Micro-Macro transition (from particles to continuum theory) Granular Materials. Approach philosophy. Model Granular Materials

Transcription

1 Micro-Macro transition (from particles to continuum theory) Stefan Luding MSM, TS, CTW, UTwente, NL Stefan Luding, MSM, TS, CTW, UTwente, NL Granular Materials Real: sand, soil, rock, grain, rice, lentils, powder, pills, granulate, micro- and nano-particles Model Granular Materials steel/aluminum spheres spheres with dissipation/friction/adhesion Approach philosophy Introduction Single particle Single Particles Particle Contacts/Interactions Contacts Many particle cooperative behavior Applications/Examples Conclusion Many particle simulation Continuum Theory 1

2 Ring geometry Ring shear cell experiment Ring shear cell experiment 2

3 Zoom into force chains 2 shear cell force chains 2 shear cell shear band 3

4 2 shear cell energy Ring geometry Ring geometry 4

5 Averaging Formalism 1 Q = Q = wv V Q p p p p V p V Any quantity: Q p - Scalar - Vector - Tensor = c Q c Averaging Formalism 1 Q = Q = wv V Q Any quantity: p p p p V p V p Q In averaging volume: V V Averaging ensity 1 p p wv V V p V Q = ν = Any quantity: p Q = 1 - Scalar: ensity/volume fraction V 5

6 ensity profile Global volume fraction Averaging Velocity 1 p p Q = ν v = wv V v V p V Any quantity: p Q = v - Scalar - Vector velocity density p p V p v Velocity field exponential 6

7 2 shear cell shear band Velocity gradient 1 vφ vφ v r φ = 2 r r s d exponential: vφ ( r) = v0 exp( ( r Ri )/ s) Velocity distribution 7

8 Averaging Stress 1 p pc c wv V p V c Q = σ = l f Any quantity: p 1 Q = σ = p l f V c - Scalar - Vector - Tensor: Stress p pc c V f c p 2 shear cell force chains Stress tensor (static) shear stress r 2 8

9 Stress tensor (dynamic) exponential 2 shear cell energy Stress equilibrium (1) 1 ( r ) 1 ( rσ r ) σ rr φ σ = σφφ e r + + σφr e φ r r r r d a = dt v = t v + ( v acceleration: ) v 2 σ rr ρa = σ 0 = ρvφ + r + ( σ rr σ φφ ), r σ rφ 0 = r + ( σ rφ + σ φr ), r ( rσ ) ( rσ r ) rr φ = σ φφ und = σ φr r 0 r 2 ( σ σ r und σ σ r ) rr φφ rφ φr 9

10 Stress equilibrium (2) 2 σ rr 0 = ρvφ + r + ( σ rr σ φφ ), r σ rφ 0 = r + ( σ rφ + σ φr ), r?? Averaging Fabric Q 1 p p pc pc wv V n n V p V c = F = Any quantity: p Q = F = n n p pc pc - Scalar: contacts - Vector: normal - Contact distribution c V c p Fabric tensor contact probability center wall 10

11 Fabric tensor (trace) contact number density Fabric tensor (deviator) an-isotropy (!) Averaging eformations π h p pc c Q = ε = wv l F V p V c eformation: 1 c pc ( ε ) 2 minimal! S = l - Scalar - Vector - Tensor: eformation V c p 11

12 Macro (contact density) Macro (bulk modulus) trσ E = trε Macro (shear modulus) devσ G = devε 12

13 Local micro-macro transition From virtual work For each single contact Stress tensor σ Stiffness matrix C (elastic) - Normal contacts - Tangential springs eformations (2): Stress changes - Isotropic compression ε V - Isotropic δσ V - eviatoric strain (=shear) ε, φ ε - eviatoric δσ, φ σ Averaging Rotations eformation: 1 p p p wv V ω V p V c Q = νω = Q p p = ω - Scalar - Vector: Spin density - Tensor V p ω c p Rotations spin density * eigen-rotation: = ω ω Wr φ 13

