Avalanches in compressed porous $\mathrm{Si}{\mathrm{O}}_2$-based materials

The failure dynamics in $\mathrm{Si}{\mathrm{O}}_2$-based porous materials under compression, namely the synthetic glass Gelsil and three natural sandstones, has been studied for slowly increasing compressive uniaxial stress with rates between 0.2 and 2.8 kPa/s. The measured collapsed dynamics is similar to Vycor, which is another synthetic porous $\mathrm{Si}{\mathrm{O}}_2$ glass similar to Gelsil but with a different porous mesostructure. Compression occurs by jerks of strain release and a major collapse at the failure point. The acoustic emission and shrinking of the samples during jerks are measured and analyzed. The energy of acoustic emission events, its duration, and waiting times between events show that the failure process follows avalanche criticality with power law statistics over ca. 4 decades with a power law exponent $\varepsilon \simeq$ 1.4 for the energy distribution. This exponent is consistent with the mean-field value for the collapse of granular media. Besides the absence of length, energy, and time scales, we demonstrate the existence of aftershock correlations during the failure process.

Figure 1

Sample height (a), the square of its time derivative (b), AE activity (c), amplitude (d), duration (e), and energy (f) of AE signals recorded during the compression experiment of the sample of Gel2.6(1). The vertical scales are logarithmic in (b), (c), (e), and (f).

Figure 2

(a) Log-log plot of the energy distribution of the AE events and (b) fitted energy exponent as a function of the lower fitting cut-off recorded for the sample of Gel2.6(1). The solid lines in (a) and (b) define the value of the fitted energy exponent, $\varepsilon =1.37\pm 0.03$, and the dashed lines in (b) represent the error limits.

Figure 3

(a) Number of aftershocks per unit time as a function of the time distance to the main shock for the sample of Gel2.6(1). Different symbols indicate different MS energy windows in aJ. The number of sequences averaged is 2343 for $1<E_{\text{MS}}<{10}^1$ (squares), 1077 for ${10}^1<E_{\text{MS}}<{10}^2$ (circles), 487 for ${10}^2<E_{\text{MS}}<{10}^3$ (triangles up), and 195 for ${10}^3<E_{\text{MS}}<{10}^4$ (triangles down). (b) Scaled plot with productivity exponent $\alpha =0.5$. The thin solid line defines the slope $p\simeq 0.71$.

Figure 4

Distribution of waiting times for different values of $E_{\text{min}}$ from the experiments with the sample of Gel2.6(1). The thin solid lines correspond to the exponents $1-\nu \simeq 1.02$ (left) and $2+\xi \simeq 2.8$ (right). The number of signals studied is 5412 for $E>{10}^{-1}$ aJ (squares), 4225 for $E>1$ aJ (circles), 1882 for $E>{10}^1$ aJ (triangles up), 805 for $E>{10}^2$ aJ (triangles down), and 318 for $E>{10}^3$ aJ (diamonds).

Figure 5

Sample height (a), the square of its derivative (b), AE activity (c), amplitude (d), duration (e), and energy (f) of AE signals recorded during the compression experiment of the sample of LGsan. The vertical scales are logarithmic in (b), (c), (e), and (f).

Figure 6

Sample height (a), the square of its derivative (b), AE activity (c), amplitude (d), duration (e), and energy (f) of AE signals recorded during the compression experiment of the sample of Rsan. The vertical scales are logarithmic in (b), (c), (e), and (f).

Figure 7

(a) Log-log plot of the energy distribution of the AE events and (b) energy exponent $\varepsilon \simeq 1.48\pm 0.08$ as a function of the lower fitting cut-off recorded for the sample of LGsan. The solid line defines the value of the effective critical exponent and the dashed lines represent the error.

Figure 8

(a) Scaled representation of number of aftershocks per unit time as a function of the time distance to the main shock for the sample LGsan. Best collapse corresponds to $\alpha =0.84$. The continuous line indicates the fitted slope $p\simeq 0.78$. The symbols indicate the different MS energy windows in aJ. The number of sequences averaged are 426 for ${10}^{-1}<E_{\text{MS}}<1$ (squares), 755 for $1<E_{\text{MS}}<{10}^1$ (circles), 136 for ${10}^1<E_{\text{MS}}<{10}^2$ (triangles up), and 46 for ${10}^2<E_{\text{MS}}<{10}^3$ (triangles down). (b) Scaled representation of the waiting times distribution. The continuous lines define the exponent $1-\nu =0.86$ (left) and $2+\xi =2.0$ (right). The number of data analyzed is 1401 for $E>0.1$ aJ (squares), 975 for $E>1$ aJ (circles), 220 for $E>10$ aJ (triangles up), and 80 for $E>100$ aJ (triangles down).

Figure 9

(a) Log-log plot of the energy distribution of the AE events. From top to bottom curves correspond to the following samples, respectively: Vycor, Gel5, Gel2.6, LGsan, Rsan, and Ysan. For the sake of clarity, except for Vycor, the curves are shifted. (b) Corresponding fitted energy exponents as a function of the lower fitting cut-off recorded for the same samples. The symbols correspond to Vycor (empty circles), Gel5 (squares), Gel2.6 (solid circles), LGsan (empty up triangle), Rsan (solid up triangle), and Ysan (down triangle). Only few representative error bars are shown.

Figure 10

(a) Log-log plot of the distribution of durations of AE events. From top to bottom data corresponds to the following samples, respectively: Vycor, Gel5, Gel2.6, LGsan, Rsan, and Ysan. Curves have been shifted vertically for clarity, except for Vycor. The continuous straight line correspond to the expected mean-field behavior. (b) Corresponding fitted duration exponents, $\tau$, as a function of the lower fitting cut-off recorded for the same samples. Symbols correspond to the different samples according to the legend in Fig. 9.

Figure 11

(a) Scaled representation of the number of aftershocks per unit time as a function of the time distance to the mainshock (Omori's plot) for all the studied Gelsil and sandstone samples. Results for Vycor are also included. The best scaling has been found with a productivity exponent $\alpha =0.6$. The slope of the thin continuous line is $p=0.75$. Energy values in the legend are expressed in aJ.

Figure 12

Scaled plot of the distribution of waiting times for different values of $E_{\text{min}}$ from Gelsil, sandstones, and Vycor samples. The continuous lines define the (average) exponents $1-\nu \simeq 1$ and $2+\xi \simeq 2.5$. Energy values in the legend are expressed in aJ.