Results for 1D BK model

In the 1D BK model there are three dimensionless parameters \ell, \alpha, and \nu: \ell determines the stiffness of the interactions from the loading plate and neighbor blocks, \alpha characterizes the velocity-weakening dynamical frictional force, and \nu is the loading velocity. The default values we used for these three parameters are 10, 2.5, and 0.01 respectively. We first use the same frictional form as Carlson and Langer in (1989 PRA) which allows the block to move in both directions. Later we used a more realistic frictional form allowing a drop (\sigma) from the maximum static frictional force to the dynamical one and moving only in one direction at the zero-loading velocity limit as they did (1991 PRA).

Results for the 1D BK model with nonzero loading velocity

• Magnitude distribution for N = 100 at R = 1, 10, 20 (see plot). Not much can be said because the system is too small (N = 100). The moment is defined the sum of the displacements of all the blocks in an event. The magnitude is the logarithm of the moment. An event is followed over time until all the blocks stop moving.
• Magnitude distribution for N = 1000 at R = 1, 5, 10 (see plot). There is only one scaling range for R = 1 with exponent b = 0.9. while there are two scaling ranges for R = 5, and 10 with exponents b1 = 1.65 (smaller s) and b2 = 0.9.
• Cluster size distributions (without taking the lifetime into account) for different N and R are shown below. In this case, a snapshot of the system is taken and all moving blocks within an interaction range are in the same cluster.

  N	R	scaling range 1	scaling range 2	scaling range 3	plot
  1000	1	0.4 ~ 0.7, b = 2.52	1.0 ~ 1.75, b = 2.66	none	plot
  1000	10	0.0 ~ 1.0, b = 1.46	1.0 ~ 1.65, b = 1.92	1.65 ~ 2.5, b = 2.34
  1000	20	0.0 ~ 1.0, b = 1.06	1.0 ~ 1.85, b = 1.69	1.85 ~ 2.8, b = 2.37
  1000	30	0.0 ~ 1.0, b = 0.99	1.0 ~ 2.00, b = 1.58	2.00 ~ 3.0, b = 2.39
  5000	20	0.0 ~ 1.0, b = 1.01	1.0 ~ 1.85, b = 1.84	1.85 ~ 2.8, b = 2.45
  5000	30	0.0 ~ 1.0, b = 0.94	1.0 ~ 2.30, b = 1.67	2.30 ~ 3.1, b = 2.44	plot

  
  

Results for the 1D BK model with zero loading velocity

• Magnitude distribution for N = 1000 at R = 1, and 100 with dt = 0.001 and v_0 = 0.001 (see plot). There is only one scaling range for R = 1 with exponent b = 0.8. For R = 100 another scaling range with b = 1.6 appears for small s. The results are similar to non-zero loading velocity case.
• Mean slip of individual blocks in an event as a function of the moment of event and the number of blocks involved in an event for  N = 1000 at R = 1, and 100 with dt = 0.001 and v_0 = 0.001. 
• Mean stress over blocks in an event just before rupture as a function of the number of blocks involved in an event for  N = 1000 at R = 1, and 100 with dt = 0.001 and v_0 = 0.001 (see log-log plot and  linear-linear plot).
• The effects of velocity cutoff to all kinds of distributions (number of blocks involved in an event, moment and magnitude) for R = 1, and R = 100 with dt = 0.001 are shown below for four different cutoff values (v_0 = 0.001, 0.0001, 0.00001, and 0.000001). 

  interaction range	number of blocks involved	moment	magnitude
  R = 1	plot	plot	plot
  R = 100	plot	plot	plot

• The effects of time step to all kinds of distributions (number of blocks involved in an event, moment and magnitude) R = 100 with v_0 = 0.0001 are shown below for three different values.

  cutoff v_0	number of blocks involved	moment	magnitude
  0.0001	plot	plot	plot
  0.00001	plot	plot	plot

  
  
• The effects of velocity cutoff to mean slip of individual blocks in an event are shown in plots for R = 1 (number of blocks involved, moment), and R = 100 (number of blocks involved, moment) with dt = 0.001 (v_0 = 0.001, 0.0001, 0.00001, and 0.000001). We can find same qualitative behavior for all v_0. For R = 1, the mean slip increases as the increase of number of involved blocks or moment of event, and  there is no constant mean slip range.  For R = 100, we can find  a constant mean slip as a function of the number of blocks and moment for about 2-3 decades. The value of constant mean slip  increases with deceasing v_0, but is the same for v_0 = 10^{-4} and 10^{-5}.
• The macroscopic quantities of an event as a function of number of  blocks involved in an event are  shown in the table below.

  	R = 1	R = 100	comments
  moment	plot	plot	With the increase of R, a range with moment proportional to number of blocks involved appears.
  magnitude	plot	plot	Similar behavior as moment
  number of blocks involved including multiple-slipping blocks	plot	plot	With the increase of R, the range of events with only one-slipping blocks increases. For larger events (N > 200) of R =100, there are much multiple-slipping blocks involved.
  radius of gyration	haven't gotten yet.	plot	For larger events of R = 100, the log of R_G is proportional to the log of number of blocks involved with slope close to 1which means the larger events are compacted. For small events of R  = 100, However,  the slope is less than 1 which means that the small events are not compacted.
  loading distance to produce an event	plot	plot	There is not too obvious difference of loading distance to produce a larger event or small event for different R.
  mean loading distance to produce an event	0.000529 for v_0 = 1^{-3},  0.000810 for v_0 = 1^{-4}, 0.000817 for v_0 = 1^{-5}, 0.001169 for v_0 = 1^{-6}.	0.000076 for v_0 = 1^{-3},  0.000099 for v_0 = 1^{-4}, 0.000098 for v_0 = 1^{-5}, 0.000090 for v_0 = 1^{-6}.	The mean loading distance will decease with the increase of R.

