Simulation of Non-Rapid, Frictional, Steady-State Shear Flow of a Ultrafine Cohesive Powder by the DEM

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1 Prolog Micro- & Macromechanics of Ultrafine Cohesive Powders Simulation of Non-Rapid, Frictional, Steady-State Shear Flow of a Ultrafine Cohesive Powder by the DEM Titania, n 7; d 5 =1 µm F N * compression F N preshear F N shear h F S F S s s h 1/ Tykhoniuk, R., S. Luding und J. Tomas, Simulation der Scherdynamik kohäsiver Pulver, Chem.- Ing.- Technik 76 (4) 1-, 59-6

2 Mechanische Verfahrenstechnik - Prof. Dr.-Ing. habil. Jürgen Tomas Micro- and Macromechanics of Ultrafine Cohesive Powders 1. Introduction, Powder Processing & Flow Problems. Particle Contact Constitutive Models. Particle Contact Failure & Cohesive Powder Flow Criteria.1 Incipient consolidation, yield & cohesive steady-state flow. Consolidation functions. DEM simulations 4. Summary & Outlook /

3 1. Introduction No adhesion, no friction DEM Simulation of Silo Flow Problems No adhesion, Clumps, rolling friction Adhesion, no friction Adhesion, friction / Tykhoniuk, R and Tomas, J., DEM simulation of cohesive powder flow, paper, CHISA, Praha 4

4 1. Introduction Processing and Handling Problems of Fine Powders 1. Assessment of adhesion potential of fine to nanoscale particles Physical principle Particle size d in µm Adhesion/gravitational force F H /F G Evaluation slightly adhesive adhesive very adhesive. Assessment of adhesion intensification F H (F N ) of fine to nanoscale particles by contact pre-loading with normal force F N Physical principle Evaluation Particle size d in µm Contact consolidation coefficient κ >.8 soft very soft extreme soft 4/ Tomas, J., Produkteigenschaften ultrafeiner Partikel - Mikromechanik, Fließ- und Kompressionsverhalten kohäsiver Pulver, Abhandlungen d. Sächsische Akademie d. Wiss., Bnd 1, Heft (9) 1-46

5 . Contact Open Questions and Challenges of Soft Adhesive Particle Contact Behaviour: 5/ Which physical particle properties determine the soft mechanical contact behaviour of ultrafine adhesive particles (d < 1 µm)? 6 degrees of freedom: 4 stressing modes of a soft adhesive contact? Contact elasticity and stiffness? Load dependent adhesion and contact stiffness? Yield limits of 4 different stressing modes? Load dependent adhesion and yield limits? Force-displacement and moment-angle behaviour? Energy absorption during elastic hysteresis? Energy absorption during frictional contact yielding? Mechano-chemical stability of contact yielding?

6 . Contact Particle Contact Stressing Modes: Force - Displacement and Moment - Angle Functions 6/

7 . Contact Contact Deformation Modes of Smooth Particles 7/

8 . Contact r rk,el F N F N h K,el r K,pl r Particle Contact Deformation in Normal Direction without Adhesion F N F N h K,pl contact normal force F N yielding loading W D = F R (h K ) dh K k N = df N dh K unloading r K,el r << 1 r K,pl r << 1 h K,f particle centre approach h K 8/ Stiffness k N = Force response F R = Deformation work W D = elastic 1 E* d h K,el 1 E* d h K,el 15 E* d 5 h K,el plastic π r p f π r p f h K,pl π r η K h K,vis π π r p f (h K,pl - h K,f ) viscoplastic π r η K /t r η K h K,vis t material data: E* effective modulus of elasticity, p f micro-yield strength, η Κ contact viscosity

9 . Contact Normal Stress Distributions at Contact of Smooth Spheres under Normal Load a) Elastic contact b) Elastic - plastic contact F N r 1 tension F N r 1 9/ r K contact radius r 1, r radii of spheres Hertzian elliptic pressure distribution: p el r K 1 = p max r K,el p max r F = π r N K,el p f p el(r K ) r K compression p f micro-yield strength p el p(r (r K K ) = p ) = p r max max r 1 r = p f K K,f r K,f r r K,f p el (r K ) p f K r K r r K,f r K K,el

10 . Contact Normal Force-Displacement Diagram of Contact Deformation of Limestone Particles 1/ Tomas, J: Assessment of mechanical properties of cohesive particulate Seminar solids part 11, 1: particle contact constitutive model, Particulate Sci. & Technol. 19 (1) 6 Dec London

11 . Contact Adhesion Force - Normal Force Diagram, Mechanical Properties of Limestone Particles adhesion force F H in nn F N F H (F N ) F N F H (F N ) t h F H = (1+ κ) F H + κ F N κ F N +F H F H = p E*a 6 π p f 6 material parameters: Particle size d 5 = 1. µm HAMAKER constant C H,sls = J Micro-yield strength p f = N/mm² Modulus of elasticity E = 15 kn/mm² POISSON ratio ν =.8 Adhesion force F H = -.64 nn 11/ normal force F N in nn Tomas, J., Product design of cohesive powders - mechanical properties, compression and flow behaviour, Chem. Engng. & Technol. 7 (4) Castellanos, A., The relationship between attractive interparticle forces and bulk behaviour in dry and uncharged fine powders, Advances in Physics 54 (5) 6-76 Plastic repulsion coefficient κ p =.15 Elastic-plastic contact consolidation coefficient κ =.4 Surface moisture X W =.5 %

