Applied-force oscillations in avalanche dynamics

Until now most studies of discrete plasticity have focused on systems that are assumed to be driven by a monotonically increasing force; in many real systems, however, the driving force includes damped oscillations or oscillations induced by the propagation of discrete events or “slip avalanches.” In both cases, these oscillations may obscure the true dynamics. Here we effectively consider both cases by investigating the effects of damped oscillations in the external driving force on avalanche dynamics. We compare model simulations of slip avalanches under mean-field dynamics with observations in slip-avalanche experiments on slowly compressed micrometer-sized Au specimens using open-loop force control. The studies show very good agreement between simulations and experiments. We find that an oscillatory external driving force changes the average avalanche shapes only for avalanches with durations close to the period of oscillation of the external force. This effect on the avalanche shapes can be addressed in experiments by choosing suitable specimen dimensions so that the mechanical resonance does not interact with the avalanche dynamics. These results are important for the interpretation of avalanche experiments with built-in oscillators, and for the prediction and analysis of avalanche dynamics in systems with resonant vibrations.

Figure 1

(a) Full experimental depth vs time curve for one of the three Au specimens (the other two specimens are qualitatively similar). The vertical jumps in the curve are a macroscopic signature of slip avalanches, in which interacting dislocations collectively move (inset shows a magnification including several slip avalanches). (b) Full experimental force vs time curve for the same Au specimen. The inset of (b) shows the damped sinusoidal oscillation of the machine at the end of the unloading process.

Figure 2

Force vs time for the same Au specimen as in Fig. 1, magnified to highlight the oscillation of the unloaded nanoindenter. The fitted damped sinusoid is overlaid on the observed oscillation, with very good agreement between the experimental measurement and the fitted function, and also very good agreement between the fitted parameters and the parameters reported in Ref. [26].

Figure 3

(a) Spring-damper model for the nanoindentation system used to compress the Au specimens. Note that for a given motion of the transducer mass, the springs add in a parallel configuration, not series. (b) Equivalent simplified spring-damper model for oscillations of the transducer mass.

Figure 4

Force vs depth for the same Au specimen as in Figs. 1 and 2. The inset is a magnification of the unloading region; the linear fit to the first 50% of the unloading region is also shown.

Figure 5

(a) Plastic region of the unfiltered depth vs time for the same Au specimen as in Figs. 1, 2, and 4. (b) Power spectral density (PSD) of the unfiltered depth shown in (a). (c) Velocity vs time of the unfiltered depth, calculated via two-point difference. (d) PSD of the velocity shown in (c). Note the peak near ∼750 Hz. (e) Velocity vs time of the filtered depth, calculated via two-point difference of the FIR-filtered depth. (f) PSD of the velocity shown in (e). All PSD curves (b), (d), and (f) are calculated via the Welch method [27] and are overlaid with a smoothed PSD curve. All PSD curves contain magnified insets spanning the mechanical resonant frequency of ∼750 Hz.

Figure 6

Average avalanche shapes for slip avalanches in Au crystals. Avalanches have been binned together by avalanche size, and the average avalanche shape in each size-bin is shown. (a) (top left) shows the average shape for avalanches in the smallest size-bin. Averages in larger bins are shown progressively in (b)–(e). Each black curve shows the predicted mean-field shape function scaled to match the largest avalanche in the figure, and the red curve represents an alternative shape function, again for comparison with the largest avalanche in the figure. Note the distortions that are similar to the distortions of the simulation with oscillation (Fig. 7), which are consistent with the machine oscillation timescale. Error bars are drawn at every third point to avoid clutter. The five smallest shapes are each an average of 35 events, while the three largest shapes are all an average of 28 events.

Figure 7

Average simulated avalanche shapes generated by the mean-field model of avalanche dynamics with a damped-oscillating external force instead of a linearly increasing external force, presented similarly to Fig. 6. Each black curve shows the predicted monotonic mean-field shape function scaled to match the largest avalanche shape in the figure, and the red curve represents an alternative shape function, again for comparison with the largest avalanche in the figure. Each plotted shape is the average of 250 avalanche events. The intermediate-size shape [Fig. 7] shows better agreement with the alternative reference shape. Almost all error bars in this figure are smaller than the marker size. Note that with the simulations, we are able to generate a very large range of avalanche size values (three orders of magnitude), wider than the range achievable with physical experiments.

Figure 8

Average avalanche shapes for slip avalanches in Au crystals from experiments, analyzed with an alternative shapes-extraction method. This alternative method first uses the same FIR filter as used in Fig. 6, but then further smooths the velocity by applying a least-squares linear fit across multiple experimental data points. This extra smoothing introduces a tradeoff: as extra smoothing is applied, the experimental noise is reduced even further, but the avalanche oscillations become smoothed out and are less visible to the point that they may not be properly recognized as oscillations.