6533b7dcfe1ef96bd1272094

RESEARCH PRODUCT

Exponential and power-law mass distributions in brittle fragmentation

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Generic arguments, a minimal numerical model, and fragmentation experiments with gypsum disk are used to investigate the fragment-size distribution that results from dynamic brittle fragmentation. Fragmentation is initiated by random nucleation of cracks due to material inhomogeneities, and its dynamics are pictured as a process of propagating cracks that are unstable against side-branch formation. The initial cracks and side branches both merge mutually to form fragments. The side branches have a finite penetration depth as a result of inherent damping. Generic arguments imply that close to the minimum strain (or impact energy) required for fragmentation, the number of fragments of size $s$ scales as ${s}^{\ensuremath{-}(2D\ensuremath{-}1)∕D}{f}_{1}(\ensuremath{-}{(2∕\ensuremath{\lambda})}^{D}s)+{f}_{2}(\ensuremath{-}{s}_{0}^{\ensuremath{-}1}{(\ensuremath{\lambda}+{s}^{1∕D})}^{D})$, where $D$ is the Euclidean dimension of the space, $\ensuremath{\lambda}$ is the penetration depth, and ${f}_{1}$ and ${f}_{2}$ can be approximated by exponential functions. Simulation results and experiments can both be described by this theoretical fragment-size distribution. The typical largest fragment size ${s}_{0}$ was found to diverge at the minimum strain required for fragmentation as it is inversely related to the density of initially formed cracks. Our results also indicate that scaling of ${s}_{0}$ close to this divergence depends on, e.g., loading conditions, and thus is not universal. At the same time, the density of fragment surface vanishes as ${L}^{\ensuremath{-}1}$, $L$ being the linear dimension of the brittle solid. The results obtained provide an explanation as to why the fragment-size distributions found in nature can have two components, an exponential as well as a power-law component, with varying relative weights.