ALTA ALTA Standard Folio Data Arrhenius-Lognormal: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
No edit summary
Line 11: Line 11:
|-
|-
| valign="middle" |
| valign="middle" |
<br>
The  <math>pdf</math>  of the lognormal distribution is given by:
<br>
::<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>
<br>
where:
<br>
<math>{T}'=\ln(T) </math>
<br>
and:
<br>
• <math>T=</math>  times-to-failure.
• <math>{T}'=</math>  mean of the natural logarithms of the times-to-failure.
• <math>T=</math>  times-to-failure.
• <math>{{\sigma }_{{{T}'}}}=</math>  standard deviation of the natural logarithms of the times-to-failure.
<br>
<br>
The median of the lognormal distribution is given by:
<br>
::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>
<br>
The Arrhenius-lognormal model  <math>pdf</math>  can be obtained first by setting <math>\breve{T}=L(V)</math>  in Eqn. (arrhenius). Therefore:  
The Arrhenius-lognormal model  <math>pdf</math>  can be obtained first by setting <math>\breve{T}=L(V)</math>  in Eqn. (arrhenius). Therefore:  


Line 53: Line 26:
Substituting Eqn. (arrh-logn-mean) into Eqn. (arrh-logn-pdf) yields the Arrhenius-lognormal model  <math>pdf</math>  or:  
Substituting Eqn. (arrh-logn-mean) into Eqn. (arrh-logn-pdf) yields the Arrhenius-lognormal model  <math>pdf</math>  or:  


::<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>
<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>


<br>
Note that in Eqn. (arrh-logn-pdf), 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).
<br>
|-
|-
|  valign="middle" | [http://reliawiki.com/index.php/Template:Alta_al#Arrhenius-Lognormal Get More Details...]
|  valign="middle" | [http://reliawiki.com/index.php/Template:Alta_al#Arrhenius-Lognormal Get More Details...]

Revision as of 17:30, 14 February 2012

Webnotes-alta.png
Standard Folio Data Arrhenius-Lognormal
ALTA

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]

Get More Details...


Docedit.png
Retrieved from "https://www.reliawiki.com/index.php?title=ALTA_ALTA_Standard_Folio_Data_Arrhenius-Lognormal&oldid=15823"