1. Energy accumulation curve
According to the first law of thermodynamics, the increment of internal energy per unit volume of an object δ U should be equal to the heat energy provided by the outside world δ The work done by Q and the outside world on it δ Sum of W:
In the case of heat exchange between the system and the outside world, when the testing machine loads the helical gears specimen, a portion of the mechanical energy is dissipated in the form of heat, while another portion of the energy is stored in the helical gears specimen in the form of internal energy.
For high cycle fatigue, the main deformation of helical gears specimens is macroscopic elastic deformation because the applied cyclic stress level is lower than the macroscopic yield stress. At the micro scale, due to the non-uniformity of material grains, the disharmony of grain boundaries, and potential defects such as pores and inclusions, there is a significant stress concentration effect in the material, which will cause irreversible microplastic deformation locally and change the microstructure inside the crystal. The energy trend that causes damage is basically the same as the internal energy trend acting on helical gears. Therefore, the energy that causes damage can be represented by the internal energy acting on the helical gears. For isotropic materials, the work required for the deformation of infinitesimal microelements dV is:
Perform fatigue tests on multiple specimens at different constant stress levels to predict cumulative fracture energy and corresponding fatigue life. The large tensile stress near the tooth surface leads to the generation of open type cracks at the tooth root, so the energy Ed consumed by a single loading in the crack initiation area at the tooth root during the test can be expressed as:
In the formula: S is the area of the dashed box area in Figure 1; σ M is the average amplitude of regional stress; Δ Li is the displacement of this region during the i-th cycle.
The energy consumed in the crack initiation area of the helical gears specimen from the beginning of the test until the nth loading can be expressed as:
Use strain gauges in the experiment to collect data, according to the formula. The energy accumulation curve of the crack initiation area at the root of the helical gears specimen under 5 different loads was calculated, as shown in Figure 2:
The energy accumulation curve can be expressed by the formula:
In the formula: n is the number of cycles, 10 ^ 4 cyc
C1, C2, and C3 were fitted based on experimental data, and C1, C2, and C3 can be represented as follows:
2. Energy growth rate curve
During the experiment, the energy consumed in the crack initiation area of the helical gears specimen is accelerating. During this acceleration process, the cumulative energy growth rate vE shows a stable growth trend, and the cumulative energy growth rate vE can be expressed as:
The cumulative energy acceleration rate aE is a constant value, which can be expressed as:
The cumulative energy growth rate vE of tooth roots under 5 different stress levels is shown in Figure 3.
From Figure 3, it can be seen that the cumulative energy growth rate vE of the tooth root is stable with the cumulative change rate of the number of cycles, and the acceleration rate aE of the cumulative energy growth of the tooth root can be regarded as a constant value. The acceleration rate aE of cumulative energy growth at different tooth root stresses is shown in Figure 4.
From Figure 4, it can be seen that the acceleration rate of cumulative energy growth at the root of the tooth, aE, approaches an exponential function with the magnitude of root stress. Based on experimental data fitting, aE can be calculated using the following equation:
In the formula, A1, B1, and t1 are material parameters that can be determined through regression analysis of experimental data.
3. Prediction model for bending fatigue life of helical gears based on energy accumulation curve
According to the cumulative energy growth rate curve and cumulative energy growth acceleration rate curve, it can be seen that during the test process, the cumulative acceleration rate of energy consumed in the crack initiation area of the helical gears specimen at the same tooth root stress level is a constant value. Therefore, the cumulative energy Ei after ni cycles can be shown as:
Once the total accumulated energy reaches the threshold, the continuous accumulation of microplasticity will lead to the initiation and propagation of macroscopic fatigue cracks, and fatigue fracture may occur quickly. When the number of cycles reaches the fatigue life of the helical gears specimen, fatigue fracture occurs, and the cumulative fracture energy E * can be expressed as:
Under different levels of tooth root stress, the cumulative fracture energy curve is shown in. It can be seen that as the tooth root stress increases, the cumulative fracture energy of the material will decrease.
The cumulative fracture energy curve can be expressed as:
In the formula, A0 and B0 are material parameters, and t0 is the fracture energy attenuation coefficient, which can be determined through regression analysis of experimental data.
Based on Palmgren Miner’s linear cumulative damage theory and the assumption of cumulative fracture energy for fatigue failure, a nonlinear energy cumulative damage model is established. Component at tooth root stress level σ The cumulative damage after ni cycles under the action of m is:
When it is considered that the cumulative damage D reaches the critical value of 1, that is, when the cumulative energy E reaches the cumulative fatigue fracture energy E *, irreversible fatigue failure occurs in the helical gears specimen. At this time, the number of cycles n is the fatigue life N of the helical gears specimen. Therefore, the fatigue life N of the helical gears specimen under single stage load can be calculated by the following equation:
