Numerical simulation of liquefaction behaviour of granular materials using Discrete Element Method

In this paper, numerical simulation of 3-dimensional assemblies of 1000 polydisperse sphere particles using Discrete Element Method (DEM) is used to study the liquefaction behaviour of granular materials. Numerical simulations of cyclic triaxial shear tests under undrained conditions are performed at different confining pressures under constant strain amplitude. Results obtained in these numerical simulations indicate that with increase in confining pressure there is an increase in liquefaction resistance.     

An AI-based model for describing cyclic characteristics of granular materials

Modelling cyclic behaviour of granular soils under both drained and undrained conditions with a good performance is still a challenge. This study presents a new way of modelling the cyclic behaviour of granular materials using deep learning. To capture the continuous cyclic behaviour in time dimension, the long short-term memory (LSTM) neural network is adopted, which is characterised by the prediction of sequential data, meaning that it provides a novel means of predicting the continuous behaviour of soils under various loading paths. Synthetic datasets of cyclic loading under drained and undrained conditions generated by an advanced soil constitutive model are first employed to explore an appropriate framework for the LSTM-based model. Then the LSTM-based model is used to estimate the cyclic behaviour of real sands, ie, the Toyoura sand under the undrained condition and the Fontainebleau sand under both undrained and drained conditions. The estimates are compared with actual experimental results, which indicates that the LSTM-based model can simultaneously simulate the cyclic behaviour of sand under both drained and undrained conditions, ie, (a) the cyclic mobility mechanism, the degradation of effective stress and large deformation under the undrained condition, and (b) shear strain accumulation and densification under the drained condition.     

Discrete element modelling of deep penetration in granular soils

This paper presents a numerical study on deep penetration mechanisms in granular materials with the focus on the effect of soil–penetrometer interface friction. A two‐dimensional discrete element method has been used to carry out simulation of deep penetration tests on a granular ground that is under an amplified gravity with a K₀ lateral stress boundary. The numerical results show that the deep penetration makes the soil near the penetrometer move in a complex displacement path, undergo an evident loading and unloading process, and a rotation of principal stresses as large as 180°. In addition, the penetration leads to significant changes in displacement and velocity fields as well as the magnitude and direction of stresses. In general, during the whole penetration process, the granular ground undergoes several kinds of failure mechanisms in sequence, and the soil of large deformation may reach a stress state slightly over the strength envelope obtained from conventional compression tests. Soil–penetrometer interface friction has clear effects on the actual penetration mechanisms. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element modelling of rate-dependent ratcheting in granular materials

The present paper introduces a comprehensive model that is capable of describing the behaviour, under cyclic loading, of the granular materials used in railway tracks and road pavement. Its main thrust is the introduction of the “Chicago” law in a continuum approach to account for the ratcheting effects. It also emphasizes rate-dependency as a dissipative mechanism that acts independently or jointly with the ratcheting effect as well as the non-associated plasticity. The numerical procedure is based on the return mapping algorithm, where Newton’s method is used to calculate the nonlinear consistency parameter of the flow rule and to obtain a consistent tangent modulus. The model was applied to specific numerical examples including multi-axial and cyclic loading conditions.     

A virtual experiment technique on the elementary behaviour of granular materials with discrete element method

Multi‐scale investigations aided by the discrete element method (DEM) play a vital role for current state‐of‐the‐art research on the elementary behaviour of granular materials. Similar to laboratory tests, there are three important aspects to be considered carefully, which are the proper stress/strain definition and measurement, the application of target loading paths and the designed experiment setup, to be addressed in the present paper. Considering the volume sensitive characteristics of granular materials, in the proposed technique, the deformation of the tested specimen is controlled and measured by deformation gradient tensor involving both the undeformed configuration and the current configuration. Definitions of Biot strain and Cauchy stress are adopted. The expressions of them in terms of contact forces and particle displacements, respectively, are derived. The boundary of the tested specimen consists of rigid massless planar units. It is suggested that the representative element uses a convex polyhedral (polygonal) shape to minimize possible boundary arching effects. General loading paths are described by directly specifying the changes in the stress/strain invariants or directions. Loading can be applied in the strain‐controlled mode by specifying the translations and rotations of the boundary units, or in the stress‐controlled mode by using a servo‐control mechanism, or in the combination of the two methods to realize mixed boundary conditions. Taking the simulation results as the natural consequences originated from a complex system, virtual experiments provide particle‐scale information database to conduct multi‐scale investigations for better understanding in granular material behaviours and possible development of the constitutive theories provided the qualitative similarity between the simulation results from virtual experiments and observations on real material behaviour. Copyright © 2011 John Wiley & Sons, Ltd.     

