Deterministic and statistical size effect during shearing of granular layer within a micro‐polar hypoplasticity

The paper deals with numerical investigations of a deterministic and statistical size effect in granular bodies during quasi‐static shearing of an infinite layer under plane strain conditions, free dilatancy and constant pressure. For a simulation of the mechanical behaviour of a cohesionless granular material during a monotonous deformation path, a micro‐polar hypoplastic constitutive relation was used which takes into account particle rotations, curvatures, non‐symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. In the paper, a deterministic and statistical size effect is analysed. The deterministic calculations were carried out with an uniform distribution of the initial void ratio for four different heights of the granular layer: 5, 50, 500 and 2000 mm. To investigate the statistical size effect, the Monte Carlo method was applied. The random distribution of the initial void ratio was assumed to be spatially correlated. Truncated Gaussian random fields were generated in a granular layer using an original conditional rejection method. The sufficient number of samples was determined by analysing the convergence of the outcomes. In order to reduce the number of realizations without losing the accuracy of the calculations, stratified and Latin hypercube methods were applied. A parametric analysis of these methods was also presented. Some general conclusions were formulated. Copyright © 2007 John Wiley & Sons, Ltd.     

FE-modeling of shear resistance degradation in granular materials during cyclic shearing under CNS condition

The behaviour of dry and cohesionless granular material during quasi-static cyclic shearing under a constant normal stiffness (CNS) condition is theoretically studied. A particular attention is laid to the volumetric strain change and the degradation of the shear resistance in the course of shearing. Numerical calculations are carried out for several shear cycles under boundary conditions which are relevant to investigate the shear interface behaviour. The global and local evolution of deformation, stress and density within the granular material is investigated with a finite element method on the basis of a hypoplastic constitutive model extended by micro-polar quantities: rotations, curvatures and couple stresses. A mean grain diameter is used as a characteristic length of micro-structure. The constitutive equations for stresses and couple stresses take also into account the effect of the evolution of the void ratio, pressure dependent relative density, direction of rate of deformation and rate of curvature. The numerical results are qualitatively compared with corresponding laboratory tests on direct wall shearing performed by DeJong, Randloph and White. In addition, the results for cyclic shearing of an infinite granular layer between two very rough boundaries under CNS conditions are also enclosed and discussed.     

Effects of periodic fluctuations of micro‐polar boundary conditions on shear localizations in granular soil–structure interaction

In this work, the interface behavior between an infinite extended narrow granular layer and a rough surface of rigid body is investigated numerically, using finite element method in the updated Lagrangian (UL) frame. In this regard, the elasto‐plastic micro‐polar (Cosserat) continuum approach is employed to remove the limitations caused by strain‐softening of materials in the classical continuum. The mechanical properties of cohesionless granular soil are described with Lade's model enhanced by polar terms, including Cosserat rotations, curvatures, and couple stresses. Furthermore, the mean grain diameter as the internal length is incorporated into the constitutive relations accordingly. Here, the evolution and location of shear band, within the granular layer in contact with the rigid body, are mainly focused. In this regard, particular attention is paid to the effects of homogeneous distribution and periodic fluctuation of micro‐polar boundary conditions, prescribed along the interface. Correspondingly, the effects of pressure level, mean grain diameter, and stratified soil are also considered. The finite element results demonstrate that the location and evolution of shear band in the granular soil layer are strongly affected by the non‐uniform micro‐polar boundary conditions, prescribed along the interface. It is found that the shear band is located closer to the boundary with less restriction of grain rotations. Furthermore, the predicted thickness of shear band is larger for higher rotation resistance of soil grains along the interface, larger mean grain diameter, and higher vertical pressure. Regarding the stratified soil, comprising a thin layer with slightly different initial void ratio, the shear band moves towards the layer with initially higher void ratio. Copyright © 2011 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.     

Shearing of a narrow granular layer with polar quantities

The paper deals with numerical investigations of the behaviour of granular bodies during shearing. Shearing of a narrow layer of sand between two very rough boundaries under constant vertical pressure is numerically modelled with a finite element method using a hypoplastic constitutive relation within a polar (Cosserat) continuum. The constitutive relation was obtained through an extension of a non‐polar one by polar quantities, viz. rotations, curvatures, couple stresses using the mean grain diameter as a characteristic length. This relation can reproduce the essential features of granular bodies during shear localization. The material constants can be easily determined from element test results and can be estimated from granulometric properties. The attention is laid on the influence of the initial void ratio, pressure level, mean grain diameter and grain roughness on the thickness of shear zones. The results of shearing are also compared to solutions without the polar extensions. The FE‐calculations demonstrate that polar effects manifested by the appearance of grain rotations and couple stresses are significant in the shear zone, and its thickness is sensitive to the initial void ratio, mean grain diameter and layer height. The effect of the pressure level is rather low within the considered range. Copyright © 2001 John Wiley & Sons, Ltd.     

