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.     

Analysis of dynamic spherical cavity expansion in undrained modified Cam Clay soil

In order to capture the influence of the cavity expansion velocity, this paper presents a semianalytical solution for dynamic spherical cavity expansion in modified Cam Clay (MCC) soil. The key problem is solving the six coupled partial differential equations (PDEs) of cavity expansion, in which the dynamic term is considered in the stress equilibrium equation. The similarity transformation technique is used to transform the PDEs into ordinary differential equations (ODEs). Subsequently, the numerical method using the function “ODE45” in MATLAB is selected to solve the ODEs, which allows the stress and excess pore pressure around the expanding spherical cavity wall to be obtained. The proposed semianalytical solution for dynamic spherical cavity expansion was validated by comparting the degenerate solution with the published quasistatic solution for the MCC model. Parametric study was then conducted to capture the influence of the cavity wall velocity on the cavity expansion response. The proposed solution has potential application to geotechnical problems such as dynamic pile driving, the dynamic cone penetration test, and so forth.     

直角域中凸起和孔洞对SH波的散射与地震动

按照波函数展开法和镜像方法,对直角域中半圆形凸起和圆形孔洞对SH波的散射进行了分析,得到其稳态解。对含孔洞和凸起的直角域做分区,等效为一个含孔洞与凹陷的直角域和一个圆域的契合,其在分界面上满足位移和应力的连续性条件,即契合条件,分别构造两个区域内的位移波函数,按照孔洞边界柱面上的应力自由和契合条件定解波函数展开式的系数。按Fourier级数展开法,得到定解条件的线性代数方程组,截断求解,进而得到问题的解析解。数值算例给出圆形孔洞边沿动应力和地表位移幅值的分布情况,得到直角域自由边界、凸起、孔洞对散射和地震动的影响。     

Application of the equations of radioactive growth and decay to geochronological models and explicit solution of the equations by Laplace transformation

A recently developed method of pore-fluid age determination assumes secular equilibrium in the ²³⁸U decay chain. The efficacy of this approximation is investigated using computer evaluations of the equations that give the time evolution of the ²³⁸U decay chain, i.e. the solution of the equations of radioactive growth and decay. This analysis is performed considering two alternative geochemical scenarios to that of secular equilibrium — only ²³⁸U present initially and ²³⁸U and ²³⁴U present initially. In addition, the effects of the ²³⁵U decay chain are also determined in a similar fashion. These particular examples were chosen to show that more sophisticated geochronological models for many dating applications can be developed using such computer calculations. To facilitate such analyses, a solution of the equations of radioactive growth and decay for an arbitrary initial condition is derived using the Laplace transformation method and matrix algebra. Other solutions — both general and special — that are found in some well-known textbooks are reviewed.     

Undrained expansion of a cylindrical cavity in clays with fabric anisotropy: theoretical solution

This paper presents a novel, exact, semi-analytical solution for the quasi-static undrained expansion of a cylindrical cavity in soft soils with fabric anisotropy. This is the first theoretical solution of the undrained expansion of a cylindrical cavity under plane strain conditions for soft soils with anisotropic behaviour of plastic nature. The solution is rigorously developed in detail, introducing a new stress invariant to deal with the soil fabric. The semi-analytical solution requires numerical evaluation of a system of six first-order ordinary differential equations. The results agree with finite element analyses and show the influence of anisotropic plastic behaviour. The effective stresses at critical state are constant, and they may be analytically related to the undrained shear strength. The initial vertical cross-anisotropy caused by soil deposition changes towards a radial cross-anisotropy after cavity expansion. The analysis of the stress paths shows that proper modelling of anisotropic plastic behaviour involves modelling not only the initial fabric anisotropy but also its evolution with plastic straining.     

Similarity solution for cavity expansion in thermoplastic soil

This paper presents an analytical solution for cavity expansion in thermoplastic soil considering non‐isothermal conditions. The constitutive relationship of thermoplasticity is described by Laloui's advanced and unified constitutive model for environmental geomechanical thermal effect (ACMEG‐T), which is based on multi‐mechanism plasticity and bounding surface theory. The problem is formulated by incorporating ACMEG‐T into the theoretical framework of cavity expansion, yielding a series of partial differential equations (PDEs). Subsequently, the PDEs are transformed into a system of first‐order ordinary differential equations (ODEs) using a similarity solution technique. Solutions to the response parameters of cavity expansion (stress, excess pore pressure, and displacement) can then be obtained by solving the ODEs numerically using mathematical software. The results suggest that soil temperature has a significant influence on the pressure‐expansion relationships and distributions of stress and excess pore pressure around the cavity wall. The proposed solution quantifies the influence of temperature on cavity expansion for the first time and provides a theoretical framework for predicting thermoplastic soil behavior around the cavity wall. The solution found in this paper can be used as a theoretical tool that can potentially be employed in geotechnical engineering problems, such as thermal cone penetration tests, and nuclear waste disposal problems.     

