Application of the Markov Process to the Chain-of-bundles Model with a Hexagonal Fiber Array.

Abstract

The Markov process is applied to the chain-of-bundles model with fibers packed in a hexagonal array, in order to obtain the analytical solution for the fracture probability of a unidirectional fiber composite. It was assumed in the process that a group consisting of fiber breakage points, the so-called cluster, evolves intermittently with an increase in stress. Then, the load sharing structure of unbroken fibers around the clusters was decided from geometric and mechanical local load sharing rules. The composite fractures if the cluster achieves a critical size. It was found that the number of possible fiber breakage paths to form a cluster is extremely increased, as the cluster size is increased. That is, a 6-break cluster exhibits eighty-two types of cluster configurations to form a 7-break critical cluster, while l- and 2-break clusters have only one type of configuration. By using the 2-parameter Weibull distribution function as a strength distribution of the fiber, the fracture probabilities of 2-break to 7-break critical clusters were analytically obtained. The results showed that the larger clusters more greatly reduce the width of the distribution, following mc= i ×mf (i : the number of broken fibers in a cluster, mc and mf : Weibull shape parameters for the fracture probabilities of a critical cluster and fiber strength, respectively). It was proved that, in addition, the analytical solutions can be approximated with a high accuracy by the one-state birth process.