Granular Materials

R. Behringer

Duke University

Durham, NC, USA

Outline

• Overview– 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– Kinetic theory– Hydrodynamics– Clustering and collapse– Simulations– Experiments

• Issues/ideas for dense granular systems– Friction and dilatancy– Force chains– Janssen model– Constant flow from a hopper– Forces under sandpiles– Texture

• Models for static force transmission– Lattice models: Q-model, 3-leg, elastic– Continuum limits of LM’s– Classical continuum models– Summary of predictions

• Experimental tests of force transmission– 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– Jamming– Percolation– Relation to other phenomena—e.g. glasses– Clustering (see gases)– Fluidization– Subharmonic Instabilities (shaken systems)– Stick-slip

• “Classical” systems– 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?

• 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: 1d < – powders

< d , 0.5cm—grains

d > 0.5 cm—pebbles, rocks, boulders…

• Size range of phenomena—packed powers (pills– m 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 densitiesthan fluids, and have Maxwell-

Boltzmann-like distributions forvelocities

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 continuousenergy input

• Subject to clustering instability

• Models (may) show granular collapse

Granular Clustering –(Luding and Herrmann)

Properties of granular solids

• Persistent contacts (contrast to collisionalpicture for gases)

• Dense slow flows or static configurations

• Force chains carry most of the force

• Force chains lead to strong spatio-temporalfluctuations

• 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 = V D2

V ~ (gD)1/2

M ~ D5/2.

Why study granular materials?

• Fundamental statistical and dynamicalchallenges

• Related to broader class of systems– e.g. foams, colloids, glasses

• Important applications:– 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

• 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 Savants7, 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– nnF ˆˆ~

nnF ˆˆ~

Fn

ˆ~

Fn

ˆ~

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?