Micro–macro homogenization of granular materials based on the average-field theory of Cosserat continuum

This paper performs further study on the micro–macro homogenization approach of granular materials (Li et al., 2010) based on the advancement of Hill’s lemma for Cosserat continuum (Liu, 2013). Firstly, the average couple stress tensor, expressed as the volume integration of quantities over the representative volume element (RVE) in the average-field theory of Cosserat continuum, is further deduced and expressed in terms of discrete quantities on the discrete particle assembly RVE of granular materials. The expression is also discussed and compared with other typical definitions of the effective couple stress tensor for granular materials in the literature. Then, rate forms of micromechanically based constitutive models consistent with different types of RVE boundary conditions are derived and discussed. Since the presented micro–macro homogenization approach is used, not only the micro–macro energy equivalence is satisfied, but also the microstructure and its evolution can be taken into account in the constitutive formulation with no need of specifying macroscopic phenomenological constitutive model.     

Development of micromechanical models for granular media

Micromechanical analysis has the potential to resolve many of the deficiencies of constitutive equations of granular continua by incorporating information obtained from particle-scale measurements. The outstanding problem in applying micromechanics to granular media is the projection scheme to relate continuum variables to particle-scale variables. Within the confines of a projection scheme that assumes affine motion, contact laws based on binary interactions do not fully capture important instabilities. Specifically, these contact laws do not consider mesoscale mechanics related to particle group behaviour such as force chains commonly seen in granular media. The implications of this are discussed in this paper by comparison of two micromechanical constitutive models to particle data observed in computer simulations using the discrete element method (DEM). The first model, in which relative deformations between isolated particle pairs are projected from continuum strain, fails to deliver the observed behaviour. The second model accounts for the contact mechanics at the mesoscale (i.e. particle group behaviour) and, accordingly, involves a nonaffine projection scheme. In contrast with the first, the second model is shown to display strain softening behaviour related to dilatancy and produce realistic shear bands in finite element simulations of a biaxial test. Importantly, the evolution of microscale variables is correctly replicated. This paper is dedicated to Professor Ching S. Chang on the occasion of his 60th birthday.     

Enhanced continua and discrete lattices for modelling granular assemblies

This article discusses the derivation of continuum models that can be used for modelling the inhomogeneous mechanical behaviour of granular assemblies. These so-called kinematically enhanced models are of the strain-gradient type and of the strain-gradient micro-polar type, and are derived by means of homogenizing the micro-structural interactions between discrete particles. By analysis of the body wave dispersion curves, the enhanced continuum models are compared to corresponding discrete lattice models. Accordingly, it can be examined up to which deformation level the continuum models are able to accurately describe the discrete particle behaviour. Further, the boundary conditions for the enhanced continuum models are formulated, and their stability is considered. It is demonstrated how to use the body wave dispersion relations for the assessment of stability.     

Patterns of shear zones in granular bodies within a polar hypoplastic continuum

Summary The paper deals with numerical investigations on the patterning of shear zones in granular bodies. The behavior of dry sand during plane strain compression tests was 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 localisation. During FE-calculations, the attention was laid on the influence of boundary conditions and the distribution of imperfections in the granular specimen on the formation of patterns of shear zones.     

颗粒流动模型研究进展

颗粒流动在自然界中和各种工业过程中广泛存在,但人们对于其机理认识的还不深入。描述颗粒运动的模型有很多,连续介质模型应用简单但准确性比较低,离散微粒学模型是近来人们研究的一个热点,以每个颗粒为考察对象,能够更准确地反应颗粒系统的性质。本文介绍了描述颗粒流动的模型,概述了各模型的理论和应用,通过对多种模型的比较可以看到,每个模型都有一定的使用范围,要更准确、更方便地描述颗粒系统的运动,还要进行深入地研究。     

