Localisation in granular media: Particle approach,homogenisation and continuum modelling

The individual motion of grains in granular material has a strong influence on the macroscopic material behaviour, which is in particular the case for the phenomena of strain localisation in shear zones and justifies the need for techniques that incorporate a micro-macro transition. In this contribution, granular media are investigated in three steps. Firstly, a microscopic particle-based modelling is set up, where individual grains are considered as rigid uncrushable particles while their motion is obtained through Newton's equations of state. The inter-particle contact forces are thereby determined via constitutive contact-force formulations, which have to account for the envisaged material behaviour. The second step is the homogenisation of the obtained particle's displacements and contact forces through a particle-centre-based strategy towards continuum quantities. Therefore, Representative Elementary Volumes (REV) are introduced on the mesoscale and the specific construction of the REV boundary leads to the understanding of granular media as a micropolar continuum. Finally, in order to verify the homogenisation strategy, a continuum based micropolar model is applied to model localisation phenomena and a comparative study of the results is carried out in a qualitative way. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From particle mechanics to microcontinuum theories

Microcontinuum theories enable the consideration of particle-based microstructures within a continuum mechanical framework. Several classes of microcontinua, such as the micromorphic, the micropolar, the microstrain or the microstrech formulation, have been successfully applied to engineering applications, although a clear physical determination and interpretation of the kinematical extensions and the resulting higher-order stresses within the formulation is frequently missing. In this regard, the present contribution focuses on establishing the physical link between discrete contact forces, stresses and deformation of particle-based microstructures and the characteristic stress states of microcontinuum theories. Representative Elementary Volumes (REVs) are therefore constructed on the mesoscale as ensembles of deformable particles from the mircoscale. Establishing the REV balance relations justifies the common generalisation of the angular momentum balance commonly applied in microcontinuum theories. It furthermore leads to the identification of the continuum stresses based on micro-quantities and enables the application of homogenisation techniques by exploitation of the equilibrium conditions of a REV. In order to investigate the hereby established link from the micro- to the macroscale, granular materials are simulated using the Discrete-Element Method (DEM). In particular, localisation phenomena in granulates, e. g. in biaxial compression tests or during ground-failure processes are studied. This implies the formulation of the contact between particles in an appropriate constitutive manner in accordance to the envisaged granular material behaviour, e. g. whether loose material, such as sand, or bonded multi-component material, such as polyurethan-sand compounds for metal casting applications are of interest. With the full solution of a particle-based initial-boundary-value problem, the homogenisation formalism is applied and enables the study of the extended continuum field quantities, essentially demonstrating the applicability of microcontinuum theories in the field of granular material. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A homogenisation strategy for micromorphic continua based on particle mechanics

Materials with a granular microstructure frequently fail in narrow zones due to strain localisation. Examplarily, one may look at the shear-zone development in dry sand during bi- and triaxial loading, where grains in the shear-zone exhibit large displacements and rotations. Furthermore, localisation is also observed in materials, where the microstructure consists of grains and a binding material, such as for example metal-casting moulds. Here, sand grains are bound together via a polyurethan-based material and macroscopic material failure originates from the deformation and breakage of the binder material. Within a continuum-based modelling approach, these microstructural effects can be accounted for by the consideration of an additional microcontinuum at each material point of the macroscopic body. These extended continuum theories, such as the micromorphic continua and its micropolar and microstrain sub-formulations, assume a characteristic microcontinuum deformation on a lower scale and have been successfully applied in the field of granular media. Exemplarily, in the framework of a micropolar continua, it is possible to contact forces to stresses and couple stresses via an appropriate homogenisation technique. This method includes the introduction of a Representative Elementary Volume (REV) on the mesoscale situated between the particle and the continuum scale. In this contribution, a homogenisation strategy based on a particle-centre-based REV definition is presented that is generally valid for micromorphic and micropolar continua. Therefore, a grain-binder microstructure is investigated, where particle rotations contribute to the micropolar part, while binder deformations yield the additional macromorphic character. Numerical examples are given, where results from discrete-element simulations are locally averaged and show the individual activation of the microcontinuum characteristics in the localised zones. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of non-cohesive resolved and coarse grain DEM models for gas flow through particle beds

