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.     

Strain localization in sand: an overview of the experimental results obtained in Grenoble using stereophotogrammetry

Experimental results are presented from the extensive program of drained plane strain compression tests on sand carried out in Grenoble over the last two decades. Systematic analysis of photographs of the deforming specimen allowed for measuring deformations and determining strain fields throughout the test, that is: prior to, at, and after the onset of strain localization. The principles, details and accuracy of the procedure are described, as well as its suitability to properly depict the patterns of deformation. Findings concerning the occurrence and progression of strain localization are discussed. The issues of shear band orientation and thickness are addressed, as well as temporary and persistent complex localization patterns, and the volumetric behaviour inside a band after its formation. The influence of such variables as initial state of the sand (effective stress and relative density), specimen size and slenderness, as well as grain size, is discussed. Copyright © 2004 John Wiley & Sons, Ltd     

基于CLoE与Gudehus-Bauer亚塑性模型的颗粒材料应变局部化对比研究

基于CLoE与Gudehus-Bauer亚塑性模型数值模拟了平面应变条件下Hostun砂的应变局部化现象。从侧向压力和初始缺陷两个方面对比研究了两种模型所预测应变局部化的产生及演化模式。结果表明：（1）两种模型均能反映Hostun砂刚度随着侧向压力提高而增大的现象。（2）相比Gudehus-Bauer亚塑性模型,CLoE亚塑性模型所得出的应变局部化形态与试验结果更加一致。（3）CLoE亚塑性模型能够反映随着荷载增加,砂的体积先膨胀后缩小的特点。（4）相比Gudehus-Bauer亚塑性模型,CLoE亚塑性模型所得到的应变-应力曲线能够更明显地反映应变局部化带中单元的软化现象。（5）CLoE亚塑性模型能够更好地模拟由初始缺陷导致的不均匀应变。总的来说,所得的数值结果表明,CLoE亚塑性模型能够较好地模拟侧向压力和初始缺陷对应变局部化的影响,在模拟应变局部化现象方面较Gudehus-Bauer更有优势。然而,现有CLoE亚塑性模型无法考虑孔隙比,也未包含颗粒材料内尺度变量,有待进一步完善。     

Phenomenological issues related to strain localization in sensitive clays

A negative second order work, strain softening, is often noticed in contractant material like sensitive clays. Failure in such clays will lead to the formation of localized deformation zone of intense inelastic strain, known as shear band. Conditions, emergence and inclination of shear band has been very well demonstrated in past decades in different manners, however a definite thickness of shear band is still an open question due to several reasons. Mesh dependency, loss of ellipticity is another challenge associated with finite element analyses for strain softening clays. This paper covers a comprehensive review of classical theories of strain localization and associated limitation. Mesh dependency, ill-possed boundary value problem is addressed using finite element simulation examples and experimental results.     

基于罚函数偶应力理论的土体应变局部化研究

与 Cosserat 理论相比,偶应力理论在一定程度上可以降低数值框架的复杂度,已逐渐应用于岩土体应变局部化分析中。然而,一般的偶应力有限元法需要满足 C¹连续性,即单元内部和单元交界面上的应变都需要具有连续性。为了避免开发较为复杂的C¹型偶应力单元,在 Cosserat 连续体理论框架下,通过借助罚函数方法对 C¹连续性进行松弛来获得偶应力理论的逼近解,建立了基于罚函数的偶应力有限元方法 PCS-FEM。通过平面应变条件下的弹性圆孔应力集中问题对 PCS-FEM 方法的有效性进行了验证,并应用于土体应变局部化分析中。通过对Ottawa砂的平面应变试验进行数值模拟,发现 PCS-FEM 方法获得的应力−应变曲线及剪切带破坏形态与试验结果基本一致,且能够克服经典连续体理论病态的网格敏感性问题,保证应变局部化问题的适定性;通过对承受偏心荷载作用下的土坡应变局部化经典算例进行分析,发现 PCS-FEM 方法同样可以克服土坡应变软化阶段的网格敏感性问题,展现土体的渐进破坏过程。     

