1. Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl Shear banding in granular materials Stefan Luding PART/DCT, TUDelft, NL Thanks to: M. Lätzel, M.-K. Müller (Stuttgart), R. P. Behringer, J. Geng, D. Howell (Duke), M. van Hecke, D. Fenistein (Leiden)

2. Introduction Results DEM Micro-Macro (global) Micro-Macro (local) Outlook - Anisotropy - Rotations Single particle Contacts Many particle simulation Continuum Theory Contents

3. Single particle Contacts Many particle simulation Continuum Theory Contents E x p e r i m e n t s Introduction Results DEM Micro-Macro (global) Micro-Macro (local) Outlook - Anisotropy - Rotations

4. Material behavior of granular media Non-Newtonian Flow behavior under slow shear inhomogeneityrotations Yield-surface Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl

5. Material behavior of granular media 3D 3-dimensional modeling of shear and sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed Universal shear zones

6. Sound propagation in granular media 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed

7. Applications Granular media (powder, sand, …) Colloidal systems (+fluid) Atomistic systems in confined geometries Many particle systems (traffic, pedestrians) Industrial scale applications ?

8. Why ? `Many particle simulations work for small systems only (10 4 - 10 6 ) Industrial scale applications rely on FEM FEM relies on continuum mechanics + constitutive relations `Micro-macro can provide those ! Homogenization/Averaging

9. Introduction Results DEM Contacts Many particle simulation Contents

10. Equations of motion Overlap Forces and torques: Normal Contacts Many particle simulation Discrete particle model

11. Contact force measurement (PIA)

12. Hysteresis (plastic deformation)

13. - (too) simple - piecewise linear - easy to implement Contact model

14. - (really too) simple - linear - very easy to implement Linear Contact model

15. - simple - non-linear - easy to implement Hertz Contact model

16. Sound Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl

17. Sound 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed

18. potential energyrotationsdisplacements Micro simulation of shear bands

19. Strain Control (top): Initial position and final position z 0 and z f Stress-control (right): With wall mass: m x, friction x and (i)Bulk material force F x (t) (ii)External force –p x z(t), and (iii)Frictional force x x(t) Model system bi-axial box

20. zz =9.1% zz =4.2% zz =1.1% zz =0.0% Simulation results

21. Bi-axial box

23. Introduction Results DEM Micro-macro (global) Contacts Many particle simulation Continuum Theory Contents

24. Bi-axial compression with p x =const.

25. geometrical friction angle kc/k20½124kc/k20½124 p x 100 zz 183 234 264 310 336 p x 500 zz 798 853 915 941 972 c 11 32 40 60 71 f min macro cohesion Cohesion – no friction

26. Sliding contact points: - Static Coulomb friction - Dynamic Coulomb friction Tangential contact model

27. - Static friction - Dynamic friction project into tangential plane compute test force sticking: sliding: Tangential contact model - spring - dashpot before contact static dynamic static

28. k c = 0 and µ = 0.5 Internal friction angle Total friction angle Friction – no cohesion

29. Bi-axial box rotations

30. Direction, amplitude, anti-symmetric (!) stress Rotations (local)

31. Rolling (mimic roughness or steady contact necks) - Static rolling resistance - Dynamic resistance Tangential contact model

32. Silo Flow with friction +rolling friction

33. Silo Flow with friction

34. Tangential contact model Torsion (large contact area) - Static torsion resistance - Dynamic resistance

35. DEM simulation macroscopic material behaviour Micro: Disks (in 2-D) Particle size a = [0.5 : 1.5] mm Stiffness k 2 = 10 5 Nm -1 and k 1 =k 2 /2 Cohesion k c = 0, ½, 1, 2, 4 k 2 Friction µ = 0 Macro: Youngs Modulus E k 1 Poisson ratio 0.66 Friction angle 13° (µ=0) and 31° (µ=0.5) Dilatancy angle 5° (p=500) and 11° (p=100) Cohesion: + Bi-axial box setup Summary: micro-macro (global)

36. An-isotropy

37. Stiffness tensor vertical horizontal shear anisotropy

38. An-isotropy Structure changes with deformation Different stiffness: More stiff against shear Less stiff perpendicular Evolution with time:

39. Stiffness tensor vertical horizontal shear shear1 shear2 shear3

40. From virtual work … For each single contact … Þ Stress tensor Þ Stiffness matrix C (elastic) - Normal contacts - Tangential springs Deformations (2D): - Isotropic compression V - Deviatoric strain (=shear) D, Þ Stress changes - Isotropic V - Deviatoric D, Local micro-macro transition

