Particle simulation of granular media and homogenisation towards continuum quantities

From a microscopic point of view, various natural and engineering materials consist of individual grains, whose motion strongly influence the macroscopic material behaviour. Exemplarily, one may look at the development of shear zones in natural granular materials, such as sand, occurring as a result of local grain dislocations and the transition of the granulate from a denser to a looser packing. The intuitive modelling approach for granular assemblies is consequently the consideration of each grain as a rigid particle. In a numerical framework, this leads to the Discrete Element Method (DEM), wherein the motion of each particle can be obtained solving Newton's equations for each particle. The present contribution discusses the basic fundaments of modelling granular material on the microscopic scale by use of the DEM. Special interest is taken to the constitutive choice of the governing particle-to-particle contact forces, as they have to account for plastic material behaviour as well as for assumptions concerning particle shape, size and distribution. As engineering problems are regularly described on the macroscale by means of continuum mechanics, a homogenisation strategy transfers the information from the microscale towards continuum quantities via volume averaging. Therefore, characteristic Representative Elementary Volumes (REV) are constructed by an ensemble of particles, where each particle can be chosen as the centre of a REV. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A rapid granular chute avalanche impinging on a small fixed obstacle: DEM modeling,experimental validation and exploration of granular stress

Granular flows are constrained by applied stresses. When a granular flow moves rapidly and impinges on an obstacle, the stress is significantly increased along the contact force networks. Granular stresses are still incompletely understood. The aim of this study is to investigate a rapid avalanche of spherical glass beads in an inclined chute with a small fixed semi-cylindrical obstacle by using particle image velocimetry (PIV) technique and discrete element method (DEM). The proposed DEM model produces good agreement with the corresponding avalanche experiment in terms of the velocity profiles. The validated DEM results are then used to explore the internal flow characteristics of a granular avalanche that are not directly observable in experiments, such as the solid fraction, the average coordination number and the granular stress. Rectangular measurement cells, similar to representative volume elements, are developed to determine the spatial variation in stresses for the granular avalanche. The internal flow characteristics of a rapid granular avalanche with and without obstacles are compared. For the unobstructed flow, the normal and shear stresses decrease in the downstream direction because the solid fraction and the average coordination number decrease, resulting from the gravitational acceleration. On the other hand, granular jamming forms in front of the semi-cylindrical obstacle and results in a significant increase in the normal and shear stresses. The unobstructed flow shows slightly anisotropic stress states, giving an earth pressure coefficient of approximately 1.0, whereas the disturbed flow exhibits strongly anisotropic stress states. The simulation results show that the corresponding earth pressure coefficient can be much higher than unity and increases to a maximum value of roughly 5.0. A shear band develops at a distance of roughly twice the particle diameter above the basal surface and a stronger shear band forms in the upstream vicinity of the obstacle.     

Finite element simulation for dynamic large elastoplastic deformation in metal powder forming

In this paper, a transient dynamic analysis of the powder compaction process is simulated by a large displacement finite element method based on a total and updated Lagrangian formulation. A combination of the Mohr–Coulomb and elliptical yield cap model, which reflects the stress state and degree of densification, is applied to describe the constitutive model of powder materials. A Coulomb friction law and a plasticity theory of friction in the context of an interface element formulation are employed in the constitutive modelling of the frictional behaviour between the die and powder. Finally, the powder behaviour during the compaction of a plain bush, a rotational flanged and a shaped tip component are analysed numerically. It is shown that the updated Lagrangian formulation, using a combination of the Mohr–Coulomb and elliptical cap model, can be effective in simulating metal powder compaction.     

Transportation of dry fine powders by coordinated friction manipulation

The transportation of dry fine powders is an emerging technologic task, as in biotechnology, pharmaceutical or coatings industry particle sizes of processed powders are getting smaller and smaller. Fine powders are primarily defined by the fact that adhesive and cohesive forces outweigh the weight forces. This leads to mostly unwanted agglomeration (clumping) and adhesion to surfaces, what makes it more difficult to use conventional conveyor systems (e. g. pneumatic or vibratory conveyors) for transport. A rather new method for transporting these fine powders is based on ultrasonic vibrations, which are used to reduce friction and adhesion between powder and the substrate. One very effective set-up consists of a pipe, which vibrates harmoniously in axial direction at low frequency combined with a pulsed radial high frequency vibration. The high frequency vibration accelerates the particles perpendicular to the surface of the pipe, which in average leads to lower normal and thereby smaller friction force. With coordinated friction manipulation the powder acceleration can be varied so that the powder may be greatly accelerated and only slightly decelerated in each excitation period of the low frequency axial vibration of the pipe. The amount of powder flow is adjustable by vibration amplitudes, frequencies, and pulse rate, which makes the device versatile for comparable high volume and fine dosing using one setup. Within this contribution an experimental set-up consisting of a pipe, a solenoid actuator for axial vibration and a piezoelectric actuator for the radial high frequency vibration is described. An analytical model is shown, that simulates the powder velocity. Finally, simulation results are validated by experimental data for different driving parameters such as amplitude of low frequency vibration, pipe material and inclination angle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanics-based Cosserat-type model for dense particulate solids

