Bounds for the time to failure of hierarchical systems of fracture

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.