Introduction to the Generalized Eyring Life-Stress Relationship

Reliability HotWire

Issue 74, April 2007

Reliability Basics

Introduction to the Generalized Eyring Life-Stress Relationship

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

When analyzing accelerated test data obtained from tests that involve multiple stress types, it is often assumed that the different stresses are independent and do not interact. The Hot Topics article in Issue 53 of Reliability HotWire described a methodology for analyzing the significance of interactions among different stress types. This article presents a life-stress relationship that is often used to analyze data obtained when temperature and a second non-thermal stress (e.g., voltage) are the accelerated stresses of a test and their interaction is also of interest. This relationship, called the generalized Eyring relationship, is available in ALTA PRO.

Life-Stress Relationship

The generalized Eyring life-stress relationship is given by:

where:

V is the temperature in absolute units (such as Rankine or Kelvin).
U is the non-thermal stress (i.e. voltage, vibration, etc.).
A, B, C, D are the parameters to be determined.

The Eyring relationship is a special case of the generalized Eyring relationship where C = D = 0 and A_Eyr = −A_GEyr.

Note that the generalized Eyring relationship includes the interaction of U and V as described by the DU/V term. In other words, this model can estimate the effect of changing one of the factors depending on the level of the other factor.

Acceleration Factor

Most life-stress models do not include interaction terms. For such models, a separate acceleration factor can be obtained for each stress by varying that stress while keeping the others constant; multiplying these individual acceleration factors yields the acceleration factor for the life-stress model. In the case of the generalized Eyring relationship, the acceleration factor is derived differently. The acceleration factor for the generalized Eyring relationship is given by:

where:

L_USEis the life at use stress level.
L_Accelerated is the life at the accelerated stress level.
V_U is the use temperature level.
V_A is the accelerated temperature level.
U_A is the accelerated non-thermal level.
U_U is the use non-thermal level.

Example

The following data set represents failure times (in hours) obtained from an accelerated life test performed on an electronic component. The objectives are to understand the synergy between temperature and voltage and to estimate the B10 life at the use conditions of Temperature = 328K and Voltage = 8. The data set is modeled using the lognormal distribution and the generalized Eyring model.

The following plot shows the probability plot at the normal use conditions.

The B10 life is calculated as shown next.

For comparison reasons, the data set was also analyzed using two alternative life-stress relationships that do not consider interactions between stresses: the temperature-nonthermal life-stress relationship and the general log-linear relationship (using a reciprocal transformation for temperature and no transformation for voltage). The two models are given by:

General Log-Linear:
Temperature-NonThermal:

The following plot shows the use level probability lines for each of the relationships. This illustrates the difference in results and interpretations that each model can lead to, and therefore the significance of choosing the appropriate life-stress relationship based on an understanding of both the effect of the stress type on the product's life and the interactions among the different stress types.