On the distribution of the breaking strain of a bundle of brittle elastic fibers

Abstract

The maximum-entropy formalism developed by E. T. Jaynes is applied to the breaking strain of a bundle of fibers of various cross-sectional areas. When the bundle is subjected to a tensile load, and it is assumed that Hooke's law applies up to the breaking strain of the fibers, it is proved that the survival strain distribution for a fiber in the bundle is restricted to a certain class consisting of generalizations of the log-logistic distribution. Since Jaynes's formalism is a generalization of statistical thermodynamics, parallels are drawn between concepts in thermodynamics and in the theory of inhomogeneous bundles of fibers. In particular, heat transfer corresponds to damage to the bundle in the form of broken fibers, and the negative reciprocal of the parameter corresponding to thermodynamic temperature is the resistance of the bundle to damage.

The odd log-logistic logarithmic generated family of distributions with applications in different areas

“…Let A(u) = u α u α +(1−u) α be the cdf of the odd log-logistic uniform (Gleaton and Lynch 2004, 2006 distribution. For β ∈ (0, 1), we have 0 < βA(u) < 1.…”

“…Recently, Gleaton and Lynch (2004, 2006 defined the cdf of the odd log-logistic family with one extra shape parameter α > 0 by whereḠ(x; ξ ) = 1 − G(x; ξ ). More precisely, they showed the following facts:…”

The Beta Odd Log-Logistic Generalized Family of Distributions

“…The odd log-logistic (OLL) family of distributions was originally developed by Gleaton and Lynch [18,19]; they called this family the generalized log-logistic (GLL) family. They showed that: -the set of GLL transformations form an Abelian group with the binary operation of composition; -the transformation group partitions the set of all lifetime distributions into equivalence classes, so that any two distributions in an equivalence class are related through a GLL transformation; -either every distribution in an equivalence class has a moment generating function, or none does; -every distribution in an equivalence class has the same number of moments; -each equivalence class is linearly ordered according to the transformation parameter, with larger values of this parameter corresponding to smaller dispersion of the distribution about the common class median; and -within an equivalence class, the Kullback-Leibler information is an increasing function of the ratio of the transformation parameters.…”

O modelo de regressão odd log-logística gama generalizada com aplicações em análise de sobrevivência

“…As distribuições foram obtidas a partir da família de distribuições odd log-logística de distribuições proposta por (Gleaton e Lynch, 2006). Como distribuição de base são utilizadas as distribuições gama generalizada apresentadas por Stacy (1962) e Stacy e Mihram (1965), respectivamente.…”

“…• Foram obtidas duas novas distribuições a partir do procedimento de (Gleaton e Lynch, 2006). Estas distribuições apresentam maior flexibilidade na modelagem de dados.…”