A two-dimensional continuum theory is presented for cohesionless granular media consisting of identical rigid disks. While the normal deformation of contacting particles is constrained, the tangential frictional contact is modelled by a line spring with a constant stiffness. To describe the static frictional system transmitting couples at contacts, a Cosserat-type continuum including rotational degrees of freedom is appropriate. Contrary to the classical elastic medium, movement of particles within a granular system in response to applied loads can give rise to localisations of force chains and large voids. In addition to relative displacement and rotation, a director governing the direction of interparticle forces and a phase field delineating density variation, are therefore introduced. Total work done involving these two order parameters for a particle is attained on an orientation average. Based on the formulation of free energy, a concentration- and anisotropy-dependent formulation for static quantities (stress and couple stress) in the rate form is derived in light of the principles of thermodynamics. It is consistent with the requirement of observer independence and material symmetry. The governing equations for two order parameters are derived, in which void concentration and stress anisotropy are related to relative displacement and rotation. As an example, the proposed model is applied to the hardening regime of deformation of a dense particle assembly with initial perfect lattice under simple shear. It is demonstrated that in the presence of dilatancy and director variation, there exists a linear relation between the shear stress and strain, in coincidence with experimental observations.     

A micromechanics-based Cosserat-type model for dense particulate solids

A two-dimensional continuum theory is presented for cohesionless granular media consisting of identical rigid disks. While the normal deformation of contacting particles is constrained, the tangential frictional contact is modelled by a line spring with a constant stiffness. To describe the static frictional system transmitting couples at contacts, a Cosserat-type continuum including rotational degrees of freedom is appropriate. Contrary to the classical elastic medium, movement of particles within a granular system in response to applied loads can give rise to localisations of force chains and large voids. In addition to relative displacement and rotation, a director governing the direction of interparticle forces and a phase field delineating density variation, are therefore introduced. Total work done involving these two order parameters for a particle is attained on an orientation average. Based on the formulation of free energy, a concentration- and anisotropy-dependent formulation for static quantities (stress and couple stress) in the rate form is derived in light of the principles of thermodynamics. It is consistent with the requirement of observer independence and material symmetry. The governing equations for two order parameters are derived, in which void concentration and stress anisotropy are related to relative displacement and rotation. As an example, the proposed model is applied to the hardening regime of deformation of a dense particle assembly with initial perfect lattice under simple shear. It is demonstrated that in the presence of dilatancy and director variation, there exists a linear relation between the shear stress and strain, in coincidence with experimental observations. Received: February 24, 2005     

Simulation of powder metallurgical coating by radial axial ring rolling

The process integrated powder coating by radial axial rolling of rings is expected to provide a new hybrid production technique to apply different kinds of powder metallurgical functional surfaces to ring–shaped work pieces. The main advantages compared with conventional manufacturing processes in this field can be found in lower costs, shorter process cycles and larger producible work pieces. In order to meet the requirements for an industrial application of this new process it is important to proof its capability particularly with regard to reproducibility and to investigate its boundaries. A reliable process simulation will provide a deeper insight into the governing parameters and reduce the money and time consuming experimental tests. Considering a numerical simulation using the FE method two challenges can be named. First one requires a material model to describe the compaction of metal powder at different elevated temperatures. As second the simulation of the ring rolling process itself is still very time consuming. A fine spatial discretization due to large deformations in the rolling gap and contact interaction between workpiece and rollers are the most prevailing factors in this context. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of PID Control Unit for System Stabilization in Process-integrated Powder Coating by Radial Axial Rolling of Rings

The process integrated powder coating by radial axial rolling of rings represents a new hybrid production technique in order to apply the functional layers on ring-shaped work pieces. Since the layer is produced in a powder metallurgical way [1], the ring volume decreases during the compaction of the layer material. In conventional ring rolling processes an isochoric plastic deformation of the ring is exploited in order to control the process. However this is not true any more for a ring exhibiting a compressible layer [2]. Consequently different control mechanisms have to be developed for the new considered process. One major aspect is the stability of the process which is governed by a stable position of the ring as well as the roundness of the ring. Therefore the finite element (FE) model has been coupled with a PID-controller unit and it will be shown that a stable process can be reached in this way. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulating solid state sintering and compaction of metal powder in the finite strain regime

