Kinematic Rupture

ERI Notes: RJA125

Conditions for Supershear Rupture“Plot of (inverse of) supershear transition
length vs. seismic S ratio (small S means highinitial stress, fault close to failure). There is niceagreement between the analytical predictionof transition length in the self-similar Dugdale
model with numerical results fromspontaneous rupture calculations in 2D.
Numerical results for unbounded faults in 3D
are also shown. The trend is similar in 3D, but
the transition length is about twice as long as
in 2D for the same value of S. In 3D, asufficiently narrow fault width suppresses thetransition; the critical width is approximately
0.8 times the transition length on anunbounded fault.”

ERI Notes: RJA

Dunham, JGR, 2007

126

Kinematic Model (Haskell, 1964)• Fault
Geometry:
Length, Width• Slip• Rise Time• Rupture
Velocity

ERI Notes: RJA

127

Representation Theorem∞

un ( x,t ) = ∫ dτ ∫∫ [ ui (ξ , τ ] cijpqν j ∂Gnp (x,t − τ ; ξ , 0) / ∂ξq dΣ−∞

ΣAki and Richards, (3.2)

Convolve slip rate function with Green’s functions of elastic medium
and sum over whole fault to get ground motion at free surface.
ERI Notes: RJA

128

Kinematic Parameters
Slip rate time functiontime

Vr=D/TrERI Notes: RJA

129

DistributionsforParameters(Cario and Nelson, 1997)
Slip
CauchyRupture Vel.
Uniform
Rise time
Beta
ERI Notes: RJA

130

Correlated Source Parameter
Slip

Spatial correlation 30%(Not true anymore.)
Average rupture
velocity

Spatial correlation 60%
Risetime
ERI Notes: RJA

Liu, Archuleta, Hartzell, BSSA, 2006

131

Flowchart for Generating
Broadband Strong Motion Synthetics

ERI Notes: RJA

Liu, Archuleta, Hartzell, BSSA, 2006132

Correlations(x i − x )(y i − y )correlation = ∑
i=1 (NM −1)std(x)std(y)1= N *M

NM

1 NM
x=xi
∑
NM i=1∑ (x − x )2istd(x) =i=1

NM −1

For each of N ruptures and each parameter pair compute the point topoint correlation. This yields N point to point correlations for eachparameter pair.
ERI Notes: RJA

133

Correlations

The correlation coefficient was computed for the regions that rupture atsubshear speed of >300 dynamic rupture models

ERI Notes: RJA

134

Correlation Matrix

ERI Notes: RJA

135

New Kinematic Model (SAL)

Schmedes, J., R. J. Archuleta, and D. Lavallée(2010), Correlation of earthquake source
parameters inferred from dynamic rupture
simulations, J. Geophys. Res., 115, B03304, doi:
10.1029/2009JB006689.

ERI Notes: RJA

136

Rupture Dynamics

K.B. Olsen, S.M. Day, L.A. Dalguer, J. Mayhew, Y. Cui, J. Zhu, V.M. Cruz-Atienza, D. Roten, P.
Maechling, T.H. Jordan, D. Okaya, and A. Chourasia,, GRL 2008
ERI Notes: RJA

137

Rupture Dynamics• Stresses: yield, initial, sliding friction and insome cases, the absolute value of friction.
• Constitutive law for friction• Fault geometry

ERI Notes: RJA

138

Anelastic Behavior at the Crack Tip
to Eliminate the Stress Singularity
Stress∞σyσ1σ1σoσf
X

Crackdc

Dislocation

ERI Notes: RJAIda, 1972

139

Shear Stress at a Point on the Fault

Stress at a point on the fault
Yield StressσyInitial Stressσy-σoσoσo-σ1σfσo-σf
Sliding Friction Stressσ1Final StressTime

S=σy-σo / σo-σf =Strength Excess/Stress Drop
ERI Notes: RJA

140

Dieterich’s Experiments

ERI Notes: RJA

141

Friction
Region where the friction
rises and the slip rate rapidly
goes to zero, locking thefault.

Friction drops rapidly
from upper limit tosliding friction over a
characteristic distanceDo.

Region where the sliding is at a nearly
constant value of friction.

ERI Notes: RJA

Madariaga, Olsen, Archuleta, 1998

142

The FV law requires more total slip to fully weaken toits minimum shear stress. Parameters were chosen forthe laws to have equal areas under the slip weakeningX - 11curve and thus have the same energy lost to frictional
dissipation. Parameters are given in Table 1.