12 . Contact a) Normal stresses at elastic - plastic contact Stress Distributions at Contact of Smooth Spheres under Normal and Tangential Loads b) Shear stresses at frictional adhesive contact F N r 1 F T F N r K r K p f micro-yield strength p el p(r (r K K ) = p ) = p r max max r 1 r = p f K,el *Mindlin, R.D. and H. Deresiewicz, Elastic spheres in contact under varying oblique forces, Trans. ASME J. Appl. Mech. (195) / K K,f r K,f r r K,f p el (r K ) p f K r K r r K,f Mindlin's* shear stress distribution (supplemented): τ µ [ F ] ( ) N + FH (FN ) r r r r i ( rk ) = K,el K K,slip π rk,el µ i τ = r K,slip = r [ F ] ( ) N + FH (FN ) r r rk,slip rk rk, el π r K,el 1 µ K,el i τ(r K ) F T K,el [ F + F (F )] N H N τ slip =µ i. (σ + σ Z (σ)) r K,slip radius of microslip K r K r K,slip K

13 . Contact Tangential Force - Displacement Diagram of Flattened Contact of Limestone Particles Particle size d 5 = 1. µm Material parameter: Contact friction coefficient µ i =.76 1/

14 . Contact Rolling Resistance Force - Rolling Angle Diagram of Flattened Contact of Limestone Particles Particle size d 5 = 1. µm Contact rolling friction coefficient µ R (F N ) = /

15 . Contact Torsional Moment - Rotation Angle Diagram of Flattened Contact of Limestone Particles Particle size d 5 = 1. µm Material parameter: Contact friction coefficient µ i =.76 15/

16 . Contact Contact Stressing Modes of Compression, Sliding, Rolling & Spinning for Load Dependent Adhesion 16/

17 . Contact Physical Basis of Constitutive Functions of Contact for Product Quality Assessment Question? Physical particle properties 4 stressing modes Reply 6 independent properties: d, E, ν, C H, p f, µ i Compression, sliding, rolling, spinning Contact elasticity and stiffness k N, k T, k R, k to = f(6 properties, F N ) Load dependent adhesion F H (F N ) = κ. F N + (1+κ). F H Adhesion dependent yield limits F N, F T, M R, M to = f(f N + F H (F N )) Force-displacement behaviour F N, F T = f(6 properties, F N ) Moment-angle behaviour M R, M to = f(6 properties, F N ) Energy absorption of hysteresis W N, W T, W R, W to = f(6 properties, F N ) Energy absorption of yielding W N, W T, W R, W to = f(6 properties, F N ) Mechano-chemical stability Problematic; but macroscopically yes! 17/

18 . Powder Open Questions & Challenges of Non-Rapid Frictional Powder Flow & Compression Behaviour Micro-macro transition from single adhesive contact to non-rapid frictional shear flow (v S < 1 m/s) of a cohesive powder (ff c < 4)? Which properties determine the mechanical behaviour of cohesive powder? Pre-consolidation dependent, cohesive powder elasticity and stiffness? Yield limits of different stressing modes? Pre-consolidation dependent yield, flow and consolidation limits? Stress-strain behaviour? Powder compressibility? Energy absorption during elastic hysteresis? Energy absorption during inelastic powder compression and shear flow? 18/

19 . Contact Micro-Macro Transition, Force and Stress Transmission in a Sheared Particle Packing 19/

20 . Powder Linear Equations of Consolidation, Yield and Stationary Flow of Cohesive Powders Yield limit Radius stress = f(centre stress) Shear stress = f(normal stress) Incipient consolidation σ (1) σr,st τ R,CL = sin ϕi ( σm,cl + σm,st ) + σ CL = tan ϕi σcl + + σm, st R, st sin ϕi (CL) Incipient yield σ R = sin ϕi ( σm σm,st ) + σr, st () σr,st (YL) τ = tan ϕ σ + σ i M, st Steady-state flow (SYL) σ ( σ + ) R,st = sin ϕst M,st σ τ sin ϕ tanϕst σst + σ (5) ( ) st = i () (4) (6) σ M,st centre stress for steady-state flow is the parameter of preconsolidation! ρ b = f(σ M,st ) compression function /

21 . Powder Shear Stress - Normal Stress Diagram of Biaxial Stress States, cont. a) The three flow parameters b) Stress states shear stress τ Stationary Yield Locus Yield Locus ϕ i End point shear stress τ Stationary Yield Locus Yield Locus σ R,st Consolidation Locus σ 1/ ϕ st normal stress σ σ M,st ϕ i - angle of internal friction, ϕ st - stationary angle of internal friction, σ - isostatic tensile strength of unconsolidated powder; and as curve parameter: σ M,st - centre stress for steady-state flow τ c σ Z1 σ Z σ σ c σ M,st normal stress σ σ VM σ 1 σ VR ϕ i σ 1 - major principal stress, σ - minor principal stress, τ c - cohesion, σ c - uniaxial compressive strength, σ Z1 - uniaxial tensile strength, σ Z - isostatic tensile strength, σ iso - isostatic pressure σ iso

