Stress- and State-Dependence
of Earthquake Occurrence: Tutorial 1
Jim Dieterich
University of California, Riverside
Constitutive formulation for earthquake rates
Approach
The formulation is based on the premise that earthquake nucleation controls
the time and place of initiation of earthquakes. Hence, processes that alter
earthquake nucleation times control changes of seismicity rates.
For faults with rate- and state-dependent friction, the relationship between
nucleation times and stress changes is highly non-linear.
Tutorial 1 reviews some features of rate- and state-dependent friction and
earthquake nucleation that form the basis of the model.
Tutorial 2 reviews the derivation of the constitutive formulation and outlines
some applications of the model.
Experimental Conditions – Rate & State
• Wide range of rocks and rock forming minerals
– Bare surfaces and gouge layers
• Also glass, wood, paper, plastic, gelatin, metals,
ceramics, Silicon in MEMs devices
• Contact times <1s - 106s (indirect ~4x107s)
• V= mm/yr - cm/s (servo-controlled tests)
• V≥100m/s (shock impact)
• T=20°C - 350°C
• Nominal s =1 MPa - 300 Mpa, Contact stresses to
12GPa
• Dry, wet, hydrothermal
Time-dependent strengthening
Response to steps in slip speed
fast
slow
fast
slow
Constitutive law
.75
µ
10µm/s
10µm/s
1µm/s
1µm/s
100 µm
.70
10µm/s
10µm/s
1µm/s
1µm/s
Granite
µ
.65
.67
#60 surface, 1mm gouge
10 MPa normal stress
100 µm
1µm/s
1µm/s
0.1µm/s
0.1µm/s
µ
.62
.85
µ
Granite
#60 surface
15 MPa normal stress
.70
1µm/s
0.1µm/s
.70
.025
#60 surface
5 MPa normal stress
100 µm
1µm/s
0.1µm/s
1µm/s
10µm/s
1µm/s
µ
.020
.70
20µm/s
µ
Teflon on steel
100 µm
2µm/s
Acrylic plastic
#60 surface
2.5 MPa normal stress
100 µm
10µm/s
Soda-lime glass
2µm/s
.2µm/s
polished surface
30 MPa normal stress
Wood
.60
100 µm
roughened surface, #40
1 MPa normal stress
Displacement-weakening at onset of rapid slip
# 240 surface
2.
0
m
Hydraulic flatjacks &
Teflon bearings
# 30 surface
1.5 m
Thickness = 0.42 m
Fault slip, m
Rate- and state-dependent formulation
Coefficient of friction:
Slip speed
Friction, µ
V1
A ln(V1/V2)
V2
V1
B ln(V1/V2)
Dc
Displacement, 

V

   0  A ln *   Bln * 
V 
 
s
State variable:
  G(t, , s , Dc ),
For example:

d  dt 
d 
ds
Dc
Bs
At steady state, d/dt=0 and
Dc
 ss 
V
ss  const.  (A  B) ln V

V 
 

Bln

 *
V* 
 
   0  A ln
ss  const.(B A)ln ss
ss 
Dc
V
Coefficient of friction 



x V1
B
B-A
Log 
ss
V 
 

Bln

 *
V* 
 
   0  A ln
ss  const.(B A)ln ss
ss 
Dc
V
Coefficient of friction 

During slip  evolves toward ss


x V1
B
ss at V1

Log 
B-A
ss
V 
 

Bln

 *
V* 
 
   0  A ln
ss  const.(B A)ln ss
ss 
Dc
V
Coefficient of friction 

During slip  evolves toward ss


x V1
B
ss at V1

Log 
B-A
ss
V 
 

Bln

 *
V* 
 
   0  A ln
ss  const.(B A)ln ss
ss 
Dc
V
Coefficient of friction 

During slip  evolves toward ss


x V1
B
ss at V1

Log 
B-A
ss
c
Time dependent strengthening 
a
d
Coefficient of friction 
b
Slip
c
ss
d
a
b
Log 
V1

