Theoretical and Theoretical and experimental investigation experimental investigation

of dynamic friction at of dynamic friction at seismic slip ratesseismic slip rates

Yuri Fialko and Kevin Brown

Institute of Geophysics and Planetary Physics Scripps Institution of Oceanography University of California San Diego, USA

Batsheva Seminar, Jan 26, 2009

€

F friction = μ ⋅N

€

F friction = τ cAcontact

€

μ =F friction

N=τ cσ n

AcontactAtotal

N N Ffriction

Rate and state dependent friction

Rate effect

State effect

Total

€

τσn

= μ0 + a lnV

V0

⎛

⎝ ⎜

⎞

⎠ ⎟+ b ln

V0θ

L

⎛

⎝ ⎜

⎞

⎠ ⎟

μ - coef. of friction at reference velocity V0

V - slip ratea, b - empirical constants O(10-3-10-2) - state variableL - critical slip distance

Dieterich, 1979; Ruina, 1983

… established for slip rates of O(10-

6-10-3 m/s). Extrapolation to seismic slip rates O(1 m/s) predicts reductions in μ of order of 10%.

Is physics different for high-Is physics different for high-speed sliding? speed sliding?

High-speed rotary shear experiments: complex evolution of shear stress on the slip interface (Tutsumi and Shimamoto, 1997; Hirose and Shimamoto, 2003; Brown and Fialko, 2008)

What are the mechanisms of frictional sliding at seismic slip rates?

How efficient is melt lubrication? Is “viscous braking” (Fialko, 1999; 2004; Koizumi et al., 2005) relevant to seismic faulting?

Hirose and Shimamoto, 2003

Is physics different for high-Is physics different for high-speed sliding? speed sliding?

High-speed rotary shear experiments: complex evolution of shear stress on the slip interface (Tutsumi and Shimamoto, 1997; Hirose and Shimamoto, 2003; Brown and Fialko, 2008)

What are the mechanisms of frictional sliding at seismic slip rates?

How efficient is melt lubrication? Is “viscous braking” (Fialko, 1999; 2004; Koizumi et al., 2005) relevant to seismic faulting?

Pseudotachylites: Field evidence for frictional melting on a fault plane

w

Vητ ∝ viscous stress

singularity at w=0

Stefan problem: start with some finite w, and solve for w(t) (Fialko, 1999; Sirono et al., 2006)

2

01

11exp),(

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−⎟⎠

⎞⎜⎝

⎛=φφφη

TB

AT

46.00 =φ critical melt fraction (Kitano et al., 1981)

- thermal conductivity

L – latent heat of melting/crystallization

– melt fraction

Rheology of melt-solid suspension:

(Fialko and Khazan, 2005)

High-speed friction experiments on argillite: Both shear stress and melt thickness increase with slip

Increases in melt viscosity, likely due to dehydration

€

τ =η⋅ dε

dt

⎛

⎝ ⎜

⎞

⎠ ⎟=η ⋅

V

w

⎛

⎝ ⎜

⎞

⎠ ⎟

τ : Shear stress

η : Viscosity of melt layer

d/dt : Shear strain rate

V : Slip rate

w : Thickness of melt layer

Shear stress, τ 1.25Shear strain rate, v/w 0.661

1Fraction of solid grain 21 19

Ujie et al., JGR (in press)

?

Flash melting (Rice, 1999; 2006)Silica gel formation (DiToro et al., 2004)

High-speed experimental apparatus

- A horizontal rotary lathe with controlled normal load, torque, and velocity-Temperature sensors within 1-2mm from the shear zone and the back of the sample to monitor frictional heating - Real-time data acquisition and display for interactive control

We use ring-shaped samples with internal diameter of 5.8 cm and external diameter of 8.1 cm to minimize variations in slip rate across the sample. An example (top): a sample of granite after a high-speed run. Note the uniform loss of gouge across the surface.

