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.     

Fabric dilatancy and the plasticity modeling of granular media

A novel conceptual model of the mechanics of sands is developed within an elastic–plastic framework. Central to this model is the realization that volume changes in anisotropic granular materials occur as a result of two fundamentally different mechanisms. The first is purely kinematic, dilative, and is the result of the changes in anisotropic fabric. There is also a second volume change in granular media that occurs as a direct response to changes in stress as in a standard elastic/plastic continuum. The inclusion of the two sources of volume change results in three important datum states. When subjected to isotropic strains, the resulting stress state in granular materials is not isotropic but lies upon the kinematic normal consolidation line. There exists a state at which the fabric‐induced volumetric strain rate becomes equal to the stress‐induced volumetric strain rate making the total plastic volumetric strain rate equal to zero. Granular response changes from contractive to dilative at this phase transformation line. The third datum state is the one in which the stress‐induced volumetric strain rate is zero. The sand, however, continues to dilate at this state with the difference between stress and dilation ratio a constant as predicted by Taylor's stress–dilatancy rule. These predictions are shown in accordance with experimental data from a series of drained tests and undrained on Ottawa sand. Copyright © 2011 John Wiley & Sons, Ltd.     

Subcritical compaction and yielding of granular quartz sand

Cylindrical samples of water-saturated, initially loose, St. Peter quartz sand were consolidated using triaxial deformation apparatus at room temperature, constant fluid pressure (12.5 MPa), and elevated confining pressures (up to 262.5 MPa). The samples were deformed along four loading paths: (1) hydrostatic stressing tests in which confining pressure was monotonically increased; (2) hydrostatic stress cycling similar to (1) except that effective pressure was periodically decreased to initial conditions; (3) triaxial deformation at constant effective pressure in which differential stress was applied after raising effective pressure to an elevated level; and (4) triaxial stress cycling similar to (3) except that the axial differential stress was periodically decreased to zero. Hydrostatic stressing at a constant rate results in a complex nonlinear consolidation response. At low pressures, large strains occur without significant acoustic emission (AE) activity. With increased pressure, the stress versus strain curve becomes quasi-linear with a corresponding nonlinear increase in AE rates. At elevated pressures, macroscopic yielding is marked by the onset of large strains, high AE rates, and significant grain failure. Stress cycling experiments show that measurable inelastic strain occurs at all stages of hydrostatic loading. The reload portions of stress cycles are characterized by a poro-elastic response and lower AE rates than during constant rate hydrostatic stressing. As the stress nears and exceeds the level that was applied during previous loading cycles, strain and AE rates increase in a manner consistent with yielding. Triaxial stressing cycles achieve greater consolidation and AE rates than hydrostatic loading at similar mean stress levels. By comparing our results with previously published studies, we construct a three-component model to describe elastic and inelastic compaction of granular sand. This model involves acoustically silent grain rearrangement that contributes significant inelastic strain at low pressures, poro-elastic (Hertzian) deformation at all pressures, and inelastic strain related to granular cracking and particle failure which increases in significance at greater pressures.     

Revisiting the existence of an effective stress for wet granular soils with micromechanics

A possible effective stress variable for wet granular materials is numerically investigated based on an adapted discrete element method (DEM) model for an ideal three‐phase system. The DEM simulations consider granular materials made of nearly monodisperse spherical particles, in the pendular regime with the pore fluid mixture consisting of distinct water menisci bridging particle pairs. The contact force‐related stress contribution to the total stresses is isolated and tested as the effective stress candidate for dense or loose systems. It is first recalled that this contact stress tensor is indeed an adequate effective stress that describes stress limit states of wet samples with the same Mohr‐Coulomb criterion associated with their dry counterparts. As for constitutive relationships, it is demonstrated that the contact stress tensor used in conjunction with dry constitutive relations does describe the strains of wet samples during an initial strain regime but not beyond. Outside this so‐called quasi‐static strain regime, whose extent is much greater for dense than loose materials, dramatic changes in the contact network prevent macroscale contact stress‐strain relationships to apply in the same manner to dry and unsaturated conditions. The presented numerical results also reveal unexpected constitutive bifurcations for the loose material, related to stick‐slip macrobehavior.     

