基于接触价键的颗粒材料微观临界状态

用颗粒离散元法,分别对二维圆形、椭圆形颗粒体进行了双轴压缩数值模拟。微观尺度的变形是基于孔隙胞元和其中的变形来计算的,而单个孔隙胞元的变形通过周围颗粒的相对运动来计算。针对该方法提出了以接触价键（每个孔隙胞元的边数）来表征颗粒材料微观临界状态的理论。为了定义临界接触价键的极限值,分别讨论了摩擦系数较大、较小时的两种情况。文中给出了微观几何织构（包括接触价键、孔隙胞元的形状、孔隙比）随压缩变形的演变过程,比较了不同颗粒形状、颗粒间摩擦系数以及颗粒体的固结压力对颗粒体的微观力学性能的影响。计算结果表明,颗粒材料的微观临界状态并不是可以唯一表征的,而是受围压、摩擦系数,颗粒形状等参数的共同影响。     

A hypoplastic model for granular soils incorporating anisotropic critical state theory

It is well known that soil is inherently anisotropic and its mechanical behavior is significantly influenced by its fabric anisotropy. Hypoplasticity is increasingly being accepted in the constitutive modeling for soils, in which many salient features, such as nonlinear stress-strain relations, dilatancy, and critical state failure, can be described by a single tensorial equation. However, within the framework of hypoplasticity, modeling fabric anisotropy remains challenging, as the fabric and its evolution are often vaguely assumed without a sound basis. This paper presents a hypoplastic constitutive model for granular soils based on the newly developed anisotropic critical state theory, in which the conditions of fabric anisotropy are concurrently satisfied along with the traditional conditions at the critical state. A deviatoric fabric tensor is introduced into the Gudehus-Bauer hypoplastic model, and a scalar-valued anisotropic state variable signifying the interplay between the fabric and the stress state is used to characterize its impact on the dilatancy and strength of the soils. In addition, fabric evolution during shearing can explicitly be addressed. Modifications have also been undertaken to improve the performance of the undrained response of the model. The anisotropic hypoplastic model can simulate experimental tests for sand under various combinations of principle stress direction, intermediate principal stress (or mode of shearing), soil densities, and confining pressures, and the associated drastic effect of different principal stress orientations in reference to the material axes of anisotropy can be well captured.     

A critical state sand plasticity model accounting for fabric evolution

Fabric and its evolution need to be fully considered for effective modeling of the anisotropic behavior of cohesionless granular sand. In this study, a three‐dimensional anisotropic model for granular material is proposed based on the anisotropic critical state theory recently proposed by Li & Dafalias [2012], in which the role of fabric evolution is highlighted. An explicit expression for the yield function is proposed in terms of the invariants and joint invariants of the normalized deviatoric stress ratio tensor and the deviatoric fabric tensor. A void‐based fabric tensor that characterizes the average void size and its orientation of a granular assembly is employed in the model. Upon plastic loading, the material fabric is assumed to evolve continuously with its principal direction tending steadily towards the loading direction. A fabric evolution law is proposed to describe this behavior. With these considerations, a non‐coaxial flow rule is naturally obtained. The model is shown to be capable of characterizing the complex anisotropic behavior of granular materials under monotonic loading conditions and meanwhile retains a relatively simple formulation for numerical implementation. The model predictions of typical behavior of both Toyoura sand and Fraser River sand compare well with experimental data. Copyright © 2013 John Wiley & Sons, Ltd.     