Velocity steps
b
V2
V1
a
a
c
Coefficient of friction 
Slip
d
V2
b
c
d
Log 
a
V1
ss
Spring-slider simulation with
rate- and state-dependent friction (blue curves)
11 microns/s
1.1 microns/s
0.8 microns/s
0.5 microns/s
Shear stress (bars)
200
180
160
200
180
160
Westerly granite, s=30 MPa
Slip (0.1 mm/division)
Imaging contacts during slip
Schematic magnified view of
contacting surfaces showing isolated
high-stress contacts. Viewed in
transmitted light, contacts appear as
bright spots against a dark
background.
Acrylic surfaces at 4MPa
applied normal stress
Contact stresses
Indentation yield stress, sy
Acrylic
400 MPa
Calcite
1,800 MPa
SL Glass 5,500 MPa
Quartz 12,000 MPa
Increase of contact area with time
Acrylic plastic
Dieterich & Kilgore, 1994, PAGEOPH
Time dependent friction & Contact area
Dieterich and Kilgore, PAGEOPH, 1994
Velocity step & Contact area

Dieterich and Kilgore, PAGEOPH, 1994
Dc
ss 
V
Interpretation of friction terms
Bowden and Tabor adhesion theory of friction
Contact area:
area = cs
c=1/ indentation yield stress
g=shear strength of contacts
Shear resistance:  = (area) (g), /s =  =cg
Time and rate dependence of contact strength terms
Indentation creep: c() = c1 + c2ln()
Shear of contacts: g(V) = g1 + g2ln(V)
 = c1 g1 + c1g2ln(V) + c2g1ln() + c2g2ln(V+)
 = 0 + A ln(V) + B ln()
(Drop the high-order term)
Contact evolution with displacement
SUMMARY – RATE AND STATE FRICTION
• Rate and state dependence is characteristic of diverse
materials under a very wide range of conditions
• Contact stresses = micro-indentation yield strength (500 MPa
– 12,000 MPa)
• State dependence represents growth of contact area caused
by indentation creep
• Other process appear to operate at low contact stresses
• Log dependence
thermally activated processes.
• Power law distribution of contact areas
• Dc correlates with contact diameter and arises from
displacement-dependent replacement of contacts
Critical stiffness and critical patch length for unstable slip
K
Kc 
Effective stiffness of slip
patch in an elastic medium
K
 G

d
l
lc 
s
Dc
G GDc

Kc
s
 crack geometry factor,  ~ 1
G shear modulus


Perturbation from steady-state sliding,
at constant s [Rice and Ruina, 1983]
  B A
Apparent stiffness of spontaneous
nucleation patch (2D) [Dieterich, 1992]
  0.4B

Large-scale biaxial test
Hydraulic flatjacks &
Teflon bearings
Minimum fault length
for unstable slip
2.
0
m
Lc =
Lc = .75 m
G  Dc
s
G = 15000 MPa
 = 0.5
D c = 2 µm
s = 5 MPa
 = .4B = 0.004
Strain gage
1.5 m
Thickness = 0.42 m
Displacement
transducer
Confined Unstable Slip
Confined stick-slip in biaxial apparatus satisfies the
relation for minimum dimension for unstable slip
Earthquake nucleation on uniform fault
Earthquake nucleation on a uniform fault
Dieterich, 1992, Tectonophysics
Earthquake nucleation – heterogeneous normal stress
4
i
3
Log ( Slip speed / D c )
2
Lc =
1
2L
Lcc
G  Dc
s
0
-1
-2
-3
-4
-5
-6
-7
0
Dieterich, 1992, Tectonophysics
300
Position ( 1 division = 200,000 Dc )
SPRING-SLIDER MODEL FOR NUCLEATION
s
K
 (t)  K
  0  A ln Ý Bln 
s
 (t)

Evolution at constant normal stress
d 1 
 Ý
d  Dc
During nucleation, slip speed accelerates and greatly exceeds steady
state slip speed
Coefficient of friction 


x
B
ss at V1
1
d

 Ý 

,   0e / Dc
Dc

d
Dc
V
1
B-A


Log 
ss
 (t)  K
B
  0  A ln Ý Bln 0  
s
Dc
SPRING SLIDER MODEL FOR NUCLEATION
 (t)  K
B
  0  A ln Ý Bln 0  
s
Dc
Re-arrange by solving for Ý


 Ýt 
exp
 dt 
0
As 
t

 H 
exp
 d
0
 A 

Where:

