Mechanics of Earthquakes and Faulting Lecture 9, 21 Sep. 2017 www.geosc.psu.edu/courses/geosc508 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
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
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
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:
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
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
Rate (v) and State (θ) Friction Constitutive Laws 1) 2) Modeling experimental data 3) Elastic Coupling Solve:
1) 2) 3) Typical Values of the RSF parameters (Marone et al., 1990)
1) 2) 3) Typical Values of the RSF parameters (Marone et al., 1990)
1) 2) Typical Values of the RSF parameters (Carpenter, Ikari & Marone 2016) 3)
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 $ ' ' %
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
0.77 1044 s hold, V s/r = 10 µm/s Coefficient of Friction 0.75 0.73 0.71 m080 Hold Reload 21 21.1 21.2 Displacement (mm) Sheared layer of quartz particles (100-150 µm), 25 MPa normal stress. Marone, 1998 0.77 m080 Hold 1044 s hold, V s/r = 10 µm/s Coefficient of Friction 0.75 0.73 0.71 1044 s Δµ Reload Time (s)
Load point 0.77 1044 s hold, V s/r = 10 µm/s Fault surface Coefficient of Friction 0.75 0.73 0.71 m080 Hold Reload Load point Fault surface 1) 21 21.1 21.2 Displacement (mm) 2) 3) Elastic Coupling
Derivation of the healing rate
Time Dependence of static friction Stressed Aging Monodisperse, angular quartz particles
Time Dependence of static friction Effect of loading velocity
0.04 1 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 0.03 0.02 0.01 0 10 0 10 1 10 2 10 3 10 4 Hold Time (s) (Marone, 1998, Nature)
0.10 Dieterich V s/r Law (µm/s) 10 100 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 = 0.001 D c = 5 µm, k= 1x10-3 µm -1 0 10-2 10-1 10 0 10 1 10 2 10 3 10 4 10 5 Hold Time (s) The rate of frictional healing depends on the rate of shearing (Marone, 1998, Nature) Rate State Friction Laws predict this behavior
Phase Plane Plots shs test: 1 µm/s 10 µm/s
shs test: 1 µm/s 10 µm/s Phase Plane Plots
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
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
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
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
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
Sheared layer of quartz particles. Marone, 1998 Time, displacement, and velocity dependence of static and dynamic friction Load point Fault surface
Time dependent yield strength: µ= τ S = σn σy Dieterich and Kilgore [1994] Time dependent growth of contact (acyrlic plastic)- true static contact
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
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
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
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 0. 3. 1. & 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