散粒介质临界状态细观力学结构特征的数值模拟与分析

基于离散单元法对不同密实度理想散粒体进行了双轴剪切试验的宏微观数值模拟,通过网格剖分将Voronoi多边形表征的loop单元作为散粒介质细观力学结构的基本单元,模拟了剪切过程中不同类型loop单元数量、几何形态和力学特征的演化过程,并重点分析了临界状态时散粒介质的细观力学结构特征。模拟结果显示,初始密实度不同的试样在向临界状态发展的过程中,高阶单元与低阶单元的发展规律完全不同,不同初始密实度试样中同阶loop单元的发展规律也不相同,但同阶loop单元的数量比例、几何形态、颗粒接触力及单元内滑动率最终均达到了各自的临界状态。从细观角度分析,散体介质的临界状态是高阶和低阶loop单元在荷载作用下相互转化的结果,是所有loop单元物理力学状态的综合平均与外在表现,临界状态时不同阶数的loop单元处于一个动态平衡状态,宏观上表现为常剪应力和常体积下剪切变形的不断发展。数值模拟结果也表明,loop细观结构单元包含了丰富的信息,其数量、几何形态、受力特征及接触稳定性的发展与散粒体的强度、剪胀以及临界状态的发展密切相关,可以将其作为散粒介质细观尺度的分析单元。     

Fabric dilatancy and the plasticity modeling of granular media

A novel conceptual model of the mechanics of sands is developed within an elastic–plastic framework. Central to this model is the realization that volume changes in anisotropic granular materials occur as a result of two fundamentally different mechanisms. The first is purely kinematic, dilative, and is the result of the changes in anisotropic fabric. There is also a second volume change in granular media that occurs as a direct response to changes in stress as in a standard elastic/plastic continuum. The inclusion of the two sources of volume change results in three important datum states. When subjected to isotropic strains, the resulting stress state in granular materials is not isotropic but lies upon the kinematic normal consolidation line. There exists a state at which the fabric‐induced volumetric strain rate becomes equal to the stress‐induced volumetric strain rate making the total plastic volumetric strain rate equal to zero. Granular response changes from contractive to dilative at this phase transformation line. The third datum state is the one in which the stress‐induced volumetric strain rate is zero. The sand, however, continues to dilate at this state with the difference between stress and dilation ratio a constant as predicted by Taylor's stress–dilatancy rule. These predictions are shown in accordance with experimental data from a series of drained tests and undrained on Ottawa sand. Copyright © 2011 John Wiley & Sons, Ltd.     

初始组构各向异性对砂土力学特性及临界状态的影响

采用离散元方法,利用半径扩展法和重力沉积法分别生成具有初始各向同性和各向异性内结构的试样,并开展三轴不排水压缩和拉伸试验,研究不同制样方法产生的初始各向异性对砂土宏微观力学特性及其临界状态的影响。运用组构张量对砂土的各向异性进行量化,分析不同初始组构各向异性对组构张量演化的影响并确定了组构张量的临界值。试验结果表明：初始组构各向异性对试样的剪胀性有重要影响,由于受重力影响形成初始各向异性,其各向异性程度越大、组构方向与加载方向越一致,剪胀性越显著;初始组构各向异性对试样的临界状态没有影响,砂土的组构张量具有唯一的临界状态值。     

Necessary and sufficient conditions for reaching and maintaining critical state

According to classical critical state theory (CST) of granular mechanics, two analytical conditions on the ratio of stress invariants and the void ratio are postulated to be necessary and sufficient for reaching and maintaining critical state (CS). The present work investigates the sufficiency of these two conditions based on the results of a virtual three-dimensional discrete element method experiment, which imposes continuous rotation of the principal axes of stress with fixed stress principal values at CS. Even though the fixity of the stress principal values satisfies the two analytical CST conditions at the initiation of rotation, contraction and abandonment of CS occur, which proves that these conditions may be necessary but are not sufficient to maintain CS. But if fixity of stress and strain rate directions in regard to the sample is considered at CS, the two analytical conditions of CST remain both necessary and sufficient. The recently proposed anisotropic critical state theory (ACST) turned this qualitative requirement of fixity into an analytical condition related to the CS value of a fabric anisotropy variable A defined in terms of an evolving fabric tensor and the plastic strain rate direction, thus, enhancing the two CST conditions by a third. In this way, the three analytical conditions of ACST become both necessary and sufficient for reaching and maintaining CS. In addition, the use of A explains the observed results by relating the stress-strain response, in particular the dilatancy, to the evolution of fabric by means of the relevant equations of ACST.     

