Statistical Properties of Granular Materials near Jamming
R.P. Behringer
STATPHYS25, 2013
July 22, 2013
Support: NSF, NASA, BSF, ARO, DTRA, IFPRI
Collaborators: Max Bi,Abe Clark, Corentin Coulais,
Julien Dervaux, Joshua Dijksman, Tyler Earnest,
Somayeh Farhadi, Junfei Geng, Dan Howell, Trush
Majmudar, Jie Ren, Guillaume Reydellet, Nelson
Sepulveda, Junyao Tang, Sarath Tennakoon, Brian
Tighe, John Wambaugh, Brian Utter, Peidong Yu, Jie
Zhang, Hu Zheng, Bulbul Chakraborty, Eric Clément,
Karin Dahmen, Karen Daniels, Olivier Dauchot, Isaac
Goldhirsch, Wolfgang Losert, Lou Kondic, Miro
Kramer, Jackie Krim, Stefan Luding, Chris Marone,
Guy Metcalfe, Konstantin Mischaikov, Corey O’Hern,
David Schaeffer, Josh Socolar, Matthias Sperl,
Antoinette Tordesillas
Roadmap
Use experiments to explore:
• Jamming of frictional systems:
compression vs. shear
• Slow relaxation in and near shear jamming
• Dynamics and fluctuations during impacts
on granular materials
• Common Theme: Granular networks
Context for Dense Granular Phases—some
identifying features
Frozen structure: forces are carried preferentially on
force chains, ‘long’ force chains  anisotropy in local
stress
Deformation, particularly shear, leads to large
spatio-temporal fluctuations
Also, shear leads to dilation and shear bands
Granular materials jam, and shear leads to
jamming below φJ, friction helps but need not be
essential--Friction and preparation history matter
1) Dilation under shear (Reynolds dilatancy) and shear
banding (new experimental technique can avoid SB’s)
Before shearing
From experiments of
Bob Hartley at Duke
After sustained shearing
Reynolds of fluid mechanics fame—discovers dilatancy
2) GM’s exhibit novel meso-scopic structures:
Force Chains:
These depend on the preparation of the material—long chains 
shear  stress and network anisotropy
2d Shear 
Experiment
Howell et al.
PRL 82, 5241 (1999)
I = Iosin2[(σ2- σ1)CT/λ]
Video of Couette shear experiment, Top View
Anisotropy in force chains can lead to surprising
results—example sand piles
2D photoelastic image
(Geng et al.)
3D heaps and pressure
at base (Vanel et al.)
In hopper flows—formation of force chains arches at
outlet leads to jamming (Junyao Tang)
One frame, showing jam and force chain arch
3) Rearrangement of networks leads to strong force fluctuations
Spectra: power-law falloff
3D
Time-varying Stress in
3D (above) and 2D
(right) Shear Flow
Miller et al. PRL 77, 3110 (1996)
Hartley & BB Nature, 421, 928 (2003)
Daniels & BB PRL 94, 168001 (2005)
2D
How to describe granular fluctuations?
• Kinetic theory approaches for granular gases
• Edwards ensemble for rigid particles: V replaces E
(Edwards, Oakeshott, Mehta), compactivity
replaces temperature, and is conjugate to V
• Real particles are deformable (elastic): stresses
are ‘conserved’ for static systems—hence
force/stress ensembles emerge—generalized
tensor (angoricity) replaces temperature, and is
conjugate to stress
4) Granular materials jam
• Jamming—how disordered N-body systems
becomes solid-like as particles are brought into
contact, or fluid-like when grains are separated—
thought to apply to many systems, including GM’s
foams, colloids, glasses…
• Isotropic case: Density is implicated as a key
parameter, expressed as packing (solid fraction) φ
• Marginal stability (isostaticity) for spherical
particles (disks in 2D) contact number, Z, attains a
critical value, Ziso at φiso
• Ziso depends on dimension, friction.
Jamming Pictures
How do disordered solids lose/gain their solidity?