A robust algorithm for ellipse-based discrete element modelling of granular materials

With recent research indicating the importance of the rolling mechanism of deformation in granular systems consisting of perfectly round particles, it has become popular to use ellipse-shaped particles in the Discrete Element Method (DEM) numerical model. Inherent in this technique is the need for accurately computing ellipse to ellipse intersection, in order to properly detect contact formation and compute relative contact velocities. However, the commonly used algorithms for computing ellipse-ellipse intersection are generally poorly conditioned and can be inaccurate. An alternate method for computing ellipse-ellipse intersection is developed and presented which results in a well-conditioned, stable and accurate contact detection method. These modification are incorporated into the general DEM algorithm.     

Discrete modeling of strain accumulation in granular soils under low amplitude cyclic loading

An advanced understanding of the strain accumulation phenomenon in granular soils subjected to low amplitude cyclic loading with relatively high frequency is needed to enhance the ability to predict the settlement of granular soils induced by vibrations. In the current study, the discrete element method is used to study this phenomenon. A loose and a medium dense sample composed of a relatively large number of spheres are considered. A series of stress controlled cyclic triaxial tests with different excitation amplitudes and frequencies is performed on these samples at different static stress states. The response of these samples at the macroscopic and microscopic scales is analyzed. The sample density, the cyclic stress amplitude and the static stress state importantly affect strain accumulation. However, the cyclic excitation frequency has a small effect on strain accumulation. At the microscopic scale, frictional sliding occurring at a few contacts continuously dissipates energy and the fraction of these contacts varies periodically during cyclic loading. The coordination number of these samples increases slightly as strain accumulates. However, the anisotropy remains almost constant during low amplitude cyclic excitation. A qualitatively good agreement between numerical and experimental results is found.     

颗粒物质在等比例应变加载下的分散性失稳模式

由地下水引起的静力液化可能是边坡失稳的隐含机制之一,松砂在不排水剪切条件下可能发生静力液化,密实的颗粒集合体在特定的应变路径下也会出现相似的现象,即试样整体发生急剧的失稳,应力状态尚处于峰值强度线以内。该种失稳模式称为分散性失稳,是为了强调失稳模式中没有出现应变局部化或者剪切带。采用连续-离散耦合分析方法,研究由不规则形状颗粒组成的密实集合体在等比例应变加载路径下的力学特性。根据Hill的材料失稳理论,当试样的应力增量 和应变增量 对应的2阶功 为负时,试样即发生不可逆的整体失稳破坏。以根据不同等比例应变路径得到 曲线为界,在 平面内将试样的应力状态分为剪缩区、剪胀-稳定区和剪胀-非稳定区,连接不同围压下试样发生分散性失稳时的应力状态形成失稳线发现,峰值强度线高于临界状态线,临界状态线高于失稳线。     

Discrete element modeling of tool‐rock interaction I: rock cutting

Tool‐rock interaction processes can be classified as indentation or cutting depending on the direction of motion of the tool with respect to the rock surface. The modes of failure induced in the rock by an indenting or a cutting tool can be ductile and/or brittle. The ductile mode is associated with the development of a damage zone, whereas the brittle mode involves the growth of macrocracks. This is the first part of a series of two papers concerned with an analysis of the cutting and the indentation processes based on using the discrete element method. In this paper, numerical simulations of the cutting process are conducted to reproduce the transition from a ductile to a brittle failure mode with increasing depth of cut, which is observed in experiments. The numerical results provide evidence that the critical depth of cut d _* controlling the failure mode transition is related to the characteristic length ? = (K_Ic ∕ σ_c)² with K_Ic denoting the material toughness and σ_c its unconfined compressive strength. The nature of frictional contact between the cutter face and the rock in the ductile failure mode is also examined. It is shown that the inclination of the total cutting force is controlled by a multi‐directional flow mechanism ahead of the cutter that is related to the formation of a wedge of failed material, intermittently adhering to the cutter. As a result, the inclination of the total cutting force varies with the rake angle of the cutter and cannot be considered an intrinsic measure of the interfacial friction between the cutter and the rock. Copyright © 2012 John Wiley & Sons, Ltd.     

