Mechanics of Earthquakes and Faulting

Transcription

1 Mechanics of Earthquakes and Faulting Lecture 9, 21 Sep Rate and State Friction Velocity step test to measure RSF parameters SHS test to measure RSF parameters Happy Fall The equinox is tomorrow at 4:02 pm EDT

2 Rate (v) and State (θ) Friction Constitutive Laws Recall (as motivation for going beyond other friction laws) Time-dependent static friction Velocity dependent sliding friction Memory effects, state dependence Repetitive stick-slip instability Key Observations log-time strengthening log-velocity dependence Application to earthquakes One set of constitutive relations to describe entire seismic cycle

3 Rate (v) and State (θ) Friction Constitutive Laws state variable, characterizes physical state of surface or shearing region reference velocity reference value of base friction critical slip distance Dieterich, aging law Ruina, slip law

4 Rate (v) and State (θ) Friction Constitutive Laws 1) 2) Implies: Vo V 1 Direct Effect µ Evolution Effect D c Steady-state sliding: => Fading memory of past state then (1) becomes:

5 Rate (v) and State (θ) Friction Constitutive Laws 1) 2) Convention is to use a, b for friction and A, B for Stress Steady-state velocity strengthening if a-b > 0, velocity weakening if a-b < 0 velocity strengthening µ velocity weakening Log V

6 Rate (v) and State (θ) Friction Constitutive Laws 1) 2) Steady-state velocity strengthening if a-b > 0, velocity weakening if a-b < 0 µ velocity strengthening velocity weakening a & b are small, dimensionless constants determined from experiments Log V Dc has units of length Modeling experimental data 3) Elastic Coupling

7 Rate (v) and State (θ) Friction Constitutive Laws 1) 2) Modeling experimental data 3) Elastic Coupling Solve:

8 1) 2) 3) Typical Values of the RSF parameters (Marone et al., 1990)

9 1) 2) 3) Typical Values of the RSF parameters (Marone et al., 1990)

10 1) 2) Typical Values of the RSF parameters (Carpenter, Ikari & Marone 2016) 3)

11 Measuring the velocity dependence of friction Frictional Instability Requires (a-b) < 0 µ " θ,v$ = µ # % 0 + aln " v v $ + bln " v o θ & & o ' & dθ dt = 1 vθ D c θ ss = D c v Δµ ss = ( a b)ln v v & o dµ dt Constitutive Modelling Rate and State Friction Law Elastic Interaction, Testing Apparatus = * " k v lp v # " # $ % # $ ' % % # D c $ ' ' %

12 Results: Velocity stepping Measuring the velocity dependence of friction Frictional Instability Requires K < K c This example shows steady-state velocity strengthening: (a-b) > 0

13 s hold, V s/r = 10 µm/s Coefficient of Friction m080 Hold Reload Displacement (mm) Sheared layer of quartz particles ( µm), 25 MPa normal stress. Marone, m080 Hold 1044 s hold, V s/r = 10 µm/s Coefficient of Friction s Δµ Reload Time (s)

14 Load point s hold, V s/r = 10 µm/s Fault surface Coefficient of Friction m080 Hold Reload Load point Fault surface 1) Displacement (mm) 2) 3) Elastic Coupling

15 Derivation of the healing rate

16 Time Dependence of static friction Stressed Aging Monodisperse, angular quartz particles

17 Time Dependence of static friction Effect of loading velocity

18 µm/s 3 µm/s 10 µm/s 30 µm/s 100 µm/s Stressed Aging Aging rate depends on the rate of shearing Frictional Healing, Δµ Hold Time (s) (Marone, 1998, Nature)

19 0.10 Dieterich V s/r Law (µm/s) Ruina Law Friction Law µ = µ o + a ln(v/v o ) + b ln(v o θ/d c ) State Evolution dθ/dt = 1 - V θ/d c dθ/dt = - V θ/d c ln( Vθ/D c ) Elastic Coupling dµ/dt = k( V lp - V) Healing, Δµ 0.05 b = 0.009, b-a = D c = 5 µm, k= 1x10-3 µm Hold Time (s) The rate of frictional healing depends on the rate of shearing (Marone, 1998, Nature) Rate State Friction Laws predict this behavior