  0 /s  0 
B / A
Ý
0  0 exp

A


H 
Initial condition

K
s

B
Dc
Model parameters

SPRING SLIDER MODEL FOR NUCLEATION
 Ýt  
A  Ý0 Hs 

ln 
1 exp
 1 , Ý 0

H  Ý 
 As  

Slip
A  Ý0 Ht  Ý
ln 1
 ,  0
H 
A 
 1 Hs   Ýt  Hs 
Ý
   Ý 
exp
 , Ý 0
 


Ý   As  Ý 
0
-1

Slip speed
 1 Ht 
Ý
   Ý   , Ý 0
 0 A 
-1

ti 

As  Ý
ln
1
 , Ý 0
Ý  HsÝ0 
Time to instability
( 1/Ý0 )
A 1  Ý
ti   Ý  ,   0
H  0 
Dieterich,Tectonophysics (1992)

Solutions for time to instability
max
2D numerical
model
Fault patch
solution
103
Slip speed (DC /s)
102
101
100
10-1
10-2
 /s=0
s
10-3
10-4
10-5
10-6
10-7

As  Ý
ti 
ln
1
Ý
Ý
  Hs0 

Ý/ s 102 / a
Ý/ s 103 / a
Ý/ s 104 / a
10-8
10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107 108 109 1010


Time to instability (s)

Dieterich, 1992,Tectonophysics
Accelerating slip prior to instability
Time to instability - Experiment and theory
Effect of stress change on nucleation time
6
Log (slip speed) m/s
4
2
0
-2
-4
-6
1 yr
-8
-10
-12
-14

As  Ý
ti 
ln
1
Ý
Ý
  Hs0 
10 yr
20 yr
Ý 0.05 MPa/yr
-16
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
 10 10 10 10 10 10 10 10 10 10 10 10 10 10
Time to instability (seconds)

Effect of stress change on nucleation time
 = 0.5 MPa
 /A
s 
 
0 
Ý
Ý
  0  exp 


s
A
s
A
s
 0

0
6
Log (slip speed) m/s
4
2
0
-2
-4
-6
5min
~1hr
~5hr
1 yr
-8
-10
-12
-14

As  Ý
ti 
ln
1
Ý
Ý
  Hs0 
10 yr
20 yr
Ý 0.05 MPa/yr
-16
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
 10 10 10 10 10 10 10 10 10 10 10 10 10 10
Time to instability (seconds)

Model for earthquake occurrence
Use the solution for time to nucleation an earthquake
(1)

As  Ý
t
ln
1
Ý
Ý
  Hs 0 
,
where
Ý const  0
sÝ 0
H
B K

DC s
and assume steady-state seismicity rate r at the stressing rate Ýr
t  n r,
n is the sequence number of the earthquake source
This defines the distribution of initial conditions
(slip speeds) for the nucleation sources
(2)
Ý0 (n) 
1
Hs   Ýr n  
exp
 1
Ý
 r   As r  
Log (slip speed)


Log (time to instability)
The distribution of slip speeds (2) can be updated at successive time steps

for any stressing history, using solutions for change of slip speed as a
function of time and stress.
Evolution of distribution of slip speeds
For example changes of Ý0 (n) with time are given by the nucleation solutions
 1 Ht 
Ý
   Ý   , Ý 0
0 A 
-1
 1 Hs   Ýt  Hs 
Ý
   Ý 
 , Ý 0
 
exp
Ý
Ý
   As   
0
-1
Ý
and change of 0 (n) with stress are given directly from the rate- and stateformulation

 /A
s 
Ý
Ý
   0 s 
 0

 
0 
exp


A
s
A
s

0 
In all cases, the final distribution has the form of the original distribution

where

Ý0 (n) 
1
Hs   Ýr n  
exp
 1
   As r  

 
1 
d 
dt  d       ds 
s
 
As 
Evolution of distribution of slip speeds
Earthquake rate is found by taking the derivative dn/dt = R
For any stressing history
R


r
Ýr

 
1 
d 
dt  d       ds 
s
 
As 
Coulomb stress formulation for earthquake rates
Earthquake rate
R
Coulomb stress
r
,
 Ýr
d 
1
As


 ds 
dt


d





s
 
dS  d   ds
Assume small stress changes (treat as constants)

Note:   and 0     .
s
Earthquake rate

R






s

, (As )


    0.3 0.4
s


Hence,  eff  
r
,
Ý
 Sr

d 
1
dt   dS
As
Dieterich, Cayol, Okubo, Nature, (2000), Dieterich and others, US Geological Survey Professional Paper - 1676 (2003)