particle technology, delftchemtech, julianalaan 136, 2628 bl delft stefan luding,...
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Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft
Stefan Luding, [email protected]
Shear banding in granular materials
Stefan LudingPART/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)
• Introduction
• Results DEM
• Micro-Macro (global)
• Micro-Macro (local)
• Outlook
- Anisotropy
- Rotations
Single particle
Contacts
Many particle simulation
Continuum Theory
Contents
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
Why ?
• `Many particle’ simulations workfor 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
• Introduction
• Results DEMContacts
Many particle simulation
Contents
Equations of motion2
2i
i i
d rmdt
f
Overlap 1
2 i j i jd d r r n
i j
ij
i j
r rn n
r r
c wii i ic w
f f f m g
Forces and torques:
i
i i
dIdt
t
44444444444444
c ci ici r ft 44444444444444
Normal
Contacts
Many particle simulation
Discrete particle model
0.0 0.2 0.4 0.6 0.80.0
0.1
0.2
0.3
0.4
0.5
Fig. 6
Nor
mal
ized
adh
esio
n fo
rce
(mN
/m)
Normalized load (mN/m)
Contact force measurement (PIA)
0( ) 1 cos2
ff
z zz t z t
Strain Control (top):
Initial position and final position z0 and zf
0
1zz
z
z
( ) ( ) ( )x xxx pm x t F t z t x Stress-control (right):
With wall mass: mx, friction x and (i) Bulk material force Fx(t)(ii) External force –pxz(t), and (iii) Frictional force xx(t)
Model system bi-axial box
zz=9.1%zz=4.2%zz=1.1%zz=0.0%
Simulation results
Bi-axial box
Bi-axial box
• Introduction
• Results DEM
• Micro-macro (global)
Contacts
Many particle simulation
Continuum Theory
Contents
Bi-axial compression with px=const.
2/ 0, 1/2, 1, 2, and 4ck k
geometrical friction angle
13
kc/k2
0½124
px
100100100100100
zz
183234264310336
px
500500500500500
zz
798853915941972
c1132406071
fmin
3 1 2
2
11.8 10
1 c
k k
kc
k
macro cohesion
Cohesion – no friction
kc = 0 and µ = 0.5
Internal friction angle 27
31 Total friction angle
Friction – no cohesion
Bi-axial box rotations
An-isotropy
Stiffness tensor
vertical
horizontal
shear
2
1
Cn c c c c
c
kaC n n n n
V
1122nC
11112222n n
D CC C
1111nC
2222nC
anisotropy
An-isotropy
• Structure changes with deformation• Different stiffness:
• More stiff against shear• Less stiff perpendicular
• Evolution with time:
maxD
DDDC C C
max1 exp
DD
D
C
C max . 0.34D conC st
,...f p
Stiffness tensor
vertical
vertical
horizontal
horizontalshear
shear1
shear2
shear3
Granular media are:- compressible- inhomogeneous (force-chains)- (almost always) an-isotropicand- micro-polar (rotations)
Summary – part 1: global view
• Introduction
• Results
• Micro-macro (global)
• Micro-macro (local)
Single particle
Contacts
Many particle simulation
Continuum Theory
Contents
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
potential energy rotations displacements
Micro informations: shear bands
Direction, amplitude, anti-symmetric (!) stress
Rotations (local)
Ring geometry (Behringer et al.)
Ring geometry
2D shear cell – energy
2D shear cell – force chains
2D shear cell – shear band
Ring geometry
Ring geometry
Averaging Formalism
Any quantity:
In averaging volume:
1p p p pV
p V
Q Q w V QV
pQV
V
Averaging Formalism
Any quantity:
- Scalar- Vector- Tensor
1p p p pV
p V
Q Q w V QV
p c
c
Q Q
Averaging Density
1 p pV
p V
Q w VV
V1pQ
Any quantity:
- Scalar: Density/volume fraction
Density profile
Global volume fraction
Any quantity:
- Scalar- Vector – velocity density
Averaging Velocity
1 p pV
p V
pQ w VV
v v
VppQ v
pv
Velocity field
exponential
Velocity gradient
exponential: 0( ) exp( ( ) )iv r v r R s
1
2r
v vD
r r
v
width of shearband: 1...2s
d
Velocity distribution
Averaging Fabric
Any quantity:
- Scalar: contacts- Vector: normal- Contact distribution
pp pc pc
c
Q F n n
1 p p pc pcV
p V c
Q w VV
F n n
p
V
c
Fabric tensorcontact probability …
center wall
Fabric tensor (trace)
contact number density
Macro (contact density)
Fabric tensor (deviator)
an-isotropy (!)
Averaging Stress
Any quantity:
- Scalar- Vector- Tensor: Stress
1pp pc cp
c
QV
σ l f
1 p pc cV
p V c
Q wV
σ l f
p
V
cf
Stress tensor (static)
shear stress 2r
Stress tensor (dynamic)
exponential
Stress equilibrium (1)
( )1 ( ) 1 rrrr r
rre e
r r r r
( )( )und rrr
r
rr
r r
0 2( und )rr r rr r
( )ddt ta v v v v
20 ( )rrrra v r
r
0 ( )rr rr
r
acceleration:
Stress equilibrium (2)20 ( )rr
rra v rr
0 ( )rr rr
r
?