22 . Powder Linear Consolidation Functions of Cohesive Powders Yield parameter Uniaxial compressive strength Uniaxial tensile strength Isostatic tensile strength ( ) Isostatic pressure Stress = f(major principal stress) ( sin ϕst sin ϕi ) sin ϕst 1+ sin ϕi σ c = σ1 + ( 1+ sin ϕst ) ( 1 sin ϕi ) 1+ sin ϕst 1 sin ϕ ( sin ϕst sin ϕi ) sin ϕst σ Z,1 = σ1 + σ ( 1+ sin ϕst ) ( 1+ sin ϕi ) 1+ sin ϕst sin ϕst sin ϕi ( 1+ sin ϕi ) sin ϕst σ Z = σ1 + σ 1+ sin ϕst sin ϕi ( 1+ sin ϕst ) sin ϕi sin ϕst + sin ϕi ( 1 sin ϕi ) sin ϕst σ iso = σ1 + σ 1+ sin ϕ sin ϕ 1+ sin ϕ sin ϕ ( ) st i ( ) σ ( ) ( ) ( ) st i i (1) () () (4) σ 1 major principal stress for steady-state flow is another parameter of pre-consolidation ρ b = f(σ 1 ) compression function /

23 . Powder Rapid-Tests to Quantify the Mechanical Properties of Cohesive Powders /

24 . Powder Testing Devices to Determine the Flow Properties of Cohesive Powders 4/

25 . Powder Simulation of Non-Rapid, Frictional, Steady-State Shear Flow by the Discrete-Element-Method Titania, n ; d 5 = µm h Titania, n 7; d 5 =1 µm h Tykhoniuk, R., S. Luding und J. Tomas, Simulation der Scherdynamik kohäsiver Pulver, Chem.- Ing.- Technik 76 (4) 1-, /

26 . Powder Shear stress τ in kpa Comparison of DEM Simulations with Shear Tests Yield Locus 4 for TiO, d 5 =.6 µm, ρ s =87 kg/m, v s = mm/min Experiment Cohesive model, D Cohesive model, higher cohesive strength, D Cohesive model, D Linear model, D τ = tan( )σ +.75 τ = tan(7 )σ +.9 τ = tan( )σ Tykhoniuk, R., Tomas, J., Luding, S., Kappl, M., Ultrafine 6/ cohesive powders Chem. Engng. Sci. 6 (7) τ = tan(1 )σ +.8 τ = tan(17 )σ Normal stress σ in kpa

27 . Powder σ 1 = kpa Simulation of Uniaxial Compression Test by the Discrete-Element-Method σ 1 = 1 kpa σ 1 σ c 7/

28 . Powder Simulation of Uniaxial Tensile Test by the Discrete- Element-Method σ n = kpa, 1, TiO particles σ 1 σ Z1 8/

29 . Powder Physical Basis of Powder for Product Quality Assessment Question? Micro-macro transition Reply Limits of compression, yield and flow Mechanical powder behaviour 5 independent properties: ϕ i, ϕ st, σ, n, ρ b, Powder elasticity and stiffness? E b, G b = f(e, ν, ϕ i, ϕ st, σ, σ M,st )* Yield limits of stressing modes Compression, CL, YL, SYL Yield & consolidation limits σ R = f(σ M, ϕ i, ϕ st, σ, σ M,st ) Stress-strain behaviour? Tensor models with dubious physical basis Powder compressibility ρ b = f(σ M,st, σ, n) Energy absorption of hysteresis? - Energy absorption of compression and preshear W b, W pre = f(ϕ i, ϕ st, σ, n, ρ b,, σ M,st ) * Has to be improved 9/

30 . Powder Physical Basis of Powder for Product Quality Assessment, cont. But a list of new problems & questions is waiting to be solved! /

31 4. Summary & Outlook National Priority Program 1486 Particles in Contact - Micromechanics, Microprocess Dynamics and Particle Collectives - Problems, Goals and Interdisciplinary Methods - Project Groups: A Physico-chemical Basic Principles within Contact Zone ( P) B Particle-Particle and Particle-Wall Contacts (16 P) C Particle Impacts and their Dynamics (1 P) D Constitutive Material Laws for Particle Collectives on Macro Level (4 P) - Subsidies (since 5/1):,7 Mill. /a 1/

32 4. Summary & Outlook Physical models for elastic plastic & Physical Basis of Constitutive Functions for Product Quality Assessment & Process Design viscoplastic particle contact deformations Load-dependent adhesion force Friction limits of sliding, rolling and torsion Load-dependent friction limits Micro-macro transition and fundamental equations for yield loci of cohesive powders, i.e. stationary yield locus instantaneous yield locus consolidation locus Consolidation functions /

33 Many Thanks For Discussion, experimental contributions, relevant information and tips of my co-workers S. Aman, S. Antoniuk, T. Gröger, B. Ebenau, L. Grossmann, T. Günther, A. Haack, W. Hintz, G. Kache, M. Khanal, T. Kollmann, C. Mendel, T. Mladenchev, P. Müller, T. Nikolov, B. Reichmann, W. Schubert, R. Tykhoniuk, Fruitful collaboration (, Physics, Materials, Mathematics, Processes) with H. Altenbach, A. Bertram, U. Gabbert, K. Kassner, D. Regener, P. Streitenberger, L. Tobiska, G. Warnecke, M. Zehn, DFG-Graduiertenkolleg 88 and 1554 Micro-Macro Interactions in Structured Media and Particle Systems ( - 8 and 1-14). / The intensive and critical discussion of physical fundamentals with S. Luding (TU Twente), H.-J. Butt and M. Kappl (MPI Mainz) during the collaboration of the project Shear Dynamics of Cohesive, Fine-Disperse Particle Systems of the joint research program Behaviour of Granular Media ( 6) as well as Contact Models, Sinter Kinetics and Sintering of our present national priority program 1486 PiKo (1 16) of German Research Association (DFG).