考虑体积塑性应变的岩石损伤本构模型研究

采用Eshelby等效夹杂方法建立岩石弹塑性损伤本构模型是一种有效方法,但相关文献目前还极少。在连续介质损伤力学框架内利用细观力学的Eshelby等效夹杂方法建立了考虑损伤相塑性体积变形的岩石的Helmholtz自由比能函数。利用连续介质损伤力学方法推导出了考虑损伤相塑性变形的岩石损伤本构关系,给出了损伤演化方程和塑性应变发展方程。并通过数值模拟证实该模型能够反映岩石体积塑性应变、损伤的变化规律和损伤部分不能承受拉应力等力学特性。     

A comparison of discrete granular material models with continuous microplane formulations

The main objective of this paper is the discussion of two different strategies of simulating the constitutive behavior of granular assemblies. For this, we will focus on discrete particle methods which are widely used in physical science and on continuum–based microplane models which are applied by the engineering community. After deriving the overall constitutive equations based on Voigt’s hypothesis, special focus will be dedicated to the comparison of the relations between the microscopic and macroscopic quantities of each model. It will be demonstrated, that the two basically different modelling techniques lead to remarkably similar results for elastic as well as elasto–plastic material behavior. Received: 19 May 1999     

Towards a theory of granular plasticity

A theory of granular plasticity based on the time-averaged rigid-plastic flow equations is presented. Slow granular flows in hoppers are often modeled as rigid-plastic flows with frictional yield conditions. However, such constitutive relations lead to systems of partial differential equations that are ill-posed: they possess instabilities in the short-wavelength limit. In addition, features of these flows clearly depend on microstructure in a way not modeled by such continuum models. Here an attempt is made to address both short-comings by splitting variables into ‘fluctuating’ plus ‘average’ parts and time-averaging the rigid-plastic flow equations to produce effective equations which depend on the ‘average’ variables and variances of the ‘fluctuating’ variables. Microstructural physics can be introduced by appealing to the kinetic theory of inelastic hard-spheres to develop a constitutive relation for the new ‘fluctuating’ variables. The equations can then be closed by a suitable consitutive equation, requiring that this system of equations be stable in the short-wavelength limit. In this way a granular length-scale is introduced to the rigid-plastic flow equations.     

Towards a theory of granular plasticity

A theory of granular plasticity based on the time-averaged rigid-plastic flow equations is presented. Slow granular flows in hoppers are often modeled as rigid-plastic flows with frictional yield conditions. However, such constitutive relations lead to systems of partial differential equations that are ill-posed: they possess instabilities in the short-wavelength limit. In addition, features of these flows clearly depend on microstructure in a way not modeled by such continuum models. Here an attempt is made to address both short-comings by splitting variables into ‘fluctuating’ plus ‘average’ parts and time-averaging the rigid-plastic flow equations to produce effective equations which depend on the ‘average’ variables and variances of the ‘fluctuating’ variables. Microstructural physics can be introduced by appealing to the kinetic theory of inelastic hard-spheres to develop a constitutive relation for the new ‘fluctuating’ variables. The equations can then be closed by a suitable consitutive equation, requiring that this system of equations be stable in the short-wavelength limit. In this way a granular length-scale is introduced to the rigid-plastic flow equations.     

Discrete modelling results of a direct shear test for granular materials versus FE results

The intention of this paper is to present a comparison of the results of discrete element and finite element simulations of a simple shear test for medium dense cohesionless sand. Such a comparison may provide useful information on the limitations and possible advantages of micro-polar continuum models for granular media as compared with discrete element models. To simulate the discrete nature of sand at the micro-level during shearing, the 3D discrete open-source model YADE developed at Grenoble University was used. Contact moments at spheres were assumed to capture the influence of force eccentricities due to grain roughness. Attention was paid to some micro-structural events (such as vortices, force chains, vortex structures, local void ratio fluctuations) appearing in a shear zone and kinetic, elastic and dissipated energies in granular specimen. The results of the discrete element simulations were compared with the corresponding finite element (FE) solutions based on a micro-polar hypoplastic constitutive model for granular material. A satisfactory agreement between discrete and FE results was achieved. Advantages and disadvantages of both approaches are outlined.     

