Predicting the fragmentation survival probability of brittle spheres upon impact from statistical distribution of material properties

Fragmentation of rocks upon impact during rockfall is a phenomenon that is poorly understood, scarcely researched and difficult to predict. However, to adequately predict the outcomes of rockfall events, it is essential to know whether a given block is likely to fragment given the impact conditions and what will be the outcome of the fragmentation process; that is, the number, size and trajectory of fragments. To date, there is no model or data that can be used to fully answer these questions. This paper presents the first theoretical model that can predict the fragmentation survival probability of brittle spherical blocks upon dynamic impact (i.e. drop tests) from the statistical distribution of material properties, determined from a range of standard quasi-static tests. Considering that survival probabilities tend to follow a Weibull distribution, the model focuses on predicting the two Weibull parameters, commonly known as the shape parameter (m) and the scale parameter (here, the critical kinetic energy). The model is based on theoretically-derived conversion factors used to turn the critical work required to fail disc samples in quasi-static indirect tension into the critical kinetic energy to cause failure of spheres at impact in drop tests. The mechanistic conversion factors specifically account for the shape and size of the specimens tested and the increase of strength under dynamic loading (strain rate effect). Three series of drop tests were conducted (on spheres of three different diameters) and complemented by extensive material characterisation testing in order to validate the novel predictive model. The variability of material properties was characterised, and it was found that the material strength found by the characterisation tests generally follows a Weibull form, but the survival probability distribution of the drop tests seems to be linear. The predicted conversion factors were first compared against their experimental counterparts before validating the prediction of survival probability of the spheres upon dynamic impact (in drop tests). It was found that it is possible to predict the survival probability of artificial rock of three different diameters (50 mm, 75 mm, 100 mm) and two different strengths upon impact solely from the statistical information coming from Brazilian tests and with an average relative error of less than 9%.