# Derivation of calculation formula for compound mode crack propagation rate of spur gear

The fatigue crack growth rate of spur gear is the basis for predicting the crack growth life. Paris formula is the most classical theory for calculating the fatigue crack growth rate of spur gear. The formula gives the mathematical expression of Paris formula:

Where C and m are material constants. It can be seen from the formula that the key to calculate the fatigue crack growth rate of spur gears is to determine the amplitude of stress intensity factor Δ K. For single mode cracks, Δ K is calculated as the difference between the maximum and minimum stress intensity factors, i.e.:

For the three-dimensional composite crack, Ki, Kii and kiII exist at the same time. When using the K criterion, there must be a theory including Ki, Kii and kiII at the same time. Ignoring any one of the crack forms will have an important impact on the results. Richard proposed that in the three-dimensional composite crack, the equivalent stress intensity factor can be used to describe the strength of the stress field at the leading edge of the fatigue crack of spur gear. For the case where three crack forms exist at the same time, Richard gave the calculation formula for calculating the equivalent stress intensity factor:

It can be seen from the formula that when calculating the fatigue crack growth rate of spur gear, the determination of the amplitude of stress intensity factor is the key. For the problem of three-dimensional composite crack, the amplitude of stress intensity factor in the formula can be replaced by the amplitude of equivalent stress intensity factor, and the amplitude of equivalent stress intensity factor is determined by the formula.

At present, there is no accurate theory to solve the problem of three-dimensional composite crack. For the problems in practical engineering, the theory introduced above can meet the needs. The formula for calculating the fatigue crack growth rate of composite spur gear is obtained:

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