Xiaoping JIA
Institut Langevin, ESPCI ParisTech, CNRS UMR & Université Paris-Est Marne-la-Vallée, France
Acknowledgments: - C. Caroli, T. Baumberger (University P.M. Curie, Paris Jussieu)
- T. Brunet, J. Laurent, Y. Yang,Y. Khidas, V. Langlois, P. Mills (Univ. Paris-Est MLV) - P. Johnson (Los Alamos National Laboratory, USA) - J. Brum (Montevideo, Uruguay) - M. Harazi, J. Léopoldès, A. Tourin, J.-L. Gennisson, M. Tanter, M. Fink (Institut Langevin)
Multiscale Acoustics of Dense Granular Media:
from probing to pumping
Tutorial Day, Physical Acoustics Group, IOP, Birmingham, UK, Sept 21 2017
1
Background
grain size 2
Motivation
Granular Matter athermal
(Andreotti, Forterre, Pouliquen, 2011)
Shaking
- Duran et al 1996 - Jaeger, Liu & Nagel 1990 - D’Anna et al 2001 - Marchal et al 2009 - van Hecke et al 2011 - Kolb, Clément et al 2011
Avalanche experiments
- Jaeger et al 1996 - Rajchenbach 2000 -GDR MIDI 2004 - …
Oscillation
3
uac < d (no rearrangement!) for d = 50µm - 1mm
Γ < 1 (fac = 0.1-100 kHz)
Investigation of the transition from solid to liquid states using sound waves:
◆Nondestructive probing (uac < 1 nm)
◆Controlled pumping (uac ~ 0.01– 10µm): NL responses & shear modulus softenning
0.64
sands
foams
colloids
emulsions
Unjamming transition (1/2): from solid to liquid states
♦ Jamming transition in soft matter (→ yield stress fluids) - Liu and Nagel, Nature (1998) - van Hecke, J. Phys.:Condens.Matter (2010)
φc ~ 0.64
4
♦ Frictional contact force networks in dense granular packings (photoelastic visualization)
under pressure (Dantu 1957) under shear (Behringer1999)
Stick-Slip Motion
◆Threshold rheology of jammed systems ◆Plasticity of amorphous materials
- Persson 1998 - Baumberger & Caroli 2006
Boundary Lubrication
10 -9 – 10 -6 m
Robbins 1990; Israelachvili 1994
10-3 - 105 m
Landsliding
Marone 1998; Scholz, 2002
Glassy Rheology
10 -9 – 10 -6 m
Barrat, Tanguy et al 1999 Maloney & Lemaitre 2000
Granular Flow
10 -4 – 10 -2 m
Jaeger, Nagel & Behringer, 1996
Liu, Nagel, 1998
5
Unjamming transition (2/2): from stick to slip
Sound in a granular material : disorder & nonlinearity C-h. Liu & S. Nagel , PRL 68 (1992) & PRB 48 (1993)
◆Vtof ≠ Vg
Vtof ≈ 250 m/s
Vg ≈ 50 m/s
(i) High-amplitude continous waves (ii) Low-amplitude pulsed waves
◆Strong intensity fluctuation
Hysteretic nonlinearity
(making/breaking of contacts?)
‘Effective Medium’ fails ? Disorder
(percolation?)
Fragile matter / unjamming transition !
P
6
Contact force networks
Outline
1. Linear ultrasound propagation in jammed granular solid ◆Coherent waves and multiple scattering ◆Material characterization (velocity and absorption) and shear banding probing
2. Nonlinear ultrasound propagation in fragile granular solid
◆Compression velocity softening: reversible → irreversible regimes } non-classic !
