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Coexistence of fast/slow slips in sliding friction of elastic bodies

弾性体のすべり摩擦における高速/低速すべりの共存

Acknowledgement: This is work is partly supported by JSPS Grant‐in‐Aid for Scientific Research on Innovative Areas （新学術領域研究）“Crustal Dynamics” （地殻ダイナミクス）and “Science of Slow Earthquakes” (スロー地震学).

Gel Gel

Asperity

Department of Mechanical Engineering

Kyushu University, Fukuoka, Japan

Tetsuo Yamaguchi

(yamaguchi@mech.kyushu-u.ac.jp)

What is Stick-slip friction ?

Stick-slip friction is a dynamical behavior repeating stick (stop) and slip (motion) during sliding friction.Example: chalk and blackboard, wiper blade, musical instrument, …

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X = V t

k

x

M

fs

U = dx/dtV

dt

dxU

FtVxkdt

dUM s

)(Spring force Friction force

1 Degree Of Freedom Spring-Block model

2 different mechanisms for occurrence of stick-slip motion

(i) (ii)

Fs

U

x

)(UFF sS

0dU

dFS

and

F

x

)(xFF sS

dx

dFk S

and

k: large

k: smallF = Fs(x)

Vt

Fs(x) = -k (Vt - x)

Velocity weakeningSlip weakening

What is Stick-slip friction ?

Stick-slip friction in a global scale: earthquake

• In plate boundary earthquakes, two tectonic plates undergoes stick-slip motions.

• When a slip occurs, the system releases elastic energy and generates seismic waves.

• Magnitude of slip is not unique, but is widely distributed. Size distribution follows power-law, known as Gutenberg-Richter law.

(http://www.sms-tsunami-warning.com/)

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Regular earthquakes, slow earthquakes

S. Ide et al. Nature, 447, 76 (2007)

log M0

Du

rati

on

[sec

]

Regular earthquakeT ~ M0

1/3

Slow earthquakeT ~ M0

Regular earthquake

Slow earthquake

There are two different classes of earthquakes. Regular earthquake: rapid event, emitting high frequency seismic waves. Slow earthquake: slow event, emitting low frequency or no seismic waves.Discovered around 2000 by geodetic observations.

Though several mechanisms are proposed, physics behind slow earthquakes

are not well understood. ⇒ Laboratory experiment

Regular earthquake

Slow earthquake

( K. Obara, NIED report )

Gel as an analogue material WHAT IS GEL and WHY WE USE GEL?

• Poro(visco)elastic material made of polymer network

• Containing a large amount of fluid (water, organic solvent)

• Soft (Young’s modulus = KPa – MPa)

• Controllable rheology

Rigid, soft, viscous, less viscous, …

• Controllable shape

Molding, 3D printing, …

• Optically transparent

Stress visualization with Particle tracking + Green’s tensor

crosslinkerpolymer

solvent

Small nucleation size

Small Vs (～ 1 – 10 m/s) (cf. rock: ～km/s)

2D rupture behavior is easily tractable

)()1(2 2 ab

EDL

n

cc

C. H. Scholz (2002)

Slip displacement [m]

(Unpublished)

500 fps

WHY NOT NUMERICAL SIMULATION?- Contact process of rough surfaces is still

difficult or impossible to simulate. - Experimental realization is needed.

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Previous studies using gels

Almost all laboratory experiments using soft materials were done with bi-material. T. Yamaguchi et al., J. Phys. (2009), JGR (2011), F. Corbi et al., JGR (2011), (2013), S. Latour et al., EPL (2011), A. Namiki et al., JGR (2014), … ⇒ Bi-material interface induces detachment (i.e., opening crack, Schallamach wave). ⇒ Detachment induces frictional weakening.

