Time‐dependent behaviour of a rigid foundation on a transversely isotropic soil layer

An analytical solution is presented in this paper to study the time‐dependent settlement behaviour of a rigid foundation resting on a transversely isotropic saturated soil layer. The governing equations for a transversely isotropic saturated soil, within Biot's poroelasticity framework, are solved by means of Laplace and Hankel transforms. The problem is subsequently formulated in the Laplace transform domain in terms of a set of dual integral equations that are further reduced to a Fredholm integral equation of the second kind and solved numerically. The developed analytical solution is validated via comparison with the existing analytical solution for an isotropic saturated soil case, and adopted as a benchmark to examine the sensitivities of the mesh refinement and the locations of truncation boundaries in the finite element simulations using ABAQUS. Particular attention is paid to the influences of the degree of soil anisotropy, boundary drainage condition, and the soil layer thickness on the consolidation settlement and contact stress of the rigid foundation. Copyright © 2009 John Wiley & Sons, Ltd.     

A contact problem for a poroelastic halfspace containing an embedded inextensible membrane

The paper examines the axisymmetric problem of the indentation of a poroelastic halfspace that is reinforced with an inextensible permeable/impermeable membrane located at a finite depth by a rigid indenter. The constitutive behavior of the poroelastic halfspace is described by the three-dimensional theory of poroelasticity proposed by M.A. Biot. The contact conditions between the indenter and the poroelastic halfspace are varied to accommodate both adhesive/frictionless contact and impermeable/permeable conditions. The formulation of the mixed boundary value problems uses the stress function approaches applicable to semi-infinite domains. Successive applications of Laplace and Hankel integral transforms are used to reduce the mixed boundary value problems to sets of coupled Fredholm integral equations of the second kind. These integral equations are solved using numerical approaches, applicable both for the solution of the systems of coupled equations and for Laplace transform inversion, to examine the time-dependent displacement of the rigid indenter. The analytical-numerical estimates for the time-dependent displacements of the rigid indenter are compared with results obtained using a finite element approach.     

The behaviour of layered soil or rock containing a decaying heat source

Solutions are presented for the behaviour of layered soil or rock deposits which contain a heat source. Such a problem arises when high level nuclear waste is placed in deep underground depositaries, as the waste continues to generate heat for many years after placement. This heating of the surrounding soil or rock may lead to expansion and cracking with subsequent contamination of ground water. Results are presented for heat soureces with different decay rates and for heat sources in layers of material with different coefficients of expansion. An example using realistic data for rock is also given. The solution method involves applying Fourier or Hankel transforms to the field quantities and this reduces the two-dimensional or axisymmetric problem to one involving a single spatial dimension. In cases where the soil or rock is horizontally layered, the method has great advantages over other numerical methods such as finite element or finite difference techniques, since little computer storage and data preparation time is required. Solution of the time-dependent problem is carried out by applying Laplace transforms to the field variables, obtaining solutions and then using numerical means to invert the transformed solutions. This enables easy solution of problems involving time-dependent (i.e. decaying) heat sources.     

Stress analysis of a deep rigid axially-loaded cylindrical anchor in an elastic medium

The elastostatic problem of an infinite elastic medium containing an axially-loaded rigid cylindrical inclusion is investigated. This problem is of interest in connection with the geotechnical study of the time-independent, load-deflection characteristics of deep rigid anchors embedded in cohesive soil or rock media. The problem is formulated by means of Hankel integral transforms and reduced to a system of four coupled sing ular integral equations, where the unknown quantities are the normal and the shear stresses acting on the entire surface of the anchor. Numerical solutions are investigated for various Poisson's ratios and several values of the aspect ratio of radius to length of the cylindrical anchor.     

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

This paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 John Wiley & Sons, Ltd.     

