## How to Calculate Fatigue Life in Product Development

This article in the series on simulation-driven product development offers deeper insights into the fatigue life, frequency response analysis, and other vital parameters that affect the life of your product.

### Fatigue Life Estimation - Strain-Based Approach: Constant Amplitude

There are copious factors that cause your material to drift into the plastic zone. For instance, when you turn on the ignition of your vehicle, some materials might undergo a higher strain-rate compared to other materials, forcing them to shift to the plastic zone, which we refer to as truck cranking.

The number of times you turn on or park your vehicle would not be as high as the vibrational number of cycles. Even when your material enters the plastic zone, it would only be there for a few thousand instances as that would sum up the total number of times you would turn on the ignition of your automobile.

However, vibration cycles would be in millions as your wagon would tread on the road and encounter bumps or other environmental factors. Thus, in the plastic zone, the material would probably last for about 5000 cycles and not for eternity. Therefore, strain-based approaches deal in high-strain zones.

Here, we emphasize the plastic strain rather than considering the stress-fatigue calculations, as represented in the formula below.

• The Δεp/2 is the plastic strain amplitude,
• The ε'f is the fatigue ductility coefficient,
• c is the fatigue ductility exponent (-0.5 to -0.7 for most metals)

Rather than the stress amplitude σa, we characterize the loading by the plastic strain amplitude Δεp/2.

Calculate the Δεp, and use the Coffin-Manson curves (available in fatigue handbooks) to determine the number of cycles you can sustain for a material.

The lengthiest topmost curve in the graph is composed of the low-cycle as-well-as high-cycle fatigue. If you wish to generate a generic fatigue-life calculation, we write it in the form -

### Static/Fatigue Life Estimation: In the Presence of Crack Fracture Mechanics - Residual Life Estimation

This section deals with crack and fracture mechanics.

If you have a crack inside the body of the product, then the life of the component gets regulated via ΔK, which is Δ(Stress Intensity Factor).

ΔK = Aσ√πa

• A is a geometrical constant
• σ is the stress level at the tip
• a is the crack length

The stress intensity factor controls the fatigue life of the material, and not the stress and strain absolute.

The formula for the rate of crack growth is -

We term this as Paris' Law.

As observed in the curve diagram above, the rate of increase is spontaneous initially, with the rate stabilizing in the middle where Paris' Law is applicable.

### Fatigue Life Estimation Packages

Here are the tools available to calculate the fatigue life of various materials -

• FESafe
• Ncode
• AFGROW
• NASGROW

AFGROW and NASGROW are more widely used in fracture analysis and residual-strength calculation if a crack emerges in a product.

The above image is a snapshot of the FESafe estimation package tool. It displays an option, 'Select an algorithm to be used.' We can observe the algorithms available that include the Biaxial Strain Life, Biaxial Stress Life, and others.

### Random Vibration

Random vibrations have varying amplitudes and diversifying patterns. We express them in terms of false spectral density.

The power spectral density displayed for a frequency is the amount of energy provided to the components. As we drift towards higher frequencies, the energy infusion into the component gradually plummets.

The accelerometers record the vibrations in your vehicle and use Fourier transform and frequency filtering to provide you with a PSD plot. The PSD plot is the input for random vibrations.

### Random Vibration Life Calculation

Considering the random vibration is a statistical parameter, you input the power spectral density and obtain an accumulative of this power spectral density on your specimen.

For instance, if you have to walk over the frequency range, the stress graph would vary profusely, because it is a white frequency range. Since you cannot consider the individual value of stress, you extract the RMS value of the stress.

RMS value helps you devise the equivalent stress value. You can compare the RMS value with the endurance limit of your material. We take the RMS value and plug it into the Basquin's equation to ultimately deduce the number of cycles, which is a conservative approach.

Since the profile of random vibrations is uncertain, a normal distribution is imminent. This uncertainty implies that stress may not be very high at all times.

We also follow the Steinberg 3-band method, which says that we must consider three times the value of RMS to cover the entire band. This way, you would imitate your random vibration cycle and consider all uncertainties into the input.

We get three different amplitudes by multiplying one, two, and three to the RMS values, which also reduces the 'percentage of occurrence' as observed in the table mentioned above.

These three different amplitudes produce different cycles; however, the distribution of these cycles would vary because of their contrasting 'percentage of occurrence'.

Thus, with a different number of cycles and different cycles to failure at each stress level, we receive varying damage values, where the contrasting figures of damage yield a net sum of less than one.

### Frequency Response Analysis (Simple Harmonic)

The table below depicts the fatigue estimation from the simple harmonic or Harmonic Analysis. We feed in the input for harmonic analysis in terms of octaves.

In simple harmonic, at every hertz, we obtain a definite value of stress, and the number of cycles for that frequency depends on the sweep time.

Thus, we take the resolution as small as viable, typically 1Hz. We then calculate the damage for each octave. As per the magnitude of the sweep time, the number of cycles to which you have to expose your maximum harmonic stress will vary.

### Final Words

As you expose your product to real-time situations where environmental factors play an influential role, it becomes paramount to deduce its fatigue life to squash errors and counter several deterrents.

It will help if you account for the random vibrations that do not get induced by engine ignition but are a derivative of environmental factors on the road.