Cumulative Damage Model for Progress Stress Profiles

Compares the results for the cumulative damage model for ramp stress.

The data set is from Table on page 232 in the book Accelerated Testing: Statistical Models, Test Plans, and Data Analysis by Dr. Nelson, John Wiley & Sons, 1990. Model of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.

An accelerated test employed a pair of parallel disk electrodes immersed in insulating oil. Voltage V across the pair was increased linearly with time t at a specified rate, and the voltage at oil breakdown was recorded. Since V = Rt(R is the ramp rate), the time to breakdown can be recorded. The breakdown time is also affected by the two electrode areas A. Three voltage linear rates and two electrode areas are used. The six stress profiles are:

The following table shows the failure data for an area of 1 square inch.

The following table shows the failure data for an area of 9 square inch.

The following life stress relationship and Weibull distribution are used for the data. At a constant stress V, the $$\eta\,\!$$ is:


 * $$ln(\eta(V, A)) = \alpha_{0}+\alpha_{1}ln(V)+\alpha_{2}ln(A)\,\!$$

where V is the voltage and A is the area.

The reliability function at time t and stress V is:


 * $$R(t,V,A) = e^{-\left(\frac{t}{\eta(V,A)} \right)^\beta}\,\!$$

When stress is varying with time, the reliability at time t is given as following:


 * $$R(t,V,A) = e^{-\left(\int_{0}^{t}\frac{1}{\eta(x,A)}dx\right)^\beta}$$

In the book, the following results are provided:


 * The ML solution for the parameters are given in Section 3 of Table 3.2 on page 511: $$1/\beta\,\!$$ = 0.07856677 ($$\beta\,\!$$=12.728027), $$\alpha_{0}\,\!$$ = 3.673202 ,   $$\alpha_{1} = \,\!$$0.05843506 ,  and  $$\alpha_{2}=\,\!$$-0.058626.
 * The maximum log likelihood is -1035.4269.

First, we create the stress profiles in ALTA. The following picture shows a stress profile with a ramp rate of 100 volts/sec.



In defining this stress profile, we add a small number to the voltage function in ALTA. This is because the log transformation is used for voltage. At time 0, the voltage value will be 0 if the profile is defined as V(t) = 100*t.

The following picture shows a plot of the stress profile.



Enter the stress profile into the ALTA standard folio.