◆Shear velocity softening: jammed → unjammed/fluidized states
3. Triggering of shear instability in granular media by acoustic fluidization
◆Interfacial sliding / Granular avalanche
◆Quicksands: ball sinking in granular sediments
Conclusion
7
Emetteur
Détecteur
D = 30 mm
Experimental set-up
Force sensor
Glass beads / sands (d ~ 0.1 – 10 mm)
Ampli Fonction generator
Pre-ampli Oscilloscope
Oedometric cell Applied stress:
P = 3 kPa – 3 MPa
Press
8
-0,4
-0,2
0
0,2
0,4
A (
V)
E S
(a)
glass beads d: 600-800 µm
« E » signal is reproducible
« S » signal is configuration
specific
-0 ,4
-0 ,2
0
0,2
◆λS ~ 2d : multiply scattered waves (S)
0,4
0 100 200 300
A (
V)
t (µs)
E S
( b)
◆λE ≥ 10 d : coherent waves (E) P
2 µs
Jia, Caroli and Velicky, PRL 82 (1999)
P = 0.75 MPa
1. Linear ultrasound propagation in jammed granular solid
1,2
1
0,8
0,6
0,4
0,2
0
Mod
ule
(a.u
.)
0 200 400 600 800 100 f (kHz)
"E " "S ource" " S "
0
9
100 200 300 t (µs)
P
P
◆Velocities of coherent waves
(Duffty & Mindlin 1957; Digby 1981)
Glass beads d: 0.4 -0.8 mm
- Goddard (1990) - de Gennes (1999) - Makse, Johnson, Schwartz (2000) - Velicky, Caroli (2002) - Coste, Gilles (2003); Roux (2000)
Coherent sound velocity V(P) versus pressure
Hertzian contact : k ~ P1/3
with Z coordination number
Bernal & Mason (1960): « close » contacts: δ ≤ 0 → F > 0 « near » contacts: δ = (0-5%)d → F = 0
L,T V (P) ∝[Z(P)]1/ 3.[k(P)]1/ 2 ∝ P1/ 6
Making of contacts (F > 0) by compression P : - buckling of the grains via slippage (Goddard 1990) - elastic deformation à la Greenwood (Pilbeam 1973)
-200
-100
0
100
200
0 100 200 300
A (m
V)
S
time (µs)
◆Effective medium theory (affine approximation)
(b) E (longitudinal wave)
(scattered waves)
2,5
2,9 2,8 2,7 2,6
3,1
3
4,5 5 6 6,5
log
V (
V in
m/s
)
5,5
log P (P in Pa)
1 /4 ~ P
1 /6 ~ P Time-of-flight VL P = 750 kPa
VL = (K + 4 / 3G) / ρ
VT = G / ρ
- Jia, Caroli, Velicky, PRL 82 (1999) - Jia, Mills, Powders & Grains (2001)
d 10
δ
P
0
-1
-2
-3
-4
-5
-6 0 100 200 300 400 500
Log
I (a
.u.)
t (µs)
1 L = 7.4 mm
2 L = 11.4 mm
3 L = 15.1 mm
2 I t τ a
= δ (z)δ (t) ∂ I − D∇ I +
Theorem of energy equipartition :
NT /NL ∝ 2 (vL/vT)3 ≈ 16
D ≈ 0.13 mm2/µs τa ≈ 64 µs
Q-1 = (2πfτa)-1 ≈ 0.005
l* ≈ 0.87 mm ~ d ξ ≤ l* ~ d (not 5-10 d !)
Short-range correlation of the force chain !
with D = (1/3) ve l* the diffusion coefficient and τa the inelastic absorption time
-800
-400
0
400
800
0 100 200 300 400 500
A (m
V)
t (µs)
S
"Dry"
Weaver, JASA (1982)
Codalike multiple scattering of shear waves in granular media (1/3)
11
Jia, PRL 93 (2004)
- Page, Weitz et al, PRE (1995) - Weaver, Sachse, JASA (1995) - Tourin, Derode, Fink, Random Media (2000)
0,0000 0,0001 0,0004 0,0005 -2,5
2,5 2,0 1,5 1,0 0,5 0,0
-0,5 -1,0 -1,5 -2,0
Q = 40 PMMA Am
plitu
de (a
.u.)
