1

Micromechanics Micromechanics of granular materials:of granular materials:

slow flowsslow flows

Niels P. Kruyt

Department of Mechanical Engineering, University of Twente,

n.p.kruyt@utwente.nlwww.ts.ctw.utwente.nl/kruyt/

2

Applications of granular materialsApplications of granular materials

• geotechnical engineering• geophysical flows• bulk solids engineering • chemical process engineering• mining• gas and oil production• food-processing industry• agriculture• pharmaceutical industry

3

FeaturesFeatures

• rate-independent• elasticity• plasticity• frictional• dilatancy• anisotropy

• viscous• multi-phase• cohesion• segregation

4

Overview talkOverview talk

• Introduction• Continuum modelling• Micromechanical modelling

• Effect of interparticle friction on macroscopic behaviour

• Validity of mean-field approaches

• Mixing in rotating cylinders

5

Slow granular flowsSlow granular flows

• Rate-independent; quasi-static

• Continuum level

• stress tensor

• strain tensor

6

Stress and strain tensorsStress and strain tensors

• Stress tensor

• Strain tensor

dAndf jiji σ=jnidf

dA

jiji dxdu ε=idx

0iu

ii duu +0

7

Constitutive relationsConstitutive relations

• Continuum theories; elasto-plasticity

• elastic and plastic strains

• yield function

• flow rule: plastic potential

• Micromechanical theories

• relation with microstructure

8

Plasticity of Plasticity of metals metals ↔↔ granular materialsgranular materials

• pressure dependence

• volume changes: dilatancy

• non-associated flowrule

• strain-softening

9

Micromechanics of slow flowsMicromechanics of slow flows

Study relation between micro and macro level

Work with Leo Rothenburg, University of Waterloo, Canada

2σ

1σ 1σ

2σ

10

ObjectiveObjective

• Micro-level → macro-level • Account for the discrete nature• Emphasis on slow flows

• Elasticity• Plasticity

• Theory and DEM simulations

11

From discrete information From discrete information →→stress and strainstress and strain

12

Averaging/Averaging/homogenisationhomogenisation

13

StressStress

jijR

i NAF σplane=

∑∈

=Cc

cj

ciij fl

V

1σ

14

CompatibilityCompatibility equationsequations

∑ =∆−=∆

S

RSi

pi

qi

pqi UU

0

pi

qi

pqi UU −=∆

0=∆+∆∑ αRi

S

RSi

[ ] [ ] [ ] [ ]0=

−+−+−+−=

∆+∆+∆+∆=∆∑ai

di

di

ci

ci

bi

bi

ai

dai

cdi

bci

abi

S

RSi

UUUUUUUU

15

StrainStrain

jijLi dXdU ε=

∑∈

∆=Cc

ci

cjij h

A

1ε

pi

qi

pqi UU −=∆

16

MicromechanicsMicromechanics

Relative displace-

ment

Stress Strain

Constitutiverelation

Microscopic level(contact)

Macroscopic level(continuum)

Ave

ragi

ng

Ave

ragi

ng

Loc

alis

atio

n

2σ

1σ 1σ

2σ

Loc

alis

atio

n

Statics Kinematics

Force

17

Topics in micromechanicsTopics in micromechanics

• Importance of geometrical anisotropy

• Effect of interparticle friction on macroscopic behaviour

• Validity of mean-field approach

18

DEM simulationsDEM simulations

• biaxial deformation

• two-dimensional

• various friction coefficients µµµµ• dense and loose initial assembly

• periodic boundary conditions

• 50,000 disks

)(2

121 σσ −=q

)(2

121 σσ +=p 21 εεε +=V

21 εεε −=S

Invariants:

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Contact constitutive relationContact constitutive relation

cnn

cn kf ∆==== c

nc

t ff ⋅≤ µctt

ct kf ∆====

Special cases:

• µ → 0• µ → ∝

µkt

kn

20

AnimationAnimation

21

Macroscopic responseMacroscopic response

22

FabricFabric

23

Induced anisotropyInduced anisotropy

24

ForceForce--fabricfabric--strength (1)strength (1)

( ) ( )[ ]02cos12

1 θθπ

θ −+= caE

( ) ( )[ ]00 2cos1 θθθ −+= nn aff

( ) ( )00 2sin θθθ −−= faf tt

25

ForceForce--fabricfabric--strength (2)strength (2)

( ) ( ) ( )∑ ∫

∑∑∑

≅=

==∈∈

gjiVgjigcij

g Cc

cj

ci

Cc

cj

ciij

dflEmflNV

flV

flV

g

θθθθθσ

σ

π2

0

, )(1

11

[ ]tnc aaa ++≅+−

2

1

21

21

σσσσ

26

Effect of interparticle friction on Effect of interparticle friction on macroscopic behaviourmacroscopic behaviour

2σ

1σ 1σ

2σ

27

Granular materials are Granular materials are frictional (1)frictional (1)

Microscopic (contact) frictional behaviour

28

Granular materials are Granular materials are frictional (2)frictional (2)

