Abstract
A group of stochastic processes akin to the Poisson process is defined in terms of rules of interactions between two types of interacting entities and in terms of a parameter corresponding to the initial relative numbers of the two types of interacting entities. One limiting value of this parameter corresponds to the Poisson process and the exponential distribution function, and the other limiting value of the parameter corresponds to a special case among the group of stochastic processes defined and a statistical distribution function used previously for incomes and fracture toughness. The related processes in between, which correspond to the intermediate values of the parameter, correspond to an intermediate statistical distribution function. The transition between the limiting cases is smooth as evinced by the change of the mean and median with change in the parameter. The scale-invariant behaviour of the fields of stress and strain at the tips of cracks is used to support the introduction of a shape parameter into the special-case function. All the distribution functions considered are found to be stable extreme-value functions, either in the sense of multiplying probabilities or in the sense of summing the variable or in a mixture of both senses.
Highlights
In a previous paper (Neville 1987), a statistical distribution function for fracture toughness was derived on the basis of the nature of the stress and strain fields near the tips of sharp cracks, with a shape parameter being included
The basis of the derivation of the function for fracture toughness is that, in a piece under increasing load, the fields of stress and strain near the tips of sharp cracks are such that they expand away from the tips of cracks so that, in a scale-invariant fashion, more material is subjected to the same stresses and strains, whereas, remote from the tips of cracks, the same material is subjected to increasing stress and strain
Balankin et al (1999, p. 2573) expressed surprise that their fracture data were fitted best by the statistical distribution function derived for fracture toughness
Summary
In a previous paper (Neville 1987), a statistical distribution function for fracture toughness was derived on the basis of the nature of the stress and strain fields near the tips of sharp cracks, with a shape parameter being included. 2573) expressed surprise that their fracture data were fitted best by the statistical distribution function derived for fracture toughness (the Fisk or log-logistic function). Support will be provided for the use of the distribution function for fracture toughness (the Fisk or log-logistic function) for failure strength by expanding the previous derivation in terms of Poisson-like stochastic processes and using the scale-invariant nature of the fields of stress and strain at the tips of cracks to justify the scale parameter. The statistical distribution function derived for fracture toughness will be regarded as an extreme-value function and its relationship to the exponential distribution function and similar extreme-value functions, as it emerges from the consideration of stochastic processes, will be investigated
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
