Laboratory experiments

Triaxial test with saw-cut Resistance of jacket Limited displacement Easy to control pore pressure High normal stress is possible

Biaxial double shear test Limited displacement (Larger than triaxial one) Limited Strength of rock No jacket

Rotary shear test Unlimited displacement (Possibility of high velocity test) Technically, challenging (misalignment of axis, confinement of gouge)

to study rock friction.

Rate- and state-dependent law

aln(10) bln(10)

~ L f

0 0 0ln( / ) ln( / )f f a V V b

Figure from Marone, 1998

f0 : Reference friction coefficientV0: Reference slip rateθ: State variable

Dieterich, 1979, Ruina, 1983

and a state-evolution equation

Velocity step test

a – b < 0 : Velocity weakening

0 /V V in various manners

Wide applicability of rate- and state-dependent frictional law

Dieterich and Kilgore 1994

Typically, decays occurs for displacement of tens of microns to 1mm.

Aging of a fault

Dieterich, 1972Sandstone

Slide-hold-slide test

Note: Cut off of this effect at low contact time is required.The strength can’t be –∞.

∂fss / ∂ln(V), “a - b value”

Shimamoto, 1986

Dieterich, 1978

f

Dry granite

NaCl

Δf ss

/ Δ

ln(V

)

Blanpied et al., 1991

Depending on the material, “a-b” is negative only in a limited range of T and V.

Wet granite

Blanpied et al., 1995

Raw data and fitting to it reported in Blanpied et al., 1991, 1998

Blanpied et al., 1998

Effect of temperature and water

∂f / ∂ln(V), “a value”

Figure from a presentation by Rice, 2007

Clayey fault gouge from Hanaore fault, Southwest Japan

Figure from Noda and Shimamoto [2009]

a-value seems nearly proportional to the absolute temperature explained by a microscopic slip process which requires thermal activation

[Nakatani 2001, Rice et al. 2001].

Friction law accounting for a slip process which requires thermal activation

Possible slip patches in an asperity

[Nakatani, 2001; Rice et al., 2001; Noda 2008]

10 exp c

B c

E

k T

10 exp c

B c

E

k T

Positive dir.

Negative dir.

E1

kB

Tc

tc

n0

::::::

Activation energyBoltzmann constantContact temperatureActivation volumeContact shear stressAttempt frequency ~ highest lattice vibration frequency

Success frequency for slipping

How aln(V) is understood.

If the strength at microscopic asperities plays an important role,

cB

cBc Tk

E

V

VTk 1

1

1 exp2

sinh

1 0( )sV r rc

Vrl

:::

Slip rateArea density of slipping sitesSlip for a single success, ~ cell size

112 exp sinh c

B c B c

EV r V

k T k T

Slip rate, V is given by

Solving for the contact shear stress,

If the negative jumps are negligible,

1

1

lnB cc

k T EV

V

Logarithmic direct effect proportional to Tc

Contact shear stress

Macroscopic shear stress

1

1lnB ck T V E

fH V H

nrc c c

n n

dA A

A A H

Ar

Real contact area

Nominal area Hardnessby definition

=>

1 1

1sinh exp

2c B c

B c

k T V Ef

H H V k T

or

∂fss / ∂(1/T)

Figure from Blanpied et al., 1995

The friction coefficient increases in a limited range of the temperature, from 25 oC to about 350 oC but depending on the slip rate.

HotCold

∂f / ∂(1/T)

Figure from Chester, 1994

Temperature step tests by Chester [1994]

On an abrupt (-/+) change in T, a (+/-) direct effect is observed followed by a (-/+) evolution effect.

A constitutive law assuming time-temperature superposition

[Chester, 1994]

(time-rate exp( / ))BF F Q k T

A phenomenological law

Q: activation energy, kB: Boltzmann constant

( , )af f Z ( , ( ))ss ss a ss bf f Z Z

ssV

L

Qa: Activation energy of a process governing the direct effectQb: Activation energy of a process governing the evolution effectZ*: Temperature-reduced rate or Zener-Hollomon parameterkB: Boltzmann constant

0 0ln( / )af f a Z Z

0ln( / )ss bb Z Z

Chester [1994] proposed a slip-law formulation

with constant a and b.

Temperature accelerates processes

exp /a a BZ V Q k T exp /b b BZ V Q k T

An assumption to explain the temperature-step tests

“Master curve”

const.F is a line in Arrhenius plot.

Development of microstructures

Logan et al., 1992

Riedel shear

Logan et al., 1992

Large (> hundreds) shear strain

Beeler et al., 1996 al.

Friction coefficient and a-b

Slip: 10 mm, a-b > 0

Early stage before localization

Locarization of strain rate on Y-plane

Slip: 65 mm, a-b < 0

Widening of the foliated gouge layer

Slip: 407 mm, a-b: positive to neutral

High velocity friction

Figure from Wibberley et al., 2008

“Byerlee’s law” [1978]

Earthquake

Weakening

Plate motion ~ 1 cm/yr

High-velocity friction experiments

High velocity friction apparatus at Kochi Core Center

Figure from Tsutsumi and Shimamoto, 1997

First weakening

Strengthening due to melt-patch generation

Weakening due to widening of molten (viscous) layer

Typical mechanical behavior at low sn

Figures from Hirose and Shimamoto, 2005

At high sn, second peak appears just after the beginning of experiment.

Gabbro

Tsutsumi and Shimamoto, 1997Tullis and Goldsby 2003

Figures from Tullis and Goldsby 2003

Flash heating

Frictional behavior at high slip rate is completely different!

Rotary shear apparatusat Brown University,

V < 0.36 m/s, sn = 5 MPa

Friction law at intermediate slip rate accounting for flash heating

First introduced in a field of dry metal friction Bowden and Thomas, 1954; Archard, 1958/1959; Ettles, 1986; Lim and Ashby, 1987; Lim et al., 1989; Molinari et al., 1999

Very high stress (~yield stress) and high slip rate.

