Mechanics of Earthquakes and Faulting · Background: How does this relate to dislocation mechanics...

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Mechanics of Earthquakes and Faulting

www.geosc.psu.edu/Courses/Geosc508

Lecture 15, 29 Nov. 2019

• Rupture propagation and the crack tip stress field• Cohesive zone; transition from ‘broken’ to ‘unbroken’ as a crack propagates.• Think of this as a method to define ‘partially broken’• Energy Balance of Dynamic Faulting• Faulting types, stress polygons• Wear and fault roughness• Thermo-mechanics of faulting• Moment, Magnitude and scaling laws for earthquake source parameters

Stress field is singular at the crack tip.

• because we assumed perfectly sharp crack(but real materials cannot support infinite stress)

) % tip

≈12πr

KI = πc σ∞

Dislocation model for fracture and earthquake rupture

Dislocation model, circular crack∆σ = (σo - σf )

Shear Stressσo

σf Δσ

Δu(x, y) = 247π Δσ

µc 2 − x 2 + y2( )

Crack tip stress fieldσ

r• Singular crack (Eshelby)x

Brittle Fracture Theory Predicts the Equation of Motion of Frictional Rupture Fronts, Ilya Svetlizky, David S. Kammer, Elsa Bayart, Gil Cohen, and Jay Fineberg, PRL 118, 125501 (2017)

Frictional Resistance within the Wake of Frictional Rupture Fronts, Ilya Svetlizky, Elsa Bayart, Gil Cohen, and Jay Fineberg PRL 118, 234301 (2017)

Classical shear cracks drive the onset of dry frictional motion, Svetlizky & Fineberg, Nature, 2014.

See also: Johnson and Scholz, JGR, 1976.

Crack tip stress field

Classical shear cracks drive the onset of dry frictional motion, Svetlizky & Fineberg, Nature, 2014.

Cohesive zone, slip weakening crack model for friction

Cracked/Slipping zone

Shear Stress

Slip, displacement

Breakdown (cohesive) zone

Intact, locked zone

Energy Balance of Dynamic Faulting

The rate of energy change is a balance between work terms, surface energy, kinetic energy, and frictonal work.

These terms operate over different regions.

Recall that in the Linear Elastic Fracture Mechanics Approach to Dynamic Crack Propagation:

LEFM assumes that cracks are cohesionless. In this case the crack tip energy term, Us, can be determined

ΣΣ�

Energy Partitioning

If we choose the bounding surface sufficiently large, work at external boundaries, W, is zero.Then the energy remaining to be radiated seismically is the dynamic change in kinetic energy(Δ implies a change in the state during dynamic rupture relative to the initial state).

ΣΣ�

Recall that in the Linear Elastic Fracture Mechanics Approach to Dynamic Crack Propagation:

LEFM assumes that cracks are cohesionless. In this case the crack tip energy term, Us, can be determined

ΣΣ�

The change in internal strain energy can be written in terms of the drop in shear stress associated with rupture Δσ = σ1 – σ2

where we assume that the initial stress, σ1, is equal to the critical stress for failure.That is, the net change in strain energy due to cracking is just equal to the work of faulting Wf.

Slip, u

If we assume that shear stress during slip is equal to a frictional stress, a dynamic friction term, σf, we can define a dynamic stress drop Δσd = (σ1 – σf). where σ1 is the initial stress –which is not necessarily equal to the yield strength σy

In this case the seismic energy is ,

if we assume that σ2 equals σf –e.g, the final stress is exactly the same as the dynamic frictional strength. (But what about dynamic overshoot, or healing pulse rupture models?)

Slip, u

The above equation for Es indicates that seismic energy depends only on the stress change, and not on the total stress. This is a big problem if you�d like to determine the complete energy balance for faulting, for example as needed for the heat flow-faulting stress problem.

Dynamic Crack Propagation: Two Approaches1) Energy balance, fracture mechanics, critical energy release rate.

Problem of stress concentrations and singularities2) Stick-slip frictional model. Critical stress needed for failure

Energy Balance: Specific fracture energy GcFor a mode III crack, we can calculate a critical half length for crack extension

note that G is shear modulus, sometimes written µLc is of order 1-2 m for Gc of 102 J/m2.

Background: How does this relate to dislocation mechanics and or energy flux to/from a crack tip? But note range of fracture energy values as a function of fracture size (Table 1.1). Earthquakes have very large values. This could imply a large process zone in those cases due to the stress concentrations associated with large ruptures.

Seismic energy is ,

if we assume that σ2 equals σf –e.g, the final stress is exactly the same as the dynamic frictional strength. (But what about dynamic overshoot, or healing pulse rupture models?)

Slip, uSeismic Efficiency

and η is generally found to be 5-10%.