Simulations of realistic granular soils in oedometer tests using physics engine

Discrete element method (DEM) has become a preeminent numerical tool for investigating the mechanical behavior of granular soils. However, traditional DEM uses sphere clusters to approximate realistic particles, which is computationally demanding when simulating many particles. This paper demonstrates the potential of using a physics engine technique to simulate realistic particles. The physics engines are originally developed for video games for simulating physical and mechanical processes that occur in the real world to produce realistic game experiences. The simulation accuracy and efficiency of physics engines have been significantly improved in the last two decades allowing them to be used as a scientific tool in many disciplines. This paper introduces modeling methodologies of physics engine including realistic particle representation and the contact model. Then, oedometer tests are simulated using realistic particles scanned by X-ray computed tomography (X-ray CT). The simulation results agree well with experimental results. This paper demonstrates that physics engines can output contact parameters for geotechnical analysis and force chains for visualization.     

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。     

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.     

Exploring the influence of interparticle friction on critical state behaviour using DEM

Understanding the extent to which discrete element method (DEM) simulations can capture the critical state characteristics of granular materials is important to legitimize the use of DEM in geomechanics. This paper documents a DEM study that considered the sensitivity of the critical state response characteristics to the coefficient of interparticle friction (μ) using samples with gradings that are representative of a real soil. Most of the features that are typically associated with sand behaviour at the critical state were seen to emerge from the DEM simulation data. An important deviation occurs when high μ values (μ ≥ 0.5) are used, as has been the case in a number of prior DEM studies. While there is a systematic variation in the critical state behaviour with μ for μ < 0.5, when μ ≥ 0.5, the behaviour at the critical state seems to be insensitive to further increases in μ. In contrast to observations of conventional soil response, when μ ≥ 0.5, the void ratio at the critical state initially increases with increasing mean effective stress (p′). Analysis of the DEM data and use of simple models of isolated force chains enabled some key observations. When ‘floating’ particles that do not transmit stress are eliminated from the void ratio calculation, the void ratio at the critical state decreases consistently with increasing p′. There is a transition from sliding to rolling behaviour at the contact points as μ increases. Beyond a limiting value of μ, further increases in μ do not increase the buckling resistance of individual strong force chains. Copyright © 2014 John Wiley & Sons, Ltd.     

A non‐coaxial critical state soil model and its application to simple shear simulations

The yield vertex non‐coaxial theory is implemented into a critical state soil model, CASM (Int. J. Numer. Anal. Meth. Geomech. 1998; 22 :621–653) to investigate the non‐coaxial influences on the stress–strain simulations of real soil behaviour in the presence of principal stress rotations. The CASM is a unified clay and sand model, developed based on the soil critical state concept and the state parameter concept. Without loss of simplicity, it is capable of simulating the behaviour of sands and clays within a wide range of densities. The non‐coaxial CASM is employed to simulate the simple shear responses of Erksak sand and Weald clay under different densities and initial stress states. Dependence of the soil behaviour on the Lode angle and different plastic flow rules in the deviatoric plane are also considered in the study of non‐coaxial influences. All the predictions indicate that the use of the non‐coaxial model makes the orientations of the principal stress and the principal strain rate different during the early stage of shearing, and they approach the same ultimate values with an increase in loading. These ultimate orientations are dependent on the density of soils, and independent of their initial stress states. The use of the non‐coaxial model also softens the shear stress evolutions, compared with the coaxial model. It is also found that the ultimate shear strengths by using the coaxial and non‐coaxial models are dependent on the plastic flow rules in the deviatoric plane. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Modelling strain localization in granular materials using micropolar theory: numerical implementation and verification

