The fracture of single, brittle, heterogeneous particle—statistical derivation of the mass distribution equation☆

Abstract Theoretical development of comminution models has come about in the last 10 years. These models assume that the material being broken is homogeneous and that the compression or energy wave is uniform throughout the specimen. Obviously, this is not the case because most comminuted materials are rocks with either many phases and/or definite grain size. The mathematical models deriving the size distribution of daughter fragments resulting from the comminution of brittle, single particles are of two schools. The first model, proposed by Gilvarry in 1961, was based on the concept of volume, surface, and edge flaw activation. The second, proposed by Meloy in 1960, was based on the concept that the crack density is a function of the energy in the comminution wave. The flaw theory has run into increasing theoretical problems and is now being abandoned. On the other hand, the crack density theory has been expanded and has successfully considered the case where there are inhomogeneities in the comminutive wave. However, the main constraint of a homogeneity in the solid being comminuted has hitherto never been overcome. In this paper, a very general derivation of the crack density model is made for inhomogeneous solids by replacing the assumption of homogeneity with an arbitrary function ϱ(y), which describes the susceptibility of the material to comminution as a function of position. The restraints on this function, ϱ(y), are very broad and, hence, the derivation is extremely general. General use of the equation derived necessitates a detailed a priori knowledge of both the fracture process and of the physical properties of the material being fractured: this knowledge is expressed in open form as, ϱ(y), the linear probability density that a plane of fracture intersects at point y on an arbitrarily oriented line segment of total length xo through the original particle. From these assumptions, a general expression for the cumulative mass distribution of particles resulting front fracture is then postulated. It is: where B(x) is a cumulative mass of material finer than size x, r is the average number of crack, ϱ(y) is defined above and Xo is the initial particle size. It is shown that suitable assumptions reduce the expression to the theoretical results of Gaudin and Meloy and to special forms of the empirical equations of Rosin—Rammleer, Gaudin and Schuhmann. Semiquantitative agreement is demonstrated by Grimshaw, with experimental results of a very complex fracture of heterogeneous sandstone blocks. A need is expressed for more theoretical development of ϱ(y) and a more extensive experimental study of both homogeneous and heterogeneous fracture. The full mathematical derivation is presented for the first time in this paper. The general implications of the equation and its application are also discussed.