Chord length distribution densities of selected elementary ge- ometric figures

2016 ◽  
pp. 71-106
Keyword(s):  
Length Distribution ◽  
Chord Length ◽  
Chord Length Distribution

scholarly journals Effect of Pre-Shear on Agglomeration and Rheological Parameters of Cement Paste

Materials ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 2173
Author(s):  
Mareike Thiedeitz ◽  
Inka Dressler ◽  
Thomas Kränkel ◽  
Christoph Gehlen ◽  
Dirk Lowke
Keyword(s):  
Yield Stress ◽  
Mixing Time ◽  
Length Distribution ◽  
Chord Length ◽  
Rheological Parameters ◽  
Reflectance Measurement ◽  
Time Dynamic ◽  
Chord Length Distribution ◽  
In Situ Investigation

Cementitious pastes are multiphase suspensions that are rheologically characterized by viscosity and yield stress. They tend to flocculate during rest due to attractive interparticle forces, and desagglomerate when shear is induced. The shear history, e.g., mixing energy and time, determines the apparent state of flocculation and accordingly the particle size distribution of the cement in the suspension, which itself affects suspension’s plastic viscosity and yield stress. Thus, it is crucial to understand the effect of the mixing procedure of cementitious suspensions before starting rheological measurements. However, the measurement of the in-situ particle agglomeration status is difficult, due to rapidly changing particle network structuration. The focused beam reflectance measurement (FBRM) technique offers an opportunity for the in-situ investigation of the chord length distribution. This enables to detect the state of flocculation of the particles during shear. Cementitious pastes differing in their solid fraction and superplasticizer content were analyzed after various pre-shear histories, i.e., mixing times. Yield stress and viscosity were measured in a parallel-plate-rheometer and related to in-situ measurements of the chord length distribution with the FBRM-probe to characterize the agglomeration status. With increasing mixing time agglomerates were increasingly broken up in dependence of pre-shear: After 300 s of pre-shear the agglomerate sizes decreased by 10 µm to 15 µm compared to a 30 s pre-shear. At the same time dynamic yield stress and viscosity decreased up to 30% until a state of equilibrium was almost reached. The investigations show a correlation between mean chord length and the corresponding rheological parameters affected by the duration of pre-shear.

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (1) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

scholarly journals Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

2008 ◽  
Vol 40 (04) ◽  
pp. 919-938 ◽  
Author(s):  
Kasra Alishahi ◽  
Mohsen Sharifitabar
Keyword(s):  
Space Dimension ◽  
Voronoi Tessellation ◽  
Length Distribution ◽  
Chord Length ◽  
Voronoi Tessellations ◽  
Constant Intensity ◽  
Chord Length Distribution ◽  
Typical Cell ◽  
Centered Ball ◽  
Contact Distribution

This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, …. First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ−1(1 − e−λu ). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

scholarly journals The chord-length distribution of a polyhedron

2020 ◽  
Vol 76 (4) ◽  
pp. 474-488
Author(s):  
Salvino Ciccariello
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Square Root ◽  
Trigonometric Functions ◽  
Chord Length Distribution ◽  
Analytic Expressions ◽  
Analytic Expression ◽  
Positive Integers ◽  
Inverse Trigonometric Functions

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (03) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (3) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

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