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Statistical Properties of Granular Materials near Jamming ESMC 2009, Lisbon September 8, 2009

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### Statistical Properties of Granular Materials near JammingESMC 2009, LisbonSeptember 8, 2009

R.P. Behringer

Duke University

Collaborators: Max Bi, Chris Bizon, Karen Daniels, Julien Dervaux, Somayeh Farhadi, Junfei Geng, Bob Hartley, Silke Henkes, Dan Howell, Trush Majmudar, Guillaume Reydellet, Trevor Shannon, Matthias Sperl, Junyao Tang, Sarath Tennakoon, Brian Tighe, John Wambaugh, Brian Utter, Peidong Yu, Jie Ren, Jie Zhang, Bulbul Chakraborty, Eric Clément, Isaac Goldhirsch, Lou Kondic, Stefan Luding, Guy Metcalfe, Corey O’Hern, David Schaeffer, Josh Socolar, Antoinette Tordesillas

Support: NSF, ARO, NASA, IFPRI, BSF

Roadmap

- What/Why granular materials?
- Behavior of disordered solids—possible universal behavior?
- Where granular materials and molecular matter part company—open questions of relevant scales

Experiments at Duke explore:

- Forces, force fluctuations
- Isotropic jamming
- Effect of shear—anisotropic stresses

What are Granular Materials?

- Collections of macroscopic ‘hard’ particles: interactions are dissipative/frictional
- Classical h 0
- Highly dissipative
- Draw energy for fluctuations from macroscopic flow
- Large collective systems, but outside normal statistical physics
- Although many-body, a-thermal in the usual sense
- Exist in phases: granular gases, fluids and solids
- Analogues to other disordered solids: glasses, colloids..
- May be dry or wet

Broader context: Disordered N-body systems—far from equilibrium

- There exist a number of such systems: glasses, foams, colloidal suspensions, granular materials,…
- For various reasons, these systems are not in ordinary thermal equilibrium (although temperature may still play a role)
- Energy may not be conserved
- Other conservation rules—e.g. stress, may apply

Standard picture of jamming

- Jamming—how disordered N-body systems becomes solid-like as particles are brought into contact, or fluid-like when grains are separated
- 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

Theoretical tools: Statistical ensembles: what to do when energy is not conserved?

- Edwards ensemble for rigid particles: V replaces E
- Real particles are deformable (elastic): forces, stresses, and torques matter, and are ‘conserved’ for static systems—hence force/stress ensembles emerge
- Snoeijer, van Saarloos et al. Tighe, Socolar et al., Henkes and Chakraborty and O’Hern, Makse et al.

Experimental tools: what to measure, and how to look inside complex systems

- Confocal techniques in 3D—with fluid-suspended particles—for colloids, emulsions, fluidized granular systems
- Bulk measurements—2D and 3D
- Measurements at boundaries—3D
- 2D measurements: particle tracking, Photoelastic techniques (this talk)
- Numerical experiments—MD/DEM

What happens when shear is applied?

Plasticity—irreversible deformation when a material is sheared

- System becomes anisotropic—e.g. long force chains form
- Shear causes irreversible (plastic) deformation. Particles move ‘around’ each other
- What is the microscopic nature of this process for granular materials?

Different types methods of applying shear

- Example1: pure shear
- Example 2: simple shear
- Example 3: steady shear

Dense Granular Material Phases-Some simple observations

Forces are carried preferentially on force chainsmultiscale phenomena

Friction matters

Howell et al.

PRL 82, 5241 (1999)

Roadmap

- What/Why granular materials?
- Where granular materials and molecular matter part company—open questions of relevant scales
- Dense granular materials: need statistical approach

Use experiments to explore:

- Forces, force fluctuations ◄
- Jamming—isotropic
- Shear and anisotropic stresses

Experiments to determine vector contact forces, distributionP1(F) is example of particle-scale statistical measure

Experiments use

biaxial tester

and photoelastic

particles

(Trush Majmudar and RPB, Nature, June 23, 2005)

Overview of Experiments

Biax schematic

Compression

Shear

Image of

Single disk

~2500 particles, bi-disperse, dL=0.9cm, dS= 0.8cm, NS /NL = 4

Basic principles of technique

- Process images to obtain particle centers and contacts
- Interparticle contact forces determine stresses within each particle, including principal stresses, σi
- Stresses determine photoelastic response:

I = Iosin2[(σ2- σ1)CT/λ]

- Now go backwards, using nonlinear inverse technique to obtain contact forces
- In the previous step, invoke force and torque balance
- Newton’s 3d law provides error checking

εxx = -εyy =0.04; Zavg = 3.1

Force distributionsfor shear and compressionCompression

εxx = -εyy =0.016; Zavg = 3.7

From T. Majmudar and RPB,

Nature, 2005

Stress ensemble models for P(f)

- Consider all possible states consistent with applied external forces, or other boundary conditions—assume all possible states occur with equal probability
- Compute Fraction where at least one contact force has value f P(f)
- E.g. Snoeier et al. PRL 92, 054302 (2004)
- Tighe et al. Phys. Rev. E, 72, 031306 (2005)

Some Typical Cases—isotropic compression and shear

Snoeijer et al. ↓

Tigue et al ↓.