The Discrete Element Method (DEM) is a widely used approach for modelling granular systems. Currently, the number of particles which can be tractably modelled using DEM is several orders of magnitude lower than the number of particles present in common large-scale industrial systems. Practical approaches to modelling such industrial system therefore usually involve modelling over a limited domain, or with larger particle diameters and a corresponding assumption of scale invariance. These assumption are, however, problematic in systems where granular material interacts with gas flow, as the dynamics of the system depends heavily on the number of particles. This has led to a number of suggested modifications for coupled gas–grain DEM to effectively increase the number of particles being simulated. One such approach is for each simulated particle to represent a cluster of smaller particles and to re-formulate DEM based on these clusters. This, known as a representative or ‘coarse grain’ method, potentially allows the number of virtual DEM particles to be approximately the same as the real number of particles at relatively low computational cost. We summarise the current approaches to coarse grain models in the literature, with emphasis on discussion of limitations and assumptions inherent in such approaches. The effectiveness of the method is investigated for gas flow through particle beds using resolved and coarse grain models with the same effective particle numbers. The pressure drop, as well as the pre and post fluidisation characteristics in the beds are measured and compared, and the relative saving in computational cost is weighed against the effectiveness of the coarse grain approach. In general, the method is found perform reasonably well, with a considerable saving of computational time, but to deviate from empirical predictions at large coarse grain ratios.     

Finite element methods for micropolar models of granular materials

We present a thermodynamically based finite element scheme for rate-independent materials and demonstrate its application in modelling the rheological behaviour of granular materials. Starting from the laws of thermodynamics, we have recently developed a new class of micropolar-type constitutive relations for two-dimensional densely packed granular media. This class of constitutive laws is expressed in terms of particle-scale properties, thus providing a direct link between observed macroscopic behaviour and the underlying particle–particle interactions. Here, we demonstrate how the connection to the underlying physics can be maintained and carried through to the finite element implementation phase of the modelling process via the same thermodynamical principles used to construct the constitutive laws. Notably, the study indicates that while the traditional Galerkin-FEM method admits a range of weighting functions, the proposed formulation provides an additional constraint that narrows the choice of admissible weighting functions via the second law of thermodynamics. Additionally, this paper presents insights into the finite element implementation of micropolar models deemed to be appropriate for modelling several classes of heterogeneous media (e.g. granular materials, cellular composites and biological materials). As the kinematics and kinetics of micropolar continua are enriched by the addition of rotational degrees of freedom to each material point, the equations governing boundary value problems for such materials differ from those of other continuum models both from the viewpoint of the constitutive law and the governing conservation laws. Analysis of elastoplastic deformation of micropolar continua is presented.     

Investigation of discrete stress distribution for dry powder compaction using Discrete Element Method and weighted Voronoi tesselation

Powder compaction of granular material plays a substantial role in the manufacturing process of ceramics industry and powder metallurgy industry. The compaction behaviour is ruled by granular flow and densification of deformable particles. Discrete element method (DEM) allows to investigate the powder compaction process numerically on the microscale by modeling the forces on the particle level and simulating the particle motion. Three-dimensional data about particle size distribution and spatial structure of the particle packing can be extracted from micro-computed tomography (µCT). An average stress tensor can be computed from DEM results, evaluating the contact forces and the distances from the particle center to the contact point with respect to an average cell volume. A weighted Voronoi tesselation of the polydisperse particle assembly is proposed for mapping a cell volume to each individual particle. With this approach all structural information of the particle system can be transferred from a discrete particle model to a heterogeneous volume model of micro-structure. Discrete stress distributions for uniaxial powder compaction are presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of microcracks in ferroelectric materials by application of a grain-boundary-motivated cohesive law

Ferroelectric materials exhibit a huge potential for engineering applications – ranging from electrical actuators (inverse piezoelectric effect) to sensor technology (direct piezoelectric effect). To give an example, lead zirconate titanate (PZT) is a typical perovskite ion crystal possessing ferroelectric properties. In this contribution, we are particularly interested in the modelling of microcracking effects in ferroelectric materials. In view of Finite-Element-based simulations, the geometry of a natural grain structure, as observed on the so-called micro-level, is represented by an appropriate mesh. While the response on the grains themselves is approximated by coupled continuum elements, grain boundaries are numerically incorporated via so-called cohesive-type elements. For the sake of simplicity, switching effects in the bulk material will be neglected. The behaviour of the grain boundaries is modelled by means of cohesive-type laws. Identifying grain boundaries as potential failure zones leading to microcracking, cohesive-type elements consequently offer a great potential for numerical simulations. As an advantage, in the case of failure they do not a priori result in ill-conditioned systems of equations as compared with the application of standard continuum elements to localised deformations. Finally, representative constitutive relations for both the bulk material and the grain boundaries, enable two-dimensional studies of low-cycle-fatigue motivated benchmark boundary value problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bridging the Length Scales: Micromechanics of Granular Media