0,0002 0,0003
Time (s)
dry PMMA beads d = 0.6 mm Q = 35
◆Dissipation mechanisms: - interfacial loss - bulk loss in grains
Liquid trapped in asperities
-8 0 0
-4 0 0
0
4 0 0
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0
A (mV)
t ( µ s )
E P
o il
" W e t " Φ = 0 . 0 1 5 %
Q = 4 0
-8 0 0
-4 0 0
0
4 0 0
◆Absorption of multiply scattered waves by added liquids 8 0 0 8 0 0
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0
A (mV)
t ( µ s )
E P
S
" D r y " Q = 2 0 0
-Bocquet, Charlaix, Crassous, Ciliberto, 1998
- Halsey & Levine, 1998 - Schiffer et al, 1999 - Herminghaus et al, 2005
Langlois & Jia (2007)
Φliquid ~ 0,05%
Mason et al, 1999
Probing the internal dissipation in dry and wet granular media with diffusively scattered waves (2/3)
12
Interfacial dissipation mechanisms in glass bead packings (3/3)
P
P
U t Ut
loss
Wstored
⊗W Q−1 = = Q−1 + Q−1 vis fric
0.003
0.004
0.005
clean
unclean
DRY
Q-1
vis
Q -1
fit
0 1.5 10-5 3.10-5 4.5 10-5 0.02
0.018 0.016 WET 0.014 0.012 COATED
0.01
0.006
Acoustic strain amplitude ε
0 5 10 15 t Tangential displacement U (nm)
Dis
sipa
tion
(Q-
1 )
K.L. Johnson (1961)
fric Q−1
t ∝ µ U P −1 −2/3
t unclean Q−1 = 0.004 +112.103 U → µunclean = 0.98
t clean Q−1 = 0.003 + 63.103 U
clean → µ = 1.75
◆Linear viscoelastic dissipation (asperities or films) (Johnson, 1955; Baumberger et al 1998)
◆Nonlinear frictional dissipation (Mindlin, 1950)
Breakdown of the Mindlin’s model: local Coulomb friction µ not legitimate !
Bureau, Baumberger, Caroli (2002)
L
‘Microslip’
L >> 100 nm (valid range)
Brunet, Jia & Mills, PRL 101 (2008)
13
Probing the shear band formation with shear wave (1/2)
S V ∝ z ⋅ φ 1/3 −1/6 •P 1/6
◆Mechanical response
Dense packing Loose packing
Decrease of the coordination number z 3D DEM simulations
Cui & O’ Sullivan 2006
Khidas & Jia, PRE 85 (2012)
◆Shear wave velocity softening before failure Dense packing
14
Loose packing
Probing intermittent behavior with scattered waves (2/2)
Loose packing
Intermittent dynamics ! (“stick-slip” events)
♦ Cross-correlation of scattered waves (i.e., acoustic speckles or coda): Γij (τ = 0) ∝ ∫ Si (t )⋅ S j (t +τ )dt
zoom
Dense packing
15
► weakly nonlinear regime: εa « ε0
0,0 0,4
-1,0
-1,2 -0,8 -0,4
-0,8
-0,6
-0,4
-0,2
Com
pres
sion
al lo
ad N
(nor
m)
Ν Normal displacement δ (norm)
0,0 Hertz contact law:
Ν N = - C(-δ )3/2
(N0 , δ0 )
◆Hertzian nonlinearity:
P
P
Ut Ut
P
P
► Shear stiffness weakening
Ut * increases kt = dFt dUt decreases
K.L. Johnson (1961)
◆Frictional nonlinearity: (Mindlin model)
microslip
a 0 a a
16
β = −1 /(4ε ) is third-order elastic constant
ο = M ε (1+ βε + ...)
