Two Methods for Analyzing Time-Varying Stress Data in ALTA
Accelerated life testing (ALT) is an essential tool in many life data analysis scenarios, due to the discrepancy between the long lifetimes of products and the short time periods allocated for product development. ALT allows you to shorten the testing time by observing failures more quickly via increasing the stress on the product. Hence, the goal of ALT is to relate the life of the product to the stresses it operates under.
Many types of products experience changing operating conditions (i.e., time-varying stresses) in the field. To accurately predict reliability for these products, you must account for this in your ALT data analysis by using the cumulative damage model. In this article, we will look at the cumulative damage model and compare two ways to implement it in ALTA.
About the Cumulative Damage Model
The cumulative damage model can be used to estimate the reliability of a product for time-varying stress profiles determined from field operating conditions. This model takes into account the cumulative effect of the applied time-varying stresses, either during testing or at the use level.
The most basic type of time-varying stress is a step-stress profile, in which each specimen is subjected to different levels of constant stress for different periods of time during the test (or while operating in the field). In addition to step-stress, there are many other types of time-dependent stress profiles that can be used in accelerated life testing. They are usually one of the following:
- A ramp stress, where stress increases linearly (from 0) with time.
- A cyclical stress, where stress follows sinusoidal, square waves, etc.
- A randomly varying stress over time (i.e., stochastic loading).
- A nonrepeating pattern.
The cumulative damage model represents the case where the stress, x(t), is a function of time and the scale parameter of the distribution, α(x,υ), is a function of time and stress:
where υ represents other possible constant variables.
Then, the cumulative exposure (damage) is a function of x(t), υ and the model parameters, and can be formulated as:
The population fraction failed by time t under varying stress, x(t), can be derived from:
where G[ ] is the assumed cumulative distribution with the scale parameter set equal to 1. 
For example, if we formulate the model using the Weibull distribution, we would obtain:
where η(x(u),υ) represents the Weibull scale parameter and can be defined by any life-stress relationship (Arrhenius, inverse power law, etc.) as a function of stress.
As an example, consider the Arrhenius life-stress relationship, which is given by the formula:
We can write the cumulative damage reliability function for the Arrhenius-Weibull model as:
We can use this cumulative damage Arrhenius-Weibull reliability function to calculate the reliability of a product at a given time under a time-varying stress with the estimated model parameters β, C and B. The model parameters are estimated using maximum likelihood estimation (MLE). 
To better understand the theory behind the cumulative damage model and how we apply that information to products that operate under changing operating conditions (time-varying stresses) in the field, we will look at a simple example. We will first analyze the data by manually combining results from multiple data sheets, each analyzed with the Arrhenius-Weibull model for a different use condition (this method can be implemented in ALTA Standard). Then we will analyze the same data set using the cumulative damage model and time-varying stress profiles that are available only in ALTA PRO.
Suppose you are a reliability engineer responsible for testing the O-rings installed on a fleet of pumps in your production plant. The O-rings are made of ethylene-propylene, which consists of a copolymer of ethylene and propylene (EPR), combined with a diene monomer (EPDM). The pumps operate in 3 different phases, where the flow rate changes depending on the production capacity. The operating temperature of the pumps, and therefore the temperature of the O-rings, also change depending on the production capacity.
- In the first phase, the pumps operate at 100% capacity for 20 days (480 hours), at a temperature of 318 K.
- In the second phase, the pumps operate at 80% capacity for 5 days (up to 600 hours), at a temperature of 298 K.
- And in the third phase, which also lasts 5 days (up to 720 hours), the capacity of production is reduced to 70%, at a temperature of 288 K.
This cycle repeats itself every 30 days.
Even though ethylene-propylene O-rings have a good resistance to heat up to 135°C (408.15 K), they are not recommended for continuous use at high temperatures. Therefore, at the end of every cycle, the O-rings installed on the pumps are replaced with brand new ones. You have 300 pumps operating in the production plant and you would like to estimate how many spare parts you need every cycle (i.e., how many O-rings will fail before the standard replacement at 30 days).
To estimate the number of spare parts you need in a month, you run an accelerated life test on 30 O-ring samples under 3 different stress levels (where each sample is tested at a single, constant stress level) and obtain the time-to-failure data given below.
