Aftershocks in slowly compressed bulk metallic glasses: Experiments and theory

We observe two distinct interevent time patterns in the slip avalanches of compressed bulk metallic glasses (BMGs). Small slip avalanches cluster together in time, but large slip avalanches recur roughly periodically. We compare the timing patterns of BMG slip avalanches with timing patterns of earthquakes and with the predictions of a mean-field model. The time clustering of small avalanches is similar to the known time clustering of earthquake foreshocks and aftershocks.

Figure 1

Interevent times for slip avalanches experimentally measured in mm-scale specimens of bulk metallic glass slowly compressed at a strain rate of ${10}^{-4}{\mathrm{s}}^{-1}$. (a) Probability density of interevent times between small slip avalanches. The probability density between 2 and 30 ms is consistent with a power law of exponent $-1.4\pm 0.2$. (b) Histogram of interevent times between large slip avalanches.

Figure 2

Bi test for small (blue ×) and large (red circle) avalanche interevent times experimentally measured in slowly compressed bulk metallic glass. The solid line represents the ideal Bi test for a local Poisson process. The curve for the small-avalanche Bi test rapidly increases near small and large values of $H_k$; this shape is typical of correlated times that occur in clusters. The curve for the large-avalanche Bi test rapidly increases near $H_k=2/3$; this shape is typical of correlated times that occur at fairly regular intervals. Small avalanches cluster together; large avalanches recur nearly periodically.

Figure 3

Average aftershock rates for small (blue ×) and large (red circle) mainshocks experimentally measured in slowly compressed bulk metallic glass. The power-law region of the aftershock rate of small mainshocks is consistent with a power-law exponent of $-1.3\pm 0.15$. The time origin is the beginning of the mainshock. After a small mainshock, the aftershock rate follows a power law as the rate decays with time; the deviation from power-law behavior at short times is likely due to the minimum detectable avalanche size. After a large mainshock, the aftershock rate follows a steep curve as the rate decays with time. The two points that have no error bars were calculated from bins in which there was only one detected aftershock nucleation and should be thought of as having relatively large uncertainty.

Figure 4

Average foreshock rates for small (blue ×) and large (red circle) mainshocks experimentally measured in slowly compressed bulk metallic glass. The power-law region of the foreshock rate of small mainshocks is consistent with a power-law exponent of $-1.15\pm 0.15$. The time origin is the beginning of the mainshock. For both small and large mainshocks, the foreshock rate preceding the mainshock follows a power law before the mainshock starts.

Figure 5

(a) Entire plastic region and (b) magnified view of the stress vs time trace for BMG specimen (1). (c) Entire plastic region and (d) magnified view of the stress vs time trace for BMG specimen (2).

Figure 6

Probability density of interevent times between small slip avalanches for (a) BMG specimen (1) and (b) BMG specimen (2). The curves labeled “first half” (“second half”) are only calculated from the interevent times from the temporal first half (second half) of the stress vs time trace. The small-avalanche interevent time distribution is relatively constant across time and from one specimen to the other. The few points in (a) and in subsequent figures that have no error bars were calculated from bins in which there was only one interevent time and should be thought of as having relatively large uncertainty.

Figure 7

Histogram of interevent times between large slip avalanches (10 MPa ≤avalanche size) for (a) BMG specimen (1) and (b) BMG specimen (2). The curves labeled “first half” (“second half”) are only calculated from the interevent times from the temporal first half (second half) of the stress vs time trace. The characteristic large-avalanche interevent time changes by about a factor of 2 from the first temporal half of specimen (1) to the second temporal half. In contrast, the characteristic interevent time is more stable for specimen (2).

Figure 8

(a) Bi test applied to the avalanches in specimen (1), separated by avalanche size and by temporal half of the stress vs time trace. (b) Bi test applied to the avalanches in specimen (2), separated by avalanche size and by temporal half of the stress vs time trace.

Figure 9

Bi test applied to the set of all detected avalanches using all interevent times aggregated from both temporal halves of both specimens. These interevent times are not separated either by temporal half of the experiment or by avalanche size into small vs large avalanches as in the main text; instead these interevent times are between any avalanches with avalanche size above the noise floor (0.2 MPa ≤avalanche size).

Figure 10

Avalanche (a) aftershock rates and (b) foreshock rates, calculated using the full time of each aftershock sequence instead of only the dead time, for small (blue ×) and large (red circle) mainshocks experimentally measured in slowly compressed bulk metallic glass. The small-avalanche aftershock rate in (a) is consistent with a power law of exponent $-1.2\pm 0.15$; the small-avalanche foreshock rate in (b) is consistent with a power law of exponent $-1.15\pm 0.2$. The power laws of the aftershock and foreshock rates of small avalanches are similar to the power laws of Figs. 3 and 4 in the main text, within the error bar ranges. The time origin is the beginning of the mainshock.

Figure 11

Collapsed average avalanche shapes, binned in size. Avalanches separated in time by less than the “minimum time” are combined into a single avalanche with the earlier start time and the later end time. The “minimum time” for these figures is (a) 0 s, (b) ${10}^{-5}$ s, (c) ${10}^{-4}$ s, (d) ${10}^{-3}$ s, (e) ${10}^{-2}$ s, and (f) ${10}^{-1}$ s. The solid line in each figure is the predicted mean-field avalanche shape [7]. The shape collapse quality becomes worse as the minimum time increases.

Figure 12

Avalanches analyzed with a “minimum time” of ${10}^{-3}$ s between the ending time of any avalanche and the beginning time of the subsequent avalanche. (a–e) show the same quantities that are reported in the main text, but with a nonzero “minimum time” for these figures. The general conclusions of the main paper, i.e., that small avalanches cluster together while large avalanches recur quasiperiodically, hold true even with this modified avalanche analysis.

Figure 13

The Bi test using a delayed Poisson process as the null hypothesis [36]. (a) The delayed Poisson process Bi test applied to the same data as in Fig. 2 (large vs small avalanches). (b) The delayed Poisson process Bi test applied to the same data as in Fig. 8. (c) The delayed Poisson process Bi test applied to the same data as in Fig. 8.