Title: Granular Materials
1Granular Materials
- R. Behringer
- Duke University
- Durham, NC, USA
2Outline
- 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
11What 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
12Numbers, 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
13Granular 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
14Distinguishing 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
15Properties 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)
17Properties 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
18Example of Force Chains from a Couette Experiment
19Solids, 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
20Granular 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
21Do 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
22Visualization in 2D by photoelasticity (more
later)
23Note method of pouring matters for the final
heap (History dependence)
24Mass flow rate vs. hopper opening diameter
25Simple argument to predict flow rate
- M rV D2
- V (gD)1/2
- M D5/2.
26Why 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
27Some technical problems
28Close to homeabout a mile from the Duke
University Campus
29Interesting phenomena
- Pattern formation
- In shaken systems
- Hopper flows
- Mixing/segregation
- Clusteringgranular gases
- Avalanches
- Rotating flows
- Granular convection
- Jamming/unjamming
30Applications
- 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
31Friction Granular and otherwise
- Two parallel/intertwined concepts
- Ordinary friction
- Granular friction
- Both referenced to Coulombs original work
- Mohr-Coulomb friction.
32C. A. Coulomb, Acad. Roy. Sci. Mem. Phys. Divers
Savants7, 343 (1773)
33Ordinary Solid Friction
34e. g. block on plane
35Indeterminacy of frictional contacts
36Hertz-Mindlin contact forces
37Reynolds Dilatancy
38Example of Reynolds dilation in before and after
images from a shear experiment
39Microscopic origin of stresses, Fabric, Anisotropy
- Fabric tensor
- Microscopic origin of stress tensor
- Shape effects
40Fabric and fragility (e.g. Cates et al. Chaos 9,
511 (1999))
41Other effects leading to anisotropy
42Aligned force chains/contacts lead to texture and
anisotropy
43Examplesimple shear creates texture
44Force chains, Spatio-temporal fluctuations
- What happens when dense materials deform?
- Strong spatio-temporal fluctuations
- Examples hopper, 2d shear, sound.
- Length scale/correlation questions
45Fluctuations during hopper flow
46Spectrum of stress time series
47Sound measurements (Liu and Nagel, PRL 68, 2301
(1992)
482D Shear Experimentstress chains break and reform
49Example of stress chains Couette shear (Bob
Hartley)
50Closeup of sheared material (Bob Hartley)
51Time series show large fluctuations (Howell et
al. PRL 82, 5241 (1999))
52Also in 3D shear experiments (Miller et al. PRL
77, 3110 (1996))
53Open 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?