Compression

Shear

Latest update—B. Tigue, this session

Correlation functions determine important scales

- C(r) = <Q(r + r’) Q(r’)>
- <> average over all vector displacements r’
- For isotropic cases, average over all directions in r.
- Angular averages should not be done for anisotropic systems

Spatial correlations of forces—angle dependent

Shear

Compression

Both directions equivalent

Chain direction

Direction

normal

To chains

New work: S. Henkes, B. Chakraborty, G. Lois, C. O’Hern, J. Zhang, RPB

Roadmap

- What/Why granular materials?
- Where granular materials and molecular matter part company—open questions of relevant scales
- Dense granular materials: need statistical approach

Use experiments to explore:

- Forces, force fluctuations ◄
- Jamming--isotropic ◄
- Shear—anisotropic stresses

The Isotropic Jamming Transition—Point J

- 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 for spherical frictionless particles (e.g. O’Hern et al. Torquato et al., Schwarz et al., Wyart et al.

β depends on force law

(= 1 for ideal disks)

Z ~ ZI +(φ – φc)ά(discontinuity)

Exponent ά ≈ 1/2

P ~(φ – φc)β

S. Henkes and B. Chakraborty: entropy-based model gives P and Z in terms of a field conjugate to entropy. Can eliminate to get P(z)

Experiment: Characterizing the Jamming Transition—Isotropic compression

- Isotropic

compression

Majmudar et al. PRL 98, 058001 (2007)

How do we obtain stresses and Z?

Fabric tensor

Rij = Sk,c ncik ncjk

Z = trace[R]

Stress tensor

:sij = (1/A) Sk,c rcik fcjk

A is system area, trace of stress tensor gives P

Roadmap

- What/Why granular materials?
- Where granular materials and molecular matter part company—open questions of relevant scales
- Dense granular materials: need statistical approach

Use experiments to explore:

- Forces, force fluctuations ◄
- Isotropic jamming ◄
- Shear—anisotropic stresses◄

A different part of the jamming diagram

σ2

σ1

What happens here or here, when shear is applied to a granular material?

Note: P = (s2 + s1)/2

:t = (s2– s1)/2

Coulomb failure:

|t|/P = m

Shear near jamming

- Example1: pure shear
- Example 2: simple shear
- Example 3: steady shear

Starting point is anisotropic, unjammed state

Isotroic jamming

point at φ = 0.82

Start here, with φ = 0.76

What happens for granular materials subject to pure shear?

Mark particles with UV-sensitive

Dye for tracking

Use biax and photoelastic particles

J. Zhang et al. to appear, Granular Matter

Apply Cyclic Pure Shear—starting from an unjammed state

And without

polarizer

Resulting state

with polarizer

Particle Displacements and Rotations

Forward shear—under UV

Deformation Field—Shear band forms

At strain = 0.085

At strain =

0.105—largest

plastic event

At strain =

0.111

Particle displacement and rotation (forward shear)

- Green arrows are displace-
- ment of particle center
- Blob size stands for
- rotation magnitude
- Blue color—clockwise
- rotation
- Brown color—
- counterclockwise rotation
- Mean displacement subtracted

Corrections for missed contacts

Estimate of missed contacts:

∫0F-min P(F)dF/∫0∞ P(F)dF

Similar approach corrects P, but

Effect is much smaller, since

P ~ ∫ F P(F) dF

How do we contemplate jamming in frictional granular materials?

σ2

Sheared granular

materials fail to

other stable states

σ1

(σ2 - σ1)

Note that Reynolds

Dilatancy weakly

confined samples

dilate under shear—

Hence, rigidly confined

materials show an

iσncrease of P under shear

J

??

??

1/φ

Conclusions

- Granular materials show important features due to friction
- Distributions of forces show sensitivity to stress state but agree reasonably with stress-ensemble approach
- Correlations for forces in sheared systems—thus, force chains can be mesoscopic
- Predictions for jamming (mostly) verified
- Granular states near jamming show jamming under shear at densities below isotropic jamming
- Friction (and also preparation history) implicated
- Z may be key variable for shear failure/jamming
- Generalized SGR explains rate-dependence

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