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MECHANICAL BEHAVIOUR OF DENSE NON-COHESIVE
GRANULAR MATERIALS MACROSCOPIC DESCRIPTION AND
MICROSCOPIC ORIGINS II. FLOW AND JAMMING
François CHEVOIR and Jean-Noël ROUX Université
Paris-EST - INSTITUT NAVIER
GDR CHANT Models and Numerical methods for
Granular Media 20 November 2007
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Various Situations
Propagation Jamming Mixing ...
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Simplifications

• Quasi-monodispersed, spherical grains
• Dry frictional grains (direct contacts)
• no interstitial fluid, no cohesion
• Simple shear geometry

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Example Flow down an Incline
Start
Propagation
Stop
Hutter et al. 90s
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Questions

• Flow regimes
• liquid or gas ?
• dependency of flow rate on
• - inclination, height
• - roughness of the wall
• - characteristics of the grains
• gt Constitutive law ?
• Flow thresholds
• gt Fluid - Solid transition

MACROSCOPIC DESCRIPTION AND MICROSCOPIC ORIGINS
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Plan
1. Macroscopic rheology
2. Constitutive law for granular flows
3. Application to inclined plane flow
4. Generalizations
5. Fluid-Solid transition
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1. Macroscopic Rheology
Continuous media
g
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Macroscopic Description
Continuous media
g
Conservation equations
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Simple shear 2D flow
y
Shear rate
x
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Simple shear 2D flow
y
g
q
x
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Mass conservation
Automatically satisfied !
Momentum conservation
y
q
g
P, t
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Constitutive Law
Shear stress
Shear rate
Newtonian
h
Viscosity h
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Viscoplastic Behavior Yield stress fluid
(Bingham)
t0
Complex fluids colloïdal or granular
suspensions, foams, emulsions
Coussot
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Consequences of the yield stress Flow down an
inclined plane
Newtonian
Bingham
y
H
g
q
t(y)
Plug flow
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2. Constitutive law for granular flows
(Steady flow)
Relations between
Shear stress t
Pressure P
Shear stress
Solid fraction F
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Quasi-Static Deformation (J-N. Roux course)
t
P
g
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Constitutive Law
First guess visco-plastic with influence of
the pressure
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Factors influencing the Viscosity ?
Microstructure gt Solid fraction
Dilute
Dense
Azanza PhD98
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Solid fraction F
d
s
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Strong agitation
Binary Collisions gt Velocity fluctuations
 granular temperature
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Kinetic theory of dense gases
Uncorrelated binary collisions, small dissipation
Jenkins, Savage, Lun, Haff 80s
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Interpretation
s
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Dense collisionnal flows
F
Fm
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Determination of w ?
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 Heat flux 
From  hot  areas ( agitated strong w) to
 cold  areas ( cool small w)
Analogous to Fouriers law
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Energy dissipation
restitution coefficient e
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Determination of w ?
2D steady and uniform flow
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Determination of w ?
2D steady and uniform flow
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Homogeneous shear
Energy balance
Work of the stress Dissipation
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Inhomogeneous shear (vicinity of a wall)
Plane shear without gravity P,t cte
 Hot  wall Dilatation Shear localization
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Dense Flows

• Solid fraction close to maximum value
• Strong geometric constraint
• Collective motions of grains
• gt Basic assumptions of kinetic theory not
  relevant

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Time scales
Inertial time
Shear time
Collision time
da Cruz et al. 03
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Flow regimes
Dilation
Quasi-Static Slow deformations ?
elasto-plastic solid
Strong agitation Collisionnal ? Dense gas
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Discrete numerical simulations

• Contact law
• elastic stiffness K
• friction coefficient m
• viscous dissipation (restitution e)

Sollicitations Velocity or stress
Roux and Chevoir BLPC05
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Periodic boundary conditions
With wall
P
V
Controlled pressure and shear rate
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Simulation of dense flows
Measurement of velocity, solid fraction,
stress Average over time and space
da Cruz et al. PRE05
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Solid fraction Law
QS
Dyn.
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Friction law
Coll.
QStat
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Influence of the grain mechanical properties
µ gt 0 Various e

