Constructing and verifying a
three-dimensional nonlocal
granular rheology
Ken Kamrin, David Henann
MIT, Department of Mechanical Engineering
Georg Koval
National Institute of Applied Sciences (INSA), Strasbourg
Stage 1: Local modeling.
Static zones and flowing zones.
Samadani & Kudrolli (2002)
(Experiment)
Kamrin, Rycroft, & Bazant (2007)
(Discrete Element Method [DEM])
Inertial Flow Rheology: Simple Shear
• Inertial number:
d = mean particle diameter
s = the density of a grain
• Inertial rheology:
[Da Cruz et al 2005, Jop et al
2006]
(Show Bagnold connection)
packing
fraction
Dense Granular Continuum?
(Blue
crosshairs
= Cauchy
stress evecs)
(Orange
crosshairs
= Strain-rate
evecs)
Rycroft, Kamrin, and Bazant (JMPS 2009)
Some evidence for continuum treatment:
• Space-averaged fields vary smoothly when coarse-grained at width 5d
• Certain deterministic relationships between fields appear in elements 5d wide
Developed packing fraction (5d scale)
Tall silo
Rycroft, Kamrin, and
Bazant (JMPS 2009)
Wide silo
Trap-door
Constitutive Process
Conservation of linear and angular momentum require
To close the system:
(1D picture)
Spring: Jiang-Liu granular
elasticity law [Jiang and
Liu, PRL, 2003]
B
(3D version,
Kroner-Lee
decomposition)
Dashpot: Inertial
rheology [Jop et. al,
Nature, 2006]
Bt
Local
region
Deformed
Reference
Relaxed
Mathematical Form
“Mandel stress”
Enforce:
Frame indifference
2nd law of thermodynamics
Coaxiality of flow and stress*
Definitions:
To close, must choose isotropic scalar functions  and dp with dp ¸ 0.
Explicit Algorithm
Create a user material subroutine
Given input: F(t), Fp(t), T(t), and F(t+t)
Compute output: Fp(t+t) and T(t+t)
1. Convert T(t) to M(t) using Fe(t)=F(t)Fp(t)-1.
1. Compute Dp(t) from M(t) using the flow rule.
.
2. Use Dp = Fp Fp-1 to solve for Fp(t+t).
1. Compute Fe(t+t) = F(t+t) Fp(t+t)-1.
1. Polar decompose Fe(t+t) and obtain Ee(t+t) = log Ue(t+t).
1. Compute M(t+t) from Ee(t+t) using the elasticity law.
1. Convert M(t+t) to T(t+t) using Fe(t+t).
Granular Elasticity Law

FHertz / 3/2
B is a stiffness modulus and  constrains the shear and
compressive stress ratio.
Under
compression
:
(Y. Jiang and M. Liu, Phys. Rev.
Lett., 2003)
Results
• Simulations performed using ABAQUS/Explicit finite element package.
• Constitutive model applied as a user material subroutine (VUMAT).
• All parameters in the model (6 in total) are from the papers of Jop et. al.
and Jiang and Liu, except for d:
K. Kamrin, Int. J Plasticity, (2010)
Rough Inclined Chute
x
y
Velocity vector
field:
Experimental
agreement:
Elasto-plastic model
Bingham fluid (no elasticity)
P. Jop et. al. Nature (2006)
Wide Flat-Bottom Slab Silo
Recall expt:
Cauchy stress field
Eigendirections
of stress:
Theory
Kamrin, Int. J
Plasticity, (2010)
DEM simulation
Rycroft, Kamrin, and
Bazant (JMPS 2009)
Flow prediction not “spreading” enough
Theory
Log of dp:
Kamrin, Int. J
Plasticity, (2010)
DEM simulation
Rycroft, Kamrin, and
Bazant (JMPS 2009)
Flow prediction not “spreading” enough
DEM data (Koval
Local model
are:
et al PRE predictions
2009)
Qualitatively ok:
- Shear band at inner wall
- Flow roughly invariant in the vertical
direction Continuum
model flawed:
Quantitatively
- No sharp flow/no-flow interface.
- Wall speed slows to zero, local law
says shear-band width
goes to zero.
v
wall
Continuum
model
Velocity field:
x
z
G.D.R. Midi, Euro. Phys. Journ. E, (2004)
Stage 2: Fixing the problem.
