Diffuse instabilities with transition to localization in loose granular materials

This paper is concerned with diffuse and other ensuing failure modes in geomaterials when tested under homogeneous states of shearing in various loading programs and drainage conditions. Material instability is indeed the basic property that accounts for the instability of an initially homogeneous deformation field leading to diffuse failure and strain localization in geomaterials. The former is normally characterized by a runaway type of failure accompanied with a sudden and violent collapse of the material in the absence of any localization phenomena. Against this backdrop, we present a brief overview of material instability in elastoplastic solids where one finds a rich source of theoretical concepts including bifurcation, strain localization, diffuse failure and second‐order work, as well as a considerable body of experiments. Some compelling laboratory experimental studies of material instability with focus to diffuse failure are then presented and interpreted based on the second‐order work. Finally, various material instability analyses using an elastoplastic constitutive and a general finite element analysis of the above‐mentioned laboratory experimental tests are presented as a boundary value problem. It is shown that instability can be captured from otherwise uniform stress, density and hydraulic states, whereas uniform deviatoric loads are being applied on the external boundaries of a specimen. Although the numerical simulations reproduce well the laboratory experimental results, they also highlight the hierarchy of failure modes where localization phenomena emerge in the post‐bifurcation regime as a result of a degradation of homogeneity starting from a diffuse mode signalled by a zero second‐order work. Copyright © 2012 John Wiley & Sons, Ltd.     

颗粒物质在微剪切振动下的能量耗散研究

采用改进的低频内耗仪研究了几种常见颗粒体系（沙子、玻璃珠）的相对能量耗散性质：能量耗散的振幅谱和频率谱。实验发现,振幅谱中随振幅增大形成一个能量耗散峰,而频率谱线中则观察到随着频率的增加依次出现四个能量耗散峰,对应着体系模量的四个衰减。随探针插入深度的增加相对能耗先增加后减小,出现一个能耗峰值,该峰值对应的深度为体系的临界深度。基于对颗粒体系的介观分析,提出一个流变模型来阐述流变耗散机制,结果表明摩擦在颗粒力学响应中除耗散能量外还起到增加体系弹性的作用。频率谱的分析表明颗粒中还存在另一种耗散机制：颗粒链的共振耗散。     

Second‐order work analysis for granular materials using a multiscale approach

It has been established that the second‐order work criterion is a general necessary condition for all instabilities by divergence in rate‐independent granular materials. The relation between the values of discrete second‐order work at the intergranular contact point level and its global macroscopic value is recalled at the beginning of this paper. Then, the basic purpose of the paper is tackled by an analysis of the main features of second‐order work criterion in relation with the granular microstructure. For that, it is considered a novel micromechanical model (the so‐called ‘H‐microdirectional model’), which has the property to involve three scales: grain scale, mesoscale with a specific granular configuration and continuum mechanics macroscale. Eventually, these exhibited features (a bifurcation stress domain and some instability cones) are qualitatively compared with the ones provided by direct numerical simulations issued from a discrete element model. The ultimate goal is to analyse what happens at the granular scale, when the macrosecond‐order work is vanishing at the macrolevel. Copyright © 2013 John Wiley & Sons, Ltd.     

A visco‐plastic constitutive model for granular soils modified according to non‐local and gradient approaches

An already available non‐associated elastic–viscoplastic constitutive model with anisotropic strain hardening is modified in order to describe both the constitutive parameter dependency on relative density and the spatio‐temporal evolution of strain localization. To achieve this latter goal, two distinct but similar approaches are introduced: one inspired by the gradient theory and one by the non‐local theory. A one‐dimensional case concerning a simple shear test for a non‐homogeneous infinitely long dense sand specimen is numerically discussed and a finite difference scheme is employed for this purpose. The results obtained by following the two different approaches are critically analysed and compared. Copyright © 2001 John Wiley & Sons, Ltd.     

