#### Vol. 2, No. 4, 2007

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Statistical strength of twisted fiber bundles with load sharing controlled by frictional length scales

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

Vol. 2 (2007), No. 4, 773–791
##### Abstract

We develop Monte Carlo simulation and theory to study the statistical strength characteristics of twisted fiber bundles. These consist of fibers that follow a Weibull distribution for strength with shape parameter $\rho$, and are arranged in an ideal helical structure with surface helix angle ${\alpha }_{s}$. Fiber interactions are considered in terms of frictional forces that control stress recovery along broken fibers away from the breaks. A twist-modified global load sharing (TM-GLS) rule is developed for stress redistribution from fibers that are slipping and thus only partially loaded near the breaks. Expressions for the radial pressure distribution in the yarn and corresponding lengths of frictional zones in broken fibers in the various layers are derived considering the discrete nature of the fibers in the bundle. Three different characteristic length scales of strength development for a twisted bundle are proposed, which depend on friction coefficient, $f$, and surface twist angle, ${\alpha }_{s}$. These are ${\delta }_{c}^{min}$, ${\delta }_{c}^{avg}$, or ${\delta }_{c}^{max}$, arising from the consideration of the minimum, average, or maximum stress recovery length among the fibers in the bundle along its axis. We show that the normalized strengths of a twisted bundle with length equal to any one of these characteristic lengths approximately follow a Gaussian distribution. Compared to a TM-ELS (twist-modified equal load sharing) bundle, the TM-GLS bundle has improved strength because through friction a broken fiber can recover its stress within the bundle length. We also show that the relationship between the normalized bundle strength and ${\alpha }_{s}$ depends on the characteristic length scale used: for ${\delta }_{c}^{min}$ the normalized strength drops quickly with ${\alpha }_{s}$; for ${\delta }_{c}^{avg}$ it decreases as well, but at a slower rate; and for ${\delta }_{c}^{max}$ the normalized strength first attains a maximum at an optimal value of ${\alpha }_{s}$ before ultimately decreasing with ${\alpha }_{s}$. Finally, we compare the simulation results for optimal twist angle with experimental data in the literature and get excellent agreement.

##### Keywords
twisted fiber bundle or yarn, Monte Carlo simulation, statistical strength, global load sharing, characteristic length scales, friction effects, pressure development