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Arrhenius-Weibull
The $$pdf$$  for the 2-parameter Weibull distribution is given by:



The scale parameter (or characteristic life) of the Weibull distribution is $$\eta $$.

The Arrhenius-Weibull model pdf can then be obtained by setting $$\eta =L(V)$$:


 * $$\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}$$

and substituting for $$\eta $$  in the 2-parameter Weibull distribution equation:


 * $$f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}$$

An illustration of the $$pdf$$  for different stresses is shown in the next figure. As expected, the $$pdf$$  at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3). This behavior is observed when the parameter $$B$$  of the Arrhenius model is positive.



The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in Chapter 5.