Cates et al. PRL 81, 1841 (1998)
Liu and Nagel, Nature 396, 21 1998
Predictions for isotropic Jamming Transition: T = 0
• Simple question:
What happens to key properties such as pressure, contact
number as a sample is isotropically compressed/dilated
through the point of mechanical stability?
Z = contacts/particle; Φ = packing fraction
Predictions (e.g. O’Hern, Silbert, Liu and
Nagel; Torquato et al., Schwarz et al.
Z ~ ZI +(φ – φc)ά
(discontinuity)
Exponent ά ≈ 1/2
P ~(φ – φc)β
β depends on force law
(= 1 for ideal disks)
Henkes and Chakraborty: entropy-based model gives P and Z in
terms of a field conjugate to entropy. Can eliminate to get P(z)
But, experiments on frictional particles show other interesting
behavior
Two kinds of state, depending on φ
1) …φS < φ < φJ—states arise under shear, |τ| > 0
2) …φ > φJ—jammed states occur at τ = 0
|τ|/P = 1
|τ|/P = μ
Original (Liu &Nagel, Nature 1998)
Bi et al. Nature, 480, 355 (2011)
Experimental tools: what to measure, and how to
look inside complex systems
• Confocal, laser scanning or tomography
techniques in 3D—with fluid-immersed
particles or x-rays
• Bulk measurements—2D and 3D
• Measurements at boundaries—3D
• 2D measurements: particle tracking,
Photoelastic techniques (this talk)
• MRI—for density, forces
• Numerical experiments—MD/DEM—theory
and experiment
I = Iosin2[(σ2- σ1)CT/λ]
Historical note
Use of photoelasticity in granular physics
dates to Wakabayashi, Dantu, later
Drescher, Joselin de Jong, …
Key new approach: obtain grain contact forces
Experiment--raw
Experiment
Color filtered
Reconstruction
From force
inverse algorithm
Basic principles of technique
• Process images to obtain particle centers and
contacts
• Invoke exact solution of stresses within a disk
subject to localized forces at circumference
• Make a nonlinear fit to (unique) photoelastic
pattern using contact forces as fit parameters
• I = Iosin2[(σ2- σ1)CT/λ]
• In the previous step, invoke force and torque
balance
• Newton’s 3d law provides error checking
Track Particle: Forces/Displacements/Rotations
Following a small
strain step we track
particle displacements
Under UV light—
bars allow us to
track particle
rotations
We now have vector forces
at contacts
Good collapse of
data for P(f) for
normal and tangential
contact forces (sheared
system)
Obtaining stresses and fabric from experimental
data
Stress
Fabric
These quantities can be coarse-grained to produce continuum fields
Stresses, fabric, force moment tensor—2D
Fabric tensor
Rij = Sk,c ncik ncjk
Z = trace[R]
Stress tensor, force moment tensor
stress: sij = (1/A) Sk,c rcik fcjk
Pressure, P and shear stress
P = Tr (s)/2 = (σ2 + σ1)/2
:τ = (σ2 - σ1)/2
Force moment Sij = Sk,c rcik fcjk = A sij
A is particle/system area
Experiment 1: Biaxial Tester
Control spacing between opposing pairs of walls
Experiments use
biaxial tester
and photoelastic
particles
Contrast Experiments with Compression and Shear
All networks are not the same
Biax schematic
Compression
Shear
Image of
Single disk
~2500 particles, bi-disperse,
dL=0.9cm, dS= 0.8cm,
(Trush Majmudar and RPB, Nature, 2005)
NS /NL = 4
Isotropic compression
Experiments on frictional grains resemble predictions for frictionless
grains
Isotropic
compression
 Pure shear
Majmudar, Sperl, Luding and BB, PRL 2007
Three predictions for the Isotropic Jamming
Transition
1) Z ~ ZI +(φ –φc)ά
(discontinuity)
Exponent ά ≈ 1/2
2) P ~(φ – φc)β
β depends on force law
(= 1 for ideal disks)
3) Henkes and Chakraborty: entropy-based model: P(Z)
Z = contacts/particle; Φ = packing fraction
Predictions (e.g. O’Hern, Silbert, Liu and
Nagel; Torquato et al., Schwarz et al.,
Henkes and Chakraborty
LSQ Fits for Z give an exponent of 0.5 to 0.6
φc ≈ 0.84
Transition is ‘rapid’ but not discontinuous—return to this
LSQ Fits for P give β ≈ 1.0 to 1.1
Comparison to Henkes and Chakraborty prediction
Jamming under Shear
How do disordered solids lose/gain their solidity
under shear?
