Mathematical identification of diffuse and localized instabilities in fluid‐saturated sands

The mathematical properties of diffuse and localized failure modes in fluid‐saturated sands are investigated. The granular medium is modeled as an elastoplastic solid, and a recently proposed set of scalar indices, here referred to as moduli of instability, is used to identify the onset of potential bifurcations of the incremental response. First, the analytical properties of these moduli are discussed, stressing their dependence on the kinematic constraints associated with the imposed deformation modes. Then, by using an elastoplastic model for sands, drained and undrained loading paths are simulated under axisymmetric, plane‐strain and simple shear conditions. For each deformation mode, the instability moduli are computed and monitored throughout the simulations, with the purpose of elucidating the consequences of changes in control conditions. In addition, it is illustrated that suitable linear transformations allow the same strategy to be used to perform drained or undrained shear band analyses and predict the interval of possible band inclinations. The final comparison against literature experiments on loose Hostun sand shows that the instability moduli are indicators of the loss of resistance against specific modes of deformation. As a result, they can be used to identify and explain a number of failure mechanisms that can be commonly observed in experiments. Copyright © 2013 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.     

Conditions for instabilities in collapsible solids including volume implosion and compaction banding

We review conditions for material instabilities in porous solids induced by a bifurcation of solution into non-unique strain rate fields. Bifurcation modes considered include jumps in the strain rate tensor of ranks one and higher representing deformation band and diffuse instability modes, respectively. Eigenmodes (e-modes) are extracted for each type of instability to fully characterize various frameworks of deformation in collapsible solids. For diffuse instability these e-modes are determined from a homogeneous system of linear equations emanating from the condition of zero jump in the stress rate tensor, which in turn demands that the tangent constitutive tensor be singular for the existence of nontrivial solutions. For isotropic materials we describe two types of singularity of the constitutive tensor: (a) singularity of the constitutive matrix in principal axes, and (b) singularity of spin. Accordingly, we derive the e-modes for each type of singularity. We utilize the singularity of the constitutive matrix in principal axes as a precursor to volume implosion in collapsible solids such as loose sands undergoing liquefaction instability and high-porosity rocks undergoing cataclastic flow. Finally, we compare conditions and e-modes for volume implosion and compaction banding, two similar failure modes ubiquitous in granular soils and rocks.Supported by U.S. Department of Energy, Grant DE-FG02-03ER15454, and U.S. National Science Foundation, Grants CMS-0201317 and CMS-0324674.     

Finite strain analysis of nonuniform deformation inside shear bands in sands

A methodology has been developed to extend the incremental (Eulerian) Digital Image Correlation (DIC) technique to enable a Lagrangian‐based large‐strain analysis framework to examine the nature of strain and kinematic nonuniformity within shear bands in sands. Plane strain compression tests are performed on dense sands in an apparatus that promotes unconstrained persistent shear band formation. DIC is used to capture incremental, grain‐scale displacements in and around shear bands. The performance of the developed accumulation algorithm is validated by comparing accumulated displacements with two sources of reference measurements. A comparison between large and infinitesimal rotation is performed, demonstrating the nature of straining within shear bands in sands and the necessity of using a finite strain formulation to characterize ensuing behavior. Volumetric strain variation along the shear band is analyzed throughout macroscopic postpeak deformation. During softening, volumetric activity within the shear band is purely dilative. During the global critical state, the shear band material is seen on the average to deform at zero volumetric strain; however, locally, the sand is seen to exhibit significant nonzero volumetric strain, putting into question the current definition of critical state. At the softening‐critical state transition, a spatially periodic pattern of alternating contraction and dilation along the shear band is evidenced, and a preliminary evaluation indicates that the periodicity appears to be a physical phenomenon dictated only in part by median grain size. Copyright © 2011 John Wiley & Sons, Ltd.     

Bifurcation and instability modelling by a multimechanism elasto‐plastic model

The bifurcation and instability conditions in geomechanics are closely related to the elasto‐plastic behaviour. In this paper the potential of a multimechanism elasto‐plastic model to predict various modes of failure is examined. First, a brief overview for the essential aspects of the constitutive model and the development of the elasto‐plastic constitutive matrix for this model are presented. Then, numerical simulations of different drained and undrained paths in the axisymmetric and plane‐strain conditions for the Hostun sand are illustrated. These examples confirm the capacity of the model to reproduce instability and strain localization phenomena. The obtained response is in agreement with experimental observations, theoretical developments and numerical analyses existing in the literature. Copyright © 2007 John Wiley & Sons, Ltd.     