Finite layer analysis of consolidation. I

A method for the analysis of the consolidation of a horizontally layered soil under plane conditions is developed. The method depends upon the transformation of the governing equations by a Fourier trasform. This transformation has the effect of reducing the partial differential equations of consolidation to ordinary differential equations. The ordinary differential equations are then solved using a finite layer or finite difference approach. Once the solution in the transformed plane has been found, the actual solution is synthesized by Fourier inversion. The method leads to a considerable reduction in the amount of core storage necessary for solution and enables the solution of quite significant problems to be obtained on a mini-computer.     

On the application of the maximum entropy meshfree method for elastoplastic geotechnical analysis

In this study, the Maximum Entropy Meshfree (MEM) method is employed for analysing geotechnical problems involving material nonlinearity, assuming small strains. The efficiency of the MEM method is evaluated through several solution schemes for the global governing equations as well as the local constitutive equations. The conventional implicit approach involving the Newton-Raphson method and an explicit adaptive dynamic relaxation technique are employed for solving the governing equations, while local constitutive equations are solved numerically as well as analytically. Two- and three-dimensional numerical experiments are performed to study the efficiency of different configurations of the solution scheme, which leads to some important conclusions about application of the MEM method in geotechnical problems.     

基于哈密顿体系辛几何算法求解空间地基问题

地基应力和位移场的求解是岩土工程中的基本问题之一,以往的求解方法是在一类变量范围内求解,属于拉格朗日体系。利用弹性力学的哈密顿理论,通过适当的变量代换,由力学的控制方程引入对偶变量,直接将方程导入到哈密顿体系,应用分离变量法求解。在哈密顿体系下,利用辛几何的性质,在完备的解空间内将方程的解用本征向量函数展开,讨论零本征值和非零本征值对应的不同本征解及其物理意义。数值算例表明,所得结果同以往结果一致。该方法不同于传统方法,为地基的研究提供了一条新途径和思路。     

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.     

多孔介质随机参数的相关性对非饱和水流的影响

基于van Genuchten-Mualem非饱和水分特征模型,建立了非饱和流运动的随机数值模型。将饱和水力传导度和孔隙大小分布参数视为服从对数正态分布的随机场,用Karhunen-Loeve展开分解,水头表示为混沌多项式展开。通过摄动方法得到一系列关于水头展开的偏微分方程,并用有限差分法进行求解。应用本文的模型分析了两随机场在统计不相关和完全相关模式下对水流随机分析的影响,结果表明两种模式下的水头均值相同,完全相关模式下的水头标准差较不相关模式下的明显偏小。     

Improved Bell's method for the stability analysis of slopes

A new procedure based on the approximation to the total normal pressure along the slip surface is developed to compute the factor of safety of slopes for slip surfaces of all shapes. By taking the whole sliding body, instead of an individual slice, as the loaded object, all the equilibrium equations are formulated according to the three‐moment equilibrium conditions rather than the two force equilibrium conditions and one‐moment equilibrium condition. The system of nonlinear equations deduced in this way is well‐scaled and enjoys excellent numerical properties such as the existence of solution with a positive factor of safety, a nearly unlimited scope of convergence and a rapid convergence rate associated with the Newton method. In the case of ?_u =0—the situation where no drainage and no consolidation are involved, furthermore, the system has a unique solution and the factor of safety has an explicit expression. Some typical examples are analyzed to demonstrate the numerical properties of the proposed procedure. Copyright © 2009 John Wiley & Sons, Ltd.     

Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

Probabilistic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell’s equations as used in the magnetotelluric method. The domain is split into non-overlapping sub-domains and the solution on the sub-domain boundaries is obtained by evaluating the stochastic form of the exact solution of Maxwell’s equations by a Monte-Carlo approach. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell’s equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function-based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.     

Elastic‐plastic solutions for expanding cavities embedded in two different cohesive‐frictional materials

An analytical solution of cavity expansion in two different concentric regions of soil is developed and investigated in this paper. The cavity is embedded within a soil with finite radial dimension and surrounded by a second soil, which extends to infinity. Large‐strain quasi‐static expansion of both spherical and cylindrical cavities in elastic‐plastic soils is considered. A non‐associated Mohr–Coulomb yield criterion is used for both soils. Closed‐form solutions are derived, which provide the stress and strain fields during the expansion of the cavity from an initial to a final radius. The analytical solution is validated against finite element simulations, and the effect of varying geometric and material parameters is studied. The influence of the two different soils during cavity expansion is discussed by using pressure–expansion curves and by studying the development of plastic regions within the soils. The analytical method may be applied to various geotechnical problems, which involve aspects of soil layering, such as cone penetration test interpretation, ground‐freezing around shafts, tunnelling, and mining. © 2014 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.     

