Localized failure of fibre‐reinforced elastic–plastic materials subjected to plane strain loading

We consider discontinuous bifurcations as the indicator of a localized failure for a class of composites that are characterized by elastic fibres reinforcing an elastic–plastic matrix. A macroscopic tangent stiffness tensor for the fibre‐reinforced composite is developed by consistently homogenizing the contribution of fibres in a spherical representative volume element. Analytical solutions are derived for the critical hardening modulus and corresponding bifurcation directions for the case of plane strain loading. Properties of the solutions are further illustrated on the example of the non‐associated Drucker–Prager model at onset of yielding. Results show that presence of fibres decreases the critical hardening modulus, thus inhibiting the onset of strain localization. The rate of decrease in the critical hardening modulus is the highest for pure shear, followed by uniaxial tension, uniaxial compression, biaxial tension and biaxial compression. The main fibre parameters that control the onset of strain localization are their volumetric content and their stiffness modulus whereby very stiff fibres can produce the most significant decrease in the critical hardening modulus, especially for the state of biaxial tension. The critical hardening modulus for the non‐associated Drucker–Prager model exhibits a full range of localization modes including compaction bands, dilation bands, and transition in the form of shear bands regardless of the presence of fibres. Presence of fibres affects bifurcation directions, except in the case when Poisson's ratio of the matrix is equal to 0.25. The results demonstrate stabilizing effects of fibres by which they provide the control against the onset of strain localization. Copyright © 2006 John Wiley & Sons, Ltd.     

Influence of Lode angle‐dependent failure criteria on shear localization analysis in sand

Geotechnical experiments show that Lode angle‐dependent constitutive formulations are appropriate to describe the failure of geomaterials. In the present study, we have adopted one such class of failure criteria along with a versatile constitutive relationship to theoretically analyze the effects of Lode angle on localized shear deformation or shear band formation in loose sand for both drained and undrained conditions. We determine the variation in the possible stress states for shear localization due to the introduction of Lode angle by considering the localized deformation as a bifurcation problem. Further, similar bifurcation analysis is performed for the stress states along a specific loading path, namely, plane strain compression at the constitutive level. In addition, the plane strain compression tests have been simulated as a boundary value finite element problem to see how Lode angle affects the post‐localization response. Results show that the inclusion of a Lode angle parameter within the failure criterion has considerable effects on the onset, plastic strain, and propagation of shear localization in loose sand specimens. For drained condition, we notice early inception of shear localization and multiple band formation when the Lode angle‐dependent failure criterion is used. Undrained localization characteristics, however, found to be independent of Lode angle consideration.     

单轴压缩下岩石应变局部化的应变梯度塑性解

通过在屈服函数中引入材料内部长度建立应变梯度塑性增量本构关系,对岩石材料在单轴压缩条件下应变局部化的二维情况进行了研究。首先,建立了在Mises屈服准则下平面应变情况的岩石材料应变局部化带的带宽与其倾角的关系式,讨论了其带宽的分布范围和最大值与最小值形成的条件,给出岩石材料破坏Ⅰ, Ⅱ类变形的条件,并推出破坏的临界角度;其次,对岩石材料Ⅰ类变形进行了详细地分析,讨论了在不同泊松比 和 条件下材料应变局部化带的倾角的变化范围;最后,利用所得的结论同材料破坏的库仑准则进行对比研究,对内摩擦系数给予了合理的解释。     

层状岩体单轴和双轴压缩蠕变特性的数值试验

以绿片岩和大理岩组成的层状岩体为研究对象,采用FLAC3D对互层状岩体进行了单轴和双轴压缩蠕变试验的数值分析,在数值分析中考虑荷载方向与层理之间的几何关系、大理岩夹层的体积分数、应力水平等的影响。研究结果表明:单轴和双轴压缩条件下,随着夹层倾角由0°增加至90°,轴向和夹层倾斜方向的应变绝对值均呈先增大后减小的变化规律;随着大理岩夹层体积分数的增加,轴向压缩变形和2个侧向方向的膨胀变形量均有所减小。单轴压缩条件下,当轴向荷载方向垂直于层理时,轴向压缩变形均大于轴向荷载方向平行于层理时的轴向压缩变形;双轴压缩条件下,当轴向荷载方向垂直于层理、侧向荷载方向平行于层理时,轴向压缩变形最大,当轴向荷载方向平行于层理、侧向荷载方向垂直于层理时,轴向压缩变形最小。     

