8.
Assessing Product Reliability
8.1. Introduction 8.1.6. What are the basic lifetime distribution models used for nonrepairable populations?


A model based on cycles of stress causing degradation or crack growth  In 1969, Birnbaum and Saunders described a life distribution model that could be derived from a physical fatigue process where crack growth causes failure. Since one of the best ways to choose a life distribution model is to derive it from a physical/statistical argument that is consistent with the failure mechanism, the BirnbaumSaunders fatigue life distribution is worth considering.  
Formulas and shapes for the fatigue life model 
Formulas
and Plots for the BirnbaumSaunders Model
The PDF, CDF, mean, and variance for the BirnbaumSaunders distribution are shown below. The parameters are: \(\gamma\), a shape parameter; and \(\mu\), a scale parameter. These are the parameters we will use in our discussion, but there are other choices also common in the literature (see the parameters used for the derivation of the model). $$ \begin{array}{ll} \mbox{PDF:} & f(t) = \frac{1}{2 \mu^2 \gamma^2\sqrt{\pi}} \left( \frac{t^2  \mu^2}{\sqrt{\frac{t}{\mu}}  \sqrt{\frac{\mu}{t}}} \right) \mbox{exp} \left[ \frac{1}{\gamma^2} \left( \frac{t}{\mu} + \frac{\mu}{t} 2 \right) \right] \\ & \\ \mbox{CDF:} & F(t) = \Phi \left( \frac{1}{\gamma} \left[ \sqrt{\frac{t}{\mu}}  \sqrt{\frac{\mu}{t}} \, \right] \right) \\ & \\ & \Phi(z) \mbox{denotes the standard normal CDF.} \\ & \\ \mbox{Mean:} & \mu \left( 1 + \frac{\gamma^2}{2} \right) \\ & \\ \mbox{Variance:} & \mu^2 \gamma^2 \left( 1 + \frac{5 \gamma^2}{4} \right) \end{array} $$ PDF shapes for the model vary from highly skewed and long tailed (small gamma values) to nearly symmetric and short tailed as gamma increases. This is shown in the figure below.
Corresponding failure rate curves are shown in the next figure.


If crack growth in each stress cycle is a random amount independent of past cycles of growth, the Fatigue Life mode model may apply. 
Derivation and Use of the BirnbaumSaunders Model:
Consider a material that continually undergoes cycles of stress loads. During each cycle, a dominant crack grows towards a critical length that will cause failure. Under repeated application of \(n\) cycles of loads, the total extension of the dominant crack can be written as $$ W_n = \sum_{j=1}^n Y_j \, , $$ and we assume the \(Y_j\) are independent and identically distributed nonnegative random variables with mean \(\mu\) and variance \(\sigma^2\). Suppose failure occurs at the \(N\)th cycle, when \(W_n\) first exceeds a constant critical value \(w\). If \(n\) is large, we can use a central limit theorem argument to conclude that $$ \mbox{Pr}(N \le n) = 1  \mbox{Pr} \left( \sum_{j=1}^n Y_j \le w \right) = \Phi \left( \frac{\mu \sqrt{n}}{\sigma}  \frac{w}{\sigma \sqrt{n}} \right) \, . $$ Since there are many cycles, each lasting a very short time, we can replace the discrete number of cycles \(N\) needed to reach failure by the continuous time \(t_f\) needed to reach failure. The CDF \(F(t)\) of \(t_f\) is given by $$ F(t) = \Phi \left\{ \frac{1}{\alpha} \left[ \sqrt{\frac{t}{\beta}}  \sqrt{\frac{\beta}{t}} \right] \right\} \, , $$ for appropriate choice of $$ \alpha = \frac{\sigma}{\sqrt{\mu w}} \,\, \mbox{ and } \,\, \beta = \frac{w}{\mu} \,\, .$$ Here \(\Phi\) denotes the standard normal CDF. Writing the model with parameters \(\alpha\) and \(\beta\) is an alternative way of writing the BirnbaumSaunders distribution that is often used (\(\alpha = \gamma\) and \(\beta = \mu\), as compared to the way the formulas were parameterized earlier in this section).
Note:


This kind of physical degradation is consistent with Miner's Rule.  The BirnbaumSaunders assumption, while physically restrictive, is consistent with a deterministic model from materials physics known as Miner's Rule (Miner's Rule implies that the damage that occurs after \(n\) cycles, at a stress level that produces a fatigue life of \(N\) cycles, is proportional to \(n/N\)). So, when the physics of failure suggests Miner's Rule applies, the BirnbaumSaunders model is a reasonable choice for a life distribution model.  
BirnbaumSaunders probability plot 
We generated 100 random numbers from a BirnbaumSaunders distribution where
\(\mu\) = 5000 and \(\gamma\) = 2,
and created a
fatigue life probability plot
of the 100 data points.
The PDF value at time \(t\) = 4000 for a BirnbaumSaunders (fatigue life) distribution with parameters \(\mu\) = 5000 and \(\gamma\) = 2 is 4.987e05 and the CDF value is 0.455. Functions for computing BirnbaumSaunders distribution PDF values, CDF values, and for producing probability plots, are found in both Dataplot code and R code. 