Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.     

Semi‐analytical solution to one‐dimensional consolidation for unsaturated soils with semi‐permeable drainage boundary under time‐dependent loading

This paper presents semi‐analytical solutions to Fredlund and Hasan's one‐dimensional consolidation of unsaturated soils with semi‐permeable drainage boundary under time‐dependent loadings. Two variables are introduced to transform two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. The pore‐water pressure, pore‐air pressure and settlement are obtained in the Laplace domain. Crump's method is adopted to perform the inverse Laplace transform in order to obtain semi‐analytical solutions in time domain. It is shown that the present solutions are more general and have a good agreement with the existing solutions from literatures. Furthermore, the current solutions can also be degenerated into conventional solutions to one‐dimensional consolidation of unsaturated soils with homogeneous boundaries. Finally, several numerical examples are provided to illustrate consolidation behavior of unsaturated soils under four types of time‐dependent loadings, including instantaneous loading, ramp loading, exponential loading and sinusoidal loading. Parametric studies are illustrated by variations of pore‐air pressure, pore‐water pressure and settlement at different values of the ratio of air–water permeability coefficient, depth and loading parameters. Copyright © 2017 John Wiley & Sons, Ltd.     

Analytical solution for one‐dimensional consolidation of unsaturated soils using eigenfunction expansion method

This paper introduces an exact analytical solution for governing flow equations for one‐dimensional consolidation in unsaturated soil stratum using the techniques of eigenfunction expansion and Laplace transformation. The homogeneous boundary conditions adopted in this study are as follows: (i) a one‐way drainage system of homogenous soils, in which the top surface is considered as permeable to air and water, whereas the base is an impervious bedrock; and (ii) a two‐way drainage system where both soil ends allow free dissipation of pore‐air and pore‐water pressures. In addition, the analytical development adopts initial conditions capturing both uniform and linear distributions of the initial excess pore pressures within the soil stratum. Eigenfunctions and eigenvalues are parts of the general solution and can be obtained based on the proposed boundary conditions. Besides, the Laplace transform method is adopted to solve the first‐order differential equations. Once equations with transformed domain are all obtained, the final solutions, which are proposed to be functions of time and depth, can be achieved by taking an inverse Laplace transform. To verify the proposed solution, two worked examples are provided to present the consolidation characteristics of unsaturated soils based on the proposed method. The validation of the recent results against other existing analytical solutions is graphically demonstrated. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A closed form analytical solution for two‐dimensional plane strain consolidation of unsaturated soil stratum

This paper discusses the excess pore‐air and pore‐water pressure dissipations and the average degree of consolidation in the 2D plane strain consolidation of an unsaturated soil stratum using eigenfunction expansion and Laplace transformation techniques. In this study, the application of a constant external loading on a soil surface is assumed to immediately generate uniformly or linearly distributed initial excess pore pressures. The general solutions consisting of eigenfunctions and eigenvalues are first proposed. The Laplace transform is then applied to convert the time variable t in partial differential equations into the Laplace complex argument s. Once the domain is obtained, a simplified set of equations with variable s can be achieved. The final analytical solutions can be computed by taking a Laplace inverse. The proposed equations predict the two‐dimensional consolidation behaviour of an unsaturated soil stratum capturing the uniformly and linearly distributed initial excess pore pressures. This study investigates the effects of isotropic and anisotropic permeability conditions on variations of excess pore pressures and the average degree of consolidation. Additionally, isochrones of excess pore pressures along vertical and horizontal directions are presented. It is found that the initial distribution of pore pressures, varying with depth, results in considerable effects on the pore‐water pressure dissipation rate whilst it has insignificant effects on the excess pore‐air pressure dissipation rate. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear coupled analysis of desiccation shrinkage in a double‐layer partially saturated medium: semi‐explicit solutions