Constitutive modelling of granular materials using a contact normal-based fabric tensor

This paper presents a fabric tensor-based bounding surface model accounting for anisotropic behaviour (e.g. the dependency of peak strength on loading direction and non-coaxial deformation) of granular materials. This model is developed based on a well-calibrated isotropic bounding surface model. The yield surface is modified by incorporating the back stress which is proportional to a contact normal-based fabric tensor for characterising fabric anisotropy. The evolution law of the fabric tensor, which is dependent on both rates of the stress ratio and the plastic strain, rules that the material fabric tends to align with the loading direction and evolves towards a unique critical state fabric tensor under monotonic shearing. The incorporation of the evolution law leads to a rotational hardening of the yield surface. The anisotropic critical state is assumed to be independent of the initial values of void ratio and fabric tensor. The critical state fabric tensor has the same intermediate stress ratio (i.e. b value) and principal directions as the critical state stress tensor. A non-associated flow rule in the deviatoric plane is adopted, which is able to predict the non-coaxial flow naturally. The stress–strain relation and fabric evolution of model predictions show a satisfactory agreement with DEM simulation results under monotonic shearing with different loading directions. The model is also validated by comparing with laboratory test results of Leighton Buzzard sand and Toyoura sand under various loading paths. The comparison results demonstrate encouraging applicability of the model for predicting the anisotropic behaviour of granular materials.

     

应力状态对粗粒料力学特性影响的大型真三轴试验

土石坝所处的应力状态较接近平面应变应力状态或三维应力状态,而常规三轴试验低估了粗粒料的力学性能。应用大型真三轴仪对常规三轴应力状态、平面应变应力状态和真三轴应力状态（ 0.25, 为中主应力系数, 、 、 分别为大、中、小主应力）下粗粒料的力学特性进行了压缩试验研究。试验结果表明：相同小主应力下,常规三轴试验、平面应变试验、真三轴试验的大、小主应力之差与大主应力方向应变的关系曲线依次变高变陡。某一试验加载条件下,体应变随球应力的增大而增大,初始剪切阶段增加较慢,随后呈线性增大,不同小主应力的体应变曲线较为接近。平面应变试验强度和真三轴试验强度比常规三轴试验强度有较大增长,真三轴试验强度增加百分比大于平面应变试验强度增加百分比。初始弹性模量随小主应力的增大呈线性增大。平面应变状态下中主应力系数随大主应力方向应变的增大而增大,初始剪切阶段增长较缓,之后近似呈线性增大。相同小主应力下,从常规三轴应力状态至平面应变应力状态再到真三轴应力状态,小主应力方向应变均为膨胀变形,且膨胀变形依次增大。同一试验加载条件下,小主应力较小时,应力比（偏应力与球应力之比）与偏应变的关系曲线位于上方。     

FE-studies on the influence of initial void ratio,pressure level and mean grain diameter on shear localization

The paper is concerned with shear localization in the form of a spontaneous shear zone inside a granular material during a plane strain compression test. The influence of an initial void ratio, pressure and a mean grain diameter on the thickness of a shear zone is investigated. A plane strain compression test with dry sand is numerically modelled with a finite element method taking into account a polar hypoplastic constitutive relation which was laid down within a polar (Cosserat) continuum. The relation was obtained through an extension of a non-polar hypoplastic constitutive law according to Gudehus and Bauer by polar quantities: rotations, curvatures, couple stresses and a characteristic length. It can reproduce the essential features of granular bodies during shear localization. The material constants can be easily calibrated. The FE-calculations demonstrate an increase in the thickness of the shear zone with increasing initial void ratio, pressure level and mean grain diameter. Polar effects manifested by the appearance of grain rotations and couple stresses are only significant in the shear zone. A comparison between numerical calculations and experimental results shows a satisfying agreement. Copyright © 1999 John Wiley & Sons, Ltd.     