A method of computing the consolidation behaviour of layered soils using direct numerical inversion of Laplace transforms

A method is presented for obtaining the consolidation behaviour of a layered soil subjected to strip, circular, or rectangular surface loadings, or subjected to fluid withdrawal due to pumping. The solution method involves applying a Fourier or Hankel transform to the field quantities along with a Laplace transformation. The effect of the Fourier or Hankel transform is to reduce a two- or three-dimensional problem or one involving axial symmetry, to one involving only a single spatial dimension. In cases where the soil is horizontally layered, this has great advantages over conventional methods, such as finite element or finite difference methods, since very little computer storage and data preparation time is required. Solution of the time dependent problem is achieved by applying a Laplace transformation to the field variables, obtaining solutions in Laplace transform space, and then numerically inverting the transformed solutions to obtain the real time behaviour. This eliminates the need for ‘marching type’ schemes where a solution is found from one at a previous time. By direct inversion of the Laplace transform, a solution may be obtained directly at any given time.     

Finite layer analysis of viscoelastic layered materials

A method is presented for obtaining the creep settlement of strip or circular loadings applied to horizontally layered soil profiles. The solution method involves applying a fourier (strip loading) or Hankel (circular loading) transform to the governing equations, which reduces the two of three dimensional problem to one involving a single spatial demension. This leads to great savings in computer storage and data preparation time, and since an exact solution may be found for each layer of material, the method has advantages over conventional finite layer techniques where field quantities must be approximated at a number of positions within each layer. The type of formulation presented hearein makies it possible to work in terms of the creep functions of the soil rather than the relaxation functions. This has distinct advantages, as it is often easier to measure the creep behaviour of a soil in the laboratory. Numerical techniques are used to invert the laplace and Hankel transforms and this means that any type of creep function (which is invertible) may be used to describe the material properties of the soil.     

Reflection–transmission matrix method for the consolidation of a multilayered saturated soil

The consolidation of the layered saturated soil is an important issue in civil engineering and has been investigated extensively during the past decades. In this study, based on the Biot's theory, the reflection–transmission matrix (RTM) method for treating the layered saturated soil under axisymmetric consolidation is developed. To decouple the governing equations of the Biot's theory, the McNamee displacement functions are introduced, and the general solution for the saturated soil is obtained using the Laplace and Hankel transforms. In order to develop the RTM method for the layered saturated soil, based on the obtained general solution, the static wave vector corresponding to the state vector of the saturated soil and the transform matrix relating the aforementioned two vectors are defined. Also, the transfer matrices corresponding to the two vectors are introduced, and the representations of the RTMs for the static wave vector of the saturated soil are presented. As the state vector, static wave vector, and the transform matrix relating the two vectors are all defined in the global coordinate system, the RTMs obtained in this study thus have a reasonable physical meaning. By using the RTMs for the layered saturated soil, the solutions for the layered saturated soil subjected to external sources are derived. Comparison of results due to the proposed RTM method with some existing results and results due to the transfer matrix method validates the developed RTM method. Some numerical results are obtained based on the proposed RTM method for the layered saturated soil. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytical solution to one-dimensional consolidation in unsaturated soils under loading varying exponentially with time

This note presents an analytical solution to one-dimensional consolidation in unsaturated soils with a finite thickness under confinement in the lateral direction and vertical loading varying exponentially with time. The boundary conditions are that the top surface is permeable to water and air and the bottom is impermeable to water and air. The transfer relationship between the state vectors at the top surface and any depth is gained by applying the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy’s law and Fick’s law. The excess pore-air and pore-water pressures and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial and boundary conditions. By performing the inverse Laplace transforms, the analytical solutions of the excess pore-air and pore-water pressures at any depth and settlement are obtained in the time domain.     

Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown

Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent induced by groundwater drawdown was studied based on Biot’s axisymmetric consolidation theory. Laplace and Hankel transforms were employed to solve the governing equation. Explicit analytical solutions are obtained in the Laplace–Hankel transform domain when groundwater drawdown is induced by a constant pumping well. Based on the solutions, numerical computations were performed to study the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer.     