Macroscopic (continuum) frictional behaviour

2σ

1σ 1σ

2σ

29

CorrelationsCorrelations betweenbetweenmicroscopic and microscopic and

macroscopic frictionmacroscopic friction

Weak dependence of macroscopic friction on microscopic friction:

• experimentally (Skinner, 1969)• numerically (Cambou, 1993)

µ

µ

φφ

φtan310

tan15sin

+=∞

Bishop (1954)

)4

tan(2)21

4(tan peak

2µφπφπ +=+

Horne (1965)

µφµ tan=

30

Influence Influence µµ

• shear strength

• dilatancy

• energy

• dissipation

31

Shear strength and dilatancyShear strength and dilatancy

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ε1

(%)

q /

p

µ→∞µ=0.5µ=0.4µ=0.3µ=0.2µ=0.1µ→0

0 5 10 15-2

0

2

4

6

8

10

12

ε1

(%)

-εV (%

)

µ→∞µ=0.5µ=0.4µ=0.3µ=0.2µ=0.1µ→0

Increasing µ Increasing µ

Shear strength Dilatancy

32

Plastic strains: unloading pathsPlastic strains: unloading paths

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

ε1

(%)

q /

p

33

Dilatancy rateDilatancy rate

Formulated in plastic strains

pS

pV

d

d

εε−

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

q / p

-dε Vp

/ d

ε Sp

µ→∞µ=0.5µ=0.3µ=0.1µ→0µ→∞ (loose)µ=0.5 (loose)

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Strength at peak & steadyStrength at peak & steady--state, state, dilatancy rate at peak strengthdilatancy rate at peak strength

0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

µ

(q/p)peak

(q/p)∞

(-dεVp / dεS

p)peak

de

nse

, µ→∞

loose

, µ→∞loose, µ=0.5

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StrengthStrength--dilatancy relationdilatancy relation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

(q/p)peak

- (q/p)∞

(-d ε

Vp /

dε Sp

) peak

DEM simulationsequation

( ) ( )[ ]∞−=

− pqpq

d

dpS

pV // peak

peak

αεε

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

µ

(q /

p )∞

DEMBishop

Strength correlations vs. Strength correlations vs. simulationssimulations

Peak strengthPeak strength SteadySteady--state strengthstate strength

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

µ

(q /

p )p

ea

k

DEMHorne

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EnergyEnergy

tn UUU +=

[ ]∑=∈ Cc

ct

t

fkVtU 2

2

11[ ]∑=∈ Cc

cn

n

fkVnU 2

2

11

0 5 10 151

2

3

4

5

6

7

ε1

(%)

U /

U0

µ=0.5µ=0.4µ=0.3µ=0.2µ=0.1µ→0

0 5 10 150

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

ε1

(%)

Ut /

Un

µ=0.5µ=0.4µ=0.3µ=0.2µ=0.1µ→0

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DissipationDissipation

dUd

dUWD

ijij −=−≅

εσδδ

DissipationWork input

Change in energy

39

0 5 10 15-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ε1 (%)

µ=0.5

1 / σ0⋅δW / δε

S1 / σ

0⋅dU / dε

S1 / σ

0⋅δD / δε

S

0 5 10 15-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ε1 (%)

µ=0.5

1 / σ0⋅δW / δε

S1 / σ

0⋅dU / dε

S1 / σ

0⋅δD / δε

S

0 5 10 15-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ε1 (%)

µ→∞

1 / σ0⋅δW / δε

S1 / σ0

⋅dU / dεS

1 / σ0⋅δD / δε

S

0 5 10 15-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

ε1.

(%)

µ→0

1 / σ0⋅δW / δε

S1 / σ

0⋅dU / dε

S1 / σ

0⋅δD / δε

S

DissipationDissipation

dUWD −≅ δδDissipation

Work input

Change in energy

40

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ε1

.

(%)

δD/[

p ⋅δε

SpF

( q/ p

)]µ→∞µ=0.5µ=0.3µ=0.1µ→0µ→∞ (loose)µ=0.5 (loose)

Dissipation functionDissipation function

( ) ( )[ ]∞−≅−

≅

pqpqd

d

dD

pS

pV

pijij

//αεε

εσδ

( ) pSpdpqFD εδ /≅

Work input dissipated:

Strength-dilatancy:

Resulting dissipation function:

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ConclusionsConclusions

• For all µ, frictional behaviour at macro-level

• Macro-level dissipation, even for µ→0 and µ→∝

• Strength-dilatancy relation

• Constant ratio of ‘tangential’ over ‘normal’ energy beyond initial range

• All work input is dissipated beyond initial range

42

DiscussionDiscussion

• Origin of macroscopic frictional behaviour:

• Interparticle friction: NO

• Contact disruption and creation: POSSIBLY

• Origin of dissipation if µ→0 or µ→ ∝:• Dynamics & restitution coefficient

• Microscopic restitution coefficient ⇒macroscopic plastic dissipation

43

SummarySummary

• Slow flows

• Stress and strain

• Frictional nature

• Geometrical anisotropy

• Validity mean-field approximation

44

45

Mixing of granular materials Mixing of granular materials in rotating cylindersin rotating cylinders

