Minimal fragmentation of regular polygonal plates

Abstract

Minimal fragmentation models intend to unveil the statistical properties of large ensembles of identical objects, each one segmented in two parts only. Contrary to what happens in the multifragmentation of a single body, minimally fragmented ensembles are often amenable to analytical treatments, while keeping key features of multifragmentation. In this work we present a study on the minimal fragmentation of regular polygonal plates with up to 100 sides. We observe in our model the typical statistical behavior of a solid teared apart by a strong impact, for example. That is to say, a robust power law, valid for several decades, in the small mass limit. In the present case we were able to analytically determine the exponent of the accumulated mass distribution to be 1 /2. Less usual, but also reported in a number of experimental and numerical references on impact fragmentation, is the presence of a sharp crossover to a second power-law regime, whose exponent we found to be 1 /3 for an isotropic model and 2 /3 for a more realistic anisotropic model.

Fragmentation of random trees

“…Our study suggests the intriguing scenario where the power-law exponent can change throughout the fragmentation process and may depend on the structure or the geometry of the fragmented object. Geometry-dependent exponents were recently reported in fragmentation of polygonal plates [54]. It will be indeed interesting to find physical fragmentation processes [55][56][57] where the size distribution becomes steeper with time.…”

Jamming and tiling in fragmentation of rectangles

“…Missing from our analysis, however, are statistics of multiple fragments such as correlations between the orientations of neighboring sticks. Both sets of statistics are relevant for characterizing the geometrical structure of planar fragmentation patterns found in martensitic transformations [26], breakage of brittle objects [47,48], cracking of soils [49], and drying of suspensions [50].…”

“…The exponent γ given by(48) versus the asymmetry parameter α. Also shown are results of numerical evaluation of the recursion equations with A = 10 8 .…”