Modeling Earthquakes Faults and StatisticsM. O. Robbins & K. M. Salerno, Johns Hopkins University

C. Maloney, Carnegie Mellon UniversityExample of simulations where follow “particles” representing atoms or chunks of solid or fluid

Routine ~106 -107 particles for 107-108 time steps

Reduce long trajectories to a few forces or simple correlation functions

Must redo many times as develop analysis methods.

When do save data, slower to read back than to analyze.

Can’t step backwards because chaotic.

Modeling Earthquakes Faults and StatisticsM. O. Robbins & K. M. Salerno, Johns Hopkins University

C. Maloney, Carnegie Mellon UniversityHow do microscopic displacements accommodate total global strain?

How are “earthquakes” distributed in energy, location and time?

What is geometry of fault and nature of sliding on rough faults?

Traditional approaches: Posit planar crack and follow dynamics

Use actual structure, but assume no stress

MotivationTextbook picture

of brittle rock failure (from C. Scholz:

The Mechanics of Earthquakes and Faulting)

Vertical compression axis\\ \\

Compression axis

• Experiment: Uniaxial compression of rock

• Otsuki and Dilov JGR 2005

• Scale ~ 20 cm

situation of interest

Using atoms to model formation, evolution of strike-slip faults

C.E. Maloney and M.O. RobbinsJohns Hopkins and KITPComputing: UCSB CNSI

Motivation

From C. Scholz: The Mechanics of Earthquakes and Faulting

•Surface trace of earthquake in Iceland

•Scale ~ 1km

•From C. ScholzTensile fissures

Push-ups

Motivation•San Andreas Fault Network

•Scale ~ 1000 km

SCEC inter-seismic velocity map (displacement since

1984)

MethodUsually 2D Molecular Dynamics:•Binary Lennard-Jones Mean diameter •Quenched at pressure p=0•Relative velocity damping (Kelvin/DPD)•Periodic boundaries•Axial, fixed area strain or simple shear•Quasi-static limit, Controlled by not t •Make system brittle by making all initial bonds 4 times stronger

Prescribed Ly(t), Lx(t) to conserve

area

Ly(t)

Lx(t)

Nonaffine Particle DisplacementsNon-affine displacement u = deviation from mean motion

Integrate affine displacement along trajectory rather than using value for initial position.

For each particle at each time step find distance moved relative to affine displacement at instantaneous position.

Eliminates terms analogous to Taylor diffusion in simple shear

Ly(t)

Lx(t)

Local rotation, ω=uFind Delaunay triangulation for initial particle centers

For each triangle:

Invariants:

ω<0ω>0

“Right Strain” “Left Strain”

Mechanism for brittleness

Ductile

Brittle

Str

ess

Strain

Brittle

Ductile

Brittle model: α=4. New bonds 4 times weaker.Ductile model: α=1. No “broken” bonds.

140 p

art

icle

s

patternsform

starts to fail

Pressure Induced DuctilityLow pressure more brittle

High pressure more ductile

Strain

Str

ess

High pressure

Low pressure

“Earthquakes” in Large Brittle Systems

10x10m AFM image of

clay

Total Shear Strain

Incremental shear

Shift to Study of Ductile SystemsS

tress

Strain

Brittle

Ductile

patternsform

starts to fail

Steady state strain accommodation

Spatial Variation of Nonaffine Displacement u

uxuy

(ux+uy)/√2 (ux-uy)/√2

Components of u during 0.2% strain+ white, - blackStrain localizes in plastic bands → step in total displacement

Largest projection of u along bands ±45ºSign → rotation senseLength up to systemTypical a ~

|r|

ω

Strain: 6.0% to 6.1%

6.0% to 6.2%

6.0% to 6.4%

Displacement and in steady stateMost analysis looks at magnitude r=| u | → smallest in plastic bandCurl sharply localized in plastic zone, +/- regions correlate along -/+45ºStrain accumulates to a~ in plastic bands through many avalanches →Displacement ~ allows all regions to find new metastable state Strain over intervals > ~/L occurs in uncorrelated locations

h~L/20

→

Displacements Diffusive with DLPlastic band formed in each strain~a/L gives |r|<a/2, add incoherently r2~/(a/L) a2/12 ~ D → D ~ L a/12Consistent with observed D=572 for a=0.7 and for L down to 40

D=572

For , no L dependence

Distribution of Vorticity Sharp elastic peak at small Exponential tails at large Weight in tails grows ~linearly with strain

Characteristic for decay ~0.1

Since ~ twice strain, * → 5% strain ~ yield strain

Plastic bands have ~10 sharper localized bands

Vorticity Correlation FunctionMean of log S scales as power of wave vector.

