Lower-tail approximations for the probability of failure of three-dimensional fibrous composites with hexagonal geometry

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The chain-of-bundles model for the strength of unidirectional fibrous materials is extended to cover 3D situations wherein the parallel filaments are arranged laterally in a hexagonal array. Within each bundle, the strengths of the fibres vary statistically and share load according to a local load-sharing rule, a rule which describes how the loads of failed fibres are redistributed on to nearby surviving fibres. We consider two idealized versions of this rule, one geometrically motivated and the other more mechanically motivated. We extend earlier asymptotic techniques for the 2D planar problem to the present 3D case, and obtain various approximations for the probability distribution for material strength. The Weibull distribution again emerges as central to the results, but the calculation of its shape and scale parameters is greatly complicated by the large number of new failure configurations that may arise in the hexagonal array of fibres. Earlier 2D results connecting the Weibull shape parameter to the critical failure sequence size do not in general hold in 3D settings. The general character of the results, however, is the same as in the 2D setting, with 3D materials being stronger because of the reduced severity of the fibre overloads in the hexagonal array. Also, the two local load-sharing rules though quite different in character yield surprisingly similar numerical results.