Growth of breakdown susceptibility in random composites and the stick-slip model of earthquakes: Prediction of dielectric breakdown and other catastrophes

The responses to short duration pulses (of electric field, of additional ''particles,'' of a mechanical ''push due to blasting'' on any ''tectonic block,'' etc.) have been studied numerically for metal-insulator composites before dielectric breakdown, the Bak-Tang-Weisenfeld (BTW) (sandpile) model before the critical avalanches, and the Burridge-Knopoff stick-slip model of earthquakes. We show that, from the response to weak pulses of appropriate external field, one can estimate the growth of local failure correlations in such systems, giving the breakdown susceptibility. The study of this breakdown susceptibility, contributed to by the correlations of microscopic local failures, indicates universal behavior near the catastrophic (global) breakdown or the self-organized critical points. Its study can thus help in accurately locating the global breakdown or disaster point (much before its occurrence) by extrapolating the inverse breakdown susceptibility to its vanishing point. We have performed numerical studies of Laplace's equation of a dielectric with random bond conductors below its percolation threshold, of the dynamics of the BTW model, and of the dynamical equations of the array of blocks in the stick-slip model of earthquakes. The breakdown susceptibility has a power law growth (with the critical interval from the global breakdown threshold) in both the electric breakdown and in the BTW model. Accurate exponent values for the growth have been obtained in the BTW case. The growth of the susceptibility, coming from stress correlations, in the Burridge-Knopoff model of earthquakes is observed to be exponential in time, as it is for pulse susceptibility in this model.