20 Phase Plane Plots shs test: 1 µm/s 10 µm/s

21 shs test: 1 µm/s 10 µm/s Phase Plane Plots

22 Stick-Slip Instability Requires Some Form of Weakening: Velocity Weakening, Slip Weakening, Thermal/hydraulic Weakening 1) 2 ) Vo V 1 Direct Effect µ Evolution Effect Stability Criterion K c = n (b a) D c (b > a), K < K c Unstable, stick-slip (a > b), K > K c Stable sliding D c [1 + mv 2 o ] nad c K/K c < 1 Fading memory of past state

23 Time (state) dependence of friction: Healing Velocity (rate) dependence of friction. Duality of time and displacement dependence of friction. Static and dynamic friction are just special cases of a more general behavior called rate and state friction

24 Stick-Slip Instability Requires Some Form of Weakening: Velocity Weakening, Slip Weakening, Thermal/hydraulic Weakening 1) 2 ) N L Stability Criterion K c = n (b a) D c [1 + mv 2 o ] nad c W Rupture area, A Slip contours, u (b > a), K < K c Unstable, stick-slip K/K c < 1 (a > b), K > K c Stable sliding

25 Dislocation model for fault slip and earthquake rupture r Relation between stress drop and slip: Δσ = 16 7π G Δu r K = K/K c < 1 Unstable, stick-slip Δσ Δu = 16 G 7π r r c = 24 7π D c G σ (b a) K/Kc > 1 Stable, aseismic slip

26 Stick-slip stress-drop amplitude varies with loading rate. Mair, Frye and Marone, JGR 2002 Duality of time and displacement dependence of friction. Static and dynamic friction are just special cases of a more general behavior called rate and state friction

27 Sheared layer of quartz particles. Marone, 1998 Time, displacement, and velocity dependence of static and dynamic friction Load point Fault surface

28 Time dependent yield strength: µ= τ S = σn σy Dieterich and Kilgore [1994] Time dependent growth of contact (acyrlic plastic)- true static contact

29 Other measures of changes in static friction, contact area, or strength hold test after Dieterich [1972] time dependent closure (westerly granite) - approximately static contact Modified from Beeler, 2003

30 Rate dependence of contact shear strength hold test µ = τ σ n = S σ y Rate dependent response S = S o + g( V) Modified from Beeler, 2003

31 Summary of friction observations: 0. Friction is to first order a constant 1. Time dependent increase in contact area (strengthening) 2. Slip dependent decrease in contact area (weakening); equivalently increase in dilatancy 3. Slip rate dependent increase in shear resistance (non-linear viscous) Modified classic theory of friction: µ = S = S o + g( V) σ y σ o + f age ( ) µ = S o + g ( V ) σ o + f age ( ) # σ o f ( age) & $ % σ o f ( age) '( Discard products of second order terms: µ = S o + g( V ) σ o σ o S o f ( age ) 2 σ o [e.g., Dieterich, 1978, 1979] Modified from Beeler, 2003

32 Summary of friction observations: 0. Friction is to first order a constant 1. Time dependent increase in contact area (strengthening) µ = S o + g( V ) σ o σ o S o f ( age) 2 σ o 2. Slip dependent decrease in contact area (weakening); equivalently increase in dilatancy 3. Slip rate dependent increase in shear resistance (non-linear viscous) 1st order term second order terms Rate and state equations: µ = µ 0 + aln V V 0 + b ln V 0θ D c & 2. θ is contact age Dieterich [1979] Rice [1983] Ruina [1983] dθ dt = 1 θv D c time dependence slip dependence dθ dt = # θ & % ( $ t ' d # + θ & % ( $ d ' t V # θ & % ( $ t ' d = 1 # θ & % ( $ d ' t = θ D c Modified from Beeler, 2003