Size effects in the distribution for strength of brittle matrix fibrous composites

Abstract

This paper addresses the effects of size (both length and width) on the probability distribution for strength of a composite consisting of brittle fibers aligned in a brittle matrix. The failure process involves quasi-periodic matrix cracking, frictional sliding of the fibers in fiber break zones, and fiber bridging of matrix cracks in a global load-sharing framework. The fiber strength follows the usual Poisson-Weibull model of random flaws along the length. We first consider a composite cross-section, and develop the probability distribution for its strength in terms of certain characteristic fiber strength and length scales and the number of fibers in the cross-section. This strength distribution turns out to be a Gaussian distribution. We also calculate refined estimates of its mean and standard deviation, taking advantage of some new results based on an exact closed form solution for fragmentation of fibers in a single filament composite. We then consider the strength of a composite having a length orders of magnitude greater than the characteristic fiber length. We develop predictions for the scaling of the strength vs composite length based on certain results from the statistical theory of extremes in Gaussian processes. For this we develop an estimate of the covariance between the strengths of two nearby cross-sections of the composite. We also develop results based on a weakest-link analysis in terms of composite links of a certain length somewhat shorter than the characteristic fiber length. We then favorably compare our analytical results to numerical results from a Monte Carlo simulation of the composite failure process. This Monte Carlo model is free of various assumptions made in the analysis. The comparison suggests that predictions of a composite strength are possible for composite lengths orders of magnitude beyond what Monte Carlo simulation programs can currently handle.