Static and flowing wet sand: Dragging Mr. Bagnold through the mud

1. Static and flowing wet sand: Dragging Mr. Bagnold through the mud Alex J. Levine Department of Physics University of Massachusetts, Amherst

2. 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)

3. 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).

4. 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.

5. No constitutive relation for a continuum treatment relating averaged properties like: to the state of deformation. A difficult problem in many body physics: Far from equilibrium A sand grain Sand and the physics of granular media Temperature is irrelevant

6. Angle of repose The stability of damp (cohesive) sand From: Hornbaker et al., Nature 387, 765 (1997). Maximum critical angle Avalanches are generically hysteretic:

7. Internal friction coefficient 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 adhesive stress

8. 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

9. The Mohr analysis: Maximum critical angle We choose in order to find the maximum angle of stability. For dry sand: Coulomb’s friction angle For wet (cohesive) sand: (Failure occurs that depth) Critical Angle depends on the depth of the pile. with:

10. R r How sticky is wet sand? Area of contact patch: A side view end on view Two grains held together by a liquid bridge Laplace pressure leads to intergrain adhesion Surface Tension

11. The contact patch and the Laplace pressure The dependence of the adhesion force on fluid volume has three qualitatively different regimes Roughness Regime Spherical Regime Asperity Regime

12. Deviations for a flat surface The asperity regime The contact patch sees a correlated landscape d Where  is the roughness exponent

13. 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

14. R The spherical regime The contact patch is affected by the global curvature 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

15. Spherical Asperity Roughness The adhesion force as a function of added fluid volume

16. The maximum critical angle vs. volume of wetting fluid where volume of nonparticipating fluid and A fitting parameter

17. DMSO C16 Comparison to experiment Independently measure: Using T.G.Mason et al., PRE 60, R5044 (1999).

18. 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

19. 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 + gnmeffvn),f(/d) = 1 or (/d)1/2 • Ft = f(/d) (-ktDst - gtmeffvt), • Dst is integral over relative displacement of two particles in contact • Coulomb proportionality | Ft |  | Fn |

20. Dry Wet The edge of the sphere Introducing intergrain adhesion Adhesion supports ~ 30 particles How sticky? A = 1

21. The phase diagram Unstable Flow Stable Flow Static A

22. 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

23. Bagnold scaling: A simple argument Collision x-momentum flux in the z direction: Momentum transfer/collision Rate of collisions

24. The breakdown of Bagnold scaling: Plug Flow Bagnold scaling A=0.0 A=0.2 A=0.4 A=0.6 Plug Flow

25. A new fitting parameter The breakdown of Bagnold scaling II: Below the plug

26. Proposal: Modified Bagnold constitutive law Collision Long-lived contacts x-momentum flux in the z direction: Bagnold mechanism Long-lived contacts

27. Evidence for modified Bagnold relation Data taken at various heights in the pile

28. II. Bagnold scaling fails in cohesive granular media and plug flow develops We propose a new constitutive law to account for long-lived interparticle contacts. Summary I. Cohesive sand fails at depth and the enhancement of the pile’s stability depends on microscopic surface details of the constituent particles.