Fracture of Brittle Solids. I. Distribution Function for Fragment Size in Single Fracture (Theoretical)

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Abstract
For single fracture of a brittle solid, the distribution function for fragment size is obtained on the basis of Griffith's theory of brittle strength (which postulates crack propagation when pre-existent flaws are activated by stress). Three assumptions are made: (a) Fracture proceeds by activation of flaws in the volume of the specimen, in fracture surfaces through the specimen, and in the edges produced by fracture surfaces, (b) the corresponding volume, facial, and edge flaws are distributed independently of each other when activated, and (c) activated flaws of a particular type are distributed at random, individually and collectively, in the sense of Fry. These assumptions yield directly and uniquely the probability dp(l,s,v) of formation of a fragment with total edge length, total face area, and total volume in the ranges l to l+dl, s to s+ds, and v to v+dv, respectively, as e−QdQ in the general case, with Q linear in l, s, and v. The derivation yielding this Poisson form requires no assumption on the shape of a fragment or the type of fracture surface. The number dn(l,s,v) of fragments with total edge length, total face area, and total volume in the ranges l to l+dl, s to s+ds, and v to v+dv, respectively, is evaluated as the product of dp(l,s,v) by the a priori number q of particles with these values of l, s, and v. The distribution function dn(l,s,v) meets the necessary physical requirement that the fracture process conserve volume independently of particle shape. By assuming that all fragments are geometrically similar, one can replace dp(l,s,v) and dn(l,s,v) by forms, p(x)dx and n(x)dx, respectively, which depend only on a mean linear dimension x of a fragment. The resulting expression for y, the cumulative fraction of the initial volume corresponding to fragments of dimension up to x, then yields rigorously forms of the empirical equations of Schuhmann and of Rosin and Rammler for this quantity, as limiting cases for x small. The conclusion follows that activation of edge flaws represents the dominant mode of fragmentation, in general. The moments of the distribution corresponding to the total number, the total edge length, and the total surface of the fragments are divergent; this anomaly is explained as the result of neglect of flaw depletion.

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