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Arrhenius-Lognormal
The $$pdf$$  of the lognormal distribution is given by:
 * $$f(T)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar}{{{\sigma }_}} \right)}^{2}}}}$$

where:


 * $${T}'=\ln(T) $$

and: •	 $$T=$$ times-to-failure.

•	 $${T}'=$$ mean of the natural logarithms of the times-to-failure.

•	 $$T=$$ times-to-failure.

•	 $${{\sigma }_}=$$ standard deviation of the natural logarithms of the times-to-failure. The median of the lognormal distribution is given by:


 * $$\breve{T}={{e}^}$$

The Arrhenius-lognormal model $$pdf$$  can be obtained first by setting $$\breve{T}=L(V)$$  in Eqn. (arrhenius). Therefore:


 * $$\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}$$

or:


 * $${{e}^}=C{{e}^{\tfrac{B}{V}}}$$

Thus:


 * $${{\overline{T}}^{\prime }}=\ln (C)+\frac{B}{V}$$

Substituting Eqn. (arrh-logn-mean) into Eqn. (arrh-logn-pdf) yields the Arrhenius-lognormal model $$pdf$$  or:


 * $$f(T,V)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_}} \right)}^{2}}}}$$

Note that in Eqn. (arrh-logn-pdf), it was assumed that the standard deviation of the natural logarithms of the times-to-failure, $${{\sigma }_},$$  is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( $${{\sigma }_}$$ is the shape parameter of the lognormal distribution).