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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.

The theoretical results of Gilvarry for the size distribution of the fragments in single fracture have been verified experimentally by fracturing spherical glass specimens under compression. The fragments were contained by a gelatin matrix to inhibit secondary fracture and thus make conditions conform as closely as possible to single fracture. Experimental values of the probability of fracture as obtained by sieve analysis show the predicted linear variation with the mean dimension x of the particles, over reasonably large intermediate ranges of the variables. It is shown that a logarithmic-normal distribution does not represent the experimental results. The over-all data exhibit three local maxima in the differential probability of fracture as a function of x, whereas the theory permits only two. Agreement in the number of peaks is obtained by subtracting the contribution to the over-all probability of those fragments containing original surface of the specimen, which yields the true probability considered in the theory. In this manner, reasonably complete agreement between theory and experiment for single fracture is obtained. For plural fracture (carried out without use of gelatin), two additional peaks exist in the curve of the over-all differential probability vs x, as compared to the case for single fracture. The theory of Gilvarry is confirmed down to a fragment dimension of at least 1 μ by means of an electrical counting instrument, and checked by direct microscopic sizing to 5 μ. The results yield numerical values of internal flaw densities, and thus provide a tool to study the distribution of Griffith flaws existing internally in a solid.

An isothermal equation of state of a solid is considered, which contains as special cases the equations of Birch, Murnaghan, Bardeen, and others. The equation is generalized to arbitrary temperature by replacing two constants of the equation by temperature-dependent parameters, whose functional form is determined by considerations of thermodynamic consistency. The thermal properties of the solid implied by this equation of state are examined. It is shown that the generalized equation is consistent with the Mie-Grüneisen relation for the thermal pressure of the lattice, and that the corresponding Grüneisen parameter is only slightly dependent on temperature, in general. The form of the generalized equation of state at low temperature is exhibited as an explicit function of volume and temperature for a solid whose heat capacity obeys the Debye law. A comparison with pressure-volume data of Swenson for potassium at low temperatures shows excellent agreement of the generalized equation of state with experiment.

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