► strongly nonlinear regime: εa > ε0
Soliton-like shock waves : 1D ordered granular chains -Nesterenko (1983); Coste, Falcon, Fauve (1997); Sen et al, 2008 - Dario, Nesterenko et al (2006); Huillard, Noblin, Rajchenbach (2011)
- Gomez, Wildenberg, van Hecke, Vitelli (2011): 2D/3D disordered packs
overlap ε0
0
(Norris & Johnson, 1997) → harmonics generation (reversible interaction)
in weakly nonlinear regime, εa < 0.1ε0
► Frictional (hysteretic) dissipation Brunet, Jia, Mills, PRL 101 (2008)
2. Nonlinear ultrasound propagation in fragile granular solid
Nonlinear acoustic resonance in granular solids (1/3)
0,3
0,4
0,5
0,6
0,7
0,8
0,9
16500 17000 18000 18500
Nor
mal
ized
am
plitu
de
17500 f (Hz)
Increasing drive amplitude
T
L
T
R
res res f = V / 2L & Q = f / ⊗f
P = 0.11 MPa and L = 18.5 mm ◆Compressional
mode P
Johnson & Jia, Nature 437 (2005)
Young modulus weakening: E = ρ V 2
◆Simulation in 2D disk packs
0,1 1
-7
-8
-6
-1
-2
-3
-4
-5
0
Longitudinal Wave
⊗E/
2E0 (%
)
10 ε (x10-6)
P
V (P) ∝ (Z )1/ 3.[k(P)]1/ 2 (∝ P1/6 )
driving
driving
17
‹Zc›
Olson, Lopatina, Jia, Johnson PRE 92 (2015)
Harmonic generation: reversible → irreversible regimes (2/3)
Ten-cycle tone burst ( f0 = 50 kHz)
L
0 100µ 300µ 400µ
-0,01
0,00
0,01
200µ
Temps (µs)
0,00 -0,04
0,04
0,0
-1,5
1,5
Filte
red
at 1
50 k
Hz
Filte
red
at 1
00 k
Hz
Filte
red
at 5
0 kH
z
-1,5
0,0
1,5 transmitted signal
3f (3rd harmonic) 0
2f (2nd harmonic) 0
f (fondamental) 0
P = 150 kPa , L = 66 mm
No
filte
ring
D = 65 mm
Brunet, Jia & Johnson, Geophys. Res. Lett. 35 (2008)
Creep-like compaction
18
◆Correlation function
Si Si+1
Ci,i+1 (τ = 0) Γi,i+1 =
Ci,i (0)Ci+1,i+1 (0)
Ci,i+1 (τ ) = ∫ Si (t )⋅ Si+1 (t +τ )dt
Compressional wave velocity softening: reversible → irreversible (3/3)
◆Mindlin model
⊗c / c0 ∝ ⊗kt / kt ∝ −εa
Jia, Brunet, Laurent, PRE 84 (2011)
0 100µ 300µ 400µ
-0.3
0.0
0.3
L = 66 mm P = 350 kPa
208µ 220µ 224µ
-0.3
0.0
0.3
212µ 216µ
Temps (µs)
Increasing source voltage
Ampl
itude
(V)
200µ
Time (µs)
⊗c/ c ~ 1-10 % (softening)
1/ 3 1/ 2 cL ∝ (Z / Rρ0 ) (kn )
with Z slipping!
19
X (m
m)
5 10 Time (ms)
15
10
20
30
40
50
60
70
80
90
◆Elastography using ultrasonic speckle interferometry
- Catheline, Gennisson, Tanter, Fink, PRL 91 (2003) - Maneville et al, PRA (2013) ● Out-of-plane surface vibration
in the linear regime Frequency : 200 Hz
● Rayleigh-like surface acosutic wave
VS ≈ 28m/s and lLF ~ 10 cm P ~ 0.6 kPa
Brum, Gennisson, Tanter, Fink, Tourin & Jia (to be submitted)
Correlation Algorithm
Shear wave velocity softening in granular sediments (1/3)
20
- Bonneau et al, PRL 101(2008) - Jacob et al, PRL 100 (2008)
Distance: x = 95.5 mm
The huge shear modulus softening is due to the contact slipping between grains
(avalanche process). jammed → unjammed states: without
packing density change ⊗φ !