考虑扩容的围岩局部破坏过程中声发射及能量释放的数值模拟

本文模拟了巷道开挖之后,围岩发生局部破坏过程中的声发射及能量释放的演变规律,其中考虑了围岩的峰后扩容特性。介绍了FLAC中扩容角的引入方法,对于剪切而言,仅第3塑性主应变的增量才与扩容角有关。阐述了统计声发射及能量释放的方法,前者仅统计事件的数目,而后者可以考虑事件的大小。研究发现,扩容角的增加会使巷道围岩中的剪破坏单元数及所释放的能量提高;会使V形坑变大、变钝、变深,这是由于第3主应力与剪切带之间的夹角增加的结果。解释了设置固定的抗拉强度,单元却发生拉伸软化及释放拉伸应变能的原因。从与能量释放有关的量的演变规律中一般能发现V形坑式破坏的前兆,而从声发射数上则不然,这反映了统计事件大小的优势。如果将扩容角置零,会低估围岩的破坏范围及释放的能量,但是对于巷道围岩的再次平衡却是有利的。     

A rational method for development of limit state for liquefaction evaluation based on shear wave velocity measurements

This papers presents a new approach for developing a limit state for liquefaction evaluation based on field performance data. As an example to illustrate the new approach, a database that consists of, among many other features, in situ shear wave velocity measurements and field observations of liquefaction/non‐liquefaction in historic earthquakes is analysed. This database is first used to train a neural network to classify liquefaction/non‐liquefaction based on soil resistance parameters and load parameters. The successfully trained and tested neural network is then used to establish a limit state, a multiple dimension boundary that separates ‘zone’ of liquefaction from ‘zone’ of non‐liquefaction. The limit state yields cyclic resistance ratio for a given set of soil resistance parameters. Examination of all cases in the database show that the developed limit state has a high degree of accuracy in predicting the occurrence of liquefaction/non‐liquefaction. The developed neural network model can accurately predict the cyclic resistance ratio of soils. Copyright © 2000 John Wiley & Sons, Ltd.     

胶结结构性土统一硬化模型

结构性土颗粒间的胶结使试样剪切破坏最终应力比高于相应重塑土,也限制了试样剪切时体积应变的自由发挥。在考虑结构垮塌为主的结构性土统一硬化(UH)模型基础上,将应力空间中静止的临界状态线扩展为动态的移动临界状态线。据此,通过建立新的屈服面方程并修正剪胀方程,将结构性土统一硬化(UH)模型扩展为胶结结构性土统一硬化(UH)模型。相对于原模型,新模型增加了1个模型参数,即初始胶结应力,反映土颗粒之间的初始胶结作用。通过4种结构性土试验数据与模型预测对照表明：所提模型能够较合理地描述结构性土等向压缩、常规三轴排水与不排水剪切等特性。     

A note on formulas for localized failure of frictional materials in compression and biaxial loading modes

The paper investigates aspects of the localization analysis of frictional materials. We derive closed formulas and diagrams for the inclination angle of critical discontinuity surfaces which develop in homogeneous compression and biaxial loading tests. The localization analysis is based on a Drucker–Prager‐type elastoplastic hardening model for non‐associated plastic flow at small strains, which we represent in spectral form. For this type of constitutive model, general analytical formulas for the so‐called critical hardening modulus and the inclination angle of critical discontinuity surfaces are derived for the plane strain case. The subsequent treatment then specializes these formulas for the analysis of compression and biaxial loading modes. The key contribution here is a detailed analysis of plane strain deformation modes where the localized failure occurs after subsequent plastic flow. The derived formulas and diagrams can be applied to the checking of an accompanying localization analysis of frictional materials in finite‐element computations. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐coaxial elasto‐plasticity model and bifurcation prediction of shear banding in sands

Numerous constitutive models built on coaxial theory and validated under axi‐symmetric condition often describe the stress–stain relationships and predict the inceptions of shear banding in sands inaccurately under true triaxial condition. By adopting an elaborated Mohr–Coulomb yield function and using non‐coaxial non‐associated flow rule, a 3D non‐coaxial elasto‐plasticity model is proposed and validated by a series of true triaxial tests on loose sands. The bifurcation analysis of true triaxial tests on dense sands predicts the influence of the intermediate principal stress ratio on the onset of shear band accurately. The failure of soils is shown to be related to the formation of shear band under most intermediate principal stress ratio conditions except for those which are close to the axi‐symmetric compression condition. Copyright © 2009 John Wiley & Sons, Ltd.     

