From particle simulations towards continuum theory including the solid-fluid transition in granular materials
COST F LOWING M ATTER
E RLANGEN , 02 M ARCH 2017
F ROM PARTICLE SIMULATIONS TOWARDS CONTINUUM THEORY
INCLUDING THE SOLID - FLUID TRANSITION IN GRANULAR MATERIALS
D. Vescovi* and S. Luding
U NIVERSITY OF T WENTE
*P OLITECNICO DI M ILANO
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
What and why
Granular material: Collection of discrete particles
characterized by a loss of energy whenever the particles interact
Industrial processes
Natural events
handling and processing in pharmaceutical
landslides and debris-flows
chemical, agricultural and mineral industries
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Main features of granular flows
Microscopic discrete nature and macroscopic behavior −→
continuum and discontinuum mechanics
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Main features of granular flows
Microscopic discrete nature and macroscopic behavior −→
continuum and discontinuum mechanics
Different mechanical behavior
Fluid-like
collisions
grains bounce in all directions
creating a dilute chaotic medium
Phase transition
Solid-like
Forterre & Pouliquen, 2008
D. Vescovi & S. Luding
force chains
extremely slow deformations
and high concentrations
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Continuum mechanics for granular systems
mass balance:
Dρ
+ ρ∇ · u = 0
Dt
momentum balance:
ρ
Du
= ∇ · σ + ρg
Dt
Continuum
Particle
diameter d
density
ρp
velocity
v
v
bulk density ρ = ν ρp
concentration ν = Wp /W
velocity u = < v >
stress tensor σ
MICRO
MACRO
−→ constitutive relations for σ
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Discontinuum mechanics for granular systems
All grains i = 1, ..., N can be soft or stiff spheres
Discrete Element Method:
Specify interactions between bodies
−→ deformations are taken into account allowing an overlap δ
Compute all forces Fi
Integrate the Newton’s equations of motion for all particles:
δ
d 2 xi
mi 2 = Fi
dt
ωi
Ft
n
Fn
vj
∀i = 1, · · · , N
xj
grain j
D. Vescovi & S. Luding
vi
ωj
March 2, 2017
grain i
diameter d i
mass m i
moment of inertia I i
xi
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Hypothesis and flow configuration
Lees-Edwards
periodic
boundaries
moving
periodic wall
y
y
y
.

u
ν
x
periodic
boundaries
Homogeneous shear flows
u = (u, 0); ∂/∂x = ∂/∂y = 0
Steady state
=⇒ σ reduces to p = pressure
and s = shear stress
... ignoring non-Newtonian eff.
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Hypothesis and flow configuration
Lees-Edwards
periodic
boundaries
moving
periodic wall
y
y
y
.

u
ν
x
periodic
boundaries
Homogeneous shear flows
Goal: investigate the phase
solid-fluid transition
u = (u, 0); ∂/∂x = ∂/∂y = 0
Steady state
−→ perform DEM simulations
=⇒ σ reduces to p = pressure
and s = shear stress
... ignoring non-Newtonian eff.
D. Vescovi & S. Luding
discontinuum mechanics
−→ derive constitutive relations
March 2, 2017
continuum mechanics
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
DEM numerical simulations
M ERCURY DPM CODE
constant volume condition
V
periodic boundary conditions
in the flow (x) and the transversal (z) directions
Lees-Edwards conditions at the shearing (y) dir.
the data are taken in the steady state
and averaged over x, y , z and time
H
parameters:
y
x
L
d =1
ρp = 1
e = 0.7
H = 20
V =H
(γ̇ = 1)
changing:
volume fraction
particle stiffness
k / ρp d 3 γ̇ 2
linear spring-dashpot model
D. Vescovi & S. Luding
ν
N = 2000
s
πNd 3
L=
6νH
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
DEM numerical simulations
6
10
5
.
p / (p d 2  2 )
10
Mitarai & Nakanishi 2007
Peyneau & Roux 2008
4
10
3
10
2
10
1
10
0
10
0.45
0.50
0.55