MODELLING STRONG DISCONTINUITIES IN SOLID MECHANICS VIA STRAIN SOFTENING CONSTITUTIVE EQUATIONS. PART 1: FUNDAMENTALS

The paper addresses some fundamental aspects about the use of standard constitutive equations to model strong discontinuities (cracks, shear bands, slip lines, etc.) in solid mechanics analyzes. The strong discontinuity analysis is introduced as a basic tool to derive a general framework, in which different families of constitutive equations can be considered, that allows to extract some outstanding aspects of the intended analysis. In particular, a link between continuum and discrete approaches to the strain localization phenomena is obtained. Applications to standard continuum damage and elastoplastic constitutive equations are presented. Relevant aspects to be considered in the numerical simulation of the problem (tackled in Part 2 of the work) are also presented.     

Mesoscale simulations of a dart penetrating sand

Historically, hydrodynamic calculations have utilized continuum constitutive models to simulate the coupled dynamic response of a solid projectile penetrating a heterogeneous target system such as concrete, foam or a granular porous medium. Continuum models fail to capture the complicated grain level response within the heterogeneous target which can result in asymmetric loading of the projectile leading to variations in projectile performance. These grain level effects can be crucial to predicting the penetration depth or overall effectiveness of the projectile. In order to assess the possibility of using mesoscale simulations to resolve the grain level dynamics, hydrodynamic simulations were performed for an 11.4 cm long, 0.9 cm diameter dart penetrating a bed of porous granular dry sand with an initial velocity of 366 m/s. Simulations were performed using the Eulerian hydrocode CTH in a two-dimensional planar configuration. The goal of the mesoscale simulations is to determine the viability of using these techniques as an alternative to continuum models and to assess the effects of grain level variability such as anisotropic material distributions and variations in the dynamic yield and fracture strength. The results indicate that variations in the size distribution of aggregate added and the fracture strength of the sand system can have a significant effect on penetration performance of the dart; whereas variations in the dynamic strength of the sand had little effect on the dart penetration.     

Mesh objective modeling of cracks using continuous linear strain and displacement interpolations

The paper addresses the problem of tensile and mixed‐mode cracking within the so‐called smeared crack approach. Because lack of point‐wise convergence on stresses is deemed as the main difficulty to be overcome in the discrete problem, a (stabilized) mixed formulation with continuous linear strain and displacement interpolations is used. The necessary convergence rate can be proved for such a formulation, at least in the linear problem. Two standard local isotropic Rankine damage models with strain‐softening, differing in the definition of the damage criteria, are used as discrete constitutive model. Numerical examples demonstrate the application of the proposed formulation using linear triangular P1P1 and bilinear quadrilateral Q1Q1 mixed elements. The results obtained do not suffer from spurious mesh‐bias dependence without the use of auxiliary tracking techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

Layerwise modeling of progressive damage in fiber-reinforced composite laminates

This paper investigates the effects of discrete layer transverse shear strain and discrete layer transverse normal strain on the predicted progressive damage response and global failure of fiber-reinforced composite laminates. These effects are isolated using a hierarchical, displacement-based 2-D finite element model that includes the first-order shear deformation model (FSD), type-I layerwise models (LW1) and type-II layerwise models (LW2) as special cases. Both the LW1 layerwise model and the more familiar FSD model use a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the LW1 model includes discrete layer transverse shear effects via in-plane displacement components that are C ⁰ continuous with respect to the thickness coordinate. The LW2 layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects by expanding all three displacement components as C ⁰ continuous functions of the thickness coordinate. The hierarchical finite element model incorporates a 3-D continuum damage mechanics (CDM) model that predicts local orthotropic damage evolution and local stiffness reduction at the geometric scale represented by the homogenized composite material ply. In modeling laminates that exhibit either widespread or localized transverse shear deformation, the results obtained in this study clearly show that the inclusion of discrete layer kinematics significantly increases the rate of local damage accumulation and significantly reduces the predicted global failure load compared to solutions obtained from first-order shear deformable models. The source of this effect can be traced to the improved resolution of local interlaminar shear stress concentrations, which results in faster local damage evolution and earlier cascading of localized failures into widespread global failure.     