14 Velocity distribution Spin distribution Velocity-spin exp. distribution exp. sim. sim. high dens. low dens. 14

15 M Macro (torque stiffness) 2 µl = κ 2 µl Summary Quantitative comparison between Experiments (2 Couette) & M simulations (soft disks) Observations: - shear band & dilation - inhomogeneous (force-chains) - (almost always) an-isotropic - micro-polar (rotations) Conclusion Qualitative agreement 100% Subjective quant. agreement 50%-80% Reasons for discrepancy: - idealized 2 disks vs. real disks - idealized 2 motion vs. out-of-plane - background friction (powder on base) - small differences in geometry - 15

16 3 shear cell H=0.016m H=0.040m H=0.062m 16

17 3 shear band center position 80% agreement up to now 3 shear band center position 80% agreement up to now 3 shear band width 80% agreement up to now 17

18 Constitutive relations shear rate γ no friction friction Constitutive relations: Mohr-Coulomb no friction friction Constitutive relations: stress-structure friction no friction 18

19 Constitutive relations: anisotropy friction no friction Constitutive relations: shear softening τ γ d0 viscosity vs. shear rate I = γ p ρ 0 γ d 0 I = p ρ0 no friction friction stress ratio vs. shear rate 19

20 stress ratio vs. I non-colinearity anisotropy vs. shear path l = t γ γ µ s γ min ( l ) s s ds α ( lγ ) = µ ( µ s 0 ) exp( lγ ) ( lγ ) α 1 = αl ( µ s ( l ) dl γ γ γ 20

21 3 Flow behavior steady state shear Obtain constitutive relations from one SINGLE simulation: - Mohr Coulomb yield stress - shear softening viscosity - compression/dilatancy - inhomogeneity (force-chains) - (almost always) an-isotropy - micro-polar effects (rotations) The End Anisotropy 21

22 Micro-macro for anisotropy rheology compression tension Micro informations: shear bands potential energy rotations displacements Rotations (local) irection, amplitude, anti-symmetric (!) stress 22

23 Anisotropy Shear? Simple shear 0 2ε s 0 ε s 0 εs ε = = ε s 0 εs 0 Rotation + symmetric shear Anisotropy Shear? Simple shear 0 2ε s 0 ε s 0 εs ε = = ε s 0 εs 0 Rotation + symmetric shear Rotate symmetric shear tensor by 45 degrees 0 εs ε 0 T s R45 R45 = ε s 0 0 ε s Biaxial shear : compression+extension An-isotropy in stress 23

24 An-isotropy (Stress) Stress: Isotropic: tr σ, and deviatoric: dev σ = σ zz -σ xx Minimal eigenvalue: σ xx Maximal eigenvalue: σ zz ev. Stress fraction s = dev σ / tr σ s = βs ( smax s ) ε Exponential approach to peak max ( ) 1 s s = exp β ε s (, p, ) β ρ µ s An-isotropy (Stress) ε s ( s s ) = β s max Stiffness tensor vertical horizontal shear ifferent moduli: against shear C 2 perpendicular C 1 one shear modulus 24

25 An-isotropy (Structure) Structure changes with deformation ifferent stiffness: More stiffness against shear C 2 Less stiffness perpendicular C 1 One (only?) shear modulus Anisotropy A = C 2 - C 1 evolution A = β F ( Amax A) ε Exponential approach to maximal anisotropy see Calvetti et al An-isotropy (Stress & Structure) ε ε s ( s s ) = β F s max ( ) A = β A A max An-isotropy (Stress & Structure) Modulus Friction ε ε s ( s s ) = β F s max ( ) A = β A A max 25