Static vs. Dynamic stress dropDynamic overshoot would make Δσ > Δσd. Consider the role of inertia and lumped mass models.Slip pulse rupture models (e.g., Scholz, 1980; Heaton, 1990) would make Δσ < Δσd.

Energy Balance: Specific fracture energy GcFor a mode III crack, we can calculate a critical half length for crack extension

note that G is shear modulus, sometimes written µLc is of order 1-2 m for Gc of 102 J/m2.

What about the stress concentration implied by such a model? How is the singularity resolved? (process zone? Irwin? )

See: • Brittle Fracture Theory Predicts the Equation of Motion of Frictional Rupture Fronts, Ilya

Svetlizky,David S. Kammer, Elsa Bayart, Gil Cohen, and Jay Fineberg, PRL 118, 125501 (2017)• Classical shear cracks drive the onset of dry frictional motion, Svetlizky & Fineberg, Nature, 2014.

Slip weakening model (Ida, 1972, 1973)

Gc is the fracture energy

This is akin to the friction-based result, in which

where C is a geometric constant and ν is poisson�s ratio.

Recall the derivation of this result:

G is Shear modulus

Concept of strength excess S

S =�y � �1

�1 � �f

Recall the derivation of this result:

Frictional Instability

Requires K < Kc

Δσ = 16 π

Δσ = 24 π

Δumax

Relation between stress drop and slip for a circular dislocation (crack) with radius r

For ν =0.25, Chinnery (1969)

Homework:Determine the minimum earthquake size (magnitude and moment) assuming: G=µ =30 GPa, normal stress = 10 MPa, b-a = 0.01, and Dc = 100 µm. Show all work, discuss any assumptions and empirical relations used.

Gc is the fracture energy

In this model, rupture propagation is highly dependent on the strength parameter S

where numerator is the strength excess and the denominator is the stress drop

Shear Stressσo

Mode II crack propagation at speed Vr

fmax scales as: fmax = Vr/w

Cracked/Slipping zoneBreakdown (cohesive) zone

Intact, locked zone

cohesive zone/slip weakening crack model for friction

�� =7⇡

24Gumax

�� =7⇡

Dc =24

��

Cohesive zone length scales as:

Shear Stressσo

Mode II crack propagation

N Propagating Rupture

w ⇡ CDcG

��

Dc⇡ C

��

= C��r3

�� =7⇡

= C��V 3r

T 3Ide et al., 2007; Peng and Gomberg, 2010

Earthquake Source Parameters and Scaling Relationsthis was the original thinking.Recent work suggests there are problems with this. See Frank and Brodsky, 2019; Michel et al., 2019

Rupture Patch Size for Slow Earthquakes

η = 0.25

Slow earthquake nucleation when K

Kc⇡ 1.0

r =GDc

�n(b� a)h* = rc

Kc ⇡�n(b� a)

K =��

Unstable ifK < Kc

= C��r3

⇡ Vr

Slow slip when effective rupture patch size is limited by heterogeneity?r

patch = Gur2

Lab data show a continuous spectrum from fast to slow slip

Ide 2014

Gomberg et al., 2016

Source Parameters and Scaling Relations for Slow Earthquakes

Michel et al., Nature, 2019

Frank and Brodsky, 2019

Anderson�s Theory of Faulting:• Free Surface and Principal Stresses

Adhesive and Abrasive Wear: Fault gouge is wear material

Chester et al., 2005

where T is gouge zone thickness, κ is a wear coefficient, D is slip, and h is material hardness

This describes steady-state wear. But wear rate is generally higher during a �run-in� period.

And what happens when the gouge zone thickness exceeds the surface roughness?

We�ll come back to this when we talk about fault growth and evolution.

Fault Growth and Development Fault gouge is wear material

Fault offset, D

�run in� and steady-state wear rate

Fault offset, D

σ2 > σ1

Fault offset, D

, T �run in� and steady-state wear rate

And what happens when the gouge zone thickness exceeds the surface roughness?

Scholz, 1987

Fault offset, D

• Fault Growth and Development• Fault Roughness

Scholz, 1990

Fault Growth and Development

Scholz, 1990

Cox and Scholz, JSG, 1988

Fault Growth and Development

Tchalenko, GSA Bull., 1970

Fault Growth and Development Fault zone width

Scholz, 1990

Fault zone roughness

Ground (lab) surface

Thermo-mechanics of faulting II

• San Andreas fault strength, heat flow. • Consider: Wf = τ v ≥ q

• If τ ~ 100 MPa and v is ~ 30 mm/year, then q is:• 1e8 (N /m2) 3e-2 (m/3e7s) = 1e-2 (J/s m2 ) ~ 100 mW/m2.

• Problem of finding very low strength materials.

• Relates to the very broad question of the state of stress in the lithosphere? • Byerlee�s Law, Rangley experiments, Bore hole stress measurements, bore hole breakouts, earthquake focal mechanisms.

• Seismic stress drop vs. fault strength.

