Social Inequality Analysis of Fiber Bundle Model Statistics and Prediction of Materials Failure

We note that the social inequality, represented by the Lorenz function obtained plotting the fraction of wealth possessed by the faction of people (starting from the poorest in an economy), or the plot or function representing the citation numbers against the respective number of papers by a scientist (starting from the highest cited paper in scientometrics), captured by the corresponding inequality indices (namely the Kolkata $k$ and the Hirsch $h$ indices respectively), are given by the fixed points of these nonlinear functions. It has been shown that under extreme competitions (in the markets or in the universities), the $k$ index approaches to an universal limiting value, as the dynamics of competition progresses. We introduce and study these indices for the inequalities of (pre-failure) avalanches (obtainable from ultrasonic emissions), given by their nonlinear size distributions in the Fiber Bundle Models (FBM) of non-brittle materials. We will show how a prior knowledge of this terminal and (almost) universal value of the $k$ index (for a range of values of the Weibull modulus characterizing the disorder, and also for uniformly dispersed disorder, in the FBM) for avalanche distributions (as the failure dynamics progresses) can help predicting the point (stress) or time (for uniform increasing rate of stress) for complete failure of the bundle. This observation has also been complemented by noting a similar (but not identical) behavior of the Hirsch index ($h$), redefined for such avalanche statistics.