| An important problem in material science is the prediction of the breaking strength of fibrous composites. Such composites are made of bundles of fibers embedded in a matrix material. Once a fiber breaks, the load carried at the break is transferred by shear stress through the matrix to neighboring fibers which can be modeled by different load sharing principles. In this dissertation, the behavior of the bundle strength for which load redistribution occurs is considered. There are two major results concerning the behavior of the bundle strength. One is the calculation of the breaking strength and the other is its increasing failure rate (IFR) character.; Harlow, Smith and Taylor (1983) have given a recursive formula for the bundle strength distribution under a general class of load sharing rules called monotone load sharing rules. As the bundle size increases, this formula becomes prohibitively complex and does not give much insight into the relationship of the load sharing constants to the distribution. It is shown how the bundle strength survival distribution can be calculated using a novel graph called the loading diagram. The graph is parallel in structure and recursive in nature and so would appear to lend itself to large-scale computations. Combinatoric properties related to the graph and other results for the bundle survival distribution are also obtained.; It is shown that under certain conditions the transition distribution function of a continuous time Markov chain with totally ordered state space is totally positive of order 2 (TP{dollar}sb2{dollar}). For such chains, it is established that first passage times have an increasing failure rate of the bundle strength for small bundles of fibers with exponentially distributed strengths.; Finally, for Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a total positivity condition on the transition matrix. Some totally positive composition results are also given. |
