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# Fatigue

# 10. Fatigue

Sierra/SD supports two forms of high cycle fatigue analysis. We will use both in this example.

1. Modal Random Vibration, which we will refer to as the “Frequency Domain” solution.

2. Modal or Direct Transient, which we will refer to as the “Time Domain” solution.

Frequency domain fatigue requires three solution cases in the input deck, and the `Fatigue` keyword in the `OUTPUTS` section:

    SOLUTION
      case eig
        eigen
        nmodes 36
        shift -1e6
      case rand
        modalRanVib
      case fat
        fatigue
    END

    OUTPUTS
      fatigue
    END

Time domain fatigue only requires a transient solution and the `Fatigue` keyword in either `OUTPUTS` or `HISTORY`:

    SOLUTION
      case trans
        transient
        nsteps 3.5e5
        time_step 1.25e-4
    END

    OUTPUTS
      fatigue
    END

Time domain and frequency domain fatigue estimates are not expected to match for several reasons:

- Time domain estimates the total accumulated damage, while frequency domain estimates the damage per second.

- Time domain can represent endurance limits and mean stresses, while frequency domain cannot.

- Frequency domain estimates the expected damage due to a random process, while time domain estimates the observed damage. Generating long enough time series for a statistically significant estimate can be costly.

From here on out, we will look at a specific example in detail.

## 10.1. Example Fatigue Model

### 10.1.1. Geometry

Figure 10.1 Generic Circuit Board geometry.

For this example we will be using a mock printed circuit board model (Figure 10.1) with all dimensions arbitrarily chosen for visual appeal. We will be driving the model with a random force on the underside of the structure while constraining all other translations and rotations to be zero at the drive point. Components are attached to each other using all-to-all contact. We will be focusing on the green electrical pins shown in Figure 10.2.

Figure 10.2 Generic Circuit Board components.

### 10.1.2. Materials

The material properties of the electrical pins are given in Sierra/SD syntax as:

    MATERIAL al_with_fatigue
        E = 1e7
        NU= 0.3
        Density = 0.1
        Fatigue_A1 = 20.68
        Fatigue_A2 = -9.84
        Fatigue_A3 =  0.63
        Fatigue_A4 =  0.0
        Fatigue_Stress_Scale = 0.001
    END

The elastic properties are a rough approximation aluminum, while the fatigue properties are specific to an un-notched 6061-T6 aluminum alloy. The 5 fatigue parameters are:

1. Fatigue_A1, complicated units, strictly positive

2. Fatigue_A2, dimensionless, strictly negative

3. Fatigue_A3, dimensionless, defines the damage contribution from mean stress, strictly positive, \(3\) is large, \(100\) is not physical

4. Fatigue_A4, units of stress, defines an endurance limit below which no damage occurs, strictly positive

5. Fatigue_Stress_Scale, optional, conversion rate between model stress units and damage function stress units, e.g. convert \(psi\) to \(ksi\)

It is not necessary to define a Fatigue_Stress_Scale, but the option exists to prevent accidental translation errors. The conversion rate of Fatigue_A1 is given by:

\[ A1_{new} = A1_{old} + A2*\log_{10}(1/C), \quad A4_{new} = A4_{old}*C, \]

where \(C\) is the conversion rate from old units to new units. Note that Sierra/SD does not attempt these conversions directly. Instead, model stresses are converted to material units before being applied to the damage function.

All together, these parameters define the number of cycles to failure \(N\) given a stress cycle with peak \(S_{max}\) and valley \(S_{min}\):

\[ \log_{10}(N) = A_1 + A_2\log_{10}(S_{max}(1-R)^{A_3}-A_4), \]

where \(R=S_{min}/S_{max}\).

In the frequency domain, we are only able to evaluate damage functions which can be represented as:

\[ N*S_{max}^m=A, \]

where \(m\) and \(A\) are material constants derived from \(A1\) to \(A4\). To reduce 4 material constants down to 2, we set \(A4=0\), and assume \(R=-1\) when doing frequency domain analyses. This limits the types of problems which can be represented accurately in the frequency domain. There will be more discussion of trade-offs later.

Since the geometry is arbitrary anyway, we don’t pay much attention to the other components. The base structure and electrical components are modeled as aluminum. The circuit board material is slightly less dense, and significantly stiffer than the aluminum, but still arbitrary.

### 10.1.3. Loads

The loading for this model is a single-point random force between 10 Hz and 2000 Hz with the autocorrelation function shown in Figure 10.3, evaluated at 0.025 Hz intervals between 10 Hz and 4000 Hz.

By sampling this random function at intervals of 1.25e-4 seconds for 40 seconds (3.2e5 time steps), we are able to generate a very close approximation in the time domain. Figure 10.4 shows a small snapshot of the time domain load, and resulting Auto Spectral Density (ASD)

Figure 10.5 shows a histogram of the force levels seen in the time domain. Note that \(4\sigma\) peaks exist in the data, and some values approach \(5\sigma\).

Figure 10.3 Frequency Domain Loading ASD.

Figure 10.4 Time Domain Load Snapshot (left), and ASD (right).

Figure 10.5 Histogram of time domain loads with vertical bars at 1-sigma intervals.

## 10.2. Results

### 10.2.1. Frequency Domain

Damage estimates in the frequency domain come in two flavors: “Narrow Band” and “Wirsching”. Both are a damage rate, representing the damage per second seen by the element. “Narrow Band” damage is intended for solutions where the stress response is occurs at a narrow band of frequencies, while “Wirsching” damage includes a correction factor for wider frequency bands. Unfortunately, Sierra/SD does not support spectral density outputs for von Mises stress, and so we have no way of knowing which we should use in this case. Narrow band damage rates are always larger than Wirsching damage rates.

Figure 10.6 Frequency Domain Damage Rate Estimates.

### 10.2.2. Time Domain

Sierra/SD supports one fatigue damage estimate in the time domain: “Damage”. This is an accumulated damage as a result of the transient environment, not a damage rate. In our case, the loading duration was 40 seconds, so the largest average damage rate is \(3.19e^{-6}\). The average damage rate has been manually calculated in Figure 10.7 for comparison to frequency domain results.

Figure 10.7 Time Domain Damage Estimate.

### 10.2.3. Comparison

The most obvious difference in these solutions is the cost. The modal transient solution took just over 3 hours to complete, while a modal random vibration solution took only 1 minute with fatigue outputs. Note: Requesting full acceleration and stress output on the pins also requires 3 hours, even in the frequency domain.

The solution quality suffers in the modal random vibration solution. In this example, we chose a material with no endurance limit so that we could make the closest comparison possible, but the frequency domain cannot account for mean stresses either. Together, these details significantly increase the predicted damage in the frequency domain. The peak time domain damage estimate was 4.5x lower than the Wirsching damage rate, and 7.4x lower than Narrow Band. This means the difference between surviving 3.6 days at these levels, and surviving 19 hours (12 for Narrow Band).

Note: The Wirsching damage estimate was not always conservative. One element in particular saw roughly 2x more damage in the time domain than the Wirsching estimate, and was in the 70th percentile of damaged elements. For that element, the Narrow Band estimate was a decent approximation of the time domain (only 13% error).