Static and flowing wet sand:
Dragging Mr. Bagnold through the mud
Alex J. Levine
Department of Physics
University of Massachusetts, Amherst
Collaborators:
Robert Brewster (University of Massachusetts, Amherst)
Deniz Ertaş and Thomas Halsey (ExxonMobil)
Gary S. Grest and James Landry (Sandia National Lab)
Thomas G. Mason (UCLA)
Leo Silbert (University of Chicago)
Sand dry and wet
We penetrated 15 miles in this way, working
north or south along a range of dunes till a
gap in the crests was found, and crossed six
ranges before camping for the night on the
top of a particularly large whaleback. ... At
the sixth range of sand the western descent
was bad, even the touring car having to be
pushed and dug downhill, so it was thought
unwise to take all three vehicles any farther
owing to the uncertainty of getting the lorries
back up the west slopes…
From: R.A. Bagnold Journeys in the Libyan desert 1929 and 1930
The Geographical Journal (1931).
Sand: wet and dry
Almost all the highways that link the
Nepalese capital with the rest of the
country have been closed by the landslides,
which are common in Nepal's summer, as
snow melts in the Himalayas and lowland
areas are hit by monsoon rains.
Herald Sun correspondent
in Katmandu
Photo: National Oceanic and Atmospheric Administration,
Department of Commerce, U.S.A.
Sand and the physics of granular media
No constitutive relation for a continuum treatment relating
to the state of deformation.
averaged properties like:
A difficult problem in many body physics:
Far from equilibrium
A sand grain
Temperature is
irrelevant
The stability of damp (cohesive) sand
From: Hornbaker et al., Nature 387, 765 (1997).
Maximum critical angle
Angle of repose
Avalanches are generically
hysteretic:
How sandcastles fall: The failure of a wet or dry sand pile
Force balance
in the bulk
Coulomb Criterion for
failure: dry and wet
Internal friction coefficient
Internal adhesive stress
Mohr-Coulomb analysis and stress indeterminacy
Solving from the force balance condition:
Stress indeterminacy
Mohr circles giving
the stress and normal
stresses across all
planes in the material
The Mohr analysis: Maximum critical angle
We choose
in order to find the maximum angle of stability.
For dry sand:
For wet (cohesive) sand:
Critical Angle depends
on the depth of the pile.
with:
Coulomb’s friction angle
(Failure occurs that depth)
How sticky is wet sand?
Area of contact patch: A
R
r
side view
end on view
Two grains held together
by a liquid bridge
Laplace pressure leads to intergrain adhesion
Surface Tension
The contact patch and the Laplace pressure
The dependence of the adhesion
force on fluid volume has three
qualitatively different regimes
Asperity Regime
Roughness Regime
Spherical Regime
The asperity regime
The contact patch
sees a correlated landscape
d
Deviations for a flat surface
Where  is the roughness exponent
The roughness regime
The contact patch sees a random
landscape
The Laplace pressure is independent of the added fluid volume
and
Force is linear
in added fluid
volume
The spherical regime
The contact
patch is affected
by the global curvature
R
An exact result:
F.P. Bowdon and D. Tabor The friction and lubrication of solids,
Oxford University Press, New York (1986).
Adhesion force
is independent of
fluid volume
The adhesion force as a function of added fluid volume
Asperity
Roughness
Spherical
The maximum critical angle vs. volume of wetting fluid
where
and
volume of
nonparticipating fluid
A fitting parameter
Comparison to experiment
Independently
measure:
Using
T.G.Mason et al.,
PRE 60, R5044 (1999).
DMSO
C16
The flowing state of wet and dry sand
Molecular dynamics: Solve Newton’s Equations using
rotational and translational degrees of freedom.
Typical Sizes: 30 X 10 X 100 particles.
Three distinct regimes of flow depend on tilt
angle :
 < r(h):
No motion
r<  <max(h): Steady state flow
 > max(h):
Avalanching flow
The force law: Dry
Repulsive forces which act only on contact: F=Fn + Ft
 coefficient of friction (Coulomb criterion)
 coefficient of restitution (determine inelasticity)
Fn = f(/d) (kndn + gnmeff vn), f(/d) = 1 or (/d)1/2
Ft = f(/d) (-ktDst - gtmeff vt),
Dst is integral over relative displacement of two particles in
contact
Coulomb proportionality | Ft |  | Fn |
Introducing intergrain adhesion
Dry
Wet
The edge of the sphere
How sticky? A = 1
Adhesion supports ~ 30 particles
The phase diagram
Unstable Flow
Stable Flow
Static
A
The constitutive relation of dry sand: Bagnold scaling
H
This is implied by dimensional analysis:
Global scaling
Local scaling
But the connection between
global scaling and local scaling is subtle
Bagnold scaling: A simple argument
Collision
x-momentum flux in the z direction:
Momentum transfer/collision
Rate of collisions
The breakdown of Bagnold scaling: Plug Flow
Bagnold scaling
A=0.0
A=0.2
A=0.4
A=0.6
Plug Flow
The breakdown of Bagnold scaling II: Below the plug
A new fitting parameter
Proposal: Modified Bagnold constitutive law
Collision
Long-lived contacts
x-momentum flux in the z direction:
Bagnold mechanism
Long-lived contacts
Evidence for modified Bagnold relation
Data taken at various heights in the pile
Summary
I.
II.
Cohesive sand fails at depth and the enhancement of the pile’s
stability depends on microscopic surface details of the constituent
particles.
Bagnold scaling fails in cohesive
granular media and plug flow develops
We propose a new constitutive
law to account for long-lived interparticle contacts.