Statistical models of brittle fragmentation

Recent developments in statistical models for fragmentation of brittle material are reviewed. The generic objective of these models is understanding the origin of the fragment size distributions (FSDs) that result from fracturing brittle material. Brittle fragmentation can be divided into two categories: (1) Instantaneous fragmentation for which breakup generations are not distinguishable and (2) continuous fragmentation for which generations of chronological fragment breakups can be identified. This categorization becomes obvious in mining industry applications where instantaneous fragmentation refers to blasting of rock and continuous fragmentation to the consequent crushing and grinding of the blasted rock fragments. A model of unstable cracks and crack-branch merging contains both of the FSDs usually related to instantaneous fragmentation: the scale invariant FSD with the power exponent (2−1/D) and the double exponential FSD which relates to Poisson process fragmentation. The FSDs commonly related to continuous fragmentation are: the lognormal FSD originating from uncorrelated breakup and the power-law FSD which can be modeled as a cascade of breakups. Various solutions to the generic rate equation of continuous fragmentation are briefly listed. Simulations of crushing experiments reveal that both cascade and uncorrelated fragmentations are possible, but that also a mechanism of maximizing packing density related to Apollonian packing may be relevant for slow compressive crushing. Contents PAGE 1. Introduction 248 2. Instantaneous fragmentation 249  2.1. Background 249  2.2. Criticality in instantaneous fragmentation 250  2.3. Numerical models of fragmentation 251  2.4. The beam model 251  2.5. Rheology of cracks 252  2.6. Instability of fast cracks 253  2.7. Side branches form fragments 256  2.8. 2D objects in 3D space 262  2.9. The transition point 262  2.10. Simulations 263  2.11. Experiments 265 3. Continuous fragmentation 268  3.1. Background 268  3.2. Uncorrelated breakup history 268  3.3. Cascade fragmentation 269  3.4. Rate equations for fragmentation 269  3.5. Fragmentation of granular packings 270 4. Summary and discussion 271 Acknowledgements 273 Appendix 273 References 276