse Fourier transform is employed. First, the governing equations in terms of the displacement components and pore fluid pressure are solved in the transformed domain by employing the Fourier transform. Next, the exact three-dimensional (3D) dynamic stiffness matrices for the TI saturated layer, as well as the TI saturated half-space, are constructed, and the global dynamic matrix of the problem is formulated by assembling the dynamic matrices of the discrete layers and the underlying half-space. Finally, solutions in the frequency-wavenumber domain of the displacement, pore pressure and stress are obtained through the dynamic stiffness method. The result in the time-space domain is recovered by the Fourier synthesis of the frequency responses which, in turn, are obtained by numerical integration over on one horizontal wavenumber. The accuracy of the developed formulations is confirmed by comparison with existing solutions for an isotropic and saturated medium that is a special case of the more general problem addressed. Numerical results for both low and high source velocities are presented, and the effects of moving speed, material anisotropy, permeability, surface drainage condition and TI saturated layer on the dynamic response are analyzed. It is observed that the dynamic responses reach their peak values when the source velocity is equal to or approaches the phase velocities of SH-, qP1-, qP2- and qSV- in the horizontal direction and the phase velocity of qRayleigh waves. Material anisotropy is very important for the accurate assessment of the dynamic response due to the moving point forces and pore pressure in a TI saturated medium.

The IBIEM Solution to the Scattering of Plane SV Waves around a Canyon in Saturated Poroelastic Half-Space

The scattering of plane SV waves by a two-dimensional canyon in saturated poroelastic half-space is investigated both in frequency and time domain by indirect boundary integrate equation method (IBIEM), and the amplification effect of displacement and fluid pore pressure are thoroughly discussed. It illustrates that the scattering characteristic strongly depends on the incident frequency and angle, soil porosity, boundary drainage condition, and canyon shape. Compared with the incidence of P waves, the amplification effect is not so large for critical porosity, but more obvious amplification feature can be found for incident SV waves at the critical angle.

Seepage consolidation of elastic half-space under an axisymmetric load

The process of seepage consolidation of elastic saturated half-space under the action of a normal load on its surface is investigated assuming both incompressibility of fluid and skeleton grains and independence of the total skeleton stresses on time. Analytic representations for the fluid pressure and the half-space surface settlement are found when the half-space is loaded by a concentrated force. The maximum settlement is also found for a uniform loading of the surface over the circle area.

Lateral Vibration of Pile Groups Partially Embedded in Layered Saturated Soils

AbstractA simplified solution procedure is proposed for estimating the dynamic response of a pile group partially embedded in a layered saturated soil and subjected to horizontal harmonic loading. In the proposed procedure, the transfer matrix method is first applied to solve the vibration equation of a single pile, from which the stiffness of the single pile is obtained. The existing lateral displacement of the saturated half-space is introduced to build the attenuation function, and the balance equation of passive pile is built accordingly. The simple Winkler model is used to derive the dynamic pile-soil-pile interaction factor. Based on the dynamic interaction factor, the horizontal impedance of the pile group is obtained by using the superposition principle. The calculated dynamic interaction factor and impedance of the pile group are in agreement with earlier results for an idealized case, verifying the correctness of the proposed method. The main parameters, such as the embedment ratio, pile separat...

Dynamic vertical interaction of a foundation–soil system generated by seismic waves

Abstract Based on Biot's dynamic poroelastic theory, a foundation–soil interaction model is established to investigate the vertical vibrations of a rigid circular foundation on poroelastic soil excited by incident plane waves, including the fast P waves and SV waves. Scattering waves caused by the foundation and fluid–solid coupling due to the pore water in the soil are also considered in the model. The solution of the vertical vibrations of the foundation subjected to seismic waves are obtained by solving two sets of dual integral equations derived from the mixed boundary-value conditions. The different vertical vibrations of foundation rest on elastic and saturated half-space are compared. The influences of incident angle, permeability of soil and foundation mass on the vertical vibrations of the foundation are then discussed. The results show that resonant phenomenon of the foundation is observed at certain excitation frequencies; the effects of the pore water on the foundation vertical vibrations are significant. In addition, significant differences are found when the foundation is excited by P waves and SV waves, respectively.

Mixed Boundary-Value Analysis of Rocking Vibrations of an Elastic Strip Foundation on Elastic Soil with Saturated Substrata

The dynamic response of an elastic strip foundation lying on elastic soil with saturated substrata is greatly affected by pore pressure induced by a rocking moment. In this paper, we explore the mixed boundary-value problem of the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata via Biot dynamic equations. First, the wave equations concerning both the single-phase elastic layer and the saturated half-space are solved using a Fourier integral transform technique. The dual integral equations of the rocking vibration of an elastic strip foundation are established according to the mixed boundary conditions. Finally, the relationship of the dynamic compliance coefficient with the dimensionless frequency is obtained by applying Simpson’s rule to conduct numerical calculation. We also analyse the influences of the elastic layer’s thickness and elastic characteristic parameters of the foundation on the rocking vibration.

Steady-state response of a saturated half-space with an overlying dry layer subjected to a moving load

In this paper, the steady-state response of a saturated half-space with an overlying dry layer subjected to a moving rectangular load is investigated. The governing partial differential equations are solved using the Fourier transform. The solutions in time-space domain are expressed in terms of infinite Fourier type integrals, which can be evaluated only by numerical quadrature. Numerical results show that the influence of a drained or undrained interface on the response is related to the permeability of the underlying saturated soil. Moreover, the effect due to the upper dry layer is associated with the thickness of the layer.

Torsional response of a rigid circular foundation on a saturated half-space to SH waves

An analytical approach is used to study the torsional vibrations of a rigid circular foundation resting on saturated soil to obliquely incident SH waves. Biot’s poroelastic dynamic theory is considered to characterize the saturated soil below the foundation, which is solved by Hankel transform later. In order to consider the scattering phenomena caused by the existence of the foundation, the total wave field in soil is classified into free-field, rigid-body scattering field and radiation scattering field. According to the classification of wave field and the mixed boundary-value conditions between the soil and the foundation, torsional vibrations of the foundation are formulated in two sets of dual integral equations. Then, the dual integral equations are reduced to Fredholm integral equation of the second kind to be solved. Combining with the dynamic equilibrium equations of the foundation, the expressions for the torsional vibrations of the foundation are obtained. Numerical results are presented to demonstrate the influence of excitation frequency, incident angle, the torsional inertia moment of the foundation and permeability of the saturated half-space on the torsional vibrations of the foundation.

Model of a porous medium with an elastic fractured skeleton

The model of a saturated porous medium with a brittle deformable skeleton is formulated in an isothermal approximation. The general form of governing equations is found, which is necessary and sufficient to fulfill the principles of objectivity and thermodynamic consistency. It is shown that the kinetics of developing the ensemble of microcracks, determined by the derivative of the elastic potential of the skeleton with respect to the fracture parameter, results in a nonnegative dissipation of the scattered destruction with any type of loading. For small deviations from the initial state, a new elastic potential is proposed which has made it possible to describe the main irregularities of the behavior of the medium in question. The solution of the problem about the consolidation of a saturated porous half-space with a brittle skeleton under a normal load has been constructed.