<< ️(AA) have introduced two types of hierarchies in a classical Gilbert tessellation. The cascading process that follows gives rise, in both cases, to a tile area distribution function that fits surprisingly well with a generalized gamma function. The latter is a truncated power function which, in turn, in the spirit of Brown’s sequential fragmentation theory, is derived from a stretched exponential hazard function that is also truncated. >>
<< ️This theoretical finding finds a natural counterpart in the experimental observations of Kooij and coworkers [S. Kooij et al., Nat. Commun. 12, 2521 (2021)], according to which the fragmentation of a brittle sample follows two distinct regimes: a random breakup that gives rise to a simple exponential distribution of mass fragments (canonical Gilbert), whereas a hierarchical crushing leads to a power-law distribution (hierarchical Gilbert, in fact g3(x) ∼ x^(c−1) for x → 0). >>
<< ️(AA) study demonstrates that the DG and HG cascade processes (i.e. Dichotomous Gilbert (DG), Hierarchical Gilbert (HG)) display a multifractal nature. Both models produce a heterogeneous distribution with different characteristics, quantitatively described by two different multifractal spectra. A direct, quantitative comparison between (Their) geometric models and specific experimental datasets remains, at present, an open challenge and would require fragmentation experiments capable of resolving a sufficiently wide range of fragment sizes to reliably probe the hierarchy-induced effect. >>
E.Consolini, B.Bonanni, M.Fanfoni. Hierarchy in Gilbert's tessellation: Multifractal behavior. Phys. Rev. E 114, 034120. Sep 11, 2026.
Also: fracture, crack, random, behav, in https://www.inkgmr.net/kwrds.html
Keywords: gst, fracture, crack, randomness, fragmentation process, fragment-size distribution, scale-free behavior, 2-D random tessellation, fractals, multifractal systems, multifractal scaling.
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