Derivation of the Weibull distribution based on physical principles and its connection to the Rosin–Rammler and lognormal distributions

Abstract

We describe a physically based derivation of the Weibull distribution with respect to fragmentation processes. In this approach we consider the result of a single‐event fragmentation leading to a branching tree of cracks that show geometric scale invariance(fractal behavior). With this approach, because the Rosin–Rammler type distribution is just the integral form of the Weibull distribution, it, too, has a physical basis. In further consideration of mass distributions developed by fragmentation processes, we show that one particular mass distribution closely resembles the empirical lognormal distribution. This result suggests that the successful use of the lognormal distribution to describe fragmentation distributions may have been simply fortuitous.