Bounds for the time to failure of hierarchical systems of fracture

TL;DR

This paper develops an algebraic method to compute and bound the failure time of hierarchical fracture systems, providing evidence that failure time remains non-zero even for infinitely large systems, which is significant for engineering and geophysics.

Contribution

It introduces an exact iterative algebraic approach to determine failure times and bounds for large hierarchical systems of fracture, confirming the non-zero failure time hypothesis.

Findings

Lower bounds for failure time are rigorously derived.

Asymptotic analysis supports non-zero failure time for infinite systems.

Method enables analysis of large-scale hierarchical fracture models.

Abstract

For years limited Monte Carlo simulations have led to the suspicion that the time to failure of hierarchically organized load-transfer models of fracture is non-zero for sets of infinite size. This fact could have a profound significance in engineering practice and also in geophysics. Here, we develop an exact algebraic iterative method to compute the successive time intervals for individual breaking in systems of height $n$ in terms of the information calculated in the previous height $n-1$. As a byproduct of this method, rigorous lower and higher bounds for the time to failure of very large systems are easily obtained. The asymptotic behavior of the resulting lower bound leads to the evidence that the above mentioned suspicion is actually true.