?
Averaging Deformations
Deformation:
- Scalar- Vector- Tensor: Deformation
1p pc cV
p V c
hQ w
V
Fl
p
V
cΔ
2 minimal !c pcS l
Macro (bulk modulus) tr
trE
Macro (shear modulus) dev
devG
d
*
iv
12
*
bulk stiffness: tr : for
shear stiffness: else
deviator orientation: 1 0
ˆ
unit-deviator operator:
ˆ
ˆ0
D DV V
V V
V
V V V
F
T
G
G
E
E FF F F
σ 1 1
F
D D
RD 1
R
Constitutive model – no rotations
Averaging Rotations
Deformation:
- Scalar- Vector: Spin density- Tensor
p
V
cp pQ
1 p p pV
p V c
Q w VV
p
Rotations – spin density*
rW eigen-rotation:
Spin distribution
Velocity-spin distribution
high dens. low dens.
sim.sim.
exp. exp.
Macro (torque stiffness) 2 Ml
2l
Deformations (2D):
- Isotropic V - Deviatoric (=shear) D, - Mean rotation (=slip) * - Difference rot. (=rolling) **
Stress changes ???
- Isotropic V - Deviatoric D, - Asymmetric A
An-isotropy and rotations
o o*
*
* *1
45 90
* **
1 2 2* *
1 1 2 2
*
**
particle eigen-rotations: = sliding component: rolling component:
ˆ
with: sliding stiffnes
ˆ
s:
ˆDV D A
V
ci i
F F
i
SE F
aa a
RG
a
σ 1 σD D D
o 45 44
and rolling stiffness:
1 0unit-deviator operator:
0 1ˆ T
FF F
S R
R RD
Constitutive model with rotations
Granular media are:- compressible- inhomogeneous (force-chains)- (almost always) an-isotropic- micro-polar (rotations)
Summary
Granular media are …… interesting
Open issues
Accounting for inhomogeneities (temperature)
Structure evolution with deformation (time)
Micropolar continuum theory
…
The End
0.0 0.2 0.4 0.6 0.80.0
0.1
0.2
0.3
0.4
0.5
Fig. 6
Nor
mal
ized
adh
esio
n fo
rce
(mN
/m)
Normalized load (mN/m)
Contact force measurement (PIA)
0 50 100 150 200 250-5
0
5
10
15
20Polystryrene
Def
orm
atio
n H
yste
resi
s in
nm
Force in nN0 1000 2000 3000
-5
0
5
10
15
20
Borosilicate glass
Def
orm
atio
n H
yste
resi
s in
nm
Force in nN
Hysteresis (plastic deformation)
1
2 0
for loading
for un-/reloading
- for unloading
hysi
c
k
f k
k
max
0 1 2 max
2 1min max
2
min min
1 /
c
c
k k
k k
k k
f k
Maximum overlap
stress-free overlap
strong attraction
attractive for
est at:
the max. : ce
- (too) simple - piecewise linear- easy to implement
Contact model
1 for un-/re-loadinghysi
k
f
- (really too) simple - linear- very easy to implement
Linear Contact model
3/ 21 for un-/re-loading
hysi
k
f
- simple - non-linear- easy to implement
Hertz Contact model
Sound
Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft
Stefan Luding, [email protected]
Sound3-dimensional modeling of sound propagation
Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft
Stefan Luding, [email protected]
P-wave shape and speed
Sliding contact points:- Static Coulomb friction - Dynamic Coulomb friction
Tangential contact model
t
t rollin
torsion
ˆ
ˆ
s
ˆˆ
g
lidingi j
i j
i j
i j
i j
i j
i j n a a
n b b
n cn c
v v
v
- Static friction - Dynamic friction
project into tangential planecompute test force
sticking:sliding:
Tangential contact model
- spring- dashpot
ˆ ˆn n 0 0 0
t t t t tˆ and f ftk f
0t s nf f
0t s nf f dt 0
t tf f
dt n t̂f f t t tf k
before contact static dynamic static
Bi-axial box rotations
Direction, amplitude, anti-symmetric (!) stress
Rotations (local)
Rolling (mimic roughnessor steady contact necks)
- Static rolling resistance- Dynamic resistance
Tangential contact model
t
t rollin
torsion
ˆ
ˆ
s
ˆˆ
g
lidingi j
i j
i j
i j
i j
i j
i j n a a
n b b
n cn c
v v
v
Silo Flow with friction +rolling friction
0.5 0.5
0.2r
Silo Flow with friction 0.5
0.2r
t
t rollin
torsion
ˆ
ˆ
s
ˆˆ
g
lidingi j
i j
i j
i j
i j
i j
i j n a a
n b b
n cn c
v v
v
Tangential contact model
Torsion (large contact area)- Static torsion resistance- Dynamic resistance
+ successful tool – few parameters
- microscopic foundations ?
- extensions & parameter identification
Continuum Theory
deformation - rotations
0.5
0.6
0.7
0.8
0.9
1.0
-0.1 0 0.10.5
0.6
0.7
0.8
0.9
1.0
u/h
e
cyclic deformations - creep
Hypoplastic FEM model
density vs. pressure & friction vs. density
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,
Local micro-macro transition
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