Gradient plasticity modelling of strain localization in granular materials

The flow stress in the yield surface of plastic constitutive equation is modified with a higher order gradient term of the effective plastic strain to model the effect of inhomogeneous deformation in granular materials. The gradient constitutive model has been incorporated into the finite element code ABAQUS and used to simulate biaxial shear tests on dry sand. It is shown that the shape of the post-peak segment of the load displacement curve predicted by the numerical analysis is dependent on the mesh size when gradient term is not used. Use of an appropriate gradient coefficient is shown to correct this and predict a unique shape of the load displacement curve regardless of the mesh size. The gradient coefficient required turns out to be approximately inversely proportional to the mesh elemental area. Use of the strain gradient term is found to diffuse the concentration of plastic strains within shear band resulting in its consistent width. The coefficient of the higher gradient term appears as a function of the grain size, the mean confining stress, and the plastic softening modulus. Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of grain crushing on shear localization in granular bodies during plane strain compression

This article deals with the effect of grain crushing on shear localization in granular materials during plane strain monotonic compression tests under constant lateral pressure. The grain diameter and the initial void ratio were stochastically distributed using a spatial correlation. To describe the mechanical behavior of cohesionless granular materials during a monotonic deformation path in plane strain compression, we used a micropolar hypoplastic constitutive model that is able to describe the salient properties of granular bodies including shear localization. The model was extended by introducing changes to the grain diameter with varying pressure using formulae from breakage mechanics proposed for crushable granulates. The initial void ratios and grain diameters took the form of correlated random spatial fields described by both symmetric and nonsymmetric random distributions using a homogeneous correlation function. The field realizations were generated with the help of an original conditional rejection method. A few representative samples of the random fields selected from the generated set were taken into account in numerical calculations. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Fabric evolution within shear bands of granular materials and its relation to critical state theory

In an effort to study the relation of fabrics to the critical states of granular aggregates, the discrete element method (DEM) is used to investigate the evolution of fabrics of virtual granular materials consisting of 2D elongated particles. Specimens with a great variety of initial fabrics in terms of void ratios, preferred particle orientations, and intensities of fabric anisotropy were fabricated and tested with direct shear and biaxial compression tests. During loading of a typical specimen, deformation naturally localizes within shear bands while the remaining of the sample stops deforming. Thus, studying the evolution of fabric requires performing continuous local fabric measurements inside these bands, a suitable task for the proposed DEM methodology. It is found that a common ultimate/critical state is eventually reached by all specimens regardless of their initial states. The ultimate/critical state is characterized by a critical void ratio e which depends on the mean stress p, while the other critical state fabric variables related to particle orientations are largely independent of p. These findings confirm the uniqueness of the critical state line in the e ? p space, and show that the critical state itself is necessarily anisotropic. Additional findings include the following: (1) shear bands are highly heterogeneous and critical states exist only in a statistical sense; (2) critical states can only be reached at very large local shear deformations, which are not always obtained by biaxial compression tests (both physical and numerical); (3) the fabric evolution processes are very complex and highly dependent on the initial fabrics. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Effects of boundary conditions and partial drainage on cyclic simple shear test results—a numerical study

Owing to imperfect boundary conditions in laboratory soil tests and the possibility of water diffusion inside the soil specimen in undrained tests, the assumption of uniform stress/strain over the sample is not valid. This study presents a qualitative assessment of the effects of non‐uniformities in stresses and strains, as well as effects of water diffusion within the soil sample on the global results of undrained cyclic simple shear tests. The possible implications of those phenomena on the results of liquefaction strength assessment are also discussed. A state‐of‐the‐art finite element code for transient analysis of multi‐phase systems is used to compare results of the so‐called ‘element tests’ (numerical constitutive experiments assuming uniform stress/strain/pore pressure distribution throughout the sample) with results of actual simulations of undrained cyclic simple shear tests using a finite element mesh and realistic boundary conditions. The finite element simulations are performed under various conditions, covering the entire range of practical situations: (1) perfectly drained soil specimen with constant volume, (2) perfectly undrained specimen, and (3) undrained test with possibility of water diffusion within the sample. The results presented here are restricted to strain‐driven tests performed for a loose uniform fine sand with relative density D_r=40%. Effects of system compliance in undrained laboratory simple shear tests are not investigated here. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐coaxiality and stress–dilatancy rule in granular materials: FE investigation within micro‐polar hypoplasticity