Fault Strength, State of Stress in the Lithosphere, and Earthquake Physics

• Thermo-mechanics of faulting…

• Fault strength, heat flow.

• Consider shear heating: Wf = τ v ≥ q• If τ ~ 100 MPa and v is ~ 30 mm/year, then q is:• 1e8 (N /m2) 3e-2 (m/3e7s) = 1e-1 (J/s m2 ) ≈ 100 mW/m2

Average(Shear(Stress

Average(slip(velocity

e.g.(Townend(&(Zoback,(2004@(Hickman(&(Zoback,(2004

• SAF

Data(from((Lachenbruch(and(Sass,(1980

Fault&Strength&and&State&of&Stress

• Heat(flow

• Stress(orientations(

Predicted Observed

σ1σ1

• Inferred(stress(directions

e.g.(Townend(&(Zoback,(2004@(Hickman(&(Zoback,(2004

• SAF

Data(from((Lachenbruch(and(Sass,(1980

Fault&Strength&and&State&of&Stress• Heat(flow

• Stress(orientations(

Have(been(used(to(imply(that(the(SAF(is(weak,((µ(≈(0.1.

• Is the San Andreas anomalously weak?

SAFOD The San Andreas Fault Observatory at Depth

SAF&2 Geology

Based(on(Zoback et(al.,(EOS,(2010

Frictional&Strength,&SAFOD&Phase&III&Core

Carpenter, Marone, and Saffer, NatureGeoscience, 2011

Carpenter, Saffer and Marone, Geology, 2012

Weak(Fault(in(a(Strong(Crust

Carpenter, Saffer, and Marone. Geology, 2012

Magnitude and Seismic Moment. Moment is a most robust measure of earthquake size because magnitude is a measure of size at only one frequency.

Mo = µ A u, where µ is shear modulus, A is fault Area and u is mean slip.

Relation to magnitude: Mw = 2/3 log Mo – 6 or Mo = 3/2 Mw +9 (for Mo in N-m)

Rupturearea, A

Slipcontours, u

Earthquakes represent failure on geologic faults. The rupture occurs on a pre-existing surface.

Faults are finite features –the Earth does not break in half every time there is an earthquake.

Earthquakes represent failure of a limited part of a fault. Most earthquakes within the crust are shallow

Definitions of Focus, Epicenter NOTE: Epicenter is also the Rancho Cucamonga Quakes� stadium –they are single-A team of the Anaheim (LA) Angles: http://www.rcquakes.com/ _____________________________________Earthquake Size (Source Properties)

Measures of earthquake size: Fault Area, Ground Shaking, Radiated Energy

Fault dimensions for some large earthquakes:L (km) W (km) U (m) Mw

Chile 1960 1000 100 >10 9.7Landers, CA 1992 70 15 5 7.3San Fran 1906 500 15 10 8.5Alaska 1964 750 180 ~12 9.3

Rupturearea, A

Slipcontours, u

Wave resulting from the interaction of P and S waves with the free surface.

Their wave motion is confined to and propagating along the surface of the body

Magnitude is a measure of earthquake size base on:• Ground shaking• Seismic wave amplitude at a given frequency

Rupturearea, A

Slipcontours, u

Magnitude is a measure of earthquake size base on:

• Ground shaking• Seismic wave

amplitude at a given frequency

Magnitude accounts for three key aspects:• Huge range of ground observed displacements --due to very

large range of earthquake sizes• Distance correction –to account for attenuation of elastic

disturbance during propagation• Site, station correction –small empirical correction to account

for local effects at source or receiver

ML = log10uT! "

# $ + q (Δ,h ) + a

Magnitude is a measure of earthquake size base on:• Ground shaking• Seismic wave amplitude at a given frequency

ML = log10uT! "

# $ + q (Δ,h ) + a

Systematic differences between Ms and Mb --due to use of different periods.

Source Spectra isn�t flat. Saturation occurs for large events, particularly saturation of Ms.

e.g: http://neic.usgs.gov/neis/nrg/bb_processing.html

ML (Richter --local-- Magnitude) & MS, based on 20-s surface wave

MB, Body-wave mag. Is based on 1-s wave p-waveMW, Moment mag. (see Hanks and Kanamori, JGR, 1979)

Magnitude and Seismic Moment. Moment is a most robust measure of earthquake size because magnitude is a measure of size at only one frequency.

Mo = µ A u, where µ is shear modulus, A is fault Area and u is mean slip.

Moment and Moment Magnitude (Hanks and Kanamori, JGR, 1979): Mw = 2/3 log Mo – 6 or Mo = 3/2 Mw +9 (for Mo in N-m)

Rupturearea, A

Slipcontours, u

Rupturearea, A

Slipcontours, u

Brune Stress drop

Mechanics of Earthquakes and Faulting · Background: How does this relate to dislocation mechanics...

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