Distributions and size scalings for strength in a one-dimensional random lattice with load redistribution to nearest...

Lattice and network models with elements that have random strength are useful tools in explaining various statistical features of failure in heterogeneous materials, including the evolution of failure clusters and overall strength distributions and size effects. Models have included random fuse and spring networks where Monte Carlo simulation coupled to scaling analysis from percolation theory has been a common approach. Unfortunately, severe computational demands have limited the network sizes that can be treated. To gain insight at large size scales, interest has returned to idealized fiber bundle models in one dimension. Many models can be solved exactly or asymptotically in increasing size n, but at the expense of major simplification of the local stress redistribution mechanism. Models have typically assumed either equal load-sharing among nonfailed elements, or nearest-neighbor, local load-sharing (LLS) where a failed element redistributes its load onto its two nearest flanking survivors. The present work considers a one-dimensional fiber bundle model under tapered load sharing (TLS), which assumes load redistribution to both the nearest and next-nearest neighbors in a two-to-one ratio. This rule reflects features found in a discrete mechanics model for load transfer in two-dimensional fiber composites and planar lattices. We assume that elements have strength 1 or 0, with probability p and q=1-p, respectively. We determine the structure and probabilities for critical configurations of broken fibers, which lead to bundle failure under a given load. We obtain rigorous asymptotic results for the strength distribution and size effect, as n-->infinity, with precisely determined constants and exponents. The results are a nontrivial extension of those under LLS in that failure clusters are combinatorially much more complicated and contain many bridging fibers. Consequently, certain probabilities are eigenvalues from recursive equations arising from the structure of TLS. Next-nearest neighbor effects weaken the material beyond what is predicted under LLS keeping only nearest neighbor overloads. Our results question the validity of scaling relationships that are based largely on Monte Carlo simulations on networks of limited size since some failure configurations appear only in extremely large bundles. The dilemma has much in common with the Petersburg paradox.