Friction Laws

DAUB AND CARLSON: A CONSTITUTIVE MODEL FOR FAULT GOUGE DEFORMATION

shear stress ττp

Table 1. Parameters used for dynamic rupture simulations.

τddc
slip ushear stress

Figure 1. Prescribed shear stress evolution for the linear
SW law. Slip initiates when the shear stress reaches thepeak strength τp . Over a slip distance dc , stress weakens
linearly to a constant dynamic sliding friction τd . Because the sliding friction τd is constant, no dynamic recovery of strength occurs for the SW law.

V = Vss + ∆VV = VssAA− BL
shear displacement
Figure 2. Shear stress versus slip for a velocity stepexperiment. The slip velocity is perturbed from steadysliding at Vss to Vss + ∆V , and the transient responseof the system is shown. The instantaneous increase infriction (A), the length scale for evolution (L), and thenew steady-state (A − B) are shown in the plot.

ERI Notes: RJA

Shared model parameters are: grid spacing dl, number of spatial grid points nx, shear modulus µ, shear wave speed cs ,Poisson’s ratio ν, Courant-Friedrichs-Lewy Ratio cfl = cs dt/dl(determines the time step dt), length of the fault permitted torupture Lfault , size of the triggering patch Ltrigg , initial shear
load τload , and stress in the triggering patch τtrigg . The frictional parameters are discussed with each model in the maintext.

Dynamic Rupture Simulation Parameters
Shared Parameters
SW Law
dl = 0.01 km
dc = 0.31 m
nx = 2048
τp = 73.85 MPaµ = 32.03812032 GPaτd = 63.375 MPa
cs = 3.464 km/s
ν = 0.25
cfl = 0.3
Lfault = 16 km
Ltrigg = 1.3 km
τload = 68 MPa
τtrigg = 74 MPa
FV Law
DR Law
χs = 0.3407
LDR = 21.5 mm
σd = 0.25 MPa
a = 0.0027
χc = 0.1145
b = 0.0077
f∗ = 222.3
f0 = 0.7
V∗ = 10−6 m/s
V0 = 10−6 m/s
Rc = 5 s−1
σ = 100 MPa
α = 0.0005 (MPa m)−1 V1 = 10−6 m/sχh = 0.0028
τ0 = 44.12 MPa
χ(t = 0) = 0.0061
θ(t = 0) = 21500 s

A Constitutive Model for Fault Gouge Deformation in Dynamic
Rupture Simulations, Eric G. Daub and Jean M. Carlson, JGR, 2008143

Slip-Weakening Curves
FV and DR laws do not explicitly depend on slip – how does
shear strength weaken with slip for these laws?

Slope of curves areinversely proportionalto L.
FV law is different fromother laws because L isa function of slip
velocity.
Parameters chosen tomatch area under thecurves (frictional work).

ERI Notes: RJA

Daub and Carlson, JGR

144

Time Histories – Slip Velocity
FV Law

SW Law

ERI Notes: RJA

DR Law

Rupture arrives earliest
for FV law, then DR, andSW law is last. Slip rate issmaller for the FV law, DRand SW are identical.

Daub and Carlson JGR145

Time Histories – Shear Traction
FV Law

SW Law

ERI Notes: RJA

DR Law

Traction evolution is very
similar for all laws. FV law
takes slightly longer todrop from peak stress tominimum stress. Stress oncrack interior is fairly
constant for DR and FVlaws.
Daub and Carlson JGR

146

Slip Weakening

Shear StressSlip Weakening Constitutive Friction, Ida (1972)

σyσoσfdc

DSlip

Area of Triangle = Fracture
EnergyArea of Triangle = Available Seismic Energy

ERI Notes: RJA

Iida, 1972147

Constraint on Slip Weakening DistanceE = 1/2 (σo - σf) a1 : where a1 is the final slip
Gc = 1/2 (σy - σf) dc: where dc is the slip weakening distanceEs = E - Gc = 1/2 (σo - σf) a1 - 1/2 (σy - σf) dc
S = (σy - σo) / (σo - σf)1+S = (σy - σf) / (σo - σf):Andrews (1976; Das and Aki, 1977)
or(σy - σf) = (1+S) (σo - σf)

Es = 1/2 (σo - σf) (a1 - [1+S] dc )For Vrup < Vs (the general case, but not always), S > 1.2
Therefore one would expect that a1 ≥ 2.2 dc for any seismic
radiation to exist