41. Constitutive model with rotations

42. + successful tool – few parameters - microscopic foundations ? - extensions & parameter identification Continuum Theory deformation - rotations cyclic deformations - creep Hypoplastic FEM model

43. density vs. pressure & friction vs. density Micro-macro transition

44. Granular media are: - compressible - inhomogeneous (force-chains) - (almost always) an-isotropic and - micro-polar (rotations) Summary – part 1: global view

45. Introduction Results Micro-macro (global) Micro-macro (local) Single particle Contacts Many particle simulation Continuum Theory Contents

46. Global average vs. Local average Global + Experimentally accessible data - Wall effects - Averaging over inhomogeneities Local - Difficult to compare to experiment + Averages away from the walls + Average over `similar volume elements

47. potential energyrotationsdisplacements Micro informations: shear bands

48. Direction, amplitude, anti-symmetric (!) stress Rotations (local)

49. Ring geometry (Behringer et al.)

50. Ring geometry

51. 2D shear cell – energy

52. 2D shear cell – force chains

53. 2D shear cell – shear band

54. Ring geometry

56. Averaging Formalism Any quantity: In averaging volume:

57. Averaging Formalism Any quantity: - Scalar - Vector - Tensor

58. Averaging Density Any quantity: - Scalar: Density/volume fraction

59. Density profile Global volume fraction

60. Any quantity: - Scalar - Vector – velocity density Averaging Velocity

61. Velocity field exponential

62. Velocity gradient exponential:

63. Velocity gradient exponential:

64. Velocity distribution

65. Averaging Fabric Any quantity: - Scalar: contacts - Vector: normal - Contact distribution

66. Fabric tensor contact probability … centerwall

67. Fabric tensor (trace) contact number density

68. Macro (contact density)

69. Fabric tensor (deviator) an-isotropy (!)

70. Averaging Stress Any quantity: - Scalar - Vector - Tensor: Stress

71. Stress tensor (static) shear stress

72. Stress tensor (dynamic) exponential

73. Stress equilibrium (1) acceleration:

74. Stress equilibrium (2) ? ?

75. Averaging Deformations Deformation: - Scalar - Vector - Tensor: Deformation

76. Macro (bulk modulus)

77. Macro (shear modulus)

78. Constitutive model – no rotations

79. Averaging Rotations Deformation: - Scalar - Vector: Spin density - Tensor

80. Rotations – spin density eigen-rotation:

81. Spin distribution

82. Velocity-spin distribution high dens.low dens. sim. exp.

83. Macro (torque stiffness)

84. Deformations (2D): - Isotropic V - Deviatoric (=shear) D, - Mean rotation (=slip) * - Difference rot. (=rolling) ** Þ Stress changes ??? - Isotropic V - Deviatoric D, - Asymmetric A An-isotropy and rotations

85. Constitutive model with rotations

86. Granular media are: - compressible - inhomogeneous (force-chains) - (almost always) an-isotropic - micro-polar (rotations) Summary Granular media are … … interesting

87. Open issues Accounting for inhomogeneities (temperature) Structure evolution with deformation (time) Micropolar continuum theory …

88. The End

90. Contact force measurement (PIA)

91. Hysteresis (plastic deformation)

92. - (too) simple - piecewise linear - easy to implement Contact model

93. - (really too) simple - linear - very easy to implement Linear Contact model

94. - simple - non-linear - easy to implement Hertz Contact model

95. Sound Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl

96. Sound 3-dimensional modeling of sound propagation Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, s.luding@tnw.tudelft.nl P-wave shape and speed

97. Sliding contact points: - Static Coulomb friction - Dynamic Coulomb friction Tangential contact model

98. - Static friction - Dynamic friction project into tangential plane compute test force sticking: sliding: Tangential contact model - spring - dashpot before contact static dynamic static

99. Bi-axial box rotations

100. Direction, amplitude, anti-symmetric (!) stress Rotations (local)

101. Rolling (mimic roughness or steady contact necks) - Static rolling resistance - Dynamic resistance Tangential contact model

102. Silo Flow with friction +rolling friction

103. Silo Flow with friction

104. Tangential contact model Torsion (large contact area) - Static torsion resistance - Dynamic resistance

106. + successful tool – few parameters - microscopic foundations ? - extensions & parameter identification Continuum Theory deformation - rotations cyclic deformations - creep Hypoplastic FEM model

107. density vs. pressure & friction vs. density Micro-macro transition

108. From virtual work … For each single contact … Þ Stress tensor Þ Stiffness matrix C (elastic) - Normal contacts - Tangential springs Deformations (2D): - Isotropic compression V - Deviatoric strain (=shear) D, Þ Stress changes - Isotropic V - Deviatoric D, Local micro-macro transition

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