Aftershock production rate of driven viscoelastic interfaces

We study analytically and by numerical simulations the statistics of the aftershocks generated after large avalanches in models of interface depinning that include viscoelastic relaxation effects. We find in all the analyzed cases that the decay law of aftershocks with time can be understood by considering the typical roughness of the interface and its evolution due to relaxation. In models where there is a single viscoelastic relaxation time there is an exponential decay of the number of aftershocks with time. In models in which viscoelastic relaxation is wave-vector dependent we typically find a power-law dependence of the decay rate that is compatible with the Omori law. The factors that determine the value of the decay exponent are analyzed.

Figure 1

(a) Mechanical representation of the quenched Edwards-Wilkinson model. [(b) and (c)] Two viscoelastic variations of the model.

Figure 2

(a) The size $S$ of the avalanches (see the definition in the appendix) as a function of time for the case of model A [Fig. 1]. The time axis is scaled by the driving velocity $V$, so all aftershocks appear on the same vertical line. In (b) and (c), we take particular clusters of events in (a) and plot them as a function of the time in units of the relaxation time constant $\eta /k_1$. We see here the decay of the aftershock rate with time. The largest event in each cluster is indicated by the small horizontal arrows. Note that in (b) the largest event is the first one, but this is not the case in (c). This last case is actually the typical situation.

Figure 3

(a) A relaxed distribution of forces $f_i$ (line), which for the present models is a constant, and the values of $f_i^{\mathrm{th}}-k_0(Vt-x_i)$ (crosses) in a one-dimensional sketch. As driving increases, crosses move down, generating a primary shock. (b) After the primary shock, the rearranged distribution of forces $f_i$ evolves due to the relaxation term, eventually triggering aftershocks.

Figure 4

Model A. Density of aftershocks as a function of time after the main shock, calculated according to Eq. (7) for three different initial interfaces with different roughness (as indicated) and the corresponding analytical estimate Eq. (12). We see the perfect accordance between the two.

Figure 5

Same as described in the caption to Fig. 4 for model B, and the corresponding analytical estimate Eq. (20). We see that the analytical estimate gives a slightly smaller decay exponent than the one calculated using Eq. (7).

Figure 6

Effect of aftershocks triggering other aftershocks. Primary aftershocks are generated with a rate $N\left(t\right)$. In (a) $N\left(t\right)\simeq exp(-t)$, in (b) $N\left(t\right)\simeq 1/{(t+c)}^p$ (we take $p=1.5$, $c=1$, for concreteness). Each aftershock can trigger successive ones with probability $\alpha$. We see that the effect of a finite $\alpha$ is to change the time decay constant in the exponential case (a), whereas in the Omori case (b) it has a minor effect on the form of the decay rate.

Figure 7

Aftershock rates obtained using a starting configuration corresponding to an equilibrium qEW surface. Open symbols correspond to the case in which each triggered aftershock is not allowed to modify the distribution of $f_i$. Full symbols instead correspond to the case in which each aftershock modifies the values of $f_i$ according to the avalanche dynamics [Eq. (4)]. Panels (a) and (b) correspond to models A and B [Eqs. (5) and (6)].

Figure 8

Aftershock rates in full simulations of the viscoelastic model studied in this paper. (a) In the single relaxation time case [model A, Eq. (5)] the aftershock rate is roughly exponential, with a decay time larger than the relaxation time in the system. (b) For $q$-dependent relaxation [model B, Eq. (6)], an Omori law is obtained, with a value of the decay exponent close to 2. (c) Aftershock rate for model C [Eq. (21)].