In many industrial applications (e.g. mills for crushing mineral goods, bearing rings) large ring-shaped workpieces with thick outer wear resistant layers can be found. The powder metallurgical production of these layers on a substrate ring using the hot isostatic pressing technology (HIP) has become the most prevailing production process in this field. Nevertheless using the HIP technology shows some important drawbacks like long process times, high logistic costs and maximum ring diameters of about 1.5 meters. A remedy to these problems is expected from the process integrated powder coating through radial axial rolling of rings. Herein, HIP is replaced by integrating the compaction and sintering of the metal powder into the rolling process of the substrate ring. For this purpose a constitutive material model is introduced in the following, which is able to describe sintering and compaction of metal powder in the finite strain regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled SPH-DEM model for micro-scale structural deformations of plant cells during drying

A single plant cell was modeled with smoothed particle hydrodynamics (SPH) and a discrete element method (DEM) to study the basic micromechanics that govern the cellular structural deformations during drying. This two-dimensional particle-based model consists of two components: a cell fluid model and a cell wall model. The cell fluid was approximated to a highly viscous Newtonian fluid and modeled with SPH. The cell wall was treated as a stiff semi-permeable solid membrane with visco-elastic properties and modeled as a neo-Hookean solid material using a DEM. Compared to existing meshfree particle-based plant cell models, we have specifically introduced cell wall–fluid attraction forces and cell wall bending stiffness effects to address the critical shrinkage characteristics of the plant cells during drying. Also, a moisture domain-based novel approach was used to simulate drying mechanisms within the particle scheme. The model performance was found to be mainly influenced by the particle resolution, initial gap between the outermost fluid particles and wall particles and number of particles in the SPH influence domain. A higher order smoothing kernel was used with adaptive smoothing length to improve the stability and accuracy of the model. Cell deformations at different states of cell dryness were qualitatively and quantitatively compared with microscopic experimental findings on apple cells and a fairly good agreement was observed with some exceptions. The wall–fluid attraction forces and cell wall bending stiffness were found to be significantly improving the model predictions. A detailed sensitivity analysis was also done to further investigate the influence of wall–fluid attraction forces, cell wall bending stiffness, cell wall stiffness and the particle resolution. This novel meshfree based modeling approach is highly applicable for cellular level deformation studies of plant food materials during drying, which characterize large deformations.     

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)     

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)     

Coupled wetting meniscus model for the mechanism of spontaneous capillary action

Capillarity plays a significant role in many natural and artificial processes, but the mechanism responsible for its dynamics is not completely understood. In this study, we consider capillary flow characteristics and propose a coupled wetting meniscus model for the mechanism of spontaneous capillary action. In this model, capillary action is considered as the dynamic coupling of two interfacial forces, i.e., the wall wetting force at the contact line and the meniscus restoring force on the free interface. The wetting force promotes the motion of the contact line directed toward an equilibrium contact angle, whereas the meniscus restoring force promotes a reduction in the interface curvature, which is more consistent with a 90° contact angle. The competing interaction between these two forces is coupled together via the evolution of the interface shape. The model is then incorporated into a finite volume method for a two-fluid flow with an interface. Capillary flow experiments were performed, including vertical and horizontal flows. Phenomena analysis and data comparisons were conducted to verify the proposed model. According to the results of our study, the model can explain the capillary flow process well and it can be also used to accurately guide capillary flow calculations.     

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.     

An integrated mechanistic-neural network modelling for granular systems

A hybrid neural network model is designed to predict the micro-macroscopic characteristics of particulate systems subjected to shearing. The network is initially trained to understand the micro-mechanical characteristics of particulate assemblies, by feeding the results based on three-dimensional discrete element simulations. Given the physical properties of the individual particles and the packing condition of the particulate assemblies under specified loading conditions, the network thus understands the way contact forces are distributed, the orientation of contact (fabric) networks and the evolution of stress tensor during the mechanical loading. These relationships are regarded as soft sensors. Using the signals received from soft sensors, a mechanistic neural network model is constructed to establish the relationship between the micro-macroscopic characteristics of granular assemblies subjected to shearing. The macroscopic results obtained form this hybrid mechanistic neural network modelling for data that were not part of the training signals, is compared with simulations based on discrete element modelling alone and in general, the agreement is good. The hybrid network responds to their inputs at a high speed and can be regarded as a real-time system for understanding the complex behaviour of particulate systems under mechanical process conditions.     

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)     

Development of a finite element model of metal powder compaction process at elevated temperature

This paper presents the finite element modelling of metal powder compaction process at elevated temperature. In the modelling, the behaviour of powder is assumed to be rate independent thermo-elastoplastic material where the material constitutive laws are derived based on a continuum mechanics approach. The deformation process of metal powder has been described by a large displacement based finite element formulation. The Elliptical Cap yield model has been used to represent the deformation behaviour of the powder mass during the compaction process. This yield model was tested and found to be appropriate to represent the compaction process. The staggered-incremental-iterative solution strategy has been established to solve the non-linearity in the systems of equations. Some numerical simulation results were validated through experimentation, where a good agreement was found between the numerical simulation results and the experimental data.