Stefan Luding

Jamming transition in granular systems

Recent simulations have predicted that near jamming for collections of spherical particles, there will be a discontinuous increase in the mean contact number Z at a critical volume fraction phi(c). Above phi(c), Z and the pressure P are predicted to increase as power laws in phi-phi(c). In experiments using photoelastic disks we corroborate a rapid increase in Z at phi(c) and power-law behavior above phi(c) for Z and P. Specifically we find a power-law increase as a function of phi-phi(c) for Z-Z(c) with an exponent beta around 0.5, and for P with an exponent psi around 1.1. These exponents are in good agreement with simulations. We also find reasonable agreement with a recent mean-field theory for frictionless particles.

Discrete element simulations of dense packings and heaps made of spherical and non-spherical particles

The discrete element method (DEM) in the simulation of static packings allows one to investigate the behavior of granular materials by modeling the forces on the particle level. No macroscopic parameters like the angle of repose enter the simulation, but they can be extracted as a result of the particle properties like friction, roughness or shape. One of the issues of static packings recently discussed is the stress distribution under granular heaps. This problem is used to highlight the possibilities of modeling at the particle level using DEM. Phenomena like arching or stress-chains are observed even for spherical particles in a regular pile in the absence of friction if the bottom is rough. The situation does not change much if polygonal, frictional particles are used without disturbing the regular piling. For more realistic situations, when the pile is built by pouring grains from above, the packing and the stresses are influenced by the creation history. The more eccentric the polygons are, the more pronounced a dip is observed in the vertical stress under the apex of the sand-pile.

Reverse Brazil Nut Problem: Competition between Percolation and Condensation

In the Brazil nut problem (BNP), hard spheres with larger diameters rise to the top. There are various explanations (percolation, reorganization, convection), but a broad understanding or control of this effect is by no means achieved. A theory is presented for the crossover from BNP to the reverse Brazil nut problem based on a competition between the percolation effect and the condensation of hard spheres. The crossover condition is determined, and theoretical predictions are compared to molecular dynamics simulations in two and three dimensions.

Footprints in Sand: The Response of a Granular Material to Local Perturbations

We experimentally determine ensemble-averaged responses of granular packings to point forces, and we compare these results to recent models for force propagation in a granular material. We use 2D granular arrays consisting of photoelastic particles: either disks or pentagons, thus spanning the range from ordered to disordered packings. A key finding is that spatial ordering of the particles is a key factor in the force response. Ordered packings have a propagative component that does not occur in disordered packings.

Continuous and discontinuous modelling of cohesive-frictional materials

Computational models for failure in cohesive-frictional materials with stochastically distributed imperfections.- Modeling of localized damage and fracture in quasibrittle materials.- Microplane modelling and particle modelling of cohesive-frictional materials.- Short-term creep of shotcrete - thermochemoplastic material modelling and nonlinear analysis of a laboratory test and of a NATM excavation by the Finite Element Method.- Thermo-poro-mechanics of rapid fault shearing.- A view on the variational setting of micropolar continua.- Macromodelling of softening in non-cohesive soils.- An experimental investigation of the relationships between grain size distribution and shear banding in sand.- Micromechanics of the elastic behaviour of granular materials.- On sticky-sphere assemblies.- Cohesive granular texture.- Micro-mechanisms of deformation in granular materials: experiments and numerical results.- Scaling properties of granular materials.- Discrete and continuum modelling of granular materials.- Difficulties and limitation of statistical homogenization in granular materials.- From discontinuous models towards a continuum description.- From solids to granulates - Discrete element simulations of fracture and fragmentation processes in geomaterials.- Microscopic modelling of granular materials taking into account particle rotations.- Microstructured materials: local constitutive equation with internal lenght, theoretical and numerical studies.- Damage in a composite material under combined mechanical and hygral load.

Role of anisotropy in the elastoplastic response of a polygonal packing

The effect of the anisotropy on the elastoplastic response of two dimensional packed samples of polygons is investigated here, using molecular dynamics simulation. We show a correlation between fabric coefficients, characterizing the anisotropy of the granular skeleton, and the anisotropy of the elastic response. We also study the anisotropy induced by shearing on the subnetwork of the sliding contacts. This anisotropy provides an explanation to some features of the plastic deformation of granular media.

STRESS DISTRIBUTION IN STATIC TWO-DIMENSIONAL GRANULAR MODEL MEDIA IN THE ABSENCE OF FRICTION

We present simulations of static model sandpiles in two dimensions (2D), and focus on the stress distribution in such arrays made of discrete particles. We use the simplest possible model, i.e., spherical particles with a linear spring and a linear dashpot active on contact and without any frictional forces. Our model is able to reproduce several recent theoretical predictions. For different boundary conditions we examine the contact network and the stresses in the array and at the bottom of the pile. In some cases we observe a dip, i.e., the relative minimum in pressure, under the center of the pile. We connect the dip to arching, and we relate arching to the structure of the contact network. Finally, we find that small polydispersity is sufficient to cause a so called stress network, i.e., strong fluctuations in stress. From these data we determine the probability distribution for the vertical stress at the bottom, and relate it to theoretical and other numerical work.

Collisions & Contacts between Two Particles

The alternative to a continuum model of granular media (see other chapters in this book) is to view the material as a collection of discrete particles. In order to simplify the description, we assume the particles to be spheres in the following. For the characterization of a system with many particles we specify only two-particle interactions, assuming many-body interactions to result from the sum of the two-particle forces. The scope of this chapter is to give a summary of frequently used approaches and to compare them.

Anisotropy in cohesive, frictional granular media

The modelling of cohesive, frictional granular materials with a discrete particle molecular dynamics is reviewed. From the structure of the quasi-static granular solid, the fabric, stress, and stiffness tensors are determined, including both normal and tangential forces. The influence of the material properties on the flow behaviour is also reported, including relations between the microscopic attractive force and the macroscopic cohesion as well as the dependence of the macroscopic friction on the microscopic contact friction coefficient. Related to the dynamics, the anisotropy of both structure and stress are exponentially approaching the maximum.

Modeling of particle size segregation: Calibration using the discrete particle method

Over the last 25 years a lot of work has been undertaken on constructing continuum models for segregation of particles of different sizes. We focus on one model that is designed to predict segregation and remixing of two differently sized particle species. This model contains two dimensionless parameters, which in general depend on both the flow and particle properties. One of the weaknesses of the model is that these dependencies are not predicted; these have to be determined by either experiments or simulations. We present steady-state simulations using the discrete particle method (DPM) for bi-disperse systems with different size ratios. The aim is to determine one parameter in the continuum model, i.e., the segregation Peclet number (ratio of the segregation velocity to diffusion) as a function of the particle size ratio. Reasonable agreement is found; but, also measurable discrepancies are reported; mainly, in the simulations a thick pure phase of large particles is formed at the top of the flow. In the DPM contact model, tangential dissipation was required to obtain strong segregation and steady states. Additionally, it was found that the Peclet number increases linearly with the size ratio for low values, but saturates to a value of approximately 7.35.