Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic–plastic soil constitutive model

Simulation of large deformation and post‐failure of geomaterial in the framework of smoothed particle hydrodynamics (SPH) are presented in this study. The Drucker–Prager model with associated and non‐associated plastic flow rules is implemented into the SPH code to describe elastic–plastic soil behavior. In contrast to previous work on SPH for solids, where the hydrostatic pressure is often estimated from density by an equation of state, this study proposes to calculate the hydrostatic pressure of soil directly from constitutive models. Results obtained in this paper show that the original SPH method, which has been successfully applied to a vast range of problems, is unable to directly solve elastic–plastic flows of soil because of the so‐called SPH tensile instability. This numerical instability may result in unrealistic fracture and particles clustering in SPH simulation. For non‐cohesive soil, the instability is not serious and can be completely removed by using a tension cracking treatment from soil constitutive model and thereby give realistic soil behavior. However, the serious tensile instability that is found in SPH application for cohesive soil requires a special treatment to overcome this problem. In this paper, an artificial stress method is applied to remove the SPH numerical instability in cohesive soil. A number of numerical tests are carried out to check the capability of SPH in the current application. Numerical results are then compared with experimental and finite element method solutions. The good agreement obtained from these comparisons suggests that SPH can be extended to general geotechnical problems. Copyright © 2008 John Wiley & Sons, Ltd.     

A two‐phase SPH model for debris flow propagation

This paper presents a model which can be used for fast landslides where coupling between solid and pore fluid plays a fundamental role. The proposed model is able to describe debris flows where the difference of velocities between solid grains and fluid is important. The approach is based on the mathematical model proposed by Zienkiewicz and Shiomi, which is similar to those of Pitman and Le and Pudasaini. The novelty of the present work is the numerical technique used, the smoothed particle hydrodynamics (SPH). We propose to use a double set of nodes for soil and water phases, the interaction between them being described by a suitable drag law. The paper presents both mathematical and numerical models, describing the main assumptions and their limitations. Then, the model is applied to (1) a simple case where shocks and expansion waves appear, (2) a dam break problem on a horizontal plane with a frictional soil phase, and (3) a debris flow which happened in Hong Kong. The main conclusions that can be drawn from the applications are:

1. Debris flows having 2 phases with important relative mobility present a rich structure of shocks and rarefaction waves, which has to be properly modeled. Otherwise, the model will have numerical damping or dispersion.
2. Dambreak exercises provide interesting information in simple and controlled situations. We can see how both phases move relative to each other.
3. Real debris flows can be simulated with the proposed model, obtaining reasonable results.

     

Estimation of a surface with known discontinuities for automatic contouring purposes

As an aid to automatic contouring, a method has been designed to estimate a surface with known discontinuities. Through a primary estimation process the surface is assumed to be known at the nodes of a regular grid. These known values are termed data points. A distance r_EI is defined between a known data point I, and any given point E to be estimated. This distance is increased, as explained, by a penalty if one discontinuity (or more) is found in between these points. At each data point I the tangential quadric to the surface is computed, using the data points K in a neighborhood of I, which will be defined. The value at E is then a linear combination of these tangential quadrics in the neighborhood of E. For this combination, a weighting p_EI is used, related as a decreasing function of r_EI. The underlying assumptions of the method are given, together with the way to compute quadrics and the properties of the weights p_EI. An application package has been written in FORTRAN and two examples are given in this paper.     

Frictional contact algorithms in SPH for the simulation of soil–structure interaction

Simulation of frictional contact between soils and rigid or deformable structure in the framework of smoothed particle hydrodynamics (SPH) is presented in this study. Two algorithms are implemented into the SPH code to describe contact behavior, where the contact forces are calculated using the law of conservation of momentum based on ideal plastic collision or using the criteria of partial penetrating. In both algorithms, the problem of boundary deficiency inherited from SPH is properly handled so that the particles located at contact boundary can have precise acceleration, which is critical for contact detection. And the movement and rotation of the rigid structure are taken into account so that it is easy to simulate the process of pile driving or movement of a retaining wall in geotechnical engineering analysis. Furthermore, the capability of modeling deformability of a structure during frictional contact simulations broadens the fields of SPH application. In contrast to previous work dealing with contact in SPH, which usually use particle‐to‐particle contact or ignoring sliding between particles and solid structure, the method proposed here is more efficient and accurate, and it is suitable to simulate interaction between soft materials and rigid or deformable structures, which are very common in geotechnical engineering. A number of numerical tests are carried out to verify the accuracy and stability of the proposed algorithms, and their results are compared with analytical solutions or results from finite element method analysis. Good agreement obtained from these comparisons suggests that the proposed algorithms are robust and can be applied to extend the capability of SPH in solving geotechnical problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A depth‐integrated,coupled SPH model for flow‐like landslides and related phenomena

