Statistical mechanics of interacting fiber bundles

Abstract

We consider quasistatic fiber bundle models with interactions. Classical load sharing rules are considered, i.e. local, global or decaying as a power-law of distance. All fibers are identically elastic, initially intact, and break at a random threshold picked from a quenched disorder (q.d.) distribution. We are interested in the probability distribution of configurations of broken fibers at a given elongation, averaged over all possible realizations of the underlying q.d.. This distribution is accessed by mapping the threshold set space onto the configurational space, each path corresponding to the evolution of a bundle corresponding to a realized q.d.. Using a perturbative approach allows to obtain this distribution to leading order in the interactions. This maps this system onto classical statistical mechanics models, i.e. percolation, standard or generalized Ising models depending on the range of the interactions chosen in the load sharing rule. This relates such q.d. based systems to standard classical mechanics, which allows to derive observables of the system, as e.g. correlation lengths. The thermodynamic parameters formally equivalent to temperature and chemical potential, are functions of the externally imposed deformation, depending on the load sharing rule and the choice of the q.d. distribution.