Shear wave velocity softening: transition from jammed to unjammed states (2/3)
→ Contact slipping by oscillatory shear leads to the softening of interfacial shear stiffness k !
♦ Multi-contact interface (Mindlin friction model)
W Fac
a ρ(a) ~ exp(-a/a0 ) à la Greenwood
0.09
0.25
0.16
0.35
0.69
1.11
2.34
10 30 20 t (ms)
F ac
(N)
Fac
W
a
10-1 100
100
ac F (N)
100 Hz 200 Hz 300 Hz 400 Hz
Friction model (µ = 0.12)
(Fac)-1
10-1
-Bureau, Caroli, Baumberger R. Soc.Proc. (2003)
Wyart, Nagel, Witten PRE (2005)
-Jia, Brunet, Laurent, PRE (2011)
S 0
Z0: mean coordination number ⊗z: excess number (to isostatic limit)
V 2 ∝ G ∝ Z * k * ⊗Z (∝ P2/3 frictionless spheres)
♦ Mean-field theory of granular media:
Time-of-flight
21
Before During After
Low oscillation: FLF ≈ 0.03 N
High oscillation: FLF ≈ 2.7 N
fLF = 100 Hz
200 400 600 800 1000 1200 1400
20
40
60
80
100
120
The shear velocity is VS ≈ 6.8 m/s → ⊗VS /VS (≈ 10/17) ≈ 55%
→ the shear modulus ⊗G/G ~ 85% !!
during unjamming transition !
Shear wave velocity softening: jammed → unjammed/flowing states (3/3)
Nonlinear elasticity is coupled with plasticity
in amorphous media !
d
Procaccia et al, PRE (2011)
22
Grain motions detected by US speckles change !
3. Triggering of shear instability by acoustic fluidization / Teff
San Andreas fault
→ Avalanche/Rheology of vibrated granular media
- Jaeger, Liu, Nagel, PRL 62 (1989)
- Dijksman, van Hecks et al, PRL 107 (2011)
- Léopoldès, Tourin, Mangeney, Jia (2016)
→ Sliding triggering of a glassy interface - Heuberger, Drummond, Israelachvili, J. Phys Chem. (1998)
- Bureau, Baumberger, Caroli, PRE 64 (2001)
- Léopoldès, Conrad, Jia, PRL 110 (2013)
Lowering the yield stress!
τ0
→ Earthquake triggering / dynamical weakening - Melosh, Nature 379 (1996) - Johnson, Jia, Nature 437 (2005) - Johnson, Gomberg, Marone et al, Nature 451 (2008) - Khidas & Jia, PRE 85 (2012)
Shear banding under high confining pressure
23
Léopoldès, Conrad & Jia, PRL 110 (2013)
Monolayers (~ 1 nm)
Static shear
Oscillatory shear
◆Elastic softening kT (interfacial stifness) under static shear
CH3 (« less-adhesive »)
24
COOH («adhesive »)
◆kT softening under oscillatory shear and triggering of sliding
COOH film
Instability triggered by acoustic fluidization (1/2): interfacial sliding
◆Sliding triggered below threshold by shear acoustic lubrication
◆Jamming transition diagram Liu & Nagel (1998)
a nd µs ≈ 2 for COOH µs ≈ 1 for CH3
* s s
* sinθ / sinθ = F / F s s ≈ c / a 2 2 ac s N ≈ 1 − (2 / 3)F / µ F < 1
s where F = σ Σ s s s with Σ : π a2 ↘ π c2
Acoustic lubrication of the stuck area !!
a c
E / B ≈ [1− (sinθ * / sinθ )]2
v s s
v T E ≈ ( A / 2)KU 2
B ∼ 1010 kT
jump N E ≈ F ⊗D ∼ 1012 kT with ⊗D ∼ 1µm (asperity)
(solid line)
and
with Vibrational energy or Teff
Sliding or shear failure is much easierly triggered by frictional oscillation mode than by opening displacement mode !