(Left): Experimental apparatus.

residual friction μr

Evolution of residual friction μr with velocity

diabase

Possible mechanisms

Flash melting (Rice 1999, 2006)

Thermally activated plasticity

Flash melting:

- extreme localization

- asperities have to be

sufficiently large

- difficulty explaining

observed strengthening

Other mechanisms:

- silica gels

- nanopowders

- ablation (for C-rich

rocks)

…

Temperature-dependent rheologywith heating/yielding distributed through asperity; peak temperatureis lower

€

μ∝1

V1/3

€

μ∝1

V

€

τ c = τ c0 −

1

10

∂G

∂TT

€

τc =G

10

Theoretical yield strength:

Elastic moduli are temperature-dependent:

Wachtman-Anderson relation for temperature dependence of elastic modulus G(T)~ (1-T) G0

€

.

= A sinh ατ c( )[ ]n

exp −B

T

⎛

⎝ ⎜

⎞

⎠ ⎟

€

.=V

a

0.1 m/s

5 μm

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

τ c =1

αsinh−1 ε

.

Aexp

B

T

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

1

n

€

τ c = τ c0 −

1

10

∂G

∂TT

(olivine!)

Mathematical modelAssumptions: The average strength of individual

asperities is a direct proxy for the coefficient of friction (area of true contact is independent of V);

Heat transfer is dominated by conduction

The asperity strength depends on two components:

1) Temperature of the shear zone (calculated based on the measured evolution of shear stress)

2) Flash heating (calculated using the temperature dependence of theoretical strength)

€

τ c =1

tcτ c T, t( )dt

0

tc

∫

T = Ta (t) + Tb

Ta

Tb

tc =a

V

τ c T( ) =

0.1G T( )

1

αsinh−1 ε

.

Aexp

B

T

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

1

n

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

asperity size

slip velocity

“flash” heatingbackground temperature

Monitoring shear zone temperature

Thermal evolution of the shear zone: measurements vs. predictions based on 1-D unsteady conduction model

€

∂T∂t

=∂

∂xκ (x)

∂T

∂x+

μ(t)σ nV

ρcw

€

∂T∂t

=∂

∂xκ (x)

∂T

∂x

in the shear zone

elsewhere

€

w = 0.2 mm

Initial surface texture Melt initiation

Melting and aggregation of small grains Striations on melt surface

SEM images of the wear products (gouge): effective asperity size O(μm)

Gouge

Example of simulated flash heating of an individual asperity

a=5 μm, Tb=200 oC, V=0.1 m/s

… non-adiabatic heating

… strain localization

“Best-fitting” models

( x 5)

ConclusionsConclusions Coefficient of friction shows a systematic and significant

decrease at slip velocities greater than 0.1 m/s, with an approximate scaling μ~V-D (D depends on normal stress)

We propose that this decrease results from temperature dependence of intrinsic strength of transient contacts

Melting is not required to explain the observed weakening … nor is the assumption of adiabatic heating (i.e.,

asperities don’t need to be >> μm) As the local asperity temperature approaches solidus,

flattening and collapse of asperities result in increases in the nominal contact area (and possibly in increases in the effective friction, depending on the normal stress)

In the post-melting regime, the dynamic shear stress is nearly independent of normal stress and is O(MPa) -> stress drops due to pseudotachylite-generating events are nearly complete

Summary of theoretical modeling Summary of theoretical modeling of macroscopic melting:of macroscopic melting:

Melt zone localization and significant weakening occurs once the melt fraction exceeds a critical value (~50%)

After the onset of weakening, melting of the wallrock is slow and inefficient -> PT layers likely formed instantaneously by bulk melting of a slip zone, rather than by progressive thermal erosion of the fault walls

DiToro et al., Science 2006

Evolution of friction with slip distance

Incipient Melting

∫ σμπ=2

1

)()(2 2R

Rn drrrrM

€

σn =F

π (R22 − R1

2)F – axial force

M – torque

21

22

31

32

3

2

RR

RRFM

−−

= μ

22 1

1

2

3

αααμ++

+=

FRM

appif constrr n =)(),( σμ

21 / RR=α

normal stress

Direct shear experiments

slip rates – 10-6-10-3 m/sambient temperatures – 20-500 oCnormal stress – 1-20 MPa

Vw= ( cth/D)[pc(Tw-Tf)/2, critical weakening rate for

flash melting (Rice, 2006)