Solutions are presented for the problem of isothermal dessiccation shrinkage in a double‐layer porous partially saturated medium. The rheological model taken into account is linear poroelastic. Hence the analysis is mainly focused on hydromechanical coupling effects and contrasts of mechanical and hydraulic properties between two materials: a low thickness skin comprised between the outer boundary and the reference porous material. Three one‐dimensional ideal structures are taken into account: a wall of finite thickness (cartesian geometry), a thick cylinder and a thick sphere. The solution of the time‐dependent problem is arrived at by applying Laplace transforms to the field variables. Exact solutions are obtained in Laplace transform space using Mathematica^© to solve the field equations whilst taking into account the continuity equations at the interface and the boundary conditions. The Talbot's modified algorithm has been performed to invert the Laplace transform solutions. A bibliographical and numerical study shows that this method is remarkably precise, stable and close to the analytical inversion. Results are presented using poroelastic data representative of a concrete material and involve a strong coupling effect between hydraulical and mechanical behaviours. A first approach elastic modelling of degradation process have been presented using a thin outer layer. Apart from emphasising the semi‐explicit solution utility due to accurate speed calculation, this paper deals with more complex problems than those which can be solved using purely analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

An explicit one-dimensional consolidation solution with semi-permeable drainage boundary for unsaturated soil

Existing solutions for analyzing one-dimensional (1-D) consolidation of unsaturated soil are only derived to cater to two extreme drainage conditions (fully drained and undrained). This study presents a new explicit solution for 1-D consolidation of unsaturated soil with semi-permeable drainage boundary. Based on the assumptions of two independent stress variables and the governing equations proposed by Fredlund, the eigenfunction expansion method is adopted to develop an explicit analytical solution to calculate excess pore-water and pore-air pressures in an unsaturated soil when it is subjected to external loads. The developed general solutions are expressed in terms of depth, z, and time, t. For the semi-permeable drainage boundary, eigenvalues and eigenfunctions in the space domain are developed. The technique of Laplace transform is used to solve the coupled ordinary differential equations in the time domain. The newly derived explicit solution is verified with the existing semi-analytical method in the literature, and an excellent agreement is obtained. Compared with the semi-analytical solution, the newly derived analytical solution is more straightforward and explicit so that this solution is relatively easier to be implemented into a computer program to carry out a preliminary assessment of 1-D consolidation of unsaturated soil.     

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.     

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.     

A quasistatic analysis of a plate on consolidating layered soils by analytical layer‐element/finite element method coupling

In this paper, a coupling method between finite element and analytical layer‐elements is utilized to analyze the time‐dependent behavior of a plate of any shape and finite rigidity resting on layered saturated soils. Based on the integral transform techniques together with the aid of an order reduction method, an analytical layer‐element solution is derived from the governing equations for three‐dimensional Biot consolidation with respect to a Cartesian coordinate system and then extended to be the fundamental solution for the layered saturated soil under a point load. The Mindlin plate is modeled by eight‐noded isoparametric elements. The governing equations of the interaction between soil and plate in the Laplace‐Fourier transformed domain are deduced by referring to the coupling theory of FEM/BEM, and the final solution is obtained by applying numerical inversion. Numerical examples concerned with the time‐dependent response of a plate are performed to demonstrate the influence of soil and plate properties on the interaction process. Copyright © 2014 John Wiley & Sons, Ltd.     

A simple analytical solution to one‐dimensional consolidation for unsaturated soils

This paper presents a simple analytical solution to Fredlund and Hasan's one‐dimensional (1‐D) consolidation theory for unsaturated soils. The coefficients of permeability and volume change for unsaturated soils are assumed to remain constant throughout the consolidation process. The mathematical expression of the present solution is much simpler compared with the previous available solutions in the literature. Two new variables are introduced to transform the two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved with standard mathematical formulas. It is shown that the present analytical solution can be degenerated into that of Terzaghi consolidation for fully saturated condition. The analytical solutions to 1‐D consolidation of an unsaturated soil subjected to instantaneous loading, ramp loading, and exponential loading, for different drainage conditions and initial pore pressure conditions, are summarized in tables for ease of use by practical engineers. In the case studies, the analytical results show good agreement with the available analytical solution in the literature. The consolidation behaviors of unsaturated soils are investigated. The average degree of consolidation at different loading patterns and drainage conditions is presented. The pore‐water pressure isochrones for two different drainage conditions and three initial pore pressure distributions are presented and discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Consolidation analysis of saturated multi‐layered soils with anisotropic permeability caused by a point sink