Coupled solution of oil slick and depth averaged tidal currents on three-dimensional geometry of Persian Gulf

In this paper, simulation of oil spill due to tidal currents in Persian Gulf is performed by coupled solution of the hydrodynamics equations and an equation for convection and diffusion of the oil. The hydrodynamic equations utilized in this work consist of depth average equations of continuity and motion in two dimensional horizontal planes. The effect of evaporation is considered in the continuity equation and the effects of bed slope and friction, as well as the Coriolis effects are considered in two equations of motion. The overlapping cell vertex finite volume method is applied for solving the governing equations on triangular unstructured meshes. Using unstructured meshes provides great flexibility for modeling the flow in arbitrary and complex geometries, such as Persian Gulf flow domain. The results of the hydrodynamic model for tidal currents in Persian Gulf domain is examined by imposing tidal fluctuations to the main flow boundary during a limited period of time. Finally, the developed model is used to simulate an accidental oil spill from a point in Persian Gulf.     

A new model for the vibration isolation via pile rows consisting of infinite number of piles

In this study, on the basis of the Floquet transform method, a numerical model for the simulation of the vibration isolation via multiple periodic pile rows with infinite number of piles is established. By means of the fictitious pile method due to Muki and Sternberg, the second kind of Fredholm integral equations for the pile rows are developed by using the fundamental solutions for the half‐space and the compatibility conditions between the piles and half‐space. Employing the Floquet transform method, integral equations for the pile rows in the wavenumber domain are then derived. Solution of the integral equations yields the wavenumber domain solution for the pile rows. The space domain solution can then be retrieved by inversion of the Floquet transform. Numerical results show that the proposed model with the Floquet transform method is in a good agreement with those of the conventional direct superposition method. On the basis of the new model, influences of the spacing between neighboring piles, the Young's modulus of the piles, and the pile length on the vibration isolation effect of the pile rows are investigated. Numerical simulations conducted in this study show that compared with the direct superposition method, the efficiency of the proposed model for simulation of the vibration isolation via pile rows is very high. Copyright © 2012 John Wiley & Sons, Ltd.     

基于同伦分析法的非饱和土层固结问题的级数解

由于非饱和土的渗透系数是基质吸力的函数,使得控制方程带有强非线性的特征,进而使得控制方程的解析求解变得十分困难。同伦分析法对级数基函数和辅助线性算子的选择具有更大的自由性、灵活性,且收敛性的控制和调节更加容易实现,求解强非线性微分方程时在选择线性算子以及辅助参数上具有明显的优势。因此,针对非饱和土固结方程的非线性特征,对于处于地表浅层的非饱和土层,假设孔隙气压力为大气压力,在Richard经验公式与非饱和土一维固结理论的基础上,推导了非饱和一维固结无量纲控制方程;应用同伦分析法,通过选取适当的初始猜测解与辅助参数,将该非线性方程转换为线性的微分方程组并求解得到固结问题的级数解。此外,以压实高岭土为研究对象,在收集相关试验参数基础之上,将由同伦分析法求得的固结问题的近似解析解与有限差分法数值结果相对比,分析结果验证了解析解的正确性。     

A local solution for deterministic and stochastic transport equations

A local method is developed for finding the hydraulic head at an arbitrary point of a soil deposit with conductivity varying randomly in space. The method is said to be local since it delivers the hydraulic head at an arbitrary point of a soil deposit directly rather than extracting it from a field solution. The local method always converges to the exact solution, is ideal for parallel computation, and is simple to implement. The method is applied to solve locally one-dimensional transport equations with mixed boundary conditions, calculate corresponding effective conductivity, and examine size effect in specimens with hydraulic conductivity varying randomly in space.     

Extended precise integration method for axisymmetric thermo‐elastic problem in transversely isotropic material

This paper presents a numerical solution for the analysis of the axisymmetric thermo‐elastic problem in transversely isotropic material due to a buried heat source by means of extended precise integral method. By virtue of the Laplace–Hankel transform applied into the basic governing equations, an ordinary differential matrix equation is achieved, which describes the relationship between the generalized stresses and displacements in transformed domain. An extended precise integration method is introduced to solve the aforementioned matrix equation, and the actual solution in the physical domain is acquired by inverting the Laplace–Hankel transform. Numerical examples are carried out to demonstrate the accuracy of the proposed method and elucidate the influence of the character of transverse isotropy, the anisotropy of linear expansion coefficient, the anisotropy of thermal diffusivity, and medium's stratification on the thermo‐elastic response. Copyright © 2015 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.