In this presentation i have given basic fault study and the important rate and state friction law. The presentation walks through the elastic rebound theory also. Thus the basic idea of why earthquake occurs, how this rate and state law came into the picture is crystal clear.

1. Study on fault basics and
Rate and state friction law
Presented By:-
RAVAL CHAITANYA
IU1751210010
Guided By:-
Assistant Professor Vasudeo Chaudhari

2. Objective
 Understanding rate and state friction law of motion to get dynamic analysis of particular
earthquake
2

3. Introduction
 Fault is a fracture/zone of fracture between two blocks of rock
 In a simple way, fault is the movement of rock on either side
 Faults are the boundaries of huge plates that make up earth’s crust
 If the rock movement is sudden, it causes earthquake
 If the rock movement is slow, it is called creep, an aseismic process
 Faults can be horizontal, vertical or oblique
 Angle of fault with respect to surface (dip) and direction of slip along fault are defined to
classify faults
3

4. Types of fault
 Faults which move along the direction of dip plane are dip-slip faults
(in vertical direction)
 If hanging wall moves downward, it is called ‘Normal fault slip’
 If hanging wall moves upward, it is called ‘Reverse (Thrust) fault slip’
 Faults which move horizontally are called strike-slip fault
 If the far block on either side moves in left, it is called ‘left lateral strike-slip
 If the far block on either side moves in right, it is called ‘right lateral strike-slip
 Sometimes fault doesn’t come up to the surface, it is known as ‘Blinded fault’
4

5. Dip-slip fault 5
Figure 1: Normal fault slip
Figure 2: Reverse( thrust ) fault slip

6. Strike-slip fault 6
Figure 4: Right lateral strike-slipFigure 3: Left lateral strike-slip

7. What is an earthquake?
 Earthquake is the energy released due to sudden movement of rocks in form of seismic
waves
 Causes of earthquake : 1 Surface causes
2 Volcanic causes
3 Tectonic causes
7

8. Elastic rebound theory
 Elastic rebound theory represents the occurrence of tectonic earthquake
 Considering an earthquake has just occurred in a block, which causes co-seismic
deformation
 After certain time many after-shocks will cause post-seismic deformation
 Considering time t = 0, when all energy is dissipated
 Slow accumulation of stress between the blocks will start from t = 0
 Pre-seismic deformation may occur between blocks in few centimeters for many years
8

9. Conti… 9
Figure 5: Elastic Rebound

10. Conti…
 The stress accumulated in the blocks are not uniformly distributed, it is localized stress at
different portion
 Localized stresses are concentrated in a narrow band in width around the fault
 Once the deformation around the fault is such that the stresses are greater than the frictional
resistance of blocks, earthquake is triggered
 At t = 𝑡0, sudden release of accumulated slip on fault occurs, which causes slip on the
fault, known as elastic rebound
 Earthquake is a cyclic process, not necessarily periodic
10

11. Nucleation procedure
 Fault rupture causes an earthquake
 The point at which the earthquake is triggered is known as ‘Nucleation’, rupture than
propagate in all direction from the nucleation point
 Fault rupture may have only one or many ruptures on the fault plane
 Many times earthquake triggered at one point causes perturbation on nearer fault, which
leads the other fault earthquake having an earthquake, called induced vibration
11

12. Asperity, slip, recurrence time, slip-rate
 Whenever two plates collide, there is chance that one plate sub-duct the other one
 At one point, the sub-ducting plate and the upper plate get locked as they both move
towards each other
 The locked area is known as ‘Asperity’ of fault
 Whenever earthquake occurs, the relative displacement occurs between two plates, it is
known as ‘Slip’
 The duration between two major events is called recurrence time interval
 Suppose major event of earthquake occurs every 1000 years and average slip per major
event is 3 meters, then the average creep of 3000/1000 = 3 mm/year occurs, called ‘slip
rate’
12

13. Stick-slip motion
 Any variation in frictional resistance during sliding causes dynamic instability, a sudden
movement occurs due to stress drop
 In repetitions, the sliding occurs when stress is recharged, followed by another instability
 Essentially, in such systems where sliding occurs due to instability, such frictional behavior
is called ‘regular stick-slip’.
13

