Fatigue and Reliability Analysis with Time-Varying Stress Using the Cumulative Damage Model
Fatigue is defined as weakness in metal or other materials caused by repeated variations of mechanical stress. Fatigue cracks are caused by cycling loading. Hence, if the loading does not change (stress stays constant), then there is no fatigue problem. On the other hand, varying stresses may lead to fatigue failure even though the magnitude of the stress is not high. In metallurgic terms, the microscopic damage caused by fatigue cyclic means there is no recovery from the fatigue failure mode.
Fatigue is a statistical problem, not a deterministic one, which means that we will not be able to calculate exactly how many cycles it takes for a product to fail. But we can estimate the reliability of that particular product at a given number of cycles via life-stress relationship models used in accelerated life testing (ALT) analysis. In this article, we will talk about two methods for analyzing fatigue failure data collected under time-varying stress, and then analyze an example data set in an ALTA folio to show how we can quickly analyze fatigue data.
Figure 1 shows that the most basic fatigue loading is through a constant amplitude.
Figure 1. Cycling fatigue loading 
The cycles to failure for a product can be estimated via its S-N curve. For example, Figure 2 shows that the number of cycles to failure for a given product at a stress amplitude of 70 ksi can be estimated as 10,000 cycles.
Figure 2. An example S-N curve for a given product 
Or we can calculate it directly from Basquin's Equation :
- Sa is the stress amplitude
- Nf is the number of cycles to failure
- σf' and b are the material constants
If the fatigue loading during the testing is more complex, we need to initially extract the cyclic content. In this article, we will see how to analyze the data collected during service loading with blocks of constant amplitude loading as shown in Figure 3b. The number of cycles to failure for each constant amplitude loading in the block loading scheme should be calculated separately (i.e., for S1, S2, and S3 in Figure 3b) from the S-N curve. Once we find the number of cycles to failure for each different amplitude loading block, we can use a linear damage rule (the Palmgren-Miner Rule) to determine failure time based on the fatigue damage accumulated from all the fatigue cycles.
Figure 3. Fatigue loading types; (a) constant amplitude loading, (b) block loading and (c) variable amplitude loading
The Palmgren-Miner Rule states that the damage fraction, C, can be calculated for k different stress amplitudes via:
- ni is the number of cycles in ith stress block
- Nfi is the cycles to failure for ith stress block's stress amplitude
In general, failure occurs when the damage fraction reaches 1. Although it is a very simple method, the cumulative damage model is still the standard approach for combining the effect of multiple cyclic loading in fatigue. For more information on the Palmgren-Miner Rule and how to run the calculations by hand, please see "Miner's Rule and Cumulative Damage Models" in HotWire Issue 116.
There are several major limitations of the Palmgren-Miner Rule. One of them is that it ignores the variation of each test unit and fails to recognize the probabilistic nature of fatigue. To overcome this shortcoming of the Palmgren-Miner Rule, we use probabilistic models in data analysis for accelerated life testing. In probabilistic models, the critical damage that causes failures is not a fixed value; instead, it follows a certain distribution. Therefore, we can calculate the reliability of the product at any given stress amplitude for any given cyclic loading.
We will see how we can apply this probabilistic models approach with an example by using the Inverse Power Law (IPL) Model. That model describes the life-stress relationship using a power function :
- L(S) is the life at a stress of S
- K and n are model parameters
Assuming L(S) is the mean life, then the Basquin's Equation can be considered as a special case of the IPL model.
Suppose you are a reliability engineer who is trying to identify the reliability of a coil spring in a vehicle suspension system. The reliability requirement you are looking for is 98% with 90% 1-sided lower confidence bounds at the end of one life. Design engineers identified that the main failure mode for this coil is fatigue. This failure occurs due to the excessive number of cycles under the weight of the vehicle, which is referred as the normal use condition. You consider that the normal use condition stress amplitude that each coil installed on the vehicle takes as 100% load.
Due to time limitations, you accelerate the testing by increasing the stress amplitude during the testing after two life cycles if there are coils still operating without failure. This is often called a step stress test, in which the stress is repeatedly increased to precipitate failure faster. This changing stress amplitude creates a challenge: how can we infer reliability at the 100% stress level when the parts are exposed to ever-increasing cyclic stress amplitude? The key is to quantify the relationship between the stress input and the resulting life. This relationship then allows us to predict reliability under any stress condition.
You run the fatigue test and collect the fatigue failure/suspension data (in thousand cycles) for the coil spring, as shown in Table 1. You estimate that one life for the coil spring as 1M cycles. During the testing of 5.5M cycles, you observe nine failures with one coil operating without failure.
Table 1 – Fatigue failure/suspension data for coil spring in a vehicle suspension system
Table 2 shows the stress amplitude profile used during fatigue testing. You consider the normal use condition as 100% and calculate the rest of the stress amplitudes accordingly.
Table 2. Stress amplitude profile followed during fatigue testing
After entering the data in an ALTA folio, you use the cumulative damage (CD) model as the life-stress relationship because the stress was non-constant during the testing. You also create a stress profile using the values given in Table 2.
Figure 4 shows the failures and the suspension data in the Stress Profile plot as red circles and blue triangle, respectively.
Figure 4. ALTA Stress Profile plot
Figure 5 shows how to apply a logarithmic stress transformation to the CD model to mimic IPL. You believe the data would follow the lognormal distribution based on your engineering judgment since the failures were due to fatigue and you confirm this decision via statistical analysis in the Distribution Wizard.
Figure 5. ALTA Stress Transformation
Figure 6 shows the results of your analysis.
Figure 6. ALTA results
Figure 7 shows that one of the best visual ways for highlighting the probabilistic nature of fatigue is the Life vs. Stress plot. You interpret that plot as how the increasing stress amplitude decreases the number of cycles to failure. The green line shows the cycles to 50% probability of failure, whereas the red line shows the number of cycles to 99% probability of failure.
Figure 7. ALTA Life vs. Stress plot
Figure 8 shows that, based on these analysis results, the reliability of the coil spring with 90% 1-sided lower confidence bound at the end of 1 life (1M cycles) is 98.6%. This meets the reliability requirement for the coil springs.
Figure 8. ALTA QCP
In this article, we presented how to analyze fatigue failure/suspension data when the stress amplitude is not constant throughout the testing by using the cumulative damage model in ALTA. Classical fatigue analysis by the Palmgren-Miner Rule ignores the variation of each unit’s fatigue strength and fails to recognize the probabilistic nature of fatigue. To overcome this shortcoming of the Palmgren-Miner Rule, we use probabilistic models in data analysis for accelerated life testing.
1. "Introduction to Fatigue Analysis Theory Reference", nCode, 2017. [Online] Available: http://www.ncode.com/en/solutions/fatigue-durability.