34 Figures for Discussion 4/

35 1. Introduction Processing and Handling Problems of Fine Powders 5/

36 1. Introduction Increasing ground water level & soaking of soil + Saturating the pores of soil with rain water Stability Problems of Fine Particles (Soil) Decreasing cohesion and shear resistance of particles Small initiation (car, wave impact or dead weight) Nachterstedt, 18 th July 9* Liquefaction of large soil masses 6/ *

37 1. Introduction Multiscale Models of Dynamics of Particle Packing a ij Searched: Cause-effectresponse functions for: I. Mechanical continua stresses σ ij = f(ε ij strain, v ij strain rates, x ij positions) II. Particle interactions m i x = F = f (x i III. Molecular interactions f ij Results: Forces, Moments and accelerations x ij, Velocities x ij, Spatial positions x ij ij ij, x ij,..) = U / a (a,t,..) ij ij ij 7/

38 . Powder t=t, x i =x,i, v i =v Forces, Moments Positions, Velocities Simulation of Steady-State Shear Flow by the Discrete-Element-Method Euler s equations of motion for each particle i m Force Balance i d ri dt ( F F NK NB (ij) (ij) = K,n + K,t ) + j= 1 j= 1 ( F (ij) B,n + F (ij) B,t ) + m i g 8/ t = t + t Moment Balance J i dω dt i NK NB (ij) (ij) = (rk FK ) + j = 1 j = 1 M (i) mg F K Contact force particle-particle or particle-wall, F B Spring force of solid bridge bond, N K, N B Number of contacts & solid bridge bonds of particle i J i Moment of inertia of particle i n, t normal, tangential (ij) B + M

39 9/ Spring-Dashpot Contact Model & Solid Bridge Model ω i F F Particle i Normal Contact Forces a ( k s + η s n F = ( k A s n (ij) K,n = ij,n ij,n ij ij,n ) (ij) K,t n ij = (ij) F k,n min ij t ij K (ij) F k,t µ ij ( a (ij) (k ) ij,t sij,t + ηij s ij,t ), µ ij FK,n tij (ij) (ij) F M B,n B (ij) σ ij = + RB σ A I (ij) B η t,ij η n,i (ij) max k ij Contact stiffness, k ij,b Area related solid bridge stiffness, s ij Centre approach or overlap, η ij Damping coefficient, µ ij Sliding friction coefficient, j k t,ij k n,ij (ij) F k,t K (ij) F k,n ij ω j Particle j Tangential Contact Forces F Failure Criterion or (ij) B,n B,ij,n ij ij,n ) = ( t (ij) B,t k B,ij,t Aij sij,t ) (ij) FB,t τ ij = τ A ij (ij) max σ and τ Normal and tangential stresses in crosssectional area of solid bridge bond, σ max and τ max Normal and shear strength of solid bridge, Moment in cross-section of solid bridge, M B A ij Particle i (ij) R b Particle j Solid bridge bond i-j Particle i (ij) R b Cross-sectional area of solid bridge, I B Moment of inertia (A ij ) ij z y (ij) A b (ij) F b,t (ij) F b,n (ij) M b,t ij x

40 . Adhesion Adhesion & Microprocesses of Particle Bonds Van der Waals forces Liquid bridge bonds Solid bridge bonds Adhesion force of rough spheres: F H CH,sls h = 6 a F= r1, 1 + 1, / h ( 1+ h / a ) r1, r1, F= Load dependent adhesion force: Elastic-plastic contact consolidation coefficient: κp κ = κ A κp p C VdW H,sls 4 σsls κ = = = p p 6 π a p a p κ F A H = ( 1+ κ) FH + κ FN f F= 1 Apl = + A + A pl el r f 5 6 F= f Range of adsorption layers: σ 1 ε = ε or: 8,88 σc = ε σ,75 Z,A π d a X X ( 1 ε) σlg sin d ( 1 sin ϕ ) Range of liquid bridges: σ c 8,5 = i W W,m ϕ ( 1 ε) ( ε) σlg ε ε d ( 1 sinϕ ) i i a a ρ ρ S l sinϕi X ρ ρ W S l X,75 W Crystallisation: σ = σ ( 1 ε) Y ( X X ) 1 exp( t t ) c,t D,s WO Chemical reaction: MS X σc,t = σds ( 1 ε) M ϑ Sintering or contact fusion: F + S W WA W WE k W t k t + 1 H,t = κt FN FH,t with tan ϕ t tan ϕ = i,t + κ t tan ϕi 4 π σzs σsg FH,t = 5 η s W κ / tan ϕ d t st [ ] t s 6 σzs t = 5 η 4/ Tomas, J.: Modellierung des Fließverhaltens von Schüttgütern auf der Grundlage der Wechselwirkungskräfte zwischen den Partikeln und Anwendung bei der Auslegung von Bunkeranlagen, (Dissertation B, Habilitation), TU Bergakademie Freiberg 1991

