  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.
