An approach to calculating large strain accumulation for discrete element simulations of granular media

For research on granular materials, establishing a method to calculate continuum strain from particle displacements is necessary for understanding the material behaviour at macro-level and developing continuum constitutive models. Existing methods are generally based on constructing a mesh or background grid to calculate strain from particle motions. These methods offer rigorous ways to measure strain for granular materials; however, they suffer from several problems such as mesh distortion and lacking grid-to-particle strain mapping procedure, which hinders their capability of calculating strain accumulation during large deformation processes of granular media. To address this issue, this study proposes a new strain calculation method for discrete element simulations of granular materials. This method describes a particle assembly as an equivalent continuum system of material points, each of which corresponds to a particle centre and represents a continuous region with its initial volume/area presumably equal to the volume/area of Voronoi cells generated in accordance with the particle assembly configuration. Smooth Particle Hydrodynamics (SPH) interpolation functions are then employed to calculate strain for these material points. This SPH-based method does not require any mesh or background grid for computation, leading to advantages in calculating strain accumulation under large deformation. Simulations of granular materials in both uniform and heterogeneous gradations were carried out, and strain results obtained by the proposed method indicate good agreements with analytical and numerical solutions. This demonstrates its potential for strain calculations in discrete element simulations of granular materials involving large deformations and/or large displacements.     

Equivalent continuum strain calculations based on 3D particle kinematic measurements of sand

Constitutive modeling of granular materials has been a subject of extensive research for many years. While the calculation of the Cauchy stress tensor using the discrete element method has been well established in the literature, the formulation and interpretation of the strain tensor are not as well documented. According to Bagi, 1 researchers mostly adopt well‐known continuum or discrete microstructural approaches to calculate strains within granular materials. However, neither of the 2 approaches can fully capture the behavior of granular materials. They are considered complementary to each other where each has its own strengths and limitations in solving granular‐mechanics problems. Zhang and Regueiro 2 proposed an equivalent continuum approach to calculating finite strain measures at the local level in granular materials subjected to large deformations. They used three‐dimensional discrete element method results to compare the proposed strains measures. This paper presents an experimental application of the Zhang and Regueiro 2 approach using three‐dimensional synchrotron microcomputed tomography images of a sheared Ottawa sand specimen. Invariant Eulerian finite strain measures were calculated for representative element volumes within the specimen. The spatial maps of Eulerian octahedral shear and volumetric strain were used to identify zones of intense shearing within the specimen and compared well with maps of incremental particle translation and rotation for the same specimen. The local Eulerian volumetric strain was compared to the global volumetric strains, which also can be considered as an averaging of all local Eulerian volumetric strains.     

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

We formulate a discrete Lagrangian model for a set of interacting grains, which is purely elastic. The considered degrees of freedom for each grain include placement of barycenter and rotation. Further, we limit the study to the case of planar systems. A representative grain radius is introduced to express the deformation energy to be associated to relative displacements and rotations of interacting grains. We distinguish inter-grains elongation/compression energy from inter-grains shear and rotations energies, and we consider an exact finite kinematics in which grain rotations are independent of grain displacements. The equilibrium configurations of the grain assembly are calculated by minimization of deformation energy for selected imposed displacements and rotations at the boundaries. Behaviours of grain assemblies arranged in regular patterns, without and with defects, and similar mechanical properties are simulated. The values of shear, rotation, and compression elastic moduli are varied to investigate the shapes and thicknesses of the layers where deformation energy, relative displacement, and rotations are concentrated. It is found that these concentration bands are close to the boundaries and in correspondence of grain voids. The obtained results question the possibility of introducing a first gradient continuum models for granular media and justify the development of both numerical and theoretical methods for including frictional, plasticity, and damage phenomena in the proposed model.     

Multiscale modeling and analysis of compaction bands in high-porosity sandstones

We present a multiscale investigation on the initiation and development of compaction bands in high-porosity sandstones based on an innovative hierarchical multiscale approach. This approach couples the finite element method and the discrete element method (DEM) to offer direct, rigorous linking of the microscopic origins and mechanisms with complex macroscopic phenomena observed in granular rocks such as strain localization and failure. To simulate compaction band in granular cementitious sandstone, we adopt a bonded contact model with normal and tangential interparticle cohesions in the DEM and propose a dual-porosity structure consisting of macro-pores and interstitial voids for the representative volume element to mimic the typical meso-structure of high-porosity sandstones. In the absence of particle crushing, our multiscale analyses identify debonding and pore collapses as two major contributors to the formation of compaction bands. The critical pressures predicted by our simulations, corresponding to surges of debonding and pore collapse events, agree well with the estimations from field data. The occurrence patterns of compaction band are found closely related to specimen heterogeneity, porosity and confining pressure. Other deformation band patterns, including shear-enhanced compaction bands and compactive shear bands, were also observed under relatively low confining pressure conditions with a rough threshold at \(0.55P^{*}\) (\(P^{*}\) is the critical pressure) on the failure envelop. Key microscopic characteristics attributable to the occurrence of these various deformation patterns, including fabric anisotropy, particle rotation, debonding and pore collapse, are examined. Shear-enhanced compaction bands and pure compaction bands bear many similarities in terms of these microscopic characteristics, whereas both differ substantially from compactive shear bands.     

