Exact Solution to an Interacting Extreme-Value Problem: The Pure-Flaw Model

Simple models play a key role in the microstructural analysis of mechanical failure in composites and other materials with complex and often disordered microstructures. Although equal load-sharing-models are amenable to rigorous statistical analysis, problems with local load enhancements near failed regions of the material have so far resisted exact analysis. Here we show for the first time, that some of the simpler of these local-load-sharing models can be solved exactly using a sub-stochastic matrix method. For diluted fiber bundles with local load sharing, it is possible to find a compact expression for the characteristic equation of the sub-stochastic matrix, and from it obtain an asymptotic expansion for the largest eigenvalue of the matrix. This in turn gives the asymptotic behavior of the size effect and statistics of the fiber-bundle models. We summarize these results, and show that the important features of the exact result can be obtained from a single scaling analysis we had developed previously. We also compare the statistics of fracture with the appropriate limiting extreme-value survival distribution (a Gumbel distribution), and, as previously indicated by Harlow and Phoenix, note that the Gumbel distribution performs quite poorly in this problem. We comment on the physical origin of this discrepancy.