Micromechanics of granular materials: slow flows Niels P. Kruyt Department of Mechanical Engineering, University of Twente, n.p.kruyt@utwente.nl www.ts.ctw.utwente.nl/kruyt/ 1
Applications of granular materials geotechnical engineering geophysical flows bulk solids engineering chemical process engineering mining gas and oil production food-processing industry agriculture pharmaceutical industry 2
Features rate-independent elasticity plasticity frictional dilatancy anisotropy viscous multi-phase cohesion segregation 3
Overview talk Introduction Continuum modelling Micromechanical modelling Effect of interparticle friction on macroscopic behaviour Validity of mean-field approaches Mixing in rotating cylinders 4
Slow granular flows Rate-independent; quasi-static Continuum level stress tensor strain tensor 5
Stress and strain tensors Stress tensor df i = n Strain tensor j σ ji da u 0 + du i i df i n j da du i = ε ij dx j dx i 6 0 u i
Constitutive relations Continuum theories; elasto-plasticity elastic and plastic strains yield function flow rule: plastic potential Micromechanical theories relation with microstructure 7
Plasticity of metals granular materials pressure dependence volume changes: dilatancy non-associated flowrule strain-softening 8
Micromechanics of slow flows Study relation between micro and macro level Work with Leo Rothenburg, University of Waterloo, Canada σ 2 σ1 σ1 σ 2 9
Objective Micro-level macro-level Account for the discrete nature Emphasis on slow flows Elasticity Plasticity Theory and DEM simulations 10
From discrete information stress and strain 11
Averaging/homogenisation 12
Stress F R i = A plane N j σ ji σ ij = 1 V c C l c i f c j 13
Compatibility equations pq i =U U q i p i S RS i + Rα i = 0 pq q i = Ui U RS i = 0 S p i S RS i = = = 0 + + [ ] + [ ] + [ ] + [ ] a b b c c d d a U U U U U U U U i ab i i i bc i i i cd i i + i da i i 14
Strain du L i = ε dx ij j ε 1 c C c = h ij A j c i pq i q i = U U p i 15
σ 2 Statics Micromechanics Kinematics σ1 σ1 Stress Macroscopic level (continuum) Strain Averaging σ 2 Force Localisation Constitutive relation Localisation Microscopic level (contact) Relative displacement Averaging 16
Topics in micromechanics Importance of geometrical anisotropy Effect of interparticle friction on macroscopic behaviour Validity of mean-field approach 17
DEM simulations biaxial deformation two-dimensional various friction coefficients µ dense and loose initial assembly periodic boundary conditions 50,000 disks Invariants: 1 p = ( σ 1 + σ 2 ) ε V = ε 1 + ε 2 2 q = 1 2 ( σ 1 σ 2 ) ε S = ε 1 ε 2 18
Contact constitutive relation f c n c c c c = k f = k f µ f n c n t t t t n Special cases: µ 0 µ k n k µ t 19
Animation 20
Macroscopic response 21
Fabric 22
Induced anisotropy 23
Force-fabric fabric-strength (1) E f f 1 2π ( θ ) [ 1+ a cos 2( θ θ )] t n = c ( θ ) f [ 1+ a cos ( θ θ )] = 0 n 2 ( θ ) = a f sin ( θ θ ) t 0 2 0 0 0 24
25 Force Force-fabric fabric-strength (2) strength (2) ( ) ( ) ( ) = = = g j i V g j i g c ij g C c c j c i C c c j c i ij d f l E m f l N V f l V f l V g θ θ θ θ θ σ σ 2π 0, ) ( 1 1 1 [ ] t n c a a a + + + 2 1 2 1 2 1 σ σ σ σ
Effect of interparticle friction on macroscopic behaviour σ 2 σ1 σ1 σ 2 26
Granular materials are Microscopic (contact) frictional behaviour frictional (1) 27
Granular materials are frictional (2) Macroscopic (continuum) frictional behaviour σ 2 σ σ 1 1 σ 2 28
