An exact closed form solution for fragmentation of Weibull fibers in a single filament composite with applications to fiber-reinforced ceramics

Abstract

Exact equations are derived governing the evolution of fiber fragments in a Weibull fiber loaded according to the “single filament composite test”. These equations differ from those formulated by others who have made a priori assumptions on the shape of the fragment distribution that are shown to be incorrect. An explicit closed form solution of the governing equations is derived for arbitrary Weibull modulus ϱ and for random initial breaks with exponentially distributed spacings of a given normalised rate α along the fiber. In particular, the special case of unique fiber strength ( ϱ = ∞), which is adapted from an exact solution of Widom [ J. Chem. Phys. 44, 3888–3894 (1966)], is a limiting case of our solution. Furthermore, the solution for the case of ϱ = 0 can be expressed in terms of elementary functions. The limiting distribution function for normalised fragment length is also obtained in closed form for all ϱ ⩾ 0. The convergence of this distribution function to that for the case of unique fiber strength, where the normalised fragment lengths x lie between 1 2 and 1, is very slow being O(ϱ −1 2 ) . For finite ϱ the lower tail asymptotics of the limiting distribution function are proportional to (2 x) 2 ϱ + 1 , x ⩾ 0, so that a Weibull plot there has a slope of 2 ϱ + 1. In fact, in the limit as ϱ → 0, the normalised fragment length follows an exponential distribution, which turns out to be a good approximation for 0 ⩽ ϱ < 1. We apply our closed form solution to the study of the strength of large fiber-reinforced ceramic composites. The ultimate strength of such composites is obtained in closed form for all ϱ. Expressions are given in terms of elementary functions which allow computation of the composite ultimate strength μ ∗ to any degree of accuracy. Our results show that the composite strength μ ∗ is unique and occurs at a bounded dimensionless stress s ∗ for all ϱ > 0, contrary to assertions made by others. In particular, s ∗ is estimated to within 2% by the formula s ∗ = [6(1 − √ 1 − 2 3ϱ )] 1 (1 + ϱ) for ϱ ⩾ 1, and we also give excellent approximations for general ϱ > 0.