[Please note that the following article — while it has been updated from our newsletter archives — may not reflect the latest software interface and plot graphics, but the original methodology and analysis steps remain applicable.]
A common question from reliability engineers performing accelerated life testing data analysis is, "Which life-stress model should I choose?" A life-stress model should be chosen based on a specific failure mechanism, and sometimes a literature search on that mechanism will yield a mathematical relationship between life and stress. As an example, in the case of high cycle mechanical fatigue, the relationship between the applied stress and the number of cycles to failure, often called the S-N curve, is known to be in the form of an inverse power law life-stress model .
As technology evolves, however, it is becoming increasingly difficult to find an established relationship between life and stress for new failure mechanisms. If no model can be found, the most direct approach to determine the appropriate analysis model is to perform life tests at many different stress levels to empirically establish the mathematical form of the relationship. The drawback to this method is that it requires many tests and consequently can be very time-consuming and resource-intensive. This article uses a fictional example to present an alternative approach for choosing an accelerated testing data analysis model in the absence of an established physics-of-failure relationship between life and stress.
The fictional XYZ Company has a highly reliable device that has been in the field for some time. Based on this success, XYZ has a potential new customer, ACME Corporation, who wants to use the device in a new application. Before ACME will purchase the device, the ACME design engineers want an assurance that the device will have sufficient reliability in the new environment. The engineers at XYZ suspect that testing the device under the new environmental conditions will not yield failures before ACME requires a reliability estimate, making it impossible to use traditional life data analysis to determine the reliability in the new environment. In addition, XYZ has only a limited number of test samples available. Therefore, the option to conduct a zero-failure reliability demonstration test is not feasible. The XYZ engineers conclude that the only way to provide ACME with the estimate of reliability they require is to perform accelerated tests and extrapolate the results to the new usage conditions.
Table 1 shows the time-to-failure data collected for three different combinations of temperature and relative humidity values. In addition, there is one set of field data available, which contains very few failures and many suspended data points. The devices in the field were operated at temperature and relative humidity values of 313K and 50%, respectively. A 1-parameter Weibull distribution was fitted to the field data using a beta (shape parameter) of 5 (based on the accelerated test data sets), and yielding an eta (scale parameter) of 129,000 hours at field conditions.
Table 1: Time-to-failure data from accelerated life testing
The failure mechanism that XYZ Company has seen in the field manifests itself when two events occur. First, high temperature causes decreased adhesion between layers of the material. Second, moisture enters the device via the void that was created by the decreased adhesion. It has also been observed that this failure mode will not occur at high temperature without moisture in the air, nor will it occur in moist air at low temperature. Thus, a two-stress model that takes into account both temperature and humidity must be used to accurately predict the failure of the device. (Note that the sequential nature of the failure mechanism cannot be considered in the analysis because it is not feasible to obtain information about the time at which temperature initiates a void in the material.)
The XYZ engineers do not know, based on physics-of-failure, the mathematical model that describes how the stresses affect the life of the device. Therefore, they decide to examine different two-stress models and choose the one that makes sense based on engineering knowledge and provides the best correlation with the results from the field data set. The specific models examined are the temperature-humidity, generalized Eyring and general log-linear models.
The first candidate model for analyzing the accelerated test data is the temperature-humidity model. The life-stress relationship for the temperature-humidity model is:
where L is the life of the device, T is temperature, RH is relative humidity, and A, b and Φ are model parameters. This model has no interaction term and therefore it assumes that the temperature and humidity stresses operate independently. Assuming a Weibull distribution, analysis of the accelerated testing data yields the parameters and 90% 2-sided confidence bounds on the parameters that are shown in Table 2.
Table 2: Calculated parameters for the temperature-humidity model
Figure 1 shows the effect of temperature on life, and Figure 2 shows the effect of humidity on life. As expected, increasing either of these stresses independently causes a decrease in life. However, Figure 3 shows a probability plot that superimposes the field data analysis against the analysis of the accelerated test data extrapolated to the use-level conditions (temperature = 313K and relative humidity = 50%). It can be seen that the temperature-humidity model predicts lifetimes that are much longer than observed in the field. For example, the B(10) life observed in the field is about 80,000 hours, while the B(10) life extrapolated to use conditions via the temperature-humidity model is around 1,400,000 hours. Therefore, XYZ Company concludes that there must be an interaction between the stresses and, therefore, the temperature-humidity model is not suitable for analysis.
Figure 1: Life-stress plot that varies temperature and holds relative humidity at 50%
Figure 2: Life-stress plot that varies relative humidity and holds temperature at 313K
Figure 3: Probability plot that compares the temperature-humidity model analysis extrapolated to use-level conditions (shown in blue) against the field data analysis (shown in black)
The second candidate model for analyzing the accelerated testing data is the generalized Eyring model. The life-stress relationship for the generalized Eyring model is:
where L is the life of the device, T is temperature, RH is relative humidity, and A, B, C and D are model parameters. This model assumes that there is an interaction between temperature and humidity. However, because the generalized Eyring model has four parameters, there must be data from at least four different combinations of temperature and humidity in order to solve for all of the model parameters. XYZ Company tested at only three combinations of temperature and humidity levels, and there are no additional units available for testing. Therefore, the generalized Eyring model cannot be used to model the available test data.
