Consolidation of viscoelastic soil by vertical drains incorporating fractional-derivative model and time-dependent loading

This paper presents a general solution to the consolidation system of viscoelastic soil by vertical drains incorporating a fractional-derivative model and arbitrary time-dependent loading. The fractional-derivative Merchant model is introduced to describe the viscoelastic behavior of saturated soil around the vertical drains. Based on this model, the governing partial differential equation of a consolidation system incorporating vertical and horizontal drainage is obtained for the equal strain condition. Then, a general solution to the consolidation system involving arbitrary time-dependent loading is derived using the Laplace transform technique and eigenfunction expansion method. Further, two comparisons are presented to verify the exactness of the proposed solution, and the consolidation behavior involving four time-dependent loadings is illustrated and discussed.     

Prediction of swelling rocks strain in tunneling

Swelling deformations leading to convergence of tunnels may result in significant difficulties during the construction, in particular for long term use of tunnels. By extracting an experimental based explicit analytical solution for formulating swelling strains as a function of time and stress, swelling strains are predicted from the beginning of excavation and during the service life of tunnel. Results obtained from the analytical model show a proper agreement with experimental results. This closed-form solution has been implemented within a numerical program using the finite element method for predicting time-dependent swelling strain around tunnels. Evaluating effects of swelling parameters on time-dependent strains and tunnel shape on swelling behavior around the tunnel according to this analytical solution is considered. The ground-support interaction and consequent swelling effect on the induced forces in tunnel lining is considered too. Effect of delay in lining installation on swelling pressure which acting on the lining and its structural integrity, is also evaluated. A MATLAB code of “SRAP” is prepared and applied to calculate all swelling analysis around tunnels based on analytical solution.     

Large strain similarity solution for a spherical or circular opening excavated in elastic‐perfectly plastic media

This paper deals with the unloading problem of a spherical or circular opening excavated in elastic‐perfectly plastic media with a nonassociated Mohr–Coulomb yield criterion. A large strain similarity solution, using incremental velocity approach, is presented by replacing partial differential equations from stress equilibrium, constitutive law, consistency condition, and displacement equation with first‐order ordinary differential equations. The classical Runge–Kutta method is used to solve the first‐order ordinary differential equations. Comparisons among small and large strain solutions are made using some data sets of soil and rock. The results show that the displacements by large strain similarity solution are smaller than those by exact small strain solution and somewhat larger than those by large strain solution using total strain approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Evaluation of an anisotropic elastoplastic–viscoplastic bounding surface model for clays

An anisotropic time-dependent bounding surface model for clays is developed by generalizing a previous time-independent model that adopts a flexible bounding surface. It is based on the framework for coupled elastoplasticity–viscoplasticity for clays and Perzyna’s overstress theory. Three viscoplastic parameters were introduced and explained in detail. The model was validated against undrained creep tests for both isotropically and anisotropically consolidated clays, undrained and drained stress relaxation tests on some undisturbed clays, and undrained triaxial tests with varying strain rates on natural Hong Kong marine deposit clay. The general agreement between the model simulations and test results was satisfactory. The varying effects of lower-level parameters were discussed on the undrained multistage stress relaxation response for normally consolidated soils which had been ignored in literature. The flexibility of the model in capturing the shear strengths, which is the unique feature of the current model, was shown in the simulations of time-dependent triaxial tests on Taipei silty clay. All the simulations show that the proposed model is a relatively practical model considering both anisotropy and time dependency of clays.     