Constitutive description of interface behavior including cyclic loading and particle breakage within the framework of critical state soil mechanics

The cyclic behavior of soil–structure interface can be very important in dynamic problems. The cyclic behavior of soil–structure interface may be nonlinear, which includes hysteresis, hardening, degradation, and particle breakage. The breakage of granular soil particles during shearing of granular soil–structure interface is associated with cyclic degradation and can be critical to the dynamic behavior of soil–structure system. The critical state soil mechanics concept formerly used to simulate the nonlinear monotonic behavior of granular soil–structure interface was modified and extended to describe the cyclic behavior, especially soil‐particle breakage and degradation during cyclic shearing. Soil‐particle breakage was assumed to relate to the energy consumption during cyclic shearing and the critical state line of the soil–structure interface was assumed to translate with the consumption of shearing energy during cyclic shearing as the threshold value is attained. The model was formulated in the framework of generalized plasticity and is capable of describing the salient features of granular soil–structure interface under cyclic loading. Most of the model parameters have straightforward physical meanings and are calibrated using monotonic or cyclic interface test results. The proposed model was calibrated and validated against published test results. The dependency of interface behavior on stress path and cyclic degradation can be successfully described by the proposed model. Copyright © 2008 John Wiley & Sons, Ltd.     

Micromechanical analysis of the failure process of brittle rock

The failure process of brittle rock submitted to a compression state of stress with different confining pressures is investigated in this paper based on discrete element method (DEM) simulations. In the DEM model, the rock sample is represented by bonding rigid particles at their contact points. The numerical model is first calibrated by comparing the macroscopic response with the macroscopic response of Beishan granite obtained from laboratory tests. After the validation of numerical model in terms of macroscopic responses, the failure process of the DEM model under unconfined and confined compression is studied in micro‐scale in detail. The contact force network and its relation to the development of micro‐cracks and evolution of major fractures are studied. Confining pressure will prohibit the development of tensile cracks and hence alter the failure patterns. An in‐depth analysis of micro‐scale response is carried out, including the orientation distribution and probability density of stress acting on parallel bonds, the effect of particle size heterogeneity on bond breakage and the evolution of fabric tensor and coordination number of parallel bond. The proposed micromechanical analysis will allow us to extract innovative features emerged from the stresses and crack evolution in brittle rock failure process. Copyright © 2014 John Wiley & Sons, Ltd.     

Biaxial test simulations using a packing of polygonal particles

The mechanical response of cohesionless granular materials under monotonic loading is studied by performing molecular dynamic simulations. The diversity of shapes of soil grains is modelled by using randomly generated convex polygons as granular particles. Results of the biaxial test obtained for dense and loose media show that samples achieve the same void ratio at large strains independent of their initial density state. This limit state resembles the so‐called critical state of soil mechanics, except for some stress fluctuations, which remain for large deformations. These fluctuations are studied at the micro‐mechanical level, by following the evolution of the co‐ordination number, force chains and the fraction of the sliding contacts of the sample. Copyright © 2007 John Wiley & Sons, Ltd.     

From solid to granular gases: the steady state for granular materials

This paper aims at extending the well‐known critical state concept, associated with quasi‐static conditions, by accounting for the role played by the strain rate when focusing on the steady, simple shear flow of a dry assembly of identical, inelastic, soft spheres. An additional state variable for the system, the granular temperature, is accounted for. The granular temperature is related to the particle velocity fluctuations and measures the agitation of the system. This state variable, as is in the context of kinetic theories of granular gases, is assumed to govern the response of the material at large strain rates and low concentrations. The stresses of the system are associated with enduring, frictional contacts among particles involved in force chains and nearly instantaneous collisions. When the first mechanism prevails, the material behaves like a solid, and constitutive models of soil mechanics hold, whereas when inelastic collisions dominate, the material flows like a granular gas, and kinetic theories apply. Considering a pressure‐imposed flow, at large values of the normal stress and small values of the shear rate, the theory predicts a nonmonotonic shear rate dependence of the stress ratio at the steady state, which is likely to govern the evolution of landslides. Copyright © 2013 John Wiley & Sons, Ltd.     

