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 axial loading of a rigid circular anchor plate embedded in an elastic half-space

The present paper examines the axisymmetric problem related to the loading of a rigid circular anchor plate which is embedded in bonded contact with an isotropic elastic half-space. A Hankel transform development of the governing equations is used to reduce the associated mixed boundary value problem to a set of coupled Fredholm integral equations of the second kind. These equatons are solved in a numerical fashion to generate results of engineering interest. In particular, the results indicate the influence of the depth of embedment on the axial stiffness of the rigid anchor plate.     

The influence of a boundary fracture on the elastic stiffness of a deeply embedded anchor plate

The problem of the axial loading of a rigid disk-shaped anchor plate embedded in an isotropic elastic medium of infinite extent is examined. At the boundary of the disk anchor plate the elastic medium contains a cracked region of finite extent. The presence of the cracked region decreases the elastic stiffness of the anchor plate. The mathematical formulation of the problem is developed, and a numerical scheme is presented which can be used to solve the resulting coupled integral equations. The numerical technique is used to evaluate the results, which illustrate the manner in which the elastic stiffness of the anchor plate is influenced by the extent of cracking. Similar results are developed for the flaw shearing mode stress intensity factor at the external boundary of the cracked region.     

A theoretical estimate of the tilt of a surface foundation due to an inclined anchor load applied at the interior of the soil medium

This paper examines the problem of the interaction between a loaded rigid circular foundation located at the surface of an isotropic elastic halfspace and an inclined concentrated anchor load which is located at a finite depth along the axis of symmetry. Such inclined loads can be induced by, for example, anchor regions supporting earth retaining structures. The loaded rigid circular foundation resting in smooth contact with the elastic soil mass experiences a displacement and a tilt due to the action of the inclined anchor load. The magnitude of the rotational settlement is evaluated in exact closed form.     

Avoiding Singularities in the Numerical Solution of the Motion of a Deformable Ellipse Immersed in a Viscous Fluid

Geological materials are largely heterogeneous and are typically comprised of approximately ellipsoidal objects immersed in a matrix with different physical properties. Methodologies for the identification of ancient regional tectonic patterns may be developed based on an understanding of the behaviour of heterogeneous materials. In this contribution, the differential equation governing the rotation of a deformable ellipse immersed in a viscous fluid is considered and is found to contain a singularity when the ellipse becomes circular in shape. This problem is avoided by reformulating the equations using the standard algebraic representation of an ellipse. Thus, the equations can be numerically solved without difficulty.     

The time–deflection behaviour of a rigid under-reamed anchor in a deep clay layer

The behaviour of an under-reamed anchor in a consolidating soil is examined by approximating the anchor as an impermeable circular plate embedded in a deep soil layer. Hankel and Laplace transforms are applied to the equations governing the consolidation process, and this greatly simplifies the equations, allowing a solution to be obtained for the transformed variables. Numerical inversion of both the Hankel and Laplace transforms is used to obtain the solution at any time. A particular feature of the solution method is that the unknown contact stress between the anchor and the soil and the unknown flows in the plane of the anchor are approximated as a series of simple functions with unknown coefficients. By determining the coefficients of the terms in the series, the complete solution may be found. Computations have been carried out using the method proposed, and results are presented for the time–settlement behaviour of an impermeable anchor. These results are compared with some published and some recomputed finite element data, and this highlights some of the difficulties encountered in using such numerical techniques.     

Finite-element modelling of folding and faulting

The paper is concerned with the finite-element modelling of geologic phenomena. More specifically, the paper presents numerical models of folding and sait doming. A (penalty-) finite-element model based on the equations governing a viscous incompressible fluid is used to simulate the folding is used to simulate the folding of a viscous layer in a less viscous matrix and the salt dome formation, and to predict the location of active faulting and deformation in prograding delta systems on continentalmargins. The initiation and development of drape folding and associated faulting are also modelled. The United States continental margin of the Gulf of Mexico is used as an example of prograding delta system. The finite-element predictions seem to agreewith the published results.     

