Granular friction in a wide rage of shear rates Takahiro Hatano - PowerPoint PPT Presentation

Granular friction in a wide rage of shear rates Takahiro Hatano (The University of Tokyo) Collaborators: Osamu Kuwano (JAMSTEC, Yokohama) Ryosuke Ando (AIST, Tsukuba) 06/27/2013 Physics of Granular Flow (Kyoto)

granular friction: examples

10 m -4 10 m 3 friction of fault 10 0 m fine rock powder “fault gouge” friction of fault friction of granular matter (microscopic basis) 3

character of “fault gouge” particle size distribution is power-law (fractal) (Heilbronner & Keulen 2006) exponent 2.5 to 3.0 (Chester et al. 1993) numerous sub-micron particles (Chester & Chester 1993) very different from industrial situation! 4

velocity range of fault motion must investigate a very wide range 10 0 earthquakes 10 − 2 “slow earthquakes” (no seismic waves) slip velocity note: [m/s] shear rate depends on shear-band thickness (thickness < 10cm) 10 − 10 plate motion 10 − 9 ≤ ˙ γ ≤ 10 1 ? 5

granular friction: an empirical law µ ( I ) = µ s + µ 2 − µ s I 0 /I + 1 positive slope � m I ≡ ˙ “inertial number” γ Pd Jop et al. JFM 2005; Jop et al. Nature 2006 ... works well for large inertial number; I=O(1) surface velocity profile flow rate vs tan θ 6

granular friction: numerical experiment consistent with Pouliquen’s law µ ( I ) = µ s + µ 2 − µ s linear I 0 /I + 1 µ ( I ) � µ s + µ 2 � µ s I I 0 I � 1 da Cruz et al. 2005 size-dependence <-- nonlocal effect (Kamrin’s talk) I ≥ 10 − 4 positive slopes only da Cruz et al. Phys. Rev. E (2005) TH, Phys. Rev. E (2007) Koval et al. 2009 Peyneau & Roux, Phys. Rev. E (2008) Koval et al. Phys. Rev. E (2009) 7

granular friction: physical experiments 0.62 a Experiment Bardassarri et al. Fit PRL 2006 Annular Plate Direction 0.61 Average torque [N m] of motion b -6 10 10 -8 S(k) [N 2 m 2 ] 0.6 10 -10 10 -12 10 -14 Force Chains Granular medium 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 0.59 k [rad -1 ] Torsion Spring 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Motor turns this axle -1 ] Velocity [rad s negative slope --> positive slope see also: Lu et al. J. Fluid Mech. 2007 Petri et al. EPJB 2008 8

for steady states constant α is generally negative! (up to ~ mm/sec) (e.g. Marone, Ann. Rev. Earth Planet. Sci. 1998) negative slope is rather ubiquitous! normal pressure ~ 100 MPa Dieterich 1979 in earthquake physics.... µ ( V ) = µ ( V ∗ ) + α log V V ∗ α ∼ − 10 − 2 to − 10 − 3

? At very low shear rates, constitutive law is still not established numerical experiment physical experiment and an example is ... current status µ ( I ) � µ s + µ 2 � µ s µ ( V ) = µ ( V ∗ ) + α log V I I 0 V ∗

what kind of constitutive law can explain this? Komatsu et al. PRL 2001 Forterre and Pouliquen, Ann. Rev. Fluid. Mech. 2008 exponential velocity profile in inclined plane flow 10 3 10 2 Gas 10 1 〈 v(h) 〉 / 〈 v(h 0 ) 〉 Liquid 10 -1 10 -2 40 10 -3 Solid 20 10 -4 0 10 -5 -10 0 10 10 -6 -10 -5 0 5 10 ( h - h 0 ) / a

1. At very low shear rates, constitutive law is still not established A. nonlocal effect B. physics of negative slope? 2. If negative slope is true, how is it compatible with Pouliquen’s law? (e.g. Kamrin & Koval 2012) (Dieterich 1979) questions µ ( I ) � µ s + µ 2 � µ s µ ( V ) = µ ( V ∗ ) + α log V I I 0 V ∗

OUR GOAL 1. Negative slope for glass beads? 2. Negative to positive crossover? How? 0.62 a Experiment Fit 0.61 Average torque [N m] b 10 -6 10 -8 S(k) [N 2 m 2 ] I = I c 0.6 10 -10 10 -12 10 -14 µ ( I c ) minimum 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 0.59 k [rad -1 ] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -1 ] Velocity [rad s 3. What if fault gouge? 13

annular channel (D1=15mm, D2=25mm) room humidity ~ 50% (dispersity ~ 10 %) mean diameter 270 μm glass beads normal stress 10 to 30 kPa (constant pressure) [m/sec] sliding velocity experimental A commercial rheometer (AR2000ex, TA Instruments) a b Torque motor Angular velocity Ω Gaps < 50µm Quartz glass Upper plate Camera Lower plate Normal force sensor transparent sidewall D 1 D 2 Ω D 2 / 2 = 10 − 4 to 3

shear rate collapse to a master curve (effective flow width) (depth)/(mean diameter) normalized by upper plate velocity velocity profile 2 c d V 0 V ( x ) � V 0 10 − x/W W � 5 d γ ≡ V 0 /W ˙ � m I = V 0 W Pd

comparable to fault gouge minimum value 10kPa 20kPa 30kPa not monotonic! At lower velocities, rate dependence of friction coefficient 0.55 0.50 0.45 µ 0.40 0.35 -4 -3 -2 -1 0 1 10 10 10 10 10 10 V (m/s) µ = µ ∗ + α log V µ min V ∗ α = − 0 . 003 ∼ − 0 . 004