A new approximation for pore pressure accumulation in marine sediment due to water waves

The residual mechanism of wave‐induced pore water pressure accumulation in marine sediments is re‐examined. An analytical approximation is derived using a linear relation for pore pressure generation in cyclic loading, and mistakes in previous solutions (Int. J. Numer. Anal. Methods Geomech. 2001; 25 :885–907; J. Offshore Mech. Arctic Eng. (ASME) 1989; 111 (1):1–11) are corrected. A numerical scheme is then employed to solve the case with a non‐linear relation for pore pressure generation. Both analytical and numerical solutions are verified with experimental data (Laboratory and field investigation of wave– sediment interaction. Joseph H. Defrees Hydraulics Laboratory, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY, 1983), and provide a better prediction of pore pressure accumulation than the previous solution (J. Offshore Mech. Arctic Eng. (ASME) 1989; 111 (1):1–11). The parametric study concludes that the pore pressure accumulation and use of full non‐linear relation of pore pressure become more important under the following conditions: (1) large wave amplitude, (2) longer wave period, (3) shallow water, (4) shallow soil and (5) softer soils with a low consolidation coefficient. 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.     

基于梯度塑性理论的岩样拉压剪破坏统一失稳判据

提出了单轴拉伸,直剪及单轴压缩剪切破坏失稳判据的统一形式。首先对de Borst R及Muhlhaus H B关于单轴拉伸试件梯度塑性理论解析解方面的开创性工作进行了回顾、评论,并指出了他们工作的缺憾。受de Borst R等工作的启发,对直接剪切试件进行了分析,得到了快速回跳条件。并发现:岩样的快速回跳条件就是系统(由剪切带及带外弹性岩石构成)的失稳判据。又对单轴压缩岩样剪切破坏进行了分析,得到了其失稳判据。通过对单轴拉伸失稳判据、直接剪切失稳判据及单轴压缩剪切破坏失稳判据的类比,得到了失稳判据的统一形式。统一失稳判据表明:岩石材料的局部化带宽度及压缩或剪切弹性模量越小,软化模量及等效试件高度越大,试件越容易发生失稳破坏。     

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.     

A non‐coaxial and unbalanced plastic flow rule for geomaterials and its application to the shear band orientation prediction

In this paper, the non‐coaxial relation between the principal plastic strain increments and the principal stresses, which results from the internal friction in geomaterials, is analyzed, and the phenomenon of the unbalanced development of plastic flow in two conjugate directions is discussed. A non‐coaxial, unbalanced plastic flow model for Coulomb frictional materials is developed and used to determine the orientation of shear band in geomaterials. It is shown that the unbalanced index r of plastic flow has important effect on the orientation of the shear band, and the orientation determined by the conventional plastic flow theory is only a special case of the proposed model when r=0. This result soundly explains the reason that the geomaterials with the same internal friction angle and dilatancy angle can have very different shear band orientations. In addition, the difference between the intrinsic and apparent dilatancy angles is analyzed, and it is emphasized that the dilatancy angle commonly used in practice is indeed the apparent dilatancy angle. Copyright © 2010 John Wiley & Sons, Ltd.     

Spoil pile instabilities with reference to a strip coal mine in Turkey: mechanisms and assessment of deformations

With the increasing adoption of the surface mining of coal, problems associated with spoil pile instability, which affects resource recovery, mining cost, and safety and presents environmental hazards, have become a matter of prime concern to mine planners and operators. The study of geotechnical aspects is thus very important in the rational planning for the disposal, reclamation, treatment and utilization of spoil material. A strip coal mine, one of the largest open pit mines in Turkey, is located in Central Anatolia and provides coal to a thermal power station. Coal production is carried out in two adjacent open pits, the Central Pit and South Pit. A large-scale spoil pile instability over an area of 0.3 km² occurred within the dumping area of the Central pit. In addition, small-scale movement occurred in the outside dumping area. This paper outlines the results of field and laboratory investigations to describe the mechanisms of the spoil pile instabilities and to assess deformations monitored over a long period following the failure. Shear test results indicate that the interface between the floor and spoil material dumped by dragline has a negligible cohesion and is the most critical plane of weakness for spoil pile instability. Back analyses based on the method of limit equilibrium and the numerical modelling technique, and observations in the pit revealed that failure occurred along a combined sliding surface consisting of a circular surface through the spoil material itself and a planar surface passing along the interface between the spoil piles and floor. The analyses also indicated that pore water pressure ratios of about 0.25 satisfy limiting equilibrium condition and that rainfall about one month before the failure may be a contributing factor to the instability. Movement monitoring data obtained following the failure over a 1.5-year period suggested that the ongoing deformations were mainly due to compaction of the spoil material. Based on the monitoring data and the results of the analyses, the failure mode of the local instability occurring at the outside dumping area was considerably similar to that of the large instability.     