基于上下负荷面的弹黏塑性本构模型及应力积分算法实现

为了描述天然软土的时间相依以及结构性特征,提出了一种能考虑土体超固结和结构性的实用弹黏塑性本构模型。它以Asaoka和Hashiguchi的上下负荷面作为某一应变速率下的参考屈服面,按照相对过应力的基本思路,新引入了两个能通过不同应变速率三轴压缩试验测定的率敏性参数 和 ,建立了以当前应力、黏塑性应变以及黏塑性应变速率为状态变量的动屈服准则函数,并给出了基于Newton-Raphson迭代的应力积分算法,且成功地将其嵌入到大型有限元软件ABAQUS中。最终通过数值算例来验证模型的正确性以及应力积分算法的可靠性。结果表明：该模型能同时描述土体的率敏性、蠕变以及结构性特征,模型参数物理意义明确、易懂可测,预测结果与试验数据吻合良好,可用于复杂边值问题的有限元计算。     

一个描述软黏土时效特性的一维弹黏塑性模型

为了描述软黏土一维应力-应变关系的时效特性,基于Bjerrum的等时间线体系,提出等黏塑性应变率线概念,推导了黏塑性应变率与黏塑性应变增量的关系,建立了软黏土的一维弹黏塑性模型;从理论上分析了新建模型与3种典型的一维弹黏塑性时效本构模型的内在联系,表明新建模型与其他3种模型在本质上是等效的,且形式更简洁,物理意义更明确;利用新建模型对软黏土的固结-蠕变耦合效应、应变率效应、应力松弛效应等时效特性进行了理论分析,并得到了相应的解析解;结合宁波软黏土的一维固结试验,阐述了模型参数的确定方法,并用新建模型对宁波软黏土的固结-蠕变试验、温州软黏土的一维多级等应变率试验、香港海相软黏土的一维应力松弛试验进行模拟,验证了新建模型的有效性。研究结果表明,新建模型能很好地模拟软黏土的一维时效特性。     

Analysis of undrained cavity expansion in elasto‐plastic soils with non‐linear elasticity

A large strain analysis of undrained expansion of a spherical/cylindrical cavity in a soil modelled as non‐linear elastic modified Cam clay material is presented. The stress–strain response of the soil is assumed to obey non‐linear elasticity until yielding. A power‐law characteristic or a hyperbolic stress–strain curve is used to describe the gradual reduction of soil stiffness with shear strain. It is assumed that, after yielding, the elasto‐plastic behaviour of the soil can be described by the modified Cam clay model. Based on a closed‐form stress–strain response in undrained condition, a numerical solution is obtained with the aid of simple numerical integration technique. The results show that the stresses and the pore pressure in the soil around an expanded cavity are significantly affected by the non‐linear elasticity, especially if the soil is overconsolidated. The difference between large strain and small strain solutions in the elastic zone is not significant. The stresses and the pore pressure at the cavity wall can be expressed as an approximate closed‐form solution. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical and Laboratory Analysis of Relaxation Tests for Determining Time-Dependent Properties of Rock

Roof falls in coal mines may occur within a few months to a few years after excavation. In this paper, we proposed the use of relaxation tests as a substitute for time-dependent tests. The relation between creep behavior and relaxation behavior was numerically investigated and demonstrates that the material assigned with creep model can show relaxation behavior. Then the relaxation model was developed by modifying the Burgers creep model. Numerical simulation of a relaxation test on a simulated rock model in 3DEC yielded results that were similar to theoretical prediction. A relaxation test was performed on two groups of specimens under varying load conditions. Results from the laboratory tests validated the approach of using relaxation test to determine time-dependent properties. Finally, time-dependent properties were investigated by performing relaxation tests at different stages of a complete stress–strain curve. The relaxation test during strain-softening was unsuccessful; however, the relaxation behavior at residual stage in post-failure region was more significant than that in pre-failure region and the sudden drop in stress indicated there was strength deterioration due to the accumulation of viscous strain.     