单层不可压缩饱和多孔介质一维瞬态响应精确解

基于Biot理论,考虑惯性、黏滞和机械耦合作用,假定固体颗粒和流体均不可压缩,得到了表面任意竖向荷载作用下单层饱和多孔介质一维瞬态响应的精确解。导出了以固体骨架位移表示的无量纲控制方程,并将边界条件齐次化。求解对应无黏滞耦合作用的特征值问题,得到一组满足齐次边界条件、关于空间坐标的正交函数基。利用变异系数法和基函数的正交性,得到一系列相互解耦的、关于时间的二阶常微分方程及相应的初始条件,并采用状态空间法求解常微分方程,得到位移分量。对整体平衡方程关于空间坐标积分,根据边界条件可确定总应力,并进而求得孔隙压力。通过算例验证所得解法的正确性     

A Velocity-Based Approach to Visco-Elastic Flow of Rock

In structural geology, viscous creep is generally recognized as the major deformation mechanism in the folding of rock layers through geological time scales of hundreds of thousands of years. Moreover, since deformation of rock salt by creep takes already place on relatively small time scales—weeks to months, say—creep is a relevant phenomenon when studying salt mining, notably the convergence of mine cavities and the land subsidence caused by it. While creep is the dominant process on relatively long time scales, elasticity plays a dominant role in processes that take place on relatively short time scales. The elastic response to a stress is a displacement; the shape of the rock is deformed instantaneously with respect to its initial shape. However, the viscous response of a rock to a stress is a relatively low velocity in the order of millimeters per months or years, say. In this paper we consider the two deformation phenomena creep and elasticity. In general, elasticity is a compressible phenomenon, while creep is incompressible. Here we approximate creep by the introduction of a negligibly small amount of compressibility, which makes creep velocity calculations similar to conventional elastic displacement calculations. Using this procedure, a standard finite element package for elasticity can be applied to viscous problems, also in combination with elasticity. The method has been demonstrated to upscaling of creep viscosities.     

Propagation of a hydraulic fracture parallel to a free surface

This paper analyses the plane strain problem of a fracture, driven by injection of an incompressible viscous Newtonian fluid, which propagates parallel to the free surface of an elastic half‐plane. The problem is governed by a hyper‐singular integral equation, which relates crack opening to net pressure according to elasticity, and by the lubrication equations which describe the laminar fluid flow inside the fracture. The challenge in solving this problem results from the changing nature of the elasticity operator with growth of the fracture, and from the existence of a lag zone of a priori unknown length between the crack tip and the fluid front. Scaling of the governing equations indicates that the evolution problem depends in general on two numbers, one which can be interpreted as a dimensionless toughness and the other as a dimensionless confining stress. The numerical method adopted to solve this non‐linear evolution problem combines the displacement discontinuity method and a finite difference scheme on a fixed grid, together with a technique to track both crack and fluid fronts. It is shown that the solution evolves in time between two asymptotic similarity solutions. The small time asymptotic solution corresponding to the solution of a hydraulic fracture in an infinite medium under zero confining stress, and the large time to a solution where the aperture of the fracture is similar to the transverse deflection of a beam clamped at both ends and subjected to a uniformly distributed load. It is shown that the size of the lag decreases (to eventually vanish) with increasing toughness and compressive confining stress. Copyright © 2005 John Wiley & Sons, Ltd.     

