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

From ReliaWiki
Jump to navigation Jump to search
No edit summary
 
(14 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Template:NoSkin}}
#REDIRECT [[Template:WebNotes/ALTAALTA_Standard_Folio_Data_Arrhenius]]
{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"
|-
| valign="middle" align="left" bgcolor=EEEEEE|[[Image: Webnotes-alta.png |center|195px]]
|}
{|  class="FCK__ShowTableBorders" border="0" cellspacing="1" cellpadding="1"
|-
|  valign="middle" |{{Font|Standard Folio Data Arrhenius-Lognormal|11|tahoma|bold|gray}}
|-
|  valign="middle" | {{Font|ALTA|10|tahoma|bold|gray}}
|-
| 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:
 
<math>\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}</math>
 
or:
 
<math>{{e}^{{{\overline{T}}^{\prime }}}}=C{{e}^{\tfrac{B}{V}}}</math>
 
Thus:
 
<math>{{\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>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>
 
<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...]
 
|}
 
<br>
[[File:docedit.png|20px|right|link=http://www.reliawiki.com/index.php?title=ALTA_ALTA_Standard_Folio_Data_Arrhenius-Lognormal&action=edit]]
 
[[Category:See example]]

Latest revision as of 23:16, 7 July 2015

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=ALTA_ALTA_Standard_Folio_Data_Arrhenius-Lognormal&oldid=58936"