   
• In order to get a good statistical result. We simulated much more events for the same system for R = 1, and R = 100 at different velocity cutoff. The interested quantities and distribution functions are shown below.

  	v_0 = 0.001	v_0 = 0.00001
  number of events	6.5 million	10 million
  magnitude distribution	plot	plot
  moment distribution	plot	plot
  distribution of total # of failed blocks	plot	plot
  distribution of # of blocks involved	plot	plot
  lifetime distribution	plot	plot
  moment .vs. mean slip	plot	plot
  # of blocks involved .vs. mean slip	plot	plot
  # of blocks involved .vs. total # of failed blocks	plot	plot
  # of blocks involved .vs. moment	plot	plot
  # of blocks involved .vs. magnitude	plot	plot
  # of blocks involved .vs. lifetime	plot	plot
  # of blocks involved .vs. R_G	plot	plot
  # of blocks involved .vs. minimum driving force of blocks involved	plot	plot
  # of blocks involved .vs. mean driving force of blocks involved	plot	plot

• The results for all kinds distributions  at dt = 0.0001and v_0 = 10^-7 are shown below the results are similar as dt = 0.001 and v_0 = 10^-5.

  	plots	comments
  number of events	100,000
  magnitude distribution	plot	For big event, All slopes  are close to 0.8.
  moment distribution	plot	All slopes  are close to 1.8.
  distribution of total # of failed blocks	plot	R = 1, slope = 2.5; R = 30, slope = 2.15, R = 100, slope = 1.90
  distribution of # of blocks involved	plot	similar as the distribution of total # of failed blocks
  lifetime distribution	plot	R = 1, slope = 2.97; R = 30, slope = 2.80, R = 100, slope = 2.60
  moment .vs. mean slip	plot	at small moment, the range with constant mean slip  increases with  R
  moment .vs. mean drop of force	plot	at small moment, the range with constant drop of force  increases with  R. Note the mean drop of force is much smaller that the RJB model.
  # of blocks involved .vs. mean slip	plot	For small event, the range with constant mean slip is close to 2R.
  # of blocks involved .vs. mean drop of force	plot	similar as # of blocks involved .vs. mean slip/
  # of blocks involved .vs. total # of failed blocks	plot	For small event, the range with multiple slips is close to 2R.
  # of blocks involved .vs. moment	plot	For small event, the linear range is close to 2R.
  # of blocks involved .vs. magnitude	plot	For small event, the linear range is close to 2R.
  # of blocks involved .vs. lifetime	plot
  # of blocks involved .vs. R_G	plot	For small event of R > 1, the events are not compact.
  # of blocks involved .vs. minimum driving force of blocks involved	plot	For small event, the range with  minimum driving force close to 1 increases as R.
  # of blocks involved .vs. mean driving force of blocks involved	plot	similar as # of blocks involved .vs. minimum driving force of blocks involved

• Main results about the BK 1D model upto June 29, 2004 have been summarized in the report.
• Besides the effects of timestep, velocity cutoff, system size, alpha, and interaction range as shown above, more results for lager systems (upto N = 10000) and  longer interaction range (upto R = 500) are shown in the table as follows. At the same time, the effects of stiffness of interaction (ell) and type of interaction are also investigated.  The default parameters are N = 1000, R = 100, ell = 10, alpha = 2.5, nu = 0.0, and sigma = 0.01 with free boundary condition.
  

  	alpha = 2.5	alpha = 0.0
  system size (N)	moment, number of failed blocks	moment, number of failed blocks	No obvious effects on scaling region except breakout event.  The slope is roughly 2.0 for  alpha  = 2.5 and 1.5 for alpha =  0.0.
  interaction range (R)		moment, number of failed blocks	Increasing the interaction range make scaling region better defined.
  interaction stiffness (ell)		moment, number of failed blocks	No obvious effects on scaling region. Larger ell induces more breakout events.
  interaction type (1/r^c)	moment, number of failed blocks	moment, number of failed blocks	If interaction type becomes more short-range-like,  the system behaves more like nearest-neighbor model.

• The table below shows the time evolution of the mean driving force of the system before an event for different alpha with N = 5000, R = 500,  nu = 0.0, and sigma = 0.01.
  

  alpha	R = 500
  2.5	plot, the system evolves from a low-stress state to a high-stress state, with a well defined quasi-periodicity T around 50000.
  2.0	plot, similar to that of 2.5,  but with a well defined quasi-periodicity T around 70000.
  1.5	plot, well defined quasi-periodicity with T around 60000. Note the system can get into the higher stress state as the alpha is decreased.
  1.1	plot, still well defined, T around 50000.
  0.9	plot, still has some kind of quasi-periodicity. After evolving into a high-stress state and the system can fluctuate around this high-stress state with some period of time.
  0.5	plot, no quasi-periodicity. The system fluctuates around the high-stress state.
  0.0	plot, same as 0.5.

  
  

Updated on 15 June, 2005.