Precursors of catastrophe in the BTW, Manna and random fiber bundle models of failure

Abstract

We have studied precursors of the global failure in some self-organised critical models of sand-pile (in 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 (τ) and an average number of topplings (Δ) spread over a radial distance (ξ). 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: (Δ) (\sim (h_{c}-h_{av}))(^{-δ}), (τ\sim (h_{c}-h_{av})^{-γ}) and (ξ) (\sim) ((h_{c}-h_{av})^{-ν}). Numerically we find (δ\simeq 2.0), (γ\simeq 1.2) and (ν\simeq 1.0) for both BTW and Manna model in two dimensions. In the strained RFB model we find that the breakdown susceptibility (χ) (giving the differential increment of the number of broken fibers due to increase in external load) and the relaxation time (τ), both diverge as the applied load or stress (σ) approaches the network failure threshold (σ_{c}) from below: (χ) (\sim) ((σ_{c}) (-)(σ)^{-1/2}) and (τ) (\sim) ((σ_{c}) (-)(σ)^{-1/2}). These self-organised dynamical models of failure therefore show some definite precursors with robust power laws long before the failure point. Such well-characterised precursors should help predicting the global failure point of the systems in advance.