Capturing strain localization behind a geosynthetic‐reinforced soil wall

This paper presents the results of finite element (FE) analyses of shear strain localization that occurred in cohesionless soils supported by a geosynthetic‐reinforced retaining wall. The innovative aspects of the analyses include capturing of the localized deformation and the accompanying collapse mechanism using a recently developed embedded strong discontinuity model. The case study analysed, reported in previous publications, consists of a 3.5‐m tall, full‐scale reinforced wall model deforming in plane strain and loaded by surcharge at the surface to failure. Results of the analysis suggest strain localization developing from the toe of the wall and propagating upward to the ground surface, forming a curved failure surface. This is in agreement with a well‐documented failure mechanism experienced by the physical wall model showing internal failure surfaces developing behind the wall as a result of the surface loading. Important features of the analyses include mesh sensitivity studies and a comparison of the localization properties predicted by different pre‐localization constitutive models, including a family of three‐invariant elastoplastic constitutive models appropriate for frictional/dilatant materials. Results of the analysis demonstrate the potential of the enhanced FE method for capturing a collapse mechanism characterized by the presence of a failure, or slip, surface through earthen materials. Copyright © 2003 John Wiley & Sons, Ltd.     

The influence of the plastic potential on plane strain failure

The influence of the shape of the plastic potential in the deviatoric plane on plane strain collapse is investigated. The most commonly employed elastic‐perfect plastic models are considered, which adopt well‐known failure criteria for defining the yield and plastic potential surfaces, namely the von Mises, the Drucker–Prager, the Tresca, the Mohr–Coulomb and the Matsuoka–Nakai criteria. Finally, the conclusions are also extended to strain hardening/softening models. For simple constitutive models based on perfect plasticity, it is shown that the value of the Lode's angle at plastic collapse in plane strain conditions strongly depends on the specific failure surface adopted for reproducing the plastic potential surface. If the value of the Lode's angle at yield coincides with the failure value prescribed by the plastic potential, the stress–strain curves exhibit the typical perfect plastic behaviour with yield coinciding with failure, otherwise the stress changes after yield and the stress‐strain curves resemble those of strain hardening/softening models. The infinite strength which is in some situations exhibited by the Drucker–Prager model in plane strain condition is investigated and explained, and it is shown that this can also affect strain hardening/softening models. Copyright © 2013 John Wiley & Sons, Ltd.     

Influences of loading direction and intermediate principal stress ratio on the initiation of strain localization in cross-anisotropic sand

Since cross-anisotropic sand behaves differently when the loading direction or the stress state changes, the influences of the loading direction and the intermediate principal stress ratio (b = (σ ₂ ? σ ₃)/(σ ₁ ? σ ₃)) on the initiation of strain localization need study. According to the loading angle (angle between the major principal stress direction and the normal of bedding plane), a 3D non-coaxial non-associated elasto-plasticity hardening model was proposed by modifying Lode angle formulation of the Mohr–Coulomb yield function and the stress–dilatancy function. By using bifurcation analysis, the model was used to predict the initiation of strain localization under plane strain and true triaxial conditions. The predictions of the plane strain tests show that the major principal strain at the bifurcation points increases with the loading angle, while the stress ratio decreases with the loading angle. According to the loading angle and the intermediate principal stress ratio, the true triaxial tests were analyzed in three sectors. The stress–strain behavior and the volumetric strain in each sector can be well captured by the proposed model. Strain localization occurs in most b value conditions in all three sectors except for those which are close to triaxial compression condition (b = 0). The difference between the peak shear strength corresponding to the strain localization and the ultimate shear strength corresponding to plastic limit becomes obvious when the b value is near 0.4. The influence of bifurcation on the shear strength becomes weak when the loading direction changes from perpendicular to the bedding plane to parallel. The bifurcation analysis based on the proposed model gives out major principal strain and peak shear strength at the initiation of strain localization; the given results are consistent with experiments.     