Geotechnical properties of 3D-printed transparent granular soil

This paper proposes a 3D-printed transparent granular soil technique based on the contour rotation interpolation method, 3D printing and transparent soil technologies. Laboratory tests, including one-dimensional compression, direct shear and triaxial compression tests, are assessed for mechanical properties of 3D-printed transparent granular soil. The results show that 3D-printed transparent granular soil can be used to consider the effect of shape on macromechanical soil properties, which is an advantage that cannot be achieved by the previously presented transparent soil techniques. Subsequently, a simple model test of rigid flat plate penetration into transparent granular soil is performed. The obtained soil displacements using the PIV technique are compared with the classical shallow strain path method solution. This comparison indicates that the proposed 3D-printed transparent granular soil can capture the soil deformation pattern, although the accuracy needs to be further improved. The proposed 3D-printed transparent granular soil technique could be used for model test that needs to consider the shape of sand particle.

     

Numerical studies of shear banding in interface shear tests using a new strain calculation method

Strain localization is closely associated with the stress–strain behaviour of an interphase system subject to quasi‐static direct interface shear, especially after peak stress state is reached. This behaviour is important because it is closely related to deformations experienced by geotechnical composite structures. This paper presents a study using two‐dimensional discrete element method (DEM) simulations on the strain localization of an idealized interphase system composed of densely packed spherical particles in contact with rough manufactured surfaces. The manufactured surface is made up of regular or irregular triangular asperities with varying slopes. A new simple method of strain calculation is used in this study to generate strain field inside a simulated direct interface shear box. This method accounts for particle rotation and captures strain localization features at high resolution. Results show that strain localization begins with the onset of non‐linear stress–strain behaviour. A distinct but discontinuous shear band emerges above the rough surface just before the peak stress state, which becomes more expansive and coherent with post‐peak strain softening. It is found that the shear bands developed by surfaces with smaller roughness are much thinner than those developed by surfaces with greater roughness. The maximum thickness of the intense shear zone is observed to be about 8–10 median particle diameters. The shear band orientations, which are mainly dominated by the rough boundary surface, are parallel with the zero extension direction, which are horizontally oriented. Published in 2007 by John Wiley & Sons, Ltd.     

空心圆柱扭转仪定向剪切试验离散元模拟

为分析砂土在复杂应力条件下的剪切力学特性,采用商业离散元软件PFC3D对单粒组中密砂的空心扭剪试验进行了仿真模拟,分析了数值试样的应力-应变关系,研究了不同剪切方向下离散介质的强度、体积应变特性以及中主应力比对它们的影响,再现了力链在加载过程中的演化,并对剪切带的倾角做了深入分析。同时,从细观上看,以颗粒接触数和纯转动率变量为中心,观察了试样内部颗粒的运动状态,对比了不同剪切方向下剪切带内外颗粒接触数与纯转动位移的变化。最后,将数值试验结果与已有的室内试验结果进行了对比。此研究实现了复杂应力条件下空心扭剪试样的三维离散元模拟,加深了对空心扭剪试验过程和结果的理解和解释。     

A thermomicromechanical approach to multiscale continuum modeling of dense granular materials

A new method is proposed for the development of a class of elastoplastic thermomicromechanical constitutive laws for granular materials. The method engenders physical transparency in the constitutive formulation of multiscale phenomena from the particle to bulk. We demonstrate this approach for dense, cohesionless granular media under quasi-static loading conditions. The resulting constitutive law—expressed solely in terms of particle scale properties—is the first of its kind. Micromechanical relations for the internal variables, tied to nonaffine deformation, and their evolution laws, are derived from a structural mechanical analysis of a particular mesoscopic event: confined, elastoplastic buckling of a force chain. It is shown that the constitutive law can reproduce the defining behavior of strain-softening under dilatation in both the mesoscopic and macroscopic scales, and reliably predict the formation and evolution of shear bands. The thickness and angle of the shear band, the distribution of particle rotation and the evolution of the normal contact force anisotropy inside the band, are consistent with those observed in discrete element simulations and physical experiments.     

