Approach to failure in porous granular materials under compression

We investigate the approach to catastrophic failure in a model porous granular material undergoing uniaxial compression. A discrete element computational model is used to simulate both the microstructure of the material and the complex dynamics and feedbacks involved in local fracturing and the production of crackling noise. Under strain-controlled loading, microcracks initially nucleate in an uncorrelated way all over the sample. As loading proceeds the damage localizes into a narrow damage band inclined at 30${}^{\circ }$–45${}^{\circ }$ to the load direction. Inside the damage band the material is crushed into a poorly sorted mixture of mainly fine powder hosting some larger fragments. The mass probability density distribution of particles in the damage zone is a power law of exponent 2.1, similar to a value of 1.87 inferred from observations of the length distribution of wear products (gouge) in natural and laboratory faults. Dynamic bursts of radiated energy, analogous to acoustic emissions observed in laboratory experiments on porous sedimentary rocks, are identified as correlated trails or cascades of local ruptures that emerge from the stress redistribution process. As the system approaches macroscopic failure consecutive bursts become progressively more correlated. Their size distribution is also a power law, with an equivalent Gutenberg-Richter $b$ value of 1.22 averaged over the whole test, ranging from 3 to 0.5 at the time of failure, all similar to those observed in laboratory tests on granular sandstone samples. The formation of the damage band itself is marked by a decrease in the average distance between consecutive bursts and an emergent power-law correlation integral of event locations with a correlation dimension of 2.55, also similar to those observed in the laboratory (between 2.75 and 2.25).

Figure 1

(a) Preparation of the sample by sedimenting spherical particles with radius $R$ sampled from a probability distribution $p\left(R\right)$. The color code corresponds to the radius of the particles such that the smallest particles are dark blue while the biggest ones have a light red color. (b) A complete cylindrical sample of $20000$ particles used in the simulations of the fracture process.

Figure 2

Analysis of the structure of the sediment. (a) The probability distribution of the particle radius in the cylindrical sample. The numerically measured distribution (symbols) has a very good agreement with the analytic curve (continuous line). (b) The average radius $\langle R\rangle$ of particles as a function of height $z$ inside the cylinder. The value of $\langle R\rangle$ is nearly constant and it is equal to the sample average. (c) Probability distribution of the contact number $n_c$ of particles. For $n_c\ge 3$ the distribution can be well described by an exponential form (green dashed line). (d) The average contact number $\langle n_c\rangle$ proved to be independent of the height along the cylinder axis.

Figure 3

Constitutive behavior $\sigma \left(\varepsilon \right)$ of a single sample (left axis) together with the accumulated damage $d\left(\varepsilon \right)$ (right axis). The large arrow and the vertical dashed line indicate the onset of beam breaking ${\varepsilon }_0$ and the position of the maximum stress ${\sigma }_c$ of the constitutive curve at ${\varepsilon }_c$, respectively. The constitutive curve is presented in a dimensionless form by rescaling the stress with its critical value ${\sigma }_c$. The inset illustrates the loading condition such that the bottom and top layers, highlighted in gold (light gray), are clamped. The top layer moves downward with velocity $v_0$ while the bottom is fixed. The visual yield point (discernible departure from linearity) is marked as ${\sigma }_Y$.

Figure 4

(a) Reassembled sample in the final state of the simulation. Particles are placed back to their original location in the cylinder and they are colored according to the size of the fragment they originally belonged to. Hence, the two big residues are dark red, and single particles are shown in yellow, while other colors indicate fragment sizes between the two limits. The damage band can clearly be observed on the surface of the cylindrical sample. (b) To have a better view on the damage band, we only present the two big pieces of the specimen without the small-sized fragments. The width and the orientation of the damage band can be better inferred from this representation.

Figure 5

Mass distribution of fragments in the final state of the simulations. Three regimes can be distinguished: the two big residues illustrated in Fig. 4 give rise to the single peak at about a few tenths of the total sample mass $m_0$. A large fraction of fragments forms the fine powder which comprises single particles and very small fragments of a few particles. In the medium range the fragments have a power-law distribution with a stretched exponential cutoff. The red line represents the fit obtained with Eq. (12).

Figure 6

Time series of burst size $\Delta$ obtained from a single model run as a function of the strain of their appearance together with the corresponding constitutive curve. The burst size $\Delta$ has strong fluctuations and on the average it increases when the maximum of $\sigma \left(\varepsilon \right)$ is approached. The moving average of $\Delta$ was calculated over 50 consecutive events.

Figure 7

Size distribution of bursts $p\left(\Delta \right)$ calculated in windows of 200 consecutive events in the time series without overlap. The thick continuous lines represent fits with Eq. (18).

Figure 8

The exponent $\xi$ of the burst size distribution evaluated for windows of 200 consecutive events and for equidistant bins of strain during the loading process. The dashed horizontal line represents the value of $\xi$ obtained by taking into account all the events. For clarity, we also show a representative example of the constitutive curve.

Figure 9

(a) The size distribution of bursts evaluated in equally sized strain bins, $\Delta \varepsilon =0.0005$. The functional form of the curves can again be described by Eq. (18). The legend indicates the middle point of the strain bin divided by the critical strain ${\varepsilon }_c$. (b) Spatial position of bursts in a single sample. Single bursts are represented by spheres with a diameter proportional to the size of the event. For clarity, only events with size $\Delta >5$ are shown. The color code illustrates the time sequence of events such that dark and light colors stand for early and late bursts, respectively.

Figure 10

(a) Average size of bursts $\langle \Delta \rangle$ as a function of strain $\varepsilon$ together with the constitutive curve $\sigma \left(\varepsilon \right)$. (b) Average waiting time between consecutive events $\langle t_w\rangle$ and the average distance between bursts normalized by the diameter of the base circle of the cylinder as function of strain.

Figure 11

Correlation integral $C\left(r\right)$ for windows of $n=200$ consecutive events. The straight line represents a power law of exponent $2.55$.