Bifurcation and second‐order work in geomaterials

In this paper, the ability of a material rate‐independent system to evolve toward another mechanical state from an equilibrium configuration, with no change in the control parameters, is investigated. From a mechanical point of view, this means that the system can spontaneously develop kinetic energy with no external disturbance from an equilibrium state, which corresponds to a particular case of bifurcation. The existence of both conjugate incremental strain and stress such that the second‐order work vanishes is established as a necessary and sufficient condition for the appearance of this bifurcation phenomenon. It is proved that this fundamental result is independent of the constitutive relation of the rate‐independent material considered. Then the case of homogeneous loading paths is investigated, and, as an illustration, the subsequent results are applied to interpret the well‐known liquefaction observed under isochoric triaxial loading conditions with loose granular materials. Copyright © 2006 John Wiley & Sons, Ltd.     

双轴压缩条件下颗粒材料中力链的演化

针对无黏性颗粒材料,开展了柔性边界双轴压缩数值试验,分析了不同围压下力链的数量、方向概率等随轴向应变的变化。研究结果表明：不断加载使得游离的高应力颗粒有所减少,荷载更多的被力链承担;统计意义上,力链最大长度为9个颗粒。轴向±30o范围内力链概率随应变的发展与偏应力一致,而与其邻接的[40o,60o]和[120o,140o]两个方向上的力链概率随应变的发展与偏应力相反;双轴试样剪切破坏时有单剪切带和双剪切带两种模式,当[40o,60o]和[120o,140o]两个角度区间的力链概率有较大差异时,剪切带出现在力链较少的角度区间;当这两个角度区间力链概率接近时,试样中将出现两个剪切带。     

颗粒接触力检测方法评述

颗粒物质是大量固体颗粒体系的集合体,其特性区别于普通固体和液体等物质的性质,颗粒物质基础力学问题的研究20年来一直是科学的前沿,从分析多尺度结构和相关物理机制的角度研究颗粒物质可能取得突破。首先介绍了微观尺度--亦即单颗粒尺度颗粒间接触力的检测方法,包括高精度电子天平称重法、显色灵敏复写纸压痕方法、光弹实验方法、荧光共聚焦显微镜法和磁共振弹性成像法等,根据检测过程中是否对颗粒带来影响划分为接触式检测和非接触式检测两大类,并对这些方法的工作原理、检测结果和优缺点进行了评述。我们指出接触式测量对颗粒体系带来了或多或少的干扰,特别是由于力链对局部力的变化反应极为敏感,轻微的变化足以使得力链结构发生很大变化,开展无干扰的光测实验是必然的发展趋势,而现阶段开展二维光弹性实验是很有意义的。最后介绍了作者正在进行的颗粒物质光弹力学实验的研究工作。     

Diffuse bifurcations engraving diverse shear bands in granular materials

Shear bands with characteristic spatial patterns observed in an experiment for a cubic or parallelepiped specimen of dry dense sand were simulated by numerical bifurcation analysis using the Cam‐clay plasticity model. By incorporating the subloading surface concept into the plasticity model, the model became capable of reproducing hardening/softening and contractive/dilative behavior observed in the experiment. The model was reformulated to be compatible with the multiplicative hyperelasto‐plasticity for finite strains. This enhanced constitutive model was implemented into a finite‐element code reinforced by a stress updating algorithm based on the return‐mapping scheme, and by an efficient numerical procedure to compute critical eigenvectors of elastoplastic tangent stiffness matrix at bifurcation points. The emergence of diamond‐ and column‐like diffuse bifurcation modes breaking uniformity of the materials, followed by the evolution of shear bands through strain localization, was observed in the analysis. In the bifurcation analysis of plane strain compression test, unexpected bifurcation modes, which broke out‐of‐plane uniformity and led to 3‐dimensional diamond‐like patterns, were detected. Diffuse bifurcations, which were difficult to observe by experiments, have thus been found as a catalyst creating diverse shear band patterns.     

光弹颗粒材料的直剪实验研究

对颗粒材料中的力链分布特征的了解,有助于从根本上了解颗粒材料基本性质。借助光弹实验,对两种直径混合的颗粒材料在二维直剪作用下的平均强度力链分布和几何结构变化进行了初步的探索。光弹实验方法是目前能够直观观察到力链分布的唯一有效的实验方法,在进行光弹实验前,设计研制了光弹直剪仪并进行了多次光弹圆盘颗粒材料退火实验。利用数字图像技术对光弹实验结果进行处理,并尝试用彩色梯度算法表征颗粒材料的平均受力情况,并获得平均强度力链。结果表明,二元颗粒材料集合体的几何结构具有各向异性。对力链方向统计分析表明,平均强度力链出现了局部化。     

