# Miner’s Rule and Cumulative Damage Models

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

Miner’s rule is one of the most widely used cumulative damage models for failures caused by fatigue. It is called "Miner’s rule" because it was popularized by M. A. Miner in 1945. In this article, we will explain what it is and how it is related to other more advanced cumulative damage models in ALTA.

### Miner’s Rule

Miner’s rule is probably the simplest cumulative damage model. It states that if there are k different stress levels and the average number of cycles to failure at the ith stress, Si, is Ni, then the damage fraction, C, is:

 (1)

where:

• ni is the number of cycles accumulated at stress Si.
• C is the fraction of life consumed by exposure to the cycles at the different stress levels. In general, when the damage fraction reaches 1, failure occurs.

The above equation can be thought of as assessing the proportion of life consumed at each stress level and then adding the proportions for all the levels together. Often an index for quantifying the damage is defined as the product of stress and the number of cycles operated under this stress, which is:

Assuming that the critical damage is the same across all the stress levels, then:

 (2)

For example, let’s say WFailure=50 for a component. So the component will fail after 10 cycles at a stress level of 5, or after 25 cycles to fail at a stress level of 2, and so on. Using Eqn. (2) as the critical value of damage that will result in failure, Eqn. (1) becomes:

 (3)

C represents the proportion of the cumulative damage to the critical value.

## Example

Let’s use a simple example to illustrate how Miner’s rule is applied to model failures caused by fatigue. Assume we are interested in the "cycles to failure" of a paper clip. As illustrated in the following figure, a cycle is defined as opening and closing a clip. It is known that the angle of the two arms of the paper clip affects the number of cycles to failure. So here the "angle" is treated as the "stress."

Figure 1: One Cycle for a Paper Clip – Opening and Closing the Clip

Assume that we know from previous testing that, at an angle of 90 degrees, the average value that causes failure is 5 cycles. So in this example, WFailure=90x5=450. According to Miner’s rule, this means that at about 30 degrees, a failure will be observed after about 15 cycles at an angle of 30 degrees; or an average of 10 cycles will be needed to induce a failure at 45 degrees.

A paper clip has been tested according to the following test profile.

 Stress (Degrees) Test Cycles Damage at Stress Level Cumulative Damage 15 4 60 60 30 4 120 180 45 4 180 360

Based on Miner’s rule, how many additional cycles at 60 degrees are required in order to break this paper clip? Using Eqn. (3):

We find that x = 1.5 more cycles are needed at 60 degrees in order to break the paper clip.

From the above example, we can see there are several major limitations of Miner’s rule:

1. Only the expected values are used. It ignores the variation of each test unit and fails to recognize the probabilistic nature of fatigue.
2. A simple linear life-stress relationship is assumed. This may not be true in many real-word applications.

### Inverse Power Law Model

To overcome the drawbacks of the Miner’s rule model, probabilistic models are used in data analysis for accelerated life testing. There are two key ingredients in probabilistic models:

1. The critical damage that causes failures is not a fixed value. It follows a certain distribution.
2. It is not necessary for the damage to accumulate linearly.

Various life-stress models have been used. [1] The Inverse Power Law model describes the life-stress relationship using a power function, which is:

 (4)

L(S) represents the life at a stress of S. K and n are model parameters. If L(S) is the average life, we can see that Miner’s rule is a special case of the Inverse Power Law model. For example, for the previous paper clip example, K=1/450 and n=1. So the life at the 90 degree angle is 5 cycles; at 60 degrees it is:

cycles

However, to consider the probabilistic nature of the failures, instead of letting L(S) be the average life at stress S, it is treated as the "life characteristic" of a failure time distribution. For example, the pdf for a Weibull distribution is:

 (5)

If we substitute L(S) for the scale parameter η, which represents the life characteristic in the Weibull distribution, we obtain the Inverse Power Law-Weibull model:

 (6)

Let’s use an example to explain the application of the Weibull distribution with the Inverse Power Law life-stress relationship. To study the number of cycles to failure of a paper clip, the following test data were obtained:

 Status (F or S) Number of Cycles Angle (Degrees) F 30 15 F 33 15 F 28 15 S 35 15 S 35 15 F 18 30 F 20 30 F 15 30 F 12 30 S 20 30 F 11 45 F 12 45 F 8 45 F 15 45 F 10 45

From the above table, we can see that 5 units are tested under each of the stress levels of 15, 30 and 45 degrees. Because of the suspensions at the stress levels of 15 and 30, it is difficult to use Miner’s rule because the average cycles to failure are not the averages of the failure and suspension times at each stress level. Therefore, a Weibull distribution with Inverse Power Law life-stress relationship is used instead.

Figure 2: Results of the Inverse Power Law Model Using ALTA

Once the model has been obtained, the reliability of the paper clip at any stress can be predicted. Figure 3 shows the reliability together with its 90% confidence bounds at an angle of 5 degrees.

Figure 3: Predicted Reliability at 5 Degrees for a Paper Clip

From the plot, we can see that the B10 life is about 70 cycles. In other words, at a stress level of 5 degrees, a paper clip has a 90% chance of surviving at least 70 cycles.

### Conclusion

In this article, we discussed the well-known Miner’s rule cumulative damage model and compared it with a more advanced probabilistic model, the Inverse Power Law-Weibull model. Although Miner’s rule is simple to use and easy to understand, it lacks the probabilistic nature that is required for the proper analysis of many fatigue failures. For more details on other cumulative damage models, please refer to http://reliawiki.org/index.php/Accelerated_Life_Testing_Data_Analysis_Reference.

## References

[1] ReliaSoft Corporation, Accelerated Life Testing Reference, Tucson, AZ: ReliaSoft Publishing, 2007.

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