In the past decades, flow‐like catastrophic landslides caused many victims and important economic damage around the world. It is therefore important to predict their path, velocity and depth in order to provide adequate mitigation and protection measures. This paper presents a model that incorporates coupling between pore pressures and the solid skeleton inside the avalanching mass. A depth‐integrated, coupled, mathematical model is derived from the velocity–pressure version of the Biot–Zienkiewicz model, which is used in soil dynamics. The equations are complemented with simple rheological equations describing soil behaviour and are discretized using the SPH method. The accuracy of the model is assessed using a series of benchmarks, and then it is applied to back‐analyse the propagation stage of some catastrophic flow‐like slope movements for which field data are available. Copyright © 2008 John Wiley & Sons, Ltd.     

样本窗口中不连续面体积密度的评价与应用

不连续面体积密度是三维网络模拟的基本参数之一,体积密度的取值合理与否直接关系到模拟结果的可信度。对不连续面体积密度的求解已经有一些方法,本文提出了一种较为简捷的求解方法,并应用于实际的模型模拟,得到了预期的效果。     

考虑物理性质参数变化的膨润土缓冲材料水-热耦合SPH数值解

为了研究热源温度和外界水压对缓冲层中水-热迁移规律的影响,以GMZ膨润土为例,从基于势能的非饱和土的水-热迁移控制方程出发,考虑了蒸发效应的影响,得到了水-热耦合的方程组,采用改进的光滑粒子流体动力学(SPH)算法,能够对每一处土体根据不同时刻的不同状态实时更新计算参数,得到参数变化的水-热耦合解。计算结果表明：土的物理性质参数与土体的温度和饱和度密切相关,是否考虑这些参数的变化会对计算结果产生较大影响;核废料释放的热量能够在较短的时间内扩散到外边界,水分迁移的速度则相对慢很多;缓冲层温度的升高会加快水分的迁移速度,外界水压对温度的分布则影响较小。     

岩体结构面新型本构模型研究

对于涉及需要单独考虑岩体结构面的工程岩体结构分析,采用能反映岩体结构面主要力学特性的合理的本构模型是取得合理解答的关键问题之一。针对在经典连续介质力学理论框架内建立岩体结构面本构模型的缺点,基于岩体结构面的实际受力变形特性,采用直接法建立了一种新型岩体结构面本构模型。所建立的模型依据岩体结构面切向应力变形曲线及剪胀曲线的实际特征,将其分为峰前线性段、峰前非线性段以及峰后软化段,并分别给出了用于描述岩体结构面变形和强度等主要力学特性的数学模型,进而推导建立了结构面各变形阶段的增量型本构模型。最后,编写相关计算程序,采用所建立的新型本构模型以及被广泛采用的Plesha模型对经典的岩体结构面直剪试验成果进行拟合分析。结果表明,所建立的新型本构模型能更为合理的描述岩体结构面的主要力学特性,且模拟能力优于Plesha模型     

A generic approach to modelling flexible confined boundary conditions in SPH and its application

In this paper, a new approach to applying confining stress to flexible boundaries in the smoothed particle hydrodynamics (SPH) method is developed to facilitate its applications in geomechanics. Unlike the conventional SPH methods that impose confining boundary conditions by creating extra boundary particles, the proposed approach makes use of kernel truncation properties of SPH approximations that occur naturally at free-surface boundaries. Therefore, it does not require extra boundary particles and, as a consequence, can be utilised to apply confining stresses onto any boundary with arbitrary geometry without the need for tracking the curvature change during the computation. This enables more complicated problems that involve moving confining boundaries, such as confining triaxial tests, to be simulated in SPH without difficulties. To further enhance SPH applications in elasto-plastic computations of geomaterials, a robust numerical procedure to implement Mohr-Coulomb plasticity model in SPH is presented for the first time to avoid difficulties associated with corner singularities in Mohr-Coulomb model. The proposed approach was first validated against two-dimensional finite element (FE) solutions for confining biaxial compression tests to demonstrate its predictive capability at small deformation range when FE solutions are still valid. It is then further extended to three-dimensional conditions and utilised to simulate triaxial compression experiments. Simulation results predicted by SPH show good agreement with experiments, FE solutions, and other numerical results available in the literature. This suggests that the proposed approach of imposing confining stress boundaries is promising and can handle complex problems that involve moving confining boundary conditions.     

