1. Presented by Chris Rollins and Voonhui Lai Ge 277 Winter 2016
Statistical physics of fracture, friction, and earthquakes Hikaru Kawamura et al. Reviews of Modern Physics, 2012
Presented by Chris Rollins and Voonhui Lai
Ge 277 Winter 2016

2. Motivation
Are the power law behaviors observed in earthquakes scale-invariant ?
How to reliably forecast large events from smaller ones based on statistical properties such as:
(i) magnitude-frequency distribution and
(ii) time evolution of earthquakes ?
What is the appropriate model to simulate earthquake faulting ?

3. Burridge-Knopoff (BK) model #1
A spring-block model used to simulate stick-slip motions along fault
Equation of motion of 2D BK model (dimensionless):
v = Loading rate; l = Stiffness parameter; = Friction force
Magnitude of event give by:

4. Burridge-Knopoff (BK) model #2
Only source of nonlinearity: Friction force
Velocity weakening friction force
decreasing function of
common form:
[Carlson and Langer]
- rate of weakening of friction force
Intrinsically discrete
Other type of friction forces:
rate and state friction law (RSF)

5. Magnitude-Frequency Relation #1
2D BK model with short range interaction
First-order transition from subcritical to supercritical
Smooth transition from supercritical to near-critical to subcritical
Fig 12
Fig 13

6. Magnitude-Frequency Relation #2
2D BK model with long range interaction
Observe similar transition from subcritical system to characteristics behavior
Fig 15
Fig 16

7. Continuum Limit of BK model
In previous model, dynamics becomes increasingly sensitive with decreasing grid spacing
To counter, introduce viscosity term in equation of motion so that magnitude distribution becomes independent of grid spacing
Subcritical
Supercritical
Fig 18

8. BK model with RSF Law #1 Friction force
Condition on frictional instability depends on: b (healing term) and l (stiffness parameter)
Strong
Weak
Aging law:

9. BK model with RSF Law #2
Characteristic behavior in weak frictional instability regime; Flat spectrum (or critical) in strong regime. 1D 2D
Strong
Strong
Weak
Weak
Fig 20
Fig 21

10. Comparison between two models
Observed similar transition of behavior (e.g. appearance of large events) which depends on friction parameter
Heterogeneity in medium essential for criticality.
2D BK Model
Amitrano, EPJ 2012

11. Take-aways
BK model is comparable to other discrete and continuous models.
There exists a transition from self-criticality to characteristic behavior, primarily dictated by friction.
“Considering fault obeying RSF, the continuum limit of system lies in limit of weak frictional instability... earthquake should exhibit characteristic properties rather than critical properties.”

12. Supplementary Figure 1
Characteristic behavior not due to finite-size effect

13. Supplementary Figure 2
Transition in view of mean displacement