Prefactor linear in →Incoherent addition of successive intervals

BUT scaling highly anisotropic

Angle Dependence of Structure Factor, Δγ=0.1%

α = a+b cos(4)A = c+d cos(2)

θ=π/8

θ=3π/8

θ=π/8 and θ=3π/8 have same shear stress, different normal stress.Mohr-Coulomb predicts shift to /8

S(q;θ)=A(θ)q-α(θ) Α: broken shear symmetrybigger for planes with low normal loadas predicted by Mohr-Coulomb

Angle Dependence of Structure Factor, Δγ=0.1%

α = a+b cos(4)A = c+d cos(2)

θ=π/8

θ=3π/8

θ=π/8 and θ=3π/8 have same shear stress, different normal stress.Mohr-Coulomb predicts shift to /8

S(q;θ)=A(θ)q-α(θ) Α: broken shear symmetrybigger for planes with low normal loadas predicted by Mohr-Coulomb

Gutenberg-Richter Law

• Empirically determined scaling between earthquake number and magnitude– log(N) = C – bM

• M = log(E) • Magnitude is the log of the energy released

– Combined gives power law scaling:

• Experimentally verified across regions, magnitude range b ~ 1

N E b

Avalanche Distribution

Deformation through series of avalanches

Find energy dissipated = change in potential energy - work done by system.

Drops have wide distribution, 0.03 to 20 here

Identify by sharp rises in dissipation rate.

Dissipation rate/Kinetic energy drops to constant as energy moves to longest wavelengths.

Ratio measures wavelength where have kinetic energy

׀

P(E) = Number of Events per Unit Strain per E Reduce strain rate so quasi-static, only affects small events.Find power law P(E)~1/E over 5 decades.P(E) and maximum E increase with system size

P(E)

Scaling Collapse of N(E) and E

E*(L/100)

Scale E by Emax ~ L . Find ~1.1N(E/Emax) must scale as L to maintain energy balance

N(E

) (L

/100

)1-

Comparison to Scaling of P(E) in Other ModelsSame system but with energy

minimization, not dynamicsExponent ~0.5 for largest systemsLargest size ~3

Overdamped 2D Discrete dislocation model N(E)~E =1.8M.-C. Miguel, A. Vespignani, S. Zapperi, J. Weiss, and J.-R. Grasso, Mater. Sci. Eng., A 309–310, 324 2001.

Why is power law different than Gutenberg-Richter?

3D amorphous metal N(E)~Exp[-E/Lx] x=1.4N. P. Bailey, J. Schiøtz, A. Lemaître, and K. W. Jacobsen, PRL 98, 095501 (2007).

Power Law Independent of Potential, Geometry and Thermostat

Our Quakes Include Fore and After ShocksHow Does this Affect Statistics?

Integrated Density of Quakes Based on Kinetic Energy Large events are broken up. Find integrated density ~ 1/E0.4

Conclusions• Strain localizes in plastic bands that extend across system

Typical slip distance along band a~particle diameterTypical thickness h scales with system size ~L/20Several sharper features in h with strain ~5 – 10%

• Non-affine displacement r2~/(a/L) a2/12 ~ D , D~La/12• Strain over short intervals has anisotropic power law correlations

S(|q|,) ~ A() |q|-() where α = a+b cos(4), A = c+d cos(2)Breaking of symmetry for A → Mohr-Coulomb

• Inertial motion seems to lead to qualitative changes in deformation statistics.

• Earthquake probability P(E) ~ 1/E over ~ 5 decadesExponent independent of geometry, interactionsMay depend on how events are broken up