Absence of a tough-brittle transition in the statistical fracture of unidirectional composite tapes under local load sharing

We show analytically that a planar, unidirectional fibrous composite, which is an idealized random heterogeneous material consisting of stiff fibers of random strength embedded in parallel in a compliant matrix, fractures in a brittle manner when the fibers engage in idealized, local load sharing. Both the fibers and matrix are assumed time independent. This brittle behavior occurs irrespective of the disorder (variability) in fiber strengths, which we represent by a power distribution function. Our result goes far toward settling a long-standing question, not resolvable by computer simulations, regarding whether or not the brittle failure regime gives way to a tough, ductile-like failure regime as the variability in fiber strengths is increased past some threshold. We establish this result by calculating upper and lower bounding distributions for composite strength using the Chen-Stein theorem of extreme value statistical theory when failure events are dependent. These bounds both have weakest link character and, by comparing them with empirical strength distributions generated by Monte Carlo simulations, we find that the upper bound is a good approximation to the actual failure probability when the fiber strength variability is large. This regime is where previous models have broken down, raising speculation about a brittle-ductile transition.