A probability model for the time to fatigue failure of a fibrous composite with local load sharing

A composite material consisting of stiff, parallel fibers embedded in a flexible bonding matrix is considered. A steady tensile load is applied to the composite, and as time passes, breaks occur at various places along the fibers. Once enough breaks have accumulated in a cross-sectional area of the composite, it fails. This failure process is modeled by viewing the composite as a chain of bundles with the loads from failed fibers being shared in a localized fashion within each bundle. Using this model, approximations to the distribution of the time to failureof the composite are derived. Two types of approximations are considered. One is an approximation to the entire distribution with uniformly small absolute error, and the other is an approximation to the lower tail of the distribution with small relative error. Both approximations are based on the fact that for an appropriate choice of K the time to failure of the bundle is asymptotically equivalent to the time of the first appearance of a set of k or more adjacent failed fibers--a crack of size k--in one of the bundles.