Stress- and State-Dependence of Earthquake Occurrence: Tutorial 1

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

2. Constitutive formulation for earthquake rates 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.

3. 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 - 106s (indirect ~4x107s) • V= mm/yr - cm/s (servo-controlled tests) • V≥100m/s (shock impact) • T=20°C - 350°C • Nominal s =1 MPa - 300 Mpa, Contact stresses to 12GPa • Dry, wet, hydrothermal

4. Time-dependent strengthening

5. Response to steps in slip speed

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

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

8. Constant V (high) x V1 B mss Coefficient of friction m B-A Constant V (low) Log q

9. During slip  evolves toward ss Constant V (high) x V1 B mss Coefficient of friction m B-A Constant V (low) Log q

10. During slip  evolves toward ss Constant V (high) x V1 B mss Coefficient of friction m B-A Constant V (low) Log q

11. During slip  evolves toward ss Constant V (high) x V1 B mss Coefficient of friction m B-A Constant V (low) Log q

12. c Time dependent strengthening m a d Slip b c constant V (high) mss Coefficient of friction m d a b constant V (low) Log q

13. b V1 V2 V1 a Velocity steps m a c Slip d constant V (high) V2 b V1 mss Coefficient of friction m c a d constant V (low) Log q

14. Spring-slider simulation withrate- and state-dependent friction (blue curves) Westerly granite, s=30 MPa

15. 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

16. Contact stresses Indentation yield stress, sy Acrylic 400 MPa Calcite 1,800 MPa SL Glass 5,500 MPa Quartz 12,000 MPa

17. Increase of contact area with time Acrylic plastic Dieterich & Kilgore, 1994, PAGEOPH

18. Time dependent friction & Contact area

19. Velocity step & Contact area

20. Bowden and Tabor adhesion theory of friction Contact area: area = cs Shear resistance: t = (area) (g), t/s = m =cg c=1/ indentation yield stress g=shear strength of contacts (Drop the high-order term) m = m0 + A ln(V) + B ln(q) Interpretation of friction terms Time and rate dependence of contact strength terms Indentation creep: c(q) = c1 + c2ln(q) Shear of contacts: g(V) = g1 + g2ln(V) m = c1 g1 + c1g2ln(V) + c2g1ln(q) + c2g2ln(V+q)

21. Contact evolution with displacement

22. 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 • Dc correlates with contact diameter and arises from displacement-dependent replacement of contacts

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

24. Large-scale biaxial test Minimum fault length for unstable slip

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

26. Earthquake nucleation on uniform fault

27. Earthquake nucleation on a uniform fault

28. Earthquake nucleation – heterogeneous normal stress Lc

29. V1 x B mss Coefficient of friction m B-A Log q SPRING-SLIDER MODEL FOR NUCLEATION  K (t)  Evolution at constant normal stress During nucleation, slip speed accelerates and greatly exceeds steady state slip speed

30. SPRING SLIDER MODEL FOR NUCLEATION Re-arrange by solving for Where: Initial condition Model parameters

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

32. Dieterich, 1992, Tectonophysics / =0 t s 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107 108 109 1010 Solutions for time to instability m 2D numerical max model 103 Fault patch 102 solution 101 100 10-1 Slip speed (DC /s) 10-2 m s 10-3 10-4 10-5 10-6 10-7 10-8 Time to instability (s)

33. Accelerating slip prior to instability

34. Time to instability - Experiment and theory

35. 6 4 2 0 -2 -4 -6 -8 -10 -12 -14 -16 10-2 10-1 100 101 102 103 104 105 106 107 108 109 1010 1011 Effect of stress change on nucleation time Log (slip speed) m/s 1 yr 10 yr 20 yr Time to instability (seconds)

36. 6 4 2 0 -2 -4 -6 -8 -10 -12 -14 -16 10-2 10-1 100 101 102 103 104 105 106 107 108 109 1010 1011 Effect of stress change on nucleation time  = 0.5 MPa 5min ~1hr Log (slip speed) m/s ~5hr 1 yr 10 yr 20 yr Time to instability (seconds)

38. Log (slip speed) Log (time to instability) Model for earthquake occurrence • 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

39. Evolution of distribution of slip speeds 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

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

41. 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 - 1676 (2003)