Discrete element modeling of tool‐rock interaction II: rock indentation

The failure mechanisms induced by a wedge‐shaped tool indenting normally against a rock surface are investigated using the discrete element method (DEM). The main focus of this study is to explore the conditions controlling the transition from a ductile to a brittle mode of failure. The development of a damage zone and the initiation and propagation of a brittle fracture is well captured by the DEM simulations. The numerical results support the conjecture that initiation of brittle fractures is governed by a scaled flaw length Λ, a ratio between the flaw size λ and the characteristic length (where K_Ic is the toughness and σ_c the uniaxial compressive strength). The size of the damage zone agrees well with analytical predictions based on the cavity expansion model. The effects of a far‐field confining stress and the existence of a relief surface near the indenter are also examined.Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of mechanical behaviour and internal stability of granular materials using discrete element method

Granular materials like sand are widely used in civil engineering. They are composed of different sizes of grains, which generate a complex behaviour, difficult to assess experimentally. Internal instability of a granular material is its inability to prevent the loss of its fine particles under flow effect. It is geometrically possible if the fine particles can migrate through the pores of the coarse soil matrix and results in a change in its mechanical properties. This paper uses the three‐dimensional Particle Flow Code (PFC3D/DEM) to study the stability/instability of granular materials and their mechanical behaviour after suffusion. Stability properties of widely graded materials are analysed by simulating the transport of smaller particles through the constrictions formed by the coarse particles under the effect of a downward flow with uniform pressure gradient. A sample made by an initially stable material according to the Kenney & Lau geometrical criterion was divided into five equal layers. The classification of these layers by this criterion before and after the test shows that even stable granular materials can lose fine particles and present local instability. The failure criterion of eroded samples, in which erosion is simulated by progressive removal of fine particles, evolves in an unexpected way. Internal friction angle increases with the initial porosity, the rate of lost fine particles and the average diameter D₅₀. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical modelling of dynamic consolidation on granular soils

The application of Pastor–Zienkiewicz constitutive model for sands to dynamic consolidation problems is presented in this paper. This model is implemented in a coupled code formulated in terms of displacements for both solid and fluid phases (u?w formulation), which is firstly compared with u?p_w formulation for some simple examples. Its range of validity, previously established for elastic problems and harmonic loading, is explored. Once the suitability of the u?w formulation has been ascertained for this kind of dynamic problems in soils, one‐ and two‐dimensional (plane strain) dynamic consolidation numerical examples are provided, aiming to give some light into the physics of this ground improvement technique. A ‘wave of dryness’, observed at the soil surface during the impact in field cases, is numerically reproduced and justified. Some hints on the influence of the loading zone size are also given. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical solution for 1D consolidation under haversine cyclic loading

Presented and discussed in this paper is an exact analytical solution of the nonhomogeneous partial differential equation governing the conventional one‐dimensional consolidation under haversine repeated loading. The derived analytical solution to the 1D consolidation equation is compared with the numerical solution of the same consolidation problem via FEM. The series solution takes into account the frequency of repeated loading through a dimensionless time factor T₀. The paper reveals that an increase in the frequency of imposed repeated haversine loading (a decrease in period of repeated loading) causes an increase in the number of cycles required to achieve the steady state, whereas the effect of frequency on the maximum excess pore water pressure at the bottom of a clay layer with permeable top and impermeable bottom for the range of frequencies studied is generally insignificant. The effective stress at the bottom of the clay deposit with permeable top and impermeable bottom increases with time but with some fluctuations without changing the sign. These fluctuations become more pronounced for increasing values of T₀. An increase in T₀ also causes an increase in maximum effective stress. Copyright © 2013 John Wiley & Sons, Ltd.     