0.60  0.65
J
0.70
Rigid particles: p −→ ∞ at the critical volume fraction νJ
(Simulations from Mitarai & Nakanishi 2007, Peyneau & Roux 2008)
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
DEM numerical simulations
6
10
5
Mitarai & Nakanishi 2007
Peyneau & Roux 2008
3
2
3
3
2
4
3
2
5
3
2
6
3
2
7
4
k /( d  ) = 10
p
.
10
3
k /(p d  ) = 10
.
10
.
k /(p d  ) = 10
2
10
k /(p d  ) = 10
.
.
p / (p d 2  2 )
.
10
k /(p d  ) = 10
k= 
1
10
0
10
0.45
0.50
0.55

0.60  0.65
J
0.70
Rigid particles: p −→ ∞ at the critical volume fraction νJ
(Simulations from Mitarai & Nakanishi 2007, Peyneau & Roux 2008)
Soft particles: softness allows the particles to deform at ν > νJ
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
DEM numerical simulations
-1
6
10
10
Mitarai & Nakanishi 2007
Peyneau & Roux 2008
2
3
3
2
4
3
2
5
3
2
6
3
2
7
k /( d  ) = 10
4
p
.
10
.
k /(p d  ) = 10
k /(p d  ) = 10
3
.
10
k /(p d  ) = 10
2
.
.
p / (p d 2  2 )
.
3
k /(p d  ) = 10
k= 
10
-2
10
-3
10
pd/k
5
10
-4
10
-5
10
-6
10
1
10
-7
10
0
-8
10
0.45
0.50
0.55

0.60  0.65
J
0.70
10
0.45
0.50
0.55

0.60  0.65
J
0.70
phase transition at the jamming volume fraction νJ
unjammed phase
ν < νJ
−→
p ∝ γ̇
jammed phase
2
ν > νJ
−→
p ∝ k (ν − νJ )
fluid-like behavior
solid-like behavior
inelastic collisions
network of persistent contacts
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Scaling laws
Pressure
4
10
2
10
p*  / | - J | 6/5
"
p∗ ν
|ν − νJ |
.0
*
0
10
6/5
-2
(
10
mpi
3
k = 10
4
k = 10
5
k = 10
6
k = 10
7
k = 10
-4
10
-6
10
.2
*
-8
10
p∗ =
-10
10
-4
10
-2
10
D. Vescovi & S. Luding
0
. 10
*/ | - J | 9/5
2
10
4
10
=
∼
2
0
#mpi
γ̇ ∗
|ν − νJ |
9/5
unjammed phase
jammed phase
pd
kr
ρp d 3
k
νJ = 0.634
γ̇ ∗ = γ̇
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Scaling laws
Pressure
4
10
2
p*  / | - J | 6/5
10
.0
*
0
10
p∗ =



 p2 (ν − ν )6/5 ,
J
ν
-2
10
3
k = 10
4
k = 10
5
k = 10
6
k = 10
7
k = 10
-4
10
-6
10
.2
*
-8
10
p∗ =
-10
10
-4
10
-2
10
D. Vescovi & S. Luding
0
. 10
*/ | - J | 9/5
2
10

γ̇ ∗2

 p1
, if ν < νJ

ν
12/5
(νJ − ν)
4
10
if ν > νJ
pd
kr
ρp d 3
k
νJ = 0.634
γ̇ ∗ = γ̇
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Scaling laws
Shear stress
10
s* / | - J | 6/5
"
s∗ ν 1/2
0
. 1/6
*
|ν − νJ |
-2
6/5
∼
#msi
γ̇ ∗
|ν − νJ |
8/5
10
(
-4
10
msi
3
k = 10
4
k = 10
5
k = 10
6
k = 10
7
k = 10
-6
10
.2
*
-8
10
s∗ =
-10
10
-4
10
-2
10
D. Vescovi & S. Luding
0
. 10
*/ | - J | 8/5
2
10
4
10
=
2
1/6
unjammed phase
jammed phase
sd
kr
ρp d 3
k
νJ = 0.634
γ̇ ∗ = γ̇
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Scaling laws
Shear stress
0
s* / | - J | 6/5
10