41 . Adhesion Adhesion & Microprocesses of Particle Bonds Tomas, J. and H. Schubert: Modelling of the strength and flow properties of moist soluble bulk materials, Proc. Intern. Symp. Powder Technology 81, , Kyoto / Tomas, J.: Modellierung des Fließverhaltens von Schüttgütern auf der Grundlage der Wechselwirkungskräfte zwischen Seminar den 11, Partikeln und Anwendung bei der Auslegung von Bunkeranlagen, (Dissertation B, Habilitation), TU Bergakademie Freiberg 6 Dec London

42 . Adhesion Liquid and Crystallisation Bridge Bonds 4/ Tomas, J.: Untersuchungen zum Fließverhalten von feuchten und leichtlöslichen Schüttgütern (Dissertation A) Freiberger Seminar Forschungsheft 11, A 6 Dec. 677 (198) London 1-1

43 . Contact Normal Force-Displacement Diagram of Contact Deformation of Smooth Spheres, Constitutive Laws a) nonlinear elastic, adhesion b) linear plastic, adhesion, dissipative force F N -F H Hertz Johnson Derjaguin displacement h K F N -F H Molerus Schubert (-) adhesion force Walton h K c) nonlinear plastic F N hardening Johnson Vu-Quoc perfect softening h K d) nonlinear viscoelastic e) nonlinear elastic, viscous (spring-dashpot) F N t= Yang F N k vis < k vis 4/ t> h K Kuwabara h K

44 . Contact Normal Force-Displacement Diagram of Contact Deformation of Smooth Spheres, Constitutive Laws f) nonlinear elastic, dissipative g) linear viscoplastic F N Sadd hysteresis h K F N -F H t= Rumpf t> h K h) nonlinear elastic-plastic, linear plastic F N Stronge h K i) linear plastic, nonlinear elastic, dissipative, adhesion F N Tomas j) linear plastic, nonlinear elastic, adhesion, dissipative, viscoelastic, viscoplastic F N t= Tomas 44/ -F H h K -F H.. h K > h K t> h K

45 . Contact 45/ normal force F N in nn 5 Normal Force-Displacement Diagram of Limestone Particles Increasing Unload Stiffness DMT a) particle approach reload Y d) particle detachment centre approach h K in nm pull-off force -F N 1-5 elastic-plastic yield limit c) elastic-plastic deformation to a nanoplate - plate contact b) elastic contact deformation HERTZ, - F H A adhesion limit U unload load Approach, Adhesion & Detachment DEM Force Law: m x Velocity: = x dt Position: = x dt Work: 1 1 = F1, v1 1 s1 1 W1, 1, 1 = F (s ) ds

46 . Contact Particle Impact, Force-Displacement Diagrams for Sticking or Rebound of Limestone, d 5 = 1. µm Contact model with history Sticking and oscillating 46/ Jasevicius, R., Tomas, J. and R. Kacianauskas, Simulation of microscopic compression-tension behaviour of cohesive Seminar dry 11, powder by applying DEM, XXXVI Summer School - Advanced Problems in, Proceedings, p. 18-1, St. Petersburg 8

47 . Contact Particle Impact, Force-Displacement Diagrams for Sticking or Rebound of Limestone, d 5 = 1. µm a) Sticking and oscillating at initial velocity v = mm/min b) Particle detachment at initial velocity v = m/min Damping coefficients acc. to Tsuji α d,i = ;.;.;.4;.5 Jasevicius, R., Tomas, J. and R. Kacianauskas, Simulation of microscopic compression-tension behaviour of cohesive Seminar dry powder 11, by 47/ applying DEM, XXXVI Summer School - Advanced Problems in, Proceedings, p. 18-1, St. Petersburg 6 Dec. 8 London F d = α d m 1, E * r 1, h K h K

48 . Contact Normal Force-Displacement Laws of Approach, Flattening, Unload, Reload & Contact Detachment Process Model Equation No Approach Van der Waals adhesion CH,sls r1, FH a F= 1 FN = FH = = - < h K force, sphere-sphere model 6 ( a ) ( ) F= h K a F= h K 1 1 Elastic deform., Hertz and DMT, F N (h K ) ν1 1 ν FN = E* r1, h K FH with: r1, = h K h K,f + r1 r, E* = + E1 E Yield limit Elastic-plastic deformation, p C VdW H,sls FN = π r1, p f ( κ A κ p ) h K FH with κ p = = h K,f h K h K,U F N (h K ) p f 6π a F= p f 1/ elastic-plastic contact area 1 A pl 1 h K,f 4 κ A = + = 1 ratio A pl A el h + K Unload Elastic recovery, F N (h K ), U-A FN = E * r1, ( h K h K,A ) π r1, κ p p f h K,A F 5 H h K,A h K h K,U Point of contact detachment h K,A,(1) = h K,U h K,f,el pl ( h K,U + κ h K,A,() ) 6 Reload F N (h K ), elastic, dissipative, FN = E * r1, ( h K,U h K ) + π p f r1, ( κ A κ p ) h K,U F 7 H h K,A h K h K,U non-linear A-U Adhesion limit Van der Waals adhesion 8 h K h K,A force, plate-plate model FN = π r1, p VdW h K FH Detachment Plate-plate model, F N (h K ) FH a π r1, pvdw h F= K,A 9 F N(hK ) = a F= - < h K h K,A ( a + h h ) ( a + h h ) / Resulting History dependent adhesion ( FN + F ) FH,() FH 1 H FH,(1) = FH + κ ( FN + FH ) π r1, κ p p f 1 force F H (F N ), non-linear r1, E * + FN F maximum + H adhesion force elastic-plastic contact κ p p VdW / p f 11 κ = = consolidation coefficient κ A κ p / + 1/ A pl /(A pl + A el ) p VdW / p f 48/ Tomas, J.: Assessment of mechanical properties of cohesive particulate solids part 1, Particulate Science & Technology 19 (1) 6 Dec London F= K,A K F= K,A K

