Rate-dependent shear bands and earthquake rupture simulations

Eric Daub

M. Lisa Manning

Constitutive laws and flowhomogeneous strain localization

Bulk metallic glasses fail along narrow shear bands

Johnson Group, Caltech

shear band thickness 10-20 nm

Prominent fracture surface

~1 mm wide, most of the slip

Fault gouge (granular)0.15-0.55 m wide

Most slip in earthquake faults occurs along narrow shear band

F. M. Chester and J. S. Chester, Tectonophys. 295, 1998.

What process(es) lead to shear bands?

Are they similar in different types of disordered solids?

Shear Transformation Zones

Spaepen (1977), Argon (1979), Falk and Langer (1998)

• continuum model for disordered solids

• tracks density and orientation of “soft spots” or STZs

• creation, annihilation, and bistable switching

• 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:

Effe

ctiv

e te

mp

era

ture

strain rate

increases with strain rate

0 is the minimum effective temperature

Haxton and Liu PRL 2007

Steady state effective temperature is rate dependent

Simple activation regime

Super-Arrhenius regime

Steady state glassy dynamics-

log

(str

ain

rate

)

Langer and MLM, PRE 2007

Constitutive model:

thermal relaxation

Q is proportional to heat generated by deformation, s

.Strain-rate dependent diffusion

from STZ theory

Three ~stationary solutions for boundary-driven shear

X=1• Homogeneous strain

=1(inside band)~0 (outside band)

• Diffusion balances shear heating

X

• Disorder-limited:

in shear band only, outside band very small

When does each type of deformation occur?

Transient linear stabilityQuestion: Are the homogeneous STZ equations unstable with respect to a perturbation in at the onset of plastic deformation?

Answer: Yes, if A22 > 0.

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:– < crit - f ( )

– small perturbations are stabilized

x

diffusion-limited shear band

homogeneous strain

disorder-limited shear band

log(imposed strain rate)

Initi

al e

ffect

ive

tem

p.

x

xunstable

stable

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)

Diffusion-limited shear bands

Fast external driving

never reaches ()

thickness ~ D1/2

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)?^

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?

^

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

Shear Strain Localization in Elastodynamic Rupture Simulations

Eric G. Daub, M. Lisa Manning, and Jean M. Carlson

Physics Department, UCSB

Rupture DynamicsLocalized Microscopic Strain STZ Friction Law

Collective Grain Motion Interfacial Friction Fault Dynamics

Earthquake Problem is Multi-Scale:

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.

Overview

Strain Localization Observed in Many Studies

SimulationsMarone, Ann. Rev. Earth Planet. Sci. 26, 1998

Chester and Chester, Tectonophys. 295, 1998.

Field Highly damaged fault gouge, further localization to narrow fracture surface

LaboratoryMorgan and Boettcher, JGR 104(B2), 1999

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.

Low Driving RateSeismic Driving Rate

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 . Higher effective temperature, more configurational disorder (entropy) in the material.

DiffusionShear heating

(Falk and Langer, Phys. Rev. E, 1998; Manning, Langer, and Carlson, Phys. Rev. E, 2007.)

Time-dependent relaxation (healing)

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.

Larger Eff. Temp.

Higher Strain Rate

Increased Shear Heating

Feedbacks leading to localization:

Strain Rate Profiles:

Homogeneous LocalizedV0

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.

Spring Slider Model

Stress vs. displacement, and representative strain rate vs. position plots.

Gouge weakens more rapidly as strain begins to localize.

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.

Spring Slider Model

Stress vs. displacement, and representative strain rate vs. position plots.

Stress doesn’t change with displacement once strain localizes to narrower shear band.

Frictional stress appears to be “steady-state,” but actually a long-lived transient effect.

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

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.

Types of Ruptures

What are the different ways that slip can propagate on a fault?

Slips and then heals shortly afterwards.

Ruptures faster than the shear wave speed.

Slips during the entire duration of the rupture.

Crack-Like Pulse-LikeSupershear

Decreasing initial shear stress

Dynamic Ruptures

Vary initial stress and transient shear band width to generate a rupture-type diagram.

HomogeneousLocalized

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.

• 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.

Recap

Friction and earthquakes

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

STZ steady state, homogeneous velocity dependence

stick slip: A – B < 0

A* is an activation energy that specifies how the plastic strain

rate q is activated by

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!

^

Stick slip instability

• Is there an analogue to “velocity weakening” when the system localizes and never reaches a steady state?

• What governs stick slip?

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?

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^

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)?

^