γ̇ ∗2
s1



 ν 1/2 (ν − ν)2 , if ν < νJ
J
s∗ =



 s2 γ̇ ∗1/6 (ν − νJ )14/15 ,
ν 1/2
if ν > νJ
. 1/6
*
-2
10
-4
10
3
k = 10
4
k = 10
5
k = 10
6
k = 10
7
k = 10
-6
10
.2
*
-8
10
s∗ =
-10
10
-4
10
-2
10
D. Vescovi & S. Luding
0
. 10
*/ | - J | 8/5
2
10
4
10
sd
kr
ρp d 3
k
νJ = 0.634
γ̇ ∗ = γ̇
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Rate dependency of the shear stress
s/p
Stress ratio = macroscopic friction
k = 10
4
k = 10
5
k = 10
6
k = 10
7
k = 10
k= ∞
(  * = 3 ∙10 )
.
-2
(  * = 1 ∙10 )
.*
-3
(  = 3 ∙10 )
.
-3
(  * = 1 ∙10 )
.
-4
(  * = 3 ∙10 )
.
-2
0.45
0.50
0.55
3
-1
10
0.40
0.60
0.65
0.70

D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Phenomenological constitutive model
Pressure

p1
γ̇ ∗2


,

ν
(νJ − ν)12/5
p∗ =



 p2 (ν − ν )6/5 ,
J
ν
ν − νJ =
D. Vescovi & S. Luding
p∗ ν
p2
5/6
−
March 2, 2017
if ν < νJ (unjammed)
if ν > νJ (jammed)
p1 γ̇ ∗2
p∗ ν
5/12
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Phenomenological constitutive model
Pressure

p1
γ̇ ∗2


,

ν
(νJ − ν)12/5
p∗ =



 p2 (ν − ν )6/5 ,
J
ν
if ν < νJ (unjammed)
if ν > νJ (jammed)
⇓
ν − νJ =





−


p1 γ̇ ∗2
p∗ ν
!5/12
,

∗ 5/6


p ν



,
p2
if ν < νJ (unjammed)
if ν > νJ (jammed)
⇓
ν − νJ =
D. Vescovi & S. Luding
p∗ ν
p2
5/6
−
March 2, 2017
p1 γ̇ ∗2
p∗ ν
5/12
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Phenomenological constitutive model
Pressure

p1
γ̇ ∗2


,

ν
(νJ − ν)12/5
p∗ =



 p2 (ν − ν )6/5 ,
J
ν
ν − νJ =
D. Vescovi & S. Luding
p∗ ν
p2
5/6
−
March 2, 2017
if ν < νJ (unjammed)
if ν > νJ (jammed)
p1 γ̇ ∗2
p∗ ν
5/12
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Phenomenological constitutive model

s1
γ̇ ∗2


if ν < νJ (unjammed)

 ν 1/2 (ν − ν)2 ,
J
s∗ =



 s2 γ̇ ∗1/6 (ν − ν )14/15 , if ν > ν (jammed)
J
J
ν 1/2
Shear stress
5
-1
10
10
-2
10
4
10
s*= s d / k
2 . 2
s / (p d  )
-3
10
3
10
2
10
1
10
-4
10
-5
10
-6
10
-7
10
0
10
-8
10
-9
-1
10
0.40
0.45
0.50
0.55

0.60 J 0.65
ν − νJ =
D. Vescovi & S. Luding
10
0.70
0.40
0.45
0.50
0.55
0.60
0.65
0.70

s∗ ν 1/2
s2 γ̇ ∗1/6
15/14
March 2, 2017
−
s1 γ̇ ∗2
s∗ ν 1/2
1/2
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Phenomenological constitutive model
Stress ratio
µ = s/p =

