Universal avalanche statistics and triggering close to failure in a mean-field model of rheological fracture

The hypothesis of critical failure relates the presence of an ultimate stability point in the structural constitutive equation of materials to a divergence of characteristic scales in the microscopic dynamics responsible for deformation. Avalanche models involving critical failure have determined common universality classes for stick-slip processes and fracture. However, not all empirical failure processes exhibit the trademarks of criticality. The rheological properties of materials introduce dissipation, usually reproduced in conceptual models as a hardening of the coarse grained elements of the system. Here, we investigate the effects of transient hardening on (i) the activity rate and (ii) the statistical properties of avalanches. We find the explicit representation of transient hardening in the presence of generalized viscoelasticity and solve the corresponding mean-field model of fracture. In the quasistatic limit, the accelerated energy release is invariant with respect to rheology and the avalanche propagation can be reinterpreted in terms of a stochastic counting process. A single universality class can be defined from such analogy, and all statistical properties depend only on the distance to criticality. We also prove that interevent correlations emerge due to the hardening—even in the quasistatic limit—that can be interpreted as “aftershocks” and “foreshocks.”

Figure 1

(a) Schematic representation of the standard DFBM constituted by an ensemble of $M$ parallel elastic-brittle elements, $N$ of which are broken because their $S_i<E\varepsilon$; (b),(c) Sketch of the strain release $E\varepsilon$ due to the breaking of a fiber $S_i$ under quasistatic stress driving, generating an avalanche of size $\mathrm{\Delta }$. (c) The avalanche stops when the system regains stability, as represented by the constitutive curve (gray line).

Figure 2

Left: schematic representation of the generalized Zener element. Right: Temporal response $E\varepsilon$ of the generalized Zener element to a stepwise increase in the load ${\sigma }_l$. Five example behaviors are represented for different values of $0\le h\le 1$ and $0\le \alpha \le 1$ (see main text for details).

Figure 3

(a) Schematic representation of the generalized rheological democratic fiber bundle model (GVE-DFBM), where the elastic element has been substituted by a generalized Zener solid element. (b)–(e) Sketch of an avalanche process in the GVE-DFBM with the same fibers considered in Fig. 1. (b) The system is hardened a factor $H\left(0\right)$ for each failed fiber (blue lines). (c) The initial avalanche, of size ${\mathrm{\Delta }}^I$, stops sooner than in the standard DFBM. (d) During the creeping phase and without driving, the hardening is relaxed just enough to activate the next failure. (e) The process is repeated until all the $\mathrm{\Delta }$ fibers of Fig. 1 are broken. All transient terms are relaxed to 0 before resuming the quasistatic driving. While part of the brittle deformation in the standard DFBM model occur as creep in the GVE-DFBM, the number of broken fibers remains the same.

Figure 4

(a) Example of an avalanche defined as the hitting times of a stationary random counting process $\xi \left(\mathrm{\Delta }\right)$ to the boundary $B\mathrm{\Delta }$. Maroon area shows the landscape $\xi \left(\mathrm{\Delta }\right)$ drawn from random Poisson increments at each elementary step $\mathrm{\Delta }$. The avalanche stops at the first $\mathrm{\Delta }$ value with $\xi \left(\mathrm{\Delta }\right)>B\mathrm{\Delta }$ represented as a dashed gray line. The salmon area represents the elements failing at the time unit, with values $\xi \left(\mathrm{\Delta }\right)<B\left[\mathrm{\Delta }\right(t-1)+1]$ (black solid line). (b) The temporal profile of the same avalanche, defining the amplitude $A$, duration $T$, and size $\mathrm{\Delta }$ (see Sec. 3b).

Figure 5

Distribution of $N={10}^7$ hitting times for the Poisson process ${\xi }_{\mathrm{\Delta }}$ to the boundary $B\mathrm{\Delta }$ tapped at $\mathrm{\Delta }={10}^4$ for different values of $B$. (a) Scaling according to Eq. (16) compared to the ansatz (grey thick line) in Eq. (17). (b) Distribution before scaling, compared to the power law expected by $B=1$ (gray thick line).

Figure 6

Scatter plots of magnitude pairs: (a) $A,T$, (b) $S,T$, (c) $E,T$ found for different values of $B$. (Since the values are discrete a uniform unitarian jitter in the $x$ and $y$ axes has been added to better visualize the density of the point clouds.) The conditional averages $\langle x|T\rangle$ are shown for $B=1$ (black error lines). The expected relations derived from Eq. (20) are consistent with an exponent value $\sigma \nu z=1/2$.

Figure 7

Complementary cumulative distribution function (CCDF) of avalanche sizes scaled by (a) their mean value ($\langle \mathrm{\Delta }\rangle$) and (b) distance to criticality ($f_B$), obtained by the numerical simulations of the GVE-DFBM with $m=1$ (standard Zener elements) and $M={10}^7$ evaluated in intervals of $\sigma$. We compare the results for $h=0.4$ (in blue) with the standard DFBM ($h=0$ in black) and the universal distribution for the analogous hitting time problem according to Eq. (17) (gray thick lines).

Figure 8

Normalized activity rate (red) as the number of events for unit of time ($N^{-1}dn/d{f}_{\sigma }$), average avalanche size ($\langle \mathrm{\Delta }\rangle$ in blue), and number of fibers failed for unit of time $d\mathrm{\Delta }/dt$ for the GVE-DFBM with values of $h=0.0$, 0.1, 0.4, and 0.8. Light lines serve as a guide to the eye with the analytical solution found for the thermodynamic limit. The temporal scale is expressed in terms to the distance to failure in stress, in order to empathize the agreement with the power-law divergences at the failure point.

Figure 9

Temporal correlations in a Zener DFBM (i.e., GVE-DFBM with $\alpha =1$) for different values of $h$. (a) Activity rate as function of the time ($t_0$) since the beginning of the cluster and (b) distribution of waiting times between consecutive events within a cluster. Time is given in units of $\tau$. Straight lines are added as a guide to the eye illustrating the power-law regimes.