Models and numerical methods for granular materials
ENPC, 1921 November 2007
Titles and abstracts
Short courses

François CHEVOIR
and JeanNoël ROUX (ENPC LMSGC and LCPC),
Mechanical behaviour of dense noncohesive 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
micromacro 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
Talks

François Bouchut (ENS Paris),
Derivation of a thin layer model for fluid/solid transition
In this work we study the modeling of onedimensional 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 thirdorder 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 nonconsistent 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 splitbottom ring
shear cell geometry show wide shear bands under slow, quasistatic
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 MohrCoulomb 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
nondimensional strainrate  indicating shear softening behavior inside the
wide shear bands.
Keywords: Quasistatic Shear Flow rheology, shear thinning, friction, shear
band, discrete element simulation (DEM), viscosity, yieldstress.

Marica Pelanti (ENS Paris),
A numerical model for twophase shallow granular flows
over variable topography
We study a depthaveraged model of gravitydriven 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 interphase
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 nonflat 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 multibody systems in virtual reality, thirdbody
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
multicontact simulations of collections of deformable bodies (Jean
1999) and the method becomes the socalled 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 multiphysics
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
lowenergy excitations characteristic of dry, disordered granular
matter. In the limit of large stiffnessto 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