Stiffness pathologies in discrete granular systems: Bifurcation,neutral equilibrium,and instability in the presence of kinematic constraints

The paper develops the stiffness relationship between the movements and forces among a system of discrete interacting grains. The approach is similar to that used in structural analysis, but the stiffness matrix of granular material is inherently nonsymmetric because of the geometrics of particle interactions and of the frictional behavior of the contacts. Internal geometric constraints are imposed by the particles' shapes, in particular, by the surface curvatures of the particles at their points of contact. Moreover, the stiffness relationship is incrementally nonlinear, and even small assemblies require the analysis of multiple stiffness branches, with each branch region being a pointed convex cone in displacement space. These aspects of the particle-level stiffness relationship give rise to three types of microscale failure: neutral equilibrium, bifurcation and path instability, and instability of equilibrium. These three pathologies are defined in the context of four types of displacement constraints, which can be readily analyzed with certain generalized inverses. That is, instability and nonuniqueness are investigated in the presence of kinematic constraints. Bifurcation paths can be either stable or unstable, as determined with the Hill–Bažant–Petryk criterion. Examples of simple granular systems of three, 16, and 64 disks are analyzed. With each system, multiple contacts were assumed to be at the friction limit. Even with these small systems, microscale failure is expressed in many different forms, with some systems having hundreds of microscale failure modes. The examples suggest that microscale failure is pervasive within granular materials, with particle arrangements being in a nearly continual state of instability.     

Experimental investigation of slip‐stick behaviour in granular materials

This paper presents the results of an experimental programme to study load oscillation in granular materials. Spherical glass beads were used in the investigation. Cylindrical specimens were compressed under axisymmetric triaxial loading condition at 25,100,250 and 400 kPa confining pressures. The test parameters included: (i) particle size; (ii) gradation (uniform versus non‐uniform specimens); (iii) confining pressure; (iv) loading rate; and (v) specimen density. In general, a slight post‐peak principal stress softening was observed as well as a continuous volume increase (dilation) even at relatively high strains. This appears to be caused by the uniform shape and smooth surface of the spherical particles. Load oscillations were observed in the very small, small, and well‐graded beads at each confining pressure. For the medium beads, the oscillations appeared at high confining pressure (250 and 400 kPa), and they did not appear in the large beads. The influences of the confining pressure, loading rate, particle size, gradation and specimen density on load oscillation are presented and discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

On incremental non‐linearity in granular media: phenomenological and multi‐scale views (Part I)

On the basis of fundamental constitutive laws such as elasticity, perfect plasticity, and pure viscosity, many elasto‐viscoplastic constitutive relations have been developed since the 1970s through phenomenological approaches. In addition, a few more recent micro‐mechanical models based on multi‐scale approaches are now able to describe the main macroscopic features of the mechanical behaviour of granular media. The purpose of this paper is to compare a phenomenological constitutive relation and a micro‐mechanical model with respect to a basic issue regularly raised about granular assemblies: the incrementally non‐linear character of their behaviour. It is shown that both phenomenological and micro‐mechanical models exhibit an incremental non‐linearity. In addition, the multi‐scale approach reveals that the macroscopic incremental non‐linearity could stem from the change in the regime of local contacts between particles (from plastic regime to elastic regime) in terms of the incremental macroscopic loading direction. Copyright © 2005 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.     

Kinematic models for non‐coaxial granular materials. Part I: theory

The purpose of this paper is to present a physically based plasticity model for non‐coaxial granular materials. The model, which we shall call the double slip and rotation rate model (DSR² model), is a pair of kinematic equations governing the velocity field. The model is based on a discrete micro‐analysis of the kinematics of particles in contact, and is formulated by introducing a quantity called the averaged micro‐pure rotation rate (APR) into the unified plasticity model which was proposed by one of the authors. Our macro–micro mechanical analysis shows that the APR is a non‐linear function of, among other quantities, the macro‐rotation rate of the major principal axis of stress taken in the opposite sense. The requirement of energy dissipation used in the double‐sliding free‐rotating model appears to be unduly restrictive as a constitutive assumption in continuum models. In the DSR² model the APR tensor and the spin tensor are directly linked with non‐coaxiality of the stress and deformation rate tensors. We also propose a simplified plasticity model based on the DSR² model for a class of dilatant materials, and analyse its material stability. Copyright © 2005 John Wiley & Sons, Ltd.     