Gelatin gel block on rough sand paper/plastic plate(F. Corbi et al., JGR 2013)

Camera

Plexiglass block on Silicone gel plate (T. Yamaguchi et al., JGR 2011, Extreme Mech. Lett. 2016)

Sand paper

Soft

Hard

Soft

Hard

Bright: contactPale: detached

Bi-material affects rupture behavior

Friction of dissimilar interfaceM. Comninou, J. Appl. Mech., 44 (4), 780 (1977)D. Andrews, Y. Ben-Zion, J. Geophys. Res. ,102, 553 (1997)K, Ranjith, J. R. Rice, J. Mech. Phys. Solids, 49, 341 (2001)E. Gerde, M. Marder, Nature 413, 285 (2001)J.- P. Ampuero, Y. Ben-Zion, Geophys. J. Int. 173, 674 (2008)

- Discontinuity in deformation velocity- Normal stress variation (reduction)- Preferred crack propagation

Schallamach waveSchallamach (1971), Roberts et al. (1975), Barquins et al. (1975, 1986), Briscoe et al. (1976), Persson (2001), Rand & Crosby (2005), S. Maegawa & K. Nakano (2007), …

- Complete detachment (formation of Schallamach wave)

D. Andrews, Y. Ben-Zion, JGR (1997)

Soft

Hard

Soft

Hard

Weakening mechanism is essentially different from that in similar interface.

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Occurrence of stick-slip in identical materials

Stick-slip cycles under steady loading has not been reported yet.⇒WHY we never observe stick-slip in soft matter?

M. Ohnaka and L-F. Shen, J. Geopys. Res. 104, 817 (1999)

S. M. Rubinstein, G. Cohen, J. Fineberg, Nature 430, 1005 (2004).

J. N. Brune, BSSA 63, 2105 (1973).

Rock-Rock Plastic-Plastic Rubber-Rubber

〇 〇 △

• To construct an identical setup using gels. • To study the relationship between motions of multiple asperities

and macroscopic frictional behavior.

Our strategy

Objectives of this study

• To conduct friction experiments with well-controlledmultiple asperities using soft polymer gels.

• To visualize breakage and formation processes of contact during sliding.

Fig: Example of polymer gel (food gel)

Real contact area

We begin with as a simple setup as possible. J. Dieterich & B. Kilgore, PAGEOPH(1994)

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Experiment

Gel Sample

Material (silicone gel)

CY 52-276 A/B (Toray Dow Corning)

A : B = 1 : 4 (in weight ratio)

Shear modulus G = 0.1 MPa

Length L = 100 mm

Height H = 100 mm

Asperity radius R = 1 and 5 mm

Quasi-2D sample

Experimental setup

iPhone cover made of silicone gel

Hemi-cylindrical asperity

V = 1 mm/s, H = 60 mm, Δz = 5mm

Movie

Force-Displacement curve

Asperity - flat ⇒ Steady sliding

Asperity - Asperity ⇒ Stick-slip

Unstable slip occurs due to SLIP WEAKENING.

Single asperity contact

12

K: Small (H: large)

f(x)

K: large (H: small)

xX

K

K

=

x

)()( xXKxf

X

f(x)

Static force balance

Displacement

Force

H

Asperity - flatAsperity - Asperity

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(R = 1 mm)10 asperities – 10 asperities (ρ = 100/m), δ = 0%V = 1mm/s

δ = 0%δ = + 50%δ = + 100%

• All asperity pairs slip rapidly by one asperity spacing. • Friction force decreases with decreasing number of asperity pairs.

Duration < 0.2 sec

Periodically located 10 asperity pairs

13

R = 1mm, δ = 0%, V = 1 mm/s

Periodically located 21 asperity pairs

After slipping by one spacing, rapid & giant slip occurs.

14

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High speed movie

Frame Rate: 4000 fps

15

Large wavelength (low energy) excitation of normal displacement causes enhancement in tangential slip.

R = 1mm, δ = 200 %, V = 1 mm/s

Randomly positioned 31 asperity pairs

V = 1 mm/s, H = 60 mm, Δz = 5 mm

δ = 0%δ = + 50%δ = + 100%

All slip events are slow and quasi-static (duration ~ sec).Force variation is smaller than that in periodic systems.

