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![Page 1: The Scientist in the Sandbox: Characterizing the Microscopic Properties of Granular Materials TUCASI Project October 5, 2006 R.P. Behringer Duke University.](https://reader035.fdocuments.us/reader035/viewer/2022062408/56649e905503460f94b94e96/html5/thumbnails/1.jpg)
The Scientist in the Sandbox:Characterizing the Microscopic Properties of
Granular MaterialsTUCASI ProjectOctober 5, 2006
R.P. Behringer
Duke University
Support: NSF, NASA
Collaborators: Karen Daniels, Junfei Geng, Dan Howell, Lou Kondic, Trush Majmudar, Guillaume Reydellet, Brian Utter, Eric Clement, Stefefan Luding
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OUTLINE
• Why granular materials?
• Where granular materials and molecular matter part company—open questions of relevant scales
• An Overview of Experiments—
Why does this require significant computational resources?
• Conclusions
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Examples of Granular Materials
• Earthquake gouge• Avalanches and mudslides• Food and other natural grains: wheat, rice,
…• Industrial materials: coal, ores,…• Soils and sands• Pharmaceutical powders• Dust• Chemical processing—e.g. fluidized beds
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What are Granular Materials?
• Collections of macroscopic ‘hard’ (but not rigid) particles: interactions are dissipative– Classical h 0
– A-thermal T 0
– Draw energy for fluctuations from macroscopic flow
– Exist in phases: granular gases, fluids and solids
– Large collective systems, but outside normal statistical physics
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Questions
• Fascinating and deep statistical questions– What is the nature of granular friction?
– What is the nature of granular fluctuations—what is their range?
– Is there a granular temperature?
– Phase transitions
– Jamming and connections to other systems: e.g. colloids, foams, glasses,…
– The continuum limit and ‘hydrodynamics—at what scales?
– What are the relevant macroscopic variables?
– Novel instabilities and pattern formation phenomena
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Practical Issues
o Massive financial costs Claim: ~$1 Trillion/year in US for granular handling
o Failures are frequent, typical facilities operate at only ~65% of design
o Soil stability is difficult to predict/assess
o How is stress/information transmitted in granular materials?
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Some Examples of Granular Catastrophes
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…And a bit further from home…
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Assessment of theoretical understanding
• Basic models for dilute granular systems are reasonably successful
• For dense granular states, theory is far from settled, and under intensive debate and scrutiny
• Current ability to predict for dense granular states is poor relative to other systems—e.g. fluids
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Granular Material Phases-Gases
Granular Gases:
Cool spontaneously, show clustering instability
Tg = (1/2)m<v2>
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Clustering in a Cooling Granular Gas(from work by S. Luding, H. Herrmann)
• Cooling simulation by Luding and Herrmann
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In the Lab: Granular Gases are sustained by vibration…
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Granular Material Phases-Dense Phases
Granular Solids and fluids much less well understood than granular gases
Forces are carried preferentially on force chainsmultiscale phenomena
Deformation leads to large spatio-temporal fluctuations
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Granular Material Phases-Dense PhasesContinued
Friction and extra contacts preparation history matters
Jamming/glassy behavior near solid-fluid transition (Liu, Nagle, O’Hern, Bouchaud et al.)
--interesting connections to plasticity in disordered solids (e.g. Falk, Langer, Lemaitre, Maloney…)
In most cases, a statistical approach may be the only possible description
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Multiple contacts => indeterminacy
Note: 5 contacts => 10 unknown forcecomponents.
