Figure 1

(Color online) (a) AE amplitude probability density functions and (b) rank ordering statistics obtained during the plastic deformation of ice and Cd single crystals. This reveals truncated power law distributions of dislocation avalanche sizes, $P\left(A_0\right)\sim A_0^{-\tau }f(A_0∕A_c)$, where the exponent $\tau =2.0\pm 0.1$ is independent of the material and/or the sample size and $f\left(x\right)$ is a cutoff function. Truncation of the distributions is not apparent on the pdf’s, but clearly detectable on rank ordering statistics. For the Cd sample, the cutoff $A_c$ is not sensitive to the jump of the strain-hardening coefficient $\theta$ from stage I to stage II. Stars: compression creep loading of an ice single crystal (sample length of $68.2\mathrm{mm}$, sample diameter of $37.4\mathrm{mm}$, $T=-10°\mathrm{C}$, applied stress of $0.76\mathrm{MPa}$, Schmid’s factor $m=0.25$, and negligible strain hardening). Open circles: tensile test under constant crosshead velocity of a Cd single crystal, stage I (initial sample length of $25\mathrm{mm}$, initial sample diameter of $3.6\mathrm{mm}$, $T=20°\mathrm{C}$, initial strain rate of $1.3\times {10}^{-3}{\mathrm{s}}^{-1}$, Schmid’s factor $m=0.46$, and $\theta \approx 0.8\mathrm{MPa}\approx E∕62500$, where $E$ is the elastic modulus). Open triangles: same test, stage II (strain hardening coefficient $\theta \approx 4.7\mathrm{MPa}\approx E∕10600$). The three different distributions have been plotted for the same number of events (3539). For the records containing more than 3539 events, we checked that the normalized rank ordering statistics were not affected by this undersampling.Reuse & Permissions

Figure 2

(Color online) AE recorded during the tensile plastic deformation of Cu single crystals (initial sample length of $20–60\mathrm{mm}$, initial sample diameter of $4\mathrm{mm}$, and Schmid’s factor $m\approx 0.3$). (a) AE amplitude probability density function obtained by collecting data recorded during six different tests under constant crosshead speed ($T=20°\mathrm{C}$, initial strain rates varying from $2.9\times {10}^{-4}\text{to}3.8\times {10}^{-2}{\mathrm{s}}^{-1}$, and maximum hardening coefficient $\theta$ varying from $E∕1000$ to $E∕3300$). (b) Correlation integral analysis calculated for one of these tests (initial strain rate of $5.1\times {10}^{-3}{\mathrm{s}}^{-1}$). The correlation integral $C\left(t\right)$ is the probability for two events to be separated by less than a time $t$ and is defined as $C\left(t\right)=[2∕N(N-1)]n(\Delta t<t)$, where $N$ is the number of considered events and $n(\Delta t<t)$ the number of pairs of events separated by less than $t$. The AE dataset (closed circles) is compared to a similar analysis performed on the same number of events $(N=61)$ randomly distributed in time (Poisson process, open circles). The random dataset follows the expected scaling $C\left(t\right)\sim t$, whereas a time clustering of the AE events is observed below $5\mathrm{s}$.Reuse & Permissions

Figure 3

Cu single crystal oriented for multislip loaded under uniaxial tension (thickness of $2\mathrm{mm}$, width of $10\mathrm{mm}$, length of $35\mathrm{mm}$, Schmid’s factor $m=0.3$, and $T=20°\mathrm{C}$. The applied strain rate is $5\times {10}^{-4}{\mathrm{s}}^{-1}$. Macroscopic force vs time curve (main graph) and displacement vs time in five different locations along the sample (inset). The initial stage II hardening rate is $\theta \approx 7E∕1000$. The displacements were recorded at intervals of $1\mathrm{mm}$ in 20 sections of the samples.Reuse & Permissions

Figure 4

Velocity signals from high-resolution extensometry derived from the displacement vs time record shown on Fig. 3. The long-term trend of the displacement-time curve on the time scale of the experiment was subtracted from the measured signal before derivation. Sequence of close-ups of the velocity signal shown in Fig. 5, suggesting scale invariance.Reuse & Permissions

Figure 5

Probability density function for peak values of the displacement velocity shown on Fig. 4. Peak values are normalized with respect to the averaged value over the time series. The dashed line indicates a power law trend with slope $\tau =2$.Reuse & Permissions

Figure 6

Multifractal behavior of the velocity signals from high-resolution extensometry in Cu single crystals. Examples of the generalized dimension spectrum $D\left(q\right)$ for the local displacement velocity signal. Dark symbols: initial strain rate of $5\times {10}^{-3}{\mathrm{s}}^{-1}$; light symbols: initial strain rate of $5\times {10}^{-4}{\mathrm{s}}^{-1}$ (signal shown in Fig. 4). The existence of a multifractal spectrum reveals a self-similar temporal structure of the signal, with an extended set of scaling exponents. The error bars were determined as least-squares estimates of the standard deviation of the slopes of the scaling dependences. For comparison, the dashed line corresponds to uniform (constant, periodic, or random) behavior [$\forall q$, $D\left(q\right)=1$]. A single fractal with fractal dimension $D_f$ would correspond to a horizontal line with $\forall q$, $D\left(q\right)=D_f$. The spectra for other sections are very similar to those on display, an evidence of the robustness of temporal correlations.Reuse & Permissions

Figure 7

(Color online) Compression creep experiment performed on an ice single crystals ($T=-10°\mathrm{C}$; and applied stress of $0.87\mathrm{MPa}$). AE amplitudes cumulated from the onset of loading, $\sum A_0$, as a function of creep time. $\sum A_0$ is a proxy of the total strain and its time evolution appears smooth at the scale of the entire test (main graph), although a staircaselike curve is observed when one focuses over a small time window (inset). This smoothing of the curve towards large time scales is the result of truncated distributions of dislocation avalanche sizes. This figure is very similar to the electromagnetic emission accompanying plastic deformation of ionic crystals that appears smooth at large time scale and staircaselike at small time scale (Ref. 33).Reuse & Permissions