Micromechanics of vortices in granular media: connection to shear bands and implications for continuum modelling of failure in geomaterials

Recent analysis of data from triaxial tests on sand and discrete element simulations indicate the final pattern of failure is encoded in grain motions during the nascent stages of loading. We study vortices that are evident from grain displacements at the start of loading and bear a direct mathematical connection to boundary conditions, uniform continuum strain and shear bands. Motions of three grains in mutual contact, that is, 3‐cycles, manifest vortices. In the initial stages of loading, 3‐cycles initiate a rotation around a region Ω^* where the shear band ultimately develops. This bias sets a course in 3‐cycle evolution, determining where they will more likely collapse. A multiscale spatial analysis of 3‐cycle temporal evolution provides quantitative evidence that the most stable, persistent 3‐cycles degrade preferentially in Ω^*, until essentially depleted when the shear band is fully formed. The transition towards a clustered distribution of persistent 3‐cycles occurs early in the loading history—and coincides with the persistent localisation of vortices in Ω^*. In 3D samples, no evidence of spatial clustering in persistent 3‐cycle deaths is found in samples undergoing diffuse failure, while early clustering manifests in a sample that ultimately failed by strain localisation. This study not only delivered insights into the possible structural origins of vortices in dense granular systems but also a tool for the early detection of the mode of failure—localised versus diffuse—a sample will ultimately undergo. Copyright © 2014 John Wiley & Sons, Ltd.     

颗粒材料的微观临界状态理论模型

存在临界状态是颗粒材料的一个重要特性。基于孔隙胞元的颗粒离散元方法对二维颗粒体进行双轴加载数值试验,在详细分析数值模拟结果的基础上,从微观几何组构的角度揭示了临界状态的存在机制。基于剪胀性原理,提出了以接触价键表征的微观临界状态理论模型,得到了接触价键与塑性剪切应变的关系表达式,理论模型的结果和二维离散元数值模拟得到的结果吻合较好。通过比较不同情况下数值结果和理论模型中的参数,得到以下结论：表征微观临界状态的参数（临界接触价键和达到临界状态所需要的塑性剪切应变）依赖于颗粒体的微观特性,如颗粒形状、表面摩擦性质、颗粒体的围压和初始孔隙比。     

Modelling strain localization in granular materials using micropolar theory: mathematical formulations

It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill‐posedness and mesh dependency. Therefore, a non‐local theory with internal length scales is needed to overcome such problems. The micropolar and high‐order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.     

基于Cosserat连续体的颗粒破碎影响研究

颗粒破碎对颗粒材料宏观力学行为有重要影响。 结合Hardin的破碎经验公式,将表征破碎程度的破碎参量与Cosserat连续体的内部长度参数相关联,形成一个基于Cosserat连续体且能考虑颗粒破碎的弹塑性模型。数值算例主要考察了颗粒破碎对颗粒材料承载能力、塑性应变及局部化行为的影响,数值结果表明,颗粒破碎主要发生在剪切带内,颗粒破碎使得剪切带明显变窄且剪切带内外等效塑性应变梯度明显增大。     

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.     

Strain localization in sandstone during embryonic stages of shear-zone evolution

Geometric analysis of nested Riedel structures was used to identify and quantify strain localization processes within faulted Navajo sandstone. The analysis shows systematic deviation from the basic Riedel geometry complying with the Mohr–Coulomb criterion. Using cross-cutting relations amongst deformation bands within the Riedel structures, and comparing the orientations of the deformation bands to theoretical strain calculations, we identify two coupling deformation mechanisms involved in the early stages of shear-zone evolution, namely, granular flow and discrete faulting. Both mechanisms localize during strain accumulation, and the granular flow facilitates considerable change in the initial geometry of the Riedel structures. The analysis demonstrates a systematic sequence, by which new Riedel structures form after a constant amount of shear strain takes place in the sandstone. Analysis further indicates that granular flow is the major deformation mechanism during early stages of shear-zone evolution and discrete faulting is the dominant mechanism during later deformation stages.     

