#### Vol. 1, No. 8, 2006

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Statistical strength of a twisted fiber bundle: an extension of daniels equal-load-sharing parallel bundle theory

### Pankaj K. Porwal, Irene J. Beyerlein and S. Leigh Phoenix

Vol. 1 (2006), No. 8, 1425–1447
##### Abstract

In this work, we extend the statistical strength model of Daniels for a parallel fiber bundle to a twisted bundle with an ideal helical structure. The bundle is clamped at each end in such a way that it has no slack fibers in the unloaded state. The fibers are linearly elastic and continuous, and have random strengths following a Weibull distribution with Weibull shape parameter $\rho$. We calculate the stress redistribution from failed to surviving fibers according to a twist-modified equal load sharing (TM-ELS) rule, introduced here. The effect of the twist is modeled analytically by two approaches, one called geometrical averaging, in which the fiber helix angles are averaged, and the other called statistical averaging, in which the fiber failure probabilities are averaged. In both probability models, the bundle strength distributions remain asymptotically Gaussian, as in Daniels’ original model; however, the associated mean and standard deviation are additionally altered by the surface twist angle. To validate these theories, a Monte Carlo model is developed to simulate fiber break initiation and progression within a cross-sectional plane under tension. For all values of surface twist angle ${\alpha }_{s}$, $\rho$ and bundle size studied, the simulated strength distributions are shown to be strongly Gaussian. Transitions in failure mode from diffuse, across the bundle cross-section, to localized near the center of the bundle occur when ${\alpha }_{s}$ and $\rho$ increase and the bundle size decreases, in spite of application of a diffuse-type loading sharing rule, TM-ELS. Both analytical models provide similar results which are in excellent agreement with the simulated results. For the most part, we consider the bundle to be short enough that interfiber friction plays no role in the stress redistribution. However, to demonstrate its importance in long bundles, we mimic the effects of interfiber friction by considering a chain of such bundles where the bundle length is chosen to approximate the characteristic length of unloading around breaks.

##### Keywords
twisted fiber bundle or yarn, Monte Carlo simulation, statistical strength, twist modified equal load sharing (TM-ELS), ideal helical structure, interfiber friction, chain-of-bundles model