Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-04T00:09:10.175Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Survival analysis and censored data Special processes Applications

Published online by Cambridge University Press:  01 July 2016

James U. Gleaton  and
James D. Lynch
James U. Gleaton*
Affiliation:
University of North Florida
James D. Lynch*
Affiliation:
University of South Carolina
*
∗ Postal address: Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA.
∗∗ Postal address: Center for Reliability and Quality Sciences, Department of Statistics, University of South Carolina, Columbia, SC 29208, USA. Email address: lynch@stat.sc.edu
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Max entropybreaking strainlog-logistic distributionthermodynamicsbundle of fiberstensile loadtemper

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62N05: Reliability and life testing 62P35: Applications to physics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 98 - 115
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by NSF Grants DMS 9877107 and NSF DMS 0243594.

References

Balakrishnan, N. (ed.) (1992). Handbook of the Logistic Distribution. Marcel Dekker, New York.Google Scholar
Boltzmann, L. (1871a). Über das Wärmegleichgewicht zwischen mehratomigen Gasmolekülen. Sitzungsber. Akad. Wiss. Math.-Natur. Klasse Wien 63, 397–418.Google Scholar
Boltzmann, L. (1871b). Einige allgemeine Sätze über Wärmegleichgewicht. Sitzungsber. Akad. Wiss. Math.-Natur. Klasse Wien 63, 679–711.Google Scholar
Boltzmann, L. (1871c). Analytischer Beweis des zweiten Hauptsatzes der mechanischen Wärm etheorie aus den Sätzen über das Gleichgewicht der lebendigen Kraft. Sitzungsber. Akad. Wiss. Math.-Natur. Klasse Wien 63, 712–732.Google Scholar
Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. John Wiley, New York.Google Scholar
Daniels, H. E. (1945). The statistical theory of the strength of bundles of threads. I. Proc. R. Soc. London A 183, 405–435.Google Scholar
Durham, S. D. and Padgett, W. J. (1997). Cumulative damage models for system failure with application to carbon fibers and composites. Technometrics 39, 34–44.CrossRefGoogle Scholar
Durham, S. D., Lynch, J. D. and Padgett, W. J. (1990). TP_2-orderings and the IFR property with applications. Prob. Eng. Inf. Sci. 4, 73–88.CrossRefGoogle Scholar
Jaynes, E. T. (1957a). Information theory and statistical mechanics. I. Physical Rev. 106, 620–630.CrossRefGoogle Scholar
Jaynes, E. T. (1957b). Information theory and statistical mechanics. II. Physical Rev. 108, 171–190.CrossRefGoogle Scholar
Jaynes, E. T. (1979). Where do we stand on maximum entropy? In Maximum Entropy Formalism (Conf. Mass. Inst. Tech., Cambridge, MA, 1978), MIT Press, Cambridge, MA, pp. 15–118.Google Scholar
Jaynes, E. T. (1982). On the rationale of maximum-entropy methods. Proc. IEEE 70, 939–982.CrossRefGoogle Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Kestin, J. and Dorfman, J. R. (1971). A Course in Statistical Thermodynamics. Academic Press, New York.Google Scholar
Mandelbrot, B. (1962). The role of sufficiency and of estimation in thermodynamics. Ann. Math. Statist. 33, 1021–1038.CrossRefGoogle Scholar
Maxwell, J. C. (1860). Illustrations of the dynamical theory of gases. Phil. Mag. 19, 19–32. Reprinted in: Collected Works, ed. Niven, W. D., Vol. I, London, 1890, pp. 377–409.Google Scholar
Shannon, C. E. (1948). A mathematical theory of communication. Bell System Tech. J. 27, 379–421.CrossRefGoogle Scholar