Minimal Fragmentation of Regular Polygonal Plates

TL;DR

This study analytically investigates the minimal fragmentation of regular polygonal plates, revealing power-law mass distribution behaviors with specific exponents and a crossover between regimes, relevant for understanding impact fragmentation.

Contribution

The paper provides an analytical characterization of the mass distribution exponents and crossover behavior in minimal fragmentation of regular polygons, including anisotropic effects.

Findings

Power-law exponent of 1/2 for small mass limit

Crossover to a second power-law regime with exponents 1/3 and 2/3

Analytical determination of mass distribution behavior in minimal fragmentation

Abstract

Minimal fragmentation models intend to unveil the statistical properties of large ensembles of identical objects, each one segmented in {\it 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 $\frac{1}{2}$. 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 $\frac{1}{3}$ for an isotropic model and $\frac{2}{3}$ for a more realistic anisotropic model.