砂岩扇形板状油藏产能预测新方法

本文基于渗流力学原理,综合运用保角变换与等值渗流阻力法推导出扇形板状油藏的产能公式,该公式可以考虑断层夹角、供给边界半径以及井在油藏中的位置等因素对产能的影响。通过与数值方法进行对比,验证了本文产能计算方法的准确性。为指导生产实践,采用本文方法对扇形油藏的产能进行敏感性分析,得到以下结论:产能指数随着供给边界半径的增大而减小,随着井到断层交点距离、断层夹角、井到断层交点连线与断层夹角的增大而增大,断层夹角对产能的影响最大,其次为井到断层交点距离、供给边界半径、井到断层交点连线与断层夹角。     

A method to predict the group fissuring and faulting caused by regional groundwater decline

Ground fissuring is a recurrent problem in many countries where water extraction surpasses the natural recharge of aquifers. Due to differential settlement, the soil layer undergoes deformation and cracks with serious consequences for civil infrastructure. Here, we propose an approximate analysis of the fissuring process that can be used to predict the location of cracks, which increasingly affect some middle- and large-sized cities in the world. For that purpose, the ground loss theory is applied to sediments overlying a sinusoidal-shaped graben. This analysis shows the existence of a tensile zone at the border of the graben with maximal values on its shoulder where tension cracks are more likely to appear. It also shows that soil deformation under differential settlement may evolve into ground faulting if water withdrawal continues. Finally, when a crack has completely developed, the tensile zone shifts towards the center of the graben, creating a new area for potential cracking and faulting.     

An improved SPH method for saturated soils and its application to investigate the mechanisms of embankment failure: Case of hydrostatic pore‐water pressure

The method of smoothed particle hydrodynamics (SPH) has recently been applied to computational geomechanics and has been shown to be a powerful alternative to the standard numerical method, that is, the finite element method, for handling large deformation and post‐failure of geomaterials. However, very few studies apply the SPH method to model saturated or submerged soil problems. Our recent studies of this matter revealed that significant errors may be made if the gradient of the pore‐water pressure is handled using the standard SPH formulation. To overcome this problem and to enhance the SPH applications to computational geomechanics, this article proposes a general SPH formulation, which can be applied straightforwardly to dry and saturated soils. For simplicity, the current work assumes hydrostatic pore‐water pressure. It is shown that the proposed formulation can remove the numerical error mentioned earlier. Moreover, this formulation automatically satisfies the dynamic boundary conditions at a submerged ground surface, thereby saving computational cost. Discussions on the applications of the standard and new SPH formulations are also given through some numerical tests. Furthermore, techniques to obtain the correct SPH solution are also proposed and discussed throughout. As an application of the proposed method, the effect of the dilatancy angle on the failure mechanism of a two‐sided embankment subjected to a high groundwater table is presented and compared with that of other solutions. Finally, the proposed formulation can be considered a basic formulation for further developments of SPH for saturated soils. Copyright © 2011 John Wiley & Sons, Ltd.     

A finite element method for localized erosion in porous media with applications to backward piping in levees

This paper presents a computational method able to effectively model both the simultaneous processes typically observed in backward erosion piping, ie, the pipe tip propagation and the conduit cross-section enlargement. The numerical method is based on the novel formulation of a problem of localized erosion along a line propagating in a multidimensional porous medium. In this line, a conduit with evolving transverse size is embedded, which conveys a multiphase flow. The two systems, porous medium and pipe, are bridged by exchange terms of multiphase fluid mass and by a shared fluid pressure field. On the contrary, different fields are considered to describe flows, which are assumed as Darcian in the porous medium and turbulent in the conduit. These two flows drive pipe propagation and enlargement, respectively, as modeled by means of proper erosion kinetic laws. The corresponding numerical formulation is based on the combination between one- and multidimensional finite elements, to model the erosion conduit and the porous medium, respectively. Several simulations are proposed to demonstrate the ability of the proposed approach in reproducing available experimental data of real-scale tests on levees. Our results point out the crucial role played by the combined influence of pipe propagation and enlargement, as well as of three-dimensional (3D) effects. We also assess the mesh independence of the proposed numerical solution, particularly as concerns the calculated pipe propagation history.