Numerical analyses of braced excavation in granular grounds: continuum and discrete element approaches

Numerical simulations have been extensively used in braced excavation design. However, previous analyses indicate that the universally adopted constitutive models such as Mohr–Coulomb (M–C) model and Drucker–Prager (D–P) model need to be further clarified due to the unsatisfactory prediction of the ground deformation. This study focuses on the features that future continuum models should capture for braced excavation in granular ground. For this purpose, a simplified braced excavation in granular ground was simulated using the distinct element method (DEM). The same excavation case was also simulated by the Finite Difference Method (FDM) using M–C and D–P model to check their applicability. The excavation was 7.5 m in depth and was braced at the level of $-1.5\,\text{ m}$ . The results indicate that the DEM simulation can reproduce the main responses of granular ground during excavation; the excavation initiates failure at the excavation depth of 5.0 m and evolves into total failure at the depth of 7.5 m; two types of stress paths in front of and behind the wall are observed, respectively; obvious principal stress rotations of soils are recognized. Compared with DEM results, M–C and D–P model can generally predict excavation responses qualitatively but under-estimate the ground deformation and internal forces of the wall. This is due to the incapability of the two continuum models to capture the mechanical behavior of granular material under complicated stress conditions in braced excavation. Based on these observations and comparisons, three features are emphasized for future continuum models: stress path dependency, non-coaxiality, and shear dilatancy.     

A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media

A hierarchical multiscale framework is proposed to model the mechanical behaviour of granular media. The framework employs a rigorous hierarchical coupling between the FEM and the discrete element method (DEM). To solve a BVP, the FEM is used to discretise the macroscopic geometric domain into an FEM mesh. A DEM assembly with memory of its loading history is embedded at each Gauss integration point of the mesh to serve as the representative volume element (RVE). The DEM assembly receives the global deformation at its Gauss point from the FEM as input boundary conditions and is solved to derive the required constitutive relation at the specific material point to advance the FEM computation. The DEM computation employs simple physically based contact laws in conjunction with Coulomb's friction for interparticle contacts to capture the loading‐history dependence and highly nonlinear dissipative response of a granular material. The hierarchical scheme helps to avoid the phenomenological assumptions on constitutive relation in conventional continuum modelling and retains the computational efficiency of FEM in solving large‐scale BVPs. The hierarchical structure also makes it ideal for distributed parallel computing to fully unleash its predictive power. Importantly, the framework offers rich information on the particle level with direct link to the macroscopic material response, which helps to shed lights on cross‐scale understanding of granular media. The developed framework is first benchmarked by a simulation of single‐element drained test and is then applied to the predictions of strain localisation for sand subject to monotonic biaxial compression, as well as the liquefaction and cyclic mobility of sand in cyclic simple shear tests. It is demonstrated that the proposed method may reproduce interesting experimental observations that are otherwise difficult to be captured by conventional FEM or pure DEM simulations, such as the inception of shear band under smooth symmetric boundary conditions, non‐coaxial granular response, large dilation and rotation at the edges of shear band and critical state reached within the shear band. Copyright © 2014 John Wiley & Sons, Ltd.     

Comparative study of granular solid-structure interaction in a uni-axial compression system

Interaction between granular solids and confining structures is an elementary problem encountered in subsurface structural design and bulk solids storing and handling. A classic scenario is uni-axial compression of granular solids in a deformable cylindrical container. Despite being apparently simple in loading condition, the understanding of this scenario remains limited, mainly due to complex interactive deformation between the two components via frictional interfaces. This paper comparatively examines such a uni-axial compression particulate system by a laboratory experiment and two different numerical approaches, namely, continuum finite element method (FEM) and linked discrete-finite element method (linked DEM-FEM). In the continuum FEM approach, two intendedly chosen simple material models, linear elastic and porous elastic models, are attempted. The comparative study reveals that the majority of resultant characteristics show satisfactory agreement amongst the numerical predictions and the experimental measurements. The simple elastic continuum FEM models can hence be a useful alternative in modelling such problems with mild structural flexibility under a monotonic loading scenario. However, precise prediction of some characteristics, such as lateral pressure ratio, may demand more elaborated material model or parameter selection. The enhancements needed for each numerical approach in order to achieve an improved result are further discussed.