Accounting for size-effects
Past nonlocal granular flow approaches:
- Self-activated process (Pouliquen & Forterre 2009)
- Cosserat continuum (Mohan et.al 2002)
- Ginzberg-Landau order parameter (Aronson & Tsimring 2001)
Deviating from a local flow law
Fix Vwall , Vary R/d
• Inertial number:
d = particle diameter
s = the density of a grain
• Local granular rheology:
R/d
[Da Cruz et al 2005, Jop et al 2006]
R/d
Koval et al. (PRE 2009)
Pout
Fix R/d, Vary Vwall
2R
Vwall
R
Fill with grains of
diameter = d.
I(x,y), (x,y)
d and Pout held fixed
Vwall
Nonlocal Fluidity Model
Existing theory for emulsions
Goyon et al., Nature (2007); Bocquet et al., PRL (2009)
Define the “fluidity”
(inverse viscosity):
Local flow law (Herschel-Bulkley):
Nonlocal add-on:
Where bulk contribution
vanishes,
we can still
Micro-level
length-scale
flow even though we’re
under theto
“yield
proportional
criterion”.
Extend to granular media
Define the “granular
fluidity” :
Local granular flow
law:
Nonlocal law:
Physical Picture
Kinetic Elasto-Plastic
(KEP) model
Bocquet et al. (PRL 2009)
Basic idea: Flow induces flow. Plastic dynamics are spatially cooperative.
1)
A local plastic rearrangement in box j causes an elastic redistribution of stress in
material nearby.
2) The stress field perturbation from the redistribution can induce material in box i
nearby to undergo a plastic rearrangement.
Granular Fluidity = g = “Relative susceptibility to flow”
Cooperativity Length
Theoretical form for emulsions is
Direct tests: From steady-flow DEM data in the 3 geometries
(annular shear, vertical chute, shear w/ gravity):
[Bocquet et
al PRL 2009]
Switching   
Extract » from
DEM using:
For our 2D DEM disks, we find:
A=0.68
Only new material constant is the
nonlocal amplitude A.
Local law constants all carry over.
Possible meaning of diverging length-scale
Lois and Carlson, EPL, (2007)
M. van Hecke, Cond.
Mat. (2009)
f>>fc
f=fc+
Staron et al,
PRL (2002)
q=5o
Pre-avalanche zone
sizes for inclined
plane flow:
q=15o
Validating the nonlocal model
Check predictions against DEM data from Koval et al.
(PRE 2009) in the annular couette cell.
Pout
2R
Vwall
R
Kamrin & Koval PRL (2012)
DEM
(symbols)
Nonlocal
Rheology
(lines)
Fix Vwall , Vary R/d
I(x,y), (x,y)
Vary Vwall , Fix R/d
R/d
Vwall
Local
Rheology
(- - -)
Wall slip? Perhaps part of joint BC.
R/d
Same model, new geometry:
Vertical Chute
Velocity fields for 3
different gravity values:
DEM
(symbols)
Theory
(lines)
Kamrin & Koval
PRL (2012)
G
G
Pwall
Same model, new geometry:
Shear with gravity
Pwall
y
G
Velocity fields for 3
different gravity values:
DEM
(symbols)
Theory
(lines)
Kamrin & Koval
PRL (2012)
Vwall
H
G
Nonlocal rheology in 3D
Solved simultaneously
alongside Newton’s laws:
Cooperativity length:
Local rheology (standard inertial flow relation):
2 local material parameters:
For glass beads:
Jop et al., JFM (2005)
+
2 grain parameters:
+
1 new nonlocal amplitude:
Our calibration for glass beads: A=0.48
Flow in the split-bottom Couette cell
No existing continuum model has been able
to rationalize flow fields in this geometry.
Schematic:
Filled with grains:
grains
H
Inner section
(fixed)
van Hecke (2003, 2004, 2006)
Outer section
rotates
split
Flow in the split-bottom
Couette cell – surface
flow
The normalized
revolution-rate:
Viewed
from the
top down:
van Hecke (2003, 2004, 2006)
Flow in the split-bottom
Couette cell – surface flow
Exp data from: Fenistein et al., Nature (2003)
Flow fields on the top surface
for five values of H:
Rescaled:
All shallow flow data
collapses onto a
near-perfect error
function!
