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Granular Flows

From: Prof. Bob Behringer, Duke University

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Lecture 3: Statics of granular media

A pile of grains that does not move…

– What is happening inside?

Statics:

Photo-elastic particles

Packing fraction revisited (bidisperse mixtures, tapping)

Force chains

Internal stresses & transmission of force

Stresses in a silo (Janssen effect)

Pressure under a pile

Effect of humidity

Angle of repose

Notes: http://www.damtp.cam.ac.uk/user/nv253/, click on “Teaching”

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Photo-elastic particles (1)

Particles from photo-elastic (birefringent) material:

Place object between two crossed circular polarizers

Light splits, has phase difference index of refraction varies

Visualizes stress (quantitatively) in sample

From: Prof. Bob Behringer, Duke University

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Photo-elastic particles (2)

From: Prof. Karen Daniels, NC State

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Photo-elastic particles (3)

From: Dr. Joshua Dijksman, Duke University

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Photo-elastic particles (4)

From: Prof. Karen Daniels, NC State

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Packing fraction – revised (1)

Ordered close packing (ocp):

Idealized perfect packing

Random close packing (rcp):

Statistical & empirical

experiments and simulations

Meta-stable packings:

Need to avoid crystallization, using bidisperse mixture

Initial state needs great care

Thinnest regular packing: = 0.5236 (lowest # neighbors)

Loose random packing: = 0.59 – 0.60 (dropped or packed)

Close random packing: = 0.625 – 0.641 (vibrating bed)

Thickest regular packing: = 0.7405 (crystallization)

Cannonballs in fcc

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Packing fraction – revised (2)

Packing fraction of a disperse granular mixture

Consider mixture of two sizes of spheres, Rl & Rs,

Assume close random packing: = max = 0.64

Total volume fraction: b = l + s

Fraction of large particles: l = Ml/(Ms+Ml)

1st case: Ml >> Ms = max/l

2nd case: Ml << Ms

= max/(1 - l (1- max))

From: Les Milieux Granulaires, O. Pouliquen

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Packing fraction – revised (3)

Idealized case: 2D Apollonian gasket

Contains ever decreasing spheres sizes

Ideal packing fraction: = 1

Practical case: high

performance concrete

Sizes over several orders

of magnitude

Very high packing fraction:

= 0.90

From: http://www.mathworks.com/

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Packing fraction – revised (4)

Tapping increases density:

Sinusoidal taps:

amplitude A

duration T

peak acceleration: ~ A/T2

Main conclusions:

Energy injected is driving

mechanism

Control parameter is:

T ~ A/T

From: Dijksman & van Hecke, EPL, 2009

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Force chains

Force chains:

“Highways” along which force travels

Inactive “rattlers” in between

Highly concentrated internal stress

Chain separation: ~ 5 grains

Evolving over time:

Not reproducible

Continuous alteration

Increasing stress: additional

force chains light up

From: Prof. Bob Behringer, Duke University

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Focus: Janssen effect in a silo (1)

Discharge of a cylindrical bucket/hopper:

Water Bernoulli predicts P = f(z):

Grains Janssen predicts P ≠ f(z):

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Focus: Janssen effect in a silo (2)

From: Prof. Bob Behringer, Duke University

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Transmission of force (1)

Distribution of forces upon loading external force:

Forces are unbounded higher loads than average loads

not a perfect lattice

Forces fall off exponentially fluctuations not arbitrary

large not a fractal network

Probability “P(w)” that force “w” occurs

Strong forces near

cylinder wall

Inhomogeneity packing

unequal distribution

forces

From: Liu, Nagel, et al., Science, 1995

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Transmission of force (2)

Nature of force transmission:

Down a regular lattice?

Shared equally below?

Continuum description?

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Transmission of force (3)

Figures from a presentation online by Bob Behringer, Duke University

Nature of transmission:

Ordered system (monodisperse disks

in hexagonal packing):

propagation through lattice

hyperbolic

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Transmission of force (4)

Figures from a presentation online by Bob Behringer, Duke University

Nature of transmission:

Disordered system (pentagonal disks

in disordered packing):

classical elasticity

elliptic

Similar to a solid sheet

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Construction history matters (1)

Pressure under a pile of granular material:

Point pour:

Localized source

Uniform pour:

Homogeneous rain

Explanation:

Flexure of base is not an issue

Geometry of contacts due to force chains are!

From: Vanel, et al., PRE, 1999

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Construction history matters (2)

Humidity-induced aging:

For higher humidity and longer time

grains become more consolidated

For longer time and stress under an angle

grains become more consolidated

Explanation:

Liquid-induced cohesion & creep

From: Bocquet, et al., CR Physique 3, 2002

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Angle of repose (1)

Methods to measure angle of repose:

“material on verge of sliding”

Funnel (point-source)

Tilting box

Rotating cylinder: Dynamic angle of repose

Static angle of repose

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Angle of repose (2)

Physical interpretation:

Static angle of repose due to cohesive forces

related to friction coefficient: s = arctan(s)

Dynamic angle of repose due to dilatation and # of contacts

difference (s - d) is “dilation angle”

Characteristic values:

Angular grains (e.g. sand, gravel):

s 40°

Rounded grains (e.g. ballotini):

s 25°

From: Santamarina & Cho, Proc. Skempton Conf., 2004

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Angle of repose (3)

Effect of reduced gravity (e.g. on Mars: a = 0.1 g):

Static angle increases: s, 0.1g = s, 1g + 5°

Dynamic angle decreases: d, 0.1g = d, 1g - 10°

Dilation angle & mobility of flow increase!

Low slopes on Mars can create large dry granular flows!

From: Kleinhans, et al., JGR, 2011 Dundas, et al., GRL, 2010