Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-30T21:19:11.060Z Has data issue: false hasContentIssue false

Recursions and limit theorems for the strength and lifetime distributions of a fibrous composite

Published online by Cambridge University Press:  14 July 2016

Chia-Chyuan Kuo
Affiliation:
Cornell University
S. Leigh Phoenix*
Affiliation:
Cornell University
*
∗∗Postal address: Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA.

Abstract

A composite material is a parallel arrangement of stiff brittle fibers in a flexible matrix. Under load fibers fail, and the loads of failed fibers are locally redistributed onto nearby survivors through the matrix. In this paper we develop a new technique for computing the probability of failure under a previously studied model of the failure process. A recursion and limit theorem are obtained which apply separately to static strength and fatigue lifetime depending on the composite loading and the probability model for the failure of individual fibers under their own loads. The limit theorem yields an approximation for the distribution function for composite lifetime which is of the form 1 – [1 – W(t)]mn where W(t) is a characteristic distribution function and mn is the composite volume, reflecting a size effect. A similar result holds also for static strength. In both cases such a result was conjectured several years ago. This limit theorem is obtained from the recursion upon applying a key theorem in the theory of the renewal equation. In the proofs three technical conditions arise which must be verified in specific applications. In the case of static strength these conditions are quite easy to verify, but in the case of fatigue lifetime the verification is generally difficult, and entails considerable numerical computation.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1987 

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

Present address: Kendall Company, 95 West St., Walpole, MA 02081, USA.

References

Coleman, B. D. (1958) Statistics and time dependence of mechanical breakdown in fibers. J. Appl. Phys. 29, 968983.Google Scholar
Gotlib, Yu Ya, El'Yashevich, A. M. and Svetlov, Yu E. (1973) Effect of microcracks on the local stress distribution in polymers and their deformation properties. Network model. Soviet Physics — Solid State 14, 26722677.Google Scholar
Harlow, D. G. (1985) The pure flaw model for chopped fibre composites. Proc. R. Soc. London A 397, 211232.Google Scholar
Harlow, D. G. and Phoenix, S. L. (1978) The chain of bundles probability model for the strength of fibrous materials II: A numerical study of convergence. J. Composite Materials 12, 314334.Google Scholar
Harlow, D. G. and Phoenix, S. L. (1981) Probability distribution for the strength of composite materials II: a convergent sequence of tight bounds. Internat. J. Fracture 17, 601630.Google Scholar
Harlow, D. G. and Phoenix, S. L. (1982) Probability distributions for the strength of fibrous materials I: Two-level failure and edge effects. Adv. Appl. Prob. 14, 6894.Google Scholar
Hedgepeth, J. M. (1961) Stress Concentrations in Filamentary Structures. NASA Technical Note D-882.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Kuo, C. C. (1983) Recursion Formulas and Limit Theorems for the Lifetime Distribution of a Model Fibrous Composite. , Cornell University.Google Scholar
Phoenix, S. L. and Tierney, L.J. (1983) A statistical model for the time dependent failure of unidirectional composite materials under local elastic load sharing among fibers. Engineering Fracture Mechanics 18, 193215.10.1016/0013-7944(83)90107-8Google Scholar
Pitt, R. E. and Phoenix, S. L. (1983) Probability distributions for the strength of composite materials IV: localized load-sharing with tapering. Internat. J. Fracture 22, 243276.Google Scholar
Smith, R. L. (1980) A probability model for fibrous materials with local load sharing. Proc. R. Soc. London A 372, 539553.Google Scholar
Smith, R. L. (1982) A note on a probability model for fibrous composites. Proc. R. Soc. London A 382, 179182.Google Scholar
Smith, R. L. (1983) Limit theorems and approximations for the reliability of load-sharing systems. Adv. Appl. Prob. 15, 304330.Google Scholar
Taylor, H. M. and Karlin, S. (1984) An Introduction to Stochastic Modeling. Academic Press, New York.Google Scholar
Tierney, L. (1982) Asymptotic bounds on the time to fatigue failure of bundles of fibers under local load sharing. Adv. Appl. Prob. 14, 95121.10.2307/1426735Google Scholar