Statistical models of brittle fragmentation

Abstract

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.

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Acknowledgements

The author would like to express his gratitude to collaborators who made this work possible: M. Kellomäki, R. Linna, B.L. Holian, F. Ouchterlony, H. Katsuragi and, in particular, J. Timonen for more than a decade of fruitful cooperation. The author also acknowledges longstanding and altruistic provision of work facilities by R. Nieminen.

Notes

¹ Breaking strain for a brittle material is typically governed by defects and micro cracks and is thus not an intrinsic material property. Breaking strain varies typically from a fraction of a percent to a few percent.

² For simulations on the atomic level there are several methods of varying accuracy such as the solution methods for the many-body Schrödinger equation, density functional theory and classical molecular dynamics.

³ A somewhat similar investigation of crack instability can be found in: M. Marder and S. Gross, J. Mech. Phys. Solids 43 1 (1995)