49 . Contact Tangential Force-Displacement Diagram of Contact Deformation of Smooth Spheres, Constitutive Laws 49/

50 . Contact 5/ Tangential stiffness Tangential Force - Displacement Laws of Load, Unload and Reload for Load Dependent Adhesion Process Model Equation No 1/ with load dependent df 1 T k T,H = = 4 G * r K 1 adhesion dδ C,H Initial tangential Load dependent dft k T,H = = 4 G * rk = 4 G * r1, h K stiffness dδ δ= Friction limits Tangential force F T,C.H = µ i ( 1+ κ) ( FH + FN ) k T,H = Tangential displacement µ i δ C,H = π pf κa ( 1+ κ) ( FN + FH ) 4 8 G * / Loading tangential force δ δ C,H displacement relation δ 5 F T = FT,C,H 1 1 δc,h / Unloading Elastic-plastic frictional δ δ -δ C,H δ δ C,H behaviour with adhesion 6 U F T = FT,U FT,C.H 1 1 δc,h / Reloading Elastic-plastic frictional δ + δ -δ C,H δ δ C,H behaviour with adhesion 7 reload F T = FT,reload + FT,C.H 1 1 δc,h Mindlin, R.D., Deresiewicz, H., Elastic Spheres in Contact Under Varying Oblique Forces, Trans. ASME J. Appl. Mech. (195) 7-44 Tomas, J., Adhesion of ultrafine particles - a micromechanical approach, Chem. Engng. Sci. 6 (7), δ δ

51 . Contact Rolling Resistance - Rolling Angle Laws of Load, Unload and Reload for Load Dependent Adhesion 51/ Process Model Equation No rolling stiffness with load dependent dfr 16 G ( 1+ κ) ( FH + FN ) k γ 1 R,H = = 1 adhesion d ( 4 ) A p γ π ν κ f γ C,H Initial rolling stiffness Load dependent dfr 16 G ( 1+ κ) ( FH + FN ) k R,H, = = dγ π ( 4 ν) κ A p γ= f Friction limits rolling resistance force ( 1+ κ) ( FH + FN ) FR,C,H = µ R ( 1+ κ) ( FH + FN ) = α R k R,H = π r1, p f κ A rolling angle α R ( 4 ν) γ = π κ p ( 1+ κ) ( F + F ) 4 Rolling friction, micro-slip Loading γ γ C,H load dependent rolling friction coeff., slip coeff. Rolling resistance force displacement relation C,H A 16 G r1, rk (FN ) µ R (FN ) = α R with 1 α R α R,max = µ i r / r 5 K r γ F R = FR,C,H 1 1 γ C,H Unloading Elastic-plastic frictional -γ C,H γ γ C,H behaviour with adhesion γ U γ F R = FR,U FR,C.H 1 1 γ C,H Reloading Elastic-plastic frictional -γ C,H γ γ C,H behaviour with adhesion γ reload + γ F R = FR,reload + FR,C.H 1 1 γ C,H Johnson, K.L., Contact, Cambridge University Press, 1985 Tomas, J., Adhesion of ultrafine particles - a micromechanical approach, Chem. Engng. Sci. 6 (7), f N H 6 7 8

52 . Contact Torsional Moment - Rotation Angle Laws of Load, Unload and Reload for Load Dependent Adhesion Process Model Equation No 1/ 1 Torsional with load dependent dm to 8 G r K M 1 to stiffness adhesion k to,h = = 1 1 Initial torsional stiffness Friction limits k to,h = 5/ Load dependent dm 8 ( 1+ κ) ( F + F ) k to,h = dφ dφ to M to = = G M π κ Torsional moment µ ( 1+ κ) ( F + F ) i N H M to,c,h = π κ A p f Rotation angle π µ i κ A pf φ C,H = 4 4 G Loading Moment-rotation angle φ φ φ φ C,H relation 5 M to = 4 M to,c,h φc,h 4 φc,h Unloading Elastic-plastic frictional ( φ φ) ( φ φ) 6 U U M to = M to,u 8 M to,c,h φ C,H φ φ C,H behaviour with adhesion φc,h 8 φc,h Reloading Elastic-plastic frictional ( φ + φ) ( φ + φ) 7 rel rel M to = M to,rel + 8 M to,c,h φ C,H φ φ C,H behaviour with adhesion φc,h 8 φc,h Deresiewicz, H. Contact of Elastic Spheres Under an Oscillating Torsional Couple, Trans. ASME J. Appl. Mech. 1 (1954) 5-56 Tomas, J., Adhesion of ultrafine particles - a micromechanical approach, Chem. Engng. Sci. 6 (7), to,c,h A H p f N /

