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Auxiliary Material for
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Acoustic emission and microslip precursors to stick-slip failure
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in sheared granular material
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Paul Johnson et al.,
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(Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los
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Alamos, NM)
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Geophysical Research Letters, 2013
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Introduction
The following is supporting information, including descriptions of supplementary
figures S1 through S7:
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1. Calculations and Supplementary Material
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1.1) Interevent (recurrence) time and stress drop.
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Fig. S1 shows an expanded view of the shear stress as a function of experimental
run time. Superimposed are the precursors. Their relative amplitudes are correct.
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1.2) Calculation of magnitude.
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AE magnitudes are calculated as M = log (u) where particle displacement u is
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determined from measured acceleration 𝑢̈ , 𝑢 = (2𝜋𝑓)2 and the measured frequency is 40.3
𝑢̈
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kHz. The distance correction term normally applied is left out of the relation because
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event distances to the accelerometer are nearly identical, and the quality factor Q for the
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propagation path through the steel is at least 104. Thus we assume no path loss of wave
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amplitudes in the steel. The calculation is imperfect but serves to provide an estimate of
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the magnitude.
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1.3) Construction of the occurrence versus time histogram.
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Fig. S2 shows the cumulative AEs for the full experiment. The rate is relatively
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high at the experiment onset where the shear rate is double (until about 500 seconds), and
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increases nonlinearly as the experiment progresses. Figure S2a shows the sum (i.e.,
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cumulative number of events) of each individual sequence of precursors as a function of
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time for the entire experiment. An expanded view of a small portion of the experiment is
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shown in Fig. S2b. There exists an increase in the event density corresponding to
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compaction and material loss as the experiment proceeds. Fig. S2c shows the cumulative
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sum of each precursor sequence again but in a different format—each characteristic slip
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event is displaced to zero time so that the relative sums can be observed. This means that
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the precursors occur in negative time leading up to time = 0. The interest of this figure is
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to show the large variation in the sum of precursor sequences. Plotting the data shown in
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S2c as a histogram gives Fig. 4b.
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1.4) Correlation of precursor stress drops with AE.
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For identification of precursor stress drops, we apply a threshold of -0.02 to the
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first order derivative of shear stress  with time, d/dt. We eliminate the characteristic
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event stress drops by applying an upper limit threshold of -0.5. We then calculate the
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precursor signal event rate from the signal that has had the two thresholds applied. A
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second event rate signal is constructed from the AE data using the inverse recurrence
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time. Both signals are interpolated for correlation, and a running average with a half
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width of 10x the sampling interval (dt = 12000/333333 = 0.0360) is applied to smooth the
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two time-series. The total width of the running average is therefore 20*dt = 20*0.0360 =
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0.72 sec. In the cross correlation, the time lag of -2.4161x10-4 sec between the two
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signals is negligible [it is less than the sampling interval of both time-series (sampling
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interval = 0.0360 seconds)].
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1.5) Block Motion as a function of experimental run time.
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Fig. S3 illustrates measured parameters that describe the behaviour of the shearing
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block as well as the precursors. At the time of a stick-slip event, the central block
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accelerates rapidly and the gouge material thins abruptly (Figure S3). Following the slip,
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the central block proceeds through three approximate regimes of behavior. The block
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first arrests, ‘sticks’, for a period of time (Regime I, Figure S3). If it is moving, the
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displacement is so small that the digitizer cannot capture it. Simultaneously, the gouge
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maintains its thickness. As the vertical RAM pushes downward at a constant velocity, the
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stress on the central block increases through the stiff steel spring, and the gouge behaves
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in an approximately linear and elastic manner, as illustrated by the linear behavior of the
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shear stress in Regime I. The block then begins to move forward very slowly (‘slow and
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silent slip’), accelerating modestly while the gouge material simultaneously dilates
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(Figure S3, Regime II). As the maximum shear stress and friction is approached, the AE
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and small stress drops commence (Region III), leading to the next slip event, after which
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the process repeats. During the entire sequence, the displacement of the forcing RAM is
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linear in time.
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1.6) Experiment conducted at increasing and decreasing normal load.
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Figure 4 provides an overview of the experimental protocol for experiment p2394,
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where applied normal load is progressively stepped up from 2 MPa to 8 MPa and
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subsequently stepped back down to 2 MPa. Figure S4b shows the friction as a function of
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experimental run time.
