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Modeling Earthquakes Faults and Statistics

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M. O. Robbins & K. M. Salerno, Johns Hopkins UniversityC. Maloney, Carnegie Mellon University

Example 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.

M. O. Robbins & K. M. Salerno, Johns Hopkins UniversityC. Maloney, Carnegie Mellon University

How 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

Compression axis

Vertical compression axis

Textbook picture of brittle rock failure (from C. Scholz: The Mechanics of Earthquakes and Faulting)

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- Experiment: Uniaxial compression of rock
- Otsuki and Dilov JGR 2005
- Scale ~ 20 cm

situation of interest

C.E. Maloney and M.O. Robbins

Johns Hopkins and KITP

Computing: UCSB CNSI

Push-ups

Tensile fissures

- Surface trace of earthquake in Iceland
- Scale ~ 1km

- From C. Scholz

From C. Scholz: The Mechanics of Earthquakes and Faulting

- San Andreas Fault Network
- Scale ~ 1000 km

SCEC inter-seismic velocity map (displacement since 1984)

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

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

Ly(t)

Lx(t)

Non-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)

ω<0

ω>0

Find Delaunay triangulation for initial particle centers

For each triangle:

Invariants:

“Left Strain”

“Right Strain”

Brittle model: α=4. New bonds 4 times weaker.

Ductile model: α=1. No “broken” bonds.

Brittle

140 particles

starts

to fail

patterns

form

Brittle

Stress

Ductile

Strain

Ductile

Low pressure more brittle

High pressure

Stress

Low pressure

Strain

High pressure more ductile

Total Shear Strain

Incremental shear

10x10mm AFM image of clay

“Earthquakes” in Large Brittle Systems

starts

to fail

patterns

form

Brittle

Stress

Ductile

Strain

Steady state strain accommodation

Components of u during 0.2% strain+s white, -s black

Strain localizes in plastic bands → step in total displacement

uy

ux

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

(ux+uy)/√2

(ux-uy)/√2

Displacement and w in steady state

→

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

|Dr|

h~L/20

ω

Strain g:

6.0% to 6.1%

6.0% to 6.2%

6.0% to 6.4%

Plastic band formed in each strain~a/L gives |Dr|<a/2, add incoherentlyDr2~Dg/(a/L) a2/12 ~ D Dg→ D ~ L a/12Consistent with observed D=57s2 for a=0.7s and for L down to 40

For w, no L dependence

D=57s2

Sharp elastic peak at small wExponential tails at large wWeight in tails grows ~linearly with strain

Characteristic w* for decay ~0.1

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

Plastic bands have ~10 sharper localized bands

Mean of log S scales as power of wave vector.

Prefactor linear in Dg→Incoherent addition of successive intervals

BUT scaling highly anisotropic

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 p/8

Α: broken shear symmetry

bigger for planes with low normal loadas predicted by Mohr-Coulomb

S(q;θ)=A(θ)q-α(θ)

α = a+b cos(4q)

A = c+d cos(2q)

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 p/8

Α: broken shear symmetry

bigger for planes with low normal loadas predicted by Mohr-Coulomb

S(q;θ)=A(θ)q-α(θ)

α = a+b cos(4q)

A = c+d cos(2q)

- 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:

- log(N) = C – bM
- Experimentally verified across regions, magnitude range b ~ 1

׀

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

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)

Scale E by Emax ~ Lb . Find b~1.1N(E/Emax) must scale as L1-b to maintain energy balance

N(E) (L/100)1-b

E*(L/100)2-b

Same system but with energy minimization, not dynamics

Exponent ~0.5 for largest systems

Largest size ~3

Overdamped 2D Discrete dislocation model N(E)~E-tt=1.8

M.-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=1.4

N. P. Bailey, J. Schiøtz, A. Lemaître, and K. W. Jacobsen, PRL 98, 095501 (2007).

Large events are broken up. Find integrated density ~ 1/E0.4

- Strain localizes in plastic bands that extend across systemTypical 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 Dr2~Dg/(a/L) a2/12 ~ D Dg, D~La/12
- Strain over short intervals has anisotropic power law correlationsS(|q|,q) ~ A(q) |q|-a(q) where α = a+b cos(4q), A = c+d cos(2q)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