26 1 2 V C 1 2 ε C C ε Constitutive model scalar (in the biaxial box eigen-system) δσ = Eε + Aε V V δσ = Aε + Bε V Constitutive model scalar ˆ ( φ ) + ( ) ( φ ) 2G δσ Aε ˆ ε = + (in the biaxial box eigen-system) δσ = Eε + Aε V V δσ = Aε + Bε V Constitutive model tensorial ( ) ( ) ˆ ( φ ) C C 6G + cos 2 ε φ 2φ (arbitrary eigen-system) ( ) δσ = Eε + Aε cos 2φ ε 2φ V V C ( 2 E) cos( 2 2 ) ˆ ( ) + ( E B) ε ˆ ( φ ) δσ = Aε V + B ε φε φc φc ε Mechanical waves in sand, 3 simulations O. Mouraille, S. Luding Particle Technology, NSMm TUelft, NL Thanks for discussions : E. Grekova, G. Herman, W. Mulder, A. Suiker, 26

27 Content Motivation The numerical method Some 3-simulation results Conclusions and Perspectives Why? - Generally only 30% to 35% of the oil contained in a reservoir is extracted. - How to take into account the complexity of granular material in the wave propagation theory? Soil investigations - What about rotations? Framework Assembly of grains (dense, frictional and non-cohesive) Compressional (P)-wave, shear (S)-wave, (R-wave?) Forces transmission, friction, rotations 3 simulations (EM) Micro-Macro theory 27

28 iscrete element method The equation of motion is solved for each particle according to the contact forces and torques. Linear contact model - Normal part : f = N k.δ repulsive and attractive forces (dissipation) - Tangential part : Sliding -(k T ), Rolling- and Torsion- resistance µ: friction coefficient Model systems Crystal structured packing Poly-disperse packing ifferent boundary conditions: with walls / periodic Sound 3-dimensional modeling of sound propagation P-wave shape and speed Stefan Luding, s.luding@tnw.tudelft.nl Particle Technology, elftchemtech, Julianalaan 136, 2628 BL elft 28

29 P-wave animation - One layer is shifted - Smooth wall motion -periodic boundaries - Fixed side walls -Black particles are fixed System 0: regular lattice a=0 P-wave animation 29

30 P-wave animation regular packing - one layer is shifted - periodic boundaries - black particles are fixed P- and S-wave Compressive (P)-wave Shear (S)-wave Stiffness tensor From the micro-macro transition theory we can derive a stiffness tensor: C C 1 2 c c c c t c c c c Cαβγφ = V a k nα nβ nγ nφ + k nα tβ nγ tφ p V c= 1 c= 1 Stiffness tensor of the regular packing imensionless tensor (only structure dependent) C = C V ka2 C 1111 = C = C 3333 = 2 C 1133 = C 2233 = C 1313 = C = C = C =

31 Waves and stiffnesses - Components of the stiffness tensor corresponding to the direction of particle-motion for both P- and S-wave - In this case : C 3333 for the P-wave and C 1313 for the S-wave. - V p and V s denote the Peak velocity. Theory : ρvp C ρvs C V 2 p 2 s V C C Waves and stiffnesses Ratios of C entries Ratios of velocities C C V 11 V C C V 33 V C C V 11 V Influence of micro properties Compressive (P)-wave Elastic Visco-Elastic How relevant is the damping coefficient in our model? 31

32 P-wave in a regular packing - Simulations varying the friction coefficient µ - Stick and slip contacts in the packing Space-Time analysis Space-Time analysis amplitude time space 32

33 +polydispersity a>0 Contacts (force intensity) Signal Analysis time-fft Stress-time signal Power-spectrum 33

34 Frequency-space iagrams a = δ/2 a = δ a = 2δ ispersion relations space-time-fft a = δ/2 a = δ a = 2δ Space-Time analysis space Fourier transform 34

35 Space-Time analysis time Fourier transform ispersion relation -Compressive and shear wave - Small amplitude - Regular packing Frequency P-wave S-wave Wave number Rotation waves? 35

36 Conclusions The signal reveals structure- and state-changes in the packing Larger friction increases the velocity The wave gets broader and accelerates while propagating The wave velocity is directly related to the corresponding stiffness Rotational waves? The model captures the particle systems interesting features Perspectives Local (point) perturbation (Greens function): spherical waves Further study of rotations, and their possible propagation Falling sandpile dense to dilute Rotational order counterclockwise clockwise 36