Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, [email protected] Shear banding in granular materials Stefan Luding

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


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


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


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


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


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



Sound3-dimensional modeling of sound propagation



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:


- 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


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


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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Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft Stefan Luding, [email protected] Shear banding in granular materials Stefan Luding