This paper presents the analytical layer‐element method to analyze the consolidation of saturated multi‐layered soils caused by a point sink by considering the anisotropy of permeability. Starting from the governing equations of the problem, the solutions of displacements and stresses for a single soil layer are obtained in the Laplace–Hankel transformed domain. Then, the analytical layer‐element method is utilized to further derive the solutions for the saturated multi‐layered soils in the transformed domain by combining with the boundary conditions of the soil system and continuity conditions between adjacent layers. The actual solutions in the physical domain can be acquired by the inversion of Laplace–Hankel transform. Numerical results are carried out to show the accuracy and stability of the proposed method and evaluate the influence of sink depth and anisotropic permeability on excess pore pressure and surface settlement. Copyright © 2011 John Wiley & Sons, Ltd.     

The fundamental solution of poroelastic plate saturated by fluid and its applications

In this paper, the numerical model of the transverse vibrations of a thin poroelastic plate saturated by a fluid was proposed. Two coupled dynamic equations of equilibrium related to the plate deflection and the equivalent moment were established for an isotropic porous medium with uniform porosity. The fundamental solutions for a porous plate were derived both in the Laplace transform domain and in the time domain. A meshless method was developed and demonstrated in the Laplace transform domain for solving two coupled dynamic equations. Numerical examples demonstrated the accuracy of the method of the fundamental solutions and comparisons were made with analytical solutions. The proposed meshless method was shown to be simple to implement and gave satisfactory results for a poroelastic plate dynamic analysis. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic responses of unsaturated half‐space soil to a moving harmonic rectangular load

The problem of the dynamic responses of a semi‐infinite unsaturated poroelastic medium subjected to a moving rectangular load is investigated analytical/numerically. The dynamic governing equations are obtained with consideration of the compressibility of solid grain and pore fluid, inertial coupling, and viscous drag as well as capillary pressure in the unsaturated soil, and they can be easily degraded to the complete Biot's theory. Using the Fourier transform, the general solution for the equations is derived in the transformed domain, and then a corresponding boundary value problem is formulated. By introducing fast Fourier transform algorithm, the unsaturated soil vertical displacements, effective stresses, and pore pressures induced by moving load are computed, and some of the calculated results are compared with those for the degenerated solution of saturated soils and confirmed. The influences of the saturation, the load speed, and excitation frequency on the response of the unsaturated half‐space soil are investigated. The numerical results reveal that the effects of these parameters on the dynamic response of the unsaturated soil are significant.     

Semi-analytical solutions of two-dimensional plane strain consolidation in unsaturated soils subjected to the lateral semipermeable drainage boundary

The study presents semi-analytical solutions of two-dimensional plane strain consolidation problem in unsaturated soils incorporating the lateral semipermeable drainage boundary by adopting Fourier sine series and Laplace transform. The two-dimensional plane strain consolidation equations in the form of two-order partial differential equations with three variables are firstly converted to two-order partial differential equations with two variables, which are similar to those of one-dimensional consolidation problem. The four-order ordinary differential equations about excess pore-air and excess pore-water pressures are got by applying Laplace transform and the substitution method. Then, the solutions of excess pore pressures and settlement are achieved in the Laplace transform domain. Afterwards, on the basis of Crump's method, the inverse Laplace transform is conducted to obtain the analytical solutions in time domain. The comparison is conducted to verify the exactness of the obtained solutions, and the two-dimensional plane strain consolidation property with the lateral semipermeable drainage boundary is illustrated and discussed. Parametric studies are demonstrated for the excess pore pressures and normalized settlement with the change of the boundary parameters, air-water and lateral-vertical permeability coefficients, and the distance and depth. It can be found that the lateral semipermeable drainage boundary impedes the consolidation rate obviously, and when different investigated parameters are adopted, the consolidation property is similar to each other under the later permeable and semipermeable drainage boundary conditions.     

