UTRIP-2016 FINAL REPORT
Development of the Boundary
Integral Equation Code for Dynamic Earthquake Rupture Simulations
MIHIR MILIND KIRLOSKAR1,2
Supervisor: Prof. Ryosuke ANDO1 and Prof. Satoshi IDE1
1. Department of Earth and Planetary Science, School of Science, The
University Of Tokyo.
2. Department of Geology and Geophysics, Exploration Geophysics, Indian
Institute of Technology, Kharagpur.
Abstract:
In this report I discuss the effects of Instantaneous Frictional Strength Drop on the
propagation of seismic rupture on active faults. I developed my own Code to solve the
Boundary Integral Equation developed by " Cochard and Madariaga" for studying Rupture
Propagation along an Anti-Plane fault in the presence of Instantaneous Frictional Strength
Drop . I studied rupture dynamics of models with single and twin asperities. All fault
parameters are assumed to be homogeneous. Asperities are regions of higher pre-stress on
the fault plane.
1. Introduction:
In the past 15 years many hypotheses have been made to study the origin of earthquake
complexity. Two extreme viewpoints of this problem have been advanced. From the first
viewpoint, complexity is due to permanent geometrical features on the fault surface, which
controls the initiation, arrest and radiation from seismic faults. Another viewpoint proposed
by " Carlson and Langer" illustrates that fault complexity arises due to Dynamic Stress Zones
on the fault surface. Most of the crucial observations required to study the origin of
heterogeneity depends on the availability of geometrical information concerning the
rupture process. Until such data is available, most of the research is based on the careful
numerical simulation using rupture models that are as realistic as possible. This was the
main aim of my research during UTRIP-2016.
In my research I have assumed a homogeneous fault surface which is 300 units long. I have
taken the propagation of the wave for 300 units of time. All quantities in my research have
been normalised according to the normalisation given by " Cochard and Madariaga".
The fault surface has been discretized and every element is 10 units long. Using the CFL
condition, the shear wave velocity for my simulation is 0.5 units.
2. Instantaneous Frictional Strength Drop Model:
This friction model, is a simple friction model wherein the Absolute Stress is a step function
where the value of the absolute friction depends on whether the given element is slipping
or not at the given point of time. A given element will undergo slip at a certain point of time
if the element satisfies the below inequality:
−
0i
j,m
i-j, n-m
≥
thresi
(1)
T0i
: Initial Traction on the element " i " .
Tthres : Threshold Traction( Assumed to be constant throughout the fault surface)
If the element satisfies the above condition then the element" i " will start slipping from the
next instance of time. Till the time the element is stationery, it will experience static
friction which is assumed to be constant at 3.5 units. As soon as the element starts slipping,
the element experiences kinetic friction which is assumed to be constant at 1.0 units. As a
result, the absolute traction on the fault surface is a step function and the value of this
absolute traction depends only on whether the element is slipping or not. It does not vary
with the slip velocity of the element. Hence, such kind of friction is also called as RateIndependent Friction.
3. Solution to Boundary Integral Equation:
,
=
( , )= −
,
∑,
−
j,m
,
+ ∑
K(x, t ; xj, tm )
−
=
∑
j,m
i-j, n-m
(2)
(3)
I solved equation 2 and 3 by developing my own code. TThe
he first equation gives us the Slip
Rate of an element " i " at a moment " n ". The double
double convolution takes into account the
effect of slip rates of other elements on the element chosen at the current moment
moment.
Tabs_i,n is determined by the friction, which is given by tthe step function described section 2 .
The numerical results that I have obtained are the result of running my own code
code to solve
the equations 2 and 3.
4. Asperity Models:
In the single asperity model, I have assumed the asperity to be of 100 units in length.
Asperity is a region of pre-stress
stress and the pre-stress
stress on the rest of the fault surface is
assumed to be zero. As a result the rupture begins to propagate from the asperity region at
speeds comparable
parable to shear wave velocity
velocity. The pre-stress
stress at the asperity is almost
almo equal to
the threshold traction. In the twin asperity model the second asperity is 50 units in length.
5. Result:
5.1. Slip Rate for Rate
Rate-Independent Friction:
Rupture begins to grow at almost shear wave velocity. Rupture propagation stops before
the rupture front reaches the edge of the fault surface because of the low pre-stress
pre stress outside
the asperity. The crack spreads over the fault plane .Slip rate is highest inside the Asperity
because of finite stress drop in that region.
5.2. Stress Field on the Fault Surface:
The pre-stress
stress stored inside the asperity is released and stored outside the asperity as the
fault propagates. The stress field inside the asperity is very smooth. The main reason for the
propagation of the wave is the release
release of stress from the asperity. The stress accumulates
outside the asperity giving rise to future asperities. The above model has a small problem
and I am working towards ironing it out.
6. Conclusion:
In the single asperity model , slip velocity naturally decreases and the rate of this decrease
depends on the value of the pre-stress outside the asperity. The more this pre-stress is
removed far from the rupture threshold, the more rapidly does the slip-rate decreases.
Dynamic rupture in the asperity propagates at almost shear wave velocity. From the figure,
we conclude that higher slip-rates occur in the regions of higher pre-stress. Rupture is
terminated off the asperity due to the low pre-stress outside the asperity.
7. Acknowledgement:
The author thanks Junki Komori for offering all kind of help during his stay at the University
of Tokyo. In addition, the author is thankful to The International Liaison Office of School of
Science, UTokyo for the help in all the administrative work. Helpful comments and
suggestions by Prof. Ryosuke ANDO and Prof. Satoshi IDE during the study to debug my
program are highly appreciated. Discussion with the research group was very helpful and
improved my understanding to great extent. The UTRIP program is funded by the GSS-UTRIP
Scholarship.
8. Reference:
Alain Cochard and Raul Madariaga (1994), Dynamic Faulting under Rate-Dependent Friction,
Vol. 142, No. 3/4