Discrete element modeling of dynamic behaviors of railway ballast under cyclic loading with dilated polyhedra

The dynamic behaviors of railway ballast under cyclic loading are simulated with discrete element method (DEM). Dilated polyhedra are constructed based on the Minkowski sum operator in order to resemble the irregular shapes of ballast particles. The polyhedral particle generation, contact detection between particles and contact laws are presented. Ballast box tests with periodic lateral boundaries are conducted to simulate the dynamics of the sleeper and ballast particles. The settlement and effective stiffness of ballast bed are investigated under cyclic loadings with five distinct frequencies. The settlement of ballast bed is significant in the first several cycles and increases with the number of cycles gradually. The higher frequency loading generates larger displacement in the same simulation time. The effective stiffness of ballast bed increases gradually. To study the effect of particles' sharpness, dilated polyhedra with different dilating radii and spherical particles are also developed. Simulation results show the sharper the ballast particles are, the smaller the produced settlement. Copyright © 2016 John Wiley & Sons, Ltd.     

循环荷载下砂土液化特性颗粒流数值模拟

利用PFC2D常体积循环双轴试验条件,对砂土在不排水循环荷载作用下的液化特性进行了颗粒流数值模拟,数值模拟按等应力幅加荷方式进行。颗粒流数值模拟的优点在于得到试样液化宏观力学表现的同时,通过不同循环加荷时刻试样内细观组构参量（包括配位数、接触法向分布、粒间法向接触力、粒间切向接触力）的演化规律,分析砂土液化过程中细观组构变化与宏观力学响应之间的内在联系,从而可进一步探讨砂土液化的细观力学机制。数值模拟研究结果表明,砂土液化现象在宏观力学表现上反映为超静孔隙水压力的累积上升和平均有效主应力的不断减小,在细观组构上对应于配位数的累积损失和粒间接触力的不断减小。砂土液化细观机制分析表明,试样配位数的减少与循环加荷过程中组构各向异性滞后于应力各向异性有关。     

Discrete element modelling of creep of asphalt mixtures

Creep tests on asphalt mixtures have been undertaken under four stress levels in the laboratory while the discrete element model (DEM) has been used to simulate the laboratory tests. A modified Burger’s model has been used to represent the time-dependent behaviour of an asphalt mixture by adding time-dependent moment and torsional resistance at contacts. Parameters were chosen to give the correct stress-strain response for constant strain rate tests in Cai et al. (2013). The stress-strain response for the laboratory creep tests and the simulations were recorded. The DEM results show reasonable agreement with the experiments. The creep simulation results proved to be dependent on both bond strength variability and positions of the particles. Bond breakage was recorded during the simulations and used to investigate the micro-mechanical deformation behaviour of the asphalt mixtures. An approach based on dimensional analysis is also presented in this paper to reduce the computational time during the creep simulation, and this analysis is also a new contribution.     

Modelling the elastic behaviour of granular materials

A review of the literature indicates that the elastic behaviour of granular materials is isotropic and that Poissony's ratio is constant, whereas Young's Modulus, the bulk modulus and the shear modulus vary with the mean normal stress and the deviatoric stress. A nonlinear, isotropic model for the elastic behaviour is developed on the basis of theoretical considerations involving the principle of conservation of energy. Energy is therefore neither generated not dissipated in closed-loop stress paths or in closed-loop strain paths. The framework for the model consists of Hooke's law, in which Poission's ratio is constant and Young's modulus is expressed as a power function invlving the first invariat of the stress tensor and the second invariant of the deviatoric stress tensor. The characteristics of the model are described, and the accuracy is evaluated by comparison with experimental results from triaxial tests and three-dimensional cubical triaxial tests with a variety of stress paths. Parameter determination from unloading–reloading cycles in conventional triaxial compression tests is demonstrated, typical parameter values are given for granular materials and extension of the model to soils with effective cohesion is described.     

A two‐mechanism elastoplastic model for shakedown of unbound granular materials and DEM simulations

An elastoplastic model has been developed for the finite elements modelling of repeated load triaxial tests. This model is based on the shakedown theory established by Zarka for metallic structures. To the previous works, which were based on the Drucker–Prager yield surface and the plastic potential of Von Mises, a compression cap has been added to each one. The model straightforwardly determines the purely elastic state or the elastic shakedown state or the plastic shakedown state and calculates the deviatoric and the volumetric plastic strains. The calibration of the elastoplastic model has been carried out with DEM simulations and an unbound granular material for roads under repeated load triaxial tests using finite element method. The calculations underline the capabilities of the model to take into account, with a unique formalism, the accumulation of the deviatoric and volumetric plastic strains along the loading cycles. Copyright © 2011 John Wiley & Sons, Ltd.