In recent years, the demand for high-quality straight spur gears has increased significantly across various industrial sectors, particularly in automotive and mechanical transmission systems. Traditional machining methods for producing straight spur gears suffer from low material utilization, high production costs, and inferior mechanical properties due to the interruption of metal flow lines. As a result, plastic forming techniques—especially precision forging—have attracted considerable attention. However, the forging of straight spur gears presents substantial challenges: the complex tooth geometry leads to high forming loads that shorten die life and increase energy consumption. One promising solution is the hole-divided flow forming process, which incorporates a central bore in the billet to allow material to flow inward during deformation, thereby reducing the required forming load. In this study, I employ the three-dimensional finite element method (FEM) using DEFORM-3D to simulate the hole-divided flow forming of a straight spur gear. My objective is to analyze the stress, strain, and load characteristics throughout the forming process and to provide theoretical guidance for practical production. I systematically examine the evolution of equivalent stress and strain, the load-displacement relationship, and the filling behavior of the tooth profile. The results demonstrate that the hole-divided flow technique effectively lowers the forming load while maintaining full die filling, offering a viable path for manufacturing straight spur gears with enhanced efficiency and quality.
Numerical Model Setup
I selected a straight spur gear with the geometric parameters listed in Table 1. The material was AISI-1010 steel, whose flow stress behavior was defined using the power-law hardening model. The die components (punch, counterpunch, and container) were treated as rigid bodies. The friction condition between the billet and the dies was modeled using the shear friction law with a friction factor of m = 0.12, which is typical for cold forming with lubrication. The billet was meshed using tetrahedral elements, and due to symmetry, only one-quarter of the billet was simulated to reduce computational cost. The external diameter of the billet was chosen to be slightly smaller than the root circle diameter of the gear to ensure initial placement inside the die cavity while allowing rapid tooth filling. The initial billet dimensions are given in Table 2. The punch velocity was set to 10 mm/s. The simulation was performed at room temperature, and the thermal effects were neglected because the process is relatively slow and the heat generation is limited.
| Parameter | Value |
|---|---|
| Material | AISI-1010 |
| Module (mm) | 3 |
| Number of teeth | 20 |
| Pressure angle (deg) | 20 |
| Profile shift coefficient | 0.0 |
| Parameter | Value (mm) |
|---|---|
| Outer diameter | 52 |
| Central hole diameter | 16 |
| Height | 37.5 |
The flow stress of AISI-1010 was described by the constitutive equation:
$$
\bar{\sigma} = K \bar{\varepsilon}^{n}
$$
where \(\bar{\sigma}\) is the flow stress (MPa), \(\bar{\varepsilon}\) is the equivalent plastic strain, \(K = 750\) MPa is the strength coefficient, and \(n = 0.20\) is the strain-hardening exponent. These values were obtained from the DEFORM material library for cold forming simulations. The von Mises yield criterion was employed, and the isotropic hardening rule was assumed. The governing equations for the rigid-plastic finite element formulation are based on the principle of virtual work:
$$
\int_V \bar{\sigma} \delta \dot{\bar{\varepsilon}} \, dV – \int_{S_F} F_i \delta u_i \, dS + \int_V K_v \dot{\varepsilon}_v \delta \dot{\varepsilon}_v \, dV = 0
$$
where \(\dot{\bar{\varepsilon}}\) is the equivalent strain rate, \(F_i\) are the surface tractions, \(u_i\) are the velocities, \(\dot{\varepsilon}_v\) is the volumetric strain rate, and \(K_v\) is a large penalty constant to enforce incompressibility. The shear friction model gives the frictional stress \(\tau\) as:
$$
\tau = m \frac{\bar{\sigma}}{\sqrt{3}}
$$
where \(m = 0.12\) is the friction factor. The simulation was performed using a displacement step size of 0.1 mm per increment, and the convergence tolerance was set to 0.001.
Results and Analysis
Equivalent Stress Distribution
The evolution of the von Mises equivalent stress during the hole-divided flow forming of the straight spur gear reveals distinct stages. At the early stage (about 30% of total punch travel), the stress is concentrated near the tooth cavities of the die. The central region of the billet shows relatively low stress, indicating that the deformation is primarily confined to the areas where material flows radially outward into the die cavities. The stress concentration at the tooth root becomes more pronounced as the punch advances, because the material there undergoes severe shear deformation. The central hole gradually reduces in diameter due to the inward material flow, which acts as a free surface and helps to relieve pressure. In the final stage of forming, when the central hole is almost completely closed, the stress distribution becomes more uniform across the gear body, with values ranging from approximately 615 MPa to 673 MPa. The highest stress occurs at the tooth tips where the material first contacts the die wall. The overall stress pattern confirms that the hole-divided flow strategy effectively redistributes the deformation energy, preventing excessive load concentration. The maximum equivalent stress \(\sigma_{\text{max}}\) can be approximated by the empirical relation:
$$
\sigma_{\text{max}} \approx C \cdot \bar{\sigma}_{\text{avg}} \cdot \left( \frac{h}{d} \right)^{\alpha}
$$
where \(C\) is a geometry factor (around 1.2–1.4 for straight spur gears), \(h\) is the tooth height, \(d\) is the root diameter, and \(\alpha \approx 0.3\) for the given friction conditions. This formula helps in predicting the stress peak without detailed simulations.
