Precursors of catastrophe in the Bak-Tang-Wiesenfeld, Manna, and random-fiber-bundle models of failure

Abstract

We have studied precursors of the global failure in some self-organized critical models of sandpile [in Bak-Tang-Wiesenfeld (BTW) and Manna models] and in the random-fiber-bundle model (RFB). In both BTW and Manna model, as one adds a small but fixed number of sand grains (heights) to any central site of the stable pile, the local dynamics starts and continues for an average relaxation time T and an average number of topplings Δ spread over a radial distance ζ. We find that these quantities all depend on the average height h_av of the pile and they all diverge as h_av approaches the critical height h_c from below: ˜(h_c-h_av)-δ, (h_c-h_av)-γ , and ζ(h_c-h_av)^-ν ;. Numerically, we find δ2.0,γ 1.2, and ν1.0 for both BTW and Manna model in two dimensions. In the strained RFB model, we find that the breakdown susceptibility X (giving the differential increment of the number of broken fibers due to increase in external load) and the relaxation time T, both diverge as the applied load or stress σ approaches the network failure threshold σ_c from below: X˜(σ_c-σ)^-1/2. These self-organized dynamical models of failure, therefore, show some definite precursors with robust power laws long before the failure point. Such well-characterized precursors should help predicting the global failure point of the systems in advance.