Reliability Models for Failure Mechanisms

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Failure Mechanism Reliability Modeling, or reliability modeling, or acceleration modeling, or simply modeling, is the mathematical representation of a failure mechanism in terms of a set of algebraic or differential equations from the perspective of its reliability implications. The term failure mechanism refers to the actual physical phenomenon behind a failure occurrence. Modeling is a means of determining and understanding the different variables or factors that bring out and accelerate a failure mechanism.

Being able to model a mechanism and quantify how it is affected by various environmental factors will allow a reliability engineer to develop appropriate reliability tests for estimating field failure rates and predicting when failures will begin to occur. Modeling is often expressed in the form of time to failure, or tf, or the acceleration factor, AF.

The Arrhenius Equation

Everything in this universe will decay or degrade with time, and the Second Law of Thermodynamics is there to make sure of this. Destruction or degradation of matter is generally due to atomic or molecular changes accelerated by external factors, one of which is temperature. The response dependence of degradation or failure mechanisms on temperature is given by the Arrhenius equation:

R = Ae^(-Ea/kT)

where R=reaction rate, A=constant, Ea=activation energy,

k= Boltzmann’s constant (8.6e-5 eV/K), T=absolute temperature

For any given reaction obeying the Arrhenius equation, R1t1=R2t2=constant, where R is the reaction rate and t is the elapsed reaction time. To illustrate this, consider a reaction process that occurs at a high temperature T1 and low temperature T2. Since temperature increases the reaction rate, then R1 is faster than R2, or R1 > R2 . However, the reaction process also takes a shorter duration at T1, or t1 < t2, such that R1t1=R2t2 =constant.

Now, let tf=time to failure, then Rtf =constant, or tf=C1/R.

Thus, tf = C1/(Ae^(-Ea/kT)) = (C)(e^(Ea/kT)).

Let the acceleration factor AF be the ratio tf_use/ tf_test .

Thus, AF=[(C)(e^(Ea/kTuse)) / (C)(e^(Ea/kTtest))]= e^{(Ea/k)
(1/Tuse-1/Ttest)}

Estimating Ea and tf using Arrhenius Plots

Recall that tf = (C)(e^(Ea/kT)). Then, ln(tf) = lnC + Ea/kT.

Thus, the plot of ln(tf) vs. 1/T yields a straight line whose slope

corresponds to Ea/k.

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