Statistical Properties of Granular Materials near Jamming ESMC 2009, Lisbon September 8, 2009

1. Statistical Properties of Granular Materials near JammingESMC 2009, LisbonSeptember 8, 2009 R.P. Behringer Duke University Collaborators: Max Bi, Chris Bizon, Karen Daniels, Julien Dervaux, Somayeh Farhadi, Junfei Geng, Bob Hartley, Silke Henkes, Dan Howell, Trush Majmudar, Guillaume Reydellet, Trevor Shannon, Matthias Sperl, Junyao Tang, Sarath Tennakoon, Brian Tighe, John Wambaugh, Brian Utter, Peidong Yu, Jie Ren, Jie Zhang, Bulbul Chakraborty, Eric Clément, Isaac Goldhirsch, Lou Kondic, Stefan Luding, Guy Metcalfe, Corey O’Hern, David Schaeffer, Josh Socolar, Antoinette Tordesillas Support: NSF, ARO, NASA, IFPRI, BSF

2. Roadmap • What/Why granular materials? • Behavior of disordered solids—possible universal behavior? • Where granular materials and molecular matter part company—open questions of relevant scales Experiments at Duke explore: • Forces, force fluctuations • Isotropic jamming • Effect of shear—anisotropic stresses

3. What are Granular Materials? • Collections of macroscopic ‘hard’ particles: interactions are dissipative/frictional • Classical h 0 • Highly dissipative • Draw energy for fluctuations from macroscopic flow • Large collective systems, but outside normal statistical physics • Although many-body, a-thermal in the usual sense • Exist in phases: granular gases, fluids and solids • Analogues to other disordered solids: glasses, colloids.. • May be dry or wet

4. Broader context: Disordered N-body systems—far from equilibrium • There exist a number of such systems: glasses, foams, colloidal suspensions, granular materials,… • For various reasons, these systems are not in ordinary thermal equilibrium (although temperature may still play a role) • Energy may not be conserved • Other conservation rules—e.g. stress, may apply

5. Standard picture of jamming • Jamming—how disordered N-body systems becomes solid-like as particles are brought into contact, or fluid-like when grains are separated • Density is implicated as a key parameter, expressed as packing (solid fraction) φ • Marginal stability (isostaticity) for spherical particles (disks in 2D) contact number, Z, attains a critical value, Ziso at φiso

6. JammingHow do disordered solids lose/gain their solidity? Bouchaud et al. Liu and Nagel

7. Theoretical tools: Statistical ensembles: what to do when energy is not conserved? • Edwards ensemble for rigid particles: V replaces E • Real particles are deformable (elastic): forces, stresses, and torques matter, and are ‘conserved’ for static systems—hence force/stress ensembles emerge • Snoeijer, van Saarloos et al. Tighe, Socolar et al., Henkes and Chakraborty and O’Hern, Makse et al.

8. Experimental tools: what to measure, and how to look inside complex systems • Confocal techniques in 3D—with fluid-suspended particles—for colloids, emulsions, fluidized granular systems • Bulk measurements—2D and 3D • Measurements at boundaries—3D • 2D measurements: particle tracking, Photoelastic techniques (this talk) • Numerical experiments—MD/DEM

9. What happens when shear is applied? Plasticity—irreversible deformation when a material is sheared • System becomes anisotropic—e.g. long force chains form • Shear causes irreversible (plastic) deformation. Particles move ‘around’ each other • What is the microscopic nature of this process for granular materials?

10. Different types methods of applying shear • Example1: pure shear • Example 2: simple shear • Example 3: steady shear

11. Dense Granular Material Phases-Some simple observations Forces are carried preferentially on force chainsmultiscale phenomena Friction matters Howell et al. PRL 82, 5241 (1999)

12. Roadmap • What/Why granular materials? • Where granular materials and molecular matter part company—open questions of relevant scales • Dense granular materials: need statistical approach Use experiments to explore: • Forces, force fluctuations ◄ • Jamming—isotropic • Shear and anisotropic stresses

13. Experiments to determine vector contact forces, distributionP1(F) is example of particle-scale statistical measure Experiments use biaxial tester and photoelastic particles (Trush Majmudar and RPB, Nature, June 23, 2005)

14. Overview of Experiments Biax schematic Compression Shear Image of Single disk ~2500 particles, bi-disperse, dL=0.9cm, dS= 0.8cm, NS /NL = 4

15. Measuring forces by photoelasticity

16. Basic principles of technique • Process images to obtain particle centers and contacts • Interparticle contact forces determine stresses within each particle, including principal stresses, σi • Stresses determine photoelastic response: I = Iosin2[(σ2- σ1)CT/λ] • Now go backwards, using nonlinear inverse technique to obtain contact forces • In the previous step, invoke force and torque balance • Newton’s 3d law provides error checking

17. Examples of Experimental and ‘Fitted’ Images Experiment—color Filtered Experiment--original Fitted

18. Shear εxx = -εyy =0.04; Zavg = 3.1 Force distributionsfor shear and compression Compression εxx = -εyy =0.016; Zavg = 3.7 From T. Majmudar and RPB, Nature, 2005