σ2
σ1
What happens here, when shear is applied to a granular material?
Note: P = (s2 + s1)/2
Coulomb failure:
Shear stress: t = (s2 – s1)/2
|t|/P = m
Return to Biaxial Experiments: Choose pure shear
mode
Biax schematic
Compression
 Shear
Image of
Single disk
~2500 particles, bi-disperse,
dL=0.9cm, dS= 0.8cm,
Zhang, Majmudar, Tordesillas, BB, Granular Matter, 2010
Bi, Zhang, Chakraborty, BB, Nature, 2011
NS /NL = 4
Different types methods of applying shear
• Example1: pure shear
• Example 2: simple shear
• Example 3: steady shear
Pure shear
Simple shear
Couette shear
Different ways to apply shear:
Common feature of different protocols for shear
One compressive and one dilational direction
(no change of area)
First: Pure Shear Experiments using Biax
Time-lapse video (one pure shear cycle) shows force
network evolution
Initial and final states
following a shear cycle—
no change in area
 Initial state, isotropic,
no stress
States at the same density
are differentiated by
properties other than
their density
Final state 
large stresses
Shear near jamming
• Example 1: pure shear
• Expt for 2: simple shear
• Example 3: steady shear
2nd apparatus: quasi-uniform simple shear
• Use bi-disperse particle:
(Joshua Dijksman and Jie Ren)
•dS=0.5”, dL=0.65”, NS/NL~3.3
•Total particle number: ~1000
• Rectangle width ~ 25dS
•Slat width=0.5”, comparable to particle size
• Shear strain ε=x/x0, increases by 0.482% per step
• Take photos of both the normal view of the
particles, and the polarized view of force chain
structures.
This new experimental approach supplies uniform shear
This new experimental approach supplies uniform shear
Time-Lapse Video of Shear-Jamming
Range of densities for which shear jamming can be
achieved
But: networks are key to shear jamming
Increasing shear strain—first unidirectional, then all-directional
percolation of strong force network
Same idea for pure and simple shear
Unjammed
not
fragile
Shear
Jammed
Fragile
Evolves
towards
more
isotropic
Jamming diagram for Frictional Particles
Three kinds of state, depending on φ and shear strain
1) …φS < φ < φJ—for small shear, fragile states
2) …then with enough strain are jammed states
3) …φ > φJ—jammed states occur at τ = 0
|τ|/P = 1
|τ|/P = μ
Original
New (Frictional)
Shear  long force chains
Fragile
Shear jammed
Increasingly isotropic
Long force chains  particles dominated by two
contacts
Force/torque-balanced case:
2 contacts on one particle
Force-moment tensor
For this case: one non-zero and
One zero eigenvalue— maximally
stress anisotropic at microscale
Need tan(θ) < μ
Lower μ’s have straighter
force chains
Note: Kumar and Luding report shear jamming in frictionless spheres
Return to stress evolution with strain—generalized
elasticity
Augment unidirectional
Shear with cyclic shear—
Probes stress evolution
And dynamics
Dynamics near shear jamming
Strobed images
Stress, position, rotation—
All evolve over many cycles
First—stresses vs. unidirectional strain below φJ
P ≈ Rγ2
Define Reynolds coefficient, R
Ren et al.
