Rate-dependent shear bands and earthquake rupture simulations

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Rate-dependent shear bands and earthquake rupture
simulations

• Eric Daub
• M. Lisa Manning

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Constitutive laws and flow
strain localization
homogeneous
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Bulk metallic glasses fail along narrow shear
bands
shear band thickness 10-20 nm
Johnson Group, Caltech
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Most slip in earthquake faults occurs along
narrow shear band
Fault gouge (granular) 0.15-0.55 m wide
Prominent fracture surface 1 mm wide, most of
the slip
F. M. Chester and J. S. Chester, Tectonophys.
295, 1998.
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What process(es) lead to shear bands?

• Are they similar in different types of disordered
  solids?

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Shear Transformation Zones

• continuum model for disordered solids
• tracks density and orientation of soft spots or
  STZs
• creation, annihilation, and bistable switching

Spaepen (1977), Argon (1979), Falk and Langer
(1998)
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• Effective temperature
• governs statistical distribution of density
  fluctuations
• describes local configurational entropy
• measured in simulations using fluctuation
  dissipation relation
• STZs are unlikely, high energy configurations
• density (?) Boltzmann factor

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Steady state effective temperature is rate
dependent
Haxton and Liu PRL 2007
increases with strain rate
Effective temperature
?0 is the minimum effective temperature
strain rate
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Steady state glassy dynamics
Super-Arrhenius regime
Langer and MLM, PRE 2007
- log (strain rate)
Simple activation regime
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Constitutive model
from STZ theory
thermal relaxation
Strain-rate dependent diffusion
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Three stationary solutions for boundary-driven
shear
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When does each type of deformation occur?
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Transient linear stability

• Question Are the homogeneous STZ equations
  unstable with respect to a perturbation in ? at
  the onset of plastic deformation?

Answer Yes, if A22 gt 0.
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Does Localization occur?

• Requires linear stability AND analysis of
  finite-size perturbations
• Localized states are transient (characterizing
  these states is difficult)
• Answer for small strain rates
• ? lt ?crit - f ( ??)
• small perturbations are stabilized

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homogeneous strain
linear stability
stable
Initial effective temp.
x
diffusion-limited shear band
x
unstable
x
disorder-limited shear band
log(imposed strain rate)
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Disorder-limited shear bands
?

• Simulations Shi et al PRL 2007
• STZ theory shear band thickness determined by
  external driving rate, STZ density
• (MLM et al., PRE 2007)

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Diffusion-limited shear bands
Fast external driving ? never reaches ?(?)

thickness D1/2
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More work on phase diagram

• Numerically integrate STZ PDE, filling in the
  phase diagram
• MATLAB just not fast enough
• Multiple shear bands?
• Width of shear bands?
• Different ?(q)?

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Experiments?

• Densely packed amorphous solid driven in simple
  shear (constant velocity)
• Look at strain rate as a function of position
• Control parameters applied strain rate and
  initial sample preparation or quench
• Can effective temperature be measured?
• If not, could we simulate the material to
  determine ?(q)?
• System bounded by slowly loaded elastic material?

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Conclusions

• Strain-rate dependent steady state effective
  temperature incorporated into STZ theory
• STZ model predicts three types of nearly
  stationary states
• homogeneous strain
• diffusion-limited shear band
• disorder-limited shear band
• Strain localization drastically changes
  constitutive laws dynamic weakening

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Shear Strain Localization in Elastodynamic
Rupture Simulations
Eric G. Daub, M. Lisa Manning, and Jean M.
Carlson Physics Department, UCSB
Earthquake Problem is Multi-Scale
Collective Grain Motion
Interfacial Friction
Fault Dynamics
Rupture Dynamics
Localized Microscopic Strain
STZ Friction Law
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Overview

• Study slip localization within cores of
  earthquake faults
• Strain localization is observed in many studies
  of faults
• We model friction using STZ Theory, a
  microscopic physical model for gouge deformation
  that allows for dynamic strain localization
  within the fault core
• Model earthquake processes with both a single
  degree of freedom spring slider and spontaneous
  elastodynamic rupture
• Spring slider model (interface scale) Strain
  spontaneously localizes and produces more
  velocity weakening than homogeneous strain
• In dynamic rupture simulations (fault scale),
  ruptures that can localize have larger stress
  drops and larger peak slip rates. Additional
  dynamic weakening allows for pulse-like rupture.