A coupled 3‐dimensional bonded discrete element and lattice Boltzmann method for fluid‐solid coupling in cohesive geomaterials

This paper presents a 3D bonded discrete element and lattice Boltzmann method for resolving the fluid‐solid interaction involving complicated fluid‐particle coupling in geomaterials. In the coupled technique, the solid material is treated as an assembly of bonded and/or granular particles. A bond model accounting for strain softening in normal contact is incorporated into the discrete element method to simulate the mechanical behaviour of geomaterials, whilst the fluid flow is solved by the lattice Boltzmann method based on kinetic theory and statistical mechanics. To provide a bridge between theory and application, a 3D algorithm of immersed moving boundary scheme was proposed for resolving fluid‐particle interaction. To demonstrate the applicability and accuracy of this coupled method, a benchmark called quicksand, in which particles become fluidised under the driving of upward fluid flow, is first carried out. The critical hydraulic gradient obtained from the numerical results matches the theoretical value. Then, numerical investigation of the performance of granular filters generated according to the well‐acknowledged design criteria is given. It is found that the proposed 3D technique is promising, and the instantaneous migration of the protected soils can be readily observed. Numerical results prove that the filters which comply with the design criteria can effectively alleviate or eliminate the appearance of particle erosion in dams.     

Discrete element modeling of direct shear tests for a granular material

A succinct 3D discrete element model, with clumps to resemble the real shapes of granular materials, is developed. The quaternion method is introduced to transform the motion and force of a clump between local and global coordinates. The Hertz–Mindlin elastic contact force model, incorporated with the nonlinear normal viscous force and the Mohr–Coulomb friction law, is used to describe the interactions between particles. The proposed discrete element model is used to simulate direct shear tests of the irregular limestone rubbles. The simulation results of vertical displacements and shear stresses with a mixture of clumps are compared well with that of laboratory tests. The bulk friction coefficients are calculated and discussed under different contact friction coefficients and normal stresses. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical investigations of shear localization in a micro‐polar hypoplastic material

In this paper a micro‐polar continuum approach is proposed to model the essential properties of cohesionless granular materials like sand. The model takes into account the influence of particle rotations, the mean grain size, the void ratio, the stresses and couple stresses. The constitutive equations for the stresses and couple stresses are incrementally non‐linear and based on the concept of hypoplasticity. For plane strain problems the implementation of the model in a finite element program is described. Numerical studies of the evolution of micro‐polar effects within a granular strip under plane shearing are presented. It is shown that the location and evolution of shear localization is strongly influenced by the initial state and the micro‐polar boundary conditions. For large shearing the state quantities tend towards a stationary state for which a certain coupling between the norm of the stress deviator and the norm of the couple stress tensor can be derived. Copyright © 2003 John Wiley & Sons, Ltd.     

Three-dimensional discrete element method parallel computation of Cauchy stress distribution over granular materials

The paper presents Cauchy stress tensor computation over parallel grids of message passing interface (MPI) parallel three-dimensional (3D) discrete element method (DEM) simulations of granular materials, considering spherical and nonspherical particles. The stress tensor computation is studied for quasi-static and dynamic conditions, and its resulting symmetry or asymmetry is discussed within the context of classical continuum mechanics (CCM), granular materials mechanics (GMM), and micropolar continuum mechanics (MCM). The average Cauchy stress tensor computation follows Bagi's and Nicot's formulations and is verified within MPI parallel 3D DEM simulations involving dynamically adaptive compute grids. These grids allow calculation of temporal and spatial distributions of stress across granular materials under static and dynamic conditions. The vertical stress component in gravitationally deposited particle assemblies exhibits nonuniform spatial distributions under static equilibrium, and its zone of maximum value changes during the process of gravitational pluviation and collapse. These phenomena reveal a microstructural effect on stress distribution within granular materials that is attributed to their discrete particulate nature (particle size, shape, gradation, boundary conditions, etc).     

FE-studies on the influence of initial void ratio,pressure level and mean grain diameter on shear localization

The paper is concerned with shear localization in the form of a spontaneous shear zone inside a granular material during a plane strain compression test. The influence of an initial void ratio, pressure and a mean grain diameter on the thickness of a shear zone is investigated. A plane strain compression test with dry sand is numerically modelled with a finite element method taking into account a polar hypoplastic constitutive relation which was laid down within a polar (Cosserat) continuum. The relation was obtained through an extension of a non-polar hypoplastic constitutive law according to Gudehus and Bauer by polar quantities: rotations, curvatures, couple stresses and a characteristic length. It can reproduce the essential features of granular bodies during shear localization. The material constants can be easily calibrated. The FE-calculations demonstrate an increase in the thickness of the shear zone with increasing initial void ratio, pressure level and mean grain diameter. Polar effects manifested by the appearance of grain rotations and couple stresses are only significant in the shear zone. A comparison between numerical calculations and experimental results shows a satisfying agreement. Copyright © 1999 John Wiley & Sons, Ltd.