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

Lattice and network models with elements that have random strength are usef ul tools in explaining various statistical features of failure in heterogen eous materials, including the evolution of failure clusters and overall str ength distributions and size effects. Models have included random fuse and spring networks where Monte Carlo simulation coupled to scaling analysis fr om percolation theory has been a common approach. Unfortunately, severe com putational demands have limited the network sizes that can be treated. To g ain insight at large size scales, interest has returned to idealized fiber bundle models in one dimension. Many models can be solved exactly or asympt otically in increasing size n, but at the expense of major simplification o f the local stress redistribution mechanism. Models have typically assumed either equal load-sharing among nonfailed elements, or nearest-neighbor, lo cal load-sharing (LLS) where a failed element redistributes its load onto i ts two nearest flanking survivors. The present work considers a one-dimensi onal fiber bundle model under tapered load sharing (TLS), which assumes loa d 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 critic al configurations of broken fibers, which lead to bundle failure under a gi ven load. We obtain rigorous asymptotic results for the strength distributi on and size effect, as n --> infinity, with precisely determined constants and exponents. The results are a nontrivial extension of those under US 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 ne ighbor effects weaken the material beyond what is predicted under LLS keepi ng only nearest neighbor overloads. Our results question the validity of sc aling relationships that are based largely on Monte Carlo simulations on ne tworks of limited size since some failure configurations appear only in ext remely large bundles. The dilemma has much in common with the Petersburg pa radox.