Micromechanics of granular materials: slow flows

Transcription

1 Micromechanics of granular materials: slow flows Niels P. Kruyt Department of Mechanical Engineering, University of Twente, 1

2 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

3 Features rate-independent elasticity plasticity frictional dilatancy anisotropy viscous multi-phase cohesion segregation 3

4 Overview talk Introduction Continuum modelling Micromechanical modelling Effect of interparticle friction on macroscopic behaviour Validity of mean-field approaches Mixing in rotating cylinders 4

5 Slow granular flows Rate-independent; quasi-static Continuum level stress tensor strain tensor 5

6 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

7 Constitutive relations Continuum theories; elasto-plasticity elastic and plastic strains yield function flow rule: plastic potential Micromechanical theories relation with microstructure 7

8 Plasticity of metals granular materials pressure dependence volume changes: dilatancy non-associated flowrule strain-softening 8

9 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

10 Objective Micro-level macro-level Account for the discrete nature Emphasis on slow flows Elasticity Plasticity Theory and DEM simulations 10

11 From discrete information stress and strain 11

12 Averaging/homogenisation 12

13 Stress F R i = A plane N j σ ji σ ij = 1 V c C l c i f c j 13

14 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 = = = [ ] + [ ] + [ ] + [ ] 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

15 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

16 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

17 Topics in micromechanics Importance of geometrical anisotropy Effect of interparticle friction on macroscopic behaviour Validity of mean-field approach 17

18 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

19 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

20 Animation 20

21 Macroscopic response 21

22 Fabric 22

23 Induced anisotropy 23

24 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

25 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, ) ( [ ] t n c a a a σ σ σ σ

26 Effect of interparticle friction on macroscopic behaviour σ 2 σ1 σ1 σ 2 26

27 Granular materials are Microscopic (contact) frictional behaviour frictional (1) 27

28 Granular materials are frictional (2) Macroscopic (continuum) frictional behaviour σ 2 σ σ 1 1 σ 2 28

29 Correlations between microscopic and macroscopic friction Horne (1965) Bishop (1954) tan 2 π ( φ 2 peak ) = π 2 tan( + 4 φ µ ) sinφ = 15 tanφ µ tanφ µ Weak dependence of macroscopic friction on microscopic friction: µ = tanφ µ experimentally (Skinner, 1969) numerically (Cambou, 1993) 29

30 Influence µ shear strength dilatancy energy dissipation 30

31 Shear strength and dilatancy Shear strength Dilatancy q / p µ µ=0.5 µ=0.4 µ=0.3 µ=0.2 µ=0.1 µ 0 -ε V (%) µ µ=0.5 µ=0.4 µ=0.3 µ=0.2 µ=0.1 µ Increasing µ Increasing µ ε (%) ε (%) 1 31

32 Plastic strains: unloading paths q / p ε 1 (%) 32

33 Dilatancy rate Formulated in plastic strains dε dε p V p S µ µ=0.5 µ=0.3 µ=0.1 µ 0 µ (loose) µ=0.5 (loose) -dε V p / dεs p q / p 33

34 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 loose, µ=0.5 loose, µ dense, µ µ 34

35 Strength-dilatancy relation (-dε V p / dεs p )peak DEM simulations equation (q/p) peak - (q/p) p dε V = α / p d / peak ε S peak [ ] ( q p ) ( q p ) 35

36 Strength correlations vs. simulations Peak strength Steady-state strength (q / p) peak (q / p) DEM Horne µ 0.1 DEM Bishop µ 36

37 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 µ=0.5 µ=0.4 µ=0.3 µ=0.2 µ=0.1 µ 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 ε (%) ε (%) 1 37

38 Dissipation Dissipation Work input Change in energy δ D δ W du = σ ij d ε ij du 38

39 Dissipation Dissipation Work input Change in energy δ D δw du µ µ=0.5 µ= µ 1 / σ 0 δw / δε S 1 / σ 0 du / dε S 1 / σ 0 δd / δε S / σ 0 δw / δε S 1 / σ 0 du / dε S 1 / σ 0 δd / δε S / σ 0 δw / δε S 1 / σ 0 du / dε S 1 / σ 0 δw / δε S ε 1. (%) / σ 0 du / dε S 1 / σ 0 δd / δε S ε 1 (%) 1 / σ 0 δd / δε S ε 1 (%) ε 1 (%) 39

40 Dissipation function 2 Work input dissipated: δd/[p δε S p F(q/p)] 1.4 Strength-dilatancy: Resulting dissipation function: 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 ε 1. (%) 40

41 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

42 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

43 Summary Slow flows Stress and strain Frictional nature Geometrical anisotropy Validity mean-field approximation 43

44 44

45 Mixing of granular materials in rotating cylinders Niels P. Kruyt Department of Mechanical Engineering, University of Twente 45

46 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

47 Mixing in rotating cylinders 47

48 Functions of rotary kilns Mixing Calcination Drying Waste incineration 48

49 Process parameters Speed of rotation, Ω Radius, R Filling degree, J Length, L Material properties Inclination Gas flow R L 49

50 Flow regimes Increasing rotational speed From: Mellmann, Powder Technology (2001) 50

51 Mixing in rotary kilns Transverse fast Longitudinal slow Fascinating patterns From: Shinbrot et al., Nature (1999) 51

52 Segregation Separation due to differences in: Size Density Transverse & longitudinal segregation Here no segregation: Equal densities Small size differences 52

53 Dependence of mixing on process parameters Non-dimensional parameters Froude number Filling degree Fr = J 2 Ω R g 53

54 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

55 Contact constitutive relation Elastic Frictional Damping 55

56 Velocities Kiln speed, Ω R Kiln speed, Ω R 56

57 Visualization: 2D 57

58 Visualizations: effect of Fr 58

59 Visualizations: effect of J 59

60 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= exp j 1 (2 j 2 2 1) π Dn (2 j 1)π n sin 2 L L 60

61 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) Dn L 2 61

62 Determination of diffusion coefficient (2) X red (n) Linear relation is indeed observed! n 62

63 Concentration profiles c(x,n) n=10 revs 0.1 n=20 revs n=30 revs n=40 revs x/l 63

64 Dependence of diffusion coefficient on process parameters 2 x D / R J=20% 0.2 J=30% J=40% Fr 64

65 Comparison with other studies Present D * Ω 1.0 D = const Dury & Ristow (1999) Rao et al. (1991) D D * * Ω Ω D = D 2π * Ω Rutgers (1965) D * Ω 0.5 Sheritt et al. (2003) D * Ω

66 Transverse mixing: initial configuration 66

67 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

68 Evolution of mixing index M(n) n M ( n) 1 exp ( Sn) Mixing speed 68

69 Dependence of mixing speed on process parameters 3D simulations 0.6 J=20% J=30% J=40% S Fr 69

70 2D vs. 3D simulations 3D simulations 2D simulations 0.6 J=20% J=30% J=40% 1 J=20% J=30% J=40% S 0.3 S Fr Fr Faster transverse mixing in 2D than in 3D 70

71 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

72 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