Fragment distributions for brittle rods with patterned breaking probabilities

We present a modeling framework for 1D fragmentation in brittle rods, in which the distribution of fragments is written explicitly in terms of the probability of breaks along the length of the rod. This work is motivated by the experimental observation of several preferred lengths in the fragment distribution of shattered brittle rods after dynamic buckling [J.R. Gladden, N.Z. Handzy, A. Belmonte, E. Villermaux, Dynamic buckling and fragmentation in brittle rods, Phys. Rev. Lett. 94 (2005) 35503]. Our approach allows for non-constant spatial breaking probabilities, which can lead to preferred fragment sizes, derived equivalently from either combinatorics or a nonhomogeneous Poisson process. The resulting relation qualitatively matches the experimentally observed fragment distribution, as well as some other common distributions, such as a power law with a cutoff.