Models of fragmentation with power law log-normal distributions

Two models of binary fragmentation are introduced in which a time-dependent transition size produces two regions of fragment sizes above and below the transition size. In the models, we consider a fixed rate of fragmentation for the largest fragment and two different rates of fragmentation for the two regions of sizes above and below the transition size. The models are solved exactly in the long time limit to reveal stable time-invariant solutions for the fragment size distributions. A rate of fragmentation proportional to the inverse of fragment size in the smaller size region produces a power-law distribution in that region. A rate of fragmentation combined of two terms, one proportional to the inverse of the fragment size and the other proportional to a logarithmic function of the fragment size, in the larger size region produces a log-normal distribution in that region. Special cases of the models with no fragmentation for the smaller fragments are also considered. The similarities between the stable distributions in our models and power-law log-normal distributions from experimental work on shock fragmentation of long thin glass rods and rupture of mercury droplets are investigated.