Finite element analysis of strain localization in frictional materials

Numerical examples are given which illustrate the poor performance of conventional finite elements in problems involving strain localization in frictional materials. In one of the cases investigated, that of granular media subjected to plane strain biaxial loading, isoparametric elements are seen to inhibit localization altogether. With these examples by way of motivation, the performance of a recently proposed finite element method in the context of strain localization in frictional materials is assessed, with particular emphasis on three-dimensional problems. In passing, some issues pertaining to the post-bifurcation response of biaxial specimens are examined. In particular, the numerical simulations suggest that the observed softening is a geometrical effect not attributable to constitutive behaviour.     

Kinematic hardening plasticity formulation of small strain behaviour of soils

An objective of this paper is to demonstrate that the small strain model developed by the authors can be incorporated into the conventional kinematic hardening plasticity framework to predict pre‐failure defor mations. The constitutive model described in this paper is constituted by three elliptical yield surfaces in triaxial stress space. Two inner surfaces are rotated ellipses of the same shape, representing the boundaries of the linear elastic and small strain regions, while the third surface is the modified Cam clay large‐scale yield surface. Within the linear elastic region, the soil behaviour is elastic with cross‐coupling between the shear and volumetric stress–strain components. Within the small strain region, the soil behaviour is elasto‐plastic, described by the kinematic hardening rule with an infinite number of loading surfaces defined by the incremental energy criterion. Within the large‐scale yield surface, the soil behaviour is elasto‐plastic, described by kinematic and isotropic hardening of the small strain region boundary. Since the yield surfaces have different shapes, the uniqueness of the plastic loading condition imposes a restriction on the ratio between their semi‐diameters. The model requires 12 parameters, which can be determined from a single consolidated undrained triaxial compression test. Copyright © 2000 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.     

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.     

基于颗粒抗转动模型的刚性挡墙被动土压力临界状态离散元分析

简要介绍了颗粒抗转动模型,并将其引入离散元程序中,通过建立挡墙地基模型和合理选取模型参数,分别考虑了地基填土不同密实度和挡墙不同位移模式（被动T模式、RB模式、RT模式）情况下,刚性挡墙被动土压力随挡墙位移增长发展到达临界状态时,土压力系数 随位移发展的变化规律及墙后填土剪切带的形成规律,并与其他学者的研究成果进行对比分析。研究结果表明,土压力系数 随着挡墙位移增长的变化规律与填土的孔隙比（或相对密实度）和挡墙的位移模式紧密相关。随着孔隙比的减小或相对密实度的增大,土压力系数 会逐渐由位移硬化特性过渡为位移软化特性。尽管中密试样在双轴压缩试验中呈现出应变软化特性,而中密样的土压力系数 随着挡墙平动位移的增长可能呈现出位移软化特性,也可能呈现位移硬化特性。随着刚性挡墙向墙后土体推移,试样中的剪应变随之增大,并会在墙后形成应变局部化,即剪切带的出现。与室内试验剪应变云图相似,离散元较好地模拟了土压力临界状态时剪切带分布规律。同时,墙后土体表面不再是光滑的平面,而是逐渐隆起的凹凸面;随着挡墙位移增长,土体表面隆起量越来越大,直至土体破坏。     

Explicit stress integration of complex soil models

In this paper, two complex critical‐state models are implemented in a displacement finite element code. The two models are used for structured clays and sands, and are characterized by multiple yield surfaces, plastic yielding within the yield surface, and complex kinematic and isotropic hardening laws. The consistent tangent operators—which lead to a quadratic convergence when used in a fully implicit algorithm—are difficult to derive or may even not exist. The stress integration scheme used in this paper is based on the explicit Euler method with automatic substepping and error control. This scheme employs the classical elastoplastic stiffness matrix and requires only the first derivatives of the yield function and plastic potential. This explicit scheme is used to integrate the two complex critical‐state models—the sub/super‐loading surfaces model (SSLSM) and the kinematic hardening structure model (KHSM). Various boundary‐value problems are then analysed. The results for the two models are compared with each other, as well with those from standard Cam‐clay models. Accuracy and efficiency of the scheme used for the complex models are also investigated. Copyright © 2005 John Wiley & Sons, Ltd.     