Modeling cyclic shearing of sands in the semifluidized state

Liquefaction is associated with the loss of mean effective stress and increase of the pore water pressure in saturated granular materials due to their contractive tendency under cyclic shear loading. The loss of mean effective stress is linked to loss of grain contacts, bringing the granular material to a “semifluidized state” and leading to development and accumulation of large cyclic shear strains. Constitutive modeling of the cyclic stress-strain response in earthquake-induced liquefaction and post-liquefaction is complex and yet very important for stress-deformation and performance-based analysis of sand deposits. A new state internal variable named strain liquefaction factor is introduced that evolves at low mean effective stresses, and its constitutive role is to reduce the plastic shear stiffness and dilatancy while maintaining the same plastic volumetric strain rate in the semifluidized state. This new constitutive ingredient is added to an existing critical state compatible, bounding surface plasticity reference model, that is well established for constitutive modeling of cyclic response of sands in the pre-liquefaction state. The roles of the key components of the proposed formulation are examined in a series of sensitivity analyses. Their combined effects in improving the performance of the reference model are examined by simulating undrained cyclic simple shear tests on Ottawa sand, with focus on reproducing the increasing shear strain amplitude as well as its saturation in the post-liquefaction response.     

Stability and bifurcation of undrained,plane rectilinear deformations on water-saturated granular soils

Stability and bifurcation of plane-strain, undrained, rectilinear deformations on water-saturated granular soil specimens are discussed. The soil, sand or normally consolidated clay, is described by a 2D-flow theory of plasticity for frictional material with non-associated flow rule. Contractant material become unstable (liquefies) at the state of maximum deviatoric stress. For dilatant material the dominant failure mode is shear banding that occurs close to the state of maximum principal effectivestress ratio. The theoretical findings are supported by and are used to explain existing experimental results.     

Evolution of induced fabric in a strain space multiple mechanism model for granular materials

The strain space multiple mechanism model idealizes the behavior of granular materials based on a multitude of virtual simple shear mechanisms oriented in arbitrary directions. Within this modeling framework, the virtual simple shear stress is defined as a quantity that depends on the contact distribution function as well as the normal and tangential components of inter‐particle contact forces, which evolve independently during the loading process. In other terms, the virtual simple shear stress is an intermediate quantity in the upscaling process from the microscopic level (characterized by the contact distribution and inter‐particle contact forces). The stress space fabric (i.e. the orientation distribution of the virtual simple shear stress) produces macroscopic stress through the tensorial average. Thus, the stress space fabric characterizes the fundamental and higher modes of anisotropy induced in granular materials. Comparing an induced fabric associated with the biaxial shear of plane granular assemblies obtained via a simulation using Discrete Element Method to the strain space multiple mechanism model suggests that the strain space multiple mechanism model has the capability to capture the essential features in the evolution of an induced fabric in granular materials. Copyright © 2012 John Wiley & Sons, Ltd.     

On the weak turbulent motions of an isothermal dry granular dense flow with incompressible grains: part I. Equilibrium turbulent closure models

The weak turbulent motions of a dry granular dense flow and the influence of the turbulent fluctuations caused by the minor short-term elastic/inelastic instantaneous collisions and the major long-term enduring frictional contacts among the grains on the mean flow characteristics are investigated. To this end, the conventional Reynolds-averaging process is applied to obtain the balance equations for the mean primitive fields associated with turbulent closure models. The thermodynamic analysis, based on the Mueller–Liu entropy principle, is carried out to derive the equilibrium formulations of the closure models. It shows that the effect of the turbulent fluctuations on the mean flow characteristics as well as the turbulent kinetic energy and dissipation can be taken into account by the granular coldness: a phenomenological measure of the fluctuating kinetic energy intensity. The implementation of the complete thermodynamically consistent turbulent closure models and the simulation of a gravity-driven stationary flow down an inclined moving plane compared with the experimental outcomes are provided in Part II of the present study.     

