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.     

Micromechanics of elastic buckling of a colloidal polymer layer on a soft substrate: experiment and theory

We study the buckling instability of a colloidal particle layer adhered to an elastic substrate using an integrated experimental and theoretical approach. Experiments using monodisperse colloid-scale spherical particles made of polystyrene and silica, show that the wavelength of the initial (critical) buckling mode is independent of particle modulus and linearly dependent on particle radius—in contradiction with the predictions of the prevailing continuum model. We developed a granular model of the particle layer using structural mechanics techniques. The granular model predicts the observed wavelength of the initial, critical buckling mode within the estimated range of parameter values for the experiment. The evolution of this mode into the post-buckling regime is examined. Results highlight the crucial role of material discreteness in the mechanical response, and the need for accurate methods of estimating parameters for the particle-scale resistances against buckling.     

Automatic solver for non‐linear partial differential equations with implicit local laws: Application to unilateral contact

In general, non‐linear continuum mechanics combine global balance equations and local constitutive laws. In this work, frictionless contact between a rigid tool and a thin elastic shell is considered. This class of boundary value problems involves two non‐linear algebraic laws: the first one gives explicitly the stress field as a function of the strain throughout the continuum part, whereas the second one is a non‐linear equation relating the contact forces and the displacement at the boundary.Given the fact that classical computational approaches sometimes require significant effort in implementation of complex non‐linear problems, a computation technique based on automatic differentiation of constitutive laws is presented in this paper. The procedure enables to compute automatically the higher‐order derivatives of these constitutive laws and thereafter to define the Taylor series that are the basis of the continuation technique called asymptotic numerical method. The algorithm is about the same with an explicit or implicit constitutive relation. In the modelling of forming processes, many tool shapes can be encountered. The presented computational technique permits an easy implementation of these complex surfaces, for instance in a finite element code: the user is only required to define the tool geometry and the computer is able to obtain the higher‐order derivatives. Copyright © 2013 John Wiley & Sons, Ltd.     

A nonlocal multiscale discrete‐continuum model for predicting mechanical behavior of granular materials

A three‐dimensional nonlocal multiscale discrete‐continuum model has been developed for modeling mechanical behavior of granular materials. In the proposed multiscale scheme, we establish an information‐passing coupling between the discrete element method, which explicitly replicates granular motion of individual particles, and a finite element continuum model, which captures nonlocal overall responses of the granular assemblies. The resulting multiscale discrete‐continuum coupling method retains the simplicity and efficiency of a continuum‐based finite element model, while circumventing mesh pathology in the post‐bifurcation regime by means of staggered nonlocal operator. We demonstrate that the multiscale coupling scheme is able to capture the plastic dilatancy and pressure‐sensitive frictional responses commonly observed inside dilatant shear bands, without employing a phenomenological plasticity model at a macroscopic level. In addition, internal variables, such as plastic dilatancy and plastic flow direction, are now inferred directly from granular physics, without introducing unnecessary empirical relations and phenomenology. The simple shear and the biaxial compression tests are used to analyze the onset and evolution of shear bands in granular materials and sensitivity to mesh density. The robustness and the accuracy of the proposed multiscale model are verified in comparisons with single‐scale benchmark discrete element method simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluation of a physical length scale for granular materials

Classical continuum mechanics considers the interaction of microstructural units of the material through stresses and displacements of material points. Therefore, conventional continuum mechanics approaches can not incorporate any intrinsic material length scale. However in reality interaction of grains may include rotations and the corresponding couple stresses as well, and real materials have a number of important length scales (e.g., grains, particles, fibers, etc.). An equation for determining the length scale is proposed. The proposed length scale equations include the effect of plastic deformation (microrotation), the effect of normal stress and contact area. The proposed length scale is implemented into elastic–elastoplastic Cosserat formulation. The effect of length scale on the finite element simulation and yield surface was evaluated by using the proposed length scale equation. The importance of length scale on the constitutive modeling of granular materials is analyzed in numerical simulations.     