Analytical layer‐element method for non‐axisymmetric consolidation of multilayered soils

A numerically efficient and stable method is developed to analyze Biot's consolidation of multilayered soils subjected to non‐axisymmetric loading in arbitrary depth. By the application of a Laplace–Hankel transform and a Fourier expansion, the governing equations are solved analytically. Then, the analytical layer‐element (i.e. a symmetric stiffness matrix) describing the relationship between generalized displacements and stresses of a layer is exactly derived in the transformed domain. Considering the continuity conditions between adjacent layers, the global stiffness matrix of multilayered soils is obtained by assembling the inter‐related layer‐elements. Once the solution in the Laplace–Hankel transformed domain that satisfies the boundary conditions has been obtained, the actual solution can be derived by the inversion of the Laplace–Hankel transform. Finally, numerical examples are presented to verify the theory and to study the influence of the layered soil properties and time history on the consolidation behavior. Copyright © 2011 John Wiley & Sons, Ltd.     

The behavior of a multilayered porous thermo-elastic medium with anisotropic thermal diffusivity and permeability

With the aid of integral transform techniques, this paper presents an extended precise integration solution for thermal consolidation problems of a multilayered porous thermo-elastic medium with anisotropic thermal diffusivity and permeability due to a heat source. From the fundamental governing equations, ordinary differential equations are derived by employing Laplace–Hankel transforms. By applying the extended precise integration method, equations in the transformed domain can be solved, and the actual solutions are further obtained by adopting a numerical inverse transformation. The accuracy and feasibility of the proposed theory is demonstrated by contrastive analysis with existing studies. Finally, several examples are carried out to investigate the influence of heat source’s type, axial distance, burial depth of heat source, ratio of thermo-permeability, permeability anisotropy, thermal diffusivity anisotropy and stratification on the thermal consolidation process.     

The response of a deep rigid anchor due to undrained elastic deformations of the surrounding soil medium

The equations governing the undrained linear elastic behaviour of a saturated soil are formally similar to the equations governing slow of an incompressible Newtonian viscous fluid. This principle of equivalence can then be effectively employed to obtain the load-deflection reiationship for a deep rigid anchor with the shape of a solid of revolution which is embedded in bonded contact with an unbounded incompressible elastic medium. It is found that the load-deflection relationship for the deep rigid anchor can be directly recovered from the expression for the drag induced on an impermeable object with the same size and shape as the anchor, which is appropriately placed in a slow viscous flow region of uniform velocity.     

Fundamental solutions for a fluid‐saturated,transversely isotropic,poroelastic solid

The fundamental solutions were obtained for step‐like point forces acting in three orthogonal directions and an instantaneous fluid point source in a fluid‐saturated, porous, infinite solid of transversely isotropic elasticity and permeability. After expressing the governing equations in the form of matrix in the Laplace space, we employed Kupradze's method together with the triple Fourier transforms. This method reduces the simultaneous partial differential equations with respect to three displacement components and a pore fluid pressure to a differential equation in terms of only one potential scalar function, which can be operationally solved in the transformed space. After the Laplace inversion of the potential, the residue theorem was applied to its Fourier inverse transform with respect to one of the transformation variables. The Fourier transforms with respect to two other variables were rewritten into the Hankel transforms. Copyright © 2002 John Wiley & Sons, Ltd.     

Influence of a point sink on a pile group in saturated multilayered soils with anisotropic permeability

The behavior of a pile group is solved using the finite element method, and the fundamental solution of saturated multilayered soils with anisotropic permeability is obtained by the analytical layer element method. Based on the supposition of no slip occurring at the pile‐soil interface, the governing equations of the interaction between the pile group and the soils due to a point sink are established in the Laplace‐Hankel transformed domain by considering the pile‐soil compatibility condition. Numerical results are presented to study the effect of point sink pumping, the properties of soils, and the geometries of piles on the behavior of the pile group.     