基于SPH方法的非饱和多孔介质相变耦合研究

在考虑相变的热能平衡方程和非饱和水分迁移质量控制方程的基础上,建立温度场-水分场的耦合模型,并采用一种无网格粒子算法（SPH）进行数值求解。其中,耦合方程中考虑了水流传热以及温度势对水流的直接驱动,在不考虑相变的情况下,该耦合模型可退化为常温下的水-热耦合模型,故可用于模拟冻融循环的相关问题。从求解热能平衡方程中的含冰量出发,实现解耦并对半无限单向冻结条件下介质内非稳态温度场和体积含水率分布场进行模拟,将耦合作用下的温度场与不耦合的解析解进行对比,反映出水分迁移对温度场存在较大影响。最后,求解了路基边坡在季节性周期温度边界下,温度场、水分场分布的演变规律,并评估了边坡阴阳面受热不均对水热两场分布的影响。计算结果基本能反映土冻结相变的实际物理过程,光滑粒子算法可以用于尝试解决冻土领域的其他相关问题。     

A depth average SPH model including μ(I) rheology and crushing for rock avalanches

Classical depth-integrated smoothed particle hydrodynamics (SPH) models for avalanches are extended in the present work to include a μ(I)− rheological model enriched with a fragmentation law. With this improvement, the basal friction becomes grain distribution dependent. Rock avalanches, where grain distribution tends to change with time while propagating, are the appropriate type of landslide to apply the new numerical proposal. The μ(I)− rheological models considered in the present work are those of Hatano and Gray, combined with two different fragmentation laws, a hyperbolic and a fractal-based law. As an application, Frank avalanche, which took place in Canada in 1903, is analyzed under the scope of the present approach, focusing in the influence of the rheological and fragmentation laws in the evolution of the avalanche.     

位移不连续模型在黏弹性节理刚度计算的应用

考虑节理质量对应力波传播的影响,运用波的位移位函数推导了谐波在厚黏弹性节理的透、反射系数计算公式,采用波形相关系数描述子波穿过黏弹性节理的波形变化,讨论具有一定厚度的黏弹性节理简化位移不连续模型的适用条件。设厚黏弹性节理模型和位移不连续模型的透射波波形相关系数为0.9时对应的节理厚度为临界厚度,岩体与节理的阻抗比对临界厚度的影响很小;临界厚度随子波中心频率增大呈负指数减小,入射角越大,临界厚度随中心频率减小得越慢。试验数据分析表明：当节理厚度为0.03 m时,采用位移不连续模型和厚黏弹性模型计算得到的节理力学参数非常接近,随节理厚度和子波中心频率增加,运用位移不连续模型的计算结果偏差越大,试验结果与理论分析是一致的。     

Influence of discontinuities and clay minerals in their filling materials on the instability of rock slopes

To assess the influence of discontinuities and clay minerals in their filling materials on the instability of rock slopes, seven rock slopes along the margin of Ganjnameh–Shahrestaneh Road, Hamedan Province, Western Iran were selected, and the physical and mechanical properties of their rocks and discontinuities were determined. By statistical studies of the discontinuities, rock slope stability analysis has been performed using kinematic and limit equilibrium methods so that safety factors for the rock slopes can be calculated. Also, sampling of filling materials and X-ray diffraction tests have been done to identify the clay minerals in the filling materials. The lithologies of the studied rock slopes are granite, diorite and hornfels. The presence of discontinuities and weakness planes with different orientations and clay minerals in filling materials of discontinuities are effective factors that cause plane, wedge and toppling failures in the rock slopes. Clay minerals as filling materials of discontinuities in the studied rock slope facilitate their instability by two different methods. First, absorption of water by infilling clay minerals causes the friction angle of discontinuity surfaces that leads to plane and wedge failures to be reduced. Second, water absorption causes the swelling of clay infilling minerals that leads to toppling failure.     

Flexible membranes anchored to the ground for slope stabilisation: Numerical modelling of soil slopes using SPH

An alternative modelling for flexible membranes anchored to the ground for soil slope stabilisation is presented using Smoothed-Particle Hydrodynamics to model the unstable ground mass in a soil slope, employing a dynamic solve engine. A regression model of pressure normal to the ground, q_sim, and also membrane deflection, f_sim, have been developed using Design of Experiment. Finally, a comparison between the pressure obtained from numerical simulation and from a limit equilibrium analysis considering infinite slope has been carried out, showing differences in the results, mainly due to the membrane stiffness.     

Influence of discontinuities on the stability of cut slopes in weathered meta-sedimentary rocks

The meta-sedimentary rocks along Pos Selim Highway in Perak State, Malaysia showed a gradational weathering profile based on differences particularly in textures, hardness, lateral changes in colour and consistency of material extension. Both large and small scale discontinuities observed in the investigated rocks reduce the physical and mechanical properties of the rocks, and provide slip surfaces for failures. Rock and soil samples were tested using established standards to determine their characteristics and responses under a wide variety of disturbances. Kinematic analyses were also carried out to determine the modes and likely modes of failures.