Stress‐dependent hardening and failure surfaces of dry sand

During several triaxial compression experiments on plastic hardening, softening, and failure properties of dense sand specimens, it was found on various stress paths that the size of the failure surface was not constant. Instead, it changed depending on the current state of hydrostatic pressure. This finding is in contrast to the standard opinion consisting of the fact that the failure surface remains constant, once it has been reached during an experiment or in situ. In general, the behaviour of cohesionless granular‐material‐like sand is somehow characterised in between fluid and solid, where the solid behaviour results from the angle of internal friction and the confining pressure. Although the friction angle is an intrinsic material property, the confining pressure varies with the boundary conditions, thus defining different solid properties like plastic hardening, softening, and also failure. Based on our findings, it was the goal of the present contribution to introduce an improved setting for the plastic strain hardening and softening behaviour including the newly found yield properties at the limit state. For the identification of the material parameters, a complete triaxial experimental analysis of the tested sand is given. The overall elasto‐plasticity concept is validated by numerical computations of several laboratory foundation‐ and slope‐failure experiments. The performance of the proposed approach is compared with the standard concept of a constant failure surface, where the corresponding yield surfaces are understood as contours of equivalent plastic work or plastic strain. Copyright © 2012 John Wiley & Sons, Ltd.     

平面应变下原状黄土的剪切带研究

针对黄土地区工程建设中的平面应变问题,使用改进的西安理工大学真三轴仪进行了不同侧向固结压力、不同含水率下原状黄土的平面应变剪切试验,得出了剪切破坏时剪切带倾角试验值,根据应力-应变关系曲线求出了剪切带倾角的Mohr-Coulomb、Roscoe理论解,分析了相应控制因素下原状黄土强度、剪切带破坏模式与倾角的变化。结果表明：应力-应变曲线随含水率、侧向固结压力的增大从原生软化型向次生硬化型转变;剪切带破坏模式随侧向固结压力的增大依次呈现单一剪切带、锥型剪切带和X型剪切带;随着含水率的增大,黏聚力、内摩擦角减小导致Mohr-Coulomb剪切带倾角减小,剪胀作用增强导致Roscoe剪切带倾角增大;随侧向固结压力增大,动内摩擦角增大及剪胀作用增强,导致Mohr-Coulomb、Roscoe剪切带倾角增大。     

三维应力状态下砂的临界状态本构模型分叉分析

就服从非关联流动法则软化型砂的临界状态本构模型,在不同应力路径下的分叉进行了理论和数值分析。理论分析表明：分叉现象强烈依赖于应力路径,即当平均应力p一定时,应力路径在洛德角在-25o～15o之间时,均会在应力-应变关系硬化阶段出现分叉现象,而其他应力路径下不会产生分叉。并且分叉时对应的应力比、主应变及剪切带倾角都随洛德角的增加而变化,其值是先增加而后减小。采用回映应力更新算法,编写了该本构模型材料子程序,借助有限元软件ABAQUS及材料子程序,通过数值计算方法预测到了分叉点所对应的应力状态,表明了分叉现象在数值计算中的存在性,并通过数值预测值和理论解的对比,两者结果基本一致。     

月壤双轴试验的剪切带离散元数值分析

土体的破坏问题一般都是从其剪切带入手进行研究的。针对真实月壤所处的环境（无水、低重力场、低气压等）和内摩擦角较大的特点,采用最近提出的1种考虑粒间抗转动作用与粒间范德华力2个因素的月壤散粒体力学接触模型,用离散单元法模拟了柔性边界条件下月球环境（含范德华力）与地球环境（不含范德华力）2种试样的双轴压缩试验,通过试样局部变形、速度场、孔隙比及转动场的变化情况研究了2种试样剪切带形成与发展。研究结果表明,剪切带的形成是试样内部应变局部化的结果,同时也伴随着试样内部颗粒相对转动的局部化,月面环境对试样的破坏形式与性状（剪切带的倾角及厚度）有显著的影响,探月工程中必须考虑到月面环境对月壤力学性质的影响。