Rate-dependent shear bands and earthquake rupture simulations Eric Daub M. Lisa Manning

Constitutive laws and flow homogeneous strain localization

Bulk metallic glasses fail along narrow shear bands Johnson Group, Caltech shear band thickness nm

Prominent fracture surface ~1 mm wide, most of the slip Fault gouge (granular) 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:

Effective temperature 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 (strain 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 stability Question: Are the homogeneous STZ equations unstable with respect to a perturbation in  at the onset of plastic deformation? Answer: Yes, if A 22 > 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) Initial effective temp. x x unstable 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 ~ D 1/2

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 StrainSTZ Friction Law Collective Grain MotionInterfacial FrictionFault 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 Simulations Marone, Ann. Rev. Earth Planet. Sci. 26, 1998 Chester and Chester, Tectonophys. 295, Field Highly damaged fault gouge, further localization to narrow fracture surface Laboratory Morgan 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 Rate Seismic 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: HomogeneousLocalized V0V0

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-LikePulse-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 s y 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)? ^