Granular Matter - Paper

This setup shows the behavior of granular material in a vibrating box. Because of the small wall, the system can break symmetry.

Not quite a hopper, but a hole in the floor.

A Novel Approach to the Simulation of Particles on a Large Size-Range.

A very tiny simulation. 36 balls are being thrown into a container. The rolling of the particles leads to a final configuration, which is virtually flat. 

By means of molecular dynamic simulation, each individual ballast stone in the ballast track can be simulated in the model.  It is investigated to what extent it is possible to define contact force laws, material parameters and geometry variables in such a way that the resulting computational model reproduces the essential behaviour of the real system. Furthermore, possible areas of application of molecular dynamic simulation are mentioned.

The walls of the hopper are not as steep as in the other examples. The flow ends when an arch forms spontaneously and the hopper is blocked.

We have a hopper, filled with particles. Additional to the repulsive forces due to collisions we have cohesive forces. You can find more information in this publications.

A piston is compressing with a certain force onto an assembly of granular materials. No gravity is acting. The Material is very soft, so there are oscillations.

Here a sand pile is build from a line source to demonstrate the importance of static friction in sandheaps. At time step 420 the friction coefficient is quickly reduced to zero. The pile melts and the granulate flows like a liquid. 

The time evolution of the force network of a sand pile which is built from a point source is shown. The width of the blue lines is proportional to the strength of the forces. The red arrows are the forces onto the ground. Note that the avalanches either go left or right and therefore lead always to slight asymmetries of the heap.

For the simulation, we have calculated the stress tensors inside the pile. In this movie, the major principal axis of the stress (crosses) and the pressure on the ground (arrows) are shown. 

This heap is build using 4500 slightly elongated Particles. The first frames shows the evolution of the heap, the bottom frame shows the pressure onto the ground. You can observe the evolution of a dip below the apex of the pile

This heap is the same as the other examples. In the top frame you can see the stress tensors inside of the pile. The second frame shows the forcenetwork; small forces are black, medium forces are red and strong forces are yellow. The 8 small frames show the axis of the major principal axis in the different layers of the pile. The last frame shows the pressure distribution.

A small movie, showing a tinkertoy-setup involving dominos and a seesaw. This was produced for the open day of the university 1998 and aims to impress non-scientists. It also demonstrates the versatility of our program.

The discrete element method allows the simulation of complex behavior of granular materials without constitutive laws. While in two dimensions shape-effects are well established, in three dimensions there is no universally applicable simulation algorithm for non-spherical particles. We will first present a force model for convex polyhedral particles, using the “overlap” of non-deformed polyhedra as a “measure” of the elastic force and explain the overlap computation algorithm.

This work is a contribution to the understanding of the mechanical properties of non-cohesive granular materials in the presence of friction and a continuation of our previous work (Roul et al. 2010) on numerical investigation of the macroscopic mechanical properties of sand piles. Besides previous numerical results obtained for sand piles that were poured from a localized source (‘‘point source’’), we here consider sand piles that were built by adopting a ‘‘line source’’ or ‘‘raining procedure’’.

The averaged stress and strain response functions of granular aggregates are investigated numerically. We use the discrete-element method (DEM) to generate granular packings consisting of soft convex polygonal particles, i.e., the simulation geometry is two-dimensional. Packings are prepared in a rectangular container. To determine the stress response of a packing, we apply an external load to a single grain from the top layer of the assembly, with a force small enough not to cause structural rearrangements.

The findings on the pressure distribution under conical piles has increased the interest in the stress distributions in static granular packings.

Since stresses inside granular systems are very difficult to access experimentally, we have performed DEM (Discret Element Model) simulations with polygonal particles. The shape and size of the particles can be specified arbitrarily in the simulation; the influence of these parameters can be determined.

The motion of a sliding particle, influenced by friction, in a rotating drum is investigated. A differential equation is formulated for general friction laws. Assuming a constant coefficient of friction, the equation is exactly solvable. For a velocity dependent coefficient of friction, perturbation methods may be used. The nonperturbed system is solved and with the help of the averaging method, the perturbed system can be examined for periodic motions.

