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 Figure from Marone, 1998 f 0 : Reference friction coefficient V 0 : Reference slip rate θ: State variable Dieterich, 1979, Ruina, 1983 and a state-evolution equation Velocity step test a – b < 0 : Velocity weakening 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, 1972 Sandstone Slide-hold-slide test Note: Cut off of this effect at low contact time is required. The strength can’t be –∞.

∂f ss / ∂ln(V), “a - b value” Shimamoto, 1986 Dieterich, 1978 f Dry granite NaCl

Δf ss / Δln(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] Positive dir. Negative dir. E 1 k B T c   c 0 :::::::::::: Activation energy Boltzmann constant Contact temperature Activation volume Contact shear stress Attempt frequency ~ highest lattice vibration frequency Success frequency for slipping How aln(V) is understood. If the strength at microscopic asperities plays an important role,

V r :::::: Slip rate Area density of slipping sites Slip for a single success, ~ cell size Slip rate, V is given by Solving for the contact shear stress, If the negative jumps are negligible, Logarithmic direct effect proportional to T c Contact shear stress

Macroscopic shear stress Real contact area Nominal area Hardness by definition => or

∂f ss / ∂(1/T) Figure from Blanpied et al., 1995 The friction coefficient increases in a limited range of the temperature, from 25 o C to about 350 o C 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] A phenomenological law Q: activation energy, k B : Boltzmann constant Q a : Activation energy of a process governing the direct effect Q b : Activation energy of a process governing the evolution effect Z * : Temperature-reduced rate or Zener-Hollomon parameter k B : Boltzmann constant Chester [1994] proposed a slip-law formulation with constant a and b. Temperature accelerates processes An assumption to explain the temperature-step tests “Master curve” 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  n Figures from Hirose and Shimamoto, 2005 At high  n, second peak appears just after the beginning of experiment.

Gabbro Tsutsumi and Shimamoto, 1997 Tullis and Goldsby 2003 Figures from Tullis and Goldsby 2003 Flash heating Frictional behavior at high slip rate is completely different! Rotary shear apparatus at Brown University, V < 0.36 m/s,  n = 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 temperature possibly 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.  c, T c Temperature: T Defined by REV >> asperities Defined by REV >> atom

High slip rate, T c > about 1000 o C [Rice, 2006] Microscopic heat conduction. with In this timescale, slip rate is constant.  : age of an asperity Assumptions: - An asperity weakens when its temperature reaches T w. - All asperities are either totally weakened of unweakened. - One-sized (D) asperities. where Rice 1999, 2006; Tullis and Goldsby 2003

Gabbro 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 slipNatural texture from Nojima falt Mizoguchi et al., 2009

Kitajima et al., 2010

Thermal pressurization of pore fluid Effective stress law For Infinitesimally thin slipping zone B.C. [Sibson, 1973; Lachenbruch, 1980; Mase and Smith, 1987; Andrews, 2002; Rice, 2006] pore pressure Conservation of energy Conservation of fluid mass 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 Thin section of Punchbowl fault, South California

Analytical solution With fixed slip rate and frictional coefficient, where, 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 National Monument in Iidaka Town, Mie Prefecture (1) Wide and complex fault zone (3) Concentrated deformation zone (Ca. 50 mm wide) (2) Very straight fault plane Tsukide Outcrop Wibberey and Shimamoto, 2003 Hydraulic properties

Hydraulic property of fault rock Wibberley and Shimamoto, 2003

How long is L * ? Intact MTL clayey gouge Accounting for “ damage ” with f = 0.25 and V = 1 m/s Rice, 2006

Predicted “ seismic ” fracture energy Definition 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, MPa initial effective normal stress. Flash heating is not included.

A sequence of earthquakes

The resulting complexity in EQ magnitude distribution Magnitude 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.

Heterogeneity in the hydraulic diffusivity Slip 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.

Stress-reduction curves and low heat generation Shear 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.

Interseismic shear stress Shear 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.

Stress-reduction curves and low heat generation Shear 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.