无锡马山抽水蓄能电站地面厂房边坡稳定性分析

以无锡马山抽水蓄能电站地面厂房建筑过程中形成的人工边坡为例,在对其工程地质剖面分析并找出可能的滑动模式后,利用极限平衡法和有限元法分析了4个典型地质剖面的稳定性安全系数以及应力应变关系。结果表明,通过对该边坡的极限平衡法得到了2-2剖面的安全系数和通过有限元法得到的计算结果比较接近,且都满足相关规范的要求;坡体的最大位移发生在高程为－20 m和－5.0 m附近,其值为－22 mm;最大主应力位于高程－45.0 m处,值为0.134 MPa;最小主应力位于坡内深处,其值为－6.22 MPa。同时,提出了下一阶段应考虑时间效应,对边坡的变形进行分析。     

Analytical solution for one-dimensional consolidation of clayey soils with a threshold gradient

General approximate analytical solutions are developed for one-dimensional consolidation with consideration of the threshold gradient under a time-dependent loading. A comparison is made between the present solution and some available numerical solutions for a particular case, and the results show that the approach employed in this article is reasonable. The influence of the threshold gradient and the loading period on consolidation behaviour is investigated, and the results show that the moving boundary of seepage moves downward gradually. The greater the threshold gradient is, the slower the boundary moves. The excess pore pressure will not be completely dissipated at the end of consolidation, and the larger the threshold gradient is, the greater the residual excess pore pressure is. The average degree of consolidation considering the threshold gradient defined by settlement is different from that defined by pore pressure. Moreover, the greater the threshold gradient is, the larger the average degree of consolidation in terms of strain, whereas the smaller the average degree of consolidation in terms of stress. It is also shown that the longer the loading period is, the longer the time moving boundary takes to reach the bottom of the layer, and the greater the average degree of consolidation is.     

基于湿度应力场理论的硬石膏岩膨胀试验研究

传统的湿度应力场理论处理硬石膏围岩膨胀问题时,未反映膨胀特性与湿度状态之间的时变关系。以该理论为基础,设计硬石膏岩膨胀试验：不同初始湿度下的膨胀,不同浸水时间的膨胀,以及膨胀后的抗拉强度测试。试验发现：随初始湿度的增加,硬石膏岩膨胀持续的时间增长,最大膨胀应变和应力增大,终止时的吸水率和结晶水率增加,而自由水率近于0;相同初始湿度下,随浸水时间增加,其吸水率先快速后缓慢增加,结晶水率逐渐增大到接近于吸水率,自由水率逐渐减小到接近于0;随吸水率的增加,其最大轴向膨胀应变、最大侧向膨胀应力和抗拉强度都呈增大趋势。根据试验结果,建立了湿度状态时变系列方程以及含时间效应的膨胀本构模型。     

单轴压缩条件下硅质粉砂岩应力阈值研究

岩石裂纹闭合应力(σ_cc)、起裂应力(σ_ci)及损伤应力(σ_cd)是岩石压缩变形破坏过程中的关键应力阈值，应力阈值的确定对分析岩石宏观力学特性及研究微观裂纹演化规律具有重要意义。本文研究了硅质粉砂岩在单轴压缩条件下应力阈值确定的方法。针对硅质粉砂岩的应力-应变曲线开展不同方法的计算分析，判定不同方法下应力阈值获取的可靠性，并采用一种优化方法获取硅质粉砂岩试样的应力阈值。结果表明，裂隙体积应变法获取闭合应力与起裂应力的求解精度易受泊松比的影响，移动点回归法易受主观因素的干扰，横向应变响应法能减小人为主观误差，但其求解方法未阐释物理意义；采用基于移动点回归技术的裂隙体积应变法减小了岩石泊松比选取对应力阈值获取的干扰，且计算结果物理意义相对明确。     

Analytical Solutions for the Construction of Deeply Buried Circular Tunnels with Two Liners in Rheological Rock

The construction of underground tunnels is a time-dependent process. The states of stress and strain in the ground vary with time due to the construction process. Stress and strain variations are heavily dependent on the rheological behavior of the hosting rock mass. In this paper, analytical closed-form solutions are developed for the excavation of a circular tunnel supported by the construction of two elastic liners in a viscoelastic surrounding rock under a hydrostatic stress field. In the solutions, the stiffness and installation times of the liners are accounted for. To simulate realistically the process of tunnel excavation, a time-dependent excavation process is considered in the development of the solutions, assuming that the radius of the tunnel grows from zero until its final value according to a time-dependent function to be specified by the designers. The integral equations for the supporting pressures between rock and first liner are derived according to the boundary conditions for linear viscoelastic rocks (unified model). Then, explicit analytical expressions are obtained by considering either the Maxwell or the Boltzmann viscoelastic model for the rheology of the rock mass. Applications of the obtained solutions are illustrated using two examples, where the response in terms of displacements and stresses caused by various combinations of excavation rate, first and second liner installation times, and the rheological properties of the rock is illustrated.     