µ1 ν 1/2 (νJ − ν)2/5 ,





µ2 ν 1/2
ν − νJ =
D. Vescovi & S. Luding
µ2 ν 1/2 γ̇ ∗1/6
µ
γ̇ ∗1/6
(ν − νJ )4/15
15/4
March 2, 2017
−
if ν < νJ (unjammed)
,
if ν > νJ (jammed)
µ
5/2
ν 1/2 µ1
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Conclusions
DEM numerical simulations of soft particles to investigate the
granular rheology below, at, and above jamming
Asymptotic scaling laws for the stresses in both unjammed
(fluid) and jammed (solid) phases
Phenomenological constitutive relations where the two phases
are merged in a unique function:
(i) continuous and differentiable at any point
(ii) able to predict the behavior even at the jamming transition
Perspectives
- Embed the new model also into the frameworks of statics,
elasticity theory and wave propagation, and kinetic theory of
granular gases, generalized for soft particles, ...
- Application to more realistic situations in practice:
(inclined flows, avalanches/landslides, silos, ...)
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Wave propagation
Ordered systems vs. disordered systems
Propagation of low frequencies only, due to disorder ξ
What if the wave triggers fluidization?
What if the wave triggers jamming?
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
What is missing?
Microstructure – isotropic
Microstructure – anisotropic
Additional non-Newtonian effects ...
Cohesion – very low and high density
Granular temperature Tg : fluctuation kinetic energy
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Microstructure – isotropic
Jamming volume fraction
νJ = 0.634
8
3. 2
7
6
C
5
k /( p d 
3.
k /( p d 
3.
k /( p d 
3.
k /( p d 
3.
k /( p d 
3
) = 10
4
) = 10
2
5
) = 10
6
2
) = 10
7
2
) = 10
2
4
3
2
1
0
0.40
0.45
0.50
0.55

0.60 J 0.65
0.70
Coordination number C: average number of contacts between particles
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Microstructure – isotropic
Jamming volume fraction
νJ = 0.634??? The jamming “point” νJ
It is a state-variable (micro-structure) and it thus depends on history!
(Kumar and Luding, Granular Matter, 2016; Luding, Nature, 2016)
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Microstructure – anisotropic
Jamming volume fraction
νJ = φJ (H) ≈ 0.634 The jamming “point” νJ
It is a state-variable (isotropic micro-structure),
complemented by the fabric (anisotropic)!
(Kumar and Luding, Granular Matter, 2016)
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Granular temperature
Jamming volume fraction
νJ = 0.634
2
-2
10
10
-3
T *= T p d / k
2 .2
T/(d  )
10
1
10
0
10
-4
10
-5
10
-6
10
-7
10
-8
10
-1
10
0.40
-9
0.45
0.50
0.55

0.60 J 0.65
0.70
10
0.40
0.45
0.50
0.55
0.60
0.65
0.70

Granular temperature Tg : fluctuation kinetic energy
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Granular temperature
Jamming volume fraction
νJ = 0.634
2
-2
10
10
-3
T *= T p d / k
2 .2
T/(d  )
10
1
10
0
10
-4
10
-5
10
-6
10
-7
10
-8
10
-1
10
0.40
-9
0.45
0.50
0.55

0.60 J 0.65
0.70
10
0.40
0.45
0.50
0.55
0.60
0.65
0.70

Granular temperature Tg : fluctuation kinetic energy
Goal: Rheology in the presence of Brownian motion and/or Tg ?
D. Vescovi & S. Luding
March 2, 2017
University of Twente
From particle simulations towards continuum theory including the solid-fluid transition in granular materials
Thanks for your attention!
D. Vescovi & S. Luding
March 2, 2017
University of Twente