Extremely high temperature at the contact(~ melting, decomposition, or oxidization point).

Abrupt weakening of asperities at a weakening temperaturepossibly because of phase transformation [Rice, 1999]

The contact temperature must be important even below the “weakening temperature”.

Aim: Derive a frictional constitutive law accounting for flash heating and the microscopic constitutive law explained so far.

tc, Tc

Temperature: T

Defined by REV >> asperities

Defined by REV >> atom

High slip rate, Tc > about 1000 oC [Rice, 2006]

Microscopic heat conduction.

2

2

z

T

t

Tth

V

z

Tc c

zth

2

1

0

with

In this timescale, slip rate is constant.

0Tc

VT

th

cc

q : age of an asperity

Assumptions: - An asperity weakens when its temperature reaches Tw. - All asperities are either totally weakened of unweakened. - One-sized (D) asperities.

LV w

w LV w w w

f V Vf

f f f V V V V

2

0 ~ 0.1 m/sth ww

c

T TV

D c

(Friction coefficient at low velocity)LVf

ywwf

where

Rice 1999, 2006; Tullis and Goldsby 2003

T

c

Tw

Tullis and Goldsby 2003

Figure from Tullis et al., 2006

Experimental evidence

Novaculite (mostly quartz)

High velocity experiments with gouge

Mizoguchi et al., 2009

Sample: natural fault gouge from Nojima fault, Southwest Japan, a source fault of 1995 Kobe Earthqake

Experimental texture at different slip Natural texture from Nojima falt

Mizoguchi et al., 2009

Kitajima et al., 2010

Kitajima et al., 2010

Thermal pressurization of pore fluid

Effective stress law

fzpf ne ))0((

2

2

z

T

t

Tth

t

T

z

p

t

phy

2

2

Vfz

Tc e

zth

2

1

0

00

zz

p

For Infinitesimally thin slipping zone

B.C.

pore pressure

Conservation of energy

Conservation of fluid mass

T

Friction

p

th < hy

Sibson, 1973; Lachenbruch, 1980; Mase and Smith, 1985, 1987; Andrews, 2002; Rice, 2006

(Also suggested for mechanism of catastrophic landslides)

Extremely concentrated deformation

Chester et al. 2005, 2003; Chester and Goldsby, 2003

Existence of principal slip plane

~100 300 m m

Thin section of Punchbowl fault, South California

Analytical solution With fixed slip rate and frictional coefficient,

0)1( ep

)1(1 0

e

th

hyT

ˆerfcˆexpwhere

22

2*

4hyth

c

VfL

*

ˆ L ,

Normalized shear stress

-Apparent evolution distance is a good fraction of total slip. (Multiple scale behavior)-Mathematically steady state shear stress is zero, regardless of slip rate.

Rice, 2006

Median Tectonic Line in Japan

Tsukide OutcropWibberey and Shimamoto, 2003

Hydraulic properties

Hydraulic property of fault rock

Wibberley and Shimamoto, 2003

Wibberley and Shimamoto, 2003

How long is L*?

/smm 450

/smm 60

4 2

22

2

*2

hyth

cVLf Intact MTL clayey gougeAccounting for “damage”

with f = 0.25 and V = 1 m/s

Rice, 2006

Predicted “seismic” fracture energyDefinition

dG0

)()(Figure from Rice, 2006

3D calculation allowing changes in temperature and pore pressure

Two patches (15 km x 15 km)

Patch I at negative x Rate-weakening friction High hydraulic diffusivity

Patch II at positive x Rate-weakening friction Potentially low hydraulic diffusivity (susceptible to thermal pressurization)

Inertial effects are included.

Noda and Lapusta, 2010

30 MPa initial effective normal stress.

5 2 20 30 MPa, 4 mm, 1 cm, 0.01, 0.006 0.014, 10 10 m /se hyL w a b

Flash heating is not included.

A sequence of earthquakes

2 2

4 2

10 m /s

in the left patch

10 m /s

in the right patch

hy

hy

The resulting complexity in EQ magnitude distributionMagnitude of the events as a function of time

Without heterogeneity, the model produces characteristic events. Heterogeneity causes long earthquake cycles that contain events of different sizes.

2 210 ,10hy 2 310 ,10hy

2 410 ,10hy 2 510 ,10hy

Heterogeneity in the hydraulic diffusivitySlip distribution at z = 0, black lines every 1 sec during EQs and gray ones every 10 years

The region more susceptible to thermal pressurization has larger displacements in model-spanning events. The slip deficit in the other region is filled with smaller and more frequent events.

210hy 210hy

0.004a b

310hy 210hy

410hy 210hy 510hy 210hy

Stress-reduction curves and low heat generationShear stress as a function of slip at x = 10 km

Apparent stress weakening distance is determined by rate- and state-law in the permeable region, and by T.P. in the less permeable region.

0.004a b

2 210 ,10hy 2 310 ,10hy

2 410 ,10hy 2 510 ,10hy

Interseismic shear stressShear stress at z = 0.

In the region of efficient thermal pressurization, shear stress is lower interseismically due to larger stress drop. That is why events that occur early in the cycle may not propagate into that region.

2 210 ,10hy 2 310 ,10hy

2 410 ,10hy 2 510 ,10hy

Stress-reduction curves and low heat generationShear stress as a function of slip at x = -10 km (black) and 10 km (gray).

Apparent stress weakening distance is determined by rate- and state-law in the permeable region, and by T.P. in the less permeable region.

0.004a b

2 210 ,10hy 2 310 ,10hy

2 410 ,10hy 2 510 ,10hy