Niels P. Kruyt

Department of Mechanical Engineering, University of Twente

n.p.kruyt@utwente.nlwww.ts.ctw.utwente.nl/kruyt/

46

CoCo--workersworkers• Gerard Finnie:

• Mineral Processing Group, University of Witwatersrand, Johannesburg, South Africa

• Mao Ye & Christiaan Zeilstra & Hans Kuipers:

• Department of Chemical Engineering, University of Twente

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Mixing in rotating cylindersMixing in rotating cylinders

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Functions of rotary kilnsFunctions of rotary kilns

• Mixing

• Calcination

• Drying

• Waste incineration

…

49

Process parametersProcess parameters

• Speed of rotation, Ω• Radius, R• Filling degree, J• Length, L

• Material properties• Inclination• Gas flow

…L

R

50

Flow regimesFlow regimes

From: Mellmann, Powder Technology (2001)

Increasing rotational speed

51

Mixing in rotary kilnsMixing in rotary kilns

• Transverse → fast

• Longitudinal → slow

• Fascinating patterns

From: Shinbrot et al., Nature (1999)

52

SegregationSegregation

• Separation due to differences in:

• Size

• Density

• Transverse & longitudinal segregation

• Here no segregation:

• Equal densities

• Small size differences

53

• Non-dimensional parameters

• Froude number

• Filling degree

Dependence of mixing on Dependence of mixing on process parametersprocess parameters

J

g

RFr

2Ω=

54

• Many-particle simulation method

• Numerics: explicit time-stepping of equations of motion

• displacements

• rotations

• simple models at micro level →complex behaviour at macro level

Approach: Discrete Element Approach: Discrete Element MethodMethod

55

Contact constitutive relationContact constitutive relation

• Elastic

• Frictional

• Damping

56

VelocitiesVelocitiesKiln speed, Ω⋅R Kiln speed, Ω⋅R

57

Visualization: 2DVisualization: 2D

58

Visualizations: effect of Visualizations: effect of FrFr

59

Visualizations: effect of Visualizations: effect of JJ

60

Longitudinal mixingLongitudinal mixing

• Diffusion process; no throughflow

L

nj

L

Dnj

jnxc

j

ππ )12(sin

)12(exp

12

1

2

1),(

12

22 −

−−−

+= ∑∞

=

l Solution

2

2

x

cD

n

c

∂∂=

∂∂

2

2*

x

cD

t

c

∂∂=

∂∂

π2* Ω= D

D

61

Determination of diffusion Determination of diffusion coefficient (1)coefficient (1)

• Mean position of “red” particles

∫+

−

=2

2

),(1

)(L

L

R dxnxxcL

nX

l For small “Fourier” number

−≅

22

4

1)(

L

DnnXR

62

Determination of diffusion Determination of diffusion coefficient (2)coefficient (2)

Linear relation is indeed observed!

0 5 10 15 20 25 30 35 400.22

0.225

0.23

0.235

0.24

0.245

0.25

n

Xre

d(n

)

63

Concentration profilesConcentration profiles

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L

c(x,

n)

n=10 revsn=20 revsn=30 revsn=40 revs

64

Dependence of diffusion Dependence of diffusion coefficient on process coefficient on process

parametersparameters

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

-3

Fr

D /

R2

J=20%J=30%J=40%

65

Comparison with other studiesComparison with other studies

• Present

• Dury & Ristow (1999)

• Rao et al. (1991)

• Rutgers (1965)

• Sheritt et al. (2003)

0.1* Ω∝D

0.2* Ω∝D

0.1* Ω∝D

5.0* Ω∝D

4.0* Ω∝D

const=D

π2* Ω= D

D

66

Transverse mixing:Transverse mixing:initial configurationinitial configuration

67

Transverse mixingTransverse mixing

• Entropy-like mixing index

( )jiijjijijiji qqppkVI ,,,,, lnln +−=

∑ ∞→=

ji

ji

nI

nInM

,

,

)(

)()( i

j

particles black"" offraction :, jip

particles gray"" offraction :, jiq

68

Evolution of mixing indexEvolution of mixing index

( )SnnM −−≅ exp1)(Mixing speed

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n

M(n

)

69

Dependence of mixing speed Dependence of mixing speed on process parameterson process parameters

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

Fr

S3D simulations

J=20%J=30%J=40%

70

2D vs. 3D simulations2D vs. 3D simulations

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

Fr

S

3D simulations

J=20%J=30%J=40%

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Fr

S

2D simulations

J=20%J=30%J=40%

Faster transverse mixing in 2D than in 3D

71

ConclusionsConclusions

• Longitudinal mixing• Diffusion equation• Diffusion coefficient

• Proportional to rotational speed• Weak dependence filling degree

• Transverse mixing • Entropy-like mixing index• Mixing speed S

Ω ↑: S ↓• J ↑: S ↓

72

Further researchFurther research

• Effect of particle properties on mixing

• Non-spherical particles

• Inclined rotary kilns

• Velocity profile in active layer

• Inclusion of gas flow

• Axial segregation