Petrographic analyses revealed associated micro-structures, and the implications of these micro-structures showed shearing components along planes of weakness. From the determined index properties of the tested soil samples, the weathered quartz mica schist is not suitable for structural support. Further study involving unconsolidated undrained direct shear box tests carried out under total stress, revealed a non-uniform response of the rocks to shearing disturbance along discontinuity planes, and the rate and depth of deformation. The shear strength components of the investigated rocks were thereby prescribed in terms of cohesion and friction angle. From the kinematic analyses across this extended cut slope, there are possibilities of wedge and planar failures.     

地下水在岩体结构面系统中运移的三维数值模拟

本文从描述流体流动的基本方程出发,推导了地下水在岩体结构面系统中运移的有限元方程,并基于这一方程编制了地下水在岩体中流动的三维数值模拟程序。     

空间坐标系下地下岩体结构面数学描述方法初探

钻孔摄像技术能够采集到大量地下岩体内的结构面数据,分析其数据有助于进一步揭示结构面的空间分布规律和组合关系。针对钻孔内获得的结构面几何参数的特殊性,提出了综合倾角、倾向和位置深度等参数的特征点描述方法,在局部坐标系下对结构面特征点的描述方法及计算过程进行了阐述,推导出了结构面几何参数与特征点坐标之间的转换公式,在全局坐标系下对特征点坐标进行了归一化处理,最后对空间坐标系下特征点的分布规律进行了探索。研究结果表明,（1）空间坐标系下的结构面特征点是结构面几何参数和空间位置的综合反映;（2）特征点在空间的分布反映了结构面的空间组合关系。为结构面的统计分析提供新的思路和方法,有助于开展结构面的空间分布规律和交切关系研究,对钻孔摄像技术的发展和以结构面几何特征为基础的岩体稳定性评价起到积极的推动作用。     

三峡库区侏罗系不同类型结构面抗剪强度对比

三峡库区侏罗系是典型的软硬相间地层,其软弱层具有亲水性强、透水性弱等特殊的工程地质特性,是库区常见的易滑地层。选取三峡库区秭归县马家沟滑坡所处地层岩体结构面为研究对象,采用室内岩体结构面直剪试验,对4种不同类型结构面的力学特性进行了研究。通过对试验数据的统计分析,绘制了各种结构面的剪应力–正应力关系曲线图及剪应力–剪切位移关系曲线图,从而获得了形成于同一地层中的4种不同结构面的抗剪强度参数,并对其剪切特性进行对比分析与研究。以结构面直剪试验结果为依据,初步分析了该地区某一典型边坡的稳定性及潜在滑动层面,为后期边坡的稳定性评价提供可靠性基础参数。     

Multiphase SPH modeling of free surface flow in porous media with variable porosity

This paper presents a numerical model for simulating free surface flow in porous media with spatially varying porosity. The governing equations are based on the mixture theory. The resistance forces between solid and fluid is assumed to be nonlinear. A multiphase SPH approach is presented to solve the governing equations. In the multiphase SPH, water is modeled as a weakly compressible fluid, and solid phase is discretized by fixed solid particles carrying information of porosity. The model is validated by several numerical examples including seepage through specimen, fast flow through rockfill dam and wave interaction with porous structure. Good agreements between numerical results and experimental data are obtained in terms of flow rate and evolution of free surface. Parameter study shows that (1) the nonlinear resistance law provides more accurate results; (2) particle size and porosity have significant influence on the porous flow.     

An improved SPH method for saturated soils and its application to investigate the mechanisms of embankment failure: Case of hydrostatic pore‐water pressure

The method of smoothed particle hydrodynamics (SPH) has recently been applied to computational geomechanics and has been shown to be a powerful alternative to the standard numerical method, that is, the finite element method, for handling large deformation and post‐failure of geomaterials. However, very few studies apply the SPH method to model saturated or submerged soil problems. Our recent studies of this matter revealed that significant errors may be made if the gradient of the pore‐water pressure is handled using the standard SPH formulation. To overcome this problem and to enhance the SPH applications to computational geomechanics, this article proposes a general SPH formulation, which can be applied straightforwardly to dry and saturated soils. For simplicity, the current work assumes hydrostatic pore‐water pressure. It is shown that the proposed formulation can remove the numerical error mentioned earlier. Moreover, this formulation automatically satisfies the dynamic boundary conditions at a submerged ground surface, thereby saving computational cost. Discussions on the applications of the standard and new SPH formulations are also given through some numerical tests. Furthermore, techniques to obtain the correct SPH solution are also proposed and discussed throughout. As an application of the proposed method, the effect of the dilatancy angle on the failure mechanism of a two‐sided embankment subjected to a high groundwater table is presented and compared with that of other solutions. Finally, the proposed formulation can be considered a basic formulation for further developments of SPH for saturated soils. Copyright © 2011 John Wiley & Sons, Ltd.