ERI Notes: RJA

Archuleta, 2001

148

Consider 2 examples of dc: Gc = 2 x 106 J/m2dc = 0.01 mdc = 1.0 m

1/2 (σy - σf) dc = 2 x 106 J/m2

1/2 (σy - σf) dc = 2 x 106 J/m2

(σy - σf) = 4 x 108 J/m3

(σy - σf) = 4 x 106 J/m386(σy - σf) = 4 x 10 Pa(σy - σf) = 4 x 10 Pa(σy - σf) = 4 x 102 MPa(σy - σf) = 4 MPa(σy - σf) = 4000 bars

(σy - σf) = 40 bars

σy = σf + 4000 bars (Not Possible)

σy = σf + 40 bars

Range of Gc = 102 - 108 J/m2
(Landers 4.8 x 106 J/m2 :Olsen et al, 1997;
Kobe 1.5 x 106 J/m2 , Ide and Takeo, 1997)

ERI Notes: RJA149

Slip Weakening Curves

ERI Notes: RJA

Guattieri and Spudich, 2000 150

Shear Stress at a Point on the Fault

Stress at a point on the fault
Yield StressσyInitial Stress

σy-σoσoσo-σ1σfσo-σf
Sliding Friction Stressσ1Final StressTime

S=σy-σo / σo-σf =Strength Excess/Stress Drop
ERI Notes: RJA151

Strength
constant S

constant strength

T p − T0
S=Andrews (1976); Das and Aki (1977)
T0 − Td•A: Use a constant S-factor to compute distribution of strength onfault. Subshear conditions are for S>1.19 (Dunham, 2007). Strengthdrops smaller than 6 MPa are not permitted to avoid triggering. In
other words we want causal rupture times.
•B: Use constant strength. No strength excess smaller than 2.5 MPa.
Note that there are local regions that have S<1.19
ERI Notes: RJA

152

Different Autocorrelation for Stress Drop
Initial stress spatial variation with power spectral decays of 1, 2 and 3.
The amplitude distribution is identical for each.

ERI Notes: RJA

Lavallée et al., GJI, 2006

153

Example: Power-law Initial Shear StressCauchy distributed white noise(Lavallée et al, 2003)(Andrews, 1980)

ERI Notes: RJA

154

Initial Shear Stress Based on Slip Maps
Slip map created using Liu et al(2006)

Static stress drop computedusing Andrews (1980)

ERI Notes: RJA

Initial shear stressfield aftersmoothing andrescaling theamplitudes

155

Example: Power-Law Initial Shear StressCauchy distributed white noise(Lavallée et al, 2003)(Andrews, 1980)

ERI Notes: RJA

156

Relation between Rupture Velocity andFracture Energy
2Andrews (1976):v
222 2(1− 2 ) = 32 G / (πµ A )βA = 2bΔτ 2L / µv
G µ(1 − 2 ) = 16 2 4 4 2βπ b Δτ L222µShear modulus

βShear wave velocity

ΔτStress dropGFracture energyv

Rupture Velocity

2LCrack lengthb(0) = 1 ≥ b(v) ≥ b(β ) = 2 / π
ERI Notes: RJA

157

Why We Expect a Correlation Between
Slip and Rupture Velocity
Going back to:

1+ S =

τy −τ fΔτ
DΔτ = C µLv
G µ(1 − 2 ) = 16 2 4 4 2βπ b Δτ L222, G = dc (τ y − τ f ) / 2 (assuming linear slip weakening),

D2SlipCconstant2c2 42vd (1+ S)= 1− 4222βπ bC DHence, larger slip corresponds to faster rupture velocity
Positive correlation between slip and rupture velocity, but Schmedes etal. JGR, 2010 find that is not the case.
ERI Notes: RJA

158

Domo Arigato
Goziamasu

Ralph Archuleta

ERI Notes: RJA

159

Interaction Between Dynamic Rupture and Off-fault Brittle Damage
Shiqing Xu1, Jean-Paul Ampuero2, Yehuda Ben-Zion1 and Vladimir Lyakhovsky3
1: USC, 2: CIT, 3: Geol. Surv. Israel

Simulated shear modulus

Simulated plastic strainstress-strain relation

Main references on the damage model

elastic moduli - damage variable

Lyakhovsky, Ben-Zion, Agnon (JGR, 1997)

evolution of damage variabledamage-related plasticity

ERI Notes: RJA

Ben-Zion and Lyakhovsky (GJI, 2006)
Lyakhovsky and Ben-Zion (GJI, 2008)160

Fault Damage: Effect on Vmax
Y=14˚

Y=56˚

Friction: Slip Weakening

7!

V (maximum slip rate)!

6

Ampuero, Ben-Zion,
Lyakhovski,SSA 2008

54
3
210ERI Notes: RJA

05

10
15
20
X (distance along strike)25161