Aristotle's statement that the whole is more than the sum of its parts aptly describes the essence of a granular material's rich and complex behaviour, which ultimately arises from internal mechanisms developed on many length scales. Recently, non-invasive experimental studies have given remarkable insight into the evolution of these mechanisms, thereby providing benchmarks and a unique opportunity for the theoretical modelling of these systems. This paper focuses on the challenges of capturing these multiscale mechanisms within the framework of continuum theory. In particular, a new approach toward developing a non-local micropolar constitutive model of granular media using micromechanics and internal variable theory is discussed. To demonstrate the predictive potential of these models, we present their application in the analysis of two fundamental problems to the mechanics of granular media: (i) formation and evolution of shear bands (the precursors of material failure), (ii) the classical Flamant problem. Finally, we briefly discuss the computational challenges in bridging the gap between micromechanical studies of granular media and the applications of continuum theory on the macro-scale via a finite element analysis of the flat punch problem. In practice, this problem is used to assess the load bearing capacity of a material and is fundamental to civil and structural engineering.     

Advanced comminution modelling: Part 2 - Mills

This second part paper explores rock breakage mechanisms, the life cycle of rocks in mills and the strong influence of end walls on charge motion within mills. We present recent advances in particle-based modelling of mills for comminution focused around wear and the effect of slurry and slurry phase grinding. Three mill scenarios are considered:

• 1.Media flow and the resulting wear evolution of the belly and end wall liners and the resulting change in mill performance for a full industrial scale dry ball mill (modelled using DEM)
• 2.Axial slurry transport and mixing in a wet overflow industrial scale ball mill (modelled using fully coupled DEM and SPH)
• 3.Effect of mill speed on slurry and solid charge motion and the resulting grinding of fine particles in a 1.8 m diameter wet Hardinge pilot mill (modelled using fully coupled DEM and SPH with advection-diffusion-population balance equations solved for the slurry size distribution for each SPH particle)

These demonstrate the nature and level of fidelity that is now possible to include in particle-scale comminution models. They provide insights into the critical importance of curtain flows generated by the end walls of tumbling mills, on wear behaviour on liners, on the structure of slurry pools and mill discharge and on the adverse effect on grinding of increasing mill speed.     

Computation of Screening Phenonema in a Vertical Tumbling Cylinder

Granular materials are an integral part of our environment. Due to their wide variety of applications in industrial and technological processes, they have captured a great interest in the recent research, see [1] and [2]. The related studies are often based on numerical simulations and it is considered as challenging to investigate computational phenomena of dense granular systems. Particle screening is an essential technology in many industrial fields and important in granular studies. The particular problem of interest is the separation of round shape particles of different geometrical sizes using a rotating tumbling vertical cylinder. The concept of discrete element method (DEM) that considers the motion of each single particle individually is applied in this study. Particle-to-particle and particle-to-wall collisions will appear under the tumbling motion of the rotating structure. The normal and frictional forces between particles themselves and particles and surrounding walls of the structure are calculated according to the rules of a penalty method, which employs spring-damper models for this purpose. As a result of collisions, the particles will dissipate kinetic energy due to the normal and frictional contact losses. Particle distribution and sifting rate of the separated particles have been studied taking into consideration different rotational speeds of the machine, various damping and frictional coefficients and different sizes of holes in the sifting plates at different levels of the structure. In an attempt to better understand the mechanism of the particle transport between the different layers of the sifting system, different computational studies have been performed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the heat flux vector for flowing granular materials—part II: derivation and special cases

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as non‐linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame‐invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourier's law), could also depend on the gradient of density (or volume fraction), and D (the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non‐linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient. Copyright © 2006 John Wiley & Sons, Ltd.     

On the heat flux vector for flowing granular materials—Part I: effective thermal conductivity and background

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as non‐linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame‐invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourier's law), could also depend on the gradient of density (or volume fraction), and D (the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non‐linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient. Copyright © 2006 John Wiley & Sons, Ltd.     