A Godunov method for localization in elastoplastic granular flow

The equations governing the elastic-plastic deformation of granular materials are typically hyperbolic, or contain small-magnitude damping or rate effects. A finite element algorithm is the standard method for the numerical integration of these systems. In particular, finite elements allow great flexibility in the design of grid geometry. However, modern finite difference methods for hyperbolic systems have been successful in aerodynamics computations, resolving wave structures more sharply than finite element schemes. In this paper we develop a finite difference scheme for granular flow problems. We report on a second-order Godunov-type scheme for the integration of hyperbolic equations for the elastoplastic deformation of a simple model of granular flow. The Godunov method includes a characteristic tracing step in the integration, providing minimal wave dispersion, and a slope limiting step, preventing unphysical oscillations. The granular flow model we consider is hyperbolic, but hyperbolicity is lost at a large value of accumulated plastic strain. This loss of hyperbolicity is a tell-tale signal for the formation of a shear band within the sample. Typically, when systems lose hyperbolicity a regularization mechanism is added to the model equations in order to maintain the well posedness of the system. These regularizations include viscosity, viscoplasticity, higher-order gradient effects or stress coupling. Here we appeal to a very different kind of regularization. When the system loses hyperbolicity and a shear band forms, we treat the band as an internal boundary, and impose jump conditions at this boundary. Away from the band, the system remains hyperbolic and the integration step proceeds as usual.     

Modeling of granular bed‐stone column‐improved soft soil

The present study pertains to the development of a mechanical model for predicting the behavior of granular bed‐stone column‐reinforced soft ground. The granular layer that has been placed over the stone column‐reinforced soft soil has been idealized by the Pasternak shear layer. The saturated soft soil has been idealized by the Kelvin–Voigt model to represent its time‐dependent behavior and the stone columns are idealized by stiffer Winkler springs. The nonlinear behavior of the granular fill has been incorporated in this study by assuming a hyperbolic variation of shear stress with shear strain as in one reported literature. Similarly, for soft soil it has also been assumed that load‐settlement variation is hyperbolic in nature. The effect of consolidation of the soft soil due to inclusion of the stone columns has also been included in the model. Plane‐strain conditions are considered for the loading and foundation soil system. The numerical solutions are obtained by a finite difference scheme and the results are presented in a non‐dimensional form. Parametric studies for a uniformly loaded strip footing have been carried out to show the effects of various parameters on the total as well as differential settlement and stress concentration ratio. It has been observed that the presence of granular bed on the top of the stone columns helps to transfer stress from soil to stone columns and reduces maximum as well as differential settlement. Copyright © 2007 John Wiley & Sons, Ltd.     

颗粒材料的微观临界状态理论模型

存在临界状态是颗粒材料的一个重要特性。基于孔隙胞元的颗粒离散元方法对二维颗粒体进行双轴加载数值试验,在详细分析数值模拟结果的基础上,从微观几何组构的角度揭示了临界状态的存在机制。基于剪胀性原理,提出了以接触价键表征的微观临界状态理论模型,得到了接触价键与塑性剪切应变的关系表达式,理论模型的结果和二维离散元数值模拟得到的结果吻合较好。通过比较不同情况下数值结果和理论模型中的参数,得到以下结论：表征微观临界状态的参数（临界接触价键和达到临界状态所需要的塑性剪切应变）依赖于颗粒体的微观特性,如颗粒形状、表面摩擦性质、颗粒体的围压和初始孔隙比。     

A virtual experiment technique on the elementary behaviour of granular materials with discrete element method

