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Granular Flows

From: Forterre & Pouliquen, Annual Review of Fluid Mech, 2008

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Lecture 4: Flow regimes – quasi-static

Which constitutive relation should we use?

- a liquid, gas or solid one?

Flow regimes:

Slow and dense quasi-static regime “solid”

Plasticity theory (Mohr-Coulomb)

Frictional contacts, dilatancy, jamming

Intermediate dense regime “fluid”

Flows between walls & with a free-surface

Collisional & frictional contacts, rheology, hydrodynamical description

Rapid and dilute collisional regime “gas”

Kinetic theory of gases, mean free path

Short-duration contacts

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

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The sinking duck (1)

Sinking, rising & floating:

Container of glass beads in a split-bottom geometry

Activation of spinning disk at the bottom

Material moves from a solid behavior into a liquid behavior

From: K. Nichol et al, PRL, 2012

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The sinking duck (2)

Vertical motion of a probe:

Liquid behavior (Archimedes’ law): density floating/sinking

Drag force scales linearly with velocity effective viscosity

Complex rheology: drag due to viscosity

From: K. Nichol et al, PRL, 2012

Linear behavior: viscous drag

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Transition: solid & liquid behavior (1)

Jamming: locking of material, no possibility of escape

Stopping of ballotini in a funnel: vibrations “unjam” again

Traffic jam: clogging of traffic on the highway

Jamming is universal:

Jammed material:

macroscopic: possesses yield stress

microscopic: disordered

not crystalline, amorphous solid

examples: sand, glass, foam

Jamming transition:

from liquid to frozen state

From: Liu and Nagel, Nature, 1998

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Transition: solid & liquid behavior (2)

Unjamming of granular material:

Increase the granular temperature T: vibrations

Decrease the density: lower volume fraction

Increase the load: deform grains

Revised phase diagram:

Reversed curvature: extremes

Point J: below certain volume

fraction, particles don’t touch

Force chains:

Dominantly around “point J”:

Below J: no contact, no force transmission

Above J: well-deformed particles are homogeneous

From: O’Hern et al., PRE, 2003

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Transition between liquid & gaseous behavior:

Criteria related to kinetic energy lost during collisions

Restitution coefficient

Granular temperature T velocity fluctuations

Transition depends on:

Mean duration of contact

Correlation length force network

Range “kinetic theory” is limited:

Dense flow: energy dissipation is efficient

Binary flows & molecular chaos breaks down

dense liquid regime

From: Forterre & Pouliquen, Annual Review of Fluid Mech, 2008

Transition: liquid & gaseous behavior (1)

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Granular quasi-solid (1)

Coulomb (1773): concepts from soil mechanics

Dry friction between two solid surfaces:

Static friction s

Dynamic friction d

Stick: Ff < s Fn

Yielding of a granular material as a frictional process

Onset of failure in a soil (not interested in dynamical process…)

Mohr-Coulomb failure criteria:

with constant material properties: cohesion “c” and friction angle “”

Continuous plastic solid with a plastic yield criterion

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Granular quasi-solid (2)

Metals – elastic-plastic behavior:

Reversible elastic region

Irreversible plastic region

Necking: thinning

Failure: rupture

Granular materials

– rigid perfectly plastic behavior:

Rigid loading: no deformation

Perfectly-plastic deformation

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Granular quasi-solid (3)

Mohr-Coulomb yield could occur in all directions

Bi-axial yield test: Initially laterally and vertically constrained by xx and zz respectively

Slowly increase the stress zz until yielding (= failure)

When does yielding occur? failure along narrow planes

Planes: “shear bands” of ~10 particles wide

Angle of shear plane failure ?

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Focus: Mohr’s circle (1)

Assume bi-axial yield test with:

Normal stress:

Tangential stress:

Expressed in:

Average normal stress:

Maximum shear stress:

Let’s draw a “Mohr circle” to help with the analysis

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Focus: Mohr’s circle (2)

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Focus: Mohr’s circle (3)

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Focus: Mohr’s circle (4)

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Focus: Mohr’s circle (5)

For internal plane:

Assuming: 30º (angle of repose)

Angle of shear plane failure: 30º

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Biaxial testing of quasi-static properties (1)

Biaxial (experimental) testing of sand :

Uses “digital image correlation”

Measures shear band properties:

Angle of shear plane failure

Thickness of shear bands

From: Rechenmacher and Finno, Geotechnical Testing Journal, 2004

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Biaxial testing of quasi-static properties (2)

Increase strain until failure:

Thickness: ~15 – 20 particles

Angle: 35º

Orientation is arbitrary!

From: Rechenmacher and Finno, Geotechnical Testing Journal, 2004

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Passive or active walls

Apply yield criteria (Rankine limits)

Passive walls (wall collapse):

Case:

Yield at:

“k” in Janssen derivation

Active walls (bulldozer):

Case:

Yield at:

From: Les Milieux Granulaires, O. Pouliquen

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Plasticity & Flow

Law of plasticity: define yield surface

Flow rule: describing quasi-static flows beyond yield

Modified plasticity models:

modified Cosserat approach: Mohan et al., 1999

modified stochastic flow rule in a Coulomb material: Kamrin et al., 2007

plasticity of amorphous solids: Lemaitre, 2002

Writing nonlocal equations:

large spatial correlations close to the flow threshold

stresses as integrals over force chains: Mills et al., 1999

shear creates fluctuations, creating shear deformations elsewhere

No uniform description of transition between quasi-static

and liquid regimes

From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008

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Reynolds dilatancy (1)

Micromechanics: grains must expand to shear

Reynolds (1885, Phil. Mag.)

expand in volume (dilatation) during deformation

independent of intergranular friction

angle of repose:

depends on particle shape & packing, not friction

Dry footprint on wet sand:

porous space increase due to dilatancy

“apparent” drying of footprint

Removing foot from wet sand:

returns to original drained state

footprints become filled with excess fluid

From: http://static.flickr.com/7/11486115_91d93738f3.jpg

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Reynolds dilatancy (2)

Rearrangement during loading:

Free rotation for high void ratios

Frustrated rotation for low void ratios:

Frictional slippage at contacts

Volume dilation to reduce # of contacts

Angle of dilatancy:

From triaxial compression or plane shear testing

Continuous interaction:

Rotation versus slippage (think bowlingball)

Dilation versus compaction “critical state”

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Reynolds dilatancy (3)

“Critical state” -- statistical balance:

Volume change due to dilation and compaction

Granular mass shears at constant volume fraction

Consolidation:

Under-consolidated increase during shear

Over-consolidated decrease during shear

Critical concentration increases only at large stress

compressibility particles

From: Les Milieux Granulaires, O. Pouliquen & Campbell, Powder Tech., 2006

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Traffic flow: 1D granular flow (1)

Particles (= cars) on a 1D highway:

Quasi-solitons near the onset of uniform flow

Experimental data from a highway (A58) in the Netherlands:

Free-flow: upward branch

Congested traffic (for car density > 30 veh/km/lane): lower branch

From: van der Weele et al., Traffic and Granular Flow ‘03, 2005

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Traffic flow: 1D granular flow (2)

Drift flux model:

Flux (flow rate) depends on velocity & density: nonlinear drift

Drivers are anticipating by looking ahead: diffusion

(Similar to Burger’s equation)

From: Wikipedia, Fundamental diagram of traffic flow