Shear banding in granular materials

1
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)

Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
2
Contents

• Introduction
• Results DEM
• Micro-Macro (global)
• Micro-Macro (local)
• Outlook
• - Anisotropy
• - Rotations

3
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
• inhomogeneity rotations
• Yield-surface

Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
5
Material behavior of granular media 3D

• 3-dimensional modeling of shear and sound
  propagation

Universal shear zones
P-wave shape and speed
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
6
Sound propagation in granular media

• 3-dimensional modeling of sound propagation

P-wave shape and speed
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
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 (104- 106)
• Industrial scale applications rely on FEM
• FEM relies on continuum mechanics
• constitutive relations
• Micro-macro can provide those !
• Homogenization/Averaging

9
Contents

• Introduction
• Results DEM

10
Discrete particle model
Equations of motion
Forces and torques
Overlap
Normal
11
Contact force measurement (PIA)
12
Hysteresis (plastic deformation)
13
Contact model
- (too) simple ? - piecewise linear - easy to
implement
14
Linear Contact model
- (really too) simple ? - linear - very easy to
implement
15
Hertz Contact model
- simple ? - non-linear - easy to implement
16
Sound
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
17
Sound

• 3-dimensional modeling of sound propagation

P-wave shape and speed
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
18
Micro simulation of shear bands


potential energy
rotations
displacements
19
Model system bi-axial box
Strain Control (top)
Initial position and final position z0 and zf
Stress-control (right)

• With wall mass mx, friction ?x and
• Bulk material force Fx(t)
• External force pxz(t), and
• Frictional force ?xx(t)

20
Simulation results
?zz9.1
?zz4.2
?zz1.1
?zz0.0
21
Bi-axial box
22
Bi-axial box
23
Contents

• Introduction
• Results DEM
• Micro-macro (global)

24
Bi-axial compression with pxconst.
25
Cohesion no friction
geometrical friction angle
kc/k2 0 ½ 1 2 4
px 100 100 100 100 100
?zz 183 234 264 310 336
px 500 500 500 500 500
?zz 798 853 915 941 972
c 11 32 40 60 71
fmin
macro cohesion
26
Tangential contact model

• Sliding contact points
• Static Coulomb friction
• Dynamic Coulomb friction

27
Tangential contact model

• - Static friction
• Dynamic friction
• project into tangential plane
• compute test force
• sticking
• sliding

• - spring
• dashpot

before contact static
dynamic static
28
Friction no cohesion
kc 0 and µ 0.5
Internal friction angle
Total friction angle
29
Bi-axial box rotations
30
Rotations (local)
Direction, amplitude, anti-symmetric (!) stress
31
Tangential contact model

• Rolling (mimic roughness
• or steady contact necks)
• Static rolling resistance
• Dynamic resistance

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
Summary micro-macro (global)
DEM simulation ?macroscopic material behaviour
Micro Disks (in 2-D) Particle size a 0.5
1.5 mm Stiffness k2 105 Nm-1and
k1k2/2 Cohesion kc 0, ½, 1, 2, 4
k2 Friction µ 0
Bi-axial box setup ?
Macro Youngs Modulus E ? k1 Poisson ratio ? ?
0.66 Friction angle ? ? 13 (µ0) and ? ? 31
(µ0.5) Dilatancy angle ? ? 5 (p500) and ?
?11 (p100) Cohesion
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
shear1
horizontal
shear2
vertical
horizontal
shear
shear3
40
Local micro-macro transition

• From virtual work
• For each single contact
• Stress tensor ?
• Stiffness matrix C (elastic)
• Normal contacts
• Tangential springs

• Deformations (2D)
• Isotropic compression ?V
• Deviatoric strain (shear) ?D, ??

• Stress changes
• Isotropic ??V
• Deviatoric ??D, ??

41
Constitutive model with rotations
42
Hypoplastic FEM model
successful tool few parameters - microscopic
foundations ? - extensions parameter
identification
deformation - rotations
cyclic deformations - creep
Continuum Theory
43
Micro-macro transition
density vs. pressure friction vs.
density
44
Summary part 1 global view

• Granular media are
• compressible
• inhomogeneous (force-chains)
• (almost always) an-isotropic
• and
• micro-polar (rotations)

45
Contents

• Introduction
• Results
• Micro-macro (global)
• Micro-macro (local)

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
Micro informations shear bands


potential energy
rotations
displacements
48
Rotations (local)
Direction, amplitude, anti-symmetric (!) stress
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
55
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
Averaging Velocity

• Any quantity
• - Scalar
• - Vector velocity density

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

center
wall
?
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
exp.
exp.
sim.
sim.
high dens.
low dens.
83
Macro (torque stiffness)
84
An-isotropy and rotations

• Deformations (2D)
• Isotropic ?V
• Deviatoric (shear) ?D, ??
• Mean rotation (slip) ?
• Difference rot. (rolling) ?

• Stress changes ???
• Isotropic ??V
• Deviatoric ??D, ??
• Asymmetric ??A

85
Constitutive model with rotations
86
Summary

• Granular media are
• compressible
• inhomogeneous (force-chains)
• (almost always) an-isotropic
• micro-polar (rotations)

Granular media are interesting
87
Open issues
Accounting for inhomogeneities (temperature) Struc
ture evolution with deformation (time) Micropolar
continuum theory
88
The End
89
(No Transcript)
90
Contact force measurement (PIA)
91
Hysteresis (plastic deformation)
92
Contact model
- (too) simple ? - piecewise linear - easy to
implement
93
Linear Contact model
- (really too) simple ? - linear - very easy to
implement
94
Hertz Contact model
- simple ? - non-linear - easy to implement
95
Sound
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
96
Sound

• 3-dimensional modeling of sound propagation

P-wave shape and speed
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
97
Tangential contact model

• Sliding contact points
• Static Coulomb friction
• Dynamic Coulomb friction

98
Tangential contact model

• - Static friction
• Dynamic friction
• project into tangential plane
• compute test force
• sticking
• sliding

• - spring
• dashpot

before contact static
dynamic static
99
Bi-axial box rotations
100
Rotations (local)
Direction, amplitude, anti-symmetric (!) stress
101
Tangential contact model

• Rolling (mimic roughness
• or steady contact necks)
• Static rolling resistance
• Dynamic resistance

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

105
(No Transcript)
106
Hypoplastic FEM model
successful tool few parameters - microscopic
foundations ? - extensions parameter
identification
deformation - rotations
cyclic deformations - creep
Continuum Theory
107
Micro-macro transition
density vs. pressure friction vs.
density
108
Local micro-macro transition

• From virtual work
• For each single contact
• Stress tensor ?
• Stiffness matrix C (elastic)
• Normal contacts
• Tangential springs

• Deformations (2D)
• Isotropic compression ?V
• Deviatoric strain (shear) ?D, ??

• Stress changes
• Isotropic ??V
• Deviatoric ??D, ??

109

110

111
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