Title: Shear banding in granular materials
1Shear 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
2Contents
- Introduction
- Results DEM
- Micro-Macro (global)
- Micro-Macro (local)
- Outlook
- - Anisotropy
- - Rotations
3Contents
E x p e r i m e n t s
- Introduction
- Results DEM
- Micro-Macro (global)
- Micro-Macro (local)
- Outlook
- - Anisotropy
- - Rotations
4Material 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
5Material 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
6Sound 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
7Applications
- Granular media (powder, sand, )
- Colloidal systems (fluid)
- Atomistic systems in confined geometries
- Many particle systems (traffic, pedestrians)
- Industrial scale applications ?
8Why ?
- 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
9Contents
10Discrete particle model
Equations of motion
Forces and torques
Overlap
Normal
11Contact force measurement (PIA)
12Hysteresis (plastic deformation)
13Contact model
- (too) simple ? - piecewise linear - easy to
implement
14Linear Contact model
- (really too) simple ? - linear - very easy to
implement
15Hertz Contact model
- simple ? - non-linear - easy to implement
16Sound
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
17Sound
- 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
18Micro simulation of shear bands
potential energy
rotations
displacements
19Model 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)
20Simulation results
?zz9.1
?zz4.2
?zz1.1
?zz0.0
21Bi-axial box
22Bi-axial box
23Contents
- Introduction
- Results DEM
- Micro-macro (global)
24Bi-axial compression with pxconst.
25Cohesion 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
26Tangential contact model
- Sliding contact points
- Static Coulomb friction
- Dynamic Coulomb friction
27Tangential contact model
- - Static friction
- Dynamic friction
- project into tangential plane
- compute test force
- sticking
- sliding
before contact static
dynamic static
28Friction no cohesion
kc 0 and µ 0.5
Internal friction angle
Total friction angle
29Bi-axial box rotations
30Rotations (local)
Direction, amplitude, anti-symmetric (!) stress
31Tangential contact model
- Rolling (mimic roughness
- or steady contact necks)
- Static rolling resistance
- Dynamic resistance
32Silo Flow with friction rolling friction
33Silo Flow with friction
34Tangential contact model
- Torsion (large contact area)
- Static torsion resistance
- Dynamic resistance
35Summary 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
36An-isotropy
37Stiffness tensor
vertical
horizontal
shear
anisotropy
38An-isotropy
- Structure changes with deformation
- Different stiffness
- More stiff against shear
- Less stiff perpendicular
- Evolution with time
39Stiffness tensor
vertical
shear1
horizontal
shear2
vertical
horizontal
shear
shear3
40Local 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, ??
41Constitutive model with rotations
42Hypoplastic FEM model
successful tool few parameters - microscopic
foundations ? - extensions parameter
identification
deformation - rotations
cyclic deformations - creep
Continuum Theory
43Micro-macro transition
density vs. pressure friction vs.
density
44Summary part 1 global view
- Granular media are
- compressible
- inhomogeneous (force-chains)
- (almost always) an-isotropic
- and
- micro-polar (rotations)
45Contents
- Introduction
- Results
- Micro-macro (global)
- Micro-macro (local)
46Global 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
47Micro informations shear bands
potential energy
rotations
displacements
48Rotations (local)
Direction, amplitude, anti-symmetric (!) stress
49Ring geometry (Behringer et al.)
50Ring geometry
512D shear cell energy
522D shear cell force chains
532D shear cell shear band
54Ring geometry
55Ring geometry
56Averaging Formalism
- Any quantity
- In averaging volume
57Averaging Formalism
- Any quantity
- - Scalar
- - Vector
- - Tensor
58Averaging Density
- Any quantity
- - Scalar Density/volume fraction
59Density profile
60Averaging Velocity
- Any quantity
- - Scalar
- - Vector velocity density
61Velocity field
62Velocity gradient
63Velocity gradient
64Velocity distribution
65Averaging Fabric
- Any quantity
- - Scalar contacts
- - Vector normal
- - Contact distribution
66Fabric tensor
center
wall
?
67Fabric tensor (trace)
?
68Macro (contact density)
69Fabric tensor (deviator)
?
70Averaging Stress
- Any quantity
- - Scalar
- - Vector
- - Tensor Stress
71Stress tensor (static)
?
72Stress tensor (dynamic)
exponential
73Stress equilibrium (1)
74Stress equilibrium (2)
?
75Averaging Deformations
- Deformation
- - Scalar
- - Vector
- - Tensor Deformation
76Macro (bulk modulus)
77Macro (shear modulus)
78Constitutive model no rotations
79Averaging Rotations
- Deformation
- - Scalar
- - Vector Spin density
- - Tensor
80Rotations spin density
eigen-rotation
81Spin distribution
82Velocity-spin distribution
exp.
exp.
sim.
sim.
high dens.
low dens.
83Macro (torque stiffness)
84An-isotropy and rotations
- Deformations (2D)
- Isotropic ?V
-
- Deviatoric (shear) ?D, ??
-
- Mean rotation (slip) ?
-
- Difference rot. (rolling) ?
- Stress changes ???
- Isotropic ??V
-
- Deviatoric ??D, ??
-
- Asymmetric ??A
85Constitutive model with rotations
86Summary
- Granular media are
- compressible
- inhomogeneous (force-chains)
- (almost always) an-isotropic
- micro-polar (rotations)
Granular media are interesting
87Open issues
Accounting for inhomogeneities (temperature) Struc
ture evolution with deformation (time) Micropolar
continuum theory
88The End
89(No Transcript)
90Contact force measurement (PIA)
91Hysteresis (plastic deformation)
92Contact model
- (too) simple ? - piecewise linear - easy to
implement
93Linear Contact model
- (really too) simple ? - linear - very easy to
implement
94Hertz Contact model
- simple ? - non-linear - easy to implement
95Sound
Stefan Luding, s.luding_at_tnw.tudelft.nl
Particle Technology, DelftChemTech, Julianalaan
136, 2628 BL Delft
96Sound
- 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
97Tangential contact model
- Sliding contact points
- Static Coulomb friction
- Dynamic Coulomb friction
98Tangential contact model
- - Static friction
- Dynamic friction
- project into tangential plane
- compute test force
- sticking
- sliding
before contact static
dynamic static
99Bi-axial box rotations
100Rotations (local)
Direction, amplitude, anti-symmetric (!) stress
101Tangential contact model
- Rolling (mimic roughness
- or steady contact necks)
- Static rolling resistance
- Dynamic resistance
102Silo Flow with friction rolling friction
103Silo Flow with friction
104Tangential contact model
- Torsion (large contact area)
- Static torsion resistance
- Dynamic resistance
105(No Transcript)
106Hypoplastic FEM model
successful tool few parameters - microscopic
foundations ? - extensions parameter
identification
deformation - rotations
cyclic deformations - creep
Continuum Theory
107Micro-macro transition
density vs. pressure friction vs.
density
108Local 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, ??
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