Rheophysicsof athermal granular

materialsP. Mills, J.-N. Roux & F. Chevoir

IMA Conference on Dense Granular FlowsCambridge - July 2013

Assembly of non brownian hard particles (d>>1 m)- Dry granular materials- Dense suspensions of particles in a viscous fluid

PressureP

Solid fraction

Shear ratediameter d mass density p

liquid viscosity lfriction coefficient g

effective friction μ= /P

* * defined by and (which depend on )g

Homogeneous shear state

Jammed state * = critical state

Shear stress

2/11

pgI d

P

Dry grains Inertial number

Dense suspensions

« Viscous number »l

sI P

Two time scales : shear time and inertial / viscous time

, ,g s g sI

Dimensionless numbers

, g sand as a function of I

3/11

Peyneau 2008 2D

Khamseh 2012 3D exp : Boyer et al. 20114/11

Dry grainsSuspension

s

Is Is

Is IsPeyneau 2008 2D g=0,3

characteristic time

),,( P )0,*,( P

cutting off the shear stresswhile maintaining the normal stress

Relaxationfrom shear state to jammed

state

5/11

0.1 to 0.5a

ad

a

Steady state :

0t * a Equation of

state

*

at

Diffusive flux

dissipative interactions

Time evolution of solid fraction

6/11

a

Relaxation

,rather than consider

where

= * +

P

g s

P

*

* a

Relaxation time

7/11

*

* *

1 ˆ

ˆwith (1 )

a

a

Da Cruz PRE 05

*

*

AI

BI

g Ref

2D 0 0,67 0,52 Peyneau 08

3D 0 0,390,42

0,380,39

Peyneau 08Khamseh 12

2D 0,3 0,87 0,81 Peyneau 08

3D 0,3 0,95 0,86 Khamseh 12

g Ref

2D 0,3 0,4 – 0,5(hmin)

0,4 – 0,2(hmin)

Peyneau 08

3D 0 0,6(hmax)

Peyneau 08

3D 0 0,5 Boyer 11

Influence of friction ! close to ?

Influence of dimension ?

Constitutive law : scaling laws ?

8/11

Depend on I range !

Dry grains Suspensions

* 2 2

*

( )

( )

g

s

g g

s l

P C d g

P C g

2

1

g

s

1( )*

g

1/C BA

Consequence for shear stress

9/11

* 2 2 f rictiona ( )1l ggP C d g

Dry grains

* (0. )5 slP C g

Dense suspensions

2 ( )!g

10/11

* 2 2 40.4f rictionles )s ( ggP C d g

Scaling laws for and ?

Relaxation of solid fraction related to viscosity ?

Microscopic interpretation of :

- equation of state : Boltzmann equation ?

- viscous shear stress ?

- strong influence of friction ?

Conclusions = Questions