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| ==Arrhenius-Weibull==
| | #REDIRECT [[Arrhenius_Relationship]] |
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| The <math>pdf</math> for the 2-parameter Weibull distribution is given by:
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| ::{{weibull2pdf}}
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| The scale parameter (or characteristic life) of the Weibull distribution is <math>\eta </math> .
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| The Arrhenius-Weibull model pdf can then be obtained by setting <math>\eta =L(V)</math> in Eqn. (arrhenius):
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| ::<math>\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}</math>
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| and substituting for <math>\eta </math> in Eqn. (Weibullpdf):
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| ::<math>f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}</math>
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| An illustration of the <math>pdf</math> for different stresses is shown in Fig. 6. As expected, the <math>pdf</math> at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3 in Fig. 6). This behavior is observed when the parameter <math>B</math> of the Arrhenius model is positive.
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| [[Image:ALTA6.6.gif|thumb|center|300px|Behavior of the probability density function at different stresses and with the parameters held constant.]] | |
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| The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in Chapter 5.
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| {{aaw stat prob sum}}
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| ===Parameter Estimation===
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| {{aaw mle}}
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