Multi‐scale investigations aided by the discrete element method (DEM) play a vital role for current state‐of‐the‐art research on the elementary behaviour of granular materials. Similar to laboratory tests, there are three important aspects to be considered carefully, which are the proper stress/strain definition and measurement, the application of target loading paths and the designed experiment setup, to be addressed in the present paper. Considering the volume sensitive characteristics of granular materials, in the proposed technique, the deformation of the tested specimen is controlled and measured by deformation gradient tensor involving both the undeformed configuration and the current configuration. Definitions of Biot strain and Cauchy stress are adopted. The expressions of them in terms of contact forces and particle displacements, respectively, are derived. The boundary of the tested specimen consists of rigid massless planar units. It is suggested that the representative element uses a convex polyhedral (polygonal) shape to minimize possible boundary arching effects. General loading paths are described by directly specifying the changes in the stress/strain invariants or directions. Loading can be applied in the strain‐controlled mode by specifying the translations and rotations of the boundary units, or in the stress‐controlled mode by using a servo‐control mechanism, or in the combination of the two methods to realize mixed boundary conditions. Taking the simulation results as the natural consequences originated from a complex system, virtual experiments provide particle‐scale information database to conduct multi‐scale investigations for better understanding in granular material behaviours and possible development of the constitutive theories provided the qualitative similarity between the simulation results from virtual experiments and observations on real material behaviour. Copyright © 2011 John Wiley & Sons, Ltd.     

Dilatancy of granular materials in a strain space multiple mechanism model

A granular material consists of an assemblage of particles with contacts newly formed or disappeared, changing the micromechanical structures during macroscopic deformation. These structures are idealized through a strain space multiple mechanism model as a twofold structure consisting of a multitude of virtual two‐dimensional mechanisms, each of which consists of a multitude of virtual simple shear mechanisms of one‐dimensional nature. In particular, a second‐order fabric tensor describes direct macroscopic stress–strain relationship, and a fourth‐order fabric tensor describes incremental relationship. In this framework of modeling, the mechanism of interlocking defined as the energy less component of macroscopic strain provides an appropriate bridge between micromechanical and macroscopic dilative component of dilatancy. Another bridge for contractive component of dilatancy is provided through an obvious hypothesis on micromechanical counterparts being associated with virtual simple shear strain. It is also postulated that the dilatancy along the stress path beyond a line slightly above the phase transformation line is only due to the mechanism of interlocking and increment in dilatancy due to this interlocking eventually vanishing for a large shear strain. These classic postulates form the basis for formulating the dilatancy in the strain space multiple mechanism model. The performance of the proposed model is demonstrated through simulation of undrained behavior of sand under monotonic and cyclic loading. Copyright © 2010 John Wiley & Sons, Ltd.     

Experimental study of the dynamic behavior and segregation of density-bidisperse granular sliding masses at the laboratory scale

From the understanding of dynamics and processes of rapid granular flows and the granular-segregation mechanism in gravity-driven flow, we can clarify the particle-composition structure in the downstream areas of avalanches in geophysical contexts, such as landslides, rock falls, and snow-slab avalanches. Such dynamics also provide a basis for geophysical studies. This study experimentally investigates the dynamic behavior and segregation phenomena of a density-bidisperse, rapid, granular flow down a quasi-2D, rough, inclined rectangular chute. Particles with two density ratios are used to investigate the mechanism of density-induced segregation, and four chute-inclination angles are tested to examine the influence of driving forces. The dynamics of the mixture flow—which includes the flow-depth evolution, stream-wise and depth-wise velocity profiles, shear rate, and granular temperature in the upper high-shear band of the flow—are obtained from particle image velocimetry (PIV) measurements. The two-dimensional concentration distributions of the particles in the stream-wise direction are also obtained using 2D image processing to determine the segregation state. In the upstream region, the variation in the concentration of heavier particles is defined as the strength of the density-induced segregation state, S_d. Our results indicate that the mixture-flow parameter—particularly the shear rate and the granular temperature in the upper high-shear band—crucially influence the strength of particle segregation in granular avalanches. In the upstream region, a higher shear rate and a higher granular temperature in the upper high-velocity band result in a smaller drag force in the mixture flow, causing stronger density-induced particle segregation. These results well describe the entire processes of dense granular flows, from upstream initiation to the downstream steady state. Therefore, they reveal the structure of the mixed flow in the depth direction and are expected to explain various gravity-driven mixture granular flows.

