(COMPUTER) Write a computer program to simulate the block-slider model of
earthquake occurrence (Fig. 10.4). Consider a model with n blocks. If the displacement of the ith block is xi and the displacement of the driving block is d, then we
may use Hooke’s law to express the force on the ith block as
where k is the spring constant (in this case assumed constant for all springs in
the model). The static and dynamic friction coefficients for the ith block may be
expressed as µS
i and µD
i , respectively. At each time step, the ith block will move
only if the force exceeds the resistance provided by the static friction, that is, if
where we have assumed a unit mass for all blocks. The moving block will assume
a new equilibrium position defined by the dynamic coefficient of friction
which follows from (10.5), substituting µD for µS.
(a) Apply your program to the case where n = 50, k = 1, µS is randomly
distributed between 0.05 and 0.55 (a different value for each block), µD is
set to 1/3 of the value of µS at each block, and the driving block moves
a distance of 0.001 at each time step. Keep things simple by assuming that
blocks move instantly to new equilibrium positions before they trigger any
adjacent blocks. If a block moves during a time step, continue computing until
no blocks are moving. Output each “quake’’and plot the results for 1000 time
steps. Figure 10.9 shows an example of how you may wish to display your
results.
(b) One characteristic of chaos is the extreme sensitivity of a system to small
changes in initial conditions. Test your program by making a small change to
one of the x values after 500 generations. How much does the output differ
after 1000 generations? After 5000 generations? Is the system more sensitive
to changes to high-friction blocks or to changes to low-friction blocks?
(c) Compute a measure of the seismic moment, M0, for each event as the sum
of the slip on all of the blocks. Define event magnitudes as
where the constant d is chosen so that the magnitudes have reasonable values.
Make a histogram of the number of events as a function of magnitude. Can
you estimate a b-value for your sequence? How could the model be changed
to obtain a larger spread in event sizes?
(d) One unrealistic aspect of this type of model is the absence of aftershock
sequences following large events. Suggest some possible modifications to
the model that might result in aftershock behavior.
(e) Hint: Here is the key part of a FORTRAN program to solve this problem:
Note that “wraparound’’ boundary conditions are imposed at the endpoints.