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
,
and are arranged in an ideal helical structure with surface helix angle
. 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,
, and surface
twist angle,
.
These are
,
, or
,
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
depends on the characteristic length scale used: for
the normalized strength
drops quickly with
;
for
it decreases as well, but at a slower rate; and for
the normalized strength first attains a maximum at an optimal value of
before ultimately
decreasing with
.
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
