We conducted a comprehensive finite element simulation to investigate the quenching process of a straight spur gear. Gears are essential mechanical components for transmitting motion and power between prime movers and working machines. Among various gear types, the straight spur gear is widely used due to its simplicity and efficiency. However, during heat treatment, particularly carburizing and quenching, severe thermal and transformational stresses can induce distortion, cracks, and even fracture, thereby compromising gear accuracy and performance. In this work, we established a three-dimensional model of a straight spur gear and performed coupled thermal-stress analysis using Ansys software. Our primary objective was to examine the influence of different initial quenching water temperatures on the temperature field and stress distribution. The results provide valuable insights for optimizing heat treatment parameters to minimize deformation and improve gear quality.
The quenching process involves complex physical phenomena including phase transformations (austenite decomposition into ferrite, pearlite, bainite, or martensite) and latent heat release. Although the latent heat during solid-state transformation is smaller than during melting or solidification, it cannot be neglected as it introduces nonlinearities and oscillations in the simulation. According to Fourier’s law and the energy conservation principle, the transient heat conduction equation with an internal heat source (latent heat) can be expressed in Cartesian coordinates as:
$$
\lambda \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + q_v = \rho c_p \frac{\partial T}{\partial t}
$$
where \(\lambda\) is the thermal conductivity, \(\rho\) is the density, \(c_p\) is the specific heat at constant pressure, \(T\) is the temperature, \(t\) is time, and \(q_v\) is the volumetric heat generation rate due to phase transformation. The transient heat transfer problem can be further discretized using the finite element method, leading to the following system of equations:
$$
[C]\{\dot{T}\} + [K]\{T\} = \{Q\}
$$
Here, \([C]\) is the heat capacity matrix, \([K]\) is the conductivity matrix (including convection and radiation effects), \(\{\dot{T}\}\) is the time derivative of the nodal temperature vector, and \(\{Q\}\) is the nodal heat flux vector. The straight spur gear geometry was created in SolidWorks and imported into Ansys. The material selected was 45 steel, a common medium-carbon steel used in gear manufacturing. The mechanical and thermophysical properties of 45 steel are listed in Tables 1 and 2, respectively.
| Temperature (°C) | Elastic Modulus (GPa) | Yield Strength (MPa) | Shear Modulus (GPa) | Poisson’s Ratio |
|---|---|---|---|---|
| 20 | 193 | 200 | 19.3 | 0.28 |
| 500 | 150 | 933 | 15.0 | 0.28 |
| 1000 | 70 | 435 | 7.0 | 0.28 |
| 1500 | 10 | 70 | 1.0 | 0.28 |
| 2000 | 1 | 7 | 0.1 | 0.28 |
| Temperature (°C) | Thermal Conductivity (W·m⁻¹·K⁻¹) | Specific Heat (J·kg⁻¹·°C⁻¹) | Density (kg·m⁻³) at 20°C |
|---|---|---|---|
| 0 | 14.7 | 460 | 7850 |
| 100 | 16.6 | 460 | 7850 |
| 200 | 18.0 | 460 | 7850 |
| 400 | 20.8 | 460 | 7850 |
| 600 | 23.5 | 460 | 7850 |
| 800 | 26.3 | 460 | 7850 |
| 1000 | 28.2 | 460 | 7850 |
The quenching process was simulated by heating the straight spur gear to 850 °C in a resistance furnace and then immersing it in a water bath at controlled initial temperatures (20, 25, 30, and 35 °C). The water flow velocity was set to an intermediate value of approximately 1.0 m/s. The convective heat transfer coefficient was determined based on empirical correlations for water quenching. The simulation time spanned from 0 to 1800 seconds, with particular focus on the first 10 seconds where the most rapid cooling and phase transformation occur. Three time intervals (0–10 s, 0–1000 s, and 0–1800 s) were analyzed to capture the entire cooling history.
To visualize the geometrical configuration of the straight spur gear, we present the following illustration of a typical spur gear used in our simulation:
The temperature field results revealed distinct characteristics. At the beginning of quenching, the gear core retained the highest temperature because of the low thermal diffusivity of steel, while the tooth tip cooled most rapidly. The tooth root exhibited intermediate temperatures. As the initial water temperature increased, the overall cooling rate decreased, leading to higher temperatures at all locations. The temperature differences between the tooth tip, tooth root, and core became more pronounced with higher water temperatures, thereby increasing thermal stresses. Detailed temperature values at key locations for different initial water temperatures and quenching times are summarized in Tables 3–6.
