Fractal models for the fragmentation of rocks and soils: a review

Fragmentation, the process of breaking apart into fragments, is caused by the propagation of multiple fractures at different length scales. Such fractures can be induced by dynamic crack growth during compressive/tensile loading or by stress waves during impact loading. Fragmentation of rocks occurs in resoonse to tectonic activity, percussive drilling, grinding and blasting. Soil fragmentation is the result of tillage and planting operations. Fractal theory, which deals with the scaling of hierarchical and irregur systems, offers new opportunities for modeling the fragmentation process. This paper reviews the literature on fractal models for the fragmentation of heterogeneous brittle earth materials. Fractal models are available for the fragmentation of: (1) classical aggregate; (2) aggregates with fractal pore space; and (3) aggregates with fractal surfaces. In each case, the aggregates are composed of building blocks of finite size. Structural failure is hierarchical in nature and takes place by multiple fracturing of the aggregated building blocks. The resulting number-size distribution of fragments depends on the probability of failure, P(1bi) at each level in the hierarchy. Models for both scale-invariant and scale-dependent P(sol1bi) are reviewed. In the case of scale-invariant P(1bi)< 1, theory predict: Df = 3 + log [P(1bi)]/log[b] for classical aggregates; Df=Dm+log[P(1bi)]/log[b] for aggregates with fractal pore space; and Df=Ds for aggregates with fractal surfaces. where b is a scaling factor and Df, Dm and Ds are the fragmentation, mass and surface fractal dimensions, respectively. The physical significance of these parameters is discussed, methods of estimating them are reviewed, and topics needing further research are identified.