Probability distributions for the strength of composite materials II: A convergent sequence of tight bounds

Abstract

A sequence of convergent upper bounds is developed for the probability distribution of strength of composite materials. The analysis is based on the well-known chain-of-bundles model, and local load sharing is assumed for the nonfailed fiber elements in each bundle. The bounds are based on the occurrence ofk or more adjacent broken fibers in a bundle, an event which is necessary but not sufficient for the failure of the material. However, we find that given the loadL on the composite, some value ofk denotedk*(L) is critical in that a group of failed elements once reaching sizek*(L) will catastrophically increase in size with virtual certainty. IfL happens to be approximately the median strength of the composite, then the bound based onk =k*(L) adjacent breaks is virtually identical to the true probability distribution of composite strength; indeed, the convergence of the sequence of bounds becomes virtually complete ask exceedsk*(L). We show that the strength distribution for the composite essentially has weakest link structure in terms of a characteristic distribution functionW(x),x ≧ 0 which depends on the load sharing and on the probability distribution for fiber element strength. Typical cases are considered under a Weibull distribution for fiber strength and under a double version which has the effect of putting a ceiling on fiber strength. We show that in typical situations, predictions using the double Weibull distribution are not as one might guess, and its use is unjustified in many cases.