Numerical simulation of elasto-plastic electro-osmosis consolidation at large strain

In this paper, a numerical solution for the electro-osmosis consolidation of clay in multi-dimensional domains at large strains is presented, with the coupling of the soil mechanical behaviour, pore water transport and electrical fields being considered. In particular, the Modified Cam Clay model is employed to describe the elasto-plastic behaviour of clay, and some empirical expressions are used to consider the nonlinear variation of the hydraulic and electrical conductivities of the soil mass during the consolidation processes. The implementation of the theoretical model in a finite element code allows for analysis of the evolution of the transient response of the clay subjected to electro-osmosis treatment. The proposed model is verified via comparison with data from a large strain electro-osmosis laboratory test, to demonstrate its accuracy and effectiveness. Various numerical examples are also investigated to study the deformation characteristics and time-dependent evolution of the excess pore pressure. Finally, a well-documented field application of electro-osmosis is simulated to provide further verification. The results show that the numerical solution is effective in predicting the nonlinear behaviour of clay during electro-osmosis consolidation.     

Approximate time-dependent analysis of tunnel support structure considering progress of tunnel face

The pressure acting on tunnel support structures generally increases with lapse in time. The pressure increase may be caused by not only the time-dependent mechanical properties of the underground medium, but also by the progression of the tunnel face. In order to clarify these two effects, three-dimensional analysis is required. In this study, however, a method which takes into account the three-dimensional effects of the tunnel face progression in two-dimensional plane strain analyses is proposedby introducing the ‘equivalent intial stress’. The proposed method can be easily applied to time-dependent analyses of the behaviour of tunnel support structures installed in a visco-elastic medium. The tunnels considered here are circular in shape, being driven in homogeneous isotropic linear visco-elastic media having hydrostatic initial stresses. In the first part of this paper, detailed discussions of the proposed equivalent initial stress are given. The second deals with the mathematical formulations for obtaining closed-form solutions for the pressure acting on tunnel support structures. In the third, numerical results and discussions are given, and special attetion is paid to the effects of both the time-dependent mechanical properties of the material and the tunnel face progression. Finally, the theoretical results obtained here are utilized for interpretation of field measurements.     

The creep of potash salt rocks from Saskatchewan

Summary The results of creep tests on the Esterhazy-and the Patience-Lake-types of potash salt rocks from Saskatchewan, Canada are presented. The investigations involved over 6 years of time-dependent experiments in uniaxial compression using potash from the Rocanville and the Lanigan mines of the Potash Corporation of Saskatchewan. A creep test at a given load would last from 2 to 8 months, with most tests conducted over a 4-month period.Since the yield stress of both types of potash lies between 9 and 11 MPa, there is very little creep below 11 MPa. Between 11 and 13 MPa, creep strain production increases sharply through plastic deformation. Above about 13 MPa, however, plastic creep is dominated by brittle creep caused by microcracking. As a result, the lateral and volume creep strain curves may then display the transient and the steady-state, or all three stages of creep, while the axial strain, which is not affected by microcracking, usually attenuates for the whole duration.Two different interpretations of the results are offered. Identifying the last (the fourth) month of testing with the steady-state model, the stress dependence of the steady-state rate has been established for both rock types. A unimodal rate model for the axial strain and a bimodal model for the lateral strain are suggested. The alternate interpretation proceeds on the assumption that under 13 MPa, both the axial and the lateral strain can be modelled through the power function formulation of transient strain.     

