This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

An energy-based damage model is proposed and applied to predict the fracture initiation of the sheet metal forming process. The fracture mechanism is investigated through the plastic energy dissipation. The concepts of the damaging work and the fracture energy are proposed for the quantitative description of damage evolution and crack initiation. The developed formulations are implemented into the finite element program ABAQUS to simulate the biaxial stretching of sheet metals and to predict the fracture strains. The material parameter needed in the damage model for fracture prediction is determined by the stress-strain history of the uniaxial tensile test. The forming limits for aluminum alloy sheets under various strain paths are predicted by the present approach and then compared to the measured data quoted from the literatures [1,2]. Good agreements are found between this study and the previous results.

The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.