The operating times and temperatures of the pumps under normal use condition are summarized in the table below. Note that times are given in hours (e.g., 20 days = 480 hours).
ALTA Standard Analysis
To perform this analysis in ALTA Standard, you enter the data in an ALTA standard folio and analyze it using the Arrhenius-Weibull model (Arrhenius as the life-stress relationship and Weibull as the life data analysis model). You create three copies of the same data sheet, with a different use stress specified for each one. For example, the parameters for the data sheet with the 318 K use stress (i.e., the first phase of operation) are calculated as 2.447, 3625.016 and 0.027 for β, B and C, respectively, as shown in the figure below.
The data passes the likelihood ratio test at 90% confidence level; therefore, the common shape parameter, β, assumption among the 3 different stress levels tested is valid.
The following figure shows the probability plot at the 3 stress levels that were applied during the test and at the use stress level of 318 K that's expected for phase 1 operation.
You perform a similar analysis in separate data sheets for the 298 K and 288 K temperatures that the pumps will experience in phase 2 and 3. The figure below shows an overlay plot from the three data sheets, which demonstrates the effect of the temperature on the reliability of the O-rings. As the temperature increases, the reliability of the O-rings decreases.
To be able to calculate the reliability of the O-rings at the end of a 30 day cycle, we need to create the use stress reliability curve for the O-rings at the time-varying use conditions.
At the point where the temperature changes from 318 K to 298 K (after 20 days of operation), we want the total percent failed to be the same immediately before and after the jump in stress level. In other words, the damage that the units have seen just before the temperature changes is equivalent to the damage they have seen just after the change. The same idea applies to the temperature change at the 25th day. Therefore, the use stress reliability curve needs to follow the high-stress reliability curve (green) until the end of the 20 day period (480 hours) and then switch to the medium-stress curve (black), keeping the accumulated damage (reliability) constant. At the end of 25 days (600 hours), the use stress reliability curve should switch from the medium-stress curve to the low-stress curve (blue), again keeping the accumulated damage constant. The use stress reliability curve should then follow the low-stress curve until the end of the 30 days (720 hours).
The following table shows the reliability values for the conditions of 318 K, 298 K and 288 K and the times of switches are highlighted.
So the true reliability curve, given below, is created by joining the three reliability curves at the highlighted points.
Therefore, the reliability is calculated as 97% (probability of failure = 3%) at the end of 30 days (720 hours). So, the number of spare O-rings needed in a month for 300 pumps is calculated as 300 × 0.03 = 9.
ALTA PRO – Cumulative Damage Analysis
In ALTA PRO, this data set can be analyzed using the cumulative damage–Weibull model. You use a Reciprocal transformation for the temperature stress, which mimics an Arrhenius relationship. In other words, the cumulative damage model is set to behave in exactly the same way as the Arrhenius life-stress relationship model that was used in the ALTA standard folio method.
The stress profile, given below, is created for the use condition as described previously.
The analysis summary is shown below. Note that the values of β are the same using both methods, and that the value of B using the manual method matches the value of α1 using the cumulative damage model and stress profiles that are available only in ALTA PRO.
The reliability at the end of the 30-day period (720 hours) is calculated as 97%, which is the same as the value calculated with the manual method. So, the number of spare O-rings needed in a month for 300 pumps is calculated as 300 × 0.03 = 9.
For comparison, please see the reliability values predicted by the 2 methods below. The minor difference between the 2 methods is due to the insufficient granularity of the time steps chosen for the calculations in the manual method.
In this article, we presented two different methods to calculate the reliability of a product that operates under a time-varying stress at the use condition. A simple example is used to illustrate the theory behind the cumulative damage model and how to perform the analysis using the different methods. The manual method (which can be performed with the ALT models available in ALTA Standard) is not hard to implement for the example presented here and it provides nearly identical results to using the cumulative damage model and stress profiles built into the ALTA Pro method. However, there are 4 important drawbacks to the manual method:
- It is time consuming.
- The accuracy of the results depends on the granularity of the time steps chosen for the calculations.
- For complicated stress profiles, such as multiple stress levels, ramp stresses, sinusoids, etc., the manual method becomes difficult or impossible to apply.
- There are no built-in plots available for results calculated with the manual method.
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