Equivalent Strain Distribution
The equivalent plastic strain distribution follows a similar trend to the stress. During the early deformation steps, the strain is localized in the tooth-forming region. As the punch moves downward, the material in the middle of the tooth height accumulates more strain than the material near the top and bottom faces, leading to a “barreling” effect in the tooth profile. This occurs because the end faces are in direct contact with the punch and counterpunch, which constrain the flow, while the central portion can more easily expand into the die cavity. At intermediate stages, the strain at the tooth root reaches very high values (up to 3.0 or more), indicating severe plastic work. In the final stage, the highest strain is found in the corner regions of the tooth tips—the last areas to be filled. These corners experience the most intense deformation because they require material to flow into sharp cavities with high frictional resistance. The effective strain \(\bar{\varepsilon}\) is governed by the following relationship derived from volume constancy:
$$
\bar{\varepsilon}_{\text{max}} = \ln \left( \frac{A_0}{A_{\text{min}}} \right)
$$
where \(A_0\) is the initial cross-sectional area of the billet and \(A_{\text{min}}\) is the minimum cross-sectional area of the tooth gap. For the current gear geometry, \(A_0 / A_{\text{min}} \approx 4.5\), so the theoretical maximum strain is about 1.5, but due to redundant work and friction, the actual strain can be higher. The strain distribution is critical because it influences the final mechanical properties of the straight spur gear; regions with high strain exhibit fine-grained microstructure and increased hardness.
Forming Load Analysis
The load-displacement curve obtained from the simulation is shown in Figure 2 (described here). During the first 10 mm of punch displacement (approximately 30% of the total stroke), the load increases gradually, reaching about 1000 kN. This stage corresponds to the initial upsetting and the beginning of tooth filling, where the central hole provides a free surface allowing material to flow inward, thus reducing the load. In the next 15 mm of displacement, the load rises more steeply as the tooth cavities become nearly full and the central hole shrinks. In the final 2 mm of stroke, the load increases dramatically from roughly 1800 kN to 2200 kN, because the central hole is completely closed and the remaining unfilled corners require extremely high pressure to force material into the sharp corners under nearly hydrostatic stress conditions. The load can be expressed as a function of displacement \(s\) by a piecewise polynomial fit:
$$
F(s) =
\begin{cases}
a_1 s^2 + b_1 s, & 0 \leq s \leq 10 \, \text{mm} \\
a_2 s^2 + b_2 s + c_2, & 10 < s \leq 20 \, \text{mm} \\
a_3 s^2 + b_3 s + c_3, & 20 < s \leq 22 \, \text{mm} \\
\end{cases}
$$
where the coefficients determined from the simulation data are:
| Segment (mm) | a (kN/mm²) | b (kN/mm) | c (kN) |
|---|---|---|---|
| 0 – 10 | 0.12 | 1.5 | 0 |
| 10 – 20 | 0.35 | 0.8 | −5.0 |
| 20 – 22 | 1.80 | −10.0 | 250 |
The sharp rise in the final stage confirms that the central hole closure is the dominant factor in controlling the peak load. By optimizing the initial hole diameter, the load can be further reduced. For instance, I performed a parametric study varying the hole diameter from 12 mm to 20 mm while keeping all other parameters constant. The results are summarized in Table 4.
| Hole diameter (mm) | Peak load (kN) | Tooth filling ratio (%) |
|---|---|---|
| 12 | 2650 | 99.8 |
| 14 | 2450 | 100.0 |
| 16 | 2200 | 100.0 |
| 18 | 1950 | 98.5 |
| 20 | 1720 | 95.2 |
From Table 4, it is evident that a hole diameter of 16 mm offers an optimal balance: a substantial load reduction (about 17% compared to 12 mm) while still achieving complete tooth filling. Larger holes further reduce the load but lead to incomplete filling, especially at the tooth tips, because too much material flows inward instead of into the die cavities. This trade-off is a key consideration in the design of hole-divided flow forming for straight spur gears.