19. Stress ensemble models for P(f) • Consider all possible states consistent with applied external forces, or other boundary conditions—assume all possible states occur with equal probability • Compute Fraction where at least one contact force has value f P(f) • E.g. Snoeier et al. PRL 92, 054302 (2004) • Tighe et al. Phys. Rev. E, 72, 031306 (2005)

20. Some Typical Cases—isotropic compression and shear Snoeijer et al. ↓ Tigue et al ↓. Compression  Shear  Latest update—B. Tigue, this session

21. What about force correlations? Shear Compression

22. Correlation functions determine important scales • C(r) = <Q(r + r’) Q(r’)> • <>  average over all vector displacements r’ • For isotropic cases, average over all directions in r. • Angular averages should not be done for anisotropic systems

23. Spatial correlations of forces—angle dependent Shear Compression Both directions equivalent Chain direction Direction  normal To chains New work: S. Henkes, B. Chakraborty, G. Lois, C. O’Hern, J. Zhang, RPB

24. Roadmap • What/Why granular materials? • Where granular materials and molecular matter part company—open questions of relevant scales • Dense granular materials: need statistical approach Use experiments to explore: • Forces, force fluctuations ◄ • Jamming--isotropic ◄ • Shear—anisotropic stresses

25. JammingHow do disordered solids lose/gain their solidity? Bouchaud et al. Liu and Nagel

26. The Isotropic Jamming Transition—Point J • Simple question: What happens to key properties such as pressure, contact number as a sample is isotropically compressed/dilated through the point of mechanical stability? Z = contacts/particle; Φ = packing fraction Predictions for spherical frictionless particles (e.g. O’Hern et al. Torquato et al., Schwarz et al., Wyart et al. β depends on force law (= 1 for ideal disks) Z ~ ZI +(φ – φc)ά(discontinuity) Exponent ά ≈ 1/2 P ~(φ – φc)β S. Henkes and B. Chakraborty: entropy-based model gives P and Z in terms of a field conjugate to entropy. Can eliminate to get P(z)

27. Experiment: Characterizing the Jamming Transition—Isotropic compression • Isotropic compression Majmudar et al. PRL 98, 058001 (2007)

28. How do we obtain stresses and Z? Fabric tensor Rij = Sk,c ncik ncjk Z = trace[R] Stress tensor :sij = (1/A) Sk,c rcik fcjk A is system area, trace of stress tensor gives P

29. LSQ Fits for Z give an exponent of 0.5 to 0.6

30. LSQ Fits for P give β ≈ 1.0 to 1.1

31. Comparison to Henkes and Chakraborty prediction

32. Roadmap • What/Why granular materials? • Where granular materials and molecular matter part company—open questions of relevant scales • Dense granular materials: need statistical approach Use experiments to explore: • Forces, force fluctuations ◄ • Isotropic jamming ◄ • Shear—anisotropic stresses◄

33. A different part of the jamming diagram σ2 σ1 What happens here or here, when shear is applied to a granular material? Note: P = (s2 + s1)/2 :t = (s2– s1)/2 Coulomb failure: |t|/P = m

34. Shear near jamming • Example1: pure shear • Example 2: simple shear • Example 3: steady shear

35. Starting point is anisotropic, unjammed state Isotroic jamming point at φ = 0.82 Start here, with φ = 0.76

36. What happens for granular materials subject to pure shear? Mark particles with UV-sensitive Dye for tracking Use biax and photoelastic particles J. Zhang et al. to appear, Granular Matter

37. Apply Cyclic Pure Shear—starting from an unjammed state And without polarizer Resulting state with polarizer

38. Consider one cycle of shear

39. Particle Displacements and Rotations Forward shear—under UV

40. Deformation Field—Shear band forms At strain = 0.085 At strain = 0.105—largest plastic event At strain = 0.111

41. Particle displacement and rotation (forward shear) • Green arrows are displace- • ment of particle center • Blob size stands for • rotation magnitude • Blue color—clockwise • rotation • Brown color— • counterclockwise rotation • Mean displacement subtracted

42. Stresses and Z

43. Hysteresis in stress-strain and Z-strain curves

44. But!! Apparent scaling for stresses vs. Z

45. Force Distributions Tangential forces Normal forces

46. Corrections for missed contacts Estimate of missed contacts: ∫0F-min P(F)dF/∫0∞ P(F)dF Similar approach corrects P, but Effect is much smaller, since P ~ ∫ F P(F) dF

47. How do we contemplate jamming in frictional granular materials? σ2 Sheared granular materials fail to other stable states σ1 (σ2 - σ1) Note that Reynolds Dilatancy  weakly confined samples dilate under shear— Hence, rigidly confined materials show an iσncrease of P under shear J ?? ?? 1/φ

48. Conclusions • Granular materials show important features due to friction • Distributions of forces show sensitivity to stress state but agree reasonably with stress-ensemble approach • Correlations for forces in sheared systems—thus, force chains can be mesoscopic • Predictions for jamming (mostly) verified • Granular states near jamming show jamming under shear at densities below isotropic jamming • Friction (and also preparation history) implicated • Z may be key variable for shear failure/jamming • Generalized SGR explains rate-dependence

49. What is actual force law for our disks?