Reports On Technological Research - Errors In Current Random Fracture Treatments Examined

This communication points out that serious errors exist in some current treatments of the random fracture of solids, including the prior treatments by Klimpel and Austin, Gilvarry, and Gaudin and Meloy. The problem considered in all of these papers is: what cumulative weight versus size distribution is expected for the fragments produced by fracture of a particle? The basic assumption made is that fracture surfaces propagate in random directions from weak flaws in the solid (these weak flaws being activated to fracture by applied stress), with the activated weak flaws distributed at random spatially within the solid. The previous treatments considered fracture in three dimensions, and all use similar physical models, although the mathematical techniques were somewhat different. We first noticed that these treatments were in error when we attempted to reduce the equations developed to the simple case of fracture of a one-dimensional particle, e.g. a long, thin fiber which breaks along its length. For this simple case, it is readily shown that the assumption of random fracture leads to the expression P(V) = (r/Vo) (1-V/Vo) r-1 dV (1) is the probability of occurrence of a fragment of length within the range V to V + dV; V being length dimension, from r random fractures passing through an original size of Vo to produce r + 1 fragments. This equation agreed with the one-dimensional form of an equivalent equation in the original Klimpel-Austin treatment. However, in this treatment it was then stated that, "The fractional weight below size V is [ ]