what sets α？(open question) a certain range of α 30kPa 10kPa 30kPa 20kPa 10kPa negative slope apparent for I ≤ 10 − 2 b c 0.06 0.04 ∆ µ ≡ µ − µ min Δμ 0.02 0 -0.02 -6 -4 -2 10 10 10 I α ∼ − 10 − 2 to − 10 − 3

chromite sand 10kPa DEM simulation (TH, PRE 2007) (c=0.6) agrees with simulations (including numerical factor!) glass beads 30kPa I 10kPa 30kPa 20kPa e.g., da Cruz et al. PRE 2005 constitutive law at high velocities a b ∆ µ ∆ µ ≡ µ − µ min ∆ µ = cI c � 0 . 6 µ = µ min + cI I > I c

agrees with simulations (including numerical factor!) obeys a master curve dilation at high velocities a b c ∆ H ≡ H ( I ) − H ( I c ) ∆ H ( I ) = c � I W s c � � 0 . 2

1. At higher shear rates, constitutive law agrees with DEM simulation. 2. Collapse to a master curve using I --> Bagnold’s regime. 3. No master curves at sufficiently lower inertial number Instead, I ≤ 10 − 2 µ = µ ∗ + α log V V ∗ α ∼ − 10 − 2 to − 10 − 3

glass beads 30 kPa constitutive law data: where is crossover point? I F2202mean c d 0.50 Err. as stdev btw run cycle (3up, 4dn) α = 0 . 0036; c = 0 . 72 0.48 0.46 µ α = 0.0036, c 1 = 0.72, µ 0 = 0.39 I c � 0 . 008 0.44 0.42 0.40 -6 -5 -4 -3 -2 -1 0 10 10 10 10 10 10 10 I � µ = µ 0 + α log(˙ γ / ˙ γ 0 ) + c ˙ m/Pd γ = O (10 − 3 ) I c = − α /c

exponential velocity profile for “creep” deformation of solid regime an application: � µ = µ 0 + α log(˙ γ / ˙ γ 0 ) + c ˙ m/Pd γ

--> Other laws may come into play Pouliquen’s law cannot reproduce this. Komatsu et al. PRL 2001 Forterre and Pouliquen, Ann. Rev. Fluid. Mech. 2008 velocity profile in inclined plane flow 10 3 10 2 Gas 10 1 〈 v(h) 〉 / 〈 v(h 0 ) 〉 Liquid 10 -1 10 -2 40 10 -3 Solid 20 10 -4 0 10 -5 -10 0 10 10 -6 -10 -5 0 5 10 ( h - h 0 ) / a

use (1) θ: angle of slope if P is independent of h (Janssen’s law), h θ P: normal pressure (1) σ: shear stress h: depth ρ: mass density force balance eq. for heap flow (along flow direction) can reproduce exponential flow profile � µ = µ 0 − α log( V/V 0 ) + c ˙ m/Pd γ d σ dh = ρ g sin θ , Gas Liquid σ = µP Solid d ˙ dµ γ = ρ g sin θ γ dh d ˙ P γ 0 e − h/h 0 h 0 ≡ α P/ ρ g sin θ γ ( h ) � ˙ ˙

underlying physics of weakening? � µ = µ 0 + α log(˙ γ / ˙ γ 0 ) + c ˙ m/Pd γ

frictional (power input) = (energy dissipation rate) i j 1. friction 2. damping first term is particle-level friction non-conservative force normal underlying physics? γσ = D ˙ D = F dis ij · v ij F dis ij = F 1 + F 2 D = ( F 1 + F 2 ) · v F 1 = | F 1 | v ( t ) + | F 2 | v ( n ) F 2 F 1 · F 2 = 0 µ = D γ P = · · · � µ p + cI | F 1 | ∝ µ p ˙

(Brechet & Estrin 1994) increase of contact area due to plasticity a, t 0 : constants t: duration of contact particle-level friction is not a constant (but time-dependent) in sheared systems, aging of grain contact A ( t ) = A 0 (1 + a log t ) t 0 γ − 1 A ( t ) = A 0 (1 − a log(˙ γ t 0 )) t � ˙ γ ) = µ 0 (1 − a � log(˙ µ p (˙ γ t 0 ))

particle-level friction is time-dependent! (in DEM, it is constant) cf. Bocquet et al. Nature 1998 aging due to moisture aging of grain contact µ p ( t ) = µ 0 (1 − a log(˙ γ t/t 0 )) γ ) = µ 0 (1 − a � log(˙ µ p (˙ γ t 0 )) b r eq e

OUR GOAL 1. How does this crossover occur? e n identify I = I c µ ( I c ) minimum o d 2. Inertial-number description valid for gouge? versus 29

experimental • Slip rate : 100 µ m/s to 0.3m/s • Normal stress: 0.1-0.9 MPa • Room temperature and humidity • Cylindrical Specimen • Westerly granite • Inner/outer diameters : 6mm/10mm. • Temperature measurement with an IR sensor • Gouge layer formed by preshearing. (sample not sealed; an open system) 30

experimental Cumulative particle-size distribution • A commercial rheometer (AR2000ex, TA Instruments) power law distribution • Normal stress: 0.1-0.9 MPa slope -2.2 • Slip rate : 100 µ m/s to 0.3m/s • Cylindrical Specimen • Westerly granite • Inner/outer diameters : 6mm/10mm. • Temperature measurement with an IR sensor Mean particle size: 2.4 µm • Presheared. Gouge layer formed. Gouge layer thickness : ~10µm (4-5 particles) 31

too large! remarkable weakening in intermediate regime ? e.g. Goldsby & Tullis 2002; di Toro et al. 2004, etc... velocity dependence of friction coefficient Weakening Strengthening Intermediate-V High-V µ = µ ∗ + α log V α � � 0 . 2 V ∗