Transient dynamic response of multilayered saturated media subjected to impulsive loadings

This paper presents a stable and efficient method for calculating the transient solution of layered saturated media subjected to impulsive loadings by means of the analytical layer element method. Starting with the field equations based on Biot's linear theory for porous, fluid‐saturated media, and the seepage continuity equation, an analytical layer element for a single layer is established by applying Laplace‐Hankel integral transform. The global stiffness matrix in the transform domain for a layered saturated half‐space subjected to a transient circular patch loading is obtained by assembling the layer elements of each layer. The displacements in the time domain are derived by Laplace‐Hankel inverse transform of the global stiffness matrix. Numerical examples are conducted to verify the accuracy of the method and to demonstrate the influences of type of transient loading, buried depth of loading, permeability, and stratification of materials on the transient response of the multilayered saturated poroelastic media.     

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.     

Three‐dimensional finite elements for the analysis of soil contamination using a multiple‐porosity approach

A three‐dimensional finite‐element model of contaminant migration in fissured clays or contaminated sand which includes multiple sources of non‐equilibrium processes is proposed. The conceptual framework can accommodate a regular network of fissures in 1D, 2D or 3D and immobile solutions in the macro‐pores of aggregated topsoils, as well as non‐equilibrium sorption. A Galerkin weighted‐residual statement for the three‐dimensional form of the equations in the Laplace domain is formulated. Equations are discretized using linear and quadratic prism elements. The system of algebraic equations is solved in the Laplace domain and solution is inverted to the time domain numerically. The model is validated and its scope is illustrated through the analysis of three problems: a waste repository deeply buried in fissured clay, a storage tank leaking into sand and a sanitary landfill leaching into fissured clay over a sand aquifer. Copyright © 2005 John Wiley & Sons, Ltd.     

Exact solutions for one‐dimensional consolidation of single‐layer unsaturated soil

Based on the Fredlund consolidation theory of unsaturated soil, exact solutions of the governing equations for one‐dimensional consolidation of single‐layer unsaturated soil are presented, in which the water permeability and air transmission are assumed to be constants. The general solution of two coupled homogeneous governing equations is first obtained. This general solution is expressed in terms of two functions psi₁ and ψ₂, where ψ₁ and ψ₂, respectively, satisfy two second‐order partial differential equations, which are in the same form. Using the method of separation of variables, the two partial differential equations are solved and exact solutions for three typical homogeneous boundary conditions are obtained. To obtain exact solutions of nonhomogeneous governing equations with three typical nonhomogeneous boundary conditions, the nonhomogeneous boundary conditions are first transformed into homogeneous boundary conditions. Then according to the method of undetermined coefficients and exact solutions of homogenous governing equations, the series form exact solutions are put forward. The validity of the proposed exact solutions is verified against other analytical solutions in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Axisymmetric thermal consolidation of multilayered porous thermoelastic media due to a heat source

This paper presents an analytical layer element solution to axisymmetric thermal consolidation of multilayered porous thermoelastic media containing a deep buried heat source. By applying the Laplace–Hankel transform to the state variables involved in the basic governing equations of porous thermoelasticity, the analytical layer elements that describe the relationship between the transformed generalized stresses and displacements of a finite layer and a half‐space are derived. The global stiffness matrix equation is obtained by assembling the interrelated layer elements, and the real solutions in the physical domain are achieved by numerical inversion of the Laplace–Hankel transform after obtaining the solutions in the transformed domain. Finally, numerical calculations are performed to demonstrate the accuracy of this method and to investigate the influence of heat source's types, layering, and the porous thermoelastic material parameters on thermal consolidation behavior. Copyright © 2015 John Wiley & Sons, Ltd.