Implementation and applications for a constitutive numerical model on F‐75 silica sand, course silica sand and two sizes of glass beads compressed under plane strain conditions are presented in this work. The numerical model is used to predict the stress versus axial strain and volumetric strain versus axial strain relationships of those materials; moreover, comparisons between measured and predicted shear band thickness and inclination angles are discussed and the numerical results compare well with the experimental measurements. The numerical model is found to respond to the changes in confining pressure and the initial relative density of a given granular material. The mean particle size is used as an internal length scale. Increasing the confining pressure and the initial density is found to decrease the shear band thickness and increase the inclination angle. The micropolar or Cosserat theory is found to be effective in capturing strain localization in granular materials. The finite element formulations and the solution method for the boundary value problem in the updated Lagrangian frame (UP) are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.     

颗粒大小对颗粒材料力学行为影响初探

利用一种特殊颗粒材料-玻璃珠进行了一系列室内直剪试验,研究颗粒大小对颗粒材料力学行为的影响。试验一共考虑了3条近乎平行的级配曲线和4种颗粒摩擦情况：干燥状态、水浸润状态、水淹没状态和油浸润状态。试验结果表明,颗粒大小对颗粒材料的力学行为有显著影响,剪胀性随着粒径的增大而增强。为考虑颗粒大小对剪胀性的影响,提出了一种新的剪胀关系式。在该剪胀关系式中,剪胀系数为依赖于颗粒大小和颗粒摩擦等颗粒基本性质的变量。试验研究同时表明临界状态摩擦角随着颗粒大小的增加而增加。此外,从颗粒细观运动的角度提出了颗粒滑动的功能模型,推导出了功能方程,并以此揭示了颗粒大小对临界状态摩擦角影响的细观机制。     

Investigation of influence of particle characteristics on the non‐coaxiality of anisotropic granular materials using DEM

As a result of deposition process and particle characteristics, granular materials can be inherently anisotropic. Many researchers have strongly suggested that the inherent anisotropy is the main reason for the deformation non‐coaxiality of granular materials. However, their relationships are not unanimous because of the limited understanding of the non‐coaxial micro‐mechanism. In this study, we investigated the influence of inherent anisotropy on the non‐coaxial angle using the discrete element method. Firstly, we developed a new discrete element method approach using rough elliptic particles and proposed a novel method to produce anisotropic specimens. Secondly, the effects of initial specimen density and particle characteristics, such as particle aspect ratio A _m, rolling resistance coefficient β , and bedding plane orientation δ , were examined by a series of biaxial tests and rotational principal axes tests. Findings from the numerical simulations are summarized as follows: (1) the peak internal friction angle ? _p and the non‐coaxial angle i both increase with the initial density, A _m and β , and they both increase initially and then decrease with δ in the range of 0–90°; (2) among the particle characteristics, the influence of A _m is the most significant; and (3) for anisotropic specimens, the non‐coaxial angle can be calculated using the double slip and rotation rate model. Then, an empirical formula was proposed based on the simulation results to depict the relationship between the non‐coaxial angle and the particle characteristics. Finally, the particle‐scale mechanism of non‐coaxiality for granular materials was discussed from the perspective of energy dissipation. Copyright © 2016 John Wiley & Sons, Ltd.     

Critical state of granular materials at mesoscale: Transition from local to global

Critical state theory is widely accepted for understanding and interpreting the behavior of granular materials such as sandy soils. Recently, motivated by the development of a multiscale analysis method, this paper presents a discrete element investigation on the critical state behavior of granular materials on two scales level, that is, the macroscale and mesoscale. Loops are taken as the basic mesostructures, and the concept of mesoscale critical state is proposed on terms of the local stress and void ratio. Based on series of numerical simulations with varying densities and confining pressures, the characters of critical state lines (CSLs) for each loop type are analyzed. An analytical relationship of the transition from local to global is then obtained for critical state. Besides, the influence of interparticle friction coefficient on CSL is also discussed on both scales. The prominent observation is that each type of loops achieves its own critical state, from which a unique global CSL can be obtained. The CSLs of different loops exhibit same shapes but move upwards with the increasing order.