A strength reliability model by Markov process of unidirectional composites with fibers placed in hexagonal arrays

This paper proposes a strength reliability model based on a Markov process for unidirectional composites with fibers in a hexagonal array. The model assumes that a group of fiber breaking points, a so-called cluster, evolves with increased stress. The cluster evolution process branches because of various fiber-breakage paths. Load-sharing structure of intact fibers around clusters was estimated from geometric and mechanical local load-sharing rules. Composites fracture if a cluster achieves a critical size, so the model expresses a fracture criterion by setting an absorbing state. Next, the author constituted a state transition diagram concerning cluster evolutions of 1-fiber to 7-fiber breaks and analytically solved simultaneous differential equations obtained from the diagram. Results showed that, as critical cluster size increases, slope of the fracture probability distribution is given in a Weibull probability scale as follows: mc=i×mf (i, the number of broken fibers in a cluster; mc and mf, Weibull shape parameters for fracture probabilities of a critical cluster and fiber strength, respectively). This relation between mc and mf had been shown by Smith et al. [Proc. R. Soc. London, A 388 (1983) 353–391], but the present study demonstrated it analytically without any lower tail of the Weibull distribution used in that paper. In addition, the present model can be approximated by a one-state birth model.