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Strain Localization Observed in Many Studies
Marone, Ann. Rev. Earth Planet. Sci. 26, 1998
Laboratory
Morgan and Boettcher, JGR 104(B2), 1999
Simulations
Field
Highly damaged fault gouge, further localization
to narrow fracture surface
Chester and Chester, Tectonophys. 295, 1998.
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Fault Gouge Experiments at High Velocities
Most rock friction experiments are done at low
driving rates (microns/sec to millimeters/sec),
but a few reach seismic velocities. There is
certainly localization occurring in these
experiments, but not clear yet exactly how much
or what the microstructures are.
Seismic Driving Rate
Low Driving Rate
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Constitutive Law
Plastic shear strain due to Shear Transformation
Zones (STZs), local regions of gouge undergoing
shear that are constantly created by shearing
Number of STZs determined by a Boltzmann
distribution, with Effective Temperature c.
Higher effective temperature, more
configurational disorder (entropy) in the
material.
Diffusion
Shear heating
Time-dependent relaxation (healing)
(Falk and Langer, Phys. Rev. E, 1998 Manning,
Langer, and Carlson, Phys. Rev. E, 2007.)
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Spring Slider Model
Start with a simple non-inertial spring slider
model, driven from rest to a seismic slip rate (1
m/s). Strain dynamically localizes unless initial
conditions are homogeneous.
Strain Rate Profiles
V0
Homogeneous
Localized
Feedbacks leading to localization
Higher Strain Rate
Larger Eff. Temp.
Increased Shear Heating
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Spring Slider Model
Stress vs. displacement, and representative
strain rate vs. position plots.
For small displacements, strain occurs throughout
the gouge.
Compare with homogeneous strain.
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Spring Slider Model
Stress vs. displacement, and representative
strain rate vs. position plots. Gouge weakens
more rapidly as strain begins to localize.
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Spring Slider Model
Stress vs. displacement, and representative
strain rate vs. position plots. Narrower,
diffusion-limited shear band begins to develop
and strain further localizes.
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Spring Slider Model
Stress vs. displacement, and representative
strain rate vs. position plots. Stress doesnt
change with displacement once strain localizes to
narrower shear band.
Frictional stress appears to be steady-state,
but actually a long-lived transient effect.
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Dynamic Ruptures
Implement localization into spontaneous
elastodynamic rupture simulations. Mode II 2D
ruptures, uniform initial shear stress and
friction parameters. Compare localized STZ
ruptures to homogeneous STZ ruptures (studied in
Daub and Carlson, JGR, submitted).
elastic rock
gouge (STZ Theory)
elastic rock
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Dynamic Ruptures
Comparisons slip rate vs. time, and stress vs.
slip.
Localized rupture has larger stress drop and less
frictional dissipation. Additional weakening
pulse-like rupture?
Peak slip rate in the localized rupture is
greater.
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Types of Ruptures
What are the different ways that slip can
propagate on a fault?
Decreasing initial shear stress
Crack-Like
Pulse-Like
Supershear
Slips and then heals shortly afterwards.
Ruptures faster than the shear wave speed.
Slips during the entire duration of the rupture.
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Dynamic Ruptures
Vary initial stress and transient shear band
width to generate a rupture-type diagram.
Homogeneous
Localized
Pulse-like rupture occurs with localization but
not for homogeneous strain. Localization reduces
the minimum stress for entire fault rupture by 10
MPa for our narrowest simulation.
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Recap

• STZ Theory, a physical model for gouge
  deformation, accounts for strain localization in
  fault zones
• Slip spontaneously localizes due effective
  temperature feedback
• Stress weakens more rapidly for localized strain
  than for homogeneous shear
• Fault-scale dynamic ruptures with localization
  have a smaller sliding stress than homogeneous
  ruptures, with higher peak slip rates.
  Small-scale physics will affect stress drops and
  ground motion in earthquakes!
• Additional dynamic weakening provided by
  localization can allow for pulse-like ruptures.
  Localization can dramatically lower the lowest
  shear stress for which a fault can fully rupture.

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Friction and earthquakes
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Dynamic weakening

• For earthquakes to propagate, require that the
  final stress state must be less than the initial
  stress state
• weakening
• In rate-and-state laws, velocity weakening is
  required for stick slip instabilities (initiating
  earthquakes)
• velocity weakening steady state stress
  decreases with increasing velocity

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STZ steady state, homogeneous velocity dependence

• stick slip A B lt 0

A is an activation energy that specifies how the
plastic strain rate q is activated by ?
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BUT

• When effective temperature localizes, the final
  states are not steady states, and ? ? ?
  everywhere
• Caveat the stress appears stationary
• Experiments on gouge shows that (A-B) evolves
  with slip
• Does a steady state analysis make sense in this
  case? NO!

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Stick slip instability

• Is there an analogue to velocity weakening when
  the system localizes and never reaches a steady
  state?
• What governs stick slip?

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So far . . .

• STZ description of fault gouge shows how a
  prominent fracture surface can spontaneously
  develop
• This is accompanied by a rapid decrease of shear
  stress on the fault dynamic weakening
• Not necessarily velocity weakening
• Strain rate in the band goes up and shear stress
  decreases
• Are final states at a lower stress?

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Friction coefficients

• Can STZ dynamics generate a friction coefficient
  that is smaller at high speeds?
• Seen in experiments
• puts a strict bound on sy in STZ theory
• the term ?(q) provides a natural time-scale for
  this crossover to occur
• Eric has seen that in localized systems, a whole
  range of final stresses are possible
• A rapidly changing R(s) leads to sensitive
  dependence on initial conditions

sy
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Future directions

• Can we match friction coefficient experiments by
  choosing right ?(q) and R(s)?
• experimental evidence that friction coefficients
  increase with v at very slow speeds?
• Can we see/characterize stick-slip in systems
  that localize (no steady state)?