Stress- and State-Dependence of Earthquake Occurrence: Tutorial 1 Jim Dieterich University of California, Riverside

Approach The formulation is based on the premise that earthquake nucleation controls the time and place of initiation of earthquakes. Hence, processes that alter earthquake nucleation times control changes of seismicity rates. For faults with rate- and state-dependent friction, the relationship between nucleation times and stress changes is highly non-linear. Tutorial 1 reviews some features of rate- and state-dependent friction and earthquake nucleation that form the basis of the model. Tutorial 2 reviews the derivation of the constitutive formulation and outlines some applications of the model. Constitutive formulation for earthquake rates

Experimental Conditions – Rate & State Wide range of rocks and rock forming minerals –Bare surfaces and gouge layers Also glass, wood, paper, plastic, gelatin, metals, ceramics, Silicon in MEMs devices Contact times <1s s (indirect ~4x10 7 s) V= mm/yr - cm/s (servo-controlled tests) V≥100m/s (shock impact) T=20°C - 350°C Nominal  =1 MPa Mpa, Contact stresses to 12GPa Dry, wet, hydrothermal

Time-dependent strengthening

Response to steps in slip speed

# 240 surface # 30 surface Fault slip,  m Displacement-weakening at onset of rapid slip

Rate- and state-dependent formulation Coefficient of friction: State variable: For example: At steady state, d  /dt=0 and

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x V1V1

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x During slip  evolves toward  ss V1V1

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x During slip  evolves toward  ss V1V1

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x During slip  evolves toward  ss V1V1

Log  Coefficient of friction  constant V (high) constant V (low)  ss Time dependent strengthening Slip  a b c d a b c d

Log  Coefficient of friction  constant V (high) constant V (low)  ss Velocity steps Slip  a b c d a b c d a V1V1 V1V1 V2V2 V1V1 V2V2

Spring-slider simulation with rate- and state-dependent friction (blue curves) Westerly granite,  =30 MPa

Imaging contacts during slip Schematic magnified view of contacting surfaces showing isolated high-stress contacts. Viewed in transmitted light, contacts appear as bright spots against a dark background. Acrylic surfaces at 4MPa applied normal stress

Contact stresses Indentation yield stress,  y Acrylic 400 MPa Calcite 1,800 MPa SL Glass 5,500 MPa Quartz 12,000 MPa

Increase of contact area with time Acrylic plasticDieterich & Kilgore, 1994, PAGEOPH

Time dependent friction & Contact area

Velocity step & Contact area

Interpretation of friction terms Time and rate dependence of contact strength terms Indentation creep: c(  ) = c 1 + c 2 ln(  ) Shear of contacts: g(V) = g 1 + g 2 ln(V)  = c 1 g 1 + c 1 g 2 ln(V) + c 2 g 1 ln(  ) + c 2 g 2 ln(V+  ) c=1/ indentation yield stress g=shear strength of contacts Bowden and Tabor adhesion theory of friction Contact area: area = c  Shear resistance:  = (area) (g),  /  =  =cg (Drop the high-order term)  =  0 + A ln(V) + B ln(  )

Contact evolution with displacement

SUMMARY – RATE AND STATE FRICTION Rate and state dependence is characteristic of diverse materials under a very wide range of conditions Contact stresses = micro-indentation yield strength (500 MPa – 12,000 MPa) State dependence represents growth of contact area caused by indentation creep Other process appear to operate at low contact stresses Log dependence thermally activated processes. Power law distribution of contact areas D c correlates with contact diameter and arises from displacement-dependent replacement of contacts

Critical stiffness and critical patch length for unstable slip K Perturbation from steady-state sliding, at constant  [Rice and Ruina, 1983] Apparent stiffness of spontaneous nucleation patch (2D) [Dieterich, 1992] Effective stiffness of slip patch in an elastic medium  crack geometry factor,  ~ 1 G shear modulus

Large-scale biaxial test Minimum fault length for unstable slip

Confined Unstable Slip Confined stick-slip in biaxial apparatus satisfies the relation for minimum dimension for unstable slip

Earthquake nucleation on uniform fault

Earthquake nucleation on a uniform fault

Earthquake nucleation – heterogeneous normal stress LcLc

SPRING-SLIDER MODEL FOR NUCLEATION  K  (t)  During nucleation, slip speed accelerates and greatly exceeds steady state slip speed Evolution at constant normal stress Log  Coefficient of friction   ss B B-A x V1V1

SPRING SLIDER MODEL FOR NUCLEATION Re-arrange by solving for Where: Model parametersInitial condition

SPRING SLIDER MODEL FOR NUCLEATION Slip Slip speed Time to instability Dieterich, Tectonophysics (1992)

Solutions for time to instability Time to instability (s) max  s    /  =0 Dieterich, 1992,Tectonophysics Slip speed (D C /s) D numerical model Fault patch solution

Accelerating slip prior to instability

Time to instability - Experiment and theory

yr 10 yr 20 yr Time to instability (seconds) Log (slip speed)  m/s Effect of stress change on nucleation time

min 1 yr 10 yr 20 yr ~1hr ~5hr Time to instability (seconds) Log (slip speed)  m/s Effect of stress change on nucleation time  = 0.5 MPa

Use the solution for time to nucleation an earthquake (1), where and assume steady-state seismicity rate r at the stressing rate This defines the distribution of initial conditions (slip speeds) for the nucleation sources (2) The distribution of slip speeds (2) can be updated at successive time steps for any stressing history, using solutions for change of slip speed as a function of time and stress., n is the sequence number of the earthquake source Model for earthquake occurrence Log (time to instability) Log (slip speed)

For example changes of with time are given by the nucleation solutions and change of with stress are given directly from the rate- and state- formulation In all cases, the final distribution has the form of the original distribution where Evolution of distribution of slip speeds

Earthquake rate is found by taking the derivative dn/dt = R For any stressing history Evolution of distribution of slip speeds

Coulomb stress formulation for earthquake rates Earthquake rate, Coulomb stress Assume small stress changes (treat as constants), Note:. Hence, Earthquake rate, Dieterich, Cayol, Okubo, Nature, (2000), Dieterich and others, US Geological Survey Professional Paper (2003)