Aftershocks and Omori's law in a modified Carlson-Langer model with nonlinear viscoelasticity

A modified Carlson-Langer model for earthquakes is proposed, which includes nonlinear viscoelasticity. Several aftershocks are generated after the main shock owing to the damping of the additional viscoelastic force. Both the Gutenberg-Richter law and Omori's law are reproduced in a numerical simulation of the modified Carlson-Langer model on a critical percolation cluster of a square lattice.

Figure 1

Schematic figure of a modified Carlson-Langer model with viscoelasticity.

Figure 2

(a) Time evolution of the sum $S$ of slips for 10-block systems at $\gamma =1/2$ ($\beta =2$), $\alpha =1.5,K=4$, and $\delta =K/{\eta }^{\beta }=1$. (b) Time evolution of the profile of $x_i$ for 10-block systems using the same parameters. The light solid lines denote the profiles of $x_i$ after main shocks and the dark dashed lines denote the profiles of $x_i$ after aftershocks. (c) Enlargement of (b) for $424.26<x<424.272$.

Figure 3

(a) Semilogarithmic plot of the probability distribution of the elapsed time $\tau$ of aftershocks at $\gamma =1,\alpha =1,K=0.5$, and $\delta =K/\eta =0.05$. (b) Double-logarithmic plot of the probability distribution of $\tau$ at $\gamma =1/2,\alpha =1,K=4$, and $\delta =K/{\eta }^2=1$. (c) Double-logarithmic plot of the probability distribution of $\tau$ at $\gamma =1/3,\alpha =0.5,K=0.4$, and $\delta =K/{\eta }^3=10000$. (d) Double-logarithmic plot of the probability distribution of $\tau$ at $\gamma =1/4,\alpha =0.5,K=0.4$, and $\delta =K/{\eta }^4=100000$.

Figure 4

Probability distribution of the magnitude at (a) $\alpha =0.3$, (b) 0.4, and (c) 0.5 for $\gamma =1/4,K=0.4$, and $\delta =K/{\eta }^4=100000$. The dashed line in Fig. 4 denotes a curve of $P\left(M\right)\propto {10}^{-1.2M}$.

Figure 5

(a) Time evolution of main shocks at $\gamma =1/2, \alpha =1.2,K=1$, and $\delta =K/{\eta }^2=0.05$. (b) Time evolution of aftershocks after the main shock at $v_{0}t=3.0812$. (c) Distribution of the epicenters of a main shock (rhombus) and the aftershocks (pluses). The critical percolation cluster is expressed using dots. (d) Relationship between the slip size $S$ of the main shock and the number $N_a$ of aftershocks after each main shock. The dashed line is $N_a=9S^{0.83}$.

Figure 6

(a) Probability distribution of the magnitude $M=(2/3){log}_{10}S$ at $\gamma =1/2, \alpha =1.2,K=1$, and $\delta =K/{\eta }^2=0.05$. The dashed line denotes an exponential distribution $P\left(M\right)\propto {10}^{-1.1M}$. (b) Probability distribution of $\tau$. The dashed line is $n\left(\tau \right)=13.3/{(680+\tau )}^{1.5}$. (c) Probability distribution of $\tau$ for $M>0$ and $M<0$. The dashed lines are $n\left(\tau \right)=7.4/{(2560+\tau )}^{1.35}$ and $n\left(\tau \right)=598/{(916+x)}^{1.95}$.

Figure 7

(a) Probability distribution of the magnitude $M=(2/3){log}_{10}S$ at $\gamma =1/4, \alpha =0.95,K=0.1$, and $\delta =K/{\eta }^4=100000$. The dashed line denotes an exponential distribution $P\left(M\right)\propto {10}^{-1.05M}$. (b) Probability distribution of $\tau$. The dashed line is $n\left(\tau \right)=0.09/{(48+\tau )}^{1.02}$. (c) Probability distribution of $\tau$ for $M>0$ and $M<0$. The dashed lines are $n\left(\tau \right)=0.0068/{(112+\tau )}^{1.1}$ and $n\left(\tau \right)=0.0068/{(215+x)}^{0.75}$.