The motion of a sliding particle, influenced by friction, in a rotating drum is investigated. A differential equation is formulated for general friction laws. Assuming a constant coefficient of friction, the equation is exactly solvable. For a velocity dependent coefficient of friction, perturbation methods may be used. The nonperturbed system is solved and with the help of the averaging method, the perturbed system can be examined for periodic motions.

Using the averaging method we

• test different friction laws
• search for periodic orbits
• investigate the structure of the phase space

At the present we are interested in better understanding the influence of friction in simulations for granular systems. The far goal is a 3-dimensional simulation for non-spherical particles.

The effect of the particle shape on the bulk-stress-strain-relations for triaxial compression of granular media is investigated via the molecular dynamics method. It is found that crucial properties exhibited by experimental granular media cannot be reproduced by round particle simulations, but only by the use of elongated particles.

The most time consuming part in molecular dynamics simulations is the collision detection. Usually, this problem is solved by restricting theshape of the particles to spheres. I will present an algorithm, originally developed for virtual reality visualizations by D.Baraff and M.C.Lin, that enables us to use complex polyhedra  (up to 920 faces and more). The expected run time is  O(N), where N is the number of particles in the simulation. Neither complexity nor shape of the particles affect the run time.

Granular materials, of which sand is the most prominent representative, are important in many fields of research. Their special properties make them important both for industrial applications and as a field of work in basic research. The present work deals with the numerical investigation of granular materials. The size scale of typical granular particles starts in the micrometre range for fine dusts. The upper limit is approximately in the range of a few kilometres particle diameter for the boulders in the rings of Saturn.

We investigate the effective material properties of sand piles of soft convex polygonal particles numerically using the discrete element method (DEM). We first construct two types of sand piles by two different procedures. We then measure averaged stress and strain, thelatter via imposing a 10% reduction of gravity, as well as the fabric tensor. Furthermore, we compare the vertical normal strain tensor between sand piles qualitatively and show how the construction history of the piles affects their strain distribution as well as the stress distribution.

Three algorithms to speed up discrete-element simulations for granular matter are presented in this paper. The first algorithm allows to determine neighborhood relations in polydisperse mixtures of particles of arbitrary shape, either discs, ellipses, or polygons. The second algorithm allows to calculate the distance of two polygons in constant time, independently of the complexity of the shape of the polygons. This makes fast simulations of polygonal assemblies possible.

Penguins, penguins, everywhere are pinguins... A movie for the former supercomputer Tina.

We have a hopper, filled with about 1300 particles. We can see the funnel-flow of the grains. At the end, two particles on the upper left are stuck.

A short movie showing pattern formation. This was produced for the ICA1 of the University of Stuttgart.

Now we have the same geometry as in the other funnel, but the shape of the particles was initialized slightly different. The hopper is not blocked.

The Galton Board is a device to explain binomial distributions. It consists of a board that has a large amount of pins fixed to it. These pins are arranged in regular horizontal rows so that the pins form a triangle with its base at the bottom of the board.

The pressure distribution under heaps has found to be dependent on the building history of the heap both in experiments and in simulations. Up to now, theoretical models and analysis assume that the packing of the heap is homogeneous. We show new experimental and simulational results which indicate that the packing is inhomogeneous and that this packing property is likely causing the pressure minimum under the heap.

This film shows the collapse of a house of cards. The setup file was created with the help of xfig. Without the implementation of a Coulomb/static friction law, the initial configuration would not be stable.

We investigate the effect of the geometry of granular heaps on the pressure distribution. For given pressure distributions under cones we compute the pressure distribution under wedges using linear superposition. For cones with a pressure minimum, the pressure minimum for the corresponding wedge vanishes. Comparisons with experimental data gives good qualitative agreement, but the total pressure is overestimated.

I want to introduce two algorithms, useful for fast collision detection in granular medium simulations. We all know that the most time consuming part in molecular dynamics simulations is the collision detection. Usually, this problem can be solved by restricting the shape of the particles to spheres. But if you want to use arbitrary convex polygons you need faster algorithms. 