The third candidate model for analyzing the accelerated testing data is the general log-linear (GLL) model. The life-stress relationship for the two-stress version of the general log-linear model is:
where L is the life of the device, X1 and X2 are stresses, and α0, α1 and α2 are model parameters. This generalized model allows the engineers to choose a transformation that describes the behavior of each stress (exponential, Arrhenius or inverse power law). The engineers know that applying the general log-linear model with Arrhenius transformations for both stresses will yield the same results as the temperature-humidity model. Therefore, they decide to try the model with an Arrhenius transformation for temperature and an inverse power law (IPL) transformation for humidity. The transformed general log-linear model is:
where L is the life of the device, T is temperature, RH is relative humidity, and α0, α1 and α2 are model parameters. Once again assuming a Weibull distribution, analysis of the accelerated testing data yields the parameters and the 90% 2-sided confidence bounds on the parameters that are shown in Table 3.
Table 3: Calculated parameters for the general log-linear model for temperature/humidity data
Figure 4 shows a probability plot that superimposes the field data analysis against the GLL analysis of the accelerated temperature/humidity data extrapolated to use-level conditions. The plot shows that the model provides fairly good correlation for unreliability values of about 25% and higher. However, since XYZ's device has very high reliability, the engineers are most concerned with very small unreliability values. For these values, the plot shows no overlap between the confidence bounds of the two analyses.
Figure 4: Probability plot that compares the general log-linear model analysis with Arrhenius transform on temperature and IPL transform on humidity (shown in blue) against the field data analysis (shown in black)
At this point, the XYZ engineers are forced to think of a creative solution to model the test data appropriately. They decide to attempt to transform the stresses themselves in order to capture the effect of the interaction between temperature and humidity. They decide to keep temperature as one stress because it is the driver for the failure mechanism. For the second stress, they will use dew point, which is the temperature to which air must be cooled at a constant pressure to become saturated . Thus, the combined effect of temperature and humidity will be captured by the second stress.
Using formulas found in a paper published by the International Association for the Properties of Water and Steam , the stresses are transformed as shown below in Table 4.
Table 4: Transformation of temperature and relative humidity to dew point
The XYZ engineers decide to use the general log-linear model again to analyze the temperature/dew point data set. Based on their experience with the other models, they select an Arrhenius transformation for temperature and an IPL transformation for dew point. Therefore, the transformed general log-linear life stress model is:
where L is the life of the device, T is temperature, DP is dew point, and α0, α1 and α2 are model parameters. Once again assuming a Weibull distribution, analysis of the accelerated testing data set yields the parameters and the 90% 2-sided confidence bounds on the parameters that are shown in Table 5.
Table 5: Calculated parameters for the general log-linear model for temperature/dew point data
Because the dew point is a function of both temperature and relative humidity, evaluating the effect of an increase in temperature or relative humidity on life must be performed using the acceleration factor. For the general log-linear analysis of the temperature/dew point data, the median life at the use-level condition (temperature = 313K, relative humidity = 50%, dew point = 300.6K) is found to be around 124,000 hours. Table 6 shows the acceleration factors for an increase in temperature while holding relative humidity constant, and also for an increase in relative humidity while holding temperature constant. As expected, the life decreases for these increased stress levels, leading to acceleration factors greater than 1.
Table 6: Acceleration factors and median life estimates
Figure 5 shows a probability plot that superimposes the field data analysis against the GLL analysis of the transformed temperature/dew point data set. It can be seen that the general log-linear model using an Arrhenius transformation for temperature and an IPL transformation for dew point predicts lifetimes that are very close to those observed in the field. Because the median lines are very close and the confidence bounds overlap for all values of unreliability (including the very low values that are of most interest to the engineers), XYZ Company concludes that this model adequately captures the interaction of temperature and relative humidity for the device. Based on their model, they are able to provide ACME Corporation with the requested reliability predictions for the device running under the new environmental conditions.
Figure 5: Probability plot that compares the general log-linear model analysis with Arrhenius transform on temperature and IPL transform on dew point (shown in blue) against the field data analysis (shown in black)
In an increasing number of real life testing scenarios, an established physical model is not available to relate applied stresses with the resulting life of a device. A systematic approach to determine a physics-of-failure model using test data alone is often not practical due to time or resource constraints, especially if interactions between the stresses are present. This article presented an approach to determine a life-stress model in which the stresses themselves were transformed to mimic the effect of the interaction between the stresses. Then, a flexible life-stress model (the general log-linear) was applied to analyze the transformed stresses. The model was validated against a set of field data to determine if it adequately captured the effects of the applied stresses on the life of the device.
For readers who are interested in more information about the underlying principles and theory of quantitative accelerated life testing data analysis, including more detailed information about the temperature-humidity, generalized Eyring and general log-linear models that were considered here, please consult the Accelerated Life Testing Analysis Reference .
 O. H. Basquin, "The Exponential Law of Endurance Tests," Am. Soc. Test. Mater. Proc., vol. 10, pp. 625-630, 1910.
 Weather Channel. "Weather Glossary."
Internet: http://www.weather.com/glossary/d.html, [March 11, 2011].
 International Association for the Properties of Water and Steam. (1997, August). Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam. [Online]. Available: www.iapws.org/relguide/IF97-Rev.pdf.
 ReliaSoft, Accelerated Life Testing Reference, ReliaSoft Publishing, 2007.