Normalized radial coordinate
Nonlocal model in the splitbottom Couette cell
• Custom-wrote an FEM subroutine to simulate the
nonlocal model via ABAQUS User-Element (UEL).
• Performed many sims varying H and d.
Simulated flow field in the
plane:
Flow field on top surface
Exp data from: Fenistein et al., Nature (2003)
H=30mm
H=10mm
Nonlocal model in the splitbottom Couette cell
Define:
Rc = Location of shear-band center
w = Width of shear-band
 = (r-Rc)/w = Normalized position
Surface flow results for all 22 combinations of H and d tried:
Collapse to error-function flow field
Non-diffusive scaling of w with H
Nonlocal model in the
split-bottom Couette cell
- beneath the surface
Exp data from: Fenistein et al., PRL (2004)
Sub-surface shear-band center
Sub-surface shear-band width
The model describes sub-surface flow as well as
surface flow.
Normalized torque
Nonlocal model in the
split-bottom Couette cell
- Torque
32
Remove the inner wall
and let H increase
Rotation of the top-center point
Exp data from: Fenistein et al., PRL (2006)
The nonlocal model correctly captures the transition
from shallow to deep behavior.
33
Existing 3D wall-shear data using glass beads
Annular shear flow:
Exp data: Losert et al., PRL (2000)
Linear shear flow with gravity:
Exp data: Siavoshi et al., PRE (2006)
Same exact model and parameters used in split-bottom cases also captures
these flows.
More on the
cooperativity length
Split-bottom cell
Suppose we try different power laws
in the cooperativity length formula:
Annular shear
Shear with gravity
Summary
• Synthesized a local elasto-viscoplastic model by uniting the inertial flow
rheology with a granular elastic response, and observed certain pros and cons.
• Have proposed a nonlocal extension to the inertial flow law, by extending the
theory of nonlocal fluidity to dry granular materials. Introduces only one new
experimentally fit material constant.
• Nonlocal model is demonstrated to reproduce the nontrivial size effect and
rate-independence seen in slow flowing granular media. Vastly improves flow
field prediction.
• Have extended the model to 3D and demonstrated a strong agreement with
experiments in many different flow geometries, including the split-bottom cell.
Relevant
papers:
2D prototype model:
K. Kamrin and G. Koval, PRL (2012)
3D constitutive relation: D. Henann and K. Kamrin, PNAS (2013)
Important to do’s on dry flow modeling
“Smaller is stronger” size effect
Need more quantitative insights on fluidity BC’s
(though much clarified by virtual power
argument). BC’s likely responsible for
observed nonlocal “thin-body” effects:
Experiments (Silbert 2001)
Nonlocal model w/ g=0 lower BC
Important to do’s on dry flow modeling
“Smaller is stronger” size effect
Focus on silo flow. Can we use our model to predict Beverloo constants that
govern silo flow outflow rates? A famous size-effect problem.
Beverloo constants
Beverloo Correlation
for drainage flow:
Outflow rate
Orifice size
Particle size
Q
D
D=kd
D
Q
Important to do’s on dry flow modeling
Merge a transient model into the steady flow response:
Propose to try:
(Anand-Gu form + Rate sensitivity)
Easiest route:
Let this term have
transients per a
critical-state-like
theory
Let steady behavior approach inertial flow law
[Rothenburg and Kruut IJSS (2004)]
Important to do’s on dry flow modeling
Non-codirectionality of stress and strain-rate tensors
Recall our usage of codirectionality
(i.e. stress-deviator proportional to
strain-rate tensor)
However, data shows codirectionality is inexact in shear flows for two reasons:
Slight non-coaxiality:
Normal stress difference N2 (for 3D):
(Shear)
Depken et al.,
EPL (2007)
(Black) Strain-rate eigen-directions
(Blue) Stress eigen-directions
Important to do’s on dry flow modeling
Material Point Method (MPM): Meshless
method for continuum simulation. Useful for
very large deformation plasticity problems.
Our own preliminary MPM
implementation:
“Phantom”
background
mesh and a set
of material
point markers
Great promise for full flow simulation
[Z. Wieckowski. (2004)]