We numerically investigate the effective material properties of aggregates consisting of soft convex polygonal particles, using the discrete element method. First, we construct two types of “sand piles” by two different procedures. Then we measure the averaged stress and strain, the latter via imposing a 10% reduction of gravity, as well as the fabric tensor. Furthermore, we compare the vertical normal strain tensor between sand piles qualitatively and show how the construction history of the piles affects their strain distribution as well as the stress distribution.

We use a discrete element method to simulate the dynamics of granulates made up from arbitrarily shaped particles. Static and dynamic friction are accounted for in our force laws, which enables us to simulate the relaxation of (two-dimensional) sand piles to their final static state. Depending on the growth history, a dip in the pressure under a heap may or may not appear. Properties of the relaxed state are measured and averaged numerically to obtain the values of field quantitities pertinent for a continuum description.

We are dealing with friction from the viewpoint of granular material research, where heaps can maintain their shape only in the presence of Coulomb friction.

The experimental motivation for this study are recent publications on cohesive granular materials. Our central question is, in which regime and by which mechanism the the movement of grains changes from movement of independent particles to a movement of small clusters with increasing cohesion. Cohesion introduces an additional length scale, so that the effects become size-dependent. The cohesive force acting on a volume element of size I x I x I is proportional to its surface, or ∝ I². The repulsive force generated by the mass of the volume element is ∝ I³.

The oscillation behaviour of a sliding particle under dry friction in a vertical rotating drum is investigated theoretically. A differential equation is set up for general friction laws. For constant friction coefficients, the equation can be solved exactly. For velocity-dependent friction, it can be treated perturbation theoretically. The unperturbed system is solved and with the help of the averaging method, the perturbed system can then be examined for periodic movements. 

This  example shows a hopper with a small outlet and a few large particles. The hopper blocks twice. The clogging is removed by tapping the hopper, so that all particles are accelerated upwards.

Sand is one of the least noticed but almost ubiquitous things in our environment. In physics, too, "simple" sand, or more precisely "granular media", is still a little-studied field. That this is so is due to the complexity hidden in the apparent "simplicity". An analytical solution is self-evident, because the nature of the interaction, friction processes and number of particles form an insurmountable obstacle. But even statistical physics does not yet offer any explanatory models for a pile of sand.

In general, collisions makes granular materials most interesting. For this movie, I tried to avoid collisions!

We present two-dimensional molecular dynamics simulations of cohesive regular polygonal particles. The cohesive part of the force-law for the particleparticle interaction is validated by the agreement with existing experimental data. We investigate microscopic parameters which are not accessible to experiments such as contact length, raggedness of the surface and correlation time. With increasing cohesion the particles move in clusters for long times.

We investigate the stresses and pressures under a 2-dimensional heap using a simulation of convex polygonal particles. Former Experiments and simulations on granular cones strongly suggest that for cones no generic pressure distribution exists but that the pressure and stress distribution is highly sensitive to the size distribution of the grains and the building history of the heap.

A rotating drum with a mixture of many small and a few large particles.

A rotating drum with a single particle. You can find more information in this publications.

In order to understand the peculiar behavior of granular matter, it is oftenelucidating to observe the physics of only a few grains. We present twosetups which fall into this class: The motion of a single particle in arotating drum, and the collective behavior of a few particles under theinfluence of a swirling m

In order to understand the peculiar behavior of granular matter, it is often elucidating to observe the physics of only a few grains. We present two setups which fall into this class: The motion of a single particle in a rotating drum, and the collective behavior of a few particles under the influence of a swirling motion.

A short talk on oscillations with dry friction.

The problem of granular materials is not alone a problem of material properties, but also a problem of structures. To examine these  interesting systems, one uses molecular dynamics simulations. The objective of the work presented here was to have a program which can run on cheap high-end shared memory workstations. Therefore we have developed  a fast thread-based simulation of polygonal particles.

The problem of granular materials is not alone a problem of material properties, but also a problem of structures. To examine these  interesting systems, one uses molecular dynamics simulations. The objective of the work presented here was to have a program which can run on cheap high-end shared memory workstations. Therefore we have developed  a fast thread-based simulation of polygonal particles.