Mechanical properties of granular materials: A variational approach to grain‐scale simulations

The mechanical properties of cohesionless granular materials are evaluated from grain‐scale simulations. A three‐dimensional pack of spherical grains is loaded by incremental displacements of its boundaries. The deformation is described as a sequence of equilibrium configurations. Each configuration is characterized by a minimum of the total potential energy. This minimum is computed using a modification of the conjugate gradient algorithm. Our simulations capture the nonlinear, path‐dependent behavior of granular materials observed in experiments. Micromechanical analysis provides valuable insight into phenomena such as hysteresis, strain hardening and stress‐induced anisotropy. Estimates of the effective bulk modulus, obtained with no adjustment of material parameters, are in agreement with published experimental data. The model is applied to evaluate the effects of hydrate dissociation in marine sediments. Weakening of the sediment is quantified as a reduction in the effective elastic moduli. Copyright © 2008 John Wiley & Sons, Ltd.     

Material stability analysis of rock joints

For prediction of rockfalls, the failure of rock joints is studied. Considering these failures as constitutive instabilities, a second‐order work criterion is used because it explains all divergence instabilities (flutter instabilities are excluded). The bifurcation domain and the loading directions of instabilities, which fulfill the criterion, are determined for any piecewise linear constitutive relation. The instability of rock joints appears to be ruled by coupling features of the behavior (e.g., dilatancy). Depending on the loading parameters, instabilities can lead to failure, even before the plastic limit criterion. Results for two given constitutive relations illustrate the approach. Some given loading paths are especially considered. Constant volume (undrained) shear and τ‐constant paths are stable or not depending on the link between the deviatoric stress and strain along undrained paths, as found for soils. Some unstable loading paths are illustrated. Along these paths, failure before the plastic limit criterion is possible. The corresponding failure rules are determined. Copyright © 2012 John Wiley & Sons, Ltd.     

Fibre-reinforcement of granular materials: DEM visualisation and analysis

Mechanical properties of granular materials can be significantly improved by the inclusion of small amounts of short synthetic fibres. This phenomenon has been experimentally studied before by many researchers who have found that the shear strength of a granular material can be significantly improved. This study presents a visualisation of the phenomenon using discrete element method (DEM) simulations of direct shear tests conducted on mixtures of an idealised granular material and randomly oriented fibres. Snapshots showing the deformation of the samples, the velocity vectors of the particles and the force chains produced inside the samples are presented at different stages of the tests. Changes in shear strength and porosity are also illustrated. It was found that the reinforcement produced depends mostly on the rigidity of the fibres.     

基于微形态理论的各向异性散体波动分析

基于散粒体微观力学理论,忽略颗粒转动引起的相对位移,考虑颗粒接触的组构各向异性,根据宏微观能量守恒推导得到了散体材料各向异性微形态本构关系,进而通过单位接触方向积分的递推公式推导出了各向异性本构张量表达式;在此基础上,根据哈密顿原理得到了各向异性散体材料的运动平衡方程和边界条件,从而求得了平面波在各向异性散粒体中的传播规律和频散关系,最后对波的频散关系和频率带隙进行了详细的参数分析。研究表明：该模型预测了散体中包含3类12种位移波：3种纵波、6种横波和3种平面内横向剪切波;横观各向同性条件下,接触各向异性参数a20越大,纵波和横波的频率越大,而平面内横向剪切波的频率越小;正交各向异性条件下,随着接触各向异性参数a22的增大,与2方向运动相关的横波频率增大,而与3方向运动相关的横波频率则减小;但a22的变化对纵波频率影响很小。材料各向异性程度对横波带宽影响不大,但对纵波带宽影响较大：a20的增大使得声?光学波间的带宽减小,而光学波间的带宽增大,当a20>0.84时,声?光学波间的带隙消失;但是a22的增大则使得声?光学波间的带宽增大,而光学波间的带宽减小。退化为各向同性模型后,预测3类波的频散曲线与其他各向同性模型的结果基本一致。