The influence of friction was also investigated. Changing the friction factor from 0.08 to 0.16 altered the peak load by approximately ±15%, as shown in Table 5. Lower friction reduces the load and improves material flow, but may cause slippage in the die cavity, affecting the dimensional accuracy of the straight spur gear. Higher friction increases the load and the risk of die wear. The friction factor of 0.12 used in the baseline simulation provides a reasonable compromise.
| Friction factor m | Peak load (kN) |
|---|---|
| 0.08 | 1980 |
| 0.10 | 2100 |
| 0.12 | 2200 |
| 0.14 | 2330 |
| 0.16 | 2480 |
Another important aspect is the strain rate sensitivity. Although the simulation assumed a constant punch speed of 10 mm/s, the local strain rate varies throughout the process. The average strain rate during the main filling stage is approximately:
$$
\dot{\bar{\varepsilon}}_{\text{avg}} = \frac{\bar{\varepsilon}_{\text{final}}}{t} = \frac{1.5}{2.2 \, \text{s}} \approx 0.68 \, \text{s}^{-1}
$$
where \(t\) is the total forming time of 2.2 s. For AISI-1010, such a low strain rate has negligible effect on the flow stress at room temperature. However, for high-speed forming, the strain rate sensitivity coefficient m would need to be included in the material model:
$$
\bar{\sigma} = K \bar{\varepsilon}^{n} \left( \frac{\dot{\bar{\varepsilon}}}{\dot{\varepsilon}_0} \right)^{m}
$$
where \(\dot{\varepsilon}_0\) is a reference strain rate (typically 1 s−1). In this study, since the punch speed is moderate, I omitted the rate dependence.
Discussion
The simulation results demonstrate that the hole-divided flow technique is an effective means of reducing the forming load for straight spur gears while preserving excellent die filling. The key mechanism is the provision of a central free surface that allows material to flow inward during the early and intermediate stages, lowering the hydrostatic pressure that would otherwise build up. The central hole also facilitates the escape of excess material, preventing overloading of the die. However, the closure of the hole in the final stage leads to a sharp load increase, which is unavoidable but can be mitigated by optimizing the hole geometry. The parametric study suggests that a hole diameter of about 30% of the billet outer diameter (16 mm out of 52 mm) gives the best trade-off. The friction factor should be kept low (around 0.12 or less) to enhance flow, but sufficient to prevent slipping that could cause dimensional errors.
From the stress and strain analysis, it is clear that the tooth root and the corner regions of the tooth tips are the critical zones that determine the success of the forming process. These areas experience the highest stress and strain, and they are the last to be filled. To ensure complete filling, additional measures such as increasing the initial billet height or using a preform with tapered ends could be considered. The results also indicate that the gear teeth are filled uniformly without defects such as folding or underfilling, confirming the robustness of the process for straight spur gears.
Compared to conventional closed-die forging without a central hole, the hole-divided flow reduces the peak load by about 30–40% for the same gear geometry. This is a significant advantage because lower loads reduce die wear, energy consumption, and machine capacity requirements. Furthermore, the metal flow lines remain continuous and aligned with the tooth profile, which enhances the fatigue strength of the straight spur gear—a critical property in power transmission applications.
Conclusions
Based on the numerical simulation of hole-divided flow forming for straight spur gears, I draw the following conclusions:
- The hole-divided flow forming process is capable of producing straight spur gears with complete tooth filling and no visible defects. The maximum equivalent stress occurs at the tooth tips in the final stage, while the peak strain appears at the tooth root corners.
- The load-displacement curve shows a gentle rise during the initial 70% of the stroke, followed by a sharp increase in the final 2 mm due to hole closure. The peak forming load reaches 2200 kN for the baseline configuration.
- Parametric studies reveal that increasing the central hole diameter reduces the peak load but may compromise filling quality. A hole diameter of 16 mm (30% of billet diameter) provides the optimal balance. The friction factor also significantly influences the load; decreasing m from 0.12 to 0.08 reduces the peak load by 10%.
- The equivalent strain distribution indicates that the most severe deformation occurs in the tooth root and the top corners. The strain values are consistent with the requirement for achieving fine-grained microstructure in the finished straight spur gear.
- The numerical methodology established in this work can serve as a reliable tool for designing and optimizing the hole-divided flow forming process for straight spur gears of various sizes and materials. Future work could extend the analysis to multi-stage forming or incorporate thermal effects to further improve process efficiency.
In conclusion, the hole-divided flow forming method represents a promising alternative to traditional machining and conventional forging for manufacturing high-quality straight spur gears. By carefully selecting the billet geometry and process parameters, manufacturers can achieve substantial reductions in forming load without sacrificing product integrity. The insights gained from this study provide a theoretical foundation for the industrial implementation of this advanced forming technology. Continued research, including experimental validation and die design optimization, will further enhance the practicality of this technique for mass production of straight spur gears.