深隧道围岩分区破裂的数学模拟

该研究为先前深隧道围岩分区破裂现象内变量梯度塑性模型的进一步发展。利用应变梯度模型研究了深隧道围岩的分区破裂现象。作为额外的状态变量,在此引入应变梯度这一新变量。利用虚功原理得到了深隧道围岩的平衡方程、边界条件和流动准则,利用Clausius-Duhem不等式获得了岩体的本构方程。对于圆形深隧道,由上述模型的一般方程得到了弹性变形情况下、具有下降段的弹塑性变形情况下和不考虑弹性变形的塑性变形情况下圆形深隧道围岩的支配方程,得到了解析解,并讨论了解析解的性质。这一模型不仅扩展了隧道围岩的经典弹塑性模型,也为下一步数值研究深隧道围岩的分区破裂现象奠定了理论基础。     

A description of nucleation of folds and boudins in terms of vorticity

For slow plane flow of an incompressible viscous fluid, vorticity is a potentially valuable interpretational aid, being simply related both to velocity (via the Biot-Savart Law) and to pressure (via the equations of motion). Using semi-qualitative arguments based on the equations of motion, the continuity conditions for velocity and traction, and an assumption compatible with the small-amplitude formulation, it is possible to determine the essential characteristics of the vorticity generated near a small deflection on an otherwise flat interface of a viscous layer subjected to compression or extension. By application of the Biot-Savart Law, this vorticity distribution is seen to correspond to a perturbing velocity field which is responsible for initiation of new deflections on both interfaces. These secondary deflections give rise, in turn, to additional vorticity. For compression of a layer which is more viscous than its surroundings, there is mutual enhancement between vorticity and deflection geometry, resulting in a “runaway” nucleation (and amplification) of folds at everincreasing distances from the original deflection. The response of a more-viscous layer under extension is more restrained, owing to a degree of destructive interference between the vorticity generated at the upper and lower interfaces; the mechanism for nucleation is essentially the same as for folding, but the resulting pinch-and-swell structure is always small in amplitude.     

The elastic response of multiple underream anchors

A technique is developed for the analysis of multiple underream anchor systems resting in an elastic soil. This technique may be used to consider anchor systems involving arbitrary anchor inclination and depth beneath the soil surface, as well as arbitrary number, shape, size and spacing of underreams. The approach is largely analytical in nature and involves only a fraction of the computation required for a finite element analysis. Consideration is given to the effects of anchor depth and inclination to the soil surface, and the spacing and number of underreams upon the elastic response of anchor systems. On the basis of the result from this study, a simple, approximate method for estimating the response of multiple underream anchors is proposed. This approach involves the use of several interaction charts, which are presented in the paper, and can be used as a hand method for estimating the load–displacement behaviour of quite general anchor systems to sufficient accuracy for most practical purposes. The use of the approximate approach is illustrated by an example.     

上覆单相弹性层饱和地基上刚性条形基础竖向振动的混合边值问题

根据Biot动力控制方程,运用Fourier积分变换技术,并按照混合边值条件和连续条件建立了上覆单相弹性层饱和地基上刚性条形基础竖向振动的对偶积分方程,并将其退化到完全饱和地基的情形。通过引进正交多项式将对偶积分方程化为线性代数方程组,从而得到了上覆单相弹性层的饱和地基上刚性条形基础的竖向振动规律。通过算例分析得到,单相弹性层的厚度对动力柔度系数有着较大的影响,在单相弹性层厚度较小时（小于条形基础半宽的0.1）,动力柔度系数曲线与完全饱和的基本重合;完全饱和地基上刚性基础的竖向振动是上覆弹性层厚度Hn=0的特例。     

An indirect boundary integral equation method for poroelasticity

This paper presents an indirect boundary integral equation method for analysis of quasi-static, time-harmonic and transient boundary value problems related to infinite and semi-infinite poroelastic domains. The present analysis is based on Biot's theory for poroelastodynamics with fluid viscous dissipation. The solution to a given boundary value problem is reduced to the determination of intensities of forces and fluid sources applied on an auxiliary surface defined interior to the surface on which the boundary conditions are specified. A coupled set of integral equations is established to determine the intensities of forces and fluid sources applied on the auxiliary surface. The integral equations are solved numerically in the Laplace domain for quasi-static and transient problems, and in the frequency domain for time-harmonic excitations. The kernel functions of the integral equation correspond to appropriate Green's functions for a poroelastic full space or half-space. The convergence and numerical stability of the present scheme are established by considering a number of bench mark problems. The versatility of the present method is demonstrated by studying the quasi-static response of a rigid spheroidal anchor, and time-harmonic and transient response of a rigid semi-circular tunnel.     

