Investigating the Local Load Sharing Fibre Bundle Model in Higher Dimensions

Abstract

We introduce a generalization of the history-independent local load sharing fibre bundle model that is valid in arbitrarily high dimensions, and study this model numerically in one to five dimensions.

Simulations show that as the dimension increases, the local load sharing model behaves more and more like the equal load sharing fibre bundle model, with the area of the difference between their averaged strain curves following a power law, indicating an infinite critical dimension where the two models converge. The exponent is estimated to be 3.4 ± 0.2 for the uniform threshold distribution P(x) = x and 3.7 ± 0.3 for P(x) = 1 - exp(-x), both in the limit of large fibre bundles.

The burst size distribution when P(x) = x has also been studied for bundle sizes of order N = 10^4 in one through five dimensions. A sudden, qualitative shift is seen when going from one to two dimensions, with the power law behaviour valid for sufficiently small bursts giving an exponent of 4.6 in one dimension and 2.6 in higher dimensions. Additionally, the higher dimensions retain a power law-like behaviour for much larger bursts than in one dimension, where the burst size distribution begins to deviate from a power law even for very small bursts.