14. Conti… 14
Figure 11: Force-displacement diagram
Figure 10: Block slider model

15. Conti…
 Condition for instability, │
𝑑𝐹
𝑑𝑢
│ > K
 Coulomb proposed that the static friction coefficient must exceed the dynamic coefficient
of friction for the unstable sliding
 Robanowicz said if two surfaces are kept stationary, given the load for time t, the static
friction coefficient increases with log t
 Robanowicz proposed Critical distance Dc, when friction value is changed from one value
to other, the instability is given by,
µ 𝑠 − µ 𝑑
𝐷 𝑐
> K
15

16. Rate and state friction
 Time and velocity both are important aspects for friction
 An increase of ∆µ 𝑠 upon the initiation of sliding, followed by previous decay of steady-
state value, the effect is called ‘healing’
 As Deiterich-Ruina stated that friction depends on instantaneous sliding velocity V and
time-dependent variable θ
 µ = µ (V, θ) = µ0 + a* ln
𝑉
𝑉0
+ b* ln
𝑉0∗𝜃
𝐷 𝑐
where, 𝜃 = 1 -
𝑉∗𝜃
𝐷 𝑐
16

17. Conti… 17
Figure 12(A): Static-dynamic model Figure 12(B): Slip weakening Figure 12(C): Deiterich-Ruina friction
model

18. Conti…
 (A) for shear stress, τ = µ*σ, where µ is coefficient of friction and σ is normal stress
 (B) shear stress is the decreasing function of slip un to 𝐷𝑐 and then remain at constant
stress, most common way is ‘piece-wise linear’
 Alternative law is Stribeck-curve, depending on a single quantity yet pure velocity
dependence fails to give the stress drop at particular point
 Aging law for healing and slip law for rapid change in slip are combined for proper result
18

19. Conti…
 The friction at steady state velocity,
µ 𝑠𝑠 = µ0 + (a-b) * ln
𝑉
𝑉0
if µ 𝑑 is defined as µ 𝑠𝑠 at velocity V, then
𝑑µ 𝑑
𝑑 ln 𝑉
= a-b
In the static case when θ = t, so for long hold times
𝑑µ 𝑠
𝑑(ln 𝑡)
= b
the friction jump of the direct effect is given by
∆µ = a* ln
𝑉2
𝑉1
19

20. A and B parameters
Seismogenic layer
If a > b, rate strengthening
If a < b, rate weakening
20
Figure 13: Parameters A and B effect in generation of
earthquake

21. References
 STICK-SLIP INSTABILITIES AND SHEAR STRAIN LOCALIZATION IN AMORPHOUS MATERIALS
EG Daub, JM Carlson - Physical Review E, 2009
 AFTERSLIP AND AFTERSHOCKS IN THE RATE‐AND‐STATE FRICTION LAW
A Helmstetter, BE Shaw - Journal of Geophysical Research: Solid Earth, 2009
 SOURCE SCALING PROPERTIES FROM FINITE-FAULT-RUPTURE MODELS
PM Mai, GC Beroza - Bulletin of the Seismological Society of America, 2000
 THE DYNAMIC ANALYSIS OF STICK-SLIP MOTION
C Gao, D Kuhlmann-Wilsdorf, DD Makel - Wear, 1994
 [HTML] TOWARDS A UNIVERSAL RULE ON THE RECURRENCE INTERVAL SCALING OF REPEATING
EARTHQUAKES?
KH Chen, RM Nadeau, RJ Rau - Geophysical Research Letters, 2007
 [PDF] SEMINAR ON B-VALUE
O Kulhanek - Dept. of Geophysics, Charles University, Prague, 2005
21

22. References
 NONLINEAR STRAIN BUILDUP AND THE EARTHQUAKE CYCLE ON THE SAN ANDREAS
FAULT
W Thatcher - Journal of Geophysical Research: Solid Earth, 1983
 THEORETICAL AND QUANTITATIVE ANALYSES OF THE FAULT SLIP RATE UNCERTAINTIES
FROM SINGLE EVENT AND EROSION OF THE ACCUMULATED OFFSET
Z Ren, Z Zhang, T Chen, W Wang - Island Arc, 2013
 ON THE VARIATION OF B-VALUES WITH EARTHQUAKE SIZE
EA Okal, BA Romanowicz - Physics of the Earth and Planetary Interiors,
1994
 SOURCE MECHANISMS OF EARTHQUAKES: THEORY AND PRACTICE
A Udías, R Madariaga, E Buforn - 2014
 “THE MECHANICS OF EARTHQUAKES AND FAULTING-” BY CHRISTOPHER H. SCHOLZ
RA Harris - Seismological Research Letters, 2003
22

23. Thank you