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