Granular Materials

1
Granular Materials

• R. Behringer
• Duke University
• Durham, NC, USA

2
Outline

• Overview
• Whats a granular material?
• Numbers, sizes and scales
• Granular phases
• Features of granular phases
• Why study granular materials?
• Special Phenomena
• Open challengeswhat we dont know

3

• Issues/ideas for granular gases
• Kinetic theory
• Hydrodynamics
• Clustering and collapse
• Simulations
• Experiments

4

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

5

• Models for static force transmission
• Lattice models Q-model, 3-leg, elastic
• Continuum limits of LMs
• Classical continuum models
• Summary of predictions

6

• Experimental tests of force transmission
• Order/disorder
• Friction
• Vector nature of force transmission
• Textured systems
• So where do we stand?

7

• Force fluctuations in dense systems
• Force chains
• Fragility
• Anisotropy

8

• Transitions
• Jamming
• Percolation
• Relation to other phenomenae.g. glasses
• Clustering (see gases)
• Fluidization
• Subharmonic Instabilities (shaken systems)
• Stick-slip

9

• Classical systems
• Shaking (convection, waves)
• Avalanches
• Rotating flows
• Hoppers and bunkers
• Shearing
• Mixing and segregation

10

• Special techniques
• Discrete element models (DEM or MD)
• Lattice models
• Special experimental techniques
• NMR
• Photoelasticity
• Carbon paper

11
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-thermalkBT ltlt Etypical mgd
• Other effects from surrounding fluid, charging
  may occur

12
Numbers, Sizes and Scales

• Sizes 1m lt d lt 100m powders
• -100m lt d , 0.5cmgrains
• d gt 0.5 cmpebbles, rocks, boulders
• Size range of phenomenapacked powers (pills mm
  to mm
• A box of cerealmm to 10 cm
• Grains in a silomm to 10s of m
• Sahara desertmm to many km
• Rings of Saturn, intergalactic dust cloudsup to
  1020m

13
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 temperaturefar
  from equilibrium
• Various possible granular temperatures proposed

14
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 Maxwell- Boltzmann-like
  distributions for velocities

15
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

16
Granular Clustering (Luding and Herrmann)
17
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 fluctu
  ations
• Interlocking of grains leads to jamming, yield
  stress, dilation on shearing

18
Example of Force Chains from a Couette Experiment
19
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

20
Granular phase transistions

• Clustering in gases
• Elastic to plastic (semi- fluid) in dense
  systemsjamming
• Jamming and fragility
• Note gravity typically compacts flowsmany
  states not easily accessible on earth

21
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
• Whyforce chains, jamming

22
Visualization in 2D by photoelasticity (more
later)
23
Note method of pouring matters for the final
heap (History dependence)
24
Mass flow rate vs. hopper opening diameter
25
Simple argument to predict flow rate

• M rV D2
• V (gD)1/2
• M D5/2.

26
Why study granular materials?

• Fundamental statistical and dynamical challenges
• 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

27
Some technical problems
28
Close to homeabout a mile from the Duke
University Campus
29
Interesting phenomena

• Pattern formation
• In shaken systems
• Hopper flows
• Mixing/segregation
• Clusteringgranular gases
• Avalanches
• Rotating flows
• Granular convection
• Jamming/unjamming

30
Applications

• Significant contribution to economy (1
  trillion per year (?) in US)
• Granular industrial facilities operate below
  designlarge financial losses result
• Large losses due to avalanches and mudslides

31
Friction Granular and otherwise

• Two parallel/intertwined concepts
• Ordinary friction
• Granular friction
• Both referenced to Coulombs original work
• Mohr-Coulomb friction.

32
C. A. Coulomb, Acad. Roy. Sci. Mem. Phys. Divers
Savants7, 343 (1773)
33
Ordinary Solid Friction
34
e. g. block on plane
35
Indeterminacy of frictional contacts
36
Hertz-Mindlin contact forces
37
Reynolds Dilatancy
38
Example of Reynolds dilation in before and after
images from a shear experiment
39
Microscopic origin of stresses, Fabric, Anisotropy

• Fabric tensor
• Microscopic origin of stress tensor
• Shape effects

40
Fabric and fragility (e.g. Cates et al. Chaos 9,
511 (1999))
41
Other effects leading to anisotropy
42
Aligned force chains/contacts lead to texture and
anisotropy
43
Examplesimple shear creates texture
44
Force chains, Spatio-temporal fluctuations

• What happens when dense materials deform?
• Strong spatio-temporal fluctuations
• Examples hopper, 2d shear, sound.
• Length scale/correlation questions

45
Fluctuations during hopper flow
46
Spectrum of stress time series
47
Sound measurements (Liu and Nagel, PRL 68, 2301
(1992)
48
2D Shear Experimentstress chains break and reform
49
Example of stress chains Couette shear (Bob
Hartley)
50
Closeup of sheared material (Bob Hartley)
51
Time series show large fluctuations (Howell et
al. PRL 82, 5241 (1999))
52
Also in 3D shear experiments (Miller et al. PRL
77, 3110 (1996))
53
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?