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The fatigue properties of materials are generally expressed in the form of uniaxial stress cycles (S-N curve. Fatigue theory based on fracture mechanics is not considered here).

The change of stress with time is also very regular, such as sine wave, square wave or pulse.

In addition, the influence of average stress on fatigue performance is rarely considered (i.e., the influence of r=S_{min}/S_{max} ≠ – 1).

But the actual stress state is mostly multiaxial stress, with poor regularity of stress change, and r ≠ – 1.

How to correspond the actual stress (irregular stress change, multiaxial, r ≠ – 1) with the fatigue performance of materials measured in the laboratory (regular stress change, uniaxial, r=1) constitutes the basis and basis for fatigue analysis.

**1. Treatment of average stress influence**

If there are S-N curves under different r values, interpolation method is generally used to determine the S-N curve under unknown r values.

If there is only S-N curve with r=- 1, the following formula can be used to calculate the equivalent stress (that is, convert the uniaxial stress of r ≠ – 1 to the uniaxial stress when r=- 1, that is, the equivalent stress):

Where, S_{a} is the half stress amplitude, S_{e} is the desired equivalent stress, S_{m} is the average stress, and different values of S_{u} and n constitute different theories:

- Soderberg yield stress (sy) 1
- Goodman ultimate tensile stress (su) 1
- Gerber ultimate tensile stress (su) 2
- Morrow true fracture stress (sf) 1

**2. Conversion of multiaxial stress to uniaxial stress**

This transformation is actually what kind of stress (or component) is used. Only the following options are available:

Von Mises equivalent stress, maximum shear stress, maximum principal stress or a certain stress component (Sx, Syz, etc.).

Sometimes, Mises stress with sign is also used, and its size is unchanged.

The sign is the sign of the maximum principal stress. The advantage is that the influence of tension or compression can be considered (reflected in the average stress or r).

Similar to the strength theory, Von Mises equivalent stress and maximum shear stress conversion are applicable to materials with good ductility, and the maximum principal stress conversion is applicable to brittle materials.

**3. Treatment of irregular stress**

Essentially, a series of simple stress cycles (Sa, Sm) and corresponding times are extracted from the equivalent uniaxial stress time curve of irregular high and low.

There are many methods to complete this counting and statistics, including path dependent method and path independent method.

The most widely used rain flow counting method is the path correlation method. Its algorithm and principle can be seen in “Downing, S., Society, D. (1982) Simplified rain flow counting algorithms. Int J Fatigue, 4, 31 – 40”).

After rain flow treatment, the irregular stress time curve is transformed into a series of simple cycles (S_{a}, S_{m }and n_{i}, and ni is the number of cycles).

In this way, the damage accumulation theory (Miner criterion) can be used to calculate and analyze: Sum (n_{i}/N_{i}) N_{i} is the life corresponding to the stress cycle (considering S_{a}, S_{m}, see above).

This can be used to measure the safety factor after a certain number of cycles, or the corresponding life of a certain complex stress cycle.

At present, commercial fatigue analysis software is mostly based on the above process.

At the same time, it should be pointed out that fatigue analysis is an empirical analysis, and there is no mature and complete theory.

**There are different views on the conversion of multiaxial stress to uniaxial stress:**

Von Mises stress is actually a quantity of stress dimension based on the concept of specific energy of shape change.

There is no concept of positive and negative, tension and compression. It is a very inaccurate method to adopt this method now.

As for which kind of stress to adopt, it also depends on the possible trend of cracks on materials or structures to determine what kind of stress is the main factor controlling fatigue failure.

Fatigue failure cases in engineering practice show that steel with good plasticity is often damaged due to repeated dynamic loading of principal stress.

**Supplement on the treatment of average stress effect:**

“If there are S-N curves under different r values, interpolation method is generally used to determine the S-N curve under unknown r values.”

This is only a method, which is useful when there are more stresses to check, but generally, this method is cumbersome when only one stress life is checked.

One method is to give the equivalent stress half amplitude under the condition of r=- 1, and then the S-N curve can be directly applied.

In case of mean stress, S-N curve cannot be used directly. Use GOODMAN CURVE or modified goodman curve.