## 2 Parametric Accelerated Life Testing for Mechanical Product
### Generalized Time to Failure Model and Sample Size Equation for Parametric ALT
Solid-state diffusion of impurities in silicon \(J\) could be stated as [30] (Figure 5):  
\[J=[aC(x-a)]\cdot exp\left[-\frac{q}{kT}\left(w-\frac{1}{2}a\xi\right)\right] \cdot v\]  
\[[\text{Density/Area}]\cdot[\text{Jump Probability}]\cdot[\text{Jump Frequency}]\]  
\[=-\left[a^{2}ve^{-qw/kT}\right]\cdot cosh\frac{qa_{\xi}^{x}}{2 kT}\frac{ \partial C}{\partial x}+\left[2 ave^{-qw/kT}\right]Csinh\frac{qa_{\xi}^{x}}{2 kT}\]  
\[=\Phi(x,t,T)sinh(a\xi)exp\left(-\frac{Q}{kT}\right) \tag{11a}\] \[=Asinh(a\xi)exp\left(-\frac{Q}{kT}\right)\]  
where \(A\) is constant, \(C\) is the concentration, \(q\) is the value of electric charge, \(\
u\) is the frequency, \(\Phi()\) is a quantity that does not change its value, \(a\) is the distance between atoms, \(\xi\) is the applied field, \(k\) is Boltzmann's constant, \(Q\) is energy, and \(T\) is temperature. \begin{table}
\begin{tabular}{c c c} \hline \hline  & Ohm’s Law of electrical conduction: \(j=-\sigma\
abla V\) & \\ \hline \(J=\text{electric current density}\), & \(X=\text{electric field, }-\
abla V\) & \(L=\text{conductivity, }\sigma=1/\rho\) \\ \(j\) (units: A/cm\({}^{2}\)) & (units: V/cm, V = electrical potential) & (units: \(\rho=\text{resistivity }(\Omega\text{ cm})\)) \\ \hline  & Fourier’s Law of heat transport: \(q=-\kappa\
abla T\) & \\ \hline \(J=\text{heat flux, }q\) (units: W/cm\({}^{2}\)) & \(X=\text{thermal force, }-\
abla T\) & \(L=\text{thermal conductivity, }\kappa\) \\  & (units: \({}^{\circ}\text{K/cm, T = temperature}\)) & (units: W/\({}^{\circ}\text{K cm}\)) \\ \hline  & [PERSON]’s Law of diffusion: \(F=-D\
abla C\) & \\ \hline \(J=\text{material flux, }F\) (units: /sec cm\({}^{2}\)) & \(X=\text{diffusion force, }-\
abla C\) & \(L=\text{diffusivity, }D\) \\  & (units: /cm\({}^{4}\), \(C=\text{concentration}\)) & (units: cm\({}^{2}\)/sec) \\ \hline  & [PERSON]’s Law of viscous fluid flow: \(F_{u}=-\mu\
abla u\) & \\ \hline \(J=\text{fluid velocity flux}\), & \(X=\text{viscous force, }-\
abla u\) & \(L=\text{viscosity, }\mu\) \\ \(F_{u}\) (units: /(sec\({}^{2}\) cm)) & (units: /sec, \(u=\text{fluid velocity}\)) & (units: /(sec cm)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Abridged linear transport phenomena [28; 29]. Conversely, the reaction process that rests on speed might be expressed as. \[K=K^{+}-K^{-}=a\frac{kT}{h}e^{-\frac{\Delta E-a\bar{S}}{kT}}-a\frac{kT}{h}e^{- \frac{\Delta E+a\bar{S}}{kT}}=2\frac{kT}{h}e^{-\frac{\Delta E}{kT}}sinh(\frac{aS }{kT}) \tag{11b}\]  
\[=Bsinh(aS)exp\left(-\frac{\Delta E}{kT}\right)\]  
The reaction rate K from Equations (11a) and (11b) might be shortened as  
\[K=Bsinh(aS)exp\left(-\frac{Q}{kT}\right) \tag{12}\]  
If Equation (12) takes an inverse function, the generalized life-stress (LS) model could be expressed as  
\[TF=A[sinh(aS)]^{-1}exp\left(\frac{E_{a}}{kT}\right) \tag{13}\]  
The sine hyperbolic formula \([sinh(aS)]^{-1}\) in Equation (13) may be stated as: (1) \((S)^{-1}\) has a little linear consequence at first, (2) \((S)^{-n}\) has what is considered as medium consequence, and (3) \((e^{aS})^{-1}\) is big in the end (Figure 5). As ALT is commonly carried out in the medium scope, Equation (13) could be redefined as  
\[TF=A(S)^{-n}exp\left(\frac{Q}{kT}\right) \tag{14}\]  
where \(n=-\left[\frac{\partial ln(TF)}{\partial ln(S)}\right]_{T^{\prime}}Q=-\left[ \frac{\partial ln(TF)}{\partial ln(\frac{T}{kT})}\right]_{S}\). For a specified crack and part geometry, Equation (14) can also be expressed as  
\[TF=B(\Delta K)^{-n}exp\left(\frac{Q}{kT}\right) \tag{15}\]  
where \(B\) is constant, stress intensity factor \(\Delta K=YS(or\ \Delta\sigma)\sqrt{\pi a}\),  
Figure 5: Meaning of hyperbolic sine stress term on Paris law. That is, a (fluctuating) stress range, \(S\), moves the crack to develop at some rate. As a certain stress range (or intensity range \(\Delta K\)) is exerted to a material for some number of cycles \(\Delta N\), it moves the crack to develop in length by a particular quantity \(\Delta a\).