A semi-discrete procedure for dynamic response analysis of saturated soils

Biot's equations of wave propagation through fluid-saturated porous elastic media are discretized spatially using the finite element method in conjunction with Galerkin's procedure. Laplace transformation of the discretized equations is used to suppress the time variable. Introducing Laplace transforms of constituent velocities at nodal points as additional variables, the quadratic set of equations in the Laplace transform parameter is reduced to a linear form. The solution in the Laplace transform space is inverted, term by term, to get the complete time history of the solid and fluid displacements and velocities. Since the solution is exact in the time domain, the error in the calculated response is entirely due to the spatial approximation. The procedure is applied to one-dimensional wave propagation in a linear elastic material and in a fluid-saturated elastic soil layer with ‘weak’, ‘strong’ as well as ‘moderate’ coupling. With refinement of the spatial mesh, convergence to the exact solution is established. The procedure can provide a useful benchmark for validation of approximate temporal discretization schemes and estimation of errors due to spatial discretization.     

渗透破坏若干问题的解析解及机理分析

讨论了柱坐标下不同介质有限区域内用分离变量法求解Laplace 方程的问题。通过一系列计算试验,给出了柱坐标下Laplace 方程在不同介质有限区域内用分离变量方法求的解,并用该解对渗透破坏现象进行了讨论和实例验证。认为渗透系数分布不均匀是发生渗透破坏的根本原因,渗透破坏朝着渗透系数差异较大的方向发展,并对管涌的复杂机理进行了引申阐述。该工作对于流土、管涌机理的更深入研究具有重要的意义。     

A semi-analytical solution to consolidation of unsaturated soils with the free drainage well

Based on Fredlund’s one-dimensional consolidation equation for unsaturated soil, Darcy’s law and Fick’s law, a semi-analytical solution was presented to the free drainage well with a finite thickness under application of uniform vertical loading and the boundary of the top and bottom surfaces impermeable to water and air. According to the polar governing equations of water and air phases and the boundary and initial conditions, the excess pore-air and pore-water pressures and the soil layer settlement in the Laplace transformed domain are obtained by performing the Laplace transform and utilizing the Bessel functions. Crump’s method is used to perform the inversion of Laplace transform in order to obtain numerical solutions in the real time domain. Finally, a typical example is given to illustrate the changes in the excess pore-air and pore-water pressures and soil layer settlement with time factor at different ratios of air–water permeability coefficient and/or different distances from the well.     

Finite element analyses of layered visco‐elastic system under vertical circular loading

Analyses for the response of a linear visco‐elastic system subjected to axi‐symmetric vertical circular loading are presented. Hankel transforms with respect to the radial spatial coordinate are used to reduce the three‐dimensional problem to that involving only a single spatial dimension, which is then discretized using the finite element method. Three techniques are employed to handle the time factor in the visco‐elastic material: (i) direct time integration; (ii) Fourier transforms; and (iii) Laplace transforms. These methods are compared and evaluated through their numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Steady‐state analytical solutions of flow and deformation coupling due to a point sink within a finite fluid‐saturated poroelastic layer

An exact steady‐state closed‐form solution is presented for coupled flow and deformation of an axisymmetric isotropic homogeneous fluid‐saturated poroelastic layer with a finite radius due to a point sink. The hydromechanical behavior of the poroelastic layer is governed by Biot's consolidation theory. Boundary conditions on the lateral surface are specifically chosen to match the appropriate finite Hankel transforms and simplify the transforms of the governing equations. Ordinary differential equations in the transformed domain are solved, and then the analytical solutions in the physical space for the pore pressure and the displacements are finally obtained by using finite Hankel inversions. The analytical solutions at some special locations such as the top and bottom surfaces, lateral surface, and the symmetrical axis are given and analyzed. And a case study for the consolidation of a water‐saturated soft clay layer due to pumping is conducted. The analytical solution is verified against the finite element solution. Meanwhile, an analysis of coupled hydromechanical behavior is carried out herein. The presented analytical solution is an exact solution to the practical poroelastic problem within an axisymmetric finite layer. It can provide us a better understanding of the poroelastic behavior of the finite layer due to fluid extraction. Besides, it can be applied to calibrate numerical schemes of axisymmetric poroelasticity within finite domains. Copyright © 2017 John Wiley & Sons, Ltd.