Mechanics of Earthquakes and Faulting

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Mechanics of Earthquakes and Faulting
21 Feb. 2007
Friction

• Rate-state friction
• State Variable
• State Evolution Equation
• Solutions
• Contact Age and Contact Growth
• Chemical Effects
• Lower Velocity Cutoff for Evolution
• Healing data Stick-slip stress drop(V),
  del_mu(t), shear stress effects

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Rate (v) and State (?) 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

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E. Rabinowicz, Sci. Amer., 109, May 1956.
External factors, such as humidity, also may play
a part. Squeaks in an automobile are apt to be
silenced on a wet day-and, perversely, almost
invariably when the car is taken to a garage to
have the squeaks located and removed.
Demonstrations of stick-slip during public
lectures are likewise hazardous undertakings.
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Rate (v) and State (?) Friction Constitutive Laws
state variable, characterizes physical state of
surface or shearing region
critical slip distance
reference velocity
reference value of base friction
Dieterich, aging law
Ruina, slip law
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Rate (v) and State (?) Friction Constitutive Laws
1)
2)
Vo
V1
Implies
Direct Effect
µ
Evolution Effect
Dc
Fading memory of past state
Steady-state sliding
gt
then (1) becomes
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Rate (v) and State (?) 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 gt 0,
velocity weakening if a-b lt 0
velocity strengthening
µ
velocity weakening
Log V
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Rate (v) and State (?) Friction Constitutive Laws
1)
2)
Steady-state velocity strengthening if a-b gt 0,
velocity weakening if a-b lt 0
velocity strengthening
a b are small, dimensionless constants
determined from experiments
µ
velocity weakening
Log V
Dc has units of length
Modeling experimental data
3)
Elastic Coupling
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Rate (v) and State (?) Friction Constitutive Laws
1)
2)
Modeling experimental data
3)
Elastic Coupling
Solve
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Sheared layer of quartz particles (100-150 µm),
25 MPa normal stress . Marone, 1998
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Load point
Fault surface
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Time Dependence of static friction Stressed
Aging
Monodisperse, angular quartz particles
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Time Dependence of static friction Effect of
loading velocity
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Stressed Aging Aging rate depends on the rate of
shearing
(Marone, 1998, Nature)
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Stresses v. Unstressed Aging
Karner Marone (GRL 1998, JGR 2001)
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100 s holds, Healing rate varies systematically
with shear stress
Karner Marone (GRL 1998 JGR 2001)
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Unstressed Aging Frictional Healing Rate is
Negative
Karner Marone (GRL 1998 JGR 2001)
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Unstressed Aging Frictional Healing Rate is
Negative
Karner Marone (GRL 1998 JGR 2001)
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Frictional Healing Depends on Shear Stress Level
Time-dependent strengthening
Time-dependent weakening
Karner Marone (GRL 1998 JGR 2001)
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Force Chains and Granular Friction

• Frictional strength
• Development of force chains, Jamming
• Aging
• Due to chemically-assisted creep at adhesive
  contact junctions
• Time-Dependent Weakening
• Time-dependent unjamming, creep

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Chemically-Assisted Frictional Aging Creep at
Adhesive Contact Junctions
In-situ Particle Comminution Production of Fresh
Surface Area
Frye and Marone, JGR 2002
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Granular quartz
Hydrolytic Weakening causes enhanced rate of
strengthening
0 20 40 60 80 100 RH
Frye and Marone, JGR 2002
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Chemically-Assisted Frictional Aging Creep at
Adhesive Contact Junctions
Hydrolytic Weakening causes enhanced rate of
strengthening, but base level frictional strength
is unchanged
Frye and Marone, JGR 2002
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Granite Surfaces
Granular Materials
Solid Surfaces Base level of frictional strength
decreases with increasing water content (cf.
Dieterich Conrad, 1984)
Granular Materials Frictional strength is
independent of water content
Interpretation Contact junctions subject to time
dependent strengthening or growth, which inhibits
sliding, but particle rolling is not affected by
these factors.
Frye and Marone, JGR 2002
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Velocity dependence of steady state friction
varies changes from positive to negative. (cf.
Tullis and co-workers) Chemically-assisted creep
at adhesive contact junctions
Frye and Marone, JGR 2002
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Modeling the effect of normal force vibration 1.
Rate and State Friction Theory
Deiterich Law
Normal Stress
Elastic Coupling
Critical Vibration Period
Critical Stiffness
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Measuring the velocity dependence of friction
Frictional Instability Requires (a-b) lt 0
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Results Velocity stepping Measuring the
velocity dependence of friction
Frictional Instability Requires K lt Kc
This example shows steady-state velocity
strengthening (a-b) gt 0