Note
ω ≫ ω0
Slider doesn’t move!
→ Ejump ≫ Ev !! FN
Instability triggered by acoustic fluidization (2/2): interfacial sliding
25
- Steel ball (intruder) of diameter 10 mm - Glass beads of 50-100 µm with packing
density ~ 0.60 saturated by water
Reflec5on from the interface
20 25 30 -0.3
-0.2
-0.1
0
0.1
0.2
35 40 45 50 time ( µs)
U(m
V)
Reflec5on from the intruder
28
30
34 32
36
38
40
42
44
46
48
0 100 200 300 400 500 600 700 800 900 1000 frame
arr
iva
l ti
me
(µ
s)
Arrival 5me increase
compac5on dila5on
dia.12 mm, fus = 2.5 MHz, λ = 0.6 mm
f = 60 Hz Γ ~ 1 m/s2
shaker
Z
0 water
suspension
Acoustic probing of a ball sinking in vibrated granular suspensions (1/3)
26
Wildenberg, Léopoldès, Tourin, and Jia (EPJ ST: Powders and Grains 2017)
♦ Acoustic monitoring of a sinking ball in quicksands
Mono-element ultrasonic transducer:
♦ Sinking dynamics under different shaking intensity Γ
Acoustic probing of a ball sinking in vibrated granular suspensions (2/3)
Agitation croissante
27
♦ Frictional rheology
Acoustic probing of a ball sinking in vibrated granular suspensions (3/3)
- Equation of motion:
m!z! = (4 / 3)π (D / 2)3 ⊗ρg − F fri
0 A0 z(t) = ( A − z*) ϒ − e−k (t −t*) ⁄
′ ( A − z*) ≤ 0
∞ ƒ
0 0 1
0 liq kη (D / 2)⊗ρ µ ρ g(D / 2)
3πρ A µ = and µ =
used to fit the exponential data z(t) yielding:
⊗ρ = 8000 - 1960 kg/m3; ρ = 1960 kg/m; g = 9.81 m/s2: ηwater = 8.9 10-4 Pa.s - Quasi-steady flow solution: !z! → 0
Ball diameters D (circle: 8mm, triangle: 10mm, square: 14mm) (a) Static friction coefficient µ0 ; (b) Viscous constant µ1
→ Rheological parameters vs shaking intensity
- Frictional rheology:
τ = ∝0 P + ∝1ηliq (z! / D)
28 Probe size dependance → Nonlocal rheology behaviour ?
µ0
µ*0 ≈ 0.5 (Γ = 0)
1 cm
1 mm
1 µm
1 nm
Granular material
Grains
Contact between grains
Asperities
Conclusion: Multiscale Acoustics of Granular Media
◆Ultrasonic interfacial rheology shear resonator & a bead layer • Resonance peaks and width
→ interfacial stiffness & dissipation Léopoldes & Jia , PRL 105 (2010)
◆λE ≥ 10 d : Coherent elastic waves
• Compressional & shear velocities → material elastic moduli K & G
Jia, Caroli and Velicky, PRL 82 (1999)
29
◆λS ~ d : Multiply scattered waves • Q-factor → dissipation at the contact
• Mean free path l* → rearrangements Jia, PRL 93 (2004); Brunet, Jia and Mills, PRL 101 (2008)
★ Time Reversal of Ultrasound in Strongly Nonlinear Granular Media - Harazi, Yang, Fink, Tourin and Jia, Euro. Phys. J. Special Topic 226, 1487 (2017)
Acoustic pumping
◆Acoustic fluidization may occur without significant packing density change
via shear modulus softening by contact slipping
◆Triggering of solid sliding / granular avalanche
via acoustic lubrication of contacts
◆Sinking ball combined with ultrasound imaging provides a useful method
for local rheological measurements in 3D granular sediments
30