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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
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
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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
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=
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)
34
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
35
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
36
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
37
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
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:
41
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
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
48
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
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
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
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 ↓
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