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decreases during longer intervals of relatively stable sliding (e.g., 0.5-1.3x104 µm).
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Figure S4c shows the AE for the stick-slip events as a function of experimental run time.
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The effects of gouge compaction are evident—as the experiment progresses, the
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amplitude of the AE become progressively larger and more frequent, just as in the
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experiment conducted at 5 MPa. Further, denser gouge is less dissipative, which is why
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amplitudes are largest in the ~ 5-8 MPa load range.
Friction is relatively independent of load level, however it
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Figure S5 shows the normalized number of event occurrences, the Gutenberg-
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Richter plot, as a function of magnitude for experiment p2394. The slope (-1.9) is similar
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to that found for the experiment conducted at 5 MPa (-1.7), suggesting that, over an
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interval of 2-8 MPa in normal load, the event statistics are similar.
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Figure S6 shows precursor time characteristics of the AEs relative to the stick-slip
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times for all load levels, excluding 2 MPa. Data at 2 MPa show no obvious precursors, as
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in this regime the material is generally stably sliding. The same is true at 3MPa (shown).
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Figure S6a shows the time to slip for the first AE in each sequence, and Figure S6b
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shows the time to slip for the last AE in each sequence. The first AE begins progressively
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later with applied load. The timing of the last event in the precursor sequence (Figure
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S6b) is unaffected by the load level. Figure S7 shows the probability density functions
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(PDF’s) for the load levels explored.
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Supplementary Figure Captions:
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Figure S1. Expanded view of a portion of experiment p2393 (see Figure 1c) showing
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shear stress (solid line) and small AE precursors (circles) as a function of time during
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stick-slip events. The AE precursors are plotted according to their relative strain
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amplitude, which varies from 3x10-9 – 5x10-8. Note that precursory AEs begin late in the
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stick-slip cycle and the rate of occurrence increases as stress increases to failure. The AE
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amplitude increases as time-to-failure approaches zero as well. Stress drop and recurrence
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interval are noted. Here the shear stress signal is left unsmoothed to show the electro-
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magnetic spikes. Spurious spikes are eliminated in Fig. 2b (inset) to illustrate only the
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stress drop behavior.
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Figure S2. Cumulative number of micro-slip events in each successive group of
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precursors preceding a slip event for (a) the entire experiment p2393 and (b) for a portion
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of the experiment. (c) All data for the full experiment shown in (a) plotted on the same
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figure, with each characteristic slip event set to time = 0.
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Figure S3. Block motion with the AE, friction, and thickness as a function of time for
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experiment p2393. (a) The block displacement is shown by the thick dark curve, the
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friction is shown as the thin line, and the AE are shown as open circles. (b) Thickness of
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the gouge layers (thick solid line, with scale). At the time of a stick-slip event, the central
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block shown in Fig. 1 stops abruptly. The frictional force builds rapidly (region I) via the
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driving RAM, and continues to force the system linearly with time via the stiff spring.
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The central block then begins to slip ‘silently and slowly’ (region II), followed eventually
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by precursor AE activity (region III) with associated stress drops. During the entire
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sequence, the displacement of the forcing RAM is linear in time.
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Figure S4. Overview of experiment p2394. (a) Experimental protocol showing applied
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normal load versus time. (b) Friction versus time. (c) Recorded AE of the stick slip
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events versus time.
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Figure S5. Relative probability of AE event occurrence plotted versus magnitude (log-
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log scale), for all AE in experiment p2394. The AE from characteristic events, the stick-
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slips, are labeled. The slope of the precursor AE is denoted by the thin, dotted line.
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Figure S6. Time characteristics of slip as a function of the applied normal load for
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experiment p2394. (a) Time to slip of the first AE in each successive sequence. (b) Time
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to slip of the last AE in each successive sequence.
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Figure S7. AE preceding slip for individual normal loads in experiment p2394. (a)
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through (f) show the results for 3-8 MPa, respectively. The data at 2 MPa did not contain
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a sufficient number of stick-slip events to obtain a clear signal. The upgoing and
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downgoing applied normal loads (at a given load level) are combined in this figure. The
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slope of the exponential increase in AE activity (linear in log-log space) is noted for loads
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greater than 3 MPa.
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