53 5/. Contact maximum specific hysteresis work W m,n,diss, W m,t,max, W m,r,max, W m,to,max in µj/g Elastic Hysteresis Work of Contact Deformation, Tangential Microslip, Rolling & Torsion F, M F(h) dh M( γ) dγ h,δ,γ W m,to,max = f(f N ) W m,n,diss = f(f N ) W m,r,max = f(f N ) normal force F N in nn Tomas, J., Adhesion of ultrafine particles - energy absorption at contact, Chem. Engng. Sci. 6 (7), W m,t,max = f(f N ) F H (F N ) M to (F N ) M R (F N ) F T (F N ) F H (F N )

54 . Contact 54/ maximum specific detachment & friction work W m,n,a, W m,t,c, W m,r,c, W m,to,c in mj/g Detachment and Friction Work of Load Dependent Tangential Contact Sliding, Rolling & Torsion F, M F(h) dh M( γ) dγ h,δ,γ W m,t,c (. r K ) W m,r,c (45 ) W m,to,c (45 ) W m,n,a (F N ) normal force F N in nn Tomas, J., Adhesion of ultrafine particles - energy absorption at contact, Chem. Engng. Sci. 6 (7), F H (F N ) M to (F N ) M R (F N ) F T (F N ) F H (F N )

55 . Contact 55/ Rolling, cycle between γ U -γ reload Elastic Hysteresis Work, Detachment and Friction Work for Load Dependent Adhesion Work Microprocess Equations No 5/ Elastic Adhesion, one closed load cycle bet E* 5 WN,diss = r1, ( h K,U h K,A ) + π r1, pf E* h K,U h K,A pf ( h K,U h ) 1 K,A Wm,N,diss = + hysteresis 1 π ρ s r 1, 16 ρs r1, work ween h K,U h K,A [ κ A h K,U κ p ( h K,U h K,A )] ( h K,U h K,A ) [ κ A h K,U κ p ( h K,U h K,A )] 5 / Elastic Microslip, cycle between δ U -δ reload F T,reload + F 4 T,U δ U + δ reload δ + δ U reload W W T,H = 8 FT,C,H δc,h T,H,max = FT,C,H δ C, H hysteresis 4 F δ C,H 5 δ 5 C,H work with R,U U + γ reload U microslip Detachment work Friction work Torsion, one closed twisting cycle between φ U -φ reload Compression & detachment, -a h K,U h K,A -a Sliding, δ > δ C,H Rolling, γ > γ C,H Torsion, φ > φ C,H T,C,H 4 M R,reload + M γ γ + γ reload W R = M R,C.H γ C,H γ M R,C.H C,H γ WR,max = M R,C.H γ C,H C,H 16 M + M φ ( φ + φ ) 16 to,reload to,u reload U reload W = M φ Wto,max = M to,c,h φ 4 C,H to to,c,h C,H 8 M 4 φ φ 9 to,c,h C,H C,H / ( ) 4 φ + φ reload U φc,h π pf W N,A = r1, pf [ κa h K,U + κp h K,U a F= ] + FH a F=, Wm,N,A = [ κ A h K,U +κ p h K,U a + 5 F= ρs r1, 1 ( 1 + κ) ( F + ) N FH W N,A = + κ( FN + FH ) a F= + FHa F= F H a F= π κ A r1, p + f π r1, pf δ * * WT,C = W T ( δ ) dδ = FT,C,H ( δ δc,h ) ( ) ( 1 + κ) ( FN + FH ) 6 WT,C,max = WT,C rk = µ i δ π p C,H f κ A γ * * WR,C = M R ( γ ) dγ = M R,C,H ( γ γ C,H ) ( ) ( 1+ κ) ( FH + FN ) 7 WR,C,max = WR,C 45 = γ π p C,H f κa φ * * Wto,C = M to ( φ ) dφ = M to,c,h ( φ φc,h ) µ i ( ) ( 1 + κ) ( FN + FH ) 8 Wto,C,max = Wto,C 45 = π κ p φc,h Tomas, J., Adhesion of ultrafine particles Energy absorption at contact, Chemical Engineering Science 6 (7), A f

56 . Contact Activation energy Geometrical activation factor Locally compressed contact zone Contact friction Microprocess Activation Energy of Contact Deformation, Tangential Contact Sliding, Rolling & Torsion Locally deformed contact zone AF Specific work Mass related W m m 64 r 1 1, 1, 6 8 = µ cap = = 1 1 m cap,1, h K,U W M in kj/mol M W m Compression and p f a FH a - 4 Wm,N,act = κ A + κ p + detachment ρs h K,U π r1, pf h K,U...h K,U...h K,A... Sliding δ > δ p r f 1, C,H W 1 - m,t,c,act = 8 µ i κa ρ h s K,U W = M ~ Rolling γ > γ C,H p r f 1, 66 - Wm,R,C,act = 4 κa ρ h Spinning φ > φ C,H i A f 1, - 1 W m,to,c,act 8 µ κ = s p ρ s K,U h r K,U 56/ For limestone powder, d 5, = 1. µm, Δh r 16 kj/mol to decompose calcite

57 . Powder Determination of Stationary and Instantaneous Yield Locus F N,pre Preshear F N Shear F S F S s s Shear force F S σ pre Preshear Shear σ pre Shear stress τ = F S / A steady-state flow incipient yielding ϕ i Endpoint Stationary yield locus Yield locus ρ b = const. τ c σ<σ pre ϕ st 57/ Shear displacement s σ Z σ σ c σ 1 Normal stress σ = F N /A σ M,st

58 . Powder Under-, Overconsolidation and Critical Consolidation at Shear Test 58/

