ALTA ALTA Standard Folio Data Arrhenius-Exponential

Standard Folio Data Arrhenius-Exponential

ALTA

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