This paper deals with FE investigations of shear localization in dilatant granular bodies. The calculations were carried out with a hypoplastic constitutive law enhanced by micro‐polar terms to properly model the shear zone evolution. The behaviour of an initially medium dense sand specimen with very smooth and very rough horizontal boundaries was analyzed during a plane strain compression test. A stochastic distribution of the initial void ratio was assumed to be spatially correlated. Attention was focused on the non‐coaxiality of the directions of the principal strain increments and principal stresses in the shear zone and on the stress–dilatancy rule. Copyright © 2008 John Wiley & Sons, Ltd.     

A distinct element method numerical investigation of compaction processes in highly porous cemented granular materials

In this paper, the results of an oedometric numerical test campaign, performed by means of a 3D Discrete Element Code on idealised cemented granular cylindrical specimens, are illustrated. The idealised microstructure taken into account is characterised by the following: (i) rigid grains bonded to one another; (ii) a high void ratio; and (iii) two different families of voids: the micro and the macro‐voids. The compaction process developing within the specimens, as well as the localization along tabular zones of pure compressive deformation (compaction banding) that in some cases takes place, are discussed. The influence on the evolution of this peculiar strain localization process of many microstructural/numerical parameters like material porosity, macro‐void size, the constitutive relationship adopted for the bonds and the bond damage rate is analysed. Tests for different values of porosity were run. Below a certain porosity threshold value, the onset of mixed modes of localisation was detected whereas the increase in the macro‐void size is observed to favour the onset of instability. Copyright © 2014 John Wiley & Sons, Ltd.     

饱和砂土局部变形带模拟的有限元数值实现

基于有限变形理论,推导了Newton-Raphson 迭代算法在k+1步增量表达的矩阵形式,实现了饱和砂土变形局部化的有限元数值计算,得到了饱和砂土发生局部化变形的准则。基于Galerkin 方法,得到了位移场和应力场的空间离散化矩阵方程;由土体局部变形带的连续性条件,引入第1切线算子,推导出了砂土等颗粒状媒介发生局部化变形的必要条件。基于此核心算法,编制了有限元计算程序,模拟了饱和砂土在不排水条件下平面压缩过程中剪切带的形成与发展;通过比较分析,研究了有限元网格粗细对于土体局部变形带的影响,结果表明,网格粗细的病态依赖只是微小的,它只与变形条带的宽度有关,对于土体所表现出来的其他力学特性没有影响。     

基于应变响应包络的颗粒材料增量力学行为研究

砂土的力学响应具有显著的增量非线性特征,这与离散颗粒的微观结构,即组构特性密不可分。采用应变响应包络可以较好地获得材料在不同加载方向上的增量力学响应。在物理试验中无法同时获得同一试样在不同加载方向下的宏、微观响应,故采用离散元方法,对具有不同应力历史、不同应力状态和不同级配的试样在Rendulic平面上的增量力学响应进行了系统研究和分析。离散元模拟结果表明,传统的塑性理论不能很好地描述具有复杂应力历史试样的增量力学响应,而基于颗粒间接触法向的组构增量与剪应变之间在多种不同的工况条件下均具有较强的线性相关性。当砂土的相对密实度相同时,该线性系数主要与围压的大小有关,对颗粒级配、应力历史以及应力比的变化均不敏感。由于组构的大小可以较好地量化砂土的内结构各向异性程度,进而表征应力历史的作用。上述模拟结果为在本构建模中引入组构演化机制,综合反映外荷载和内结构各向异性对砂土增量本构关系的影响提供了较好的微观物理依据。     

