Avalanche Behavior in Creep Failure of Disordered Materials

We present a mesoscale elastoplastic model of creep in disordered materials, which considers temperature-dependent stochastic activation of localized deformation events that are coupled by internal stresses, leading to collective avalanche dynamics. We generalize this stochastic plasticity model by introducing damage in terms of a local strength that decreases, on statistical average, with increasing local plastic strain. The model captures failure in terms of strain localization in a catastrophic shear band concomitant with a finite-time singularity of the creep rate. The statistics of avalanches in the run-up to failure is characterized by a decreasing avalanche exponent $\tau$ that, at failure, approaches the value $\tau =1.5$ typical of a critical branching process. The average avalanche rate exhibits an inverse Omori law as a function of time to failure. The distribution of interavalanche times turns out to be consistent with the epidemic-type aftershock sequences (ETAS) model of earthquake statistics.

Figure 1

Top: average creep-strain vs time curve for the default set of parameters, ${\mathrm{\Sigma }}_T=0.0075$; background: strain increments in individual avalanches during a single realization; center: plastic strain localization patterns at different stages of the creep curve; bottom: strain evolution of the plastic event correlation and localization parameters; failure occurs, in system units, at strain ${\epsilon }_{\mathrm{f}}=1.22$ and time $t_{\mathrm{f}}=1.73\times {10}^{12}$.

Figure 2

Top row (a) and (b): average avalanche size $⟨S⟩$; bottom row (c) and (d): avalanche rate $\dot{n}$ rescaled by failure time $t_{\mathrm{f}}$, both as functions of reduced time-to-failure $t^{\prime }$; left column: data for varying external stress, right column: data for varying effective temperature.

Figure 3

left: probability distribution $P(\mathrm{\Delta }t)$ of interavalanche waiting times, in units of mean waiting time, for different strains (different avalanche rates); right: probability distribution of waiting times scaled by mean avalanche rate as obtained from ETAS model for different rates $\mu$ of spontaneously triggered avalanches, figure taken from Touati et al. [26].

Figure 4

(a) Avalanche size distribution $P(S)\sim S^{-\tau }$ at different strains, inset: $\tau$ vs strain; (b) local stability distribution $P(\mathrm{\Phi })\sim {\mathrm{\Phi }}^{\theta }$ at different strains.

Figure 5

Avalanche size exponent $\tau$ vs time to failure; evolution depends on simulation parameters (external stress $\mathrm{\Sigma }$, disorder $k$, system size $L$ and effective temperature ${\mathrm{\Sigma }}_T$), exponents converge towards $\tau =1.5$ at failure.

Figure 6

Avalanche size distributions near failure ($\epsilon /{\epsilon }_{\mathrm{f}}=[0.995-1.0]$) for different system sizes $L$ rescaled with $L^{d_{\mathrm{f}}}$ and $d_{\mathrm{f}}=1.25$; inset: original distributions, the dashed lines mark the sizes of the final avalanches at which the simulations are terminated.