Summary of Current Research and Research Plans

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Summary of Current Research and Research Plans

James U. Gleaton

Current Research:

I am currently working with a co-author, Dr. Mahbubur Rahman, on a paper which follows up another paper (2) that was recently submitted for publication, and an earlier paper (3) published in the Journal of Probability and Statistical Science. The earlier papers established properties of extended generalized log-logistic (EGLL) families of lifetime distributions. A distribution from such a family is generated from a given lifetime distribution function by an EGLL transformation:

,

where  > 0 is the parameter of a proportional odds (p.o.) transformation

;  > 0 is the parameter of a generalized log-logistic (g.l.l.) transformation: ; and is the vector-valued parameter of the generating distribution. The earlier papers showed that the EGLL transformation partitions the family of all lifetime distributions. Some common characteristics of all distributions in an EGLL equivalence class were established, including the existence of moments and moment generating functions, the linear ordering of the distributions in the class according to the g.l.l. transformation parameter, the stochastic ordering of the distributions in the class according to the p.o. transformation parameter, and the fact that the Kullback-Leibler information function relating two members of the class is an increasing function of either the ratio of the g.l.l. parameters, or of the ratio of the p.o. parameters. The EGLL transformation was used to attempt to fit data that are not well fit by a Weibull distribution. The M.L.E.’s of the parameters of the distribution were estimated by a process involving the Nelder-Mead simplex method, followed by the Newton-Raphson algorithm.

The current paper examines the problem of asymptotic inference for the M.L.E.’s of the parameters of an EGLL distribution, for two types of distributional families, one in which the generating member is a Weibull distribution, and the other in which the generating member is an inverse Gaussian distribution.

These three papers follow up a paper written with J. D. Lynch, which appeared in the March, 2004 issue of Advances in Applied Probability (4). This paper examined the problem of the reliability of inhomogeneous bundles of brittle elastic fibers under tensile stress. Here, “inhomogeneous” means that the fibers have differing cross-sectional areas. “Elastic” means that the relative increase in length of a fiber, the strain, is proportional to the applied tensile force per unit area, the stress. “Brittle” means that when a transverse crack begins to form in a fiber, the work of fracture is not dissipated into microscopic dislocations that would inhibit crack growth. Examples of such fibers are the glass fibers embedded in fiberglass, and the carbon fibers embedded in carbon-carbon composite materials.

We examined the problem from the perspective of E. T. Jaynes’ Maximum Entropy Principle (5), which gives a method of finding an appropriate probability distribution for situations in which a random variable has a finite number of possible values. The principle, based on Shannon’s information function (7), says that, in making inferences about a system based on incomplete information, the inferences should be made using the probability distribution that has the most uncertainty (largest entropy) consistent with whatever information is actually available about the system.

When applied to the system under study, the principle leads to restrictions on the form of the survival distribution of individual fibers in the bundle. The distribution is found to be a member of the generalized log-logistic class of distributions.

Research Plans:

1. One line of my research will extend the current work by looking at the issue of uniqueness of the M.L.E.’s. The current paper establishes sufficient conditions for the existence of sequences of solutions of the likelihood equations that have the properties of consistency and asymptotic unbiasedness, normality, and efficiency, but does not establish that the solutions are unique. Uniqueness must be established, not for the general case, but for each specific choice of a generating distribution..

2. Another line of my research is a concerned with developing a model for the reliability of fiber composite materials under tensile loads. The model is based on an experiment conducted by B. W. Rosen (6). Rosen examined the effect of transmission of shearing stresses by the matrix of a fiber composite material, using a sample consisting of glass fibers embedded in a plastic matrix. If the fibers were not embedded in a matrix, there would be no transmission of shearing stresses, and when a fiber broke, that fiber would no longer make any contribution to the tensile strength of the bundle of fibers.

The first step toward a developing a model consisted of examining a bundle of fibers, for two cases: 1) all fibers have the same cross-sectional area, and 2) there are fibers of a finite number of different cross-sectional areas. It was found that in the second case, the log of the odds for survival of a fiber, as a function of the induced strain, would be proportional to the fiber cross-sectional area.

In the next step in developing the model I am considering a chain of bundles of brittle elastic fibers, under the assumption that, when a fiber breaks, its share of the load is redistributed equally among the remaining intact fibers in the bundle. Conditional on the applied tensile load, the bundles are independent, and the reliability of the system is defined by the breaking strength of the weakest bundle. The breaking strength distribution for the bundle may be found by embedding the breaking strength distribution for an isolated fiber, which has been found to have an inverse Gaussian distribution (1), into the probability distribution for the number of intact fibers in the bundle.

The next stage in the development of the model will consider the issue of load sharing among the fibers. In a fiber composite, the fibers are embedded in a matrix, which tends to limit load sharing around a broken fiber to the neighborhood of that fiber. Nearest neighbor load sharing will be assumed. When a fiber breaks, its share of the load is redistributed equally among its nearest neighbors (from 1 to 6, depending on the number of intact nearest neighbors). If there are no nearest neighbors, then the load is redistributed equally among all remaining intact fibers in the bundle.

3. For an inhomogeneous bundle (differing cross-sectional areas) of fibers, the log of the odds of survival of a fiber is proportional to the cross-sectional area, so that fibers of smaller cross-section tend to fail more rapidly. At some point, it may be that the remaining intact fibers in the bundle have the same cross-section, so that the bundle becomes homogeneous. The Maximum Entropy Principle places no restriction on the survival distribution of fibers in homogeneous bundles. Hence there may be a transition, such that in the inhomogeneous phase, the survival distribution of a fiber in the bundle is generalized log-logistic, but after the transition, there is a different survival distribution.

This phase transition yields another line of research. It may be that fibers in the homogeneous bundle will fit an inverse Gaussian survival distribution, as found for isolated fibers (1).

4. Since the breaking strength distribution of a fiber in a bundle depends on whether the bundle is homogeneous or inhomogeneous, another line of research is to assume that the bundle of fibers is a random sample from a distribution of cross-sections, in which the manufacturer intends the fibers to have a certain average diameter. It is reasonable to assume that the distribution of fiber diameters will be normal, provided the variability in diameters is not too large. Then the distribution of cross-sectional areas, suitably standardized, will be a chi-square distribution with 1 degree of freedom.

The reliability of the bundle depends both on the Young’s modulus for the material and on the fiber cross-sectional areas, and so will be a function of the random sample of cross-sections.

References:

1.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.

2. Gleaton, J. U. and Lynch, J. D. (submitted for publication). “Extended generalized log-logistic families of lifetime distributions, with an application.”

3. Gleaton, J. U. and Lynch, J. D. (2006). “Properties of generalized log-logistic families of lifetime distributions,”Journal of Probability and Statistical Science, 4, 1, 51-64.

4. Gleaton, J. U. and Lynch, J. D.(2004). "On the breaking strain of a bundle of brittle elastic fibers," Advances in Applied Probability,, 36, 98-115.

5. Jaynes, E. T. (1982). On the Rationale of Maximum-Entropy Methods.Proc. IEEE, 70, pp. 939-982.

6. Rosen, B. W. (1964). "Tensile failure of fibrous composites," AIAA Journal, 2, 11, 1985-1991.

7. Shannon, C. E. (1948). “A mathematical theory of communication,” The Bell System Technical Journal, 27, 3, 379-421.