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Statistical Properties of Granular Materials near Jamming
ESMC 2009, LisbonSeptember 8, 2009
R.P. BehringerDuke 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, distribution
P1(F) is example of particle-scale statistical measure
(Trush Majmudar and RPB, Nature, June 23, 2005)
Experiments usebiaxial testerand photoelasticparticles
Overview of Experiments
Biax schematic Compression
ShearImage 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
Force distributionsfor shear and compressionShear
Compressionεxx = -εyy =0.04; Zavg = 3.1
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
Chain direction
Direction normalTo chains
Both directions equivalent
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?
Predictions for spherical frictionless particles (e.g. O’Hern et al. Torquato et al., Schwarz et al., Wyart et al.
Z ~ ZI +(φ – φc)ά
(discontinuity)Exponent ά ≈ 1/2
P ~(φ – φc)β
Z = contacts/particle; Φ = packing fraction
β depends on force law(= 1 for ideal disks)
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
Isotropiccompression
Majmudar et al. PRL 98, 058001 (2007)
How do we obtain stresses and Z?
Fabric tensorRij = k,c nc
ik ncjk
Z = trace[R]
Stress tensor:ij = (1/A) k,c rc
ik 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
Note: P = ( –
|P = Coulomb failure:
What happens here or here, when shear is applied to a granular material?
σ2
σ1
Starting point is anisotropic, unjammed state
Isotroic jammingpoint at φ = 0.82
Start here, with φ = 0.76
What happens for granular materials subject to pure shear?
J. Zhang et al. to appear, Granular Matter
Use biax and photoelastic particlesMark particles with UV-sensitiveDye for tracking
Apply Cyclic Pure Shear—starting from an unjammed state
Resulting state with polarizer
And without polarizer
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, butEffect is much smaller, since
P ~ ∫ F P(F) dF
How do we contemplate jamming in frictional granular materials?
Sheared granularmaterials fail to other stable states
Note that ReynoldsDilatancy weaklyconfined samples dilate under shear—Hence, rigidly confinedmaterials show an iσncrease of P under shear
σ2
σ1(σ2 - σ1)
1/φ ????
J
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