A method of analysis for horizontally embedded anchors in an elastic soil

A general method is presented for the analysis of horizontally embedded anchors in an elastic soil. Provision is made in the analysis for the consideration of anchor shape, layer depth, anchor–soil interface condition, breakaway of the anchor from the underlying soil and interaction between groups of anchors. Application of the analytical technique is illustrated for strip and circular anchors, and these solutions are presented in the form of influence charts which may be used directly in hand calculations to predict the elastic load deflection behaviour, of anchor plates for a wide variety of material and geometric conditions.     

Response of circular footings and anchor plates in non-homogeneous elastic soils

The axisymmetric elastic response of circular footings and anchor plates in a linearly non-homogeneous elastic soil is analysed. It is assumed that footings/anchors are flexible and subjected to axisymmetric vertical loads. The response of the footing/anchor is modelled by using the classical Poisson–Kirchhoff thin plate theory. A variational technique is used to analyse the interaction problem. A representation for the contact stress is established by using a fundamental solution corresponding to a unit vertical pressure acting over an annular region in the interior of the non-homogeneous soil. The fundamental solution can be derived by using rigorous analytical procedures. The influence of the footing flexibility and the degree of soil non-homogeneity on the displacements, bending moments and contact stresses of a surface footing is examined over a wide range of governing parameters. In the case of anchor plates the influence of depth of embedment, degree of soil non-homogeneity and anchor flexibility on the anchor displacement is investigated.     

层状饱和地基上刚性圆板的扭转振动及参数分析

用解析的方法首次研究了层状饱和地基上刚性圆板的扭转振动特性.运用Hankel变换求解饱和介质动力问题的控制方程,按混合边值条件建立了层状饱和地基上刚性圆板扭转振动的第二类Fredholm积分方程.数值算例给出了动力柔度系数和扭转角幅值随无量纲频率的变化曲线,与单相弹性及匀质饱和介质情况进行了对比分析,并进行了参数研究.     

Analysis of fully coupled thermomechanical behaviour around a rigid cylindrical heat source buried in clay

The problem of fully coupled consolidation and heat flow around a rigid cylindrical heat source buried in clay has been studied. The governing equations of the problem are summarized in the paper and a finite element time-marching scheme to obtain an approximate solution to the governing equations is described. The stress–strain behaviour of the skeleton of the saturated soil has been represented by both a linear elastic model and the modified Cam clay soil model. The results of a limited parametric study are presented with the aim of understanding the major mechanisms of soil behaviour close to buried canisters of hot radioactive waste. A range of soil properties has been included in the study, and the effects of soil disturbance during canister emplacement have also been considered.     

A finite-element treatment of sea ice dynamics for different ice rheologies

In this paper the effects of four different rheologies on the evolution of a large-scale sea ice pack are determined and compared. Two rheologies are of viscous-plastic form, and two are viscous fluid relations. The initial pack domain is rectangular, and the motion is driven by wind stress and resisted by ocean drag. Two adjacent edges are rigid shore boundaries, and the other two are free boundaries at open water which move during the pack motion, so that the pack domain changes in time. Two different forms of boundary conditions at the rigid shore edges are considered, which also influence the evolution. The governing equations are solved numerically using a finite-element method, and, unlike previous numerical treatments, no artificial viscosity is incorporated to stabilise the algorithm near interfaces between converging and diverging flow. Instability arises when any tensile stress is abruptly cut-off when diverging flow is initiated, and an alternative view is offered. © 1998 John Wiley & Sons, Ltd.