Inelastic constitutive properties and shear localization in Tennessee marble

The inelastic response of Tennessee marble is modelled by an elastic plastic constitutive relation that includes pressure dependence of yield, strain‐softening and inelastic volume strain (dilatancy). Data from 12 axisymmetric compression tests at confining pressures from 0 to 100 MPa are used to determine the dependence of the yield function and plastic potential, which are different, on the first and second stress invariants and the accumulated inelastic shear strain. Because the data requires that the strain at peak stress depends on the mean stress, the locus of peak stresses is neither a yield surface nor a failure envelope, as is often assumed. Based on the constitutive model and Rudnicki and Rice criterion, localization is not predicted to occur in axisymmetric compression although faulting is observed in the tests. The discrepancy is likely due to the overly stiff response of a smooth yield surface model to abrupt changes in the pattern of straining. The constitutive model determined from the axisymmetric compression data describes well the variation of the in‐plane stress observed in a plane strain experiment. The out‐of‐plane stress is not modelled well, apparently because the inelastic normal strain in this direction is overpredicted. In plane strain, localization is predicted to occur close to peak stress, in good agreement with the experiment. Observation of localization on the rising portion of the stress–strain curve in plane strain does not, however, indicate prepeak localization. Because of the rapid increase of mean stress in plane strain, the stress–strain curve can be rising while the shear stress versus shear strain curve at constant mean stress is falling (negative hardening modulus). Copyright © 2001 John Wiley & Sons, Ltd.     

Strain localization in clay: plane strain versus triaxial loading conditions

The paper presents a comparison between the behavior of slurry-consolidated Kaolin specimens tested under axisymmetric Conventional Triaxial Compression (CTC) and Plane Strain (PS) loading conditions. The PS experiments were conducted on an instrumented apparatus capable of capturing the onset of shear bands. The specimens were consolidated and then sheared under undrained condition. The PS specimens failed via a well-defined shear band that began to develop during the hardening stress regime whereas CTC specimens failed through a diffuse bulging mode. The undrained shear strength (S _u) of CTC experiments is smaller than the S _u of PS experiments. However, the normalized S _u of CTC experiments is very close to PS1, which was consolidated under similar K _o condition.     

A novel coupled FDM-DEM modelling method for flexible membrane boundary in laboratory tests

Difficulties are involved in discrete element method (DEM) modelling of the flexible boundary, that is, the membranes covering the soil sample, which can be commonly found in contemporary laboratory soil tests. In this paper, a novel method is proposed wherein the finite difference method (FDM) and DEM are coupled to simulate the rubber membrane and soil body, respectively. Numerical plane strain and triaxial tests, served by the flexible membrane, are implemented and analysed later. The effect of the membrane modulus on the measurement accuracy is considered, with analytical formulae derived to judge the significance of this effect. Based on an analysis of stress-strain responses and the grain rotation field, the mechanical performances produced by the flexible and rigid lateral boundaries are compared for the plane strain test. The results show that (1) the effect of the membrane on the test result becomes more significant at larger strain level because the membrane applies additional lateral confining pressure to the soil body; (2) the tested models reproduce typical stress and volumetric paths for specimens with shear bands; (3) for the plane strain test, the rigid lateral boundary derives a much higher peak strength and larger bulk dilatation, but a similar residual strength, compared with the flexible boundary. The latter produces a more uniform (or ‘diffuse') rotation field and more mobilised local kinematics than does the former. All simulations show that the proposed FDM-DEM coupling method is able to simulate laboratory tests with a flexible boundary membrane.     

Strain localization in soft rock—a typical rate‐dependent solid: experimental and numerical studies

Strain localization developing inside soft rock specimens is examined through experimental observation and numerical simulation. In the experimental study, soft rock specimens are sheared at different strain rates under plane strain conditions and deformation and strain localization characteristics are analysed. Transition of localization mode from highly localized mode for higher strain rate to distributed and diffused mode of strain localization for lower strain rates was observed. In the numerical study, simulations of plane strain compression tests are carried out at different strain rates by using an overstressed‐type elasto‐viscoplastic model in finite element computations. The role of strain rates on setting gradients of strain fields across shear band is clarified. The probable mechanism for transition of localization mode is discussed. Copyright © 2005 John Wiley & Sons, Ltd.