Limits to the Jaumann stress rate

In this paper it is shown that the Jaumann stress rate is not accurate for moderate deformations with deviatoric strains of more than 10 per cent; in such cases the objective stress rate is better calculated using the material rotation rate as calculated by polar decomposition. For very large deformations of granular materials this approach becomes invalid too because then the fabric tends to rotate together with principal stresses. Apparently in general the fabric rotation of a granular material cannot be described by continuum mechanics alone. For the numerical calculation of the material rotation for moderate deformations a numerical subroutine, based on polar decomposition, has been developed.     

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.     

Simple shear of isotropic elasto–plastic soil

The implications of assuming isotropic elasto–plasticity to model the behaviour of soil under simple shear conditions are considered. For small strains, use of such a model implies the following three consequences: (1) strains and strain increments at any stage of shearing may be expressed as the sum of elastic and plastic components; (2) principal directions of stress and of plastic strain increment are collinear; (3) principal directions of stress increment and of elastic strain increment are collinear. These consequences are used in order to establish relationships between the stresses, stress increments and strains which develop in a simple shear test. No additional assumptions with regards the form of the yield function, the flow rule or the hardening function are required for this development. By defining the ratio of the plastic to the total shear strain increment on the horizontal plane (the plane of zero extension) as λ, it is possible to define the horizontal normal stress σ_x in terms of λ and other stresses and strains which are normally known during simple shear loading. As a result, all components of the stress tensor in the simple shear plane may be defined. Results of some direct simple shear tests on soft clay have been interpreted using the model and found to be generally consistent with some of the observations reported in the literature from tests in which boundary stresses were measured.     

A visco‐elasto‐plastic model for granular materials under simple shear conditions

The numerical simulation of rapid landslides is quite complex mainly because constitutive models capable of simulating the mechanical behaviour of granular materials in the pre‐collapse and post‐collapse regimes are still missing. The goal of this paper is to introduce a constitutive model capable of capturing the response of dry granular flows from quasi‐static to dynamic conditions, in particular when the material experiences a sort of solid‐to‐fluid phase transition. An ideal assembly of identical spheres under simple shear conditions is considered. In the constitutive model, void ratio and granular temperature have been chosen as state variables, and both shear and normal stresses are computed as the sum of two contributions: the quasi‐static one and the collisional one. The former is determined by using a perfect elasto‐plastic model including the critical state concept, while the latter is derived from the kinetic theory of granular gases. The evolution of the granular temperature, fundamentally governing the material phase transition, is obtained by imposing the kinetic fluctuating energy balance. The constitutive relationship has been integrated, under both constant pressure and constant volume conditions, and the influence of shear strain rate, initial void ratio and normal pressure on the mechanical response has been investigated. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Influence of initial density of cohesionless soil on evolution of passive earth pressure

This paper focuses on the influence of the initial void ratio on the evolution of the passive earth pressure and the formation of shear zones in a dry sand body behind a retaining wall. For the numerical simulation a rigid and very rough retaining wall undergoing a horizontal translation against the backfill is considered. The essential mechanical properties of cohesionless granular soil are described with a micro-polar hypoplastic model which takes into account stresses and couple stresses, pressure dependent limit void ratios and the mean grain size as a characteristic length. Numerical investigations are carried out with an initially medium dense and initially loose sand using a homogeneous and random distribution of the initial void ratio. The geometry of calculated shear zones is discussed and compared with a corresponding laboratory model test.     

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.     

含软弱夹层板岩抗剪强度试验及表征方法

在微机控制岩石剪切试验机上,针对预裂的含薄层黏土充填结构面板岩试件,进行不同法向应力下的结构面直剪试验,发现:结构面直剪试验中,并不是整个结构面都参与抗剪工作,只有部分结构面发挥抗剪作用,而且发挥抗剪作用的有效摩擦面积随法向应力的增大而增大,但是增大速率逐渐减小。基于整个结构面发挥抗剪作用的假设,获得了结构面表观剪应力-剪位移关系曲线,拟合计算了结构面表观抗剪强度指标;同时基于每次直剪试验中结构面部分区域发挥抗剪作用的假设,获得了结构面有效剪应力-剪位移关系曲线,拟合计算了结构面有效抗剪强度指标。对比两者发现:结构面抗剪强度指标中,内摩擦角与发挥抗剪作用面积的选取无关,黏聚力则受发挥抗剪作用面积选取的影响;基于结构面有效摩擦面积计算得到的有效黏聚力,相比表观黏聚力,提高了95%。本文研究成果对于滑坡工程治理方案的评估与选择具有重要的意义和价值。