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| ==Arrhenius-Lognormal==
| | #REDIRECT [[Arrhenius_Relationship#Arrhenius-Lognormal]] |
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| The <math>pdf</math> of the lognormal distribution is given by:
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| ::<math>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>
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| where:
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| ::<math>{T}'=\ln(T) </math>
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| and:
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| • <math>T=</math> times-to-failure.
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| • <math>{T}'=</math> mean of the natural logarithms of the times-to-failure.
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| • <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.
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| The median of the lognormal distribution is given by:
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| ::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>
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| The Arrhenius-lognormal model <math>pdf</math> can be obtained first by setting <math>\breve{T}=L(V)</math>. Therefore:
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| ::<math>\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}</math>
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| or:
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| ::<math>{{e}^{{{\overline{T}}^{\prime }}}}=C{{e}^{\tfrac{B}{V}}}</math>
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| Thus:
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| ::<math>{{\overline{T}}^{\prime }}=\ln (C)+\frac{B}{V}</math>
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| Substituting the above equation into the lognormal <math>pdf</math> yields the Arrhenius-lognormal model <math>pdf</math> or:
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| ::<math>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>
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| Note that in the Arrhenius-lognormal <math>pdf</math>, it was assumed that the standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}},</math> is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( <math>{{\sigma }_{{{T}'}}}</math> is the shape parameter of the lognormal distribution).
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| {{alta al stat prop sum}}
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| ===Parameter Estimation===
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| {{arrhenius-log mle}}
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