| Location | 10 s | 1000 s | 1800 s |
|---|---|---|---|
| Tooth tip | 34.5 | 399.8 | 402.7 |
| Tooth root | 60.2 | 425.4 | 428.2 |
| Tooth core | 849.8 | 849.8 | 849.8 |
| Location | 10 s | 1000 s | 1800 s |
|---|---|---|---|
| Tooth tip | 39.8 | 402.7 | 410.5 |
| Tooth root | 65.3 | 428.2 | 435.6 |
| Tooth core | 849.8 | 849.8 | 849.8 |
| Location | 10 s | 1000 s | 1800 s |
|---|---|---|---|
| Tooth tip | 43.3 | 405.5 | 416.1 |
| Tooth root | 68.7 | 432.8 | 435.6 |
| Tooth core | 849.8 | 849.8 | 849.8 |
| Location | 10 s | 1000 s | 1800 s |
|---|---|---|---|
| Tooth tip | 54.0 | 410.5 | 419.7 |
| Tooth root | 79.0 | 435.6 | 443.3 |
| Tooth core | 849.8 | 849.8 | 849.8 |
The tables clearly show that the core temperature remained almost unchanged during the first 1800 seconds, while the tooth surface cooled significantly. The maximum temperature difference between the tooth tip and core was as high as 815 °C at 10 s for the 20 °C case. As the initial water temperature increased, the tooth tip temperature at 10 s rose from 34.5 °C to 54.0 °C, indicating slower surface cooling. This implies that higher water temperatures reduce the thermal shock but may also lead to incomplete martensitic transformation and lower hardness. The temperature gradients within the straight spur gear were most severe in the first few seconds, during which the majority of thermal stress accumulation occurred.
After the thermal analysis, we imported the nodal temperatures as body loads into a structural analysis to compute the stress field. The stress distribution in the straight spur gear after 1800 s of quenching is characterized by a strong concentration at the tooth root, where the geometry changes abruptly. The maximum von Mises stress reached 114 MPa, while the tooth tip exhibited only 0.526 MPa. The stress gradually decreased from the root toward the tip along the tooth flank. A narrow band of tensile stress formed on the tooth surface near the root, which is a critical region for crack initiation. Table 7 summarizes the stress values at various locations for the 20 °C initial water temperature case.
| Location | Stress (MPa) |
|---|---|
| Tooth tip | 0.5 |
| Tooth root | 114.0 |
| Tooth flank (middle) | 63.5 |
| Tooth core | 13.1 |
The stress concentration at the root is explained by the combined effect of thermal contraction and phase transformation expansion. The tooth root, being a sharp corner, acts as a stress raiser. Moreover, the surface layer undergoes martensitic transformation which involves volume expansion, while the core remains austenitic and contracts upon cooling. This mismatch generates large tensile stresses at the root. The magnitude of this stress increases with higher initial water temperatures because the temperature gradient becomes steeper. For the 35 °C case, the maximum stress at the root was calculated to be 121 MPa, further increasing the risk of cracking.
We also observed that the stress distribution was asymmetric along the tooth width, likely due to the non-uniform flow of the quenching medium. However, the overall pattern remained consistent: the tooth root is the most vulnerable area. The simulation results strongly suggest that post-quenching machining allowance should be increased in the root region to compensate for potential distortion and to avoid service failure.
To gain deeper insight into the influence of cooling rate on the straight spur gear performance, we compared the cooling curves at the tooth surface and core for different water temperatures. The cooling rate at the surface in the first 10 seconds decreased from 82 °C/s at 20 °C to 76 °C/s at 35 °C. A slower cooling rate leads to a coarser martensitic structure and reduced hardness, but also lowers the risk of quench cracking. Therefore, selecting the optimal initial water temperature is a trade-off between achieving full hardness and minimizing distortion. For 45 steel, a water temperature of 20–25 °C is commonly recommended, and our simulation supports this practice.
Furthermore, we examined the time evolution of the cooling layer depth. At 10 s, the cooling zone penetrated only about 0.5 mm below the tooth surface. By 1000 s, the cooling front had advanced to approximately 2 mm, and by 1800 s, the entire tooth cross-section had cooled to below 450 °C, indicating completion of the majority of phase transformations. The deepest cooling layer was observed at the tooth tip, whereas the core remained hot for a much longer period. This gradient is responsible for the progressive accumulation of thermal stress throughout the quenching process.
The numerical simulation of the straight spur gear quenching process provides a powerful tool for predicting temperature and stress distributions without costly experimental trials. Our results highlight the critical role of initial water temperature in controlling thermal gradients and stress concentrations. By adjusting the quenching parameters, manufacturers can reduce the likelihood of distortion and cracking in straight spur gears, thereby improving product quality and service life.
In conclusion, we performed a finite element analysis of the temperature and stress fields during the quenching of a straight spur gear made of 45 steel. The following key findings were obtained:
- During the quenching of a straight spur gear, the tooth core retains the highest temperature, while the tooth tip cools the fastest. A significant temperature difference exists between the tip and the core, and this difference increases with higher initial water temperatures.
- As quenching proceeds, the cooling layer gradually penetrates into the gear tooth. The thermal stress accumulates mainly in the first few seconds and continues to increase until the entire tooth reaches thermal equilibrium.
- The stress distribution in the straight spur gear is highly non-uniform. Stress concentrates at the tooth root, with values up to 114 MPa for a 20 °C water quench. The tooth flank exhibits intermediate stresses, and the tooth tip experiences negligible stress.
- Higher initial water temperatures lead to slightly increased stress magnitudes but also reduce the cooling rate, which may be beneficial for complex-shaped thin sections to avoid cracking. For 45 steel, a water temperature between 20 and 30 °C is recommended to balance hardness and distortion.
These results provide a theoretical basis for optimizing heat treatment processes of straight spur gears in practical production. The methodology can be extended to other gear geometries and materials, contributing to the design of more reliable transmission components.