We study the averaged macroscopic strain tensor for a sand pile consisting of soft convex polygonal particles numerically, using the discrete-element method (DEM). First, we construct two types of “sand piles” by two different pouring protocols. Afterwards, we deform the sand piles, relaxing them under a 10% reduction of gravity. Four different types of methods, three best-fit strains and a derivative strain, are adopted for determining the strain distribution under a sand pile. The results of four different versions of strains obtained from DEM simulation are compared with each other.

We investigate numerically the micro and macro mechanical behaviour of non-cohesive granular materials, especially in the static limit. To achieve this goal we performed numerical simulations generating twodimensional “sand piles” from several thousands of convex polygonal particles with varying shapes, sizes and corner numbers, using a discrete element approach based on soft particles. We emphasize that the displacement (strain) fields inside sand piles have not been measured in experiments on sand piles.

We are interested in the stress distribution in static granular matter. Experiments have found a minimum of the vertical normal stress beneath the apex of a sandpile. Because of the indeterminacy of static friction force even in the simplest sandpile and the ensuing absence of a constitutive relation between stress and strain (Hooke's law) there is no closed set of equations.

We show how Differential Algebraic Systems (Ordinary Differential Equations with algebraic constraints) in mechanics are affected by stability issues and we implement Lubich’s projection method to reduce the error to practically zero. Then, we explain how the “numerically exact” implementation for static friction by Differential Algebraic Systems can be stabilized. We conclude by comparing the corresponding steps in the “Contact mechanics” introduced by Moreau.

Granular matter hides a very complex behaviour behind its apparent simplicity ("... is just sand"). Typical properties of granulates are, for example, the discrete structure and the inhomogeneity. This leads to the fact that backfills far away from thermal equilibrium can be very "stable" after all. The question now arises as to what consequences this has for the behaviour of sand accumulations.

Granular media conceal a very complex behaviour behind their apparent simplicity ("... is just sand"). Typical properties of granulates are, for example, the discrete structure and the inhomogeneity. This leads to the fact that backfills far away from thermal equilibrium can be very "stable" after all. The question now arises as to what consequences this has for the behaviour of sand accumulations.

Die Neigung dieses Trichters ist viel steiler als bei den anderen Beispielen. Wir sehen hier das Massenflussregime, bei dem die Partikel wie eine Flüssigkeit durch den Auslass strömen.

We are interested in the stress distribution in static granular matter. Experiments have found a minimum of the vertical normal stress beneath the apex of a sandpile. Because of the indeterminacy of static friction force even in the simplest sandpile and the ensuing absence of a constitutive relation between stress and strain (Hooke's law) there is no closed set of equations. Continuum theories, trying to describe the dip, have to make assumptions on the existence of constitutive relations among the components of the stress tensor itself.

• What is the pressure distribution below sandpiles? → DIP
• How can I get information from the inside ?
• What can be stated about continuum theories now ?

We are interested in the stress distribution in static granular matter. Experiments have found a minimum of the vertical normal stress beneath the apex of a sandpile.

Because of the indeterminacy of static friction force even in the simplest sandpile and the ensuing absence of a constitutive relation between stress and strain (Hooke's law) there is no closed set of equations. Continuum theories, trying to describe the "dip", have to make assumptions on the existence of constitutive relations among the components of the stress tensor itself.

A Poster on stress propagation in sand beds.

The pressure distribution under heaps has found to be dependent on the builing hostory of the heap both in experiments and simulations. Up to now, theoretical models and analysis assume that the packing of the heap is homogeneous. We show new experimental and simulational results which indicate that the packing is inhomogeneous and that this packing property is likley causing the pressure minimum under the heap.

Ein kleiner Film, der die Musterbildung zeigt. Er wurde für den Tag der offenen Tür der Universität 1999 produziert und soll Nicht-Wissenschaftler beeindrucken. Der Kopf, der sich bildet, ist das Porträt von Otto von Guericke, Bürgermeister von Magdeburg, der als erster die Wirkung des Luftdrucks über 2 evakuierte Halbkugeln aus Eisen demonstrierte. Das Porträt ist das Siegel der Universität Magdeburg. Es verdeutlicht auch die Vielseitigkeit unseres Programms.