     

Bifurcation and second‐order work in geomaterials

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

Deformation mechanism of strain localization in 2D numerical interface tests

Heterogeneity arises in soil subjected to interface shearing, with the strain gradually localizing into a band area. How the strain localization accumulates and develops to form the structure is crucial in explaining some significant constitutive behaviors of the soil–structural interface during shearing, for example, stress hardening, softening, and shear-dilatancy. Using DEM simulation, interface shear tests with a periodic boundary condition are performed to investigate the strain localization process in densely and loosely packed granular soils. Based on the velocity field given by grains’ translational and rotational velocities, several kinematic quantities are analyzed during the loading history to demonstrate the evolution of strain localization. Results suggest that tiny concentrations in the shear deformation have already been observed in the very early stage of the shear test. The degree of the strain localization, quantified by a proposed new indicator, α, steadily ascends during the stress-hardening regime, dramatically jumps prior to the stress peak, and stabilizes at the stress steady state. Loose specimen does not develop a steady pattern at the large strain, as the deformation pattern transforms between localized and diffused failure modes. During the stress steady state of both specimens, remarkable correlations are observed between α and the shear stress, as well as between α and the volumetric strain rate.     

Dynamic stability analysis of undrained simple shear on water-saturated granular soils

Dynamic stability of undrained, simple-shear deformations on water-saturated granular soil specimens is discussed. The soil is described by a 2D-flow theory of plasticity for frictional and dilatant material. Contractant material becomes unstable (liquefies) at the state of maximum shear stress, whereas dilatant material becomes unstable after the state of maximum effective stress obliquity is reached in the softening regime of the background drained behaviour. In both cases the correct evolution equations for the growth coefficient of the instability are derived and compared with the results of the inertia-free formulation of the problem.     

非均匀颗粒流应力关系实验研究

颗粒流问题在自然界和许多工程领域都可能遇到。采用特殊的内外筒均可单独旋转的同心圆筒颗粒剪切实验装置,测量了非均匀离散颗粒在旋转剪切流动下的切应力变化,分析了颗粒浓度、粒径与边界条件等因素对颗粒流动应力的影响。结果表明,颗粒浓度和颗粒粒径均对剪切应力产生影响,Savage实验（内筒旋转）和Bagnold实验（外筒旋转）的结果产生差异的主要原因是边界滑移和旋转离心力。同时,基于已有成果,深入研究了颗粒应力关系变化的特点和规律。     

散粒介质临界状态细观力学结构特征的数值模拟与分析

基于离散单元法对不同密实度理想散粒体进行了双轴剪切试验的宏微观数值模拟,通过网格剖分将Voronoi多边形表征的loop单元作为散粒介质细观力学结构的基本单元,模拟了剪切过程中不同类型loop单元数量、几何形态和力学特征的演化过程,并重点分析了临界状态时散粒介质的细观力学结构特征。模拟结果显示,初始密实度不同的试样在向临界状态发展的过程中,高阶单元与低阶单元的发展规律完全不同,不同初始密实度试样中同阶loop单元的发展规律也不相同,但同阶loop单元的数量比例、几何形态、颗粒接触力及单元内滑动率最终均达到了各自的临界状态。从细观角度分析,散体介质的临界状态是高阶和低阶loop单元在荷载作用下相互转化的结果,是所有loop单元物理力学状态的综合平均与外在表现,临界状态时不同阶数的loop单元处于一个动态平衡状态,宏观上表现为常剪应力和常体积下剪切变形的不断发展。数值模拟结果也表明,loop细观结构单元包含了丰富的信息,其数量、几何形态、受力特征及接触稳定性的发展与散粒体的强度、剪胀以及临界状态的发展密切相关,可以将其作为散粒介质细观尺度的分析单元。