Models and numerical methods for granular materials
ENPC, 19-21 November 2007

Titles and abstracts

• François CHEVOIR and Jean-Noël ROUX (ENPC LMSGC and LCPC), Mechanical behaviour of dense non-cohesive granular materials: macroscopic description and microscopic origins
  
  Granular materials can behave like solids or liquids, depending on the mechanical loads they are submitted to.
  
  In these two regimes, we will describe the main macroscopic mechanical behaviour features, and we will briefly overview the available micro-macro models. We will focus on simple materials (sets of spherical particles, interacting one with another by direct contact; we will take into account friction and hence dissipation; we will not consider the case of an interestitial fluid) submitted to simple mechanical loads.
  
  In the first part of the course, we will consider the solid regime. The main rheological features, such as internal friction, dilatancy, critical state, as well as simple elastoplastic approaches, will be recalled. Next, we will present contact laws and, more generally, microscopic modelling at the grain scale. We will define important dimensionless parameters, which determine the state of the material and its behaviour: inertial number, stiffness, friction coefficient. Next, we will present some results of the micromechanical approach that highlight some aspects of quasistatic granular materials rheology: static state characterization (solid fraction, coordination number, fabric and other geometrical quantities); influence of the assembly process; definition and extension of an elastic response domain; stability of a contact network; deformation by reorganization and evolution towards a steady state. Questions related to the localization of the deformation will be briefly addressed.
  
  In the second part, we will consider the fluid regime (this is typically the regime for avalanche flows). We will first describe the main macroscopic properties. We will then show how rheological laws have been understood from the 80s, by means of either a theoretical approach (the kinetic approach in the dilute regime), or discrete numerical simulations (in the dense regime), as a complement to physical experiments. We will begin with homogeneous shear, and we will apply the obtained conclusions to the case of flows on inclined planes. Recent results on 3D flows will also be described.
  
  We will conclude by summarizing the influence of micromechanical parameters of the macroscopic behaviour. We will finally discuss the fluid/solid transition, which has been recently discussed in the literature.



• Bertrand MAURY (Université Paris Sud Orsay), Numerical modelling of dry and wet contacts between rigid bodies
  
  

• François Bouchut (ENS Paris), Derivation of a thin layer model for fluid/solid transition
  
  In this work we study the modeling of one-dimensional avalanche flows made of a thin moving layer over a static base, where the interface between the two can be time dependent. We propose a general thin layer model, obtained by looking for an approximate solution with constant velocity profile to the incompressible Euler equations. This model has an energy dissipation equation that is consistent with the depth integrated energy equation of the Euler system. It has physically relevant steady state solutions, and, for constant slope, it gives a particular exact solution to the incompressible hydrostatic Euler equations. Then, we propose a simplified model, for which the energy conservation holds only up to third-order terms. Its associated eigenvalues depend on the mass exchange velocity between the static and moving layers. We show that a simplification used in some previously proposed models gives a non-consistent energy equation. Our models do not use, nor provide, any equation for the moving interface (i.e. for the fluid/solid transition law), thus other arguments have to be used in order to close the system.



• Denis Duhamel (ENPC LAMI, Marne la Vallée), Vibration of granular materials
  
  When granular materials are submitted to vibrations, several phenomena can occur, depending on the intensity of these vibrations. For low frequency or low amplitude, the material behaves as a solid. When submitted to large accelerations (of the order of gravity), some important reorganization takes place. The talk will present experiments and numerical computations results related to these questions. The application we have in mind is the behaviour of the ballast of railway tracks used for high speed trains.



• Chay Goldenberg (ESPCI Paris), Systematic coarse graining: from particle dynamics to continuum mechanics
  
  I will present a systematic spatial averaging (coarse graining) method which relates particle dynamics (as obtained, e.g., from simulations or experimental particle tracking data) and a corresponding continuum description. It provides expressions, valid for any averaging scale, for macroscopic fields (such as the stress and strain tensors) in terms of microscopic quantities. The method therefore enables a systematic study of the scale dependence of continuum fields and constitutive relations, as well as the characterization of inhomogeneities and fluctuations. Specific applications to simulations and experiments on granular materials, as well as simulations of model glasses, will be presented.



