|
|
| (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]]
| |