The linear fatigue damage theory refers to the fact that fatigue damage can be accumulated linearly under cyclic loading, and the stresses are independent of each other. The average damage caused by each cycle is 1/N, and the damage caused by n constant amplitude loads equal to its cycle ratio C = n/N. The damage D of the amplitude load is equal to the sum of its cycle ratios.

where; k: Stress level of variable amplitude load, n_i: Number of cycles of the i-th load, N_i: Fatigue life under the i-th load.

When the damage has accumulated to the critical value D_f, that is, when D = D_f, fatigue failure occurs. D_f is the critical damage sum, referred to as damage sum. When D_f = 1, it is the typical Miner’s rule in linear cumulative damage.

Miner's rule is simple in form and easy to use and is widely used in engineering practice. However, Miner's rule does not consider issues such as the interaction between stresses, the order of loads, and the assumption that stress below the fatigue limit does not cause damage. Therefore, scholars have proposed the theory of non-linear fatigue cumulative damage, which the Carten-Dolan theory is more commonly used in engineering.

In Carten-Dolan theory, the damage caused by n cycles under the same amplitude load is:

Under variable amplitude load, the damage caused by n loads is:

In the formula, n_i is the number of cycles of the i-th load; ∑i=1kni=n,m is the number of material damage cores, the greater the stress, the greater m; r is the damage development rate proportional to the stress level S; c and d are the material constants.

A fixed constraint is applied to the left spline of the transmission shaft, and a torque of 434 Nm is applied to the right spline, which is twice the usual working torque. After the static analysis, the equivalent stress distribution cloud diagram is obtained which is shown in Fig. 5. The maximum stress is located at the root of the left spline, which is consistent with the results of the static strength test. The static strength test is broken at the root of the left spline, and large stress is also distributed in the large oil hole in the middle. The maximum stress value is 569 MPa, which is less than the yield strength of the material, so no strength failure occurs.

Fig. 5 
Equivalent (von-Mises) stress

The fatigue life of transmission parts (Such as Gears) is often simulated by Miner’s rule.¹¹ By analyzing a large number of test data, it is found that the damage of input shaft is linear superposition. Therefore, performing fatigue analysis by ANSYS in this paper, the damage of the input shaft is calculated under cyclic load through Miner’s rule. First, the average stress through static calculation is obtained, and then the fatigue life of the analyzed object through interpolation calculation of the material life curve is calculated. If no material life curve is available, the average stress theory is used for approximate calculations, Goodman's equation is often used for approximate estimation.

Fatigue analysis is performed by applying about twice the working torque on the shaft, and loading is carried out with symmetrical cycle torque. The torque amplitude is 434 Nm and the average torque is zero. When the stress life analysis is performed in the software, the equivalent stress is selected for calculation.

Since the shaft is mainly subjected to shear stress only under the action of torque, the calculation using shear stress can also obtain similar results. The fatigue failure of this shaft usually occurs at the large oil hole in the middle. The fatigue life analysis is performed for the large oil hole. The stress life nephogram of the oil hole is shown in Fig. 6. The minimum life is 112,660 times, which meets the minimum life.

Fig. 6 
The stress life nephogram of the oil hole

Various factors such as the torque fluctuation of the engine and different usage conditions will affect the magnitude of the torque received by the transmission input shaft. The torque of the above analysis is proportionally scaled between 0.9 and 1.1 times, and the fatigue life of the input shaft is calculated based on the scaled torque, and the sensitivity of the fatigue life to the torque amplitude is obtained¹² as shown in Fig. 7. From the Fig. 7, the amplitude of torque has a greater impact on fatigue life. When the torque increases by 10%, the fatigue life decreases by 77.8%.

Fig. 7 
The sensitivity of life to torque amplitude

The test sample is an actual processed product, and the sample material is 42CrMo. The machining of the sample includes turning, grinding, drilling, spline splicing, and straightening. The heat treatment is subjected to intermediate frequency quenching and tempering.¹³ After the processing is completed, it can be used as a test sample for torsional fatigue only after passing the magnetic particle inspection and the roughness at the oil hole ≤ Ra0.32.

The torsional fatigue test is performed according to the standard ISO 1352: 2011 Metallic materials-Torque-controlled fatigue testing.¹⁴ The testing machine shall have the ability to load torque clockwise/counterclockwise. The accuracy of the torque sensor shall not be greater than 1%. When the sample breaks, the testing machine shall be able to stop automatically. The equipment used in this test is a torsional fatigue tester produced by MTS, as shown in Fig. 8. It is mainly composed of oil, twist cylinder, servo valve, torque sensor, angle sensor, controller and test software. The equipment is powered by hydraulic oil, and the torque control is stable, which can meet the requirements of the torque fatigue test.

Fig. 8 
MTS torsional fatigue testing machine

The test sample is a transmission input shaft that is qualified in the same batch and is processed according to the actual load spectrum of the vehicle to obtain the torque T of the input shaft under common working conditions. and is loaded at twice the torque T.

The loading waveform is a sine wave, as shown in Fig. 9, where the load ratio is -1, the amplitude is 434 Nm, and the test frequency is 4 Hz. During the test, the loading end of the sample swings cyclically within a certain angle [φ₁, φ₂]. When the angle is less than φ₁ - 1 or greater than φ₂ + 1, the sample is considered to be broken, and the equipment automatically stops, and the number of cycles is recorded.

Fig. 9 
Torque loading waveform

This test mainly obtains the fatigue life at a specific torque amplitude. The test carried out a torsional fatigue test on 6 input shafts of a certain gearbox. The test results are shown in Table 2. The test data are consistent and all the test data are valid. All the samples fractured at the large oil hole in the middle of the shaft, and the fracture direction was 45 degrees from the axis, as shown in Fig. 10. According to the fracture analysis, it can be seen that the fracture is divided into two obvious parts. As shown in Fig. 11, the fracture near the oil hole is smoother, the area far from the oil hole is rough and there are obvious tear marks. The two parts are crack growth zone and fault zone.

Fig. 10 
Fracture position of the specimen

Fig. 11 
Fracture of specimen

Fatigue failure usually goes through three stages, which are divided into crack initiation, crack propagation and fracture. For components made of high-strength materials, the fatigue life is mainly concentrated in the crack initiation stage.^15,16 A fracture will occur within a period of time. The data recorded 10 minutes before the fracture of the axial fatigue test were analyzed. As shown in Fig. 12, the abscissa is time and the ordinate is the amplitude of the torsional angle. It can be seen that the angle amplitude starts to increase at 530 minutes and at 534 minutes The shaft fractured, and the crack propagation time to complete fracture was only 4 minutes, which only accounted for 0.8% of the fatigue life.

Fig. 12 
Chart of time-angle