Correlations between microscopic and macroscopic friction Horne (1965) Bishop (1954) tan 2 π ( + 4 1 φ 2 peak ) = π 2 tan( + 4 φ µ ) sinφ = 15 tanφ µ 10 + 3tanφ µ Weak dependence of macroscopic friction on microscopic friction: µ = tanφ µ experimentally (Skinner, 1969) numerically (Cambou, 1993) 29
Influence µ shear strength dilatancy energy dissipation 30
Shear strength and dilatancy Shear strength Dilatancy q / p 0.7 0.6 0.5 0.4 0.3 µ µ=0.5 µ=0.4 µ=0.3 µ=0.2 µ=0.1 µ 0 -ε V (%) 12 10 8 6 4 µ µ=0.5 µ=0.4 µ=0.3 µ=0.2 µ=0.1 µ 0 0.2 2 0.1 Increasing µ Increasing µ 0 0 5 10 15 ε (%) 1 0-2 0 5 10 15 ε (%) 1 31
Plastic strains: unloading paths 0.6 0.5 0.4 q / p 0.3 0.2 0.1 0 0 5 10 15 ε 1 (%) 32
Dilatancy rate Formulated in plastic strains dε dε p V p S 0.5 0.45 0.4 0.35 µ µ=0.5 µ=0.3 µ=0.1 µ 0 µ (loose) µ=0.5 (loose) -dε V p / dεs p 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 q / p 33
Strength at peak & steady-state, state, dilatancy rate at peak strength 0.7 (q/p) peak 0.6 (q/p) (-dε V p / dε S p ) peak 0.5 0.4 0.3 0.2 0.1 loose, µ=0.5 loose, µ dense, µ 0 0 0.1 0.2 0.3 0.4 0.5 0.6 µ 34
Strength-dilatancy relation 0.6 0.5 (-dε V p / dεs p )peak 0.4 0.3 0.2 0.1 DEM simulations equation 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (q/p) peak - (q/p) p dε V = α / p d / peak ε S peak [ ] ( q p ) ( q p ) 35
Strength correlations vs. simulations Peak strength Steady-state strength 0.7 0.7 0.6 0.6 0.5 0.5 (q / p) peak 0.4 0.3 (q / p) 0.4 0.3 0.2 0.2 0.1 DEM Horne 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 µ 0.1 DEM Bishop 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 µ 36
Energy U n = 1 V c C 1 2k n [ c ] f n 2 U t = 1 V c C 1 2k t [ c ] f t 2 U = U n + U t 7 6 5 µ=0.5 µ=0.4 µ=0.3 µ=0.2 µ=0.1 µ 0 0.1 0.09 0.08 0.07 0.06 µ=0.5 µ=0.4 µ=0.3 µ=0.2 µ=0.1 µ 0 U / U 0 4 U t / U n 0.05 0.04 3 0.03 2 0.02 0.01 1 0 5 10 15 ε (%) 1 0 0 5 10 15 ε (%) 1 37
Dissipation Dissipation Work input Change in energy δ D δ W du = σ ij d ε ij du 38
Dissipation Dissipation Work input Change in energy δ D δw du 0.3 0.25 µ 0 0.8 0.7 0.6 0.8 0.7 0.6 µ=0.5 µ=0.5 1.6 1.4 1.2 µ 1 / σ 0 δw / δε S 1 / σ 0 du / dε S 1 / σ 0 δd / δε S 0.2 0.15 0.1 1 / σ 0 δw / δε S 1 / σ 0 du / dε S 1 / σ 0 δd / δε S 0.5 0.4 0.5 0.4 0.3 1 / σ 0 δw / δε S 1 / σ 0 du / dε S 1 / σ 0 δw / δε S 1 0.8 0.6 0.05 0-0.05 0 5 10 15 ε 1. (%) 0.3 0.2 0.1 0 0.2 0.1 0 1 / σ 0 du / dε S 1 / σ 0 δd / δε S -0.1 0 5 10 15 ε 1 (%) 1 / σ 0 δd / δε S 0.4 0.2 0-0.2 0 5 10 15 ε 1 (%) -0.1 0 5 10 15 ε 1 (%) 39
Dissipation function 2 Work input dissipated: 1.8 1.6 δd/[p δε S p F(q/p)] 1.4 Strength-dilatancy: 1.2 1 0.8 0.6 Resulting dissipation function: 0.4 0.2 δ D d d ε ε p V p S σ ij d ε α p ij δ D F / µ µ=0.5 µ=0.3 µ=0.1 µ 0 p µ (loose) µ=0.5 (loose) [( q / ) ( q / p) ] ( q p) pdε p S 0 0 5 10 15 ε 1. (%) 40
Conclusions 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 41
Discussion 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 42
Summary Slow flows Stress and strain Frictional nature Geometrical anisotropy Validity mean-field approximation 43
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Mixing of granular materials in rotating cylinders Niels P. Kruyt Department of Mechanical Engineering, University of Twente n.p.kruyt@utwente.nl www.ts.ctw.utwente.nl/kruyt/ 45