59 .. Powder Simulation of Non-Rapid, Frictional, Steady-State Shear Flow by the Discrete-Element-Method ~ TiO particles, d 5 = µm, σ = kpa, v s = mm/min Underconsolidation Shear force Shear displacement ε s,d ε s,d /

60 . Powder Shear Stress - Normal Stress D. of Yield Loci (YL) and Stationary Yield Locus (SYL) for Limestone 15 SYL YL 4 6/ shear stress τ in kpa 1 5 YL 1 YL YL σ normal stress σ in kpa particle size d 5 = 1. µm, surface moisture X W =.5 % solid density ρ s = 74 kg/m shear rate v S = mm/min straight line regression fit.98

61 . Powder Isentropic Compression of Cohesive Powders 1) Uniaxial powder compression ) Isentropic compression function Powder pressure σ h a) elastic σ loading Y W V = σ(h) d(h/h ) plastic compression b) elastic-plastic elastic recovery unloading displacement h Bulk density ρ b ρ b, Isostatic tensile strength -σ n = 1 Compressibility index of ideal gas n = incompressible σ M,st σ ρ b = ρ b, (1 + ) < n < 1 compressible Centre stress of steady-state flow σ M,st n 61/ Adiabatic gas compression: dv 1 V = dp κ ad p (1) Isentropic powder compression: ρ σ b M,st dρ dσ b M,st = n ρ σ + σ () ρb, b M,st Compressibility index of cohesive powders for small (1 < σ < 5 kpa) and medium pressures (5 < σ < 1 kpa) Index n Evaluation Examples Flowabiliy.1 incompressible gravel free flowing.1.5 low compressibility fine sand compressible dry powder cohesive.1-1 very compressible moist powder very cohesive

62 . Powder Compression Functions of Cohesive Powders for Small, Medium and Large Pressures Compression parameter Compression rate Compression function Compression work Equations for small to medium pressures dρ b ρb (1) = n dσ σ + σ M,st M,st n ρ σ b M,st () ρ W b, m,b = 1+ σ n σ = 1 n ρ b, σ 1+ σ M,st 1 n 1 Equations for medium to large pressures (ρ b ρ s ) dρ b ρs ρb () = n dσ σ + σ M,st M,st n σm,st ρ b = ρs ( ρs ρb, ) 1+ (4) σ p (5) max ρs ρb ( σm,st ) m,b = n ρ ( σ ) W dσ (6) b M,st M,st 6/

63 . Powder a) Shear force-displacement diagram of preshear Mass Related Preshear and Compression Work F N b) Mass related work versus centre stress of steady-state flow for compression and preshear shear force F S 6/ s pre F S s τ pre, YL τ pre, YL τ pre, YL1 W b, pre = F S (s) ds displacement s specific compression and preshear work W m,b, W m,b,pre isostatic tensile strength -σ W m,b,pre =. cosϕ.. i sinϕ st (1 + ) s pre. σ h Sz. ρ b, s pre. σ h Sz. ρ b,. cosϕ i. sinϕ st Compression n 1 - n σ σ M,st < n < 1 W m,b =.. (1 + ) - 1 ρ b, σ M,st σ σ 1-n 1-n average pressure at steady-state flow σ M,st

64 . Powder Mass Related Preshear and Compression Work and Power Consumption for Limestone Powder mass related preshear and compression work W m,b,pre, W m,b,com in J/kg 64/ shear stress τ τ pred γ pre distortion γ σ centre stress for steady-state flow σ M,st in kpa bulk density ρ b dσ ρ ( σ centre stress σ M,st b P m,b,pre W m,b,pre M,st M,st ) W m,b,com Tomas, J., Product design of cohesive powders - mechanical properties, compression and flow behaviour, Chem. Engng. & Technol. 7 (4), specific power consumption P m,b,pre in mw/kg angle of internal friction ϕ i = 7 stationary angle of internal friction ϕ st = 4 isostatic tensile strength σ =.65 kpa preshear displacement s pre = mm shear rate v S = mm/min curve regression fit =.98 σ* σ pre compression preshear h τpre s

65 65/. Contact maximum specific detachment & friction work W m,n,a, W m,t,c, W m,r,c, W m,to,c in J/kg Detachment and Friction Work of Load Dependent Tangential Contact Sliding, Rolling & Torsion F, M F(h) dh M( γ) dγ h,δ,γ W m,t,c (. r K ) W m,r,c (45 ) W m,to,c (45 ) W m,n,a (F N ) normal force F N in nn Tomas, J., Adhesion of ultrafine particles - energy absorption at contact, Chem. Engng. Sci. 6 (7), F H (F N ) M to (F N ) M R (F N ) F T (F N ) F H (F N )

66 . Powder Constitutive Functions of Stiff/Compliant Particle Contacts and Cohesive, Compressible Powders a) particle contact deformation b) particle adhesion c) powder yield loci force F N -F H stiff compliant displacement h K adhesion force F H compliant stiff normal force F N shear stress τ σ cohesive YL YL SYL free flowing SYL normal stress σ 66/ d) consolidation functions uniaxial compressive/ tensile strength σ c, σ Z1 compliant cohesive stiff, free flowing consolidation stress σ 1 e) powder constitutive models radius stress σ R YL σ cohesive YL SYL CL SYL CL free flowing average pressure σ Μ σ iso f) compression function bulk density ρ b ρ b, compliant compressible stiff, incompressible consolidation stress σ 1