mathematical details are as follows. Let X(t) denote cumulative damage until time t, and assume that X(t) ≥ 0. Assuming that as t grows, X(t) becomes approximately normal, and assuming that μ E(X(t)) = μt and Var(X((t)) = o2t, the distribution of the first t at which X(t) exceeds ∆ can be shown to be:
There are many generalizations of this argument, including those (1) for means and variances that are other functions of t, (2) for means and variances that are functions of other factors, and (3) where the distribution of X(t)is substantially non-normal.
Saunders described a current application derived from the generalized inverse Gaussian distribution applied to waiting times to failure for polymer coatings. These coatings often have requirements that they last for 30– 40 years. Modeling this requires some understanding of the chemical process of degradation, which in turn entails understanding how sun, rain, ultraviolet exposure, temperature, and humidity combine to affect coatings. Further, the chemistry must be linked to observable degradation, such as loss of gloss, fading, and discoloration. (For more details, see Saunders .)
Joe Padgett outlined several currently used approaches he has been researching that can be applied to either the modeling of the failure of material or systems due to cumulative damage or the modeling of crack growth due to fatigue. A good motivating example is the modeling of the tensile strength of carbon fibers and composite materials. In fibrous composite materials, it is essentially the brittle fibers that determine the material’s properties. To design better composites, it is important to obtain good estimates of fibers’ tensile strength. Numerous experiments have been conducted to provide information on the tensile strength of various single-filament and composite fibers. Failure of the fibers due to cumulative damage can be related to times to “first passage,” that is, times at which a sto-