Random Distribution of Lines in a Plane

Goudsmit, S.

doi:10.1103/revmodphys.17.321

Cited by 84 publications

(53 citation statements)

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“…Note that the same fragment distribution is also obtained by a different geometrical fragmentation procedure described in [18] that can be interpreted as fragmentation by intersecting cracks. The plane is covered by two sets of lines parallel to x and y axes, respectively.…”

Section: Perimeter Fragmentationmentioning

confidence: 97%

“…The distribution of the rectangular fragments cut by the two sets of lines is given by Eq. (24) (derived in [18] for unit density of lines). Fragment area distribution also attains a scaling form…”

Section: Perimeter Fragmentationmentioning

confidence: 99%

“…(10), to fragmentation rate equation (18), recursive relation for the moments is obtained vMðs 1 ; s 2 ; tÞ vt Z K2 s 1 K 2 s 1 Mðs 1 C 1; s 2 ; tÞ C s 2 K 2 s 2 Mðs 1 ; s 2 C 1; tÞ…”

Section: Perimeter Fragmentationmentioning

confidence: 99%

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The effect of defect location on coating fragmentation patterns under biaxial tension

Andersons

Leterrier

2005

Probabilistic Engineering Mechanics

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Section: Perimeter Fragmentationmentioning

confidence: 97%

Section: Perimeter Fragmentationmentioning

confidence: 99%

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The effect of defect location on coating fragmentation patterns under biaxial tension

Andersons

Leterrier

2005

Probabilistic Engineering Mechanics

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“…To solve that, models of fibrous arrays as 2D networks of infinite length fibers were used. Those models were first studied by Goudsmit, who investigated the problem of infinite lines randomly distributed in a plane [18]. Miles later examined the statistical properties of the polygons formed by such a system [19].…”

Section: Introductionmentioning

confidence: 99%

Calibration and adjustment of mechanical property prediction model for poly(vinyl alcohol)-enhanced carbon nanotube buckypaper manufacturing

Wang

Vanli²,

Zhang

et al. 2016

Int J Adv Manuf Technol

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Buckypaper is a free-standing carbon nanotube (CNT) sheet used to improve handling and manufacturability of CNT-based nanocomposites. To enhance the mechanical properties and manufacturing efficiency of buckypaper, polymeric binders, such as poly(vinyl alcohol) (PVA), are often added into the CNT network during the manufacturing process. This paper describes a physics-based model to predict the elastic modulus of PVA-enhanced buckypaper and a statistical approach to calibrate and adjust the model based on physical experiments. Compared to the physics-based model alone, the hybrid model can provide more accurate predictions for the Young's modulus of PVA-enhanced buckypaper and give a 95 % confidence interval on the prediction. One of the inputs for this model, the average length of carbon nanotubes, was calibrated using maximum likelihood estimation (MLE). The bias of this model was adjusted by estimating a bias function. Both the calibration parameter and adjustment function were estimated from a set of experimental measurements. The improvement in making a prediction was validated by comparing the performance of the physics-based model, statistical model, and statistics-enhanced physics model at a new experiment point. The hybrid model provides a more accurate prediction than either the physics-based model or statistical model does. This model calibration technique provides an effective tool for nanomanufacturing process design and material property prediction.

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“…Goudsmit [2], carried out at the suggestion of Niels Bohr. The question in that paper arose from cloud chamber experiments: when three lines seemingly originate from the same point, then what is the probability that they do not result from the same event?…”

Section: Poisson-voronoi and Poisson Line Tessellationsmentioning

confidence: 99%

Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

Hilhorst¹,

Calka²

2008

J Stat Phys

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We consider a family of random line tessellations of the Euclidean plane introduced in a much more formal context by Hug and Schneider [Geom. Funct. Anal. 17, 156 (2007)] and described by a parameter α ≥ 1. For α = 1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for α = 2 it coincides with the typical Poisson-Voronoi cell. Let p n (α) be the probability for the zero-cell to have n sides. By the methods of statistical mechanics we construct the asymptotic expansion of log p n (α) up to terms that vanish as n → ∞. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when α > 1, but gets delocalized for the Crofton cell, α = 1, which is a singular point of the parameter range. The large-n expansion of log p n (1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the typical n-sided cell of a Poisson line tessellation. Keywords: random line tessellations, Crofton cell, exact resultsLPT Orsay 08-17 1 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000 000000000000000000000

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Random Distribution of Lines in a Plane

Cited by 84 publications

References 0 publications

The effect of defect location on coating fragmentation patterns under biaxial tension

The effect of defect location on coating fragmentation patterns under biaxial tension

Calibration and adjustment of mechanical property prediction model for poly(vinyl alcohol)-enhanced carbon nanotube buckypaper manufacturing

Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

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