Issue 23, January 2003

A Look Under the Hood at the Cumulative Damage Model

Past issues of the HotWire and Reliability Edge have presented examples (see Analyzing Step-Stress Data and Analyzing Accelerated Test Data with Time-Varying Test and Use Stress Profiles) that utilize the cumulative damage model. But what is happening behind the scenes? This article will present the cumulative damage model and how it is formulated.

To formulate the cumulative exposure/damage model, consider a simple step-stress experiment where an electronic component was subjected to a voltage stress, starting at 2V (use stress level) and increased to 7V in stepwise increments, as shown in Figure 1. The following steps, in hours, were used to apply stress to the products under test: 0 to 250, 2V; 250 to 350, 3V; 350 to 370, 4V; 370 to 380, 5V; 380 to 390, 6V; and 390 to 400, 7V.

Figure 1: Step-stress profile of voltage stress

In this example, eleven units were available for the test. All eleven units were tested using this same stress profile. Units that failed were removed from the test and their total times on test were recorded. The following times-to-failure were observed in the test, in hours: 280, 310, 330, 352, 360, 366, 371, 374, 378, 381 and 385. The first failure in this test occurred at 280 hours when the stress was 3V. During the test, this unit experienced a period of time at 2V before failing at 3V. If the stress were 2V, one would expect the unit to fail at a time later than 280 hrs, while if the unit were always at 3V, one would expect that failure time to be sooner than 280 hrs. The problem faced by the analyst in this case is to determine some equivalency between the stresses. In other words, what is the equivalent of 280 hours (with 250 hours spent at 2V and 30 spent at 3V) at a constant 2V stress or at a constant 3V stress?

Mathematical Formulation

To mathematically formulate the model, consider the following step-stress test, shown in Figure 2, with stresses S₁, S₂ and S₃. Furthermore, assume that the underlying life distribution is the Weibull distribution and also assume an inverse power law relationship between the Weibull scale parameter and the applied stress.

Figure 2: Step-stress profile and corresponding life distributions

From the inverse power law relationship, the scale parameter of the Weibull distribution, η, can be expressed as an inverse power function of the stress, V.

where K and n are the model parameters.

The fraction of the units failing by time under a constant stress, V = S₁, is given by:

Where:

Combining Eqns. (1), (2) and (3) yields the cdf for each constant stress level.

	(4)
	(5)
	(6)

The above equations would suffice if the units did not experience different stresses during the test, as they did in this case. To analyze the data from this step-stress test, a cumulative exposure model is needed. Such a model will relate the life distribution of the units, in this case the Weibull distribution, at one stress level to the distribution at the next stress level. In formulating this model, it is assumed that the remaining life of the test units depends only on the cumulative exposure the units have seen and that the units do not "remember" how such exposure was accumulated. Moreover, since the units are held at a constant stress at each step, the surviving units will fail according to the distribution at the current step, but with a starting age corresponding to the total accumulated time up to the beginning of the current step. This model can be formulated as follows:

• Units failing during the first step have not experienced any other stresses and will fail according to Eqn. (4). Units that made it to the second step will fail according to Eqn. (5), but will have accumulated some equivalent age, ε₁, at this stress level (given the fact that they have spent t₁ hours at S₁).

In other words, the probability that the units will fail at a time t, while at S₂ and between t₁ and t₂ is equivalent to the probability that the units will fail after accumulating (t -t₁) plus some equivalent time ε₁ (to account for the exposure the units have seen at S₁).

• The equivalent time, ε ₁, will be the time by which the probability of failure at S₂ is equal to the probability of failure at S₁ after an exposure of t₁:

• One would repeat this for step 3 taking into account the accumulated exposure during steps 1 and 2, or in more general terms and for the ith step:

And where:

• Once the cdf for each step has been obtained, the pdf can also be determined utilizing:

Once the model has been formulated, model parameters (i.e. K, n and β) can be computed utilizing maximum likelihood estimation methods.

Generalized Formulation for Time-Varying Stress

The example presented above can be expanded for any time-varying stress. ALTA 6 PRO allows you to define any stress profile. For example, the stress can be a ramp stress, a monotonically increasing stress, a sinusoidal stress, etc. This section presents a generalized formulation of the cumulative damage model, where stress can be any function of time.

General Weibull-Power Model

Given a time-varying stress x(t), then the reliability function of the unit under a single stress is given by:

and assuming the inverse power law relationship:

Therefore, the pdf is:

Parameter estimation can be accomplished via maximum likelihood estimation methods and confidence intervals can be approximated using the Fisher Matrix approach. Once the parameters have been determined, all other characteristics of interest can be obtained utilizing the statistical properties definitions for mean life, failure rate, etc.

General Weibull-Exponential Model

This model can be extended to include other distributions and relationships. ALTA 6 PRO includes this model (cumulative damage) for both the Weibull and exponential distributions and utilizing either an exponential (i.e. Arrhenius) or inverse power law (IPL) relationship. The formulation of the Weibull-exponential model is as follows:

Where:

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