Issue 75, May 2007

Expanded Models for Accelerated Life Test Analysis with Time-Varying Stresses

Some accelerated life tests may use a time-varying stress application (such as step-stress or ramp-stress) to assure the observation of failures more quickly in order to shorten the product development cycle. Likewise, some products are expected to be operated under normal use conditions where the stress exposure varies with time. When performing data analysis for these types of situations, the life-stress relationship model must take into account the cumulative effect of the applied stresses. Such a model is commonly referred to as a cumulative damage or "cumulative exposure" model. ReliaSoft's ALTA PRO software supports the use of the cumulative damage model for data with time-varying stresses and with the recent release of Version 7, the software is now capable of analyzing data with multiple time-varying stresses. This article presents a summary of the four versions of the cumulative damage model available in ALTA 7 PRO, along with information about how these models may be applied.

Background
Analysis of accelerated life test data consists of an underlying life distribution (such as Weibull, lognormal and exponential) that describes the product at different stress levels and a life-stress relationship that quantifies the manner in which the life distribution changes across different stress levels. The cumulative damage life-stress relationship, which is used when the stress is time-dependent, has different versions that depend on the nature of the stress. ALTA 7 PRO features the following cumulative damage relationships:

• Cumulative damage Arrhenius relationship (for analysis involving one time-varying stress)
• Cumulative damage power relationship (for analysis involving one time-varying stress)
• Cumulative damage exponential relationship (for analysis involving one time-varying stress)
• Cumulative damage general log-linear relationship (for analysis involving multiple time-varying stresses)

These relationships are combined with the different distributions to create the "complete" model to be used for analyzing accelerated life test data with time-varying stresses. The following is a summary of these models. It should be noted that there is more uncertainty in the results from such time-varying stress tests than from traditional constant stress tests of the same length and sample size.

Cumulative Damage Arrhenius Relationship

This relationship is typically used when the stress type is thermal in nature (i.e. temperature) and is based on the Arrhenius relationship, which is perhaps the most common life-stress relationship utilized in accelerated life testing data analysis. Given a time-varying stress x(t), the cumulative damage Arrhenius relationship is given by:

Note: In ALTA 7 PRO, the above relationship is actually presented in a format consistent with the Arrhenius general log-linear (GLL) relationship:

Combining the cumulative damage Arrhenius relationship with an underlying life distribution assumption leads to the following models:

Cumulative Damage Arrhenius-Exponential	Cumulative Damage Arrhenius-Weibull	Cumulative Damage Arrhenius-Lognormal

where:

and:

Cumulative Damage Power Relationship

This relationship is typically used when the stress type is non-thermal in nature (e.g. voltage, vibration, shock, load, pressure, etc.) and is based on the inverse power law model. Given a time-varying stress x(t), the cumulative damage power relationship is given by:

Note: In ALTA 7 PRO, the above relationship is actually presented in a format consistent with the power general log-linear (GLL) relationship:

Combining the cumulative damage power relationship with an underlying life distribution assumption leads to the following models:

Cumulative Damage Power-Exponential	Cumulative Damage Power-Weibull	Cumulative Damage Power-Lognormal

where:

and:

Cumulative Damage Exponential Relationship

This relationship can be used when the stress is a categorical type that can take on discrete values, such as the lot designation for products from different manufacturing lots or made of different materials, etc. It is also sometimes used when the stress type is corrosion (humidity) or voltage. Given a time-varying stress x(t), the cumulative damage exponential relationship is given by:

Note: In ALTA 7 PRO, the above relationship is actually presented in a format consistent with the exponential general log-linear (GLL) relationship:

Combining the cumulative damage exponential relationship with an underlying life distribution assumption leads to the following models:

Cumulative Damage Exponential-Exponential	Cumulative Damage Exponential-Weibull	Cumulative Damage Exponential-Lognormal

where:

and:

Cumulative Damage General Log-Linear Relationship

This relationship is used when analyzing accelerated life data where multiple stress types are applied simultaneously and the application of one or more of those stresses varies with time. It is based on the general log-linear relationship. Given n time-varying stresses X₁(t), X₂(t),..., X_n(t), the cumulative damage general log-linear relationship is given by:

Combining the cumulative damage general log-linear relationship with an underlying life distribution assumption leads to the following models:

Cumulative Damage General Log-Linear-Exponential	Cumulative Damage General Log-Linear-Weibull

where:

The following example demonstrates a potential application for one version of the cumulative damage model in ALTA PRO.

Example
24 units of an electronic component were subjected to temperature and voltage stresses over a period of 1000 hours. The temperature was initially set at 400K for 500 hours and was then increased to 415K for the remainder of the test. The normal use condition is defined as 390K. The voltage stress was held constant during the entire test, with half of the samples tested at 6V and the other half tested at 8V. The normal use condition is defined as 4V.

The stress profiles are set up in ALTA 7 PRO as follows (by selecting Add Stress Profile from the Project menu).

The failure times obtained in the test, as entered in ALTA PRO, are shown in the next figure, where the labels in the Temp and Volt stress columns represent the three stress profiles shown above.

The components failure behavior is assumed to follow a Weibull distribution. For the temperature stress, the Arrhenius relationship is used.  For the voltage stress, the power relationship is used. These relationships are specified in the Stress Transformation window, as shown next.

The estimated model parameters are shown next.

The use level reliability plot is shown in the next figure.

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