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

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The  <math>pdf</math>  of the 1-parameter exponential distribution is given by:
 
<br>
<math>f(t)=\lambda {{e}^{-\lambda t}}</math>
<br>
It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:
<br>
<math>\lambda =\frac{1}{m}</math>
<br>
thus:
<br>
<math>f(t)=\frac{1}{m}{{e}^{-\tfrac{t}{m}}}</math>
<br>
The Arrhenius-exponential model  <math>pdf</math>  can then be obtained by setting  <math>m=L(V)</math>  in Eqn. (arrhenius).
<br>
Therefore:
<br>
<math>m=L(V)=C{{e}^{\tfrac{B}{V}}}</math>
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Substituting for  <math>m</math>  in Eqn. (pdfexpm) yields a  <math>pdf</math>  that is both a function of time and stress or:
<br>
<math>f(t,V)=\frac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot {{e}^{-\tfrac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot t}}</math>
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