• 0
• e 0.9
• ? e 0.1

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Friction law in 3D (simple shear)
m 0
m
I
Peyneau 07
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Comparison (homogeneous shear)
Dilute Flow
Dense Flow
F
m
Fm
mK
mS
I
I
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3. Application to inclined plane flows
Steady flows
V
?
H
Flow law V(q,H) ?
Experiments and Discrete Simulations
Test of the constitutive law
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Prediction of the constitutive law (measured in
homogenous shear)
F
m
m(I)
F(I)
q
F(q)
I
I
I(q)
I(q)
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Steady flows Kinematics
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Experiment
Extension to large I
m
Pouliquen PF99, JFM05 Azanza PhD98
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4. Generalization other geometries
Confined
Vertical chute
Annular Shear
g
Plane shear
Free surface
Heap
Rotating drum
Experiments and Discrete simulations
GDR MIDI EPJE04
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Stress Heterogeneity
gt Shear localization (around 5 grains)
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Flow down a Heap
Liquid
Solid
Creeping regime exponential localization
Incompatible with m(I)
Komatsu PRL01
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Influence of lateral walls
Flow rate Q
Thin channels Discrete simulations Taberlet et
al. PRL03
Wide channels (W/d -gt 600) Experiment Jop et al.
JFM05
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Internal friction m(I) Wall friction mw
Steady flow
Jop et al. JFM05
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Inclined plane versus heap
(From Pouliquen IHP05)
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3D flow Tensorial formulation of the friction
law
Jop et al. Nat06
Rough lateral walls
3D velocity profile (num. sim.)
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Unsteady, non uniform flows Saint-Venant
approach
L gtgt H
L
depth-averaged conservation equations
cf. F. Bouchut
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Shape of the front
Pouliquen 99
H0
Steady flow
x
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Spreading of a granular mass down a slope
Comparison Experiment Saint Venant simulation
Pouliquen and Forterre JFM02
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5. Fluid-Solid Transition
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Jamming down an inclined plane
m(I)
When q decreases
Progressive slowing
I
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Consequence on the flow law
Pouliquen PF99
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Hysteresis
Flow
Stop
Pouliquen 99, Daerr 00
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Origin of jamming
Mechanism 1 Roughness Mechanism 2
Correlations

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Roughness Single grain model
Douady et al. 2000
1. GRAVITY COLLISIONS
Abrupt and hysteretic transition
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Dynamical system
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Generalization to a layer
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Generalization to a layer
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Correlations motion, forces
Correlation length of velocity fluctuations
Measurements Pouliquen, Rognon Lemaître, Model
s Mills, Ertas,
Lc
(mixing length)
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Initiation of a surface avalanche
q increases
Percolation of sliding contacts areas
Staron and Radjaï PRL02
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Influence of Micromechanical Parameters
Initial State I k (gt 103) e or z m
Assembling Preparation Yes No Yes Yes
Very small deformation (e ? 10-5) Yes No Yes No Yes
Small deformations (e ? 10-2) Yes No (if I ? 10-3) No No (if I ? 10-3) Yes
Critical state (e ? 1) No No (if I ? 10-4 ?) No No (if I ? 10-4 ?) Yes
Dense flows (10-3 lt I ? 0.1) No Yes No No Yes
Coll. flows (I gt 0.1) No Yes No ? Yes Yes
Roux and Chevoir BLPC05
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Perspectives
Influence of grain shape cohesion interstitia
l fluid Behavior near an interface Mixing/Segrega
tion Frontier with the quasi-static
regime Complex geometry Large number of grains
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Reading tips
Proceedings Powders and Grains 2005 Edited by S.
McNamara et al., Balkema. GDR MIDI On dense
granular flows European Journal of Physics E
(2004). Ph. Coussot Rheometry of pastes,
suspensions and granular materials Wiley
(2005). F. Chevoir et al. Ecoulements
granulaires physique et applications, in
Rhéologie des pâtes et des matériaux granulaires,
Coll. Etude et Recherche des Laboratoires des
Ponts et Chaussées (2006). F. da
Cruz Ecoulements de grains secs Frottement et
Blocage PhD ENPC (2004). http//pastel.paristech.o
rg/946/ P. Jop et al. A constitutive law for
dense granular flows Nature (2006).
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THANK YOU FOR YOUR ATTENTION
Ackowledgments
E. AZANZA, F. BERTRAND, A. CORFDIR, Ph. COUSSOT,
F. DA CRUZ, I. IORDANOFF , J. JENKINS, G. KOVAL,
P. MILLS, J-J. MOREAU, P. MOUCHERONT, M.
NAAIM, O. POULIQUEN, M. PROCHNOW, F. RADJAI, P.
ROGNON, D. WOLF