3 particles => 9 constraints
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Frictional indeterminacy => history dependence
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Dilation under shear
Before shearing After sustained shearing
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Example of Force Chains—Shear ExperimentHowell et al. PRL 82, 5241 (1999)
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Stress Fluctuations in 3D Shear FlowMiller et al. PRL 77, 3110 (1996)
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Video of 2D shear flow
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A computational model of shear: Lou Kondic (NJIT)
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Understanding Static Stress Balance—Ideally from Micromechanics
• Four unknown stress components (2D)• Three balance equations
– Horizontal forces – Vertical forces– Torques
• Need a constitutive equation
σxx
x
σxz
z0
σxz
x
σzz
z0 σ
xz=Ï ƒ
zx
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Some approaches to describing stresses
• Elasto-plastic models (Elliptic, then hyperbolic)
• Lattice models– Q-model (parabolic in continuum limit)
– 3-leg model (hyperbolic (elliptic) in cont. limit)
– Anisotropic elastic spring model
• OSL model (hyperbolic)
• Telegraph model (hyperbolic)
• Double-Y model (type not known in general)
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Experiments to determine vector contact forces(Trush Majmudar and RPB, Nature, June 23, 2005)
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Experiments Use Photoelasticity:
Biax schematic Compression
ShearImage of Single disk
~2500 particles, bi-disperse, dL=0.9cm, dS= 0.8cm, NS /NL = 4
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Measuring forces by photoelasticity
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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 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
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Examples of Experimental and ‘Fitted’ Images
Experiment Fit
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Current Image Size
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Track Particle Displacements Too
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Edwards Entropy-Inspired Models for P(f)
• Consider all possible states consistent with applied forces
• 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. preprint (Duke University)
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Granular friction and dynamics in a 2D sheared system
B.Utter and RPB PRE 69, 031308 (2004) Eur. Phys. J. E 14, 373 (2004)
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Schematic of apparatus
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Photo of Couette apparatus
~ 1 m
~50,000 particles, some have dark bars for tracking
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Videos at different shear rates
γ = 0.0027Hz γ = 0.027Hz
γ = 0.27Hz
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Stress Avalanches
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Granular Rheology—a slider experiment
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What is the relation between stick slip and granular force structure?
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Videos of force evolution
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Order-disorder:Transition from solid to dense fluid
Jamming/unjammingK. Daniels and RPB, PRL 94 168001 (2005)
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Videos of ordered/disordered states
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Freezing by Heating—Competition between shearing and vibration (Γ = 2.0)
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Conclusions
• Granular materials are extremely important in applications
• Our understanding compared to other materials is poor
• Experiments and simulations increasingly need to probe the microscopic details
• There will be an ever-increasing need for imaging and other computational resources
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Some approaches to describing stresses
• Elasto-plastic models (Elliptic, then hyperbolic)
• Lattice models– Q-model (parabolic in continuum limit)
– 3-leg model (hyperbolic (elliptic) in cont. limit)
– Anisotropic elastic spring model
• OSL model (hyperbolic)
• Telegraph model (hyperbolic)
• Double-Y model (type not known in general)
![Page 45: The Scientist in the Sandbox: Characterizing the Microscopic Properties of Granular Materials TUCASI Project October 5, 2006 R.P. Behringer Duke University.](https://reader035.fdocuments.us/reader035/viewer/2022062408/56649e905503460f94b94e96/html5/thumbnails/45.jpg)
A gradient technique to obtain grain-scale forces
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calibration
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Disks-single response
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Before-after
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disk response mean
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Large variance of distribution
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Pentagon response
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Rectangular packing reduces contact disorder
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Hexagonal vs. square, data
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Hexagonal vs. square packing
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Square packs, varying friction
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Conclusions
• Normal force distributions are sensitive to stress state• Long-range correlations for forces in sheared systems—
thus, force chains can be mesoscopic at least• Diffusion in sheared systems: insights into microscopic
statistics of driven granular materials• Logarithmic rate dependence is seen in sheared granular
systems• Interesting connections to avalanches/earthquakes…• Order-disorder transition—first order characterizes jamming-unjamming, contradictions notions
of vibrationtemperature in granular systems• Strong effects on transmission from order/disorder (spatial
and force-contact)—overall response is mostly elastic
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What are important questions?(Dense materials)
• What are statistical properties/variability of granular systems?
• What is the nature of spatio-temporal correlations/fluctuations?
The answer to this requires addressing the relevant multi-scale phenomena involved—something that is just now being considered
• Is there a universal description for stress, deformation, etc?