Duration ～ sec

16

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We observed crossover from fast slip to slow slip by changing asperity density and its configuration randomness.

Slip size – Duration relation

Slow

Fast

Experiment

Observation

17

Asperity number – Randomness phase diagram

◎： Giant unstable slip○： Unstable slip△：Boundary×：Quasi-static slip

100 200 300 Density [1/m]

Rapid slip

Phase diagram

Mechanisms

1818 18

Tension Compression

BulkBulk

Case A. Small configuration randomness• (Almost) simultaneous slip• Spring constant becomes small⇒ Unstable slip is possible

Case B. Large configuration randomness + local pre-slip• Remaining asperities are locked• Spring constant becomes large⇒ Unstable slip is difficult

Case A Case B

Bulk

Case C

C. Large configuration randomness + local afterslip• Remaining asperities are unlocked• Spring constant is kept small⇒ Unstable slip is easy

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What happens if R becomes larger?

1

9

Material: Silicone gel (CY52-276 & Silpot 184) R = 4, 12.5, 30, 50, 100, 200 [mm]FN = 2, 4, 6 [N]V = 0.1, 0.2, 0.5, 1, 2, 5, 10 [mm/s]

R = 4, 12.5, 30, 50, 100, 200 mm

h = 4 mm

R = 4mm

5

10

15

20

25

30

0.1 1 10

F[N

]

V [mm/s]

FN = 6N

FN = 4N

FN = 2N

R = 12.5 mm

5

10

15

20

25

30

0.1 1 10

F[N

]

V [mm/s]

FN = 6N

FN = 4N

FN = 2N

R = 30 mm

5

10

15

20

25

30

0.1 1 10

F[N

]

V [mm/s]

FN = 6N

FN = 4N

FN = 2N

R = 50 mm

5

10

15

20

25

30

0.1 1 10

F[N

]

V [mm/s]

FN = 6N

FN = 4N

FN = 2N

2 mm

R = 200 mm

R = 200mm

h = 4 mmWhat happens if R becomes larger?

VELOCITY WEAKENING successfully appears!

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Normalization of curves

2

1

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.001 0.01 0.1 1 10 100

V/Vpeak

F/Fpeak

),( Npeak FRgF

)(RhVpeak )/(/ peakpeak VVfFF

)(xf : Universal function?

Why a peak appears?

SF

Steady state friction

)exp(0

Tk

UVV

B

a )log(

0

0V

VTkB

)log(10

c

c

VV

VBSS

)log()log(0

0

c

c

VV

Vb

V

VaFF

S

V

S0

VcOSaturation of asperity contact area

Stress assisted Creep

F

log V

Slope: a - b Slope: a

Vc

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Stick-slip experiment

23Stick-slip with asperity-flat contact

R = 200 mm, H = 50 mm

Summary & future plan

24Summary

We conducted friction experiments between gels having multiple asperities on both surfaces.

• Slip-weakening friction was observed for asperity – asperity contact.

• Slow slip as well as fast slip was observed.

• Velocity-weakening friction was reproduced by enlarging curvature radius of asperities. Healing process

seems to play an important role in constitutive law.

Future plan (what to do next)

• Writing papers

• Spatial heterogeneity and frictional behavior

• Fractal (hierarchy) structure and nucleation/growth of rupture

• Numerical simulation

Multiple sized asperity

Weakening

Strengthening

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Regular earthquake

Size-duration relation for regular earthquakes

3/1

0~ MT

T: Duration

DSM 0: Earthquake moment

μ: Rigidity (shear modulus)

D: Slip

S = L2: Rupture area

L = c T (c: propagation velocity)S ∝ L2∝ T2

D ∝ L ∝ T (Δγstrain drop = const.)(Kanamori & Anderson, 1975)

L

DCrack

plate

plateCrack

- How can we explain scaling for slow earthquakes?- Is it possible to reproduce these behaviors in laboratory experiments ?

Side view Top view

L

Seismic wave

3

0 ~ TM