R.P. Behringer, Lou Kondic , Konstantin Mischaikov October 8, 2015

1. Building a Theory of Granular Materials Statics and Dynamics of Granular Force Networks R.P. Behringer, Lou Kondic, Konstantin Mischaikov October 8, 2015

2. Key Ideas • Granular materials are of great strategic and technical importance—at least $1T spent/yr in US on GM’s • The current basic understanding is limited by our ability to address the large range of relevant scales in space and time • New developments from theory and experiment provide an exciting new direction for evolving basic understanding • Granular networks are the link between micro and macro scales • Experiments and simulations are now at a level that allows us access to the full range of scales—in 2D, 3D, time • Modeling challenge: need mathematical tools to extract the key aspects from experimental/numerical data over unprecedented range of scales

4. Technical Importance • Large volumes of particulate solids handling in many industries~$1 Trillion/year in US alone for granular handling—vast amounts of energy needed • Failures are not uncommon—typically costly, sometimes catestrophic • Great need for (and general lack of) predictive models Failures are frequent, typical facilities operate at only ~65% of design

5. Strategic Importance • Projectile impacts—penetrability • Off-road mobility • Soil stability (also earthquakes/avalanches) • NASA lunar and Martian mission vehicles—in situ resource utilisation

6. Modeling and scientific Issues • Traditional empirical soil mechanics models: limited predictability, mathematically problematic, lack a connection to the underlying multi-scale physics—particularly granular networks • Dynamics of force and contact networks are an intrinsic part of granular behavior • Jamming is key manifestation of network formation • New experiments and models point the way to basic understanding of granular physics • Modeling challenge: need mathematical tools to extract the key aspects from experimental/numerical data over unprecedented range of scales

7. Granular Material:Dense Phases Howell, Veje, and RB PRL 82, 5241 (1999) Forces are carried preferentially on force chains (Networks) multiscale phenomena Deformation leads to large spatio-temporal fluctuations Granular materials jam —fluid  solid transition

8. Measuring contact forces by photoelasticity—quantiative experiments from smallest scales

9. Key new approach: obtain grain contact forces Measure all system properties Experiment--raw Reconstruction from force inverse algorithm Experiment Color filtered

11. Force networks are ubiquitous for grains with friction

12. Force networks appear dynamically: formation of force chains arches at outlet leads to clogging (J. Tang & RB)

13. One frame, showing jam and force chain arch 2D hopper flow

14. Also at very high speeds photoelastic video of impact

15. Many processes live near Jamming:fluid-solid transitionWhy? Frictionless particles Frictional particles |τ|/P = 1 |τ|/P = μ Liu &Nagel, Nature, 1998 Bi, Zhang, Chakraborty, RB Nature, 480, 355 (2011)

16. What is the density doing during flow and after jam? Low φ even after jamming— Hence in shear jamming regime T=0 Liu-Nagel jamming diagram Jam in a hopper Experiment

17. Time-lapse video (one shear cycle) shows force network evolution—Shear Jamming— incompatible with LN picture

18. Shear-Jamming—very clean experiment—states well characterized

19. Initial and final states following a shear cycle— no change in area— Density cannot distinguish --but networks can  Initial state, isotropic, no stress Works between φS < φ < φJ Final state  large stresses jammed

20. Networks are at core of evolving granular systems—e.g. Strobed images-shear cycles: stress activated process stresses fluctuate Stress, position, rotation— All evolve over many cycles Positions are nearly frozen 

21. New approaches now provide similar data in 3DIndex match surrounding fluid-particles add fluorscent dye to particles (hydrogels)

22. 3Dlaser scanning experiments—simple principle N. Brodu, J. Dijksman, RB, Nature Comm. (2015)

23. A typical slice image

24. ‘Flats’ at contacts  deformation, δHertz contact force law: δ3/2

25. Tomographic Reconstruction—including forces

26. Reconstruct particle surfaces contact forces

27. What are strongest forces in 3D network?

28. How can we bridge microscopic to macroscopic scales—must involve networksCH provides a powerful tool to reduce vast amounts of dataExamples of network evolution and relation to dynamical and statistical properties

29. Return to shear jamming—and simulation vs. experiment

32. Zn evolves like a nonlinear chemical rate equation

33. How do we track/characterize structural evolution?

34. How can we compare results to see if similar behavior is present in two realizations of a system— must involve networksCH provides a powerful tool to reduce vast amounts of dataTest case comparing experiments to simulations that try to replicate exact experimental conditions—Do simulations and experiments agree?How can we be sure?

35. Contrast experimental (Left) and computed (Right) force networks Experiment DEM-Simulations Are these the ‘same’ ? How can we compare?

36. Zn = fraction of particles with n contacts vs. fraction of non-rattlers---nonrattler fraction + are from experiments, Δ are from simulations

37. Stresses are macroscopic measures—and reflect network properties—but how? P = Tr (σ) σ = stress tensor τ = (σ2 – σ1 )/(σ2 + σ1 )—stress anisotropy

38. Distributions of contact forces—a microscopic statistical measure Experiment DEM

39. Use Betti numbers to characterize networks Experiment DEM

40. Experimental errors—about 0.1N at contacts Adding noise to DEM leads to improved agreement in betti nos. Expt.vs. DEM

41. How can we bridge microscopic to macroscopic scales—Goal: construct model that incorporates simple measures of network propertiesNeed: more data—experimental and DEMUse network measures as ‘state variables’Unique possibilities: experiments that for the first time access microscopic detailsincreasing computational powerpowerful/simple tools to characterize networks