Accelerated Testing Data Analysis
Without a Known Physical Failure Model
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 [1].
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.
Introduction
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.
Temperature-Humidity Model
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)
Generalized Eyring Model
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.
General Log-Linear Model
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)
Transforming the Second Stress to Dew Point
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 [2]. 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 [3], 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)
Conclusion
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 [4].
References
[1]
O. H. Basquin, "The Exponential Law of Endurance Tests," Am. Soc. Test.
Mater. Proc., vol. 10, pp. 625-630, 1910.
[2]
Weather Channel. "Weather Glossary."
Internet: http://www.weather.com/glossary/d.html, [March 11, 2011].
[3]
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.
[4]
ReliaSoft Corporation, Accelerated Life Testing Reference, ReliaSoft
Publishing, 2007.
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