PRL 110, 018302
(2013)
Shear jamming dynamics below φJ
Define Reynolds coefficient, R
P ≈ Rγ2
R ~ (φJ - φ)3
Apply symmetric cyclic shear—rapid relaxation to
limit cycle
Apply asymmetric cyclic shear: note slow relaxation
Pmax
Cycle 1
Cycle 29
ΔP = Pmax - Pmin
Pmin
Asymmetric shear:
log-relaxation-simple φ and γ dependence
ΔP = -ß ln(n/no)
Universal relaxation—suggests activated process in a
stress ensemble
Roadmap
Let’s end with a ‘bang’, or at least a thump
• Dynamics of granular impact
Past work by: Durian, Debruyn, Swinney, Goldman
See Clark and BB, Phys. Rev. Lett. 109, 238302 (2012)
2D Granular Impact Dynamics with Photoelastic Particles:
Interesting questions of coarse-graining and scales
How is energy lost? What is the
force/deceleration during the impact process?
How does the material respond?
Using photoelastic particles—reveals
effect of persistent force networks
Schematic of Experiment
1Mp frames at rates to ~50,000fps
reduced resolution to much higher rates
High speed visualization
with photoelasticity
Trajectory of
Ogive or disk
Intruder
Photoelastic
particles allow us
to visualize the
forces in the
granular material.
4 points plus outline
allow tracking of intruder
We also track
Individual
particles. 
kinematics
Particle Tracking
Schematic and Video of
Experiment
1Mp frames at rates to ~50,000fps
reduced resolution to much higher rates
High speed photoelastic video of impact
Use photoelastic response to probe micro-scale
response—depth time plot of photoelastic response
Note ‘acoustic’ pulses
Acoustic Propagation
• Sound speed consistent with typical
granular speeds—about 1/10 of speed in
solid bulk material
• Dependence on impact velocity/depth
(higher pressure = faster speeds)
• Use space-time plots to show intruder
motion and photoelastic response
underneath
Also, track intruder motion/deceleration
Note physical fluctuations in acceleration
Measured from tracking
(time scale ~ 5 particles /
velocity)
Expect something like
this from macro-scale
(Poncelet) model
Compare 1) original photoelastic signal, 2) the timefiltered photoelastic signal,and 3) the intruder
deceleration
Compare to slow-time, macro-scale Models
• Most previous descriptions can be
generalized into:
mz = mg  f1 ( z )  f 2 ( z ) z
[1], [2],
[3]
2
Gravity
Depth-dependence
Inertialdrag
[1] Poncelet, J.V. Cours de M´ecanique
Industrielle. Paris, 1829.
These work pretty [2] Tsimring, Volfson. Powders and
well (f1~linear, Grains, 2005.
[3] Katsuragi, Durian. Nature Physics,
f2~const)
2007.
Fitting to a Macro-scale Model
• Use meso-scale data from whole data set (many different runs, many
different impact velocities)
• At fixed depth, plot acceleration vs. velocity squared
• f1~offset, f2~slope
mz = mg  f1 ( z0 )  f 2 ( z0 ) z
y = a + bx
2
Slope = f 2(z0)
Offset = f 1(z0)
Measured f1(z)
Measured f2(z) = h(z)
Acceleration ↔ Photoelasticity
Fluctuations decay, but not when normalized by v2
mz = mg  f1 ( z )  f 2 ( z ) z (t )
2
From calibrated photoelastic
response
PDF of fluctuations has exponential tail
Autocorrelation function of
fluctuations has nearly cnst form
Summing up impacts
• Macro-scale: Poncelet description, works fairly well
• Meso-scale: substantial fluctuations, can be explained in
terms of photoelastic measurements
• Micro-scale: force fluctuations up to 5X of mean; “forcelaw” almost never true instantaneously
* Need stochastic description, random variable X to capture
fluctuations:
mz = mg  f1 ( z )  f 2 ( z ) z (t )
2
•Under shear: jamming occurs for frictional particles where it
it was not expected
Summing
•Fragile and fully jammed states/networks
•Key question: how to characterize states
•Strain and density are not best variables
•fNR is a step in the right direction
up
Slow expts
•Fragile and shear jammed states need a network description
•Novel dynamics under cyclic shear stress-activated process
Summing up slow flows and jamming
•Under shear: jamming occurs for frictional particles where it
it was not expected
•Fragile and fully jammed states/networks
•With continued strain past shear jamming: more isotropic
•Key question: how top characterize states
•Strain and density are not ‘good’ variables
•fNR is a step in the right direction
•Fragile and shear jammed states need a network description