Discrete models for dynamic fragmentation with mass consumption are considered. Simple inequalities for N(t), the total number of fragments, and D(t), the diversity are derived. For free fragmentation processes it is shown that the maxima of the number of fragments N and of the diversity D, which occur at times tN and tD, respectively, are related to the total mass M(t) by N(tN) ~ M(tN) and D(tD)
... [Show full abstract] ~ √(M(tD)), independently of the dimension and topology of the lattice. A one-dimensional model with position-dependent consumption rates is analysed. Exact solutions for N(t), M(t), the average fragment size s, and the average number of fragments of size s, n(s, t) are obtained. The existence of scaling regimes for n(s, t) is investigated.