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The [math]\displaystyle{ pdf }[/math] of the lognormal distribution is given by:
- [math]\displaystyle{ f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}} }[/math]
where:
[math]\displaystyle{ {T}'=\ln(T) }[/math]
and:
• [math]\displaystyle{ T= }[/math] times-to-failure.
• [math]\displaystyle{ {T}'= }[/math] mean of the natural logarithms of the times-to-failure.
• [math]\displaystyle{ T= }[/math] times-to-failure.
• [math]\displaystyle{ {{\sigma }_{{{T}'}}}= }[/math] standard deviation of the natural logarithms of the times-to-failure.
The median of the lognormal distribution is given by:
- [math]\displaystyle{ \breve{T}={{e}^{{{\overline{T}}^{\prime }}}} }[/math]
The Arrhenius-lognormal model [math]\displaystyle{ pdf }[/math] can be obtained first by setting [math]\displaystyle{ \breve{T}=L(V) }[/math] in Eqn. (arrhenius). Therefore:
[math]\displaystyle{ \breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}} }[/math]
or:
[math]\displaystyle{ {{e}^{{{\overline{T}}^{\prime }}}}=C{{e}^{\tfrac{B}{V}}} }[/math]
Thus:
[math]\displaystyle{ {{\overline{T}}^{\prime }}=\ln (C)+\frac{B}{V} }[/math]
Substituting Eqn. (arrh-logn-mean) into Eqn. (arrh-logn-pdf) yields the Arrhenius-lognormal model [math]\displaystyle{ pdf }[/math] or:
- [math]\displaystyle{ f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}} }[/math]
Note that in Eqn. (arrh-logn-pdf), it was assumed that the standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\sigma }_{{{T}'}}}, }[/math] is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( [math]\displaystyle{ {{\sigma }_{{{T}'}}} }[/math] is the shape parameter of the lognormal distribution).
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