Transcript Granular Materials

Granular Materials
R. Behringer
Duke University
Durham, NC, USA
Outline
• Overview
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What’s a granular material?
Numbers, sizes and scales
Granular phases
Features of granular phases
Why study granular materials?
Special Phenomena
Open challenges—what we don’t know
• Issues/ideas for granular gases
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Kinetic theory
Hydrodynamics
Clustering and collapse
Simulations
Experiments
• Issues/ideas for dense granular systems
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Friction and dilatancy
Force chains
Janssen model
Constant flow from a hopper
Forces under sandpiles
Texture
• Models for static force transmission
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Lattice models: Q-model, 3-leg, elastic
Continuum limits of LM’s
Classical continuum models
Summary of predictions
• Experimental tests of force transmission
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Order/disorder
Friction
Vector nature of force transmission
Textured systems
So where do we stand?
• Force fluctuations in dense systems
– Force chains
– Fragility
– Anisotropy
• Transitions
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Jamming
Percolation
Relation to other phenomena—e.g. glasses
Clustering (see gases)
Fluidization
Subharmonic Instabilities (shaken systems)
Stick-slip
• “Classical” systems
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Shaking (convection, waves…)
Avalanches
Rotating flows
Hoppers and bunkers
Shearing
Mixing and segregation
• Special techniques
– Discrete element models (DEM or MD)
– Lattice models
– Special experimental techniques
• NMR
• Photoelasticity
• “Carbon paper”
What is a granular material?
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Large number of individual solid particles
Classical interactions between particles
Inter-particle forces only during contact
Interaction forces are dissipative
– Friction, restitutional losses from collisions
• Interaction forces are dissipative
• A-thermal—kBT << Etypical ~ mgd
• Other effects from surrounding fluid, charging may occur
Numbers, Sizes and Scales
• Sizes: 1m < d < 100m– powders
-100m < d , 0.5cm—grains
d > 0.5 cm—pebbles, rocks, boulders…
• Size range of phenomena—packed powers (pills– mm to
mm
– A box of cereal—mm to 10 cm
– Grains in a silo—mm to 10’s of m
– Sahara desert—mm to many km
– Rings of Saturn, intergalactic dust clouds—up to 1020m
Granular Phases and Statistical Properties
• Qualitative similarity of fluid, gas and solid states
for granular and molecular systems
• Difficult question: how do granular phase changes
occur?
• Open question: what are the statistical properties
of granular systems?
• Caveat: No true thermodynamic temperature—far
from equilibrium
• Various possible granular ‘temperatures’ proposed
Distinguishing properties of phases
• Solids resist shear
• Fluids are viscous, i.e. shear stresses
scale with the velocity gradients
• Gases are also viscous, have lower densities
than fluids, and have MaxwellBoltzmann-like distributions for
velocities
Properties of granular gases
• Characterized by pair-wise grain
collisions
• Kinetic theory works reasonably well
• Velocity distributions are modified M-B
• Gases can only persist with continuous
energy input
• Subject to clustering instability
• Models (may) show granular collapse
Granular Clustering –(Luding and Herrmann)
Properties of granular solids
• Persistent contacts (contrast to collisional
picture for gases)
• Dense slow flows or static configurations
• Force chains carry most of the force
• Force chains lead to strong spatio-temporal
fluctuations
• Interlocking of grains leads to jamming,
yield stress, dilation on shearing
Example of Force Chains from a Couette Experiment
Solids, continued
• Dilation under shear (Reynolds)
• Grains interact via friction (Coulomb)
Note frictional indeterminacy history
dependence
– Persistent contacts may limit sampling of
phase space
• Conventionally modeled as continuum
– Strong fluctuations raise questions of
appropriate continuum limit
Granular ‘phase’ transistions
• Clustering in gases
• Elastic to plastic (semi- ‘fluid’) in dense
systems—jamming
• Jamming and fragility
• Note: gravity typically compacts flows—
many states not easily accessible on earth
Do granular materials flow like water?
• Example: sand flowing from a hopper:
– Mass flow, M, independent of fill height
– M ~ Da a ~ 2.5 to 3.0
– Why—force chains, jamming…
Visualization in 2D by photoelasticity (more later)
Note: method of pouring matters for the final heap
(History dependence)
Mass flow rate vs. hopper opening diameter
Simple argument to predict flow rate
M = rV D2
V ~ (gD)1/2
M ~ D5/2.
Why study granular materials?
• Fundamental statistical and dynamical
challenges
• Related to broader class of systems
– e.g. foams, colloids, glasses
• Important applications:
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Coal and grain handling
Chemical processing
Pharmaceuticals
Xerography
Mixing
Avalanche phenomena
Earthquakes and mudslides
Some technical ‘problems’
Close to home—about a mile from the Duke
University Campus
Interesting phenomena
• Pattern formation
– In shaken systems
– Hopper flows
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Mixing/segregation
Clustering—granular gases
Avalanches
Rotating flows
Granular convection
Jamming/unjamming
Applications
• Significant contribution to economy
(~1$ trillion per year (?) – in US)
• Granular industrial facilities operate below
design—large financial losses result
• Large losses due to avalanches and
mudslides
Friction: Granular and otherwise
• Two parallel/intertwined concepts:
– ‘Ordinary’ friction
– Granular friction
• Both referenced to Coulomb’s original work
• Mohr-Coulomb friction.
C. A. Coulomb, Acad. Roy. Sci. Mem. Phys. Divers Savants
7, 343 (1773)
Ordinary Solid Friction
e. g. block on plane
Indeterminacy of frictional contacts
Hertz-Mindlin contact forces
Reynolds Dilatancy
Example of Reynolds dilation in before and after
images from a shear experiment
Microscopic origin of stresses, Fabric,
Anisotropy
• Fabric tensor
• Microscopic origin of stress tensor
• Shape effects–

F
nˆ
F ~~ nˆnˆ 
F ~ nˆ  nˆ
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 ~ nˆ  F
Fabric and fragility (e.g. Cates et al. Chaos 9, 511
(1999))
Other effects leading to anisotropy
Aligned force chains/contacts lead to texture and anisotropy
Example—simple shear creates texture
Force chains, Spatio-temporal fluctuations
• What happens when dense materials
deform?
– Strong spatio-temporal fluctuations
– Examples: hopper, 2d shear, sound.
• Length scale/correlation questions
Fluctuations during hopper flow
Spectrum of stress time series
Sound measurements (Liu and Nagel, PRL 68, 2301
(1992)
2D Shear Experiment—stress chains break and
reform
Example of stress chains: Couette shear (Bob
Hartley)
Closeup of sheared material (Bob Hartley)
Time series show large fluctuations (Howell et al.
PRL 82, 5241 (1999))
Also in 3D shear experiments (Miller et al. PRL 77,
3110 (1996))
Open Questions: what we do not
know
• What are the statistical properties of
granular materials?
• What is their relation, if any, to broader
classes of materials?
• What are the limits on predictability?
• What are the optimum continuum models?
• When do they apply?