Anisotropy in particle contacts associated with shearing in granular media

Summary Work on directional properties in granular media is briefly reviewed with an emphasis on the angular frequency of normals to particle contact tangents. Data from a two dimensional model material is compared with that of sand sheared in plane strain with a similar stress path. This stress path included a chosen sudden change in major principal stress direction.Directional stress-strain behaviour in granular media is related to the changing angular frequency of particle contacts. Measurements of this changing frequency are related to new model predictions and actual experimental data for sand sheared in plane strain.With 5 Figures     

A thermomechanical approach to the development of micropolar constitutive models of granular media

Summary. In this paper, we lay the groundwork for the development of micropolar (Cosserat) constitutive relations for granular media within the framework of the theory of thermomechanics. Expressions for the free energy and the dissipation function have been derived using a micromechanical analysis of a cluster consisting of a particle and its immediate neighbors (i.e., the first ring). Fluctuations in particle displacements and rotations within this mesoscale assembly as well as fluctuations in strain and curvature are represented by internal variables. Using thermomechanical techniques previously employed for classical materials, a non-local micropolar model is constructed and then subsequently applied to a granular material undergoing simple shear. The effects of the boundaries through particle rotations are discussed.     

Development and validation of calibration methods for discrete element modelling

Discrete element method (DEM) is proving to be a reliable and increasingly used tool to study and predict the behaviour of granular materials. Numerous particle-scale mechanisms influence the bulk behaviour and flow of bulk materials. It is important that the relevant measurable input parameters for discrete element models be measured by laboratory equipment or determined by physical calibration experiments for rational results. This paper describes some of the bench-scale experiments that have been developed to calibrate the DEM simulations to reflect actual dynamic behaviour. Relevant parameters such as static and rolling coefficients of friction, coefficient of restitution and inter-particle cohesion forces from the presence of liquid bridges have been investigated to model the bulk behaviour of dry and moist granular materials. To validate the DEM models, the results have been checked against experimental slump tests and hopper discharge experiments to quantitatively compare the poured and drained angles of repose and solids mass flow rate. The calibration techniques presented have the capability to be scaled to model and fine tune DEM parameters of granular materials of varying length scales to obtain equivalent static and dynamic behaviour.     

Extension of the node‐to‐segment contact element for surface‐expansion‐dependent contact laws

A class of friction laws depending on the measure of contact surface expansion is defined in the paper within the continuum contact mechanics framework. The nominal and spatial forms of constitutive relations are discussed, including incremental penalty relations. Further, an extended node‐to‐segment element is derived which is capable of treating surface‐expansion‐dependent contact laws in a consistent way. The approach is suitable for any kind of node‐to‐segment contact elements. Finally, the computational efficiency of the extended element as well as other possible approaches are illustrated by numerical examples relevant to metal forming applications. Copyright © 2001 John Wiley & Sons, Ltd.     

Two-dimensional discrete element modelling of bender element tests on an idealised granular material

The small-strain (elastic) shear stiffness of soil is an important parameter in geotechnics. It is required as an input parameter to predict deformations and to carry out site response analysis to predict levels of shaking during earthquakes. Bender element testing is often used in experimental soil mechanics to determine the shear (S-) wave velocity in a given soil and hence the shear stiffness. In a bender element test a small perturbation is input at a point source and the propagation of the perturbation through the system is measured at a single measurement point. The mechanics and dynamics of the system response are non-trivial, complicating interpretation of the measured signal. This paper presents the results of a series of discrete element method (DEM) simulations of bender element tests on a simple, idealised granular material. DEM simulations provide the opportunity to study the mechanics of this testing approach in detail. The DEM model is shown to be capable of capturing features of the system response that had previously been identified in continuum-type analyses of the system. The propagation of the wave through the sample can be monitored at the particle-scale in the DEM simulation. In particular, the particle velocity data indicated the migration of a central S-wave accompanied by P-waves moving along the sides of the sample. The elastic stiffness of the system was compared with the stiffness calculated using different approaches to interpreting bender element test data. An approach based upon direct decomposition of the signal using a fast-Fourier transform yielded the most accurate results.     

A micromechanics-based gradient model and the effect of high-order stress and particle rolling on localizations for granular materials

This study develops a gradient model to consider heterogeneity of granular materials by means of a thermo-micromechanics method and the hypothesis of kinematics. In this model, the macroscopic variables are related to the micro information by a non-affine projection scheme. Numerical examples illustrated the capability and performance of the presented model in modeling deformation patterns in case of considering particle rolling and not considering particle rolling, and investigated the validity for the stress–strain relation of the presented model.     

