Minimal fragmentation problem

Parisio, Fernando; Dias, Laércio

doi:10.1103/physreve.84.035101

Cited by 2 publications

(7 citation statements)

References 15 publications

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“…In this section we will recast some analytical results previously obtained in [7]. We consider this review to be necessary in order to clarify our procedure in the case of regular polygons with arbitrary number of sides.…”

Section: Squarementioning

confidence: 99%

“…We consider this review to be necessary in order to clarify our procedure in the case of regular polygons with arbitrary number of sides. In Equation (9) of [7] the mass distribution for rectangles of arbitrary aspect ratios γ is given within the same MF model. To obtain the corresponding distribution P (4) (µ) for an ensemble of squares we simply set γ = 1.…”

Section: Squarementioning

confidence: 99%

“…There we derive in detail exact results for P(µ). In section IV we review some results found in [7] on the MF of squared plates. Section V presents a study of circular plates minimally fragmented.…”

Section: Introductionmentioning

confidence: 99%

“…The minimal fragmentation (MF) model of planar objects consists in considering a large collection of identical bodies split in two fragments only, instead of a single body cracked in a large number of pieces [7]. In our minimal fragmentation model for a polygon a crack is represented by a straight segment that is fully characterized by two random variables: l ∈ [0, L] representing one of the crack limits intersecting the border of the plate, with L being the polygon perimeter, and, φ ∈ [−π/2, π/2] being the angle between the segment and the normal direction to the side selected by the variable l. Initially we don't consider any directional or positional bias such that l and φ, have uniform distributions.…”

Section: Introductionmentioning

confidence: 99%

“…Very few statistical fragmentation models are amenable to a fully analytical approach, among them, the random fragmentation of a line [4], the model by Mott to describe the fragmentation of a flat surface into rectangles with random side lengths [5] (for a recent account see [6]), and the minimal fragmentation of an ensemble of rectangular plates [7]. Although all these constructions in the realm of geometrical probability [8] are highly idealized, they do shed some light into more realistic aspects of multifragmentation problems, for instance, the existence of power-law regimes in the fragment size distribution in the limit of small mass [7].…”

Section: Introductionmentioning

confidence: 99%

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Minimal fragmentation of regular polygonal plates

Dias

Parisio

2014

Phys. Rev. E

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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.

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Section: Squarementioning

confidence: 99%

Section: Squarementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimal fragmentation of regular polygonal plates

Dias

Parisio

2014

Phys. Rev. E

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Fragmentation of random trees

Kalay

Ben‐Naim

2014

J. Phys. A: Math. Theor.

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We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected existing node. As nodes are removed from the tree, one at a time, the tree dissolves into an ensemble of separate trees, namely, a forest. We study statistical properties of trees and nodes in this heterogeneous forest, and find that the fraction of remaining nodes m characterizes the system in the limit N → ∞. We obtain analytically the size density φs of trees of size s. The size density has power-law tail φs ∼ s −α with exponent α = 1 + 1 m . Therefore, the tail becomes steeper as further nodes are removed, and the fragmentation process is unusual in that exponent α increases continuously with time. We also extend our analysis to the case where nodes are added as well as removed, and obtain the asymptotic size density for growing trees.

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Minimal fragmentation problem

Cited by 2 publications

References 15 publications

Minimal fragmentation of regular polygonal plates

Minimal fragmentation of regular polygonal plates

Fragmentation of random trees

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