1. Force Transmission in Granular Materials R.P. Behringer Duke University Support: US NSF, NASA Collaborators: Junfei Geng, Guillaume Reydellet, Eric Clement, Stefan Luding

2. OUTLINE Introduction –Overview –Important issues for force propagation –Models Experimental approach Exploration or order/disorder and friction Conclusion

3. Friction and frictional indeterminacy Condition for static friction:

4. Multiple contacts => indeterminacy Note: 5 contacts => 10 unknown force components. 3 particles => 9 constraints

5. Frictional indeterminacy => history dependence

6. Stress balance

7. Stress balance, Continued Four unknown stress components (2D) Three balance equations –Horizontal forces –Vertical forces –Torques Need a constitutive equation

8. Some approaches to describing stresses Elasto-plastic models (Elliptic, then hyperbolic) Lattice models –Q-model (parabolic in continuum limit) –3-leg model (hyperbolic (elliptic) in cont. limit) –Anisotropic elastic spring model OSL model (hyperbolic) Telegraph model (hyperbolic) Double-Y model (type not known in general)

9. Features of elasto-plastic models Conserve mass: (Energy: lost by friction) Conserve momentum:

10. Concept of yield and rate-independence  shear stress,  normal stress Stable up to yield surface

11. Example of stress-strain relationship for deformation (Strain rate tensor with minus) |V| = norm of V Contrast to a Newtonian fluid:

12. OSL model  phemonological parameters

13. q-model (e.g. in 2D) q’s chosen from uniform distribution on [0,1] Predicts force distributions ~ exp(-F/F o )

14. Long wavelength description is a diffusion equation Expected stress variation with depth

15. Convection-diffusion/3-leg model Applies for weak disorder Expected response to a point force:

16. Double-Y model Assumes Boltzmann equation for force chains For shallow depths: One or two peaks Intermediate depths: single peak-elastic-like Largest depths: 2 peaks, propagative, with diffusive widening

17. Anisotropic elastic lattice model Expect progagation along lattice directions Linear widening with depth

18. Schematic of greens function apparatus

19. Measuring forces by photoelasticity

20. Diametrically opposed forces on a disk

21. A gradient technique to obtain grain-scale forces

22. calibration

23. Disks-single response

24. Before-after

25. disk response mean

26. Large variance of distribution

27. Organization of Results Strong disorder: pentagons Varying order/disorder –Bidisperse disks –Reducing contact number: square packing –Reducing friction Comparison to convection-diffusion model Non-normal loading: vector/tensor effects Effects of texture

28. Pentagon response

29. Elastic response, point force on a semi-infinite sheet In Cartesian coordinates:

30. Example: solid photoelastic sheet

31. Moment test (See Reydellet and Clement, PRL, 2001)

32. Pentagons, width vs. depth

33. Variance of particle diameters to distinguish disorder

34. Spectra of particle density

35. Bidisperse responses vs. A

36. Weakly bi-disperse: two-peak structure remains

37. Bidisperse, data

38. Rectangular packing reduces contact disorder

39. Hexagonal vs. square packing

40. Hexagonal vs. square, data

41. Square packs, varying friction

42. Data for rectangular packings

43. Fits to convection-diffusion model

44. Variation on CD model--CW

45. Fits to CD- and CW models

46. Non-normal response, disks, various angles

47. Non-normal response vs. angle of applied force

48. Non-normal responses, pentagons

49. Non-normal response, pentagons, rescaled

50. Creating a texture by shearing

51. Evolution of force network– 5 degree deformation

52. Force correlation function

53. Correlation functions along specific directions

54. Response in textured system

55. Response, textured system, data

56. Fabric in textured system

57. “Fabric” from strong network

58. Conclusions Strong effects from order/disorder (spatial and force-contact) Ordered systems: propagation along lattice Disorderd: roughly elastic response Textured systems –Power law correlation along preferred direction –Forces tend toward preferred direction Broad distribution of local response