Probabilistic calibration of discrete element simulations using the sequential quasi-Monte Carlo filter

The calibration of discrete element method (DEM) simulations is typically accomplished in a trial-and-error manner. It generally lacks objectivity and is filled with uncertainties. To deal with these issues, the sequential quasi-Monte Carlo (SQMC) filter is employed as a novel approach to calibrating the DEM models of granular materials. Within the sequential Bayesian framework, the posterior probability density functions (PDFs) of micromechanical parameters, conditioned to the experimentally obtained stress–strain behavior of granular soils, are approximated by independent model trajectories. In this work, two different contact laws are employed in DEM simulations and a granular soil specimen is modeled as polydisperse packing using various numbers of spherical grains. Knowing the evolution of physical states of the material, the proposed probabilistic calibration method can recursively update the posterior PDFs in a five-dimensional parameter space based on the Bayes’ rule. Both the identified parameters and posterior PDFs are analyzed to understand the effect of grain configuration and loading conditions. Numerical predictions using parameter sets with the highest posterior probabilities agree well with the experimental results. The advantage of the SQMC filter lies in the estimation of posterior PDFs, from which the robustness of the selected contact laws, the uncertainties of the micromechanical parameters and their interactions are all analyzed. The micro–macro correlations, which are byproducts of the probabilistic calibration, are extracted to provide insights into the multiscale mechanics of dense granular materials.     

Implementation and validation of a triaxiality dependent cohesive model: experiments and simulations

A recently formulated triaxiality dependent cohesive model for plane strain is implemented and its versatility is tested in simulation of ductile fracture of mild steel at different states of stress. The triaxiality dependent model was implemented as linear displacement formulation based elements whose constitutive behaviour was dependent on the stress-state of the neighbouring continuum element. By comparing the experimental data and predictions of corresponding plane strain simulations, the model parameters are estimated. The model is shown to be effective in reproducing characteristic features of the macroscopic response of both pre-cracked as well as geometries without a preexisting nominal defect. Since the model parameters are held constant for simulations at different stress-states, they are effectively material constants.     

Thermodynamically consistent coupled viscoplastic damage model for perforation and penetration in metal matrix composite materials

Accurate modeling and efficient analysis of the metal matrix composite materials failure mechanism during high velocity impact conditions is still the ultimate goal for many researchers. The objective is to develop a micromechanical constitutive model that can effectively simulate the high impact damage problem of the metal matrix composite materials. Therefore in this paper, a multiscale micromechanical constitutive model that couples the anisotropic damage mechanism with the viscoplastic deformation is presented here as a solution to this situation. This coupled viscoplastic damage model is formulated based on thermodynamic laws. Nonlinear continuum mechanics is used for this heterogeneous media that assesses a strong coupling between viscoplasticity and anisotropic damage. It includes the strong directional effect of the fiber on the evolution of the back stress and the development of the viscoplastic strain in the material behavior for high velocity impact damage related problems.     

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。     

基于弹塑性损伤本构模型的复合材料层合板破坏荷载预测

在连续损伤力学和塑性力学框架内,建立一个同时考虑塑性效应和损伤累积导致材料属性退化的复合材料弹塑性损伤本构模型。基于最近点投影回映算法,开发本构模型的应变驱动隐式积分算法以更新应力及与解答相关的状态变量,并推导与所开发算法相应的数值一致性切线刚度矩阵,保证有限元分析采用NewtonRaphson迭代法解答非线性问题的计算效率。采用断裂带模型对已开发的本构模型软化段进行规则化,以减轻有限元分析结果的网格相关性问题。对损伤变量进行粘滞规则化,并推导出相应的粘滞规则化数值一致性切线刚度张量,解决了在有限元隐式计算程序中采用含应变软化段本构关系的数值分析由于计算困难而提前终止的问题。开发包含数值积分算法的用户材料子程序UMAT,并嵌于有限元程序Abaqus v6.14中。通过对力学行为展现显著塑性效应的AS4/3501-6V型开口复合材料层合板的渐进失效分析,验证本文提出的材料本构模型的有效性。结果显示,预测结果与已报道的试验结果吻合良好,并且预测精度高于其他已有弹性损伤模型。表明已建立的弹塑性损伤本构模型能够准确预测力学行为,展现显著塑性效应的复合材料层合板的破坏荷载,为其构件和结构设计提供一种有效的分析方法。