Co-workers Gerard Finnie: Mineral Processing Group, University of Witwatersrand, Johannesburg, South Africa Mao Ye & Christiaan Zeilstra & Hans Kuipers: Department of Chemical Engineering, University of Twente 46
Mixing in rotating cylinders 47
Functions of rotary kilns Mixing Calcination Drying Waste incineration 48
Process parameters Speed of rotation, Ω Radius, R Filling degree, J Length, L Material properties Inclination Gas flow R L 49
Flow regimes Increasing rotational speed From: Mellmann, Powder Technology (2001) 50
Mixing in rotary kilns Transverse fast Longitudinal slow Fascinating patterns From: Shinbrot et al., Nature (1999) 51
Segregation Separation due to differences in: Size Density Transverse & longitudinal segregation Here no segregation: Equal densities Small size differences 52
Dependence of mixing on process parameters Non-dimensional parameters Froude number Filling degree Fr = J 2 Ω R g 53
Approach: Discrete Element Method Many-particle simulation method Numerics: explicit time-stepping of equations of motion displacements rotations simple models at micro level complex behaviour at macro level 54
Contact constitutive relation Elastic Frictional Damping 55
Velocities Kiln speed, Ω R Kiln speed, Ω R 56
Visualization: 2D 57
Visualizations: effect of Fr 58
Visualizations: effect of J 59
Longitudinal mixing Diffusion process; no throughflow c n = 2 c D 2 x c t = D * 2 c 2 x D = D 2π * Ω l Solution c( x, n) 1 = + 2 j= 1 2 1 exp j 1 (2 j 2 2 1) π Dn (2 j 1)π n sin 2 L L 60
Determination of diffusion coefficient (1) Mean position of red particles X R ( n) = 1 L + L 2 L 2 xc( x, n) dx l For small Fourier number X R ( n) 1 2 4 Dn L 2 61
Determination of diffusion coefficient (2) 0.25 0.245 0.24 X red (n) 0.235 0.23 0.225 0.22 0 5 10 15 20 25 30 35 40 Linear relation is indeed observed! n 62
Concentration profiles 1 0.9 0.8 0.7 0.6 c(x,n) 0.5 0.4 0.3 0.2 n=10 revs 0.1 n=20 revs n=30 revs n=40 revs 0-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 x/l 63
Dependence of diffusion coefficient on process parameters 2 x 10-3 1.8 1.6 1.4 1.2 D / R 2 1 0.8 0.6 0.4 J=20% 0.2 J=30% J=40% 0 0 0.5 1 1.5 2 2.5 Fr 64
Comparison with other studies Present D * Ω 1.0 D = const Dury & Ristow (1999) Rao et al. (1991) D D * * Ω Ω 2.0 1.0 D = D 2π * Ω Rutgers (1965) D * Ω 0.5 Sheritt et al. (2003) D * Ω 0.4 65
Transverse mixing: initial configuration 66
Transverse mixing Entropy-like mixing index p i, j : fraction of "black"particles q i, j : fraction of "gray" particles ( p ln p + q q ) I i, j = kvi, j i, j i, j ij ln i, j j I i, j( n) M( n) = i I( n ) i, j 67
Evolution of mixing index 1 0.9 0.8 0.7 0.6 M(n) 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 n M ( n) 1 exp ( Sn) Mixing speed 68
Dependence of mixing speed on process parameters 3D simulations 0.6 J=20% J=30% J=40% 0.5 0.4 S 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Fr 69
2D vs. 3D simulations 3D simulations 2D simulations 0.6 J=20% J=30% J=40% 1 J=20% J=30% J=40% 0.5 0.8 0.4 0.6 S 0.3 S 0.2 0.4 0.1 0.2 0 0 0.5 1 1.5 2 2.5 Fr 0 0 0.5 1 1.5 2 2.5 Fr Faster transverse mixing in 2D than in 3D 70
Conclusions 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 71
Further research Effect of particle properties on mixing Non-spherical particles Inclined rotary kilns Velocity profile in active layer Inclusion of gas flow Axial segregation 72