This chapter includes the following subchapters:
The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e. temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887. The Arrhenius reaction rate equation is given by:
R is the speed of reaction.
A is an unknown nonthermal constant.
EA is the activation energy (eV).
K is the Boltzman’s constant (8.617385 ´ 10-5 eV K-1).
T is the absolute temperature (Kelvin).
The activation energy is the energy that a molecule must have to participate in the reaction. In other words, the activation energy is a measure of the effect that temperature has on the reaction.
The Arrhenius life-stress model is formulated by assuming that life is proportional to the inverse reaction rate of the process, thus the Arrhenius life-stress relationship is given by:
L represents a quantifiable life measure, such as mean life, characteristic life, median life or B(x) life, etc.
V represents the stress level (formulated for temperature and temperature values in absolute units i.e. degrees Kelvin or degrees Rankine).
C is one of the model parameters to be determined, (C > 0).
B is another model parameter to be determined.
Fig. 1: Graphical look at the Arrhenius life-stress relationship (linear scale) for different life characteristics, assuming a Weibull distribution.
Since the Arrhenius is a physics-based model derived for temperature dependence, it is strongly recommended that the model be used for temperature accelerated tests. For the same reason, temperature values must be in absolute units (Kelvin or Rankine), even though Eqn. (1) is unitless.
The Arrhenius relationship can be linearized and plotted on a life vs. stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in Eqn. (1) or:
Fig. 2: Arrhenius plot for Weibull life distribution.
In Eqn. (2) ln (C) is the intercept of the line and B is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In Figure 2, life is plotted versus stress and not versus the inverse stress. This is because Eqn. (2) was plotted on a reciprocal scale. On such a scale, the slope B appears to be negative even though it has a positive value. This is because B is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( is decreasing as V is increasing). The two different axes are shown in Figure 3.
Fig. 3. An illustration of both reciprocal and non-reciprocal scales.
The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in Figure 3 it is more convenient to locate the life corresponding to a stress level of 370K rather than to take the reciprocal of 370K (0.0027) first and then locate the corresponding life.
The shaded areas shown in Figure 3 are the imposed pdfs at each test stress level. From such imposed pdfs one can see the range of the life at each test stress level, as well as the scatter in life. The next figure (Figure 4) illustrates a case in which there is a significant scatter in life at each of the test stress levels.
Fig. 4: An example of scatter in life at each test stress level.
A Look at the Parameter B
Depending on the application (and where the stress is exclusively thermal), the parameter B can be replaced by:
Note that in this formulation, the activation energy must be known a priori. If the activation energy is known then there is only one model parameter remaining, C. Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat B as one of the model parameters. As it can be seen in Eqn. (1), B has the same properties as the activation energy. In other words, B is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of B, the higher the dependency of the life on the specific stress (see Figure 5). Parameter B may also take negative values. In that case, life is increasing with increasing stress (see Figure 5). An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.
Fig. 5: Behavior of the parameter B.
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