A micromechanically motivated model for ferroelectrics combined with polygonal finite elements

Piezoelectric materials are one of the most prominent smart materials due to their strong electromechanical coupling behaviour. Ferroelectric ceramics behave like piezoelectric materials under low electrical and mechanical loads, but exhibit pronounced nonlinear response at higher loads due to microscopic domain switching. Modern smart devices consist of complex geometries that may force the ferroelectrics employed within them to experience higher fields than they were originally designed for, so that the material responds within its nonlinear region. Hence, models predicting the nonlinear effects of ferroelectrics under complex loading cases are important from the design point of view. Within standard finite element models dealing with electromechanical problems, each grain may be subdiscretized by several finite elements. This problem can be approximated or rather overcome by a polygonal finite element method, where each grain is modelled by solely one single finite element. In this contribution, a micromechanically motivated switching model for ferroelectric ceramics, as based on volume fraction concepts, is combined with polygonal finite element approach. Related representative numerical examples allow to further study and understand the nonlinear response of this material under complex loading cases. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling subgroup behaviour in crowd dynamics DEM simulation

This paper investigates the behaviour of subgroups in crowd dynamics by means of filming and observation. An existing crowd modelling program, CrowdDMX, based on a discrete element model (DEM) has been modified on the basis of observations made in this paper and literature. Each person is represented as three overlapping circles and motion is modelled in a Newtonian manner. It incorporates psychological forces as well as physical forces in a 2D time-stepping environment. The DEM model was modified to include realistic subgroup behaviour, representing people in the crowd desiring to stay together (families, friends, etc.). Subgroup psychological forces were incorporated. The previous model only simulated individuals moving independently, which was unrealistic in some situations as shown by the observation and filming part of the study. The revised program models subgroups realistically including the tendency to avoid subgroup division in cases of contra-flow.     

Failure of granular materials at different scales - microscale approach

In recent years there has been an increase of interest in packed granular and discontinuous media. Due to the properties of such materials, a simple continuum approach is not appropriate to describe the material behavior. Breaking and forming of new contacts between grains, distinguishing for granular media, cannot be captured. We apply a computational homogenization method [6] which is based on a two scale concept to complete the task of modeling packed granular materials. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional modelling of discrete particles by superellipsoids

Discrete Elements are used for the simulation of granular materials (sand, ballast) as well as for molecular assemblies. Circles (2D) and spheres (3D) are often used in literature on the Discrete Element Method (DEM) however they represent a strong idealisation of the real geometry. Superellipsoids provide the opportunity to generate a wide variety of three-dimensional geometrical shapes (e.g. sphere, cube, cylinder). The motion of each particle is described by means of rigid body dynamics. Suitable numerical integration methods are necessary which are able to conserve the essential physical quantities like momentum energy etc.. Possible choices are e.g. the explicit Verlet-Leapfrog method for the translation and the explicit fourth order Runge-Kutta method for the rotation. The implemented contact formulation takes damping as well as friction into account. Efficient implementation of the contact search is the main aim of this part of the work. It is subdivided into the neighbourhood search and the local search. A bisection algorithm is used to calculate the gap between two superellipsoids within the search. For the neighbourhood search two binning algorithms were implemented and compared for several packages of particles. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress-dependent Failure Surface of Granular Materials

Numerical computations of geotechnical problems will become increasingly important because of the growing complexity of geotechnical applications. Therefore, a well-founded prediction of stability statements requires appropriate models, which are able to realistically depict the stress-strain behaviour of non-cohesive-frictional granular materials. On several stress paths, drained triaxial compression experiments on compact dense sand specimen exhibited that the size of the failure surface is not independent from the hydrostatic pressure. The failure surface and, thus, the maximal shear stresses at a specific confining pressure σ₃^exp can be increased by a compression preload at a level higher than σ₃^exp. This means that granular materials own several failure surfaces in dependence of the hydrostatic pressure. Consequently, the failure criteria based on the assumption of a compression stress-path-independent single-failure surface cannot recover the newly detected plastic yielding behaviour of granular materials. An improved approach for modelling the plastic hardening and softening behaviour coupled with the new yield properties at the limit state will be presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the effective permeability of microcracked materials using continuum and lattice micromechanics

Using the framework of the mean-field homogenization method, we propose a continuum and a lattice version of the cascade micromechanics model for the estimation of the effective permeability of microcracked materials. Estimates for the critical microcrack density below which the REV becomes effectively impermeable are derived for both the continuum and the lattice idealizations. A validation example that compares model predictions with direct numerical simulations for the effective permeability as a function of the microcrack volume fraction is provided. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)