Acceleration and localization of subcritical crack growth in a natural composite material

Catastrophic failure of natural and engineered materials is often preceded by an acceleration and localization of damage that can be observed indirectly from acoustic emissions (AE) generated by the nucleation and growth of microcracks. In this paper we present a detailed investigation of the statistical properties and spatiotemporal characteristics of AE signals generated during triaxial compression of a sandstone sample. We demonstrate that the AE event amplitudes and interevent times are characterized by scaling distributions with shapes that remain invariant during most of the loading sequence. Localization of the AE activity on an incipient fault plane is associated with growth in AE rate in the form of a time-reversed Omori law with an exponent near 1. The experimental findings are interpreted using a model that assumes scale-invariant growth of the dominating crack or fault zone, consistent with the Dugdale-Barenblatt “process zone” model. We determine formal relationships between fault size, fault growth rate, and AE event rate, which are found to be consistent with the experimental observations. From these relations, we conclude that relatively slow growth of a subcritical fault may be associated with a significantly more rapid increase of the AE rate and that monitoring AE rate may therefore provide more reliable predictors of incipient failure than direct monitoring of the growing fault.

Figure 1

Sketch of a typical evolution of (sub-) critical crack size $C\left(t\right)$ (full black line) and event rate $\dot{n}\left(t\right)$ (dashed red line) according to Eqs. (3) and (9), respectively; $\nu =1/20$, time is normalized by the failure time $t_f$.

Figure 2

(a) Histogram of the perpendicular distances of the event locations to the final failure plane for different subsequences of the record. The inset shows the curves for the first and last 5 000 events in a double logarithmic presentation. The black straight line is a reference curve with a power law exponent of $-2$. (b) Location of acoustic emissions (dots), projected onto a cross-section of the the x-z plane, where $z=0$ defines the eventual fault plane. Events numbered $i=12000$–17 000 are indicated in dark yellow (grey) and $i=20001$–25 000 in black. In both plots the red (dark gray) lines indicate the damage zone of width $\pm 2.5$ mm around the failure plane. The green dashed lines in plot (b) indicate the boundary and the green dash-dotted line indicates the middle axis of the specimen, which is also the direction of the applied stress.

Figure 3

Evolution of various experimental parameters as a function of the AE event number $i$; on the left-hand side, event number refers to all events; on the right-hand side, only events located within the damage zone surrounding the final failure plane are counted. (a) and (b) Interevent times: black, all interevent times ${\tau }_i$; brown (gray), interevent times averaged over windows of 50 events; in the regions R1–R4, marked by red (dark gray) boxes, the interevent times can be fitted by exponential functions. (c) and (d) Differential stress ${\sigma }_i$ in units of MPa. (e) and (f) Average spatial separation of nearest neighbor events (mean nearest-neighbor distance, $\langle \mathrm{nnd}\rangle )$, evaluated over a moving window of 1 000 events.

Figure 4

Signatures of the scaled interevent time statistics in the different exponential relaxation regimes: top row, covariance of interevent times separated by $s$ events; bottom row, mean scaled interevent time $M_{50}\left(k\right)$ (black) and standard deviation $S_{50}\left(k\right)$ [red (gray)], evaluated over disjoint windows of 50 events starting at event $k$. The last column shows the results for random surrogate data (2000 time values drawn independently from an exponential distribution with mean value 0.1 s).

Figure 5

Statistics of the scaled interevent times $\tilde{\tau }=\tau exp\left(ai\right)$ for the different exponential relaxation regimes R1–R4; the corresponding mean values are $\langle \tilde{\tau }\rangle =(0.052\pm 0.0008)$ s for R1, $\langle \tilde{\tau }\rangle =(0.0071\pm 0.00011)$ s for R2, $\langle \tilde{\tau }\rangle =(0.536\pm 0.011)$ s for R3, $\langle \tilde{\tau }\rangle =(0.0036\pm 0.00014)$ s for R4.

Figure 6

Black solid curve, evolution of the mean event number $n$ as a function of time-to-failure $t/t_f$; the straight line on the semilogarithmic plot follows exactly the prediction of Eq. (8); red dash-dotted curve, coefficient of variation of $n$.

Figure 7

Distributions of event amplitudes for different AE event subsequences; the right graphs represent events contained within the damage zone. All amplitudes have been normalized by the mean values for the respective subsequence.

Figure 8

Top: evolution of event amplitude in the run-up to failure; left, all events; right, only events in damage zone around final failure plane; all amplitudes have been normalized by the average amplitude over all events. Bottom: evolution of event rate in the run-up to failure; left, all events; right, only events in damage zone around final failure plane; black curves, raw data; green (light gray) curves, data pertaining to second acceleration regime; blue (dark gray) curves, data from first acceleration regime with shifted failure time $t_f^{\prime }=t_f-(t_1-t_2)$; black lines in the top figures, Eq. (3) with AE amplitude $\propto C$ and $\nu =1/20$; black straight lines in the bottom figures, Eq. (9).

Figure 9

Plan view of the final failure plane of the Clashach sandstone specimen. The symbols indicate the location of acoustic emissions measured around the final plane for (a) events 500–1 500, (b) 1 500–2 500, (c) 5 000–6 000, (d) 6 000–7 000, and (e) 6 500–7 500. Events with small amplitudes $\left(m/\langle m\rangle \le 1.1\right)$ are indicated by circles, medium-sized events $\left(1.1<m/\langle m\rangle \le 2.2\right)$ are indicated by squares, and triangles indicate events with large amplitudes $\left(2.2<m/\langle m\rangle \right)$.