Inequality indices for heterogeneous systems: a tool for failure prediction

Abstract: We have numerically studied a mean-field fiber bundle model of fracture at a non-zero temperature and acted by a constant external tensile stress. The individual fibers fail (local damage) due to creep-like dynamics that lead up to a catastrophic breakdown (global failure). We quantify the variations in sizes of the resulting avalanches by calculating the Lorenz function and two inequality indices -- Gini ($g$) and Kolkata ($k$) indices -- derived from the Lorenz function. We show that the two indices cross just prior to the failure point when the dynamics goes through intermittent avalanches. For a continuous failure dynamics (finite numbers of fibers breaking at each time step), the crossing does not happen. However, in that phase, the usual prediction method i.e., linear relation between the time of minimum strain-rate and failure time, holds. The boundary between continuous and intermittent dynamics is very close to the boundary between crossing and non-crossing of the two indices in the temperature-stress phase space.