• Stefan Luding (Twente Univ., the Netherlands), From particles simulations to constitutive laws for granular flow
  
  Frictional and frictionless granular materials in a split-bottom ring shear cell geometry show wide shear bands under slow, quasi-static deformation. Here, the differences between frictional and frictionless materials are elaborated using discrete element simulations (DEM). Several continuum fields like the density, the velocity field, the deformation gradient and the stress are evaluated and used here for comparison.
  
  Interestingly, the shear stress intensity, i.e., the shear stress divided by the pressure, is approximately constant throughout the wide shear band, as long as the strain rate is large enough - indicating a Mohr-Coulomb type yield stress fluid. The "viscosity", i.e., the shear stress divided by the strain rate, is proportional to the pressure - which is proportional to the contact number density. Furthermore, the viscosity is inversely proportional to the non-dimensional strain-rate - indicating shear softening behavior inside the wide shear bands.
  
  Keywords: Quasi-static Shear Flow rheology, shear thinning, friction, shear band, discrete element simulation (DEM), viscosity, yield-stress.



• Marica Pelanti (ENS Paris), A numerical model for two-phase shallow granular flows over variable topography
  
  We study a depth-averaged model of gravity-driven flows made of solid granular material and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid components and interstitial fluid.
  
  The model system consists of mass and momentum balance equations for the solid and fluid constituents, and it includes inter-phase drag effects. The system can be shown to be hyperbolic at least for flow regimes characterized by sufficiently small phase velocity differences.
  
  We numerically solve the model equations in one dimension by a finite volume scheme based on an approximate Riemann solver. Several numerical experiments are presented, including problems of perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.



• Mathieu Renouf (INSA Lyon), Non Smooth approaches for the simulation of divided media
  
  Discrete Element Methods appear as the most appropriate tool to represent the evolution of a media considered as a collection of particles. Rigid multi-body systems in virtual reality, third-body particle flows in rheology or granular material in soil mechanics or geophysics refer to the same concept of particle systems. This diversity of application fields and related topics motivate the permanent improvements of DEMs.
  
  Their developments start with the pioneer works of Cundall (1971) who developed the Distinct Element Method. Initially used to simulate rock systems, the method is extended to the simulation of granular media. Contact interactions are described by compliant model related to an admissible numerical penetration. Then, improvements of the method are proposed by many authors.
  
  The Contact Dynamics method initially developed by Moreau (1988) based on the convex analysis framework appears as a different approach. Contrary to compliant models, no regularisation scheme is used to describe particle interactions: the non smooth contact feature is preserved according to an implicit formulation of the global contact problem solved classically using a projected block splitting algorithm. Further works lead to the extension of the method to multi-contact simulations of collections of deformable bodies (Jean 1999) and the method becomes the so-called Non Smooth Contact Dynamics method (NSCD).
  
  After an overview of the initial frame, different improvements of the method will be presented as well as some extension to multi-physics applications.



• Nicolas Rivier (IPCMS, Université de Strasbourg), Stability and jamming transition in hard granular materials: Algebraic graph theory
  
  Dry granular matter is modelled as a graph of grains linked by purely repulsive contacts. Its stability (jamming) is insured by odd circuits that prevent the grains from rolling on each other. A topological dynamical matrix is associated with the graph; it has a spectrum of low-energy excitations characteristic of dry, disordered granular matter. In the limit of large stiffness-to load ratio, dry granular matter has two possible dynamical states, dry fluid and jammed, rigid but fragile solid.



• Fabrice Toussaint (Lafarge, Pole technologique), Concrete: a challenging material for granular packing and granular flows
  
  Concrete is a highly multiscale granular material with particles ranging from centimetres to hundreds of nanometres. Each scale is affected by specific phenomena, ranging from colloidal interactions, hydrodynamic, friction or collisions.
  
  Concrete mix designers try to get specific properties and generally want to reach good mechanical compressive strength with a small amount of cement. One method is to decrease water dosage in order to get strong hydrates. On the contrary, one could improve flowing capability of fresh concrete by increasing water dosage. Granular packing is a key to optimize both properties.
  
  At the aggregate scale, the optimization procedure leads to decrease the amount of cement paste which operates as a glue. Looking at scales of microns, say cement paste, improvement of packing in connection with defloculation leads to better mechanical properties of the glue without affecting rheology.
  
  Different tools have been developed using previous theoretical works. Global modelling of concrete rheology remains a challenge.



• Dietrich Wolf (Duisburg, Germany), Properties of shear zones

Last update: November 7th, 2007