A method of time-dependent analysis using elastic solutions for nonlinear materials

The formulation of viscoelastic solutions from elastic equations using the ‘correspondence principle’ and an inverse Laplace transform has been discussed extensively in the literature. Because this method has been developed, many time-dependent solutions can be obtained from closed form elastic solutions and conditions have been delineated in which the ‘quasi-elastic’ approximation of the viscoelastic solution is within acceptable tolerance. This communication shows the feasibility of the application of these methods to formulate approximate nonlinear viscoelastic solutions with nonlinear stress-strain materials, and for want of a specific nonlinear model to demonstrate this, the hyperbolic model was selected. The ‘power law’ is used to model the relaxation modulus of the viscoelastic materials. There are five related development that are discussed here using a simple numerical example to illustrate each of them and they are: (1) a linear elastic solution, (2) a linear viscoelastic solution, (3) a nonlinear elastic solution, (4) a nonlinear viscoelastic solution and finally, (5) a ‘regression’ approximation of the nonlinear viscoelastic solution which is suggested by the series form of the elastic solution. All of these are related to one another and each provides an acceptably accurate solution of the problem it addresses. The latter is of particular practical interest since it can be used to provide answers to problems involving nonlinear viscoelastic materials while requiring only very small calculation times. The problem used as an example is the calculation of the displacement of a circular hole in an infinite plate made of a material with a nonlinear time-dependent stress-strain relationship. The nonlinear elastic form of the solution was developed by matching results from nonlinear finite element analysis.     

Non‐linear analysis of shear band formation in sand

The paper investigates the incipience of shear band with an incrementally non‐linear constitutive equation. Necessary conditions for the emergence of shear band are derived. The lower bound solution is obtained by taking the strain rate inside and outside the shear band into consideration. Numerical results of localized bifurcation for general stress and strain are presented and compared with experiments. In the principal stress space, the stresses at the onset of shear band form a surface, which is partially enclosed by the failure surface for homogeneous straining. The significance of the analysis for identification of the material parameters and verification of the constitutive model against experiments is discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

Characterising and modelling the excavation damaged zone in crystalline rock in the context of radioactive waste disposal

This paper describes current knowledge about the nature of and potential for thermo–hydro–mechanical–chemical modelling of the excavation damaged zone (EDZ) around the excavations for an underground radioactive waste repository. In the first part of the paper, the disturbances associated with excavation are explained, together with reviews of Workshops that have been held on the subject. In the second part of the paper, the results of a DECOVALEX [DEmonstration of COupled models and their VALidation against EXperiment: research funded by an international consortium of radioactive waste regulators and implementers (http://www.decovalex.com)] research programme on modelling the EDZ are presented. Four research teams used four different models to simulate the complete stress–strain curve for Avro granite from the Swedish Äspö Hard Rock Laboratory. Subsequent research extended the work to computer simulation of the evolution of the repository using a ‘wall-block model’ and a ‘near-field model’. This included assessing the evolution of stress, failure and permeability and time-dependent effects during repository evolution. As discussed, all the computer models are well suited to sensitivity studies for evaluating the influence of their respective supporting parameters on the complete stress–strain curve for rock and for modelling the EDZ.     

Formulation and validation of a constitutive model for sands in monotonic shear

A constitutive model for sands in monotonic shear is presented. The model is designed to simulate the behavior of sands in the whole stress and strain range of engineering interest with enough accuracy for practical usage. Material parameters were chosen to be state independent and easy to calibrate using conventional testing procedures. The formulation is based on effective stresses, pressure-dependent hyperelasticity, non-associative elastoplasticity, an isotropic hardening law and Rowe’s stress-dilatancy theory. The implementation of Rowe’s stress-dilatancy theory within the framework of elastoplasticity theory is discussed. It is found that Rowe’s theory produces a volumetric plastic strain rate function that has a discontinuity in its first derivative w.r.t. stress, and a smoothed form is proposed instead. Finally, some experimental tests are simulated and the results are briefly discussed.